Introduction

Cavity optomechanics, as a unique platform to explore the radiation-pressure interaction, has attracted extensive experimental and theoretical interests owing to its wide applications in various classical and quantum information processing, weak signal measurement, and development of new devices1,2,3,4,5,6,7. All kinds of optomechanical systems have been proposed and manufactured to research abundant physical phenomena and applications, such as entanglements8,9,10, optomechanical chaos11,12, high-order sideband generation13,14,15,16,17, cooling of mechanical oscillators18,19,20, optomechanically induced transparency (OMIT)21,22,23,24,25. Due to the sharp transmission features of OMIT regulated by the control laser beam, it makes many applications ranging from force sensor to quantum communication possible. In recent years, except for the simple cavity optomechanical systems, various hybrid optomechanical systems have been proposed to study OMIT in numerous different physical mechanisms, such as parity-time symmetric OMIT26,27, nonlinear OMIT28,29 and double OMIT30,31. Compared to the conventional OMIT, double and multiple OMIT32 own many favorable features except multichannel transparency windows, including extensive integration and performance. Moreover, OMIT in integrated resonators provides us with a practical method for achieving optical nonreciprocity.

Optical nonreciprocity is a phenomenon in some devices that only allows optical signals to be transmitted in one direction, but prevents their transmission in the opposite direction, which is essential in a wide range of applications such as stealth or noiseless information processing. In the early days, breaking reciprocity or time-reversal symmetry is usually achieved through the magneto-optical effects33,34,35, but its disadvantage were large, high cost, and not suitable for on-chip integration. With the development of nanophotonics as well as the improvement of nano-fabrication techniques, non-magnetic approaches for realizing optical nonreciprocity have been extensively studied with significant advances, such as nonlinear effect36,37,38, parametric modulation39,40,41 and optomechanical coupling42,43,44,45,46. Until now, nonreciprocal effects have been studied extensively in the context of optomechanical systems, such as by tuning the phase difference between the two optomechanical coupling rates to breaking the time-reversal symmetry of the system47,48, by optically pumping the ring resonator49 or by resonant Brillouin scattering50,51 to enhance the optomechanical coupling in one direction and suppress in the other one, by breaking of the algebraic duality symmetry52 to give rise to different behaviors between the forward and the backward transmissions, and so on.

In this paper, a scheme about optical nonreciprocal phenomena in a double-cavity optomechanical system with nonreciprocal coupling is proposed. First, we investigate the steady-state dynamic processes of the double-cavity system in the red-detuning regime by using the perturbation method. We find that the system gives rise to bistability due to the strong optomechanical interaction driven by the control field, and the coupling strengths between two cavities have a regulating effect on the number of photons, which plays an important role in the subsequent study of optical nonreciprocal transmission. Second, we show the transmission spectrum of the probe field and analyze the physical mechanism of the induced transparency windows in detail. It is found that the coupling strengths play a decisive role in the nonreciprocal response of the probe field transmission, which breaks the spatial symmetry to lead to optical nonreciprocal transmission. Finally, the effects of the parameters including the coupling strengths and the dissipation rates of cavity fields on the nonreciprocal response of the system are discussed numerically, and with selecting the appropriate parameters, we can achieve tunable optical nonreciprocity. The theoretical scheme to realize optical nonreciprocal transmission in double-cavity optomechanical systems provides a theoretical basis for applications of optical circulators, isolators and directional amplifiers and so on.

The paper is organized as follows. In Sec. 2, we describe the model of the double-cavity optomechanical system and present the corresponding Hamiltonian. Based on the Hamiltonian, we introduce the linearization of the optomechanical interaction, and give the steady state of optomechanical dynamics. In Sec. 3, we show the transmission spectrum of the probe field with different characteristics, and discuss the optical nonreciprocity adjusting by both the coupling strengths and the dissipation rates of cavity fields. In Sec. 4, a conclusion of the results is summarized.

Theoretical model and equations

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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Schematic of a double-cavity optomechanical system that consists of two cavities, in which one of cavities combines with a mechanical oscillator to form an optomechanical cavity. The coupling strengths between the two cavities represent \(\lambda _1\) and \(\lambda _2\). The system is driven by a strong control field with frequency \(\omega _c\) input from cavity \(a_1\), and two weak probe fields with frequency \(\omega _p\) input from both cavity \(a_1\) and cavity \(a_2\).

Here, we study the optical nonreciprocal transmission based on the optomechanically induced transparency in a double-cavity optomechanical system. As shown in Fig. 1, we consider the two cavities consisting of an optomechanical cavity and an empty cavity. It is worth noting that the coupling strength between two cavities in our proposal is nonreciprocal (\(\lambda _1\ne \lambda _2\)), which can be achieved by making use of an imaginary gauge field53,54, impurity55, or an auxiliary nonreciprocal transition device56,57,58,59. In addition, we provide a possible method for achieving nonreciprocal coupling in the Appendix 5. We demonstrate that coupling two cavities to a common additional mode and adiabatic elimination of the additional mode can result in the unequal coupling (\(\lambda _1 \ne \lambda _2\)) expressed in Eq.(1). The double-cavity optomechanical system is driven by a strong control field with frequency \(\omega _c\) input from cavity \(a_1\), and two weak probe fields with frequency \(\omega _p\) input from both cavity \(a_1\) and cavity \(a_2\). In case one (as forward direction), the probe field is only input from cavity \(a_1\), and the output field is obtained from cavity \(a_2\). In case two (as backward direction), the probe field is only input from cavity \(a_2\) and the output field is obtained from cavity \(a_1\). We will next study the anti-Stokes field (probe field) output from the two opposite directions. The total Hamiltonian of the system can be written as follows60:

$$\begin{aligned} \hat{H}=&\hbar \omega _1{\hat{a}_1}^{\dagger }\hat{a}_1+\hbar \omega _2{\hat{a}_2}^{\dagger }\hat{a}_2+\hbar \omega _m{\hat{b}}^{\dagger }\hat{b}+\hbar \lambda _1{\hat{a}_1}^{\dagger }\hat{a}_2 +\hbar \lambda _2{\hat{a}_1\hat{a}_2}^{\dagger }\nonumber \\&-\hbar g\hat{a}_1^{\dagger }\hat{a}_1(\hat{b}^{\dagger }+\hat{b})+\hat{H}_{driven}, \end{aligned}$$
(1)

where \(\hat{a}_{1,2}\) (\(\hat{a}_{1,2}^{\dagger }\)) indicates the annihilation (creation) operator of the cavity mode with resonance frequency \(\omega _{1,2}\), and \(\hat{b}\) (\(\hat{b}^{\dagger }\)) denotes the annihilation (creation) operator of the mechanical oscillator mode with vibration frequency \(\omega _m\). \(\lambda _1\) and \(\lambda _2\) are the coupling strengths between the two optical cavities respectively, and g is the single photon coupling rate. \(\hat{H}_{driven}\) is the driving term, and in case one , \(\hat{H}_{driven}=i\hbar \sqrt{\eta _1\kappa _1}\varepsilon _c(\hat{a}_1^{\dagger }e^{- i\omega _c t}-\hat{a}_1 e^{i\omega _c t})+i\hbar \sqrt{\eta _1\kappa _1}\varepsilon _p(\hat{a}_1^{\dagger }e^{- i\omega _p t}-\hat{a}_1e^{i\omega _p t})\), and in case two, \(\hat{H}_{driven}=i\hbar \sqrt{\eta _1\kappa _1}\varepsilon _c(\hat{a}_1^{\dagger }e^{- i\omega _c t}-\hat{a}_1 e^{i\omega _c t})+i\hbar \sqrt{\eta _2\kappa _2}\varepsilon _p(\hat{a}_2^{\dagger }e^{- i\omega _p t}-\hat{a}_2e^{i\omega _p t})\). The amplitudes of the control field and the probe field are \(\varepsilon _{c}=\sqrt{\mathrm {P_c}/{\hbar \omega _c}}\) and \(\varepsilon _{p}=\sqrt{\mathrm {P_p}/{\hbar \omega _p}}\), respectively, where \(\mathrm {P_c}\) is the power of the control field, and \(\mathrm {P_p}\) is the power of the probe field. \(\kappa _1\) and \(\kappa _2\) are the total dissipation rates of cavity \(a_1\) and cavity \(a_2\). We have furthermore used the coupling parameter \(\eta _{1,2}\equiv \kappa _{ex}/(\kappa _o+\kappa _{ex})\) where \(\kappa _o\) denotes the intrinsic loss rate, which depends on the Q factor of the cavity and \(\kappa _{ex}\) is the external loss rate (i.e. wave guide coupling). Experimentally, the parameter \(\eta _{1,2}\) can be continuously adjusted by tuning the taper-resonator gap61. Here the coupling parameter is chosen to be the critical coupling 1/221.

In the rotating frame of the frequency \(\omega _c\), by introducing the dissipations, the dynamic behavior of the system can be described by the following Heisenberg-Langevin equations:

$$\begin{aligned} \dot{\hat{a}}_1=&[-i\Delta _{1}+i g(\hat{b}^{\dagger }+\hat{b})-\kappa _{1}/2]\hat{a}_{1}-i\lambda _{1}\hat{a}_{2}\nonumber \\&+\sqrt{\kappa _1/2}(\varepsilon _{c}+\varepsilon _p e^{- i\Omega t}) +\sqrt{\kappa _{1}/2}\hat{a}_{\mathrm {1.in}},\end{aligned}$$
(2)
$$\begin{aligned} \dot{\hat{a}}_2=&(-i\Delta _{2}-\kappa _{2}/2)\hat{a}_{2}-i\lambda _{2}\hat{a}_{1}+\sqrt{\kappa _2/2}\varepsilon _p e^{- i\Omega t}\nonumber \\&+\sqrt{\kappa _{2}/2}\hat{a}_{\mathrm {2.in}},\end{aligned}$$
(3)
$$\begin{aligned} \dot{\hat{b}}=&(-i\omega _{m}-\gamma _{m}/2)\hat{b}+i g\hat{a}_1^{\dagger }\hat{a}_1+\sqrt{\gamma _m/2}\hat{b}_{\textrm{in}}, \end{aligned}$$
(4)

where \(\Delta _{i}=\omega _{i}-\omega _{c}\) (\(i=1,2\)) is the frequency detuning of the control field from the cavity field, and \(\Omega =\omega _{p}-\omega _{c}\) is the frequency detuning of the control field from the probe field. The dissipation rate of the mechanical oscillator is \(\gamma _m\). In addition, \(\hat{a}_{\mathrm {1.in}}\), \(\hat{a}_{\mathrm {2.in}}\), and \(\hat{b}_{\textrm{in}}\) represent the quantum noise of the two cavity fields and the thermal noise of the mechanical oscillator, respectively, and they are characterized by the following temperature-dependent correlation functions: \(\langle \hat{a}_{\mathrm {1.in}}(t)\hat{a}_\mathrm {1.in}^{\dagger }(t^{'})\rangle =[n_{th}(\omega _1)+1]\delta (t-t^{'})\), \(\langle \hat{a}_{\mathrm {1.in}}^{\dagger }(t)\hat{a}_{\mathrm {1.in}}(t^{'})\rangle =[n_{th}(\omega _1)]\delta (t-t^{'})\), \(\langle \hat{a}_{\mathrm {2.in}}(t)\hat{a}_{\mathrm {2.in}}^{\dagger }(t^{'})\rangle =[n_{th}(\omega _2)+1]\delta (t-t^{'})\), \(\langle \hat{a}_{\mathrm {2.in}}^{\dagger }(t)\hat{a}_{\mathrm {2.in}}(t^{'})\rangle =[n_{th}(\omega _2)]\delta (t-t^{'})\), \(\langle \hat{b}_{\textrm{in}}(t)\hat{b}_{\textrm{in}}^{\dagger }(t^{'})\rangle =[b_{th}(\omega _m)+1]\delta (t-t^{'})\), \(\langle \hat{b}_{\textrm{in}}^{\dagger }(t)\hat{b}_{\textrm{in}}(t^{'})\rangle =[b_{th}(\omega _m)]\delta (t-t^{'})\). Here, \(n_{th}(\omega _{1,2})=[\exp (\frac{\hbar \omega _{1,2}}{K_{B}T})-1]^{-1}\) and \(b_{th}(\omega _{m})=[\exp (\frac{\hbar \omega _{m}}{K_{B}T})-1]^{-1}\) are the equilibrium average thermal photon and phonon number respectively, where \(K_{B}\) is the Boltzmann constant and T is the ambient temperature.

In this work, we focus on the mean response of the system, and thus the evolution of the system operators can be reduced to their expectation values, viz. \(\langle \hat{a}_{1}\rangle =a_1\), \(\langle \hat{a}_{2}\rangle =a_2\), \(\langle \hat{b}\rangle =b\). Moreover, the quantum and thermal noise terms (\(\langle \hat{a}_{\mathrm {1.in}}(t)\rangle\), \(\langle \hat{a}_{\mathrm {2.in}}(t)\rangle\), \(\langle \hat{b}_{\textrm{in}}(t)\rangle\)) can be dropped safely because their expectation values are zero in the semiclassical approximation. Therefore, the mean value equations can be further written as follows:

$$\begin{aligned} \dot{a}_1=&[-i\Delta _{1}+i g(b^{*}+b)-\kappa _{1}/2]a_{1}-i\lambda _{1}a_{2}\nonumber \\&+\sqrt{\kappa _1/2}(\varepsilon _{c}+\varepsilon _p e^{- i\Omega t}) ,\end{aligned}$$
(5)
$$\begin{aligned} \dot{a}_2=&(-i\Delta _{2}-\kappa _{2}/2)a_{2}-i\lambda _{2}a_{1}+\sqrt{\kappa _2/2}\varepsilon _p e^{- i\Omega t},\end{aligned}$$
(6)
$$\begin{aligned} \dot{b}=&(-i\omega _{m}-\gamma _{m}/2)b+i g a_{1}^{*}a_{1}. \end{aligned}$$
(7)

Here, we consider the control field is significantly stronger than the probe field, and thus the perturbation approach can be used to deal with the above equations of motion. The influence of the control field on the system can be regarded as a steady state, and the probe field is treated as a perturbation of the steady state due to the relatively weak probe field. Based on this research method, we can reformulate the operators as: \(a_{1,2}=\bar{a}_{1,2}+\delta a_{1,2}\), \(a_{1,2}^*=\bar{a}_{1,2}^*+\delta a_{1,2}^*\), \(b=\bar{b}+\delta b\), \(b^*=\bar{b}^*+\delta b^*\). By substituting the redefined operators into Eqs. (5)-(7), the steady-state solutions of the system can be obtained as:

$$\begin{aligned} \bar{a}_1=&\frac{i\lambda _{1}\bar{a}_{2}-\sqrt{\kappa _{1}/2}\varepsilon _{c}}{-i\Delta _{1}+i g(\bar{b}^{*}+\bar{b})-\kappa _{1}/2},\end{aligned}$$
(8)
$$\begin{aligned} \bar{a}_2=&\frac{i\lambda _{2}\bar{a}_{1}}{-i\Delta _{2}-\kappa _{2}/2},\end{aligned}$$
(9)
$$\begin{aligned} \bar{b}=&\frac{i g\bar{a}_{1}^{*}\bar{a}_{1}}{i\omega _{m}+\gamma _{m}/2}. \end{aligned}$$
(10)

According to Eqs.(8-10), we have a third-order nonlinear equation for the photon number \(|\bar{a}_1|^2\),

$$\begin{aligned} t_3|\bar{a}_1|^6+t_2|\bar{a}_1|^4+t_1|\bar{a}_1|^2-d_1=0, \end{aligned}$$
(11)

with

$$\begin{aligned} t_3=&g^4\omega _m^2(\Delta _2^2+\kappa _2^2),\end{aligned}$$
(12)
$$\begin{aligned} t_2=&2g^2\omega _m(\omega _m^2+\gamma _m^2/4)[2(\varsigma +\lambda _1\lambda _2)\Delta _2-\beta \kappa _2],\end{aligned}$$
(13)
$$\begin{aligned} t_1=&[(\varsigma +\lambda _1\lambda _2)^2+\beta ^2](\omega _m^2+\gamma _m^2/4)^2,\end{aligned}$$
(14)
$$\begin{aligned} d_1=&(\omega _m^2+\gamma _m^2/4)^2(\Delta _2^2+\kappa _2^2/4)\kappa _{1}\varepsilon _{c}^2/2, \end{aligned}$$
(15)

in which \(\varsigma =\kappa _1\kappa _2/4+J^2-\Delta _1\Delta _2\), \(\beta =\Delta _1\kappa _2/2+\Delta _2\kappa _1/2\). In this way, we can easily obtain the photon number \(|\bar{a}_1|^2\), and further obtain the photon number \(|\bar{a}_2|^2\) and the phonon number \(|\bar{b}|^2\).

Through the above Eqs.(8-10), we find that the coupling strengths \(\lambda _1\) and \(\lambda _2\) between the two cavities are closely related to the photon number of the two cavities. Then, we plot Fig. 2 to show the steady-state average photon number \(|\bar{a}_{1}|^2\) varies with the power \(\mathrm {P_c}\) of the control field and the coupling strength \(\lambda _1\). It is obvious that the steady-state photon number \(|\bar{a}_{1}|^2\) increases with the increase of the power of the control field, and when the power reaches the range of around 1.95 mW to 5.08 mW, three solutions exist and the middle part is an unstable solution, where the system gives rise to bistability in this case. The result is consistent with the bistable properties22 caused by the optomechanical interation in a general optomechanical system. In addition, it is found that when the power is small, the coupling strength \(\lambda _1\) has almost no modulation effect on the photon number, but with the increase of the power, the coupling strength begins to modulate the number of photons. The coupling strength \(\lambda _2\) has a similar regulatory function. Here, we focus on the low-excitation regime where the control field is too weak to induce bistability (the power of the control field is kept below 1.95 mW throughout the analysis to avoid any bistability in the system).

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Three dimensional functional diagram of coupling strength \(\lambda _{1}\), pump power \(\mathrm {P_c}\) and average photon number \(|\bar{a}_{1}|^2\) of cavity \(a_1\). The parameters are \(\lambda _{2}=\lambda _{1}\), \(\gamma _{m}/2\pi =410\) KHz, \(\kappa _{1}/2\pi =50\) MHz, \(\omega _{m}/2\pi =100\) MHz, \(g=1.2\times 10^{-4}\omega _m\), \(\Delta _{1}=\Delta _{2}=\Omega =\omega _m\), \(\kappa _{2}=\kappa _{1}\) and \(\varepsilon _{p}=0.05\varepsilon _c\), which are adopted from recent relevant studies62 and are available in the experiments.

Next, we further consider the perturbations generated by the probe field on the system. Submitting the expressions \(a_{1,2}=\bar{a}_{1,2}+\delta a_{1,2}\), \(a_{1,2}^*=\bar{a}_{1,2}^*+\delta a_{1,2}^*\), \(b=\bar{b}+\delta b\), \(b^*=\bar{b}^*+\delta b^*\) into Eqs. (5)-(7), the equations of fluctuation terms can be written as:

$$\begin{aligned} \frac{d}{dt}\delta a_{1}=&[-i\Delta _{1}+i g(\bar{b}^{*}+\bar{b})-\kappa _{1}/2]\delta a_{1}-i\lambda _1\delta a_2\nonumber \\&+i g(\delta b^{*}+\delta b)(\bar{a}_{1}+\delta a_1)+\sqrt{\kappa _{1}/2}\varepsilon _{p}e^{-i\Omega t},\end{aligned}$$
(16)
$$\begin{aligned} \frac{d}{dt}\delta a_{2}=&(-i\Delta _2-\kappa _{2}/2)\delta a_{2}-i\lambda _{2}\delta a_{1}+\sqrt{\kappa _{2}/2}\varepsilon _{p}e^{-i\Omega t},\end{aligned}$$
(17)
$$\begin{aligned} \frac{d}{dt}\delta b=&(-i\omega _{m}-\gamma _{m}/2)\delta b+i g(\bar{a}_{1}^{*}\delta a_{1}+\bar{a}_{1}\delta a_{1}^{*}+\delta a_{1}\delta a_{1}^{*}). \end{aligned}$$
(18)

When we consider the nonlinear terms \(ig(\delta b^{*}+\delta b)\delta a_1\) and \(ig\delta a_{1}\delta a_{1}^{*}\), some interesting effects appear, one of which is a series of spectroscopic sideband with frequency \(\omega _c\pm n\Omega\). In this work, we mainly focus on the first-order sideband generation, and the spectral components of second-order and high-order sidebands are ignored, so the perturbation solutions can be further ansatz as: \(\delta a_{1}=A^{-}e^{-i\Omega t}+A^{+}e^{i\Omega t}\), \(\delta a_{2}=B^{-}e^{-i\Omega t}+B^{+}e^{i\Omega t}\), \(\delta b=C^{-}e^{-i\Omega t}+C^{+}e^{i\Omega t}\), where \(\Omega =\omega _{p}-\omega _{c}\). The frequency components with \(-\Omega\) and \(+\Omega\) represent the upper and lower first-order sideband generation, respectively, corresponding to the anti-Stokes field and Stokes field. In what follows, we solve Eqs. (16)-(18) by using above ansatz and give the amplitudes of the first-order sideband, leading to six algebraic equations:

$$\begin{aligned} (-i\Omega -\Theta )A^{-}=&i g[C^{-}+(C^{+})^{*}]\bar{a}_{1}-i\lambda _{1}B^{-}+\sqrt{\kappa _{1}/2}\varepsilon _{p},\end{aligned}$$
(19)
$$\begin{aligned} (i\Omega -\Theta )A^{+}=&i g[C^{+}+(C^{-})^{*}]\bar{a}_{1}-i\lambda _{1}B^{+},\end{aligned}$$
(20)
$$\begin{aligned} (-i\Omega -D)B^{-}=&-i\lambda _{2}A^{-}+\sqrt{\kappa _{2}/2}\varepsilon _{p},\end{aligned}$$
(21)
$$\begin{aligned} (i\Omega -D)B^{+}=&-i\lambda _{2}A^{+},\end{aligned}$$
(22)
$$\begin{aligned} (-i\Omega -E)C^{-}=&i g[A^{-}\bar{a}_{1}^{*}+(A^{+})^{*}\bar{a}_{1}],\end{aligned}$$
(23)
$$\begin{aligned} (i\Omega -E)C^{+}=&i g[A^{+}\bar{a}_{1}^{*}+(A^{-})^{*}\bar{a}_{1}], \end{aligned}$$
(24)

where \(\Theta =-i\Delta _{1}+i g(\bar{b}^{*}+\bar{b})-\kappa _{1}/2\), \(D=-i\Delta _{2}-\kappa _{2}/2\), \(E=-i\omega _{m}-\gamma _{m}/2\). \(A^\pm\) represents the amplitude of the lower (upper) first-order sideband of cavity \(a_1\); \(B^\pm\) represents the amplitude of the lower (upper) first-order sideband of cavity \(a_2\) and \(C^\pm\) represents the amplitude of the lower (upper) first-order sideband of the mechanical oscillator. In order to study the optical response of light transmission when the probe field enters the cavity field from two opposite directions (forward and backward directions), we divide them into two cases in the following discussion:

In case one (as forward direction), the probe field is input from the left cavity \(a_1\) and we study the optical response output from right cavity \(a_2\). Here \(A_{1}^-\) represents the amplitude of the upper first-order sideband of cavity \(a_1\), and \(B_{1}^-\) represents the amplitude of the upper first-order sideband of cavity \(a_2\). The sideband amplitudes in this case can be obtained by solving Eqs.(19-24) where \(\sqrt{\kappa _{2}/2}\varepsilon _{p}\) is dropped:

$$\begin{aligned} A_{1}^{-}=&\frac{\Lambda Q(i\Omega +D)\sqrt{\kappa _{1}/2}\varepsilon _{p}}{\Lambda \Xi -4\omega _{m}^{2}g^{4}|\bar{a}_{1}|^{4}(i\Omega +D)(-i\Omega -D^{*})},\end{aligned}$$
(25)
$$\begin{aligned} B_{1}^{-}=&\frac{i \Lambda Q\lambda _{2}\sqrt{\kappa _{1}/2}\varepsilon _{p}}{\Lambda \Xi -4\omega _{m}^{2}g^{4}|\bar{a}_{1}|^{4}(i\Omega +D)(-i\Omega -D^{*})}, \end{aligned}$$
(26)

where \(Q=(\gamma _{m}/2-i\Omega )^{2}+\omega _{m}^{2}\), \(\Xi =Q(-i\Omega -\Theta )(i\Omega +D)-Q\lambda _{1}\lambda _{2}-2i\omega _{m}g^{2}|\bar{a}_1|^{2}(i\Omega +D)\), \(\Lambda =Q(-i\Omega -\Theta ^{*})(-i\Omega -D^{*})+Q\lambda _{1}\lambda _{2}+2i\omega _{m}g^{2}|\bar{a}_1|^{2}(-i\Omega -D^{*})\).

In case two (as backward direction), the probe field is input from the right cavity \(a_2\) and we study the optical response output from left cavity \(a_1\). Here \(A_{2}^-\) represents the amplitude of the upper first-order sideband of cavity \(a_1\), and \(B_{2}^-\) represents the amplitude of the upper first-order sideband of cavity \(a_2\). The sideband amplitudes in this case can also be obtained by solving Eqs.(19-24) where \(\sqrt{\kappa _{1}/2}\varepsilon _{p}\) is dropped:

$$\begin{aligned} A_{2}^{-}=&\frac{i \Lambda Q\lambda _{1}\sqrt{\kappa _{2}/2}\varepsilon _{p}}{\Lambda \Xi -4\omega _{m}^{2}g^{4}|\bar{a}_{1}|^{4}(i\Omega +D)(-i\Omega -D^{*})},\end{aligned}$$
(27)
$$\begin{aligned} B_{2}^{-}=&-\sqrt{\kappa _{2}/2}\varepsilon _{p}(i\Omega +D)\nonumber \\&-\frac{\Lambda Q\lambda _{1}\lambda _{2}(i\Omega +D)\sqrt{\kappa _{2}/2}\varepsilon _{p}}{\Lambda \Xi -4\omega _{m}^{2}g^{4}|\bar{a}_{1}|^{4}(i\Omega +D)(-i\Omega -D^{*})}. \end{aligned}$$
(28)

According to the standard input-output notation63,64, the total output in the forward and backward directions (in case one and in case two) at the probe field frequency \(\omega _p\) can be obtained as:

$$\begin{aligned} c_{p_1}=&-\sqrt{\kappa _{1}/2}B_{1}^{-},\end{aligned}$$
(29)
$$\begin{aligned} c_{p_2}=&-\sqrt{\kappa _{2}/2}A_{2}^{-}. \end{aligned}$$
(30)

In addition, we define \(t_{p_1}\) and \(t_{p_2}\) as the corresponding transmission amplitudes:

$$\begin{aligned} t_{p_{1}}=\frac{c_{p_{1}}}{\varepsilon _{p}}, \; t_{p_2}=\frac{c_{p_2}}{\varepsilon _{p}}. \end{aligned}$$
(31)

Here \(|t_{p_{1}}|^2\) and \(|t_{p_{2}}|^2\) are dimensionless, as the efficiency of the upper first-order sideband (the probe field) process. In what follows, we will present a detailed discussion of the transmission characteristics of the probe field.

The features of optical nonreciprocal transmission

In this section, we discuss in detail the transmission characteristics of the probe field output from cavity \(a_2\) (in case one) and cavity \(a_1\) (in case two). The transmission efficiency \(|t_{p_{1}}|^2\) and \(|t_{p_{2}}|^2\) of the probe field as a function of the driven frequency \(\Omega /\omega _m\) are shown in Fig. 3 under different coupling strengths (\(\lambda _1\) and \(\lambda _2\)) and powers \(\mathrm {P_c}\) of the control field. When the coupling strength is weak, seeing in Fig. 3(a) (\(\lambda _1=\lambda _2=0.4\kappa _1\)), a Lorentzian curve appears in the transmission spectrum of the probe field. The Lorentzian peak in the transmission spectrum represents an absorption of the probe field, because the transmission amplitude is actually the case of the probe field in the cavity. However, by increasing the coupling strength to \(1.2\kappa _1\), a transparency window with a certain width (a dip corresponds to a transparent window) starts appearing, as shown in Fig. 3(c). The reason for the result is that the strong coupling strength of two cavities leads to the splitting of the two cavity modes, thus forming two absorption points (two peaks presented in this work). Moreover, the transparency dip of the transmission spectrum of the probe field will increase with increasing the coupling strength.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Calculation results of \(|t_p|^2\) (including \(|t_{p_{1}}|^2\) and \(|t_{p_{2}}|^2\)) are plotted as functions of \(\Omega /\omega _m\) under different powers \(\mathrm {P_c}\) and coupling strengths \(\lambda\) (including \(\lambda _{1}\) and \(\lambda _{2}\)). We use \(\Delta _{1}=\Delta _{2}=1.5\omega _m\) in panels (c) and (d), and the other parameters are the same as in Fig. 2.

When we consider the driving field (control field) applied to the optomechanical cavity, the effective optomechanical coupling constant g begins to play a major role in the probe field transmission process, as shown in Fig. 3b, where the power of the control field is 100 \(\mu\)W. It is found that there is a transparent window near the resonance condition \(\Omega =\omega _m\), and as the power increases, the transparent window will be more obvious and much deeper, which is similar to the contents of previous studies22. Its physical essence is that when the resonant condition of the mechanical oscillator is the same as that of the cavity, the destructive interference between the input probe field and the sideband excitations of the control field occurs, leading to a tunable transparent window. Therefore, the transparent windows in Figs. 3(b) and 3(c) are caused by different physical mechanisms. Further, when we use \(\lambda _1=\lambda _2=1.2\kappa _1\) and \(\mathrm {P_c}=100\) \(\mu\)W in Fig. 3d, the result was predictable that there are double transparent windows, one of which is induced due to the strong coupling strength of two cavities, and the other of which is caused by the effective optomechanical interaction.

In order to investigate the nonreciprocal transmission of the probe field along two opposite directions, we plot Fig. 4. Based on Figs. 3(b), 3(c), and 3(d), we have only changed \(\kappa _2\) to \(0.5\kappa _1\) in Figs. 4(a), 4(b), and 4(c). In contrast to Fig. 3, Fig. 4 shows obvious nonreciprocal transmissions of the probe field, but upon closer inspection, there are no significant change in the overall graphical characteristics. From the point of view of the transmission amplitudes, the value of \(|t_{p_{1}}|^2\) is significantly increased, while the value of \(|t_{p_{2}}|^2\) is relatively reduced. For example, when \(\Omega =\omega _m\), the amplitudes of both \(|t_{p_{1}}|^2\) and \(|t_{p_{2}}|^2\) are about 0.03 in Fig. 3(b), but the amplitude of \(|t_{p_{1}}|^2\) is 0.09 and the amplitude of \(|t_{p_{2}}|^2\) is about 0.02 in Fig. 4(a), which shows a clearly nonreciprocal response of the probe field transmission. Especially for the comparison of Fig. 3(c) and Fig. 4(b), in the absence of the optomechanical interaction (\(\mathrm {P_c}=0\) \(\mu\)W), the optical nonreciprocal transmission is also obtained by changing the dissipation rate of cavity \(a_2\). This indicates the algebraic duality symmetry does not rely on the specific optomechanical interaction. However, the drawback is that in Figs. 4(a), 4(b) and 4(c), although we can intuitively see the difference between the amplitudes of \(|t_{p_{1}}|^2\) and \(|t_{p_{2}}|^2\) , the ratio of the amplitude difference does not change with the change of the driven frequency \(\Omega /\omega _m\), that is to say, \(\Omega /\omega _m\) can not modulate the optical nonreciprocal transmission.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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The transmission efficiency \(|t_{p_{1}}|^2\) (blue solid curve) and \(|t_{p_{2}}|^2\) (red dashed curve) are plotted as functions of the \(\Omega /\omega _m\) under different powers \(\mathrm {P_c}\) and coupling strengths \(\lambda\) (including \(\lambda _{1}\) and \(\lambda _{2}\)) in panels (a), (b) and (c) with \(\kappa _2=0.5\kappa _1\). In panel (d), the isolation ratio \(\textrm{I}\) of the probe field nonreciprocal transmission is plotted as a function of the \(\kappa _1/\kappa _2\). The other parameters are the same as in Fig. 3.

We define a dimensionless quantity \(\textrm{I}\) to describe the optical nonreciprocal transmission in the optomechanical system:

$$\begin{aligned} \textrm{I}=&10\bigg |\log _{10}\frac{|t_{p_1}|^2}{|t_{p_2}|^2}\bigg |=10\bigg |\log _{10}\frac{|c_{p_1}|^2}{|c_{p_2}|^2}\bigg |, \end{aligned}$$
(32)

where the unit of \(\textrm{I}\) is dB. We label the dimensionless quantity \(\textrm{I}\) as the isolation rate to describe the degree of nonreciprocal transmission of the forward and backward input of the probe field. By judging the value of \(\textrm{I}\), we can determine the degree of the optical nonreciprocal transmission. When \(|t_{p_1}|^2/|t_{p_2}|^2=1\), that is \(\textrm{I}=0\), indicating that the transmission of first-order sidebands (the probe field output) in two opposite directions is reciprocal. Conversely, a nonzero \(\textrm{I}\) presents the optical nonreciprocal transmission, and the greater the value of \(\textrm{I}\) the higher the degree of nonreciprocity of the transmission. For example, \(\textrm{I} \ge 1\), the transmission coefficient in one direction is at least 10 times that of the other direction, which indicates a strong nonreciprocity in the optical transmission along two opposite directions. By substituting Eq. (31) into Eq. (32), then the isolation rate can be further simplified as:

$$\begin{aligned} \textrm{I}=10\bigg |2\log _{10}\frac{\kappa _{1}\lambda _{2}}{\kappa _{2}\lambda _{1}}\bigg |. \end{aligned}$$
(33)

According to Eq. (33), we find that the isolation rate is independent of the driven frequency \(\Omega /\omega _m\), but related to the coupling strengths and the dissipation rates of cavity fields. Therefore, in Figs. 4(a), 4(b), and 4(c) it can be found that the nonreciprocal transmission of the probe field occurs, but the corresponding isolation rate remains constant with the change in frequency, because the dissipation rate changes only once. For the sake of further observing the influence of the dissipation rates on the isolation rate, we draw Fig. 4(d) to show the value of \(\textrm{I}\) as functions of the \(\kappa _1/\kappa _2\). It can be observed that the isolation rate \(\textrm{I}\) varies linearly with the dissipation rate \(\kappa _1/\kappa _2\). When \(\lambda _1=\lambda _2=0.4\kappa _1\) in Fig. 4(d), the value of \(\textrm{I}\) becomes \(10\bigg |2\log _{10}\frac{\kappa _{1}}{\kappa _{2}}\bigg |\), which is perfectly consistent with the evolution of image features. Especially when the value \(\kappa _1/\kappa _2\) is smaller, not only the value of \(\textrm{I}\) is significantly improved, but also it enables us to obtain a high isolation rate (\(>20\) dB)65.

Fig.5
Fig.5The alternative text for this image may have been generated using AI.
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The transmission efficiency \(|t_{p_{1}}|^2\) (blue solid curve) and \(|t_{p_{2}}|^2\) (red dashed curve) are plotted as functions of the \(\Omega /\omega _m\) under different coupling strengths \(\lambda _{2}\). We use \(\mathrm {P_c}=0\) \(\mu\)W, \(\lambda _{1}=0.4\kappa _1\), and the other parameters are the same as in Fig. 2.

On the basis of Eq. (33), we can know that in addition to the dissipation rates of cavity fields, the coupling strengths \(\lambda _{1}\) and \(\lambda _{2}\) between two cavities also affect the nonreciprocal transmission of the probe field. We give the numerical simulation of the transmission efficiency in Fig. 5, with different \(\lambda _{2}\). We make the power value 0 \(\mu\)W in Fig. 5, and therefore, the curve formed in the figure has nothing to do with the optomechanical interaction. When \(\lambda _{1}=0.4\kappa _1\) and \(\lambda _{2}=0.5\kappa _1\) in Fig. 5(a), there are two Lorentz curve lines. Using Eq. (33), the value of \(\textrm{I}\) becomes \(10\bigg |2\log _{10}\frac{\lambda _{2}}{\lambda _{1}}\bigg |\), and once \(\lambda _{2}\) is unequal to \(\lambda _{1}\), \(\textrm{I}\ne 0\). These two transmission curves are inevitably unequal, as shown in Fig. 5(a). With the increase of \(\lambda _{2}\), the amplitude value of the transmission efficiency \(|t_{p_{1}}|^2\) remains increasing, as shown in Figs. 5(b), 5(c), 5(d), and when the value of \(\lambda _{2}\) reaches \(1.2\kappa _1\) in Fig. 5(c), the blue solid curve due to the strong coupling strength begins to split sharply and a transparent window is formed. However, the amplitude value of the transmission efficiency \(|t_{p_{2}}|^2\) does not change much because the coupling strength \(\lambda _{1}\) keeps constant. The variation of these two curves with respect to the coupling strengths can be concluded from Eqs. (26)-(27). Such a result can also be seen in Fig. 5(d). According to the above exhibition of the nonreciprocal transmission of the probe field regulated by the frequency detuning \(\Omega /\omega _m\) shown in Figs. 4 and 5, it is found that the essentially achieving non-reciprocity originates from the the dissipation rates of cavity fields and the coupling strengths between two cavities. If you want to realize the optical nonreciprocal transmission modulated by the frequency detuning \(\Delta _1-\Delta _2\), it is needed to study the transmission of steady-state photon number52. In this work, we consider the transmission of the probe field under the linear effect of the optomechanical interaction, so the frequency detunings \(\Omega /\omega _m\) and \(\Delta _1-\Delta _2\) have no regulating effect on the nonreciprocal transmission.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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The isolation ratio \(\textrm{I}\) of the probe field nonreciprocal transmission is plotted as a function of the coupling strengths \(\lambda _{1}\) and \(\lambda _{2}\). We use \(\mathrm {P_c}=0\) \(\mu\)W, \(\kappa _2=0.5\kappa _1\), and the other parameters are the same as in Fig. 2.

Therefore, for the sake of achieving the regulation of nonreciprocal transmission, we try to change other variables, i.e., the coupling strengths (\(\lambda _{1}\) and \(\lambda _{2}\)). Figure 6 shows the isolation ratio \(\textrm{I}\) of the probe field nonreciprocal transmission as a function of the coupling strengths \(\lambda _{1}\) and \(\lambda _{2}\). As shown the yellow color parts in Fig. 6, the smaller the value of \(\lambda _{1}\) or \(\lambda _{2}\), the greater the value of the isolation rate \(\textrm{I}\). In order to effectively raise the isolation ratio \(\textrm{I}\) of the optical nonreciprocal transmission, we make \(\kappa _2\) equal to 0.5 times \(\kappa _1\), or we can increase the ratio further. According to Eq. (33), the value of \(\textrm{I}\) here becomes \(10\bigg |2\log _{10}\frac{2\lambda _{2}}{\lambda _{1}}\bigg |\), so the value of \(\textrm{I}\) is still a linear relationship to the ratio between \(\lambda _{2}\) and \(\lambda _{1}\). When \(\lambda _{2}\) is equal to 0.5 times \(\lambda _{1}\), \(\textrm{I}=0\), as shown by the black line in the figure, indicating reciprocal transmission. However, as the ratio of \(\lambda _{1}\) to \(\lambda _{2}\) increases, we can obtain a high isolation rate \(\textrm{I}\), which is clearly shown in the small plot in Fig. 6. This is the same as the result in Fig. 4 (d), since the isolation rate is linear with the coupling strengths and the dissipation rates, and with appropriate parameter selection, the isolation ratio \(\textrm{I}\) of optical nonreciprocal transmission can be significantly improved.

Conclusions

In summary, we study the optical nonreciprocal transmission of the probe field in a double-cavity optomechanical system with nonreciprocal coupling, and find that the tunable optical nonreciprocity can be obtained by adjusting the dissipation rates of cavity fields and the coupling strengths between two cavities. Using the linearization of the Heisenberg-Langevin equations, we have discussed the steady-state dynamic behavior of the system, and shown the transmission spectrum of the probe field with different characteristics, and their corresponding physical interpretation is also given. There are two different types of transparent windows, one of which is due to strong coupling strengths and the other is due to the optomechanical interaction. It is found that the dissipation rates and the coupling strengths play a decisive role in the optical nonreciprocal transmission, which is exactly consistent with the results of theoretical calculations. The greater the ratio of \(\kappa _1\lambda _2\) to \(\kappa _2\lambda _1\), the higher the isolation rate \(\textrm{I}\) obtained. The weak tunability of the driven frequency \(\Omega /\omega _m\) is the disadvantage of the present mechanism, which can be transformed by studying the steady-state photon number. The double-cavity system with nonreciprocal coupling has been proposed in many other studies60 and we hope that this theoretical study can provide some guidance for the construction of non-reciprocal devices, such as optical isolators, diodes, and transducers.