Abstract

Magnetostrictive interaction, as a nonlinear effect, provides a great opportunity to establish a tunable cavity magnomechanical system and may bring many interesting physical phenomena and potential applications. Here, we theoretically investigate the generation and control of higher-order sideband phenomena in a hybrid cavity electro–opto–magnomechanical system, in which the geometrical deformation of yttrium iron garnet can be treated as an excellent mechanical resonator. We show that the amplitude of the sideband can be considerably enhanced in the case of blue detuning of the microwave cavity field, and we also find an interesting pump-field-detuning-controlled optical switch effect. Furthermore, numerical calculations of the system’s dynamic equations show excellent agreement with our analytical results. Our results will contribute to the understanding of nonlinear coherent phenomena and have the potential to greatly advance the fields of cavity electro–opto–magnomechanical systems and nonlinear optics.

© 2020 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Generation and enhancement of sum sideband in a quadratically coupled optomechanical system with parametric interactions

Xiao-Yun Wang, Liu-Gang Si, Xiao-Hu Lu, and Ying Wu
Opt. Express 27(20) 29297-29308 (2019)

Ground-state cooling of a magnomechanical resonator induced by magnetic damping

Ming-Song Ding, Li Zheng, and Chong Li
J. Opt. Soc. Am. B 37(3) 627-634 (2020)

Magnetically controllable slow light based on magnetostrictive forces

Cui Kong, Bao Wang, Zeng-Xing Liu, Hao Xiong, and Ying Wu
Opt. Express 27(4) 5544-5556 (2019)

References

  • View by:
  • |
  • |
  • |

  1. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
    [Crossref]
  2. H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
    [Crossref]
  3. X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
    [Crossref]
  4. C. Bekker, R. Kalra, C. Baker, and W. P. Bowen, “Injection locking of an electro-optomechanical device,” Optica 4, 1196–1204 (2017).
    [Crossref]
  5. H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
    [Crossref]
  6. Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97, 013801 (2018).
    [Crossref]
  7. H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
    [Crossref]
  8. T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
    [Crossref]
  9. D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
    [Crossref]
  10. S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref]
  11. P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
    [Crossref]
  12. H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
    [Crossref]
  13. H. Xiong and Y. Wu, “Optomechanical Akhmediev breathers,” Laser Photon. Rev. 12, 1700305 (2018).
    [Crossref]
  14. L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
    [Crossref]
  15. X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
    [Crossref]
  16. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
    [Crossref]
  17. H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
    [Crossref]
  18. H. Xiong, Z.-X. Liu, and Y. Wu, “Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation,” Opt. Lett. 42, 3630–3633 (2017).
    [Crossref]
  19. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
    [Crossref]
  20. X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
    [Crossref]
  21. M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
    [Crossref]
  22. D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
    [Crossref]
  23. B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
    [Crossref]
  24. D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
    [Crossref]
  25. Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
    [Crossref]
  26. Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
    [Crossref]
  27. C. Kittel, “Interaction of spin waves and ultrasonic waves in ferromagnetic crystals,” Phys. Rev. 110, 836–841 (1958).
    [Crossref]
  28. J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
    [Crossref]
  29. M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
    [Crossref]
  30. J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
    [Crossref]
  31. C. Kong, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “Magnetically controllable slow light based on magnetostrictive forces,” Opt. Express 27, 5544–5556 (2019).
    [Crossref]
  32. Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
    [Crossref]
  33. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
    [Crossref]
  34. S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
    [Crossref]
  35. L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
    [Crossref]
  36. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2004).
  37. B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
    [Crossref]
  38. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref]
  39. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley-Interscience, 1987).
  40. Z. X. Liu, B. Wang, H. Xiong, and Y. Wu, “Magnon-induced high-order sideband generation,” Opt. Lett. 43, 3698–3701 (2018).
    [Crossref]
  41. X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
    [Crossref]

2019 (4)

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
[Crossref]

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

C. Kong, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “Magnetically controllable slow light based on magnetostrictive forces,” Opt. Express 27, 5544–5556 (2019).
[Crossref]

2018 (8)

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

Z. X. Liu, B. Wang, H. Xiong, and Y. Wu, “Magnon-induced high-order sideband generation,” Opt. Lett. 43, 3698–3701 (2018).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
[Crossref]

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref]

Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97, 013801 (2018).
[Crossref]

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

H. Xiong and Y. Wu, “Optomechanical Akhmediev breathers,” Laser Photon. Rev. 12, 1700305 (2018).
[Crossref]

2017 (5)

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
[Crossref]

H. Xiong, Z.-X. Liu, and Y. Wu, “Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation,” Opt. Lett. 42, 3630–3633 (2017).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref]

C. Bekker, R. Kalra, C. Baker, and W. P. Bowen, “Injection locking of an electro-optomechanical device,” Optica 4, 1196–1204 (2017).
[Crossref]

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

2016 (4)

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

2015 (4)

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref]

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

2014 (4)

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

2013 (1)

2012 (1)

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

2011 (2)

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

2010 (1)

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

2009 (1)

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

2008 (1)

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

1978 (1)

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
[Crossref]

1975 (1)

T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
[Crossref]

1958 (1)

C. Kittel, “Interaction of spin waves and ultrasonic waves in ferromagnetic crystals,” Phys. Rev. 110, 836–841 (1958).
[Crossref]

Agarwal, G. S.

J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Alvermann, A.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Bakemeier, L.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref]

Baker, C.

Barzanjeh, S.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Bekker, C.

Bowen, W. P.

Butcher, J. C.

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley-Interscience, 1987).

Chan, J.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Chen, A. X.

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Chen, B.

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Creedon, D. L.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Delèglise, S.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Drullinger, R. E.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
[Crossref]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Fan, Y.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Farr, W. G.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Fehske, H.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref]

Feng, M.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Gan, J.-H.

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2004).

Gavartin, E.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Goryachev, M.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Guo, L. X.

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

Hammerer, K.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

Hansch, T. W.

T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Hu, C. M.

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Ishikawa, T.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Ishino, S.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Jiang, L.

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

Kalra, R.

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Kittel, C.

C. Kittel, “Interaction of spin waves and ultrasonic waves in ferromagnetic crystals,” Phys. Rev. 110, 836–841 (1958).
[Crossref]

Kong, C.

Kostylev, M.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Li, J.

J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
[Crossref]

Li, T. F.

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Li, T.-F.

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Li, X. H.

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Li, Y.

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Liu, Y.

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Liu, Z. X.

Liu, Z.-X.

Lü, X. Y.

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
[Crossref]

Lukin, M.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

Luo, X. Q.

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Luo, X.-Q.

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Ma, P. C.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Milburn, G. J.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Nakamura, Y.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Nori, F.

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Painter, O.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Rabl, P.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

Rivière, R.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Schawlow, A. L.

T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
[Crossref]

Schliesser, A.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Si, L. G.

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Si, L.-G.

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
[Crossref]

Sun, Z. Y.

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Tabuchi, Y.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Tang, H. X.

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

Tobar, M. E.

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Tombesi, P.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Usami, K.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Vitali, D.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Wallquist, M.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

Walls, F. L.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
[Crossref]

Wang, B.

Wang, L. D.

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Wang, M.

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Wang, S. P.

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Wang, X. M.

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Wang, Y. P.

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Wang, Y.-P.

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Wineland, D. J.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
[Crossref]

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Wu, W.

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Wu, Y.

C. Kong, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “Magnetically controllable slow light based on magnetostrictive forces,” Opt. Express 27, 5544–5556 (2019).
[Crossref]

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref]

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref]

Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97, 013801 (2018).
[Crossref]

H. Xiong and Y. Wu, “Optomechanical Akhmediev breathers,” Laser Photon. Rev. 12, 1700305 (2018).
[Crossref]

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

Z. X. Liu, B. Wang, H. Xiong, and Y. Wu, “Magnon-induced high-order sideband generation,” Opt. Lett. 43, 3698–3701 (2018).
[Crossref]

H. Xiong, Z.-X. Liu, and Y. Wu, “Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation,” Opt. Lett. 42, 3630–3633 (2017).
[Crossref]

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref]

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Wu, Y. Y.

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Xiao, Y.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Xiong, H.

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref]

C. Kong, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “Magnetically controllable slow light based on magnetostrictive forces,” Opt. Express 27, 5544–5556 (2019).
[Crossref]

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref]

H. Xiong and Y. Wu, “Optomechanical Akhmediev breathers,” Laser Photon. Rev. 12, 1700305 (2018).
[Crossref]

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97, 013801 (2018).
[Crossref]

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

Z. X. Liu, B. Wang, H. Xiong, and Y. Wu, “Magnon-induced high-order sideband generation,” Opt. Lett. 43, 3698–3701 (2018).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref]

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
[Crossref]

H. Xiong, Z.-X. Liu, and Y. Wu, “Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation,” Opt. Lett. 42, 3630–3633 (2017).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Xiong, W.

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Xu, X. W.

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Xue, H. B.

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Yamazaki, R.

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

Yang, X.

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Yang, X. X.

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38, 353–355 (2013).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

You, C.

You, J. Q.

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Zhai, A. P.

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Zhang, D.

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Zhang, G. Q.

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Zhang, J.

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Zhang, J. Q.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Zhang, X.

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

Zhang, X.-F.

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

Zhang, Z. M.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Zheng, A. S.

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Zhu, S. Y.

J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
[Crossref]

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
[Crossref]

Zoller, P.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2004).

Zou, C. L.

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

Zou, C.-L.

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

Appl. Phys. Lett. (1)

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Appl. Phys. Lett. 110, 171102 (2017).
[Crossref]

Appl. Phys. Rev. (1)

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

IEEE Photon. J. (1)

M. Wang, D. Zhang, X. H. Li, Y. Y. Wu, and Z. Y. Sun, “Magnon chaos in PT-symmetric cavity magnomechanics,” IEEE Photon. J. 11, 1–8 (2019).
[Crossref]

Laser Photon. Rev. (1)

H. Xiong and Y. Wu, “Optomechanical Akhmediev breathers,” Laser Photon. Rev. 12, 1700305 (2018).
[Crossref]

Nat. Commun. (1)

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nat. Commun. 8, 1368 (2017).
[Crossref]

Nature (2)

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[Crossref]

npj Quantum Inf. (1)

D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere,” npj Quantum Inf. 1, 15014 (2015).
[Crossref]

Opt. Commun. (1)

T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Optica (1)

Phys. Lett. A (1)

B. Chen, L. D. Wang, J. Zhang, A. P. Zhai, and H. B. Xue, “Second-order sideband effects mediated by microwave in hybrid electro-optomechanical systems,” Phys. Lett. A 380, 798–802 (2016).
[Crossref]

Phys. Rev. (1)

C. Kittel, “Interaction of spin waves and ultrasonic waves in ferromagnetic crystals,” Phys. Rev. 110, 836–841 (1958).
[Crossref]

Phys. Rev. A (7)

J. Li, S. Y. Zhu, and G. S. Agarwal, “Squeezed states of magnons and phonons in cavity magnomechanics,” Phys. Rev. A 99, 021801 (2019).
[Crossref]

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

L. G. Si, L. X. Guo, H. Xiong, and Y. Wu, “Tunable high-order-sideband generation and carrier-envelope-phase-dependent effects via microwave fields in hybrid electro-optomechanical systems,” Phys. Rev. A 97, 023805 (2018).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

X. W. Xu, Y. Li, A. X. Chen, and Y. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97, 013801 (2018).
[Crossref]

Phys. Rev. Appl. (1)

M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, “High-cooperativity cavity QED with magnons at microwave frequencies,” Phys. Rev. Appl. 2, 054002 (2014).
[Crossref]

Phys. Rev. B (1)

Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, “Magnon Kerr effect in a strongly coupled cavity-magnon system,” Phys. Rev. B 94, 224410 (2016).
[Crossref]

Phys. Rev. Lett. (8)

J. Li, S. Y. Zhu, and G. S. Agarwal, “Magnon-photon-phonon entanglement in cavity magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018).
[Crossref]

Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, “Bistability of cavity magnon polaritons,” Phys. Rev. Lett. 120, 057202 (2018).
[Crossref]

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, “Strongly coupled magnons and cavity microwave photons,” Phys. Rev. Lett. 113, 156401 (2014).
[Crossref]

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to chaos in optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).
[Crossref]

X. Y. Lü, L. G. Si, X. Yang, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma soliton dynamics based on the mechanical effect of light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref]

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
[Crossref]

Phys. Scr. (1)

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009).
[Crossref]

Sci. Adv. (1)

X.-F. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Cavity magnomechanics,” Sci. Adv. 2, e1501286 (2016).
[Crossref]

Sci. China Phys. Mech. Astron. (1)

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 050302 (2015).
[Crossref]

Science (1)

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref]

Other (2)

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2004).

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley-Interscience, 1987).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of a hybrid cavity electro–opto–magnomechatically system in which the micro deformation of YIG’s geometry structure is treated as an excellent mechanical resonator. The optical cavity and microwave cavity are simultaneously coupled to the same mechanical resonator via the magnetostrictive (radiation-pressure-like) interaction. The optical cavity is driven coherently by a pump field with frequency $ {w_{c1}} $ as well as a weak probe field with frequency $ {w_p} $, while the microwave cavity is driven by an ac voltage bias $ U(t)$ with frequency $ {w_{c2}} $. Meanwhile, the YIG sphere is directly driven by a microwave source with frequency $ {w_d} $. The instantaneous micro deformation of the YIG sphere from its center position is denoted by $ r + x $, where $ r $ is the radius of the YIG sphere. $ l $ and $ d $ give, respectively, the length of the optical cavity and the distance of the parallel plate capacitor $ C $ when $ x = 0 $.
Fig. 2.
Fig. 2. Calculation results of the transmission coefficient $ |{t_p}{|^2} $ and amplitude $ \eta $ of second-order sideband as a function of the power (i) $ {P_{c2}} $, (ii) $ {P_{c1}} $, and (iii) $ {P_d} $ at the resonance condition $ \delta = {\Omega _m} $. We use $ {\kappa _1}/2\pi = 250\,\,{\rm KHz} $, $ {\kappa _2}/2\pi = 170 \,\,{\rm KHz}$, $ {w_{c1}}/2\pi = 1.77 \,\, {\rm PHz}$, $ {w_{c2}}/2\pi = 7.47 \,\, {\rm GHz} $, $ {g_1}/2\pi = 46\,\,{\rm Hz} $, $ {g_2}/2\pi = 230\,\,{\rm Hz} $, $ {\Omega _m}/2\pi = 10 \,\,{\rm GHz} $, $ {\Gamma _m}/2\pi = 100\,\,{\rm Hz} $, $ {\gamma _m}/2\pi = 1 \,\,{\rm GHz} $, $ {g_m}/2\pi = 0.2\,\,{\rm Hz} $, $ {\varepsilon _p} = 0.05\,\,{\varepsilon _{c1}} $, and $ {\Delta _m} = {\Omega _m} $, $ {\Delta _{c1}} = {\Omega _m} $.
Fig. 3.
Fig. 3. Transmission coefficient $ |{t_p}{|^2} $ and amplitude $ \eta $ of second-order sideband versus the probe–pump detuning $ \delta $ for different values of $ {P_{c2}} $ and $ {P_d} $, with $ {P_{c1}} = 6 \,\,{\rm mW} $ and $ {\Delta _{c2}} = {\Omega _m} $. The other parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. Contour maps of the transmission intensity of probe field $ |{t_p}{|^2} $ as a function of the probe–pump detuning $ \delta $ and driving field detuning $ {\Delta _m} $ and pump field detuning $ {\Delta _{c1}} $, respectively. We use $ {P_d} = 12 \,\, \unicode{x0B5} {\rm W} $, $ {P_{c1}} = 6 \,\, {\rm mW} $, $ {P_{c2}} = 0.6 \,\,{\rm nW} $, and $ {\Delta _{c2}} = {\Omega _m} $. The other parameters are the same as in Fig. 2.
Fig. 5.
Fig. 5. (a), (b) Real and imaginary parts of $ {A_c} \equiv \sqrt {{\eta _1}{\kappa _1}} A_{12}^ - /{\varepsilon _p}{e^{ - i\psi }} $ under different phases. (c) Calculation results of the amplitude $ \eta $ of second-order sideband as a function of the probe–pump detuning $ \delta $ and the phase $ \psi $ of probe field. We use $ {\Delta _m} = {\Omega _m} $ and $ {\Delta _{c1}} = {\Omega _m} $. The other parameters are the same as in Fig. 4.
Fig. 6.
Fig. 6. Output higher-order sidebands spectra (in logarithmic scale). We use $ \psi = 0 $. The other parameters are the same as in Fig. 5.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

H t o t a l = H 0 + H i n t + H m e c h + H d , H 0 = w 1 a ^ 1 a ^ 1 + ϕ 2 L + e 2 2 C 2 + w m m ^ m ^ , H i n t = g 1 a ^ 1 a ^ 1 x C 0 2 d C 2 q 2 x + g m m ^ m ^ x , H m e c h = Ω m 2 ( x 2 + p 2 ) , H d = i η 1 κ 1 ε c 1 ( a ^ 1 e i w c 1 t a ^ 1 e i w c 1 t ) + i η 1 κ 1 ε p ( a ^ 1 e i ( w p t + φ ) a ^ 1 e i ( w p t + ψ ) ) + i ε d ( m ^ e i w d t m ^ e i w d t ) + q U ( t ) ,
q 2 2 [ C 1 + C 0 ( x ) ] q 2 2 C C 0 2 d C 2 q 2 x .
H t o t a l = w 1 a ^ 1 a ^ 1 + w m m ^ m ^ + Ω m 2 ( x 2 + p 2 ) + w 2 a ^ 2 a ^ 2 + g 1 a ^ 1 a ^ 1 x + g m m ^ m ^ x g 2 a ^ 2 a ^ 2 x + i η 1 κ 1 ε c 1 ( a ^ 1 e i w c 1 t a ^ 1 e i w c 1 t ) + i η 1 κ 1 ε p ( a ^ 1 e i ( w p t + ψ ) a ^ 1 e i ( w p t + ψ ) ) + i ε d ( m ^ e i w d t m ^ e i w d t ) + i η 2 κ 2 ε c 2 ( a ^ 2 e i w c 2 t a ^ 2 e i w c 2 t ) .
a ^ ˙ 1 = ( i Δ c 1 + κ 1 2 ) a ^ 1 i g 1 a ^ 1 x ^ + η 1 κ 1 ε p e i δ t i ψ + η 1 κ 1 ε c 1 + a ^ 1 i n , a ^ ˙ 2 = ( i Δ c 2 + κ 2 2 ) a ^ 2 + i g 2 a ^ 2 x ^ + η 2 κ 2 ε c 2 + a ^ 2 i n , m ^ ˙ = ( i Δ m + γ m ) m ^ i g m m ^ x ^ + ε d + m ^ i n , x ^ ˙ = Ω m p ^ , p ^ ˙ = i Ω m x ^ g m m ^ m ^ + g 2 a ^ 2 a ^ 2 g 1 a ^ 1 a ^ 1 Γ m p ^ + Γ ^ t h ,
a 10 = η 1 κ 1 ε c 1 i Δ 1 + κ 1 / 2 ,
a 20 = η 2 κ 2 ε c 2 i Δ 2 + κ 2 / 2 ,
m 0 = ε d i Δ + γ m ,
x 0 = g 2 | a 20 | 2 g 1 | a 10 | 2 g m | m 0 | 2 Ω m ,
d d t δ a 1 = ( i Δ 1 + κ 1 / 2 ) δ a 1 i g 1 a 10 δ x i g 1 δ a 1 δ x + η 1 κ 1 ε p e i ( δ t + ψ ) ,
d d t δ a 2 = ( i Δ 2 + κ 2 / 2 ) δ a 2 + i g 2 a 20 δ x + i g 2 δ a 2 δ x ,
d d t δ m = ( i Δ + γ m ) δ m i g m m 0 δ x i g m δ m δ x ,
Θ ^ δ x = Ω m g 2 ( a 20 δ a 2 + a 20 δ a 2 + δ a 2 δ a 2 ) Ω m g 1 ( a 10 δ a 1 + a 10 δ a 1 + δ a 1 δ a 1 ) + Ω m g m ( m 0 δ m + m 0 δ m + δ m δ m ,
δ a 1 ( 2 ) = n ( A 1 ( 2 ) n e i n δ t + A 1 ( 2 ) n + e i n δ t ) , δ m = n ( B n e i n δ t + B n + e i n δ t ) , δ x = n ( X n e i n δ t + X n e i n δ t ) ,
α 1 ( δ ) A 11 = i g 1 a 10 X 1 + η 1 κ 1 ε p e i ψ ,
β 1 ( δ ) A 11 + = i g 1 a 10 X 1 ,
α 2 ( δ ) A 21 = i g 2 a 20 X 1 , β 2 ( δ ) A 21 + = i g 2 a 20 X 1 ,
f 1 ( δ ) B 1 = i g m m 0 X 1 , f 2 ( δ ) B 1 + = i g m m 0 X 1 ,
ζ ( δ ) X 1 = Ω m g 2 [ a 20 ( A 21 + ) + a 20 A 21 ] Ω m g 1 [ a 10 ( A 11 + ) + a 10 A 11 ] Ω m g 1 [ m 0 ( B 1 + ) + m 0 B 1 ] ,
X 1 = X 1 ,
α 1 ( 2 δ ) A 12 = i g 1 ( a 10 X 2 + A 11 X 1 ) ,
β 1 ( 2 δ ) A 12 + = i g 1 ( a 10 X 2 + A 11 + X 1 ) ,
α 2 ( 2 δ ) A 22 = i g 2 ( a 20 X 2 + A 21 X 1 ) ,
β 2 ( 2 δ ) A 22 + = i g 2 ( a 20 X 2 + A 21 + X 1 ) ,
f 1 ( 2 δ ) B 2 = i g m ( m 0 X 2 + B 1 X 1 ) ,
f 2 ( 2 δ ) B 2 + = i g m ( m 0 X 2 + B 1 + X 1 ) ,
ζ ( 2 δ ) X 2 = Ω m g 2 [ a 20 ( A 22 + ) + a 20 A 22 + ( A 21 + ) A 21 ] Ω m g 1 [ a 10 ( A 12 + ) + a 10 A 12 + ( A 11 + ) A 11 ] Ω m g 1 [ m 0 ( B 2 + ) + m 0 B 2 + ( B 1 + ) B 1 ] ,
X 2 = X 2 ,
α 1 ( 2 ) ( n δ ) = κ 1 ( 2 ) / 2 + i Δ 1 ( 2 ) i n δ , β 1 ( 2 ) ( n δ ) = κ 1 ( 2 ) / 2 + i Δ 1 ( 2 ) + i n δ , f 1 ( n δ ) = i Δ + γ m i n δ , f 2 ( n δ ) = i Δ + γ m + i n δ , ζ ( n δ ) = Ω m 2 ( n δ ) 2 i n δ Γ m .
A 11 = [ i g 1 2 | a 10 | 2 α 1 2 ( δ ) M ( δ ) + 1 α 1 ( δ ) ] η 1 κ 1 ε p e i ψ ,
A 11 + = i g 1 2 | a 10 | 2 [ ( κ 1 / 2 + i δ ) 2 + Δ 1 2 ] [ M ( δ ) ] η 1 κ 1 ε p e i ψ ,
A 21 = i g 1 g 2 a 10 a 20 α 1 ( δ ) α 2 ( δ ) η 1 κ 1 ε p e i ψ ,
A 21 + = i g 1 g 2 a 10 a 20 β 1 ( δ ) ( κ 2 / 2 i Δ 2 + i δ ) [ M ( δ ) ] η 1 κ 1 ε p e i ψ ,
B 1 = i g 1 g m a 10 m 0 α 1 ( δ ) f 1 ( δ ) η 1 κ 1 ε p e i ψ ,
B 1 + = i a 10 m 0 g 1 g m f 2 ( δ ) ( κ 1 i Δ 1 + i δ ) [ M ( δ ) ] η 1 κ 1 ε p e i ψ ,
X 1 = g 1 a 10 α 1 ( δ ) M ( δ ) η 1 κ 1 ε p e i ψ ,
A 12 = A 11 X 1 g 1 3 | a 10 | 2 α 1 2 ( 2 δ ) M ( 2 δ ) + i a 10 g 1 E α 1 ( 2 δ ) M ( 2 δ ) i g 1 A 11 X 1 α 1 ( 2 δ ) ,
M ( n δ ) = ζ ( n δ ) Ω m 2 | a 20 | 2 Δ 2 g 2 2 ( κ 2 / 2 i n δ ) 2 + Δ 2 2 2 | m 0 | 2 Δ g m 2 ( γ m i n δ ) 2 + Δ 2 2 Δ 1 | a 10 | 2 g 1 2 ( κ 1 / 2 i n δ ) 2 + Δ 1 2 , E = g m B 1 ( B 1 + ) + g 1 A 11 ( A 11 + ) g 2 A 21 ( A 21 + ) + i X 1 [ a 10 g 1 2 ( A 11 + ) β 1 ( 2 δ ) + a 20 g 2 2 ( A 21 + ) κ 2 / 2 i Δ 2 2 i δ + g m 2 m 0 ( B 1 + ) f 2 ( 2 δ ) a 20 g 2 2 A 21 α 2 ( 2 δ ) ] .
l S o u t = c 1 e i w c 1 t + c p e i w p t η 1 κ 1 A 11 + e i ( w c 1 δ ) t η 1 κ 1 A 12 e i ( w c 1 + 2 δ ) t η 1 κ 1 A 12 + e i ( w c 1 2 δ ) t .
t p = 1 η 1 κ 1 A 11 ε p e i ψ .
η = | η 1 κ 1 A 12 / ε p e i ψ | ,

Metrics