1. Introduction

As well known, analogy to photon blockade [1–8], Coulomb blockade [9], and magnon blockade [10,11], PB is a pure quantum phenomenon which refers to the anti-bunching effect of the phonon in a system. It provides the possibility to develop novel phonon devices at the single-phonon level [12]. So far, there are two kinds of PB mechanisms: one is based on the destructive interference between different excitation paths, i.e., unconventional PB, which has been proposed in nanomechanical resonator [13,14] and optomechanical systems [15–17]; another is based on the anharmonicity of the energy levels, known as conventional PB, which has been realized by coupling a mechanical oscillator to a qubit [18,19] or a two-level defect [20]. On the contrary, PIT is a completely different phenomenon from PB, that is, absorption of the first phonon is conducive to the absorption of subsequent phonons, and it is often regarded as a potential tool for generating specific multi-phonon states. Both of these two physical effects are important for developing the phonon devices.

Nonreciprocal devices have garnered significant attention due to their ability to enable unidirectional light transmission, which is crucial for various information processing applications [21–24]. It has been demonstrated in various physical systems such as atomic gases [25,26], non-Hermitian optics [27,28], the nonlinear cavity [29–32]. Different from the classical nonreciprocity, quantum nonreciprocal devices are used to realize nonreciprocal quantum effects based on quantum regime [33–35]. Very recently, an optical diode with $99.6\%$ isolation has been observed by using a spinning resonator [36]. By spinning a shaped-deformed resonator, a variety of works have been studied such as nonreciprocal entanglement [37,38], nonreciprocal phonon laser [39], nonreciprocal optical soliton [40], nonreciprocal slow light [41,42], and enhanced sensing [43,44]. Nonreciprocal photon blockade and magnon blockade have been proposed in a spinning Kerr cavity [45], a quadratic optomechanical coupling system [46] and a magnon-based hybrid system [47], respectively. Such devices can be applied to the quantum control of light in chiral quantum computing and topological quantum techniques. Xue et al. have proposed to achieve nonreciprocal photon blockade in a spinning resonator in which the Fizeau shifts can be flexibly adjusted [48]. Li et al. have investigated the unconventional photon blockade in a spinning optomechanical resonator [49]. However, the nonreciprocal phonon blockade effect induced by a rotating optomechanical resonator is rarely studied. The optomechanical system is usually composed of the optical cavity coupled with the mechanical oscillator through radiation pressure interaction, which can be used to change the properties of the optical field or control the behavior of the mechanical oscillator. It has been found that many unique and interesting phenomena can occur in the optomechanical resonator, such as enhancement of nonlinearity [50,51], phonon laser [52–54], super-resolution mass sensing [55], etc. In addition, improving the anti-noise ability of quantum systems is always a concern in development of quantum devices and quantum information processing. Some strategies have been proposed to resist optical noise or avoid backscattering, e.g., by introducing additional scattering [56], synthetic gague fields [57], phased-controlled asymmetric optomechanical system [58], and using rotating cavity [37]. These anti-noise quantum technologies can improve the stability and reliability of quantum systems. However, it should be noted that noise-resistant quantum devices do not completely eliminate noise, but minimize the effect of noise through effective control and suppression. Therefore, the exploration of noise-resistant quantum nonreciprocal devices is worthy of further study.

Inspired by these contributes, here we study nonreciprocal conventional PB in a spinning optomechanical resonator assisted by a two-level atom by considering the radiation pressure interaction between the breathing mode and optical mode. The analytical energy spectrum of the atom-phonon subsystem and the optimal condition of PB are obtained by adiabatically eliminating the optical mode with large detuning. We show that PB effects and PIT can be realized in the spinning resonator in a nonreciprocal manner. Furthermore, either the nonreciprocal 1PB or 2PB can be achieved by varying both the amplitude and frequency of the driving field applied to the breathing mode. Since the optical mode is adiabatically eliminated, the PB is insensitive to the cavity decay, which enables the scheme more robust to optical noise. There is no strict restriction on the quality of the resonator, that is, the scheme is still feasible even in a low-Q cavity. It can be used as a robust unidirectional phonon source in one-way quantum computing networks.

The remainder of the paper is organized as follows. In Sec. 2, we describe the physical model and derive the effective Hamiltonian of the system. In Sec.3, the optimal condition of PB is obtained by calculating the eigenenergy spectrum. In Sec.4, we investigate the nonreciprocal PB effect including 1PB and 2PB with the equal-time second- and third-order correlation functions by using numerical simulation of master equation. In Sec.5, we discuss the validity of present scheme and the influence of thermal phonons on PB, and conclude the results finally.

2. Model and effective Hamiltonian

The hybrid system considered here consists of a spinning toroidal resonator supporting a breathing mode, and a two-level atom with frequency $\omega _a$ is arranged near the resonator so that coupling with the optical modes, as shown in Fig. 1. In the spinning resonator, the clockwise (CW) or counterclockwise (CCW) optical mode undergoes different refractive indices. Additionally, a weak optomechanical coupling between the optical mode and the breathing mode is considered. For a resonator spinning at an angular velocity $\Omega$, the optical mode experiences a Fizeau shift $\Delta _F$, leading to the optical resonance frequency of the optomechanical resonator $\omega _c$ changed, that is, $\omega _c\rightarrow \omega _c+\Delta _F$ [59], with

 

Fig. 1. A spinning optomechanical resonator coupled to a two-level atom. Spinning the resonator results in different Fizeau drag $\Delta _F$ for engineering the clockwise or counterclockwise optical mode undergoing different refractive indices with an angular velocity $\Omega$. (a) denotes driving the resonator from left side ($\Delta _F>0$), and (b) from right side ($\Delta _F<0$).

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In a rotating frame at the driving laser frequency $\omega _{L}$, the system Hamiltonian can be described as ($\hbar =1$)

3. Optimal condition of 1PB

In the subspace spanned by base vectors $\{|m,g\rangle,|m-1,e\rangle \}$, where $|m\rangle$ $(m=1,2,3,\ldots )$ represents Fock state of the phonon and $|g\rangle$ ($|e\rangle$) the ground (excited) state of the two-level atom, the effective Hamiltonian without involvement of the weak driving term can be written as a matrix form

The energy eigenvalues are calculated as

In order to obtain the optimal parametric condition of 1PB, we truncate the largest phonon excitation number to $m=2$, as shown in the Fig. 2. Assuming that the system is initially in its ground state $|0,g\rangle$, we only need to consider the transitions involving the states $|0,g\rangle$, $|1,\pm \rangle$ and $|2,\pm \rangle$ due to the weak driving on mechanical mode. If the driving light resonantly excites the transition $|0,g\rangle \leftrightarrow |1,\pm \rangle$, the condition $\omega _p=E_{1,\pm }$ should be satisfied. Then we get the optimal condition for 1PB as

 

Fig. 2. The energy spectrum of the phonon-atom subsystem in the few-phonon subspace when the angular velocity $\Omega =0$. $|m,g(e)\rangle$ represents the state with $m$ phonons and the atom in ground (excited) state. The green and purple arrows represent the frequency of driving laser.

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With this optimal condition in Eq. (10), the driving field with frequency $\omega _p$ will excite the system to a single-phonon state $|1,\pm \rangle$, but cannot further excite the system to higher phonon state due to the large detuning. In other words, once a phonon is excited in the system, it will prevent the second phonon with the same frequency from entering the hybrid system. Therefore, the strong 1PB can be implemented. Moreover, the optimal condition in Eq. (10) is dependent on Fizeau shift, which indicates the spinning angular velocity $\Omega$ can be used for tuning the PB effect.

4. Nonreciprocal PB effects

To investigate the effect of thermal noise on the quantum statistics of the phononic field, the quantum master equation of the system reads [62,63]

In general, the value of $g^{(2)}{(0)}$ characterizes the probability for detecting two phonons simultaneously. The case of $g^{(2)}{(0)}>1$ implies a super-Poissonian statistics of the phonons, which is also referred to as bunching effect. The case of $g^{(2)}{(0)}<1$ implies the sub-Poissonian statistics of the phonons, which is referred to as phonon antibunching effect, i.e., detecting the phonons one by one. Especially, $g^{(2)}{(0)}=0$ denotes a perfect PB effect.

4.1 Nonreciprocal 1PB effect

We firstly study the 1PB effect for a stationary resonator (i.e., $\Omega =0$) by numerically calculating the equal-time second-order correlation function and average phonon number of the mechanical mode. Firstly, the logarithmic second-order correlation function $\lg [g^{(2)}(0)]$ and the average phonon number $N_{\rm b}$ are plotted as the functions of ($J/\kappa$, $\Delta /\kappa$) in Fig. 3(a) and 3(c), and as the functions of ($G/\kappa$, $\Delta /\kappa$) in Fig. 3(b) and 3(d), respectively, for absolute temperature $T$=0K. Here, the red contour delineates the case of second-order correlation function $g^{(2)}(0)=1$, which is the boundary between the super-Poissonian light ($\lg g^{(2)}(0)>0$) and sub-Poissonian light ($\lg g^{(2)}(0)<0$). The results of numerical simulation show that when the logarithmic second-order correlation function is less than 0 in Fig. 3(a) and 3(b), the average phonon number has the maximum value in Fig. 3(c) and 3(d), the parameters fully conform to the optimal condition in Eq. (10). However, there is no PB effect for $J=0$ or $G=0$ in that the anharmonicity of eigenenergy spectrum disappears, which can be seen from Eqs. (7)–(8).

 

Fig. 3. Logarithmic second-order correlation function $\lg {g^{(2)}(0)}$ in (a) and the mean phonon number $N_{b}$ in (c) as functions of $J/\kappa$ and $\Delta /\kappa$ for $G/\kappa =10$. Logarithmic second-order correlation function $\lg {g^{(2)}(0)}$ in (b) and mean phonon number $N_{b}$ in (d) as functions of $G/\kappa$ and $\Delta /\kappa$ for $J/\kappa =10$. The white dotted line denotes the optimal conditions of the conventional PB in Eq. (10). The red contour denotes the case of second-order correlation function $g^{(2)}(0)=1$. The other parameters are taken as $\Omega =0$, $E/\kappa =0.01$, $\delta /\kappa =50$, $n_{th}=0$, $\gamma /\kappa =0.03$.

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Next, we discuss nonreciprocal PB effect for a spinning optomechanical resonator by taking the experimentally feasible parameters [64]: $Q=5\times 10^{9}$, $\lambda =1550nm$, $r=0.3mm$, $n=1.4$. Moreover, spinning objects have achieved significantly higher velocities, reaching the GHz range [65,66]; such systems also can be used to research the nonreciprocal PB through quadratic optomechanical coupling [67,68] and Kerr-like optomechanical interactions [69]. In order to observe the nonreciprocal PB effect more clearly, we assume that 1PB and two-phonon tunneling occur when the driving laser input from the left and right side, respectively. Considering this limitation and the eigenenergy spectrum in Eq. (8), an optimal Fizeau shift $|\Delta _F/\kappa |=3.95043$ can be calculated by the method in Ref. [48], and the corresponding detuning $\Delta /\kappa =3.70711$ can be derived. In Fig. 4(a), the second-order correlation function $g^{(2)}{(0)}$ is plotted versus the detuning $\Delta /\kappa$ for $\Delta _F>0$ and $\Delta _F<0$, respectively. It has been shown that by taking $\Delta /\kappa =3.70711$, a dip “a” with a negative value appears when the driving laser is input from the left side (the blue solid curve), implying that the 1PB effect can be observed. While a peak “b” with a positive value appears when the driving laser is input from the right side (the red dashed curve), i.e., the PIT occurs. This is agree with our expectation. This quantum nonreciprocity can be found with a difference of up to 5 orders of magnitude between two second-order correlation functions in opposite directions.

 

Fig. 4. (a) Second-order correlation function $g^{(2)}{(0)}$ versus the detuning $\Delta /\kappa$ for different driving directions of the cavity. The nonreciprocal 1PB occurs in solved detuning region at $\Delta /\kappa =3.70711$ and $\Delta _{F}/\kappa =\pm 3.95043$. (b) The energy levels when driving the system from left and right side is nonequidistant for $\Delta /\kappa =3.70711$. Other parameters are the same as those in Fig. 3.

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Interestingly, the enhancement of PB effect can be observed for $\Delta _F<0$ in Fig. 4(a) by comparing to the cases of $\Delta _F>0$ and $\Delta _F=0$. This can be explained by the fact that for $J=G$, the excitation $\vert 1,+\rangle _{L} \rightarrow |2,+\rangle _{L}$ is far from resonance since the detuning $\dfrac {(2-\sqrt {2})JG}{\delta +\Delta _{F}}\gg \{\kappa,\gamma \}$ calculated from Eq. (8). Consequently, a larger detuning is induced for $\Delta _{F}<0$, resulting in a lower occupancy of the two-phonon state $\vert 2,+\rangle _{L}$. As a result, PB effect is enhanced. This indicates that a spinning cavity may be used to enhance PB effect.

The energy eigenstates are shown in Fig. 4(b), where $|m,+\rangle _{L(R)}$ denotes the related state of $m$-phonon when the optomechanical resonator is driven from the left (right) side. We can see that when the driving frequency on the phonon is resonantly coupled to the transition $|0,g\rangle _{L} \rightarrow |1,+\rangle _{L}$, the transition $|1,+\rangle _{L} \rightarrow |2,+\rangle _{L}$ will be suppressed for the large detuning. On the other hand, there will be a two-phonon resonance $|0,g\rangle _{R} \rightarrow |2,+\rangle _{R}$ under the same laser frequency, thus a process of PIT can be induced when the resonator is driven from the right side. Therefore, the nonreciprocal conventional 1PB in such spinning device is achieved.

4.2 Nonreciprocal 2PB effect

Now we further discuss the nonreciprocity of 2PB and PIT effect, which can be intuitively understood from the anharmonicity of energy structure as shown in Fig. 5. When the cavity is driven from the left side ($\Delta _{F}>0$), it is required that the light resonants with the $|0\rangle \rightarrow |2\rangle$ transition, but the transition $|2\rangle \rightarrow |3\rangle$ is suppressed, which features the 2PB effect; in contrast, the three-phonon resonance happens for the transition $|0\rangle \rightarrow |3\rangle$ when the cavity is driven from the right side ($\Delta _{F}<0$), which leads to three-phonon tunneling. This nonreciprocity can be understood by numerically analyzing the equal-time second- and three-order correlation functions of mechanical mode $g^{(2)}(0)$ and $g^{(3)}(0)$, i.e., [70]

 

Fig. 5. The energy level spectrum of nonreciprocal 2PB. With enhanced driving strength $E/\kappa =0.2$ and the same Fizeau drag $\Delta _{F}/\kappa =\pm 3.95043$, 2PB occurs when the resonator is driven from left side ($\Delta _{F}>0$), while three-phonon induced PIT happens when the resonator is driven from right side ($\Delta _{F}<0$).

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In the meantime, the phonon-number probability ${P}(m)=\langle m|\rho |m\rangle$ can be obtained from the steady-state solution $\rho$ of the master equation in Eq. (11). A relative deviation of a given phonon-number distribution from the corresponding Poissonian distribution $\mathcal {P}(m)$ is used for describing the statistical property of the phonon distribution, i.e.,

Under the limit of the condition in Eq. (13), the nonreciprocal 2PB effect can be achieved by increasing the driving strength $E$ and tuning driving frequency $\omega _p$ of the mechanical mode. The correlation function is depicted versus the detuning $\Delta /\kappa$ for $E/\kappa =0.2$ and $\Delta _{F}/\kappa =\pm 3.95043$ in the Fig. 6(a), where the blue curves represent $g^{(n)}_{L}(0)$ under the left driving ($\Delta _{F}>0$), and the red curves represent $g^{(n)}_{R}(0)$ under the right driving ($\Delta _{F}<0$) $(n=2,3)$. It can be clearly observed from the enlarged picture 6(b) that 2PB occurs in the area of light blue strip around detuning $\Delta /\kappa =3.25$ for the left driving, that is, the correlation functions $g^{(2)}(0)$ and $g^{(3)}(0)$ satisfy the criteria given in Eq. (13). Besides, the nonreciprocal 2PB can also be confirmed by comparing the phonon-number distribution ${P}(n)$ with the Poissonian distribution $\mathcal {P}(n)$. The Fig. 6(c) shows that two-phonon probability ${P}(2)$ is significantly enhanced while other-phonon probabilities are suppressed for left driving ($\Delta _{F}>0$), leading to 2PB. In contrast, the Fig. 6(d) shows that there is no 2PB occurs for the right driving ($\Delta _{F}<0$) but a three-phonon resonance, resulting in the three-phonon tunneling.

 

Fig. 6. (a) Second-order correlation function $g_{L,R}^{(2)}(0)$ (solid curves) and three-order correlation function $g_{L,R}^{(3)}(0)$ (dashed curves) versus the frequency detuning for different driving directions, and the nonreciprocal 2PB can be observed clearly by the enlarged figure in (b). Relative deviation of the phonon population from the Poisson distribution is shown in (c) when the nonreciprocal 2PB occurs and in (d) when the PIT occurs.

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Here, we should note that the energy level separation between $|1,-\rangle$ and $|1,+\rangle$ and that between $|2,-\rangle$ and $|2,+\rangle$ should be large enough to ensure the level anharmonicity. Here, the driving frequency of cavity can be adjusted to enhance cavity detuning and increase the energy level spacing. Thus ensures the transition $|1,+\rangle \rightarrow |2,+\rangle$ cannot be influenced by the other transitions $|1,+\rangle \rightarrow |2,-\rangle$, and $|1,-\rangle \rightarrow |2,\pm \rangle$.

5. Discussions and conclusions

To verify the validity of the theoretical scheme, the equal-time second-order correlation function $g^{(2)}(0)$ is simulated by using the linearized Hamiltonian $H_2$ and the effective Hamiltonian $H_{\rm {eff}}$, respectively. Considering the driving field with frequency $\omega _p$ of the mechanical mode, the Hamiltonian $H_2$ can be rewritten in a rotating frame defined by $U_{1}=\exp {[-i\omega _pt(a^{\dagger }a+b^{\dagger }b+\sigma _{+}\sigma _{-})]}$ as

 

Fig. 7. The numerical results of second-order correlation function $g^{(2)}(0)$ simulated by the linearized Hamiltonian $H_l$ and the effective Hamiltonian $H_{\rm {eff}}$ coincide well with each other. The parameters are taken as $\tilde {\Delta }_{c}/\kappa =50$, $J/\kappa =G/\kappa =10$, $E/\kappa =0.01$, $\gamma /\kappa$=0.03.

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Additionally, the effect of thermal phonons on PB is considered. The $g^{(2)}(0)$ is plotted as a function of $\Delta /\kappa$ for different thermal phonon numbers in Fig. 8. It can be seen that the thermal phonons has an adverse effect on PB with the temperature increasing. Therefore, in order to observe the PB effect, the influence of thermal phonons must be overcome, for instance, by lowering the ambient temperature or cooling the mechanical mode to its ground state.

 

Fig. 8. The second-order correlation function $g^{(2)}(0)$ as a function of the detuning $\Delta /\kappa$ for different thermal phonon numbers $n_{th}$. Other parameters are the same as those in Fig. 3.

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Finally, the current scheme of non-reciprocal PB is implemented by adjusting the frequency and amplitude of the mechanical driving field. The mechanical driving for a ring resonator can be realized by leveraging the experimental technique outlined in the Ref. [71]. In this study, researchers designed and fabricated a suspended ring resonator using piezoelectric material aluminum nitride (AlN), which simultaneously functions as an optical cavity for sensitive displacement readout. The mechanical mode can be driven by electrical signal.

In conclusion, we have presented a scheme to realize the nonreciprocal conventional PB in a spinning optomechanical resonator assisted by a two-level atom. In this scheme, by adjusting both the amplitude and frequency of the driving field applied to the mechanical mode, the 1PB and 2PB can be implemented in the nonreciprocal way. The validity of the scheme is confirmed by comparison between the results of numerical simulation governed, respectively, by the linearized and effective Hamiltonian. Since the optical mode is adiabatically eliminated, the PB effect is insensitive to the cavity decay, which makes the scheme more robust to optical noise, i.e., it is feasible even in a low-Q cavity. The scheme provides a flexible way for implementing nonreciprocal PB effects, and can be used as an unidirectional phonon source or memory in a chiral quantum network.

Appendix A: Derivation of the effective Hamiltonian

For a hybrid system consisting of a two-level atom and the mechanical mode both coupled to a cavity, the total Hamiltonian after linearization of the system can be written as

(17)$$\begin{aligned} H_{2} & =H_{0}+H_{I},\cr\cr H_{0} & =(\Delta_c+\Delta_F)a^{\dagger}a+\Delta_a\sigma_{+}\sigma_{-}+\omega_mb^{\dagger}b,\cr\cr H_{I} & =J(a^{\dagger}\sigma_{-}+a\sigma_{+})+G(a^{\dagger}b+ab^{\dagger}). \end{aligned}$$

In the case of $|\delta _a|\gg J$ and $|\delta _b|\gg G$, with $\delta _a=\Delta _a-\Delta _c$, $\delta _b=\omega _m-\Delta _c$, the effective coupling between the two-level atom and the mechanical mode can be obtained by applying the Fröhlich-Nakajima transformation onto the Hamiltonian $H_{2}$ in Eq. (17).

We consider a unitary transformation $V$=exp($S$), where

(18)$$S=\dfrac{J}{\delta_a+\Delta_{F}}(a^{\dagger}\sigma_{-}-a\sigma_{+})+\dfrac{G}{\delta_b+\Delta_{F}}(a^{\dagger}b-ab^{\dagger})$$

is anti-Hermitian and satisfies $H_{I}+[H_0,S]=0$. Up to the second order, the transformed Hamiltonian, $H=V^{\dagger }H_{2}V$, can be approximatively written as [60,61]

(19)$$\begin{aligned} H & \approx H_{0}+\dfrac{1}{2}[H_{I},V]\cr\cr & =\left(\Delta_a+\dfrac{J^2}{\delta_a+\Delta_{F}}\right)\sigma_{+}\sigma_{-}+\left(\omega_m+\dfrac{G^2}{\delta_b+\Delta_{F}}\right)b^{\dagger}b\cr\cr & +\left(\dfrac{1}{2}\dfrac{JG}{\delta_a+\Delta_{F}}+\dfrac{1}{2}\dfrac{JG}{\delta_b+\Delta_{F}}\right)(b^{\dagger}\sigma_{-}+b\sigma_{+}) +\left(\Delta_c+\Delta_{F}-\dfrac{G^2}{\delta_b+\Delta_{F}}+\dfrac{J^2}{\delta_a+\Delta_{F}}\sigma{z}\right)a^{\dagger}a. \end{aligned}$$

In this expression the cavity has been decoupled from the atom-phonon system. Therefore, it is reasonable to assume that the cavity mode with large detuning always remains in the ground state, i.e., $\langle a^{\dagger }a\rangle \approx 0$. Under this approximation of $\langle a^{\dagger }a\rangle \approx 0$, the effective Hamiltonian in Eq. (19) can be reduced to Eq. (5) in the main text.

Appendix B: Effect of cavity driving on average photon number

In the above Eq. (19), since the cavity mode is decoupled from the system, it cannot influence the evolution of phonon-atom system. To demonstrate the validity of approximation $\langle a^{\dagger }a\rangle \approx 0$, we investigate the impact of optical driving field on the mean photon number.

By the standard linearization procedure with Hamiltonian in Eq. (2), the Heisenberg-Langevin equations for the average amplitudes of the system are given by

(20)$$\begin{aligned} \dot{\bar{a}} & ={-}i(\Delta_{c}+\Delta_{F})\bar{a}-iJ\bar{\sigma}_{-}-ig\bar{a}(\bar{b}^{{\ast}}+\bar{b})-i\varepsilon-\kappa\bar{a}/2,\cr\cr \dot{\bar{b}} & ={-}i\omega_{m}\bar{b}-ig\vert{\bar{a}}\vert^2-\gamma \bar{b}/2,\cr\cr \dot{\bar{\sigma}}_{-} & ={-}i\Delta_a\bar{\sigma}_{-}-iJ\bar{\sigma}_{z}\bar{a}, \end{aligned}$$

where we have taken $\Delta _a=\omega _m$ so that $\delta _a=\delta _b=\delta$. Letting $\bar {\sigma }_{z}=-1$ (the atom in its ground state) for simplicity, and optomechanical coupling $g\ll \kappa$, the average amplitude of cavity mode can be calculated as

(21)$$\bar{a}=\dfrac{i\varepsilon}{-i(\Delta_{c}+\Delta_{F})+iJ^2/\Delta_a-\kappa/2}.$$

The mean photon number $\langle a^{\dagger }a\rangle$ can be depicted in Fig. 9 using Eq. (21), where it tends to approach zero for large detuning. The approximation $\langle a^{\dagger }a\rangle \approx 0$ is valid. Nevertheless, it should be noted that in Fig. 9, the average photon number exhibits a monotonic increase as the drive amplitude of the cavity mode increases. Therefore, the drive amplitude of the cavity mode should not be too large in order to maintain the vacuum state of the cavity.

 

Fig. 9. The driving amplitude of the cavity mode $\varepsilon /\kappa$ as a function of cavity photon number $\langle a^{\dagger }a\rangle$ for different detuning frequency $\Delta _{c}$. The parameters are taken as $J/\kappa =5$, $\Delta _a/\kappa =\omega _{m}/\kappa =10$, $\Delta _{F}/\kappa =0$, $\gamma /\kappa =0.03$.

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Funding

National Natural Science Foundation of China (12264051).

Disclosures

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

Data availability

No data were generated or analyzed in the presented research.

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