1. Introduction

Nonreciprocal transmission, analogy to the diode in classical circuit, allows the one direction transmission and prohibits the opposite direction, which means that nonreciprocal device is a very important control element in quantum information processing. Cavity optomechanical system (COMS) is a promising candidate to generate optical nonreciprocity [1–3], besides its abundant physical phenomena such as ground-state cooling [4–6], phonon lasing [7–9], squeezing [10–12], weak-force sensing [13,14], blockade [15,16] and quantum entanglement [17–20]. Nonreciprocal devices, including isolators [21], circulators [22] and amplifiers [23], serve as fundamental elements in integrated quantum circuits [24]. Generally, nonreciprocity can be achieved to employ quantum interference [22,25–28], Sagnac effect [29,30], and optical nonlinearity [31,32]. Up to now, most of nonreciprocal schemes have been investigated under the Born-Markov approximation for COMS [21–23,25–32].

Every quantum system inevitably interacts with its environment, and the quantum properties could be lost. For protecting the coherence of quantum system and avoiding dissipation, it is believed that the memory effect of non-Markovian environment can be useful when the system strongly coupled to the environment. In order to treat the non-Markovian problems, people have developed several successful methods, such as quantum state diffusion [33–35], path-integral [36], the stochastic Liouville [37–39] and the Heisenberg-Langevin equation [40–44]. All of the methods require the spectral structure of the environment [45–48]. Therefore, spectral density of the structured bath has attracted significant attentions both theoretically [49–52] and experimentally [45]. In the optomechanical system, it has been reported that the non-Markovian environment can enhance the sensitivity of weak-force detection [41], obtain strong squeezing [42], lift the frequency degeneracy of the optomechanically induced transparency (OMIT) [43,53], protect quantum entanglement [40,54,55], improve cooling [56,57] and generate nonreciprocal state transfer [58]. In [59], single-photon blockade has been carefully studied by comparing the results under the Markovian approximation with that in the non-Markovian regime, and the transition from the non-Markovian regime to the Markovian regime also has been shown. However, optical nonreciprocity under the structured environment has not been investigated. The information back-flow process from the structured bath of the mechanical oscillator may have a significant impact on optical transmission. It is necessary to study the effect of the structured environment on quantum information unidirectional transmission.

In this paper, we aim to study optical-unidirectional transmission in the non-Markovian regime in COMS. We consider a nanosphere trapped in a single (dual)-mode cavity [60–62] couples to a non-Markovian structured bath with XX-type interaction. In order to make clear the effect of the non-Markovian environment on the optical nonreciprocity transmission, we first investigate the mechanical response function and the optical transmission in the case of the single optomechanical coupling. Our results show that the non-Markovian environment can shift the effective frequency and enhance the mechanical response function. In this way, there is a small difference in the transmission rate in the Markovian regime, but unidirectional amplification can be obtained in the non-Markovian regime. At some degree, the mechanical mode with its bath is regarded as a controller for unidirectional amplification. In the double-optomechanical coupling system, the phase difference can determine the direction of signal transmission as in the general COMS [22] and contrast to the case of Markovian regime, the non-Markovian environment offers us a noticeable unidirectional amplification.

The paper is organized as follows. In Sec. 2, under strong classical driving condition, we solve the linearized Heisenberg-Langevin equations under the non-Markovian environment. In Sec. 3, we discuss detailed the response function in frequency domain for several structured baths, then the unidirectional transmission is discussed for single and double-optomechanical coupling systems. Finally, the conclusion is given in Sec. 4.

2. Model and its solution

A levitated nanosphere with center-of-mass position $x$ is trapped near the antinode $x=0$ in a dual-mode cavity, shown in Fig. 1(a) and the corresponding schematic diagram is (b). We consider the amplitude of the cavity mode depending on the spatial profile, i.e., $E_{1}\propto \mathtt {cos}k_{1}X$ and $E_{2}\propto \mathtt {cos}(k_{2}X+\theta )$ ($k_{1,2}$ is the wavenumber of the optical field, $\theta$ is the phase difference between optical potentials, $X$ is the coordinate of the nanosphere). The nanosphere interacts with the two cavity modes with the Hamiltonian ($\hbar =1$)

We consider the nanosphere coupling to a structured bath with XX-type interaction. The Hamiltonian can be given by

The first two terms describe the energy of the $l$th environmental oscillator with frequency $\Omega _{l}$ and mass $m_{l}$. The last term represents the interaction between nanosphere and bath, where $\gamma _{l}$ is the coupling strength between nanosphere and the $l$th environmental oscillator. This kind of environment may be realized by the Coulomb force in quantum electromechanical system (QEMS) [66–68] or a high-reflectivity mirror fixed in the center of a doubly clamped beam in the cavity of optomechanical system [45]. The total Hamiltonian can be expressed by $H=H_s+H_d+H_e$. $X, P, X_{l}, P_{l}$ can be expressed as $X=\sqrt {\frac {1}{2m\omega _{m}}}(b+b^{\dagger })$, $P=-i \sqrt {\frac {m\omega _{m}}{2}}(b-b^{\dagger })$, $X_{l}=\sqrt {\frac {1}{2m_{l}\Omega _{l}}}(b_{l}+b_{l}^{\dagger })$ and $P_{l}=-i\sqrt {\frac {m_{l}\Omega _{l}}{2}}(b_{l}-b_{l}^{\dagger })$. Considering the cavity is driven by the strong classical fields, we can linearize quantum Langevin equations (QLEs) by $a_{j}=\delta a_{j}+|\alpha _j|e^{i \varphi _j}$, $b=\delta b+\beta$ [40,41,56,57]. As revealed in [40], the classical part evolves to a limit cycle in the phase space, and the mechanical oscillator can arrive at a new equilibrium position in the long time limit, which mean that the amplitudes of the classical parts achieve steady values. Although the steady value of the classical part with an oscillating phase is not the same as that in Markovian regime, the linearization approximation still can be used. For simplicity, we will ignore $\delta$ and write $\delta a_{j}$ ($\delta b)$ as $a_{j}$ ($b$). In the rotating frame with $H_0=\sum _{j=1,2}\omega _{L}a_j^\dagger a_j$, the linearized QLEs can be written as

 

Fig. 1. Sketch of the optomechanical system. (a) The case of double-optomechanical coupling, where a nanosphere surrounded by a structure bath is trapped in a dual-mode cavity by the optical potentials ($\theta$ is the phase between the optical potentials) and interacts with two cavity modes. The interaction between the nanosphere and its bath is pictured by the green arrows. (b) The simple model diagram of double-optomechanical coupling system, $\varphi$ is the phase difference between two effective optomechanical couplings. (c) The case of single-optomechanical coupling. The system consists of two coupled cavities and a nanosphere trapped in one of cavities. The corresponding simple model diagrams is depicted in (d).

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We are interested in the scattering properties in the optical system. Therefore, the mechanical oscillator can be regarded as a connector between two optical modes. By introducing the Fourier transformation $\tilde {o}(\omega )=\frac {1}{\sqrt {2\pi }}\int o(t)e^{i\omega t}\mathtt {d}t$ (in following, $\tilde {o}(\omega )$ is replaced by $o(\omega )$ for simplicity), from Eqs. (3) we can derive

The impact of the mechanical oscillator on optical modes is almost entirely reflected in the response function. If the mechanical response is very small and can be neglected, the mechanical oscillator has little effect on optical modes, so the system becomes a pure optical system. When the response is strong, the oscillator as well as its environment does strongly affect the transport properties between optical modes.

By substituting Eq. (5) into the optical operators, we can obtain a two-mode optical system modulated by the mechanical response function with a compact form as

Operator vector $O(\omega )\equiv \lbrack a_{1}(\omega ),a_{2}(\omega ),a_{1}^{\dagger }(-\omega ),a_{2}^{\dagger }(-\omega )]^{T}$ and the corresponding input operator vector $O_{in}(\omega )\equiv \lbrack a_{1,in}(\omega ),a_{2,in}(\omega ),a_{1,in}^{\dagger }(-\omega ),a_{2,in}^{\dagger }(-\omega )]^{T}$, dissipative matrix $\sqrt {\varGamma }\equiv \mathtt {diag}[\sqrt {\kappa _{1}},\sqrt {\kappa _{2}},\sqrt {\kappa _{1}},\sqrt {\kappa _{2}}]$. The modified optical susceptibility matrix $\chi \equiv \mathtt {diag}[\chi _{1+}^{\prime },\chi _{2+}^{\prime },\chi _{1-}^{\prime },\chi _{2-}^{^{\prime }}]$ where $\chi _{j\pm }^{\prime }\equiv \frac {1}{\chi _{j+}^{-1}\pm i\tilde {A}_{j}^{2}\Upsilon _{m}}$ with $\chi _{j\pm }^{-1}=-i\omega \pm i\tilde {\Delta }_{j}+\frac {\kappa _{j}}{2}$ ($j=1,2$). Thus the induced self-energy $\pm i\tilde {A}_{k}^{2}\Upsilon _{m}$ emerges in $\chi _{j\pm }^{\prime }$ leading to damping (or anti-damping) as well as a frequency shift. The thermal noise of the mechanical oscillator in Eq. (8) is neglected because we are interested in the transmitted rate between two optical signals. The drift matrix

With the input-output relation $a_{j,out}=\sqrt {\kappa }a_{j}-a_{j,in}$ ($j=1,2$), we define output (input) spectrum by $P_{out(in)}=\int \mathtt {d}\omega ^{\prime }\langle a_{out(in)}^{\dagger }(\omega ^{\prime })a_{out(in)}(\omega )\rangle$. Then, we obtain

3. Unidirectional amplification under non-Markovian environment

We are now in the position to show the unidirectional amplification under the non-Markovian environment. In order to make clear the effect of the non-Markovian environment, we first investigate the mechanical response function modulated by the non-Markovian environment. Then, we will show the unidirectional amplification in our system.

3.1 Mechanical response function

The mechanical oscillator as a connecting medium has an impact on two cavity modes regardless of Markovian or non-Markovian regimes. The non-Markovian environment of the mechanical oscillator mainly contributes to the modulation of the global response function of mechanical oscillator $\Upsilon _{m}$, by which the unidirectional amplification is affected by the non-Markovian environment.

Substituting $\bar {\chi }_{m+}$ , $\bar {\chi }_{m-}$ and $\Sigma _{m}$ into Eqs. (6) and (7), we can respectively rewrite the response functions in non-Markovian and Markovian regimes as

The real and imaginary parts of the response function as a function of $\omega$ shown in Fig. 2 (a) and (b) (only shown the parts $\omega >0$). For non-Markovian bath, the real part of response function exhibits a sharp change around the effective frequency, and the imaginary part of the response function obviously shows the frequency shift far away from $\omega _m$, which equals to the effective frequency in the Markovian regime. The mechanical response functions for structured bath are amplified than that in the Markovian regime. Physically, the amplitude of response function is magnified by virtue of the coupling between the nanosphere and its environment with $XX$-type interaction. Mathematically, we can also understand it by approximately estimating the extrema of Eqs. (11) and (12). In Eq. (12), extrema of $\mathrm {Re}[\Upsilon _{m}^{(M)}]$ are located at $\omega ^{2}=\omega _{eff}^{(M)2}+\frac {\kappa _{m}^{2}}{4} \pm \sqrt {\kappa _{m}^{2}(\frac {\kappa _{m}^{2}}{4}+\omega _{eff}^{(M)2})}$, and then the absolute value (two extrema have the same absolute value and opposite sign) $|\mathrm {Re}[\Upsilon _{m}^{(M)}]|=\omega _{m} /[\frac {\kappa _{m}^{2}}{2}+\sqrt {\kappa _{m}^{2}(\frac {\kappa _{m}^{2}}{4}+\omega _{eff}^{(M)2})}]$. When the decay rate of the mechanical oscillator is small, i.e., $\kappa _{m}\ll \omega _{m}\sim \omega _{eff}^{(M)}$, the denominator of $|\mathrm {Re}[\Upsilon _{m}^{(M)}]|$ is mainly contributed by $\kappa _{m}\omega _{eff}^{(M)}$, then $|\mathrm {Re}[\Upsilon _{m}^{(M)}]| \approx \omega _{m}/\kappa _{m}\omega _{eff}^{(M)}$. Similarly, in the non-Markovian regime, extrema is $|\mathrm {Re}[\Upsilon _{m}^{(N)}]|=\frac {\omega _{m}} {|2\tilde {J}^{2}(\omega _{e})+2\tilde {J}(\omega _{e})\omega _{eff}^{(N)}|}$ located $\omega _{e}^{2}=\omega _{eff}^{(N)2}\pm 2\tilde {J}(\omega _{e}) \omega _{eff}^{(N)}$ (for estimation, $\omega _{e}$ is derived by ignoring $\tilde {J}(\omega )$ depending on $\omega$). And its denominator is mainly affected by $|2\tilde {J}(\omega )\omega _{eff}^{(N)}|$ due to $|\tilde {J}(\omega _{e})| \sim \kappa _m \ll \omega _{eff}^{(M)}\sim \omega _{eff}^{(N)}$. Then $|\mathrm {Re}[\Upsilon _{m}^{(N)}]| \approx \omega _{m}/|2\tilde {J}(\omega _{e})\omega _{eff}^{(N)}|$ can be obtained. Moreover, $\omega _{eff}^{(M)}>\omega _{eff}^{(N)}$ and $\kappa _{m}>|2\tilde {J}(\omega )|$ give rise to $\kappa _{m}\omega _{eff}^{(M)}>|2\tilde {J}(\omega )\omega _{eff}^{(N)}|$ and $|\mathrm {Re}[\Upsilon _{m}^{(N)}]|>|\mathrm {Re}[\Upsilon _{m}^{(M)}]|$. That is to say, the mechanical response can be magnified with the help of structured baths compared with that in the Markovian regime. Since $\tilde {J}(\omega _{e})$ and $\omega _{eff}^{(N)}$ depend on the parameters $s$ and $\eta$, the mechanical response is sensitive to the different form of Ohmic spectral environment. Taking into account the effect of mechanical oscillator on optical modes, the induced coupling $G$ in matrix (9) should be modulated strongly by the mechanical response. Consequently, the optical transmission should be greatly affected by the structure bath.

 

Fig. 2. Real (a) and imaginary (b) parts of mechanical response $\Upsilon _{m}$ as functions of $\omega$ within the Markovian and non-Markovian regimes, where $\Lambda =0, \omega _c/\omega _m=15$, and $\{s=0.5,\eta =5.485\times 10^{-3}\}$, $\{s=1,\eta =2.132\times 10^{-2}\}$, $\{s=1.4,\eta =6.308\times 10^{-2}\}$ for three kinds of Ohmic-type spectra, respectively. In terms of the mechanical damping rate, $\kappa _{m}=\pi \times J(\omega _{m})$ is chosen for comparing non-Markovian with Markovian regimes.

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3.2 Unidirectional amplification in the non-Markovian regime

To see the pure effect of non-Markovian bath on the unidirectional amplification transmission, we ignore the optomechanical coupling for mode 2, the schematic shown in Fig. 1(c) and (d). Then the drift matrix becomes

It indicates that the role of mechanical oscillator in the optomechanical system is to produce the parametric term of cavity mode $a_{1}$. Assuming that $\tilde {\Delta }_{1,2}=\tilde {\Delta }$ and $\kappa _{1,2}=\kappa$, then $\chi _{1,2\pm }=\chi _{\pm }$. The scattering rates between optical modes are given

The scattering rates as a function of $\omega$ within the Markovian regime for $\Lambda =0$ are plotted in Fig. 3 (a). It is obvious that nonreciprocity is invisible because the mechanical response is so small and has little effect, displayed in Fig. 2(b). The two peaks around $-0.5\omega _{m}$ and $1.5\omega _{m}$ originate from a pair of hybrid modes $a_{\pm }=\frac {1}{\sqrt {2}}(a_{1}\pm a_{2})$ with frequencies $\tilde {\Delta }\pm g$ , which can be understood from the effective Hamiltonian Eq. (20). In the super-Ohmic spectral environment, the effective mechanical frequency is shifted to $0.66\omega _{m}$ shown in Fig. 2(b). From Fig. 3(b), the unidirectional amplification can be achieved based on the magnification of the mechanical response around $\omega \approx 0.66\omega _m$. Therefore, non-Markovian structured bath can support the nonreciprocal-amplified transmission.

 

Fig. 3. The scattering rate $|S_{12(21)}|$ as a function of $\omega$ in the Markovian (a) and super-Ohmic spectral environment (b) with $\Lambda /\omega _{m}=0$. (c) The difference $|S_{12}|-|S_{21}|$ as a function of $\omega$ with $\Lambda /\omega _{m}=-0.02$ (solid lines), 0 (dashed lines). (d) The difference $|S_{12}|-|S_{21}|$ as a function of $\omega$ with two group of parameters of the super-Ohmic spectral bath for $\Lambda =0$. The other parameter $\tilde {\Delta }/\omega _{m}=0.5, \kappa /\omega _{m}=0.5, g/\omega _{m}=1, \tilde {A}_{1}/\omega _{m}=0.1$.

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With $\Lambda =-0.02\omega _{m}$, asymmetry is still inconspicuous in the Markovian regime with difference $|S_{12}|-|S_{21}|\approx \pm 0.04$, as the subplot displayed in Fig. 3(c). In the case of super-Ohmic spectral environment, comparing the green-dashed curve and the black-solid curve in Fig. 3(c), one can see that $|S_{12}|-|S_{21}|$ becomes greater with nonzero $\Lambda$ than that with $\Lambda =0$ around each effective mechanical frequency. When the parameters change to {$s=2,\eta =3.5\times 10^{-2}$}, the effective frequency is undoubtedly changed. Significantly, there is an outstanding difference of scattering rates displayed in Fig. 3(d) by the green-solid curve, where a peak and a dip of $|S_{12}|-|S_{21}|$ distribute at different frequency, which means that one can determine the transmission direction $1\rightarrow 2$ or $2\rightarrow 1$ by choosing detection frequency in peak or dip position. In a word, unidirectional amplification can be achieved due to the modulation of the structured bath, even if nonreciprocity is inconspicuous in the Markovian regime.

On the other hand, multipath interference is a common way to reverse transmitted direction. Now, including the mode $a_{2}$, the system is shown in Fig. 1(a). See equivalent schematic Fig. 1(b), two cavity modes and mechanical mode consist of a two-path interference for the mutual signals transmission $a_{1}\longleftrightarrow a_{2}$. Consequently, the phase difference $\Delta \varphi =\varphi _{1}-\varphi _{2}$ may play a key role in enhancing or damaging the interference paths.

For the sake of symmetry, the equal coupling strengths are assumed, i.e., $\mu =1$ in the drift matrix (9). Figure 4 (a) and (d) clearly show that the isolation ratio $\mathcal {I}=10\mathtt {log}_{10}\frac {|S_{12}|}{|S_{21}|}$ is sensitive to the phase difference $\Delta \varphi$. Furthermore, the direction of nonreciprocity can be reversed for $n\pi \pm \Delta \varphi$. In the Markovian regime, around $\pm \omega _{m}$, $|S_{12}|>|S_{21}|$ is achieved in the condition of $\Delta \varphi \in (0,\pi )$ and $|S_{12}|<|S_{21}|$ can be reached when $\Delta \varphi \in (-\pi,0)$, shown in Fig. 4(a). From Fig. 4(b) and (c), when $\Delta \varphi =\pm 0.38\pi$, nonreciprocity can be distinguishable with the maximal difference $0.18$ around $\omega _{m}$. From Fig. 4(d), in the case of the super-Ohmic spectral bath, around its effective frequency $\pm 0.66\omega _{m}$, the maximal isolation is larger than that in the Markovian regime, indicated that there is more significant nonreciprocity, which can be approved in Fig. 4(e-f) when $\Delta \varphi =\pm 0.38\pi$. Comparing to Fig. 3, with the help of interference, nonreciprocity is more obvious in both the Markovian regime and the super-Ohmic spectral bath.

 

Fig. 4. The isolation as a function of $ \omega$ and phase difference $\Delta \varphi$ in the Markovian regime (a) and the super-Ohmic spectral environment (d) for $\Lambda =0$. The scattering rate as a function of $ \omega$ in the Markovian (b-c) and super-Ohmic spectral environment (e-f) for two values of $\Delta \varphi$. The other parameters $g/ \omega _{m}=1$, $\tilde {A}_{1,2}/ \omega _{m}=0.1$, $\Lambda =0$, and $\tilde {\Delta }_{1,2}/ \omega _{m}=0.5, \kappa _{1,2}/ \omega _{m}=0.5$.

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4. Conclusion

In conclusion, we have studied optical nonreciprocity in a multi-mode optomechanical system, where the mechanical oscillator couples to a structured bath. In the case of single-optomechanical coupling system, we focused on the role of the non-Markovian bath in nonreciprocity and demonstrated that the mechanical response function is enhanced by the structured bath due to the mechanical gain and the information back-flow. As a result, unidirectional amplification is achieved around its effective frequency in the non-Markovian regime, but obvious nonreciprocity is hard to realized in the Markovian regime. For the double-optomechanical coupling system, significant amplification is shown in the non-Markovian regime. Further, we can switch the direction of transmission and adjust the difference between two scattering rates by modifying the phase difference of effective optomechanical couplings.

Appendix A. The derivations of Langevin equations

With the definition of the quadrature operators $X=\sqrt {\frac {1}{2m\omega _{m}}}(b+b^{\dagger })$, $P=-i\sqrt {\frac {m\omega _{m}}{2}}(b-b^{\dagger })$, $X_{l}=\sqrt {\frac {1}{2m_{l}\Omega _{l}}}(b_{l}+b_{l}^{\dagger })$, $P_{l}=-i\sqrt {\frac {m_{l}\Omega _{l}}{2}}(b_{l}-b_{l}^{\dagger })$, the Hamiltonian $H_s$ of the levitated optomechanical system Eq. (1) and the Hamiltonian $H_e$ related to the mechanical bath Eq. (2) can be expressed as

(15)$$\begin{aligned} H_{s}=& \sum_{j=1,2}\omega_j^\prime a_j^\dagger a_j +\frac{\omega_{m}}{2}b^{{\dagger}}b-\frac{\omega_{m}}{4}\left(b^{{\dagger} 2}+b^{2}\right) +g(a_1^\dagger a_2+a_1a_2^\dagger)\\ &+A_{1}a_{1}^{{\dagger}}a_{1}\left(b+b^{{\dagger}}\right)^{2}+A_{2}a_{2}^{{\dagger}}a_{2}\left[ \mathtt{cos} 2\theta \left(b^{{\dagger}}+b\right)^{2}-\mathtt{sin} 2\theta \left(b^{{\dagger}}+b\right) \right] ,\\ H_{e}=&\sum_{l}[\Omega_{l}b_{l}^{{\dagger}}b_{l}+\Gamma_{l}(b_{l}+b_{l}^{{\dagger}})(b+b^{{\dagger}})], \end{aligned}$$

where $\omega _{1}^{\prime }=\omega _{1}+A_{1}$, $\omega _{2}^{\prime }=\omega _{2}+\frac {1}{2}A_{2}(1+\mathtt {cos}2\theta - \mathtt {sin}2\theta )$, and the effective coupling between nanosphere and the $l$th oscillator in the reservoir $\Gamma _{l}=\frac {\gamma _{l}}{2}\sqrt {\frac {1}{m_{l}m\omega _{m}\Omega _{l}}}$.

The quantum Langevin equations (QLEs) with total Hamiltonian $H=H_{s}+H_{e}+H_{d}$ in a rotating frame with respect to driving frequency $\omega _{L}$ can be written as:

(16)$$ \dot{a_{1}}=({-}i\Delta _{1}-\frac{\kappa _{1}}{2})a_{1}-iga_{2}-iA_{1}a_{1}(b^{{\dagger} }+b)^{2}+\varepsilon _{1}+\sqrt{\kappa _{1}}a_{1,in},$$

(17)$$\begin{aligned} \dot{a_{2}}=&({-}i\Delta _{2}-\frac{\kappa _{2}}{2})a_{2}-iga_{1}+\varepsilon _{2}+\sqrt{\kappa _{2}}a_{2,in}\\ &-iA_{2}a_{2}[\mathtt{cos}2\theta (b^{{\dagger} }+b)^{2}-\mathtt{sin}2\theta (b^{{\dagger} }+b)], \end{aligned}$$

(18)$$\begin{aligned} \dot{b}=&-i\frac{\omega_{m}}{2}(b-b^{{\dagger}})-2iA_{1}a_{1}^{{\dagger}}a_{1}(b+b^{{\dagger}})+iA_{2}\mathtt{sin}2\theta a_{2}^{{\dagger} }a_{2}\\ &-2iA_{2}\mathtt{cos}2\theta a_{2}^{{\dagger}}a_{2}(b+b^{{\dagger}})-i\sum_{l}\Gamma_{l}(b_{l}+b_{l}^{{\dagger}}), \end{aligned}$$

(19)$$ \dot{b_{l}}=-i\Omega_{l}b_{l}-i\Gamma_{l}(b+b^{{\dagger}}), $$

where $\Delta _{j}=\omega _{j}-\omega _{L}$. Equation (19) can be integrated formally

(20)$$ b_{l}(t)=b_{l}(0)e^{{-}i\Omega _{l}t}-i\Gamma _{l}\int_{0}^{t}\mathtt{d}s(b(s)+b^{{\dagger} }(s))e^{{-}i\Omega _{l}(t-s)}, $$

and then the last term of Eq. (18) is

(21)$$\begin{aligned} -i\sum_{l}\Gamma_{l}(b_{l}^{{\dagger}}+b_{l})=& -i\sum_{l}\Gamma_{l}[b_{l}^{{\dagger}}(0)e^{i\Omega_{l}t}+b_{l}(0)e^{{-}i\Omega _{l}t}]\\ &+i\int_{0}^{t}\mathtt{d}s(b(s)+b^{{\dagger} }(s))[\sum_{j}\Gamma_{l}^{2}2\mathtt{sin}\Omega_{j}(t-s)], \end{aligned}$$

where the first term represents the noise of the nanosphere, the second term indicates the memory effect of the environmental oscillator on the nanosphere. Thus the memory kernel function can be obtained $f(t)=-2\sum _{l}\Gamma _{l}^{2}\mathtt {sin}(\Omega _{l}t)$. Due to the continuity of energy levels of the reservoir composed of massive particles, we can replace the sum of modes by the integral of wave vector $\mathbf {k}$

(22)$$ \sum_{l}\Gamma_{l}^{2}2\mathtt{sin} \Omega_{l}(t-s)=\int_{0}^{\infty}\mathtt{d}\Omega D(\Omega)2\Gamma ^{2}(\Omega)\mathtt{sin} \Omega (t-s), $$

where $D(\Omega )$ is the density of state, $\Gamma (\Omega )$ depends on $\Omega$ and is proportional to $\sqrt {\Omega }$ as mentioned in Refs. [46,48]. Introducing the spectral density of the reservoir $J(\Omega )=2\pi D(\Omega )\Gamma ^{2}(\Omega )$, for the Ohmic dissipation, the spectral density of the reservoir is given as $J(\Omega )=\eta \Omega (\frac {\Omega }{\omega _{c}})^{s-1}e^{-\frac {\Omega }{\omega _{c}}}$ [46,69], where $\eta$ is a dimensionless measure of the strength of the nanosphere-environment coupling [46]. Then we have $f(t)=-\frac {1}{\pi }\int _{0}^{\infty } \mathtt {d}\Omega J(\Omega )\mathtt {sin}(\Omega t)$ and rewrite Eq. (18) as

(23)$$\begin{aligned} \dot{b}=&-i\frac{\omega_{m}}{2}(b-b^{{\dagger}}) -2iA_{1}a_{1}^{{\dagger}}a_{1}(b+b^{{\dagger}})+iA_{2}\mathtt{sin}2\theta a_{2}^{{\dagger} }a_{2}\\ &-2iA_{2}\mathtt{cos}2\theta a_{2}^{{\dagger} }a_{2}(b+b^{{\dagger} })-i\int_{0}^{t}\mathtt{d}sf(t-s)(b(s)+b^{{\dagger} }(s))-if_{in}(t), \end{aligned}$$

In order to solve the nonlinear Langevin equations, the operator can be divided into the classical value and its fluctuation, i.e., $a_{j}= |\alpha _j|e^{i\varphi _j}+\delta a_{j}$ and $b=\beta +\delta b$. When the driving field is strong enough that $a_{j}\ll |\alpha _{j}|$, then the QLEs can be linearized by neglecting the high-order small terms, given by

(24)$$\begin{aligned} \dot{a_{j}}=&({-}i\tilde{\Delta}_{j}-\frac{\kappa _{j}}{2})a_{j}-iga_{3-j}-i\tilde{A}_{j}e^{i\varphi _{j}}(b^{{\dagger} }+b)+\sqrt{\kappa _{j}}a_{j,in},(j=1,2),\\ \dot{b}=&-i\tilde{\omega}_{m}b+i\Lambda b^{{\dagger} }-i\int_{0}^{t}\mathtt{d}sf(t-s)(b(s)+b^{{\dagger} }(s))\\ &-i(\tilde{A}_{1}e^{i\varphi _{1}}a_{1}^{{\dagger} }+\tilde{A}_{2}e^{i\varphi_{2}}a_{2}^{{\dagger} }+h.c.)-if_{in}(t), \end{aligned}$$

where $\delta$ in $\delta a_j$ and $\delta b$ are ignored for simplicity, $\tilde {A}_1=2A_1|\alpha _1|(\beta +\beta ^*)$ and $\tilde {A}_2=A_2|\alpha _2|[2(\beta +\beta ^*)\mathtt {cos}2\theta -\mathtt {sin}2\theta ]$. The corresponding effective non-Hermitian Hamiltonian of the optomechanical system by adding to the optical decay rates and the optical noise operators can be given as

(25)$$\begin{aligned} H_{sys}^{eff}=&\sum_{j=1,2}[ (\tilde{\Delta}_j-i\frac{\kappa_j}{2}) a_j^\dagger a_j+(i\sqrt{\kappa_j}a_j^\dagger a_{j,in}+h.c.)] + \tilde{\omega}_{m}b^{{\dagger} }b - \frac{\Lambda }{2}(b^{{\dagger} 2}+b^{2})\\ &+\sum_{j=1,2} (\tilde{A}_je^{i\varphi_j}a_j^\dagger{+}h.c.)(b^\dagger{+}b) +{+}g(a_{1}^{{\dagger}}a_{2}+a_{1}a_{2}^{{\dagger}}), \end{aligned}$$

where $\tilde {\omega }_{m}=\frac {\omega _{m}}{2}+2A_{1}\alpha _{1}^{2}+2A_{2}\mathtt {cos}2\theta \alpha _{2}^{2}$ is the modified frequency of mechanical oscillator. $\Lambda =\frac {\omega _{m}}{2}-2A_{1}\alpha _{1}^{2}-2A_{2}\mathtt {cos}2\theta \alpha _{2}^{2}$ is the nonlinear amplification strength of mechanical oscillator, where the first term is arisen from the potential energy of nanosphere, and the last two terms are resulted from the quadratic optomechanical coupling. $\tilde {\Delta }_{j}=\Delta _{j}+A_{j}(\beta ^{\ast }+\beta )^{2}$ ($j=1,2$) is the effective optical detuning. In the main text, $\theta =\pi$ is assumed.

Appendix B. The stable condition of the effective optomechanical system

For the effective optomechanical system, corresponding to the effective Hamiltonian Eq. (25), it is necessary to discuss the stable condition due to the mechanical gain and XX-type optomechanical coupling. The XX-type coupling between the structure bath and the nanosphere also may result in unstable. Since we consider the coupling strength of non-Markovian is still in weak regime, the stability is still determined by other interaction rather than the non-Markovian environment [40–42,45]. The dynamical matrix M can be written as

(26)$$ M=\begin{bmatrix} -i\tilde{\Delta}_1-\frac{\kappa_1}{2} & -ig & -i\tilde{A}_1e^{i\varphi_1} & 0 & 0 & -i\tilde{A}_1e^{i\varphi_1} \\ -ig & -i\tilde{\Delta}_2-\frac{\kappa_2}{2} & -i\tilde{A}_2e^{i\varphi_2} & 0 & 0 & -i\tilde{A}_2e^{i\varphi_2} \\ -i\tilde{A}_1e^{{-}i\varphi_1} & -i\tilde{A}_2e^{{-}i\varphi_2} & -i\tilde{\Delta}_m-\frac{\kappa_m}{2} & -i\tilde{A}_1e^{i\varphi_1} & -i\tilde{A}_2e^{i\varphi_2} & i\frac{\Lambda}{2} \\ 0 & 0 & i\tilde{A}_1e^{{-}i\varphi_1} & i\tilde{\Delta}_1-\frac{\kappa_1}{2} & ig & i\tilde{A}_1e^{{-}i\varphi_1} \\ 0 & 0 & i\tilde{A}_2e^{{-}i\varphi_2} & ig & i\tilde{\Delta}_2-\frac{\kappa_2}{2} & i\tilde{A}_2e^{{-}i\varphi_2} \\ i\tilde{A}_1e^{{-}i\varphi_1} & i\tilde{A}_2e^{{-}i\varphi_2} & -i\frac{\Lambda}{2} & i\tilde{A}_1e^{i\varphi_1} & i\tilde{A}_2e^{i\varphi_2} & i\tilde{\Delta}_m-\frac{\kappa_m}{2} \end{bmatrix}. $$

According to the Routh-Hurwitz criterion [70], the system is stable if and only if the real parts of all eigenvalues of dynamical matrix M are negative. The stable regime (the gray shading region in Fig. 5) varying with mechanical decay and gain is shown.

 

Fig. 5. The stable condition of the effective optomechanical system in the Markovian regime. The red and blue lines correspond to $ \kappa _{m}/ \omega _{m}= \pi \times J( \omega _{m})$ and $\Lambda / \omega _{m}=-0.02$, respectively. The other parameters $\tilde {\Delta }/ \omega _{m}=0.5, \kappa / \omega _{m}=0.5, g/ \omega _{m}=1, \tilde {A}_{1}/ \omega _{m}=0.1, \varphi _{1}=\pm 0.38 \pi, \varphi _{2}=0$.

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Appendix C. Expressions of scattering rates

From Eq. (8), we can obtain $O_{out}(\omega )=(\frac {\Gamma }{\chi ^{-1}-T}-I)O_{in}(\omega )$ where the input-output relation $a_{out}=\sqrt {\kappa }a-a_{in}$ have been employed. The spectrum of output field is defined by $P_{out}=\int \mathtt {d}\omega ^{^{\prime }}\langle a_{out}^\dagger (\omega ^{^{\prime }}) a_{out}(\omega )\rangle$. The spectra of the input field $P_{j,in}(\omega )$ is defined via $\langle a_{j,in}^{\dagger }(\omega ^{\prime })a_{j,in}(\omega )\rangle =P_{j,in}(\omega )\delta (\omega +\omega ^{\prime })$ and $\langle a_{j,in}(\omega ^{\prime })a_{j,in}^\dagger (\omega )\rangle =(1+P_{j,in}(\omega ))\delta (\omega +\omega ^{\prime })$ $(j=1,2)$, where the term 1 arising from the effect of vacuum noise. Then the relationship of the output and input spectra vector is given by Eq. (10), where the elements of $S(\omega )$ are given by

(27)$$\begin{aligned}|S_{jj}(\omega)|=&\kappa_j^2(|(D_{3-j}F_{jj}+C_{j(3-j)}F_{(3-j)j})-1|^2+|D_{3-j}E_{jj}+C_{j(3-j)}E_{(3-j)j}|^2)/D_s,\\ |S_{ij}(\omega)|=&\kappa_i\kappa_j(|D_jF_{ij}+C_{ij}F_{jj}|^2+|D_jE_{ij}+C_{ij}E_{jj}|^2)/D_s, \qquad (i\neq j), \end{aligned}$$

where $D_s=|D_1D_2-C_{12}C_{21}|^2$, and the others are given as ($i, j=1, 2$)

(28)$$\begin{aligned}C_{ij}=&M_{i3}M_{3j}+M_{i4}M_{4j}, \qquad (i\neq j),\\ D_i=&\chi_{+}^{{-}1} \chi_{-}^{{-}1}-M_{i3}M_{3i}-M_{i4}M_{4i},\\ E_{i1}=&M_{i3}\chi_{2-}^{\prime-1} +M_{i4}T_{43}, \qquad E_{i2}=M_{i3}T_{34}+M_{i4}\chi_{1-}^{\prime-1},\\ F_{ii}=&\chi_{-}^{{-}1} \chi_{(3-i)+}^{\prime-1}, \qquad F_{ij}=\chi_{-}^{{-}1}T_{ij}, \qquad (i\neq j),\\ M_{i3}=&T_{i3} \chi_{(3-i)+}^{\prime-1}+T_{i(3-i)}T_{(3-i)3},\\ M_{i4}=&T_{i4}\chi_{(3-i)+}^{\prime-1}+T_{i(3-i)}T_{(3-i)4},\\ M_{3i}=&T_{3i} \chi_{2-}^{\prime-1}+T_{34}T_{4i},\\ M_{4i}=&T_{4i}\chi_{1-}^{\prime-1}+T_{3i}T_{43}, \end{aligned}$$

and $\chi _{-}^{-1}= \chi _{1-}^{\prime -1}\chi _{2-}^{\prime -1}-T_{34} T_{43}$, $\chi _{+}^{-1}=\chi _{1+}^{\prime -1}\chi _{2+}^{\prime -1}-T_{12} T_{21}$. In general, the perfect nonreciprocity can be achieved when $|S_{12}(\omega )|=0$ ($E_{12}=F_{12}=C_{12}=0$) and $|S_{21}(\omega )|\neq 0$ ($E_{21},F_{21},C_{21}\neq 0$) at a same frequency. Therefore, for the elements of drift matrix (9), $T_{12}=T_{14}=T_{13}T_{32}=0$ and $T_{21},T_{23},T_{24}T_{41} \neq 0$ are requested. For our system, such drift matrix can not be achieved, then there is missing the perfect nonreciprocity.

Funding

National Natural Science Foundation of China (11874099); National Key Research and Development Program of China (2021YFE0193500).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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