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本研究提出了一种在腔光磁机械混合系统中实现可调谐的磁振子与光子双稳态的理论框架. 系统由微波腔、磁振子、机械振子和光学腔组成. 其中, 磁振子模通过磁偶极相互作用与微波腔模耦合, 同时经磁致伸缩效应与声子模相互作用; 声子模则通过辐射压力以光机械耦合方式与光学腔模作用. 通过数值求解量子朗之万方程, 分析了耦合强度、失谐参数及耗散率对双稳态阈值与迟滞特性的影响. 并进一步研究了不同系统参数下稳态间量子态跃迁的动力学行为. 结果显示, 该双稳态特性可通过系统参数进行精确调控, 其阈值及迟滞宽度对耦合效率与耗散机制呈现非线性关系. 该研究为可调谐光学开关及量子信息处理器件的设计提供了理论依据.This study establishes a theoretical framework for realizing and dynamically controlling magnon and optical bistability in a hybrid cavity optomagnomechanical system composed of microwave cavity mode, magnon mode, phonon mode, and optical cavity mode. The objective is to investigate the synergistic interplay among self-Kerr nonlinearity, magnetostrictive effect, and radiation pressure induced optomechanical coupling in generating and modulating bistable behavior. Furthermore, this work aims to reveal the transient quantum state transition dynamics between bistable states. The system Hamiltonian includes magnetic dipole interaction between the magnon mode and microwave cavity mode, magnomechanical interaction between the magnon mode and phonon mode, and optomechanical interaction between the phonon mode and optical cavity mode. In addition, the self-Kerr nonlinearity of the magnon mode is considered. Numerical analysis of the system dynamics is conducted using quantum Langevin equations that include dissipation and input noise terms. Steady-state analytical solutions for the average magnon number and optical photon number are derived, revealing a bistable characteristic with three possible solutions. Numerical simulations are performed using experimentally feasible parameters, including coupling strengths, frequency detunings, and dissipation rates. The results indicate that both magnon and optical bistabilities are tunable. Specifically, adjusting the microwave cavity–magnon coupling efficiency enables modulation of the energy transfer efficiency from microwave to magnon, thereby altering the hysteresis window and excitation threshold of the magnon bistability. Tuning the magnon-phonon interaction can influence the energy transfer from magnon to phonon. A larger magnon-pump detuning enhances nonlinear frequency shifts, alters energy transfer pathways, broadens the hysteresis loop, and increases the magnon population on the upper branch of the bistable curve. Higher magnon dissipation rate hinders the accumulation of nonlinear effect, narrowing the bistability window and shifting the threshold to higher pump powers. For optical bistability, stronger optomechanical interaction reduces the effective cavity loss and weakens the nonlinear response to the pump field, leading the amplitude of bistability to decrease and the hysteresis loop to narrow. The increase of the optical cavity–pump detuning suppresses energy transfer efficiency, necessitating higher pump power to achieve the same photon number, thereby enhancing the prominence of the bistability. Elevating the optical cavity dissipation rate requires stronger driving to compensate for photon losses, resulting in a narrower hysteresis loop and a rightward shift of the threshold. Sharp vertical jumps observed in the bistability curves correspond to instantaneous transitions at critical driving points, enabling switch-like behavior. Moreover, transient dynamics obtained by numerically solving the Langevin equations reveal the time evolution of magnon and photon numbers under nonequilibrium initial conditions. Within the bistability regime, the system exhibits quantum state transitions between low and high steady states. The transition rates are determined collectively by the system parameters. Therefore, this study provides a theoretical platform for the multi-parameter cooperative control of magnon and optical bistability. The tunability mechanism is governed by the joint action of coupling strength, detuning, and dissipation rate. The controllability of the bistability thresholds, hysteresis widths, and transient quantum state transition dynamics demonstrated in this work highlights the significant potential for applications such as tunable optical switches, quantum information processing devices, and fundamental studies of nonlinear quantum dynamics in hybrid systems.
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图 1 腔光磁机系统的示意图 (a) 钇铁石榴石晶体中的磁振子模m通过磁偶极相互作用与微波腔模a耦合, 声子模b通过色散型磁致伸缩相互作用与磁振子模m耦合, 同时通过辐射压力与光学腔模c耦合; (b) 等效物理模型
Fig. 1. Schematic diagram of the cavity opto-magnomechanical system: (a) The magnon mode m in the yttrium iron garnet crystal is coupled to the microwave cavity mode a via magnetic dipole interaction, the phonon mode b is coupled to the magnon mode m through dispersive magnetostrictive interactions and simultaneously coupled to the optical cavity mode c via radiation pressure; (b) the equivalent physical model.
图 2 稳态磁振子数$ {n_{\text{m}}} $随磁振子驱动场的归一化振幅$ {{{\varepsilon _{\text{d}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{d}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $的变化图像 (a) 不同微波-磁振子耦合强度$ {g_a} $下, 蓝色曲线表现为更宽的双稳态窗口, 表明强微波-磁振子耦合能够增强能量转移效率; (b) 不同磁机械耦合强度$ {g_m} $下, 蓝色曲线所在的磁子数最低, 表明磁机械耦合增大导致磁振子能量向声子耗散增强, 固定光学腔驱动强度$ {\varepsilon _{\text{l}}} $ = 100 GHz
Fig. 2. The variation of steady-state photon number $ {n_{\text{m}}} $ as a function of the normalized amplitude of the optical cavity driving field $ {{{\varepsilon _{\text{d}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{d}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $: (a) Under different microwave-magnon coupling strength $ {g_a} $, the blue curve exhibits a wider bistable window, indicating that strong microwave-magnon coupling enhances energy transfer efficiency; (b) under different magnomechanical coupling strength $ {g_m} $, the blue curve corresponds to the lowest magnon number, suggesting that increased magnomechanical coupling leads to enhanced dissipation of magnon energy into phonons, assuming a fixed magnon driving strength of $ {\varepsilon _{\text{l}}} $ = 100 GHz.
图 3 稳态磁振子数$ {n_m} $随磁振子驱动场的归一化振幅$ {{{\varepsilon _{\text{d}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{d}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $的变化图像 (a) 不同磁振子-泵浦失谐$ {\varDelta _m} $下, 蓝色曲线表现为更高的磁子数和更宽的双稳态窗口, 源于失谐增强非线性频移并改变能量转移路径; (b) 不同磁振子耗散率$ {\kappa _m} $下, 耗散率的增大使得双稳态回线宽度收窄且阈值右移, 表明更高的耗散抑制了非线性效应积累
Fig. 3. The variation of steady-state photon number $ {n_m} $ as a function of the normalized amplitude of the optical cavity driving field $ {{{\varepsilon _{\text{d}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{d}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $: (a) Under different magnon-pump detuning $ {\varDelta _m} $, the blue curve exhibits a higher magnon number and a wider bistable window, this is attributed to the enhanced nonlinear frequency shift caused by the detuning, which alters the energy transfer pathways; (b) under different magnon dissipation rate $ {\kappa _m} $, an increase in the decay rate results in a narrower bistable hysteresis loop and a rightward shift of the threshold, this indicates that higher decay rates suppress the accumulation of nonlinear effects.
图 4 稳态光子数$ {n_c} $随光学腔驱动场的归一化振幅$ {{{\varepsilon _{\text{l}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{l}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $的变化图像 (a) 不同光机械耦合强度$ {g_c} $下, 蓝色曲线所在的双稳态幅度最小、回线最窄, 表明强耦合增大抑制双稳态; (b) 不同光学腔-泵浦失谐$ {\varDelta _c} $下, 蓝色曲线双稳态最显著, 表明较大的失谐抑制能量转移效率; (c) 不同光学腔耗散率$ {\kappa _c} $下, 耗散率的增大导致双稳态回线宽度变窄且阈值右移, 表明高损耗需更强泵浦补偿能量, 固定磁振子驱动强度$ {\varepsilon _{\text{d}}} = {10^4} $ GHz
Fig. 4. The variation of steady-state photon number $ {n_c} $ as a function of the normalized amplitude of the optical cavity driving field $ {{{\varepsilon _{\text{l}}}} \mathord{\left/ {\vphantom {{{\varepsilon _{\text{l}}}} {{\kappa _m}}}} \right. } {{\kappa _m}}} $: (a) Under different optomechanical coupling strength $ {g_c} $, the blue curve exhibits the smallest bistable amplitude and narrowest hysteresis loop, indicating that stronger coupling enhances the suppression of bistability; (b) under different optical cavity-pump detuning $ {\varDelta _c} $, the blue curve exhibits the most pronounced bistability, demonstrating that larger detuning suppresses energy transfer efficiency; (c) under different optical cavity damping rate $ {\kappa _c} $, an increase in the decay rate leads to a narrower bistable hysteresis loop and a rightward shift of the threshold, this indicates that higher losses require stronger pumping to compensate for energy dissipation, assuming a fixed magnon driving strength of $ {\varepsilon _{\text{d}}} = {10^4} $ GHz.
图 5 (a) 磁振子数$ {n_m}\left( t \right) $随时间的演化曲线; (b) 光学腔光子数$ {n_c}\left( t \right) $随时间的演化曲线, 固定参数 $ {\kappa /{{{2\pi = 1{\text{ MHz}}}}}} $, 初始条件$ \left\langle {m\left( {0} \right)} \right\rangle = \left\langle {c\left( {0} \right)} \right\rangle = {0} $
Fig. 5. (a) Temporal evolution of magnon population $ {n_m}\left( t \right) $; (b) temporal evolution of optical cavity photon population $ {n_c}\left( t \right) $, parameter is fixed at $ {\kappa /{{{2\pi = 1{\text{ MHz}}}}}} $ with initial condition $ \left\langle {m\left( {0} \right)} \right\rangle = \left\langle {c\left( {0} \right)} \right\rangle = {0} $.
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[1] Abraham E, Smith S D 1982 Rep. Prog. Phys. 45 815
Google Scholar
[2] Gibbs H M, McCall S L, Venkatesan T N C 1976 Phys. Rev. Lett. 36 1135
Google Scholar
[3] Chen Y Y, Li Y N, Wan R G 2018 J. Opt. Soc. Am. B 35 1240
Google Scholar
[4] Kubytskyi V, Biehs S A, Ben-Abdallah P 2014 Phys. Rev. Lett. 113 074301
Google Scholar
[5] Anton M A, Calderón O G, Melle S, Gonzalo I, Carreno F 2006 Opt. Commun. 268 146
Google Scholar
[6] Chen S W, Zeng Y X, Li Z F, Mao Y, Dai X Y, Xiang Y J 2023 Nanophotonics 12 3613
Google Scholar
[7] Sete E A, Eleuch H 2012 Phys. Rev. A 85 043824
Google Scholar
[8] Wang Z P, Zhen S L, Yu B L 2015 Laser Phys. Lett. 12 046004
Google Scholar
[9] Yan D, Wang Z H, Ren C N, Gao H, Li Y, Wu J H 2015 Phys. Rev. A 91 023813
Google Scholar
[10] Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M, You J Q 2018 Phys. Rev. Lett. 120 057202
Google Scholar
[11] Yang Z B, Jin H, Jin J W, Liu J Y, Liu H Y, Yang R C 2021 Phys. Rev. Research 3 023126
Google Scholar
[12] Wang Y P, Zhang G Q, Zhang D, Luo X Q, Xiong W, Wang S P, Li T F, Hu C M, You J Q 2016 Phys. Rev. B 94 224410
Google Scholar
[13] Chen Z C, Kong D Y, Wang F 2024 Results Phys. 61 107762
Google Scholar
[14] Shen R C, Li J, Fan Z Y, Wang Y P, You J Q 2022 Phys. Rev. Lett. 129 123601
Google Scholar
[15] Wang Q, Verba R, Davídková K, Heinz B, Tian S X, Rao Y H, Guo M Y, Guo X Y, Dubs C, Pirro P, Chumak A V 2024 Nat. Commun. 15 7577
Google Scholar
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Google Scholar
[17] Kuo D M T, Chang Y C 2009 Jpn. J. Appl. Phys. 48 104504
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[21] Zhang X, Zou C L, Jiang L, Tang H X 2016 Sci. Adv. 2 e1501286
Google Scholar
[22] Fan Z Y, Qian H, Zuo X, Li J 2023 Phys. Rev. A 108 023501
Google Scholar
[23] Martinis J M, Nam S, Aumentado J, Urbina C 2002 Phys. Rev. Lett. 89 117901
Google Scholar
[24] Fan Z Y, Shen R C, Wang Y P, Li J, You J Q 2022 Phys. Rev. A 105 033507
Google Scholar
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Google Scholar
[26] Zuo X, Fan Z Y, Qian H, Ding M S, Tan H, Xiong H, Li J 2024 New J. Phys. 26 031201
Google Scholar
[27] Fan Z Y, Qian H, Li J 2022 Quantum Sci. Technol. 8 015014
[28] Fan Z Y, Qiu L, Gröblacher S, Li J 2023 Laser Photonics Rev. 17 2200866
Google Scholar
[29] Engelhardt F, Bittencourt V A, Huebl H, Klein O, Kusminskiy S V 2022 Phys. Rev. Appl. 18 044059
Google Scholar
[30] Di K, Tan S, Wang L Y, Cheng A Y, Wang X, Liu Y, Du J J 2023 Opt. Express 31 29491
Google Scholar
[31] Di K, Wang X, Xia H R, Zhao Y X, Liu Y, Cheng A Y, Du J J 2024 Opt. Lett. 49 2878
Google Scholar
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Google Scholar
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Google Scholar
[34] Bi M X, Yan X H, Xiao Y, Dai C J 2020 J. Appl. Phys. 127 084501
Google Scholar
[35] Rameshti B Z, Kusminskiy S V, Haigh J A, Usami K, Lachance-Quirion D, Nakamura Y, Hu C M, Tang H X, Bauer G E W, Blanter Y M 2022 Phys. Rep. 979 1
Google Scholar
[36] Wu Q, Hu Y H, Ma P C 2017 Int. J. Theor. Phys. 56 1635
Google Scholar
[37] Barbhuiya S A, Bhattacherjee A B 2022 J. Appl. Phys. 132 123104
Google Scholar
[38] Yeasmin S, Yadav S, Bhattacherjee A B, Banerjee S 2021 J. Mod. Opt. 68 975
Google Scholar
[39] Kumar-Singh M, Mahajan S, Bhatt V, Yadav S, Jha P K, Bhattacherjee A B 2024 J. Appl. Phys. 136 214401
Google Scholar
[40] Zhang G Q, Chen Z, Xiong W, Lam C H, You J Q 2021 Phys. Rev. B 104 064423
Google Scholar
[41] Wu W J, Xu D, Qian J, Li J, Wang Y P, You J Q 2022 Chin. Phys. B 31 127503
Google Scholar
[42] 张高见, 王逸璞 2020 物理学报 69 047103
Google Scholar
Zhang G J, Wang Y P 2020 Acta Phys. Sin. 69 047103
Google Scholar
[43] Gröblacher S, Hammerer K, Vanner M R, Aspelmeyer M 2009 Nature 460 724
Google Scholar
[44] Weis S, Rivière R, Deléglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520
Google Scholar
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