1. Introduction

Methods to create and manipulate single-photon sources [1] have attracted great interest over the past few decades, for their importance in most quantum optical techniques. In the field of modern quantum optics, a high efficiency and low error single-photon sources delivering light pulses in well-demarcated polarization and spatiotemporal modes and containing exactly one photon [2,3]. The photon blockade (PB) effect [4–10] is a quantum antibunching effect that corresponds to a sub-Poissonian distribution, as one of the effective ways for the generation of a single-photon. It plays a key role in quantum key distribution [11–14] and quantum repeater [15,16]. So far, there are two general ways to achieve strong PB effects depending on the underlying physical mechanism, namely, the conventional photon blockade (CPB) effect [17–21] and the unconventional photon blockade (UPB) effect [22–28]. Physically, the CPB effect relies on the anharmonicity of the eigenenergy spectrum coming from kinds of nonlinearities, while the UPB effect is induced by the destructive quantum interference among different transition paths in the system [29].

By far, the CPB has been extensively researched in cavity quantum electrodynamical or circuit quantum electrodynamical systems [30,31], which depicts the interaction between optical field and emitters. As one of the conditions capable of achieving CPB, significant nonlinear optical interactions have been realized via graphene plasmons [32] and ultrastrong Kerr nonlinearities has also been found in self-similar nanocavity with ultrasmall mode volume [33]. Moreover, exciton-polaritons can exhibit single-photon nonlinearity undergoing bosonic stimulation which have managed to trigger at the single-photon level [34]. In addition, a larger anharmonic eigenenergy spectrum can be constructed via the dispersive coupling between the optical field and the two-level qubit to realize the CPB effect [35,36]. It has also been predicted in various optical systems with Kerr-type nonlinear dielectrics and optomechanical devices [18]. However, the perfect CPB effect can only appear in optomechanical devices with strong single-photon nonlinearity according to the CPB physical mechanism, which is a major obstacle in practice [37].

In order to overcome this obstacle, an additional component is usually required to be coupled to the original system to construct two or more quantum transition paths between the one-photon state and the two-photon state to achieve the UPB effect even for arbitrarily weak nonlinearities, such as atom [38,39], auxiliary cavity [40–42], parametric amplifier [43,44] and so on. Due to the different quantum transition pathways, the destructive quantum interference between the transition paths can be found in a specific parameter regime, which leads to the probability of detecting two photons simultaneously being almost zero, so-called UPB. By contrast to CPB, UPB usually does not require a strong nonlinearity to achieve the photon antibunching and optimally sub-Poissonian in the second order statistics $g^{(2)}(0)\approx 0$ by artificially selecting the best interference conditions. According to authoritative report [45], the classical light generated via UPB sometimes produces super-Poisson distribution in the high-order statistics, that is, $g^{(\mu )}(0)>1$ ($\mu =3,4,\ldots$), which means UPB may not a perfect single photon source even if $g^{(2)}(0)\approx 0$. So the term UPB is fundamentally different from CPB, as evidenced by their physical mechanisms and the nature of light. Moreover, UPB has experimental advantages and successfully achieved experimentally via two independent teams, in orthogonally polarized coupled optical and superconducting resonators [8,10].

Reciprocity is critical for the function and analysis of optical systems [46,47], but breaking it can also be beneficial for some practical situations. For example, nonreciprocal devices can protect laser sources from noise (isolators) [48–50] and allow the signal to be transmitted in a single rotation between its ports (circulators) [51]. Therefore, how to achieve a strong nonreciprocal PB effect is beneficial for the study of signal processing and invisible sensing [52]. Currently, nonreciprocal devices have been studied in nonlinear optics [53,54], atomic gases [55,56], and non-Hermitian optics [57–60]. Expressly, a rotating resonator with broken time-reversal symmetry has been realized in an optomechanical system [29,61–63], which provides an ideal platform for nonreciprocal PB. Similar to reciprocal PB, nonreciprocal PB can also be roughly divided into two types: conventional nonreciprocal PB and unconventional nonreciprocal PB. The former depends on the anharmonicity between the energy levels of the system, while the latter relies on the destructive quantum interference between distinct driven dissipative pathways. Indeed, conventional nonreciprocal PB was first predicted by using a single spinning Kerr-type resonator [60] and has been experimentally confirmed [64,65]. On the other hand, the unconventional nonreciprocal PB was first predicted in a spinning optomechanical system, as reported in Ref. [66]. Moreover, nonreciprocal quantum entanglement has also been predicted in a similar spinning resonator [67,68].

At the same time, the quantum non-Hermitian system with its own non-Hermitian properties has also attracted extensive research interest in the field of quantum optics [69]. It is well known that the non-Hermitian system can be broadly divided into two categories, namely, the gain system [70,71] and the nonreciprocal coupling system [72,73]. The exploration of the statistical properties of photons in the non-Hermitian system has been reported in Refs. [37,74–76]. Up to now, the PB effect in the non-Hermitian system and nonreciprocal photon antibunching are separately developed. Whether they are closely related has not been widely reported. Naturally, we expect to further explore the nonreciprocal photon statistical properties in the non-Hermitian system. Inspired by researches on photon blockade in optomechanical systems [37,66,77,78], we focus on investigating the effects of nonreciprocal PB in a coupled system with an optical harmonic cavity and a spinning optomechanical resonator, in which the hopping between photons is nonreciprocal. Combined with the Sagnac effect [61,62] originating from the spinning of an optomechanics resonator, the nonreciprocal PB can emerge even with weak single-photon nonlinearity and linear coupling. In other words, the strong PB effect can only appear by driving the ordinary cavity from one side but not from the other side.

It is worth noting that what we study in this paper is essentially different from Ref. [37]. The contrast with Ref. [37] is only somewhat similar in the model but the nature of the study is different, which focuses on the nonclassical antibunching effect and satisfies the sub-Poisson optical statistics. However, we care about a new route to achieve quantum nonreciprocal devices, which are crucial elements in topological photonics [79] and chiral quantum technologies [80], that is, nonreciprocal photon blockade effect. Compared with the recent nonreciprocal photon blockade schemes [23,29,81–85], our work has significant advantages in both weak linear and nonlinear couplings, which may provide a new way to experimentally fabricate high-purity nonreciprocal single-photon sources and the experimental restrictions might be relaxed. On the other hand, the authors in Ref. [22] showed that the photon blockade under the weak coupling regime needs a dual-driving mechanism, which is also different from our scheme.

The paper is organized as follows: In Sec. 2, we derive the effective Hamiltonian of the spinning optomechanical system with nonreciprocal coupling. In Sec. 3, we give the optimal conditions for photon blockade and discuss the perfect nonreciprocal photon blockade in detail. In Sec. 4, we discuss the experimental feasibility of our scheme. Finally, we give a brief conclusion in Sec. 5.

2. System and Hamiltonian

Recently, the nonreciprocal coupling has been realized by using synthetic gauge field [86,87], quantum impurity [88,89], irregular resonant cavity [90], imaginary gauge field [91–93], nonlinear meta-atoms [94], atomic array [95], the auxiliary nonreciprocal device [96], and photonic crystal optomechanical setup [97]. In particular, nonreciprocal coupling between optical lattices has been extensively studied in non-Hermitian topological systems [73,98–104]. Here, we focus on the controllable nonreciprocal PB in a spinning optomechanical system, where the mechanical mode $b$ is coupled to the optical mode $a_{1}$ via radiation pressure and the interaction between the optical modes is nonreciprocal ($J_1\neq J_2$), as shown in Fig. 1. The external classical laser field is coupled into and out of the ordinary optical mode $a_2$ through a tapered fiber with frequency $\omega _{l}$. The Hamiltonian can be written as ($\hbar =1$)

 

Fig. 1. Schematic diagram of the spinning optomechanical system with nonreciprocal coupling. The external driving field is coupled in and out of the original cavity $a_2$ through an optical tapered fiber. The hopping strength from $a_1$ to $a_2$ is $J_2$, and $a_2$ to $a_1$ is $J_1$, that is, the coupling strength between the original cavity mode $a_2$ and the optomechanical cavity mode $a_1$ is nonreciprocal. (a) The resonator spins at a certain angular velocity $\Omega$ and the device is driven from the left side ($\Delta _{\textrm{F}}<0$). (b) For the case of driving the device from the right side ($\Delta _{\textrm{F}}>0$).

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In the rotating frame with respect to the optical field $H_0=\omega _l(a^{\dagger}_1a_1+a^{\dagger}_2a_2)$, the full Hamiltonian of the system reads [66]

3. Nonreciprocal photon blockade

The dynamics of the non-Hermitian system can be described with the quantum master equation [105]

On the other hand, ignoring the mechanical parts of the system as shown in Eq. (4), the non-Hermitian Hamiltonian has the following form

The equal-time second-order correlation function can be given by

Before studying nonreciprocal photon blockade, in order to check the rationality of the approximation from Eq. (3) to Eq. (4), we plot the time evolution of $P_{01}$, $P_{02}$, and $g_{2}^{(2)}(0)$ in Fig. 2, where $P_{01}$, $P_{02}$, and $g_{2}^{(2)}(0)$ are numerically calculated by the quantum master equation for both full Hamiltonian $H^{'}$ and effective Hamiltonian $H_{eff}$. Obviously, as shown in Fig. 2, the simulation results corresponding to $H_{eff}$ (solid red curve) agree well with the solution in view of the full Hamiltonian $H^{'}$ (blue squares), that is, the approximation of the full Hamiltonian $H^{'}$ is reasonable. In other words, the full Hamiltonian $H^{'}$ and effective Hamiltonian $H_{eff}$ are equivalent in physics. Thus, in the following discussion, the effective Hamiltonian $H_{eff}$ will be used when we employ Eq. (5) for numerical simulation. Moreover, Fig. 2 also shows that the steady-state values of $P_{01}$, $P_{02}$, and $g_{2}^{(2)}(0)$ can be quickly reached at around $\gamma t=10$, which means that the relaxation time of the system is about 10 $\mathrm{\mu}$s if the dissipation rates $\kappa _{1}$ and $\kappa _{2}$ of optical mode are chosen to be on the order of megahertz.

 

Fig. 2. The time evolution of probabilities $P_{01}$ (a), $P_{02}$ (b), and the equal-time second-order correlation function $g_2^{(2)}(0)$ (c) with full Hamiltonian $H^{'}$ (solid red curve) and effective Hamiltonian $H_{eff}$ (blue squares), respectively. Both solid red curve and blue squares are simulated by solving Monte Carlo wave function method [106] (the mcsolve function runs 500 trajectories with the time step 0.01), where $P_{01}=|C_{01}|^{2}$ and $P_{02}=|C_{02}|^{2}$. The system parameters are set as $\kappa _1=\kappa _2=\gamma =2\pi \times 0.15$ MHz, $g=2\pi \times 0.7$ MHz, $\Delta _1=\Delta _2=\Delta =\Delta _{\textrm{opt}}^{(1)}$, $J_1=0.9\gamma$, $J_2=J_{2\textrm{opt}}^{(1)}$, $\chi =0.73\gamma$, $\omega _m=30\gamma$, $\gamma _m=10^{-6}\omega _m$, $\Delta _{\textrm{F}}=0$, $n_\mathrm {th}=0$, and $E=0.05\gamma$.

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Furthermore, in order to verify the validity of the states truncation in the two-excitation subspace, the fidelity can be defined by the sum of occupying probabilities, which reads

The state truncation is reasonable if $F\approx 1$. Thus we plot the steady-state solution of the fidelity as a function of the scaled amplitude of the driven laser field $E$ as shown in Fig. 3. We find that fidelity $F$ decreases with the increase of the amplitude of the driven laser field $E/\gamma$, and $F$ is almost equal to 1 regardless of the value of $\Delta _{\textrm{F}}$ when $E/\gamma <0.1$. This is because the probability of the system being in a high-excitation subspace (more than 2) becomes higher for a strong driven laser field, that is, the states truncation applied in the derivation of $g_{2}^{(2)}(0)$ is violated. At the same time, photon blockade cannot occur due to the increased probability of photons being in highly excited states, which means that the weak driven laser field ($E/\gamma <0.1$) is necessary.

 

Fig. 3. A plot of the fidelity $F$ given by Eq. (10) as a function of the driven laser field $E/\gamma$. The inset is the zoomed-in plot of $F$ as a function of $E/\gamma$ from 0 to 0.1. $|\Delta _{\textrm{F}}|=0.3\gamma$ is chosen. The other parameters are the same as in Fig. 2.

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After verifying the validity of the approximation of the Hamiltonian and the rationality of the states truncation, the equal-time second-order correlation function $g_{2}^{(2)}(0)$ has been shown in Fig. 4 as a function of optical detuning $\Delta /\gamma$ with different Fizeau shift $\Delta _{\textrm{F}}$. We see that the analytical results agree well with the numerical simulations in (a)-(e). As shown in Figs. 4(a) and 4(b), perfect reciprocal photon blockade occurs when Eq. (9) is satisfied regardless of the direction of the driven laser field in the case of a nonspinning resonator $(\Delta _{\textrm{F}}=0)$. Physically, the origin of strong unconventional photon blockade can be traced to destructive quantum interference between two different excitation paths, that is, $|01\rangle \xrightarrow {\sqrt {2}E}|02\rangle$ and $|01\rangle \xrightarrow {J_1}|10\rangle \xrightarrow {E}|11\rangle \xrightarrow {\sqrt {2}J_2}|02\rangle$. The contributions of $|01\rangle$ and $|11\rangle$ to $|02\rangle$ cancel each other, resulting in a low probability distribution on $|02\rangle$ when $\Delta =\Delta _{\textrm{opt}}^{(1)}$ or $\Delta =\Delta _{\textrm{opt}}^{(2)}$. Such destructive quantum interference essentially leads to the probability of simultaneous observation of two photons in $a_2$ mode being almost zero, which means the emergence of photon antibunching.

 

Fig. 4. The equal-time second-order correlation function $g_{2}^{(2)}(0)$ versus the optical detuning $\Delta /\gamma$ with different Fizeau shift $\Delta _{\textrm{F}}$. Here $J_2=J_{2\textrm{opt}}^{(1)}$ for (a) and (c), $J_2=J_{2\textrm{opt}}^{(2)}$ for (b) and (e). (d) and (f) correspond to the local magnifications of (c) and (e), respectively, which enables a clearer observation of the statistical properties of photons. In (a) and (b), $\Delta _{\textrm{F}}=0$, which leads to the reciprocal photon blockade. In (c) and (e), $|\Delta _{\textrm{F}}|=0.3\gamma$, which leads to the nonreciprocal photon blockade. Analytical and numerical simulation results represented by red (black) curve and red (black) diamonds, respectively, in (c) and (e). The other parameters are the same as in Fig. 2.

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In particular, the equal-time second-order correlation function $g_{2}^{(2)}(0)$ exhibits strong nonreciprocity in the case of the spinning case, which can be found in Figs. 4(c) and 4(e). In Fig. 4(d), more clearly, the nonreciprocal photon blockade can be generated, that is, $g_{2}^{(2)}(0)\approx 0.002$ when $\Delta _{\textrm{F}}<0$ and optical detuning $\Delta /\gamma \approx 0.7$ [corresponds to the analytically optimal optical detuning $\Delta _{\textrm{opt}}^{(1)}$ in Eq. (9)]. On the contrary, the strong photon bunching occurs for $g_{2}^{(2)}(0)\approx 4.36$ in the case of $\Delta _{\textrm{F}}>0$ when the optical detuning $\Delta /\gamma \approx 0.7$. In other words, when the spinning resonator rotates at a fixed angular velocity $\Omega$ and the device is driven from the left side ($\Delta _{\textrm{F}}<0$), photon antibunching occurs. However, photon bunching occurs when the device is driven from the right-hand side ($\Delta _{\textrm{F}}>0$) with the same parameter regime. More interestingly, there is a different statistical distribution of photons from Fig. 4(c) when $J_2=J_{2\textrm{opt}}^{(2)}$. Clearly, as shown in Figs. 4(e) and 4(f), when the system is driven from the right side, it exhibits a photon antibunching effect ($g_{2}^{(2)}(0)\approx 0.002$), whereas it is significantly suppressed, exhibiting a strong photon bunching effect ($g_{2}^{(2)}(0)\approx 36.91$) when $\Delta /\gamma \approx 1.2$ [corresponds to the analytically optimal optical detuning $\Delta _{\textrm{opt}}^{(2)}$ in Eq. (9)].

The nonreciprocal photon blockade induced by the Sagnec effect, driving in opposite directions makes $g_{2}^{(2)}(0)$ differ by three or even four orders of magnitude even if the system is under weak nonlinear and linear coupling regime, namely, $J_1<\gamma$ and $g\ll \omega _m$. Physically, due to the weak nonlinearity of the spinning system, the destructive quantum interference between different pathways is the main reason for inducing the strong photon antibunching (unconventional photon blockade) as established by our analytical calculations. Here the anharmonicity between system energy levels is ignored due to the lack of a sufficiently large nonlinearity ($g\ll \omega _{m}$). The spinning resonator leads to the opposite Fizeau light-dragging effect ($\Delta _{\textrm{F}}>0$ or $\Delta _{\textrm{F}}<0$), which causes a certain transition path to be forbidden under the appropriate linear coupling strength and optical detuning. In detail, direct excitation from state $|01\rangle$ to state $|02\rangle$ will be closed by destructive quantum interference with the indirect paths of two-photon excitations ($|11\rangle \xrightarrow {\sqrt {2}J_2}|02\rangle$) when driving the system from one side. For example, driving the spinning system from the right side result in the forbidden path of $|01\rangle$ to $|02\rangle$, that is, complete destructive quantum interference, which makes the probability of photons in the $|02\rangle$ state almost zero, so there will be photon blockade phenomenon when we select $\Delta /\gamma \approx 1.2$ and $J_2=J_{2\textrm{opt}}^{(2)}$ (please see the black curve in Fig. 4(e)). However, in the case of driving in the opposite direction, under the same parameter regime, due to the lack of destructive quantum interference [107], the probability of photons in the $|02\rangle$ state are greatly improved, resulting in the photon bunching effect (please see the red curve in Fig. 4(e)). Furthermore, the probability of photon distribution in a high-excited state is discussed in Appendix B. In essence, the main reason for the difference between Figs. 4(c) and 4(e) is the difference of completely destructive quantum interference conditions caused by different optical detuning and nonreciprocal coupling strength.

In addition, the results of the equal-time second-order correlation function $g_{2}^{(2)}(0)$ versus $\Delta /\gamma$ has shown in Fig. 5 when $\Delta _{\textrm{F}}$ is severally taken as different values. Increasing the Fizeau shift $\Delta _{\textrm{F}}$ leads to the detuning for the optimal photon blockade shifts linearly with $\Delta _{\textrm{F}}$ under both two perfect blockade conditions, which coincides with the analytical results in Eq. (9). Furthermore, different driving directions can induce a blue-shift [Figs. 5(a) and 5(c)] or a red-shift [Figs. 5(b) and 5(d)] of the optical detuning, that is, an additional degree of freedom to manipulate and regulate the nonreciprocal photon blockade can be found.

 

Fig. 5. The equal-time second-order correlation function $g_{2}^{(2)}(0)$ versus optical detuning $\Delta /\gamma$ at various Fizeau shift $\Delta _{\textrm{F}}$ upon driving the device from [(a) and (c)] the right-hand side or [(b) and (d)] the left-hand side. Here $J_2=J_{2\textrm{opt}}^{(1)}$ for (a) and (b), $J_2=J_{2\textrm{opt}}^{(2)}$ for (c) and (d). The other parameters are the same as in Fig. 2.

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So far, we have only considered the case of completely ignoring the mechanical part. Although this approximation has been proved to be reasonable in Fig. 2, the mechanical thermal noise induced by low-frequency phonon modes cannot be ignored in actual manipulation. Figure 6 displays the equal-time second-order correlation function $g_{2}^{(2)}(0)$ as a function of $\Delta /\gamma$ with different mean thermal phonon numbers $n_{\textrm{th}}$. When the detuning $\Delta =\Delta _{\textrm{opt}}^{(j)}$, the quality of the photon blockade decreases with the increase of $n_{\textrm{th}}$ but still survives as shown in Figs. 6(a) and 6(c). This is the result of the single-photon excited state being destroyed by the unwanted thermal noise [108]. More importantly, the spinning resonator induced nonreciprocal photon blockade can still exist even in the presence of thermal noise, which can be found in Figs. 6(b) and 6(d).

 

Fig. 6. The equal-time second-order correlation function $g_{2}^{(2)}(0)$ versus optical detuning $\Delta /\gamma$ at various mechanical thermal phonon number $n_{\textrm{th}}$. Here $J_2=J_{2\textrm{opt}}^{(1)}$ for (a) and (b), $J_2=J_{2\textrm{opt}}^{(2)}$ for (c) and (d). (b) and (d) correspond to the local magnifications of (a) and (c), respectively, which enables a clearer observation of the statistical properties of photons. The other parameters are the same as in Fig. 2.

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In order to further demonstrate the effect of nonreciprocal coupling on photon blockade, we show that the equal-time second-order correlation function $g_{2}^{(2)}(0)$ as a function of nonreciprocal coupling strength $J_1$ and $J_2$ in Figs. 7(a) and 7(b). These plots show that the excellent photon blockade effect appears exactly at the exceptional values, which have been predicted by our analytical calculations in Eq. (9). More notably, the optimal nonreciprocal coupling marked by the red dashed line exhibits an inversely proportional function $J_1J_2/\gamma ^2\approx 0.81$, which means that both strong photon blockade and excellent nonreciprocal photon antibunching can occur with weak linear and nonlinear coupling. Such novel phenomena, especially nonreciprocal photon blockade, break the traditional cognition of the statistical properties of photons.

 

Fig. 7. The equal-time second-order correlation function $g_{2}^{(2)}(0)$ in logarithmic scale versus the nonreciprocal coupling strength $J_1$ and $J_2$. The red dashed curve in (a) and (b) correspond to the optimal nonreciprocal coupling given by Eq. (9). Here $\Delta =\Delta _{\textrm{opt}}^{(1)}$ and $\Delta _{\textrm{F}}=-0.3\gamma$ for (a), $\Delta =\Delta _{\textrm{opt}}^{(2)}$ and $\Delta _{\textrm{F}}=0.3\gamma$ for (b). The other parameters are the same as in Fig. 2.

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4. Experimental feasibility

We now discuss the feasibility of the proposed scheme in experiment. The intensity of nonreciprocal transitions between photons is not difficult to achieve, for example, a nonreciprocal coupling in a imaginary gauge field will scale by exp($\pm h$) in the opposite transmission directions, where $h$ describes the influence of the imaginary vector potential that can be manipulated [37]. On the other hand, the nonreciprocal hopping in the electromechanical system can be indirectly induced by coupling to a charge qubit [109,110]. In addition, it can be expected to be implemented through asymmetric structural engineering. The nonreciprocal coupling $J_1$ and $J_2$ can be adjusted by manipulating auxiliary nonreciprocal transition device [86,96], imaginary gauge field, synthetic gauge field, imaginary vector potential $h$ and so on. The Fizeau shift $\Delta _{\textrm{F}}$ is able to be regulated by changing the value of angular velocity $\Omega$ and manipulating the direction of light propagation, which is independent of the nonreciprocal coupling.

With the experimentally feasible parameter values for the spinning resonator [58,111–120], that is, $\lambda =1550$ nm, $r=0.3$ mm, and $n=1.44$. In the current scheme, the dissipation rate of the two optical modes can be chosen as $\kappa _1=\kappa _{2}=\gamma =2\pi \times 0.15$ MHz [112,114,121,122], then the maximum required Fizeau shift in Fig. 5 is approximately $\Delta _{\textrm{F}}\approx 0.6$ MHz ($\Delta _{\textrm{F}}=0.6\gamma$). There is one more key parameter to implement our proposed scheme, that is, the controllable angular velocity of the spinning resonator $\Omega$. Based on Refs. [66,81,83], the value of the angular velocity $\Omega$ is probably in the order of kilohertz to megahertz, and even reaches the order of magnitude of gigahertz in some special experimental devices [62,123], which meets the requirement of our proposal. The bare optomechanical coupling strength is approximately $g/2\pi \approx 0.7$ MHz [111–113], which ensures that the nonlinear coefficient $\chi \approx 0.73\gamma$. Last but not least, the current scheme is robust to the bath temperature and still achieves a strong nonreciprocal photon blockade effect even for the thermal mean phonon number $n_{\textrm{th}}=1$ (see Fig. 6).

5. Conclusions

In conclusion, the perfect nonreciprocal photon blockade has been studied in a spinning optomechanical system, where the hopping between two optical modes is nonreciprocal. Through analytical calculations and numerical simulations under the weak driving limit, we obtain the equal-time second-order correlation functions describing the photon blockade effect, respectively, and they are in good agreement with each other. Due to the ideal destructive quantum interference between two different excitation paths and the Fizeau Sagnac effects, the photon blockade can be generated nonreciprocally in our scheme, that is, the photon blockade can occur when the hybrid system is driven from an arbitrary direction (left side or right side) but not from the other under appropriate parameters regime, even under weak nonlinear and linear interactions. Meanwhile, the dynamical evolutions of probabilities $P_{01}$ and $P_{02}$ are respectively plotted by solving the quantum master equation, which further verifies the suppression of the high-excited state ($|02\rangle$) via destructive quantum interference between different transition paths when the external driving field is in different directions. Furthermore, we show that the nonreciprocal photon blockade still survives when thermal phonon noises are considered. Benefiting from the rapid development of micro-machining and manufacturing technology, our proposal may have potential applications in achieving, e.g., generating the few-photon quantum states, back-action–free quantum sensing, high-quality nonreciprocal single-photon source, and may become an important one in the application of chiral quantum information processing.

Appendix A: Photon statistics in optomechanical cavity

In this appendix, we derive the optimal analytical conditions for nonreciprocal photon blockade according to the method in Refs. [124–126]. Based on the quantum-trajectory method, the non-Hermitian Hamiltonian can be written by adding phenomenologically the imaginary decays into the original Hamiltonian, which can be given by

(12)$$\begin{aligned} H_{\mathrm{NM}}&=(\Delta_{1}^{'}-i\frac{\kappa_{1}}{2})a_{1}^{\dagger}a_{1}+(\Delta_{2}-i\frac{\kappa_{2}}{2})a_{2}^{\dagger}a_{2}-\chi(a_{1}^{\dagger}a_{1})^{2}\\ &\quad+J_1a_{1}^{\dagger}a_{2}+J_2a_{1}a_{2}^{\dagger}+E(a_{2}^{\dagger}+a_{2}), \end{aligned}$$

where $\Delta _{1}^{'}=\Delta _{1}+\Delta _{\textrm{F}}$ and $\kappa _j$ denotes the decay rate of the corresponding optical mode. In order to obtain the analytical expression of photon statistics, the standard Schrödinger equation can be introduced as

(13)$$\frac{i\partial|\psi(t)\rangle}{\partial t}=H_{\textrm{NM}}|\psi(t)\rangle,$$

where $|\psi (t)\rangle$ is the wave function of the system. Because of the non-conservative excitation numbers, we cannot establish a closed subspace composed of Fock-state basis $|nm\rangle =|n\rangle \otimes |m\rangle$ [81], with the number $n$ ($m$) representing the photons being in optical mode $a_1$ ($a_2$). Accordingly, the Hamiltonian matrix cannot be diagonalized exactly in this subspace. In order to understand the origin of photon blockade, the wave function of the system at any time can be written as

(14)$$|\psi(t)\rangle=\sum_{n,m}^{n+m}C_{nm}(t)|nm\rangle,$$

where $C_{nm}(t)$ denotes the probability amplitude of the system being in the state $|nm\rangle$. In the case of $E/\kappa _{j}\ll 1$, the weak driving term in Eq. (1) can be regarded as a perturbation, i.e., the evolution space can be truncated to a two-excitation subspace with $n+m\leqslant 2$ for the photon statistical. Therefore the general state of the system can be written as

(15)$$\begin{aligned} |\psi(t)\rangle&=C_{00}(t)|00\rangle+C_{01}(t)|01\rangle+C_{10}(t)|10\rangle\\ &\quad+C_{11}(t)|11\rangle+C_{02}(t)|02\rangle+C_{20}(t)|20\rangle, \end{aligned}$$

with initial probability amplitudes satisfy $\left \{C_{11}(0),C_{20}(0),C_{02}(0)\right \} \ll \left \{C_{10}(0),C_{01}(0)\right \}\ll C_{00}(0)\approx 1$, that is, the probability of the system initially in low-excited states is much larger than high-excited states when $E/\kappa _{j}\ll 1$. By substituting Eq. (12) and the above states into standard Schrödinger equation, a set of linear differential equations about the probability amplitudes of the system can be given by

(16)$$\begin{aligned} i\frac{\partial C_{10}}{\partial t}&=(\Delta_{1}+\Delta_{\textrm{F}}-i\frac{\kappa_{1}}{2}-\chi)C_{10}+J_1C_{01}+EC_{11},\\ i\frac{\partial C_{01}}{\partial t}&=(\Delta_{2}-i\frac{\kappa_{1}}{2})C_{01}+J_2C_{10}+EC_{00}+\sqrt{2}EC_{02},\\ i\frac{\partial C_{20}}{\partial t}&=[2(\Delta_{1}+\Delta_{\textrm{F}})-i\kappa_1-4\chi)]C_{20}+\sqrt{2}J_1C_{11},\\ i\frac{\partial C_{11}}{\partial t}&=(\Delta_{1}+\Delta_{\textrm{F}}-i\frac{\kappa_1}{2}+\Delta_{2}-i\frac{\kappa_{2}}{2}-\chi)C_{11}\\ &\quad+\sqrt{2}(J_1C_{02}+J_2C_{20})+EC_{10},\\ i\frac{\partial C_{02}}{\partial t}&=(2\Delta_{2}-i\kappa_2)C_{02}+\sqrt{2}J_2C_{11}+\sqrt{2}EC_{01}. \end{aligned}$$

Under the conditions of $\Delta _{1}=\Delta _{2}=\Delta$, $\kappa _{1}=\kappa _2=\gamma$, and drop those higher-order terms, the approximate steady-state solution of the one- and two-particle states can be calculated as

(17)$$\begin{aligned} C_{10}&=\frac{EJ_1}{A_1},\\ C_{01}&=\frac{E(B-2\chi)}{2A_1},\\ C_{20}&=8\sqrt{2}A_9J_1^2(2\Delta+\Delta_{\textrm{F}}-i\gamma-\chi),\\ C_{11}&=8J_1A_5A_6A_9,\\ C_{02}&=2\sqrt{2}(A_7+A_8)A_9, \end{aligned}$$

with

(18)$$\begin{aligned} A_1&=J_1J_2-(\Delta-i\frac{\gamma}{2})(\Delta+\Delta_{\textrm{F}}-i\gamma-2\chi),\\ A_2&=-4J_1J_2+(2\Delta-i\gamma)(B-2\chi),\\ A_3&=\chi[8J_1J_2-(2\Delta-i\gamma)(10\Delta+6\Delta_{\textrm{F}}-5i\gamma)],\\ A_4&=[{-}4J_1J_2+(2\Delta-i\gamma)B] (2\Delta+\Delta_{\textrm{F}}-i\gamma)+4\chi^2(2\Delta-i\gamma),\\ A_5&=B-4\chi,\\ A_6&=2\Delta+\Delta_{\textrm{F}}-i\gamma-\chi,\\ A_7&=-B\chi(14\Delta+8\Delta_{\textrm{F}}-7i\gamma) +2\chi^2(14\Delta+10\Delta_{\textrm{F}}-7i\gamma),\\ A_8&=(2\Delta+\Delta_{\textrm{F}}-i\gamma)B^2 -4J_1J_2\chi-8\chi^3,\\ A_9&=\frac{E^2}{A_2A_3+A_4},\\ B&=2\Delta+2\Delta_{\textrm{F}}-i\gamma. \end{aligned}$$

The final analytical representations can be obtained by the steady state wave function as

(19)$$\begin{aligned} g_{1}^{(2)}(0)&=\frac{2|C_{20}|^{2}}{(|C_{10}|^{2}+|C_{11}|^{2}+2|C_{20}|^{2})^{2}}\simeq\frac{2|C_{20}|^{2}}{|C_{10}|^{4}},\\ g_{2}^{(2)}(0)&=\frac{2|C_{02}|^{2}}{(|C_{01}|^{2}+|C_{11}|^{2}+2|C_{02}|^{2})^{2}}\simeq\frac{2|C_{02}|^{2}}{|C_{01}|^{4}}. \end{aligned}$$

It is worth noting that the analytical results of the equal-time second-order correlation function $g_{j}^{(2)}(0)$ in Eq. (19) is semiclassical descriptions, which completely ignores the effect of quantum jumps [105]. However, the numerical simulation approach based on the master equation in Eq. (5) is fully quantum, which contains the effect of quantum jumps. The differences between these two methods can be explained via the hybrid-Liouvillian formalism, which has been well discussed in Ref. [127]. Moreover, the equal-time second-order correlation function obtained by these two methods is proved to be equivalent, that is, the reported semiclassical predictions can be confirmed fully.

One can make $|C_{02}|=0$ to achieve the perfect photon blockade and the corresponding parameter relations can be given as

(20)$$\begin{aligned} \Delta^{(n)}_{\rm opt}&=\frac{1}{6}[7\chi-5\Delta_\textrm{F}+({-}1)^n A_0],\\ J^{(n)}_{2{\rm opt}}&=\frac{1}{54J_1\chi}[-\Delta_\textrm{F}^3+6\chi\Delta_\textrm{F}^{2}-3\Delta_\textrm{F}\chi^2 -10\chi^3-({-}1)^n (\Delta_\textrm{F}^2 A_0-4\Delta_\textrm{F}\chi A_0\\ &\quad+12\gamma^2 A_0+7\chi^2A_0)], \end{aligned}$$

with

(21)$$A_0=\sqrt{\Delta_{\textrm{F}}^2+3\gamma^2-4\Delta_{\textrm{F}}\chi+7\chi^2},$$

where $n=1,2$. On other hand, to obtain the perfect antibunching effect of $a_1$ photons, the parameter conditions corresponding to $|C_{20}|=0$ (perfect photon blockade) is

(22)$$\begin{aligned} \begin{cases} J_1^2(2\Delta+\Delta_{\textrm{F}}-\chi)=0,\\ J_1^2\gamma =0, \end{cases} \end{aligned}$$

which again proves the perfect photon blockade is impossible, that is, $J_1^2\gamma \neq 0$. However, it is worth noting that nonreciprocal photon blockade can still occur in the $a_1$ mode since only $g_{1}^{(2)}(0)<1$ is required. Here, we would not discuss them in detail.

Appendix B: The probability in high-excited state

In order to adequately demonstrate the probability of photons in state $|02\rangle$ with different driving directions, we plot the probability of photons distribution $P_{02}$ as a function of optical detuning $\Delta /\gamma$ with different $\Delta _{\textrm{F}}$ as shown in Fig. 8. These plots are consistent with discussions of Fig. 4. Specifically, when driven from the left side ($\Delta _{\textrm{F}}<0$) the probability of photon distribution on $|02\rangle$ state is much lower than that from the other side at $\Delta =\Delta _{\textrm{opt}}^{(1)}$ and $J_2=J_{2\textrm{opt}}^{(1)}$, which is clearly shown in Fig. 8(a). On the contrary, the probability of photons in $|02\rangle$ state driven from the left side is greater than that driven from the other side if the parameters are selected as $\Delta =\Delta _{\textrm{opt}}^{(2)}$ and $J_2=J_{2\textrm{opt}}^{(2)}$, which can be found in Fig. 8(b). It is because of the change of photons distribution probability on high-excited state ($|02\rangle$) after $\Delta _{\textrm{F}}$ changes the sign that leads to the effect of photon bunching or antibunching. The physical reason can be explained as the lack of complete destructive quantum interference between the energy levels inducing a higher probability photon distribution on $|02\rangle$ state, which is due to the change of the energy level structure of the system after the sign change of $\Delta _{\textrm{F}}$.

 

Fig. 8. The photon occupy probability $P_{02}$ as a function of optical $\Delta /\gamma$ with different $\Delta _{\textrm{F}}$. The blue dashed lines in (a) and (b) correspond to the optimal optical detuning, which defined by Eq. (9). Here $J_2=J_{2\textrm{opt}}^{(1)}$ for (a), $J_2=J_{2\textrm{opt}}^{(2)}$ for (b). For both (a) and (b) $|\Delta _{\textrm{F}}|=0.3\gamma$. The other parameters are the same as in Fig. 2.

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Funding

National Natural Science Foundation of China (12074330, 12175033).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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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