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PT-symmetric phonon laser under gain saturation effect

Yi Fei Xie, Zhen Cao, Bing He, and Qing Lin

Open Access Open Access

Abstract

As an analog of optical laser, phonon laser has potential applications in various areas. We study a type of phonon laser implemented by two coupled micro-cavities, one of which carries optical gain medium. The phonon laser operation is under a blue detuned external drive leading to dynamical instability. The saturation of the optical gain is considered, and its induced nonlinearity results in more complicated behaviors in stimulated phonon emission. To deal with such complex dynamics, we apply a composite numerical approach, in addition to a previously used dynamical approach, to the time evolution of the system. The workable phonon laser operation is found to be achievable by choosing the proper system parameters. Moreover, low threshold for the phonon laser operation is possible with the suitable coupling between the cavities and an optimum damping rate in one cavity.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Properly stimulated mechanical oscillation in certain medium makes a phonon laser. Thanks to the ultra-short wavelength of sound waves compared to the light waves of the same frequency, phonon laser is expected to find applications in highly precise sensing, imaging or switching [1–8]. Many physical systems (see, e.g. [9–33]) have been considered for the construction of phonon laser setups. These systems include heterostructures, trapped ions, quantum wells, and coupled micro-cavities. In principle any experimentally realized device that coherently magnifies phonon field (quanta of mechanical vibration) is called phonon laser, but the performance of various such devices involving nonlinear mechanisms should be understood further.

In this work we focus on a type of phonon laser realized via the process of cavity optomechanics. Early in 2010, a compound two-microcavity system was applied to implement phonon laser operation [21]. It was also experimentally demonstrated that broadened linewidth and low threshold of such phonon laser can be realized when the system is operated around the exceptional point (EP), at which the eigen-modes of the system coalesce [30]. Optical gain medium can be introduced into one of the cavities in the similar systems, forming a category of coupled cavities with the parity-time (PT) symmetry or its generalized symmetry (see, e.g. [34–40]). Low threshold for pumping a PT-symmetric phonon laser was found to be achievable with the balanced gain and loss, if such a device works at the PT-symmetry regime where the coupling between two microcavities is sufficiently strong [25]. This theoretical conclusion is based on the constant optical gain and steady state of the system. A later study [26] based on the dynamical process of the system shows that more efficient phonon laser operation can be implemented under a blue detuned pump drive, which excludes the possible existence of time-independent steady state. It was also found that type of phonon laser can be better operated in the PT-symmetry broken regime rather than in the PT-symmetry regime [26,30]. However, the requirements for having the realistic operation of the PT-symmetric phonon laser have not been completely clear yet. A main factor is that any optical gain medium carries saturation; if the light field intensity in the medium becomes high, the effective gain rate will drop according to a nonlinear law. The gain saturation thus induces another type of nonlinearity to the system, in addition the nonlinearity due to the radiation pressure on a microcavity. The extra nonlinearity can bring about considerable difference to the phonon laser operation. Then, in this situation, will the phonon laser be still be implementable under the realistic conditions?

In what follows we will study the dynamics of the coupled optomechanical system, in which the gain saturation effect plays an important role. Due to the nonlinearity induced by the gain saturation, the stimulated phonon emission totally becomes variable. We will show that, to lower the threshold for phonon lasing, the mutual coupling between the cavities should be enhanced. The relation between the threshold and the damping rate of the active cavity (containing the gain material) is complicated; it will first decrease under increased damping rate, but will increase with such damping rate if its value is beyond a certain value. Unlike a phonon laser without optical gain [30], the minimum threshold does not occur around the EP, indicating that the realistic gain with saturation can impact a phonon laser operation significantly.

The rest of the paper is organized as follows. In Sec. 2., the system model and dynamical equations are presented, with which the dynamical evolutions of the population inversion under different conditions are studied in Sec. 3. The amplification of stimulated phonon field intensity and the threshold for phonon lasing are investigated in Sec. 4. The final part is left for the concluding remarks. There are two appendices about the derivation of Eq. (7) and the description of the numerical calculations, respectively.

2. System dynamics

The concerned system consists of two micro-cavities, one passive cavity with the damping rate $\kappa$ and one active cavity with both gain rate $g$ and damping rate $\gamma$, as shown in Fig. 1. The passive cavity also carries a breathing mode of mechanical oscillation that is excited under the radiation pressure. A blue detuned field by the mechanical frequency $\omega _m$ pumps the passive cavity through fiber 1. In terms of the annihilation (creation) operator $\hat {a}_{1(2)}$ ($\hat {a}^\dagger _{1(2)}$) of the cavity field and the corresponding one $\hat {b}$ ($\hat {b}^\dagger$) of the mechanical mode, the Hamiltonian of the whole system can be constructed and consists of three parts:

$$H(t)=H_S(t)+H_{OM}+H_{SR}(t).$$
The first part includes the free oscillations with the cavity resonant frequency $\omega _{c_1}=\omega _{c_2}=\omega _c$ and the mechanical resonant frequency $\omega _m$, and the coupling of the two cavities with the coupling strength $J$ and the coupling of the passive cavity to the external drive with its amplitude $E$ and frequency $\omega _L$ ($\hbar =1$):
$$H_S(t)=\omega_c \hat{a}_1^{\dagger}\hat{a}_1+\omega_c \hat{a}_2^{\dagger}\hat{a}_2+\omega_m \hat{b}^{\dagger}\hat{b}+J(\hat{a}_1\hat{a}_2^{\dagger}+\hat{a}_1^{\dagger}\hat{a}_2)+iE(\hat{a}_1^{\dagger}e^{{-}i\omega_L t}-\hat{a}_1e^{i\omega_L t}).$$
The second is the optomechanical interaction with the single-photon coupling strength $g_m$ due to the radiation pressure:
$$H_{OM}={-}g_{m}\hat{a}_1^{\dagger}\hat{a}_1(\hat{b}+\hat{b}^{\dagger}).$$
The third is the coupling of the system to the reservoirs with the damping rates $\kappa$, $\gamma$ of the passive cavity and active cavity, respectively, as well as the mechanical damping rate $\gamma _m$ and the noise associated the gain rate $g(t)$:
$$\begin{aligned} H_{SR}(t) = &i\sqrt{2\kappa}\{\hat{a}_1^{\dagger} \hat{\xi}_p(t)-\hat{a}_1\hat{\xi}^{\dagger}_p(t)\}+i\sqrt{2\gamma_m}\{\hat{b}^{\dagger} \hat{\xi}_m(t)-\hat{b}\hat{\xi}^{\dagger}_m(t)\}\\ &+i\sqrt{2g(t)}\{\hat{a}_2^{\dagger} \hat{\xi}_a^{\dagger}(t)-\hat{a}_2\hat{\xi}_a(t)\}+i\sqrt{2\gamma}\{\hat{a}_2^{\dagger}\hat{\xi}_a(t) -\hat{a}_2\hat{\xi}_a^{\dagger}(t)\}, \end{aligned}$$
where $\langle \hat {\xi }_p(t)\hat {\xi }^{\dagger}_p(t')\rangle =\langle \hat {\xi }_a(t)\hat {\xi }^{\dagger}_a(t')\rangle =\delta (t-t')$, and $\langle \hat {\xi }^\dagger _m(t)\hat {\xi }_m(t')=n_{m}\delta (t-t')$ ($n_m$ is the thermal occupation). The system evolution is determined by the evolution operator as the time-ordered exponential [41]
$$U(t)=\mathcal{T}\exp\left\{{-}i\int^t_0d\tau H(\tau)\right\}=\mathcal{T}\exp\left\{{-}i\int^t_0d\tau H_S(\tau)+H_{OM}+H_{SR}(\tau)\right\},$$
which leads to the nonlinear Heisenberg-Langevin equations
$$\begin{aligned} \dot{\hat{a}}_1&={-}(\kappa-ig_m(\hat{b}+\hat{b}^{\dagger}))\hat{a}_1-iJ\hat{a}_2+Ee^{{-}i\Delta t}+\sqrt{2\kappa}\hat{\xi}_p,\\ \dot{\hat{a}}_2&=(g(t)-\gamma)\hat{a}_2-iJ\hat{a}_1+\sqrt{2g(t)}\hat{\xi}^{\dagger}_a,\\ \dot{\hat{b}}&={-}\gamma_m \hat{b}-i\omega_m \hat{b}+ig_m\hat{a}^{\dagger}_1\hat{a}_1+\sqrt{2\gamma_m}\hat{\xi}_m \end{aligned}$$
by using the proper Ito’s rules for the stochastic Hamiltonian in Eq. (4).

 figure: Fig. 1.

Fig. 1. The model of a coupled micro-cavities system. The side of the cavity with the damping rate $\kappa$ is elastic, and the associated mechanical oscillation $X_m(t)$ with the frequency $\omega _m$ is induced by the radiation pressure of the cavity field. The other cavity carries optical gain material, which has the initial gain rate $g_0$ and also damps at the rate $\gamma$. The gain saturation leads to a time-dependent gain rate $g(t)$. Such coupled system realizes the optical supermodes separated by the energy difference $2J$. The stimulated emission of phonons takes place when the two supermods has a population inversion, i.e., the population of the upper level is higher than that of the lower, magnifying the phonon field propagating perpendicular to the mechanical oscillation (the device is assumed to be placed in vacuum).

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The above nonlinear Heisenberg-Langevin equations are not straightforwardly solvable. Here we adopt the approach developed in [26] to deal with the quantum dynamics of the system. The variations of the method have been applied to a number of other physical systems [42–51]. By that approach, the nonlinear dynamics described by Eq. (6) can be approximated by the following equations (see Appendix A for its derivation)

$$\begin{aligned} \frac{d}{dt}\hat{o}_1=&\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}_1-\frac{1}{2}(g(t)+\kappa-\gamma)e^{2iJt}\hat{o}_2+ig_mE_1(t)e^{iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ +&(-\kappa E_1(t)+g(t)E_2(t)-\gamma E_2(t))e^{iJt}+e^{iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)+\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)+\sqrt{\gamma}\hat{\xi}_a(t)),\\ \frac{d}{dt}\hat{o}_2=&-\frac{1}{2}(g(t)+\kappa-\gamma)e^{{-}2iJt}\hat{o}_1+\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}_2+ig_mE_1(t)e^{{-}iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ &-(\kappa E_1(t)+g(t)E_2(t)-\gamma E_2(t))e^{{-}iJt}+e^{{-}iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)-\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)-\sqrt{\gamma}\hat{\xi}_a(t)),\\ \frac{d}{dt}\hat{b}=&-\gamma_m \hat{b}+ig_mE^*_1(t)e^{i\omega_mt}(\hat{o}_1e^{{-}iJt}+\hat{o}_2e^{iJt})+ig_mE_1(t)e^{i\omega_mt}(\hat{o}^{\dagger}_1e^{iJt}+\hat{o}^{\dagger}_2e^{{-}iJt})\\ +&\sqrt{2\gamma_m}e^{i\omega_mt}\hat{\xi}_m(t)+2ig_m|E_1(t)|^2e^{i\omega_mt} \end{aligned}$$
under the condition $g_m/\omega _m\ll 1$, where the optical supermodes and the effective coherent drive terms are defined as
$$\begin{aligned} \hat{o}_{1,2}=&(\hat{a}_1\pm\hat{a}_2)/\sqrt{2}\\ E_1(t)=&\frac{iE}{2\sqrt{2}}(\frac{1}{\Delta+J}e^{{-}iJt}+\frac{1}{\Delta-J}e^{iJt}-\frac{2\Delta}{\Delta^2-J^2}e^{i\Delta t})\\ E_2(t)=&\frac{iE}{\sqrt{2}}(\frac{J}{\Delta^2-J^2}e^{i\Delta t}-\frac{J}{\Delta^2-J^2}\cos(Jt)-i\frac{\Delta}{\Delta^2-J^2}\sin(J t)), \end{aligned}$$
The numerical calculations we perform are based on the above equations, together with Eq. (6), to find the real-time evolutions of the optical supermodes, as well as the corresponding population inversion and amplification of the stimulated phonon field intensity.

3. Achievable population inversion

The stimulated phonon emission exists only when there exists the population inversion

$$\Delta N(t)=\langle\hat{o}^{\dagger}_1\hat{o}_1(t) \rangle-\langle\hat{o}^{\dagger}_2\hat{o}_2(t)\rangle>0$$
between the two supermodes $\hat {o}_1$ and $\hat {o}_2$; see Fig. 1. We will find this quantity in various situations.

3.1 Difference between optical gains without and under saturation

We first take a look at how the population inversion could be if the gain rate is fixed. The numerical calculations are the same as those in [26], by treating Eq. (7) (the more complete version is Eq. (A5) in Appendix A) as a system of linear differential equations with the constant $g$. In Fig. 2(a1), the examples are obtained with $g/\kappa =1, 3$ and drive intensity $E/\kappa =5000$. When the gain rate $g/\kappa =1$ is the same as the damping rate $\gamma /\kappa =1$ of the active cavity, the population inversion will become stable oscillation (blue line), but the population inversion is too low to have stimulated phonon emission. If the gain rate is increased to $g/\kappa =3$, larger than the damping rate, the population inversion will quickly increase to high value (red line), being beneficial to phonon laser operation. If the gain rate is fixed as $g/\kappa =2$, on the other hand, the population inversion evolves under different drive intensity $E/\kappa =5000, 10^4$ as in Fig. 2(b1). Obviously, a higher drive intensity is better for the stimulated phonon emission. A high population inversion (e.g., $10^{16}$ in Fig. 2(a1)) can be achieved easily with the relatively high gain rate or drive intensity. In this simplest case of constant gain, the effective dynamical equations, Eq. (7), are the linear differential equations with the drive terms.

 figure: Fig. 2.

Fig. 2. Comparison between the evolved population inversions with and without gain saturation. (a1) Without the gain saturation and under the drive intensity $E/\kappa =5000$. (a2) With the gain saturation of $I_0=10^{10}$ and under the drive intensity $E/\kappa =5000$. (b1) Without the gain saturation and with a constant gain rate $g/\kappa =2$. (b2) With the gain saturation of $I_0=10^{10}$ and the initial gain rate $g_0/\kappa =2$.(Insets) The corresponding gain rates. The fixed parameters for the system are $g_m/\kappa =5\times 10^{-5}$, $\gamma _m/\kappa =0.037$, $J/\kappa =1$, $\gamma /\kappa =1$, $\omega _m=2J$, $\Delta =-3J$.

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However, in the realistic situations, the optical gain cannot keep constant due to the saturation effect. Its rate lowers with the field intensity in the cavity, following the law (see, e.g. [52])

$$g(t)=\frac{g_0}{1+\frac{\left\langle\hat{a}^{\dagger}_2\hat{a}_2(t) \right\rangle}{I_0}}.$$
The saturation intensity $I_0$ usually refers to the single-photon energy per excited life time of the used atoms for amplifying the light field and per stimulated cross section in the gain material. Here we adopt the dimensionless saturation intensity corresponding to the photon number $\langle \hat {a}^{\dagger}_2\hat {a}_2(t) \rangle$ (see the definition in the supplementary material of [53]). Starting from the initial empty cavity, the photon number $\langle \hat {a}^{\dagger}_2\hat {a}_2(t)\rangle$ will increase after turning on the pumping field, to have the effective gain rate dropping according to the nonlinear law in Eq. (10). For this reason the dynamical equations, Eq. (7), are no longer linear since they carry the nonlinear terms of the gain rate $g(t)$.

As comparison, the evolutions of population inversion under the gain saturation with $I_0=10^{10}$ are depicted in Figs. 2(a2) and (b2). The numerical calculations leading to the results are performed through a composite procedure explained in Appendix B. From these simulations, one sees that the effective gain rate illustrated in the insets of the figures quickly decreases to the level of the damping rate in the same cavity. As a result, the population inversions due to the lower drive intensity $E/\kappa =5000$ and lower gain rate $g/\kappa =1, 2$ (blue line) change little, but the population inversions from the higher gain rate $g/\kappa =3$ or higher drive intensity $E/\kappa =10^4$ (red line) significantly deviate from the results of the constant gain. Moreover, the stabilized population inversions for two different initial gain rates $g_0/\kappa =1, 3$ in Fig. 2(a2) and $g_0/\kappa =2$ in Fig. 2(b2) become almost identical, as determined by the quick dropping of the gain rates to the same value. Since the initial gain rate does not affect the final population inversion so much, we will fix the initial gain rate as $g_0/\kappa =2$ in following calculations. A peculiar feature in these examples is that the finally stabilized gain rates, as well as the corresponding population inversions, keep oscillating with time.

3.2 Population inversion under various conditions

We now examine what factors determine the concerned phonon laser operation. As we have shown that the supermode populations are usually time-dependent, we will illustrate the time evolutions for the population inversion under different conditions. To achieve the best population inversion, there should be the resonant squeezing effect for the mode $\hat {o}_1$ and $\hat {b}$, i.e. the effective Hamiltonian has a form $\hat {o}_1\hat {b}+h.c$, so that the detuning should be chosen as $\Delta =-\omega _m-J$ in Eq. (A3). According to the transition between two supermodes Fig. 1 (the energy levels are the eigenvalues of the supermodes), we have the relation $\omega _m=2J$. Then the used drive detuning is fixed as $\Delta =-3J$, a blue detuned value that can be realized by choosing the drive frequency.

There are two quantities, the drive intensity $E$ and the cavity coupling $J$, which can be adjusted for the system. We illustrate the dependence of the population inversion on these quantities in Fig. 3. First we consider a lower drive amplitude $E/\kappa =10^3$. In this case the intensity of the effective coupling, which is proportional to $g_mE_1(t)$ in Eq. (A3), can be enhanced only by reducing the mutual coupling $J$, since the used optomechanical coupling is rather small as $g_m/\kappa \sim 10^{-5}$. Then, a high population inversion is realizable with a very small $J/\kappa =0.03$ as seen from Fig. 3(a1). Since the drive detuning is chosen at $\Delta =-3J$, the pump becomes close to the resonance with the cavity with a small $J$, and the cavity field thus enhanced is beneficial to the effective optomechanical coupling that is to realize the phonon laser operation. A lower pumping power can thus be compensated by choosing the proper frequency of the pump laser, so that a good phonon laser operation can be possible. Here the squeezing effect increases the population inversion quickly, though the effective optical gain rate $g(t)$ is reduced to a lower value as in Fig. 2.

 figure: Fig. 3.

Fig. 3. Evolved population inversion under different drive intensities $E$ and cavity coupling strengths $J$. The overall tendency is that increasing the drive intensity and decreasing the cavity coupling can enhance the population inversion value. The used system parameters are the same as those in Fig. 2.

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On the other hand, if choosing the high mutual coupling as in Fig. 3(a2), one will not have good population inversion. It is more understandable from the original Hamiltonian in Eqs. (2) and (3). Due to the small optomechanical coupling constant $g_m$, the effect of the interaction term in Eq. (3) is insignificant when the drive amplitude $E$ is low and the two cavities are coupled strongly with a high $J$, i.e. compared to the mutual coupling $J(\hat {a}_1\hat {a}_2^{\dagger}+\hat {a}_1^{\dagger}\hat {a}_2)$, the effect of the optomechanical interaction $-g_{m}\hat {a}_1^{\dagger}\hat {a}_1(\hat {b}+\hat {b}^{\dagger})$ is much more like a negligible small perturbation because the radiation pressure is not much enhanced by the cavity field created by a low drive amplitude $E$. In such cases of high $J$, the dynamical behavior of the system is more like the one without optomechanical coupling ($g_m=0$), and is therefore hard to implement phonon laser operation.

The results in Figs. 3(b1)-(b2) and 3(c1)-(c2) validate the above observation further. As the drive amplitude $E$ is increased, the performance with the higher coupling $J$ can be improved. With a high drive amplitude $E/\kappa =10^5$, the system with strong cavity coupling like $J/\kappa =1.5$ can also achieve high population inversion. In this sense, the choice of the system parameters can be flexible to make phonon laser operation with the concerned system. However, as the drive amplitude $E$ is increased, the squeezing effect for the lower supermode $\hat {o}_2$ and the mechanical one $\hat {b}$ will also be enhanced, destroying the population inversion $\Delta N(t)$ to the negative values; see Figs. 3(c1)-(c2).

A high population inversion is an important condition for stimulated phonon emission, so it is natural to have better phonon lasing with low cavity coupling strength but high drive intensity. The lowest pump drive intensity for phonon laser operation is its threshold. As we will see below, very low phonon laser threshold can be achieved by decreasing the mutual coupling between the cavities, and is consistent with the conclusion in [30].

4. Phonon field amplification and phonon laser threshold

To a laser, its threshold power is one of the most important figures-of-merit, being essential in its design. In analogue to an optical laser [54], the threshold of our concerned phonon laser can be found from the amplified phonon field intensity according to the following equations [21,26]:

$$\begin{aligned}\dot{b}_s&=(-\gamma_m-i\omega_m)b_s-ig_mp/2,\\ \dot{p}&=ig_m\Delta N(t)b_s/2+\left[(g(t)-\gamma-\kappa)-2iJ\right]p, \end{aligned}$$
where $b_s=\langle \hat {b}_s\rangle$ is the stimulated phonon field and $p=\langle \hat {o}_2^{\dagger}\hat {o}_1\rangle$. Different from the previous works [25,26], the gain rate $g(t)$ here is unstable due to the gain saturation, and the corresponding population inversion $\Delta N(t)$ is time-dependent as well. As a result, the above equations should be solved numerically, by substituting the gain rate $g(t)$ and population inversion $\Delta N(t)$ found previously into the above equations.

4.1 Relevance of cavity coupling strength

As it has been discussed above, a low cavity coupling strength $J$ is beneficial to achieving high population inversion, so it helps to amplify the stimulated phonon field as well. In Fig. 4, the evolved amplification rate $|b_s(t)|^2/|b_s(0)|^2$ for the phonon field is illustrated for different cavity couplings $J$ and drive intensity $E$. When the cavity coupling is as low as $J/\kappa =0.03$ or $0.1$, the amplification rate will be higher than one, to have the phonon field amplified even with a drive amplitude as low as $E/\kappa =10^3$. When the drive amplitude is increased to $E/\kappa =10^4$, the phonon lasing can be operated with the cavity coupling $J/\kappa =0.5$, but it cannot exist with the higher cavity couplings $J/\kappa =1$ and $J/\kappa =1.5$. When the drive amplitude is further increased to $E/\kappa =10^5$, the phonon lasing can be realized with a relatively large cavity coupling $J/\kappa =1$ or $J/\kappa =1.5$. Therefore, the lower cavity coupling strength is, the lower the threshold drive power can be. This feature has been experimentally demonstrated in [30] as well.

 figure: Fig. 4.

Fig. 4. Amplified phonon field due to different cavity coupling strength $J$ and drive intensity $E$. The threshold drive intensity for the pump is seen to decrease with the cavity coupling strength $J$. The system parameters are the same as those in Fig. 2.

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4.2 Role of damping rate in active cavity

There exists a non-trivial relation between the threshold and the damping rate $\gamma$ in the active cavity, which is found to affect the threshold drive intensity under different conditions. The evolutions of amplified phonon field due to different cavity coupling strength $J$, drive intensity $E$, as well as the damping rate $\gamma$ are depicted in Fig. 5. Judged from these evolutions, the phonon laser operates best within a particular range of the damping rate $1.6<\gamma /\kappa <1.7$, when $J/\kappa =0.5$. However, if the cavity coupling is increased to $J/\kappa =1$, the range for the optimum pumping threshold will be changed to $1.5<\gamma /\kappa <1.6$. Thus it can be seen that the damping rate determines the threshold for the phonon laser in a complex way.

 figure: Fig. 5.

Fig. 5. Optimal phonon laser threshold changed by the damping rate $\gamma$ in the active cavity. The lowest threshold in the four sets of evolutions occurs to the quickly increasing curve indicated by the corresponding damping rate $\gamma$. The other system parameters are the same as those in Fig. 2.

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For the phonon laser without optical gain, the pump threshold can be significantly lowered around its EP [30]. For a PT-symmetry system composed by a pair of gain $g$ and loss $\kappa$ cavities, the EP is at $J=(g+\kappa )/2$. If the damping rate $\gamma$ of the active cavity is taken into account, the EP is changed to $J=(g-\gamma +\kappa )/2$. Given the initial gain rate $g_0/\kappa =2$, the corresponding damping rates at the EP are $\gamma /\kappa =2$ for $J/\kappa =0.5$ and $\gamma /\kappa =1$ for $J/\kappa =1$, respectively. The optimum values $\gamma$ for the lowest threshold in Fig. 5 obviously deviate from the damping rates at the EPs, and such result is very different from that in [30]. In fact, due to the gain saturation, the effective gain rate is not fixed as it changes nonlinearly with the photon number in the active cavity; see the insets in Fig. 2. Considering the varying $g(t)$, one will not have a fixed EP during the evolution course. As a result, the lowest threshold will never occur at the EP, and can only be found with the numerical simulation that involves the actual gain saturation as in Fig. 5. In any case, the introduction of optical gain with saturation brings about more non-trivial features to a PT-symmetric phonon laser.

5. Conclusion

In this paper, we reconsider a type of PT-symmetric phonon laser realized by two coupled micro-cavities with optical gain and loss, respectively. The detailed dynamical evolutions of the system under the effect of gain saturation are investigated, to find the evolved population inversions and amplified phonon fields under different conditions. Different from a more simplified picture of constant optical gain, which would monotonously increase photon and phonon number together through the coupling of two cavities, the population inversion for the phonon laser and the correspondingly amplified phonon field are unstable after the gain saturation effect is taken into account. For this reason a dynamical approach is indispensable, so we develop a composite numerical approach to study the nonlinear system dynamics. With a particular choice of system parameters, the phonon laser threshold can be simply lowered with the reduced mutual coupling between two cavities. However, there also exists an optimum value for the used damping rate in the active cavity, at which the threshold pumping power will be the lowest. The desired population inversion and amplified phonon emission are expected to exist with the low cavity coupling and a proper damping rate of active cavity, when the gain saturation effect is considerable in the phonon laser operation. These understandings could be useful to the development of the concerned phonon laser setups.

Appendix A: the derivation of Eq. (7)

First, we decompose the evolution operator $U(t)$ in Eq. (5) as follows [42,43]

$$U(t)=\underbrace{\mathcal{T}\exp\left\{{-}i\int^t_0d\tau H_S(\tau)\right\}}_{U_0(t)}\times~\mathcal{T}\exp\left\{{-}i\int^t_0d\tau U^{\dagger}_0(\tau)(H_{OM}+H_{SR}(\tau))U_0(\tau)\right\}.$$
Using the transformations
$$\begin{aligned}\left(\begin{array}{c}U^{\dagger}_0\hat{a_1}U_0 \\U^{\dagger}_0\hat{a_2}U_0\end{array}\right)=&\frac{1}{\sqrt{2}}e^{{-}i\omega_c t}\left(\begin{array}{c}\hat{o}_1e^{{-}iJt}+\hat{o}_2e^{iJt} \\\hat{o}_1e^{{-}iJt}-\hat{o}_2e^{iJt}\end{array}\right)+\sqrt{2}e^{{-}i\omega_c t}\left(\begin{array}{c}E_1(t) \\E_2(t)\end{array}\right)\\ U^{\dagger}_0\hat{b}U_0=&e^{{-}i\omega_m t}\hat{b} \end{aligned}$$
with the definitions in Eq. (8), one can separate the transformed Hamiltonian $U^{\dagger}_0(\tau )(H_{OM}+H_{SR}(\tau ))U_0(\tau ))$ into the quadratic part
$$\begin{aligned} H_1(t)=&-g_m\{[E_1(t)(\hat{o}_1^{\dagger}e^{iJt}+\hat{o}_2^{\dagger}e^{{-}iJt})+H.c.]+2|E_1(t)|^2\}\\ &\times(e^{{-}i\omega_mt}\hat{b}+e^{i\omega_mt}\hat{b}^{\dagger})\\ &+i\sqrt{\kappa}\{(\hat{o}_1^{\dagger}e^{iJt}+\hat{o}_2^{\dagger}e^{{-}iJt}+2E_1^*(t))e^{i\omega_c t}\hat{\xi}_p-H.c.\}\\ &+i\sqrt{2\gamma_m}(e^{i\omega_mt}\hat{b}^{\dagger}\hat{\xi}_m(t)-e^{{-}i\omega_mt}\hat{b}\hat{\xi}^{\dagger}_m(t))\\ &+i\sqrt{g(t)}\{(\hat{o}_1^{\dagger}e^{iJt}-\hat{o}_2^{\dagger}e^{{-}iJt}+2E_2^*(t))e^{i\omega_c t}\hat{\xi}_a^{\dagger}-H.c.\}\\ &+i\sqrt{\gamma}\{(\hat{o}_1^{\dagger}e^{iJt}-\hat{o}_2^{\dagger}e^{{-}iJt}+2E_2^*(t))e^{i\omega_c t}\hat{\xi}_a-H.c.\}, \end{aligned}$$
and the purely nonlinear part
$$H_2(t)={-}\frac{1}{2}g_m(\hat{o}^{\dagger}_1e^{iJt}+\hat{o}^{\dagger}_2e^{{-}iJt})(\hat{o}_1e^{{-}iJt}+\hat{o}_2e^{iJt})\times(\hat{b}e^{{-}i\omega_m t}+\hat{b}^{\dagger}e^{i\omega_m t}).$$
Because we start from the empty cavity with the initial vacuum state $|0\rangle _c$ before pumping, the nonlinear Hamiltonian itself does not affect the system evolution since $H_2(t)|0\rangle _c=0$. Moreover, the realistic single-photon coupling strength satisfies $g_m\ll \omega _m$, so that the interplay of $H_2(t)$ with the quadratic part $H_1(t)$ due to their non-commutativity can be neglected as well. Then the quantum dynamics of the system is solely determined by the quadratic Hamiltonian, which leads to the following equations:
$$\begin{aligned} \frac{d}{dt}\hat{o}_1=&\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}_1-\frac{1}{2}(g(t)+\kappa-\gamma)e^{2iJt}\hat{o}_2+ig_mE_1(t)e^{iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ +&(-\kappa E_1(t)+g(t)E_2(t)-\gamma E_2(t))e^{iJt}+e^{iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)+\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)+\sqrt{\gamma}\hat{\xi}_a(t)),\\ \frac{d}{dt}\hat{o}^{\dagger}_1=&\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}^{\dagger}_1-\frac{1}{2}(g(t)+\kappa-\gamma)e^{{-}2iJt}\hat{o}^{\dagger}_2-ig_mE^*_1(t)e^{{-}iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ +&(-\kappa E^*_1(t)-\gamma E^*_2(t)+g(t)E^*_2(t))e^{{-}iJt}+e^{{-}iJt}e^{{-}i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p^{\dagger}(t)+\sqrt{g(t)}\hat{\xi}_a(t)+\sqrt{\gamma}\hat{\xi}_a^{\dagger}(t)),\\ \frac{d}{dt}\hat{o}_2=&-\frac{1}{2}(g(t)+\kappa-\gamma)e^{{-}2iJt}\hat{o}_1+\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}_2+ig_mE_1(t)e^{{-}iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ &-(\kappa E_1(t)+g(t)E_2(t)-\gamma E_2(t))e^{{-}iJt}+e^{{-}iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)-\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)-\sqrt{\gamma}\hat{\xi}_a(t)),\\ \frac{d}{dt}\hat{o}^{\dagger}_2=&-\frac{1}{2}(g(t)+\kappa-\gamma)e^{2iJt}\hat{o}^{\dagger}_1+\frac{1}{2}(g(t)-\kappa-\gamma)\hat{o}^{\dagger}_2-ig_mE^*_1(t)e^{iJt}(\hat{b}e^{{-}i\omega_mt}+\hat{b}^{\dagger}e^{i\omega_mt})\\ &-(\kappa E^*_1(t)+g(t)E^*_2(t)-\gamma E^*_2(t))e^{iJt}+e^{iJt}e^{{-}i\omega_ct}(\sqrt{\kappa}\hat{\xi}^{\dagger}_p(t)-\sqrt{g(t)}\hat{\xi}_a(t)-\sqrt{\gamma}\hat{\xi}_a^{\dagger}(t)),\\ \frac{d}{dt}\hat{b}=&-\gamma_m \hat{b}+ig_mE^*_1(t)e^{i\omega_mt}(\hat{o}_1e^{{-}iJt}+\hat{o}_2e^{iJt})+ig_mE_1(t)e^{i\omega_mt}(\hat{o}^{\dagger}_1e^{iJt}+\hat{o}^{\dagger}_2e^{{-}iJt})\\ +&\sqrt{2\gamma_m}e^{i\omega_mt}\hat{\xi}_m(t)+2ig_m|E_1(t)|^2e^{i\omega_mt},\\ \frac{d}{dt}\hat{b}^{\dagger}=&-\gamma_m \hat{b}^{\dagger}-ig_mE^*_1(t)e^{{-}i\omega_mt}(\hat{o}_1e^{{-}iJt}+\hat{o}_2e^{iJt})-ig_mE_1(t)e^{{-}i\omega_mt}(\hat{o}^{\dagger}_1e^{iJt}+\hat{o}^{\dagger}_2e^{{-}iJt})\\ +&\sqrt{2\gamma_m}e^{{-}i\omega_mt}\hat{\xi}^{\dagger}_m(t)+{-}2ig_m|E_1(t)|^2e^{{-}i\omega_mt}. \end{aligned}$$

Appendix B: the numerical calculation procedure in brief

To find the inversion in Eq. (9), we adopt a composite numerical method to deal with the nonlinear dynamics. In both Eq. (6) and Eq. (7) there are two types of drives, the coherent ones proportional to $E$ and the stochastic ones as the noises, which act simultaneously. The numerical calculation consists of two parts. One is to find the contribution of the coherent drive terms to the evolved supermode populations. It is performed by taking the mean-field average of each term in Eq. (6); in this way the effect of the noise drives are averaged out to have

$$\begin{aligned} \dot{\alpha}_1&={-}(\kappa-ig_m(\beta+\beta^{{\ast}}))\alpha_1-iJ\alpha_2+Ee^{{-}i\Delta t},\\ \dot{\alpha}_2&=\left(\frac{g_0}{1+\frac{\left\langle\hat{a}^{\dagger}_2\hat{a}_2(t) \right\rangle}{I_0}}-\gamma \right)\alpha_2-iJ\alpha_1,\\ \dot{\beta}&={-}\gamma_m \beta-i\omega_m \beta+ig_m|\alpha_1|^2, \end{aligned}$$
where $\alpha _i=\langle \hat {a}_i\rangle$ ($i=1,2$) and $\beta =\langle \hat {b}\rangle$. The supermode populations induced by the pumping drive $Ee^{-i\Delta t}$ alone are from the solution of this set of equations as
$$\langle \hat{o}_{1,2}^\dagger\hat{o}_{1,2}\rangle_c=\frac{1}{2}\left|\alpha_1\pm\alpha_2\right|^2.$$
The contribution to the supermode populations by the noise drive terms are important too. It is calculated through Eq. (A5) having the following noise drive terms and their conjugates:
$$\begin{aligned}\hat{n}_1(t)&=e^{iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)+\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)+\sqrt{\gamma}\hat{\xi}_a(t)),\\ \hat{n}_2(t)&=e^{{-}iJt}e^{i\omega_ct}(\sqrt{\kappa}\hat{\xi}_p(t)-\sqrt{g(t)}\hat{\xi}_a^{\dagger}(t)-\sqrt{\gamma}\hat{\xi}_a(t)),\\ \hat{n}_3(t)&=\sqrt{2\gamma_m}e^{i\omega_mt}\hat{\xi}_m(t). \end{aligned}$$
The effective gain rate $g(t)$ is treated as a time-dependent coefficient in Eq. (A5). The contribution of these noise drive terms to the evolved mode operators
$$\hat{\vec{c}}=(\hat{o}_1,\hat{o}^{\dagger}_1,\hat{o}_2,\hat{o}^{\dagger}_2,\hat{b},\hat{b}^{\dagger})^T$$
takes the form
$$\hat{\vec{c}}_n(t)=\int_0^t d\tau~ \mathcal{T}\exp\{\int_\tau^t dt' M(t')\}\hat{\vec{n}}(\tau),$$
where the matrix $M(t)$ is constructed with the coefficients in Eq. (A5). The time-order exponential here can be numerically calculated as the product of the matrices $(I+M(t_i)\Delta t)$ with a small step $\Delta t$ for $i=0,1,2,\ldots n$, which discretize the range between $\tau$ and $t$ [26]. Such contribution $\langle \hat {o}_{1,2}^\dagger \hat {o}_{1,2}\rangle _n$ of the noise drives is based on the understanding that, for the systems satisfying $g_m\ll \omega _m$, the quantum nonlinear dynamics described by Eq. (6) is well approximated by Eq. (A5).

Equation (B1) and Eq. (A5) should be integrated simultaneously, because the cavity photon number $\langle \hat {a}^{\dagger}_2\hat {a}_2(t) \rangle$ in Eq. (10) consists of the contributions from both coherent drive and noises too. In each iterative step of the numerical integrations, the varying gain rate $g(t_i)$ at the time $t_i$ is determined by such total photon number $\langle \hat {a}^{\dagger}_2\hat {a}_2(t_i)\rangle =\langle \hat {a}^{\dagger}_2\hat {a}_2(t_i)\rangle _c+\langle \hat {a}^{\dagger}_2\hat {a}_2(t_i)\rangle _n$ at this moment, and it is plugged into the discretized Eq. (A5) and Eq. (B1) for the evolution over a small step to the next time point $t_{i+1}$.

Funding

National Natural Science Foundation of China (11574093).

Acknowledgment

The authors thank Dr. Liu Yang for helpful discussion on the numerical algorithm.

Disclosures

The authors declare no conflicts of interest.

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Figures (5)
Fig. 1.
Fig. 1. The model of a coupled micro-cavities system. The side of the cavity with the damping rate $\kappa$ is elastic, and the associated mechanical oscillation $X_m(t)$ with the frequency $\omega _m$ is induced by the radiation pressure of the cavity field. The other cavity carries optical gain material, which has the initial gain rate $g_0$ and also damps at the rate $\gamma$. The gain saturation leads to a time-dependent gain rate $g(t)$. Such coupled system realizes the optical supermodes separated by the energy difference $2J$. The stimulated emission of phonons takes place when the two supermods has a population inversion, i.e., the population of the upper level is higher than that of the lower, magnifying the phonon field propagating perpendicular to the mechanical oscillation (the device is assumed to be placed in vacuum).

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Fig. 2.
Fig. 2. Comparison between the evolved population inversions with and without gain saturation. (a1) Without the gain saturation and under the drive intensity $E/\kappa =5000$. (a2) With the gain saturation of $I_0=10^{10}$ and under the drive intensity $E/\kappa =5000$. (b1) Without the gain saturation and with a constant gain rate $g/\kappa =2$. (b2) With the gain saturation of $I_0=10^{10}$ and the initial gain rate $g_0/\kappa =2$.(Insets) The corresponding gain rates. The fixed parameters for the system are $g_m/\kappa =5\times 10^{-5}$, $\gamma _m/\kappa =0.037$, $J/\kappa =1$, $\gamma /\kappa =1$, $\omega _m=2J$, $\Delta =-3J$.

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Fig. 3.
Fig. 3. Evolved population inversion under different drive intensities $E$ and cavity coupling strengths $J$. The overall tendency is that increasing the drive intensity and decreasing the cavity coupling can enhance the population inversion value. The used system parameters are the same as those in Fig. 2.

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Fig. 4.
Fig. 4. Amplified phonon field due to different cavity coupling strength $J$ and drive intensity $E$. The threshold drive intensity for the pump is seen to decrease with the cavity coupling strength $J$. The system parameters are the same as those in Fig. 2.

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Fig. 5.
Fig. 5. Optimal phonon laser threshold changed by the damping rate $\gamma$ in the active cavity. The lowest threshold in the four sets of evolutions occurs to the quickly increasing curve indicated by the corresponding damping rate $\gamma$. The other system parameters are the same as those in Fig. 2.

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Equations (21)

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H ( t ) = H S ( t ) + H O M + H S R ( t ) .
H S ( t ) = ω c a ^ 1 † a ^ 1 + ω c a ^ 2 † a ^ 2 + ω m b ^ † b ^ + J ( a ^ 1 a ^ 2 † + a ^ 1 † a ^ 2 ) + i E ( a ^ 1 † e − i ω L t − a ^ 1 e i ω L t ) .
H O M = − g m a ^ 1 † a ^ 1 ( b ^ + b ^ † ) .
H S R ( t ) = i 2 κ { a ^ 1 † ξ ^ p ( t ) − a ^ 1 ξ ^ p † ( t ) } + i 2 γ m { b ^ † ξ ^ m ( t ) − b ^ ξ ^ m † ( t ) } + i 2 g ( t ) { a ^ 2 † ξ ^ a † ( t ) − a ^ 2 ξ ^ a ( t ) } + i 2 γ { a ^ 2 † ξ ^ a ( t ) − a ^ 2 ξ ^ a † ( t ) } ,
U ( t ) = T exp ⁡ { − i ∫ 0 t d τ H ( τ ) } = T exp ⁡ { − i ∫ 0 t d τ H S ( τ ) + H O M + H S R ( τ ) } ,
a ^ ˙ 1 = − ( κ − i g m ( b ^ + b ^ † ) ) a ^ 1 − i J a ^ 2 + E e − i Δ t + 2 κ ξ ^ p , a ^ ˙ 2 = ( g ( t ) − γ ) a ^ 2 − i J a ^ 1 + 2 g ( t ) ξ ^ a † , b ^ ˙ = − γ m b ^ − i ω m b ^ + i g m a ^ 1 † a ^ 1 + 2 γ m ξ ^ m
d d t o ^ 1 = 1 2 ( g ( t ) − κ − γ ) o ^ 1 − 1 2 ( g ( t ) + κ − γ ) e 2 i J t o ^ 2 + i g m E 1 ( t ) e i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) + ( − κ E 1 ( t ) + g ( t ) E 2 ( t ) − γ E 2 ( t ) ) e i J t + e i J t e i ω c t ( κ ξ ^ p ( t ) + g ( t ) ξ ^ a † ( t ) + γ ξ ^ a ( t ) ) , d d t o ^ 2 = − 1 2 ( g ( t ) + κ − γ ) e − 2 i J t o ^ 1 + 1 2 ( g ( t ) − κ − γ ) o ^ 2 + i g m E 1 ( t ) e − i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) − ( κ E 1 ( t ) + g ( t ) E 2 ( t ) − γ E 2 ( t ) ) e − i J t + e − i J t e i ω c t ( κ ξ ^ p ( t ) − g ( t ) ξ ^ a † ( t ) − γ ξ ^ a ( t ) ) , d d t b ^ = − γ m b ^ + i g m E 1 ∗ ( t ) e i ω m t ( o ^ 1 e − i J t + o ^ 2 e i J t ) + i g m E 1 ( t ) e i ω m t ( o ^ 1 † e i J t + o ^ 2 † e − i J t ) + 2 γ m e i ω m t ξ ^ m ( t ) + 2 i g m | E 1 ( t ) | 2 e i ω m t
o ^ 1 , 2 = ( a ^ 1 ± a ^ 2 ) / 2 E 1 ( t ) = i E 2 2 ( 1 Δ + J e − i J t + 1 Δ − J e i J t − 2 Δ Δ 2 − J 2 e i Δ t ) E 2 ( t ) = i E 2 ( J Δ 2 − J 2 e i Δ t − J Δ 2 − J 2 cos ⁡ ( J t ) − i Δ Δ 2 − J 2 sin ⁡ ( J t ) ) ,
Δ N ( t ) = ⟨ o ^ 1 † o ^ 1 ( t ) ⟩ − ⟨ o ^ 2 † o ^ 2 ( t ) ⟩ > 0
g ( t ) = g 0 1 + ⟨ a ^ 2 † a ^ 2 ( t ) ⟩ I 0 .
b ˙ s = ( − γ m − i ω m ) b s − i g m p / 2 , p ˙ = i g m Δ N ( t ) b s / 2 + [ ( g ( t ) − γ − κ ) − 2 i J ] p ,
U ( t ) = T exp ⁡ { − i ∫ 0 t d τ H S ( τ ) } ⏟ U 0 ( t ) ×   T exp ⁡ { − i ∫ 0 t d τ U 0 † ( τ ) ( H O M + H S R ( τ ) ) U 0 ( τ ) } .
( U 0 † a 1 ^ U 0 U 0 † a 2 ^ U 0 ) = 1 2 e − i ω c t ( o ^ 1 e − i J t + o ^ 2 e i J t o ^ 1 e − i J t − o ^ 2 e i J t ) + 2 e − i ω c t ( E 1 ( t ) E 2 ( t ) ) U 0 † b ^ U 0 = e − i ω m t b ^
H 1 ( t ) = − g m { [ E 1 ( t ) ( o ^ 1 † e i J t + o ^ 2 † e − i J t ) + H . c . ] + 2 | E 1 ( t ) | 2 } × ( e − i ω m t b ^ + e i ω m t b ^ † ) + i κ { ( o ^ 1 † e i J t + o ^ 2 † e − i J t + 2 E 1 ∗ ( t ) ) e i ω c t ξ ^ p − H . c . } + i 2 γ m ( e i ω m t b ^ † ξ ^ m ( t ) − e − i ω m t b ^ ξ ^ m † ( t ) ) + i g ( t ) { ( o ^ 1 † e i J t − o ^ 2 † e − i J t + 2 E 2 ∗ ( t ) ) e i ω c t ξ ^ a † − H . c . } + i γ { ( o ^ 1 † e i J t − o ^ 2 † e − i J t + 2 E 2 ∗ ( t ) ) e i ω c t ξ ^ a − H . c . } ,
H 2 ( t ) = − 1 2 g m ( o ^ 1 † e i J t + o ^ 2 † e − i J t ) ( o ^ 1 e − i J t + o ^ 2 e i J t ) × ( b ^ e − i ω m t + b ^ † e i ω m t ) .
d d t o ^ 1 = 1 2 ( g ( t ) − κ − γ ) o ^ 1 − 1 2 ( g ( t ) + κ − γ ) e 2 i J t o ^ 2 + i g m E 1 ( t ) e i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) + ( − κ E 1 ( t ) + g ( t ) E 2 ( t ) − γ E 2 ( t ) ) e i J t + e i J t e i ω c t ( κ ξ ^ p ( t ) + g ( t ) ξ ^ a † ( t ) + γ ξ ^ a ( t ) ) , d d t o ^ 1 † = 1 2 ( g ( t ) − κ − γ ) o ^ 1 † − 1 2 ( g ( t ) + κ − γ ) e − 2 i J t o ^ 2 † − i g m E 1 ∗ ( t ) e − i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) + ( − κ E 1 ∗ ( t ) − γ E 2 ∗ ( t ) + g ( t ) E 2 ∗ ( t ) ) e − i J t + e − i J t e − i ω c t ( κ ξ ^ p † ( t ) + g ( t ) ξ ^ a ( t ) + γ ξ ^ a † ( t ) ) , d d t o ^ 2 = − 1 2 ( g ( t ) + κ − γ ) e − 2 i J t o ^ 1 + 1 2 ( g ( t ) − κ − γ ) o ^ 2 + i g m E 1 ( t ) e − i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) − ( κ E 1 ( t ) + g ( t ) E 2 ( t ) − γ E 2 ( t ) ) e − i J t + e − i J t e i ω c t ( κ ξ ^ p ( t ) − g ( t ) ξ ^ a † ( t ) − γ ξ ^ a ( t ) ) , d d t o ^ 2 † = − 1 2 ( g ( t ) + κ − γ ) e 2 i J t o ^ 1 † + 1 2 ( g ( t ) − κ − γ ) o ^ 2 † − i g m E 1 ∗ ( t ) e i J t ( b ^ e − i ω m t + b ^ † e i ω m t ) − ( κ E 1 ∗ ( t ) + g ( t ) E 2 ∗ ( t ) − γ E 2 ∗ ( t ) ) e i J t + e i J t e − i ω c t ( κ ξ ^ p † ( t ) − g ( t ) ξ ^ a ( t ) − γ ξ ^ a † ( t ) ) , d d t b ^ = − γ m b ^ + i g m E 1 ∗ ( t ) e i ω m t ( o ^ 1 e − i J t + o ^ 2 e i J t ) + i g m E 1 ( t ) e i ω m t ( o ^ 1 † e i J t + o ^ 2 † e − i J t ) + 2 γ m e i ω m t ξ ^ m ( t ) + 2 i g m | E 1 ( t ) | 2 e i ω m t , d d t b ^ † = − γ m b ^ † − i g m E 1 ∗ ( t ) e − i ω m t ( o ^ 1 e − i J t + o ^ 2 e i J t ) − i g m E 1 ( t ) e − i ω m t ( o ^ 1 † e i J t + o ^ 2 † e − i J t ) + 2 γ m e − i ω m t ξ ^ m † ( t ) + − 2 i g m | E 1 ( t ) | 2 e − i ω m t .
α ˙ 1 = − ( κ − i g m ( β + β ∗ ) ) α 1 − i J α 2 + E e − i Δ t , α ˙ 2 = ( g 0 1 + ⟨ a ^ 2 † a ^ 2 ( t ) ⟩ I 0 − γ ) α 2 − i J α 1 , β ˙ = − γ m β − i ω m β + i g m | α 1 | 2 ,
⟨ o ^ 1 , 2 † o ^ 1 , 2 ⟩ c = 1 2 | α 1 ± α 2 | 2 .
n ^ 1 ( t ) = e i J t e i ω c t ( κ ξ ^ p ( t ) + g ( t ) ξ ^ a † ( t ) + γ ξ ^ a ( t ) ) , n ^ 2 ( t ) = e − i J t e i ω c t ( κ ξ ^ p ( t ) − g ( t ) ξ ^ a † ( t ) − γ ξ ^ a ( t ) ) , n ^ 3 ( t ) = 2 γ m e i ω m t ξ ^ m ( t ) .
c → ^ = ( o ^ 1 , o ^ 1 † , o ^ 2 , o ^ 2 † , b ^ , b ^ † ) T
c → ^ n ( t ) = ∫ 0 t d τ   T exp ⁡ { ∫ τ t d t ′ M ( t ′ ) } n → ^ ( τ ) ,
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