Cavity optomechanical systems have attracted a great deal of attention from both theorists and experimentalists in recent years.^[1–36] The systems demonstrate the interaction between the movable mechanical resonator and the optical field in the cavity by radiation pressure, and becomes a platform for studying optomechanically induced transparency (OMIT),^[2–12] optomechanically induced amplification,^[13,14] ground-state cooling of the mechanical resonator,^[15–18] normal-mode splitting,^[19] parity-time-symmetry-breaking chaos,^[20] and so on. Recently, OMIT has been studied in various cavity optomechanical systems.^[6–12] For example, Zhang et al. proposed a potentially practical scheme to precisely measure the charge number of small charged objects using OMIT in an optomechanical system, where the charged mechanical mode is subject to the Coulomb force due to the charged body nearby.^[34] Wu investigated the properties of the tunable ponderomotive squeezing by Coulomb interaction in a cavity optomechanical system.^[22]

In the past decades, a phenomenon called nonreciprocity effect has been predicted theoretically and observed experimentally.^[37–41] The nonreciprocity effect is the basis of directional amplifiers, isolators, circulators,^[37] and an important device for information processing. This effect is usually caused by breaking the time-reversal symmetry of photons. There are four main ways to break the time-reversal symmetry of photons: (i) using magneto-optical effects,^[37] (ii) using non-magnetic strategies of optical nonlinearity,^[38,39] (iii) using dynamic modulation,^[40] and (iv) using angular momentum biasing.^[41] Recently, it is noteworthy that some researchers have used the radiation-pressure-induced optomechanical coupling to break the time-reversal symmetry and to realize the nonreciprocal effects for light.^[42–56] The significant achievements have been made in this field, including nonreciprocal amplification,^[42] nonreciprocal transmission,^[43,44] nonreciprocal single-photon effects,^[45] and nonreciprocal slow light,^[9,47] etc. Jiang et al. proposed a scheme for realizing optical directional amplification between optical and microwave fields based on an optomechanical system with optical gain.^[48] They found that the direction of amplification can be controlled by the phase differences between the effective optomechanical couplings. Xu et al. showed the optical nonreciprocal response in a three-mode optomechanical system consisting of two optical modes and one mechanical mode.^[53] They proved that the optical nonreciprocal response is achieved by adjusting the phase difference between the optomechanical coupling rates to cause breaking of time-reversal symmetry of the system. Xu et al. demonstrated the possibility of a three-port circulator in a three-mode optomechanical system, which is referred to as an optomechanical circulator.^[51] Li et al. studied a three-mode optomechanical system, where the mechanical mode is subject to the additional mechanical drive.^[46] When the driving frequency of the mechanical mode is equal to the frequency difference between the optical probe and pump fields, the system will get directional amplification of an optical probe field.

Inspired by Ref. [48], we naturally ask whether the optical nonreciprocity can be realized by Coulomb interaction in optomechanical system or not. If possible, what are new insights that the Coulomb interaction will bring about for the optical nonreciprocal response? Based on the above problems, we theoretically study how to realize the optical nonreciprocal response based on a four-mode optomechanical system. As shown in Fig. 1, two charged mechanical mode are coupled by Coulomb interaction. One mechanical mode is coupled with two directly coupled optical modes respectively. In addition, two optical modes are driven by two coupling fields respectively. The optical nonreciprocal response can be achieved when the four-mode optomechanical system works under certain conditions. We will also clarify the role of each physical quantity in the system. The main differences between our work and Ref. [48] are: (1) The systems that form the optical nonreciprocal response are different. The system in Ref. [48] consists of two optical modes and one mechanical mode. The system we consider contains two optical modes and two charged mechanical modes coupled by Coulomb interaction. As far as optomechanically induced transparency is considered, there have been schemes utilizing charged objects in cavity optomechanical systems.^[22,33,34] However, no proposals have been found to combine both mechanical modes and Coulomb interaction for studying the optical nonreciprocal response in quantum optomechanics though much richer nonlinear phenomena are expected. (2) We will discuss the optimal parameters of observing the optical nonreciprocal response. Furthermore, Compared with Refs. [46,51], there is no additional mechanical drive in the mechanical mode of our system, so our system is more stable.

Fig. 1.

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	Fig. 1. The schematic of the four-mode cavity optomechanical system consisting of two optical modes (c1 and c2) and two charged mechanical modes (b3 and b4). The optical modes and the mechanical mode b3 are coupled via radiation pressure. The mechanical modes b3 and b4 are coupled via Coulomb interaction.

The paper is organized as follows. In Section 2, we introduce the theoretical model, the detailed analytical expressions of the optomechanical system, and obtain the transmission probability of the probe field. In Section 3, we discuss in detail the effect of the phase difference, the Coulomb coupling, and the effective optomechanical coupling strengths on the optical nonreciprocal response. Finally, in Section 4, we summarize our work.

The four-mode optomechanical system under consideration is schematically shown in Fig. 1, including two optical modes (c_i, frequency ω_ci, length L_i (i = 1,2)) and two charged mechanical modes (b_j, effective mass m_j, frequency ω_mj and damping rate γ_mj (j = 3,4)). Two optical modes couple directly with coupling strength J. Furthermore, the two optical modes couple with the charged mechanical mode b₃ respectively, and the optomechanical coupling strength is (i = 1,2). The charged mechanical mode b₃ is coupled to the other charged mechanical mode b₄ by a tunable Coulomb interaction V. The b₃ and b₄ take the charges Q₃ = C₃V₃ and Q₄ = C₄V₄, with C₃ (C₄) and V₃ (V₄) being the capacitance and the voltage of the bias gate,^[22] respectively. The two optical modes are driven by the strong coupling field with frequency ω_Li = 2π c/λ_i (λ_i (i = 1,2) represents the wavelength, c is the speed of light in vacuum), in which P_i (i = 1,2) denotes its power and κ_i is the total damping rate of optical mode c_i (i = 1,2). Without loss of generality, here we have assumed that J, g_0i, V and E_Li (i = 1,2) are real numbers. The Hamiltonian of the four-mode optomechanical system is (ħ = 1)

Starting from the Heisenberg equation of motion , taking into account the dissipations of the optical modes and the mechanical modes, and adding quantum noise and thermal noise, we can give the time evolutions of the expectation values of the system operators as follows:

The stability conditions of the system require all the real parts of the eigenvalues of matrix A to be negative, which can be analyzed by the Routh–Hurwitz criterion.^[57,58] The pushing process is too complicated, so we omitted it here.

The Fourier transform of the operators are given by

The input-output theorem is^[2]

In this section, we numerically evaluate the transmission probability of the probe field to prove the possibility of the optical nonreciprocal response in the four-mode optomechanical system. According to the numerical results, the optimum parameters for observing the optical nonreciprocal response are obtained. Relevant parameters are taken as κ₀₁ = π × 10⁶ Hz, κ₀₂ = 2π × 10⁶ Hz, κ_ex1 = κ_ex2 = 4π × 10⁶ Hz, γ_m3 = γ_m4 = 4π × 10⁶ Hz, G₁ = 4π × 10⁶ Hz, G₂ = 6π × 10⁶ Hz, θ = π/3, J = (2κ₁ κ₂)^1/2, V = 0.1γ_m3.^[48]

In Fig. 2, we plot the transmission probability of the probe field |T_ij|² (i,j = 1,2) as a function of the ω/2π with ϕ = π/2 (red dashed lines) and ϕ = −π/2 (green solid lines), where the parameters are κ₁ = 5π × 10⁶ Hz, κ₂ = 6π × 10⁶ Hz, γ_m3 = γ_m4 = 4π × 10⁶ Hz, G₁ = 4π × 10⁶ Hz, G₂ = 6π × 10⁶ Hz, θ = π/3, J = (2κ₁ κ₂)^1/2, V = 0.1γ_m3. Under the condition of the phase difference ϕ = ± π/2, there are two peaks near the two frequencies with the probe detuning ω ≃ ± 5 π × 10⁶ Hz and dip near the frequency with the probe detuning ω ≃ 0 in the spectrum line of |T_ij|² (i = j = 1,2). At the same time, |T₁₂|² is always not equal to |T₂₁|² in the process of probe detuning ω from −8 π × 10⁶ Hz to 8 π × 10⁶ Hz, that is to say, transmission probability of the probe field does not satisfy the Lorentz reciprocal theorem, and there appears the optical nonreciprocal response. When the phase difference ϕ = −π/2, we have |T₁₂|² < |T₂₁|². When the phase difference ϕ = π/2, we have |T₁₂|² > |T₂₁|². For example, when the phase difference ϕ = −π/2, |T₂₁|² ≃ 0.3 while |T₁₂|² = 0 at ω = 0. This shows that the probe field can be transmitted from cavity c₁ to cavity c₂, but the transmission from cavity c₂ to cavity c₁ is almost prohibited. The optical nonreciprocal response arises due to interference between two possible paths, where one path is along c₁ → b₃ → c₂ and the other path is along c₁ → c₂. When ϕ = −π/2, constructive interference between the two paths leads to the optical transmission from cavity c₁ to cavity c₂, but oppositely the optical transmission cavity c₂ to cavity c₁ is forbidden due to destructive interference.^[48] The case of ϕ = π/2 is exactly opposite to that of ϕ = −π/2, |T₁₂|² ≃ 0.3 whereas |T₂₁|² = 0 at ω = 0. Consequently, the optical nonreciprocal response can be realized based on the four-mode optomechanical system.

Fig. 2.

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	Fig. 2. The transmission probabilities |Tij|2 (i,j = 1,2) as a function of the ω/2π. (i) The red dashed lines show the ϕ = π/2, (ii) green solid lines show the ϕ = −π/2. The values of the other parameters are set as κ01 = π × 106 Hz, κ02 = 2 π × 106 Hz, κex1 = κex2 = 4π × 106 Hz, γm3 = γm4 = 4π × 106 Hz, G1 = 4π × 106 Hz, G2 = 6π × 106 Hz, θ = π/3, J = (2κ1 κ2)1/2, V = 0.1γm1.

Figure 3 shows the transmission probability of the probe field |T₁₂|² (blue dashed line) and |T₂₁|² (red solid line) as a function of the phase difference ϕ/π. As shown in Fig. 3, the transmission probability of the probe field shows regular periodic oscillation. The |T₁₂|² = |T₂₁|² when ω = 0 and ϕ = ± n π (n is an integer). When ϕ ≠ ± nπ (n is an integer), the |T₁₂|² ≠ |T₂₁|², the time-reversal symmetry is broken, and the four-mode optomechanical system presents the optical nonreciprocal response. The optimal optical nonreciprocal response is obtained as ϕ = −π/2 ± 2nπ [|T₁₂|² ≃ 0 and |T₂₁|² ≃ 0.3] and ϕ = π/2 ± 2nπ [|T₁₂|² ≃ 0.3 and |T₂₁|² ≃ 0] (n is an integer).

Fig. 3.

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	Fig. 3. The transmission probabilities |T12|2 (blue dashed line) and |T21|2 (red solid line) versus ϕ/π with ω = 0. The other parameters are the same as those in Fig. 2.

The Coulomb interaction is also an important parameter. Therefore, the transmission probability of the probe field |T₁₂|² (blue dashed lines) and |T₂₁|² (red solid lines) plotted as a function of the ω/2π are shown in Fig. 4 for different Coulomb interaction at the phase ϕ = −π/2: (a) V = 0.1γ_m3, (b) V = 0.5γ_m3, (c) V = γ_m3, (d) V = 12γ_m3. We can see that the Coulomb interaction has a significant influence on transmission probability of the probe field |T₁₂|² and |T₂₁|². When the Coulomb interaction is much smaller than the mechanical damping rate γ_m3, e.g., V = 0.1γ_m3 [see Fig. 4(a)], |T₁₂|² ≠ |T₂₁|², and the optical nonreciprocal response is pronounced with the ω = 0. However, as the Coulomb interaction increases, the optical nonreciprocal response becomes invisible with the ω = 0 [see Figs. 4(b) and 4(c)]. When V = 12γ_m3, |T₁₂|² is equal to and |T₂₁|², as shown in Fig. 4(d), the optical nonreciprocal response disappears. From Figs. 4(a)–4(d), we can see that the smaller the Coulomb interaction, the more obvious the optical nonreciprocal response. The optimal optical nonreciprocal response is obtained as V = 0.1γ_m3. From Eq. (1), we can obtain the steady-state mean value of mechanical mode b₃ as follows:

Fig. 4.

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	Fig. 4. The transmission probabilities |T12|2 (blue dashed lines) and |T21|2 (red solid lines) versus ω/2π with ϕ = − π/2. (a) V = 0.1γm3, (b) V = 0.5 γm3, (c) V = γm3, (d) V = 12γm3. The other parameters are the same as those in Fig. 2.

To further understand the effect of the Coulomb interaction on the transmission probability of the probe field, we plot the probability of the transmission |T₁₂|² (blue dashed line) and |T₂₁|² (red solid line) as a function of Coulomb interaction V/γ_m3 in Fig. 5. We assume that the parameters ϕ = −π/2, ω = 0 and other parameters are the same as those in Fig. 2. It can be seen from Fig. 5 that |T₂₁|² is a decreasing function of V/γ_m3, whereas |T₁₂|² is opposite. It can be easily observed that the optical nonreciprocal response only exists in a limited value range of V ≤ 5γ_m3. When the Coulomb interaction V is stronger than 5γ_m3, the transmission probability from cavity c₂ to cavity c₁ is the same as the transmission probability from cavity c₁ to cavity c₂, |T₁₂|² ≈ |T₂₁|². The values of |T₁₂|² and |T₂₁|² tend to be closer and closer to 0.21 as the Coulomb interaction is gradually increased, and the optical nonreciprocal response cannot be observed.

Fig. 5.

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	Fig. 5. The transmission probabilities |T12|2 (blue dashed line) and |T21|2 (red solid line) versus V/γm3 with ϕ = −π/2, ω = 0. The other parameters are the same as those in Fig. 2.

The effective optomechanical coupling strengths G_i (i = 1,2) are also a critical parameters in the optomechanical system. Next, we discuss the effect of the effective optomechanical coupling strengths G_i (i = 1,2) on the transmission probability of the probe field. The transmission probability of the probe field |T₁₂|² (blue dashed line) and |T₂₁|² (red solid line) plotted as a function of the G₂/G₁ are shown in Fig. 6. When G₂/G₁ < 1.5, |T₂₁|² increases with increasing G₂/G₁, and |T₁₂|² is suppressed by the increasing G₂/G₁. By contrast, when G₂/G₁ > 1.5, with the further increase of G₂/G₁, |T₂₁|² gradually decreases and |T₁₂|² gradually increases. In the neighborhood of G₂/G₁ = 1.5, |T₂₁|² reaches the maximum |T₂₁|² = 0.33, |T₁₂|² reaches the minimum |T₁₂|² = 0, the optical nonreciprocal response is the most obvious. The results show that G₂/G₁ can influence the optical nonreciprocal response, which agrees with our theoretical prediction. The optimal optical nonreciprocal response is obtained as G₂/G₁ = 1.5.

Fig. 6.

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	Fig. 6. The transmission probabilities |T12|2 (blue dashed line) and |T21|2 (red solid line) versus G2/G1 with ϕ = −π/2, ω = 0. The other parameters are the same as those in Fig. 2.

In summary, we have investigated the optical nonreciprocal response in a four-mode optomechanical system. This system involves the Coulomb interaction and the optomechanical interaction. Under the condition of the strong coupling field, the phase difference between the optomechanical couplings, the Coulomb interaction, the effective optomechanical coupling strengths on the optical nonreciprocal response are analyzed. The results show that the optical nonreciprocal response is caused by the quantum interference between the optomechanical couplings and the linearly coupled interaction that breaks the time-reversal symmetry. In addition, we can obtain the optimum parameters for observing optical nonreciprocal response. We also see that the Coulomb effect has a negative effect on the optical nonreciprocal response.