Annals of Physics 318 (2005) 308–315 www.elsevier.com/locate/aop

Entanglement dynamics for two interacting spins Marcel Novaes* Instituto de Fı´sica ‘‘Gleb Wataghin’’, Universidade Estadual de Campinas, 13083-970 Campinas-SP, Brazil Received 26 October 2004; accepted 17 January 2005 Available online 14 March 2005

Abstract We study the dynamical generation of entanglement for a very simple system: a pair of interacting spins, s1 and s2, in a constant magnetic field. Two different situations are considered: (a) s1 fi 1, s2 = 1/2 and (b) s1 = s2 fi 1, corresponding, respectively, to a quantum degree of freedom coupled to a semiclassical one (a qubit in contact with an environment) and a fully semiclassical system, which in this case displays chaotic behavior. Relations between quantum entanglement and classical dynamics are investigated.  2005 Elsevier Inc. All rights reserved. PACS: 03.67; 05.45.Mt; 75.10.Jm Keywords: Entanglement; Chaos; Decoherence

Recent technological advances have made it possible to create and manipulate individual quantum states in the laboratory, thus allowing direct observations of entanglement and decoherence [1–8], concepts that are central to quantum computation and quantum information [9,10]. Given a system that consists of two subsystems S1 and S2, a pure state |wæ is said to be an entangled state if it is not possible to write it as a product of individual states belonging to the subsystems jwi 6¼ j/i1  j/i2 : *

Fax: +551933885496. E-mail address: [email protected]fi.unicamp.br.

0003-4916/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2005.01.003

ð1Þ

M. Novaes / Annals of Physics 318 (2005) 308–315

309

In this case one cannot ascribe a definite quantum state to any of the subsystems and if they are to be described individually, it must be through a density matrix. But if q2, the density matrix of S2, is different from a projector, q2 „ |/æÆ/|, then it is different from its square, q22 6¼ q2 , and since Tr q2 = 1 it happens that Tr q22 6¼ 1. This opens a possiblity for quantifying entanglement, which in general is a difficult task, by defining an easy-to-handle measure: the linear entropy (or idempotency defect) d¼

d ð1  Tr q22 Þ: d 1

ð2Þ

In this equation d is the dimension of the Hilbert space of S2. A similar definition can be made for S1 and in fact they are exactly the same. When one considers S2 to be initially in a pure state (as we do here), then how fast its entropy grows indicates how fast it suffers decoherence due to entanglement with S1. Decoherence is believed to be responsible for the suppression of quantum effects at macroscopic scale. Since no physical system can be perfectly isolated, residual interaction with the environment is always present. This interaction induces decoherence upon the system, selecting the quantum states that may be realized in a real experiment and forbidding macroscopic superpositions [11]. Such superpositions are precisely the advantage of an eventual quantum computer over the classical ones, and therefore the question of understanding decoherence and how to avoid it is a very important one. Another interesting point concerns the dependence of quantum entanglement upon the classical dynamics of the system, more specifically what is the role of chaos in the decoherence process. This has been the subject of many recent investigations. In a seminal work Furuya et al. [12] have studied the Jaynes–Cummings model without the rotating-wave approximation, and found the entanglement rate to be greater for chaotic initial conditions. In later works [13,14] they have shown that recoherences were related to the shape of the spin orbits, and therefore were related to the compactness of the spin phase space. In the past few years much work has been done in this area, considering for instance continuous variables [15,16], tops [17–20], and spin chains [21–23]. In the present work, we wish to present a very simple system in which one can study entanglement dynamics. First, we analyze the interaction of a qubit with a semiclassical environment and study the dependence of entanglement with the initial conditions. Very different qualitative behaviors are seen to be possible, indicating that care must be taken when trying to generalize numerical results. Then we study the interaction of two semiclassical systems, and investigate the role of chaos and of the geometry of phase space. We consider two spins, S1 and S2, in a constant magnetic field ~ B ¼ B0^z, interacting according to the Hamiltonian H ¼ 1 B0 S z1 þ 2 B0 S z2 þ aS x1 S x2 ;

ð3Þ

which is somewhat reminiscent of the usual XXZ Hamiltonian widely used in statistical mechanics. Spin Hamiltonians have been considered in connection to quantum computation proposals using magnetic resonance [24,25] (see also [10]) and cold

310

M. Novaes / Annals of Physics 318 (2005) 308–315

atoms in optical lattices [26–28]. Entanglement in spin chains at finite temperature was considered for example in [29,30]. For simplicity we shall assume 1B0 = 2B0 = 1 and choose a = 2.3 (there is nothing special about this value of a; it was chosen to give a nice picture of the phase space, see Fig. 4) . Even this very simple case already shows a rich behavior depending on the initial conditions and the spin magnitudes s1 and s2. To simulate the interaction of a quantum system with a semiclassical one, let us use s2 = 1/2 and s1 = 200 (we take  h = 1). In quantum information theory a two-level system is often called a ‘‘qubit.’’ Such a huge value for spin S1 is not intended to be realistic, but only to simulate an environment for dissipation, when infinitely many degrees of freedom are required. The state of the environment has an important role in the decoherence process. It has been said that if its Wigner function has subPlanck scale structure, as happens for chaotic systems, its ability to induce decoherence is enhanced [33]. We consider spin s1 to be, at t = 0, in three different conditions: in a coherent state [31,32] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi s1 X 2s1 zs11 þm1 ð4Þ jm1 i; jz1 i ¼ s1 þ m1 ð1 þ jz1 j2 Þs1 m1 ¼s1 in a non-localized state jwi ¼ ð2s1 þ 1Þ

1=2

s1 X

jm1 i

ð5Þ

m1 ¼s1

and in a thermal mixture, in which its density matrix is q1 ¼

s1 1 X em1 =T jm1 ihm1 j; N m1 ¼s1

ð6Þ

where N is the normalization factor. Any coherent state |z1æ can be obtained from the lowest state (m1 = s1 or spin ‘‘pointing down’’) by the rotation |z1æ = exp{ih (sin / Jx  cos /Jy)}|s1æ. The relation between the variable z1 and the usual spherical coordinates (h, /) is z1 = ei/ tan(h/2), and thus one obtains a spin ‘‘pointing up’’ in the limit z1 fi 1, and eigenstates of Jx for z1 = ±1. The numerical simulations are not difficult to carry out for the system we are considering. Since the pertinent Hilbert space is always of finite dimension, a direct diagonalization of the Hamiltonian is possible with great accuracy. The partial traces involved in the calculation of the linear entropy (2) are simple to obtain and accurate for any instant of time. We take at first both spins to be initially in coherent states with z1 = z2 = 0, which means they are both ‘‘pointing down.’’ The linear entropy is shown in Fig. 1. We see that even though the environment is initially in a very simple state, the entanglement reaches its maximum value very fast and remains saturated for some time. However, strong and localized recoherences take place in a nearly periodic way. That the spin S2 is very close to having a definite quantum state at these instants can be

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Fig. 1. (s1 = 200, s2 = 1/2) Linear entropy (solid line) and angular momentum (dashed line) as functions of time for z1 = z2 = 0. We see that d saturates very fast, but strong recoherences appear, related to hS 2z i.

appreciated from the figure: the recoherences occur when hS z2 i ’ 1=2, which means it is almost pointing ‘‘down’’ again. In Fig. 2A we see the results for z1 = 0, z2 = 1, which corresponds to the environment pointing ‘‘down’’ and the qubit pointing ‘‘left.’’ Even though the state of the environment is the same as in the previous case, the entanglement is completely different. Entanglement increases with time, but much slower than before, and instead of raising immediately to its maximum value it increases by jumps, alternating fast increments with plateaus. We also consider z1 = z2 = 1, shown in Fig. 2B. The behavior of the entropy is quite similar to Fig. 2A, but in a very different scale: entanglement is nearly a hundred times smaller in this case, i.e., the system is practically entanglement-free. It is clear that the entanglement rate strongly depends on the initial state of the environment, even if this is a coherent state. The time evolution of the entanglement can again be related to the average value of S z2 , but the relation is not as direct as it was in Fig. 1. We see from the dashed lines in Figs. 2A and B that hS z2 i behaves roughly as a multiple of the derivative of d (t).

Fig. 2. (s1 = 200, s2 = 1/2) Linear entropy for z1 = 0,z2 = 1 (A) and for z1 = z2 = 1 (B). In both cases we see plateaus followed by jumps, but there is a large difference in magnitude. We also plot a re-scaled average value of S z2 .

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M. Novaes / Annals of Physics 318 (2005) 308–315

That means the entanglement rate dd/dt is greater for large absolute values of the spin than for small ones. The plateaus therefore correspond to periods of time during which hS z2 i remains practically constant. Finally, we have considered the environment to be initially in the non-localized state (5) and in the thermal mixture (6). In all cases we have used z2 = 1, and in Fig. 3 we show the evolution of the linear entropy (the solid line is the same as in Fig. 2A). The qualitative behavior is the same for all initial states (in the thermal mixture we have used T = s1/10), but somewhat surprisingly the ability to induce decoherence is considerably greater for the coherent state. This shows that the environment does not need to have much structure, nor a high temperature, to be a powerful source of decoherence. We now consider both spins to be large (semiclassical), namely we use s1 = s2 = 15. For any spin Hamiltonian it is possible to obtain a semiclassical dynamics using su (2) coherent states (4), in which the quantum dynamics is then approximated in the semiclassical limit  h fi 0 by the Hamilton equations of motion [31] 2 2

z_ ¼ i

ð1 þ jzj Þ oH ; oz 2sh

ð7Þ

where H ¼ hzjH jzi plays the part of a classical Hamiltonian. In our case this is given by [34] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a ð8Þ H ¼ ðA1 þ A2  31Þ þ q1 q2 ð4s1  A1 Þð4s2  A2 Þ; 2 4 where Ai ¼ q2i þ p2i

ð9Þ

and the canonical coordinates are defined by qj þ ipj zj pffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 4sj 1 þ jz j

j ¼ 1; 2:

ð10Þ

j

Fig. 3. (s1 = 200, s2 = 1/2) Linear entropy evolution, for the environment initially in a coherent state (solid line), a thermal mixture (dashed line), and a non-localized state (dotted line).

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It is important to note that the spin phase space, unlike the more usual one associated with particles, is compact. Each pair of canonical coordinates is restricted to the interior of a disk q2j þ p2j 6 4sj ;

j ¼ 1; 2:

ð11Þ

This has important consequences for the entanglement process, as already observed in [13]. In Fig. 4, we see a Poincare´ section showing the (q1, p1) coordinates at p2 = 0. We use the marked points as initial conditions for our coherent states to be evolved. The trajectory marked with a triangle is chaotic, the one marked with a square is regular and the circle denotes a periodic orbit. Note that we do not evolve the state by using Eq. (7), because that would preserve coherence, and thus produce no entanglement. Instead we consider the exact quantum evolution and use Fig. 4 as a guide to intuition. In Fig. 5, we see the linear entropy as a function of time. The rate of entanglement for short times is very similar for all initial conditions, contrary to what is observed for example in the Jaynes–Cummings system [12]. We also note that the maximal value of d is larger for the periodic orbit than for the more generic regular trajectory. Numerical analysis indicate that both the short-time entanglement rate and the maximum entanglement value are related not to the classical dynamics but to strictly quantum properties of the wave packet, such as energy uncertainty. Further work is clearly necessary in order to determine what is precisely the role of the classical dynamics. There is however a strong difference between the regular and chaotic initial conditions: the former present strong recoherences that are not present in the latter. As we see in Fig. 6 these recoherences are again related to the mean value of the z-component of the spin, but in this case the relevant spin is S1. The minima of the functions coincide, and at these points hS z1 i returns to its initial value. The maximum value of d, on the other hand, is associated with r1  0.5, which corresponds to

Fig. 4. (s1 = s2 = 15) Poincare´ section of the classical Hamiltonian. We use the marked points as initial conditions for our coherent states to be evolved.

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M. Novaes / Annals of Physics 318 (2005) 308–315

Fig. 5. (s1 = s2 = 15) Linear entropy d (t) for three different initial conditions. The regular cases present strong recoherences, while these are suppressed in the chaotic case.

Fig. 6. (s1 = s2 = 15) Linear entropy and angular momentum r1 ¼ hS z1 i þ s1 =2s1 for the periodic initial condition. Recoherences are related to the compactness of phase space.

hS z1 i  0, and therefore once again we conclude that the entanglement behavior is in close relation with the classical dynamics. Note that coherent states have localized Husimi distributions [35] and thus we may expect that more general states would be less likely to display such recoherences. Summarizing, we have used a very simple system to study several interesting properties of entanglement dynamics. We have seen for example that the decohering power of the environment strongly depends on its initial state. In particular, we saw that for some initial states the entanglement increases by jumps, and also that a coherent state may induce more decoherence than a thermal mixture. In the case of two semiclassical spins we have analyzed the influence of chaos upon the entanglement process. We saw that the short-time behavior of the linear entropy was insensitive to the nature of the classical dynamics, but rather determined by strictly quantum properties of the wave packet. On the other hand, chaotic initial conditions presented no recoherences, while regular ones did, and this was related to the

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classical motion in spin phase space. We conclude that not only the time evolution of quantum entanglement is highly non-trivial, but also its relation with classical dynamics. Both need further studies, and we believe that simple model systems such as the one proposed here may prove to be useful.

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Entanglement dynamics for two interacting spins

fully semiclassical system, which in this case displays chaotic behavior. Relations .... degrees of freedom are required. The state .... [10] The Quantum Information Science and Technology Roadmap can be found at http://qist.lanl.gov. [11] W.H. ...

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