Mechanical squeezing by light modulation A. Mari 1 and J. Eisert 1,2 1Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam, Germany 2Institute for Advanced Study Berlin, D-14193 Berlin, Germany

We introduce a framework of optomechanical systems that are driven with a mildly amplitude-modulated light field, but that are not subject to classical feedback or squeezed input light. We find that in such a system one can achieve large degrees of squeezing of a mechanical micromirror signifying quantum properties within parameters reasonable in experimental settings. Entanglement dynamics is shown of states following classical quasiperiodic orbits in their first moments. We discuss the complex time dependence of the modes of a cavity-light field and a mechanical mode in phase space. Such settings give rise to certifiable quantum properties within experimental conditions feasible with present technology.

How to squeeze opto-mechanical systems? • QND measurement + feedback [1]

Asymptotic solution

Mechanical squeezing

• We define a vector containing the mechanical and optical quadrature operators:

A weak sinusoidal modulation of the input amplitude with a frequency Ω = 2ωm causes the emergence of significant genuine quantum squeezing of the mechanical oscillator quantum state.

u(t) = [q(t), p(t), x(t), y(t)], √ √ † † where x = (a + a )/ 2 and y = (a − a )/i 2. If ha†ai ≫ 1, the light-mirror quantum state converges to a Gaussian state.

• Squeezing of the optical bath [2]

Position uncertainty

<δq2>

Heisenberg limit

• Displacement vector d(t) (1st moments)

tτ

di(t) = hui(t)i

Wigner function

• Correlation matrix V (t) (2nd moments) hui(t)uj (t) + uj (t)ui(t)i Vi,j (t) = − di(t)dj (t) 2

We propose a different solution • Modulation of the input light [3]

q

p

q

p

p q

Experimental parameters ωm = 2πMHz, γm = 10−6ωm , ∆0 = ωm , κ = 0.214ωm , m = 150ng, L = 25mm, λ = 1064nm, T = 0.1 K, P0 = 10mW, P±1 = 2mW, Ω = 2ωm . ⇐= modulation

Displacement vector • Asymptotic expansion of the displacement vector The simple application of an amplitude modulator to the well known self-cooling setup [4] can produce significant mechanical squeezing and opto-mechanical entanglement oscillations.

d(t) =

Opto-mechanical entanglement

j dj,neinΩtG0

• The same setup without any modulation has been shown to generate quantum correlations between the cavity mode and the mechanical mode [5, 6].

j=0 n=−∞

• Analytical solution

• These correlations can be quantified by the entanglement negativity [7], defined as

inΩ pn,0 = 0 pn,j = qn,j , ωm j−1 X ∞ X qn,0 = 0 qn,j = ωm

a∗m,k an+m,j−k−1 , 2 − nΩ2 + iγ nΩ ω m m=−∞ m

Mathematical background • Given a linear ODE with time dependent coefficients

∞ ∞ X X

an,0 =

E−n κ + i(∆0 + nΩ)

an,j = i

k=0 j−1 X ∞ X k=0

am,k qn−m,j−k−1 . κ + i(∆0 + nΩ) m=−∞

EN (t) = log kρP T (t)k1,

P T = partial transposition

x(t) ˙ = B(t)x(t) + b(t) • Phase space orbits

the solution can be written as x(t) = P (t, t0)x(t0) +

Z t

Opto-mechanical entanglement can be periodically enhanced by driving with amplitude modulated light. (a)

P (t, s)b(s)ds.

(b)

t0

• Floquet theorem If B(t + τ ) = B(t) and b(t + τ ) = b(t) then, P (t, t0) = X(t, t0)e(t−t0)Y (t0), X(t + τ, t0) = X(t, t0).

MECHANICAL MODE

OPTICAL MODE

The eigenvalues λi of Y (t0) are called Floquet exponents.

Correlation matrix • Asymptotically periodic solutions If λ = maxi{re λi} < 0, then kx(t + τ ) − x(t)k = eλ(t−t0)P oly(t − t0) −→ 0

• If hai ≫ 1 we can linearize the Langevin equations around the time dependent first moments [4]. • The correlation matrix V (t) is the solution of the following linear differential equation,

[1] Clerk, Marquardt, Jacobs, New J. Phys. 10 (2008).

V˙ (t) = A(t)V (t) + V (t)AT (t) + D

Light-mirror dynamics

[3] Mari, Eisert, Phys. Rev. Lett. 103 (2009). 

0 ωm 0 0   −γ G (t) G (t) −ω   m m x y A(t) =   −κ ∆(t)   −Gy (t) 0 0 −∆(t) −κ Gx(t) n + 1), κ, κ). D = diag(0, γm(2¯

q˙ = ωmp p˙ = −ωmq − γmp + G0a†a + ξ √ P+∞ −inΩt + 2κain a˙ = −(κ + i∆0)a + iG0aq + n=−∞ Ene 2π ∆0 = ωc − ωL, Ω= , τ where τ is the period of modulation. • Correlation functions of noise operators: n + 1)δ(t − t′), hξ(t)ξ(t′) + ξ(t′)ξ(t)i/2 ≃ γm(2¯ hain(t)ain†(t′)i ≃ δ(t − t′),

[2] Jaehne, Genes, Hammerer, Wallquist, Polzik, Zoller, Phys. Rev. A 79 (2009).

where 

• Quantum Langevin equations

hain†(t)ain(t′)i ≃ 0.

References

The real matrix A(t) contains the time-modulated coupling constants and the detuning: Gx(t) = hx(t)iG0, Gy (t) = hy(t)iG0, ∆(t) = ∆0 − G0hq(t)i. • Asymptotic expansion of the correlation matrix A(t + τ ) = A(t) =⇒ V (t) = V (t + τ ) V (t) =

∞ X

VneinΩt

n=−∞

• Each component Vn can be calculated with a perturbation theory in frequency space.

[4] Genes, Vitali, Tombesi, Gigan, Aspelmeyer, Phys. Rev. A 77 (2008), Phys. Rev. A 79 (2009). [5] Vitali, Gigan, Ferreira, Boehm, Tombesi, Guerreiro, Vedral, Zeilinger, Aspelmeyer, Phys. Rev. Lett. 98 (2007). [6] Paternostro, Vitali, Gigan, Kim, Brukner, Eisert, Aspelmeyer Phys. Rev. Lett. 98 (2007). [7] Eisert, PhD thesis (Potsdam, February 2001); Vidal, Werner, Phys. Rev. A 65 (2002); M.B. Plenio, Phys. Rev. Lett. 95 (2005).