1. Introduction

In recent years, cavity optomagnonical system has garnered significant attention in quantum physics due to the strong or even ultrastrong couplings of collective spin excitations in a millimeter-scale yttrium iron garnet (YIG) crystal and the cavity photons [1–7]. The utilization of YIG sphere is universal in hybrid systems based on magnonics, primarily owing to their remarkable advantages of high spin density and low damping rate [8]. Strong and even ultrastrong coupling between the magnon mode in the ferromagnet YIG sphere and the cavity photons has been proposed in Refs. [9–13]. The cavity optomagnonical system has become a promising platform for the implementation of various interesting phenomena, such as magnon dark modes memory [14], chaos [15], exceptional point [16–18], dissipative magnon-photon coupling [19–21] and stationary one-way quantum steering [22,23].

The concept of parity-time ($\mathcal {PT}$)-symmetry has recently attracted a lot of attention. It is well-understood that when system parameters in the Hamiltonian are appropriately adjusted, a transition occurs from an unbroken $\mathcal {PT}$-symmetric phase (with a real eigenvalue spectrum) to a spontaneously broken $\mathcal {PT}$-symmetric phase (characterized by a complex eigenvalue spectrum) [24]. This shift usually occurs around the exceptional points (EPs) [25,26], i.e., $\mathcal {PT}$-symmetric phase transition point [27–29]. This phenomenon attracted remarkable interests and rapidly emerged as a forefront topic in the area of non-Hermitian physics. In the field of cavity optomagnonics, these advancements have resulted in a range of important applications, including the realization of magnon-induced chaos in a $\mathcal {PT}$-symmetric cavity optomagnonical system [30]. Moreover, the observations of EPs in non-Hermitian systems [16,31–33], as well as the controlled EPs and bound states in anti-$\mathcal {PT}$-symmetric cavity optomagnonical systems have been already investigated [34]. Importantly, a $\mathcal {PT}$-symmetric magnon laser in a cavity optomagnonical system has been proposed in Ref. [35] recently. This development signifies that $\mathcal {PT}$-symmetric systems have indeed emerged as a promising platform for investigating magnon laser effects.

Nonreciprocal devices have the ability to allow incident light to exhibit different properties in different propagation directions, which play a crucial role in a variety of applications, for instance signal processing and invisibility sensing [36]. Several physical mechanisms such as optomechanics [37,38], non-Hermitian optics [39,40], and spinning resonators [41–44] have become great platforms for studying optical nonreciprocity. Furthermore, the nonreciprocal phonon laser by coupling optomechanical resonators with spinning optical resonators has been proposed in Ref. [45]. A scheme for generating nonreciprocal phonon lasers in a single spinning mechanical resonator is realized [46], extending nonreciprocal phonon laser to cavity magnomechanical system.

So far, based on the Brillouin scattering process in YIG sphere, magnon laser have been theoretically proposed as a new type of laser similar to optical laser and phonon laser, and the experimental feasibility under current experimental conditions has been evaluated theoretically [47]. The magnon laser has quickly become an important platform for research the intersections of quantum optics and magnon spintronics. Note that nonreciprocal magnon laser was recently achieved by coupling to a spinning resonator [48]. In addition, the nonreciprocal magnon laser realized by introducing parametric amplification further improved the nonreciprocity [49,50]. Recently, a $\mathcal {PT}$-symmetric magnon laser has been achieved in a cavity optomagnonical system through the introduction of gain [35]. This achievement significantly reduces the threshold power of the magnon laser and extends its applicability to non-Hermitian domains. Notably, the exploration of nonreciprocal magnon lasers within $\mathcal {PT}$-symmetric systems also remains an underexplored area, despite the substantial of efforts devoted to magnon lasers.

Inspired by the implications of these studies, in this paper, we investigate the nonreciprocal $\mathcal {PT}$-symmetric magnon laser consists of active and passive optical whispering gallery mode (WGM) spinning resonators and a YIG sphere, where the magnon mode in YIG sphere couples with the cavity mode $a_2$ in passive resonator. We demonstrate that when driving the system from one direction enhances the magnon laser, while driving it from the opposite direction suppresses the magnon laser. The Fizeau light-dragging effect, generated by the resonator spinning, adjusts the steady-state photon population, magnon gain, and threshold power in the two resonators. Therefore, by tuning the spinning speed of the resonators and driving direction, a highly tunable nonreciprocal $\mathcal {PT}$-symmetric magnon laser can be achieved. Finally, by increasing the active cavity gain coefficient, we obtain a nonreciprocal magnon laser with ultra-low threshold power, and one-way magnon laser can be realized across a range of input power. Compared with earlier studies on $\mathcal {PT}$-symmetric magnon lasers [35], our work achieves nonreciprocity in the device by introducing spinning cavities of gain and loss, which allows our work to realize a unidirectional magnon laser in $\mathcal {PT}$-symmetric systems. Besides, in contrast to previous nonreciprocal magnon laser scheme [49], our work significantly increases the maximum stimulated emission magnons number of the system and decreases the threshold power. Our scheme provides a new way to explore various nonreciprocal effects in non-Hermitian cavity magnonics systems and holds potential applications in various spintronics devices.

The paper is organized as follows: In Sec. 2, we introduce the physical model and the Hamiltonian of the $\mathcal {PT}$-symmetric cavity optomagnonical system, and describe the system dynamics using the Heisenberg-Langevin equations. We analyze the position of the EP, determine the optimal parameters of the system, and obtain the magnon gain using an adiabatic elimination method. In Sec. 3, we extensively discuss the nonreciprocity of the system, including the variations of the steady-state photon populations, the magnon gain, the stimulated emitted magnon number, and the threshold power for different driving directions. Finally, a conclusion is given in Sec. 4.

2. System and Hamiltonian

The $\mathcal {PT}$-symmetric cavity optomagnonical system, which consists of a active WGM spinning resonator with doped gain medium, a passive WGM spinning resonator and a ferrimagnet YIG sphere as shown in Fig. 1. The YIG sphere is uniformly magnetized along the $\mathit {z}$ direction by an external static magnetic field $\textbf {B}_\mathit {z}$ = $B\textbf {e}_\mathit {z}$. Two optical WGMs in spinning resonators sharing the same resonant frequency $\omega _\textrm {c}$, refractive index, radius. Both resonators are equipped with spinning bases that rotate in opposite directions. Therefore, the clockwise and counter-clockwise optical WGMs in the spinning resonators have different refractive indices, which causes them to experience a Fizeau shift, the key factor leading to nonreciprocal behavior. The Fizeau shift $\Delta _{\mathrm {F}}$ can be written as [51]

 

Fig. 1. Schematic diagram of a nonreciprocal $\mathcal {PT}$-symmetric cavity optomagnonical system. The ferrimagnet YIG sphere supports optical WGM $a_{2}$ and magnon mode $m$ and optical WGM $a_{2}$ couples to active optical WGM $a_{1}$. The input light evanescent couples to the active optical WGM via a tapered fiber. Driving the system from the left ($\Delta _{\mathrm {F}}>0$) enhances the magnon laser, while driving from the right ($\Delta _{\mathrm {F}}<0$) suppresses or even blocks magnon laser emission.

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The total Hamiltonian of the system in the frame rotating at the cavity driving frequency $\omega _\textrm {d}$ can be written as

The Heisenberg-Langevin equations of motion of the system can be written as

 

Fig. 2. (a) Eigenfrequencies splitting of supermodes $(\omega _{\pm }-\tilde {\Delta })/\gamma$ and (b) system dissipation of the eigenmodes $\gamma _{\pm }/\gamma$ vary with the optical tunneling rate $J/\gamma$ under the condition of $\kappa /\gamma$ = 1.

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By introducing the supermode operators, and applying the rotating-wave approximation, the Hamiltonian of the system can be rewritten as

According the Hamiltonian as aforementioned, the dynamical equations for the supermodes $\Psi _{\pm }$ and magnon mode $m$ can be obtained from Eq. (6):

We can define population-inversion operator $\Delta _N=\Psi _{+}^{\dagger }\Psi _{+}-\Psi _{-}^{\dagger } \Psi _{-}$ and the ladder operator $p=\Psi _{-}^{\dagger }\Psi _{+}$ of the optical supermodes, the above dynamical equations can be rewritten as

The equations of motion of the system then become

Due to the optical decay is more rapid than the magnon decay, i.e., $\gamma, \gamma ^{\prime }\gg \gamma _{m}$, $\tilde {m}$ can be viewed as a constant compared to $\tilde {p}$ and since $\gamma ^{\prime }\gg G \sqrt {S}$, the time-dependent term in $\tilde {p}$ can be safely ignored [27,35]. Therefore, we can solve Eq. (10a) analytically with a first-order ordinary differential equation, obtaining the first-order approximate solution as:

By substituting $\tilde {p}$ into Eq. (10b) , the amplitude of magnon $\tilde {p}$ can be obtained as

From Eq. (12), the effective magnon gain $\mathcal {G}$ is given by

The magnon gain $\mathcal {G}$ represents the extent to which magnons are amplified through their interaction with stimulated emission when they transition from a higher energy level ($\omega _{+}$) to a lower energy level ($\omega _{-}$). Furthermore, because $\Delta _{\mathrm {F}}$ is included in $\Delta _N$, in cases where $\Delta _{\mathrm {F}}$ > 0 and $\Delta _{\mathrm {F}}$ < 0, different magnon gain $\mathcal {G}$ can be obtained, enabling the realization of nonreciprocal magnon laser. We define stimulated emission of magnon number is $M=\exp {[2(\mathcal {G}-\gamma _\textrm {m}/2)/(\gamma _\textrm {m}/2)]}$, which characterizes the performance of the magnon laser. It indicates the occurrence of magnon amplification when $M=1$, thus the threshold condition for magnon laser is $\mathcal {G}=\gamma _\textrm {m}/2$. From the threshold condition $\mathcal {G}=\gamma _\textrm {m}/2$, the threshold power of magnon laser can be derived as

3. Analysis and discussion

In above discussion, we have already determined the position of the EP and found the optimal coupling strength $J$. Therefore, we can research the maximum achievable gain and nonreciprocal effects of the system. We plot the magnon gain $\lg {(\mathcal {G})}$ as a function of normalized detuning $\Delta /\omega _\textrm {m}$, as shown in Fig. 3(a). It can be observed that when the fixed resonators case, i.e., $\Delta _{\mathrm {F}}=0$, the system can achieve maximum gain at $\Delta$ = $0.5~\omega _\textrm {m}$, and the maximum gain can reach magnitudes of $10^9$. In this case, regardless of the direction from which the pumping is input into the system, the resulting magnon gain $\mathcal {G}$ remains the same, indicating that the system is reciprocal at this point. On the other hand, when $\Delta _{\mathrm {F}}>0$ ($\Delta _{\mathrm {F}}<0$), it causes the maximum value of $\mathcal {G}$ to shift to the left (right), this results in directional correlation of our system. By selecting an appropriate detuning, it is possible to enhance the magnon gain when driving the resonator from one direction, while suppressing the magnon gain when driving the resonator from the opposite direction. This indicates that the system is nonreciprocal under these conditions. In order to verify that the system produces nonreciprocity, we plot the steady-state photon numbers $\left |a_{1s}\right |^2$ of the active optical WGM $a_1$ as functions of normalized detuning in Fig. 3(b) and the steady-state photon numbers $\left |a_{2s}\right |^2$ of the passive optical WGM $a_2$ as functions of normalized detuning in Fig. 3(c), respectively. Clearly can be seen, in different driving directions, the position of maximum steady-state photon number in the two spinning resonators experience a frequency shift. Therefore, by selecting specific detunings, significant differences in the steady-state photon number occur for two driving directions. These differences in steady-state photon number lead to variations in the magnetic dipole interaction strength. Thus, the magnon gain exhibits a substantial difference when the system is driven in opposite directions.

 

Fig. 3. (a) The magnon gain $\lg {(\mathcal {G})}$, (b) steady-state photon number $\left |a_{1s}\right |^2$ of active WGM mode, and (c) steady-state photon number $\left |a_{2s}\right |^2$ of passive WGM mode as a function of the normalized detuning $\Delta /\omega _\textrm {m}$. The parameters are used $\kappa /2\pi =15$ MHz, $\gamma /2\pi =0.1$ MHz, $G=1$ Hz [54,55], $S=1\times 10^{10}$, $\omega _\textrm {m}/2\pi =1$ GHz, $\Delta _{\mathrm {F}}=0.1\omega _m$, $\Delta \omega =\omega _\textrm {m}$, $\kappa /\gamma =0.8$, $\omega _\textrm {d}=193$ THz, $\gamma _\textrm {m}/2\pi =0.1$ MHz, $J=\sqrt {(\kappa +\gamma )^2/16+\omega _{\textrm {m}}^2/4}\approx \pi$ GHz [35,47,53,56].

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To better explain the influence of resonators rotation on magnon lasers, we introduce the isolation parameter $\mathfrak {R}$ = $10 \log _{10} [\mathcal {G}\left (\Delta _{\mathrm {F}}<0\right )]/[\mathcal {G}\left (\Delta _{\mathrm {F}}>0\right )]$. For the fixed resonators case, implying a reciprocal system, the isolation parameter $\mathfrak {R}=0$, however when the isolation parameter $\mathfrak {R}\neq 0$, it signifies the emergence of nonreciprocity in the system. Consequently, in Fig. 4, we plot the isolation parameter as a function of normalized detuning $\Delta /\omega _{\textrm {m}}$ and the Fizeau shift $\Delta _{\mathrm {F}}$. As the Fizeau shift $\Delta _{\mathrm {F}}$ increases, the nonreciprocity of the spin resonator becomes more obvious. By selecting an appropriate normalized detuning, we can achieve a pronounced nonreciprocal magnon laser phenomenon. Therefore, adjusting the driving direction of the resonator can be used as a switch to activate or deactivate the laser. This is helpful to understand and control of nonreciprocal behavior in magnon lasers, and it has significant implications for their practical applications in physics research.

 

Fig. 4. Isolation parameter $\mathfrak {R}$ versus the normalized detuning $\Delta /\omega _\textrm {m}$ and the Fizeau shift $\Delta _{\mathrm {F}}$ with the input power $P_{\textrm {in}}=2$ mW. The other parameters are the same as those in Fig. 3

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We further demonstrate the nonreciprocal characteristics of the system and the advantages of $\mathcal {PT}$-symmetric systems, which provide attractive opportunities for our system to generate one-way magnon laser across a range of input power while achieving ultra-low threshold power. Figure 5 shows the stimulated emitted magnon number $M$ versus the input power $P_{\textrm {in}}$ in the presence of $\Delta _{\mathrm {F}}<0$, $\Delta _{\mathrm {F}}=0$ and $\Delta _{\mathrm {F}}>0$ with different values of the active optical WGM gain coefficient $\kappa$. The shaded part of the figure indicates that magnon laser is below the threshold condition $M=1$. Based on previous research, it is known that laser generation occurs only when the system reaches the threshold condition, and the input power at the threshold condition is called the threshold power, and the three colored dots denote the threshold power $P_{\textrm {th}}$, respectively. We find that the stimulated emission magnon number $M$ exhibits strong nonlinearity with the input power and is directional dependent. When $\Delta _{\mathrm {F}}>0$, i.e., pumping light is input from the left, the threshold power $P_{\textrm {th}}$ is significantly lower than in the case of the opposite input direction and under the fixed resonators case. Taking Fig. 5(a) as an example, in the case of fixed resonator, the threshold power of the system $P_{\textrm {th}}\approx 17$ $\mu \mathrm {W}$. For $\Delta _{\mathrm {F}}>0$, the threshold power $P_{\textrm {th}}$ can be reduced to $2.3$ $\mu \mathrm {W}$, which is attributed to the enhancement of the magnon gain. For the case of $\Delta _{\mathrm {F}}<0$, the threshold power is as high as $49.2$ $\mu \mathrm {W}$. It is because the suppression of magnon gain at $\Delta = 0.34$ $\omega _\textrm {m}$, hence requiring a higher input power. Therefore, $\Delta _{\mathrm {F}}>0$ produces magnon laser when the input power is greater than $2.3$ $\mu \mathrm {W}$ and less than $49.2$ $\mu \mathrm {W}$, while when $\Delta _{\mathrm {F}}<0$ no magnon laser occurs at all because the threshold power is not reached. A one-way magnon laser is realized in the input power range. Figures 5(b)-(d) show the effects on the system threshold power as the optical gain-loss ratio increases to 0.85, 0.9, 0.95, respectively. Obviously, as the gain-loss ratio gradually approaches 1, the system can reach ultra-low threshold power, and even for $\Delta _{\mathrm {F}}>0$, the system can approach zero threshold power. This is because for a $\mathcal {PT}$-symmetric system, the dissipation $\gamma _{\pm }$ of the system can approach 0 when the gain-loss ratio is close to 1, so the energy exchange between the supermodes has a very low internal loss and the gain can be effectively introduced, thus achieving ultra-low threshold power.

 

Fig. 5. The stimulated emitted magnon number $M$ as a function of the driving power $P_{\textrm {in}}$ at the detuning $\Delta /\omega _\textrm {m}$ = 0.34 in the context of gain-loss ratio (a) $\kappa /\gamma =0.8$ (b) $\kappa /\gamma =0.85$ (c) $\kappa /\gamma =0.9$ (d) $\kappa /\gamma =0.95$. The other parameters are the same as those in Fig. 3.

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4. Conclusions

In conclusion, we have proposed the nonreciprocal $\mathcal {PT}$-symmetric magnon laser in a $\mathcal {PT}$-symmetric cavity optomagnonical system consists of active and passive optical spinning resonators as well as a YIG sphere. By introducing the Fizeau light-dragging induced by the rotation of resonators, we observe the significant variations in the steady-state photon populations of active optical WGM and passive optical WGM as well as magnon gain. It is verified that the trend of the magnon gain is the same as the trend of the photon populations number in both resonators. We also show that the Fizeau shift can further reduce the threshold power of the system, allowing the system to approach 0 threshold power without reaching gain-loss balance, and achieve higher isolation rates within experimentally feasible parameter ranges. Furthermore, by controlling the rotational speed of the spinning resonators and driving direction, the magnon gain and threshold power can be flexible adjusted. Notably, we successfully demonstrate a one-way magnon laser can be realized across a range of input power. Thus a nonreciprocal $\mathcal {PT}$-symmetric magnon laser can be achieved. This provides a flexible method for manipulating non-Hermitian cavity magnon systems and holds potential applications in various spin-electronics devices.