Baijun Li Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China    Ran Huang Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China    Xun-Wei Xu Department of Applied Physics, East China Jiaotong University, Nanchang, 330013, China    Adam Miranowicz Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland    Hui Jing Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
###### Abstract

We propose how to achieve quantum nonreciprocity via unconventional photon blockade (UPB) in a compound device consisting of an optical harmonic resonator and a spinning optomechanical resonator. We show that, even with an extremely weak single-photon nonlinearity, nonreciprocal UPB can emerge in this system, i.e., strong photon antibunching can emerge only by driving the device from one side, but not from the other side. This nonreciprocity results from the Fizeau drag, leading to different splitting of the resonance frequencies for the counter-circulating modes. Such nonreciprocal quantum UPB devices can be particularly useful in achieving e.g., few-photon diodes or circulators, and quantum chiral photonic engineering.

## I Introduction

Photon blockade (PB) Tian92Quantum (); Leonski94Possibility (); Imamoglu97Strongly (); Birnbaum05Photon (), i.e., the generation of a single photon in a nonlinear cavity can diminish (almost to zero) the probability of generating another photon in the cavity Muller15Coherent (), plays a key role in achieving single-photon sources Gu17Microwave (), which are crucial element for quantum information science Scarani09The (); Buluta11Natural (). In experiments, PB has been demonstrated in cavity-QED systems Birnbaum05Photon (); Peyronel12Quantum (); Muller15Coherent (), circuit-QED systems Lang11Observation (); Hoffman11Dispersive (), and quantum dot in a photonic crystal resonator Faraon08Coherent (). PB has also been predicted in various nonlinear optical systems, e.g., Kerr-type cavities Imamoglu97Strongly (); Ferretti10Photon (); Liao10Correlated (); Miranowicz13Two (), and optomechanical (OM) devices Rabl11Photon (); Nunnenkamp11Single (); Liao13Photon (); Xie16Photon (). Conventional PB happens under the stringent condition of strong single-photon nonlinearities, which turns out to be highly challenging in practice.

To overcome this obstacle, coupled nonlinear resonators Leonski04Two (); Miranowicz06Kerr (); Liew10Single (); Bamba11Origin () have been proposed to achieve unconventional photon blockade (UPB). In contrast to conventional PB, UPB relying on the destructive quantum interferences between different dissipative pathways, can be achieved even with arbitrarily weak nonlinearities Liew10Single (); Bamba11Origin (); Majumdar12Loss (); Komar13Single (); Xu13Antibunching (); Ferretti13Optimal (); Xu14Strong (); Zhang14Optimal (); Shen15Tunable (); Flayac17Unconventional (); Flayac17Nonclassical (); Snijders18Observation (); Vaneph18Observation (). Recently, UPB has been successfully achieved in coupled optical Snijders18Observation () or superconducting resonators Vaneph18Observation (). In addition to be a powerful tool to generate optimally sub-Poissonian light, UPB also provides a unique way to reveal exotic quantum correlations in devices with a weak single-particle nonlinearity Flayac17Unconventional (); Flayac17Nonclassical ().

Here, we propose how to achieve and control nonreciprocal UPB with spinning devices. We note that nonreciprocal optical devices, allowing the flow of light from one side but blocking it from the other, have been realized in OM devices Manipatruni09Optical (); Shen16Experimental (); Bernier17Nonreciprocal (), Kerr resonators Cao17Experimental (); Bino18Microresonator (); Shi15Limitations (), thermo systems Fan11An (); Zhang18Thermal (); Xia18Cavity (), devices with temporal modulation Sounas17Non-reciprocal (); Caloz18Electromagnetic (), and non-Hermitian systems Bender13Observation (); Peng14Parity (); Chang14Parity (). In a very recent experiment Maayani18Flying (), 99.6% optical isolation in a spinning resonator has been achieved based on the optical Sagnac effect. However, these studies have mainly focused on the classical regimes; that is, unidirectional control of transmission rates instead of quantum noises. We also note that in recent works, single-photon optical diodes Xia14Reversible (); Tang18An (), quantum circulators Scheucher16Quantum (), unidirectional quantum amplifiers Abdo14Josephson (); Metelmann15Nonreciprocal (); Malz18Quantum (); Shen18Reconfigurable (); Song18Direction (), and one-way quantum routers of thermal noises Barzanjeh18Manipulating () have been explored. In particular, nonreciprocal PB was predicted in a spinning Kerr resonator Huang18Nonreciprocal () and quadratic OM system Xu18arXiv (), which, however, relies on the conventional condition of strong single-photon nonlinearity.

It should be stressed PB and UPB are very different phenomena, thus also their nonreciprocal generalizations are also different, as can be seen by comparing the present manuscript with Ref. Huang18Nonreciprocal (). Indeed PB refers to a process, when a single photon is blocking the entry (or generation) of more photons in a strongly nonlinear cavity. Thus, PB refers to state truncation, also referred to as nonlinear quantum scissors Leonski01 (). PB can be used as a source of single photons, since the PB light is sub-Poissonian (or photon antibunched) in second- and higher-orders, as characterized by the correlation functions for . By contrast to PB, UPB refers to the light, which is optimally sub-Poissonian in second order, , and is generated in a weakly-nonlinear system allowing for multi-path interference (e.g., two linearly-coupled cavities, when one of them is also weakly coupled to a two-level atom). Thus, PB and UPB are induced by different effects: PB due to a large system nonlinearity and UPB via multi-path interference assuming even an extremely-weak system nonlinearity. Note that light generated via UPB can exhibit higher-order super-Poissonian photon-number statistics, for some . Thus, UPB is, in general, not a good source of single photons. This short comparison of PB and UPB indicates that the term UPB, as coined in Ref. Carusotto13 () and now commonly accepted, is fundamentally different from PB concerning their physical mechanisms and properties of their generated light.

In this paper, we study nonreciprocal UPB in a coupled system composed of an optical harmonic cavity and a spinning OM resonator. Coupled-cavity systems have been extensively studied in experiments Vaneph18Observation (); Zhang18A (); Konotop16Nonlinear (); Ganainy18Non (), providing not only a unique way to achieve UPB, but also a versatile platform to study, e.g., phonon lasing Grudinin10Phonon (); Jing14PT (); Zhang18A (), loss-induced transparency Zhang18Loss (), and quantum sensing Liu16Metrology (); Konotop16Nonlinear (); Ganainy18Non (). We find that, by the spinning of an OM resonator, UPB can emerge in a nonreciprocal way even with a weak single-photon nonlinearity; that is, strongly antibunched photons can emerge only by driving the device from one side, but not the other side. Our work opens up a new route to engineer quantum chiral UPB devices, which can have practical applications in achieving, for example, photonic diodes or circulators, and nonreciprocal quantum communications at the few-photon level.

## Ii Model and Solutions

We consider a compound system consisting of an optical harmonic resonator (with the resonance frequency of the cavity field and the decay rate ) and a spinning anharmonic resonator (with the resonance frequency of the cavity field and the decay rate ), as shown in Fig. 1. An external classical light is coupled into and out of the resonator through a tapered fiber of frequency and these two whispering-gallery-mode (WGM) resonators are evanescently coupled to each other with coupling strength  Spillane03Ideality (). Note that in the previous proposal Huang18Nonreciprocal (), requiring the strong Kerr nonlinearity, (where is the cavity linewidth), is challenging for current experiments. In our device, we can use experimentally feasible Kerr-nonlinear strength to realize nonreciprocal PB; that is,  Vaneph18Observation (), which is about two orders of magnitude smaller than the former work Huang18Nonreciprocal (). The Kerr interactions can also be achieved in cavity-atom systems Schmidt96Giant () or magnon devices Wang18Bistability (), and in OM systems Gong09Effective () which we focus on here. We consider a weakly OM coupling strength () in an auxiliary cavity which is well within current experimental abilities Ding11Wavelength (); Snijders16Purification (); Enzian19Observation (). In a spinning resonator, the refractive indices associated with the clockwise () and anticlockwise () optical modes are given as , where is the tangential velocity with the angular velocity and radius  Maayani18Flying (). For light propagating in the spinning resonator, optical mode experiences a Fizeau shift  Malykin00The (); that is, , with

 ΔF =±nrΩωRc(1−1n2−λndndλ)=±ηΩ, (1)

where is the optical resonance frequency for the nonspinning OM resonator, () is the speed (wavelength) of light in the vacuum, and is the refractive index of the cavity. The dispersion term , characterizing the relativistic origin of the Sagnac effect, is relatively small in typical materials (Malykin00The (); Maayani18Flying (). The denote the light propagating against () and along () the direction of the spinning OM resonator, respectively.

In a rotating frame with respect to , the effective Hamiltonian of the system can be written as (see Appendix A for more details)

 H= ℏΔLa†LaL+ℏ(ΔR+ΔF)a†RaR+ℏωmb†b +ℏJ(a†LaR+a†RaL)+ℏga†RaR(b†+b) +iℏϵd(a†L−aL), (2)

where () and () are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted by subscript ) and the OM cavity (denoted by subscript ), respectively; () is the annihilation (creation) operator for the mechanical mode of the OM cavity. The frequency detuning between the cavity field in the left (right) cavity and the driving field is denoted by where ; The parameter denotes the strength of the photon hopping interaction between the two cavity modes; describes the radiation-pressure coupling between the optical and vibrative modes in the OM resonator with frequency and effective mass ; denotes the driving strength that is coupled into the compound system through the optical fiber waveguide with cavity loss rate and driving power .

The Heisenberg equations of motion of the system are then written as

 ddtq= ωmp, ddtp= −ωmq−gba†RaR−γm2p+ξ, ddtaL= −(κL2+iΔL)aL−iJaR+ϵd+√κLaL,in, ddtaR= −(κR2+iΔ′R)aR−iJaL−igbqaR +√κLaR,in, (3)

where , and ; and are dimensionless canonical position and momentum with and , respectively; () is the dissipation rate and () is the quality factor of the left (right) cavity; is damping rate with the quality factor of the mechanical mode. Moreover, is the zero-mean Brownian stochastic operator, , resulting from the coupling of the mechanical resonator with corresponding thermal environment and satisfying the following correlation function Ford88Quantum ():

 ⟨ξ(t)ξ(t′)⟩=12π∫dωe−iω(t−t′)Γm(ω), (4)

where the auxiliary function are . is effective temperature of the environment of the mechanical resonator and is the Boltzmann constant. The annihilation operators and are the input vacuum noise operators of the optical cavity and the OM cavity with zero mean value, respectively, i.e., , and comply with following nonzero time-domain correlation functions Gardiner00Quantum (); Walls94Quantum ():

 ⟨a†K,in(t)aK,in(t′)⟩= 0, ⟨aK,in(t)a†K,in(t′)⟩= δ(t−t′), (5)

for . Because the whole system interacts with a low-temperature environment (in this paper we consider 0.1 ), we neglect the mean thermal photon numbers at optical frequencies in the two cavities. In order to linearize the dynamics around the steady state of the system, we expend the operators as the sum of its steady-state mean values and a small fluctuations with zero mean value around it; that is, , , , and . By neglecting higher-order terms, for example, , we can solve fluctuation equations in the frequency domain (see Appendix B for more details)

 δaL(ω)= E(ω)aL,in(ω)+F(ω)a†L,in(ω)+G(ω)aR,in(ω) +H(ω)a†R,in(ω)+Q(ω)ξ(ω), (6)

where

 E(ω)= √κLA1(ω)A5(ω), F(ω)= −√κLA2(ω)A5(ω), G(ω)= √κRA3(ω)A5(ω), H(ω)= −√κRA4(ω)A5(ω), Q(ω)= −igbχ(ω)ωmA5(ω)[βA3(ω)+β∗A4(ω)], (7)

and

 A1(ω)= [(κR2+iω)2+Δ′′2R]V−1(ω) −g4b|β|4(χ(ω)ωm)2V−1(ω)+J2V+2, A2(ω)= −iJ2g2bβ2χ(ω)ωm, A3(ω)= −iJV−1(ω)V−2−iJ3, A4(ω)= −Jg2bβ2χ(ω)ωmV−1(ω), A5(ω)= V+1A1(ω)+iJA3(ω), (8)

where we introduced

 Δ′′R= Δ′R+gbqs−g2b|β|2χ(ω), χ(ω)= ω2m/(ω2m−ω2+iωγm2), V±1(ω)= κL2±i(ΔL−ω), V±2(ω)= κL2±i(ΔL−ω). (9)

## Iii Nonreciprocal Optical Correlations

Now, we focus on the statistical properties of photons in optical cavity, which are described quantitatively via normalized zero-time delay second-order correlation function  Walls94Quantum (); Xu13Antibunching (). By taking the semiclassical approximation, i.e., , the correlation function can be given as Xu13Antibunching ()

 g(2)L(0)= |α|4+4|α|2R1+2Re[α∗2R2]+R3(|α|2+R1)2, (10)

where , , and .

From Eq. (II), the correlation between and can be calculated as

 ⟨δa±L(t)δaL(t)⟩= 12π∫+∞−∞Xa±LaLdω, (11)

where , , and

 Xa†LaL= |Q(−ω)|2Γm(−ω)+|F(−ω)|2+|H(−ω)|2, XaLaL= Q(ω)Q(−ω)Γm(−ω)+E(ω)F(−ω) +G(ω)H(−ω). (12)

To obtain more accurate results, we introduce the density operator and numerically calculate normalized zero-time delay second-order correlation by the Lindblad master equation Johansson13Qutip ()

 ˙ρ= 1iℏ[H,ρ]+L[aL](ρ)+L[aR](ρ)+L[b](ρ)+L[b†](ρ), (13)

where are the Lindblad superoperators Walls94Quantum (), for , , , , and is the mean thermal phonon numbers of the mechanical mode at temperature .

The second-order correlation function is shown in Fig. 2 as function of the optical detuning and the angular velocity . We assume , and use experimentally feasible parameters Vahala03Optical (); Peng14Parity (); Teufel11Sideband (); Ding11Wavelength (); Verhagen12Quantum (); Aspelmeyer14Cavity (); Huet16Millisecond (); that is, , , , , , , , , , , . is typically  Vahala03Optical (); Aspelmeyer14Cavity (); Huet16Millisecond (), is typically  Ding11Wavelength (); Verhagen12Quantum (); Aspelmeyer14Cavity () in optical microresonators, and  Snijders18Observation (); Vaneph18Observation () was experimentally achieved. In a recent experiment, autocorrelation measurements range from to have been achieved with average fidelity in a photon-number-resolving detector Hlousek18Accurate (). Moreover, we set , which is experimentally feasible. The resonator with a radius of can spin at an angular velocity  Maayani18Flying (). By use of a levitated OM system Reimann18GHz (); Ahn18Optically (), can be increased even up to values.

Our analytical results agree well with the numerical one. In the familiarly nonspinning-resonator case, as shown in Fig. 2(a), is reciprocal regardless the direction of the driving light, and always has a dip at and a peak at , corresponding to strong photon antibunching and photon bunching, respectively Xu13Antibunching (). The physical origin of strong photon antibunching is the destructive interference between different paths of two-photon excitations. Specifically, the direct and indirect excitation paths are respectively given by

 |1,0⟩\lx@stackrel√2ϵd⟶|2,0⟩, |1,0⟩\lx@stackrelJ⟶|0,1⟩\lx@stackrelϵd⟶|1,1⟩\lx@stackrel√2J⟶|2,0⟩.

In contrast, for a spinning device, exhibits giant nonreciprocity, which can be seen in Fig. 2(b). The PB can be generated, i.e., , for , while significantly suppressed, i.e., , for , which can be seen more clearly in Fig. 2(c). The nonreciprocal UPB induced by Fizeau light-dragging effect, with up to two orders of magnitude difference of for opposite directions, can be achieved even with a weak nonlinearity and, to our knowledge, has not been studied in coupled-cavity systems.

Spinning the OM resonator results in different Fizeau drag for the counter-circulating whispering-gallery modes of the resonator. By driving the system from the left side, the direct excitation from state to state will be forbidden by the destructive quantum interference with the indirect paths of two-photon excitations, which leads to photon antibunching. In contrast to this, photon bunching occurs by driving the system from the right side, due to the lack of the complete destructive quantum interference between the indicated levels. As shown in Fig. 3(a), increasing the angular velocity results in an opposing frequency linear shift of for light coming from opposite directions. Similarly, the dependence of corresponding to the minimum value of on is clearly seen in Fig. 3(b). From an overall perspective, also experiences linearly shifts with , but with different directions for or ; that is, we observe either a blueshift [see Fig. 3(c)] or a redshift [see Fig. 3(d)] with or , respectively. This indicates a highly-tunable nonreciprocal UPB device. Because of these shifts, we can achieve unidirectional UPB by flexible tuning of the angular velocity and the optical detuning (e.g., , = 12 ). The correlation function is sensitive to the angular velocity [see Fig. 3(b)], and this feature may be used in accurate velocity measurements.

## Iv Optimal Parameters for Strong Antibunching

As discussed above, UPB can be generated nonreciprocally. In this section, we analytically derive the optimal conditions of strong antibunching as follows: because of the fact that the phonon states can be decoupled from the photon states by using the unitary operator (see Appendix A for more details), the states of the system can be expressed as , where and are the photon states and the phonon states, respectively. Under the weak-driving conditions, one can make the ansatz Bamba11Origin ()

 |φ⟩= C00|0,0⟩+C10|1,0⟩+C01|0,1⟩+C20|2,0⟩ +C11|1,1⟩+C02|0,2⟩, (14)

and consider that for and , the steady-state coefficients of one-particle states can given as (see Appendix C for more details)

 0= δLC10+JC01+iϵdC00, 0= δRC01+JC10, (15)

where , and . We have introduced the dissipative terms (proportional to and ) and neglected the higher-order terms, as justified under the weak-driving conditions.

Similarly, we can obtain steady-state coefficients of the two-particle states (see Appendix C for more details)

 0= 2δLC20+√2JC11+i√2ϵdC10, 0= (δL+δR)C11+√2JC20+√2JC02+iϵdC01, 0= 2(δR−δ)C02+√2JC11. (16)

By considering , , and the condition of , we can obtain

 0= κ2(2δ−6Δ−5Δ2F)+4Δ2(2Δ−2δ−5δΔ2F) +4ΔF(4ΔΔF−3δΔ−δΔF+Δ2F)−4J2δ, 0= 8δΔ−12Δ2+κ2+ΔF(6δ−20Δ−8ΔF). (17)

Then we can obtain

 a4Δ4+a3Δ3+a2Δ2+a1Δ+a0=0, (18)

where are defined in Appendix C. For fixed and , the optimal conditions are given by

 Δopt≈ −a3+sgn(E)√λ1−√λ24a4, gopt=   ⎷−ωm[Δopt(4Δ2opt+5κ2)+ΔFλ3]2(2J2−κ2)+2ΔFλ4, (19)

where is the signal function, and are defined in Appendix C.

In order to visualize the UPB effects more clearly, we show the contour plots of the correlation function in logarithmic scale [i.e., ] as function of the radiation-pressure coupling and the optical detuning in Fig. 4(a). By fixing , we obtain the function of in logarithmic scale versus the coupling strength of the resonators and , as shown in Fig. 4(b). These plots show that strong photon antibunching occurs exactly at the values predicted from our analytical calculations (e.g., , ), in Eq. (IV). Moreover, the correlation function as the function of the optical detuning and the angular velocity with different mean thermal phonon numbers as shown in Fig. 5. It is seen that rotation-induced nonreciprocity can still exist by considering thermal phonon noises.

Finally, we note that a state (generated via UPB or another effect) with vanishing (or almost vanishing) second-order photon-number correlations, , is not necessarily a good single-photon source, i.e., the state might not be a (partially-incoherent) superposition of only the vacuum and single-photon states. A good single-photon source is characterized not only by , but also by vanishing higher-order photon-number correlation functions, for . In UPB, for can be greater than , or even greater than 1 Radulaski17 (). Indeed a standard analytical method for analyzing UPB, as proposed by Bamba et al. Bamba11Origin () and applied here in Appendix C, is based on expanding the wave function of a two-resonator system in power series up to the terms () only, as given in Eq. (IV). To obtain the optimal system parameters, which minimize in UPB, this method requires to set as set in Appendix C. Actually, the same expansion of and same ansatz are made in Ref. Bamba11Origin (). These assumptions imply that higher-order correlation functions with vanish too. However, the truncation of the above expansion at the terms is often not justified for a system exhibiting UPB. Indeed, we find parameters for our system, for which and simultaneously . This can be shown by a precise numerical calculation of the steady states of our system based on the non-Hermitian Hamiltonian, given in Eq. (C), in the Hilbert space larger than .

## V Conclusions

In summary, we studied theoretically nonreciprocal UPB in a compound system consisting of a purely optical resonator and a spinning OM resonator. Due to the interference between two-photon excitations paths and the Sagnac effect, UPB can be generated nonreciprocally in our system; that is, UPB can occur when the system is driven from one direction but not from the other, even under the weak OM interactions. The optimal conditions for one-way UPB were given analytically. Moreover, we found this quantum nonreciprocity can still exist by considering thermal phonon noises. Our proposal provides a feasible method to control the behavior of one-way photons, with the potential applications in achieving, e.g., photonic diodes or circulators, quantum chiral communications, and nonreciprocal light engineering in deep quantum regime.

## Appendix A Derivation of Effective Hamiltonian

The coupled system can be described by the following Hamiltonian

 H= H0+Hin+Hdr, H0= ℏωLa†LaL+ℏ(ωR+ΔF)a†RaR+ℏωmb†b, Hin= ℏJ(a†LaR+a†RaL)+ℏga†RaR(b†+b), Hdr= iℏϵd(a†Le−iωdt−aLeiωdt), (20)

where () and () are the photon annihilation (creation) operators for the cavity modes of the optical cavity (denoted by subscript ) and the OM cavity (denoted by subscript ), respectively; () is the annihilation (creation) operator for the mechanical mode of the OM cavity. The frequencies of the cavity fields are denoted by and . is the coupling strength between the two resonators, is the OM coupling strength between the optical mode and the mechanical mode in the OM cavity, denotes the driving strength which is coupled into the compound system through the optical fiber waveguide.

Using the unitary operator to Hamiltonian (A), we obtain a Kerr-type one Gong09Effective ()

 Heff= U†HU = ℏωLa†LaL+ℏ(ωR+ΔF)a†RaR−ℏδ(a†RaR)2 +ℏJ[a†LaRe−δ(b†−b)+aLa†Reδ(b†−b)] +iℏϵd(a†Le−iωdt−aLeiωdt), (21)

where . Under the conditions, and , the Hamiltonian (A) can be read as

 H′eff= ℏωLa†LaL+ℏ(ωR+ΔF)a†RaR−ℏδ(a†RaR)2 +ℏJ[a†LaR+aLa†R] +iℏϵd(a†Le−iωdt−aLeiωdt). (22)

## Appendix B The Fourier Analysis of Fluctuation Terms

According to the Heisenberg equations of motion of Hamiltonian (II), and using semiclassical approximation method, i.e., , , , and , the steady-state values of the system satisfy the following equations:

 0= (κL2+iΔL)α+iJβ−ϵd, 0= [κR2+i(Δ′R+gbqs)]β−iJα, 0= ωmqs−gb|β|2. (23)

Then we can obtain

 b3q3s+b2q2s+b1qs+b0=0, (24)

where

 b0= gbJ2ϵ2d, b1= ωm(κLκR4+J2)2+ωm(κLΔ′R2+κRΔL2)2 −ωmΔLΔ′R(κLκR2+2J2−ΔLΔ′R), b2= 2ωmgb[κ2LΔ′R4+ΔL(ΔLΔ′R−J2)], b3= ωmg2b(κ2L4+Δ2L). (25)

The fluctuation terms of the system can be written as

 ddtδq= ωmδp, ddtδp= −ωmδq−gb(β∗δaR+βδa†R)−γm2δp+ξ, ddtδaL= −(κL2+iΔL)δaL−iJδaR+√κLaL,in, ddtδaR= −(κR2+iΔ′R)δaR−iJδaL−igbqsδaR −igbβδq+√κRaR,in, (26)

where we have neglected higher-order terms, . Here, the steady-state mean value is numerically solved from Eqs. (24) and (B).

By introducing the Fourier transform to the fluctuation equations, we can obtain

 iωδaL(ω)= −(κL2+iΔL)δaL(ω)−iJδaR(ω) +√κLaL,in(ω), iωδaR(ω)= −(κR2+iΔ′′′R)δaL(ω)−iJδaR(ω) −igbβδq(ω)+√κRaR,in(ω), iωδq(ω)= ωmδp(ω), iωδp(ω)= −ωmδq(ω)−gb[β∗δaR(ω)+βδa†R(ω)] −γm2δp(ω)+ξ(ω), (27)

where , then we obtain

 δq(ω)= −gbβ∗χ(ω)δaR(ω)−gbβχ(ω)δa†R(ω) +χ(ω)ξ(ω), (28)

where

 χ(ω)=ωmω2m−ω2+iωγm/2. (29)

Substituting Eq. (B) into Eq. (B), we can obtain

 M(ω)δaR(ω)= ig2bβ2χ(ω)δa†R(ω)−igbβχ(ω)ξ(ω) −iJδL(ω)+√κRaR,in(ω), (30)

where

 M(ω)=κR2+iω+iΔ′′′R−i|β|2g2bχ(ω). (31)

According to Eq. (B), we can obtain

 iωδa†L(ω)= −(κL2−iΔL)δa†L(ω)+iJδa†R(ω) +√κLa†L,in(ω), iωδa