1. Introduction

Optomechanical systems provide a powerful platform for studying quantum mechanics because they may exhibit quantum effects on the macroscopic scale [1,2]. In optomechanics, mechanical motions can interact with optical modes in a large range of frequencies via radiation pressure. Such interactions, which are intrinsically nonlinear but can be linearized and enhanced via strong optical drivings, lead to a host of important applications such as ground-state cooling of mechanical resonators [3,4], precise sensing [1,5–8], and entanglement generation [9–11]. On one hand, one can manipulate mechanical motions and prepare quantum states for mechanical modes via optomechanical interactions [12–15]. On the other hand, the mechanical motion can in turn affect the optical properties and thus results in some interesting interference effects. For example, optomechanically induced transparency (OMIT) [16–20], which is the optomechanical analogue of electromagnetically induced transparency (EIT) [21,22], arises from the destructive interference between the probe signal and anti-Stokes field. Compared with simple two-mode cases, multimode optomechanical systems may exhibit richer exotic behaviors due to the multiple interference paths, such as topological energy transfer [23], coherent perfect absorption (CPA) and synthesis (CPS) [24–26], ultra-narrow linewidth [27], and two-mode squeezing [28,29]. The great progress in optomechanics further stimulates investigations of other solid-state quantum systems with optimechanical-like interactions, such as cavity magnonics systems [30–34].

Meanwhile, nonreciprocal optics has attracted growing interest due to its crucial applications in optical communication and quantum information processing. In particular, optomechanical systems provide a promising platform for realizing nonreciprocal devices. As is well known, the basic idea to realize nonreciprocity is to break the time-reversal symmetry of the system [35], which can be readily achieved in optomechanics without using the traditional magneto-optical effects [36,37]. In general, nonreciprocity can be realized by tuning the relative phase between different interference paths. With this method, optical isolators [38–42], circulators [38,43–45], and directional amplifiers [46–50] have been theoretically studied and experimentally realized. Moreover, nonreciprocal photon transmissions can also be realized by spinning a whispering-gallery-mode (WGM) resonator due to the Sagnac effect [51,52]. This technology was then extended to some other purposes such as nonreciprocal photon blockade [53] and phonon lasing [54]. Most recently, Metelmann and Clerk proposed a nonreciprocal transmission scheme based on reservoir engineering [55,56]. With the same method, nonreciprocal transition of nondegenerate energy levels was realized, which may serve as the kernel of a nonreciprocal single-photon transporter [57]. In this context, it is natural to consider realizing more nonreciprocal optical phenomena via the seminal methods, such as nonreciprocal interference.

In this paper, we propose a feasible scheme of nonreciprocal interference and coherent photon routing based on a three-port optomechanical system, which consists of three indirectly coupled optical modes and one intermediate mechanical mode. By regarding one cavity as a control port and perturbing it continuously by a control signal, the system may exhibit quite different interference effects if we inject another signal upon different target ports. The isolation ratio can be controlled by adjusting the phase difference between the control and target signals. With specific parameters, one can observe unidirectional or bidirectional transmission blockade over the full frequency domain. Moreover, coherent photon routing can be realized if we perturb all ports simultaneously. In this case, the synthetic signal only outputs from the desired port. Finally, we extend the results to a four-port optomechanical system to show the generalizability of our scheme.

2. Model and equations

We consider in Fig. 1 a three-port optomechanical system, which consists of three cavity modes $a_{j}$ $(j=1,2,3)$ and one mechanical mode $b$. In this model, the resonance frequencies of the three cavity modes can be vastly different since there is no direct interaction between them, i.e., the mechanical mode serves as an intermediate bridge between the cavity modes. In addition, cavity mode $a_{j}$ of resonance frequency $\omega _{j}$ is driven by a strong control field of amplitude $\varepsilon _{c,j}$, phase $\vartheta _{j}$, and frequency $\omega _{c,j}$. In the interaction picture with respect to $H_{0}=\sum _{j}\omega _{c,j}a^{\dagger }_{j}a_{j}$, the Hamiltonian of the unperturbed system can be given by $(\hbar =1)$

 

Fig. 1. Schematic illustration of the three-port optomechanical system under consideration. The cavity mode $a_{j}$ $(j=1,2,3)$ of decay rate $\kappa _{j}$ is driven by the red-sideband control field $\varepsilon _{c,j}$ and perturbed by the weak signal $\varepsilon _{s,j}$. Each cavity mode couples with the same mechanical mode $b$ of damping rate $\gamma _{m}$, with $G_{j}$ denoting the effective optomechanical coupling strength between $a_{j}$ and $b$.

Download Full Size | PDF

Now we consider two scenarios under which the three-port optomechanical system is perturbed by weak signals in different ways. In both situations, we regard cavity $a_{3}$ as a control port and always inject upon it a control signal of amplitude $\varepsilon _{s,3}$. Cavities $a_{1}$ and $a_{2}$, however, serve as two target ports of our interest. For the first scenario, one can select cavity mode $a_{1}$ or $a_{2}$ to be perturbed by a target signal of amplitude $\varepsilon _{s,1}$ or $\varepsilon _{s,2}$ to investigate the interference phenomena in different directions. In this case, the Hamiltonian of signals (in the interaction picture) is given by

Taking relevant dissipations and noises into account, one can attain the Heisenberg-Langevin equations of the mode operators according to Eqs. (1) and (4), i.e.,

In this paper, we assume that all three cavity modes are driven on the mechanical red sidebands with $\Delta _{1}'=\Delta _{2}'=\Delta _{3}'=\omega _{m}$, and our system works in the resolved sideband regime with $\omega _{m}\gg \{\kappa _{j},\,\gamma _{m},\,G_{j}\}$. This is because in this regime, photons impinging by the control fields will primarily scatter upward in frequency by absorbing a phonon from the mechanical mode. Controllable optical responses to weak signals are thus allowed due to the interference between the resulting anti-Stokes fields and the signals [1]. With these assumptions, one can perform the transformation

To study the steady-state optical response to the weak signals, we can express the mean values of the perturbation parts as $\langle \delta o\rangle =o_{-}e^{-i\xi t}+o_{+}e^{i\xi t}$ $(o=a_{j},\,b)$. Note the anti-Stokes component $o_{-}$ corresponds to the signal frequency $\omega _{s,j}$ while the Stokes component $o_{+}$ corresponds to the frequency $2\omega _{c,j}-\omega _{s,j}$ arising from a nonlinear wave mixing process.

For the first scenario, when the target signal is injected upon cavity mode $a_{1}$ $(\varepsilon _{s,2}=0)$, the anti-Stokes component of cavity mode $a_{2}$ is attained as

For the second scenario, with the assumption $\varepsilon _{s,1}=\varepsilon _{s,2}=\varepsilon _{s,3}=\varepsilon _{s}$ and $\varphi _{1}=\varphi _{2}=\varphi _{3}=0$, we attain

3. Nonreciprocal interference

In this section, we consider the first scenario with $\kappa _{1}=\kappa _{2}=\kappa _{3}=\kappa$ $(f_{1}=f_{2}=f_{3}=f=\kappa /2-i\xi )$ for simplicity. As mentioned in Sec. 2., the modulus and phase of $G_{j}$ can be readily controlled by adjusting $\varepsilon _{c,j}$ and $\vartheta _{j}$, respectively. Thus we assume the moduli as $|G_{1}|=|G_{2}|=G$, $|G_{3}|=G'$ and the phases as $\theta _{1}=\theta _{3}=0$, $\theta _{2}=\theta$. Moreover, we consider in this paper the so-called overcoupled regime [61,62], i.e., $\kappa _{\textrm {ex},j}=\kappa _{j}=\kappa$. In this way, the transmission coefficients can be simplified as

We first show in Figs. 2(a)–2(c) the impact of the third (control) port on the transmission rates $T_{j\rightarrow j'}=|t_{j\rightarrow j'}|^{2}$ with no phase difference ($\theta =\phi =0$). In the absence of cavity $a_{3}$ ($G'=0$), typical OMIT with three transparency windows can be observed, as shown in Fig. 2(a). Such a phenomenon is reciprocal ($T_{1\rightarrow 2}=T_{2\rightarrow 1}$) because the Hamiltonian is time-reversal symmetric in this case. Figure 2(b) shows that in the presence of cavity $a_{3}$ ($G'\neq 0$) and with proper parameters, the transmission rate may become larger than unity. Note that there is no signal amplification in our system, the enhanced transmission rate is attributed to the constructive interference between different transmission paths, i.e., $a_{1(2)}\rightarrow b \rightarrow a_{2(1)}$ and $a_{3}\rightarrow b \rightarrow a_{2(1)}$, rather than parametric amplification. The same is true for other situations discussed below. The interference effect can be tuned by adjusting the effective optomechanical coupling strength between cavity $a_{3}$ and the mechanical mode $b$. As shown in Fig. 2(c), one can observe constructive interference ($T_{1\rightarrow 2}=T_{2\rightarrow 1}>1$) with $0<G'<2\kappa$ but destructive interference ($T_{1\rightarrow 2}=T_{2\rightarrow 1}<1$) with $G'>2\kappa$. Physically, this is because according to Eqs. (27) and (28), $T_{1\rightarrow 2}(0)=T_{2\rightarrow 1}(0)\approx 4(\kappa ^2+\kappa G')^2/(2\kappa ^2+G'^{2})^{2}=T$ with the parameters in Figs. 2(a)–2(c). It is clear that the transmission rates on resonance can be tuned to be either larger or smaller than unity via $G'$, whereas $T\equiv 1$ in the absence of the third cavity ($G'=0$). That is to say, the control port (signal) provides the tunability by means of the interference effect. In particular, we have $T=1$ for $G'=2\kappa$, which indicates the critical value between the constructive and destructive interference regimes. Note completely destructive interference requires a quite large $G'$ with which the system may enter the bistable region (not shown here).

 

Fig. 2. Transmission rates $T_{1\rightarrow 2}$ (blue solid) and $T_{2\rightarrow 1}$ (red dashed) versus $\xi$ with $G'=0$ (a), versus $\xi$ with $G'=\kappa$ (b), and versus $G'$ with $\xi =0$ (c). Transmission rates $T_{1\rightarrow 2}$ (d) and $T_{2\rightarrow 1}$ (e) versus $\xi$ with different values of $\theta$. We assume $\theta =0$ in (a)-(c) while $G'=\kappa$ in (d) and (e). Other parameters are $\gamma _{m}=10^{-3}\kappa$, $G=\kappa$, $\eta =1$, and $\phi =0$.

Download Full Size | PDF

It is clear from Eqs. (27) and (28) that nonreciprocal interference can be achieved by adjusting the phase difference $\theta$ ($T_{2\rightarrow 1}$ is sensitive to $\theta$ while $T_{1\rightarrow 2}$ is $\theta$-independent). This can be verified by the numerical results shown in Figs. 2(d) and 2(e). We find that by changing $\theta$ from $0$ to $\pi$, $T_{1\rightarrow 2}$ in Fig. 2(d) remains invariable while $T_{2\rightarrow 1}$ in Fig. 2(e) decreases gradually. In particular, $T_{2\rightarrow 1}$ vanishes over the full frequency domain with $\theta =\pi$. We refer to such a phenomenon as frequency-independent perfect blockade (FIPB). In this case, one can achieve within a large frequency range constructive interference in one direction but destructive interference in the opposite direction. Similarly, a contrary nonreciprocal phenomenon ($T_{2\rightarrow 1}\neq 0$ and $T_{1\rightarrow 2}\equiv 0$) can be achieved by assuming $\theta _{2}=\theta _{3}=0$ and $\theta _{1}=\theta$ instead with $\theta \neq 2n\pi$ $(n=0,\,\pm 1,\,\ldots )$.

The nonreciprocal phenomenon, however, is not available by adjusting only the phase difference $\phi$ in the case of $\theta =2n\pi$. As shown in Figs. 3(a) and 3(b), $T_{1\rightarrow 2}$ and $T_{2\rightarrow 1}$ are always the same with the peak values decreasing gradually as $\phi$ increases from $0$ to $\pi$. In the case of $\phi =\pi$, both $T_{1\rightarrow 2}$ and $T_{2\rightarrow 1}$ vanish over the full frequency domain ($T_{1\rightarrow 2}=T_{2\rightarrow 1}\equiv 0$), implying that bidirectional FIPB is achieved. This result can also be understood from Eqs. (27) and (28) that $T_{1\rightarrow 2}\equiv T_{2\rightarrow 1}$ in the case of $\theta =2n\pi$ while the phase difference $\phi$ only contributes to the magnitudes of the transmission rates. In Fig. 3(c), we find that the transmission rates can be tuned continuously by adjusting $\phi$. Compared with the control scheme in Fig. 2(c), the scheme here shows the advantage in terms of keeping the system away from its bistable region, i.e., it does not require large enough effective coupling strengths. Moreover, we point out that the isolation ratio $I=T_{1\rightarrow 2}/T_{2\rightarrow 1}$ can also be tuned flexibly via $\phi$ in the nonreciprocal case, as shown in Fig. 3(d). In particular, the isolation ratio tends to infinity and negative infinity for $\phi =0$ and $\phi =\pi$, respectively, which implies FIPB in opposite directions.

 

Fig. 3. Transmission rates $T_{1\rightarrow 2}$ (a) and $T_{2\rightarrow 1}$ (b) versus $\xi$ with different values of $\phi$. (c) Transmission rates $T_{1\rightarrow 2}$ (blue solid) and $T_{2\rightarrow 1}$ (red dashed) versus $\phi$ with $\xi =0$. (d) Isolation ratio versus $\phi$ with $\xi =0$. We assume $\theta =0$ in (a)-(c) and $\theta =\pi$ in (d). Other parameters are $\gamma _{m}=10^{-3}\kappa$, $G=G'=\kappa$, and $\eta =1$.

Download Full Size | PDF

Note we have assumed $\varepsilon _{s,1(2)}=\varepsilon _{s,3}=\varepsilon _{s}$ $(\eta =1)$ in Figs. 2 and 3 for simplicity. It is worth pointing out, however, that the nonreciprocity can be enhanced accompanied with FIPB persisting in one direction by adjusting $\varepsilon _{s,3}$ and $G'$ properly. According to Eqs. (27) and (28), $G'\eta =G$ is the necessary condition of FIPB $(T_{2\rightarrow 1}\equiv 0)$ in the case of $\theta =\pi$ and $\phi =0$. With this condition, one can enhance the transmission in the opposite direction $(T_{1\rightarrow 2})$ by decreasing $G'$. Figure 4(a) shows the logarithm of the isolation ratio $log_{10}(I)$ as a function of $G'$ and $\eta$. As expected, we can find the maximum of the isolation ratio around $G'\eta =G$, which corresponds to the optimal nonreciprocal region with enhanced transmission in one direction and FIPB in the other one. As an example, we plot in Fig. 4(b) the transmission rates versus the detuning $\xi$ with $G'=0.1\kappa$ and $\eta =10$. In this case, the transmission rate in one direction ($T_{1\rightarrow 2}$) is increased by more than double while that in the opposite direction ($T_{2\rightarrow 1}$) remains vanishing over the full frequency domain. Furthermore, we point out that the nonreciprocity can be significantly enhanced by driving cavity $a_{3}$ on the blue mechanical sideband (not shown here). The enhancement arises from the fact that in the blue-sideband case, the two-mode-squeezing interactions lead to an exponential energy growth in both cavity and mechanical modes, which lies at the heart of parametric amplification. Note there is a trade-off that the blue-sideband driving may come at the cost of shrinking the stable region.

 

Fig. 4. (a) Logarithm of the isolation ratio $log_{10}(I)$ versus $G'$ and $\eta$ with $\xi =0$. (b) Transmission rates $T_{1\rightarrow 2}$ (blue solid) and $T_{2\rightarrow 1}$ (red dashed) versus $\xi$ with $G'=0.1\kappa$ and $\eta =10$. Other parameters are $\gamma _{m}=10^{-3}\kappa$, $G=\kappa$, $\theta =\pi$, and $\phi =0$.

Download Full Size | PDF

4. Coherent photon routing

Now we consider the second scenario with $\varepsilon _{s,1}=\varepsilon _{s,2}=\varepsilon _{s,3}=\varepsilon _{s}$ and $\varphi _{1}=\varphi _{2}=\varphi _{3}=0$, which implies that the three input signals can be generated by different light sources with identical intensity and locked phases. For simplicity, we still assume $\kappa _{\textrm {ex},j}=\kappa _{j}=\kappa$ in this case. Moreover, we assume $|G_{3}|=G$ and $\theta _{3}=0$ ($G_{3}$ is purely real) without loss of generality.

Instead of focusing on the cumbersome analytical expressions in Eqs. (24)–(26), we plot in Fig. 5 the numerical results to show coherent photon routing. To achieve this in a more simple way, we first assume $G_{2}=G_{3}$ ($|G_{2}|=G$ and $\theta _{2}=0$) and $\theta _{1}=\pi$. Figure 5(a) shows the normalized output energies corresponding to resonant signals $(\xi =0)$, from which we find that $S_{2}(0)$ and $S_{3}(0)$ are always identical while $S_{1}(0)$ exhibits a quite different behavior in this case. Clearly, this tunable scheme allows us to adjust the ratio of output energies between the desired port and other ports. In particular, $S_{2}(0)$ and $S_{3}(0)$ can be completely suppressed while $S_{1}(0)$ is enhanced around $|G_{1}|=0.75\kappa$. As shown in Fig. 5(b), the three signals undergo CPS and only output from cavity $a_{1}$ in this case. Moreover, we can find from Fig. 5(a) that the three output fields become equal when $|G_{1}|=2\kappa$. Interestingly, all three output fields are frequency-independent with $S_{j}\equiv 1$ ($j=1,\,2,\,3,\,4$) in this case, as shown in Fig. 5(c). Such a phenomenon is reminiscent of the frequency-independent perfect reflection (FIPR) proposed in [26,27], which is attributed to the destructive interference between different phonon excitation paths (three paths for our system, i.e., $a_{1-,2-,3-}\rightarrow b_{-}$). This can be verified by the inset in Fig. 5(c), where the maximal excitation number $|b_{-,\textrm {max}}|^{2}=\textrm {max}[|b_{-}(\xi )|^{2}]$ of the anti-Stokes component of $\delta b$ vanishes around $|G_{1}|=2\kappa$. Figure 5(d) shows that the synthetic signal can also output only from cavity $a_{2}$ if $G_{1}=G_{3}=G$ and $G_{2}=-0.75G$. In this way, the three-port optomechanical system can serve as a controllable photon router with which the synthetic signal can output from the desired target port. As mentioned above, the three cavities here can be far-off-resonant and there is no restriction on signal frequencies as well except $\xi _{s,1}=\xi _{s,2}=\xi _{s,3}=\xi$. Therefore our system shows the ability to convert and synthesize signals with different frequencies.

 

Fig. 5. Normalized output energies $S_{1}$ (blue solid), $S_{2}$ (red dashed), and $S_{3}$ (yellow dotted) versus $|G_{1}|$ with $\xi =0$ (a), versus $\xi$ with $|G_{1}|=0.75\kappa$ (b), versus $\xi$ with $|G_{1}|=2\kappa$ (c), and versus $\xi$ with $|G_{2}|=0.75\kappa$ (d). The inset in (c) depicts the maximal excitation number $|b_{-,\textrm {max}}|^{2}$ versus $|G_{1}|$. We assume $\{|G_{2}|,\,\theta _{1},\,\theta _{2}\}=\{\kappa ,\,\pi ,\,0\}$ in (a)-(c) and $\{|G_{1}|,\,\theta _{1},\,\theta _{2}\}=\{\kappa ,\,0,\,\pi \}$ in (d). Other parameters are $\gamma _{m}=10^{-3}\kappa$ and $G=\kappa$.

Download Full Size | PDF

Finally, we reveal that the above results can be extended to a more-port optomechanical system with one control port and more than two target ports. Considering for example a four-port model with an additional cavity mode $a_{4}$ (total decay rate $\kappa _{4}=\kappa _{\textrm {ex},4}+\kappa _{0,4}$ with $\kappa _{\textrm {ex},4}$ and $\kappa _{0,4}$ being the external and intrinsic losses, respectively), in which $a_{4}$ is also driven on the red mechanical sideband with the effective optomechanical coupling strength $G_{4}$. We perturb all cavity modes by weak signals of identical amplitude $\varepsilon _{s}$ and vanishing phase. Once again, for strong driving each mode operator can be expressed as the sum of its steady-state value and the perturbation part. By assuming the mean values of the perturbation parts as $\langle \delta o\rangle =o_{-}e^{-i\xi t}+o_{+}e^{i\xi t}$ $(o=a_{j},\,b)$, one can obtain

Figure 6 shows the numerical results of the normalized output energies versus $\xi$ in this case. On one hand, we reveal that the synthetic signal can be totally routed to a desired port $a_{j}$ with $\theta _{j}=\pi$ and $\theta _{j'\neq j}=0$. As shown in Fig. 6(a) for instance, the signals undergo CPS and output only from cavity $a_{1}$ when $\theta _{1}=\pi$ and $\theta _{2}=\theta _{3}=\theta _{4}=0$. On the other hand, Fig. 6(b) shows that one can also split the synthetic signal equally into two desired ports: assuming $\theta _{1}=\theta _{2}=\pi /2$ and $\theta _{3}=\theta _{4}=0$, the output spectra become asymmetric with $S_{1(3)}=S_{2(4)}=2$ and $S_{3(1)}=S_{4(2)}=0$ within certain frequency ranges. At the resonance point ($\xi =0$), however, all output energies become identical with $S_{1}(0)=S_{2}(0)=S_{3}(0)=S_{4}(0)=1$, implying no constructive or destructive interference in this case. Owing to the strong frequency dependence, the present scheme allows us to split signals of different frequencies to different target ports, which is of potential applications in optical communications.

 

Fig. 6. Normalized output energies $S_{1}$ (blue solid), $S_{2}$ (red dashed), $S_{3}$ (yellow dotted), and $S_{4}$ (black circle) versus $\xi$ with (a) $\theta _{1}=\pi$ and $\theta _{2}=0$; (b) $\theta _{1}=\theta _{2}=\pi /2$. Other parameters are $\gamma _{m}=10^{-3}\kappa$, $G=\kappa$, and $\theta _{3}=\theta _{4}=0$.

Download Full Size | PDF

5. Conclusions

In summary, we have proposed a three-port optomechanical system including three indirectly coupled cavity modes and one mechanical mode. While one cavity serves as a control port and is perturbed continuously by a control signal, the other two cavities serve as target ports of our interest. Based on the system, two scenarios have been considered under which the system is perturbed in different ways. For the first scenario, we have revealed that the transmission behaviors may be different if another signal is injected upon the two target ports respectively. Such a nonreciprocal phenomenon is attributed to different interference effects in opposite directions. In particular, the transmission can be completely suppressed over the full frequency domain, which is referred to as frequency-independent perfect blockade. For the second scenario, all cavity modes are perturbed simultaneously. We have achieved coherent photon routing, with which the synthetic signal only outputs from the desired port. The results can be extended to more-port optomechanical systems, which may provide a feasible scheme of multi-port quantum node and quantum network.