Nonreciprocity in a strongly coupled three-mode optomechanical circulatory system

Cheng Shang; H. Z. Shen; H. Z. Shen; X. X. Yi; X. X. Yi

doi:10.1364/OE.27.025882

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 ω_a₁ and ω_a₂), both of them are coupled to another optical mode (a₃, with frequency ω_a₃) through an CK nonlinear interaction, see Fig. 1. The optical mode with CK medium is strongly driven by a monochromatic field of frequency ω_a₃ = ω_d at rate ε_a₃, and another two optical modes are driven by external laser fields with frequencies ω_a₁ = ω_a₂ = ω_d at rates ε_a₁ and ε_a₂, respectively. The total Hamiltonian of this field-reservoir system can be written as (ħ = 1)

{\hat{H}}_{T} = {\hat{H}}_{0} + {\hat{H}}_{I} + {\hat{H}}_{d},

with

\begin{array}{l} {\hat{H}}_{0} = \sum_{i = 1}^{3} ω_{a_{i}} {\hat{a}}_{i}^{†} {\hat{a}}_{i} + \sum_{k} \sum_{i = 1}^{3} ω_{i k} {\hat{Γ}}_{i k}^{†} {\hat{Γ}}_{i k}, {\hat{H}}_{d} = i \sum_{i = 1}^{3} (ε_{a_{i}} {\hat{a}}_{i}^{†} e^{i (φ_{a_{i}} - ω_{d} t)} - H . c), \\ {\hat{H}}_{I} = \sum_{j = 1}^{2} χ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} {\hat{a}}_{3}^{†} {\hat{a}}_{3} + J ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) + \sum_{k} \sum_{i = 1}^{3} g_{i k} ({\hat{a}}_{i}^{†} {\hat{Γ}}_{i k} + {\hat{Γ}}_{i k}^{†} {\hat{a}}_{i}), \end{array}

where a reservoir that couples to the system is introduced. Ĥ₀ denotes the Hamiltonian of the system and the reservoir with frequencies ω_{a_i} and ω_ik, respectively.

{\hat{a}}_{i}^{†}

(â_i) and

{\hat{Γ}}_{i k}^{†}

(Γ̂_ik) for i = 1, 2, 3 are the corresponding creation (annihilation) operators of the optical modes and environment modes. Hamiltonian Ĥ_I describes the CK interactions between the optical modes with coupling strength χ_a₁ and χ_a₂, the linear interaction between the optical modes with coupling J, and the couplings between the system and the reservoir. The Hamiltonian Ĥ_d represents the pump fields applied to the system with phases ϕ_{a_i}. Without loss of generality, we assume that the parameters χ_a₁, χ_a₂, g_ik, ε_{a_i}, 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 ε_a₁, ε_a₂, and ε_a₃ with frequency ω_d. 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 ω_d of the driving field, we apply a unitary transformation $\hat{U} = exp (i ω_{d} \sum_{i = 1}^{3} {\hat{a}}_{i}^{†} {\hat{a}}_{i} t)$ to Eq. (1), and the total Hamiltonian in the rotating frame becomes Ĥ_rot = ÛĤ_TÛ^† − iÛdÛ^†/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 ${\hat{D}}_{a_{3}} (α_{3}) = exp (α_{3} {\hat{a}}_{3}^{†} - {α_{3}}^{*} {\hat{a}}_{3})$ , defining ${\hat{H}}^{'} = {\hat{D}}_{a_{3}} (α_{3}) {\hat{H}}_{rot} {\hat{D}}_{a_{3}}^{†} (α_{3})$ , we have

{\hat{H}}^{'} = {\hat{H}}_{0}^{'} + {\hat{H}}_{I}^{'} + {\hat{H}}_{d}^{'} - ({\hat{B}}_{3} + {\hat{B}}_{3}^{†}) + F,

with

\begin{array}{l} {\hat{H}}_{0}^{'} = \sum_{j = 1}^{2} {Δ^{'}}_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + Δ_{a_{3}} {\hat{a}}_{3}^{†} {\hat{a}}_{3} + \sum_{k} \sum_{i = 1}^{3} ω_{i k} {\hat{Γ}}_{i k}^{†} {\hat{Γ}}_{i k}, \\ {\hat{H}}_{I}^{'} = {\hat{H}}_{I} - \sum_{j = 1}^{2} χ_{a_{j}} | α_{3} | ({\hat{a}}_{3}^{†} e^{i θ} + {\hat{a}}_{3} e^{- i θ}) {\hat{a}}_{j}^{†} {\hat{a}}_{j}, \\ {\hat{H}}_{d}^{'} = i \sum_{i = 1}^{3} (ε_{a_{i}} {\hat{a}}_{i}^{†} e^{i φ_{a_{i}}} - H . c) - Δ a_{3} (α_{3} {\hat{a}}_{3}^{†} + α_{3}^{*} {\hat{a}}_{3}), \end{array}

where Δ_{a_i} = ω_{a_i} − ω_d 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 = Δ_a₁ + χ_a₁ |α₃|² and Δ′_a₂ = Δ_a₂ + χ_a₂ |α₃|² are the detunings including the frequency shifts caused by the strong coupling. The enhanced coupling strength is g_{a_j} = χ_{a_j} |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|eⁱ^θ. The environment operator is

{\hat{B}}_{3} = \sum_{k} g_{3 k} α_{3}^{*} {\hat{Γ}}_{3 k}

. We introduced the normalized driven field F = |α₃|²Δ_a₃ + iε_a₃(α₃e^−iϕ_a₃ − H.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[m_{a_j}χ_{a_j}] ≪ Δ_a₃, then the CK nonlinear interaction term in Ĥ_I can be safely ignored [56], and when we choose the laser parameters ω_d and ε_a₃ to satisfy equation

ε_{a_{3}} e^{i φ_{a_{3}}} + (i Δ_{a_{3}} - γ_{a_{3}} / 2) α_{3}^{s} = 0 .

Eq. (3) becomes

\hat{H} = {\hat{H}}_{0 s}^{'} + {\hat{H}}_{I s}^{'} + {\hat{H}}_{d s}^{'} - ({\hat{B}}_{3 s} + {\hat{B}}_{3 s}^{†}) + F_{s},

where γ_a₃ is the damping rate of the mechanical mode â₃, and

α_{3}^{s}

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 ω_d and ε_a₃ 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),

\frac{d}{d t} {\hat{a}}_{3} = - k {\hat{a}}_{3} + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i θ} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + \sqrt{γ_{a_{3}}} {\hat{a}}_{3}^{in} + (i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) α_{3}^{s} + ε_{a_{3}} e^{i φ_{a_{3}}} .

For details, see Appendix B [61]. The single-photon optomechanical coupling strength is

g_{a_{j}}^{s} = χ_{a_{j}} | α_{3}^{s} |

, and

{\hat{a}}_{3}^{in}

is the noise operator with zero mean, k = γ_a₃/2 + iΔ_a₃.

As aforementioned, the average of a₃ can be determined in the displacement representation when the coherent displacement amplitude $α_{3}^{s}$ obeys Eq. (5). To analyze the influence of the driving laser, we write $α_{3}^{s} = | α_{3}^{s} | e^{i θ}$ with

θ = arctan (\frac{\frac{γ_{a_{3}}}{2} sin φ_{a_{3}} + Δ_{a_{3}} cos φ_{a_{3}}}{\frac{γ_{a_{3}}}{2} cos φ_{a_{3}} - Δ_{a_{3}} sin φ_{a_{3}}}), | α_{3}^{s} | = \frac{ε_{a_{3}}}{| k |} .

It can be seen from Eq. (8) that

α_{3}^{s}

is tunable by ε_a₃ and Δ_a₃. The value of

| α_{3}^{s} |

could be large enough in the strong-driving case ε_a₃ ≫ {Δ_{a_j}, γ_{a_j}}. In particular, the interaction strength

g_{a_{j}}^{s}

is controllable because we can obtain a desired

α_{3}^{s}

by designing a proper driving amplitude ε_a₃.

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

\begin{array}{l} \frac{d}{d t} {\hat{a}}_{1} = + {\hat{Γ}}_{1} {\hat{a}}_{1} - i J {\hat{a}}_{2} + ε_{a_{1}} e^{i φ_{a_{1}}} + \sqrt{γ_{a_{1}}} {\hat{a}}_{1}^{in}, \\ \frac{d}{d t} {\hat{a}}_{2} = + {\hat{Γ}}_{2} {\hat{a}}_{2} - i J {\hat{a}}_{1} + ε_{a_{2}} e^{i φ_{a_{2}}} + \sqrt{γ_{a_{2}}} {\hat{a}}_{2}^{in}, \\ \frac{d}{d t} {\hat{a}}_{3} = - k {\hat{a}}_{3} + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i θ} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + \sqrt{γ_{a_{3}}} {\hat{a}}_{3}^{in}, \end{array}

where γ_{a_j} is the damping rate of the optical mode a_j, and

{\hat{a}}_{j}^{in}

is the noise operator with zero mean value. In the derivation, a normalized dampling rate Γ̂_j = −γ_{a_j} /2 + iκ̂_j was defined with

{\hat{κ}}_{j} = g_{a_{j}}^{s} ({\hat{a}}_{3}^{†} e^{i θ} + {\hat{a}}_{3} e^{- i θ}) - Δ_{a_{j}}^{' s}

, and

Δ_{a_{j}}^{' s} = Δ_{a_{j}} + χ_{a_{j}} {| α_{3}^{s} |}^{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., â_i = β_i + δâ_i, the steady mean values of β_i can be obtained by setting the (first order) derivative of those operator with respect to time to zero,

η_{1} = \frac{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} = \frac{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} = \frac{i \sum_{j = 1}^{2} g_{a_{j}}^{s} {| η_{j} |}^{2} e^{i θ}}{k},

where the auxiliary parameter

k_{j} = γ_{a_{j}} / 2 + i Δ_{a_{j}}^{σ s}

with

Δ_{a_{j}}^{σ s} = Δ_{a_{j}}^{' s} - 2 g_{a_{j}}^{s} Re [β exp (- i θ)]

. Substituting Eq. (10) into Eq. (9) and omitting the high-order terms in small quantum fluctuations δâ_i, we obtain the linearized QLEs

\begin{array}{l} \frac{d}{d t} δ {\hat{a}}_{1} = - k_{1} δ {\hat{a}}_{1} + i G_{a_{1}} \hat{σ} - i J δ {\hat{a}}_{2} + \sqrt{γ_{a_{1}}} δ {\hat{a}}_{1}^{in}, \\ \frac{d}{d t} δ {\hat{a}}_{2} = - k_{2} δ {\hat{a}}_{2} + i G_{a_{2}} \hat{σ} - i J δ {\hat{a}}_{1} + \sqrt{γ_{a_{2}}} δ {\hat{a}}_{2}^{in}, \\ \frac{d}{d t} δ {\hat{a}}_{3} = - k δ {\hat{a}}_{3} + i \sum_{j = 1}^{2} {\hat{τ}}_{j} e^{i θ} + \sqrt{γ_{a_{3}}} δ {\hat{a}}_{3}^{in}, \end{array}

where

\hat{σ} = δ {\hat{a}}_{3}^{†} exp [i θ] + δ {\hat{a}}_{3} exp [- i θ]

and

{\hat{τ}}_{j} = G_{a_{j}}^{*} δ {\hat{a}}_{j} + G_{a_{j}} δ {\hat{a}}_{j}^{†}

.

G_{a_{j}} \equiv g_{a_{j}}^{s} η_{j} \equiv | G_{a_{j}} | exp [i ϕ_{a_{j}}]

is the effective single-photon-optomechanical coupling.

For convenience, we assume that the system works in the resolved sideband regime, i.e., {γ_{a_i}, G_{a_j}, J} ≪ Δ_a₃ and $Δ_{a_{j}}^{σ s} ~ Δ_{a_{3}}$ . 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

\frac{d}{d t} \hat{υ} = - μ \hat{υ} + L {\hat{υ}}_{in},

where the fluctuation vector υ̂ = (δâ₁, δâ₂, δâ₃)^T, the input field vector

{\hat{υ}}_{in} = {(δ {\hat{a}}_{1}^{in}, δ {\hat{a}}_{2}^{in}, δ {\hat{a}}_{3}^{in})}^{T}

, the damping vector

L = diag (\sqrt{γ_{a_{1}}}, \sqrt{γ_{a_{2}}}, \sqrt{γ_{a_{3}}})

, and the coefficient vector μ in terms of system parameters takes

μ = [\begin{array}{l} 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 \end{array}],

The characteristic equation |μ − λI| = 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) = ℱ [τ (t)] = \int_{- \infty}^{+ \infty} τ (t) e^{i ω t} d t,

(for any operator τ) and using the derivation role ℱ[τ̇(t)] = −iωℱ[τ(t)], the solution to Eq. (12) in the frequency domain is

υ (ω) = {(μ - i ω I)}^{- 1} L υ^{in} (ω),

where I denotes the identity vector.

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

υ_{o u t} (ω) = Γ (ω) υ_{in} (ω),

where the corresponding output vector

υ_{out} (ω) = {(δ {\hat{a}}_{1}^{out}, δ {\hat{a}}_{2}^{out}, δ {\hat{a}}_{3}^{out})}^{T}

and the scattering matrix

Γ (ω) = L^{T} {[μ - i ω I]}^{- 1} L - I .

The specific form of Γ(ω) is given as

\frac{1}{| A |} [\begin{array}{l} 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 | \end{array}],

where

| A | = \sum_{j = 1}^{2} ξ_{j} exp [- 2 i θ] + \prod_{j = 1}^{2} ζ_{j} (k - i ω)

is the value of the determinant of A with

ξ_{j} = {| G_{a_{j}} |}^{2} (k_{j + 1} - i ω) - 2 i J Re [G_{a_{1}}^{*} G_{a_{2}}]

and ζ_j= (k_j − iω) + J². The remaining companion matrix elements are given as

\begin{array}{l} 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} . \end{array}

The spectrum of the output fields is defined by

S_{o u t} (ω) = \int d ω^{'} 〈 υ_{o u t}^{†} (ω^{'}) υ_{o u t} (ω) 〉 .

Substituting Eq. (16) into Eq. (20), we get

S_{out} (ω) = T (ω) S_{in} (ω) + S_{Q} (ω) + S_{T} (ω),

where T(ω) is the contribution arising from the presence of a single photon in the input field. S_Q (ω) is the contribution from the incoming vacuum field, while S_T (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 S_Q (ω) and vacuum noise S_T (ω) is much smaller than the input field S_in (ω) [66], we can neglect S_Q (ω) and S_T (ω) and obtain

S_{out} (ω) = T (ω) S_{in} (ω),

where

S_{in} (ω) = {(S_{a_{1}}^{in}, S_{a_{2}}^{in}, S_{a_{3}}^{in})}^{T}

,

S_{out} (ω) = {(S_{a_{1}}^{out}, S_{a_{2}}^{out}, S_{a_{3}}^{out})}^{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 $T_{a_{3}}^{a_{2}} (ω) = A_{23}^{*} A_{23}^{* †} / {| A |}^{2}$ . The effective part in the physical system can be written as

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}]) .

Similarly, the valid part of a₃ to a₂ as follows

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}]) .

From Eqs. (23) and (24), keeping the other parameters fixed, we can see that

T_{a_{2}}^{a_{3}} (ω) = C_{1}

and

T_{a_{3}}^{a_{2}} (ω) = C_{2}

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 $T_{a_{2}}^{a_{3}} (ω)$ and $T_{a_{3}}^{a_{2}} (ω)$ as functions of the incoming signal ω for different phase difference φ is shown in Fig. 3, where the parameters chosen are γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = G_a₂ = G_a₁e^iφ = 0.5γ_a₃. It is shown that the photon transmission satisfies the Lorentz reciprocal theorem provided φ = nπ (n is an integer) [68]. When φ ≠ nπ, the Lorentz reciprocal theorem is not satisfied, and the OMCS enters the nonreciprocal regime. The perfect nonreciprocity appears at the point φ = 2nπ + π/2 or φ = 2nπ − π/2(ω = 10γ_a₃), which is in good agreement with the analytical result in 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 γ_a₃) for different phase different φ from 0 to 1 (in units of π). The other parameters are γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = G_a₂ = G_a₁e^iφ = 0.5γ_a₃. 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 $T_{a_{2}}^{a_{3}} (ω)$ and $T_{a_{3}}^{a_{2}} (ω)$ as functions of the incoming signal ω for different optomechanical coupling constant G_a₂ with parameters γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = 0.5γ_a₃, G_a₁ = −iG_a₂. In the regime G_a₂ ≪ γ_a₃, we can see that $T_{a_{3}}^{a_{2}} (ω) \approx T_{a_{2}}^{a_{3}} (ω) \approx 0$ . When the optomechanical coupling strength G_a₂ increase, the scattering probability increase, too, and it can reaches the best value $T_{a_{2}}^{a_{3}} (ω) = 1$ (ω = 10γ_a₃) at G_a₂ = 0.5γ_a₃.

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 γ_a₃) for different optomechanical coupling constant G_a₂(units of γ_a₃). The other parameters chosen are γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = 0.5γ_a₃, G_a₁ = −iG_a₂, G_a₂ ∈ [0, 2γ_a₃]. 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 $T_{a_{2}}^{a_{1}} (ω) = T_{a_{3}}^{a_{2}} (ω) = T_{a_{1}}^{a_{3}} (ω) = 1$

or $T_{a_{3}}^{a_{1}} (ω) = T_{a_{2}}^{a_{3}} (ω) = T_{a_{1}}^{a_{2}} (ω) = 1$ with all of the other scattering probabilities equal to zero [69]. By numerical calculations and careful examination of the scattering probabilities $T_{a_{3}}^{a_{2}} (ω)$ and $T_{a_{2}}^{a_{3}} (ω)$ , we find that the optimal conditions are met if the parameter satisfying the following conditions γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = G_a₂ = G_a₁e^iφ = 0.5γ_a₃, φ = 2nπ − π/2 or γ_{a_j} = γ_a₃, $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , J = G_a₂ = G_a₁e^iφ = 0.5γ_a₃, φ = 2nπ + π/2. In this case, the scattering matrix Eq. (17) takes

Γ_{+} = [\begin{matrix} 0 & 0 & - i \\ - i & 0 & 0 \\ 0 & - 1 & 0 \end{matrix}] or Γ_{-} = [\begin{matrix} 0 & - i & 0 \\ 0 & 0 & - 1 \\ - i & 0 & 0 \end{matrix}],

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 (a₁ → a₂ → a₃ → a₁) for φ = 2nπ + π/2. While Γ₋ describes an ideal optomechanical circulator with the signal is transferred from one OMCS to another clockwise (a₁ → a₃ → a₂ → a₁) for φ = 2nπ − π/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 $Δ_{a_{j}}^{σ s} = Δ_{a_{3}} = 10 γ_{a_{3}}$ , G_a₁ = −iG_a₂ ≪ Δ_a₃, γ_{a_j} = γ_a₃, J = 0.5γ_a₃, we are delighted to find that when the frequency of incident photon is tuned to be ω/γ_a₃ = 10, $A_{23}^{*} A_{23}^{* †} = 0$ , $A_{32}^{*} A_{32}^{* †} = 1$ are satisfied. The transmission probabilities a₂ → a₃ and a₃ → a₂ are then independent upon the optomechanical coupling constant G_a₁. 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 G_a₁ on photon transmission bandwidth. Under the RWA, Fig. 5(a) and Fig. 5(c) plots the transmission probability between the two optical cavities a₂ → a₁ and optomechanical system a₃ → a₂ as a function of the detuning ω/γ_a₃ for different optomechanical coupling constant G_a₁. We observe that when the optomechanical coupling constant G_a₁ is less than γ_a₃, both the optical system and optomechanical system of bandwidth increases slowly as the coupling strength increases. But when the optomechanical coupling constant G_a₁ is greater than γ_a₃, 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 γ_a₃) for different optomechanical coupling constant G_a₁. (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(ε_pa₁ exp iω_pt − H.c) (see the dotted line in Fig. 1) and assume that the amplitude of the probe field ε_P ≪ ε_{a_1,2,3}, it is easy to find the steady-state solutions of the operators a_i that would not be affected by the probe field. Hence, the dynamics equation describing this system differs from Eq. (11) only at a ε_p exp i(ω_d − ω_p)t term, which corresponds to a₁.

Finding the steady-state solutions and using the input-output relationship (with $〈 δ a_{1}^{in} 〉 = ε_{P} / \sqrt{γ_{a_{1}}}$ and $〈 δ a_{j}^{in} 〉 = 0$ (forj=2,3)) [73], we obtain the optical response of the system to the probe field,

T_{a_{2}}^{a_{1}} = \frac{γ_{a_{1}} γ_{a_{2}}}{{| β |}^{2}} {{| G_{a_{1}} |}^{2} {| G_{a_{2}} |}^{2} + {| J k^{'} |}^{2} - 2 J σ},

the optomechanical response of the system to the probe field can be derived as

T_{a_{3}}^{a_{2}} = \frac{γ_{a_{3}}}{γ_{a_{2}}} \frac{{| 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},

with the denominator

β = {k^{'}}_{2} ({k^{'}}_{1} k^{'} + J^{2}) + \sum_{j = 1}^{2} {k^{'}}_{j} {| G_{a_{j + 1}} |}^{2} - 2 i J Re (G_{a_{1}} G_{a_{2}}^{*}),

where

σ = Im (k^{'} G_{a_{1}} G_{a_{2}}^{*})

and

σ^{'} = Im ({k^{'}}_{2} G_{a_{2}} G_{a_{1}}^{*})

with k′ = γ_a₃/2 + iΔ′_a₃ and

{k^{'}}_{j} = γ_{a_{j}} / 2 + i Δ_{a_{j}}^{\bar{s}}

. The Δ′_a₃ = Δ_a₃ + (ω_d − ω_p) and

Δ_{a_{j}}^{\bar{s}} = Δ_{a_{j}}^{σ s} + (ω_{d} - ω_{p})

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 $T_{a_{3}}^{a_{2}} = 1 - ε_{T}$ . Re[ε_T] characterizes the absorption, and Im[ε_T] describes the dispersion [74]. In Fig. 6 we plot the absorption probability as a function of coupling constant G_a₁. When Ω = ω_p − ω_d = 10γ_a₃, the switch is always off. Conversely, in the condition G_a₁ = 0.5γ_a₃ with Ω ≤ 0, Ω ≥ 20 and $T_{a_{3}}^{a_{2}} ≐ 1$ , the detection field exhibits OMIT behavior and the switch is fully open.

Fig. 6 Numerical calculation Re[ε_T ] as a function of Ω (units of γ_a₃) for different optomechanical coupling constant: G_a₁ = 0.1γ_a₃ (blue solid line with stars), G_a₁ = 0.2γ_a₃ (red solid line with circles), G_a₁ = 0.3γ_a₃ (orange solid line with squares), G_a₁ = 0.4γ_a₃ (solid brown line), G_a₁ = 0.5γ_a₃ (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

{\hat{H}}_{a} = {\hat{H}}_{a}^{0} + {\hat{H}}_{a}^{I} + {\hat{H}}_{a}^{d},

with

\begin{array}{l} {\hat{H}}_{a}^{0} = \sum_{τ = 1}^{N} {Δ^{'}}_{a_{τ}} {\hat{a}}_{τ}^{†} {\hat{a}}_{τ} + Δ_{a_{m}} {\hat{a}}_{m}^{†} {\hat{a}}_{m}, \\ {\hat{H}}_{a}^{I} = \sum_{τ = 1}^{N} J ({\hat{a}}_{τ}^{†} {\hat{a}}_{τ + 1} + {\hat{a}}_{τ + 1}^{†} {\hat{a}}_{τ}) - \sum_{τ = 1}^{N} g_{a_{m}}^{s} {\hat{ℓ}}_{m} {\hat{a}}_{τ}^{†} {\hat{a}}_{τ}, \\ {\hat{H}}_{a}^{d} = \sum_{τ = 1}^{N} i ε_{a_{τ}} {\hat{a}}_{τ}^{†} e^{i φ_{a_{τ}}} + κ_{m} {\hat{a}}_{m}^{†} + H . c, \end{array}

where m represent an equivalent mechanical mode, and τ stands for the optical modes (changing from 1 to N).

{Δ^{'}}_{a_{τ}} = Δ_{a_{τ}} + {| α_{m}^{s} |}^{2} χ_{a_{τ}}

is the corresponding detuning of optical mode, Δ_{a_m} is the detuning of mechanical mode, and

κ_{m} = i ε_{a_{m}} e^{i φ_{a_{m}}} - Δ_{a_{m}} α_{m}^{s}

. The corresponding single-photon optomechanical coupling constant is

g_{a_{m}}^{s} = χ_{a_{m}} | α_{m}^{s} |

, and

{\hat{ℓ}}_{m} = {\hat{a}}_{m}^{†} e^{i θ_{m}} + {\hat{a}}_{m} e^{- i θ_{m}}

. The

α_{m}^{s}

satisfies the equation

0 = ε_{a_{m}} e^{i φ_{a_{m}}} + (i Δ_{a_{m}} - \frac{γ_{a_{m}}}{2}) α_{m}^{s},

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,

{\hat{H}}_{2} = {\hat{H}}_{2}^{0} + {\hat{H}}_{2}^{I} + {\hat{H}}_{2}^{d},

with

\begin{array}{l} {\hat{H}}_{2}^{0} = \sum_{j = 1}^{2} Δ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + Δ_{a_{3}}^{' s s} {\hat{a}}_{3}^{†} {\hat{a}}_{3}, \\ {\hat{H}}_{2}^{I} = J ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) - \sum_{j = 1}^{2} g_{a_{j}}^{s s} {\hat{ℓ}}_{j} {\hat{a}}_{3}^{†} {\hat{a}}_{3}, \\ {\hat{H}}_{2}^{d} = \sum_{i = 1}^{3} i ε_{a_{i}} {\hat{a}}_{i}^{†} e^{i φ_{a_{i}}} - \sum_{j = 1}^{2} o_{j} {\hat{a}}_{j}^{†} + H . c, \end{array}

where j varies from 1 to 3,

Δ_{a_{3}}^{' s s} = Δ_{a_{3}} + \sum_{j = 1}^{2} χ_{a_{j}} {| α_{j}^{s s} |}^{2}

is the detuning of optical mode, Δ_{a_j} is the detuning of mechanical mode, and

{\hat{ℓ}}_{j} = {\hat{a}}_{j}^{†} e^{i θ_{j}} + {\hat{a}}_{j} e^{- i θ_{j}}

. The corresponding single-photon optomechanical coupling constant is

g_{a_{j}}^{s s} = χ_{a_{j}} | α_{j}^{s s} |

, and

o_{j} = Δ_{a_{j}} α_{j}^{s s} + J α_{j + 1}^{s s}

. Following the same procedure as in the last section,

{\hat{a}}_{1} \to {\hat{a}}_{1} + α_{1}^{s s}

, and

{\hat{a}}_{2} \to {\hat{a}}_{2} + α_{2}^{s s}

. We can obtain equations that

α_{1}^{s s}

and

α_{2}^{s s}

satisfy

\begin{array}{l} 0 = + ε_{a_{1}} e^{i φ_{a_{1}}} + (i Δ_{a_{1}} - \frac{γ_{a_{1}}}{2}) α_{1}^{s s}, \\ 0 = + ε_{a_{2}} e^{i φ_{a_{2}}} + (i Δ_{a_{2}} - \frac{γ_{a_{2}}}{2}) α_{2}^{s s} + i J α_{1}^{s s} . \end{array}

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

{\hat{H}}_{s} = \sum_{i = 1}^{3} ω_{a_{i}} {\hat{a}}_{i}^{†} {\hat{a}}_{i} + \sum_{j = 1}^{2} χ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} {\hat{a}}_{3} {\hat{a}}_{3} + J ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) + i \sum_{i = 1}^{3} (ε_{a_{i}} {\hat{a}}_{i}^{†} e^{i (φ_{a_{i}} - ω_{d})} - 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

\hat{U} (t) = exp [i ω_{d} \sum_{i = 1}^{3} {\hat{a}}_{i}^{†} {\hat{a}}_{i}]

to Eq. (35), we get

{\hat{H}}_{s}^{'} = \sum_{i = 1}^{3} Δ_{a_{i}} {\hat{a}}_{i}^{†} {\hat{a}}_{i} + \sum_{j = 1}^{2} χ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} {\hat{a}}_{3} {\hat{a}}_{3} + J ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) + i \sum_{i = 1}^{3} (ε_{a_{i}} {\hat{a}}_{i}^{†} e^{i φ_{a_{i}}} - H . c .),

where Δ_{a_i} = ω_{a_i} − ω_d 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

\dot{ρ} = i [ρ, {\hat{H}}_{s}^{'}] + \sum_{i = 1}^{3} γ_{a_{i}} {({\bar{n}}_{a_{i}} + 1) \hat{ℒ} [{\hat{a}}_{3}] + {\bar{n}}_{a_{i}} \hat{ℒ} [{\hat{a}}_{i}^{†}]} ρ .

where γ_{a_i} is the damping rate of the corresponding cavity mode a₃, and n̄_{a_i} is the ith environment thermal excitation number. The Lindblad operator is

\hat{ℒ} [{\hat{a}}_{i}] = {\hat{a}}_{i} ρ {\hat{a}}_{i}^{†} - ({\hat{a}}_{i}^{†} {\hat{a}}_{i} ρ + ρ {\hat{a}}_{i}^{†} {\hat{a}}_{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^{'} = {\hat{𝒟}}_{a_{3}} (α_{3}) ρ {\hat{𝒟}}_{a_{3}}^{†} (α_{3}),

where ρ′ is the density matrix of the system after the displacement. The displacement operator is

{\hat{D}}_{a_{3}} (α_{3}) = exp (α {\hat{a}}_{3}^{†} - α^{*} {\hat{a}}_{3})

.

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

\dot{ρ} = \partial_{t} {\hat{𝒟}}_{a_{3}} (α_{3}) ρ^{'} {\hat{𝒟}}_{a_{3}}^{†} (α_{3}) + {\hat{𝒟}}_{a_{3}} (α_{3}) ρ^{'} \partial_{t} {\hat{𝒟}}_{a_{3}}^{†} (α_{3}) + {\hat{𝒟}}_{a_{3}} (α_{3}) {\dot{ρ}}^{'} {\hat{𝒟}}_{a_{3}}^{†} (α_{3}) .

Substituting

{\hat{𝒟}}_{a_{3}} (α_{3}) = exp ({| α_{3} |}^{2} / 2) exp (- {\hat{a}}_{3} α_{3}^{*}) exp ({\hat{a}}_{3}^{†} α_{3})

into Eq. (39), we can obtain

\partial_{t} {\hat{𝒟}}_{a_{3}} (α_{3}) = - \frac{1}{2} (α_{3} {\dot{α}}_{3}^{*} - {\dot{α}}_{3} α_{3}^{*}) {\hat{𝒟}}_{a_{3}} (α_{3}) - {\hat{𝒟}}_{a_{3}} (α_{3}) ({\hat{a}}_{3} {\dot{α}}_{3}^{*} - {\hat{a}}_{3}^{†} {\dot{α}}_{3}),

thus, Eq. (39) can be reduced to

{\dot{ρ}}^{'} = [{\hat{𝒟}}_{a_{3}}^{†} (α_{3}) {\dot{ρ}}^{'} {\hat{𝒟}}_{a_{3}} (α_{3}), ({\hat{a}}_{3}^{†} {\dot{α}}_{3} - {\hat{a}}_{3} {\dot{α}}_{3}^{*})] + {\hat{𝒟}}_{a_{3}}^{†} (α_{3}) \dot{ρ} {\hat{𝒟}}_{a_{3}} (α_{3}) .

Considering

{\hat{𝒟}}_{a_{3}} (α_{3}) {\hat{a}}_{3} {\hat{𝒟}}_{a_{3}}^{†} (α_{3}) = {\hat{a}}_{3} - α_{3}

, we obtain the transformed master equation as follow

\begin{array}{l} {\dot{ρ}}^{'} = & + i [ρ^{'}, {\hat{H}}_{s}^{ℓ}] + \sum_{i = 1}^{3} γ_{a_{i}} {({\bar{n}}_{a_{i}} + 1) \hat{ℒ} [{\hat{a}}_{i}] + {\bar{n}}_{a_{i}} \hat{ℒ} [{\hat{a}}_{i}^{†}]} ρ^{'} \\ + {{\dot{α}}_{3} + (i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) α_{3} + ε_{a_{3}} e^{i φ_{a_{3}}}} [{\hat{a}}_{3}^{†}, ρ^{'}] \\ - {{\dot{α}}_{3}^{*} + (- i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) α_{3}^{*} + ε_{a_{3}} e^{- i φ_{a_{3}}}} [{\hat{a}}_{3}, ρ^{'}], \end{array}

where the transformed Hamiltonian in the displacement representation becomes

\begin{array}{l} {\hat{H}}_{s}^{ℓ} = & + \sum_{j = 1}^{2} {Δ^{'}}_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + Δ_{a_{3}} {\hat{a}}_{3}^{†} {\hat{a}}_{3} + \sum_{j = 1}^{2} χ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} {\hat{a}}_{3}^{†} {\hat{a}}_{3} + J ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) - \sum_{j = 1}^{2} χ_{a_{j}} {\hat{a}}_{j}^{†} {\hat{a}}_{j} {\hat{a}}_{3}^{†} {\hat{a}}_{3} \\ - \sum_{j = 1}^{2} g_{a_{j}}^{s} ({\hat{a}}_{3}^{†} e^{i θ} + {\hat{a}}_{3} e^{- i θ}) {\hat{a}}_{j}^{†} {\hat{a}}_{j} . \end{array}

The average of a₃ can be determined in the displacement representation when the coherent displacement amplitude

α_{3}^{s}

obeying the equation

{\dot{α}}_{3}^{s} + (i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) α_{3}^{s} + ε_{a_{3}} e^{i φ_{a_{3}}} = 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

\frac{d}{d t} {\hat{a}}_{3} = - i Δ_{a_{3}} {\hat{a}}_{3} - i \sum_{k} g_{3 k} {\hat{Γ}}_{3 k} + i Δ_{a_{3}} α_{3}^{s} + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i θ} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + ε_{a_{3}} e^{i ϕ_{a_{3}}},

and environment operator Γ̂_3k,

\frac{d}{d t} {\hat{Γ}}_{3 k} = - i ω_{3 k} {\hat{Γ}}_{3 k} - i g_{3 k} {\hat{a}}_{3} - i g_{3 k} α_{3}^{s} .

Solving Eq. (45) for Γ̂_3k, we get

{\hat{Γ}}_{3 k} = - i g_{3 k} \int_{0}^{t} {{\hat{a}}_{3} (τ) + α_{3}^{s}} e^{- i ω_{3 k} (t - τ)} d τ + {\hat{Γ}}_{3 k} (0) e^{- i ω_{3 k} t} .

Inserting Eq. (46) into Eq. (44), we have

\begin{array}{l} \frac{d}{d t} {\hat{a}}_{3} = & + {\hat{f}}_{a_{3}} - \sum_{k} g_{3 k}^{2} \int_{0}^{t} {{\hat{a}}_{3} (τ) + α_{3}^{s}} e^{- i ω_{3 k} (t - τ)} d τ + i Δ_{a_{3}} (α_{3}^{s} - {\hat{a}}_{3}) \\ + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i θ} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + ε_{a_{3}} e^{i ϕ_{a_{3}}}, \end{array}

with

{\hat{f}}_{a_{3}} (t) = - i \sum_{k} g_{3 k} {\hat{Γ}}_{3 k} (0) e^{- i ω_{3 k} t},

here f̂_a₃(t) is a noise operator because it depends upon the environment operators Γ̂_3k (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(t − τ) function and the integration can then be carried out. We obtain

\begin{array}{l} \frac{d}{d t} {\hat{a}}_{3} = & + {\hat{f}}_{a_{3}}^{'} (t) - \int_{0}^{t} g_{3}^{2} (ω) π δ (t - τ) {{\hat{a}}_{3} (τ) + α_{3}^{s}} d τ \\ + i Δ_{a_{3}} (α_{3}^{s} - {\hat{a}}_{3}) + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i θ} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + ε_{a_{3}} e^{i ϕ_{a_{3}}}, \end{array}

with

{\hat{f}}_{a_{3}}^{'} (t) = - i \int_{0}^{\infty} g_{3} (ω) {\hat{Γ}}_{30} (ω) e^{- i ω t} d ω .

Hereafter, we assume that g₃(ω) is dependent of frequency over a wide range of frequencies around ω = 0, by Markov approximation we get

g_{3}^{2} (ω) = γ_{a_{3}} / 2 π

, and Eq. (50) becomes

\frac{d}{d t} {\hat{a}}_{3} = (- i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) {\hat{a}}_{3} + i \sum_{j = 1}^{2} g_{a_{j}}^{s} e^{i ω} {\hat{a}}_{j}^{†} {\hat{a}}_{j} + ε_{a_{3}} e^{i ϕ_{a_{3}}} + (i Δ_{a_{3}} - \frac{γ_{a_{3}}}{2}) α_{3}^{s} + \sqrt{γ_{a_{3}}} {\hat{a}}_{3}^{in},

with

{\hat{a}}_{3}^{in} = - \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} i g_{3} (ω) {\hat{Γ}}_{30} (ω) e^{- i ω t} d ω .

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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Nonreciprocity in a strongly coupled three-mode optomechanical circulatory system

Abstract

1. Introduction

2. Formalism

2.1. Model

2.2. Main theory

2.3. Experimental implementation

2.4. Quantum langevin equations

3. Applications

3.1. Optomechanical circulator

3.2. Optomechanical diode

3.3. Optomechanical transistor

4. General expansion

4.1. 1D optomechanical loop model

4.2. 2D OMCS model

4.3. Generalized planer optomechanical quantum network

5. Conclusion

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

Appendix B: The derivation of Eq. (7)

Funding

Acknowledgments

References

Cited By

Figures (7)

Equations (53)

Optics Express