Mechanical Squeezing in Quadratically-coupled Optomechanical Systems Priyankar Banerjee Sampreet Kalita and Amarendra K. Sarma Department of Physics Indian Institute of Technology Guwahati Guwahati-781039 India

2025-05-02 0 0 900.67KB 9 页 10玖币

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Mechanical Squeezing in Quadratically-coupled Optomechanical Systems

Priyankar Banerjee, Sampreet Kalita, and Amarendra K. Sarma∗

Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India

We demonstrate the generation of a strong mechanical squeezing in a dissipative optomechanical

system by introducing a periodic modulation in the amplitude of a single-tone laser driving the

system. The mechanical oscillator is quadratically coupled to the optical mode, which contributes

to a strong squeezing exceeding the 3-dB standard quantum limit. The Bogoliubov mode of the

mechanical oscillator also cools down to its ground state due to sideband cooling. We further

optimize this ratio of sideband strengths to introduce enhanced squeezing. We also compare our

results with the analytical (under adiabatic approximation) and the exact numerical solution.

Even for a thermal occupancy of 104phonons, mechanical squeezing beyond 3 dB and a strong

optomechanical entanglement is observed.

I. INTRODUCTION

In recent years, the amount of research on macroscopic

quantum systems for the generation of nonclassical states

such as squeezed and entangled states has increased

manifold [1–6]. Such states have numerous applications

in the development of quantum technologies for quantum

computation [7], quantum information processing [8, 9],

and quantum-enhanced force sensing [10, 11]. In the past

decades, micro-mechanical systems relying on position

measurements have been extensively used to sense,

store and process information. However, the precision

of position-measurements of a mechanical oscillator is

limited by its zero-point ﬂuctuations [12]. In addition

to that, measuring the mechanical position may itself

induce noise into the system. In fact, most position

measuring systems have noise contributions that exceed

the standard quantum limit (SQL) of 3 dB [13]. To

circumvent this, the motion of the mechanical oscillator

can be squeezed beyond the SQL, by reducing the

variance in one of its quadratures at the expense of

increased variance in the other quadrature [14]. Also,

the preparation and control of a mechanical element in

a quantum state of motion at a mesoscopic level allows

us to test quantum mechanics’ fundamental hypotheses

at the quantum-classical boundary [15]. A suitable

platform to generate such quantum states is a cavity

optomechanical system [16]. Such a system relies on

the radiation-pressure interaction of a mechanical system

with the electromagnetic radiation inside a cavity. The

high sensitivity of the phase of the cavity photons to

small displacements of the mechanical oscillator makes

them an ideal candidate for force-sensing applications

and are routinely used in interferometric detectors such

as the ones in LIGO [17], and VIRGO [18] experiments.

Recently, many optomechanical models have been

proposed to induce squeezing below the 3 dB level by

using squeezed light [19, 20], two-tone driving [4, 13, 21],

reservoir engineering [22–25] and frequency modulation

[5]. It has also been shown that periodically modulating

∗aksarma@iitg.ac.in

the amplitude of the external ﬁeld can induce a high

degree of mechanical squeezing and optomechanical

entanglement in a cavity optomechanical system [24]. A

prerequisite of such a scheme is the ground-state cooling

of the mechanical mode, usually achieved by sideband

cooling. Dissipative optmechanical systems, where the

mechanical mode modulates the decay rate of the cavity

mode [26] enables this in the unresolved sideband limit

[26–30]. Such a scheme can be thought as an example

of reservoir engineering [31] where the cavity acts as a

reservoir whose force noise is squeezed.

Most schemes implemented to achieve mechanical

squeezing using reservoir engineering technique are based

on two-tone driving [5, 6, 13, 21]. Only recently,

Bai et. al. [1] proposed a scheme to squeeze

the mechanical position using single-tone driving in a

standard optomechanical system. In this work, we

study the dissipative generation of mechanical squeezing

in a quadratically-coupled optomechanical system by

periodically modulating the driving amplitude and show

that along with a robust squeezing, a high degree of

optomechanical entanglement can be produced. We

also show that the steady-state squeezing generated

using this scheme is sensitive to the ratio of the cavity

drive amplitudes. The paper is organised as follows.

In Sec. II, we introduce the membrane-in-the-middle

cavity optomechanical system driven by a amplitude-

modulated single-tone laser, analyzing the dynamical

behavior of the system with and without the rotating-

wave approximation (RWA). In Sec. III, we present the

squeezing in the position quadrature of the mechanical

mode and discuss the squeezing eﬀect as a competing

behaviour between two conﬂicting tendencies, and obtain

an optimal ratio of the sidebands to maximize squeezing

by balancing these eﬀects. In Sec. IV, we derive

an analytical solution for squeezing of the mechanical

position under adiabatic approximation, and compare it

with the exact numerical solution. We then examine the

robustness of the squeezing achieved and the behaviour of

entanglement between the optical and mechanical modes

in Sec. V. Finally, we summarise our paper in Sec. VI.

arXiv:2210.00510v1 [quant-ph] 2 Oct 2022

II. SYSTEM AND DYNAMICS

FIG. 1. A membrane-in-the-middle optomechanical system,

driven by an amplitude-modulated external laser.

The system under consideration is illustrated in Fig.

1. An optomechanical system is driven by an external

laser (frequency ωl) whose amplitude lis periodically

modulated. The optical mode (frequency ωo) is coupled

to the mechanical mode (frequency ωm) via a radiation-

pressure interaction of strength g. In a frame rotating at

the frequency ωl, the system Hamiltonian takes the form

(in units of ~) [16]

H= ∆0a†a+ωmb†b−ga†ab†+b2

+ila†−a+ηωmb†+b,(1)

where ∆0=ωo−ωlis the detuning of the optical

cavity and a(b) is the annihilation operator of the

optical (mechanical) mode. Here, the ﬁrst and the second

terms represent the individual energies of the optical

cavity and the mechanical membrane. The third term

is the optomechanical interaction energy with quadratic

coupling. The fourth term represents the time-dependent

laser whose driving amplitude is such that l(t) = l(t+

τ) = Pn=∞

n=−∞ ne−inΩt, where τis the modulation period

with Ω = 2π/τ, and n’s are the sideband modulation

strengths.The ﬁnal term represents a constant impulsive

force of η~ωmacting on the mechanical membrane.

The time evolution of the mode operators of the system

follow the Heisenberg equations of motion [12]. Taking

into account the dissipation of the cavity mode (rate κ)

and the decay of the mechanical resonator (rate γ), along

with the eﬀect of the vacuum and thermal ﬂuctuations

entering the system, we obtain the quantum Langevin

equations (QLEs) given by [32]

˙a=−i∆0a−κ

2a+iga b†+b2+l+√κain,(2a)

b=−iωmb−γ

2b+ 2iga†ab†+b−iηωm

+√γbin.(2b)

Here, ain and bin are the noise operators associated

with the vacuum and thermal ﬂuctuations, which are

assumed to have a Gaussian nature, given as [15, 33]

ha†

in(t)ain(t0)i=naδ(t−t0),(3a)

hain(t)a†

in(t0)i= (na+ 1)δ(t−t0),(3b)

hb†

in(t)bin(t0)i=nbδ(t−t0),(3c)

hbin(t)b†

in(t0)i= (nb+ 1)δ(t−t0),(3d)

where naand nbare the thermal occupancies of the

optical and mechanical modes respectively, given by nj=

{exp[~ωj/(kBT)] −1}−1for j∈ {a, b}at temperature T.

For suﬃciently strong drive amplitudes, the QLEs

obtained in Eqs. (2) can be approximated using

a linearized description by assuming that the mode

operators (O) are equal to the sum of their classical

expectation values and their quantum ﬂuctuations, i.e.,

O=hOi +δO[12]. This gives us the classical equations

for α=haiand β=hbias

˙α=−i∆0α−κ

2α+igα (β∗+β)2+l,(4a)

β=−iωmβ−γ

2β+ 2ig |α|2(β∗+β)−iηωm,(4b)

and their quantum counterparts (δO → O)

˙a≈ −i∆a−κ

2a+ 2igα (β∗+β)b†+b+√κain,(5a)

b≈ −iωmb−γ

2b+ 2ig α∗a+αa†(β∗+β)

+2ig |α|2b†+b+√γbin,(5b)

where, the eﬀective detuning of the linearized dynamics

is given by ∆ = ∆0−g(β∗+β)2.

As the primary contribution of the modulated drive

comes from the oﬀset strength and the ﬁrst-order

modulations, for our analysis, we assume that l(t)≈

−1eiΩt+0+1e−iΩt. Then, according to the

Floquet theorem, at a long-time limit, the cavity mode

and the mechanical mode amplitude would show the

same behaviour as the modulated external ﬁeld, i.e.,

limt→∞ α(t) = α(t+τ) and limt→∞ β(t) = β(t+τ)

[1, 2, 24]. We therefore redeﬁne these classical amplitudes

α=a−1eiΩat+a0+a1e−iΩat,(6a)

β=b−1eiΩbt+b0+b1e−iΩbt.(6b)

A. Dynamics under RWA

We now analyze the dynamics of the slowly varying

ﬂuctuations in the rotating frame of their oscillations. We

rewrite the ﬂuctuation operators and their corresponding

input noises as a= ˜ae−i∆t,b=˜

be−iωmtand ain =

˜aine−i∆t,bin =˜

bine−iωmtrespectively. Setting the

eﬀective cavity detuning at ωmand the external driving

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MechanicalSqueezinginQuadratically-coupledOptomechanicalSystemsPriyankarBanerjee,SampreetKalita,andAmarendraK.SarmaDepartmentofPhysics,IndianInstituteofTechnologyGuwahati,Guwahati-781039,IndiaWedemonstratethegenerationofastrongmechanicalsqueezinginadissipativeoptomechanicalsystembyintroducingaperio...

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Mechanical Squeezing in Quadratically-coupled Optomechanical Systems Priyankar Banerjee Sampreet Kalita and Amarendra K. Sarma Department of Physics Indian Institute of Technology Guwahati Guwahati-781039 India

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