1. Introduction

Nonreciprocal devices have attracted a great deal of attention from physicists in quantum communication and signal processing due to many wide potential applications in invisible sensing and signal processing such as isolators and directional amplifiers [1]. Nonreciprocal effect appears when the time-reversal symmetry is broken in these devices. Traditional nonreciprocal devices are build on magnetic bias field by magneto-optic material responses [2,3]. However, they not only have weak magneto-optic coefficients, but are also hard to integrate on chips . To overcome all these drawbacks, recently, many alternative methods have been proposed to replace traditional schemes, including angular momentum biasing in photonic [4] or phononic crystals, the parity-time-symmetric structures [5], dynamic spatiotemporal modulation of refractive-index [6], quantum hall effect [7], synthetic magnetism [8], optical nonlinearity [9], optomechanical interactions [10] and so on.

In recent years, the cavity optomechanics [11] with rapidly growing speed has shown promising potential for applications in quantum information processing and communication, such as optomechanically induced transparency (OMIT) [12–14], normal-mode splitting in the strong coupling regime [15], ground state cooling of the mechanical resonator [16], optical nonlinear effects [17] and so on. It is promising to obtain nonreciprocal transmission in cavity optomechanical system based on spatiotemporal modulation of the refractive index to break time-reversal symmetry [18]. Nonreciprocity in multimode cavity optomechanical devices has been investigated theoretically [19] and experimentally [20]. At present, most researches focus on the nonreciprocity in the same frequency domain, while across-frequency study of nonreciprocity is relatively rare. Nonreciprocity between optical photons transmission is demonstrated in Refs. [21–25]. In addition, some works have shown the nonreciprocal transmission between microwave photons in a microwave optomechanical device [26,27]. The models mentioned above have one common feature, that is, the two cavities are directly coupled to the same mechanical oscillator. The advantage of this model is that there are two different paths for photon transmission in the two optical cavities, and the interference between the different paths will induce the nonreciprocity. In order to obtain nonreciprocity across the microwave and optical domain, Lan et al. poposed a scheme including two mechanical oscillators coupled with the same microwave and optical cavities, where two mechanical oscillators can mediate coherent and dissipative interferences on two transfer paths respectively to induce nonreciprocity [28], and L. Tian and Z. Li have achieved nonreciprocal quantum-conversion between microwave and optical photons by introducing an auxiliary cavity to influence the interference between two transfer paths [29].

The quantum states of microwave frequency photons coding can be manipulated efficiently and form local information processing unit with qubits in quantum network [30,31]. However, microwave frequency photons are difficult to transmit over long distances due to the thermal noise [30,32]. Fortunately, photons of optical frequency can be coherently transmitted over long distances through optical fibers and waveguides due to low decoherence and dissipation rates [33,34]. In this paper, based on the characteristics of mechanical oscillator, which can be coupled to arbitrary frequency optical field [35], we integrated microwave and optical frequency fields into a hybrid cavity optomechanical system to study nonreciprocity of two across-frequency domains.

Compared with previous research, the main advantage of our system are given as follows. 1. The nonreciprocity of photons in the same frequency domain and different frequency domains can be realized simultaneously in the same optomechanical system. 2. Electrically controlled reciprocity and non-reciprocity. This electrical processing technology promotes the development of nonreciprocity towards integration and miniaturization, especially on-chip technology. 3. It has important applications in respects of quantum information processing in microwave domain and long-distance information transmission in optical wave domain.

2. System model

A hybrid electro-optomechanical system we considered here is comprised of four cavity modes $c_{i}$ $(i=1,2,3$ and $4)$ and two charged mechanical modes $b_{j}$ $(j=1,2)$ schematically shown in Fig. 1. The two microwave cavity fields $c_{1}$ and $c_{2}$ are coupled directly to each other with the tunneling coupling strength $J_{1}$, and coupled simultaneously to a same mechanical mode $b_{1}$ via radiation pressure. Similarly, the optical cavity fields $c_{3}$ and $c_{4}$ are coupled to each other directly with the tunneling coupling strength $J_{2}$, and simultaneously coupled to the same mechanical mode $b_{2}$ via radiation pressure. Two charged mechanical modes $b_{1}$ (charged $Q_{1}$, position $q$) and $b_{2}$ (charged $Q_{2}$) are coupled to each other via a tunable Coulomb interaction , which is a bond between two types of frequency cavity fields. Two charged mechanical modes may be given by two charged mechanical oscillators (for example, two conductive plates), as shown in Fig. 2(a). Two mechanical oscillators can be charged by injecting the error function of the voltage (or current) waveforms $A(1-erf(\mp \frac {t}{\Gamma } )$ with the characteristic time $\Gamma$ in the external circuit. The accumulated charge $Q$ is presented on the mechanical oscillator, as shown in Fig. 2(b). The equilibrium distance between two charged mechanical modes is expressed by $r_{0}$ in the absence of the Coulomb interaction and optomechanical coupling. Under the combined action of Coulomb force and optomechanical force, the small displacements of the two mechanical oscillators are given as $q_{1}$ and $q_{2}$, which is much smaller than the distance between two conductive plates, i.e., $q_{1},q_{2}\ll r_{0}$ . The Coulomb potential between two conductive plates can be can be given by [36]

 

Fig. 1. Schematic of the hybrid optomechanical system. Two mechanical modes ($b_{1}$ and $b_{2}$) are coupled optomechanically to two microwave modes ($c_{1}$ and $c_{2}$) on the left and two optical modes ($c_{3}$ and $c_{4}$) on the right. The mechanical modes $b_{1}$ and $b_{2}$ are coupled to each other via Coulomb interaction simultaneously. Each cavity is driven by a strong coupling field $\varepsilon _{Li}$ and a weak field $\varepsilon _{pi}$.

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Fig. 2. (a) The Coulomb interaction between two charged mechanical modes. (b) The Charge on a mechanical oscillator.

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In the case of $q_{1},q_{2}\ll r_{0}$, with the second-order expansion, it is rewritten as

Through the method in Refs. [36–38], Eq. (2) can be simplified to $H_{C}=\hbar V(b_{1}^{\dagger }b_{2}+b_{2}b_{1}^{\dagger })$, where $V$ is called as the Coulomb interaction coupling strength. Each cavity $c_{i}$ is driven by a strong coupling field $\varepsilon _{Li}$ and a weak probe field $\varepsilon _{Pi}$. The total Hamiltonian of the hybrid optomechanical system reads ($\hbar =1$)

Here $\Delta _{ci}=\omega _{ci}-\omega _{Li}$ $(\delta _{Pi}=\omega _{pi}-\omega _{Li})$ accounts for the detuning between cavity modes (probe fields) and coupling fields. Considering the damping and fluctuation terms, the quantunm Heisenberg-Langevin equations of dynamics can be given as

From Eqs. (8)-(13), we can find that there are two kinds of the noises, i.e., cavity input noise $c_{in}^{i}$ and thermal noise from the mechanical resonators $b_{in}^{j}$. The cavity dynamics depend on the cavity input noise, which is very smaller than the strong input field and has the zero mean value $\left \langle c_{in}^{i}(t){c_{in}^{i}}^{\dagger }(t') \right \rangle =\delta (t-t')$ [39,40]. Thermal noise from the mechanical resonator is affected by a Brownian noise with the correlation function [39,41,42] $\left \langle b_{in}^{j}(t){c_{in}^{i}}^{\dagger }(t') \right \rangle =\frac {\gamma _{j}}{\omega _{mj} }\int \frac {d\omega }{2\pi }e^{-i\omega (t-t')}\omega \left [ coth(\frac {\hbar \omega }{2\kappa _{B}T} )+1 \right ] =\gamma _{j}(2\bar {n}+1 )\delta (t-t')$ where $\kappa _{B}$ is the Boltzmann constant, $T$ is the temperature of the reservoir of mechanical oscillator and $\bar {n}=\left [ exp(\frac {\hbar \omega _{j}}{\kappa _{B}T} )-1 \right ]^{-1}$ is the mean thermal excitation number of the resonator. In the experiment, this thermal noise can be reduced through additional sideband cooling and is suppressed by increasing the amplitudes in terms of cooperativities, which have studied in Ref. [27]. Here, we are only interested in the mean response of the system, therefore the Heisenberg-Langevin noise operators can be reduced to their expectation values, i.e., the quantum and thermal noise terms are ignorable. Under the condition of $\varepsilon _{Li}\gg \varepsilon _{Pi}$, the Heisenberg operators can be written as the sum of average values for the steady-state and small fluctuation, i.e., $O=O_{s}+\delta {O}$ $(O=c_{i}, b_{j})$, where $\delta {O}$ can be also viewed as a perturbation [43]. In addition, the mean-field approximation $\left \langle ab \right \rangle \approx \left \langle a \right \rangle \left \langle b \right \rangle$ [44] is considered here. Then we can obtain the steady-state mean values of $O$ as

Using input-output theory $\left \langle \varepsilon _{i}^{out} \right \rangle +\varepsilon _{Li}+\varepsilon _{Pi}e^{-i\delta _{Pi} t}=\sqrt {\kappa _{i}} \left \langle c_{i} \right \rangle$, we can obtain the output of the cavity field:

To study nonreciprocity at the probe frequency $\omega _{pi}$, we can obtain output fields $\varepsilon _{Pi}^{out}$ (i=1,2,3 and 4) according to the input-output relation [46]

3. Stability analysis

The stability of the system can be given by Eqs. (14)-(19). Because of the symmetry, without loss of generality, we only analyze Eq. (14) $c_{1s}=\frac {\varepsilon _{L1 }-iJ_{1}c_{2s}}{\frac {\kappa _{1}}{2}+i\Delta _{1}}$, where $\Delta _{1}$ is the effective optomechanical detuning between cavity mode and coupling field due to the optomechanical interactions. For simplicity, we assume that the incident strong field $\varepsilon _{L1}$ is much larger than the tunneling coupling interaction term, i.e., $\varepsilon _{L1}\gg \left | J_{1}c_{2s} \right |$. Therefore, we can rewrite as $c_{1s}=\frac {\varepsilon _{L1}}{\frac {\kappa _{1}}{2}+i\Delta _{1} }$. The steady-state photon number $n_{1}$ of the first cavity field satisfies the following equation:

 

Fig. 3. The stable-state behavior of first cavity field with $\Delta _{1}=\omega _{m1}/2\pi =10$ MHz, $\kappa _{1}=0.4\omega _{m1}$, $g_{1}=1.347$ kHz, $\gamma _{1}=100$ Hz.

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4. Nonreciprocal state conversion

In this section, we discuss the respective nonreciprocal responses between two cavity fields with the same type frequencies and between two cavity fields with different type frequencies.

4.1 Same type frequencies

First, we study the nonreciprocal responses between the same type frequencies (microwave or optical frequency). Without loss of generality, we discuss the nonreciprocity between two microwave frequency cavities (i.e., $c_{1}$ and $c_{2}$) and the nonreciprocity between optical cavities ($c_{3}$ and $c_{4}$) is similar. The transmission amplitudes can be obtained as

It can be seen from Eqs. (31), (32) that the two output fields are equal when $\theta =n\pi$ ($n$ is an integer), which indicates the microwave photons transmission is reciprocal. When $\theta \ne n\pi$, the time-reversal symmetry is broken and the system exhibits a nonreciprocal response. For simplicity, we discuss the case of $\theta =\varphi =\pi /2$, $\kappa _{1}=\kappa _{2}=\kappa _{a}$, $\kappa _{3}=\kappa _{4}=\kappa _{b}$, $\gamma _{1}=\gamma _{2}=\gamma$, $G_{1}=G_{2}=G_{a}$ and $G_{3}=G_{4}=G_{b}$, then Eqs. (31), (32) can be simplified as

From Eqs. (33), (34), the perfect nonreciprocity ($T_{12}=0$, $T_{21}=1$) can be achieved, when

It can be found from Eq. (35) that the effects of the system parameters on the properties of tranmission amplitudes $T_{12}$ and $T_{21}$. In Fig. 4(a), we plot transmission amplitudes $T_{12}$ (blue dashed line) and $T_{21}$ (red soild line) as a function of detuning $x/2\pi$ with $V=V_{0}$ , where the other parameters are $\theta =\varphi =\pi /2$, $\kappa _{a,b}/2\pi =5$ MHz, $\gamma /2\pi =1$ MHz, $G_{a}/2\pi =3$ MHz, $G_{b}/2\pi =2$ MHz, $J_{1}=\kappa _{a}/{2}$ and $J_{2}=\kappa _{b}/{2}$ [48]. It can be clearly seen from Fig. 4(a) that the perfect nonreciprocity occurs at the resonant frequency ($x=0$), which shows that the probe field can be transmitted from cavity $c_{2}$ to cavity $c_{1}$, but not vice versa. The nonreciprocity arises due to interference between two possible paths, where one path is along $c_{1}\to b_{1}\to c_{2}$ and the other path is along $c_{1}\stackrel {J_{1}}\to c_{2}$. When the phase differences $\theta =\pi /2$ and $\varphi =\pi /2$, the constructive interference between the two paths leads to the transmission from cavity $c_{2}$ to $c_{1}$, whereas the disappear of $T_{12}$ results from distructive interference. Figure 4(b) shows transmission amlpitudes $T_{12}$ (blue dashed line) and $T_{21}$ (red soild line) vs Coulomb interaction $V/\gamma$ with $J_{1}=\kappa _{a}/2$, $x=0$ and the other parameters are the same as those in Fig. 4(a). It can be found from Fig. 4(b) that when $J_{1}=\kappa _{a}/2$, $T_{21}$ is always equal to unity, even if $V$ changes. The perfect nonreciprocity occurs at $V=V_{0}$, which is consistent with Fig. 4(a). Figure 4(b) further approves the modulation capability induced by the Coulomb interaction in perfect nonreciprocity.

 

Fig. 4. The transmission probabilities $T_{12}$ (blue dashed line) and $T_{21}$ (red soild line) as a function of (a) detuning $x/2\pi$ with $V=V_{0}$ and (b) Coulomb interaction $V/\gamma$ with $x=0$. The other parameters are $\theta =\varphi =\pi /2$, $\kappa _{a,b}/2\pi =5$ MHz, $\gamma /2\pi =1$ MHz, $G_{a}/2\pi =3$ MHz, $G_{b}/2\pi =2$ MHz, $J_{1}=\kappa _{a}/{2}$, $J_{2}=\kappa _{b}/{2}$.

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Eq. (35) reveals the importance of the nonreciprocity due to the tunable Coulomb interaction. Figure 5(a) shows transmission amlpitudes $T_{12}$(blue dashed line) and $T_{21}$(red soild line) vs Coulomb interaction $V/\gamma$ with $J_{1}=(2\kappa _{1}\kappa _{2})^{1/2}$, $J_{2}=(2\kappa _{3}\kappa _{4})^{1/2}$ [48], $x=0$, and other parameters are the same as those in Fig. 4. It can be seen from Fig. 5(a) that $T_{21}$ is a decreasing function of $V/\gamma$ and finally approches a stable value. However, $T_{12}$ firstly decreases when $V\le \gamma$, then increases when $V> \gamma$. It can be easily observed that nonreciprocal response exists when $V\le 12\gamma$. When the Coulomb interaction $V$ is stronger than $12\gamma$, $T_{12}\approx T_{21}$, which means that the transmission probability from cavity $c_{1}$ to cavity $c_{2}$ is the same as the transmission probability from cavity $c_{2}$ to cavity $c_{1}$, close to 0.6, i.e., reciprocity can be observed. The reciprocity arises due to the increases of the Coulomb interaction $V$, which makes the interference effect between the two paths weaker [49]. Thus one can find that the photon transmission can be properly and electrically controlled by the Coulomb interaction based on different purposes. The isolationb ratio is plotted in Fig. 5(b), which is defined as $IR=-10log10(\frac {T_{12}}{T_{21}})$ [50]. It is seen that the isolation ratio maximum exceeds 20 dB at the detuning $x=0$.

 

Fig. 5. (a) The transmission probabilities $T_{12}$ (blue dashed line) and $T_{21}$ (red solid line) vs Coulomb interaction $V/\gamma$ with $J_{1}=(2\kappa _{1}\kappa _{2})^{1/2}$, $J_{2}=(2\kappa _{3}\kappa _{4})^{1/2}$, $x=0$. (b) Isolation ratio with $J_{1}=(\kappa _{1}\kappa _{2})^{1/2}/2$, $J_{2}=(\kappa _{3}\kappa _{4})^{1/2}/2$. Other parameters are the same as those in Fig. 4.

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4.2 Different type frequencies

Now we study the nonreciprocity between the microwave and optical frequency field. Analogously, the microwave and optical output fields can be obtained as

It can be seen from Eqs. (36), (37) that reciprocity response arises when $\theta =n\pi$ (n is an integer). To study nonreciprocity, we dicuss the case of phase different $\theta =\varphi =\pi /2$, then Eqs. (36), (37) can be simplified as

 

Fig. 6. (a) The transmission probability of the probe field $T_{14}$ as a function of the coupling constant $G_{b}$ at various $V$ values. (b) Isolation ratio. $G_{a}/2\pi =1$ MHz, $\gamma /2\pi =0.005$ MHz and other parameters are the same as those in Fig. 4.

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With Eqs. (38), (39), we study the detuning $x$ and $V$ dependence of the nonreciprocal state conversion. An intuitive illustration of perfect nonreciprocity is given in Fig. 7, where the transmission probability is plotted versus the detuning. Here $x=0$ corresponds to the resonant frequency of the respective cavities. It is obvious that the hybrid system shows nonreciprocal response between the optical and microwave modes. We can get the perfect nonreciprocity when $T_{14}=1, T_{41}=0$ at $x=0$, $V/2\pi =2$ MHz.

 

Fig. 7. The transmission probability (a) $T_{14}$ and (b) $T_{41}$ versus detuning $x/2\pi$ with $V/2\pi =(0.1,1,2,10)$ MHz where the other parameters are the same as those in Fig. 6.

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5. Conclusion

In summary, we have investigated the nonreciprocal conversion between optical and microwave photons in an optomechanical system, which involves the optomechanical, tunneling and Coulomb interactions. By manipulating the phase differences between the couplings and the Coulomb interaction, the perfect nonreciprocal conversion between the two same and different frequency photons can be achieved. In addition, conversion between optical reciprocity and nonreciprocity can be electrically turned on by adjusting the Coulomb interaction. Our scheme is general and can serve as an isolator, a circulator, a directional amplifier, a photon switching and a router in quantum information processing and quantum networks.