Generating quantum entanglement between macroscopic objects with continuous measurement and feedback control Daisuke Miki1Nobuyuki Matsumoto2Akira Matsumura1Tomoya Shichijo1

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Generating quantum entanglement between macroscopic objects

with continuous measurement and feedback control

Daisuke Miki,1Nobuyuki Matsumoto,2Akira Matsumura,1Tomoya Shichijo,1

Yuuki Sugiyama,1Kazuhiro Yamamoto,1,3Naoki Yamamoto,4,5

1Department of Physics, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan

2Department of Physics, Faculty of Science, Gakushuin University, 1-5-1, Mejiro, Toshima, Tokyo, 171-8588 Japan

3Research Center for Advanced Particle Physics, Kyushu University,

744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

4Quantum Computing Center, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama 223-8522, Japan and

5Department of Applied Physics and Physico-Informatics,

Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama 223- 8522, Japan ∗

(Dated: March 15, 2023)

This paper is aimed at investigating the feasibility of generating quantum conditional entangle-

ment between macroscopic mechanical mirrors in optomechanical systems while under continuous

measurement and feedback control. We consider the squeezing of the states of the mechanical com-

mon and the diﬀerential motions of the mirrors by the action of measuring the common and the

diﬀerential output light beams in the Fabry-Perot-Michelson interferometer. We carefully derive

a covariance matrix for the mechanical mirrors in a steady state, employing the Kalman ﬁltering

problem with dissipative cavities. We demonstrate that Gaussian entanglement between the me-

chanical mirrors is generated when the states of the mechanical common and diﬀerential modes of

the mirrors are squeezed with high purity in an asymmetric manner. Our results also show that

quantum entanglement between 7 mg mirrors is achievable in the short term.

I. INTRODUCTION

Cavity optomechanics deals with the coupled dynamics of the oscillating end mirrors of cavities (mechanical oscil-

lators) and the optical mode therein. This ﬁeld has the potential to reveal the boundary between the classical and

the quantum world [1–6]. The quantum states of mechanical oscillators can be achieved by quantum control through

interaction with optical cavity modes, whereas mechanical oscillators lose quantum coherence owing to thermal ﬂuc-

tuations. The technique of continuous measurement cooling shows the potential to achieve the quantum states of

macroscopic mechanical oscillators [4, 6, 7]. Ref. [8] demonstrated cooling a mechanical oscillator to the ground

state through cavity detuning and feedback control. Moreover, optomechanical systems are helpful in generating

entanglements. Ref. [9] discussed the role of feedback cooling; the authors showed that the entanglement between two

levitated nanospheres due to the Coulomb force could be measured experimentally with the feedback-based setup.

The authors in Refs. [10, 11] considered the detectability of entanglement between the optical cavity mode and the

mechanical oscillator in the ground state. Refs. [12–14] showed that the generation of quantum entanglement between

nanoscale objects was realized experimentally. Recently, cavity optomechanics has attracted signiﬁcant interest as a

possible ﬁeld for investigating the quantum nature of gravity through tabletop experiments [15–21]. Entanglement

generation due to gravitational interaction can be considered as evidence of the quantum nature of gravity [22, 23],

which has sparked several investigations [24–30]. Moreover, related to gravitational entanglement, the quantum nature

of gravity has been discussed in gravitons and quantum ﬁeld theory [31–38]. However, verifying the quantum nature

of gravity requires entanglement between heavier objects [16, 19]. The realization of macroscopic quantum systems is

pivotal for investigating the unexplored areas between the quantum world and gravity.

In this paper, we consider the feasibility of realizing Gaussian entanglement between macroscopic oscillators via

optomechanical coupling. It is known that entanglement between two squeezed light beams with diﬀerent squeezing

angles is generated by passing them through the beam splitter (e.g., Ref. [39]). The authors of Ref. [40] analyzed

the entanglement in a comparable situation where the power-recycled mirror squeezed the oscillators’ common and

diﬀerential modes asymmetrically. However, their analysis was limited to high-frequency regions, where the oscillators

were regarded as free mass. Namely, they only demonstrated entanglement generation between Fourier modes of the

∗miki.daisuke@phys.kyushu-u.ac.jp,

nobuyuki.matsumoto@gakushuin.ac.jp,

matsumura.akira@phys.kyushu-u.ac.jp,

shichijo.tomoya.351@s.kyushu-u.ac.jp,

sugiyama.yuki@phys.kyushu-u.ac.jp,

yamamoto@phys.kyushu-u.ac.jp,

yamamoto@appi.keio.ac.jp

arXiv:2210.13169v2 [quant-ph] 14 Mar 2023

macroscopic oscillator’s motions in high-frequency regions. Therefore the previous work is not enough to include

the analysis around resonant frequencies. Quantum control of macroscopic oscillators around resonant frequencies is

important for entanglement generation (e.g., Refs. [16, 19]). Then, our analysis here is not limited to high-frequency

regions.

We revisit the realization of entanglement between macroscopic oscillators with the Kalman ﬁlter’s formalism in a

wide range of parameter spaces. We employ feedback control, which decreases the eﬀective temperature, and detunes

enabling us to trap the mechanical oscillator stably with the optical spring, as discussed in Ref. [6]. To clarify the

diﬀerence between the previous [40] and present paper, we note that the detuning was not considered in the previous

work [40]. By using these quantum controls in an optomechanical system with a power-recycled mirror, we clariﬁed

the relationship between the entanglement and squeezing of states. Our results show that quantum cooperativity

and detuning characterize the entanglement behavior, quantum squeezing, and purity. The entanglement generation

requires quantum squeezing of both the common and diﬀerential modes of the oscillators. Squeezing, however, does

not always result in entanglement generation, as high-purity squeezed states are also required. We demonstrate that

the entanglement occurs for the quantum cooperativity C±/n±

th >

∼3 with the experimentally achievable parameters

in amplitude quadrature measurement (Xmeasurement).

The remainder of this paper is organized as follows: In Section II, we present a brief review of optomechanical sys-

tems while under continuous measurement and feedback control. In Section III, we provide a mathematical formula

for the Riccati equation to describe the covariance matrix using a quantum Kalman ﬁlter to minimize the correlation.

In Section IV, we extended the formulations in the previous sections to those with two optomechanical systems, in

which we consider the entanglement between them through a beam splitter in a power-recycled interferometer. We

determined the feasibility of preparing entanglements between the mirror oscillators in the space of the model pa-

rameter, depending on the amplitude quadrature measurement (Xmeasurement) and phase quadrature measurement

(Ymeasurement), respectively. Finally, Section V presents our conclusions. The derivation of the input-output rela-

tion in interferometer is presented in Appendix A. In Appendix B, we describe the details of logarithmic negativity

for estimating the entanglement developed in this paper. In Appendix C, we describe the details of computing the

squeezing angle.

II. FORMULAS

In this section, we consider a driven optical cavity mode that interacts with an oscillating mirror, which is regarded

as a mechanical harmonic oscillator. The Hamiltonian of our system is as follows:

H=P2

2m+1

2mΩ2Q2+~ωca†a+~ωc

`Qa†a+i~E(a†e−iωLt−aeiωLt),(1)

where Qand Pare the canonical position and momentum operators of the oscillator, satisfying the commutation

relation [Q, P ] = i~, while mand Ω are the mass and resonance frequency of the oscillator, respectively; aand a†are

the annihilation and creation operators of the optical modes in the cavity, `is the cavity length, and ωcis the cavity

frequency. The last term describes the input laser with frequency ωLand amplitude E=pPinκ/~ωL, where Pin is

the input laser power and κis the optical decay rate. Here, we introduce non-dimensional variables

q=r2mΩ

~Q, p =r2

m~ΩP, (2)

that satisfy the commutation relation [q, p]=2i.

The Langevin equations are given by

˙q= Ωp,

˙p=−Ωq−2Ga0†a0−Γp+√2Γpin,(3)

˙a0=i(ωL−ωc)a0−iGqa0+E−κ

2a0+√κain,

where a0=eiωLtadenotes the redeﬁned annihilation operator and G= (ωc/`)p~/2mΩ is the optomechanical coupling.

Γ denotes the mechanical decay rate and pin is the mechanical noise input with a variance of hp2

ini= 2kBT/~Ω + 1.

Similarly, ain is the optical noise input speciﬁed by ha2

ini= (2Nth + 1)/2 with thermal photon occupation number

Nth. Considering the linearization q→¯q+δq,p→¯p+δp, and a0→¯a0+δa0, we derive the following equations for

the steady state:

¯q= Ω¯p,

¯p=−Ω¯q−2G|¯a0|2−Γ¯p, (4)

¯a0=i(ωL−ωc−G¯q)¯a+E−κ

2¯a0.

Here, considering ˙

¯q=˙

¯p=˙

¯a0= 0, we have

¯q=−2G

Ω|¯a0|2,

¯p= 0,(5)

¯a0=2E

κ−2i∆,

where we deﬁne the detuning ∆ = ωL−ωc+ 2(G|¯a0|)2/Ω. The perturbation equations are as follows:

δq = Ωδp, (6)

δp =−Ωδq −2g(e−iφδa0+eiφδa0†)−Γδp +√2Γpin −Zt

−∞

dsgF B (t−s)X(s),(7)

δa0=i∆δa0−igeiφδq −κ

2δa0+√κain,(8)

where ¯a0=eiφ|¯a0|and g= (|¯a0|ωc/`)p~/2mΩ denotes the redeﬁned optomechanical coupling. We add that the last

term in Eq. (7) described the feedback eﬀects [2, 6] and we henceforth simply represent (δq, δp, δa0) as (q, p, a0). By

introducing the amplitude quadrature x=e−iφa0+eiφa0† and the phase quadrature y= (e−iφa0−eiφa0†)/i, Eq. (8)

yields

˙x=−κ

2x−∆y+√κxin,(9)

˙y=−κ

2y+ ∆x+√κyin −2gq, (10)

where xin and yin are the corresponding input noises similarly deﬁned as ain, whose variance is speciﬁed by: hx2

ini=

hy2

ini= 2Nth + 1.

Here, we consider the adiabatic limit κΩ, which allows the continuous measurement of the oscillator position

because the cavity photon dissipation is suﬃciently larger than the frequency of the oscillator. The adiabatic limit is

rephrased as the limit of the dissipation dominant regime where the time derivative term of the optical ﬁeld ˙a0is much

smaller than the terms of the right-hand side of Eq. (8). Then, the time derivatives of the optical amplitude quadrature

xand the phase quadrature yare also negligible in Eqs. (9) and (10), which leads to the following equations:

x=8∆g

κ2+ 4∆2q+2κ√κ

κ2+ 4∆2xin −4∆√κ

κ2+ 4∆2yin,(11)

y=−4κg

κ2+ 4∆2q+4∆√κ

κ2+ 4∆2xin +2κ√κ

κ2+ 4∆2yin,(12)

Introducing the rescaled variables

q=q0rΩ

ωm

, p =p0rωm

Ω, ωm=rΩ2+ Ω 16∆g2

κ2+ 4∆2, gm=grΩ

ωm

,(13)

we rewrite the equation of motion as:

˙q0=ωmp0,(14)

˙p0=−ωmq0−γmp0+p2γmp0

in −4gmκ√κ

κ2+ 4∆2xin +8gm∆√κ

κ2+ 4∆2yin,(15)

x=8∆gm

κ2+ 4∆2q0+2κ√κ

κ2+ 4∆2xin −4∆√κ

κ2+ 4∆2yin,(16)

y=−4κgm

κ2+ 4∆2q0+4∆√κ

κ2+ 4∆2xin +2κ√κ

κ2+ 4∆2yin,(17)

where γmis the eﬀective mechanical decay rate under feedback control, and the thermal noise input changes to

hp02

ini= 2nth + 1 with nth =kBTΓ/~γmωm.

The quadratures of the optical cavity modes xand ycontain information regarding the position of the mechanical

oscillator q0in Eqs. (49) and (50). To estimate the oscillator position, we either consider the measurement of the

amplitude quadrature x, or the measurement of the phase quadrature y. The amplitude quadrature of the output

optical ﬁeld is obtained by the input-output relation xout =xin −√κx [2, 41]. However, we need to consider the

additional noise input due to the imperfect measurement. Thus, the observation signal of amplitude quadrature xis

described by the following equation:

X=√ηxout +p1−ηx0

in,(18)

where η∈[0,1] is the detection eﬃciency and x0

in is the additional vacuum noise for the imperfect measurement,

which satisﬁes hx0

2i= 1. Under the limit of the dissipation domination, we have

X=−8gm∆√ηκ

κ2+ 4∆2q0−√ηκ2−4∆2

κ2+ 4∆2xin +√η4κ∆

κ2+ 4∆2yin +p1−ηx0

in.(19)

On the other hand, the observation signal of the phase quadrature yis also described by the output equation

Y=√ηyout +p1−ηy0

in,with yout =yin −√κy, (20)

which reduces to

Y=4gmκ√ηκ

κ2+ 4∆2q0−√η4κ∆

κ2+ 4∆2xin −√ηκ2−4∆2

κ2+ 4∆2yin +p1−ηy0

in.(21)

III. RICCATI EQUATION

Because the observation signals in Eqs. (19) and (21) include noise information, we employed the quantum ﬁlter

for optimal estimation. Here, we consider the quantum Kalman ﬁlter, which allows us to minimize the mean-squared

error between the canonical operators r= (q0, p0)Tand the estimated values ˜

r= (˜q0,˜p)T, i.e., each component of

the covariance matrix V=h{r−˜

r,(r−˜

r)T}i is minimized. The quantum ﬁlter is essential to reduce the thermal

ﬂuctuations and increase the squeezing level. With the quantum Kalman ﬁlter, we can track the behavior of r

conditioned on the measurement result, and its ﬂuctuation is represented by the conditional covariance matrix following

the Riccati equation. On the other hand, without the quantum ﬁlter, we only have the average behavior of r.

Importantly, the covariance matrix without the ﬁlter is always larger than the covariance matrix conditioned on the

measurement. Hence, in the absence of the ﬁlter, the squeezing level and, accordingly, the entanglement level must

decrease. This is essential for the entanglement between mechanical mirrors in the next section.

We rewrite the Langevin equation in matrix form as follows:

r=Ar +0

w,(22)

X=CXr+vX,(23)

Y=CYr+vY,(24)

where,

A=0ωm

−ωm−γm, w =p2γmp0

in −4gmκ3/2

κ2+ 4∆2xin +8gmκ1/2∆

κ2+ 4∆2yin,(25)

CX=−8gm∆√ηκ

κ2+4∆20, vX=−κ2−4∆2

κ2+ 4∆2√ηxin +4κ∆

κ2+ 4∆2√ηyin +p1−ηx0

in,(26)

CY=4gmκ√ηκ

κ2+4∆20, vY=−4κ∆

κ2+ 4∆2√ηxin −κ2−4∆2

κ2+ 4∆2√ηyin +p1−ηy0

in.(27)

We use Eq. (23) or Eq. (24) for the optical amplitude measurement and the phase measurement, respectively.

For the Kalman ﬁlter [42, 43], we obtained the time evolution of the optimized covariance matrix as the following

Riccati equation:

V=AV +V AT+N−(V CT

I+LI)M−1(V CT

I+LI)T(28)

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GeneratingquantumentanglementbetweenmacroscopicobjectswithcontinuousmeasurementandfeedbackcontrolDaisukeMiki,1NobuyukiMatsumoto,2AkiraMatsumura,1TomoyaShichijo,1YuukiSugiyama,1KazuhiroYamamoto,1;3NaokiYamamoto,4;51DepartmentofPhysics,KyushuUniversity,744Motooka,Nishi-Ku,Fukuoka819-0395,Japan2Departm...

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Generating quantum entanglement between macroscopic objects with continuous measurement and feedback control Daisuke Miki1Nobuyuki Matsumoto2Akira Matsumura1Tomoya Shichijo1

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