1. Introduction

As a valuable phenomenon in various fields of optics, optical non-reciprocity, of which the key element of nonreciprocal devices is the time-reversal symmetry breaking [1,2], provides the possibility that signals can exhibit different transport behaviors along opposite directions and can be used to prevent the unwanted or useless signals getting into the network, thus playing an essential role in the quantum information processing and information protection. Nowadays, nonreciprocal components such as isolators, circulators and directional amplifiers are emerging [3–6]. The relatively mature ways to implement nonreciprocal devices include magneto-optical effects [7–10], angular momentum biasing [11–13], dynamic modulation [14–17], chiral light-matter interaction [18,19], optical nonlinearity [20–22]. More surprising, the nonreciprocal effect can also be realized with topological energy transfer by encircling exceptional points, where the adiabatic topological operations is used to transfer energy between mechanical modes [23].

On the other hand, in optomechanical systems, the interaction achieved between optical and mechanical degrees of freedom provides a promising platform for studying quantum sciences at the macroscopic scale [24]. Up to now, a host of breakthroughs have been obtained in optomechanics, such as ground-state cooling of mechanical resonators [25,26], ultralow threshold chaos [27,28], quantum synchronization [29–31] and optomechanically induced transparency (OMIT) [32–34]. In particular, based on the optomechanical system, the nonreciprocal effect, which can be realized via the quantum interference [35–37], Sagnac effect [38–41], reservoir engineering [42,43], and so on, has come into view of researchers. The non-reciprocity induced by optomechanics can be modulated by the mechanical driving, the coupling between cavities and so on, of which the relevant schemes have been proposed theoretically [35,44–46] and demonstrated experimentally [47–50]. Under optomechanical nonreciprocal conditions, directional amplification can be realized with blue-sideband lasers [45,46], coherent mechanical driving [51,52], and active objects (gain) [53,54]. In addition, the optomechanical system is also applied to realize the tunable slow/fast light effect by adjusting relevant parameters [55–57] and narrow response windows [58,59].

Motivated by seminal works, we propose a tunable dual-gate optical transistor scheme of directional amplification, which is made up of three indirectly coupled cavities and an intermediate mechanical resonator. In this system, by driving one (two) cavity mode(s) on the blue (red) mechanical sideband and pumping the mechanical resonator with a frequency-matched field, the dual-gate optical transistor will be achieved, where the blue-sideband cavity and the pumped mechanical resonator serve as two gates/poles controlling the photon transmission. We reveal that directional amplification can be achieved and easily controlled by phase modulation. Moreover, the width of the spectral window and the nonreciprocal slow/fast light can be controlled by the blue-sideband driving field. This work provides a new method of controlling the amplification in optomechanical systems, which may inspire follow-up works.

2. Model and transmission coefficient

As shown in Fig. 1(a), we consider an optomechanical system, where three cavity modes $a_k$ ($k= 1, 2, 3$) of frequency $\omega _k$ without inter-coupling, respectively, couple to a common mechanical resonator $b$ of frequency $\omega _m$ via radiation-pressure forces. The red sideband control fields of frequencies $\omega _{c_{1}}$ and $\omega _{c_{2}}$ drive cavities $a_{1}$ and $a_{2}$, respectively, and a weak field of frequency $\omega _{p}$ (amplitude $\varepsilon _{p}$) probes either of the two cavities. As an auxiliary cavity, $a_{3}$ is only driven on the blue sideband at frequency $\omega _{c_{3}}$. A mechanical driving field of frequency $\omega _{b}$ (amplitude $\varepsilon _{b}$) is applied to the resonator $b$ . A kind of implementation scheme is depicted in Fig. 1(b), and such a model can also be realized based on other experimental systems, such as microwave superconducting circuits [48,60] and optomechanical crystal systems [61]. Figure 1(c) shows that there are two closed-loop structures so that relevant transition frequencies should satisfy the frequency-matching condition. The system Hamiltonian has the form $(\hbar =1)$,

 

Fig. 1. (a) Schematic diagram of an optomechanical system composed of three indirectly coupled cavities and a mechanical resonator. (b) Schematic illustration of this system composed of three Fabry-Pérot-type resonators with a common movable mirror driven by a mechanical field. (c) Transition relation diagram of internal states in this multi-mode system where $n_{ak}$ and $n_{b}$ are the excitation numbers of the three cavity modes and the mechanical mode, respectively.

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Here $H_{0}$ represents the free Hamiltonian of the system,

In the rotating frame with respect to the control field frequency, respectively, the Heisenberg-Langevin equations for the operators in the system can be obtained as [62],

Taking relevant dissipations and noises in account, we introduce the damping rate $\kappa$ ($\gamma _{m}$) of the intrinsic cavity (resonator) and the noise operator $a_{k}^{in}$ ($b^{in}$) of zero mean value associated with $\kappa$ ($\gamma _{m}$). Aiming to linearization, each operator is written as the sum of its steady-state value and a small fluctuation, i.e., $a_{k}=\langle a_{k}\rangle +\delta a_{k}$ and $b=\langle b\rangle +\delta b$. Afterwards, the steady-state solutions are written as,

By neglecting the nonlinear terms in Eq. (5), a series of linearized equations can be obtained,

Assuming that the system is operated in the resolved sideband regime with $\omega _{m}\gg \kappa , \gamma _{m}, G_{k}$, we do the following substitution: $\delta a_{k}\rightarrow \delta a_{k}e^{-i\Delta _{k}^{\prime }t}$, $\delta a_{k}^{in}\rightarrow \delta a_{k}^{in}e^{-i\Delta _{k}^{\prime }t}$, $\delta b\rightarrow \delta be^{-i\omega _{m}t}$, $\delta b^{in}\rightarrow \delta b^{in}e^{-i\omega _{m}t}$ [35,51]. Equation (7) becomes,

To solve Eq. (8), the equations are transformed into another interaction picture with $\delta a_{k}\rightarrow \delta a_{k}e^{-i\delta t}, \delta b\rightarrow \delta be^{-i\delta t}$. Neglecting the noise terms, substituting gives

Then the cavity output fields $\langle \delta a_{1,2}^{out}\rangle$ can be obtained by the standard input-output relation $\langle \delta a_{1,2}^{out}\rangle +\langle \delta a_{1,2}^{in}\rangle =\sqrt { 2\kappa }\langle \delta a_{1,2}\rangle$ [63] in the overcoupling regime [64,65]. When $\langle \delta a_{1}^{in}\rangle =\varepsilon _{p}/\sqrt {2\kappa }$ and $\langle \delta a_{2}^{in}\rangle =0$, the transmission coefficient $t_{21}\equiv \partial \langle \delta a_{2}^{out}\rangle /\partial \langle \delta a_{1}^{in}\rangle$ [51,66] can be derived

On account of the gain arising from the blue-sideband laser, the stability of the system should be tested. The coupled Eq. (8) can be solved self-consistently, and then they can be rewritten into the matrix form

 

Fig. 2. (a) Stability diagram of the system on the plane of coupling strengths $G_{1}$ and $G_{3}$ for $\phi =0$ (The diagram is the same as those with $\phi$ of any value.). (b) Stability diagram of the system on the plane of phase difference $\phi$ and the coupling strength $G_{1}$ with $n=2$. The other parameter is $\gamma _{m}=\kappa /100$. The yellow regions are stable and the green regions are unstable.

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3. Results and discussion

In this section, we will study optical responses of the system as a dual-gate optical transistor with amplification due to the gain arising from the auxiliary cavity $a_{3}$ on blue sideband and the mechanical resonator $b$ pumped by field $\varepsilon _{b}$ [51], including the manipulation of the direction (nonreciprocal amplification and unidirectionality) and group velocity (slow/fast light effect) of photon transport. There being no need for the frequency matching between cavities due to their decoupling, the cavity can be either microwave or optical one. Thus, the modulation of two photon flows, two microwave fields or both of them all can be realized. Clearly, such a scheme proposed is an open platform. Here we regard the cavity $a_{1}$ ($a_{2}$ and $a_{3}$) in the microwave (optical) domain. Thus, our system shows the ability to convert the frequencies of input signals under the frequency-mixing process, e.g., when inputting an optical signal into $a_{1}$, the retrieval signal from $a_{2}$ can be microwave with proper parameters. Similar to electricity, the cavity can be treated as a gate/grid of the transistor (like electrical triode), where the base poles are $a_{3}$ and $b$ for controlling the amplification and other gates are $a_{1}$ and $a_{2}$.

The undirectional amplification, which can be investigated by comparing two transmissivities $T_{12}$ and $T_{21}$, may be more obvious when $a_{3}$ is driven by a blue-sideband field, and can be modulated by adjusting the coupling strength $G_{3}$ from Eqs. (11) and (12). Figure 3(a) shows that $T_{12}$ is very sensitive to $G_{3}$, i.e., it is enhanced obviously with the increasing $G_{3}$ around resonant point $\delta =0$. However, Fig. 3(b) shows that $T_{21}$ without obvious gain around resonance becomes zero at $\delta =0$. That means the transmission of $a_{1}\rightarrow a_{2}$ is suppressed completely at the resonance point. Thus, the nonreciprocity between $T_{12}$ and $T_{21}$ can be induced in the near-resonance range. For the sake of clarity, in Fig. 3(c), we plot $T_{12}$ and $T_{21}$ as a function of $\delta$ with $G_{3}=2.8\kappa$ (in the stable region). It is found that, at the resonance point, the transmission of $a_{1}\rightarrow a_{2}$ is suppressed but the opposite transmission is amplified near $10^{4}$ times. Obviously, there is a strong unidirectional amplification for the probe field in the vicinity of $\delta =0$. Note the blue-sideband field driving $a_{3}$ may play a important role in this unidirectional amplification. For comparison, in Fig. 3(d) the nonreciprocity of the system with $a_{3}$ on the red sideband will also be observed, whose tranmissivities can be given just with the substitution $-|G_{3}|^{2}\rightarrow +|G_{3}|^{2}$ in the denominator $N$. With the same $G_{3}$, there is no gain for both transmission at the resonance point. The physics mechanism of the amplification is that the cavity being driven by the blue-sideband field will induce a anti-Stokes process in this system, which can be viewed as a heat bath. The unidirectional amplification is tunable by modulating $G_{3}$, thus the cavity $a_{3}$ can be regarded as one basic pole/gate of the optical transistor for controlling the amplification magnitude.

 

Fig. 3. Logarithms of transmissivities (a) $T_{12}$ and (b) $T_{21}$ versus detuning $\delta$ and coupling strength $G_{3}$ for cavity $a_{3}$ on blue sideband with $y=2$. Logarithms of the transmissivities $T_{12}$ and $T_{21}$ with $G_{3}=2.8\kappa$ and $y=2$ (c) for $a_{3}$ on blue sideband; (d) for $a_{3}$ on red sideband. The transmissivity $T_{12}$ versus $\delta$ with (e) $G_{3}=0$ and (f) $G_3=2.8\kappa$ for cavity $a_{3}$ on blue sideband. Other parameters are $\phi =\pi$, $\beta =\pi /2$, $\gamma _{m}=\kappa /100$ and $G=2\kappa$.

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In addition, another effect of the basic pole/gate $a_{3}$ is narrowing the linewidth of the spectral window by increasing $G_{3}$, still caused by its operation in the blue-sideband domain. As shown in Figs. 3(e) and 3(f), increasing $G_{3}$ from 0 to $2.8\kappa$ with $a_{3}$ on the blue sideband, the linewidth of the window becomes narrow, where $G_3=0$ corresponds to the common three-mode optomechanical system. It is clear that the number of peaks is tunable by the coupling strength. In our system, the increasing of $G_3$ merges the three peaks into only one with the narrower window. The phenomenon can be well explained by the effective mechanical damping rate determining the linewidth of the transparency (amplification or absorption) window in optomechanics [24,58,59]. Through the steady-state solution of $\langle \delta b\rangle$ in Eq. (9),

As the other basic pole/gate of this optical transistor amplifier, the driven mechanical resonator can not only control amplification but the direction of the nonreciprocity by the modulation of the phase difference $\beta$, to adjust which we vary the ratio $y$. Figure 4(a) shows logarithms of $T_{21}$ and $T_{12}$ as a function of $y$ with $G_{3}=2.8\kappa$. When $y=0$ corresponding to $\varepsilon _{b}=0$, $T_{21}=T_{12}$ implies the reciprocity. With the increasing of $y$, $T_{12}$ is amplified gradually but $T_{21}$ becomes zero at $y=2$, which shows strong unidirectional amplification. Obviously, the amplification surely gets larger with increasing $y$. With Fig. 1(c), the underlying physics can be expressed. As depicted in Fig. 1(c), such a system of five states involving excitation numbers $n_{ak}$ and $n_{b}$ forms the closed-loop transition structure sensitive to the phase. The transition $|n_{a1}, n_{a2}, n_{a3}+1, n_{b}\rangle \leftrightarrow |n_{a1}, n_{a2}, n_{a3}, n_{b}+1\rangle$ can be regarded to be the association of this structure, thus $a_{3}$ is a basic pole only for controlling the amplification magnitude. In the absence of the transition $|n_{a1}, n_{a2}, n_{a3}, n_{b}\rangle \leftrightarrow |n_{a1}, n_{a2}, n_{a3}, n_{b}+1\rangle$, there is only one loop $|n_{a1}, n_{a2}, n_{a3}, n_{b}\rangle \leftrightarrow |n_{a1}+1, n_{a2}, n_{a3}, n_{b}\rangle \leftrightarrow |n_{a1}, n_{a2}, n_{a3}, n_{b}+1\rangle \leftrightarrow |n_{a1}, n_{a2}+1, n_{a3}, n_{b}\rangle \leftrightarrow |n_{a1}, n_{a2}, n_{a3}, n_{b}\rangle$. When inputting probe field $\varepsilon _{p}$ from $a_{1}$, a phase-matched filed out of $a_{2}$ can arise from the four-wave mixing independent of phase differences, and vice verse. Therefore, the transmissions of $a_{1}\rightarrow a_{2}$ and $a_{2}\rightarrow a_{1}$ are reciprocal. However, in the presence of that transition, two different closed loops made up will show different phase-dependent behaviors, respectively, thus the phase difference introduced plays an essential role in the quantum interference between two transition paths of the probe fields and then optical nonreciprocity. The importance of $\phi$ for the nonreciprocity can also seen from Eqs. (11) and (12).

 

Fig. 4. (a) Logarithms of the transmissivities $T_{21}$ and $T_{12}$ and (b) isolation ratio $I$ versus relative ratio $y$ with $\delta =0$. Other parameters are the same as those in Fig. 3.

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However, to examine the quality of nonreciprocity (like for a optical switching device), the isolation ratio

As mentioned above, the phase is vital and both phase differences $\phi$ and $\beta$ can modulate the transport behavior from Eqs. (11) and (12). In addition, from those equations, the nonreciprocity should need $\phi \neq \pm 2k\pi$ ($k=0,1,2,3\cdots$) to satisfy the Lorentz reciprocal theorem no matter what $\beta$ value. As in Figs. 5(a)–5(b) with $\phi =\pi /2$, the unidirectional amplification can be obtained with the broken time-reversal symmetry of the system. With $\beta =0$ in Fig. 5(a), $log_{10}(T_{21})\simeq 4$ and $T_{12}\simeq 0$ at $\delta =0$ means unidirectional transmission $a_{1}\rightarrow a_{2}$; tuning $\beta$ to $\pi /2$, the direction of unidirectional transmission is reversed. Figures 5(c)–5(d) show that the nonreciprocal transmission behavior is changed by varying $\beta$. With a few certain parameters (like $\beta =0$ and $\phi =\pi$ in Fig. 5(d)), $T_{21}=T_{12}$ appearing indicates the reciprocal transmission. It is noteworthy that, by modulating the phase $\beta$, the direction of the amplified (suppressed) output probe field can be switch, i.e., the direction of amplification of the optical transistor, for expressing which we plot isolation ratio $I$ against $\beta$ in Fig. 5(g). It is more intuitive that the isolation ratio varies with $\beta$ periodically and continuously between 1 and −1, which indicates the two opposite unidirectional amplification, respectively. As shown in Figs. 5(e)–5(f), with a given $\beta$, $T_{21}$ is independent on $\phi$, which is also be deduced from Eq. (11). Thus, $\phi$ is just essential for nonreciprocity but cannot change the direction. Strikingly, when $\beta =\pi /2, \phi =0, \pm 2k\pi$ both probe fields can be suppressed simultaneously, i.e., bidirectional opaque, as in Fig. 5(h). In this regard, we can realize tunable switching scheme among the bidirectional opaque and transmission, and unidirectional transmission with controllable direction.

 

Fig. 5. Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\delta$ for $\phi =\pi /2$ and (a) $\beta =0$; (b) $\beta =\pi /2$. Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\beta$ for $\delta =0$ and (c) $\phi =\pi /2$; (d) $\phi =\pi$. Logarithms of transmissivities $T_{21}$ and $T_{12}$ verse $\phi$ for $\delta =0$ and (e) $\beta =3\pi /2$; (f) $\beta =\pi$. (g) Isolation ratio $I$ verse $\beta$ for $\phi =\pi /2$ and $\delta =0$. (h) Transmissivities $T_{21}$ and $T_{12}$ verse $\phi$ for $\beta =\pi /2$. Other parameters are the same as those in Fig. 3.

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Moreover, besides the direction, the group velocity of the probe field in this transistor may be also controlled, i.e., the slow/fast light effect. Here we focus on the tunable fast and slow light effect with undirectional amplification, and the way to the switching between slow and fast lights by adjusting $G_{3}$ arising from the auxiliary cavity $a_{3}$. The optical transmission group delay is defined as [53,56]

In the situation of directional transmission of $T_{12} \neq T_{21}=0$ for $\phi =\beta =\pi /2$, Fig. 6(a) shows the group delay $\tau _{12}$ as a function of the coupling strength $G_{3}$ at the resonance, where $\tau _{12}=0$ with $G_{3}=2.65\kappa$. When $G_{3}\;<\;2.65\kappa$ ($>\;2.65\kappa$), $\tau _{12}\;<\;0$ ($>\;0$) corresponding to the fast (slow) light. That is to say, the light group velocity of the directional transmission can be modulated and switched between slow and fast light effects by adjusting $G_{3}$. Thus, both the amplitude and direction of the directional amplification can be easily controlled, involving the slow/fast light effect in this system. In the situation of bidirectional amplification for $\phi =\pi$ and $\beta =0$, the system may show asymmetric light propagations of different group velocities. Then, near the resonance (in the vicinity of $\delta =-1.1\kappa$), the group velocity $a_{1}\rightarrow a_{2}$ is slowed down, whereas the transmission $a_{2}\rightarrow a_{1}$ shows the fast light. At the symmetric position around $\delta =1.1\kappa$, the transmissions along two directions have inverse fast/slow light behaviors. Generally, slow light can be observed along with the standard OMIT, owing to the anomalous dispersion. Essentially, the fast and slow light effects are physically governed by the anti-rotating-wave and rotating-wave interaction terms [56,57], respectively. There are both anti-rotating-wave (two-mode-squeezing interaction induced by the blue-sideband cavity $a_3$) and rotating-wave (beam-splitter interactions induced by the red-sideband cavities $a_1$ and $a_2$) terms, which may have competition with each other. Therefore, in such a system, we can observe fast/slow light effect and the switch between them. The group velocity difference between two transmissions may imply opportunities for precise measurement.

 

Fig. 6. (a) Group delay $\tau _{12}$ versus coupling strength $G_{3}$ for $\phi =\pi /2$, $\beta =\pi /2$ and $\delta =0$. (b) Group delays $\tau _{12}$ and $\tau _{21}$ versus detuning $\delta$ for $\phi =\pi$ and $\beta =0$. $\tau _{12(21)}=0$ is the transition point of slow and fast lights. Other parameters are the same as those in Fig. 3.

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4. Conclusions

In summary, we proposed a dual-gate optical transistor in a multimode optomechanical system, composed of three microwave/optical cavities without direct interaction and an intermediate driven mechanical resonator $b$, where cavities $a_{1}$ and $a_{2}$ are driven on red sidebands and cavity $a_{3}$ on blue sideband. Thus, we fabricate a closed-loop mode structure sensitive to the phase. We can adjust the amplitude of the unidirectional amplification by modifying the amplitude of driving fields on $a_{3}$ and $b$, which act as two basic poles/gates of the optical transistor, while the direction of the unidirectional transmission can be turned by phase modulation. The narrower windows induced by the blue-sideband field can be observed in our system. In addition, this transistor has the ability to switch transmission between slow and fast lights and then shows nonreciprocal group velocities. Such an optical transistor amplifier with many prominent functions may provide potential applications for quantum optical devices and has extraordinary promising prospects.