Open quantum systems in spatially correlated regimes

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Open quantum systems in spatially correlated regimes Dara P. S. McCutcheon

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A thesis submitted in partial fulfilment of the requirements for the degree of doctor of philosophy Ph.D. October 2010

Open quantum systems in spatially correlated regimes Dara P. S. McCutcheon University College London

Abstract Almost all quantum systems are open; interactions with the surrounding environment generally lead to complex dissipative behaviour with a sensitive dependence on the details of the system-environment coupling. This thesis presents results from theoretical investigations into such behaviour in single and two-site quantum systems with a particular emphasis on strong system-environment coupling regimes, and also the eﬀects of spatial correlations in the environment fluctuations. Within a weak system-environment coupling framework, it is found that an increased level of correlation is able to protect coherence shared between two spatially separated two-level quantum systems. Moreover, it is found that these correlations are in fact able to generate coherence between the two systems, and in certain regimes, cause the systems to become entangled. Using a polaron transform strong coupling master equation technique, the discussion is extended to the strong system-environment coupling or high temperature regime. To assess the validity of this approach in an experimentally relevant system, it is applied to the description of excitonic Rabi oscillations in a resonantly driven quantum dot. For most of the parameters of interest, the strong coupling theory is found to be valid over a far greater range of temperatures and coupling strengths than the standard weak-coupling theory. The coherent or incoherent nature of energy transfer dynamics is then studied by applying the strong coupling theory to a donor-acceptor pair model. Increased spatial correlations are found to extend the range of temperatures which allow coherent energy transfer to take place. Finally, a variational theory is introduced which allows for exploration of certain parameter regimes where both the weak-coupling and strongcoupling theories become invalid. The variational theory is then used to investigate the ground state properties of a double two-level impurity model. High levels of spatial correlation are found to suppress the tunnelling amplitude within each impurity.

Acknowledgements I would first like to express my profound gratitude to my supervisors, Andrew Fisher and Sougato Bose. I’m immensely grateful for having the opportunity to work with them and for all of the support they have given to me. It is hard to imagine a duo who both share such a fantastic combination of frightening ability and genuine modesty. I have also been incredibly lucky to work closely with Ahsan Nazir. The value of the help, both academic and personal, that Ahsan has given to me throughout is immeasurable. Ahsan has been a fantastic colleague, friend, and as he likes to put it, ‘superior’ ! Furthermore, Ahsan has been perhaps my biggest single source of inspiration to become a fully-fledged physicist. I thank Alexandra Oyla-Castro for many great discussions, particularly on the subject of energy transfer. Alexandra is a great role model and I’m pleased to have had a chance to work along side her. On the subject of energy transfer I also thank Avinash Kolli. For many stimulating discussions, often in the Royal Oak, I thank Brendon Lovett and Erik Gauger. It’s been a great privilege to enjoy their hospitality in Oxford and their contribution to my understanding of quantum dots is greatly appreciated. Throughout my time at UCL I have also had the great honour of having many friends who have all contributed to my scientific understanding, more often than not in the relaxed atmosphere of a public house. Doug Lazenby, David Western, James Burnett, Pete Johnson, Adam Harman-Clarke and Christian Klettner, thank you all for making UCL such a fantastic place to be. Lastly I’d like to thank my friends and family not already mentioned. None of this could have been achieved without your support.

Contents 1 Introduction

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2 Open quantum systems and master equations 2.1 Basic concepts in quantum mechanics . . . . . . . . . 2.1.1 Postulates of quantum mechanics . . . . . . . 2.1.2 Open and closed systems . . . . . . . . . . . 2.1.3 Properties of the density operator . . . . . . . 2.1.4 The Heisenberg and interaction picture . . . . 2.2 Markovian master equations . . . . . . . . . . . . . . 2.2.1 Hamiltonian decomposition . . . . . . . . . . 2.2.2 Rotating wave approximation (RWA) . . . . . 2.2.3 Rates and energy shifts . . . . . . . . . . . . 2.2.4 Weak coupling master equation . . . . . . . . 2.3 The spin-boson model . . . . . . . . . . . . . . . . . 2.3.1 Independent boson model . . . . . . . . . . . 2.3.2 Weak coupling limit . . . . . . . . . . . . . . 2.3.3 The noninteracting blip approximation (NIBA) 2.3.4 Ground state properties . . . . . . . . . . . .

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5 7 7 12 14 17 20 22 24 25 27 29 30 33 38 41

3 Entanglement induced by a spatially correlated thermal 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . 3.2 Model and master equation . . . . . . . . . . . . . . . 3.2.1 Bath correlation functions and rates . . . . . . . 3.3 State Dynamics . . . . . . . . . . . . . . . . . . . . . 3.3.1 Generalisation of the Bloch vector . . . . . . . . 3.3.2 Liouvilian spectrum . . . . . . . . . . . . . . . 3.4 Entanglement Dynamics . . . . . . . . . . . . . . . . . 3.4.1 Entanglement generated through the Lamb-shift 3.5 Experimental Realisation . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44 44 46 47 49 49 50 52 55 57 61

4 Development of a strong coupling theory 4.1 Introduction . . . . . . . . . . . . . . . 4.1.1 Relaxation of the RWA . . . . . 4.2 Non-interacting blip approximation . . . 4.2.1 Polaron transformation . . . . . . 4.2.2 NIBA master equation . . . . . . 4.2.3 NIBA results . . . . . . . . . . .

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62 63 63 65 65 66 70

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Contents 4.3 4.4

Projection operator techniques . . . . 4.3.1 Nakajima-Zwanzig equation . 4.3.2 Time-convolutionless method Summary . . . . . . . . . . . . . . .

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74 74 78 84

5 Excitonic Rabi oscillations of a resonantly driven quantum dot 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 Background . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Model and polaron transformation . . . . . . . . . . . . . . . 88 5.2.1 Exciton-phonon spectral density . . . . . . . . . . . . 91 5.3 Master equation derivation . . . . . . . . . . . . . . . . . . . 92 5.3.1 Markov approximation . . . . . . . . . . . . . . . . . 94 5.3.2 Regimes of validity . . . . . . . . . . . . . . . . . . . 95 5.4 Resonant excitation dynamics . . . . . . . . . . . . . . . . . 98 5.4.1 Time-dependent driving . . . . . . . . . . . . . . . . 100 5.4.2 Constant driving . . . . . . . . . . . . . . . . . . . . 104 5.4.3 Non-Markovian eﬀects . . . . . . . . . . . . . . . . . 109 5.5 Discussion and summary . . . . . . . . . . . . . . . . . . . . 113 6 Coherent and incoherent dynamics in excitonic energy transfer115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . 116 6.2 Polaron transform master equation . . . . . . . . . . . . . . 118 6.2.1 The system and polaron transformation . . . . . . . . 118 6.2.2 Markovian master equation . . . . . . . . . . . . . . 121 6.2.3 Evolution of the Bloch vector . . . . . . . . . . . . . 122 6.3 Resonant energy transfer . . . . . . . . . . . . . . . . . . . . 124 6.3.1 Coherent to incoherent transition . . . . . . . . . . . 128 6.4 Oﬀ resonance . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.4.1 Near Resonance . . . . . . . . . . . . . . . . . . . . 134 6.4.2 Weak coupling limit . . . . . . . . . . . . . . . . . . 135 6.4.3 High temperature or far from resonance limit . . . . . 137 6.4.4 Correlated fluctuations . . . . . . . . . . . . . . . . . 141 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7 Variational theory 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Single impurity spin-boson model . . . . . . . . . . . 7.2 Two-impurity spin-boson model . . . . . . . . . . . . . . . . 7.3 Variational calculation . . . . . . . . . . . . . . . . . . . . . 7.3.1 Crude Ising approximation . . . . . . . . . . . . . . . 7.3.2 Free energy minimisation . . . . . . . . . . . . . . . 7.3.3 Separation-dependent localisation . . . . . . . . . . . 7.4 Full variational treatment . . . . . . . . . . . . . . . . . . . 7.4.1 Free energy minimisation and self-consistent equations 7.4.2 Comparison of full and crude Ising strengths . . . . . 7.5 Variational ground state . . . . . . . . . . . . . . . . . . . . 7.5.1 Two-impurity spin-boson Hamiltonian in the displaced oscillator basis . . . . . . . . . . . . . . . . . . . . .

145 146 146 148 152 154 154 156 158 162 162 163 167 167

Contents 7.5.2

7.6

Experimental signatures of localisation tion crossover . . . . . . . . . . . . . 7.5.3 System-bath entanglement . . . . . . . Summary . . . . . . . . . . . . . . . . . . . .

vi to delocalisa. . . . . . . . 168 . . . . . . . . 170 . . . . . . . . 171

A Strong coupling correlation functions

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B Summations in k-space

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C High temperature rates

180

Index

184

List of Figures 2.1

Pure dephasing in the independent-boson model . . . . . . .

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3.1

Dissipatively induced entanglement as a function of temperature and time . . . . . . . . . . . . . . . . . . . . . . . . . Entanglement dynamics for diﬀerent initial states . . . . . . .

54 56

Argand diagram relevant to the necessary inverse Laplace transform for non-interacting blip calculations . . . . . . . . .

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3.2 4.1 5.1 5.2 5.3 5.4 5.5 5.6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 7.1 7.2 7.3 7.4 7.5

Excitonic Rabi rotations for a guassian pulse shape . . . . . . 101 Temperature dependence of the polaron, weak coupling and single phonon rates . . . . . . . . . . . . . . . . . . . . . . 105 Dependence of the polaron and weak coupling rates on the driving strength . . . . . . . . . . . . . . . . . . . . . . . . 106 Exciton population as a function of pulse area for constant driving108 Low temperature Markovain and non-Markovian excited state dynamics in the time domain . . . . . . . . . . . . . . . . . 110 Markovian and non-Markovian dynamics in a high temperature regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Energy transfer dynamics in the resonant case . . . . . . . . Correlation eﬀects in resonant energy transfer dynamics . . . Coherent to incoherent crossover temperature as a function of cut-oﬀ frequency and correlation level . . . . . . . . . . . . . Energy transfer dynamics for a small and large energy mismatch and a number of temperatures . . . . . . . . . . . . . . . . . Coherence dynamics for resonant and oﬀ-resonant transfer at high temperature . . . . . . . . . . . . . . . . . . . . . . . . Coherence dynamics in the far from resonant regime for diﬀerent temperatures . . . . . . . . . . . . . . . . . . . . . . . . Correlation eﬀects on energy transfer in the oﬀ-resonant case

126 127 129 132 140 141 142

Expectation value of σz for the variationally determined spinboson ground state . . . . . . . . . . . . . . . . . . . . . . . 151 Distance dependence of the measure of correlation in bath fluctuations for diﬀerent system-bath coupling dimensionalities . . 155 A graphical aid to solving the self-consistent condition for the renormalised tunnelling strength . . . . . . . . . . . . . . . . 160 Separation dependent renormalised tunnelling strength as a function of the system-bath coupling strength . . . . . . . . 161 Induced Ising strength as a function of the system-bath coupling164

List of Figures 7.6 7.7 7.8 7.9

Induced Ising strength as a function of spin separation . . . . Expectation value of σx1 (or σx2 ) as a function of the scaled spin separation . . . . . . . . . . . . . . . . . . . . . . . . . . . Expectation value of σx1 as a function of the external bare tunnelling strength . . . . . . . . . . . . . . . . . . . . . . . System-bath entanglement as a function of the system-bath coupling strength for various spin separations . . . . . . . . .

viii 165 168 169 171

List of Publications This thesis is based on the following publications: • Long-lived spin entanglement induced by a spatially correlated thermal bath, D. P. S. McCutcheon, A. Nazir, S. Bose and A. J. Fisher, Phys. Rev. A, 80, 022337 (2009) • Quantum dot Rabi rotations beyond the weak exciton-phonon cou-

pling regime, D. P. S. McCutcheon and A. Nazir, New J. Phys., 12, 113042 (2010)

• Coherent and incoherent dynamics in excitonic energy transfer: correlated fluctuations and oﬀ-resonance eﬀects, D. P. S. McCutcheon and A. Nazir, Phys. Rev. B, 83, 165101 (2011) • Separation-dependent localization in a two-impurity spin-boson model, D. P. S. McCutcheon, A. Nazir, S. Bose and A. J. Fisher, Phys. Rev. B, 81, 235321 (2010)

Chapter 1 Introduction This thesis presents theoretical results of investigations into dissipative quantum phenomena in a number of settings. Here we focus of the eﬀects of spatial correlations in bath fluctuations and regimes inaccessible to weak system-bath coupling theories. While the theoretical approaches developed are applied to a variety of important physical systems, it could be said that the main concern of this work is the development and evaluation of techniques which can go beyond more common approaches. As such, much of the relevant background information and motivation behind each physical system explored is left until the relevant chapters. Here we give two examples of fields in which spatial correlations and strong coupling eﬀects are expected to be important. Quantum dissipative processes are of utmost importance in the field of quantum information processing. The eﬀect of the environment in this context is often detrimental, since it generally results in the rapid loss of quantum coherence. Coherence is the essential feature of quantum systems allowing for the speed-up of certain computational tasks in a quantum computer. A thorough understanding of the system-environment coupling, and theories to account for it accurately, are therefore of great importance if the promise of viable quantum technologies is to be realised. Recent experiments performed on biological light-harvesting molecules isolated from marine algae and sulphur bacteria have also demanded a theoretical description at a quantum mechanical level. In these experiments, clear signatures of quantum mechanical eﬀects have been observed at elevated temperatures and on relatively long timescales. There is therefore currently a great interest in how it is fragile quantum mechanical phenom-

3 ena are able to survive at these temperatures. Perhaps more intriguing still is the question of whether these phenomena play a functional role in the photosynthetic process. While little is known about the quantum nature of the environments in these systems, it is believed that spatial correlations may play an important role. Additionally, given the high temperatures involved, it is unlikely that weak-coupling theories are equipped to describe them accurately. Theories which extend into the high temperature regime and correctly account for spatial correlations will therefore constitute a valuable tool if the questions these experiments have raised are to be answered.

Plan of the thesis Broadly speaking this thesis contains two ‘background’ chapters and four ‘results’ chapters. In chapter 2 a brief introduction to the basic concepts in quantum mechanics and open quantum systems is given, as well as a summary of the important known features of the spin-boson model. A majority of the terminology and a portion of the important formalism used throughout this thesis can be found there. In particular, the standard Born-Markov weak-coupling approach to open systems is introduced. Chapter 3 then uses this weak-coupling approach to investigate the entanglement dynamics in a two-spin-boson model [1]. Within the weakcoupling formalism it is found that entanglement between two uncoupled spins can be generated through their mutual interaction with a common environment. The timescale on which this survives as a function of the level of environment spatial correlations is analysed. In chapter 4 a theory capable of exploring strong coupling regimes is developed. This uses a polaron transformation combined with the timeconvolutionless projection operator technique. The latter provides a rigorous formalism from which to derive perturbative master equations, as well as including non-Markovian eﬀects. The former permits an identification of a smaller parameter to use as a perturbation. The next two chapters then go on to use the strong coupling techniques developed. Excitonic Rabi rotations in quantum dots are explored in chapter 5 where known experimental parameters are used for a direct comparison of the strong and weak coupling theories [2]. In chapter 6 the

4 coherent and incoherent nature of energy transfer dynamics is investigated in a donor-acceptor pair model [3]. The nature of the energy transfer process is explored as a function of the correlation in bath fluctuations at each site. The emphasis is then shifted slightly in chapter 7 where the ground state properties of a two-spin-boson model are investigated [4]. To do this a variational technique is employed which allows for an investigation of the full range of system-environment coupling strengths.

Chapter 2 Open quantum systems and master equations Contents 2.1

Basic concepts in quantum mechanics . . . . 2.1.1 Postulates of quantum mechanics . . . . Quantum states and state space . . . . . State evolution . . . . . . . . . . . . . . Measurements in quantum mechanics . . Composite systems . . . . . . . . . . . . 2.1.2 Open and closed systems . . . . . . . . . Closed systems . . . . . . . . . . . . . . Open systems . . . . . . . . . . . . . . . 2.1.3 Properties of the density operator . . . . The Bloch vector . . . . . . . . . . . . . Density operator for composite states . . 2.1.4 The Heisenberg and interaction picture . Heisenberg picture . . . . . . . . . . . . . Interaction picture . . . . . . . . . . . . . 2.2 Markovian master equations . . . . . . . . . 2.2.1 Hamiltonian decomposition . . . . . . . . 2.2.2 Rotating wave approximation (RWA) . . . 2.2.3 Rates and energy shifts . . . . . . . . . . 2.2.4 Weak coupling master equation . . . . . . 2.3 The spin-boson model . . . . . . . . . . . . . 2.3.1 Independent boson model . . . . . . . . . 2.3.2 Weak coupling limit . . . . . . . . . . . . Bath correlation functions . . . . . . . . . Weak coupling rates and Lamb-shift terms

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2.3.4

Bloch vector evolution . . . . . . . . . . . . . The noninteracting blip approximation (NIBA) Dynamics . . . . . . . . . . . . . . . . . . . . Steady state in the NIBA . . . . . . . . . . . . Ground state properties . . . . . . . . . . . . .

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37 38 39 40 41

2.1 Basic concepts in quantum mechanics

2.1

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Basic concepts in quantum mechanics

In this section we introduce some of the basic concepts of quantum mechanics which will be used throughout this thesis. The approach used here follows closely that found in Ref. [5], and a knowledge of linear algebra is assumed. We set � = 1.

2.1.1

Postulates of quantum mechanics

We naturally begin by stating the postulates of quantum mechanics. Quantum states and state space The first postulate of quantum mechanics is concerned with the quantum states themselves. • Postulate 1: Any isolated physical system has an associated Hilbert

space known as the state space of the system. The system is completely described by its state vector which is a unit vector in this state space.

A important type of state which will be often studied is the state of a twolevel system. Such a two-level system could be a spin one-half particle, where the two levels correspond to spin “up” and spin “down”. Alternatively the two levels could correspond to the ground and excited states of an atom. State vectors will be denoted in their “bra-ket” notation. A general state vector in a two dimensional Hilbert space is therefore written |Ψ� = a |0� + b |1�

(2.1)

where the states |0� and |1� are some orthonormal basis in the state space

and a and b are possibly complex coeﬃcients. The inner product of two states |Ψ� and |Ψ� � is written �Ψ|Ψ� � where �Ψ| = (|Ψ�)† . The condition that |Ψ� has unit length then requires |a|2 + |b|2 = 1.

This much studied two-level system has great relevance in the field of

quantum information processing where it represents the basic quantum bit, or qubit. In this context the labels 0 and 1 become apparent since they are analogous to the 0 and 1 states of a classical computational bit. The fundamental diﬀerence is that the qubit can exist in a superposition

2.1 Basic concepts in quantum mechanics

8

of states simultaneously, as in Eq. (2.1). Superposition states such as this have no classical analogue and constitute one of the defining diﬀerences between quantum and classical physics. State evolution • Postulate 2: The evolution of a closed quantum system is described by the Schr¨odinger equation, i

d |Ψ� = H |Ψ� . dt

(2.2)

The operator H is a Hermitian operator known as the Hamiltonian of the system. With knowledge of H one can, in principle, determine entirely the evolution of the quantum state |Ψ�. For a time-independent Hamiltonian Eq. (2.2) has solution

|Ψ(t)� = exp[−iH(t − t0 )] |Ψ(t0 )�

(2.3)

where |Ψ(t0 )� is the state of the system at t = t0 .

We now consider two very basic examples of state evolution. Most

simple of all is the case where the initial state is an eigenstate of the Hamiltonian. Consider the initial state |E(t0 )� which satisfies H |E(t0 )� =

E |E(t0 )�. Upon exponentiation of the Hamiltonian, from Eq. (2.3) we then find

|E(t)� = exp[−iH(t − t0 )] |E(t0 )� = e−iE(t−t0 ) |E(t0 )�

(2.4)

and it is seen that the state simply accrues a phase factor in time. We now consider a slightly more interesting example. We consider the evolution of a two-level system or qubit initially in the state |Ψ(t0 )� = |0�

and the Hamiltonian is now chosen such that |0� is not an eigenstate of H. To write down the Hamiltonian is it convenient to introduce the Pauli

2.1 Basic concepts in quantum mechanics

9

operators. In the {|0� , |1�} basis these operators are defined as σx = |0��1| + |1��0|

(2.5)

σy = −i |0��1| + i |1��0|

(2.6)

σz = |0��0| − |1��1|

(2.7)

and we also define the identity operator σ0 = |0��0|+|1��1|. In this example we take a Hamiltonian of the form H = (∆/2)σx . For the initial state |0� we therefore find the subsequent evolution

|Ψ(t)� = cos(∆t/2) |0� − i sin(∆t/2) |1� .

(2.8)

It can be seen that the system periodically returns to its initial state with a frequency ∆. This sort of oscillatory behaviour is a common feature of quantum systems. Importantly, this oscillatory behaviour, in principle, extends to situations where the two-level system is actually a macroscopic body. It will be seen, however, that inevitable interactions with the environment surrounding the quantum system tend to suppress these oscillations. The suppression of such oscillations and the‘washing’ out of quantum mechanical eﬀects will be a central theme in this thesis. Measurements in quantum mechanics The third postulate of quantum mechanics regards measurements. • Postulate 3: Measurements are described by a set of measurement operators for which we write {Mm }. The label m here indicates which of all the possible measurement outcomes the operator Mm

refers to. For a state |Ψ� the probability of measurement outcome m is given by

† p(m) = �Ψ| Mm Mm |Ψ�

and after the state of the system is |Ψ� � = (1/

(2.9) � p(m))Mm |Ψ�. The

measurement operators satisfy the completeness condition �

† Mm Mm = I

m

which ensures that probabilities sum to one.

(2.10)

2.1 Basic concepts in quantum mechanics

10

As an example we consider the probability of a two-level system, as described by Eq. (2.8), being in the state |0�. The corresponding measurement operator for this is M0 = |0��0|. We find

p(0) = �Ψ|0��0|0��0|Ψ� = cos2 (∆t/2).

(2.11)

The only alternative measurement outcome in this example is that of the two-level system being in state |1�. This has probability p(1) = sin2 (∆t/2). � � We see that m=0,1 p(m) = 1 and m=0,1 = |0��0|+|1��1| = I as expected. � Similarly, for a general superposition state |Ψ� = n an |ψn �, the third postulate allows the squared modulus of the coeﬃcient an to be identified as the probability of the state being in the corresponding state |ψn �.

We will often be interested in quantities of the form �Ψ| σi |Ψ� which

we will write as �σi �. These quantities clearly have something to do with measurement outcomes since they can be written in the form of Eq. (2.9).

To see what they represent we consider, for example, the superposition √ state |Ψ� = (1/ 2)(|0� + |1�). For this state we have �σx � = 1 while √ �σy � = 0 and �σz � = 0. If instead we have the state |Ψ� = (1/ 2)(|0�−|1�) we find �σx � = −1 while �σy � = 0 and �σz � = 0. The quantities �σi �

therefore represent the diﬀerence in probability of finding the state in the two diﬀerent eigenstates of the operator σi . We will call �σi � the expectation value of the operator σi .

In general, the quantity �O� for some arbitrary Hermitian operator

O will be called the expectation value of O. To see why, we write the � operator O in its eigenvector basis, O = i λi |λi ��λi |. We then expand � an arbitrary state in this basis, |Ψ� = n an |λn �. We then find that �O� = �Ψ| O |Ψ� =

� n

|an |2 λn .

(2.12)

Recalling that |an |2 is the probability of |Ψ� being found in the state |λn �,

we see that the quantity �O� is indeed the expectation value of the operator O, given the state of the system |Ψ�. Composite systems The last postulate of quantum mechanics gives a prescription of how to construct the combined state of a number of subsystems.

2.1 Basic concepts in quantum mechanics

11

• Postulate 4: The state space of a composite physical system is the tensor product of the state spaces of the component systems.

Additionally, for component system states |ψi �, with i running from

1 to n, the state of the combined system is given by |ψ1 � ⊗ |ψ2 � ⊗

· · · ⊗ |ψn �.

Thus if we have two two-level systems described by the states |ψ� and

|φ�, their composite state is given by |Ψ� = |ψ� ⊗ |φ�. The shorthand |ψ� ⊗ |φ� = |ψφ� will often be used.

An important concept in the discussion of composite systems is that of

entanglement. To see what is meant by this, we consider the evolution of a composite system consisting of two two-level systems under the action of the Hamiltonian H = Xσz1 σz2 , where the superscripts here indicate on what system, and in which space, the operator acts. Considering both √ two-level systems to be in the σx eigenstate (1/ 2)(|0� + |1�), we have the composite initial state |Ψ(0)� = (1/2)(|0� + |1�) ⊗ (|0� + |1�). From Eq. (2.3) we then find the state has evolved at time t into |Ψ(t)� =

� 1 � iXt e (|00� + |11�) + e−iXt (|01� + |10�) . 2

(2.13)

At time t = π/4X we find the state can be written � � 1 |Ψ(t = π/4X)� = eiπ/4 |0� ⊗ (|0� − i |1�) − i |1� ⊗ (|0� + i |1�) 2 � � 1 = √ eiπ/4 |0� ⊗ |γ− � − i |1� ⊗ |γ+ � (2.14) 2

√ where we have just defined the states |γ± � = 1/ 2(|0� ± i |1�) in the second line. The interesting feature of the state described by Eq. (2.14) is that it cannot be written as a single tensor product of two states, as the initial condition was. The action of the Hamiltonian on the system has led their combined state vector to become inseparable. Such states are called entangled , and are said to possess entanglement. An important point to also notice is that this entangling of the combined state is oscillatory in nature. Without digressing into a discussion of measures of entanglement (for a review see Plenio and Virmani [6]), this can be seen by observing that for times t = nπ/X with n = 1, 2, 3 . . . the state will have returned to its original state, up to an unimportant phase factor. The initial state was in a form allowing it to be written as a tensor product and therefore

2.1 Basic concepts in quantum mechanics

12

possessed no entanglement, by definition. The oscillatory nature of this entanglement is typical of entanglement induced by coherent interactions as studied here. However, we shall see in chapter 3 that this is not the only sort of behaviour entanglement can exhibit.

2.1.2

Open and closed systems

Having introduced the main principles of quantum mechanics, we now consider the time evolution of states in more detail. In particular, we establish the concepts of open and closed quantum systems. Closed systems As we have seen already, according to quantum mechanics, the state vector |Ψ(t)� evolves in time according to the Schr¨odinger equation i

d |Ψ(t)� = H(t) |Ψ(t)� dt

(2.15)

where H(t) is the Hamiltonian operator of the total system under consideration. We now generalise the discussion slightly by allowing for the possibility that the Hamiltonian operator be time-dependent. The solution to Eq. (2.15) can be written in terms of the unitary time-evolution operator U (t, t0 ), which transforms a state at time t0 into that at time t: |Ψ(t)� = U (t, t0 ) |Ψ(t0 )� .

(2.16)

Inserting Eq. (2.16) into Eq. (2.15) one obtains i

∂U (t, t0 ) = H(t)U (t, t0 ) ∂t

(2.17)

which is the defining diﬀerential equation for the time-evolution operator and subject to the initial condition U (t0 , t0 ) = I. In general Eq. (2.17) has formal solution [7] �

U (t, t0 ) = T← exp − i

�

t

dsH(s) t0

�

(2.18)

where T← is the time ordering operator which orders products of timedependent operators such that their time arguments increase from right

2.1 Basic concepts in quantum mechanics

13

to left. For a completely isolated system we expect the Hamiltonian operator to be time-independent. In this case we obtain the simpler form � � U (t, t0 ) = exp − iH(t − t0 )

(2.19)

which is the form of time evolution operator we considered in section 2.1. Physical examples of such systems might include an atom in free space or an entire solid sample. Such systems will be said to isolated. However, it is often the case that the system under consideration is not completely isolated and the time dependence in H remains. An example of such a system could be the evolution of a particle subject to some varying external classical field. If it is still possible that the system be formulated in terms of a time-dependent Hamiltonian then it will be said to be closed. Importantly, for a closed or isolated system the time evolution is always unitary. This can be seen by diﬀerentiating U (t, t0 )U −1 (t, t0 ) = I with respect to time, and with the use of Eq. (2.17) observing that U −1 (t, t0 ) obeys the same diﬀerential equation as that of U † (t, t0 ). Open systems We now consider systems which are subject to the influence of some external macroscopic environment. In such cases it is generally not possible to solve for the Hamiltonian unitary dynamics directly. These systems will be called open. Throughout this thesis we shall frequently refer to the ‘system’ and the ‘environment’ (or often ‘bath’). The exact distinction between the two may not always be well defined. However, in general, the ‘system’ will be the part in which we are interested while the ‘environment’ will refer to whatever surrounds the system and influences its behaviour. Examples of open systems might include a molecule (the system) interacting with a solvent (the environment) or a small nanostructure imbedded and interacting with a bulk solid sample. In the quantum computation setting one often considers a two-level system or qubit (the system) interacting with some general dissipative environment. The aim of open quantum system theories is to determine the evolution of the system under the influence of its environment. Ideally one is able to derive equations governing the system only, where the environmental

2.1 Basic concepts in quantum mechanics

14

influence enters the problem only in the form of parameters. Importantly, for an open system, its interaction with the environment introduces some classical uncertainty. In such cases we must formulate the problem in terms of the density operator ρ rather than the state vector itself. If it is known that a system is described by some ensemble of states {|Ψn �} with corresponding probabilities {pn } then the density operator is given by ρ(t) =

� n

pn |Ψn (t)��Ψn (t)| .

(2.20)

If pn �= 1 for any n we say that ρ is a mixed state. If pn = 1 for some n and all other pn = 0 then ρ is said to be pure. Note that only pure states have a single corresponding state vector |Ψ(t)�. The states |Ψn (t)�

evolve according to the Schr¨odinger equation. The evolution of the density operator is then given by ρ(t) =

� n

pn U (t, t0 ) |Ψn (t0 )��Ψn (t0 )| U † (t, t0 )

= U (t, t0 )ρ(t0 )U † (t, t0 ).

(2.21)

Diﬀerentiation with respect to time gives the equation of motion dρ(t) = −i[H(t), ρ(t)] dt

(2.22)

known as the von Neumann equation. Eq. (2.22) is often written dρ(t) = L(t)ρ(t) dt

(2.23)

where L(t) is known as the Liouvillian super-operator (or just the Liou-

villian) and is defined in such a way that the right hand side of Eq. (2.22) is correctly recovered. The term super-operator is used since L maps operators to operators.

2.1.3

Properties of the density operator

Before we consider open systems in more detail it is important to establish some properties of the density operator and to see how quantum mechanics can be reformulated in terms of it.

2.1 Basic concepts in quantum mechanics

15

• From Eq. (2.20) it is possible to see that Tr(ρ(t)) = 1

(2.24)

where Tr means to trace over some complete set of states in the relevant state space. Eq. (2.24) is ensured if the probabilities sum to one, as they should. • For an arbitrary vector |φ� in state space we have �φ| ρ |φ� =

� n

pn |�φ|Ψn �|2

≥ 0,

(2.25)

thus the density operator is positive. • Expectation values can be calculated from the density operator. For a classical ensemble of quantum states {|Ψn �} with corresponding probabilities {pn } the expectation value of an operator A is �A� = = =

� n

pn �Ψn | A |Ψn �

�� i

n

i

n

��

pn �Ψn |i� �i| A |Ψn � pn �i| A |Ψn � �Ψn |i�

= Tr(Aρ)

where we have used the completeness relation

(2.26) �

i

|i��i| = I.

• For a pure state Tr(ρ2 ) = 1 whereas for a mixed state Tr(ρ2 ) < 1.

This identification gives a criterion to determine whether a state is pure or mixed.

The Bloch vector At this point it is convenient to introduce what is known as the Bloch vector. This is a useful way in which to think about the density operator of a single two-level system. Consider the density operator of a two-level

2.1 Basic concepts in quantum mechanics

16

system of the form 1 ρ = (I + αx σx + αy σy + αz σz ). 2

(2.27)

It is straightforward to show that a density operator as described by Eq. (2.27) satisfies all of the properties listed above provided |αx |2 +|αy |2 +

|αz |2 ≤ 1. For example, the traceless property of the Pauli operators ensures that Tr(ρ) = 1. Moreover, since the Pauli operators all share the property σi2 = I, it can be seen that �σi � = Tr(σi ρ) = αi .

(2.28)

Therefore, the coeﬃcients appearing in Eq. (2.27) are the expectation values of the Pauli operators. Eq. (2.27) can be written ρ = 12 (I + α · σ) where σ = (σx , σy , σz )T is a

vector of Pauli operators and α = (αx , αy , αz )T is a vector containing the expectation values and is known as the Bloch vector. Note that the Bloch vector completely defines the state of the two-level system, the state of which can now be thought of as a point in three dimensional space. For a pure state we require Tr(ρ2 ) = 1. This condition can be shown to be equivalent to the condition that the Bloch vector has unit length. Pure states therefore correspond to Bloch vectors that lie on the surface of a sphere with unit radius. Mixed states on the other hand correspond to Bloch vectors whose end points lie within this sphere. Density operator for composite states If we have subsystems numbered 1 to n, and subsystem i is described by a density operator ρi , then the joint state of the total system is described by the density operator ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρn . For pure states this is implied by postulate 4 from section 2.1.

A very useful consequence of working in terms of the density operator is that it provides one with a way in which to describe subsystems of a larger composite system. In this thesis we will be interested in describing small systems interacting with a much larger system (the environment) so this is of great interest to us. For physical systems labelled A and B whose joint density operator is ρAB , the correct description of the system

2.1 Basic concepts in quantum mechanics

17

A is given by the density operator ρA = TrB (ρAB )

(2.29)

where TrB means the partial trace over system B. The partial trace is defined as TrB (|a1 ��a2 | ⊗ |b1 ��b2 |) = |a1 ��a2 | Tr(|b1 ��b2 |)

(2.30)

where |a1 � and |a2 � are any two vectors in the state space of A, and simi-

larly |b1 � and |b2 � are any two vectors in the state space of B. The operator ρA is called the reduced density operator.

Suppose we have a joint system described by ρAB and we wish to know the expectation value of an operator OA which acts only on the space of the subsystem A. The correct operator to find the expectation value of in the total space is therefore OA ⊗ IB where IB is an identity operator in the space of state B. We then have

�OA ⊗ IB � = Tr(OA ⊗ IB ρAB ) = TrA (OA ρA ).

(2.31)

From this we see that given a reduced density operator ρA and an operator OA acting in the state space of ρA , expectation values can be calculated in the usual way. Moreover, the reduced density operator ρA contains all of the necessary information regarding the whole state ρAB to find the expectation value of any operator acting only the state space of A.

2.1.4

The Heisenberg and interaction picture

Heisenberg picture So far we have exclusively been working in the Schr¨odinger picture in which the time dependence of the density operator is governed by the von Neumann equation (2.22). However, an equivalent description of quantum mechanics can be obtained by transferring the time dependence from the density operator to the operators in Hilbert space. To see how this works, we consider the expectation value of some arbitrary, possibly time

2.1 Basic concepts in quantum mechanics

18

dependent, operator O(t) at time t. We have � � �O(t)� = Tr Oρ(t) � � = Tr OU (t, t0 )ρ(t0 )U † (t, t0 ) .

(2.32)

Using the cyclic invariance of the trace it can be seen that the expectation value is left unchanged if we instead compute Tr(OH (t)ρH (t0 )) where OH (t) = U † (t, t0 )O(t)U (t, t0 )

(2.33)

is the operator in the Heisenberg picture and ρH (t0 ) is the fixed density operator in the Heisenberg picture. It is clear that the Heisenberg and Schr¨odinger pictures coincide at time t = t0 since U (t0 , t0 ) = I. Diﬀerentiating Eq. (2.33) with respect to time gives the equation of motion for the operator dOH (t) ∂OH (t) = i[HH (t), OH (t)] + dt ∂t

(2.34)

where HH (t) is the Hamiltonian in the Heisenberg picture. The last term represents the partial derivative with respect to the explicit time dependence of the Schr¨odinger picture operator, moved into the Heisenberg picture, ∂OH (t) ∂O(t) = U † (t, t0 ) U (t, t0 ). ∂t ∂t

(2.35)

Interaction picture The picture used extensively in this thesis is the interaction picture. Suppose we have a Hamiltonian in the Schr¨odinger picture which we separate into two terms, H(t) = H0 (t) + HI (t).

(2.36)

Generally speaking, the possibly time dependent free Hamiltonian H0 (t) will describe the evolution of quantum systems in the absence of interactions, while the Hamiltonian HI (t) describes the coupling between systems. However, the exact partition between the two Hamiltonians is in general arbitrary. Defining the free time evolution operator �

U0 (t, t0 ) = T← exp − i

�

t

H0 (s)ds t0

�

(2.37)

2.1 Basic concepts in quantum mechanics

19

which is the solution of the diﬀerential equation ∂U0 (t, t0 ) = H0 (t)U0 (t, t0 ) ∂t

(2.38)

� � �O(t)� = Tr U0† (t, t0 )O(t)U0 (t, t0 )UI (t, t0 )ρ(t0 )UI† (t, t0 ) � � ˜ ρ(t) = Tr O(t)˜

(2.39)

˜ = U0† (t, t0 )O(t)U0 (t, t0 ) O(t)

(2.40)

i we then write

where we have now introduced the interaction picture operator

and the density operator is now in its interaction picture form ρ˜(t) = UI (t, t0 )ρ(t0 )UI† (t, t0 )

(2.41)

with UI (t, t0 ) = U0† (t, t0 )U (t, t0 ). Note that throughout this thesis tildes will be used to denote operators in the interaction picture. With use of the defining diﬀerential equations for U (t, t0 ) and U0 (t, t0 ), Eqs. (2.17) and (2.38), it can be shown that UI (t, t0 ) obeys a diﬀerential equation of the same form, i

∂UI (t, t0 ) ˜ I (t)UI (t, t0 ) =H ∂t

(2.42)

where we have now introduced the interaction Hamiltonian in the interaction picture, ˜ I (t) = U0† (t, t0 )HI (t)U0 (t, t0 ). H

(2.43)

Diﬀerentiation of Eq. (2.41) then leads to the von Neumann equation in the interaction picture d˜ ρ(t) ˜ I (t), ρ˜(t)], = −i[H dt

(2.44)

which will often be our preferred starting point from which to derive a master equation describing a subsystem of ρ.

2.2 Markovian master equations

2.2

20

Markovian master equations

While the concept of an open quantum system has been introduced, we have not considered how such a system is expected to evolve with time. In general this is a very diﬃcult task and greatly depends on the type of system under consideration, as well as the type and details of environment with which it interacts. It is the case that the combined state of the open system and its environment evolves according to the Schr¨odinger equation. However, only fairly simple examples of little practical interest allow for a direct integration and exponentiation of the Hamiltonian to obtain the corresponding time evolution operator. Instead one generally uses a number of approximations in an eﬀort to determine the evolution of the open system. In this section we outline the derivation of the weak-coupling BornMarkov master equation. The central approximation used in this derivation is that the interaction of the open system with its environment is weak. The procedure given here largely follows that of Ref. [8]. We start by considering a time independent Hamiltonian consisting of three parts. The first is the system Hamiltonian, HS , which acts only in the Hilbert space of the system in which we are interested. The second is the bath or environment Hamiltonian, HB , acting in the (possibly infinite) Hilbert space of the environment. Lastly, we have an interaction Hamiltonian, HI which describes how the system and bath are coupled to one another and which acts in the total Hilbert space. Therefore, in the Schr¨odinger picture our total Hamiltonian is written H = HS + HI + HB .

(2.45)

Our aim is to derive an equation of motion for the reduced density operator ρ by tracing out the environmental degrees of freedom. We will work in the interaction picture in which, χ, the density operator of the whole system plus bath satisfies the von Neumann equation dχ(t) ˜ ˜ I (t), χ(t)] = −i[H ˜ dt

(2.46)

˜ I (t) is the interaction Hamiltonian in the interaction picture with where H

2.2 Markovian master equations

21

respect to HS + HB . Equation (2.46) has formal solution χ(t) ˜ = χ(t ˜ 0) − i

�

t

˜ I (s), χ(s)]. ds[H ˜

(2.47)

t0

Eq. (2.47) can then be substituted back into Eq. (2.46) to yield an integrodiﬀerential equation for the time evolution of the total density operator. Doing so and then tracing over the bath degrees of freedom gives d˜ ρ ˜ I (t), χ(0)] − = −iTrB [H dt

�

t 0

� � ˜ I (t), [H ˜ I (s), χ(s)] dsTrB H ˜

(2.48)

where ρ(t) = TrB (χ(t)) is the reduced density operator describing the system of interest, TrB denotes a trace over the bath degrees of freedom and we have set t0 equal to zero. A number of approximations are now made. We first make the Born approximation which is to assume we may factorise the total system-plusbath density operator into its system and bath components at all times. This approximation is reliant on the system-bath coupling strength being suﬃciently weak that their corresponding density operators do not become appreciably entangled. We write χ(s) ≈ ρ(s) ⊗ ρB

(2.49)

where ρB is the density operator of the bath which is assumed to be stationary for all times. Again, this is typically justified provided the systembath interaction is weak in the sense that the influence of the system on the bath is small. The Born approximation gives d˜ ρ =− dt

�

t 0

˜ I (t), [H ˜ I (s), ρ˜(s) ⊗ ρB ]]. dsTrB [H

(2.50)

˜ I (t)ρB ) = 0. This assumption is not where we have also assumed TrB (H restrictive since one can always absorb terms into HS to ensure that it is indeed the case. We now replace ρ˜(s) in the integrand with ρ˜(t); we approximate the state of the system at time s by its future state at time t. This replacement assumes that the system does not evolve appreciably on the timescale associated with the relaxation of the bath, which in turn

2.2 Markovian master equations

22

determines the times for which the integrand is finite. We now have d˜ ρ =− dt

�

t 0

˜ I (t), [H ˜ I (s), ρ˜(t) ⊗ ρB ]]. dsTrB [H

(2.51)

Lastly, to make the evolution Markovian, we push the upper limit of the integration to infinity. This is justified provided the integrand dies oﬀ suﬃciently fast, an assumption discussed in what follows. With a change of integration variable t − s → τ we arrive at the Markovian quantum master equation in the interation picture, d˜ ρ =− dt

�

∞ 0

˜ I (t), [H ˜ I (t − τ ), ρ˜(t) ⊗ ρB ]] dτ TrB [H

(2.52)

valid to second order in the interaction Hamiltonian HI . The assumptions that have led to Eq. (2.52) will be collectively referred to as the BornMarkov approximation.

2.2.1

Hamiltonian decomposition

The operators appearing in equation (2.52) are written in the interaction picture. To proceed it will be useful to decompose the interaction Hamiltonian into eigenoperators of the system Hamiltonian so that their time dependencies have a simple exponential form. In the Schr¨odinger picture, the interaction Hamiltonian has the general form HI =

� α

Aα ⊗ B α

(2.53)

where Aα are system operators, Bα are bath operators, and the summation ensures that an arbitrary HI can be written as such. The required decomposition into eigenoperators is achieved with Aα (ω) =

�

�� −�=ω

|��| Aα |�� |

(2.54)

where |�� and |�� are eigenstates of the system Hamiltonian with eigenvalues � and �� respectively and the sum runs over all eigenvalues with fixed

diﬀerence ω. Defining Aα in this way ensures the following relations are

2.2 Markovian master equations

23

immediately satisfied, [HS , Aα (ω)] = −ωAα (ω)

(2.55)

[HS , A†α (ω)] = +ωA†α (ω).

(2.56)

Moving the interaction Hamiltonian into the interaction picture now becomes particularly simple since eiHS t Aα (ω)e−iHS t = e−iωt Aα (ω)

(2.57)

and hence the interaction Hamiltonian in the interaction picture at time t becomes ˜ I (t) = H

� α,ω

˜α (t) e−iωt Aα (ω) ⊗ B

(2.58)

˜α (t) = eiHB t Bα e−iHB t is a bath operator in the interaction picture where B at time t. Noting that for Hermitian operators A, B and C we can write [A, [B, C]] = ABC − BCA + H.c.

(2.59)

allows us to rewrite Eq. (2.52) as d˜ ρ = dt

�

∞ 0

� ˜ I (t − τ )˜ ˜ I (t) dτ TrB [H ρ(t)ρB H

˜ IH ˜ I (t − τ )˜ − H(t) ρ(t)ρB ] + H.c.

�

(2.60)

where H.c. denotes the Hermitian conjugate. Inserting Eq. (2.58) for ˜ − τ ) (summing over β) and its Hermitian conjugated expression for H(t ˜ H(t) (summing over α), and using the cyclic invariance of the trace, we arrive at the following expression for the time evolution of the system density matrix, d˜ ρ �� = dt α,β ω,ω � ×e

�

∞ 0

i(ω � −ω)t

�

˜ † (t)B ˜β (t − τ )�B dτ eiωτ �B α

Aβ (ω)˜ ρ(t)A†α (ω � )

−

A†α (ω � )Aβ (ω)˜ ρ(t)

�

+ H.c.

(2.61)

2.2 Markovian master equations

24

which can be written d˜ ρ �� = Kαβ (ω)ei(ω −ω)t dt α,β ω,ω � � � × Aβ (ω)˜ ρ(t)A†β (ω � ) − A†α (ω � )Aβ (ω)˜ ρ(t) + H.c.

(2.62)

where Kαβ (ω) = =

�

∞

�0 ∞ 0

� † � ˜α (t)B ˜β (t − τ )ρB dτ eiωτ TrB B ˜α† (t)B ˜β (t − τ )�B dτ eiωτ �B

(2.63)

are the one sided Fourier transforms of the bath correlation functions, and the angular brackets indicate an expectation value with respect to the unperturbed bath state: �O� = TrB {OρB }.

For a stationary state of the bath as is considered here we have

[HB , ρB ] = 0. Utilising the cyclic invariance of the trace once again we can write � † � � � ˜ (t)B ˜β (t − τ )˜ TrB B ρB (t) = TrB eiHB t Bα† e−iHB t eiHB (t−τ ) Bβ e−iHB (t−τ ) ρB α � � = TrB eiHB τ Bα† e−iHB τ Bβ ρB ˜ † (τ )B ˜β (0)�B , = �B α

(2.64)

and we define the correlation functions � † � ˜α† (τ )B ˜β (0)�B = TrB B ˜α (τ )B ˜β (0) . Λαβ (τ ) = �B

(2.65)

The correlation functions depend only on the diﬀerence in time of the bath operators. This identification also allows the Markovian approximation made on Eq. (2.61) to be put on a more rigorous footing. Pushing the limits of integration in Eq. (2.61) to infinity and making the Markovian approximation is justified provided the correlation functions Λαβ (τ ) die oﬀ suﬃciently fast.

2.2.2

Rotating wave approximation (RWA)

To further simplify Eq. (2.62) we consider the diﬀerent terms appearing in the frequency summations. In general there will be three diﬀerent

2.2 Markovian master equations

25

timescales associated with any quantum dissipative process. Typically, for a large enough bath, the shortest timescale will be the bath relaxation time which we denote τB . The assumption that this timescale is short is what allowed us to make a Markovian approximation. For any non-zero system Hamiltonian, there will also be a typical timescale associated with the internal dynamics of the system τS . In our treatment this will typically be of the order |ω − ω � |−1 for ω �= ω � . Lastly, for a system interacting with a dissipative environment, there will be a timescale associated with the relaxation of the system which we denote τR . If the relaxation of the system is much slower than its Hamiltonian dynamics, i.e. if τS � τR , then we expect that the terms appearing in

Eq. (2.62) for which ω �= ω � will perform many oscillations over the time that ρ varies appreciably in the interaction picture. If this is indeed the case then, to some level of approximation, they may be neglected since their oscillations will average to zero. This leaves � � d˜ ρ �� = Kαβ (ω) Aβ (ω)˜ ρ(t)A†β (ω) − A†α (ω � )Aβ (ω)˜ ρ(t) + H.c. (2.66) dt α,β ω Neglecting these fast oscillating terms is often called the rotating wave or secular approximation.

2.2.3

Rates and energy shifts

To proceed we now decompose the Fourier transforms of the bath correlation functions into their real and imaginary parts. We write 1 Kαβ (ω) = γαβ (ω) + iSαβ (ω) 2

(2.67)

so that ∗ γαβ (ω) = Kαβ (ω) + Kβα (ω)

(2.68)

while Sαβ (ω) =

1 ∗ (Kαβ (ω) − Kβα (ω)). 2i

(2.69)

2.2 Markovian master equations

26

˜ † (τ )B ˜α (0)ρB )∗ = TrB (ρB B ˜ † (0)B ˜β (τ )) = By using Λ∗βα (τ ) = TrB (B α β Λαβ (−τ ) it can be shown that γαβ (ω) =

�

+∞

eiωτ Λαβ (τ )dτ

(2.70)

−∞

and 1 Sαβ (ω) = 2i

��

+∞

e

iωτ

0

Λαβ (τ )dτ −

�

+∞

e

−iωτ

Λαβ (−τ )dτ

0

�

(2.71)

where the stationarity of the bath has again been used. We shall see shortly that the full Fourier transforms γαβ (ω) take on the role of rates in the corresponding master equation. It will be useful at this stage to establish a relationship between rates which correspond to emission (γαβ (+|ω|)) and those which correspond to absorption (γαβ (−|ω|)). To find this relationship we first assume that the bath is in thermal equilibrium and therefore has corresponding density operator ρB = (1/Z) exp[−βHB ] where Z = Tr(exp[−βHB ]) is the partition function. For a bath of this sort the correlation functions satisfy the invariance � � ˜ † (τ )B ˜β (0)� = 1 TrB e−βHB eiHB τ B † e−iHB τ Bβ �B α α Z � � 1 = TrB Bβ eiHB (τ +iβ) Bα† e−iHB (τ +iβ) e−βHB Z ˜β (0)B ˜ † (τ + iβ)�B = �B α

(2.72)

Using this result one can show γαβ (−ω) = e

−ωβ

�

∞−iβ −∞−iβ

� ˜β (0)B ˜α† (−τ � )�dτ � eiωτ �B

where a change of variables τ � → −τ − iβ has been made.

(2.73) Assum-

ing the integrand has no poles in the rectangle bounded by the points {±∞, 0}, {±∞, −β} in the complex plane, Eq. (2.73) can be put in the form

γαβ (−|ω|) = e−βω γβ † α† (|ω|)

(2.74)

where the notation implies that the bath operators appearing in Eqs. (2.70) and (2.65) are Hermitian conjugated if their indices carries a dagger, i.e. Λβ † α† (s) = �(Bβ† (s))† Bα (0)† �.

2.2 Markovian master equations

2.2.4

27

Weak coupling master equation

We are now in a position to write down the Markovian master equation for the reduced density operator ρ˜ in the interaction picture. Swapping the α and β indices in the Hermitian conjugated terms in Eq. (2.66) and inserting Eq. (2.67), we arrive at d˜ ρ(t) = −i[HLS , ρ˜(t)] + D(˜ ρ(t)) dt where HLS =

�� ω

(2.75)

Sαβ (ω)A†α (ω)Aβ (ω)

(2.76)

αβ

is called the Lamb-shift Hamiltonian and provides a unitary Hamiltonianlike contribution to the dynamics. We shall see in what follows that this contribution can manifest itself as a renormalisation of the energy levels in the system due to the interaction with the bath. In more complicated systems involving more than one particle it can amount to a direct interaction between the particles. The decohering and dissipative eﬀects of the environment are contained within the dissipator, D(˜ ρ(t)), given by D(˜ ρ(t)) =

�� ω

α,β

γαβ (ω)

�

Aβ (ω)˜ ρ(t)A†α (ω)

� 1 † − {Aα (ω)Aβ (ω), ρ˜(t)} . 2

(2.77)

We now wish to bring Eq. (2.75) into the Schr¨odinger picture. Since a Schr¨odinger picture density operator ρ is related to its interaction picture counterpart, ρ˜, via ρ(t) = e−iHS t ρ˜(t)eiHS t , it follows that their equations of motion are related through dρ(t) = −i[HS , ρ(t)] + e−iHS t dt

�

d˜ ρ(t) dt

�

eiHS t .

(2.78)

Noting that the operators in Eq. (2.75) appear only accompanied by operators with the opposite frequency dependence (see relations (2.56)), it is possible to see that the Schr¨odinger picture master equation is obtained simply by adding the system Hamiltonian HS to the Lamb-shift Hamiltonian and replacing all of the interaction picture density operators on the right hand side by the corresponding Schr¨odinger picture density opera-

2.2 Markovian master equations

28

tors. The Schr¨odinger picture master equation is then dρ(t) = −i[HS + HLS , ρ(t)] + D(ρ(t)). dt

(2.79)

with HLS and D defined as before in Eqs. (2.76) and (2.77). We note that with the appropriate diagonalisation of the matrix γαβ (ω), the master equation (2.79) can be put in what is known as Lindblad form [9], meaning that the density operator it describes remains physical for all times. A number of times in this thesis we will be interested in how to approximately treat a part of the system Hamiltonian whose associated dynamics is slow compared to that associated with the rest of the system Hamiltonian. To see how this is done, we take a component of the system Hamiltonian into the interaction Hamiltonian and treat it perturbatively. That is, we let HS + HI = HS� + HI�

(2.80)

where HS� = HS − HIS and HI� = HI + HIS and the Hamiltonian HIS

contains no bath operators. Importantly, we now use HS� +HB to move into the interaction picture. Following the procedure outlined in this section we obtain instead of Eq. (2.48) d˜ ρ ˜ I (t), χ(0)] − = − iTrB [H dt

�

t

�0

� � ˜ I (t), [H ˜ I (s), χ(s)] dsTrB H ˜

t � � ˜ IS (t), χ(0)] − ˜ IS (t), [H ˜ IS (s), χ(s)] − iTrB [H dsTrB H ˜ 0 � t � � ˜ IS (t), [H ˜ I (s), χ(s)] − dsTrB H ˜ �0 t � � ˜ I (t), [H ˜ IS (s), χ(s)] − dsTrB H ˜ . (2.81) 0

Noting that the terms on the second line represent the exact evolution of the density operator under the influence of the interaction Hamiltonian ˜ IS (t), χ(t)]. HIS (t), we replace them with the commutator −iTrB [H ˜ Upon

˜ I (t)ρB ) = making the Born-Markov approximation and again using TrB (H 0, instead of Eq. (2.52) we obtain d˜ ρ ˜ IS (t), ρ˜(t)] − = −i[H dt

�

∞ 0

˜ I (t), [H ˜ I (t − τ ), ρ˜(t) ⊗ ρB ]]. (2.82) dτ TrB [H

The point to notice is that is that the first term in Eq. (2.82) does not aﬀect

2.3 The spin-boson model

29

any of the procedures used in this section to derive the final Markovian master equation (2.79). We end up with an master equation of exactly the same form but with one important diﬀerence. The Hamiltonian HIS was not used to move into the interaction picture. While it certainly will influence the Hamiltonian evolution of the reduced density operator, its eigenvalues will appear nowhere in the rates and energy shifts. This procedure is approximate of course and is only valid provided the evolution generated by HIS is far slower than that of HS� = HS − HIS .

2.3

The spin-boson model

Many physical systems encountered in physics and chemistry can be described by a generalised coordinate with which there is an associated potential energy profile containing two minima with approximately the same energy. If the thermal energy is small in comparison to the level spacing in both of the potential minima, then the dynamics of the system evolves in what is essentially a two-dimensional Hilbert space. If the system is also subject to the dissipative eﬀects of a bath of bosons, it is possible to show that the problem can be truncated to that of a two-level system interacting with a bath of oscillators [10, 11]. With this truncation procedure the analogy of a spin one-half particle in a dissipative environment becomes apparent. From this point onwards we shall refer then not to the potential minima themselves but to the two (pseudo)-spin states. The corresponding Hamiltonian is generally written in the form � � � ∆ H = σz − σx + σz (gk b†k + gk∗ bk ) + ωk b†k bk . 2 2 k k

(2.83)

The energy splitting diﬀerence between the spin “up” and spin “down” states (or the diﬀerence between the potential energy minima) is � and the tunnelling matrix element ∆. The bath of oscillators is described by creation (annihilation) operators b†k (bk ) with corresponding frequencies ωk and wavevectors k. The coupling of the spin with the oscillators is given by the coupling constants gk . Despite its seemingly simple form, the spin-boson Hamiltonian (2.83), provides one with an excellent starting point from which to investigate a wide range of phenomena in open quantum systems. This section is there-

2.3 The spin-boson model

30

fore devoted to giving an overview of some of the well-known results for the spin-boson model. We will be primarily interested in the expectation value of the σz operator, which we will denote �σz � = αz . In the double

potential well picture this quantity represents the diﬀerence in probability that the particle be found in the left or right well. For the spin one-half particle, it represents the diﬀerence in probability of a z-measurement finding the spin in an up or down state. For the spin this quantity could also be considered a polarisation.

2.3.1

Independent boson model

We start the overview by investigating a limit where the spin-boson model is exactly solvable. This arrises when the tunnelling parameter in Eq. (2.83) is set equal to zero. With ∆ = 0 we have � � � HIB = σz + σz (gk b†k + gk∗ bk ) + ωk b†k bk 2 k k

(2.84)

which is known as the independent boson model. To solve Eq. (2.84) we use a canonical transformation as shown in Ref. [12], or in a spirit more similar to that here in Ref. [13]. With ∆ = 0 the independent boson Hamiltonian is exactly diagonalised by the unitary transformation HD = e+S HIB e−S where � � � e±S = exp ± σz (αk b†k − αk∗ bk )

(2.85)

k

with αk = gk /ωk . One finds HD = H0 − r where r = constant and

� � H0 = σ z + ωk b†k bk . 2 k

�

k

|gk |2 ωk−1 is a (2.86)

We note that in the limit � → 0 we can write the ground state of the total system in the diagonalised basis as simply

1 |GD � = √ |B0 � ⊗ (|0� ± |1�) 2

(2.87)

where |B0 � is the vacuum state of the bath. To find the ground state in

2.3 The spin-boson model

31

the original basis we simply invert the transformation to give 1 |G� = e−S |GD � = √ 2

�

� k

D

�g � k

ωk

|B0 � |1� ±

� k

� � g � k D − |B0 � |0� , ωk (2.88)

where we have introduced the displacement operators [14] � �� g � g k † � g k �∗ k D ± = exp ± b − bk . ωk ωk k ωk

(2.89)

The ground state (2.88) describes a superposition of the spin projection with the bath oscillators displaced accordingly. It is also interesting to note that Eq. (2.88) describes an entangled state of the spin and the oscillator environment. We now reintroduce the system Hamiltonian, � �= 0, and consider the

dynamics of the independent boson model. To do so we utilise the diagonalisation described above by writing the time evolution operator from time t = 0 to t, U (t) = exp[−iHIB t], in its interaction picture form: U˜ (t) = eiH0 t e−iHIB t

(2.90)

and then introduce copies of the identity, e−S e+S = I, to give U˜ (t) = eiH0 t e−S e+S e−iHIB t e−S e+S = eirt eiH0 t e−S e−iH0 t e+S ˜

= eirt e−S(t) e+S

(2.91)

� ˜ where S(t) = eiH0 t Se−iH0 t = σz k (αk b†k eiωk t − αk∗ bk e−iωk t ). This then allows us to write

� � � � U˜ (t) = eiCt |0��0| D(uk )eiφ + |1��1| D(−uk )eiφ k

(2.92)

k

where uk = αk (1 − eiωk t ), and we have used D(u)D(u� ) = D(u +

u� ) exp[iIm(uu�∗ )] to find the phase φ = −|αk |2 eiωk t . The time evolution operator U˜ (t) clearly commutes with both σz eigenstates. An initial state of this sort is therefore stationary and does not change with time. Let us then consider an initial state of the spin in a σx eigenstate and the bath in thermal equilibrium at inverse temperature β.

2.3 The spin-boson model

32

1.0 1. 0.75 0.5 0.25 0. !0.25 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.8 0.6 "Σx $

0.4 0.2 0.0 !0.2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

t

Figure 2.1: Decay of the coherence �σx � in the independent boson model for temperatures of kB T /ωc = 0, kB T /ωc = 0.5 and kB T /ωc = 1 where the arrow indicates increasing temperature. In the main part of the plot �/ωc = 1 whereas in the inset �/ωc = 2.

We can write this as 1 χ(0) = ρ(0) ⊗ ρB = (I2 + σx ) ⊗ ρB 2 where ρB = e−βHB /Tr[e−βHB ] with HB =

�

k

(2.93)

ωk b†k bk and I2 is an identity

operator in the space of the spin only. The general expression for the evolution of a system operator σi given the initial system-plus-bath state χ(0) is

� � �σi (t)� = TrS+B σ ˜i (t)U˜ (t)χ(0)U˜ † (t)

(2.94)

where σ ˜i (t) is σi in the interaction picture at time t. From Eq. (2.94) with the initial state given by Eq. (2.93) and using σ ˜x (t) = σx cos(�t)−σy sin(�t) we find �σx (t)� =

�� 1 cos(�t) � D(2uk )� + � D(−2uk )� . 2 k k

(2.95)

Details of how to evaluate the bath expectation values on the right hand side of Eq. (2.95) can be found in appendix A. Using the results given

2.3 The spin-boson model

33

there we find �

�σx (t)� = cos(�t) exp − 4 �

= cos(�t) exp − 4

� |gk |2 ωk2

k

�

∞

0

(1 − cos(ωk t)) coth(βωk /2)

�

� J(ω) (1 − cos(ωt)) coth(βω/2)dω (2.96) ω2

where we have introduced the spectral density J(ω) =

�

k

|gk |2 δ(ω − ωk )

which is discussed in more detail in the rest of this section and throughout this thesis. To illustrate the typical behaviour of Eq. (2.96), in Fig. 2.1 we plot �σx (t)� having used J(ω) = ωe−ω/ωc . The frequency ωc is introduced to

ensure the integral remains finite. We see that the coherence drops from 1 to 0, performing small oscillations if the system Hamiltonian strength � is strong enough to overcome the dephasing eﬀect of the bath. Moreover, we see the coherence dies oﬀ faster with increasing temperature.

2.3.2

Weak coupling limit

We now consider the weak system-bath coupling limit of the spin-boson model. As well as providing useful insight into the dynamics, this limit provides us with a perfect setting to apply some of the formalism presented in section 2.2. Our aim is to derive an equation of motion for the reduced density operator ρ describing the spin under the influence of the Hamiltonian given by Eq. (2.83). In this section we assume the zero bias condition � = 0. As in section 2.2 we split the Hamiltonian such that H = HS + HI + HB where now the Hamiltonian acting only on the system is given by HS = − ∆2 σx and the free bath Hamiltonian given by � HB = k ωk b†k bk . This leaves the interaction Hamiltonian HI = σ z

�

(gk b†k + gk∗ bk )

(2.97)

k

which we will treat as the perturbation. Decomposition into eigenoperators of HS is relatively straightforward since Eq. (2.97) contains just one

2.3 The spin-boson model

34

operator acting on the system. We find 1 A(+∆) = (σz − iσy ) 2 1 A(−∆) = (σz + iσy ) 2

(2.98) (2.99)

where we have used Eq. (2.54). It can be verified that exp[iHS t]A(±∆) exp[−iHS t] = A(±∆) exp[∓i∆t]

(2.100)

in accordance with Eq. (2.58). Bath correlation functions Having established the system operators that will appear in the master equation we must now determine the decay rates and energy shifts, both of which are found through the bath correlation functions. For the spinboson model in the form we are considering, the interaction term contains just one bath operator Bz = Bz† =

�

(gk b†k + gk∗ bk )

(2.101)

k

and we therefore have one corresponding correlation function that must be calculated: ˜z (τ )B ˜z (0)�. Λzz (τ ) = �B

(2.102)

Bringing Bz into the interaction picture is performed by recalling that [bk� , b†k ] = δkk� and one arrives at the double summation Λzz (τ ) =

� k,k�

�(gk b†k eiωk τ + gk∗ bk e−iωk τ )(gk� b†k� + gk∗ � bk� )�.

(2.103)

For independent bath modes we have �bk bk� � = �b†k b†k� � = 0 while �b†k bk� � =

δkk� n(ωk ) and �bk b†k� � = δkk� (n(ωk ) + 1) where n(ωk ) = (eβωk − 1)−1 is the

Bose occupation number [7]. Using these relations we obtain Λzz (τ ) =

� k

� � |gk |2 n(ωk )eiωk τ + (n(ωk ) + 1)e−iωk τ

(2.104)

2.3 The spin-boson model

35

As in section 2.3.1, we now define the system-bath spectral density [11] J(ω) =

� k

|gk |2 δ(ωk − ω)

(2.105)

which allows the summation in Eq. (2.104) to be converted into an integral, Λzz (τ ) =

�

∞ 0

� � dωJ(ω) n(ω)eiωτ + (n(ω) + 1)e−iωτ .

(2.106)

The system-bath spectral density plays a crucial role in any models described by a Hamiltonian having the form of Eq. (2.83). The spectral density completely captures the nature of the system-bath coupling under consideration. In practice it is often assumed that the oscillators in the bath are dense enough such that it can be well approximated by a smooth function of ω. Its functional form will then be determined by the specific system and bath which one wishes to model. We note that from Eq. (2.105) it is possible to see that the spectral density contains information regarding both how strongly each mode couples to the system of interest, through |gk |2 , and also the density of oscillators in the bath with that particular frequency, implicitly through the summation over k. For our current purposes it is not necessary to specify J(ω) and we leave the integration in Eq. (2.106) unevaluated. Weak coupling rates and Lamb-shift terms We are now in a position to evaluate the rates appearing in the weak coupling master equation, Eq. (2.79). With Eq. (2.106) we find �

γzz (ω ) =

�

+∞

�

eiω τ Λzz (τ )dτ −∞

= 2π

�

0

+∞

� � dωJ(ω) n(ω)δ(ω + ω � ) + (n(ω) + 1)δ(ω − ω � )

(2.107)

where use of the identity

�∞

−∞

dτ eiωτ = δ(ω) has been made. Evaluating

this expression at the relevant frequencies gives γzz (+∆) = γ0 (n(∆) + 1)

(2.108)

γzz (−∆) = γ0 n(∆)

(2.109)

2.3 The spin-boson model

36

where γ0 = 2πJ(∆) is the single spin decoherence rate at zero temperature. As expected from Eq. (2.74) these weak coupling rates satisfy γzz (−∆) = e−β∆ γzz (∆). At this stage we can identify γzz (+∆) as the emission rate since it remains finite at zero temperature owing to spontaneous emission processes. The absorption rate, γzz (−∆), on the other hand, vanishes in the zero temperature limit reflecting that the environment cannot put energy into the system. An important point regarding both of these rates is that they contain the spectral density evaluated at, and only at, the bare tunnelling strength, ∆. This can be attributed to the Markovian assumption made in deriving the master equation. Without this the delta functions in Eq. (2.107) would not be present and potentially the full frequency spectrum of the system-bath interaction could influence γzz (ω). With Eqs. (2.99) and (2.76) it is possible to see that the Lamb-shift Hamiltonian for the spin-boson model within the current weak-coupling treatment is given by � 1 � � 1 � HLS = I S(∆) + S(−∆) − σx S(∆) − S(−∆) . 2 2

(2.110)

The first term in Eq. (2.110) is proportional to the identity and as such has no eﬀect on the dynamics of our spin since it commutes with ρ. The second term however represents a renormalisation of the tunnelling strength due to the interaction of our spin with the bath. This can be seen from Eq. (2.79) where it is clear that HLS is added to HS when the master equation is written in the Schr¨odinger picture. This creates an eﬀective system Hamiltonian Heﬀ , � 1 � ∆eﬀ Heﬀ = HS + HLS = − σx ∆ + (S(∆) − S(−∆)) = − σx 2 2

(2.111)

where ∆eﬀ = ∆ + λ is the renormalised tunnelling strength with λ = S(∆) − S(−∆). The renormalisation of the tunnelling strength can be intuitively understood as a consequence of the oscillators in the bath ‘dragging’ behind the spin down as it tunnels. Using Eq. (2.71)] we can write the frequency renormalisation in integral form as λ = 2∆P

�

∞ 0

dω

J(ω) coth(βω/2) , ∆2 − ω 2

(2.112)

2.3 The spin-boson model

37

where we have used the identity �

∞

dτ eiωτ = πδ(ω) + P

0

i ω

(2.113)

and P indicates that the Cauchy principal value should be taken. Bloch vector evolution We are now in a position to write down the equation of motion for the reduced density operator in the Schr¨odinger picture. Owing to the somewhat complicated form of the master equation, we shall not write it out explicitly here. Instead we shall write down the Bloch equations which represent the equations of motion for the Bloch vector introduced in section 2.1.3. Substituting ρ = (1/2)(I + σ · α) in the right hand side of Eq. (2.79)

the Bloch equations can be found by multiplication by the appropriate

Pauli operator and taking the trace: α˙ i = Tr(σi ρ). ˙ We find an equation of motion for α of the form ˙ =M ·α+b α where



 M =

−ΓW 0 0

(2.114)

0

0

−(1/2)ΓW

∆eﬀ

−∆eﬀ

−(1/2)ΓW

  

(2.115)

with the weak coupling rate given by ΓW = γzz (∆) + γzz (−∆) and b = (κ, 0, 0) with κ = γzz (∆) − γzz (−∆).

Let us first consider the spin evolution in the absence of its coupling

to the environment. This corresponds to setting all gk appearing in the original Hamiltonian (2.83), to zero. With this we find ΓW ∼ γ0 = 0 and λ = 0 (which sets ∆eﬀ = ∆). The terms remaining correspond to the parametric forms for circular motion with angular frequency ∆ in the y-z plane. We conclude that any initial Bloch vector state will simply precess about the x-axis. Reintroducing the coupling to the bath, the steady state can be found ˙ = 0. One finds αy (∞) = αz (∞) = 0 while αx (∞) = by solving α tanh(β∆/2). The dynamics itself is obtained using Eq. (2.114) to con-

2.3 The spin-boson model

38

struct a second order equation for the evolution of αz . Doing so gives � � 1 α ¨ z + ΓW α˙ z + ∆2eﬀ + Γ2W αz = 0, 4

(2.116)

and it can be seen that the introduction of the system-bath coupling has changed our system from an undamped to a damped oscillator. A natural question to ask is whether the system-bath coupling can be strong enough such that the system becomes over-damped and incapable of showing coherent oscillations. Solving Eq. (2.116) gives −(1/2)ΓW t

αz (t) = e

�

C1 e

i∆eﬀ t

+ C2 e

−i∆eﬀ t

�

(2.117)

where C1 and C2 are constants dependent on the initial conditions. It can be seen that the coeﬃcients in Eq. (2.116) are such that its (non-trivial) solutions always contain oscillatory components. Only when ∆eﬀ = 0 (∆ = −λ) does the weak-coupling treatment presented here predict purely

incoherent relaxation for αz and this can be attributed to the rotating wave approximation made in derivation of the general master equation (2.79). For the typical initial condition α = (0, 0, 1)T , Eq. (2.117) becomes αz (t) = e−(1/2)ΓW t cos(∆eﬀ t),

(2.118)

showing clearly the population diﬀerence performing damped oscillations at a renormalised frequency. Generally speaking, behaviour of this sort will often be referred to as coherent relaxation whereas behaviour showing no signs of oscillation will be referred to as incoherent. A more thorough discussion of diﬀerent types of behaviour constitutes a large part of chapter 6.

2.3.3

The noninteracting blip approximation (NIBA)

Having investigated the spin-boson model within the Born-Markov weakcoupling limit, we now give a brief account of results obtained using what is known as the noninteracting blip approximation (NIBA), as pioneered by Leggett et al [11]. The starting point of this analysis is the exact expression for the population diﬀerence (αz ) as a function of time in its path integral form. In the path integral formalism the amplitude for a process depends on the action of all paths connecting the start and end

2.3 The spin-boson model

39

points in configuration space [15]. Since the population is a probability, it therefore depends on the action of all pairs of paths. In the terminology used in Ref. [11], a “sojourn” corresponds to a period of time that the pair of spin states are the same, while a “blip” describes a period of time that the spin states are diﬀerent. The NIBA assumes that on average blips will be much shorter lived than sojourns which allows the resulting expressions to be enormously simplified. The justification for the NIBA and the associated level of error greatly depends on the specific parameters of the problem under consideration. We will not give a full account here. Instead we will consider a few examples which highlight both the power of the NIBA and one of its failings. Despite having its roots elsewhere, it is possible to derive the NIBA expression for αz (t) from master equation techniques similar to those already seen in this chapter. The derivation itself will be left until chapter 4 since it uses many of the ideas developed there. For now we simply summarise the results obtained within the NIBA. Results obtained within the NIBA (and elsewhere) depend critically on the form of spectral density. We will generally consider spectral densities of the form J(ω) = (As /2)ω s ω01−s e−ω/ωc

(2.119)

where As is a dimensionless quantity capturing the strength of the system bath interaction, ω0 is a is typical phonon frequency introduced to ensure As is indeed dimensionless. The frequency ωc provides a high frequency cut-oﬀ ensuring vacuum contributions to correlation functions remain finite. Dynamics The special case of s = 1 is known as Ohmic damping since the corresponding equation of motion in the position basis has a frequency independent damping coeﬃcient. With Ohmic damping, zero bias and within the scaling limit (ωc → ∞) the NIBA provides results that are known to be exact

in certain limits. At zero temperature it is found that the population diﬀerence goes through a crossover from coherent damped oscillations to incoherent relaxation as the coupling strength A1 moves through the critical value A1 = 0.5. At finite temperature this crossover moves down to a

2.3 The spin-boson model

40

value 0 ≤ A1 < 0.5. The first of these results is derived in section 4.2.3.

For 1 < s ≤ 3 (super-Ohmic) and zero bias the behaviour of the popu-

lation diﬀerence is that of damped oscillations for low temperatures, with a crossover to incoherent relaxation as the temperature is increased. For 0 < s < 1 (sub-Ohmic), at zero temperature the eﬀective field experienced by the spin aligns along the z-axis and the spin is said to be localised. At finite temperatures the spin relaxes incoherently. Steady state in the NIBA For finite � the NIBA predicts the steady state value αz (∞) = − tanh(β�/2)

(2.120)

valid for an arbitrary spectral density. The steady state value of αz given by Eq. (2.120) thus shows a failure of the NIBA which arises for nonzero � at low temperatures. In the limit that T → 0 the NIBA predicts symmetry breaking and strict localisation of the spin, even for infinitesimal �. Moreover, this prediction is independent of the strength of the system bath interaction. This seems intuitively incorrect since one would expect αz (∞) at zero temperature to depend on both � and ∆. This intuition can be put on a more quantitative footing by considering the steady state value of αz in the weak coupling limit. If we take HS = (�/2)σz − (∆/2)σx and assume that in the long time limit the eigenstates

of HS will be thermally occupied, we have for the system steady state density matrix ρ(∞) =

1 exp[−βHS ] Z

(2.121)

where Z = Tr{exp[−βHS ]} is the system partition function. The steady state value of αz is then αz (∞) = �0| ρ(∞) |0�−�1| ρ(∞) |1�. By evaluating

the eigensystem of HS one then finds

� αz (∞) = − tanh(βη/2) η with η =

√

(2.122)

�2 + ∆2 being the resultant field formed by vector addition of

the bias and the tunnelling strength. Although Eq. (2.122) was obtained through physical arguments only, it has the same form as the expression obtained through more rigorous techniques found in Ref. [10].

2.3 The spin-boson model

2.3.4

Ground state properties

We now give a brief account of the ground state properties of the spinboson model. The central feature here is that the coupling of the spin to the bath causes a reduction in the eﬀective tunnelling element. The question is then how the reduced tunnelling element behaves as a function of coupling strength and temperature. For an Ohmic bath, as described by Eq. (2.119) with s = 1, it is known that for an unbiased system (� = 0), at zero temperature the spin-boson model goes through a quantum (rather than thermodynamic) phase transition of Kosterlitz-Thouless type as the coupling strength moves through A1 = 1 + ∆/ωc [16,17]. For coupling strengths below this value the ground state as a superposition of spin up and spin down eigenstates and is said to be delocalised. For coupling strengths above this value there exists a broken symmetry localised phase where the tunnelling element is renormalised to zero and the spin becomes locked in either of its σz eigenstates: �σz � = � 0. This localisation phenomena for a single spin, as well as for two spins in a common bath will be considered in greater detail in chapter 7.

The entropy of entanglement between the spin and the environment has also been studied in this context for a biased system [18]. It is found that the entanglement increases as the coupling is increased from zero and then begins to decline as the system approaches the localised regime. Once in the localised regime the spin state becomes pure and entirely disentangled from the oscillator bath. The inclusion of a tunnelling term in Ref. [18] discussed here means the system-environment entanglement shows far more interesting features than that discussed in section 2.3.1 in the context of the independent boson model. For sub-Ohmic baths (s < 1) the spin-boson model is also predicted to exhibit a quantum phase transition. This has been shown using an mean field type variational approach [19], as well as by numerical renormalisation group [20] and density matrix renormalisation group methods [21]. The exact nature of the phase transition in this sub-Ohmic case has however proved to be somewhat more subtle than in the Ohmic case. Only recently have the critical exponents at the phase transition been identified by quantum Monte-Carlo [22] and exact diagonalisation techniques [23]. Work presented in this thesis is exclusively concerned with Ohmic and super-Ohmic spectral densities so we do not give any further details of the

41

2.3 The spin-boson model sub-Ohmic case here.

42

Chapter 3 Entanglement induced by a spatially correlated thermal bath Contents 3.1 3.2 3.3

3.4 3.5 3.6

Introduction . . . . . . . . . . . . . . . 3.1.1 Motivation . . . . . . . . . . . . . . Model and master equation . . . . . . . 3.2.1 Bath correlation functions and rates State Dynamics . . . . . . . . . . . . . 3.3.1 Generalisation of the Bloch vector . 3.3.2 Liouvilian spectrum . . . . . . . . . Entanglement Dynamics . . . . . . . . 3.4.1 Entanglement generated through the Experimental Realisation . . . . . . . . Summary . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lamb-shift . . . . . . . . . . . . . . .

. . . . .

44 44 46 47 49 49 50 52 55 57 61

3.1 Introduction

3.1

Introduction

In this chapter the results of a weak-coupling treatment applied to a twospin-boson model are presented [1]. As well as being of general interest, this chapter provides us with a good example of an application of the weak-coupling theory to a system more complicated than the single spinboson model, as studied in section 2.3.2. As we shall see, the analytic forms of various quantities which the weak-coupling assumption allows one to find makes much of the resulting dynamics easier to understand and approximate. The main result found in this investigation is that two non-interacting, spatially separated spins of the type studied in section 2.3.2 can become entangled through their mutual interaction with a common environment. We therefore first give a brief summary of why this constitutes an interesting result, particularly in the field on quantum computing.

3.1.1

Motivation

Entanglement is the hallmark of correlations in quantum theory and has come to be seen as a precious resource essential for many quantum information processing protocols [5]. An entangled state can show correlations stronger than those allowed classically, however, these correlations are often extremely fragile. In chapter 2 we saw how interactions with the environment surrounding a quantum system tend to cause a loss of quantum coherence. This generally also leads to entanglement within the system being destroyed on a short timescale [24]. Various techniques have thus been developed to protect entangled states from their surroundings, such as constructing decoherence free subspaces [25,26], dynamical decoupling [27, 28], and exploiting the quantum Zeno eﬀect [29]. As well as avoiding decoherence, entanglement must also be generated. This can be achieved by a number of means; for example, by harnessing intrinsic system couplings [30] (this was seen in section 2.1.1), through projective measurements [31], or via a quantum bus [32]. Recently, it was shown that entanglement between a pair of two-level systems can in fact be generated by the same processes that are usually considered to be detrimental, if the systems are allowed to interact with a common bath [33, 34]. This oﬀers a potential way to explore the interplay between

44

3.1 Introduction coherent and incoherent multi-spin dynamics with a significantly reduced level of external system control. In general, immersing a pair of otherwise non-interacting spins in a common heat bath will give rise to two terms in the subsequent master equation: a unitary, Hamiltonian-like Lamb-shift term leading to coherent spin evolution, and a dissipative term. In section 2.2.4 this was seen within a weak-coupling treatment. It has been shown that both of these terms may generate spin entanglement [33–37]. While it is known that in the idealised case of unseparated spins entanglement induced by the dissipative term can persist indefinitely [38], less attention has been paid to the dynamics of its generation and decay in the more realistic setting of finite inter-spin separation. In this experimentally relevant case, it is important to establish what timescale the generated entanglement persists, and whether its observation is feasible at all. It has been shown that the entangling capability of the Lamb-shift is highly sensitive to the inter-qubit separation [35], with dissipative processes destroying any generated entanglement more rapidly as separation increases. Furthermore, it has recently been shown that the level of entanglement generated between two harmonic oscillators suﬀers from a similar critical dependence on oscillator separation [39]. Hence, a comparison of the timescales associated with conventional decoherence dynamics to those for dissipatively-induced spin-entanglement generation and decay is needed. In this chapter, the above issues are addressed by studying the dynamics of bath-induced entanglement in the context of the two-spin-boson model, which has wide applications in the solid state and elsewhere [11,40]. The novel result found here is that for small but finite spin separation, the timescale on which dissipatively-induced spin entanglement survives can be far larger than the corresponding single spin decoherence time. It also exceeds the timescale on which entanglement induced by either a direct exchange interaction or the Lamb-shift persists. In particular, we obtain an approximate analytic expression for the long-time dynamics of the twospin concurrence and from this determine its survival time. In the final section an ion trap realisation of our model is suggested in which observation of the generated entanglement should be feasible.

45

3.2 Model and master equation

3.2

46

Model and master equation

Throughout this chapter we study the two-spin-boson model.

In the

present context this consists of two spatially separated, identical, noninteracting spins, each subject to a static tunnelling strength ∆/2 causing transitions between the states within each spin. The spins are coupled to a common bath of harmonic oscillators. Owing to the particular relevance of the results obtained in this chapter to the field of quantum computation, the spins will often be referred to as qubits. The Hamiltonian is given by H = HS + HI + HB 2 2 � ∆� n � n = − σx + σz ⊗ B n + ωk b†k bk . 2 n=1 n=1 k

(3.1)

Here, σin is a Pauli operator acting on the nth qubit (n = 1, 2; i = x, y, z), Bn is the bath operator coupling qubit n to the bath, and as usual ωk is the angular frequency, and b†k (bk ) the creation (annihilation) operator of the bath mode of wave vector k. To investigate the dynamics of the reduced two-spin density operator ρ we assume that the qubit-bath coupling is weak compared to ∆ and follow the standard Born-Markov and rotating-wave approximation approach, precisely as presented in section 2.2. Setting �HI �B = 0, we obtain a Schr¨odinger-picture master equation of the usual form

dρ(t) = −i[HS + HLS , ρ(t)] + D(ρ(t)), dt

(3.2)

valid to second order in HI . Here, the Lamb-shift provides a Hamiltonianlike contribution and is of the form HLS = A(σx1 + σx2 ) + B(σz1 σz2 + σy1 σy2 ),

(3.3)

where a constant has been omitted. The precise expressions for A and B depend on the details of the qubit-bath interactions and will be given in section 3.2.1. For now we will comment on their likely qualitative eﬀects. The first term of HLS simply renormalises the static field strength due to the presence of the bath modes. The entangling capability of the second term deserves some attention since it represents an induced interaction

3.2 Model and master equation

47

between the two qubits. However, we will show that in general this entanglement decays at a rate far quicker than that generated by the dissipator, here given by D(ρ(t)) =

2 � � ω

� � 1 γnm (ω) Am (ω)ρ(t)A†n (ω) − {A†n (ω)Am (ω), ρ(t)} 2 n,m=1

(3.4)

with frequency summation over the eigenvalue diﬀerences of HS (ω = ±∆) and corresponding eigenoperators An (±∆) = (1/2)(σzn ∓ iσyn ).

3.2.1

Bath correlation functions and rates

The key quantities in the present discussion are the separation dependent rates, defined in our usual notation as γnm (ω) =

�

∞ −∞

˜n† (τ )B ˜m (0)�, dτ eiωτ �B

(3.5)

and for which we need a specific form for Bn . As usual in the spin-boson model, we consider linear coupling between the qubits and the coordi˜n (τ ) = � (g n b† eiωk τ + nate of each bath mode [11, 12, 40], such that B k k k gkn∗ bk e−iωk τ ), with coupling constants gkn . Note that with this form of Bn

our assumption of �HI �B = 0 leading to Eq. (3.2) is justified. An important aspect of this work is that the qubits have an explicit spatial separation

and thus gk1 �= gk2 . To make this evident, we consider our spins to be separated by a distance d along the z-axis such that gk1 = gk e(idkcosθ)/2

and gk2 = gk e−(idkcosθ)/2 , where θ is the polar angle measured against the z-axis in k-space, and |gk1 | = |gk2 | = gk . Taking the bath to be in thermal equilibrium, we find γ12 (ω) =

�

∞ −∞

dτ eiωτ

� k

� |gk |2 n(ωk )eiωk τ eikd cos θ + (n(ωk ) + 1)e

−iωk s −ikd cos θ

e

�

, (3.6)

where n(ωk ) = [exp(ωk /kB TB ) − 1]−1 is the thermal occupation of mode k, kB is Boltzmann’s constant, TB the temperature of the bath, γ12 (ω) = γ21 (ω) (once the summation is performed), and γnn (ω) is obtained by setting d = 0.

3.2 Model and master equation

48

Performing the summation over k in Eq. (3.6) requires some care since ˜1 (τ )B ˜2 (0)� contains an angular depenthe correlation function Λ12 (τ ) = �B

dence in k-space and will therefore depend on the dimensionality of the system-bath interaction. Details of how to deal with summations of this type can be found in appendix B. Defining the bath spectral density to � be J(ω) = k |gk |2 δ(ω − ωk ), and an inverse dispersion relation (assumed isotropic) k = |k| = κ(ω), we take the continuum limit of the summation over k above to find

γ12 (ω) = FD (κ(ω)d)γ11 (ω),

(3.7)

with γ11 (+∆) = γ22 (+∆) = (n(∆) + 1)γ0 and γ11 (−∆) = γ22 (−∆) = n(∆)γ0 , where γ0 = 2πJ(∆) is the single-spin decoherence rate at zero temperature already encountered in section 2.3.2 in the discussion of the decay of a single spin. Here, FD (x) describes the bath’s spatial correlations and is determined by its dimensionality D(= 1, 2, 3). For D = 1 we have F1 (x) = cos(x); for D = 2, F2 (x) = J0 (x), where J0 is a Bessel function of the first kind; and for D = 3, F3 (x) = sinc(x). We thus write γ12 (±∆) = (1 − δ)γ11 (±∆),

(3.8)

where 1 − δ captures the degree of correlation between the baths seen by each qubit, becoming unity at d = δ = 0 (completely correlated) and, for

D > 1, zero as d → ∞ (δ → 1, completely independent). When D = 1, the Markovian assumption constrains γ12 to be periodic with respect to

d. However, we will concentrate here on the limit where d is small enough such that δ ≈ (κ(∆)d)2 /2D, for all D.

From the general expression for the Lamb-shift Hamiltonian within a

Born-Markov weak-coupling treatment, Eq. (2.76), the strength of the two terms in the Lamb-shift Hamiltonian are found to be � � � ∞ � � ∆ A=2 J(ω)coth ω/(2kB TB ) dω ∆2 − ω 2 0 and B=

�

∞ 0

J(ω)f (κ(ω)d)

�

ω 2 ∆ − ω2

�

dω

(3.9)

(3.10)

where principal values are assumed. Note that for system-bath coupling

3.3 State Dynamics

49

in 2 or 3 dimensions B → 0 as the qubit separation is increased to infin-

ity, expressing the fact that an uncorrelated bath cannot give rise to any coherent coupling between the qubits. In contrast, A contains no distance dependence since it represents a renormalisation of the single-qubit energy levels in each spin, independent of any bath correlations. The relative strength of the coherent terms in the evolution, A and B, compared to the strength of the dissipative terms, given by γ11 and γ12 , dictates whether the bath is capable of generating entanglement through the Lamb-shift [35]. Evaluation of the relevant integrals involved necessitates a specific form of the bath spectral density. In this chapter we shall focus on the entanglement generated through dissipative processes and as such we may leave A and B unevaluated. In fact, we shall show in section 3.4.1 that the dissipatively induced entanglement can persist for times far larger than entanglement generated through the Lamb-shift, regardless of its strength.

3.3 3.3.1

State Dynamics Generalisation of the Bloch vector

Rather than working directly with the reduced density operator ρ, it is more instructive to work in terms of a 16 dimensional vector α, which is a generalisation of the Bloch vector for two-qubit states. It is constructed by flattening the matrix whose elements αij satisfy 3 1� ρ= αij σi1 ⊗ σj2 , 4 i,j=0

(3.11)

where σ01 = σ02 = I. In analogy with the discussion of a single spin Bloch vector in section 2.1.3, we see that the traceless property of the Pauli matrices ensures that in the present case we have αij = �σi1 σj2 �,

and conservation of probability demands that α00 = 1. To describe the evolution of our system, we consider the eigensystem of the Liouvillian super-operator L defined in this case to operate on α rather than the ˙ density operator itself: α(t) = Lα(t). The linearity of the transformation

between ρ and α ensures that the dynamics generated by L is equivalent to that of Eq. (3.2). Clearly, a state α initially in an eigenstate of L, say

3.3 State Dynamics

50

αl (with eigenvalue λl ), will evolve according to α(t) = αl eλl t . Hence, a general state evolves such that α(t) =

15 �

al α l e λ l t ,

(3.12)

l=0

where the coeﬃcients al are determined from the initial conditions, and the sum runs over all eigenstates of L.

3.3.2

Liouvilian spectrum

In this chapter we are primarily interested in the long-time dynamics of our system. Hence, it makes sense to search for eigenvalues of L with small (or zero) real parts since, assuming these parts are all negative, the corre-

sponding eigenvectors will contribute towards the total state Eq. (3.12) on the largest timescale. We evaluate the full eigensystem of L analytically, though this leads to cumbersome expressions which we shall not give here.

However, it is possible identify a number of important features for the subsequent analysis. For any non-zero qubit separation (δ �= 0) there exists a

single eigenvector, α0 , with zero eigenvalue, and all other eigenvalues have negative real parts. We therefore associate α0 with the thermal state since it is that to which all states tend towards as t → ∞. Of the remaining 15, there is a single eigenvalue, λ1 (with corresponding eigenvector α1 ), which

vanishes as δ → 0 at all temperatures. Expanding the exact expression to first order in δ and second order in n(∆) gives the simple form λ1 = −(1 + 3n(∆))δγ0 .

(3.13)

It is also possible to show graphically that λ1 varies approximately linearly with n(∆) at all temperatures. For reasons that should become clear, we shall refer to the eigenvector corresponding to the null eigenvalue, α0 , and the eigenvector corresponding to the vanishing eigenvalue, α1 , as our eigenvectors of interest. The eigenvector corresponding to the thermal state, α0 , is expressible solely in terms of R = (1 + 2n(∆))−1 = tanh(∆/2kB TB ), and is given by α0 : {α00 , α01 , α11 , α22 } = {1, R, R2 , 0},

(3.14)

3.3 State Dynamics

51

where the notation implies that the eigenvector has the elements specified, and that α22 = α33 , α01 = α10 , with all other elements being zero. From a numerical analysis of α1 we find that for δ � 1, to a very good approxi-

mation α1 can also be expressed just in terms of R, with corrections being of the order of δ: α1 : {α00 , α01 , α11 , α22 } ≈ {0, R, 1 + R2 , 1},

(3.15)

where the notation is the same as in Eq. (3.14). There are two further notable eigenvalues, λ2 and λ3 = λ∗2 . Once again, expanding the exact eigenvalues to first order in δ, and to first order in R about R = 1, gives the expression λ2 = λ∗3 = −(1/2)γ0 (1 − R + 2δ − δR) − i∆.

(3.16)

Note that these eigenvalues have vanishing real parts only in the limit that both the separation and temperature go to zero (δ → 0 and R → 1). In either the zero temperature limit (R → 1) or the zero separation limit (δ → 0) the corresponding eigenvectors, α2 and α3 = α∗2 , are given by α2 : {α02 , α03 , α12 , α13 } = {i, 1, i, 1},

(3.17)

where once again the notation implies the eigenvector has the elements specified but this time with α20 = −α02 , α30 = −α03 , α21 = −α12 and α31 = −α13 , and all other elements being zero.

This leaves us with 12 eigenvalues to consider. Plotting their real parts

as a function of R it becomes clear that they are all ∼ −γ0 /R, for all values of δ. As such, the corresponding eigenvectors contribute towards the total state significantly only for times t < γ0−1 regardless of the temperature. These eigenvalues and eigenvectors shall be referred to as those with l > 3. With the relative size of the real parts of the various eigenvalues in mind, we see that for small enough δ, at times suﬃciently greater than γ0−1 , a general state can be approximated by ∗

α(t) = α0 + a1 α1 eλ1 t + a2 (α2 eλ2 t ± α∗2 eλ2 t ),

(3.18)

where we have normalised α0 , set a0 = 1, and set a3 = ±a2 to ensure pos-

3.4 Entanglement Dynamics

52

itivity of the corresponding density operator. The coeﬃcient a1 is found, by setting t = 0 in Eq. (3.12), to be a1 = (Λ − R2 )/(3 + R2 ), where

Λ = �σ 1 · σ 2 � = α11 + α22 + α33 . For separable pure states Λ represents a

scalar product of single qubit Bloch vectors. Positivity of the corresponding density operator limits the range of Λ to −3 ≤ Λ ≤ +1.

To gain insight into the general features of a state described by

Eq. (3.18), it is useful to write down the corresponding density operator in the zero temperature and separation limit, in which there are no decoherent processes due to the real parts of the relevant eigenvalues vanishing. Using Eq. (3.11) we find � �� ρ(t) = (1 + a1 ) |↑↑�x�↑↑|x − a1 �Ψ− x Ψ− �x √ � � � � � � + a2 2 e−i∆t �Ψ− x�↑↑|x ± ei∆t |↑↑�x Ψ− �x ,

(3.19)

√ √ where |Ψ− �x = (1/ 2)(|↑↓�x − |↓↑�x ) = (1/ 2)(|↑↓�z − |↓↑�z ) is the Bell singlet, and |↑↑�x = (1/2)(|↑↑�z + |↓↓�z + |↑↓�z + |↓↑�z ). The subscripts

x and z refer to the relevant basis, with our qubits defined with respect to z. Written in this way, we can see that our state is a coherent mixture of the singlet and the state |↑↑�x . At zero temperature, the state

|↑↑�x is the ground state since it minimises the energy associated with the system Hamiltonian and energy cannot be absorbed from the envi-

ronment. Also, at zero separation, the Bell singlet is stable since it is the totally anti-symmetric state, while the Hamiltonian is totally symmetric [41]. Therefore, in a combination of these limits coherent mixtures of these states are stable. However, the states are at diﬀerent energies which gives rise to the exponential factors in Eq. (3.19).

3.4

Entanglement Dynamics

We are now in a position to say that, provided our qubits are suﬃciently close together such that δ � 1, for times t > γ0−1 contributions from eigenvectors with l > 3 and their associated dynamics will have all but

vanished, and to a good approximation (and a better approximation as time increases), our two-qubit state will be well described by Eq. (3.18). To quantify the resulting entanglement dynamics we use Wootters concurrence [42], which ranges from 0 for separable states to 1 for maximally

3.4 Entanglement Dynamics

53

entangled states. This is calculated as follows. For a general two-qubit state ρ the spin-flipped state is defined: ρ¯ = (σy ⊗ σy )ρ∗ (σy ⊗ σy ).

(3.20)

Using this the eigenvalues ci of the non-hermitian matrix ρ¯ ρ are found. The concurrence of the state ρ is then defined as √ √ √ √ C = max( c1 − c2 − c3 − c4 , 0)

(3.21)

where the eigenvalues are in decreasing order. It can be shown numerically that the concurrence of a state described by Eq. (3.18) depends only very weakly on the magnitude of a2 , and in the zero temperature and separation limit has completely vanishing dependence. We may therefore set a2 = 0 to derive a simple concurrence expression, and from Eqs. (3.14), (3.15) and (3.18) find C = max

� (R2 − 1)(R2 + 3) + (R2 − Λ)(3 − R2 )eλ1 t 2(R2 + 3)

� ,0 ,

(3.22)

valid (with increasing accuracy) for timescales t > γ0−1 . The legitimacy of setting a2 = 0 will be demonstrated towards the end of this section, where comparisons with numerics using the full eigensystem of L are made.

Whether the bath is capable of inducing spin entanglement, and if

so how it subsequently decays, is now clear. Firstly, no entanglement is generated unless Λ<

5R2 − 3 , 3 − R2

(3.23)

in agreement with Ref. [38]. Secondly, provided this inequality is satisfied, we see from Eq. (3.22) that the induced entanglement will decay exponentially until (R2 − Λ)(3 − R2 )eλ1 t = −(R2 − 1)(R2 + 3) is satisfied, after which time the entanglement is zero. This occurs at

� 2 � 1 (R − Λ)(R2 − 3) tc = ln , |λ1 | (R2 + 3)(R2 − 1)

(3.24)

as demonstrated in Fig. 3.1. Note that Eqs. (3.22) and (3.24) are valid for a range of temperatures, however, in view of the inequality of Eq. (3.23), we will focus on small temperatures since these maximise the amount and life-

3.4 Entanglement Dynamics

54

1.0 R 0.5 0.0 0.4 0.3 C 0.2 0.1 0.0

0.0 0.5 !Λ1 !t

1.0 1.5

Figure 3.1: Concurrence of the initially separable state |↑↓� in either x, y, or z (Λ = −1) as a function of time (scaled by λ1 ) and R = tanh(∆/2kB TB ) calculated from the full Liouvillian, though ignoring the Lamb-shift.

time of any induced entanglement. Interestingly, in the limit of vanishing temperature (R → 1), the entanglement reaches zero only asymptotically.

Also, as the qubit separation d → 0 the level of entanglement becomes a

function of the initial state only and tc → ∞, i.e. the steady-state becomes entangled [38]. In general, tc varies as (δγ0 )−1 for given R and Λ, hence

it can be lengthened by increasing the ratio ∆/TB , or by decreasing the separation d. We would naturally like to know which initially separable states result in the largest generated concurrence. For fixed temperature and spin separation the only parameter to consider is Λ, and from inspection of Eq. (3.22) we see that it should be minimised. This corresponds to an initial state that is as anti-symmetric as possible; hence, Λ is minimised by anti-aligning the single spin Bloch vectors, giving Λ = −1 for pure states.

Such a state corresponds to |↑↓� in x, y or z. As the states become more mixed, the Bloch vectors decrease in length and Λ → 0. Interestingly,

even a maximally mixed state (Λ = 0) can become entangled provided � ∆ > 2 coth−1 ( 5/3)kB TB . In general, we expect an initially separable state to reach its maximum entanglement after a time ∼ γ0−1 , typical of the decay of eigenvectors αl for l > 3. Note that if we do not restrict our

initial state to a separable state, Λ is minimised by the Bell singlet, for

3.4 Entanglement Dynamics which Λ = −3. As we have mentioned, in the limit that the qubit separation goes to zero the singlet is able to maintain its full entanglement for all times. To illustrate these points, in the main part of Fig. 3.2 we plot the time evolution of the concurrence for various initial states, calculated both from Eq. (3.22) (dashed lines) and numerically using the full Liouvillian (solid lines), though here ignoring the Lamb-shift. As claimed, on timescales > γ0−1 ∼ δ in the scaled time units, the entanglement dynamics is well approximated by the analytic form. Note also that neglecting the eigenvectors α2 and α3 has had no discernible eﬀect on the accuracy of Eq. (3.22) on the timescales it is expected to be valid. For the initially separable states (Λ = −1, 0), we clearly see that the entanglement reaches its max-

imum on a timescale > γ0−1 ∼ δ, decaying subsequently at a rate ∼ δγ0 . For the Bell singlet (Λ = −3), the analytic approximation becomes almost

exact since this state is simply a linear combination of the vectors α0 and α1 in Eqs. (3.14) and (3.15). Also shown is the behaviour of the Bell √ state (1/ 2)(|↑↓�z + |↓↑�z ), for which Λ = 1. Unlike the singlet, all other

Bell states have the maximum possible value of Λ, and as such lose their entanglement rapidly. We could consider the maximally mixed state (Λ = 0) as being the infinite temperature thermal state since it represents a state for which thermal fluctuations completely overcome any external fields. We see that as the bath “cools” this state towards the thermal state at TB , it does so via entangled states. Of course, after a time tc the state of the qubits becomes separable once more, and will eventually reach the thermal state at TB . There is in fact a well defined condition for TB and (initially prepared) qubit temperature TQ such that the bath has the ability to entangle the qubits. In the limit of small bath and qubit temperature this condition becomes θB − θQ > 21 ln3, where θB = (∆/2kb TB ) and θQ = (∆/2kb TQ ).

3.4.1

Entanglement generated through the Lambshift

It is important to consider the role of the Lamb-shift Hamiltonian HLS , thus far ignored. Within the limits of our derivation, namely that ∆ is large and hence rotations about x are so rapid that the y- and z- directions

55

3.4 Entanglement Dynamics

56

1.0 0.6 0.5

0.8

!"#3

0.4 0.3 0.2

!"1 0.6

0.1 0.0 0.0

C

0.1

0.2

0.3

0.4

0.5

0.4 !"#1 0.2 !"0 0.0 0.0

0.5

1.0 !Λ1 !t

1.5

2.0

Figure 3.2: Main: Concurrence as a function of (scaled) time calculated analytically (dashed lines, valid for t > γ0−1 ) and numerically (solid lines). √ We consider four initial states, the Bell singlet (1/ 2)(|↑↓� − |↓↑�) (blue, Λ = −3), the pure state |↑↓� (red, Λ = −1), the√ maximally mixed state ρ = (1/4)I (green, Λ = 0) and the Bell state (1/ 2)(|↑↓� + |↓↑�) (black, Λ = 1). Parameters: δ = 0.05, R = 0.9. Inset: Behaviour of the initial state |↑↓� where, in the numerical calculations, the Lamb-shift and exchange interactions have been included at an arbitrarily chosen strength B = ξ = 1/(2|λ1 |). are eﬀectively equivalent, its form is determined by symmetry. With this form it commutes with our eigenvectors of most interest, α0 and α1 . It can also be shown that the eigenvectors α2 and α3 are eigenoperators of HLS , with eigenvalues 2(A + B) and −2(A + B), respectively. Therefore,

the Lamb-shift Hamiltonian can influence only the eigenvectors with l > 3, the imaginary parts of their eigenvalues, and the imaginary parts of λ2 and λ3 . Hence, despite the fact that HLS can entangle our spins, its eﬀect is restricted to a timescale of order γ0−1 (after which the other eigenvectors have decayed) regardless of its amplitude, and it will therefore have no eﬀect on the long-time entanglement dynamics or the analytic expressions we have derived. Furthermore, in section 2.2 we saw how a term from the system Hamiltonian treated perturbatively in a Born-Markov weak-coupling approach ends up simply back in the system Hamiltonian in the final master equation. Therefore we can further account for the eﬀect of a direct spin exchange interaction simply by adding a term of the form HE = ξσ 1 · σ 2

into HS in Eq. (3.2), provided that the evolution it generates occurs on

3.5 Experimental Realisation timescales much slower than the bath correlation time. This procedure is valid in the regime of ∆ � ξ, such that ∆ sets the relevant frequency scale for the system-bath interaction. In this case, exactly the same conclusions hold as for the Lamb-shift term since α0 and α1 again commute with this form of interaction, and α2 and α3 are also eigenoperators of HE . Its influence will thus similarly be restricted to short timescales ∼ γ0−1 . This

point is illustrated in the inset of Fig. 3.2 where we plot the analytically and numerically calculated concurrence of the initial state |↑↓� in x, y or

z, including both HLS and HE with arbitrarily chosen strengths. As expected, their impact is seen only on a timescale ∼ γ0−1 , much shorter than that over which the dissipatively induced entanglement survives.

With these conclusions, it seems entanglement induced by dissipative dynamics operates on a time-scale quite diﬀerent to the usual timescales associated with dissipative processes for a single spin. Perhaps the most fundamental timescale is the inverse of the single qubit decoherence rate at zero temperature, (γ0 )−1 . We saw in section 2.3.2 that it is this time-scale that determines the life-time of any coherence in a single spin system. We have now seen that this quantity also sets the time-scale for both the generation of dissipitively induced entanglement in two qubit systems and, importantly, the time-scale that entanglement induced by either Lamb-shift terms or (weak) direct exchange interactions can survive. Also, provided the qubits are suﬃciently close together, the system now has with it an entirely new time-scale, the time-scale that the dissipitively induced entanglement survives. The proximity of two qubits can preserve coherence in a system that would have otherwise decayed.

3.5

Experimental Realisation

We now propose a possible experimental realisation in which the entanglement phenomena described in this chapter should be observable. While one might expect the spin-boson model to most commonly apply in the solid state, more controlled realisations have recently been proposed in ion traps and we consider such an experimental set-up here. In this section, we rely heavily on the theoretical results found by Porras et al [43]. We consider a 1D chain of N ions in a trap. The collective vibrational modes of the ions provide our (finite) bosonic bath with Hamiltonian HB =

57

3.5 Experimental Realisation �N n

58

ωn b†n bn (note that we label bath modes with an “n” as opposed to a “k”

do distinguish the bath as finite). We create the system and interaction Hamiltonians as follows. Using two travelling wave light sources with frequency ωL and wave-vector k, we individually address two ions in the chain, which couples two of their hyperfine levels. We shall call these ions the central ions. The interaction of each central ion with its laser field produces, in a frame rotating at ωL , an interaction Hamiltonian, HI =

2 � Ω� i=1

2

(σxi + iσyi )eikZi + (σxi − iσyi )e−ikZi

�

(3.25)

where i labels the two central ions, Ω is the Rabi frequency of the laserion interaction and Zi is the coordinate operator of the ith ion. Since the central ions are themselves part of the bosonic bath, their positions are the appropriately weighted contributions from each of the bath modes. Therefore

N �

Zi =

Mni z¯n (b†n + bn )

(3.26)

n

√ where z¯n = 1/ 2ωn m with m the mass of the ions and Mni is the amplitude of mode n at the ith ion. Note that [Z1 , Z2 ] = 2i

� n

� � z¯n2 |Mn1 ||Mn2 | sin kn dcosθ → 0

(3.27)

since the implicit angular integration gives zero regardless of the dimensionally, provided the dispersion relation is isotropic. Applying the unitary transformation U = exp to HB + HI gives

�

−(i/2)k(Z1 σz1

+

Z2 σz2 )

�

(3.28)

� � � � Ω U (HB + HI )U † = ωn b†n bn + (σx1 + σx2 ) + σzi (gni b†n + gni∗ bn ) 2 n n i � � g 1 g 2∗ + g 1∗ g 2 � n n n n + σz1 σz2 (3.29) ω n n �

where gni = −(i/2)kωn z¯n Mni and we have omitted a constant. The last term in the Hamiltonian (3.29) represents a direct interaction between the

ions we have chosen to address. To understand this term better let us

3.5 Experimental Realisation

59

consider the form of the spectral density produced when addressing the atoms in the way described above. As always our spectral density is defined as J(ω) =

� n

|gn |2 δ(ω − ωn )

(3.30)

where we have assumed that our two ions are identical and that the lasers addressing each are suﬃciently matched such that |gn1 | = |gn2 | = |gn |.

Although we are dealing with a finite bath, to a good approximation its qualitative features can be found by taking the continuum limit: �

∞

dn|gn |2 δ(ω − ωn ) 0 � ∞ 1 = dnk 2 |Mn |2 ωn δ(ω − ωn ). 8m 0

J(ω) →

(3.31)

We now need to know how the frequency of the bath modes ωn depends on n. For N ∼ 102 it has been shown that ωn ≈ ωt n where ωt is the trapping frequency [43]. With this we find

� ∞ 1 J(ω) = dnk 2 |Mn |2 ωt nδ(ω − ωt n) 8m 0 � �2 k¯ zt = (1/N )ω 2 = (α/2)ω

(3.32)

√ where z¯t = 1/ 2mωt and we have assumed |Mn |2 = 1/N since in the limit of large N the vibrational modes are given by plane waves. The factor of two appearing in the last line has been introduced to ensure consistency with the conventions found in the literature [11]. The point of all this is to see that, addressing the ions in the way we have, gives a spectral density which is proportional to ω. We are now in a better position to consider the strength of the unwanted last term in the Hamiltonian (3.29). We will call it Hz and write it as follows, Hz = σz1 σz2

� |gn |2 n

ωn

(mn1 m∗n2 + m∗n1 mn2 )

(3.33)

where mni is a matrix containing only phase information regarding the

3.5 Experimental Realisation

60

amplitude of mode n at the ith ion. We have shown that the spectral density is proportional to ω and as such the first factor in the summation above has no ω dependence. Therefore, the summation above essentially corresponds to the matrix product, �

(m† )in (m)nj = δij

(3.34)

n

due to the orthogonality of normal modes [44]. We conclude that for any finite bath with a spectral density proportional to ω, Hz = 0. With this, the Hamiltonian (3.29) reduces to the two-spin-boson model from which our results have been derived. We have only applied a unitary transformation to the original Hamiltonian and as such, the same quantum dynamics should be produced. Owing to the finite size of the bath, the system evolves as if it were coupled to a continuum for only short times, after which a quantum revival is seen. For example, with N = 102 this revival is seen after approximately 2π/ωt . For the dynamics which we predict to be observable, this time must be both greater than our entanglement generation time, γ0−1 and the timescale which we expect our entanglement to decay appreciably, (δγ0 )−1 . The experimental set-up allows us to tune many of our parameters. We choose ∆ = 25ωt by adjusting the laser Rabi frequency. Our results are valid only in the weak coupling regime and as such we choose J(ω) = (α/2)ω = (0.1/2)ω, which can be achieved by adjusting the parameters that constitute α. This gives γ0 = 2πJ(∆) = (5/2)πωt

(3.35)

and as such our revival time is ∼ 50γ0−1 . Now, assuming that the wavelength associated with ∆ is approximately N/(∆/ωt ) in units of the ion spacing, we choose to address two neighbouring ions and we find, 1 δ= 2

�

∆ N ωt

�2

≈ 0.03.

(3.36)

Lastly, we must require that the temperature of the ion chain is low enough such that the inequality (3.23) is satisfied and a finite level of entanglement is reached. For example, if we require R = 0.5 we find that temperatures

3.6 Summary in the mK range are required for a typical trapping frequency in the MHz range. With these specifications, we predict that after a time ∼ γ0−1 = (25παωt )−1 an entanglement of ≈ 0.15 should be reached from an initial

state |↑↓�. This should then decay by at least a factor of e−1 before the dynamics associated with the finite size of the bath are seen.

3.6

Summary

To summarise, we have shown that the mechanisms normally associated with dissipative processes can lead to long-lived entanglement in noninteracting, spatially separated two-qubit systems. We have highlighted two important timescales. The first, shorter timescale γ0−1 is that with which we expect a single qubit to dephase. When a second qubit is introduced close to the first, we find dissipatively-induced entanglement is generated on this short timescale, and further that there is a second larger timescale (δγ0 )−1 on which the induced entanglement decays. Importantly, the influence of both the bath-induced Lamb-shift or a direct spin exchange interaction is still restricted to the original shorter timescale. Hence, the presence of a second qubit within the bath induces coherences in the overall system state that can persist on timescales far larger than either the corresponding single qubit decoherence time, or timescales associated with the influence of direct exchange or the Lamb-shift.

61

Chapter 4 Development of a strong coupling theory Contents 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Relaxation of the RWA . . . . . . . . . . . . . . . 4.2 Non-interacting blip approximation . . . . . . . . . . 4.2.1 Polaron transformation . . . . . . . . . . . . . . . 4.2.2 NIBA master equation . . . . . . . . . . . . . . . 4.2.3 NIBA results . . . . . . . . . . . . . . . . . . . . . Steady state in the NIBA . . . . . . . . . . . . . . Ohmic damping, symmetric case at zero temperature 4.3 Projection operator techniques . . . . . . . . . . . . 4.3.1 Nakajima-Zwanzig equation . . . . . . . . . . . . . Weak-coupling form . . . . . . . . . . . . . . . . . 4.3.2 Time-convolutionless method . . . . . . . . . . . . Systematic expansion in the interaction strength . . Second order form . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 65 66 70 70 71 74 74 77 78 79 82 84

4.1 Introduction

4.1

Introduction

While much can be gained from a weak-coupling analysis, the parameter range over which it is valid is intrinsically limited. In this chapter we introduce a combined polaron transform time-local master equation technique. For many parameters of interest, this technique allows for exploration of strong coupling and/or high temperature regimes where the weak-coupling theory becomes invalid. To motivate the technique, in section 4.1.1 we first give a brief description of the qualitative changes in prediction that a weak-coupling approach can have when the rotating wave approximation is relaxed. In section 4.2 we then derive the NIBA expressions discussed in section 2.3.3 within a time non-local polaron transform formalism. The purpose being to introduce the polaron transformation and to show how it can be used to derive some of the well known (and accurate) results from the NIBA. Section 4.3 then introduces projection operator techniques which are used to derive rigorously a time-local master equation, which when combined with the polaron transformation constitutes the strong coupling theory which is used in chapters 5 and 6.

4.1.1

Relaxation of the RWA

Perhaps the most straightforward way to modify the standard weakcoupling Born-Markov approach is to relax the rotating wave approximation (RWA). In section 2.2 we saw how to derive a master equation of Lindblad form from the microscopic Hamiltonian determining the evolution of the total system-plus-bath state. In particular, section 2.2.2 outlined the motivation behind the RWA which is essentially to assume that the system relaxes or dissipates on a timescale far slower than the timescale associated with the dynamics generated by its free Hamiltonian. Intuitively, one can therefore identify the RWA as natural extension of a perturbative weak coupling approximation. Although a weak coupling master equation is ultimately limited in its validity owing to the perturbative expansion used in its derivation, we nevertheless now relax the RWA and consider what diﬀerences this makes to the resulting Bloch equations. Since the RWA is an approximation, we expect that Bloch equations derived in its absence should be more accurate,

63

4.1 Introduction

64

even if they do not necessarily preserve positivity of the corresponding density operator. Just as in section 2.3.2 our starting point is the spinboson Hamiltonian, Eq. (2.83), with zero bias term, � = 0. The master equation derivation follows exactly that in section 2.2 with the exception that a RWA is not made at any stage. In exact analogy with Eqs. (2.114) and (2.115) we find the Bloch vector describing the spin has equation of motion ˙ =M ·α+b α where



 M =

−ΓW 0 0

0

(4.1) 0



 −ΓW (∆ + 2λ)  −∆ 0

(4.2)

and as before the weak coupling rate is given by ΓW = γzz (∆) + γzz (−∆), the renormalisation of the tunnelling rate is λ = Szz (∆) − Szz (−∆), and

b = (κ, 0, 0) with κ = γzz (∆) − γzz (−∆). The rotating wave approximation, in this context, assumes rotations about the x-axis are so rapid that the y and z directions are eﬀectively equivalent. Having made no such approximation here, we see that the Bloch equations of y and z have a qualitatively diﬀerent form. Solving for αz we find � � αz (t) = e−ΓW t/2 C1 e+iξt/2 + C2 e−iξt/2

(4.3)

where the constants C1 and C2 are determined by the initial conditions and the oscillation frequency is now given by � ξ = 4∆(∆ + 2λ) − Γ2W .

(4.4)

In contrast to the corresponding expression found using a RWA, we see that it is now possible for αz (t) to become over-damped. With use of Eq. (2.108) and (2.109) for the rates γzz (±∆) we have ΓW = 2πJ(∆) coth(β∆/2).

(4.5)

Taking, for example, an Ohmic spectral density in the scaling limit, J(ω) = (α/2)ω, and assuming λ to be negligible, we find a zero temperature crossover from coherent to incoherent dynamics when the coupling

4.2 Non-interacting blip approximation

65

strength reaches the critical value αc =

2 ≈ 0.64, π

(4.6)

which we contrast with the well known value of αc = 0.5 found using the NIBA [11]. While this approach has an appealing simplicity, and seems to predict a reasonable value, it should be noted that expressions such as (4.6) should be viewed with extreme caution. Although a RWA approximation was not made in derivation of the Bloch equations (4.1), a weak coupling BornMarkov approximation was made. One therefore expects that they are valid for low temperatures and/or weak system bath coupling only. In the regime defined by Eq. (4.6) it may well be the case that one (or many) of the approximations made to obtain it become invalid.

4.2

Non-interacting blip approximation

In this section we derive the NIBA expressions for the population diﬀerence as well as some of the results discussed in section 2.3.3.

4.2.1

Polaron transformation

Ultimately, the weak-coupling theory is limited because it relies on a perturbative expansion in the system-bath coupling strength. To go beyond the weak-coupling limit and to derive the NIBA expressions, here we apply a unitary transformation to the spin-boson Hamiltonian which allows for a perturbative expansion in a diﬀerent parameter in a method similar to that of Aslangul et al [45] and Dekker [46]. While there is no a priori reason to believe this new parameter to be smaller than the system-bath coupling strength, we shall see that in many cases this is indeed the case. We begin by considering the full spin-boson Hamiltonian in the Schr¨odinger picture � � � ∆ H = σz − σx + σz (gk b†k + gk∗ bk ) + ωk b†k bk 2 2 k k

(4.7)

where all symbols have their usual meaning. The interaction term in Eq. (4.7) causes the oscillators in the bath to adjust their equilibrium

4.2 Non-interacting blip approximation

66

positions in response to the state of the spin. The idea of the polaron transformation is to move into a frame where this eﬀect is accounted for. To this Hamiltonian we now apply the unitary transformation defined by HP = e+S He−S where e

±S

�

= exp ± σz

�

(αk b†k

k

−

αk∗ bk )

�

,

(4.8)

with αk = gk /ωk . The result is HP = HSP + HIP + HBP with system Hamiltonian in the polaron frame HSP = (�/2)σz , bath Hamiltonian � HBP = k ωk b†k bk , and new interaction term HIP = −

� ∆� B x σx + B y σy , 2

(4.9)

where the bath operators are given by

1 Bx = (B+ + B− ), 2 1 By = (B− − B+ ) 2i with B± =

�

k

D(±2αk ) =

�

k

(4.10) (4.11)

exp[±2(αk b†k − αk∗ bk )].

The polaron transformation has removed the coupling through σz in

Eq. (4.7), but at the expense of introducing a coupling term now through the oﬀ-diagonal system operators σx and σy . We note that with ∆ = 0 the polaron transformation exactly diagonalises the Hamiltonian, precisely as was seen in section 2.3.1.

4.2.2

NIBA master equation

In a manner similar to that in section 2.2, we now proceed by deriving an equation of motion for the reduced density operator. However, our present treatment is diﬀerent in two important respects. Firstly, the Hamiltonian from which the master equation will be derived is in the polaron frame. In the interaction picture the density operator has the equation of motion dχ˜ ˜ I (t), χ(t)]. = −i[H ˜ dt

(4.12)

4.2 Non-interacting blip approximation

67

where as usual tildes indicate operators in the interaction picture. It follows then that in the polaron frame the density operator satisfies dχ˜P ˜ IP (t), χ˜P (t)]. = −i[H dt

(4.13)

where the subscripts indicate an operator in the polaron frame: OP = eS Oe−S . Thus, an equation of motion derived from HP must be that of a density operator in the polaron frame. Our second point to notice is that our perturbation term, HIP , is now quite diﬀerent from that which we used in deriving a weak-coupling master equation. Inserting the formal solution of Eq. (4.13) back into itself and tracing over the bath gives the polaron frame equation of motion d˜ ρP ˜ IP (t), χP (0)] − = −iTrB [H dt

�

t

˜ IP (t), [H ˜ IP (s), χ˜P (s)]] (4.14) dsTrB [H

0

˜ IP (t) = exp[i(HSP + HBP )t]HIP exp[−i(HSP + HBP )t] is the inwhere H teraction Hamiltonian, in the interaction picture and polaron frame, and χP (0) is the initial total density operator in the polaron frame at time t = 0. To proceed we now make the Born approximation, in the polaron frame, which is to assume that the total density operator χ˜P factorises into its system and bath components at all times, χ˜P (s) = ρ˜P (s) ⊗ ρBP , and we assume a stationary thermal equilibrium state of the bath in the polaron

frame, ρBP = exp[−βHBP ]/Tr{exp[−βHBP }. We note that this approxi-

mation does not assume that the total density operator factorises in the original frame. Expanding the double commutator we find d˜ ρP ∆ = i B [˜ σx (t),ρP (0)] dt 2 � �2 � ∞ � ∆ − ds [˜ σx (t), σ ˜x (s)˜ ρP (s)]Λxx (t − s) 2 0

�

+ [˜ σy (t), σ ˜y (s)˜ ρP (s)]Λyy (t − s) + H.c. .

(4.15)

The bath state specified above has allowed us to partially evaluate the first term in Eq. (4.15) since for a thermal bath state it is found that ˜y (t)) = 0 and TrB (ρBP B ˜x (t)) = B where B = TrB (ρBP B± (t)). TrB (ρBP B

We have also now introduced the bath correlation functions Λxx (t − s) =

4.2 Non-interacting blip approximation

68

TrB (ρBP Bx (t)Bx (s)) and Λxx (t−s) = TrB (ρBP Bx (t)Bx (s)). Details of how these correlation functions and the expectation value B are calculated can be found in appendix A. Moving back into the Schr¨odinger picture at time t and making a change of variables, τ = t − s, we have dρP ∆ = Bσy − i[HSP ,ρP (t)] dt 2 � �2 � ∞ � ∆ − dτ [σx , σ ˜x (−τ )ρP (t, τ )]Λxx (τ ) 2 0

+[σy , σ ˜y (−τ )ρP (t, τ )]Λyy (τ ) + H.c.

�

(4.16)

where ρP (t, τ ) = e−iHSP τ ρP (t − τ )eiHSP τ with ρP (t − τ ) the system density

operator at time t − τ in the Schr¨odinger picture and we have assumed the typical initial state ρP = |0��0| which commutes with both HSP and the polaron transformation.

We wish to calculate the equation of motion for the expectation value of σz . Normally this would be found from simply α˙ z = Tr(σz ρ). ˙ To see how this quantity can be calculated from a master equation in the polaron frame, we write α˙ z = TrS+B (σz χ) ˙ = TrS+B (σz e−S χ˙ P e+S ) = TrS+B (σz ρ˙ P ) where we have used the cyclic invariance of the trace and observed that σz commutes with the polaron transformation, eS σz e−S = σz . Thus we see that expectation values of σz can be calculated from density operators in the polaron or lab frames. From Eq. (4.16) we obtain � �2 � ∞ � ∆ dαz =− dτ Tr i{˜ σy (τ ), σx ρP (t − τ )Λxx (τ ) dt 2 0 −ρP (t − τ )σx Λ∗xx (τ )}

+i{˜ σx (τ ), −σy ρP (t − τ )Λyy (τ ) � +ρP (t − τ )σy Λ∗yy (τ )}

(4.17)

where curly brackets indicate anti-commutators and we have made use of simplifications such as � � � � Tr σi σ ˜j (−τ )ρP (t, τ ) =Tr σi e−iHSP τ σj eiHSP τ e−iHSP τ ρP (t − τ )eiHSP τ � � =Tr eiHSP τ σi e−iHSP τ σj ρP (t − τ ) � � =Tr σ ˜i (τ )σj ρP (t − τ ) . (4.18)

4.2 Non-interacting blip approximation

69

Inserting σ ˜x (τ ) = σx cos(�τ )−σy sin(�τ ) and σ ˜y (τ ) = σy cos(�τ )+σx sin(�τ ) into Eq. (4.17) one then reaches the result dαz = −∆2 dt

�

t 0

� dτ αz (t − τ ) cos(�τ )Re[Λxx (τ )

� + Λyy (τ )] − sin(�τ )Im[Λxx (τ ) + Λyy (τ )] .

(4.19)

From appendix A we have that Λxx (τ ) + Λyy (τ ) = C(τ ) = exp[−Q� (τ ) − iQ�� (τ )] with

�

Q (τ ) =4 ��

Q (τ ) =4

� �

∞ 0

0

∞

J(ω) coth(βω/2)(1 − cos(ωτ ))dω ω2 J(ω) sin(ωτ )dω ω2

(4.20) (4.21)

and Eq. (4.19) can be put in the form dαz = −∆2 dt

�

t 0

� � dτ αz (t − τ ) cos(�τ )e−Q (τ ) cos[Q�� (τ )] �

+ sin(�τ )e−Q (τ ) sin[Q�� (τ )]

�

(4.22)

which agrees exactly with the corresponding expression found using the NIBA [10, 11]. Moving into Laplace space, the expression for the spin population difference under the influence of the full spin-boson Hamiltonian given by �∞ Eq. (2.83) becomes (αz (p) = 0 e−pt αz (t)dt) αz (p) =

ˆ (a) (p)/p 1−K ˆ (s) (p) p+K

where we have made use of the convolution theorem: � ∞ � t −pt G(p)H(p) = e g(τ )h(t − τ )dtdτ t=0

(4.23)

(4.24)

τ =0

where g(t) and h(t) are general functions of time with corresponding ˆ (s) (p) Laplace transforms G(p) and H(p) respectively. The functions K ˆ (a) (p) are the Laplace transforms of the kernels symmetric and anitand K

4.2 Non-interacting blip approximation

70

symmetric under bias inversion (� → −�) respectively: ˆ (s)

K (p) = ∆

2

ˆ (a) (p) = ∆2 K

� �

∞ 0

∞

�

e−pτ dτ cos(�τ )e−Q (τ ) cos[Q�� (τ )] �

e−pτ dτ sin(�τ )e−Q (τ ) sin[Q�� (τ )].

(4.25) (4.26)

0

To find the evolution of αz in the time domain the Laplace transform used to obtain Eq. (4.23) must be inverted giving the usual contour integral 1 αz (t) = 2πi

�

+i∞+c

ept αz (p)dp,

(4.27)

−i∞+c

where c is some real number such that all poles of the integrand lie to the left of the integration path in the complex plane.

4.2.3

NIBA results

We now use the expressions derived in the sections above to obtain a small selection of the many known results of the NIBA. Steady state in the NIBA From Eq. (4.27) it is possible to see that the steady state value of αz will be given by the residue of the pole of αz (p) at p = 0, since in the long time limit all other contributions will have died away. Taking the limit ˆ (a) (0)/K ˆ (s) (0). From p → 0 in Eq. (4.23) gives αz (∞) = αz (t → ∞) = −K Eqs. (4.25) and (4.26) we find

2 ˆ (s) (0) = ∆ (γC (�) + γC (−�)) K 4 2 ˆ (a) (0) = ∆ (γC (�) − γC (−�)) K 4

where γC (±�) =

� +∞ −∞

(4.28) (4.29)

e±i�τ C(τ )dτ are rates in our usual notation. Since

C(τ ) is a bath correlation function of the sort introduced in section 2.2, the corresponding rates satisfy the detailed balance relation, Eq. (2.74). The steady state is therefore found to read αz (∞) = − tanh(β�/2)

(4.30)

4.2 Non-interacting blip approximation

71

valid for an arbitrary spectral density. As discussed in section 2.3.3, this results represents one of the main failings of the NIBA for a biased system since it predicts the qualitatively incorrect result of localisation of the spin projection at zero temperature, even for infinitesimal bias. Ohmic damping, symmetric case at zero temperature Having considered the steady state predicted by the NIBA, we now focus on the special case of Ohmic damping and a symmetric or unbiased system. The latter condition corresponds to the spin up and spin down states being degenerate, � = 0. The former corresponds to the particular form of the spectral density J(ω) =

α −ω/ωc ωe . 2

(4.31)

We will also focus on the limit ωc → ∞ which is commonly called the

scaling limit. The reason for studying this particular region of parameter space is that, as discussed in section 2.3.3, the NIBA approximation predicts a crossover from coherent to incoherent relaxation as the coupling strength is increased through a special value. This is in contrast to what was found using the standard Born-Markov approach with RWA and can also be compared with the crossover found when the RWA was relaxed in section 4.1.1. Additionally, the methods used in this section, although only applied to a particular case, are a good example of how the NIBA expressions presented in Eqs. (4.23), (4.25) and (4.26) are used to analyse the dynamics in the time domain. With Eq. (4.31) in Eq. (4.21) we find Q�� (τ ) = 2αarctan(ωc τ )

(4.32)

which in the scaling limit approximates to Q�� (τ ) ≈ απsgn(τ ). The scaling limit form for Q� (τ ) is found to read [10] Q� (τ ) = 2α ln

� βω π

c

sinh

� π|τ | �� β

.

(4.33)

With the scaling forms for Q� (τ ) and Q�� (τ ) the Laplace transform of the symmetric kernel can be found in analytic form. From Eq. (4.25) with

4.2 Non-interacting blip approximation

72

Im

x

Re x

Figure 4.1: Position of the poles and branch cut of αz (p) for 0 < α < 1/2. For 1/2 < α < 1 the poles do not contribute since they cross the branch cut.

ˆ (a) (p) = 0 while � = 0 we find K � β∆ �1−2α h(p) R (s) ˆ K (p) = ∆R 2π α + pβ/2π

(4.34)

where h(p) = Γ[1 + α + pβ/2π]/Γ[1 − α + pβ/2π] and the eﬀective renormalised tunnelling element is given by

� �1/(2(1−α)) � ∆ �α/(1−α) ∆R = Γ[1 − 2α] cos(πα) ∆ ωc with Γ[z] =

�∞ 0

(4.35)

tz−1 e−t dt the Euler gamma function. Taking the zero

temperature limit one finds � �1−2α ˆ (s) (p) = ∆R ∆R K p

(4.36)

and from Eq. (4.23) our problem reduces to finding the inverse Laplace transform of αz (p) =

�

p + ∆R

� ∆ �1−2α �−1 R

p

.

(4.37)

We first consider the regime 0 < α < 1/2. For these values of α the Laplace transform αz (p) has a branch point at p = 0. Furthermore, there

4.2 Non-interacting blip approximation

73

exists a conjugate pair of simple poles on the principal sheet at positions � p± = ∆R exp ±

iπ � . 2(1 − α)

(4.38)

Choosing the branch cut to be along the negative real p axis [see Fig 4.1], from Eq. (4.27) we see that αz (t) can be written αz (t) = αzINC (t) + αzCOH (t)

(4.39)

where the coherent contribution comes from the closed contour integral, αzCOH (t)

1 = 2πi

�

C

αz (p)ept dp = A+ ep+ t + A− ep− t

(4.40)

where A± is the residue of the pole at p = p± . The incoherent component, αzINC (t), arrises since we must subtract the finite contribution to the closed

integral coming from the branch cut. Written in terms of the dimensionless time argument y = ∆R t we have αzINC (t)

− sin(2πα) = π

�

∞ 0

e−uy u2α−1 du . u2 + u2α cos(2πα) + u4α−2

(4.41)

Evaluating the residues A± gives the coherent contribution αzCOH (t) =

1 e−γt cos(Ωt) 1−α

(4.42)

where from Eq. (4.38) γ = ∆R sin

�

πα � , 2(1 − α)

Ω = ∆R cos

�

πα � . 2(1 − α)

(4.43)

The corresponding quality factor then has the frequency independent form Ω/γ = cot(πα/2(1 − α)). It has been shown that this quality factor

is exact [47]. Only the frequency scale ∆R itself contains a slight error inherited from the NIBA made. For α = 1/2 there is no branch point in Eq. (4.37) and the incoherent contribution to αz (t) is not present. Moreover Eq. (4.37) reduces to αz (p) = (p + ∆R )−1 which is readily inverted to given the well known result [11] αz (t) = exp[−(π∆2 /2ωc )t]

(4.44)

4.3 Projection operator techniques

74

valid for α = 1/2 and we have used Eq. (4.35) for ∆R . For 1/2 < α < 1 the poles of αz (p) no longer lie on the principal p-sheet since they have crossed the branch cut. We are left with only the incoherent contribution to αz (t) given by Eq. (4.41). We conclude therefore that as the coupling strength is increased through the special value α = 1/2 the dynamics at zero temperature changes from damped coherent oscillations to incoherent relaxation.

4.3

Projection operator techniques

In this section we use projection operator techniques to derive a time-local master equation describing the reduced dynamics of a system subject to the dissipative influence of some environment. This is not the first time in this thesis that this has been done. However, the advantage of the procedure which we use here is that it allows for a systematic way in which to include non-Markovian eﬀects. Furthermore, the various approximations used to derive a time-local master equation in section 2.2 can be put on a much more formal footing.

4.3.1

Nakajima-Zwanzig equation

To begin with we derive a time non-local master equation using the Nakajima-Zwanzig projection operator technique [48, 49]. The approach we use here follows closely that found in Ref. [8]. As usual we consider a physical system S coupled to an environment or bath B. In the Schr¨odinger picture we have the total Hamiltonian H(t) = H0 (t) + αHI

(4.45)

where we allow for explicit time dependence in the free Hamiltonian H0 and HI describes the interaction between the system and the environment. The dimensionless parameter α is introduced here to keep track of powers of the interaction Hamiltonian. In the interaction picture the total density operator χ(t) ˜ satisfies the von Neumann equation ∂ χ(t) ˜ ˜ I (t), χ(t)] = −iα[H ˜ = αL˜χ(t) ˜ ∂t

(4.46)

4.3 Projection operator techniques

75

˜ I (t) = U0 (t)† HI U0 (t) being the interaction picture interaction with H Hamiltonian. The Liouvillian super-operator in the interaction picture ˜ is L.

We now define the projection operators which are used to derive an

exact equation of motion for the reduced density operator. The projection operator P is defined by Pχ = TrB (χ) ⊗ ρB ≡ ρ ⊗ ρB

(4.47)

where ρB is some fixed reference state of the environment. In a sense the projection operator projects the necessary information onto the relevant part of the density operator so that the reduced density operator of the system ρ can be constructed. The complimentary projection operator Q is defined through

Qχ = χ − Pχ

(4.48)

and which maps onto the irrelevant part of the density operator. Note that with Eqs. (4.47) and (4.48) it is readily verified that P + Q = I,

P 2 = P,

Q2 = Q, PQ = 0.

(4.49) (4.50) (4.51) (4.52)

We note here that the reference state ρB is an arbitrary fixed operator in the Hilbert space of the environment. The form that is chosen for ρB will depend on the particular problem we wish to investigate. It is usually convenient to choose ρB to be a thermal state since it simplifies the calculation of many quantities. Also, it is often the case that odd moments of the interaction Hamiltonian with respect to reference states of this sort are zero, ˜ I (t1 )H ˜ I (t2 ) . . . H ˜ I (t2n+1 )ρB ) = 0 Tr(H

(4.53)

˜ 1 )L(t ˜ 2 ) . . . L(t ˜ 2n+1 )P = 0. P L(t

(4.54)

which leads to

for n = 1, 2, . . . . We shall see that this greatly simplifies many of the

4.3 Projection operator techniques

76

resulting expressions which we obtain. Applying both projection operators to Eq. (4.46) we arrive at the following coupled equations for the evolution of the relevant and irrelevant parts of the density operator, ∂P χ˜ ˜ ˜ = αP L(t)P χ(t) ˜ + αP L(t)Q χ(t) ˜ ∂t ∂Qχ˜ ˜ ˜ = αQL(t)P χ(t) ˜ + αQL(t)Q χ(t) ˜ ∂t

(4.55) (4.56)

where we have used P + Q = I. To proceed we solve Eq. (4.56) for the time evolution of the irrelevant part of the density operator and insert the

solution into Eq. (4.55). For an initial state of the total system χ(t0 ) we can solve Eq. (4.56) using an integrating factor. Multiplication from the left by the, as yet to be determined function F (t, t0 ), gives F (t, t0 )

∂Qχ˜ ˜ ˜ − αF (t, t0 )QL(t)Q χ(t) ˜ = αF (t, t0 )QL(t)P χ(t) ˜ ∂t

(4.57)

and we find Qχ(t) ˜ =F

−1

(t, t0 )C + α

�

t t0

˜ F −1 (t, t0 )F (s, t0 )QL(s)P χ(s) ˜

(4.58)

with C a constant, provided F (t, t0 ) is the solution to the diﬀerential equation ∂F (t, t0 ) ˜ Qχ(t) ˜ = −αF (t, t0 )QL(t)Q χ(t). ˜ ∂t

(4.59)

The function F (t, t0 ) therefore has the form of a time evolution operator: �

F (t, t0 ) = T→ exp − α

�

t t0

� ˜ QL(s)ds ,

(4.60)

with T→ is the anti-chronological time-ordering operator. Inserting this into Eq. (4.58) and finding the constant C by setting t = t0 , we arrive at the formal solution Qχ(t) ˜ = GF (t, t0 )Qχ(t0 ) + α

�

t t0

˜ dsGF (t, s)QL(s)P χ(s) ˜

(4.61)

4.3 Projection operator techniques

77

where we have introduced the forwards in time propagator � � t � � � ˜ (t, s) = T← exp α ds QL(s ) .

(4.62)

with T← the usual chronological time-ordering operator.

Inserting

GF (t, s) ≡ F

−1

s

Eq. (4.61) into Eq. (4.55) gives the exact expression for the relevant part of the density operator ∂P χ˜ ˜ ˜ = αP L(t)G ˜ F (t, t0 )Qχ(t0 ) + αP L(t)P χ(t) ∂t � t

+α

2

t0

˜ ˜ dsP L(t)G ˜ F (t, s)QL(s)P χ(s),

(4.63)

known as the Nakajima Zwanzig equation. Weak-coupling form With the use of Eq. (4.54) we can instead write ∂P χ˜ ˜ = αP L(t)G F (t, t0 )Qχ(t0 ) + ∂t

�

t

dsΦ(t, s)P χ(s) ˜

(4.64)

t0

with the convolution kernel given by ˜ ˜ Φ(t, s) = α2 P L(t)G F (t, s)QL(s)P.

(4.65)

To further simplify Eq. (4.64) we now make the assumption that the initial conditions are such that the total density operator factorises into a system part and the bath reference state. With χ(t0 ) = ρ(t0 ) ⊗ ρB we have Pχ(t0 ) = χ(t0 ) and Eq. (4.64) takes on the simple form ∂P χ˜ = ∂t

�

t

dsΦ(t, s)P χ(s). ˜

(4.66)

t0

To obtain the weak-coupling form of the Nakajima-Zwanzig equation we expand Φ(t, s) to second order in α. Replacing GF (t, s) with the identity in Eq. (4.65) gives ˜ ˜ Φ(t, s) ≈ α2 P L(t)Q L(s)P which leads to

∂P χ˜ = α2 ∂t

�

t t0

˜ L(s)P ˜ dsP L(t) χ(s) ˜

(4.67)

(4.68)

4.3 Projection operator techniques

78

˜ where we have again used P L(t)P = 0. From Eq. (4.46) we have that ˜ ˜ L(t)χ(t) ˜ = −i[HI (t), χ(t)] ˜ for an arbitrary total system density operator

˜ ˜ I (t), P χ(t)]. at time t, χ(t). ˜ It follows that L(t)P χ(t) ˜ = −i[H ˜ Eq. (4.68)

can therefore be put in the form ∂ ρ˜(t) = −α2 ∂t

�

t t0

� � ˜ I (t), [H ˜ I (s), ρ˜(s) ⊗ ρB ]] . dsTrB [H

(4.69)

which has the same form as Eq. (2.50) obtained using the Born approximation to second order in section 2.2. We note, however, that in derivation of Eq. (4.69) we assumed that the total system-bath density operator was separable at the initial time t0 , but it was not necessary to assume that this holds true for all times from t0 to t.

4.3.2

Time-convolutionless method

While Eq. (4.69) is a good starting point to derive a more detailed master equation describing the reduced density operator, the appearance of ρ˜(s) in the integrand can lead to diﬃculties. Our aim now is to find a rigorous way to replace ρ˜(s) instead with its value at time t, ρ˜(t). To do so we use the time-convolutionless projection operator technique originally developed by Shibata et al [50, 51]. The reduced density operator ρ˜(s) appearing in Eq. (4.69) can be traced back to the appearance of χ(s) ˜ in Eq. (4.61). To procede we write χ(s) ˜ = GB (t, s)(P + Q)χ(t) ˜

(4.70)

where we have introduced the reverse evolution operator �

GB (t, s) = T→ exp − α

�

t s

� � ˜ ds L(s ) . �

(4.71)

With this replacement Eq. (4.61) becomes Qχ(t) ˜ = GF (t, t0 )Qχ(t ˜ 0) + α

�

t t0

˜ dsGF (t, s)QL(s)PG ˜ B (t, s)(P + Q)χ(t). (4.72)

4.3 Projection operator techniques

79

If we now introduce the super-operator Σ(t) = α

�

t t0

˜ dsGF (t, s)QL(s)PG B (t, s)

(4.73)

we can write (1 − Σ(t))Qχ(t) ˜ = GF (t, t0 )Qχ(t ˜ 0 ) + Σ(t)P χ(t). ˜

(4.74)

Provided the system-bath coupling strength α and/or the time interval t − t0 are not too large the operator (1 − Σ(t)) can be inverted to give the expression

Qχ(t) ˜ = (1 − Σ(t))−1 Σ(t)P χ(t) ˜ + (1 − Σ(t))−1 GF (t, t0 )Qχ(t ˜ 0 ).

(4.75)

We now insert Eq. (4.75) into Eq. (4.55) for the evolution of the relevant part of the density operator to give ∂P χ(t) ˜ = K(t)P χ(t) ˜ + I(t)Qχ(t ˜ 0) ∂t

(4.76)

where we have introduced the time-local generator ˜ K(t) = αP L(t)(1 − Σ(t))−1 P

(4.77)

and the inhomogeneous term is given by ˜ I(t) = αP L(t)(1 − Σ(t))−1 GF (t, t0 )Q.

(4.78)

Systematic expansion in the interaction strength While Eq. (4.76) is exact and time-local, the super-operators K(t) and

I(t) are, in general, very complicated objects. To put Eq. (4.76) in a more tangible form, we now perform a systematic expansion in powers of the

system-bath coupling strength α. We start by recognising that provided (1 − Σ(t))−1 exists we can use the geometric series (1 − Σ(t))

−1

=

∞ � n=0

[Σ(t)]n

(4.79)

4.3 Projection operator techniques

80

which upon insertion into Eq. (4.77) for the time-local generator gives K(t) = α

∞ � n=0

n ˜ P L(t)[Σ(t)] P=

∞ � n=1

αn Kn (t).

(4.80)

To find the nth order contributions Kn (t) we now also expand Σ(t) in powers of α,

Σ(t) =

∞ �

αn Σn (t).

(4.81)

n=1

Equating powers of α in Eq. (4.80) gives the first to fourth order terms ˜ K1 (t) = P L(t)P

(4.82)

˜ K2 (t) = P L(t)Σ 1 (t)P � � ˜ K3 (t) = P L(t) [Σ1 (t)]2 + Σ2 (t) P � � ˜ K4 (t) = P L(t) [Σ1 (t)]2 + Σ1 (t)Σ2 (t) + Σ2 (t)Σ1 (t) + Σ3 (t) P.

(4.83) (4.84) (4.85)

It can be seen immediately that K1 (t) = 0 owing to Eq. (4.54). To

find the higher order contributions to the kernel K(t) we must determine

the diﬀerent terms in the expansion of the super-operator Σ(t), defined in Eq. (4.73). These in turn are found with the use of the expansions GF (t, s) = 1 + α

GB (t, s) = 1 − α

�

t s

�

t s

˜ � ) + α2 ds QL(s �

˜ � ) + α2 ds� L(s

�

� t

s

t

ds

�

s

ds�

�

�

s� s

s� s

˜ � )QL(s ˜ �� ) + . . . ds�� QL(s (4.86)

˜ �� )L(s ˜ � ) + . . . (4.87) ds�� L(s

where we note the anti-chronological time-ordering in the expression for GB (t, s). From the expansions above we find Σ1 (t) =

�

t 0

˜ 1 )P dt1 QL(t

(4.88)

where t0 has been set equal to zero for simplicity and a trivial change of variables s → t1 has been made. Eq. (4.88) leads to the second order contribution

K2 (t) =

�

t 0

˜ L(t ˜ 1 )P. dt1 P L(t)

(4.89)

4.3 Projection operator techniques

81

The second order term Σ2 (t) is found to be Σ2 (t) = =

� �

t

ds 0 t

�

dt1 0

t s

�

�

˜ � )QL(s)P ˜ ˜ ˜ �) ds QL(s − QL(s)P L(s �

t1 0

�

� � ˜ 1 )QL(t ˜ 2 )P − QL(t ˜ 2 )P L(t ˜ 1) . dt2 QL(t

(4.90) (4.91)

Using PQ = 0 we find [Σ1 (t)]2 = 0 and ˜ K3 (t) = P L(t)Σ 2 (t)P =

�

t

dt1 0

�

t1 0

˜ L(t ˜ 1 )L(t ˜ 2 )P = 0. dt2 P L(t)

(4.92)

To find the fourth order contribution K4 (t) we note that [Σ1 (t)]3 =

[Σ1 (t)]2 Σ1 (t) = 0 and Σ1 (t)Σ2 (t) = 0 because of PQ = 0. This leaves � � ˜ K4 (t) = P L(t) Σ2 (t)Σ1 (t) + Σ3 (t) P

(4.93)

which has as its first term

� t � t1 � t2 ˜ ˜ P L(t)Σ2 (t)Σ1 (t)P = − dt1 dt2 dt3 P L(t) 0 0 0 � � ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ × L(t2 )P L(t1 )L(t3 ) + L(t3 )P L(t1 )L(t2 ) + L(t3 )P L(t2 )L(t1 ) P

(4.94)

which we have brought into time-ordered form: t ≥ t1 ≥ t2 ≥ t3 ≥ 0. With similar methods we find � t � t1 � t2 ˜ ˜ P L(t)Σ3 (t)P = dt1 dt2 dt3 P L(t) 0 0 0 � � ˜ 1 )QL(t ˜ 2 )L(t ˜ 3 ) + L(t ˜ 3 )P L(t ˜ 2 )L(t ˜ 1 ) P. × L(t

(4.95)

Putting all this together we have the forth order term K4 (t) =

�

t

dt1 0

�

t1

dt2 0

�

� ˜ ˜ 1 )L(t ˜ 2 )L(t ˜ 3 ) − L(t ˜ 1 )P L(t ˜ 2 )L(t ˜ 3) dt3 P L(t) L(t 0 � ˜ 2 )P L(t ˜ 1 )L(t ˜ 3 ) − L(t ˜ 3 )P L(t ˜ 1 )L(t ˜ 2 ) P. −L(t t2

(4.96)

4.3 Projection operator techniques

82

Similarly, expanding the inhomogeneous term in powers of α: I(t) =

∞ � n=1

αn In (t)

(4.97)

we find the first and second order contributions ˜ I1 (t) = P L(t)Q � t ˜ L(t ˜ 1 )Q. I2 (t) = dt1 P L(t)

(4.98) (4.99)

0

Second order form In the following two chapters we will use the time-local master equation derived in the previous sections, including terms up to second order in the interaction Hamiltonian HI . From Eqs. (4.76), (4.89), (4.98) and (4.99) we find that to second order in α the system density operator, in the interaction picture, evolves according to ∂ ρ˜(t) = − α2 ∂t

�

t

˜ I (t), [H ˜ I (s), ρ˜(t) ⊗ ρB ]] dsTrB [H � t 2 ˜ ˜ I (t), [H ˜ I (s), Qχ(0)]]. − iαTrB [HI (t), Qχ(0)] ˜ −α dsTrB [H ˜ 0

0

(4.100)

It is important to note than in deriving Eq. (4.100) no assumptions about the state of either the bath or system have been made. In particular it has not been assumed that the total system-plus-bath state is factorisable at any time. However, because of this our equation of motion has a somewhat complicated form due to the second two inhomogeneous terms in Eq. (4.100) involving the initial state. Let us consider these terms in more detail. The important quantity to consider is the projection onto the irrelevant part of the density operator at time zero. We have Qχ(0) = χ(0) − TrB (χ(0)) ⊗ ρB = χ(0) − ρ(0) ⊗ ρB

(4.101)

where we have now dropped the tildes on the initial density operator since it is equal in the Schr¨odinger and interaction pictures. If we now suppose that our initial condition is such that the total density operator factorises

4.3 Projection operator techniques

83

into system part, ρ(0), and bath part, σB , we find Qχ(0) = ρ(0) ⊗ (σB − ρB ).

(4.102)

If at this stage we choose our reference state ρB to be equal to the initial state of the bath we find Qχ(0) = 0 and the inhomogenous terms in Eq. (4.100) vanish. In fact, for Qχ(0) = 0 the inhomogeneous terms in

the exact equation of motion (4.76) vanish to all orders in α. Importantly, it is with respect to the reference state that the expectation values and correlation functions implicitly contained in Eq. (4.100) must be calculated. Thus, letting ρB = σB may not be a convenient choice and it may be easier to instead keep the inhomogeneous terms in favour of an easier to use reference state. Let us lastly consider the inhomogeneous terms when we have derived a master equation such as Eq. (4.100) from a polaron transformed Hamiltonian. In this case, all density operators (and Liouvillians) are in the polaron frame. We then have QχP (0) = χP (0) − TrB (χP (0)) ⊗ ρB

= eS χ(0)e−S − TrB (eS χ(0)e−S ) ⊗ ρB .

(4.103)

We now see that for the inhomogeneous terms to be zero it is not enough that the initial state factorises into a system part and the reference state. Instead, we now require that this is again the case and that the initial state commutes with the polaron transformation. Another way for the inhomogeneous terms to vanish is to assume factorising initial conditions and that the system part only commutes with the polaron transformation. We then have QχP (0) = ρS ⊗ (eS σB e−S − ρB )

(4.104)

where again σB is the initial state of the bath and ρB the reference state. It can be seen that the inhomogeneous terms now vanish if the initial bath state or the reference state are appropriately chosen with respect to one and other.

4.4 Summary

4.4

Summary

In this chapter we have investigated three important considerations regarding the development of a strong coupling theory. Firstly, we saw how a relaxation of the rotating wave (secular) approximation led to qualitatively diﬀerent predictions in behaviour for the spin-boson model in a weak-coupling analysis. Though one should be cautious not to take too seriously predictions a weak-coupling theory makes in a strong coupling regime, we can expect that if we wish to explore incoherent regimes at all, it makes sense to relax the RWA. Secondly, in section 4.2 we used a polaron transform time-non-local master equation technique to derive some of the accurate (and inaccurate) results known from the non-interacting blip approximation in the intermediate to strong coupling regime. Although the master equation was still approximate, the polaron transformation allowed for it to be derived using a diﬀerent perturbation parameter. We can therefore expect that, for some parameters at least, the polaron transformed Hamiltonian provides us with a better starting point from which to derive a master equation. Lastly, in section 4.3 projection operator techniques were introduced. These were used to derive a time-local master equation valid to second order in some parameter multiplying the ‘interaction’ Hamiltonian. We note that the perturbative parameter could be the system-bath coupling strength, as is commonly used, or, a new parameter facilitated by the polaron transformation. Although a perturbative master equation of this type was also derived in section 2.2, the advantage of the derivation presented in this chapter is that it allowed the Born-approximation to be put on a much more rigorous footing; we saw that it is only necessary to assume a factorising initial state, together with an appropriately chosen reference state, to obtain a homogenous master equation. In chapters 5 and 6 we use a combination of the polaron transformation together with the time-convolutionless projection operator technique to investigate dissipative dynamics beyond the weak-coupling regime. In neither chapter do we make a rotating wave (secular) approximation. NonMarkovian eﬀects are also considered in chapter 5. The combination of methods used in both chapters will be collectively referred as the strong coupling theory.

84

Chapter 5 Excitonic Rabi oscillations of a resonantly driven quantum dot Contents 5.1 5.2 5.3

5.4

5.5

Introduction . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Background . . . . . . . . . . . . . . . . . . . . Model and polaron transformation . . . . . . . . . . 5.2.1 Exciton-phonon spectral density . . . . . . . . . Master equation derivation . . . . . . . . . . . . . . 5.3.1 Markov approximation . . . . . . . . . . . . . . . 5.3.2 Regimes of validity . . . . . . . . . . . . . . . . Resonant excitation dynamics . . . . . . . . . . . . 5.4.1 Time-dependent driving . . . . . . . . . . . . . . 5.4.2 Constant driving . . . . . . . . . . . . . . . . . . Equivalence of the polaron and weak coupling driving renormalisation . . . . . . . . . . . 5.4.3 Non-Markovian eﬀects . . . . . . . . . . . . . . Discussion and summary . . . . . . . . . . . . . . .

. . . . . .

86 86 88 91 92 94 95 98 100 104

. 108 . 109 113

5.1 Introduction

5.1

Introduction

In this chapter we use the strong coupling theory developed in chapter 4 to investigate the excitonic dynamics of a resonantly driven semiconductor quantum dot (QD), beyond the weak system-bath coupling regime. While a weak-coupling theory has been successfully applied to such a system by Ramsay et al [52, 53], the accessible experimental parameters are pushing such an approach to its limits. QD systems of the type studied here therefore provide us with a perfect setting in which to asses the validity of the strong coupling theory since many of the parameters occurring in the model have been experimentally determined. Furthermore, we will see that the strong coupling theory is consistent with some of the measured experimental trends which the weak coupling theory cannot explain. Since the work in this chapter is very much in the context of QD experiments, and particularly those reported in Refs. [52, 53], we first give a brief introduction to QDs and the important sources of decoherence in those systems.

5.1.1

Background

A semiconductor quantum dot of the type studied here consists of a hetrostructure of a lower band-gap material surrounded by a higher band-gap material. Spatial confinement of charge carriers within the hetrostructure gives rise to an atomic-like discrete energy level structure within the dot region [54–56], which allows for the selective probing of particular excitonic (electron-hole pair) transitions [57, 58]. This has led to demonstrations of fundamentally quantum mechanical eﬀects, such as laser-driven excitonic Rabi rotations [52, 53, 59–67] and two-photon interference in QD emission [68–71]. Moreover, optical preparation, control, and readout of a single self-assembled QD spin has been achieved [72–77], while various forms of coupling between closely spaced dots have been observed and characterised [78–82]. Such experimental progress clearly demonstrates the feasibility of creating and manipulating both excitonic and spin quantum coherence in QD samples. However, despite this, QD charge carriers are often still strongly influenced by their surrounding solid-state environment. Though the resulting decoherence processes must generally be mitigated in or-

86

5.1 Introduction der for QDs to be used, for example, in quantum information processing devices [83–93], they also open up intriguing opportunities for exploring system-environment interactions in the solid-state. The combination of strong optical-dipole transitions, well developed control techniques, and relatively pronounced environmental interactions allows QDs to be used to study important open system eﬀects that may be more diﬃcult to observe, for instance, in atomic systems. As an example, the damping of excitonic Rabi rotations in single self-assembled semiconductor QDs has been demonstrated to be drivingdependent [52, 53, 59–63, 66]. Owing to the large variation in QD type, growth strategy, and experimental set-up, a number of possible decoherence channels may be responsible for such behaviour. Two prominent mechanisms are oﬀ-resonant excitation of the wetting layer [66, 94, 95], and coupling to lattice vibrations (phonons) [52, 53, 96–99]. In particular, recent experiments have provided compelling evidence that interactions with longitudinal acoustic (LA) phonons via deformation potential coupling dominates the damping of (ground-state) excitonic Rabi rotations in optically driven InGaAs/GaAs QDs [52, 53]. Furthermore, while it might be hoped that some potential decoherence sources could be suppressed by careful selection of samples and experimental techniques, ultimately selfassembled QDs are embedded in a host matrix. Interactions with phonons thus constitute an intrinsic limitation on the level of coherence seen in their excitonic transitions [12, 96–103]. As such, a range of theoretical approaches have previously been developed to investigate the eﬀects of phonon interactions on the coherent manipulation of excitons in QDs. Examples include perturbative expansions of the QD-phonon coupling, resulting in master equation descriptions of both Markovian [52, 53, 99, 104] and non-Markovian [96, 105, 106] nature, correlation expansions [97, 107, 108], and non-perturbative, numerically exact techniques which rely on calculation of the path integral [98]. The aim of the work presented in this chapter is to extend the master equation approach to QD exciton-phonon interactions beyond the weakcoupling regime. With the tools developed in chapter 4 we present theoretical results describing the phonon-induced damping of a resonantly driven QD using a polaron transform [12, 109] plus time-local master equation technique [8, 110]. The theory exploits a perturbative expansion in the

87

5.2 Model and polaron transformation

88

polaron transformed representation, rather than in the system-bath interaction itself. As we shall show, under certain conditions this allows us to identify a perturbation term that is small over a much larger range of parameters than in the weak-coupling approach [109]. In particular, our master equation is able to account for “nonperturbative” eﬀects not captured in a weak-coupling treatment, such as multiphonon processes and phonon-induced renormalisation of the driving pulse. This is particularly important in exploring the exciton dynamics at elevated temperatures (above 30 K for the parameters we consider), where such eﬀects may become important. Furthermore, we also extend the master equation to the non-Markovian regime.

5.2

Model and polaron transformation

We consider a single QD modelled (as in Refs. [52, 53]) as a two-level system with ground-state |G� and single-exciton state |X�, separated by an energy ω0 . The dot is driven by a laser of frequency ωl and with Rabi frequency Ω(t), set by the strength of the interaction between the dot and the laser field. The dot is coupled to a phonon bath represented by an infinite collection of harmonic oscillators with frequencies ωk and creation (annihilation) operators b†k (bk ). The system-plus-bath Hamiltonian takes the form H(t) =ω0 |X��X| + Ω(t) cos(ωl t)(|G��X| + |X��G|) � � + ωk b†k bk + |X��X| (gk b†k + gk∗ bk ), k

(5.1)

k

where the exciton-phonon couplings are denoted by gk . We now move into a frame rotating at frequency ωl by applying a time-dependent unitary transformation to Eq. (5.1) defined by �

†

H (t) = U (t)H(t)U (t) + i

�

∂U (t) ∂t

�

U † (t)

(5.2)

with U (t) = exp[iωl tσz /2]. We note that the second term in Eq. (5.2) is necessary to ensure that a transformed state U (t) |Ψ(t)� obeys the

Schr¨odinger equation with Hamiltonian H � (t). After the transformation

5.2 Model and polaron transformation

89

we have � ω0 − ωl � H (t) = |X��X| − |G��G| 2 � Ω(t) � + (1 + e−2iωl t )|G��X| + (1 + e2iωl t )|X��G| 2 � � + ωk b†k bk + |X��X| (gk b†k + gk∗ bk ), �

k

(5.3)

k

where a term proportional to the identity has been neglected. In a spirit similar to that of section 2.2.2, we now assume that the laser frequency ωl is larger than both the corresponding Rabi frequency Ω(t), and the detuning ω0 − ωl . In such a case, the terms multiplied by exponentials involving ωl

in Eq. (5.3) may be neglecting since they are expected to average to zero on the timescales that we are interested. We then arrive at the rotating wave Hamiltonian HRWA

� Ω(t) � � δ� |G��X| + |X��G| = |X��X| − |G��G| + 2 � 2 � + ωk b†k bk + |X��X| (gk b†k + gk∗ bk ), k

(5.4)

k

where δ = ω0 − ωl . We note that the rotating wave approximation made here is similar, but diﬀerent from that of section 2.2.2. In the present

case we neglect rapidly oscillating terms in the Hamiltonian itself, in the Sch¨odinger picture. In section 2.2.2, however, terms appearing in the master equation describing the reduced density operator in the interaction picture were neglected. We do not therefore expect the rotating wave approximation made here to have the same consequences for the Bloch vector evolution as that of section 2.2.2 To move into the appropriate basis for the subsequent perturbation theory, we now apply a polaron transformation to HRWA in much the same way as in section 4.2. The transformation displaces the bath oscillators when the QD is in its excited state only since there is no coupling to the ground state. The transformed Hamiltonian is defined by HP = eS HRWA e−S , where now S = |X��X|

� k

(αk b†k − αk∗ bk ),

(5.5)

5.2 Model and polaron transformation

90

with αk = gk /ωk . Hence, we may write e±S = |G��G| + |X��X| where

�

± k D(±αk ) = e

�

† ∗ k (αk bk −αk bk )

�

D(±αk ),

(5.6)

k

is a product of displacement op-

erators. Defining the Pauli matrices in the {|G�, |X�} basis as σx = |X��G| + |G��X|, σy = i(|G��X| − |X��G|), and σz = |X��X| − |G��G|, we find that our polaron-transformed Hamiltonian reads [109] HP =

� δ� Ωr (t) Ω(t) σz + σx + ωk b†k bk + (σx Bx + σy By ) , 2 2 2 k

(5.7)

where the detuning is now δ � = ω0� − ωl , defined in terms of the bath-shifted � QD transition energy ω0� = ω0 − k ωk |αk |2 , and we have ignored a term proportional to the identity. Bath-induced fluctuations are now described by the Hermitian combinations 1 Bx = (B+ + B− − 2B), 2 and By = where B± =

�

k

(5.8)

1 (B− − B+ ), 2i

(5.9)

D(±αk ), and B = �B± � is the expectation value of the

bath displacement operators with respect to the bath reference state.

An important diﬀerence between the approach used in derivation of the NIBA [section 2.3.3] and that used here is that we have subtracted from the bath operators above their expectation values. In doing this we ensure that the expectation value of the interaction term in Eq. (5.7) is zero, and the time-convolutionless projection operator technique from section 4.3.2 can be used. Furthermore, this also causes a reintroduction of the driving term in Eq. (5.7) but now at a strength renormalised by a factor equal to the bath operator expectation value: Ωr (t) = Ω(t)B. For a phonon bath in thermal equilibrium at inverse temperature β = 1/kB T , from appendix A we find

�

B ≡ �B± � = exp − (1/2)

� k

�

2

|αk | coth (βωk /2) .

Defining the spectral density in the usual way, J(ω) =

�

k

(5.10)

|gk |2 δ(ω − ωk ),

5.2 Model and polaron transformation

91

allows the summation in Eq. (5.10) to be converted into an integral, �

B = exp − (1/2)

�

∞ 0

� J(ω) dω 2 coth (βω/2) . ω

(5.11)

In the following subsection we consider the appropriate functional form of J(ω).

5.2.1

Exciton-phonon spectral density

We are specifically interested in the coupling of bulk longitudinal acoustic phonons (LA-phonons) to our QD exciton by means of deformation potential coupling, shown to dominate the dephasing dynamics in Ref. [52]. √ For coupling of this type it has been shown that gk = kD(k)/ 2µωk V , where µ is the mass density and V is the unit cell volume [104, 111]. The form factor is given by D(k) =

�

� � dr Dh |ψh (r)|2 − De |ψe (r)|2 e−ik·r ,

(5.12)

where ψe (r) (ψh (r)) is the electron (hole) ground state wavefunction and De (Dh ) the bulk deformation-potential coupling for electrons (holes). For simplicity we assume a spherically symmetric confining potential for the QD which gives

√

ψe (r) = (de π)

−3/2

�

r2 � exp − 2 2de

(5.13)

and an identical expression for ψh with the electron localisation length de replaced with that for a hole, dh . If we further assume de = dh = d and a linear dispersion relation, k = ωk /c with c the excitation speed, we find D(k) = (Dh − De ) exp[−(ωk /ωc )2 ], with the naturally arising cut-oﬀ √ frequency ωc = 2c/d. The summation over k in J(ω) must be done in three dimensions and with the help of appendix B we find in the continuum limit J(ω) =

|Dh − De |2 3 −(ω/ωc )2 2 ω e = αω 3 e−(ω/ωc ) . 2 5 4π c µ

(5.14)

The coupling constant α (here having units of s2 ) captures the strength of the exciton-phonon interaction and can be seen to be dependent upon bulk quantities of the QD sample [52]. The exponential cut-oﬀ with frequency

5.3 Master equation derivation

92

ωc is proportional to the inverse of the carrier localisation length. It is important to note that, apart from the rotating-wave approximation on the driving, we have made no further approximations in our manipulations leading from Eq. (5.4) to Eq. (5.7). We have simply put the Hamiltonian into a form that clearly separates the eﬀects of the QDphonon coupling into renormalisation of QD parameters, through Ωr (t) and ω0� , and bath-induced fluctuations, through the last term in HP .

5.3

Master equation derivation

Utilising our transformed representation of the QD Hamiltonian, we shall now derive a time-local master equation describing the driven QD exciton dynamics under the influence of the acoustic phonon environment. Since this is the first time we have done so some detail will be presented. Furthermore, the time dependence in the driving field introduces some subtleties which must be considered. To proceed we separate the polaron-transformed Hamiltonian such that HP (t) = H0P (t) + HIP (t). Here, H0P (t) = HSP (t) + HBP , with bath Hamil� tonian HBP = k ωk b†k bk , and time-dependent system part δ� Ωr (t) σz + σx , 2 2

(5.15)

Ω(t) (σx Bx + σy By ) 2

(5.16)

HSP (t) = while HIP (t) =

is the interaction Hamiltonian, to be treated as a perturbation. Moving into the interaction picture with respect to H0P (t) yields an interaction Hamiltonian in the (polaron-transformed) interaction picture of the form ˜ IP (t) = U † (t)HIP (t)U0P (t), H 0P

(5.17)

where U0P (t) = USP (t)e−iHBP t , with �

USP (t) = T← exp −i

�

t

�

dvHSP (v) . 0

(5.18)

Here, the Schr¨odinger and interaction pictures have been chosen to coincide at time t = 0, while the time-ordering operator T← is necessary as, in

5.3 Master equation derivation

93

general, HSP (t) does not commute with itself at two diﬀerent times. We therefore write the interaction Hamiltonian as � � ˜ IP (t) = Ω(t) σ ˜x (t) + σ ˜y (t) , H ˜x (t)B ˜y (t)B 2

(5.19)

† ˜l (t) = eiHBP t Bl e−iHBP t , for l = x, y. where σ ˜l (t) = USP (t)σl USP (t) and B

We now follow the standard projection-operator procedure, outlined in section 4.3.2, to derive a time-local master equation for the reduced system density operator, ρ˜SP (t), in the polaron frame interaction picture. Considering the QD to be initialised in its ground state, with the bath initially in thermal equilibrium, ρB (0) = ρB = e−β

�

k

ωk b†k bk

/trB (e−β

�

k

ωk b†k bk

), we

see that the initial system-bath density operator χ(0) = |0��0|ρB is un-

aﬀected by transformation into the polaron representation, i.e. χP (0) = eS χ(0)e−S = |0��0|ρB = χ(0). Hence, taking a thermal equilibrium state ˜ IP (t) to second order, we of the bath as a reference state and treating H find a homogeneous equation [8, 110] ∂ ρ˜SP (t) =− ∂t

�

t 0

� � ˜ IP (t), [H ˜ IP (s), ρ˜SP (t)ρB ]] , dstrB [H

(5.20)

describing the dynamics of the excitonic system in the polaron frame, under the influence of the phonon bath. Substituting in from Eq. (5.19), we obtain ∂ ρ˜SP (t) Ω(t) =− ∂t 4

�

t

�

dsΩ(s) [˜ σx (t), σ ˜x (s)˜ ρSP (t)]Λxx (τ ) 0

�

+ [˜ σy (t), σ ˜y (s)˜ ρSP (t)]Λyy (τ ) + H.c. , (5.21)

where H.c. refers to the Hermitian conjugate, and we have made use of the stationarity of the bath reference state to write ˜l (t)B ˜l (s)�B = �B ˜l (t − s)B ˜l (0)�B = Λll (τ ), �B with τ = t − s. Using J(ω) =

�

k

(5.22)

|gk |2 δ(ω − ωk ) allows us to write the

5.3 Master equation derivation

94

relevant correlation functions in the continuum limit as B 2 φ(τ ) (e + e−φ(τ ) − 2), 2 B 2 φ(τ ) Λyy (τ ) = (e − e−φ(τ ) ), 2

Λxx (τ ) =

(5.23) (5.24)

where φ(τ ) =

�

∞ 0

� J(ω) � dω 2 cos ωτ coth(βω/2) − i sin ωτ . ω

(5.25)

Now, moving back into the Schr¨odinger picture, and making the change of variables s → t − τ , we obtain i ρ˙ SP (t) = − [δ � σz + Ωr (t)σx , ρSP (t)] 2 � � Ω(t) t − dτ Ω(t − τ ) [σx , σx (t − τ, t)ρSP (t)]Λxx (τ ) 4 0 � + [σy , σy (t − τ, t)ρSP (t)]Λyy (τ ) + H.c. , (5.26)

† † where σl (s, t) = USP (t)USP (s)σl USP (s)USP (t).

Eq. (5.26) is a non-

Markovian master equation describing the QD exciton dynamics in the polaron frame for a time-dependent laser-driving pulse envelope Ω(t), and valid to second order in HIP (t).

5.3.1

Markov approximation

While we could directly use Eq. (5.26) as a basis for numerical simulation of the exciton dynamics (and we shall in fact do so in Section 5.4.3), a great deal of insight into the system behaviour can be gained through the simplifications allowed by the Markov approximation. To make a Markov approximation in the present case, we let the upper limit of integration in Eq. (5.26) go to infinity under the assumption that the bath correlation functions Λll (τ ) decay on a timescale that is short compared to that of the system dynamics we would like to capture. Given this, we may also approximate † USP (t − τ )USP (t) ≈ exp [iHSP (t)τ ],

(5.27)

5.3 Master equation derivation

95

while replacing Ω(t − τ ) by Ω(t) in the integral in Eq. (5.26). We may then write

σx (t − τ, t) ≈

δ �2 cos ητ + Ωr (t)2 δ � sin ητ δ � Ωr (t)(1 − cos ητ ) σ + σ + σz , x y η2 η η2 (5.28)

and σy (t − τ, t) ≈ − where η =

δ � sin ητ Ωr (t) sin ητ σx + cos ητ σy + σz , η η

(5.29)

� δ �2 + Ωr (t)2 .

In the following, we shall consider the case of resonant excitation, δ � = 0, which simplifies Eqs. (5.28) and (5.29) to σx (t − τ, t) = σx and σy (t −

τ, t) ≈ cos(Ωr (t)τ )σy + sin(Ωr (t)τ )σz , respectively. We then arrive at a Markovian, polaron transformed master equation i Ω(t)2 ρ˙ SP (t) = − [Ωr (t)σx , ρSP (t)] − 2 4

�

∞ 0

�

dτ [σx , σx ρSP (t)]Λxx (τ )

+ cos(Ωr (t)τ )[σy , σy ρSP (t)]Λyy (τ ) + sin(Ωr (t)τ )[σy , σz ρSP (t)]Λyy (τ ) � + H.c. , (5.30) which we shall use to explore the dynamics of a resonantly driven QD beyond the weak exciton-phonon coupling regime. We note that although Eq. (5.30) is now of the form of a Born-Markov master equation in the polaron frame, upon transformation back into the original (or “lab”) frame changes in the QD state can have an influence on the phonon bath. Specifically, the bath state is not stationary in the original frame, and the systembath state is not generally separable.

5.3.2

Regimes of validity

In this section we use a simple analysis to give an estimate of the accuracy of the polaron transform master equation we have derived. Having derived Eqs. (5.26) and (5.30) perturbatively in the polaron frame, we should certainly expect their validity to be limited in some manner. Recall that the polaron transformation displaces the bath oscillators in reaction to a change of state of the QD. Intuitively, we would therefore expect the polaron transformed representation to be applicable when the bath is able

5.3 Master equation derivation

96

to react on a timescale shorter than or similar to that on which the QD exciton itself evolves. Since the timescale on which the bath reacts is set approximately by the inverse of the cut-oﬀ frequency (τB ∼ 1/ωc ), we would therefore expect Eq. (5.26) to work best in the regime Ω/ωc < 1.

Additionally, the Markov approximation made in deriving Eq. (5.30) limits its validity to timescales greater than τB . To put these considerations on a slightly more quantitative footing, we can make a rough estimate of the regime of validity of our perturbative expansion by considering the magnitude of the perturbative terms in the master equation, namely (Ω2 /4)Λll (τ ) [8, 110], for constant driving Ω(t) = Ω. For example, consider the upper bound on the magnitude of Λyy (τ ), given by |Λyy (0)| = (1/2)(1 − B 4 ). Bearing in mind that

Λyy (τ ) tends to zero on a timescale of order 1/ωc , we see that we want (Ω2 /4)(|Λyy (0)|/ωc ) = (Ω2 /8ωc )(1 − B 4 ) to be small in the sense that

terms higher than second order in HIP may be neglected in the master equation expansion. Since �HIP � = 0, we know from section 4.100 that the

next term is of fourth order, and its magnitude can be estimated in a similar manner by (Ω2 /4)2 (|Λyy (0)|2 /ωc3 ). Thus, ignoring numerical factors, we find that the fourth order term is small in comparison to the second order, provided that the condition �

Ω ωc

�2

(1 − B 4 ) � 1,

(5.31)

is satisfied 1 . In line with our previous intuition, this condition tells us that in the scaling limit (Ω/ωc � 1) we expect our treatment to be valid well

beyond a standard weak system-bath coupling approach, such that we can explore both the weak (B ≈ 1, small α and/or low T ) and strong (B � 1,

large α and/or high T ) system-bath coupling regimes, as well as reliably interpolate between these two extremes [109]. Outside the scaling limit, our approach should remain valid provided that the system-bath coupling is small enough, or the temperature low enough, such that the inequality of Eq. (5.31) is still satisfied. To demonstrate how the polaron approach generally allows a larger regime of parameter space to be explored than a standard weak-coupling 1

Considering instead the magnitude of Λxx (τ ) leads to a similar condition, which gives essentially the same regime of validity.

5.3 Master equation derivation

97

treatment, we can also apply the above reasoning to assess the regime of validity of such a weak-coupling master equation. In this case, we have a weak-coupling correlation function [99], ΛW (τ ) =

�

∞ 0

dωJ(ω)(cos (ωτ ) coth (βω/2) − i sin (ωτ ))

(5.32)

which was seen in section 2.3.2. The weak coupling correlation function again falls to zero on a timescale of order 1/ωc . Hence, in a similar manner to before, we estimate the second order perturbation to be roughly of magnitude |ΛW (0)|/ωc , while the fourth order is then |ΛW (0)|2 /ωc3 . We then find the condition

|ΛW (0)| � 1, ωc2

(5.33)

as an estimate of the range of validity of the weak-coupling approach. Considering first the zero temperature limit, we find ΛW (0) ∼ αωc4 ,

where we use the QD spectral density given in Eq. (5.14). Hence, at zero temperature, our condition implies that a weak exciton-phonon coupling treatment should be adequate to describe the QD excitonic dynamics provided that αωc2 � 1.

(5.34)

However, as temperature is increased, the magnitude of ΛW (0) does too, and we therefore expect the weak-coupling treatment to worsen. Approximating coth (βω/2) ≈ 2/(βω), we find αωc � 1, β

(5.35)

or αωc2 /(βωc ) � 1, which is clearly a harder criterion to fulfill than the zero temperature condition. Hence, for a given system-bath coupling strength

α and cutoﬀ frequency ωc , as the temperature of the bath is increased, a weak-coupling treatment of the system-bath interaction becomes a worse approximation. Though we should be wary of reading too much into numerical values obtained from these rough validity conditions, for the system studied in Refs. [52, 53] we can take α = 0.027 ps2 and ωc = 2.2 ps−1 extracted through fits to the data, which gives αωc2 ≈ 0.1. Hence, we might expect

a weak-coupling treatment to be valid at low temperatures for this QD

5.4 Resonant excitation dynamics system, as borne out by the excellent agreement between experimental data and theory in Refs. [52, 53]. However, by a temperature of 50 K, we find |ΛW (0)|/ωc2 ≈ 0.4, such that the weak-coupling approximation is now

becoming dubious. In contrast, for the same parameters, and taking Ω = 1 ps−1 , we find that the polaron condition [Eq. (5.31)] gives (Ω/ωc )2 (1 −

B 4 ) ≈ 0.03 at T = 0, increasing up to (Ω/ωc )2 (1 − B 4 ) ≈ 0.15 at T = 50 K. In fact, we shall show below that it is around temperatures of 30 K

and above that we begin to see significant diﬀerences between the weakcoupling and polaron treatments of our driven QD, signifying (in this case) that the system is beginning to move out of the weak-coupling regime, and both driving-renormalisation and multiphonon processes are starting to become important.

5.4

Resonant excitation dynamics

We now proceed to explore the excitonic dynamics of our QD system, focusing in particular on comparing how the polaron and weak-coupling theories capture the interplay between the driving-induced coherent population oscillations and the phonon environment as we vary the temperature. Interestingly, we shall see that as the phonon-induced damping rate naturally depends upon the renormalised Rabi frequency Ωr in the polaron theory, but on the original Rabi frequency Ω in the weak-coupling theory, the weak-coupling approach can actually overestimate the damping rate even in the high-temperature regime where mulitphonon eﬀects are important. Experimentally, it is generally the excitonic population ρXX that is measured, for example through photocurrent detection [52, 61] or microcavity-asissted photon emission [64, 65]. Here, we express the solutions to our Markovian master equation through the Bloch vector introduced in section 2.1.3, but now defined in the polaron frame as αP = (αxP , αyP , αzP )T = (�σx �P , �σy �P , �σz �P )T , where �σi �P = trS+B (σi χP (t)),

for i = x, y, z. Since σz is invariant under the polaron transformation, eS σz e−S = σz , we see that in the original (lab) frame αz = trS+B (σz χ(t)) = trS+B (σz χP (t)) = αzP , and the Bloch vector elements along z are equivalent in the two representations. Hence, ρXX = (1+αz )/2 = (1+αzP )/2, and we may work entirely in the polaron frame provided we are only interested

98

5.4 Resonant excitation dynamics

99

in population dynamics. From Eq. (5.30) we find that the polaron frame Bloch vector evolves according to ˙ P = M (t) · αP + b(t), α where



 M (t) = 

−(Γz − Γy )

0

(5.36) 0



 −Ωr (t)  , (Ωr (t) + λ) −Γz −Γy

0 0

and b(t) = (−κx , 0, 0)T . Here,

Ω(t)2 γxx (0), 2 Ω(t)2 Γz = (γyy (Ωr (t)) + γyy (−Ωr (t)) + 2γxx (0)) , 4 Ω(t)2 λ= (Syy (Ωr (t)) − Syy (−Ωr (t))) , 2 Ω(t)2 κ= (γyy (Ωr (t)) − γyy (−Ωr (t))) , 4

Γy =

(5.37)

(5.38) (5.39) (5.40) (5.41)

where as usual the rates (P)

(5.42)

(P)

(5.43)

γll (ω) = 2Re[Kll (ω)], and energy shifts Sli (ω) = Im[Kli (ω)],

are written in terms of the one-sided Fourier transforms of the correlation functions [see section 2.2.3] (P) Kll (ω)

=

�

∞

dτ eiωτ Λll (τ ).

(5.44)

0

Note that as we are solely interested in the exciton population dynamics it suﬃces to consider only the Bloch equations for αy and αz since, in the resonant case, that for αx becomes decoupled. In comparison, the weak coupling theory from Refs. [52,53], also found in section 4.1.1 (although the notation is slightly diﬀerent there) finds

5.4 Resonant excitation dynamics

100

equations of motion for the relevant elements of the Bloch vector α˙ y

=

−ΓW αy − (Ω(t) + λW )αz ,

(5.45)

α˙ z

=

Ω(t)αy .

(5.46)

The damping rate and energy shift can be expressed in our current notation as 1 (γW (Ω(t)) + γW (−Ω(t))) , 4 1 = (SW (Ω(t)) − SW (−Ω(t))) , 2

ΓW =

(5.47)

λW

(5.48)

respectively, with the weak-coupling correlation function given by ΛW (τ ) =

�

∞ 0

J(ω)dω(cos ωτ coth (βω/2) − i sin ωτ ).

(5.49)

We can evaluate ΓW in closed form, giving ΓW =

π J(Ω(t)) coth(βΩ(t)/2). 2

(5.50)

Hence, the weak-coupling rate displays a linear temperature dependence in the high-temperature regime [52] and, as mentioned previously, is dependent upon the original Rabi frequency Ω(t) as opposed to the bathrenormalised value. Furthermore, we see that there is no frequency independent or pure-dephasing contribution to the weak-coupling rate in the Born-Markov approximation, as was also seen in section 4.1.1. This is in contrast to the terms γxx (0) appearing in the polaron theory through Γy and Γz .

5.4.1

Time-dependent driving

Having outlined some of the similarities and diﬀerences between the weakcoupling and polaron transform approaches, we shall now compare their respective predictions in the case of resonant driving with a Gaussian pulse envelope. Rather than looking at the dynamics in the time domain, we shall instead explore oscillations in the excitonic population (Rabi rota� +∞ tions) as a function of varying pulse area, Θ = −∞ Ω(t)dt, for fixed pulse duration, as is common experimentally. We therefore consider a Gaus-

5.4 Resonant excitation dynamics

101

T!75

14

T!70 T!65

12

T!60 T!55

10

T!50 T!45

8 Ρxx

T!40 T!35

6

T!30 T!25

4

T!20 T!15

2

T!10

0 0

T!5

2

4

6

8

10

12

"!Π"

Figure 5.1: Excitonic population as a function of driving pulse area (in units of π), for temperatures ranging from 5 K to 75 K, where each curve has been oﬀset by an increasing integer for clarity. Blue solid lines are calculated using the polaron approach, while red dashed lines are calculated using weak-coupling theory. Note that due to phonon-induced frequency shifts the maxima and minima of the curves are not expected to occur at integer multiples of π. Parameters: α = 0.027 ps2 and ωc = 2.2 ps−1 .

5.4 Resonant excitation dynamics

102

sian pulse of fixed width τ but varying peak magnitude, centred around √ t = 0, and described by Ω(t) = (Θ/2τ π)exp[−(t/2τ )2 ]. Starting at a time −t0 well before the pulse (i.e. t0 � τ ), we initialise the QD in its ground state: αP = α = (0, 0, −1)T . We then numerically solve the Bloch equations (Eq. (5.36) in the polaron case, Eqs. (5.45) and (5.46) in the

weak-coupling case) to find the state of the system at any time t satisfying t � τ , such that the pulse has eﬀectively ended.

Fig. 5.1 shows the final excitonic population, ρXX , calculated from the

polaron and weak-coupling theories as described above, as a function of total pulse area, Θ (in units of π), for temperatures ranging from T = 5 K to T = 75 K (each plot has been oﬀset by an increasing integer for clarity). We use experimentally determined values of the exciton-phonon coupling strength and cut-oﬀ frequency, α = 0.027 ps2 and ωc = 2.2 ps−1 , respectively [53], and a Gaussian driving pulse of width τ = 4 ps. Note that for the largest pulse areas studied here the ratio Ω/ωc has a maximum value of ∼ 1.2.

At low to intermediate temperatures (T < 30 K), we see that the weak-

coupling and polaron theories agree very closely in their predictions for the population dynamics, consistent with the excellent agreement found previously between experimental observations and the weak-coupling theory in this regime [52, 53]. Importantly, the two theories predict almost exactly the same dependence of the Rabi rotation damping rate and frequency shift on increasing temperature and pulse area, provided the temperature does not increase much above 30 K. As expected, the phonon-induced damping is strongly driving-dependent, with oscillations becoming almost totally suppressed at high pulse areas for all but the lowest temperatures. Perhaps the most striking feature apparent from Fig. 5.1, however, is that the weak-coupling theory tends to overestimate the damping eﬀect of the phonons at higher temperatures, when compared to the polaron theory. In fact, as we shall see below in the case of constant driving, provided Ω/ωc ∼ 0.7 or smaller, the weak-coupling theory predicts a larger

damping rate than the polaron theory at all temperatures for the realistic parameters studied here. At the single-phonon level, this diﬀerence can be attributed directly to the temperature-dependent suppression of the driving pulse that occurs in the polaron transformed Hamiltonian (see Eq. (5.7)). Among other

5.4 Resonant excitation dynamics

103

things, this has the consequence that the rates appearing in the polaron Bloch equations are to be evaluated at the (smaller) renormalised pulse strength Ωr (t), rather than at the bare pulse strength Ω(t) as in the weakcoupling theory. The resulting eﬀect can be seen clearly by expanding the relevant polaron rates Γy and Γz (in Eqs. (5.38) and (5.39), respectively) up to their single-phonon terms. We then find a damping rate of precisely the same form as in the weak-coupling theory, Γ1−ph =

π J(Ωr (t)) coth (βΩr (t)/2) , 2

(5.51)

though evaluated at the renormalised Ωr (t), as expected. For low pulse areas, we can approximate Γ1−ph ≈ (απ/2)Ωr (t)3 coth (βΩr (t)/2). Hence, for single-phonon processes at least, the lessening of the damping rate in the polaron theory is simply due to the fact that we are sampling the spectral density at a lower frequency, since Ωr (t) < Ω(t). Any diﬀerences would then become more pronounced at higher temperatures, since this is when Ωr most diﬀers from Ω. In the full polaron theory, however, the situation is of course much more complicated than this simple analysis would suggest. To begin with, we have no particular reason to expect the single-phonon rate of Eq. (5.51) to be valid over a larger temperature range than the weak-coupling rate of Eq. (5.50), so the sampling of the spectral density at diﬀerent frequencies in the two theories cannot be the whole story. Looking again, for example, at the full polaron rate Γy (which in fact disappears in the single-phonon approximation), we see from Eqs. (5.23), (5.38), (5.42) and (5.44) that it may be written Γy

= =

2

Ω(t) Re

�

Ωr (t)2 Re 2

∞

dτ Λxx (τ ),

0

�

∞ 0

dτ (eφ(τ ) + e−φ(τ ) − 2).

(5.52)

Thus, in determining the overall size of the full polaron rates at higher temperatures, there additionally exists a competition between the multiphonon eﬀects accounted for by the exponentiation of the phonon propagator φ(τ ), which increases the rate in comparison to the single-phonon approximation, and the overall factor proportional to Ωr (t)2 , which again acts to decrease it with increasing temperature.

5.4 Resonant excitation dynamics

104

A further feature to draw out from the comparison presented in Fig. 5.1 is that while the polaron theory predicts physical behaviour at all temperatures considered, for the highest temperature (T = 75 K) the weakcoupling theory actually predicts unphysical behaviour, since ρXX becomes negative for pulse areas Θ ∼ π. This behaviour can be related to an overes-

timate of the phonon-induced frequency shift in the weak-coupling analysis at high temperatures, and will again be discussed in more detail below for the case of constant driving.

5.4.2

Constant driving

In order to put the arguments outlined in the previous section on a more formal footing, it is helpful to consider the dynamics of the QD system for constant driving, in which case an analytic form can be given for the population diﬀerence, αz = ρXX − ρ00 . We construct a second-order dif-

ferential equation for the time evolution of αz in both the polaron and weak-coupling theories. From Eq. (5.36) we find for the polaron theory (using αzP = αz ) α ¨ z + (Γy + Γz )α˙ z + (Ωr (Ωr + λ) + Γy Γz )αz = 0,

(5.53)

which has solution (for αz (0) = −1) αz (t) = −e

−ΓP t/2

�

� ΓP cos(ξP t/2) + sin(ξP t/2) , ξP

(5.54)

with damping rate Γ P = Γy + Γ z =

Ω2 (γyy (Ωr ) + γyy (−Ωr ) + 4γxx (0)), 4

(5.55)

and oscillation frequency ξP =

�

4Ωr (Ωr + λ) − (Γz − Γy )2 .

(5.56)

On the other hand, Eqs. (5.45) and (5.46) give for the weak-coupling theory α ¨ z + ΓW α˙ z + Ω(Ω + λW )αz = 0,

(5.57)

5.4 Resonant excitation dynamics

105

!!W ,P,1"ph" !ps"1 "

0.20

0.15

0.10

0.05

0.00 0

10

20

30

40

50

60

70

T!K"

Figure 5.2: Temperature dependence of the weak-coupling rate ΓW (red dashed curve), polaron rate ΓP (blue solid curve), and single-phonon expansion of the polaron rate Γ1−ph (green dotted curve). Parameters: α = 0.027 ps2 , ωc = 2.2 ps−1 , and we have evaluated each rate at Ω = 0.5 ps−1 .

which has a solution of exactly the same form, αzW (t) = −e

−ΓW t/2

�

� ΓW cos(ξW t/2) + sin(ξW t/2) , ξW

(5.58)

though this time with the weak-coupling damping rate ΓW of Eq. (5.50), and oscillation frequency ξW

� = 4Ω(Ω + λW ) − Γ2W .

(5.59)

In the constant driving case, we may therefore directly compare the rate ΓP and frequency ξP in the polaron theory to the weak-coupling expressions ΓW and ξW , respectively. In Fig. 5.2 we plot the damping rates ΓP and ΓW , along with the singlephonon expansion of the polaron rate (Γ1−ph of Eq. (5.51)), as a function of temperature for an arbitrarily chosen value of the constant driving, Ω = 0.5 ps−1 . For these parameters, the weak-coupling approximation is indeed shown to predict a larger rate for all values of T . Furthermore, there is a significant diﬀerence between the full polaron rate (ΓP ) and its single-phonon expansion (Γ1−ph ) above temperatures of about 10 − 15 K, indicating that multiphonon eﬀects are becoming important. Hence, in this regime, even though the weak-coupling rate is still too large, we cannot

5.4 Resonant excitation dynamics

106

2.0

$"W ,P# "ps#%1

1.5

1.0 T!75K

0.5 T!5K

0.0 0.0

0.2

0.4

0.6

0.8

1.0

"!Ωc

Figure 5.3: Dependence of the polaron damping rate ΓP (blue solid curves) and weak-coupling damping rate ΓW (red dashed curves) on the driving frequency Ω. The two sets of curves correspond to temperatures of T = 5 K and T = 75 K, as indicated. Parameters: α = 0.027 ps2 , ωc = 2.2 ps−1

simply fix it by replacing Ω → Ωr in ΓW (i.e. taking ΓW → Γ1−ph ) as this neglects important multiphonon processes. Notice also that while the

weak-coupling rate varies linearly with temperature above a few Kelvin, the single-phonon expansion does not, despite having a very similar form, due to the temperature dependence inherent to Ωr . In Fig. 5.3 we show how the polaron and weak-coupling rates vary with the strength of the driving frequency Ω, for low and high temperatures. As observed experimentally [52, 53], we see a clear and strong dependence on the driving strength for both temperature regimes, and in both the weakcoupling and polaron theories. It is also interesting to note that around Ω/ωc ∼ 0.7 in the high temperature case, the polaron and weak-coupling rates cross, indicating that above this value the hierarchy of rates discussed in reference to Fig. 5.2 no longer holds. We emphasise again that, as in the case of time-dependent driving, there are two important eﬀects present in the polaron theory which are not captured by the weak-coupling treatment, and which become increasingly relevant as the temperature is increased. Firstly, there are multiphonon contributions, which tend to increase the damping rate, as can clearly be seen by comparing the full polaron rate to its single-phonon expansion in Fig. 5.2. Secondly, the interaction of the QD exciton with the phonon bath causes a reduction in the eﬀective driving field. For Ω/ωc < 1, this tends

5.4 Resonant excitation dynamics

107

to decrease the damping rate, as can be seen from Fig. 5.3. We are also now in a position to explain the origin of the unphysical behaviour predicted by the weak-coupling theory at 75 K (see Fig. 5.1). In Fig. 5.4 we again plot the excitonic population as a function of pulse area, but this time for a constant driving pulse of 14 ps duration, which is roughly equal to the full-width-half-maximum (FWHM) of the Gaussian pulse used in Fig. 5.1. Notice that for T = 75 K, the excited state again takes on unphysical negative values in the weak-coupling theory for pulse areas Θ ∼ π (plot (b)). To see how this comes about, we must

consider the weak-coupling oscillation frequency ξW of Eq. (5.59). For Γ2W > 4Ω(Ω + λW ), we find that ρXX can take on negative values when λW < −Ω, i.e. when the correction to the driving frequency is larger than the frequency itself, the weak-coupling theory discussed here breaks down.

This can ultimately be attributed to the fact that no secular (rotating wave) approximation of the sort discussed in section 2.2.2 was made in the derivation of the weak-coupling Bloch equations [8]. The corresponding master equation is therefore not of Lindblad form. Let us now consider the frequency shift in slightly more detail. In the weak-coupling theory we find from Eqs. (5.48) and (5.49) that λW = Ω P

�

∞

dω

0

J(ω) coth(βω/2) , Ω2 − ω 2

(5.60)

where the P indicates that the Cauchy principal value should be taken, �∞ and we have made use of the identity 0 eiωs ds = πδ(ω) + P(i/ω). Since we expect the weak-coupling theory to break down in the high temperature limit, we can evaluate λW analytically by approximating coth(x) ≈ x−1 in the integrand of Eq. (5.60). In doing so, we find λW

√ � Ω παωc � ≈− 1 − 2(Ω/ωc )F (Ω/ωc ) , β

where F (x) = exp[−x2 ]

�x 0

(5.61)

exp[y 2 ]dy is the Dawson integral. The condition

λW < −Ω, which determines when we expect unphysical behaviour from the weak-coupling theory, then becomes β<

√

παωc ,

(5.62)

5.4 Resonant excitation dynamics

108

1.0 !a"

0.8 0.6 ΡXX 0.4 0.2 0.0 0

2

4

6

8

10

12

!!Π" 1.0 !b"

0.8 0.6 ΡXX 0.4 0.2 0.0 0

2

4

6

8

10

12

!!Π"

Figure 5.4: Exciton population as a function of pulse area for constant driving. Plot (a) is calculated using the polaron theory while plot (b) uses weakcoupling theory. The diﬀerent curves in each plot correspond to temperatures ranging from 5 K to 75 K in steps of 10 K, with lower temperatures coloured blue and higher temperatures red (the arrows indicate increasing temperature). Parameters: pulse duration = 14 ps, α = 0.027 ps2 and ωc = 2.2 ps−1 .

where we take the limit Ω/ωc � 1. For the parameters of Fig. 5.4, we

then expect to obtain unphysical behaviour when T > 72 K, in good agreement with the actual dynamics. We note that while Eq. (5.62) may give a bound on when the limits of the weak-coupling theory are met, its degree of accuracy may become poor well before this condition is satisfied (see Eq. (5.35) and discussion there). Equivalence of the polaron and weak coupling driving renormalisation Turning now to the low temperature regime where we expect the weak coupling theory to be accurate, we have seen previously that by expanding the full polaron damping rate to its single-phonon terms we may recover the weak-coupling damping rate, though evaluated at a renormalised frequency (compare Eqs. (5.50) and (5.51)). In order to complete the picture, we shall now show a similar equivalence between the polaron and weak-

5.4 Resonant excitation dynamics

109

coupling frequency shifts at the single-phonon level. Expanding Eq. (5.40) to first order in J(ω) we find the single-phonon approximation to the polaron frequency shift, λ → λ1−ph , where λ1−ph =

Ω3r P

�

∞

dω

0

J(ω) coth(βω/2) . ω 2 (Ω2r − ω 2 )

(5.63)

However, this is not quite the whole story since, in the polaron theory, the driving frequency is shifted both by λ, and also at the Hamiltonian level through Ω → Ωr . Ultimately, it is the observables, such as the population diﬀerence, that are the physically meaningful quantities to consider. Inspection of the frequencies ξP in Eq. (5.56) and ξW in Eq. (5.59) therefore tells us that we should compare Ωr (Ωr + λ) in the polaron theory to Ω(Ω + λW ) in the weak-couping theory, as we know that Γz − Γy ≈ ΓW

at the single-phonon level. Expanding Ωr to first order in J(ω), we find Ωr (Ωr + λ) ≈ Ω(Ω + ∆Ω) where ∆Ω = Ω

�

∞ 0

J(ω) dω 2 coth(βω/2) ω

�

ω 2 + Ω2r (B 2 − 1) Ω2r − ω 2

�

.

(5.64)

Expanding the remaining factors of B and occurrences of Ωr to first order in J(ω) we find that Eq. (5.64) reduces to Eq. (5.60) (i.e. ∆Ω →

λW ); in the weak-coupling limit, the polaron and weak-coupling theories therefore predict the same correction to the driving frequency.

5.4.3

Non-Markovian eﬀects

Finally, we shall relax the Markov approximation made in section 5.3.1 to investigate non-Markovian eﬀects on the QD exciton dynamics [96–98, 105–108], within the polaron frame [110]. Referring to Eq. (5.26), and considering the case of constant driving for simplicity, we find that our non-Markovian master equation may be written �

t

� dτ [σx , σx (t − τ, t)ρSP (t)]Λxx (τ ) 0 � + [σy , σy (t − τ, t)ρSP (t)]Λyy (τ ) + H.c. .

i � Ω2 ρ˙ SP (t) = − [δ σz + Ωr σx , ρSP (t)] − 2 4

(5.65)

5.4 Resonant excitation dynamics

110

1.2 "P !t"!ps"#1

1.5

1.0 0.8

1 0.5 0 0

ΡXX 0.6

0.5

1.

1.5

2

0.4 0.2 0.0 0

5

10

15

20

t!ps"

Figure 5.5: Exciton population dynamics in the time domain calculated with (solid blue curve) and without (dashed black curve) a Markov approximation. The inset shows the non-Markovian decay rate (dashed red curve) approaching its constant Markovian value (solid orange line). Parameters: α = 0.027ps2 , ωc = 2.2ps−1 , T = 10 K and Ω = 2ps−1 .

Avoiding the Markov approximation thus corresponds to the introduction of time-dependent rates and energy shifts in our master equation [8]. We note that having obtained Eq. (5.65) from a rigorously derived timelocal master equation, non-Markovian eﬀects can be explored with the restriction that we are limited by the validity of the perturbative expansion used, but not by any assumptions about the separability of the system and bath density operators. We determine Bloch equations from Eq. (5.65) in exactly the same way as the Markovian case. Considering resonant excitation, δ � = 0, and inserting σx (t − τ, t) = σx and σy (t − τ, t) = cos(Ωr τ )σy + sin(Ωr τ ) (which are exact for constant driving),

we find an equation of motion for the polaron frame Bloch vector identical to Eqs. (5.36)-(5.41) but with Ω(t) → Ω, Ωr (t) → Ωr , and all γll (ω) and Sll (ω) replaced with the time-dependent quantities γll (ω, t) = 2Re and Sll (ω, t) = Im

��

��

t

e

iωτ

0

t

e 0

iωτ

�

(5.66)

�

(5.67)

Λll (τ )dτ ,

Λll (τ )dτ ,

5.4 Resonant excitation dynamics

111

1.0 "P !t"!ps"#1

1.5

0.8

1 0.5

0.6

0 0

ΡXX

0.5

1.

1.5

2

0.4 0.2 0.0 0

5

10

15

20

t!ps"

Figure 5.6: Exciton population dynamics in the time domain for parameters identical to Fig. 5.5 but with a higher temperature of T = 50 K. Calculations with (solid blue curve) and without (dashed black curve) a Markov approximation are again shown. The inset shows the corresponding Markov (solid orange line) and non-Markov (dashed red curve) rates.

respectively. For the model we consider here, the diﬀerence between the non-Markovian and Markovian polaron frame dynamics is entirely captured in Eqs. (5.66) and (5.67). The Markov approximation simply corresponds to pushing the upper integration limits to infinity. We can therefore make the immediate observation that we should expect the Markovian and non-Markovian dynamics to deviate most at short times, since this is when γll (ω, t) diﬀers significantly from γll (ω, ∞) (and similarly for Sll (ω, t)

and Sll (ω, ∞)). These deviations should be most pronounced when Λll (0) is greatest in magnitude, since this maximises the diﬀerence between the

Markovian and non-Markovian rates (and energy shifts). Also, when Λll (0) decays on a long timescale (set by 1/ωc ) we also expect an increase in nonMarkovian eﬀects as this increases the time over which the non-Markovian rates (and energy shifts) reach their Markovian limits. To show that this is indeed the case, in Fig. 5.5 we plot the excitonic population of our QD as a function of time (rather than pulse area). For this figure we take the relatively large value of Ω = 2 ps−1 , so that the excitonic system evolves appreciably within the phonon bath correlation time, and consider a low temperature regime of T = 10 K. For these parameters non-Markovian eﬀects are most pronounced at short times, as expected, though it is generally fairly diﬃcult to distinguish between the two theories, especially

5.4 Resonant excitation dynamics

112

beyond t ∼ 10 ps. In the inset we plot the non-Markovian generalisation of the polaron theory decay rate (see Eq. (5.55)) ΓP (t) =

� Ω2 � γyy (Ωr , t) + γyy (−Ωr , t) + 4γxx (0, t) , 4

(5.68)

which rapidly approaches its Markovian limit on a timescale ∼ 1 ps.

We can enhance short-time non-Markovian eﬀects by considering

higher temperatures, as this increases the diﬀerence between the Markov and non-Markov rates and energy shifts on the bath correlation timescale (though it does not change the timescale on which the Markov limit is reached). This is shown in Fig. 5.6, where we again compare Markovian and non-Markovian dynamics, but now at the higher temperature of T = 50 K. Non-Markovian eﬀects are indeed more pronounced at short times in this case, and, as shown in the inset, the Markov and non-Markov rates do diﬀer more significantly at short times, though the Markov limit is again reached on a similar timescale to that at 10 K. Once more, beyond 5 − 10 ps there is very little to distinguish the Markovian and nonMarkovian dynamics.

The inclusion of non-Markovian eﬀects within the polaron frame master equation can therefore aﬀect the population dynamics at short times (∼ 5 ps and below), but makes very little diﬀerence on longer timescales. When plotting excitonic Rabi rotations as a function of pulse area, as in Figs. 5.1 and 5.4, it is only the final exciton population which is measured. For the parameters of Fig. 5.4, for example, this corresponds to reading out the excited state population after 14 ps. Even for the relatively large Rabi frequencies used in Figs. 5.5 and 5.6, we see that non-Markovian eﬀects are almost negligible on this timescale. Furthermore, at larger temperatures, for which short-time non-Markovian eﬀects seem to be more noticeable, the damping is more pronounced, so that the steady state is reached sooner. Hence, since short time behaviour is not captured in the pulse area plots of Figs. 5.1 and 5.4, neither are non-Markovian eﬀects (remember that in Fig. 5.1 the pulse FWHM is close to 14 ps). In fact, if we plot the exciton population as a function of pulse area using our non-Markovian polaron master equation [Eq. (5.65)] for the same parameters as Fig. 5.4, we find that it is almost indistinguishable from the Markov version on the scale shown there. Hence, our theory predicts that (for the parameters

5.5 Discussion and summary

113

considered here) in order for polaron frame non-Markovian signatures to be evident in pulse-area plots, FWHM pulse durations on the sub 5 ps timescale should be used, much shorter than those in the experiments performed in Refs. [52, 53].

5.5

Discussion and summary

In order to assess the validity of the strong coupling theory introduced in chapter 4 when applied to an experimentally relevant system, we have investigated the excitonic dynamics of a resonantly driven QD under the influence of dephasing due to its interactions with an acoustic phonon environment. The strong coupling theory accounts for non-perturbative eﬀects such as multiphonon processes and phonon-induced driving renormalisation. It is found that for low temperatures (< 30 K), the weak-coupling theory presented in Refs. [52,53] is in excellent agreement with the polaron master equation dynamics. However, as the temperature is increased, we find that the weak-coupling treatment begins to overestimate the damping rate, compared to the polaron theory prediction. In fact, it is interesting to note that in Ref. [53] it was reported that a weak-coupling fit to the data slightly overestimates the damping for temperatures > 40 K, consistent with our findings. For these temperatures, the non-perturbative aspects of the polaron theory are becoming important. Renormalisation of the Rabi frequency tends to decrease the damping rate, while multiphonon processes act to increase it above the single-phonon level (see Eqs. (5.50), (5.51) and (5.55) for the weak-coupling rate, the single-phonon approximation to the polaron rate, and the full polaron rate, respectively). Deviations from the weak-coupling theory should be even more pronounced at higher temperatures (above the highest temperature of ∼ 50 K explored in Ref. [53]),

though other decoherence mechanisms could also come into play in this regime. We also considered the important role of the energy-shift terms in the weak-coupling and polaron theories. These terms, analogous to the Lamb-shift in energy levels of atomic physics, are responsible for driving and temperature dependent shifts in the exciton population oscillation frequency, as also reported in Ref. [53]. While, in general, the energy shifts are necessary for a full description of the dynamics, at high temperatures

5.5 Discussion and summary

114

(> 70 K for the parameters studied here) we find that in the weak-coupling theory they give rise to unphysical behaviour, and therefore set a bound on the applicability of this approach. On the other hand, the polaron theory suﬀers no such limitation in this regime. Finally, we explored the role of non-Markovian eﬀects within the polaron frame, and found that they are predominantly a short-time phenomenon for our experimentally relevant parameters. Hence, they should have little bearing on pulse area plots of Rabi rotations if the pulse duration is long on the bath correlation timescale [52, 59, 61, 62]. Note this implies that, under the same excitation conditions, non-Markovian eﬀects are also negligible in pulse area plots in the weak-coupling theory at low temperatures, since the polaron and weak-coupling approaches agree well in this regime. In order to enhance non-Markovian eﬀects, shorter duration pulses or longer bath correlation times are required.

Chapter 6 Coherent and incoherent dynamics in excitonic energy transfer Contents 6.1 6.2

6.3 6.4

6.5

Introduction . . . . . . . . . . . . . . . . . . . . 6.1.1 Background . . . . . . . . . . . . . . . . . . Polaron transform master equation . . . . . . . 6.2.1 The system and polaron transformation . . . 6.2.2 Markovian master equation . . . . . . . . . . 6.2.3 Evolution of the Bloch vector . . . . . . . . . Resonant energy transfer . . . . . . . . . . . . . 6.3.1 Coherent to incoherent transition . . . . . . . Oﬀ resonance . . . . . . . . . . . . . . . . . . . 6.4.1 Near Resonance . . . . . . . . . . . . . . . . 6.4.2 Weak coupling limit . . . . . . . . . . . . . . 6.4.3 High temperature or far from resonance limit 6.4.4 Correlated fluctuations . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

116 . 116 118 . 118 . 121 . 122 124 . 128 131 . 134 . 135 . 137 . 141 143

6.1 Introduction

6.1

116

Introduction

Having applied the strong coupling theory to a model quantum dot system, we now use it to investigate energy transfer dynamics in a two-site donor-acceptor pair model. A theory that allows for exploration of strong coupling and/or high temperatures is particularly important in this context owing to recent observations of quantum coherence at elevated temperatures in a variety of systems [112–120]. An understanding of the transition from coherent to incoherent energy transfer should shed light on the mechanisms at play in these systems. Also, returning to the theme of this thesis, correlations in bath fluctuations are often cited as being responsible for allowing coherent transfer to take place at relatively high temperatures. Recent theoretical [121] and experimental [122] work has supported this claim. In this chapter we use the methods that have been developed to explore weak-to-strong coupling regimes, including the eﬀects of bath correlations. We find an interesting interplay between the bath correlations, the cut-oﬀ frequency of the bath, and the energy mismatch between the donor-acceptor pair, all of which can have a bearing to the nature of the energy transfer process. To put this work in context, we first give a brief summary of theoretical methods that have previously been used to investigate energy transfer.

6.1.1

Background

The observation of quantum coherence in the energy transfer dynamics, as mentioned above, has sparked renewed interest in modelling excitation energy transfer beyond standard approaches [110, 121, 123–129]. This process, which occurs when energy absorbed at one site (the donor) is transferred to another nearby site (the acceptor) via a virtual photon, [130] is often considered to be incoherent; the result of weak donor-acceptor interactions, treated perturbatively using Fermi’s golden rule [131, 132]. However, while this approach has proved to be immensely successful when applied in many situations [133, 134], accounting for quantum coherence within the energy transfer dynamics requires an analysis beyond straightforward perturbation theory in the donor-acceptor interaction. An alternative starting point for investigations into coherent energy transfer is to treat the system-environment interaction as a perturbation

6.1 Introduction

117

instead. Such weak-coupling theories, as introduced in section 2.2, have been successfully applied to elucidate a number of eﬀects that could be at play in multi-site donor-acceptor complexes. Examples include studying the interplay of coherent dynamics and dephasing in promoting eﬃcient energy transfer in quantum aggregates [135–141], exploring the role of environmental correlations in tuning the energy transfer process [142], and extensions to assess the potential importance of non-Markovian dynamics [143]. Nevertheless, in order to properly understand the transition from coherent to incoherent transfer between these two regimes, which occurs as the system-environment coupling or temperature is increased [121,144–146], it is necessary to be able to describe energy transfer dynamics beyond either of these limiting cases [134,147]. Building on earlier work [144,148–150], a number of methods have been put forward for this purpose. For example, modifications to both Redfield [151–153] and F¨orster [154–156] theory have extended the range of validity of both approaches. Moreover, as we have seen in chapters 4 and 5, it is possible to define a new perturbation term through the polaron transformation which under certain conditions then allows interpolation between the Redfield and F¨orster limits [110,121,124], as can the hierarchical equations of motion technique [127, 157]. Numerically exact calculations, based, for example, on path integral [126,158] and density matrix renormalisation group [125] methods, have also been applied to study energy transfer dynamics beyond perturbative approaches. In this chapter, we examine the conditions under which coherent or incoherent motion is expected to dominate the energy transfer dynamics of a model donor-acceptor pair. We employ a Markovian master equation derived within the polaron representation for this purpose, since it allows for a consistent analysis of the dynamics from weak to strong systembath coupling (or, equivalently, low to high temperatures). We explore in detail the important eﬀects of donor-acceptor energy mismatch on the energy transfer dynamics, as well as moving beyond the scaling limit [10] to consider the role of the high-frequency cut-oﬀ in the bath spectral density.

6.2 Polaron transform master equation

6.2

118

Polaron transform master equation

In this section we outline the model studied here and give details of the master equation derivation. Many of the methods used here are similar to those in chapter 5. As such, only the main steps are presented and technical details given only when necessary.

6.2.1

The system and polaron transformation

We consider a donor-acceptor pair (j = 1, 2), each site of which is modelled as a two-level system with ground state |G�j , excited state |X�j , and en-

ergy splitting �j . The pair interact via a Coulombic energy transfer with strength V , which is responsible for the transfer of excitation from one site to the other. The environment surrounding the donor-acceptor pair is modelled as a common bath of harmonic oscillators, which couples linearly to the excited state of each site. The total system-bath Hamiltonian is therefore written H=

2 � j=1

+

�j |X�j�X| + V (|XG��GX| + |GX��XG|)

2 � j=1

|X�j�X|

�

(j)

(j)∗

(gk b†k + gk bk ) +

k

�

ωk b†k bk ,

(6.1)

k

where as usual the bath is described by creation (annihilation) operators b†k (bk ) with corresponding angular frequency ωk , and wavevector (j)

k. The system-bath couplings are given by gk . As in Ref. [121], we shall consider the case in which each site is coupled to the bosonic bath with the same magnitude |gk |, but make the separation between the sites

explicit through position-dependent phases in the coupling constants of (j)

the form gk

= |gk |eik·rj , with rj being the position of site j.

In a

similar manner to that seen in chapter 3, this form of coupling gives rise to correlations in the bath influences seen at each site, allowing the full range of totally correlated, partially correlated, and completely uncorrelated fluctuations to be explored [121, 158, 159].

Inspection of

Eq. (6.1) reveals that it generates dynamics in three decoupled subspaces, spanned by {|GG� , {|GX� , |XG�}, |XX�}. We are interested here in ex-

citation energy transfer and therefore focus on the single-excitation sub-

6.2 Polaron transform master equation

119

space, {|GX� , |XG�}, in which this occurs. Relabelling |XG� → |0� and

|GX� → |1�, the Hamiltonian of the single-excitation subspace may be written

HSUB =�1 |0��0| + �2 |1��1| + V (|0��1| + |1��0|) � + |0��0| Bz(1) + |1��1| Bz(2) + ωk b†k bk ,

(6.2)

k

(j)

where we have defined the bath operators Bz =

�

(j) † k (gk bk

(j)∗

+ gk bk ). A

standard weak-coupling approach to the system dynamics would now be to derive a master equation for the evolution of the reduced system density operator under the assumption that the system-bath interaction terms, as written in Eq. (6.2), can be treated as weak perturbations [127, 136, 139, 142, 143, 160]. In this chapter, we shall instead derive a master equation describing the donor-acceptor energy transfer dynamics using the polaron representation in a manner similar to that in chapter 5. To proceed, we thus apply a unitary transformation which displaces the oscillators in the bath according to the location of the excitation. We note that the polaron transformation used here is slightly diﬀerent to that used in chapter 5, since here the bath couples to both states within the single excitation subspace. Defining HP = eS HSUB e−S , where now we have the diﬀerent form of transformation (1)

(2)

S = |0��0| P (gk /ωk ) + |1��1| P (gk /ωk ), with bath operators P (αk ) =

�

† k (αk bk

(6.3)

− αk∗ bk ), results in the polaron

transformed spin-boson Hamiltonian [109] HP = H0 + HI , with � � H0 = σ z + V R σ x + ωk b†k bk , 2 k

(6.4)

HI = V (Bx σx + By σy ).

(6.5)

and

Here, the bias � = �1 − �2 , gives the energy diﬀerence between the donor and acceptor, while the Pauli operators are defined in a basis in which σz =

|0��0| − |1��1| = |XG��XG| − |GX��GX|. The bath operators appearing in Eq. (6.4) are constructed as Bx = (1/2)(B+ + B− − 2B) and By =

(i/2)(B+ − B− ), exactly as in chapter 5. However, they now contain the

6.2 Polaron transform master equation

120

(i)

implicitly (through gk ) separation dependent operators B± =

�

D

k

�

±

(1) (gk

− ωk

(2) gk )

�

,

(6.6)

with displacement operators D(±αk ) = exp[±(αk b†k − αk∗ bk )]. Note that the interaction terms [Eq. (6.5)] therefore depend upon the diﬀerence (1)

(2)

in donor and acceptor system-bath couplings gk and gk , respectively. Importantly, the term driving the coherent energy transfer process in Eq. (6.4) will not be treated perturbatively, but it does now have a bathrenormalised strength, VR = BV , where �

B = exp −

� |gk |2 ωk2

k

(1 − cos(k · d)) coth(βωk /2)

�

(6.7)

is the expectation value of the bath operators with respect to the free Hamiltonian: B = �B± �H0 . The donor-acceptor separation is given by d = r1 − r2 .

As usual, we define the bath spectral density as J(ω) =

ωk ), and from appendices A and B we find �

B = exp −

�

∞ 0

�

k

� J(ω) (1 − FD (ω, d)) coth(βω/2) . ω2

|gk |2 δ(ω −

(6.8)

Here, β = 1/kB T , while the function FD (ω, d) captures the degree of correlation in bath fluctuations seen at each site, and is dependent upon the dimensionality of the system-bath interaction (D = 1, 2, 3). We note that the FD (ω, d) is precisely the function encountered in chapter 3 and plays a similar role in the dynamics here. We recall that in one dimension F1 (ω, d) = cos(ωd/c), with c the bosonic excitation speed, in two dimensions F2 (ω, d) = J0 (ωd/c), where J0 (x) is a Bessel function of the first kind, and in three dimensions F3 (ω, d) = sinc(ωd/c). In all cases FD (ω, d) → 1 as d → 0, i.e. when the donor and acceptor are at the same position, bath

fluctuations are perfectly correlated, and the energy transfer strength is not renormalised (VR → V ). In fact, in this perfectly-correlated limit dissi(1)

(2)

pative process are entirely suppressed (provided |gk | = |gk |) and energy

transfer remains coherent for all times and in all parameter regimes in our model (the single-excitation subspace is then decoherence-free. [25, 26]). This protection of coherence in the single excitation sub-space is in exact

6.2 Polaron transform master equation

121

analogy to the stability of the singlet state which was seen in chapter 3. In two and three dimensions, as d → ∞, FD (ω, d) → 0, and the renormal-

isation takes on the value that would be obtained by considering separate, completely uncorrelated baths surrounding the donor and acceptor. In the following, we shall characterise the degree of correlation in terms of the dimensionless parameter µ = c/dω0 , where ω0 is a typical bath frequency scale (see Eq. (6.32) below). We therefore have µ = 0 in the absence of correlations, µ < 1 for weak correlations, and µ > 1 for strong correlations.

6.2.2

Markovian master equation

We now derive an equation of motion of the reduced density operator ρ describing the donor-acceptor pair up to second order in HI . In this chapter we will not be concerned with non-Markovian eﬀects. We are interested in the general behaviour of the energy transfer dynamics and not in the short time behaviour where non-Markovian eﬀects are expected to be most pronounced. We note that this expectation has been confirmed by Jang et al where non-Markovian as well as non-equilibrium bath eﬀects have been explored [110]. Using the combined polaron transform time-local master equation technique described in chapter 4 yields a homogeneous, polaron frame, interaction picture master equation of the form ∂ ρ˜(t) =− ∂t

�

∞ 0

� � ˜ I (t), [H ˜ I (t − τ ), ρ˜(t) ⊗ ρB ] , dτ trB [H

(6.9)

˜ = where as usual tildes indicate operators in the interaction picture, O(t) eiH0 t Oe−iH0 t . With reference to section 4.3.2, in deriving Eq. (6.9) we have assumed factorising initial conditions, that the initial system state commutes with the polaron transformation, and that the initial state of the bath is a polaron transformed equilibrium state. We take as our reference state the thermal equilibrium state ρB = e−βHB /trB (e−βHB ). We further assume that the interaction is weak in the polaron frame so that we may factorise the joint density operator as χ(t) ˜ = ρ˜(t)ρB at all times. Inserting Eq. (6.5) into Eq. (6.9), and moving back into the Schr¨odinger picture, we arrive at our Markovian master equation describing the dynamics within the single-excitation subspace, and written in the polaron frame

6.2 Polaron transform master equation

122

as ∂ρ(t) = −i[(�/2)σz + VR σx , ρ(t)]−V 2 ∂t

�

∞

�

dτ [σx , σ ˜x (−τ )ρ(t)]Λxx (τ ) � + [σy , σ ˜y (−τ )ρ(t)]Λyy (τ ) + H.c. , 0

(6.10)

where H.c. denotes Hermitian conjugation. The eﬀect of the bath is now contained within the correlation functions Λll (τ ) = �Bl (t)Bl (s)�H0 , which are given explicitly by

Λxx (τ ) = (B 2 /2)(eφ(τ ) + e−φ(τ ) − 2), Λyy (τ ) = (B 2 /2)(eφ(τ ) − e−φ(τ ) ),

(6.11) (6.12)

where φ(τ ) = 2

�

∞ 0

�

� J(ω) dω (1 − FD (ω, d)) (cos ωτ coth(βω/2) − i sin ωτ ) . ω2 (6.13)

Notice that the phonon propagator, φ(τ ), is correlation-dependent due to the factor (1 − FD (ω, d)), and so clearly the dissipative eﬀect of the

bath will be highly dependent upon the degree of correlation too. For example, as d → 0, FD (ω, d) → 1 and the dissipative contribution to

Eq. (6.10) vanishes. As mentioned previously, for a donor-acceptor pair with complete fluctuation correlations, the single excitation sub-space in which we are interested becomes decoherence free, resulting in undamped coherent population oscillations between the sites.

6.2.3

Evolution of the Bloch vector

As in chapter 5 we solve our master equation in terms of the Bloch vector, defined in section 2.1.3 as α = (αx , αy , αz )T = (�σx �, �σy �, �σz �)T . As we are working exclusively in the single-excitation subspace, αx and αy

describe the coherences between the states |0� ≡ |XG� and |1� ≡ |GX�,

while αz captures the donor-acceptor population transfer dynamics generated by the coupling V . Though Eq. (6.10) is written in the Sch¨odinger picture, it is still in the polaron frame, and so we must determine how expectation values in the polaron frame are related to those in the original, or “lab” frame.

6.2 Polaron transform master equation

123

S We can see this by writing α˙ i = TrS+B (σi χ˙ L (t)) = TrS+B (σi e−S χ(t)e ˙ )= −S TrS+B (eS σi e−S ρ(t)ρ ˙ χ(t)eS is the lab frame total denB ), where χL (t) = e

sity operator, and we have made use of the Born approximation in the polaron frame to write χ(t) = ρ(t)ρB . Since eS σx e−S = |1��0| B+ + |0��1| B− ,

eS σy e−S = i(|1��0| B+ −|0��1| B− ), and eS σz e−S = σz , this implies that the lab Bloch vector elements are αi = BαiP , for i = x, y, and αz = αzP , where

αiP is an expectation value in the polaron frame: α˙ iP = TrS (σi ρ(t)). ˙ Alternatively, we can define a matrix L which maps the polaron frame Bloch vector to its lab frame counterpart: α = L · αP , where L = diag(B, B, 1).

Working in terms of the Bloch vector, we arrive at an equation of

motion of the form ˙ α(t) = M · α(t) + b.

(6.14)

In the following, we shall often be interested in determining the coherent or incoherent nature of the energy transfer process. It is then helpful to write Eq. (6.14) as ˙ � (t) = M · α(t)� , α

(6.15)

with α� (t) = α(t) − α(∞), where α(∞) = −M −1 · b is the steady state.

This makes clear that the nature of the energy transfer process lies solely in the matrix M , while the inhomogeneous term b is needed only in determining the steady state. Equipped with the eigensystem of M , we may determine the corresponding time evolution as follows. An eigenvector of M , say mi , has equa˙ i = qi mi , where qi is the corresponding eigenvalue. Its tion of motion m subsequent evolution then has the simple exponential form mi (t) = mi eqi t . More generally, we can say that any initial state α� (0) will have subsequent evolution �

α (t) =

3 �

ai m i e q i t ,

(6.16)

i=1

where the coeﬃcients ai are determined by the initial conditions (i.e. the � solutions of α� (0) = i ai mi ). The solution to the full inhomogeneous equation is then found simply by addition of the steady state: α(t) = α� (t) + α(∞).

6.3 Resonant energy transfer

6.3

124

Resonant energy transfer

We start by considering the important special case of resonant energy transfer, in which the interplay of coherent and incoherent eﬀects is particularly pronounced. As we shall see, in this case it is relatively straightforward within our formalism to give a strict criterion for when we expect the energy transfer dynamics to be able to display signatures of coherence [109, 121]. Hence, the resonant case provides a natural situation in which to begin to understand, for example, the role of bath correlations [121, 158, 159] or the range of the bath frequency distribution in determining the nature of the energy transfer process. Setting the donor-acceptor energy mismatch to zero, � = 0, we find ˙ = from Eq. (6.10) dynamics generated by an expression of the form α MR · α + bR , with 

 MR = 

−(Γz − Γy ) 0 0

0

0



 −2BVR  , B −1 (2VR + λ3 ) −Γz −Γy

(6.17)

and bR = (−Bκx , 0, 0)T , where Γy = 2V 2 γxx (0),

(6.18)

Γz = V 2 (γyy (2VR ) + γyy (−2VR )) + 2V 2 γxx (0),

(6.19)

λ3 = 2V 2 (Syy (2VR ) − Syy (−2VR )),

(6.20)

κx = V 2 (γyy (2VR ) − γyy (−2VR )).

(6.21)

The rates and energy shifts are related to the one-sided Fourier transforms of the correlation functions in the usual way, Kll (ω) =

�

∞ 0

1 dτ eiωτ Λll (τ )= γll (ω) + iSii (ω), 2

such that γll (ω) = 2Re[Kll (ω)] =

�

(6.22)

+∞

dτ eiωτ Λll (τ ),

(6.23)

−∞

and Sll (ω) = Im[Kll (ω)]. The resonant steady-state is straightforwardly found to be αx (∞) = −B tanh(βVR ),

(6.24)

6.3 Resonant energy transfer

125

while αy (∞) = αz (∞) = 0. Notice that while this is of the same form as the steady-state that would be obtained from a weak system-bath coupling treatment (see section 2.3.2), αx (∞) is determined here by VR , rather than the original coupling V , and there is also an extra factor of B suppressing its magnitude. The procedure described in section 6.2.3 to determine the time evolution of α is somewhat unnecessary here, since the equation of motion of αx is decoupled from that of αy and αz . However, with some foresight we calculate the eigenvectors of MR in any case, finding m1 = m2 = m∗3 =

� �

1, 0, 0 0,

�T

,

B(Γz − Γy − iξR ) �T ,1 , 2(2VR + λ3 )

(6.25) (6.26)

with corresponding eigenvalues q1 = Γy − Γz , and q2 = q3∗ = −(1/2)(Γy + Γz + iξR ). Thus, referring to Eq. (6.16), we see that

� ξR = 8VR (2VR + λ3 ) − (Γy − Γz )2

(6.27)

determines whether any coherence exists within the energy transfer dynamics. Considering the initial state α(0) = (0, 0, 1)T , corresponding to excitation of the donor, ρ(0) = |0��0| = |XG��XG|, we find analytical forms for the evolution of the Bloch vector components:

αx (t) = −B tanh(βVR )(1 − e−(Γy −Γz )t ), �ξ t� 2BVR −(Γy +Γz )t/2 R αy (t) = − e sin , ξR 2 � �ξ t� Γ − Γ � ξ t �� R y z R αz (t) = e−(Γy +Γz )t/2 cos + sin . 2 ξR 2

(6.28) (6.29) (6.30)

Inspection of Eqs. (6.27) and (6.30) allows us to identify a crossover from coherent to incoherent motion in the energy transfer dynamics as the point at which oscillations in the population diﬀerence vanish: (Γy − Γz )2 = 8VR (2VR + λ3 ).

(6.31)

For (Γy − Γz )2 < 8VR (2VR + λ3 ), ξR is real and both the population diﬀerence and coherence αy describe damped oscillations, while for

(Γy − Γz )2 ≥ 8VR (2VR + λ3 ), ξR is either zero or imaginary, with the

6.3 Resonant energy transfer

126

pop. difference !Αz "

1.0

0.5

0.0

!0.5

!1.0 0

20

40

60

80

100

Ω0 t

Figure 6.1: Population diﬀerence as a function of scaled time ω0 t for temperatures of kB T /ω0 = 1 (blue dashed curve), kB T /ω0 = 5 (green dotted curve), kB T /ω0 = 12 (orange solid curve) and kB T /ω0 = 20 (red dot-dashed curve). Parameters: α = 0.05, V /ω0 = 0.5, ωc /ω0 = 4, � = 0 and µ = c/ω0 d = 0.5.

dynamics then being entirely incoherent. To further analyse the behaviour of αz (t), and the conditions for which the boundary defined by Eq. (6.31) is crossed, we now take a specific form for the system-bath spectral density. In this chapter we consider a spectral density of the form J(ω) = α

ω 3 −ω/ωc e , ω02

(6.32)

where α is a dimensionless quantity capturing the strength of the systembath interaction. As we have seen in chapter 5, the cubic frequency dependence in Eq. (6.32) is typical, for example, in describing dephasing in quantum dots due to coupling to acoustic phonons, but can also be used to elucidate the behaviour in which we are interested in general [158]. The inverse cut-oﬀ frequency also sets a typical relaxation timescale for the bath [8]. To illustrate the dynamics and crossover behaviour in the resonant case, in Fig. 6.1 we plot the population diﬀerence (αz ) as a function of the scaled time ω0 t for a range of temperatures, showing the transition from coherent to incoherent transfer as the temperature is increased. In this plot, and all the following, we consider the case of three-dimensional coupling, F3 (ω, d) = sinc(ωd/c). The role of bath spatial correlations in protecting coherence can be seen in Fig. 6.2, where we again plot the evolution of the

6.3 Resonant energy transfer

127

coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t

coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t

coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t

Figure 6.2: Population diﬀerence as a function of scaled time ω0 t for temperatures of kB T /ω0 = 5 (blue dashed curves) and kB T /ω0 = 10 (red dotted curves), and for separations corresponding to no correlation, µ = c/ω0 d = 0 (top), weak correlations, µ = 0.5 (middle), and strong correlations µ = 2 (bottom). The insets show the evolution of the corresponding coherence αy . Parameters: α = 0.05, V /ω0 = 0.5, and ωc /ω0 = 4.

6.3 Resonant energy transfer

128

population diﬀerence (the insets show the corresponding coherence αy ), this time for representative intermediate and high-temperature cases. The diﬀerent plots in Fig. 6.2 correspond to zero correlations, characterised by µ = c/ω0 d = 0 (d → ∞, top), weak correlations, µ = 0.5 (middle), and

strong correlations µ = 2 (bottom). Progressing from the uppermost plot to the lowest, we clearly see that an increase in correlation strength prolongs the timescale over which oscillations in both the population diﬀerence and coherence persist. Moreover, by looking at the curves corresponding to the higher temperature, we can see that as the degree of correlation is increased from zero, the dynamics moves from a regime showing purely incoherent relaxation, to a regime which displays coherent oscillations at the same temperature. The increase in correlations is thus able to extend the region of parameter space which permits coherence [121], as we shall now explore in greater detail.

6.3.1

Coherent to incoherent transition

We now return our attention to the crossover from coherent to incoherent transfer, defined by Eq. (6.31). Intuitively, we might expect the dynamics in the low-temperature (or weak-coupling) regime to be coherent; for example, in Fig. 6.1 incoherent relaxation only occurs in the high-temperature limit. If we therefore assume that the crossover itself occurs in the hightemperature regime, it is possible to derive an analytic expression governing the crossover temperature by approximating the rates Γy and Γz . Details of this approximation, and its range of validity, can be found in the appendix C. Generally, for high enough temperatures and/or strong enough system-bath coupling (such that βVR � 1) we can approximate γxx (η) ≈ γyy (η) ≈ γyy (0) in Γy and Γz , where

βB 2 eφ0 C0 (x,y) γyy (0) ≈ � , 2 πC2 (x, y)φ0

(6.33)

with φ0 = 2π 2 α/ω02 β 2 , x = πd/cβ and y = ωc β. The functions C0 (x, y) and C2 (x, y) are given by Eqs. (C.8) and (C.9), and the renormalisation factor B by the product of Eqs. (C.15) and (C.16). If we further assume that the energy shift λ3 vanishes in the high-temperature limit, Eq. (6.31)

6.3 Resonant energy transfer

129

14 Increasing Μ 12 10 kB Tc !Ω0

8

30 25 20 15 10 5 0 0

6 4 2 0 1

5

10

Increasing Ωc 2

4

6

1!Μ#Ω0 d!c

15

8

10

20

Ωc !Ω0

Figure 6.3: Crossover temperature separating the coherent and incoherent regimes against cut-oﬀ frequency, for levels of correlation given by µ = c/ω0 d = 0, µ = 0.5 and µ = 1, increasing as shown. The solid blue curves have been calculated from Eq. (6.31) (using the full rates), while the dashed red curves are solutions to the high-temperature approximation, Eq. (6.35). The inset shows the dependence on the level of correlation for diﬀerent cutoﬀs, ωc /ω0 = 2, ωc /ω0 = 3, and ωc /ω0 = 4, again increasing as shown. Parameters: α = 0.05 and V /ω0 = 0.5.

reduces to (Γz − Γy ) = 4VR ,

(6.34)

and we arrive at the expression �

kB T ω0

�2

=

V Beφ0 C0 (x,y) � , ω0 4 2π 3 αC2 (x, y)

(6.35)

with solution, Tc , giving the crossover temperature separating the coherent and incoherent regimes. The dependence of Tc on the various parameters involved in the problem is not straightforward, owing to the temperature dependence in the renormalisation factor B, in the functions C0 and C2 , and in φ0 . In fact, there are three distinct and important temperature scales which deter√ mine whether the transfer is coherent or incoherent: T0 = ω0 /( 2απkB ), which depends upon the system-bath coupling strength; Tx = c/dπkB , which arises due to the fluctuation correlations and becomes unimportant in the uncorrelated case (Tx → 0 as d → ∞); and Ty = ωc /kB , dependent

6.3 Resonant energy transfer

130

upon the cut-oﬀ frequency, and irrelevant in the scaling limit (y → ∞).

Hence, changes in any of α, d, or ωc can have an eﬀect on the crossover temperature. For example, the main part of Fig. 6.3 shows the solution to Eq. (6.35), i.e. the crosover temperature Tc , as a function of the dimensionless cut-oﬀ frequency ωc /ω0 . A calculation using Eq. (6.31) with the full rates, and including λ3 , is also shown for comparison. The three pairs of curves correspond to increasing levels of correlation, ordered as indicated. We see that, except for small ωc /ω0 in the case µ = 0, where λ3 becomes important, solutions to Eq. (6.35) give an excellent approximation to the crossover temperature calculated using the full rates. This confirms that the coherent-incoherent crossover does indeed occur in the high-temperature (multi-phonon) regime, and consequently could not be captured by a weak system-bath coupling treatment. As the cut-oﬀ frequency is increased from its minimum value, the crossover temperature begins to decrease. This behaviour can be understood qualitatively by examining Eq. (6.34), and considering the competition this condition captures between the rate Γz − Γy and the coherent

interaction VR in defining the nature of the dynamics. Larger values of the cut-oﬀ frequency correspond to smaller values of the renormalised interaction strength VR (see e.g. Eq. (C.15)), while the rates Γy and Γz vary less strongly with ωc in this regime. Thus, increasing ωc from its minimum value decreases VR , and therefore reduces the range of temperatures for which 4VR > Γz − Γy and coherent transfer can take place.

Thus, the crossover temperature falls. Physically, this can be understood by noting that as the cut-oﬀ frequency is increased, so too is the eﬀective frequency range and peak magnitude of the system-bath interaction, characterised by the spectral density [Eq. (5.14)]. Hence, increasing from small ωc /ω0 , the environment begins to exert an enhanced influence on the system behaviour, and so coherent dynamics no longer survives to such high temperatures. As ωc continues to increase, however, we see the crossover temperature begins to rise. The renormalisation factor B tends to zero with increasing ωc and here becomes the dominating quantity, thus causing the rate Γz − Γy ∼ O(B 2 ) to vanish faster than the renormalised donor-acceptor coupling VR = BV .

The interplay between the size of ωc and the level of spatial correlation is best understood by considering the inset of Fig. 6.3. For all curves

6.4 Oﬀ resonance

131

shown the crossover temperature increases as the distance d is reduced, since the level of correlation µ increases correspondingly. As we have seen previously in Fig. 6.2, stronger correlations allow coherent dynamics to be observed at higher temperatures; since environmental eﬀects are suppressed, so the crossover temperature Tc must rise accordingly. This behaviour can be attributed to an increase in the renormalised interaction strength, VR , in relation to the rate Γz − Γy , this time with variations in the correlation level µ. Interestingly, as the cut-oﬀ frequency is increased

up to ωc /ω0 = 4 (lowest curve), we see that not only does the crossover temperature decrease, but also that the degree of correlation necessary to show a marked rise in Tc increases. As can be seen by comparing the separation between the diﬀerent curves in the main part of the figure, increasing the cut-oﬀ frequency tends to suppress the extent to which correlations are able to protect coherence in the system. This tallies with the dynamics shown in Fig. 6.2, for which ωc /ω0 = 4, and correlations as high as µ = 2 were needed before a significant change in behaviour was seen. Finally, since the renormalisation factor B tends to a constant nonzero value as the correlations vanish at large d (as opposed to B → 0 as ωc → ∞), the dependence of the crossover temperature on µ is monotonic, in contrast to its dependence on ωc .

6.4

Oﬀ resonance

It is often the case in practice that the donor and acceptor will have diﬀerent excited state energies, �1 −�2 = � �= 0, and so we now turn our attention to the energy transfer dynamics in oﬀ-resonant conditions. Regarding the

coherent to incoherent transition, in the resonant case we were able to identify this transition with a pair of conjugate eigenvalues converging on the real axis, thus changing oscillatory terms into relaxation. We might hope that in the oﬀ-resonant case we are able to identify a similar crossover criterion, and again use this to investigate the eﬀects of bath correlations and the cut-oﬀ frequency. However, we shall see that identification of such a crossover is less straightforward in the oﬀ-resonant regime. We first present the full Bloch equations describing the evolution of our donor-acceptor pair for arbitrary energy mismatch. As in the resonant ˙ = M · α + b, but now case, we have an equation of motion of the form α

6.4 Oﬀ resonance

132

pop. difference !Αz "

1.0

!a" 0.5 0.0 !0.5

!1.0 0

20

40

60

80

100

Ω0 t

pop. difference !Αz "

1.0

!b" 0.5 0.0 !0.5

!1.0 0

20

40

60

80

100

Ω0 t

Figure 6.4: Population diﬀerence for (a) small energy mismatch (�/ω0 = 0.1) and (b) larger energy mismatch (�/ω0 = 1) as a function of scaled time ω0 t. Temperatures kB T /ω0 = 1 (blue dashed curve), kB T /ω0 = 5 (green dotted curve), kB T /ω0 = 12 (orange solid curve) and kB T /ω0 = 20 (red dot-dashed curve) are shown. Parameters: α = 0.05, V /ω0 = 0.5, ωc /ω0 = 4, and µ = 0.5.

the matrix M is given by 

−Γx

 M =  (� + λ2 ) −1

B ζ

−(� + λ1 )

0



 −2BVR  , B −1 (2VR + λ3 ) −Γz −Γy

(6.36)

with b = (−Bκx , −Bκy , −κz )T . The rates now take on the form Γx = V 2 (γyy (η) + γyy (−η)), � 2 � 4VR �2 2 Γy = 2V γxx (0) + 2 (γxx (η) + γxx (−η)) , η2 2η Γ z = Γx + Γ y ,

(6.37) (6.38) (6.39)

6.4 Oﬀ resonance

133

while the energy shifts are given by 2V 2 � (Syy (η) − Syy (−η)), η 2V 2 � λ2 = (Sxx (η) − Sxx (−η)), η 4V 2 VR λ3 = (Syy (η) − Syy (−η)), η λ1 =

(6.40) (6.41) (6.42)

and the remaining quantities are ζ= κx = κy = κz = Here, η =

�

� � 4V 2 VR � 1 γxx (0) − (γxx (η) + γxx (−η)) , η2 2 2 2V VR (γyy (η) − γyy (−η)), η � � 8V 2 VR � 1 Sxx (0) − (Sxx (η) + Sxx (−η)) , η2 2 V 2� ((γxx (η) − γxx (−η)) + (γyy (η) − γyy (−η))) . η

(6.43) (6.44) (6.45) (6.46)

�2 + 4VR2 is the system Hamiltonian eigenstate splitting in the

polaron frame. To exemplify the dynamics generated by the full Bloch equations, in Fig. 6.4 we plot the evolution of the population diﬀerence in the case of (a) a small donor-acceptor energy mismatch, � = 0.2V , and (b) a more substantial mismatch, � = 2V . By comparison of Fig. 6.1 (plotted in the resonant case) and Fig. 6.4(a), we see that the introduction of a small energy mismatch has only a marginal eﬀect on the dynamics, and most importantly the coherent or incoherent nature of the energy transfer process seems unaﬀected. In contrast, in Fig. 6.4(b) the presence of a larger energy mismatch causes the low-temperature oscillations to increase in frequency but decrease markedly in amplitude, such that for kB T /ω0 = 5 the oscillations are now almost imperceptible. We also see that the population diﬀerence now tends to a non-zero steady-state at low temperatures, as we might expect from simple thermodynamic arguments, since the states αz = 1 and αz = −1 now have diﬀerent energies. As the temperature is raised, however, the dynamics still looks to be approaching the resonant case shown in Fig. 6.1. Finding the eigensystem of M [Eq. (6.36)] in the oﬀ-resonant case is

6.4 Oﬀ resonance

134

not straightforward and analytical solutions to the full Bloch equations are consequently lengthy, and therefore of little direct use in gaining an understanding of the behaviour seen in Fig. 6.4. A large part of this section is thus devoted to deriving simplified expressions for the energy transfer dynamics in a number of limits. These expressions not only provide insight into the oﬀ-resonant behaviour of the system, but also serve to highlight the diﬃcultly in now defining a simple crossover criterion, as was possible in the resonant case.

6.4.1

Near Resonance

We have seen that when we introduced only a small donor-acceptor energy mismatch, very little diﬀerence to the resonant dynamics was observed. This is to be expected, as the interaction V still dominates over � in this regime, i.e. �/V � 1. As discussed in section 2.2, provided we are suﬃciently within this limit, the dynamics generated by � is also much slower than the dissipative processes we wish to capture, and we can simply add a term



0 −� 0

 E= �

0

0

0



 0  0

(6.47)

˙ � = (MR + E) · α� . Hence, the to MR , reducing our problem to solving α arguments presented in Section 6.2.3 tell us that the characteristics of the

energy transfer process can be found simply by analysing the spectrum of MR + E. On resonance (� = 0) the eigenvectors and eigenvalues of MR are given by Eqs. (6.25), (6.26) and (6.27), together with the accompanying text. By diagonalisation of MR + E for small energy mismatch, we find that to first order in � these eigenvalues are unchanged. To second order in � the eigenvalues have corrections 4�2 Γy , (3Γy − Γz )2 + ξR2 � � i�2 (Γy − Γz )(3Γy − Γz ) + 2iξR Γy + ξR2 ∆q2 = − , ξR (3Γy − Γz )2 + ξR2 ∆q1 = −

(6.48) (6.49)

and ∆q3 = ∆q2∗ . The change ∆q1 is always real since the rates Γz and Γy are real by construction and ξR can be either purely real or purely imaginary. Inspection of the expression for ∆q2 reveals that is has real

6.4 Oﬀ resonance

135

and imaginary components when ξR is real (in which case q2 also has both real and imaginary parts), and is entirely real if ξR is imaginary (in which case q2 is also real). Put another way, the second order (in �) correction to the eigenvalues q2 and q3 does not change whether they lie on the real axis or otherwise. Recalling that a complex q2 and q3 (or real ξR ) corresponds to coherent dynamics, we confirm that the introduction of a small energy diﬀerence between the two sites does not aﬀect whether the energy transfer process is of coherent or incoherent nature.

6.4.2

Weak coupling limit

We now move beyond the small � approximation and look instead at the weak system-bath coupling limit, which we obtain by expanding all relevant quantities to first order in J(ω). With reference to our expressions for the correlation functions [Eqs. (6.11) and (6.12)], we see that within this approximation Λxx (τ ) → 0 while Λyy (τ ) remains finite. We may then

set all rates and energy shifts which are functions of only Λxx (τ ) to zero in Eq. (6.36), which results in the far simpler form 

 MW = 

−ΓW �

0

−(� + λ1 )

0



 −2BVR  , B −1 (2VR + λ3 ) −ΓW 0

(6.50)

where the weak coupling rate is given by [160] ΓW

�

VR = 4π η

�2

J(η)(1 − F (η, d)) coth(βη/2),

(6.51)

and the two energy shifts may be written λ1 = (�/η)Λ and λ3 = (2VR /η)Λ, with Λ = 2V 2 (Syy (η) − Syy (−η)).

(6.52)

6.4 Oﬀ resonance

136

The inhomogeneous term becomes bW = {−Bκx , 0, −(�/2VR )κx }T in the same limit, which leads to the weak-coupling steady state αx (∞) = −

2BVR tanh(βη/2), η

(6.53)

αy (∞) = 0,

(6.54)

� αz (∞) = − tanh(βη/2). η

(6.55)

As in the resonant case but where no weak-coupling approximation was made, this steady-state has precisely the form that would be expected from a standard weak-coupling approach, though with the replacement V → VR , and the extra factor of B suppressing the coherence αx (∞). As the energy

mismatch increases in relation to V , the weak-coupling steady state therefore becomes increasingly localised in the lower energy state |1� ≡ |GX�.

Interestingly, this contrasts with the qualitatively incorrect form (at low temperatures at least) given by the non-interacting blip approximation (NIBA) discussed in section 2.3.3, αzNIBA (∞) = − tanh (β�/2) [10, 11],

which predicts complete localisation in the lower energy state at zero temperature, regardless of the size of �/V . We should thus expect the present theory to fair far better than the NIBA for low-temperatures (or weakcoupling) in the oﬀ-resonant case, � �= 0. Also, the rate given in Eq. (6.51)

is again of the form expected from a weak-coupling treatment, though once more with V → VR . In fact, such a replacement is sometimes made by

hand in weak-coupling theories to provide agreement with numerics over a larger range of parameters [160], though it arises naturally in the polaron formalism here. We can therefore conclude that, in addition to allowing for the exploration of multiphonon eﬀects [109, 110, 121], the polaron master equation provides a rigorous way to explore the (single-phonon) weak-coupling regime for spectral densities of the type in Eq. (6.32) [109]. As before, to find the time evolution of α we evaluate the eigensystem of MW . For � �= 0, we find eigenvectors m1 = m2 = m∗3 =

� 2BV �B �

��

R

, 0, 1

�T

2VR −

,

η2 2VR

(6.56) �

,

B(ΓW − iξW ) �T ,1 , 2(2VR + λ3 )

(6.57)

with corresponding eigenvalues q1 = −ΓW , q2 = q3∗ = −(1/2)(ΓW + iξW ),

6.4 Oﬀ resonance

137

and weak-coupling oscillation frequency ξW

� = 4η(η + Λ) − Γ2W ,

(6.58)

which we should expect to be real to be consistent with our original expansion. We immediately see that, in general, the oﬀ-resonant dynamics in the weak-coupling limit should have a diﬀerent form to the resonant dynamics. In fact, Eqs. (6.56) and (6.57) show that, in contrast to the resonant case, the population evolution should have two distinct contributions. This can be made explicit by considering the initial state α(0) = {0, 0, 1}T , for which we obtain � � � � � −ΓW t � −ΓW t αz (t) = e − 1−e tanh(βη/2) η η � � � ξ t �� 4VR2 − ΓW t ξW t � ΓW W + 2 e 2 cos − sin . (6.59) η 2 ξW 2 Here, the first term, proportional to (�/η) and present nowhere in the resonant case, describes incoherent relaxation towards the steady state value given by Eq. (6.55). The second term, proportional to (VR /η)2 and having a similar form to the resonant dynamics, describes damped oscillations with frequency ξW . Importantly, these oscillations have a temporal maximum amplitude of 4VR2 /η 2 ≤ 1, compared to 1 in the resonant case. The

eﬀect of the energy mismatch in this limit is thus to suppress the magnitude of any oscillations in the population diﬀerence, while increasing their frequency due to the form of Eq. (6.58), exactly as in Fig. 6.4(b).

6.4.3

High temperature or far from resonance limit

By taking appropriate limits, we have now been able to explain the behaviour seen at small � and at low temperatures in Fig. 6.4. However, at higher temperatures, we find something quite diﬀerent; the population dynamics appears to be relatively insensitive to the size of the energy mismatch. In order to investigate this eﬀect in more detail, we shall now make a high temperature (or strong system-bath coupling) approximation to the full energy transfer dynamics. Specifically, we consider the regime VR /� � 1. This limit can in fact

be achieved in two possible ways. Firstly, recalling that VR = V B, we

6.4 Oﬀ resonance

138

see that VR can be made small by increasing the system-bath coupling strength or temperature, such that B � 1. Alternatively, if the donor-

acceptor pair are far from resonance, the ratio V /� will be small, and hence VR /� smaller still. Observing that the correlation functions given by Eqs. (6.11) and (6.12) are both proportional to B 2 , we can see that all ˙ = M · α + b, are at least dissipative terms in the equation of motion, α of order VR2 . We proceed by keeping only terms up to order (VR /�)2 in the full oﬀ-resonant M and b. This allows us to set λ3 , ζ, κx and κy to zero, while the remaining quantities reduce to Γy = V 2 (γxx (η) + γxx (−η)), � � Γz = V 2 γxx (η) + γxx (−η) + γyy (η) + γyy (−η) ,

(6.60)

λ2 = 2V 2 (Sxx (η) − Sxx (−η)), � � κz = V 2 γxx (η) − γxx (−η) + γyy (η) − γyy (−η) .

(6.63)

λ1 = 2V 2 (Syy (η) − Syy (−η)),

(6.61) (6.62)

(6.64)

Hence, in the high temperature limit, Eq. (6.36) takes on the simpler form 

 MHT = 

−(Γz − Γy ) −(� + λ1 ) −Γy

(� + λ2 )

2B −1 VR

0

0



 −2BVR  , −Γz

(6.65)

while the inhomogeneous term reduces to bHT = {0, 0, −κz }T . We then find the approximate steady-state αx (∞) = − αy (∞) = 0,

2BVR tanh(βη/2), � �

αz (∞) = − 1 +

� 2

4VR �2

Γy −1 Γz

(6.66) ��

(6.67) tanh(βη/2),

(6.68)

valid up to second order in VR /�. For VR � �, the steady-state is strongly

localised in the low energy state (αz (∞) ≈ −1) if � � kB T , though for � � kB T , thermal eﬀects dominate, and αz (∞) ≈ 0 as in the resonant case

(however, since VR is small, αx (∞) ≈ 0 as well in this situation). Again, this behaviour tallies with Fig. 6.4.

To obtain the corresponding population dynamics, we note in reference to Eq. (6.16) that the coeﬃcients ai , the eigenvectors mi , and the eigen-

6.4 Oﬀ resonance

139

values qi will contain powers of our expansion parameter VR /�. Expanding both qi and the products ai mi to second order, we find �

αz (t) = e

4V 2 1 − 2R �

�

4V 2 + 2R e−Γz t/2 cos(¯�t) � � �� βη 4VR2 Γy −Γz t −(1 − e ) tanh 1+ 2 −1 2 � Γz −Γz t

(6.69)

where the shifted oscillation frequency is �¯ = � + (1/2)(λ1 + λ2 ) + 2�(VR /�)2 .

(6.70)

As in the weak coupling case [Eq. (6.59)] the evolution of the donoracceptor population diﬀerence consists of two contributions, incoherent relaxation towards the steady-state, and an oscillatory component with vanishing amplitude as VR /� → 0. The energy mismatch again serves to suppress oscillations in the population diﬀerence.

The most striking feature, however, of Eq. (6.69) is that there is an oscillatory component at frequency �¯ at all. In the high temperature limit, we might expect that this frequency would reach a point where it becomes imaginary and αz (t) displays purely incoherent relaxation, as in the equivalent resonant case. However, we can see that this is not the case since �¯ is always real by definition. Furthermore, at very high temperatures �¯ → �, and it therefore also remains finite. Eq. (6.69) thus highlights an important diﬀerence between the energy transfer dynamics in the res-

onant and oﬀ-resonant cases. In the resonant case, as temperature is increased, the energy transfer process becomes less coherent through a reduction in oscillation frequency (i.e. VR becomes small in comparison to Γz − Γy ), eventually reaching a point at which population simply relaxes

incoherently towards the steady state. In the oﬀ-resonant case, the transfer process becomes less coherent predominately through a reduction in oscillation amplitude. For high temperatures, an oscillatory component is still (in theory) present in the system, although it becomes ever more dominated by incoherent relaxation towards the steady state. These features are clearly seen in Fig. 6.4. Only to first order in VR /� do our expressions predict purely incoherent

6.4 Oﬀ resonance

140

0.0000

!0.0005

1 pop. diff. Αz

coherence !Αy "

0.0005

!0.0010

0.5 0. !0.5 0

!0.0015 0

20

40

60

80

20

100

40

60

120

80

140

Ω0 t

Figure 6.5: Coherence (αy ) as a function of scaled time ω0 t for resonant (� = 0, solid curve) and oﬀ-resonant (�/ω0 = 0.2, dashed curve) cases. The temperature, kB T /ω0 = 13, is chosen to be above the relevant crossover Tc in the resonant case, such that the resonant dynamics is guaranteed to be incoherent. Parameters: α = 0.05, V /ω0 = 0.5, ωc /ω0 = 4, and µ = 0.5. The inset shows the corresponding population dynamics.

population transfer in the oﬀ-resonant case: αz (t) = e−Γz t − (1 − e−Γz t ) tanh(βη/2).

(6.71)

Let us also consider the evolution of αx and αy in the same limit: � � 2BVR −Γz t −Γz t −(1/2)Γz t αx (t) = e − (1 − e ) tanh(βη/2) − e cos(¯�t) , � (6.72) 2BVR −(1/2)Γz t αy (t) = − e sin(¯�t). (6.73) � Hence, although the donor-acceptor population itself evolves entirely incoherently in this limit, the coherences may still perform oscillations due to the energy mismatch. To illustrate the diﬀerence in the transition to incoherent population transfer on- and oﬀ-resonance, in the main part of Fig. 6.5 we plot the evolution of the coherence αy (t) in both cases. The parameters have been chosen such that the resonant dynamics is in the incoherent regime (T > Tc ), hence the resonant αy displays no oscillations [see Eq. (6.29)]. In accordance with Eq. (6.73), however, the introduction of an energy mismatch induces (small magnitude) oscillations in the donor-

6.4 Oﬀ resonance

141

1.0 pop. diff. !Αz "

coherence !Αy "

0.2

0.1

0.5 0.0 !0.5 !1.0 0

20

40

60

80 100

0.0

!0.1

!0.2 0

10

20

30

40

50

Ω0 t

Figure 6.6: Coherence (αy ) as a function of scaled time ω0 t for temperatures of kB T /ω0 = 1 (blue dashed curve), kB T /ω0 = 5 (green dotted curve), kB T /ω0 = 12 (orange solid curve) and kB T /ω0 = 20 (red dot-dashed curve). Parameters: α = 0.05, V /ω0 = 0.5, ωc /ω0 = 4, �/ω0 = 2 and µ = 0.5. The inset shows the corresponding population dynamics.

acceptor coherence. While these oscillations have an almost negligible amplitude, this behaviour serves to illustrate the subtlety in defining a strict crossover from coherent to incoherent dynamics in the oﬀ-resonant case. In particular, despite the diﬀerent forms of coherence behaviour, the corresponding (essentially incoherent) population dynamics shown in the inset is almost indistinguishable in the two cases, even though there should still be a strongly suppressed coherent contribution in the oﬀ-resonant curve. An alternative way to obtain coherence oscillations in a regime of predominantly incoherent population transfer is to introduce a large energy mismatch (i.e. make V /� small) at low temperature, as shown in Fig. 6.6. Here, for the lowest temperature considered the population relaxes towards its steady state value with negligible, if any, sign of oscillation, while the coherence performs oscillations with a significant amplitude and a considerable lifetime. This behaviour is strongly suppressed, however, as temperature increases, such that kB T > �.

6.4.4

Correlated fluctuations

We have seen in the previous sections that for oﬀ-resonant energy transfer the distinction between coherent and incoherent dynamics is less easily defined than in the resonant case. However, we should still expect changes

6.4 Oﬀ resonance

142

coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t coherence !Αy "

pop. difference !Αz "

1.0

0.5

0.4 0.2 0.0 !0.2 !0.4 0

10 20 30 40 50

0.0

!0.5

!1.0 0

20

40

60

80

100

120

140

Ω0 t

Figure 6.7: Population diﬀerence as a function of scaled time ω0 t for temperatures of kB T /ω0 = 5 (blue dashed curve) and kB T /ω0 = 10 (red dotted curve), and for separations corresponding to no fluctuation correlations, µ = c/dω0 = 0 (top), weak correlations, µ = 0.5 (middle) and strong correlations µ = 2 (botttom). The insets show the evolution of the corresponding coherence αy . Parameters: α = 0.05, V /ω0 = 0.5, �/ω0 = 0.5, and ωc /ω0 = 4.

6.5 Summary

143

in the level of correlation in the donor and acceptor fluctuations to have a similar eﬀect on the transfer process as in the resonant case. To illustrate that this is indeed the case, in Fig. 6.7 we plot the donoracceptor population dynamics in oﬀ-resonant conditions for three diﬀerent levels of fluctuation correlation (increasing from top to bottom). Just as we found in the resonant case of Fig. 6.2, an increase in correlations enhances the lifetime of coherence present in the energy transfer process. In addition, in the oﬀ-resonant case stronger correlations also serve to amplify the coherent contribution to the full energy transfer dynamics, since the renormalised interaction strength VR increases in relation to the energy mismatch �.

6.5

Summary

In summary, we have investigated various factors that determine the nature of the energy transfer dynamics in a model donor-acceptor pair. To do this, we used a polaron transform, Markovian master equation technique. We are able to describe consistently oﬀ-resonant eﬀects, unlike in the NIBA [10, 11], and the influence of bath correlations. In the resonant case we were able to identify a crossover temperature separating coherent and incoherent energy transfer. We found a non-trivial dependence of this temperature on the level of correlations in bath fluctuations and the cut-oﬀ frequency of the bath. Smaller cut-oﬀ frequencies increase the extent to which correlations in the bath protect coherence in the system, and therefore increase the crossover temperature. In the oﬀ-resonant case it was found that coherent and incoherent regimes are less easily defined. In particular, for a suﬃciently large energy mismatch between the sites (although still comparable to the coherent site interaction strength), coherence in the system is present at all finite temperatures, albeit with an ever decreasing amplitude. Although a crossover temperature was not able to be defined in this oﬀ-resonant regime, using analytic expressions in various limits and numerical investigations, we were able to show, for example, that correlations have qualitatively similar eﬀects on the transfer process as in the resonant case. While we have concentrated in this chapter on elucidating general features of donor-accpetor energy transfer dynamics using a simple model

6.5 Summary

144

system, the insight gained could be relevant to a variety of systems. In addition to those already mentioned [112–120], closely-spaced pairs of semiconductor quantum dots could provide a solid-state implementation of the model studied here [79]. In particular, our polaron master equation theory provides a bridge between the weak [160] and strong [161] system-bath coupling approximations already explored in this context. It would also be interesting to analyse the energy transfer dynamics of larger donoracceptor complexes within the polaron formalism, to see if further insights into the interplay of coherent and incoherent processes in such systems can be gained.

Chapter 7 Variational theory Contents 7.1

7.2 7.3

7.4

7.5

7.6

Introduction . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . 7.1.2 Single impurity spin-boson model . . . . . . . . . Two-impurity spin-boson model . . . . . . . . . . . Variational calculation . . . . . . . . . . . . . . . . . 7.3.1 Crude Ising approximation . . . . . . . . . . . . 7.3.2 Free energy minimisation . . . . . . . . . . . . . 7.3.3 Separation-dependent localisation . . . . . . . . Section summary . . . . . . . . . . . . . . . . . Full variational treatment . . . . . . . . . . . . . . . 7.4.1 Free energy minimisation and self-consistent equations . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Comparison of full and crude Ising strengths . . . Variational ground state . . . . . . . . . . . . . . . . 7.5.1 Two-impurity spin-boson Hamiltonian in the displaced oscillator basis . . . . . . . . . . . . . . . 7.5.2 Experimental signatures of localisation to delocalisation crossover . . . . . . . . . . . . . . . . . . 7.5.3 System-bath entanglement . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . .

146 . 146 . 148 152 154 . 154 . 156 . 158 . 162 162 . 162 . 163 167 . 167 . 168 . 170 171

7.1 Introduction

7.1

146

Introduction

The work presented in this chapter represents a slight deviation from that presented elsewhere in this thesis. So far we have been primarily interested in the dynamics of open systems, with emphasis on correlations in bath fluctuations and exploration of regimes which standard weak-coupling theories cannot probe. In this chapter, we now investigate interesting ground state properties of the single spin and two-spin-boson model.

7.1.1

Motivation

The spin-boson model introduced in section 2.3 is known to exhibit nontrivial ground-state behaviour [10, 11, 19–21, 162, 163], displaying a zerotemperature (quantum) phase transition [164,165] as a function of systembath coupling strength, attributed to zero-point (rather than thermal) fluctuations within the bath. Besides being of general theoretical interest, many physical systems in the solid state, and elsewhere, are well described by models of a spin-boson type [10–12, 40, 43, 52, 166–169]. Specific experimentally relevant examples include large arrays of trapped ions [43], the persistent current in a metal ring threaded by an Aharonov-Bohm flux [166, 167], and atomic dots coupled to a Bose-Einstein condensate bath [168]. These systems are of particular importance, since it is predicted that they show qualitative and detectable changes in ground-state properties as a function of accessible external parameters. For ease of reference we once again write down the spin-boson Hamiltonian in its usual form, � � � ∆ H = σz − σx + σz (gk b†k + gk∗ bk ) + ωk b†k bk , 2 2 k k

(7.1)

where � is the energy bias between the system states, ∆ is the (bare) tunnelling strength, and σi (i = x, y, z) is the usual ith-Pauli operator in a basis where σz = |0��0| − |1��1|. As always the bath is represented by the creation (annihilation) operators b†k (bk ) for each bath mode, with

wave-vector k and corresponding angular frequency ωk . The system-bath interaction is captured by the coupling constants gk . As we have now seen many times, the interaction of a quantum system with an environment of the type given in Eq. (7.1) causes a renormalisation

7.1 Introduction

147

of the bare system energy levels and, in particular, a suppression of any tunnelling probability the system may possess [8, 10, 11]. For the spinboson model, the system-bath interaction can be completely characterised � by the spectral density J(ω) = k |gk |2 δ(ω −ωk ), which we shall take here to be of the paradigmatic Ohmic form J(ω) = (α/2)ω for ω < ωc , where α

is a dimensionless coupling strength and ωc a high-frequency cutoﬀ [8, 10, 11]. In this case, and as mentioned in section 2.3.4, it has been found that above a certain critical system-bath coupling strength, αc , the tunnelling probability is completely suppressed (∆ → 0) [11]. For small �/∆, as

the parameter α is increased through αc , the ground state of the twolevel system shows a crossover from being dominated by the tunnelling term (∆/2)σx , and hence delocalised, to being dominated by the bias term (�/2)σz , and therefore localszed in either |0� or |1� [18, 162]. At zero temperature and for � = 0 this localisation phenomenon has been identified as a Kosterlitz-Thouless quantum (rather than thermodynamic) phase transition [164, 165]. Calculation of the Ohmic critical coupling strength in this regime has found αc ≈ 1 for small ∆/ωc [11].

In this chapter, we shall investigate the delocalised-localised crossover

in the ground state of a pair of non-interacting two-level systems in a common bath of harmonic oscillators, termed here the two-impurity spinboson model and whose dynamical entanglement properties we studied in chapter 3, and also in Refs. [1, 34, 35, 39, 170]. In particular, we elucidate how this crossover depends upon the separation between the impurities through a bath-induced inter-spin interaction. Aside from being a natural extension of the single impurity model, the two-impurity case represents perhaps the simplest dissipative model in which to explore the interplay of coherent system interactions and the dissipative influence of the bath. This has relevance, for example, in the field of quantum computation [5], where the two-impurity model could be thought of as the basic unit of a dissipative spin chain [171–173] or as two quantum bits in a dissipative register [40, 174, 175]. To perform our analysis we employ a variational technique originally developed by Silbey and Harris [176]. The method consists of assuming a particular variational form of the ground-state wavefunction of the combined system and bath, and a subsequent optimisation based upon a minimisation of the associated free energy (or ground state energy at

7.1 Introduction

148

zero temperature). While this technique might be vulnerable to errors in certain limits, it has proven to be relatively robust when applied to single spin-boson systems described by Ohmic spectral densities [176]. Furthermore, conclusions drawn from the method have also been verified by path integral [11,17], flow equation [177] and scaling techniques [178], as well as by Bethe-ansatz [18, 179, 180], Numerical Renormalisation Group [18, 20], Density Matrix Renormalisation Group [21], exact diagonalisation [23], and Monte Carlo calculations [22]. Besides its relative simplicity, the variational technique is also attractive from the point of view of gaining insight into the form of the ground-state of the model, and how this varies through the delocalised to localised crossover. Furthermore, as we shall show below, it can be used to provide analytical calculations of bath-induced spin interaction terms and tunnelling renormalisations inherent to the model.

7.1.2

Single impurity spin-boson model

Before we go on to study a pair of two-level systems in a common bath, it is instructive to apply the variational technique to the (single) spin-boson model, Eq. (7.1), in an eﬀort both to understand the variational method employed, and also to assess its validity. Let us start by considering the ground state of the Hamiltonian with � = 0: H=−

� � ∆ σx + σz (gk b†k + gk∗ bk ) + ωk b†k bk . 2 k k

(7.2)

In the limit gk → 0 with ∆ �= 0, the spin is entirely decoupled from the √ bath and its ground state will be the σx eigenstate (1/ 2)(|0� + |1�). The

state of the bath will be some superposition of number states (eigenstates of b†k bk ) that depends upon the temperature. In the opposite limit, ∆ → 0 with gk �= 0, the Hamiltonian reduces the the independent boson model

studied in section 2.3.1. The system-bath interaction now dominates and the oscillators constituting the bath will be displaced from their equilibrium positions to minimise the corresponding interaction energy. We may write the ground state of the combined system-plus-bath in this case as 1 |Ψ� = √ 2

�

� k

D

�g � k

ωk

|B0 � |1� +

� k

� � g � k D − |B0 � |0� , ωk

(7.3)

7.1 Introduction

149

where |B0 � is the ground state of the bath for vanishing system-bath coupling, and the displacement operators [14] are defined in the usual way � �� g � g k † � g k �∗ k D ± = exp ± b − bk . ωk ωk k ωk

(7.4)

In the general case, when neither limit is met, the spin-boson Hamiltonian is not straightforwardly diagonalisable. Note, however, that in the state described by Eq. (7.3) each oscillator is displaced by an amount determined by the the ratio gk /ωk , and that as gk → 0 the correct uncoupled

ground state is recovered. The variational theory thus assumes that the ground state of the spin-boson Hamiltonian for non-zero gk and ∆ is always of the form of Eq. (7.3), but allows for the possibility that the amount a given mode is displaced may have a more complicated dependence on the Hamiltonian parameters. With these considerations in mind, we now reintroduce the energy bias between the spin states and proceed by writing down the total Hamiltonian, Eq. (7.1), in a basis {|B− � |0� , |B+ � |1�}, where |B± � = � k D(±fk /ωk ) |B0 �, and we have introduced the as yet to be determined variational parameters fk . These will be found by minimising the free energy of the total system [176]. At zero temperature, we obtain � ∆r H= σ ˜z − σ ˜x + R, 2 2

(7.5)

where in the new basis σ ˜z = |B− � |0� �0| �B+ | − |B+ � |1� �1| �B− | R=

� k

ωk−1 fk (fk − 2gk ).

1

and (7.6)

Importantly, the tunnelling matrix element has now been renormalised due to the system-bath interaction: ∆r = ∆�B�, where �

�B� = �B± |B± � = exp − 2

� k

2

�

(fk /ωk ) .

(7.7)

Diagonalisation of H in the transformed basis then gives a ground state � energy of λ0 = 12 (2R − η), where η = �2 + ∆2r , and the corresponding 1 Note that the tildes here do not refer to operators in the interaction picture as elsewhere in this thesis.

7.1 Introduction

150

ground state |φ0 � = n0

�

� η−� |B− � |0� + |B+ � |1� , ∆r

(7.8)

where n0 = ((η − �)2 /∆2r + 1)−1/2 is a normalisation factor.

The task now is to find the variational parameters fk , which in turn will

allow us to evaluate the renormalised tunnelling strength ∆r , and hence obtain the approximate ground state. To do so, we naturally impose the condition that the ground-state energy, λ0 , should be minimised. This leads straightforwardly to fk = g k

�

∆2 1+ r ωk η

�−1

,

(7.9)

and our expression for the renormalised tunnelling strength becomes �

∆r = ∆ exp −2

� k

� gk2 . (ωk + ∆2r /η)2

(7.10)

We now take the continuum limit to convert the summation over k into an integral with respect to ω, and recall the definition of the (Ohmic) � system-bath spectral density, J(ω) = k |gk |2 δ(ω − ωk ) = (α/2)ω. With these replacements, we find

�

∆r = ∆ exp −α

�

ωc 0

� ω dω . (ω + ∆2r /η)2

(7.11)

Note that had we written the original Hamiltonian in a basis defined with the displacement operators of Eq. (7.4), i.e. functions of gk rather than fk , the integral in Eq. (7.11) would suﬀer from an infra-red divergence, and we would conclude (incorrectly) that ∆r = 0 for all values of α. That is, the full polaron transformation is limited to incoherent regimes for an Ohmic bath as considered here. In the present case, the integration can be performed straightforwardly and leads to the following equation which one must solve self-consistently for ∆r : ∆r

�

∆2r ∆2r + ωc η

�−α

�

� −α ωc η exp 2 = ∆, ∆r + ωc η

(7.12)

where ∆r takes on values between ∆ and 0 as α is increased from zero. For ∆/ωc � 1 and � = 0, Eq. (7.12) gives the well-known behaviour

7.1 Introduction

151

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0.6

0.8

Α

Figure 7.1: Expectation value of σz for the variationally determined spinboson ground state (plot points) and using Bethe-ansatz techniques (solid lines) as a function of α, plotted for various values of � (in units of ωc ). Blue circles: � = 0.005; red squares: � = 10−3 ; yellow diamonds: � = 10−4 ; green upright triangles: � = 10−6 ; and purple inverted triangles: � = 10−8 . In all cases ∆/ωc = 0.01.

∆r ∼ ∆(∆/ωc )α/(1−α) [10, 11, 176].

In order to assess the validity of the variational technique, in Fig. 7.1

we plot the ground state magnetisation, �σz � = �φ0 | σz |φ0 � = −�/η, as a function of α for various values of �, where we have set ∆/ωc = 0.01 and

∆r has been found by numerically solving Eq. (7.12). Shown also are the corresponding plots generated by mapping the spin-boson model to the Kondo model and using Bethe-ansatz solution techniques, details of which can be found in Refs. [162, 179, 180]. For all values of � the methods show good qualitative agreement. Most importantly, the variational calculation correctly identifies the region of α over which the ground state becomes dominated by the bias (localisation) rather than the tunnelling (delocalisation), though it should be noted that �σz � reaches its minimum value

(�σz � → −1 as ∆r → 0) somewhat more sharply than in the Bethe-ansatz

calculations. We can therefore be confident that the variational method does capture the localisation crossover in the ground state behaviour in which we are interested.

7.2 Two-impurity spin-boson model

7.2

152

Two-impurity spin-boson model

We now return our attention to the main subject of this chapter, determining the ground state behaviour of a pair of impurity spins interacting with a common bosonic bath. Since we expect the bath to mediate a separationdependent coherent interaction between the spins [1, 34, 35, 40, 181] we make their spatial separation explicit by placing them at positions r1 and r2 . The total Hamiltonian which we also considered in chapter 3 is then given by H=−

� � � � † � ∆ 1 (σx +σx2 )+ ωk b†k bk + σzn gk bk eik·rn +bk e−ik·rn , (7.13) 2 n k k

where σin (n = 1, 2; i = x, y, z) is the usual ith-Pauli operator acting on the relevant spin, and we have assumed that the system-bath coupling constants for each spin diﬀer only in a position-dependent phase factor. For simplicity, we now limit our analysis to the case in which there is no bias on either spin. We proceed in a slightly diﬀerent manner to Section 7.1.2 and apply a unitary transformation to the Hamiltonian which displaces each oscillator by an amount parameterised by the variational parameter fk . We note, however, that at zero temperature this procedure gives the same ground state as one would obtain following the method of the previous section. The variationally transformed Hamiltonian is written HV = eS1 +S2 He−(S1 +S2 ) = HSV + HBV + HIV with �

e±Sn = exp ±σzn

� k

�

αk b†k eik·rn − bk e

� −ik·rn

�

,

(7.14)

where αk = fk /ωk is assumed real. The transformation is aided by the observation that, provided the dispersion relation of the bath is isotropic and the variational parameters satisfy fk = f−k , the commutator [S1 , S2 ] = 2iσz1 σz2

� k

αk2 sin (k · (r1 − r2 ))

(7.15)

vanishes once the summation has been performed, regardless of the dimensionality or frequency spectrum of the system-bath interaction. The variational technique now relies on a careful choice of HSV , HBV

7.2 Two-impurity spin-boson model

153

and HIV from the various terms available after the transformation. We define the new unperturbed Hamiltonian as H0V = HSV + HBV , with HSV = −

� � ∆r � 1 σx + σx2 − 2Xσz1 σz2 + 2 fk (fk − 2gk )/ωk , 2 k

(7.16)

and HBV = HB . Here, ∆r is now determined by the finite temperature generalisation of Eq. (7.7), and is given by �

∆r = ∆�B� = ∆ exp − 2

�

αk2 coth(βωk /2)

k

�

,

(7.17)

where the inverse temperature is β = 1/kB T , while the form of HSV has been chosen such that the expectation value of HIV with respect to H0V vanishes. We shall see that this significantly simplifies the determination of the {fk } below.

There then remain two forms of system-bath interaction, HIV = Hz +

H⊥ , where Hz =

� n

�

σzn

� k

�

(gk − fk ) b†k eik·rn + bk e−ik·rn

�

�

,

(7.18)

and � � ∆� (n) (n) n n H⊥ = − (B+ − �B�)σ+ + (B− − �B�)σ− , 2 n

(7.19)

n with σ± = (1/2)(σxn ± iσyn ), and bath operators again given by products of

the displacement operators: (n) B±

�

= exp ±2

� k

�

αk (b†k eik·rn − bk e−ik·rn ) .

(7.20)

Note that if we assume the bath to be in thermal equilibrium, the latter four bath operators all have same expectation value with respect to H0V : � (n) �B� = �B± �H0V = exp [−2 k αk2 coth(βωk /2)].

The unperturbed Hamiltonian, H0V , has two important features.

Firstly, the tunnelling strength, ∆r , has been renormalised. Secondly, the two spins are now coupled via a bath mediated, separation-dependent,

7.3 Variational calculation

154

Ising-like interaction, with a strength X=

� k

ωk−1 fk (2gk − fk ) cos(k · (r1 − r2 )).

(7.21)

Evaluation of both ∆r and X requires knowledge of the set of variational parameters {fk }. The variational procedure determines these by free energy minimisation arguments. However, before we continue the analysis, we outline a significant simplification which can be made.

7.3 7.3.1

Variational calculation Crude Ising approximation

The variational parameters {fk }, appearing in both ∆r [Eq. (7.17)] and in the induced Ising strength X [Eq. (7.21)], were introduced to overcome an infra-red divergence in ∆r that occurs for an Ohmic spectral density when applying a polaron transformation to our Hamiltonian, since it fully displaces the bath modes as in Eq. (7.3) [176]. As mentioned previously, this divergence would lead to a complete suppression of the tunnelling probability, ∆r → 0, and can be seen by making the replacement fk → gk in Eq. (7.17), and using J(ω) = (α/2)ω.

However, we may make the replacement fk → gk in the definition

of the Ising strength [Eq. (7.21)] and find that it suﬀers from no such

divergence. Therefore, to some level of approximation at least, we can make this replacement (in Eq. (7.21) only) and evaluate X outside the variational calculation. We know that in the limit that the coupling of the system to the bath completely dominates, the oscillators are fully displaced, i.e. fk → gk anyway. Hence, we can identify this replacement as a kind of strong coupling approximation on X, as will be discussed in

more detail in Section 7.4.2. We shall refer to the Ising term evaluated within this approximation, and to the approximation itself, as “crude” since it does not take into account deviations of fk from gk in X. Assuming a linear dispersion relation |k| = ω/c, where c is the excita-

tion speed, we find

XC =

αωc fD (td ωc ), 2

(7.22)

where the subscript C indicates a crude value. The impurity distance

7.3 Variational calculation

155

1.0 0.8 0.6 0.4 fD!td Ωc " 0.2 0.0 !0.2 !0.4 0

5

10 td Ωc

15

20

Figure 7.2: Measure of the correlation between the bath-induced fluctuations experienced at each impurity spin, plotted as a function of the scaled impurity separation td ωc = |r1 − r2 |ωc /c. The diﬀerent curves correspond to system bath coupling in one dimension (blue dotted curve), two dimensions (dashed red curve), and three dimensions (solid green curve). In all cases the correlation is maximised for zero separation (fD = 1, complete correlation) and tends to zero as the separation goes to infinity (fD = 0, no correlation).

dependence enters through td = |r1 − r2 |/c, which is the time bosonic

excitations take to travel between the spins, and determines the value of the function fD (x) (D = 1, 2, 3), which is a measure of the (separationdependent) correlation between the bath influences seen at each spin, and is therefore dependent on the dimensionality D of the system-bath interaction. We note that the function fD is not the same as the function FD encountered in chapters 3 and 6, although it does share some properties. We find f1 (x) = sinc(x) in one dimension, f2 (x) = 1 F2 ({1/2}, {1, 3/2}, −x2 /4)

is a generalised hypergeometic function in two dimensions, and f3 (x) = �x Si(x)/x in three dimensions, Si(x) = 0 (sin t/t)dt being the sine integral function. As shown in Fig. 7.2, in all cases fD (x) has a maximum value of fD (0) = 1, and in two and three dimensions has a minimum value fD (∞) = 0. Additionally, in one dimension f1 (x) displays decaying oscillations, becoming zero whenever td = nπωc , for n = 1, 2, 3, . . . . (1)

(2)

Note that, ignoring any spatial correlations in B± and B± , our trans˜ with the replacement X → XC now has exactly formed Hamiltonian H the same form as that which would be obtained if we transformed a Hamil-

tonian describing two spins in separate baths, each subject to a transverse field of strength ∆, and coupled via a ferromagnetic Ising field of strength

7.3 Variational calculation

156

2XC [182, 183]. That is, had we transformed the Hamiltonian HT I

=

∆ 1 (σx + σx2 ) − 2XC σz1 σz2 2 � � +σz1 (gk b†k + gk∗ bk ) + σz2 (gk a†k + gk∗ ak )

−

+

�

k

ωk b†k bk

k

+

�

k

νk a†k ak ,

(7.23)

k

where we have introduced a second bath which couples only to the second spin and is described by creation (annihilation) operators a†k (ak ), with corresponding frequencies νk .

7.3.2

Free energy minimisation

Precisely as in the single-spin case, our task is now to find the set of variational parameters {fk }, which will then allow us to find the renormalised tunnelling strength ∆r . If, for a given α, we find that ∆r → 0 (i.e. fk → gk ), the system will be dominated by the induced Ising interac-

tion, forming a ferromagnetic or antiferromagnetic pair. The spins will be unable tunnel between their states |0� and |1� and will be said to be in a localised regime. On the other hand, if ∆r �= 0, the tunnelling probability

remains finite and the spins are delocalised. We expect that as α → αc , ∆r → 0, and we enter a regime in which the renormalised tunnelling has a negligible influence on the ground state.

To find the set {fk } we follow Refs. [19] and [176] and compute the

Bogoliubov-Feynman upper bound on the free energy of the total systemplus-bath, AB , which is related to the true free energy, A, via AB ≥ A [184], where

� 2 � ˜ 0 ]} + �H ˜ I � ˜ + O �H ˜ �˜ . AB = −β −1 lnTr{exp[−β H I H0 H0

(7.24)

We have constructed our perturbation terms HIV and free Hamiltonian H0V such that �HIV �H0V = 0 by definition. We shall assume that terms of

2 order �HIV �H0V are small, as shown in Ref. [19], and approximate the free

energy using only the first term of Eq. (7.24). Neglecting the free energy of the bath, since it does not depend on the variational parameters, we

7.3 Variational calculation

157

find AB ≈ 2

� k

where EC =

ωk−1 fk (fk −2gk )−β −1 ln

� � �� 2 cosh(2βXC )+cosh(βEC ) , (7.25)

� 4XC2 + ∆2r . Minimising AB with respect to the variational

parameters yields the choice fk = g k

�

∆2r 1+ ωk EC

�

sinh(βEC ) coth(βωk /2) cosh(2βXC ) + cosh(βEC )

��−1

.

(7.26)

As we are interested here in the ground state (zero temperature) behaviour of the system we take the limit β → ∞ to find fk = g k

�

∆2r 1+ ωk EC

�−1

.

(7.27)

Having found the optimal choice for each fk in Eq. (7.27), we can now insert this into our expression for the renormalised tunnelling strength, Eq. (7.17). Taking the continuum limit and using the same form of Ohmic spectral density as before, we obtain the following self-consistent equation ∆r

�

∆2r ∆2r + ωc EC

�−α

�

� −α ωc EC exp 2 = ∆. ∆r + ωc EC

(7.28)

Note that with the replacement EC → η (or 2XC → �) this equation is

identical to Eq. (7.12) derived in Section 7.1.2 when considering a single spin with finite bias. This stems from the observation that, from the point of view of one of the spins, its Ising-like coupling to the other spin can be thought of as providing an eﬀective energy diﬀerence between its σz eigenstates. The solutions of Eq. (7.28) give values of ∆r that correspond to stationary points of the free energy approximation AB . That a given solution exists does not necessarily mean it is appropriate to assume the system will adopt this value. Rather, within our approximate treatment, the system will adopt the value of ∆r that gives the lowest AB [185]. To see which solution for ∆r will be favored, we compute the free energy at zero temper-

7.3 Variational calculation

158

ature using the variational parameters we have just derived in Eq. (7.27): AB ≈ −EC − αEC

�

∆2r EC + ωc2 ωc

�−1

.

(7.29)

Since we are working in the limit ∆/ωc � 1, it must also be true that

(∆r /ωc )2 � EC /ωc , regardless of the value of XC , and we can further approximate

AB ≈ −αωc − EC

�

� ∆2r 1−α 2 . EC

(7.30)

The system will adopt the value of ∆r which makes the second term in Eq. (7.30) most negative, i.e. that value which most strongly satisfies the condition 4XC2 > ∆2r (α − 1). For α < 1 it is clear that this will correspond

to the greatest positive value of ∆r . Therefore, where multiple solutions to Eq. (7.28) exist, for α < 1 we should choose the largest value of ∆r .

7.3.3

Separation-dependent localisation

In general, solving Eq. (7.28) for ∆r analytically is not possible and it must be solved for numerically instead. However, to begin with, note that ∆r = 0 is always a solution, regardless of the value of α or XC . Now, we can look for other analytical solutions in certain limits. Perhaps the simplest of these is the limit XC → 0, corresponding to two infinitely separated

spins in a common bath or two uncoupled spins in separate baths. From either interpretation, we should recover the well-known single spin-boson results. Setting XC = 0 in Eq. (7.28) gives ∆r

�

∆r ∆r + ωc

�−α

�

� −αωc exp = ∆, ∆r + ωc

(7.31)

which in the limit ∆/ωc � 1 (∆r ≤ ∆) gives the well-established form [10, 11, 176]

�

e∆ ∆r ≈ ∆ ωc

�α/(1−α)

.

(7.32)

Hence, for XC = 0 and ∆/ωc � 1, the renormalised tunnelling strength

smoothly reaches zero as α → 1, and we predict that the critical cou-

pling strength separating the delocalised and localised phases is given by αc (XC = 0) = 1, precisely as in the single spin-boson case. Let us now consider the opposite limit, XC /∆ � 1, corresponding

7.3 Variational calculation

159

either to closely spaced spins in a common bath with intermediate or strong dissipation (so that XC is large), or two spins in separate baths coupled via a relatively strong Ising interaction. Since ∆r ≤ ∆, we may

also assume XC /∆r � 1. Setting EC ≈ 2XC and neglecting ∆r in the denominators in both bracketed factors in Eq. (7.28), we find �

e∆2 ∆r ≈ ∆ 2ωc XC

�α/(1−2α)

.

(7.33)

Within the limits this expression has been derived, the bracketed factor is small and we observe that ∆r → 0 as α → 0.5. We conclude that

for an Ising strength XC much larger than the bare tunnelling strength, the critical system-bath coupling strength is no longer given by the single spin-boson value (αc ≈ 1), but instead by αc (XC /∆ � 1) ≈ 0.5. This can also be seen by expanding our expression for the minimised free energy, Eq. (7.30), to lowest order in ∆r /XC , which gives AB ≈ −αωc − 2XC

�

� ∆2r 1+ (1 − 2α) . 8XC2

(7.34)

It is clear from this expression that in the limit XC � ∆r a finite ∆r will be favoured only if α < 0.5.

When neither of these conditions are met, i.e. when XC /∆ ∼ 1,

Eq. (7.28) is best studied graphically. To do so, we define the left hand

side of Eq. (7.28) as a function Θ(∆r ). Any points at which Θ(∆r ) crosses the line ∆ will then give non-zero solutions for ∆r . In the main part of Fig. 7.3 we plot Θ(∆r ) for a fixed spin separation, corresponding to a value of fD (td ωc ) = 0.05, and for various values of α. Also shown is the dashed line at 0.2, which represents the value of the bare tunnelling strength ∆ (in units of ωc ) taken here. The first feature to notice is that the curve shows a dramatic change in behavior as the coupling strength α moves through the value 0.5, developing a minimum in the first quadrant for α > 0.5. Therefore, when α < 0.5, we can always expect a single finite solution for ∆r . On the other hand, when α > 0.5, depending on the specific values of the ratio XC /∆ and α, the curve Θ(∆r ) may not cross the line ∆ at all, just touch it, or dip low enough to cross it twice. In the main part of Fig. 7.3 the ratio XC /∆ is small enough such that Θ(∆r ) does dip below ∆ for α ≥ 0.5, and the critical coupling strength is

7.3 Variational calculation

160

0.4 Increasing Α

0.3

Α#Αc !fD " Α$Αc !fD "

&!%r " 0.2

0.6

Α#0.5

0.4

0.1

0.2

Α"0.5

0.0 0.00

0. 0.

0.02

0.04

0.06

0.05

0.08

0.1

0.10

%r

Figure 7.3: The renormalised tunnelling strength is the solution to Θ(∆r ) = ∆ (see Eq. (7.28)). Main: Here we plot Θ(∆r ) for various values of α and for a fixed spin separation corresponding to fD (td ωc ) = 0.05. Setting ωc = 1 then gives XC = 0.025α. We see that as α is increased the solution for ∆r decreases until a point at which the curve Θ(∆r ) just touches the line ∆ = 0.2. After this point, the only solution is ∆r = 0. Inset: Here we plot the same curves but with fD (td ωc ) = 0.3. In this case fD (and hence XC ) is so large that αc = 0.5, since above this value Θ(∆r ) never crosses ∆.

then given when Θ(∆r ) just touches the line ∆. We see that 0.5 < αc < 1 in such cases. In the inset we show the same set of plots, but for a smaller spin separation corresponding to a higher value of fD (td ωc ) = 0.3. In this case, the ratio XC /∆ is large enough that once Θ(∆r ) changes its qualitative behavior (i.e. when α > 0.5), it never crosses the line ∆ and the only solution to Eq. (7.28) is ∆r = 0. This confirms the limiting behavior, αc ≈ 0.5 for XC /∆r � 1, discussed earlier in reference to Eqs. (7.33) and (7.34).

Knowing how we expect ∆r to behave in certain limits we may now numerically solve Eq. (7.28) to find the renormalised tunnelling strength as a function of α, for various spin separations characterised by the function fD (td ωc ) measuring the bath correlations. We shall restrict ourselves here to the two and three dimensional cases, and the results can be seen in Fig. 7.4. The solid black curve shows the renormalised tunnelling for infinitely separated spins, i.e. fD = 0 (no bath correlations). As expected, in this regime of zero Ising strength, ∆r → 0 as the system-bath coupling strength α → 1, precisely as in the single spin case. The dashed blue curve

shows the variation of the renormalised tunnelling with α for a spin sep-

7.3 Variational calculation

161

0.20 1.

0.15

Αc !fD " 0.5 Increasing fD

0. 0.

"r 0.10

0.25

0.5

fD

0.05 Αc !fD " 0.00 0.0

0.2

0.4

0.6

0.8

1.0

Α

Figure 7.4: Main: Numerically evaluated renormalised tunnelling strength as a function of the system bath coupling strength for various spin separations, captured by the value of the function fD measuring the bath correlations. The solid black curve corresponds to infinitely separated spins (no Ising interaction), fD = 0. The dashed blue curve corresponds to an intermediate spin separation (or Ising interaction), fD = 0.05. The dotted red curve corresponds to small spin separation (or large Ising interaction), fD = 0.3. Inset: Here we show how the critical coupling strength varies with the spin separation. For all curves, ∆/ωc = 0.2.

aration corresponding to fD = 0.05. For this intermediate separation (or, equivalently, Ising strength) we see that ∆r discontinuously approaches zero as α reaches a critical value somewhere between 0.5 and 1.0 (αc ≈ 0.65 for the values of ∆/ωc and fD used here). This agrees with the intuition

we gained previously from Fig. 7.3. The red dotted curve corresponds to a small spin separation giving fD = 0.3 (or large Ising strength). Here, ∆r → 0 continuously as α → 0.5, again in agreement with our analysis

of Fig. 7.3. In the inset of Fig. 7.4, we show explicitly how the critical coupling strength depends on the spin separation. As expected, for large spin separations (fD → 0) αc tends to 1, while as the spins are brought closer together and fD increases, αc approaches its minimum value of 0.5.

The last piece of information needed to complete our picture is the value of fD , say fD0 , after which the crossover always occurs around αc = 0.5 (i.e. beyond fD0 the bath correlations are large and XC quickly dominates with increasing α). Finding where the minimum of Θ(∆r ) just crosses the

7.4 Full variational treatment

162

line ∆ yields the simple result fD0

� �2 ∆ = 2e . ωc

(7.35)

For ∆/ωc = 0.2, we get fD0 ≈ 0.22, in agreement with Fig. 7.4. From this

expression it can be seen that the further a given system lies within the scaling limit (∆/ωc � 1), the larger the range of spin separations which diﬀer from the single-spin case (αc ≈ 1). Section summary We conclude this section with a brief summary. For distantly separated spins or negligible Ising strengths, the delocalised-localised crossover corresponds to the well studied single spin-boson model [19, 22, 162, 185]. The critical coupling strength after which the tunnelling element is renormalised to zero is predicted to be αc = 1, for ∆/ωc � 1. As the spins are brought closer together within their common bath, they become cou-

pled via an Ising-like interaction. This causes the crossover region to drop from αc = 1, as can be seen by tracing from left to right in the inset of Fig. 7.4. At a certain spin separation, the function scaling the Ising strength, fD (td ωc ), reaches a special value, fD0 , given by Eq. (7.35). For this spin separation, and all smaller separations, the crossover is predicted to occur around αc = 0.5.

7.4 7.4.1

Full variational treatment Free energy minimisation and self-consistent equations

The results presented in the previous section were obtained by approximating the induced Ising strength, X, by a value XC , through the replacement fk → gk . This significantly simplified the task of finding the set of variational parameters {fk }, which then allowed us to determine the renormalised tunnelling strength in a straightforward manner. To go be-

yond this approximation, we shall now perform the variational calculation making no such simplification, and hence use the full fk -dependent Ising strength given in Eq. (7.21).

7.4 Full variational treatment

163

As before, we calculate the free energy associated with the Hamiltonian HV = H0V + HIV , given by Eqs. (7.16), (7.18) and (7.19). This leads to an expression for AB identical to Eq. (7.25), but with XC replaced by X: AB ≈ 2

� k

where E =

� � �� ωk−1 fk (fk − 2gk ) − β −1 ln 2 cosh(2βX) + cosh(βE) , (7.36)

� 4X 2 + ∆2r . Minimisation with respect to the variational

parameters {fk } gives us the zero-temperature condition fk = g k

�

� � � E + 2X cos k · (r1 − r2 ) � � , E + 2X cos k · (r1 − r2 ) + ∆2r /ωk

(7.37)

which is consistent with our assumption fk = f−k , used with reference

to Eq. (7.15) in our derivation of the transformed Hamiltonian HV . We proceed by inserting Eq. (7.37) into our expressions for the renormalised tunnelling strength, Eq. (7.17), and the full Ising strength, Eq. (7.21). For simplicity, we now restrict our discussion to system-bath coupling in three dimensions, in which we may write k · (r1 − r2 ) = ωtd cos(θ), where

θ is a polar angle in k-space over which we must integrate. We then obtain the following two equations which we must simultaneously solve self-consistently: �

α ∆r = ∆ exp − 2

�

+1

dx −1

�

ωc

−1

2

�

ω G (ω, x)dω , 0

(7.38)

and α X= 4

�

+1

dx −1

�

ωc 0

� � G(ω, x) 2 − G(ω, x) cos(ωxtd )dω,

(7.39)

where x = cos(θ), and we have defined the function G(ω, x) =

7.4.2

�

� E + 2X cos(ωxtd ) . E + 2X cos(ωxtd ) + ∆2r /ω

(7.40)

Comparison of full and crude Ising strengths

Extracting useful analytic expressions from Eqs. (7.38) and (7.39) is not easily achieved. However, we note that the values ∆r = 0 and X =

7.4 Full variational treatment

0.10

164

0.20 #!" #!"

# " !

!

0.15

# " !# " " !# # " !

# "

# "

# "

!

0.08

# "

!

#r 0.10

!

!

!

# "# "# "# "# "# # " # #

! !

0.00 0.0

!

# #

!

X!Ωc

#

"

!

0.05

0.06 0.2

# # # # # ## !!!!! "! "! "! "! "! "! "! "! "! "! "! "# ! "# ! "# ! "# ! "

0.4

0.6

0.8

0.04 ! "

0.02

"

"

1.0 "

Α

"

"

"

"

"

" "

"

"

" !

" 0.00#!" # 0.0

"

" " " # # #

"

"

"

"

"

"

# # # # # # # # # # # # # # # # # # # #

0.2

0.4

0.6

0.8

1.0

Α

Figure 7.5: Main: Comparison of the numerically calculated Ising strength (points) and the crude Ising strength (solid lines) as a function of α, with ∆/ωc = 0.2. This is done for td ωc = 1 (red line, circular markers), td ωc = 15 (blue line, square markers) and in the limit td ωc → ∞ (black line, triangular markers, close to the x-axis). After the system-bath coupling strength α = αc , ∆r → 0 (see inset) and the crude Ising strength matches the full value. The dashed line shows the small Ising strength approximation of Eq. (7.42). Inset: Renormalised tunnelling strength calculated using the full Ising strength (points) and crude Ising approximation (lines) as a function of α, for the same parameters as the main figure.

XC solve these equations exactly. That is, in the localised regime, where the ground state becomes completely dominated by the Ising term, the Ising strength is given by its crude value. This tallies with our earlier assertion that the crude Ising approximation is essentially a strong systembath coupling approximation on the induced interaction strength. Let us also consider the regime in which the spins are distantly separated. On physical grounds, we expect that X → 0 as |r1 − r2 | → ∞, since it seems

inappropriate that the bath could mediate an interaction between spins separated by a large distance (certainly, we know that XC → 0 as the spin

separation is increased to infinity). This can be seen in the present case by making the assumption that for large td (i.e. large spin separation), X will be small (which we shall justify numerically in the following) and expand the integrand of Eq. (7.39) to second order in X. Having done so, the integrations with respect to x and ω can be performed analytically, leaving a quadratic equation for X which we write as 0 = h0 (td ) + X(h1 (td ) − 1) + X 2 h2 (td ),

(7.41)

7.4 Full variational treatment

165

0.30" 0.25

0.04 "

0.03

"

0.20!

"

"

"

0.02

X!Ωc 0.15

"

!

!

0.01

"

"

!

!

"

!

"

!

!

!

!

0.00

0.10

16 18 20 22 24 26 28 30

! " "

0.05

"

! !

0.00 0

5

"

"

!

!

!

10

" !

15

" !

" !

20

" !

" !

" !

25

" !

" !

30

td Ωc

Figure 7.6: Main: Induced Ising strength as a function of the scaled spin separation, td ωc , for two values of the system-bath coupling strength, α = 0.4 (blue circular markers) and α = 0.6 (red square markers). The markers indicate values calculated numerically from Eqs. (7.38) and (7.39) and the solid lines represent XC values calculated using Eq. (7.22). Inset: Magnification of the lower right corner, revealing how the discrepancy between X and XC increases when the Ising strength is small enough such that ∆r �= 0. where h0 , h1 , and h2 are cumbersome expressions (proportional to α) which we shall not give here. Taking the limit td → ∞, we find that h0 → 0 and h1 → αωc2 /2(∆r + ωc )2 . Applying the same limit to h2 is less

straightforward, although it is easy to see graphically that h2 → 0 as td → ∞. Hence, as expected, we have confirmed that X → 0 as td → ∞. Further, when X = 0, the self-consistent equation for the renormalised tunnelling strength, Eq. (7.38), reduces to that for a single spin given by Eq. (7.31). When neither the spin separation nor the system-bath coupling strength are large enough such that the above arguments apply, we must solve the self-consistent equations by numerical iteration. Solutions found in this way are shown in Fig. 7.5, where the plot points are calculated iteratively from Eqs. (7.38) and (7.39), and the solid lines calculated using the crude Ising approximation of the previous section. Red circular points correspond here to a small spin separation, td ωc = 1, blue squares to an intermediate separation, td ωc = 15, and black triangles to the limit td → ∞ (for which there is no discrepancy between the full and crude

Ising strengths). From the main part of the figure we can see that the crude value of the Ising strength generally gives a reasonably good ap-

7.4 Full variational treatment

166

proximation to the full expression. As the system-bath coupling strength is increased, there comes a point at which the tunnelling strength becomes entirely suppressed, ∆r → 0 (see figure inset), in which case Eq. (7.39) for

X reduces to the simpler form of Eq. (7.22). Hence, in the localised regime X = XC , as expected. From the inset of Fig. 7.5 we see that the behaviour of the renormalised tunnelling strength is well approximated across a range of diﬀerent parameter regimes by replacing X by XC in the self-consistent equations. Hence, our analysis of the localisation crossover in the twoimpurity spin-boson model given in the previous section is expected to hold true, even when the full bath-induced Ising form is used. In order in reproduce the behaviour of X for small values of α and moderate spin separations, where it diﬀers most markedly from XC in Fig. 7.5, we can expand the solution to Eq. (7.41) to first order in α. In doing so, we find X ≈ h0 (td ), with α∆r sin(td ωc ) 2td µ α + (Ci(td µ) − Ci(td ∆r ))(td ∆r cos(td ∆r ) − sin(td ∆r )) 2td α + (Si(td µ) − Si(td ∆r ))(td ∆r sin(td ∆r ) + cos(td ∆r )), (7.42) 2td

h0 (td ) = −

where Ci(x) = −

�∞ x

cos(t)/tdt is the cosine integral function, and we

have made the substitution µ = ∆r + ωc . The dashed curve in Fig. 7.5 shows this function plotted for td ωc = 15, where we also approximate the renormalised tunnelling strength as ∆r ≈ ∆(∆/ωc )α/(1−α) .

Lastly, in Fig. 7.6 we plot a comparison of the behaviour of X and

XC with varying (scaled) spin separation, td ωc . Recall that when ∆r = 0, X = XC , as can be seen in the majority of the plot for α = 0.6. For this value of the system-bath coupling, α > αc over almost the full range of separations considered, and the tunnelling is consequently renormalised to zero for most values of td ωc too. As the spin separation is increased, X decreases, and there comes a point at which ∆r �= 0 (td ωc ≈ 26). Here, we begin to see deviations of X from XC . When α = 0.4 the system is always in the delocalised regime (∆r �= 0) and we therefore see deviations of X from XC for all spin separations.

7.5 Variational ground state

7.5 7.5.1

167

Variational ground state Two-impurity spin-boson Hamiltonian in the displaced oscillator basis

In the preceding sections, we have used a variational treatment to establish how both the renormalised tunnelling strength and bath-induced Ising interaction vary as a function of system-bath coupling strength and spinseparation in the two-impurity spin-boson model. We shall now use this information to explore the interplay of these two quantities in determining how the form of the ground state of the system changes in diﬀerent parameter regimes. From this, we shall identify a physical indicator of the delocalised to localised crossover in the dissipative two-spin system. To obtain the variational ground state we generalise the procedure given in section 7.1.2 to two spins.

We write the total Hamiltonian

[Eq. (7.13)] in a displaced oscillator basis, this time defined by the four states {|B−− � |00� , |B−+ � |01� , |B+− � |10� , |B++ � |11�}, with |B±± � =

�

D(±αk eik·r1 )

|B±∓ � =

�

D(±αk eik·r1 )

and

k

�

D(±αk eik·r2 ) |B0 � ,

(7.43)

�

D(∓αk eik·r2 ) |B0 � ,

(7.44)

k

k

k

where once again αk = fk /ωk and |B0 � is the state of the bath for vanishing system-bath coupling. In this basis, the two-impurity spin-boson Hamiltonian becomes H=−

∆r 1 (˜ σ +σ ˜x2 ) − 2X σ ˜z1 σ ˜z2 + 2R, 2 x

(7.45)

where the zero temperature limit has been taken, and R, ∆r , and X are defined in Eqs. (7.6), (7.17), and (7.21), respectively. Diagonalising this Hamiltonian gives a ground state energy of Λ0 = 2R − E, and corresponding ground state

� � �� |Φ0 � = N0 |B−− � |00� + |B++ � |11� − ξ |B+− � |10� + |B−+ � |01� , (7.46)

where N0 = (2(1 + ξ 2 ))−1/2 , ξ = (2X − E)/∆r , and E =

� 4X 2 + ∆2r as

7.5 Variational ground state

168

1.0 Increasing Α 0.8 0.6 #Σx % 0.4 0.2 0.0 0

20

40

60

80

100

td Ωc

Figure 7.7: Expectation value of σx1 (or σx2 ) as a function of the scaled spin separation for diﬀerent values of the system bath coupling strength α (=0.20, 0.30, 0.55, 0.65 ordered as indicated). For α = 0.55 and α = 0.65 we see that at a particular spin separation there emerges a non-zero expectation value, signifying the crossover from localisation to delocalisation.

before. Minimising Λ0 with respect to the variational parameters leads to exactly the same condition [Eq. (7.37)] as derived in section 7.4. Therefore, we shall make the crude Ising approximation to evaluate X and ∆r , giving all of the required information relating to the variational ground state.

7.5.2

Experimental signatures of localisation to delocalisation crossover

To show how evidence for the localisation crossover might be observed experimentally, in Fig. 7.7 we plot the ground-state expectation value of the single-spin operator σx1 (or equivalently σx2 ), �σx1 � = �Φ0 | σx1 |Φ0 � = −2ξ�B�/(1 + ξ 2 ), as a function of the scaled spin separation for various

values of the system-bath coupling strength. For small values of α (α = 0.2, 0.3) the tunnelling element is renormalised to a finite value (delocalised regime) and �σx1 � is predominantly determined by the relative size of the bare tunnelling element to the Ising strength, saturating at a value �σx1 � ≈ �B� at large spin separations (small X). There is no qualitative change in

the ground-state form as the relative size of ∆r and X varies, in this case through increasing the spin separation. For larger values of α, lying between 0.5 and 1, the Ising strength at small spin separations is large enough such that the renormalised tun-

7.5 Variational ground state

169

0.7 0.6 0.5 0.4 "Σx 1 $

0.3 0.2 0.1 0.0 0

20

40

60

80

100

120

!!X

Figure 7.8: Expectation value of σx1 (or σx2 ) as a function of ∆ (measured in units of X) for α = 0.35 (black solid line), α = 0.6 (red dotted line), and α = 0.75 (blue dashed line). For these plots the induced Ising strength was kept at X = 0.0015ωc for each α, with ωc = 1.

nelling strength is completely suppressed, and �σx1 � → 0 (localised regime).

As the spin separation increases, the Ising strength decreases, and there comes a point at which X is small enough such that �σx1 � can now take on non-zero values (delocalised) for the same value of α. Therefore, if it

is possible to engineer a pair of Ising-coupled spins for which the Ising strength can be varied, and 0.5 < α < 1, the crossover region should be identifiable by the emergence of a non-zero value for �σx1 � (or �σx2 �) as the Ising interaction is decreased.

It is also possible to observe the crossover behaviour without the need for varying the Ising strength, by instead altering the bare tunnelling frequency due to the applied field. In Fig. 7.8 we again plot �σx1 � but this time as a function of the bare tunnelling strength with fixed bath mediated Ising strength X. For α = 0.35 we expect no crossover in ground state behaviour and we see �σx1 � → 0 only as ∆ → 0. For the curves corresponding to α > 0.5, when ∆/X is small we are in the regime in which

Eq. (7.33) is valid. As such, αc = 0.5 and we see �σx1 � = 0. As the ratio

∆/X is increased, we eventually move into a regime in which Eq. (7.32) is valid and αc → 1. For α = 0.6 and α = 0.75 we must therefore enter the delocalised regime as ∆/X increases, and �σx1 � thus begins to take on non-zero values.

7.5 Variational ground state

7.5.3

170

System-bath entanglement

Quantum phase transitions are associated with non-analyticity in the entanglement present in the total system-plus-bath state [186–189]. Although the variational treatment may not identify a true quantum phase transition, it is expected that the change in ground-state properties that are identified will have a manifestation in the entanglement as shown by Le Hur [162]. Since, within the variational approach, the total state [Eq. (7.46)] is a pure state, we can investigate such behaviour in our model simply by tracing out the bath degrees of freedom and calculating the von Nuemann entropy of the two-spin state. This will give a measure of the degree to which the spins are entangled with the bath [5]. Note that this entanglement is diﬀerent to that studied in chapter 3. There we were interested in the entanglement shared between the spins, whose joint state was mixed and the concurrence was used to quantify the entanglement. Here we are interested in the entanglement between the impurities and the bath, and as the joint impurity-bath state is pure, it suﬃces to use the von Nuemann entropy, giving a measure of how mixed the state of the impurities is. We define the reduced two-spin state as ρ = trB (|Φ0 ��Φ0 |), where trB denotes a trace over the bath degrees of freedom. The von Neumann entropy is then defined as S = −ρln(ρ) = −

4 �

τi ln(τi ),

(7.47)

i=1

where the τi are the four eigenvalues of ρ [5]. In Fig. 7.9 we plot the entropy as defined above (normalised by its maximum possible value) for three diﬀerent spin separations, corresponding to fD = 0 (black solid line), fD = 0.05 (blue dashed line) and fD = 0.3 (red dotted line). For fD = 0 the situation is identical to the single spin case. As α is increased, the extent to which the spins and the bath interact increases and their state becomes ever more entangled. For the curves corresponding to fD = 0.05 and fD = 0.3 we see a similar situation for small values of α. However, for moderate values of α we see that the entanglement reaches a maximum and then begins to fall. This corresponds to the onset of the crossover between delocalisation and localisation in the ground state. At the critical values of α for these spin separations

7.6 Summary

171

1.0 0.8 0.6 S!Ln4 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

Α

Figure 7.9: Normalised von Neumann entropy of the two-spin system as a function of α for fD = 0 (black solid line), fD = 0.05, (blue dashed line) and fD = 0.3 (red dotted line), with ∆/ωc = 0.2.

(αc = 0.65 and αc = 0.5, respectively, for these parameters), the entanglement sharply drops to a value of 0.5 as ∆r → 0. For a single spin S = 0

in its localised regime [162]. In the present case we find S/ ln 4 = 0.5 since there is nothing in our model to lift the degeneracy between the states |00� and |11� in the localised regime.

7.6

Summary

To summarise, in this chapter we have investigated the delocalised to localised crossover for a pair of two-level systems in a common bosonic bath. Our analysis followed closely that introduced for single spins by Silbey and Harris [176] which used a variational approach. The crossover region is identified by a complete suppression of the tunnelling element (∆r → 0) as the system-bath coupling is increased (α → αc ). We find an interesting

interplay between the magnitude of an environment-induced Ising spin interaction (X) and the applied tunnelling field (∆) in determining αc . In particular, our analysis suggests that the presence of the Ising term encourages the spins to enter the localised regime at a smaller value of the system-bath interaction than in the single spin case. Specifically, only for infinitely separated spins do we recover αc = 1, as in the single spin-boson model. On reducing the spin separation from infinity, αc → 0.5. Interestingly, αc reaches this minimum value at a finite spin separation, and

7.6 Summary

172

retains this value for all smaller separations. We also obtained the variational ground state, and from this showed that a signature of the ground state crossover could be found in the emergence of a finite single-spin expectation value �σx � as either the spin separation or the ratio of tunnelling

strength to Ising interaction is increased. The crossover should also be evident in the entanglement shared between the system and bath.

Appendix A Strong coupling correlation functions Here we show how to calculate the correlation functions Λxx (τ ) and Λyy (τ ) which naturally arise when a master equation is derived from a polaron transformed spin-boson model. The method used here follows that presented in Ref. [12]. The bath operators are given by the hermitian combinations 1 Bx (τ ) = (B+ (τ ) + B− (τ )) 2 1 By (τ ) = (B− (τ ) − B+ (τ )) 2i where B± (τ ) =

�

k

(A.1) (A.2)

exp[±2(αk b†k (τ ) − αk∗ bk (τ ))] with αk = gk /ωk while

b†k (τ ) = b†k eiωk τ and bk (τ ) = bk e−iωk τ . Thus

1 Λxx (τ ) = �Bx (τ )Bx (0)� = (C(τ ) + G(τ )) 2 1 Λyy (τ ) = �By (τ )By (0)� = (C(τ ) − G(τ )) 2

(A.3) (A.4)

where we have anticipated C(τ ) = �B+ (τ )B− (0)� = �B− (τ )B+ (0)� and G(τ ) = �B+ (τ )B+ (0)� = �B− (τ )B− (0)�

Focus will be put upon evaluating C(τ ) since the procedure used to

obtain G(τ ) is almost identical. Independence of the bath modes allows

174 us to write C(τ ) =

� k

Ck (τ ) =

� k

�exp[2(−αk b†k (τ ) + αk∗ bk (τ ))]exp[2(αk b†k − αk∗ bk )]� (A.5)

The angular brackets refer to an expectation value with respect to the bath. Assuming the bath to be in thermal equilibrium then we can express this by writing ∞ 1� Ck (τ ) = �n| e−βnωk exp[2(−αk b†k (τ ) + αk∗ bk (τ ))] Z n=0 k

×exp[2(αk b†k − αk∗ bk )] |n�k

(A.6)

where Z = (1−e−βω ) is the partition function of the bath and |n�k labels a

state of the bath with n excitations in mode k. From this point onwards we shall omit the k subscript with the understanding that we shall concentrate on evaluating a single Ck (τ ), taking the product once we have finished.

To proceed we would like to move all of the annihilation operators in equation (A.6) to the right leaving all the creation operators on the left. Recalling that for any two operators A and B whose commutator is a c-number we can write 1

eA+B = eA eB e− 2 [A,B]

(A.7)

which allows Eq. (A.6) to be put in the following form ∞ 1� 2 † ∗ † ∗ C(τ ) = �n| e−βnω e−4|α| e−2αb (τ ) e2α b(τ ) e2αb e−2α b |n� . Z n=0

(A.8)

We would now like to interchange the fourth and fifth exponentials. To do so we write e2α

∗ b(τ )

†

†

†

e2αb = e2αb (e−2αb e2α

∗ b(τ )

†

e2αb ).

(A.9)

The last three exponentials in the above expression now look like a canonical transformation. Using †

†

e−2αb b(τ )e2αb = e−iωτ (b + 2α)

(A.10)

175 allows Eq. (A.9) to be written e2α

∗ b(τ )

†

e2αb = e4|α|

2 exp[−iωτ ]

†

e2αb e2α

∗ b(τ )

(A.11)

which completes the reordering of the creation and annihilation operators and we have C(τ ) =

∞ 1� 2 −iωτ ) −2αb† (τ ) 2αb† 2α∗ b(τ ) −2α∗ b �n| e−βnω e−4|α| (1−e e e e e |n� (A.12) Z n=0

∞ 1� 2 −iωτ ) u∗ b† −ub = �n| e−βnω e−4|α| (1−e e e |n� Z n=0

with u = 2α∗ (1 − e−iωτ ).

We now consider what eﬀect the operators eu

∗ b†

(A.13)

and e−ub have on the

bath states |n�. The annihilation operators remove an excitation from a state and, with the correct normalisation condition satisfy bl |n� =

�

n! �1/2 |n − l� (n − l)!

(A.14)

and hence we have e

−ub

|n� =

n � (−u)l � l=0

l!

n! �1/2 |n − l� (n − l)!

(A.15)

n! �1/2 �n − l| (n − l)!

(A.16)

and similarly, �n| eu

∗ b†

=

n � (u∗ )l � l=0

l!

where the summations have been terminated since the annihilation operator acting on the vacuum state |0� gives zero. Putting these results together we find

�n| e

u∗ b† −ub

e

|n� =

n � (−|u|2 )l l=o

(l!)2

= Ln (|u|2 )

n! (n − l)!

(A.17) (A.18)

176 where Ln is the nth Laguerre polynomial [190]. With the result ∞ � n=0

Ln (|u|2 )e−βωn = (1 − eβω )−1 e−n(ω)|u|

2

(A.19)

we arrive, after some algebra at � � �� Ck (τ ) = exp −4|αk |2 isin(ωk τ ) + (1 − cos(ωk τ ))coth(βωk /2) (A.20)

which in turn gives �

C(τ ) = exp −4

� k

2

��

�

|αk | +isin(ωk τ ) + (1 − cos(ωk τ ))coth(βωk /2) . (A.21)

Note that Eq. (A.21) justifies the assumption C(τ ) = �B+ (τ )B− (0)� =

�B− (τ )B+ (0)� since C(τ ) is invariant under αk → −αk . Using similar methods it is possible to show �

G(τ ) = exp −4

� k

2

��

�

|αk | −isin(ωk τ ) + (1 + cos(ωk τ ))coth(βωk /2) . (A.22)

With a similar but less involved procedure the expectation value of the bath operators is found to be � � � |gk |2 B = �B± (τ )� = exp − 2 coth(βω /2) k ωk2 k where as before B± (τ ) =

�

k

(A.23)

exp[±2(αk b†k (τ )−αk∗ bk (τ ))] with αk = gk /ωk .

We note that Eq. (A.23) is valid for αk with arbitrary magnitude and phase. In section 2.3.1 the independent boson model was discussed and the expectation value �

� k

D(±2uk )� = �

� k

exp[±2(uk b†k − u∗k bk ]�

(A.24)

with αk (1−eiωk t ) was required. From Eq. (A.23) we can immediately write down the result �

� k

�

D(±2uk )� = exp −4

� |gk |2 k

ωk2

�

(1 − cos(ωk t)) coth(βωk /2) . (A.25)

Appendix B Summations in k-space Here we give details of how to perform summations over k in multiple dimensions by using the system-bath spectral density. As motivation we compute the correlation dependent rates which were first encountered in chapter 3. The key quantity to consider is the cross-correlation function ˜1 (τ )B ˜2 (0)� Λ12 (τ ) = �B

(B.1)

˜n (τ ) = � (g n b† eiωk τ + g n∗ bk e−iωk τ ) and the spatial separation where B k k k k

between the spins is made explicit by letting gk1 = gk e(ikdcosθ)/2 and gk2 = gk e−(ikdcosθ)/2 where θ is the polar angle measured against the z-axis in ˜n (s) and using usual methods we find k-space. Inserting the form of B Λ12 (τ ) =

� k

� � |gk |2 n(ωk )eiωk τ eikdcosθ + (n(ωk ) + 1)e−iωk τ e−ikdcosθ . (B.2)

We would now like to convert the summation over k in Eq. (B.2) into � an integral. Usually we would simply make the replacement k |gk |2 → �∞ J(ω)dω. However, Eq. (B.2) contains an angular dependence which 0

must be integrated out first. To see how to do this consistently for systembath coupling in one, two and three dimensions, it is helpful to consider how a summation over k might be done without prior knowledge of the specific form of the system-bath spectral density. For system bath coupling in one dimension, taking the continuum limit we could write � k

L |gk | → 2π 2

�

L dk|gk | = 2 2π 2

�

∞ 0

κ� (ω)|g(ω)|2 dω

(B.3)

178 where |k| = κ(ω) is the inverse dispersion relation, κ� (ω) is its derivative

with respect to ω and L is the sample length. The factor of two appears in the second line because we must sum over positive and negative k = |k|. For system-bath coupling in two dimensions we write � k

A |gk | → (2π)2 2

�

� 2π � ∞ A dk|gk | = dθ κ(ω)κ� (ω)|g(ω)|2 dω (2π)2 0 0 � ∞ A = κ(ω)κ� (ω)|g(ω)|2 dω (B.4) 2π 0 2

where A is the area of the sample. With three dimensional coupling to a sample with volume V we have � k

V |gk | → (2π)3 2

�

dk|gk |2

� 2π � 1 � ∞ V = dφ d(cosθ) κ(ω)2 κ� (ω)|g(ω)|2 dω (2π)3 0 −1 0 � ∞ V = 2 κ(ω)2 κ� (ω)|g(ω)|2 dω (B.5) 2π 0

where φ is the azimuthal angle in k-space. With these relations in mind we can see that the spectral density is related to the system-bath coupling strength and the details of the sample with which our spin is interacting through L � κ (ω)|g(ω)|2 π A J 2D (ω) = κ(ω)κ� (ω)|g(ω)|2 2π V 3D J (ω) = 2 κ(ω)2 κ� (ω)|g(ω)|2 2π J 1D (ω) =

(B.6) (B.7) (B.8)

for coupling in one, two and three dimensions respectively. We are now in a better position to evaluate the correlation function Λ12 (τ ). In one dimension θ must equal zero and we have Λ1D 12 (τ )

L = π � =

�

0

0 ∞

∞

2 �

�

|g(ω)| κ (ω)dωcos(κ(ω)d) n(ω)e �

J(ω)dωcos(κ(ω)d) n(ω)e

iωτ

iωτ

+ (n(ω) + 1)e

+ (n(ω) + 1)e

−iωτ

�

−iωτ

�

(B.9)

179 In two dimensions we have Λ2D 12 (τ )

� 2π � ∞ A = dθ |g(ω)|2 κ(ω)κ� (ω)dω (2π)2 0 0 � � iωτ iκ(ω)dcosθ × n(ω)e e + (n(ω) + 1)e−iωτ e−iκ(ω)dcosθ � ∞ � � = J(ω)dωJ0 (κ(ω)d) n(ω)eiωτ + (n(ω) + 1)e−iωτ

(B.10)

0

since

� 2π 0

dθe±iacosθ = 2πJ0 (a) where J0 is a Bessel function of the first

kind of order 1. Lastly, for coupling in three dimensions we have Λ3D 12 (τ )

� 2π � 1 � ∞ V = dφ d(cosθ) |g(ω)|2 κ(ω)2 κ� (ω)dω (2π)3 0 −1 0 � � iωτ iκ(ω)dcosθ × n(ω)e e + (n(ω) + 1)e−iωτ e−iκ(ω)dcosθ � ∞ � � = J(ω)dωsinc(κ(ω)d) n(ω)eiωτ + (n(ω) + 1)e−iωτ .

(B.11)

0

Note that the position dependence has manifested itself in the form of an addition function in the integrand. In general we can write Λ12 (τ ) =

�

∞ 0

� � J(ω)dωFD (κ(ω)d) n(ω)eiωτ + (n(ω) + 1)e−iωτ .

(B.12)

where the function FD (κ(ω)d) captures the level of correlation and is dependent on the dimensionality of the system-bath interaction.

Appendix C High temperature rates Here we show how to obtain analytic approximations for the decoherence rates at high temperatures by use of a saddle point integration. Within our formalism there are two rates which need to be evaluated: � B 2 +∞ γxx (η) = dτ eiτ η (eφ(τ ) + e−φ(τ ) − 2), 2 −∞ � B 2 +∞ γyy (η) = dτ eiτ η (eφ(τ ) − e−φ(τ ) ), 2 −∞ where φ(τ ) is given by Eq. (5.25).

(C.1) (C.2)

With the appropriate manipula-

tions [109] it is possible to show that � B 2 βη/2 +∞ ˜ ˜ γxx (η) = e dτ eiτ η (eφ(τ ) + e−φ(τ ) − 2), 2 −∞ � +∞ 2 B βη/2 ˜ ˜ γyy (η) = e dτ eiτ η (eφ(τ ) − e−φ(τ ) ), 2 −∞

(C.3) (C.4)

˜ ) = φ(−τ ˜ where now φ(τ ) = φ(τ − iβ/2), and is given explicitly in integral

form by

˜ )=2 φ(τ

�

∞ 0

dω

J(ω) cos(ωτ ) (1 − F (ω, d)) . 2 ω sinh(βω/2)

(C.5)

Using a super-Ohmic form of spectral density, J(ω) = αω 3 ω0−2 e−ω/ωc , and assuming system-bath coupling in three dimensions such that F (ω, d) = ˜ ) to be found analytically. We find φ(τ ˜ ) = sinc(ωd/c), allows φ(τ

181 φ0 C(x, y, τ � ), where � � � � � −i 1 1 i � 1 1 i � C(x, y, τ ) = ψ + − (τ + x) − ψ + − (τ − x) 2πx 2 y π 2 y π � � � �� 1 1 i � 1 1 i � +ψ + + (τ − x) − ψ + + (τ + x) 2 y π 2 y π � � � � �� 1 1 1 i � 1 1 i � � � + 2 ψ + − τ −ψ + + τ . (C.6) π 2 y π 2 y π �

Here, φ0 = 2π 2 α/(ω02 β 2 ), x = πd/cβ, y = ωc β, τ � = πτ /β, ψ(z) is the digamma function, and ψ � (z) its first derivative. To proceed, we assume a high-temperature or strong-coupling regime, such that the dominant contributions to the integrals in Eqs. (C.3) and ˜ ) at τ = 0. More specifically, inspec(C.4) will come from the peak in φ(τ tion of C(x, y, τ � ) reveals that for y � 1 (the scaling limit of large ωc ), we

require φ0 � 1 for large x (weak correlations), or φ0 x2 � 1 for small x ˜ ) around τ = 0 to (strong correlations), in order for an expansion of φ(τ be valid. These definitions of the high-temperature (or strong-coupling) regime tally with the expansion parameters identified in Ref. [121]. In the opposite limit, y � 1, we generally need φ0 x2 y 3 /π 4 � 1, except in

the limit of very large separations (vanishing correlations), x → ∞, where φ0 y � 1 is the relevant condition.

˜ ) to second With these conditions in mind, we therefore expand φ(τ

order in τ � = πτ /β, which gives ˜ ) ≈ φ0 (C0 (x, y) − τ �2 C2 (x, y)), φ(τ

(C.7)

where C0 (x, y) =

� 1 1 ix �� i � � 1 1 ix � 2 �1 1� ψ + + −ψ + − + 2 ψ� + , πx 2 y π 2 y π π 2 y (C.8)

and � i � �� 1 1 ix � 1 ix �� 1 �� 1 1 � �� 1 C2 (x, y) = 3 ψ + + −ψ + − + 4ψ + . 2π x 2 y π 2 y π π 2 y (C.9) ˜ )] in Eqs. (C.3) and (C.4) In this limit terms containing a factor of exp[φ(τ

182 dominate which allows us to write B 2 eβη/2 β φ0 C0 γll (η) ≈ e 2π

�

+∞

dτ � eiτ

� ηβ/π

e−τ

�2 φ

0 C2

.

(C.10)

−∞

The integral is Gaussian and we arrive at the result βB 2 eφ0 C0 (x,y) βη/2 −β 2 η2 /(4π2 C2 (x,y)φ0 ) γll (η) = � e e , 2 πC2 (x, y)φ0

(C.11)

which reduces to

βB 2 eφ0 C0 (x,y) γll (η) ≈ γll (0) = � , 2 πC2 (x, y)φ0

(C.12)

if the temperature is high enough such that 1/ηβ � 1.

It remains now to determine the bath renormalisation factor B, given

by Eq. (6.8). To do so it is helpful to separate vacuum and thermal contributions. We write B = B0 Bth , where �

B0 = exp −

�

∞ 0

� J(ω) dω 2 (1 − FD (ω, d)) , ω

(C.13)

and �

Bth = exp −

�

∞ 0

� J(ω) dω 2 (1 − FD (ω, d))(coth(βω/2) − 1) . ω

(C.14)

Inserting the spectral density, and again assuming system-bath coupling in three dimensions, we find �

� �� ωc2 (dωc /c)2 B0 = exp − α 2 , ω0 1 + (dωc /c)2

(C.15)

and �

� �� φ0 iπ � −1 −1 � −1 Bth = exp H(y − ix/π) − H(y + ix/π) − 2ψ (1 + y ) , 2π 2 x (C.16) where H(m) =

�m

i=1 (1/i)

is the mth harmonic number. We note that

in the infinite separation (uncorrelated) limit, one finds B0 (d → ∞) =

183 exp[−α(ωc2 /ω02 )], and Bth = exp[α(ωc2 /ω02 )(1 − y −2 (ζ(2, 1 + y −1 ) + ζ(2, y −1 )))], where ζ(s, a) is the generalised Riemann zeta function.

(C.17)

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