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Abstract
We propose a scheme to achieve a tunable nonreciprocal magnon laser with parametric amplification in a hybrid cavity optomagnonical system, which consists a yttrium iron garnet (YIG) sphere and a spinning resonator. We demonstrate the control of magnon laser can be enhanced via parametric amplification, which make easier and more convenient to control the magnon laser. Moreover, we analyze and evaluate the effects of pump light input direction and amplification amplitude on the magnon gain and laser threshold power. The results indicate that we can obtian a higher magnon gain and a broader range of threshold power of the magnon laser. In our scheme both the nonreciprocity and magnon gain of the magnon laser can be increased significantly. Our proposal provides a way to obtain a novel nonreciprocal magnon laser and offers new possibilities for both nonreciprocal optics and spin-electronics applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The discovery of radiation pressure has promoted both theorists and experimentalists of cavity optomechanical systems [1,2]. These systems not only provide an excellent platform for studying quantum mechanical effects in macroscopic mechanical objects but also have important applications in quantum information processing [3–5] and precision measurements [6,7]. Laser has opened up a new field of optical research, while also injecting fresh vitality into traditional domains. The laser can achieve these transformations since its exceptionally high spectral purity and spatial coherence [8]. Recently, researchers have found that laser can be generated by the Brillouin scattering process [9–11], an inelastic scattering process that leads to frequency shifts. For example, it involves the annihilation of a pump photon, simultaneously creating a scattered photon and an phonon, where the phonon plays the role traditionally associated with a Stokes wave. In this context, phonon laser were first proposed in cavity optomechanical system [12], and then, phonon laser devices were experimentally realized in various physical systems [13–17]. Due to the similarity of the classical models between cavity optomagnonical compared to cavity optomechanical, as well as the advantages of high spin density and low damping rate [18–23], it has attracted a great deal of researchers’ attention as soon as it was proposed attention of researchers [24]. In particular, the YIG sphere supports optical whispering gallery modes (WGMs) and magnon mode [22,25]. The cavity magnomechanical system has emerged as a burgeoning platform for achieving quantum coherence and coupling among magnons, cavity photons, and phonons, offering wide-ranging potential applications in quantum communication [26,27], magnetic sensing [28–30], spintronics [31], and quantum computing [32]. Magnons are collective spin-excited quasiparticles in ordered magnetic materials that assume an increasingly important part in the fabrication and control of chip devices [33–35]. They can be coherently coupled to superconducting qubits [26,36,37], microwave photons [38–41], and phonons [16,17,42,43]. Moreover, several quantum phenomena in cavity magnomechanical system have been reported, such as squeezed states [43–45], magnon blockade [46,47], entanglement [27,42,46,48], and one-way steering [27,49]. Analogous to the phonon laser, researchers propose magnon laser in a cavity optomagnonical system [50]. In their research, the excited magnon emission occurs as a coherent amplification of the Brillouin scattering process, with the magnon resembles a traditional Stokes wave, ultimately giving rise to the magnon laser. Magnon laser research contributes to the development of new techniques for dealing with spin-wave excitation, and is critical for the preparation of coherent magnon sources and the manipulation of on-chip magnetic devices.
In order to protect laser from the detrimental effects of reflections, researchers have shifted their focus on the optical nonreciprocity. The magneto-optical Faraday effect is the traditional method for implementing optical nonreciprocal devices [51,52]. However, the integration of large material volumes onto microchips and the generation of strong magnetic fields pose significant challenge. To overcome this challenge, optical nonreciprocal devices have been implemented using alternative approaches, such as optical nonlinearities [53–58], the chiral interaction [59–61], the Fizeau light-dragging effect in spinning resonators, and squeezing effects in specific direction. Previous research has demonstrated that nonreciprocal amplification of phonon can be achieved by coupling an optomechanical resonator and a spinning resonator [62,63]. Subsequently, the concept of magnon laser in hybrid cavity optomagnonical system consisting of a YIG sphere and a spinning resonator was introduced [64]. In the spinning resonator cavity rotating clockwise(CW) or counterclockwise(CCW), the optical mode in the YIG sphere and the optical mode in the spinning resonator undergo abrupt coupling [62–67]. The optical pathways within the spinning resonator differ due to rotation, resulting in irreversibly distinct refractive indices for clockwise and counterclockwise modes, i.e., $n_{\mathrm {F}}=n[1\pm nR \Omega (n^{-2}-1)/c ]$. Here $n$ is the refractive index of the material, $R$ is the radius of the resonator, $\Omega$ is the angular velocity of the resonator, and $c$ is the speed of light in the vacuum. Consequently, these two modes also undergo opposite Sagnac-Fizeau shift [68]. The form is as follows:
Recently, researchers concentrate on enhancing the nonreciprocity of magnon laser [64]. Inspired by the proposal of squeezed induced optical nonreciprocity, researchers introduced parametrically induced nonreciprocal magnon laser [70]. Comparing to spinning cavity nonreciprocal magnon laser, their proposed approach significantly improves nonreciprocity. However, they have not obtained a higher magnon gain. Based on previous work, we propose a novel approach to realize nonreciprocal magnon laser in a hybrid cavity optomagnonical system. Our work can not only achieve a higher nonreciprocity but also obtain a higher the gain effect of magnon laser. By adjusting the pump light input direction and the parametric amplification amplitude, we can modulate the magnon gain and the threshold power for laser. Different from the conventional nonreciprocal magnon laser, the controllability of our scheme may open up exciting possibilities for achieving highly tunable and efficient magnon laser. Futhermore, we explore the fundamental principles of the parametrically amplified nonreciprocal magnon laser by comprehensive theoretical analysis and numerical simulations. According to the phase matching conditions and Fizeau light-dragging effect, we investigate the impact of parametric amplification on the magnon gain and laser threshold power. The results indicate that the proposed scheme significantly contributes to the fundamental comprehension of magnon-photon interactions in a hybrid cavity magnetic system. Simultaneously, it provides invaluable insights for the development of advanced optomagnetic devices and spintronic technologies. The remarkable adjustability inherent in the parametrically amplified nonreciprocal magnon laser may offer significant potential for utilization in nonreciprocal optics, spin-based information processing, and quantum technologies.
The organization of the paper is as follows: We describe the physical model and Hamiltonian of the hybrid cavity optomagnonical system in Sec. 2. Subsequently, we provide the expressions for the magnon gain and threshold power under two scenarios forward and backward pump light input. In Sec. 3, we discuss the influences of various parameters for the nonreciprocal magnon laser in detail. Finally, a conclusion is given in Sec. 4.
2. Model and Hamiltonian
We propose a scheme to achieve nonreciprocal magnon laser induced by parametric amplification in a hybrid cavity optomagnonical system which contains a YIG sphere coupled to a spinning resonator. Unlike the traditional approach of controlling a nonreciprocal magnon laser consisting of a spinning resonator by changing the direction of the pump light input, our scheme involves controlling the magnon laser by changing the direction of the pump light input and the parametric amplification amplitude. In addition, the propose scheme can obtain superior isolation parameters, demonstrate excellent nonreciprocity, and provide enhanced laser protection. The schematic diagram of the system model is shown in Fig. 1. The YIG sphere simultaneously supports optical WGMs and magnon mode. Injecting a pump light with the same polarization $\sigma ^{+}$ as the TM mode, which undergoes Brillouin scattering processes and excites a low-frequency TE mode with $\pi$ polarization and a magnon mode, i.e., $\omega _\mathrm {TM}$ $\rightarrow$ $\omega _\mathrm {M}$ + $\omega _\mathrm {TE}$, here $\omega _\mathrm {TM}(\omega _\mathrm {TE})$ and $\omega _\mathrm {M}$ are frequencies of TM (TE) mode and magnon mode, respectively [22,71,72]. The spinning resonator can be made of high-quality factor thin film with $\chi ^{(2)}$-nonlinearity to achieve the parametric nonlinear optical process, such as lithium niobate or aluminum nitride [73]. The spinning resonator is driven by a continuous coherent laser field with frequency $\omega _\mathrm {p}$, amplitude $\Lambda _\mathrm {p}$, and phase $\theta _\mathrm {p}$. Here the amplitude $\Lambda _\mathrm {p}$ is determined by the driving power $P_\mathrm {p}$ [70]. According to the phase-matching condition in the parameter nonlinear process, when the pump light is input in the forward direction, after undergoing the Fizeau shift the $c_\mathrm {F}$ mode will be squeezed into the $c_\mathrm {Fs}$ mode. In this situation, the magnon laser can be controlled by changing the amplification amplitude $\Lambda _\mathrm {p}$. On the contrary, the pump light is input in the backward direction, the $c_\mathrm {F}$ mode will propagate opposite to the direction of the coherent light input, and the $c_\mathrm {F}$ mode could not be squeezed. Significantly, to avoid the parametric nonlinear process driven by the pump field in the YIG sphere affecting the spinning resonator, the YIG sphere and the spinning resonator are slightly different in size. Therefore, our primary focus lies on mode squeezing in the spinning resonator [70]. To better investigate the process of nonreciprocal magnon lasing, we introduce the following initial Hamiltonian ($\hbar$=1):
Fig. 1. Schematic of the considered hybrid system. The YIG sphere supporting a magnon mode and two optical WGMs is coupled to a spinning resonator cavity. (a) The forward input pump field excites a CW mode $a_\mathrm {TM}$ in the YIG sphere, and then the TE mode $a_\mathrm {TE}$ continues to couple to a compressed mode $c_\mathrm {Fs}$ in the spinning resonator cavity after a Fizeau shift has occurred. (b) The backward input pump field excites a CCW mode $a_\mathrm {TM}$. The TE mode $a_\mathrm {TE}$ is then coupled to the mode $c_\mathrm {F}$ after a Fizeau shift has occurred.
We drive the TM mode in the YIG sphere by a strong coherent driving field $H_\mathrm {d}=i \epsilon _\mathrm {d}\left (a_\mathrm {TM}^{\dagger } e^{-i \omega _\mathrm {TM} t}-a_\mathrm {TM} e^{i \omega _\mathrm {TM} t}\right )$ with the pump frequency of $\omega _\mathrm {TM}$ and the input intensity of $\epsilon _\mathrm {d}=\sqrt {2 \kappa _\mathrm {TM} P_\mathrm {in} /\left (\hbar \omega _\mathrm {TM}\right )}$, where $\kappa _\mathrm {TM}$ and $P_\mathrm {in}$ are decay rate and input light power. With a classical field $\sqrt {n_{\mathrm {TM}}} e^{-i \omega _{\mathrm {TM}}t}$, $a_\mathrm {TM}$ can be replaced where $n_{\mathrm {TM}}$ = 2$\kappa _{\mathrm {TM}} P_{\mathrm {in}} /\left [\hbar \omega _{\mathrm {TM}}\left (\Delta _{\mathrm {TM}}^2+\kappa _{\mathrm {TM}}^2\right )\right ]$ and the detuning $\Delta _{\mathrm {TM}}$ = $\omega _{\mathrm {TM}} - \Omega _{\mathrm {TM}}$. According to the frequency-matching condition, we set $\omega _{\mathrm {TM}}-\omega _{\mathrm {TE}}-\omega _{\mathrm {m}}=0$ and $\omega _{\mathrm {TE}}=\omega _{\mathrm {c}}=\omega _{\mathrm {p}}/2$, the Hamiltonian can be written in the rotating frame with $U_\mathrm 0={\mathrm {exp}[(-i\omega _{\mathrm {TM}} a_{\mathrm {TM}}^{\dagger } a_{\mathrm {TM}}-i\omega _\mathrm {m} m^{\dagger } m-i\omega _{\mathrm {TE}} a_{\mathrm {TE}}^{\dagger } a_{\mathrm {TE}}-i\omega _{\mathrm {TE}} c^{\dagger } c)t]}$ as
As shown in Fig. 1, when the forward light drives the YIG sphere, the mode $c_\mathrm {F}$ in the spinning resonator can be squeezed due to the phase-matching condition. According to the squeezing transformation [74,75] $c_\mathrm {F} = \cosh (r_\mathrm {p})c_\mathrm {Fs}-e^{-i\theta _\mathrm {p}}\sinh (r_\mathrm {p})c_\mathrm {Fs}^{\dagger }$, with $r_\mathrm {p}=\ln [(1-\beta )(1+\beta )]/4$ and $\beta =\Lambda _\mathrm {p}/\Delta _\mathrm {cF}$, we obtain effective Hamiltonian $H_\mathrm {eff}^s$ in the squeezing picture. Using the rotating-wave approximation $\Delta _{\mathrm {TE}}+\Delta _{\mathrm {cs}}\gg g_\mathrm {c}\sinh (r_\mathrm {p})$ and ignoring the counterrotating terms, the effective Hamiltonian can be described as
By introducing the supermode operators $\Psi _{\pm }=(a_\mathrm {TE} \pm c_\mathrm {Fs})/\sqrt 2$, we can rewrite the Hamiltonian of the system as
In the case of $\kappa _\mathrm {m} \ll \kappa$, we can adiabatically eliminate [76] the supermode degrees of freedom. Substituting the steady-state solutions of $a_\mathrm {\pm }$ into $m$ as
The effective damping rate of the magnon mode $\kappa _\mathrm {eff}=(1-G)\kappa _\mathrm {m}$ can be decreased by the nonnegative magnon gain $G$.
In order to evaluate the efficiency of the magnon laser, we can calculate the stimulated emission magnon number, denoted as $N_\mathrm {m}$, by employing the threshold condition $G = 1$, which signifies the generation of magnon lasing. The form depicts as follows:
We can also obtain the threshold power of the magnon laser in the situation of the threshold condition $G = 1$:Next, we consider the case of backward input of the pump light, where the $c$ mode does not interact with the coherent light, referred to as $\Lambda _\mathrm {p} = 0$. In this case, the dominant influence on the magnon laser is the Fizeau shift caused by the Fizeau light-dragging effect, denoted as $\Delta _\mathrm {F}$. The Hamiltonian can be rewritten as
3. Numerical simulation analysis
In this section, we give a detailed discussion of the influences with different parameters for the nonreciprocal magnon laser in the hybrid cavity optomagnonical system.
For the sake of simplicity, we set $\Delta _\mathrm {m} =0$ to investigate the fluctuation of the magnon gain $G$ versus a function of normalized detuning $\Delta$ under different conditions, which can be seen in Fig. 2. The black dashed line represents the threshold condition $G=1$, below which the magnon laser could not be achieved. Here, the red, green, and blue dots represent the corresponding magnon gain values at $\Delta = 3\kappa$. In Fig. 2(a), the spinning resonator rotates set at a fixed rotational velocity and direction in the absence of coherent light input. The blue (red) solid line represents the forward (backward) input of the pump light. While maintaining forward input of the pump light, we increase the parameter amplification amplitude $\Lambda _\mathrm {p}$ by adjusting the driving power of the coherent light. As shown in Fig. 2(b), the red solid line, green dashed line, and blue solid line represent the case of $\Lambda _\mathrm {p}$= 0, 1.5$\kappa$, and 3$\kappa$, respectively. It is evident that the magnon gain $G$ is notably influenced, given that the magnon laser can be deftly manipulated by adjusting the parameter amplification amplitude. Subsequently, we take into account the situation where coherent light is continuously supplied, along with either forward or backward injection of the pump light, as depicted in Fig. 2(c). The detailed description of the physical mechanism is as follows: when the pump light is input in the forward direction, the motion direction of the $c$ mode, excited by the TE mode in the spinning resonator, is opposite to the spin direction of the cavity. This results in a frequency detuning of the $c$ mode, transforming it into the $c_\mathrm {F}$ mode due to the Fizeau shift. Conversely, when the $c_\mathrm {F}$ mode propagates in the same direction as the input of coherent light, the $c_\mathrm {F}$ mode is squeezed to the $c_\mathrm {Fs}$ mode. In this situation, the resonance condition between the TE mode and the $c$ mode is broken, which suppresses the generation of stimulated magnon and reduces in the magnon laser effect. However, in the case of backward input of the pump light, the $c$ mode becomes the $c_\mathrm {F}$ mode and the motion direction of the $c$ mode in the spinning resonator is consistent with the spinning direction of the cavity. In this situation, there is no squeezing because of the opposite input direction of the coherent light. The $c_\mathrm {F}$ mode remains coupled with the TE mode owing to the Fizeau shift, which is conducive to the generation of stimulated magnons. It is evident that the magnon gain is significantly enhanced in the case of backward input in contrast to the previous work [70]. The increased excitation intensity of the magnon laser paves a promising path for designing magnon-laser amplifiers and creating coherent magnon sources.
Fig. 2. The magnon gain $G$ versus a function of normalized detuning $\Delta$ with the pump light input power $P_\mathrm {in}$ = $2\,\mathrm {mW}$. (a) $\Delta _\mathrm {F}$ = $\pm 1\kappa$; (b) $\Lambda _\mathrm {p}$ = 0, 1.5$\kappa$, and 3$\kappa$; (c) $\Delta _\mathrm {F} = \pm 1\kappa$ and $\Lambda _\mathrm {p}$ = 0, 1$\kappa$, and 2$\kappa$. The black dashed line represents the threshold condition $G$ = 1. The other parameters are used [22,25]: $\kappa _{\mathrm {TM}} / 2 \pi$ = $\kappa _{\mathrm {TE}} /2\pi$ = $\kappa _\mathrm {c}/2\pi$ = $1.7 \,\mathrm {MHz}$, $\kappa _\mathrm {m}/2\pi$ = $0.56 \,\mathrm {MHz}$, $g_\mathrm {c}$ = $3.5\kappa$, $g_\mathrm {m} / 2 \pi$ = $39.2 \,\mathrm {Hz}$, $\omega _\mathrm {d} / 2 \pi$ = $193 \,\mathrm {THz}$, $\Delta _{\mathrm {TM}} = \Delta _\mathrm {m} = 0$.
In order to make a straightforward comparison of the impacts of the pump direction and the amplitude of parametric amplification $\Lambda _\mathrm {p}$ on the magnon gain, we plot the magnon gain $G$ as a function of $\Delta _\mathrm {F}$ and $\Lambda _\mathrm {p}$ in Fig. 3. Here, the normalized detuning $\Delta$ sets as $3 \kappa$ and the white line represents the threshold condition of the $G=1$. Under the circumstance of forward input of the pump light, as indicated by the blue dot in the illustration, it corresponds to the situation of blue solid line in Fig. 2(c). As shown in Fig. 3, magnon gain $G$ of the blue dot reaches its minimum, making the generation of a magnon laser unattainable. In contrast, when the pump light is introduced in the backward direction, as indicated by the red dot and corresponding to the red solid line in Fig. 2(c), the magnon gain reaches its maximum, which is beneficial for promoting the excitation of the magnon laser. Furthermore, we also consider the situation in which the spinning resonator remains stationary without any input of coherent light, depicted by the green dot in Fig. 3. By comparing the red dot with the green dot, it is evident that our scheme has generated significantly enhanced magnon gain effects. It is clear that compared to previous work [64], the proposed scheme not only enhances the intensity of the magnon laser but also maintains the flexibility of the parameter amplification amplitude for adjusting the magnon laser.
Fig. 3. Magnon gain $G$ as a function of the Fizeau shifts $\Delta _\mathrm {F}$ and the parametric amplification amplitude $\Lambda _\mathrm {p}$ with the normalized detuning $\Delta$ = 3$\kappa$. The other parameters are the same as those in Fig. 2.
To investigate the threshold power of the magnon laser, we plot the diagram of the stimulated emission of magnons $N_\mathrm {m}$ versus the pump light power $P_\mathrm {in}$ with $\Delta =3\kappa$, as shown in Fig. 4. The black dashed line represents the threshold condition $N_\mathrm {m}=1$. $P_\mathrm {th}$ with different colors represent the threshold power in different situations. Firstly, we consider the situation of the system without parameter driving, as shown in Fig. 4(a). The red (blue) solid line represents the case where the $c_\mathrm {F}$ mode rotates in the CW (CCW) direction, corresponding to the input of the pump light in the backward (forward) direction. While the green dashed line represents the case of spinning resonator remains stationary. We can observed that the threshold power with $\Delta _\mathrm {F}=1\kappa$ is significantly lower than the other two situations. While the threshold power with $\Delta _\mathrm {F}=-1\kappa$ is notably higher than the other two situations. It is evident that the rotation direction of the resonator has an impact on the pump light power which is necessary for exciting the magnon laser. In other words, the threshold power is influenced by the Fizeau shift. Next, we consider the situation in the presence of parametric amplification while the spinning resonator is fixed and stationary, as shown in Fig. 4(b). The red, green, and blue lines correspond to the cases with parameter driving amplitudes $\Lambda _\mathrm {p}$ = 0, 1.5$\kappa$, and 2.8$\kappa$, respectively. Comparing the threshold powers with $\Lambda _\mathrm {p}$ = 0, 1.5$\kappa$, and 2.8$\kappa$, we can find that the threshold powers influenced by parameter driving obviously. In the two situations discussed above, we now take into account the situation of the combined effect of the spinning resonator and parameter driving, as shown in Fig. 4(c). The red (blue) line represents the case of backward (forward) input of the pump light with a parameter driving amplitude $\Lambda _\mathrm {p}$ = 0 ($\Lambda _\mathrm {p}$ = 1.9$\kappa$). The corresponding threshold power are 0.73 $\mathrm {mW}$ and 9.82 $\mathrm {mW}$, respectively. The green line represents the most trivial the situation in the absence of parametric amplification while the spinning resonator is fixed and stationary. In this case, the threshold power is 1.1 $\mathrm {mW}$. Clearly, the threshold power $P_\mathrm {th}$ of the magnon laser is strongly influenced by the parametric amplification. Compared with the cases in Fig. 4(b), we not only achieve lower threshold power but also obtain a larger threshold interval. Thus, the proposed scheme can control the magnon laser more flexibly and conveniently.
Fig. 4. The stimulated emitted magnon number $N_\mathrm {m}$ as a function of the pump power $P_\mathrm {in}$ with the detuning $\Delta =3\kappa$. The other parameters are the same as those in Fig. 2.
We introduce an isolation parameter to analyze the influence of input direction of the pump light and parametric amplification on the nonreciprocal magnon laser, which is given by
Figure 5 illustrates the relationship of isolation parameter $\mathfrak {R}$ between normalized detuning $\Delta$ and parameter amplification amplitude $\Lambda _\mathrm {p}$. The white line represents the situation in which the system lacks parameter amplification and the spinning resonator remains stationary. By using a combination of spin resonators and parametric amplification, we can successfully obtain a significantly higher isolation parameter up to $\mathfrak {R}=23.7\,\mathrm {dB}$. Therefore, we can achieve excellent nonreciprocity of magnon laser. Moreover, compared to nonreciprocal magnon laser induced by parametric amplification and nonreciprocal magnon laser without parametric amplification in traditional spinning resonator cavities, our scheme enables convenient optical-level control of the magnon laser while maintaining the magnon gain effect since the direct influence of the driving power $P_\mathrm {p}$ on the amplification amplitude $\Lambda _\mathrm {p}$. A higher isolation parameter indicates enhanced nonreciprocity, which plays a crucial role in protecting the laser against harmful reflections. This characteristic is especially vital for nonreciprocal devices and magnon lasers.
Fig. 5. Isolation parameter $\mathfrak {R}$ versus the optical parametric amplitude $\Lambda _\mathrm {p}/\kappa$ and the normalized detuning $\Delta /\kappa$. Here the input power $P_\mathrm {in}=2\,\mathrm {mW}$, and the other parameters are the same as those in Fig. 2.
4. Conclusion
In conclusion, we theoretically demonstrate a parametrically amplified nonreciprocal magnon laser in a hybrid cavity optomagnonical system consisting of a YIG sphere and a spinning resonator. According to the phase matching conditions, we find that both the pump light input direction and the parametric amplification amplitude can significantly affect the magnon gain and the threshold power for magnon laser. Furthermore, compared to the conventional nonreciprocal magnon laser in a spinning resonator, the propose scheme can be more conveniently controlled with the help of parametric amplification. The isolation parameter of the proposed scheme can reach 23.7 dB, which has a remarkable enhancement of nonreciprocity. Therefore, we can obtain a highly tunable nonreciprocal magnon laser. The results of our work may provide a novel approach for utilizing parametric amplification to control optomagnetic devices and hold the potential for engaging applications in the fields of nonreciprocal optics and spintronics.
Funding
National Natural Science Foundation of China (62071412, 62101479, 12074330, 12375020).
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.
References
1. T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef]
2. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
3. X. B. Yan, Z. J. Deng, X. D. Tian, et al., “Entanglement optimization of filtered output fields in cavity optomechanics,” Opt. Express 27(17), 24393–24402 (2019). [CrossRef]
4. C. H. Bai, D. Y. Wang, S. Zhang, et al., “Generation of strong mechanical-mechanical entanglement by pump modulation,” Adv. Quantum Technol. 4(5), 2000149 (2021). [CrossRef]
5. D. Y. Wang, C. H. Bai, Y. Xing, et al., “Enhanced photon blockade via driving a trapped Λ-type atom in a hybrid optomechanical system,” Phys. Rev. A 102(4), 043705 (2020). [CrossRef]
6. M. J. Woolley and A. A. Clerk, “Two-mode back-action-evading measurements in cavity optomechanics,” Phys. Rev. A 87(6), 063846 (2013). [CrossRef]
7. D. A. Long, B. J. Reschovsky, F. Zhou, et al., “Electro-optic frequency combs for rapid interrogation in cavity optomechanics,” Opt. Lett. 46(3), 645–648 (2021). [CrossRef]
8. R. Chriki, M. Nixon, V. Pal, et al., “Manipulating the spatial coherence of a laser source,” Opt. Express 23(10), 12989–12997 (2015). [CrossRef]
9. G. Valley, “A review of stimulated brillouin scattering excited with a broad-band pump laser,” IEEE J. Quantum Electron. 22(5), 704–712 (1986). [CrossRef]
10. A. Osada, A. Gloppe, R. Hisatomi, et al., “Brillouin light scattering by magnetic quasivortices in cavity optomagnonics,” Phys. Rev. Lett. 120(13), 133602 (2018). [CrossRef]
11. A. K. Zamzuri, M. I. M. Ali, A. Ahmad, et al., “Brillouin-raman comb fiber laser with cooperative rayleigh scattering in a linear cavity,” Opt. Lett. 31(7), 918–920 (2006). [CrossRef]
12. I. S. Grudinin, H. Lee, O. Painter, et al., “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010). [CrossRef]
13. J. Chen and J. B. Khurgin, “Feasibility analysis of phonon lasers,” IEEE J. Quantum Electron. 39(4), 600–607 (2003). [CrossRef]
14. A. Kent, R. Kini, N. Stanton, et al., “Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance,” Phys. Rev. Lett. 96(21), 215504 (2006). [CrossRef]
15. R. Beardsley, A. Akimov, M. Henini, et al., “Coherent terahertz sound amplification and spectral line narrowing in a stark ladder superlattice,” Phys. Rev. Lett. 104(8), 085501 (2010). [CrossRef]
16. M. S. Ding, L. Zheng, and C. Li, “Phonon laser in a cavity magnomechanical system,” Sci. Rep. 9(1), 15723 (2019). [CrossRef]
17. T. Wang, W. Zhang, J. Cao, et al., “Exceptional-point-engineered phonon laser in a cavity magnomechanical system,” New J. Phys. 25(8), 083045 (2023). [CrossRef]
18. Z. X. Liu, C. You, B. Wang, et al., “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44(3), 507–510 (2019). [CrossRef]
19. Z. X. Liu and Y. Q. Li, “Optomagnonic frequency combs,” Photonics Res. 10(12), 2786–2793 (2022). [CrossRef]
20. H. Xie, L. W. He, C. G. Liao, et al., “Generation of robust optical entanglement in cavity optomagnonics,” Opt. Express 31(5), 7994–8004 (2023). [CrossRef]
21. V. Wachter, V. A. S. V. Bittencourt, S. Xie, et al., “Optical signatures of the coupled spin-mechanics of a levitated magnetic microparticle,” J. Opt. Soc. Am. B 38(12), 3858–3871 (2021). [CrossRef]
22. A. Osada, R. Hisatomi, A. Noguchi, et al., “Cavity optomagnonics with spin-orbit coupled photons,” Phys. Rev. Lett. 116(22), 223601 (2016). [CrossRef]
23. E. Almpanis, G. Zouros, P. Pantazopoulos, et al., “Spherical optomagnonic microresonators: triple-resonant photon transitions between zeeman-split mie modes,” Phys. Rev. B 101(5), 054412 (2020). [CrossRef]
24. B. Wang, C. Kong, Z. X. Liu, et al., “Magnetic-field-controlled magnon chaos in an active cavity-magnon system,” Laser Phys. Lett. 16(4), 045208 (2019). [CrossRef]
25. X. Zhang, N. Zhu, C. L. Zou, et al., “Optomagnonic whispering gallery microresonators,” Phys. Rev. Lett. 117(12), 123605 (2016). [CrossRef]
26. D. W. Luo, X. F. Qian, and T. Yu, “Nonlocal magnon entanglement generation in coupled hybrid cavity systems,” Opt. Lett. 46(5), 1073–1076 (2021). [CrossRef]
27. W. Zhang, T. Wang, X. Han, et al., “Quantum entanglement and one-way steering in a cavity magnomechanical system via a squeezed vacuum field,” Opt. Express 30(7), 10969–10980 (2022). [CrossRef]
28. L. Du, Z. Wang, and Y. Li, “Controllable optical response and tunable sensing based on self interference in waveguide qed systems,” Opt. Express 29(3), 3038–3054 (2021). [CrossRef]
29. T. T. Pang, Y. P. Gao, W. L. Xu, et al., “Optical magnetic sensing using whispering-gallery mode optomagnonical cavity,” IEEE Sens. J. 21(3), 2742–2748 (2020). [CrossRef]
30. S. P. Wolski, D. Lachance-Quirion, Y. Tabuchi, et al., “Dissipation-based quantum sensing of magnons with a superconducting qubit,” Phys. Rev. Lett. 125(11), 117701 (2020). [CrossRef]
31. M. Harder and C. M. Hu, “Cavity spintronics: An early review of recent progress in the study of magnon–photon level repulsion,” Solid State Phys. 69, 47–121 (2018). [CrossRef]
32. S. Andrianov and S. Moiseev, “Magnon qubit and quantum computing on magnon bose-einstein condensates,” Phys. Rev. A 90(4), 042303 (2014). [CrossRef]
33. J. Barker and G. E. Bauer, “Thermal spin dynamics of yttrium iron garnet,” Phys. Rev. Lett. 117(21), 217201 (2016). [CrossRef]
34. N. Arakawa, “Stabilizing mechanism for bose-einstein condensation of interacting magnons in ferrimagnets and ferromagnets,” Phys. Rev. Lett. 121(18), 187202 (2018). [CrossRef]
35. Y. Li, W. Cao, V. P. Amin, et al., “Coherent spin pumping in a strongly coupled magnon-magnon hybrid system,” Phys. Rev. Lett. 124(11), 117202 (2020). [CrossRef]
36. C. X. Ning and M. Yin, “Entangling magnon and superconducting qubit by using a two-mode squeezed-vacuum microwave field,” J. Opt. Soc. Am. B 38(10), 3020–3026 (2021). [CrossRef]
37. D. Xu, X. K. Gu, H. K. Li, et al., “Quantum control of a single magnon in a macroscopic spin system,” Phys. Rev. Lett. 130(19), 193603 (2023). [CrossRef]
38. B. Wang, Z. X. Liu, C. Kong, et al., “Magnon-induced transparency and amplification in $\mathcal {PT}$-symmetric cavity-magnon system,” Opt. Express 26(16), 20248–20257 (2018). [CrossRef]
39. W. Zhang, D. Y. Wang, C. H. Bai, et al., “Generation and transfer of squeezed states in a cavity magnomechanical system by two-tone microwave fields,” Opt. Express 29(8), 11773–11783 (2021). [CrossRef]
40. J. Cheng, Y. M. Liu, X. Yi, et al., “Generation and enhancement of mechanical squeezing in a hybrid cavity magnomechanical system,” Ann. Phys. 534(5), 2100493 (2022). [CrossRef]
41. C. M. Zheng, W. Zhang, D. Y. Wang, et al., “Simultaneously enhanced photon blockades in two microwave cavities via driving a giant atom,” New J. Phys. 25(4), 043030 (2023). [CrossRef]
42. J. Li, S. Y. Zhu, and G. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121(20), 203601 (2018). [CrossRef]
43. J. Li, S. Y. Zhu, and G. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99(2), 021801 (2019). [CrossRef]
44. X. Han, D. Y. Wang, C. H. Bai, et al., “Mechanical squeezing beyond resolved sideband and weak-coupling limits with frequency modulation,” Phys. Rev. A 100(3), 033812 (2019). [CrossRef]
45. M. Asjad, J. Li, S. Y. Zhu, et al., “Magnon squeezing enhanced ground-state cooling in cavity magnomechanics,” Fundam. Res. 3(1), 3–7 (2023). [CrossRef]
46. L. Wang, Z. X. Yang, Y. M. Liu, et al., “Magnon blockade in a pt-symmetric-like cavity magnomechanical system,” Ann. Phys. 532(7), 2000028 (2020). [CrossRef]
47. Y. J. Xu, T. L. Yang, L. Lin, et al., “Conventional and unconventional magnon blockades in a qubit-magnon hybrid quantum system,” J. Opt. Soc. Am. B 38(3), 876–884 (2021). [CrossRef]
48. J. Cheng, Y. M. Liu, H. F. Wang, et al., “Entanglement and asymmetric steering between distant magnon and mechanical modes in an optomagnonic-mechanical system,” Ann. Phys. 534(12), 2200315 (2022). [CrossRef]
49. S. Y. Guan, H. F. Wang, and X. Yi, “Cooperative-effect-induced one-way steering in open cavity magnonics,” npj Quantum Inf. 8(1), 102 (2022). [CrossRef]
50. Z. X. Liu and H. Xiong, “Magnon laser based on brillouin light scattering,” Opt. Lett. 45(19), 5452–5455 (2020). [CrossRef]
51. H. Dötsch, N. Bahlmann, O. Zhuromskyy, et al., “Applications of magneto-optical waveguides in integrated optics,” J. Opt. Soc. Am. B 22(1), 240–253 (2005). [CrossRef]
52. T. Wang, X. Han, S. Liu, et al., “Optomechanically induced faraday effect with mechanical driving and coulomb coupling,” Adv. Quantum Technol. 6(7), 2200162 (2023). [CrossRef]
53. Q. Guo, J. Bai, L. Y. Cheng, et al., “Simplified optical quantum-information processing via weak cross-kerr nonlinearities,” Phys. Rev. A 83(5), 054303 (2011). [CrossRef]
54. S. Hua, J. Wen, X. Jiang, et al., “Demonstration of a chip-based optical isolator with parametric amplification,” Nat. Commun. 7(1), 13657 (2016). [CrossRef]
55. C. H. Bai, D. Y. Wang, S. Zhang, et al., “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019). [CrossRef]
56. C. Kong, H. Xiong, and Y. Wu, “Magnon-induced nonreciprocity based on the magnon kerr effect,” Phys. Rev. Appl. 12(3), 034001 (2019). [CrossRef]
57. L. Tang, J. Tang, M. Chen, et al., “Quantum squeezing induced optical nonreciprocity,” Phys. Rev. Lett. 128(8), 083604 (2022). [CrossRef]
58. W. Zhang, T. Wang, X. Han, et al., “Mechanical squeezing induced by duffing nonlinearity and two driving tones in an optomechanical system,” Phys. Lett. A 424, 127824 (2022). [CrossRef]
59. B. Peng, Ş. K. Özdemir, F. Lei, et al., “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
60. T. Yu, X. Zhang, S. Sharma, et al., “Chiral coupling of magnons in waveguides,” Phys. Rev. B 101(9), 094414 (2020). [CrossRef]
61. X. Wang, W. X. Yang, A. X. Chen, et al., “Phase-modulated single-photon nonreciprocal transport and directional router in a waveguide–cavity–emitter system beyond the chiral coupling,” Quantum Sci. Technol. 7(1), 015025 (2022). [CrossRef]
62. Y. Jiang, S. Maayani, T. Carmon, et al., “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018). [CrossRef]
63. Y. Xu, J. Y. Liu, W. Liu, et al., “Nonreciprocal phonon laser in a spinning microwave magnomechanical system,” Phys. Rev. A 103(5), 053501 (2021). [CrossRef]
64. Y. J. Xu and J. Song, “Nonreciprocal magnon laser,” Opt. Lett. 46(20), 5276–5279 (2021). [CrossRef]
65. Y. F. Jiao, S. D. Zhang, Y. L. Zhang, et al., “Nonreciprocal optomechanical entanglement against backscattering losses,” Phys. Rev. Lett. 125(14), 143605 (2020). [CrossRef]
66. D. W. Zhang, L. L. Zheng, C. You, et al., “Nonreciprocal chaos in a spinning optomechanical resonator,” Phys. Rev. A 104(3), 033522 (2021). [CrossRef]
67. W. Zhang, T. Wang, S. Liu, et al., “Nonreciprocal photon blockade in a spinning resonator coupled to two two-level atoms,” Sci. China Phys. Mech. Astron. 66(4), 240313 (2023). [CrossRef]
68. G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000). [CrossRef]
69. G. C. Righini, Y. Dumeige, P. Feron, et al., “Whispering gallery mode microresonators: fundamentals and applications,” La Rivista Del Nuovo Cimento 34(7), 435–488 (2011). [CrossRef]
70. K. W. Huang, Y. Wu, and L. G. Si, “Parametric-amplification-induced nonreciprocal magnon laser,” Opt. Lett. 47(13), 3311–3314 (2022). [CrossRef]
71. Q. Cai, J. Liao, and Q. Zhou, “Stationary entanglement between light and microwave via ferromagnetic magnons,” Ann. Phys. 532(12), 2000250 (2020). [CrossRef]
72. J. Haigh, A. Nunnenkamp, A. Ramsay, et al., “Triple-resonant brillouin light scattering in magneto-optical cavities,” Phys. Rev. Lett. 117(13), 133602 (2016). [CrossRef]
73. I. Breunig, “Three-wave mixing in whispering gallery resonators,” Laser Photonics Rev. 10(4), 569–587 (2016). [CrossRef]
74. X. Y. Lü, Y. Wu, J. Johansson, et al., “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114(9), 093602 (2015). [CrossRef]
75. W. Qin, A. Miranowicz, P. B. Li, et al., “Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification,” Phys. Rev. Lett. 120(9), 093601 (2018). [CrossRef]
76. C. H. Bai, D. Y. Wang, S. Zhang, et al., “Qubit-assisted squeezing of mirror motion in a dissipative cavity optomechanical system,” Sci. China Phys. Mech. Astron. 62(7), 970311 (2019). [CrossRef]
