Normal-mode splitting in the optomechanical system with an optical parametric ampliﬁer and coherent feedback Yue Li and Yijian Wang

2025-05-02 0 0 878.78KB 8 页 10玖币

侵权投诉

Normal-mode splitting in the optomechanical system with an optical parametric

ampliﬁer and coherent feedback

Yue Li and Yijian Wang

State Key Laboratory of Quantum Optics and Quantum Optics Devices,

Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China

Hengxin Sun,∗Kui Liu, and Jiangrui Gao

State Key Laboratory of Quantum Optics and Quantum Optics Devices,

Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China and

Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

(Dated: October 11, 2022)

Strong coupling in optomechanical systems is the basic condition for observing many quantum

phenomena such as optomechanical squeezing and entanglement. Normal-mode splitting (NMS) is

the most evident signature of strong coupling systems. Here we show the NMS in the spectra of

the movable mirror and the output ﬁeld in an optomechanical system can be ﬂexibly engineered by

a combination of optical parametric ampliﬁer (OPA) and coherent feedback (CF). Moreover, the

NMS could be enhanced by optimizing the parameters such as input optical power, OPA gain and

phase, CF strength in terms of amplitude reﬂectivity of beam splitter.

I. INTRODUCTION

In recent decades, cavity optomechanics composed of

coupled cavity ﬁeld and movable mirror, has become

an important ﬁeld due to its potential applications in

quantum optics [1–4]. A prerequisite for these applica-

tions is ground state cooling of the movable mirror. Re-

cently, great progress has been made in achieving ground

state cooling of oscillators with various methods, such

as dispersive coupling [5–8], dissipative coupling [9, 10],

dynamic cooling [11, 12], atom-assisted cooling [13–15],

and external cavity cooling [16], which sets the stage for

us to observe the quantum behavior such as mechanical

squeezing [17–19], mechanical entanglement [20], and op-

tomechancal squeezing [21–25] and entanglement [26–29].

Normal-mode splitting (NMS) is the most evident sig-

nature in strong coupling optomechanical systems [30–

39]. The NMS generally occurs when the energy exchang-

ing rate between two coupled subsystems is much faster

than their energy-dissipating into the environment. The

concept of NMS originally comes from the vacuum Rabi

splitting in a coupled atom-cavity system in 1980s [40–

43]. The NMS exhibits two-peak spectra of the position

of movable mirror and the noise spectra of output optical

ﬁeld in cavity optomechanical systems [30, 31], basically

due to strong coupling. There are several methods to en-

hance the NMS eﬀect. Particularly, enhancement could

be realized by adding a degenerate optical parametric

ampliﬁer (OPA) [32] in the cavity, or introducing a sin-

gle coherent feedback (CF) [38] outside the cavity. Here,

we combine the two schemes of OPA and CF and analyze

the NMS. Compared to the previous scheme with OPA

or CF alone, more ﬂexible degrees of freedom could be

utilized to control the optomechanical coupling strength

∗hxsun@sxu.edu.cn

and NMS. Strong coupling and more obvious NMS could

be achieved by optimizing the parameters such as input

optical power, OPA gain and phase, and amplitude re-

ﬂectivity of the beam splitter of CF.

The layout of the paper is presented below. In Sec. II

we introduce the theoretical model, present the Hamilto-

nian of the system, give the Langevin equations of motion

for the movable mirror and the cavity ﬁeld, and obtain

the steady-state mean values. In Sec. III we linearize

the quantum Langevin equations, give the stability con-

ditions of the system, derive the spectrum of position

ﬂuctuation of movable mirror. In Sec. IV we analyze the

behavior of the NMS in terms of location and linewidth

of two normal modes by varying the following parame-

ters: amplitude reﬂectivity of beam splitter, input laser

power, OPA gain and phase, and compare it with the

case only OPA or CF is added. In Sec. V we get the

spectrum of output ﬁeld and show the two-peak spectra

of movable mirror and output ﬁeld.

II. MODEL

As shown in Fig. 1, we consider an optical cavity con-

sisting of two mirrors separated by a distance L, com-

posed of one ﬁxed mirror with partial power reﬂectivity

and one movable mirror with total power reﬂectivity. A

second-order nonlinear OPA device is placed in the cav-

ity. The cavity output ﬁeld from the ﬁxed mirror is par-

tially sent back into the cavity via a totally reﬂecting

mirror and a beam splitter (BS), forming an optical co-

herent feedback. The movable mirror is in a thermal bath

at temperature Tand regarded as a quantum mechani-

cal harmonic oscillator with eﬀective mass m, resonance

frequency ωm, and damping rate γm. An input laser

beam with frequency ωland an amplitude εlrelated to

a power of Pin by εl=q2κPin

~ωl, is split into two parts by

arXiv:2210.04197v1 [quant-ph] 9 Oct 2022

OPA

ˆout

a′

ˆin

a′

ˆout

ˆin

aBS

Totally reflecting mirror

Fixed mirror Movable mirror

,,,

(

)

FIG. 1. Opto-mechanical system with an OPA and CF. The

transmitted part of the input laser is sent into the cavity by a

ﬁxed mirror. Then a part of output ﬁeld from the cavity ﬁeld

is fed back into the cavity through a totally reﬂecting mirror

and a partially reﬂecting beam splitter (BS).

the BS with amplitude reﬂectivity rand transmissivity

t,κis the cavity ﬁeld decay rate from the ﬁxed mirror

and ~is Planck constant divided by 2π. No extra opti-

cal loss is assumed. The cavity ﬁeld exerts a radiation

pressure force on the movable mirror due to momentum

transfer from the photons in the cavity. The position of

the movable mirror oscillates around its equilibrium posi-

tion under the thermal Langevin force and the radiation

pressure force.

The adiabatic limit, ωmπc/L is assumed, where c

is the light speed in vacuum and Lis the cavity length.

Hence, we can consider the model to the case of single-

cavity and mechanical mode [44, 45]. In the frame ro-

tating at the laser frequency ωl, the total Hamiltonian

describing the coupled system is given by

H=~(ωc−ωl)ˆa†ˆa−~g0ˆa†ˆaˆ

Q+~ωm

4(ˆ

Q2+ˆ

P2)

+i~tεl(ˆa†−ˆa) + i~G(eiθ ˆa†2−e−iθ ˆa2),(1)

where ˆa(ˆa†) is the annihilation (creation) operator of the

fundamental cavity ﬁeld, Qand Pare the dimensionless

position and momentum operators of the movable mir-

ror with ˆ

Q=q2mωm

~ˆq,ˆ

P=q2

m~ωmˆp, and they obey

the relationship [ˆ

Q, ˆ

P]=2i. In Eq. (1), the ﬁrst term

represents the energy of the cavity ﬁeld, ˆna= ˆa†ˆais the

number of the photons inside the cavity. The second

term describes the optomechanical interaction between

cavity ﬁeld and movable mirror via radiation pressure,

g0=ωc

Lq~

2mωmis the single-photon optomechanical cou-

pling constant. The third term describes the energy of

movable mirror. The fourth term corresponds to the cav-

ity ﬁeld driven by the external ﬁeld. The last term de-

notes the second-order nonlinear interaction energy, Gis

the OPA gain related to the power of second harmonic

ﬁeld, θis the relative phase between the fundamental and

second harmonic ﬁelds.

Using the Heisenberg equations of motion, adding the

noise and damping terms, and also taking the feedback

term into account, we obtain the following Langevin

equations of motion:

Q=ωmˆ

P , (2a)

P= 2g0ˆna−ωmˆ

Q−γmˆ

P+ˆ

ξ, (2b)

ˆa=−i(ωc−ωl−g0ˆ

Q)ˆa+ 2Geiθ ˆa†−κˆa

+tεl+√2κ(tδˆain +rˆ

a0out).(2c)

where the ﬁrst two terms are related to the position and

momentum of the movable mirror, respectively, while the

third one corresponds to the intracavity ﬁeld.

The force ˆ

ξis related to the thermal noise of the movable

mirror in thermal equilibrium, which has zero mean value

and nonzero time domain correlation function [46]

hξ(t)ξ(t0)i=~γm

2πmZdωωe−iω(t−t0)

×hcoth ~ωm

2kBT+ 1 i,(3)

where kBis the Boltzmann constant and Tis the envi-

ronment temperature.

In Eq. (2c), δˆain is the optical vacuum noise operator

with zero mean value and its δ-correlated function in the

time domain [47] is given by

hδˆain(t)δˆa†

in(t0)i=δ(t−t0),(4a)

hδˆain(t)δˆain(t0)i=hδˆa†

in(t)δˆain(t0)i= 0.(4b)

According to the input-output relation of the cavity [48]

a0out =√2κˆa−ˆ

a0in,(5)

and the beam splitter model

a0in =tˆain +rˆ

a0out,(6)

we get

a0out =√2κ

1 + rˆa−t

1 + rˆain.(7)

We get the input-output relationship similar with Ref.

[49], in which the CF is used to manipulate entanglement

from a nondegenerate OPA. Note Eq. (7) is diﬀerent from

the general input-output relation of the cavity as Eq.

(5). Appendix A gives detailed derivation of this relation

by solving equations of ﬁeld relations. Substituting Eq.

(7) into Eq. (2c), we get the ﬁnal motional equation of

intracavity ﬁeld:

ˆa=−[κef f +i(ωc−ωl−g0ˆ

Q)]ˆa+tεl1−r

r+ 1

+2Geiθ ˆa†+√2κ1−r

r+ 1tδˆain.(8)

Due to the CF, the eﬀective cavity decay rate (or cavity

linewidth) becomes κeff =κ(1 −r)/(1 + r), which de-

creases with increasing rwithin a scale of [−1,1]. Set-

ting the time derivative terms in Eqs. (2a), (2b) and (8)

to be zero, the steady-state solutions are given by

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Normal-modesplittingintheoptomechanicalsystemwithanopticalparametricamplierandcoherentfeedbackYueLiandYijianWangStateKeyLaboratoryofQuantumOpticsandQuantumOpticsDevices,InstituteofOpto-Electronics,ShanxiUniversity,Taiyuan030006,ChinaHengxinSun,KuiLiu,andJiangruiGaoStateKeyLaboratoryofQuantumOptics...

展开>> 收起<<

Normal-mode splitting in the optomechanical system with an optical parametric ampliﬁer and coherent feedback Yue Li and Yijian Wang.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Normal-mode splitting in the optomechanical system with an optical parametric ampliﬁer and coherent feedback Yue Li and Yijian Wang

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: