Abstract

In this work, we propose a scheme in three-mode optical systems to simulate a strongly coupled optomechanical system. The nonreciprocity observed in such a three-mode optomechanical circulatory system (OMCS) is explored. To be specific, we first derive a quantum Langevin equation (QLE) for the strongly coupled OMCS by suitably choosing the laser field, then we give a condition for the frequency of the laser and the mechanical decay rate, beyond which the optomechanical system has a unidirectional transmission regardless of how strong the optomechanical coupling is. The optomechanically induced transparency is also studied. The present results can be extended to a more general two-dimensional optomechanical system and a planar quantum network, and the prediction is possible to be observed in an optomechanical crystal or integrated quantum superconducting circuit. This scheme paves a way for the construction of various quantum devices that are necessary for quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Single-photon sources which can be prepared by photon blockade [1–9] are considered as the ideal carriers for quantum information processing such as long-distance quantum communication and large-scale quantum computing [10–16]. In recent years, in order to build a quantum network to manipulate the single-photon non-reciprocal transmission, considerable efforts have been devoted to the development of optomechanical systems [17–19], because these systems are easy to modulate and allow the signal transmission in one direction while block the propagation in the opposite one [20–25]. Due to this special property, it has applications in many aspects, such as isolators [26] and cirulators [27]. Moreover, it has been realized in optomechanical systems [28, 29] and Kerr resonators [30, 31]. In addition, the optomechanical system with nonlinear radiation-pressure induced couplings between a movable mirror and an optical cavity [32, 33] have shown to have an enormous advantage for applications in building optomechanical circular [34] and optomechanical isolator [35]. However, for such systems to work well, it requires that the single-photon coupling strength can approach the order of damping rate of the cavity mode and the frequency of the mechanical mode [36–40]. This is unfortunately not met for the single-photon coupling strength in optomechanical systems [41] for a typical quantum cavity optomechanical system. This stimulates the study of how to enhance the single-photon optomechanical coupling strength.

A number of efforts have been devoted in the enhancement of strong coupling optomechanical system in the single-photon regime [42–46]. Most recently, much attention has been paid to the enhancement of the single-photon optomechanical coupling strength via nonlinear Kerr media and driving laser [47, 48], and an similarity between the optomechanical system and all-optical system with Kerr nonlinearity was found [49].

It is well known that when two electric fields are applied simultaneously on a Kerr nonlinear medium, the cross-Kerr (CK) effect might occur, which can change the refractive index of the Kerr nonlinear medium [50,51]. Hence, in such systems the single-photon optical non-reciprocal transmission can be achieved by adjusting the driving field. Stimulating by the aforementioned studies, we are aware of that an effective CK medium for the optical mode can act as a mechanical resonator, and the coupling strength might be modulated by the driving laser.

In this paper, taking an pure optical system with CK coupling driven by a strong laser into account, we come up with a scheme to realize strong optomechanical nonreciprocity in a three-mode optomechanical circulatory system (OMCS), which consists of one equivalent mechanical mode(simulated by an optical mode) and two optical modes. Here the equivalent mechanical mode is coupled to the two optical modes simultaneously. We show that the strong optomechanical interaction can be realized which originates from the CK nonlinear medium driven by a strong laser field. The optomechanical coupling is enhanced by the CK terms [52]. As an application, we propose typical non-reciprocal quantum devices, such as circulators, diodes, and transistors with this system. We find that there exists an analytical optimal condition for the frequency of an incident photon and mechanical decay rate, beyond which optomechanical system has a directional transmission regardless of how strong the optomechanical coupling is. We also show that the optomechanically induced transparency might provide us with a practical way to achieve optical nonreciprocity [53–55]. Numerical simulations are also performed and the results are in good agreement with the analytical one. Furthermore, we have extended the theory to a two-dimensional optomechanical system and a planar quantum network.

The remainder of this paper is organized as follows. In Sec. 2, we introduce the model and discuss briefly its experimental implementation. The equations of motion are also derived. In Sec. 3, we analyze the how strong coupling effects influence the non-reciprocity of single photons in the OMCS. In Sec. 4, we generalize the theory to a complicated system. And finally in Sec. 5, we conclude and present an outlook.

2. Formalism

2.1. Model

To begin with, we consider an all-optical system consisting of two linearly coupled optical modes (a1 and a2, with frequencies ωa1 and ωa2), both of them are coupled to another optical mode (a3, with frequency ωa3) through an CK nonlinear interaction, see Fig. 1. The optical mode with CK medium is strongly driven by a monochromatic field of frequency ωa3 = ωd at rate εa3, and another two optical modes are driven by external laser fields with frequencies ωa1 = ωa2 = ωd at rates εa1 and εa2, respectively. The total Hamiltonian of this field-reservoir system can be written as (ħ = 1)

H^T=H^0+H^I+H^d,
with
H^0=i=13ωaia^ia^i+ki=13ωikΓ^ikΓ^ik,H^d=ii=13(εaia^iei(φaiωdt)H.c),H^I=j=12χaja^ja^ja^3a^3+J(a^1a^2+a^2a^1)+ki=13gik(a^iΓ^ik+Γ^ika^i),
where a reservoir that couples to the system is introduced. Ĥ0 denotes the Hamiltonian of the system and the reservoir with frequencies ωai and ωik, respectively. a^i (âi) and Γ^ik (Γ̂ik) for i = 1, 2, 3 are the corresponding creation (annihilation) operators of the optical modes and environment modes. Hamiltonian ĤI describes the CK interactions between the optical modes with coupling strength χa1 and χa2, the linear interaction between the optical modes with coupling J, and the couplings between the system and the reservoir. The Hamiltonian Ĥd represents the pump fields applied to the system with phases ϕai. Without loss of generality, we assume that the parameters χa1, χa2, gik, εai, and J are real.

 

Fig. 1 The schematic illustration of the model. A cyclic three-mode optical system driven by three pump fields of amplitudes εa1, εa2, and εa3 with frequency ωd. Two of the optical modes are linearly coupled with each other, while they are coupled to another described by cross-Kerr (CK) nonlinear terms.

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2.2. Main theory

In this section, we present an efficient approach to realize conversion from CK nonlinearly coupling to strong optomechanical coupling in the few-photon regime. The simulated mechanical mode is realized by applying strong coherent drive on one of the two optical modes.

First, in the rotating reference frame at the frequency ωd of the driving field, we apply a unitary transformation U^=exp(iωdi=13a^ia^it) to Eq. (1), and the total Hamiltonian in the rotating frame becomes Ĥrot = ÛĤTÛiÛdÛ/dt. With a strong drive, the operator â3 can be written as a summation of its mean value and a quantum fluctuation. Such a treatment is equivalent to introduce a displacement operator D^a3(α3)=exp(α3a^3α3*a^3), defining H^=D^a3(α3)H^rotD^a3(α3), we have

H^=H^0+H^I+H^d(B^3+B^3)+F,
with
H^0=j=12Δaja^ja^j+Δa3a^3a^3+ki=13ωikΓ^ikΓ^ik,H^I=H^Ij=12χaj|α3|(a^3eiθ+a^3eiθ)a^ja^j,H^d=ii=13(εaia^ieiφaiH.c)Δa3(α3a^3+α3*a^3),
where Δai = ωaiωd is the detuning of the pump field from its driven mode. α3 is the coherent displacement amplitude, which needs to be determined in the transformed quantum Langevin equations (QLEs). Δ′a1 = Δa1 + χa1 |α3|2 and Δ′a2 = Δa2 + χa2 |α3|2 are the detunings including the frequency shifts caused by the strong coupling. The enhanced coupling strength is gaj = χaj |α3| (j = 1, 2). θ is the quadrature phase angle of the mode a3, i.e., the coherent displacement can be written as α3 = |α3|eiθ. The environment operator is B^3=kg3kα3*Γ^3k. We introduced the normalized driven field F = |α3|2Δa3 + a3(α3ea3H.c.), which can be formally seen as a generalized plane wave. Clearly, the system described by Eqs. (3) and (4) are equivalent to a system consisting two optical modes (â1 and â2) and one mechanical mode (â3).

In this work, we focus on few-photon OMCS and hence consider the regime max[majχaj] ≪ Δa3, then the CK nonlinear interaction term in ĤI can be safely ignored [56], and when we choose the laser parameters ωd and εa3 to satisfy equation

εa3eiφa3+(iΔa3γa3/2)α3s=0.
Eq. (3) becomes
H^=H^0s+H^Is+H^ds(B^3s+B^3s)+Fs,
where γa3 is the damping rate of the mechanical mode â3, and α3s is a coherent displacement that satisfies the above laser condition. Eq. (6) is generalized Hamiltonian of strong coupling OMCS. Here the subscript s stands for the parameters that ωd and εa3 satisfy Eq. (5). An alternative viewpoint of Eq. (5) can be found in Appendix A.

2.3. Experimental implementation

In this section, we present briefly an possible experimental implementation of Eq. (2). The present model has been achieved recently in the superconducting circuit experiment, which consists of three cavities with CK nonlinear medium. The mirrors on both sides of the cavity are fixed, and the mirrors in the middle can move. The CK effect can be achieved via a two-level system or a superconducting charge qubit coupled to the two cavity modes [57,58], as illustrated in Fig. 2(a). An equivalent quantum superconducting circuit with a superconducting charge qubit coupled to two on-chip cavities is shown in Fig. 2(b). In addition, such a system can be implemented in an optomechanical crystal coupled to the other optomechanical crystal [59], as illustrated in Fig. 2(c). The geometry of the optomechanical crystal structure consists of a silicon nanobeam with rectangular holes formed by thin cross-bars connected on both sides to thin rails [60], and the nodes of the two optomechanical crystals are coupled by a nonlinear Kerr medium and are driven by a strong laser. The optical modes at both ends are linearly coupled.

 

Fig. 2 Schematic diagram of typical experimental configurations of the all-optical CK nonlinear coupling system. (a) The mirrors on both sides of the cavity are fixed, and the mirrors in the middle can move. The assisting qubits are used to induce the CK effect between the two cavity modes. (b) An equivalent quantum circuit with superconducting charge qubits. (c) Two optomechanical crystals are coupled to each other.

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2.4. Quantum langevin equations

Substituting Eq. (6) into the Heisenberg equation and taking the damping and corresponding noise term into account, we can obtain a closed integro-differential equation for â3(t),

ddta^3=ka^3+ij=12gajseiθa^ja^j+γa3a^3in+(iΔa3γa32)α3s+εa3eiφa3.
For details, see Appendix B [61]. The single-photon optomechanical coupling strength is gajs=χaj|α3s|, and a^3in is the noise operator with zero mean, k = γa3/2 + iΔa3.

As aforementioned, the average of a3 can be determined in the displacement representation when the coherent displacement amplitude α3s obeys Eq. (5). To analyze the influence of the driving laser, we write α3s=|α3s|eiθ with

θ=arctan(γa32sinφa3+Δa3cosφa3γa32cosφa3Δa3sinφa3),|α3s|=εa3|k|.
It can be seen from Eq. (8) that α3s is tunable by εa3 and Δa3. The value of |α3s| could be large enough in the strong-driving case εa3 ≫ {Δaj, γaj}. In particular, the interaction strength gajs is controllable because we can obtain a desired α3s by designing a proper driving amplitude εa3.

Repeating the above process, we get the QLEs for the operators of the optical and equivalent mechanical modes

ddta^1=+Γ^1a^1iJa^2+εa1eiφa1+γa1a^1in,ddta^2=+Γ^2a^2iJa^1+εa2eiφa2+γa2a^2in,ddta^3=ka^3+ij=12gajseiθa^ja^j+γa3a^3in,
where γaj is the damping rate of the optical mode aj, and a^jin is the noise operator with zero mean value. In the derivation, a normalized dampling rate Γ̂j = −γaj /2 + iκ̂j was defined with κ^j=gajs(a^3eiθ+a^3eiθ)Δajs, and Δajs=Δaj+χaj|α3s|2.

We are now at the position derive the dynamics matrix of the OMCS. For this purpose, we write each operator of the modes as a sum of a mean value and a small quantum fluctuation, i.e., âi = βi + δâi, the steady mean values of βi can be obtained by setting the (first order) derivative of those operator with respect to time to zero,

η1=k2εa1eiφa1iJεa2eiφa2k1k2+J2,η2=k1εa2eiφa2iJεa1eiφa1k1k2+J2,η3=ij=12gajs|ηj|2eiθk,
where the auxiliary parameter kj=γaj/2+iΔajσs with Δajσs=Δajs2gajsRe[βexp(iθ)]. Substituting Eq. (10) into Eq. (9) and omitting the high-order terms in small quantum fluctuations δâi, we obtain the linearized QLEs
ddtδa^1=k1δa^1+iGa1σ^iJδa^2+γa1δa^1in,ddtδa^2=k2δa^2+iGa2σ^iJδa^1+γa2δa^2in,ddtδa^3=kδa^3+ij=12τ^jeiθ+γa3δa^3in,
where σ^=δa^3exp[iθ]+δa^3exp[iθ] and τ^j=Gaj*δa^j+Gajδa^j. Gajgajsηj|Gaj|exp[iϕaj] is the effective single-photon-optomechanical coupling.

For convenience, we assume that the system works in the resolved sideband regime, i.e., {γai, Gaj, J} ≪ Δa3 and Δajσs~Δa3. Under these assumptions, we can apply the rotating-wave approximation (RWA) to the above QLEs and neglect the counter-rotating terms. Then Eq. (11) can be simplified to be

ddtυ^=μυ^+Lυ^in,
where the fluctuation vector υ̂ = (δâ1, δâ2, δâ3)T, the input field vector υ^in=(δa^1in,δa^2in,δa^3in)T, the damping vector L=diag(γa1,γa2,γa3), and the coefficient vector μ in terms of system parameters takes
μ=[k1iJiGa1eiθiJk2iGa2eiθiGa1*eiθiGa2*eiθk],
The characteristic equation |μλI| = 0 can be reduced to C3λ3+C2λ2+C1λ1+C0=0, where the coefficients can be derived by straightforward but tedious algebra. For the third-order polynomial, according to the Routh-Hurwitz criterion [62], the stability conditions can be obtained as C2, C0 < 0 and C2C1 < C0 [63]. In the following discussion, the stability conditions are satisfied by choosing properly the parameters [64].

Defining the Fourier transform for operators

τ(ω)=[τ(t)]=+τ(t)eiωtdt,
(for any operator τ) and using the derivation role [τ̇(t)] = −iωℱ[τ(t)], the solution to Eq. (12) in the frequency domain is
υ(ω)=(μiωI)1Lυin(ω),
where I denotes the identity vector.

Substituting Eq. (15) into the input-output relation υout(ω) + υin(ω) = υ(ω) [65], we obtain

υout(ω)=Γ(ω)υin(ω),
where the corresponding output vector υout(ω)=(δa^1out,δa^2out,δa^3out)T and the scattering matrix
Γ(ω)=LT[μiωI]1LI.
The specific form of Γ(ω) is given as
1|A|[γa12A11*|A|γa1γa2A12*γa1γa3A13*γa2γa1A21*γa22A22*|A|γa2γa3A23*γa3γa1A31*γa3γa2A32*γa32A33*|A|],
where |A|=j=12ξjexp[2iθ]+j=12ζj(kiω) is the value of the determinant of A with ξj=|Gaj|2(kj+1iω)2iJRe[Ga1*Ga2] and ζj= (kj) + J2. The remaining companion matrix elements are given as
Ajj*=(kjiω)(kiω)+|Gaj|2e2iθ,A33*=(k1iω)(k2iω)+J2,A12*=J(ik+ω)Ga1*Ga2e2iθ,A21*=J(ik+ω)Ga1Ga2*e2iθ,A13*={Ga1*(ik2+ω)+JGa2*}eiθ,A31*={Ga1(ik2+ω)+JGa2}eiθ,A23*={Ga2*(ik1+ω)JGa1*}eiθ,A32*={Ga2(ik1+ω)+JGa1}eiθ.
The spectrum of the output fields is defined by
Sout(ω)=dωυout(ω)υout(ω).
Substituting Eq. (16) into Eq. (20), we get
Sout(ω)=T(ω)Sin(ω)+SQ(ω)+ST(ω),
where T(ω) is the contribution arising from the presence of a single photon in the input field. SQ (ω) is the contribution from the incoming vacuum field, while ST (ω) is the contribution from the fluctuations of the mirror.

To build a single-photon quantum device, we now examine T(ω). Since the contribution of thermal noise SQ (ω) and vacuum noise ST (ω) is much smaller than the input field Sin (ω) [66], we can neglect SQ (ω) and ST (ω) and obtain

Sout(ω)=T(ω)Sin(ω),
where Sin(ω)=(Sa1in,Sa2in,Sa3in)T, Sout(ω)=(Sa1out,Sa2out,Sa3out)T, and the scattering probability is T(ω) = |Γ(ω)|2.

3. Applications

In order to show the application of our theoretical model in building non-reciprocal quantum devices, in the following we demonstrate how various typical devices work with the OMCS of strong coupling.

3.1. Optomechanical circulator

In this section, we study the optimal conditions for single-photon nonreciprocity and optomechanical circulation in a three-mode OMCS.

According to Eq. (22), the analytical solution of the transmission probability from a2 to a3 is Ta3a2(ω)=A23*A23*/|A|2. The effective part in the physical system can be written as

A23*A23*=J2|Ga1|22JRe[Ga2*Ga1(ik1+ω)]+|Ga2|2(|k1|2+ω22ωIm[k1]).
Similarly, the valid part of a3 to a2 as follows
A32*A32*=J2|Ga1|22JRe[Ga1*Ga2(ik1+ω)]+|Ga2|2(|k1|2+ω22ωIm[k1]).
From Eqs. (23) and (24), keeping the other parameters fixed, we can see that Ta2a3(ω)=C1 and Ta3a2(ω)=C2 for any θ. It is shown that the phase, which introduced by the strong coupling is trivial to the optomechanical transmission between the a2 and a3 modes.

The optomechanical nonreciprocal behavior is induced by the phase difference φ of the two optomechanical coupling rates, which break the time-reversal symmetry of the OMCS [67]. Scattering probabilities Ta2a3(ω) and Ta3a2(ω) as functions of the incoming signal ω for different phase difference φ is shown in Fig. 3, where the parameters chosen are γaj = γa3, Δajσs=Δa3=10γa3, J = Ga2 = Ga1e = 0.5γa3. It is shown that the photon transmission satisfies the Lorentz reciprocal theorem provided φ = (n is an integer) [68]. When φ, the Lorentz reciprocal theorem is not satisfied, and the OMCS enters the nonreciprocal regime. The perfect nonreciprocity appears at the point φ = 2 + π/2 or φ = 2π/2(ω = 10γa3), which is in good agreement with the analytical result in Fig. 3.

 

Fig. 3 Scattering probabilities Ta2a3(ω) and Ta3a2(ω) as functions of the frequency of the incoming signal ω (in units of γa3) for different phase different φ from 0 to 1 (in units of π). The other parameters are γaj = γa3, Δajσs=Δa3=10γa3, J = Ga2 = Ga1e = 0.5γa3. The right-hand panel is a contour plot for the left-hand panel, and the purple portion of the figure is replaced by white.

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In Fig. 4, we show the scattering probabilities Ta2a3(ω) and Ta3a2(ω) as functions of the incoming signal ω for different optomechanical coupling constant Ga2 with parameters γaj = γa3, Δajσs=Δa3=10γa3, J = 0.5γa3, Ga1 = −iGa2. In the regime Ga2γa3, we can see that Ta3a2(ω)Ta2a3(ω)0. When the optomechanical coupling strength Ga2 increase, the scattering probability increase, too, and it can reaches the best value Ta2a3(ω)=1 (ω = 10γa3) at Ga2 = 0.5γa3.

 

Fig. 4 Scattering probabilities Ta2a3(ω) and Ta3a2(ω) as a function of the frequency of the incoming signal ω (units of γa3) for different optomechanical coupling constant Ga2(units of γa3). The other parameters chosen are γaj = γa3, Δajσs=Δa3=10γa3, J = 0.5γa3, Ga1 = −iGa2, Ga2 ∈ [0, 2γa3]. The right-hand panel is a contour plot for the left-hand panel, and the dark blue part of the left figure corresponds to the white in the right figure.

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It is instructive to find the optimal conditions for observing perfect optomechanical circulators. A perfect optomechanical circulator is obtained when we have Ta2a1(ω)=Ta3a2(ω)=Ta1a3(ω)=1

or Ta3a1(ω)=Ta2a3(ω)=Ta1a2(ω)=1 with all of the other scattering probabilities equal to zero [69]. By numerical calculations and careful examination of the scattering probabilities Ta3a2(ω) and Ta2a3(ω), we find that the optimal conditions are met if the parameter satisfying the following conditions γaj = γa3, Δajσs=Δa3=10γa3, J = Ga2 = Ga1e = 0.5γa3, φ = 2π/2 or γaj = γa3, Δajσs=Δa3=10γa3, J = Ga2 = Ga1e = 0.5γa3, φ = 2 + π/2. In this case, the scattering matrix Eq. (17) takes

Γ+=[00ii00010]orΓ=[0i0001i00],
which are consistent with the condition for perfect optomechanical circulators. Γ+ represents a perfect optomechanical circulator with the signal is transferred from one OMCS to another counterclockwise (a1a2a3a1) for φ = 2 + π/2. While Γ describes an ideal optomechanical circulator with the signal is transferred from one OMCS to another clockwise (a1a3a2a1) for φ = 2π/2.

3.2. Optomechanical diode

The optomechanical diode is a device that restricts the photon transmission in only one direction [70]. According to Eqs. (23) and (24), under the optimal parameter conditions Δajσs=Δa3=10γa3, Ga1 = −iGa2 ≪ Δa3, γaj = γa3, J = 0.5γa3, we are delighted to find that when the frequency of incident photon is tuned to be ω/γa3 = 10, A23*A23*=0, A32*A32*=1 are satisfied. The transmission probabilities a2a3 and a3a2 are then independent upon the optomechanical coupling constant Ga1. The transmission probability is constantly equal to zero and one, respectively. Thus, light traveling through an optical cavity to a resonator is mostly prohibited, while the light that travels in the reverse direction can pass freely.

It is well known that the width of the transmission window(bandwith) is the other parameter to evaluate the photon transmission performance of quantum devices [71]. In the following, under the best parameter conditions, we study the effect of strong coupling Ga1 on photon transmission bandwidth. Under the RWA, Fig. 5(a) and Fig. 5(c) plots the transmission probability between the two optical cavities a2a1 and optomechanical system a3a2 as a function of the detuning ω/γa3 for different optomechanical coupling constant Ga1. We observe that when the optomechanical coupling constant Ga1 is less than γa3, both the optical system and optomechanical system of bandwidth increases slowly as the coupling strength increases. But when the optomechanical coupling constant Ga1 is greater than γa3, bandwidth surge appears. The bandwidth surge is caused by the splitting of the formant. When the critical point is reaching, the energy spectrum of the system will produce an energy level staggering phenomenon, therefore inducing energy level splitting.

 

Fig. 5 Numerical calculations for the transmission rates. Fig. (a)–(d) shows the transmission probability of optical and optomechanical system as a function of the frequency of the incoming signal ω(units of γa3) for different optomechanical coupling constant Ga1. (a) shows the optical transmission given by calculations with RWA. (b) is for the transmission considering the anti-rotational terms. (c) shows the optomechanical transmission with RWA. (d) shows the optomechanical transmission with anti-rotational wave terms. The other parameters chosen are set to meet the optimal parameter conditions.

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Further, we point out that in the case of weak coupling, the effect of anti-rotational wave terms on bandwidth is negligible. But under strong coupling, both of the optical subsystem and optomechanical subsystem show that the anti-rotational wave term would result in significant asymmetric vacuum splitting, see Fig. 5(b) and Fig. 5(c).

3.3. Optomechanical transistor

Electromagnetically induced transparency or optomechanical induced transparency in OMCS provides us with a practical way to achieve optical switch or optomechanical switch, the physics behind these devices is the optical nonreciprocity or optomechanical nonreciprocity [72].

Apply a probe field to cavity a1 in the form of i(εpa1 exp ptH.c) (see the dotted line in Fig. 1) and assume that the amplitude of the probe field εPεa1,2,3, it is easy to find the steady-state solutions of the operators ai that would not be affected by the probe field. Hence, the dynamics equation describing this system differs from Eq. (11) only at a εp exp i(ωdωp)t term, which corresponds to a1.

Finding the steady-state solutions and using the input-output relationship (with δa1in=εP/γa1 and δajin=0 (forj=2,3)) [73], we obtain the optical response of the system to the probe field,

Ta2a1=γa1γa2|β|2{|Ga1|2|Ga2|2+|Jk|22Jσ},
the optomechanical response of the system to the probe field can be derived as
Ta3a2=γa3γa2|Ga1|2|k|2+|JGa2|22Jσ|Ga1|2|Ga2|2+|Jk|22Jσ,
with the denominator
β=k2(k1k+J2)+j=12kj|Gaj+1|22iJRe(Ga1Ga2*),
where σ=Im(kGa1Ga2*) and σ=Im(k2Ga2Ga1*) with k′ = γa3/2 + iΔ′a3 and kj=γaj/2+iΔajs¯. The Δ′a3 = Δa3 + (ωdωp) and Δajs¯=Δajσs+(ωdωp) being the detunings in the new frame.

Keeping the optimal system parameters fixed, we can clearly see from Eq. (26) that the numerator is much small than the denominator whereby the transmission of the probe field between two optical modes is prohibited. This indicates that the switch of two optical modes is always off.

With regard to Eq. (27), the transmission coefficient between optical mode and equivalent mechanical mode can be written as Ta3a2=1εT. Re[εT] characterizes the absorption, and Im[εT] describes the dispersion [74]. In Fig. 6 we plot the absorption probability as a function of coupling constant Ga1. When Ω = ωpωd = 10γa3, the switch is always off. Conversely, in the condition Ga1 = 0.5γa3 with Ω ≤ 0, Ω ≥ 20 and Ta3a21, the detection field exhibits OMIT behavior and the switch is fully open.

 

Fig. 6 Numerical calculation Re[εT ] as a function of Ω (units of γa3) for different optomechanical coupling constant: Ga1 = 0.1γa3 (blue solid line with stars), Ga1 = 0.2γa3 (red solid line with circles), Ga1 = 0.3γa3 (orange solid line with squares), Ga1 = 0.4γa3 (solid brown line), Ga1 = 0.5γa3 (green dotted line).

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4. General expansion

In this section, we extend the present theory an optical cavity array that each cavity couples to an (simulated) mechanical mode. The theory can also be extended to a system with two equivalent mechanical modes and one optical mode.

4.1. 1D optomechanical loop model

Now we discuss the case where an equivalent mechanical mode is coupled to an optical cavity array. As shown in Fig. 7(a), the Hamiltonian of the system can be written as

H^a=H^a0+H^aI+H^ad,
with
H^a0=τ=1NΔaτa^τa^τ+Δama^ma^m,H^aI=τ=1NJ(a^τa^τ+1+a^τ+1a^τ)τ=1Ngams^ma^τa^τ,H^ad=τ=1Niεaτa^τeiφaτ+κma^m+H.c,
where m represent an equivalent mechanical mode, and τ stands for the optical modes (changing from 1 to N). Δaτ=Δaτ+|αms|2χaτ is the corresponding detuning of optical mode, Δam is the detuning of mechanical mode, and κm=iεameiφamΔamαms. The corresponding single-photon optomechanical coupling constant is gams=χam|αms|, and ^m=a^meiθm+a^meiθm. The αms satisfies the equation
0=εameiφam+(iΔamγam2)αms,
further discussion related to Eq. (29) can be found in Refs. [75,76].

 

Fig. 7 (a) A loop system where an equivalent mechanical mode is coupled to each cavity in an optical cavity-array. (b) The three-mode optomechanical system consists of two mechanical modes and one optical mode. (c) The general form of a planar quantum network, which is composed of arbitrary optical mode and a mechanical mode.

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4.2. 2D OMCS model

In the last section, we have discussed the case where only one equivalent mechanical mode is coupled to a single cavity. The theory can be easily extend to the situation where two equivalent mechanical modes coupled to an optical mode, as schematically shown in Fig. 7(b). The system with two simulated mechanical modes are called two-dimensional OMCS. The two equivalent mechanical modes are linearly coupled, while each equivalent mechanical mode is coupled to the optical mode by Kerr medium. The Hamiltonian of such a system is,

H^2=H^20+H^2I+H^2d,
with
H^20=j=12Δaja^ja^j+Δa3ssa^3a^3,H^2I=J(a^1a^2+a^2a^1)j=12gajss^ja^3a^3,H^2d=i=13iεaia^ieiφaij=12oja^j+H.c,
where j varies from 1 to 3, Δa3ss=Δa3+j=12χaj|αjss|2 is the detuning of optical mode, Δaj is the detuning of mechanical mode, and ^j=a^jeiθj+a^jeiθj. The corresponding single-photon optomechanical coupling constant is gajss=χaj|αjss|, and oj=Δajαjss+Jαj+1ss. Following the same procedure as in the last section, a^1a^1+α1ss, and a^2a^2+α2ss. We can obtain equations that α1ss and α2ss satisfy
0=+εa1eiφa1+(iΔa1γa12)α1ss,0=+εa2eiφa2+(iΔa2γa22)α2ss+iJα1ss.
Eq. (32) contains all information about the 2D optomechanical system, see Ref. [77].

4.3. Generalized planer optomechanical quantum network

As shown, when optical modes are coupled via cross-Kerr couplings, the optical mode that is driven by a strong coherent laser is equivalent to a mechanical mode. This virtue merit can be generalized to N-dimensional planar optomechanical quantum network by designing the couplings between the optical modes of an optical network and selecting the driving laser field intensity. This generalization could be implemented with the existing optomechanical quantum network, see Fig. 7(c) [78], and it benefits for building a quantum network, and achieving long-distance quantum communication and large-scale quantum computing.

5. Conclusion

In summary, we proposed a scheme to simulate the strongly coupled optomechanical circulatory system by driving one of the bosonic modes that is coupled through a cross-Kerr interaction with others. The QLEs for this system is derived and the nonreciprocal feature of the system and its application is studied.

We find that cross-Kerr nonlinear medium together with a strong drive can induce strong optomechanical couplings between the optical mode and the mechanical mode. As an application, we propose typical non-reciprocal quantum devices, such as circulators, diodes, and transistors with this system. We find that there exists an analytical optimal condition for the frequency of an incident photon and mechanical decay rate, beyond which optomechanical system has a directional transmission regardless of how strong the optomechanical coupling is. We also find that the optomechanically induced transparency might provide us with a practical way to achieve optical nonreciprocity. The above results are in good agreement with the results of numerical simulation. Further, the theory has been extended to a two-dimensional optomechanical system and a planar quantum network.

Appendix A: The specific derivation of Eq. (5)

The Hamiltonian of the system can be written as

H^s=i=13ωaia^ia^i+j=12χaja^ja^ja^3a^3+J(a^1a^2+a^2a^1)+ii=13(εaia^iei(φaiωd)H.c.).
In order to study the dynamics of the system, we work with the rotating frame with respect to the frequency of the pump field. Applying a unitary transformation U^(t)=exp[iωdi=13a^ia^i] to Eq. (35), we get
H^s=i=13Δaia^ia^i+j=12χaja^ja^ja^3a^3+J(a^1a^2+a^2a^1)+ii=13(εaia^ieiφaiH.c.),
where Δai = ωaiωd is the detuning of the pump field from the cavity mode.

In the Schrödinger picture and consider the effect of environment, the evolution of the system is governed by the quantum Markov master equation

ρ˙=i[ρ,H^s]+i=13γai{(n¯ai+1)^[a^3]+n¯ai^[a^i]}ρ.
where γai is the damping rate of the corresponding cavity mode a3, and ai is the ith environment thermal excitation number. The Lindblad operator is ^[a^i]=a^iρa^i(a^ia^iρ+ρa^ia^i)/2.

In the strong-driving limit, the photon number in mode a3 is large. This suggests to write operator a3 as a sum of an average and a fluctuation. Alternatively, This can be treated by performing the following displacement transformation,

ρ=𝒟^a3(α3)ρ𝒟^a3(α3),
where ρ′ is the density matrix of the system after the displacement. The displacement operator is D^a3(α3)=exp(αa^3α*a^3).

In the displacement transformation, the left-hand side of the master equation becomes

ρ˙=t𝒟^a3(α3)ρ𝒟^a3(α3)+𝒟^a3(α3)ρt𝒟^a3(α3)+𝒟^a3(α3)ρ˙𝒟^a3(α3).
Substituting 𝒟^a3(α3)=exp(|α3|2/2)exp(a^3α3*)exp(a^3α3) into Eq. (39), we can obtain
t𝒟^a3(α3)=12(α3α˙3*α˙3α3*)𝒟^a3(α3)𝒟^a3(α3)(a^3α˙3*a^3α˙3),
thus, Eq. (39) can be reduced to
ρ˙=[𝒟^a3(α3)ρ˙𝒟^a3(α3),(a^3α˙3a^3α˙3*)]+𝒟^a3(α3)ρ˙𝒟^a3(α3).
Considering 𝒟^a3(α3)a^3𝒟^a3(α3)=a^3α3, we obtain the transformed master equation as follow
ρ˙=+i[ρ,H^s]+i=13γai{(n¯ai+1)^[a^i]+n¯ai^[a^i]}ρ+{α˙3+(iΔa3γa32)α3+εa3eiφa3}[a^3,ρ]{α˙3*+(iΔa3γa32)α3*+εa3eiφa3}[a^3,ρ],
where the transformed Hamiltonian in the displacement representation becomes
H^s=+j=12Δaja^ja^j+Δa3a^3a^3+j=12χaja^ja^ja^3a^3+J(a^1a^2+a^2a^1)j=12χaja^ja^ja^3a^3j=12gajs(a^3eiθ+a^3eiθ)a^ja^j.
The average of a3 can be determined in the displacement representation when the coherent displacement amplitude α3s obeying the equation
α˙3s+(iΔa3γa32)α3s+εa3eiφa3=0.

Appendix B: The derivation of Eq. (7)

Taking the damping term into account and substituting Eq. (6) into the Heisenberg equation, we obtain a set of equations for the system operator â3

ddta^3=iΔa3a^3ikg3kΓ^3k+iΔa3α3s+ij=12gajseiθa^ja^j+εa3eiϕa3,
and environment operator Γ̂3k,
ddtΓ^3k=iω3kΓ^3kig3ka^3ig3kα3s.
Solving Eq. (45) for Γ̂3k, we get
Γ^3k=ig3k0t{a^3(τ)+α3s}eiω3k(tτ)dτ+Γ^3k(0)eiω3kt.
Inserting Eq. (46) into Eq. (44), we have
ddta^3=+f^a3kg3k20t{a^3(τ)+α3s}eiω3k(tτ)dτ+iΔa3(α3sa^3)+ij=12gajseiθa^ja^j+εa3eiϕa3,
with
f^a3(t)=ikg3kΓ^3k(0)eiω3kt,
here a3(t) is a noise operator because it depends upon the environment operators Γ̂3k (0).

First, assuming that the modes of the field are closely spaced in frequency, we can replace the summation over k by an integral of ω. As in the Weisskopf-Wigner approximation, the summation in Eq. (47) yields a δ(tτ) function and the integration can then be carried out. We obtain

ddta^3=+f^a3(t)0tg32(ω)πδ(tτ){a^3(τ)+α3s}dτ+iΔa3(α3sa^3)+ij=12gajseiθa^ja^j+εa3eiϕa3,
with
f^a3(t)=i0g3(ω)Γ^30(ω)eiωtdω.
Hereafter, we assume that g3(ω) is dependent of frequency over a wide range of frequencies around ω = 0, by Markov approximation we get g32(ω)=γa3/2π, and Eq. (50) becomes
ddta^3=(iΔa3γa32)a^3+ij=12gajseiωa^ja^j+εa3eiϕa3+(iΔa3γa32)α3s+γa3a^3in,
with
a^3in=12π0ig3(ω)Γ^30(ω)eiωtdω.

Funding

National Natural Science Foundation of China (NSFC) (11534002, 11775048, 11705025); Fundmental Research Funds for the Central Universities (2412019FZ044).

Acknowledgments

Cheng shang thanks Shuang Xu for helpful discussions. This work is supported by National Natural Science Foundation of China (NSFC) under Grants No. 11534002, No. 11775048, and No. 11705025, the Fundamental Research Funds for the Central Universities under No. 2412019FZ044.

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References

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  1. A. Imamođlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev. Lett. 79, 1467 (1997).
    [Crossref]
  2. M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
    [Crossref]
  3. H. Flayac, D. Gerace, and V. Savona, “An all-silicon single photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).
    [Crossref]
  4. H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015)
    [Crossref]
  5. H. Z. Shen, Y. H. Zhou, H. D. Liu, G. C. Wang, and X. X. Yi, “Exact optimal control of photon blockade with weakly nonlinear coupled cavities,” Opt. Express 23, 32835 (2015).
    [Crossref] [PubMed]
  6. Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
    [Crossref]
  7. H. Z. Shen, S. Xu, Y. H. Zhou, G. C. Wang, and X. X. Yi, “Unconventional photon blockade from bimodal driving and dissipations in coupled semiconductor microcavities,” J. Phys. B 51, 035503 (2018).
    [Crossref]
  8. Y. H. Zhou, H. Z. Shen, X. Q. Shao, and X. X. Yi, “Strong photon antibunching with weak second-order nonlinearity under dissipation and coherent driving,” Opt. Express 24, 17332 (2016).
    [Crossref]
  9. Y. H. Zhou, H. Z. Shen, X. Y. Zhang, and X. X. Yi,“ Zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian with a gain cavity,” Phys. Rev. A 97, 043819 (2018).
    [Crossref]
  10. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
    [Crossref] [PubMed]
  11. I. A. Walmsley and J. Nunn, “Editorial: Building Quantum Networks,” Phys. Rev. Appl. 6, 040001 (2016).
    [Crossref]
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    [Crossref]
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2019 (2)

B. J. Li, R. Huang, X. W. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photon. Res. 7, 630 (2019).
[Crossref]

L. Tang, J. S. Tang, W. D. Zhang, G. W. Lu, H. Zhang, Y. Zhang, K. Y. Xia, and M. Xiao, “On-chip chiral single photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99, 043833 (2019).
[Crossref]

2018 (13)

R. Huang, A. Miranowicz, J. Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

G. L. Li, X. Xiao, Y. Li, and X. G. Wang, “Tunable optical nonreciprocity and a phonon-photon router in an optomechanical system with coupled mechanical and optical modes,” Phys. Rev. A 97, 023801 (2018).
[Crossref]

B. He, L. Yang, X. S. Jiang, and M. Xiao, “Transmission Nonreciprocity in a Mutually Coupled Circulating Structure,” Phys. Rev. Lett. 120, 203904 (2018).
[Crossref] [PubMed]

L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect, Optica 5, 279 (2018).
[Crossref]

S. Zippilli, N. Kralj, M. Rossi, G. D. Giuseppe, and D. Vitali, “Cavity optomechanics with feedback-controlled in-loop light,” Phys. Rev. A 98, 023828 (2018).
[Crossref]

H. Z. Shen, S. Xu, Y. H. Zhou, G. C. Wang, and X. X. Yi, “Unconventional photon blockade from bimodal driving and dissipations in coupled semiconductor microcavities,” J. Phys. B 51, 035503 (2018).
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Y. H. Zhou, H. Z. Shen, X. Y. Zhang, and X. X. Yi,“ Zero eigenvalues of a photon blockade induced by a non-Hermitian Hamiltonian with a gain cavity,” Phys. Rev. A 97, 043819 (2018).
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I. Cohen and K. Mølmer, “Deterministic quantum network for distributed entanglement and quantum computation,” Phys. Rev. A 98, 030302 (R) (2018).
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Z. C. Shi, D. Ran, L. T. Shen, Y. Xia, and X. X. Yi, “Quantum state engineering by periodical two-step modulation in an atomic system,” Optics Express 26(26), 34789 (2018).
[Crossref]

H. Z. Shen, C. Shang, Y. H. Zhou, and X. X. Yi, “Unconventional single-photon blockade in non-Markovian systems,” Phys. Rev. A 98, 023856 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, and F. Nori, “Flying couplers above spinning resonators generate irreversible refraction,” Nature (London) 558, 569 (2018).
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X. W Xu, L. N. Song, Qiang Zheng, Z. H. Wang, and Yong Li, “Optomechanically induced nonreciprocity in a three-mode optomechanical system,” Phys. Rev. A 98, 063845 (2018).
[Crossref]

X. Z. Zhang, Lin Tian, and Yong Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

2017 (4)

X. W. Xu, A. X. Chen, Y. Li, and Y. X Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
[Crossref]

Y. H. Zhou, S. S. Zhang, H. Z. Shen, and X. X. Yi, “Second-order nonlinearity induced transparency,” Opt. Lett. 42(7), 1289 (2017).
[Crossref] [PubMed]

Q. T. Cao, H. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. Gong, and Y. F. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118, 033901 (2017)
[Crossref] [PubMed]

2016 (8)

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1578 (2016).
[Crossref] [PubMed]

I. A. Walmsley and J. Nunn, “Editorial: Building Quantum Networks,” Phys. Rev. Appl. 6, 040001 (2016).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal Photon Laser,” Phys. Rev. Appl. 6, 040001 (2016).

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577 (2016).
[Crossref] [PubMed]

Z. Shen, Y. L. Zhang, Y. Chen, C. L. Zou, Y. F. Xiao, X. B. Zou, F. W. Sun, G. C. Guo, and C. H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657 (2016).
[Crossref]

Q. C. Wu, Y. H. Chen, B. H. Huang, J. Song, Y. Xia, and S. B. Zheng, “Improving the stimulated Raman adiabatic passage via dissipative quantum dynamics,” Optics Express 20(7), 22847 (2016).
[Crossref]

Y. H. Zhou, H. Z. Shen, X. Q. Shao, and X. X. Yi, “Strong photon antibunching with weak second-order nonlinearity under dissipation and coherent driving,” Opt. Express 24, 17332 (2016).
[Crossref]

W. Xiong, D. Y. Jin, Y. Y. Qiu, C. H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93, 023844 (2016).
[Crossref]

2015 (11)

T. K. Paraïso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, and O. Painter, “Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity,” Phys. Rev. X 5, 041024 (2015).

R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
[Crossref]

X. W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

J. M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref] [PubMed]

H. Flayac, D. Gerace, and V. Savona, “An all-silicon single photon source by unconventional photon blockade,” Sci. Rep. 5, 11223 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Tunable photon blockade in coupled semiconductor cavities,” Phys. Rev. A 91, 063808 (2015)
[Crossref]

H. Z. Shen, Y. H. Zhou, H. D. Liu, G. C. Wang, and X. X. Yi, “Exact optimal control of photon blockade with weakly nonlinear coupled cavities,” Opt. Express 23, 32835 (2015).
[Crossref] [PubMed]

Y. H. Zhou, H. Z. Shen, and X. X. Yi, “Unconventional photon blockade with second-order nonlinearity,” Phys. Rev. A 92, 023838 (2015).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical Nonreciprocity in Asymmetric Optomechanical Couplers,” Sci. Rep. 5, 8657 (2015).
[Crossref] [PubMed]

Z. Q. Yin, W. L. Yang, L. Sun, and L. M. Duan, “Quantum network of superconducting qubits through an optomechanical interface,” Phys. Rev. A 91, 012333 (2015).
[Crossref]

2014 (4)

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Effcient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

H. Z. Shen, Y. H. Zhou, and X. X. Yi, “Quantum optical diode with semiconductor microcavities,” Phys. Rev. A 90, 023849 (2014).
[Crossref]

J. Gough, “Feedback network models for quantum transport,” Phys. Rev. E 90, 062109 (2014).
[Crossref]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing Optomechanical Coupling via the Josephson Effect,” Phys. Rev. Lett. 112, 203603 (2014).
[Crossref]

2013 (7)

J. Q. Liao and C. K. Law, “Correlated two-photon scattering in cavity optomechanics,” Phys. Rev. A 87, 043809 (2013).
[Crossref]

X. W. Xu, Y. J. Li, and Y. X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
[Crossref]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity,” Phys. Rev. Lett. 111, 053602 (2013).
[Crossref] [PubMed]

S. Aldana, C. Bruder, and A. Nunnenkamp, “Equivalence between an optomechanical system and a Kerr medium,” Phys. Rev. A 88, 043826 (2013).
[Crossref]

M. Ludwig and F. Marquardt, “Quantum Many-Body Dynamics in Optomechanical Arrays,” Phys. Rev. Lett. 111, 073603 (2013).
[Crossref] [PubMed]

T. Hong, H. Yang, H. X. Miao, and Y. Chen, “Open quantum dynamics of single-photon optomechanical devices,” Phys. Rev. A 88, 023812 (2013).
[Crossref]

2012 (4)

J. Q. Liao, H. K. Cheung, and C. K. Law, “Spectrum of single-photon emission and scattering in cavity optomechanics,” Phys. Rev. A 85, 025803 (2012).
[Crossref]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801 (R) (2012).
[Crossref]

M. Hafezi and P. Rabl, “Optomechanically induced nonreciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
[Crossref] [PubMed]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012).
[Crossref]

2011 (3)

P. Rabl, “Photon Blockade Effect in Optomechanical Systems,” Phys. Rev. Lett. 107, 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-Photon Optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[Crossref] [PubMed]

M. Bamba, A. Imamođlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802(R) (2011).
[Crossref]

2010 (1)

S. Weis, R. Rivière, S. Deléglise, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

2009 (2)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical Nonreciprocity in Optomechanical Structures,” Phys. Rev. Lett. 102, 213903 (2009).
[Crossref] [PubMed]

2008 (6)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4(7), 561–564 (2008).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322(5899), 235–238 (2008).
[Crossref] [PubMed]

J. K. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

T. R. Zaman, X. Y. Guo, and R. J. Ram, “Semiconductor Waveguide Isolators,” J. Lightwave Technol.,” 26(2), 291–301 (2008).
[Crossref]

2006 (1)

K. Koshini, “Semiclassical evaluation of two-photon cross-Kerr effect,” Phys. Rev. A 74, 053818 (2006).
[Crossref]

2000 (1)

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H. Z. Shen, S. Xu, Y. H. Zhou, G. C. Wang, and X. X. Yi, “Unconventional photon blockade from bimodal driving and dissipations in coupled semiconductor microcavities,” J. Phys. B 51, 035503 (2018).
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Figures (7)

Fig. 1
Fig. 1 The schematic illustration of the model. A cyclic three-mode optical system driven by three pump fields of amplitudes εa1, εa2, and εa3 with frequency ωd. Two of the optical modes are linearly coupled with each other, while they are coupled to another described by cross-Kerr (CK) nonlinear terms.
Fig. 2
Fig. 2 Schematic diagram of typical experimental configurations of the all-optical CK nonlinear coupling system. (a) The mirrors on both sides of the cavity are fixed, and the mirrors in the middle can move. The assisting qubits are used to induce the CK effect between the two cavity modes. (b) An equivalent quantum circuit with superconducting charge qubits. (c) Two optomechanical crystals are coupled to each other.
Fig. 3
Fig. 3 Scattering probabilities T a 2 a 3 ( ω ) and T a 3 a 2 ( ω ) as functions of the frequency of the incoming signal ω (in units of γa3) for different phase different φ from 0 to 1 (in units of π). The other parameters are γaj = γa3, Δ a j σ s = Δ a 3 = 10 γ a 3, J = Ga2 = Ga1e = 0.5γa3. The right-hand panel is a contour plot for the left-hand panel, and the purple portion of the figure is replaced by white.
Fig. 4
Fig. 4 Scattering probabilities T a 2 a 3 ( ω ) and T a 3 a 2 ( ω ) as a function of the frequency of the incoming signal ω (units of γa3) for different optomechanical coupling constant Ga2(units of γa3). The other parameters chosen are γaj = γa3, Δ a j σ s = Δ a 3 = 10 γ a 3, J = 0.5γa3, Ga1 = −iGa2, Ga2 ∈ [0, 2γa3]. The right-hand panel is a contour plot for the left-hand panel, and the dark blue part of the left figure corresponds to the white in the right figure.
Fig. 5
Fig. 5 Numerical calculations for the transmission rates. Fig. (a)–(d) shows the transmission probability of optical and optomechanical system as a function of the frequency of the incoming signal ω(units of γa3) for different optomechanical coupling constant Ga1. (a) shows the optical transmission given by calculations with RWA. (b) is for the transmission considering the anti-rotational terms. (c) shows the optomechanical transmission with RWA. (d) shows the optomechanical transmission with anti-rotational wave terms. The other parameters chosen are set to meet the optimal parameter conditions.
Fig. 6
Fig. 6 Numerical calculation Re[εT ] as a function of Ω (units of γa3) for different optomechanical coupling constant: Ga1 = 0.1γa3 (blue solid line with stars), Ga1 = 0.2γa3 (red solid line with circles), Ga1 = 0.3γa3 (orange solid line with squares), Ga1 = 0.4γa3 (solid brown line), Ga1 = 0.5γa3 (green dotted line).
Fig. 7
Fig. 7 (a) A loop system where an equivalent mechanical mode is coupled to each cavity in an optical cavity-array. (b) The three-mode optomechanical system consists of two mechanical modes and one optical mode. (c) The general form of a planar quantum network, which is composed of arbitrary optical mode and a mechanical mode.

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

H ^ T = H ^ 0 + H ^ I + H ^ d ,
H ^ 0 = i = 1 3 ω a i a ^ i a ^ i + k i = 1 3 ω i k Γ ^ i k Γ ^ i k , H ^ d = i i = 1 3 ( ε a i a ^ i e i ( φ a i ω d t ) H . c ) , H ^ I = j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + k i = 1 3 g i k ( a ^ i Γ ^ i k + Γ ^ i k a ^ i ) ,
H ^ = H ^ 0 + H ^ I + H ^ d ( B ^ 3 + B ^ 3 ) + F ,
H ^ 0 = j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 a ^ 3 a ^ 3 + k i = 1 3 ω i k Γ ^ i k Γ ^ i k , H ^ I = H ^ I j = 1 2 χ a j | α 3 | ( a ^ 3 e i θ + a ^ 3 e i θ ) a ^ j a ^ j , H ^ d = i i = 1 3 ( ε a i a ^ i e i φ a i H . c ) Δ a 3 ( α 3 a ^ 3 + α 3 * a ^ 3 ) ,
ε a 3 e i φ a 3 + ( i Δ a 3 γ a 3 / 2 ) α 3 s = 0 .
H ^ = H ^ 0 s + H ^ I s + H ^ d s ( B ^ 3 s + B ^ 3 s ) + F s ,
d d t a ^ 3 = k a ^ 3 + i j = 1 2 g a j s e i θ a ^ j a ^ j + γ a 3 a ^ 3 in + ( i Δ a 3 γ a 3 2 ) α 3 s + ε a 3 e i φ a 3 .
θ = arctan ( γ a 3 2 sin φ a 3 + Δ a 3 cos φ a 3 γ a 3 2 cos φ a 3 Δ a 3 sin φ a 3 ) , | α 3 s | = ε a 3 | k | .
d d t a ^ 1 = + Γ ^ 1 a ^ 1 i J a ^ 2 + ε a 1 e i φ a 1 + γ a 1 a ^ 1 in , d d t a ^ 2 = + Γ ^ 2 a ^ 2 i J a ^ 1 + ε a 2 e i φ a 2 + γ a 2 a ^ 2 in , d d t a ^ 3 = k a ^ 3 + i j = 1 2 g a j s e i θ a ^ j a ^ j + γ a 3 a ^ 3 in ,
η 1 = k 2 ε a 1 e i φ a 1 i J ε a 2 e i φ a 2 k 1 k 2 + J 2 , η 2 = k 1 ε a 2 e i φ a 2 i J ε a 1 e i φ a 1 k 1 k 2 + J 2 , η 3 = i j = 1 2 g a j s | η j | 2 e i θ k ,
d d t δ a ^ 1 = k 1 δ a ^ 1 + i G a 1 σ ^ i J δ a ^ 2 + γ a 1 δ a ^ 1 in , d d t δ a ^ 2 = k 2 δ a ^ 2 + i G a 2 σ ^ i J δ a ^ 1 + γ a 2 δ a ^ 2 in , d d t δ a ^ 3 = k δ a ^ 3 + i j = 1 2 τ ^ j e i θ + γ a 3 δ a ^ 3 in ,
d d t υ ^ = μ υ ^ + L υ ^ in ,
μ = [ k 1 i J i G a 1 e i θ i J k 2 i G a 2 e i θ i G a 1 * e i θ i G a 2 * e i θ k ] ,
τ ( ω ) = [ τ ( t ) ] = + τ ( t ) e i ω t d t ,
υ ( ω ) = ( μ i ω I ) 1 L υ in ( ω ) ,
υ o u t ( ω ) = Γ ( ω ) υ in ( ω ) ,
Γ ( ω ) = L T [ μ i ω I ] 1 L I .
1 | A | [ γ a 1 2 A 11 * | A | γ a 1 γ a 2 A 12 * γ a 1 γ a 3 A 13 * γ a 2 γ a 1 A 21 * γ a 2 2 A 22 * | A | γ a 2 γ a 3 A 23 * γ a 3 γ a 1 A 31 * γ a 3 γ a 2 A 32 * γ a 3 2 A 33 * | A | ] ,
A j j * = ( k j i ω ) ( k i ω ) + | G a j | 2 e 2 i θ , A 33 * = ( k 1 i ω ) ( k 2 i ω ) + J 2 , A 12 * = J ( i k + ω ) G a 1 * G a 2 e 2 i θ , A 21 * = J ( i k + ω ) G a 1 G a 2 * e 2 i θ , A 13 * = { G a 1 * ( i k 2 + ω ) + J G a 2 * } e i θ , A 31 * = { G a 1 ( i k 2 + ω ) + J G a 2 } e i θ , A 23 * = { G a 2 * ( i k 1 + ω ) J G a 1 * } e i θ , A 32 * = { G a 2 ( i k 1 + ω ) + J G a 1 } e i θ .
S o u t ( ω ) = d ω υ o u t ( ω ) υ o u t ( ω ) .
S out ( ω ) = T ( ω ) S in ( ω ) + S Q ( ω ) + S T ( ω ) ,
S out ( ω ) = T ( ω ) S in ( ω ) ,
A 23 * A 23 * = J 2 | G a 1 | 2 2 J Re [ G a 2 * G a 1 ( i k 1 + ω ) ] + | G a 2 | 2 ( | k 1 | 2 + ω 2 2 ω Im [ k 1 ] ) .
A 32 * A 32 * = J 2 | G a 1 | 2 2 J Re [ G a 1 * G a 2 ( i k 1 + ω ) ] + | G a 2 | 2 ( | k 1 | 2 + ω 2 2 ω Im [ k 1 ] ) .
Γ + = [ 0 0 i i 0 0 0 1 0 ] or Γ = [ 0 i 0 0 0 1 i 0 0 ] ,
T a 2 a 1 = γ a 1 γ a 2 | β | 2 { | G a 1 | 2 | G a 2 | 2 + | J k | 2 2 J σ } ,
T a 3 a 2 = γ a 3 γ a 2 | G a 1 | 2 | k | 2 + | J G a 2 | 2 2 J σ | G a 1 | 2 | G a 2 | 2 + | J k | 2 2 J σ ,
β = k 2 ( k 1 k + J 2 ) + j = 1 2 k j | G a j + 1 | 2 2 i J Re ( G a 1 G a 2 * ) ,
H ^ a = H ^ a 0 + H ^ a I + H ^ a d ,
H ^ a 0 = τ = 1 N Δ a τ a ^ τ a ^ τ + Δ a m a ^ m a ^ m , H ^ a I = τ = 1 N J ( a ^ τ a ^ τ + 1 + a ^ τ + 1 a ^ τ ) τ = 1 N g a m s ^ m a ^ τ a ^ τ , H ^ a d = τ = 1 N i ε a τ a ^ τ e i φ a τ + κ m a ^ m + H . c ,
0 = ε a m e i φ a m + ( i Δ a m γ a m 2 ) α m s ,
H ^ 2 = H ^ 2 0 + H ^ 2 I + H ^ 2 d ,
H ^ 2 0 = j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 s s a ^ 3 a ^ 3 , H ^ 2 I = J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) j = 1 2 g a j s s ^ j a ^ 3 a ^ 3 , H ^ 2 d = i = 1 3 i ε a i a ^ i e i φ a i j = 1 2 o j a ^ j + H . c ,
0 = + ε a 1 e i φ a 1 + ( i Δ a 1 γ a 1 2 ) α 1 s s , 0 = + ε a 2 e i φ a 2 + ( i Δ a 2 γ a 2 2 ) α 2 s s + i J α 1 s s .
H ^ s = i = 1 3 ω a i a ^ i a ^ i + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + i i = 1 3 ( ε a i a ^ i e i ( φ a i ω d ) H . c . ) .
H ^ s = i = 1 3 Δ a i a ^ i a ^ i + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + i i = 1 3 ( ε a i a ^ i e i φ a i H . c . ) ,
ρ ˙ = i [ ρ , H ^ s ] + i = 1 3 γ a i { ( n ¯ a i + 1 ) ^ [ a ^ 3 ] + n ¯ a i ^ [ a ^ i ] } ρ .
ρ = 𝒟 ^ a 3 ( α 3 ) ρ 𝒟 ^ a 3 ( α 3 ) ,
ρ ˙ = t 𝒟 ^ a 3 ( α 3 ) ρ 𝒟 ^ a 3 ( α 3 ) + 𝒟 ^ a 3 ( α 3 ) ρ t 𝒟 ^ a 3 ( α 3 ) + 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) .
t 𝒟 ^ a 3 ( α 3 ) = 1 2 ( α 3 α ˙ 3 * α ˙ 3 α 3 * ) 𝒟 ^ a 3 ( α 3 ) 𝒟 ^ a 3 ( α 3 ) ( a ^ 3 α ˙ 3 * a ^ 3 α ˙ 3 ) ,
ρ ˙ = [ 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) , ( a ^ 3 α ˙ 3 a ^ 3 α ˙ 3 * ) ] + 𝒟 ^ a 3 ( α 3 ) ρ ˙ 𝒟 ^ a 3 ( α 3 ) .
ρ ˙ = + i [ ρ , H ^ s ] + i = 1 3 γ a i { ( n ¯ a i + 1 ) ^ [ a ^ i ] + n ¯ a i ^ [ a ^ i ] } ρ + { α ˙ 3 + ( i Δ a 3 γ a 3 2 ) α 3 + ε a 3 e i φ a 3 } [ a ^ 3 , ρ ] { α ˙ 3 * + ( i Δ a 3 γ a 3 2 ) α 3 * + ε a 3 e i φ a 3 } [ a ^ 3 , ρ ] ,
H ^ s = + j = 1 2 Δ a j a ^ j a ^ j + Δ a 3 a ^ 3 a ^ 3 + j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 + J ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) j = 1 2 χ a j a ^ j a ^ j a ^ 3 a ^ 3 j = 1 2 g a j s ( a ^ 3 e i θ + a ^ 3 e i θ ) a ^ j a ^ j .
α ˙ 3 s + ( i Δ a 3 γ a 3 2 ) α 3 s + ε a 3 e i φ a 3 = 0 .
d d t a ^ 3 = i Δ a 3 a ^ 3 i k g 3 k Γ ^ 3 k + i Δ a 3 α 3 s + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
d d t Γ ^ 3 k = i ω 3 k Γ ^ 3 k i g 3 k a ^ 3 i g 3 k α 3 s .
Γ ^ 3 k = i g 3 k 0 t { a ^ 3 ( τ ) + α 3 s } e i ω 3 k ( t τ ) d τ + Γ ^ 3 k ( 0 ) e i ω 3 k t .
d d t a ^ 3 = + f ^ a 3 k g 3 k 2 0 t { a ^ 3 ( τ ) + α 3 s } e i ω 3 k ( t τ ) d τ + i Δ a 3 ( α 3 s a ^ 3 ) + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
f ^ a 3 ( t ) = i k g 3 k Γ ^ 3 k ( 0 ) e i ω 3 k t ,
d d t a ^ 3 = + f ^ a 3 ( t ) 0 t g 3 2 ( ω ) π δ ( t τ ) { a ^ 3 ( τ ) + α 3 s } d τ + i Δ a 3 ( α 3 s a ^ 3 ) + i j = 1 2 g a j s e i θ a ^ j a ^ j + ε a 3 e i ϕ a 3 ,
f ^ a 3 ( t ) = i 0 g 3 ( ω ) Γ ^ 30 ( ω ) e i ω t d ω .
d d t a ^ 3 = ( i Δ a 3 γ a 3 2 ) a ^ 3 + i j = 1 2 g a j s e i ω a ^ j a ^ j + ε a 3 e i ϕ a 3 + ( i Δ a 3 γ a 3 2 ) α 3 s + γ a 3 a ^ 3 in ,
a ^ 3 in = 1 2 π 0 i g 3 ( ω ) Γ ^ 30 ( ω ) e i ω t d ω .

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