Multiterminal nonreciprocal routing in an optomechanical plaquette via synthetic magnetism

Optomechanical systems with parametric coupling between optical (photon) and mechanical (phonon) modes provide a useful platform to realize various magnetic-free nonreciprocal devices, such as isolators, circulators, and directional amplifiers. However, nonreciprocal router with multiaccess channels has not been extensively studied yet. Here, we propose a nonreciprocal router with one transmitter, one receiver, and two output terminals, based on an optomechanical plaquette composing of two optical modes and two mechanical modes. The time-reversal symmetry of the system is broken via synthetic magnetism induced by driving the two optical modes with phase-correlated laser fields. The prerequisites for nonreciprocal routing are obtained both analytically and numerically, and the robustness of the nonreciprocity is demonstrated numerically. Multiterminal nonreciprocal router in optomechanical plaquette provides a useful quantum node for development of quantum network information security and realization of quantum secure communication.

In this paper, we introduce a nonreciprocal router with one transmitter, one receiver, and two output terminals, and propose a scheme to realized the nonreciprocal router based on an optomechanical plaquette composing of two optical modes and two mechanical modes.We demonstrate that the optomechanical plaquette can serve as a nonreciprocal router with one optical mode as transmitter, another optical mode as receiver, and two mechanical modes as output terminals.We will investigate the optimal conditions for nonreciprocal router and discuss its robustness against the imperfectness.Our work may inspire the study of nonreciprocal quantum nodes for scalable quantum information processing in quantum secure network and communication.
The remainder of the paper is organized as follows.In Sec.II, we introduces the model of nonreciprocal router and show how to realize it in an optomechanical plaquette.In Sec.III, we show the nonreciprocal scattering matrix of the optomechanical plaquette and discuss the transmitter receiver terminal terminal influence of experimental parameters on the nonreciprocal routing effect.This work is summarized in Sec.IV.

II. NONRECIPROCAL ROUTER AND OPTOMECHANICAL PLAQUETTE
In this section, we will introduce the basic ideal of a nonreciprocal router, and then propose an optomechanical model to realize the nonreciprocal router.

A. Nonreciprocal router
We consider a nonreciprocal router consisting of one transmitter, one receiver, and two terminals.The basic ideal for the nonreciprocal router considered in the paper is shown in Fig. 1(a).Let's introduce the characteristics of the nonreciprocal router.The nonreciprocity indicates that the signals transport nonreciprocal between different ports, which are shown in the following three aspects: Firstly, the signal is transmitted from the transmitter to the terminals but cannot return from the terminals to the transmitter.Secondly, the signal from the terminals can be transmitted to the receiver, but the signal from the receiver cannot transport to the terminals.Thirdly, the nonreciprocity between the transmitter and the receiver is manifested as the signal can only be transmitted from the receiver to the transmitter but not from the transmitter to the receiver.In order to realize such functions in an integrated platform, we consider an optomechanical plaquette, which has been used to realize non-reciprocal photon transport with high isolation and directional optical amplification via synthetic magnetism [54][55][56][57][58][59][60][61][62].

B. Optomechanical plaquette
We consider an optomechanical plaquette composing of two optical modes a j (j = 1, 2) with frequency ω c,j and two mechanical modes b j with frequency ω j , as shown in Fig. 1(b).The two optical (mechanical) modes couple to each other with strength J c (J m ), and the optical mode (a j ) interacts with the mechanical mode (b j ) via radiation pressure with single-photon optomechanical coupling rate g j .To enhance the optomechanical coupling strength g j , each optical mode is pumped by a strong optical field with strength ε j and frequency ω p,j relatively detuned from the resonant frequency of optical mode ω c,j by the mechanical frequency ω j , ∆ 0,j ≡ ω c,j − ω p,j ≈ ω j .In a rotating frame with respect to the unitary transformation R(t) = exp(−i j=1,2 ω p,j ta † j a j ), the optomechanical plaquette can be described by a Hamiltonian as (ℏ = 1) where φ 1 and φ 2 are the phases of the pumping fields and they are correlated, as an essential ingredient for nonreciprocity.
According to the Heisenberg equation of motion and taking into account the damping and corresponding input noise, we get the quantum Langevin equations (QLEs) as where κ e and κ 0 (γ e and γ 0 ) are the external and internal decay rates of optical (mechanical) modes, with the corresponding input optical fields a j,in and a j,vac (mechanical fields b j,in and b j,th ).κ = κ e + κ 0 and γ = γ e + γ 0 are the total decay rates of the optical and mechanical modes, respectively.
To obtain the linear response of the system to weak optical and mechanical signals, we will solve the QLEs by the standard process of the linearization [5].We solve the QLEs in steady state based on the mean field approximation first, then the linearization of the system near the steady-state values yields to the corresponding linearized QLEs.The steady-state values α j ≡ ⟨a j ⟩ and β j ≡ ⟨b j ⟩ are obtained from the QLEs by setting the time derivative terms to zero and using the factorization assumption ⟨AB⟩ = ⟨A⟩⟨B⟩ for any two operators A and B, as where which means that the amplitude |α j | and phase ϕ j of α j can be freely adjusted by the external field ε j e −iφj .The linearized QLEs for the quantum fluctuation operators δa j ≡ a j − α j and δb j ≡ b j − β j (j = 1, 2) are obtained as Here G 1 = |α 1 | g 1 and G 2 = |α 2 | g 2 are the linearized optomechancial coupling strengths; the nonlinear terms are negligible for the assumption |α j | 2 ≫ ⟨δa † j δa j ⟩, and the counter-rotating terms are neglected based on rotationwave approximation under conditions ∆ j ≈ ω j ≫ G j .
The linearized QLEs can be concisely expressed as where V (t) is the vector for the quantum fluctuation operators defined by T , Γ e ≡ diag(κ e , κ e , γ e , γ e ), Γ 0 ≡ diag(κ 0 , κ 0 , γ 0 , γ 0 ), and the linearized coefficient matrix M is given by In order to ensure the stability of the system, we need to ensure that the real part of all eigenvalues of the coefficient matrix M are positive in the following discussions.
The linearized QLEs can be solved analytically in the frequency domain by the method of Fourier transform, with the definition of Fourier transform for operator o as The solution of the linearized QLEs in the frequency domain is obtained as where I denotes the identity matrix.Based on the inputoutput relation [84] the output fields from the optomechanical system are obtained as where are the scattering matrices.So the scattering probabilities from Port j to Port i are given by where U ij (ω) (i, j = 1, 2, 3, 4) denotes the element of U (ω) at ith row and jth column, given analytically in Appendix.

C. Parameter conditions for nonreciprocal routing
The optomechanical plaquette can work as a nonreciprocal router with two optical modes acting as the transmitter and receiver, and two mechanical modes as two terminals.For example, we can use the optical mode a 1 as the transmitter, a 2 as the receiver, and the two mechanical modes b 1 and b 2 as two terminals.In order to realize the nonreciprocal router shown in Fig. 1(a), the scattering probabilities from Port 1 to 2, from Ports 3 and 4 to 1, and from Port 2 to 3 and 4, should be significantly inhibited, i.e., approximately equal to zero, Besides that, we can also use the optical mode a 2 as the transmitter, a 1 as the receiver, and the two mechanical modes as two terminals.Then the scattering probabilities from Port 2 to 1, from Ports 3 and 4 to 2, and from Port 1 to 3 and 4, should be significantly inhibited as The parameter conditions for these scattering probabilities given in Eqs. ( 23) and ( 24) can be derived from the analytical expression of U ij (ω) given in Appendix.
Before the numerical simulation given in the next section, let us derive the parameters conditions for nonreciprocal router analytically based on some assumptions.Without loss of generality, here we set ω 1 = ω 2 = ω m , ∆ 1 = ∆ 2 = ω m , and G 1 = G 2 = G for simplicity.Moreover, we introduce the synthetic flux (relative phase) Φ ≡ ϕ 2 − ϕ 1 threading the four-mode plaquette to break time-reversal symmetry of the system, leading to nonreciprocal transport of signals between different ports.Based on the Eqs.(A.1)-(A.16),we find that the optomechanical plaquette can work as a nonreciprocal router with a 1 as the transmitter and a 2 as the receiver around the frequency ω = ω m − J m for Φ = π/2 or around the frequency ω = ω m +J m for Φ = 3π/2.Conversely, the optomechanical plaquette can also work as a nonreciprocal router with a 1 as the receiver and a 2 as the transmitter around the frequency ω = ω m + J m for Φ = π/2 or around the frequency ω = ω m − J m for Φ = 3π/2.The other parameter conditions are given by: (i) to realize S 12 (ω) ≈ 0 or S 21 (ω) ≈ 0, the parameters need to satisfy the conditions (ii) S 14 (ω) ≈ S 32 (ω) ≈ 0 or S 41 (ω) ≈ S 23 (ω) ≈ 0, are expected with the parameters satisfying the conditions (iii) S 13 (ω) ≈ S 42 (ω) ≈ 0 or S 31 (ω) ≈ S 24 (ω) ≈ 0, are obtained under the conditions The conditions for nonreciprocal routing are summarized in Table I.
The conditions required to observe nonreciprocal routing can be reached for the current state-of-the-art optomechanical crystal circuit systems.For example, Ref. [54] demonstrated nonreciprocal photon transport and amplification by an optomechanical plaquette with the parameters: mechanical frequency ω m /2π ≈ 5.8 GHz, external optical damping rate κ e /2π ≈ 0.75 ∼ 1 GHz, intrinsic optical damping rate κ 0 /2π ≈ 0.3 GHz, single-photon optomechanical coupling rate g i /2π ≈ 0.8 MHz, mechanical dissipation rate γ/2π ≈ 4 ∼ 5 MHz, optical hopping rate J c /2π ≈ 0.1 ∼ 1.4 GHz, mechancial hopping rate J m /2π ≈ 2.8 MHz.The coupling rates J c and J m can be varied by changing the number and shape of the holes that make up the optomechanical crystal between the two optomechanical cavities [85].The linearized optomechanical coupling rates G j = |α j | g j can be enhanced by pumping the optomechanical cavities with phase-correlated lasers.

III. NONRECIPROCAL ROUTING
In this section, we will show the nonreciprocal routing effect in the optomechanical plaquette by numerical calculations.As a simple example, we show the nonreciprocal scattering matrix first.Then, we discuss the optimal parameters (synthetic flux Φ, optomechanical coupling rate G, and external decay rate κ e ) for nonreciprocal routing, and the influences of the detuning between the two mechanical modes (δ = ω 1 − ω 2 ) and internal optical and mechanical decays (κ 0 and γ 0 ) on nonreciprocal routing.

A. Nonreciprocal scattering matrix
The scattering probabilities S ij (ω) between different ports are shown as functions of the frequency (ω −ω m )/γ In contrast, at frequency ω = ω m + J m , the optomechanical plaquette exhibits nonreciprocal routing behavior with signals transporting in different direction.As shown in Figs.2(c) and 2(e), the optical mode a 2 is taken as a transmitter and the optical mode a 1 is taken as a receiver.As summarized in Table I, the optomechanical plaquette show nonreciprocal routing behavior in revers direction for Φ = π/2 and Φ = 3π/2.Thus we can steer the direction of the nonreciprocal routing by adjusting either the synthetic flux Φ or the frequency of the input signals.

B. Numerical results for optimal parameters
In order to demonstrate the optimal conditions for nonreciprocal routing quantitatively, i.e., κ = 2J c , G = √ J c γ, and Φ = π/2 or 3π/2, we show the scattering probabilities S ij (ω) as functions of synthetic flux Φ, linearized optomechancial coupling strength G, and external optical decay κ e in Figs.3(a Synthetic flux Φ is one ingredient for breaking the time-reversal symmetry, and plays a key role in controlling the nonreciprocal routing [see Fig. 3

C. Effects of detuning and internal decays
In the discussions above, we assume that the two mechanical modes have the same resonant frequency ω 1 = ω 2 = ω m and ignore the intrinsic decay κ 0 = γ 0 = 0 for simplicity.However, there are unpredictable uncertainty in devices fabrication and we may have the frequency difference of two mechanical modes δ = ω 1 −ω 2 and intrinsic decay κ 0 and γ 0 in experiments.In this subsection, we will show numerically that the nonreciprocal routing is robust against the mechanical frequency difference δ and intrinsic decay κ 0 and γ 0 .
In order to study the influences of the intrinsic optical decay on the nonreciprocal routing, the scattering probabilities are plotted as functions of κ 0 /κ e in Fig. 4(b).Both S 12 (ω) and S 21 (ω) decrease with the creasing of κ 0 , but the isolation of S 12 (ω)/S 21 (ω) is not sensitive to the change of κ 0 [see Fig. slowly with the increasing of γ 0 , and S 13 (ω), S 14 (ω), S 42 (ω), and S 32 (ω) are not sensitive to γ 0 .

D. Nonreciprocal routing with normal mechanical modes
Under the strong coupling condition J m ≫ γ, we can introduce the normal modes (b ± ) of the two coupled mechanical modes (b 1 and b 2 ) as with resonant frequency ω ± = ω m ± J m .The resonant frequency of ω ± provide us a clue to understand the scattering probabilities S ij (ω) change dramatically round the splitting frequencies ω m ± J m .The linearized QLEs (13) for the system can be rewritten with normal mechanical modes (b ± ) as T , and The scattering probabilities S ij (ω) between modes j and i (i, j = 1, 2, +, −, for a 1 , a 2 , b + , b − ) can be obtained by following the standard method as shown in the last paragraph of subsection II B.
Scattering probabilities

IV. CONCLUSIONS
In summary, we have introduced a nonreciprocal router composing of a transmitter, a receiver, and two terminals.We have also proposed a scheme to realize the nonreciprocal router based on an optomechanical plaquette consisting of two optical and two mechanical modes, with one optical mode as transmitter, one optical mode as receiver, and two mechanical modes as terminals.We demonstrated that the transport direction of the signals in the router can be steered by the synthetic flux induced by the external driving fields, and the frequency of the input signals.We shew that the nonreciprocal router based on optomechanical plaquette are robust against the experimental imperfectness and within the reach of current experimental conditions.Our work lays the foundation for useful applications of nonreciprocal routers in quantum network and quantum secure communication.

FIG. 1 .
FIG. 1. (Color online) (a) Schematic diagram of a nonreciprocal router with one transmitter, one receiver, and two output terminals.The signals transport unidirectionally from the transmitter to the two terminals, from the terminals to the receiver, and from the receiver to the transmitter.(b)Schematic diagram of a four-mode optomechanical plaquette comprising of two optical modes (a1 and a2) and two mechanical modes (b1 and b2).The optical (mechanical) modes are coupled to each other with strength Jc (Jm), and the optical and mechanical modes (a1 and b1, a2 and b2) are coupled via radiation pressure, with a synthetic flux Φ = ϕ2 − ϕ1 threading the four-mode plaquette for the linearized optomechancial coupling strengths G1e iϕ 1 and G2e iϕ 2 .