Abstract

We consider the probe absorption properties in a mechanically coupled optomechanical system in which the two coupled nanomechanical oscillators are driven by the time-dependent forces, respectively. It is found that the mechanical interaction splits the transparency window for a usual single-mode optomechanical system into two parts and then leads to appearance of the double optomechanically induced transparency. The distance between the two transparency positions (the frequency for the maximal transparency) is determined by the mechanical interaction amplitude. This can be explained by using optomechanical dressed-mode picture which is analogue to the interacting dark resonances in coherent atoms. When the mechanical resonators are driven by the external forces, the transparencies in the double-transparency spectrum can be increased into amplifications or be suppressed by tuning the amplitude of the forces. Additionally, it is shown that the double transparencies or the amplifications oscillate with the initial phases of the forces with a period of 2π. These investigations will be useful for more flexible controllability of multi-channel optical communication based on the optomechanical systems.

© 2017 Optical Society of America

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  1. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
    [Crossref]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
    [Crossref] [PubMed]
  3. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
    [Crossref]
  4. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [Crossref] [PubMed]
  5. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
    [Crossref] [PubMed]
  6. Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).
  7. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
    [Crossref] [PubMed]
  8. M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
    [Crossref]
  9. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
    [Crossref]
  10. G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
    [Crossref]
  11. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
    [Crossref] [PubMed]
  12. A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
    [Crossref] [PubMed]
  13. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
    [Crossref] [PubMed]
  14. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
    [Crossref] [PubMed]
  15. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
    [Crossref] [PubMed]
  16. X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
    [Crossref] [PubMed]
  17. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
    [Crossref] [PubMed]
  18. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
    [Crossref] [PubMed]
  19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
    [Crossref] [PubMed]
  20. L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
    [Crossref]
  21. J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
    [Crossref] [PubMed]
  22. X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
    [Crossref]
  23. Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
    [Crossref]
  24. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
    [Crossref] [PubMed]
  25. J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
    [Crossref] [PubMed]
  26. K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
    [Crossref]
  27. Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
    [Crossref]
  28. Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
    [Crossref]
  29. F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
    [Crossref] [PubMed]
  30. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
    [Crossref]
  31. H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
    [Crossref]
  32. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
    [Crossref]
  33. C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
    [Crossref]
  34. M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
    [Crossref] [PubMed]
  35. K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
    [Crossref] [PubMed]
  36. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
    [Crossref] [PubMed]
  37. H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
    [Crossref]
  38. 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(4), 043825 (2014).
    [Crossref]
  39. Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
    [Crossref]
  40. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
    [Crossref]
  41. J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
    [Crossref]
  42. J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
    [Crossref] [PubMed]
  43. W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
    [Crossref]
  44. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
    [Crossref]
  45. K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
    [Crossref]
  46. H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
    [Crossref]
  47. H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
    [Crossref]
  48. X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
    [Crossref]
  49. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
    [Crossref]
  50. C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
    [Crossref]
  51. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
    [Crossref] [PubMed]
  52. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [Crossref]
  53. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
    [Crossref]

2017 (1)

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

2016 (6)

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

2015 (10)

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

2014 (5)

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[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(4), 043825 (2014).
[Crossref]

2013 (9)

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref] [PubMed]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref] [PubMed]

2012 (7)

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

2011 (6)

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

2010 (3)

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

2009 (2)

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

2007 (2)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref] [PubMed]

1999 (1)

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Agarwal, G. S.

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Alegre, T. P.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Asano, T.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Ashhab, S.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Bai, C.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Bian, X. T.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Børkje, K.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Brown, E.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Chang, C. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Chang, E. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chen, B.

Chen, G.

Chen, G. B.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Chen, Q.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

Cui, Y.

Cui, Y. S.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

Dong, C. H.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Du, C.

Du, C. G.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Fang, J. X.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Fang, K. J.

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Feng, M.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[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(4), 043825 (2014).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Fleischhauer, M.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

Gao, M.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Girvin, S. M.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Gourgout, A.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Gröblacher, S.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Guo, G. C.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Guo, Y. J.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

Habraken, S. J. M.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Hammerer, K.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

He, Y.

Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
[Crossref]

Hill, J. T.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Hou, B. P.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

Huang, S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Huang, Y. M.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

Jia, W. Z.

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

Jiang, C.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref] [PubMed]

Jing, Q. L.

Kamada, T.

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref] [PubMed]

Komar, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Krause, A.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Kronwald, A.

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref] [PubMed]

Lai, D. G.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

Lei, F. C.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

Lemonde, M. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

Li, J.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

Li, K.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

Li, X.

Li, Y.

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Liao, J. Q.

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Liu, H.

Liu, Y. C.

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Liu, Y. X.

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

Long, G. L.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

Lü, X. Y.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Luan, X.

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Ludwig, M.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

Lukin, M. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

Ma, J.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

Ma, P. C.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[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(4), 043825 (2014).
[Crossref]

Mahboob, I.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

Matheny, M. H.

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Mayer Alegre, T. P.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Meystre, P.

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

Naderi, M. H.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

Nie, W. J.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

Noda, S.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Nori, F.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Nunnenkamp, A.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Okamoto, H.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Onomitsu, K.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Painter, O.

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Qu, K.

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

Rabl, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Sato, Y.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Scully, M. O.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

Shahidani, S.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

Shen, Z.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Shu, F. J.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Si, L. G.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Soltanolkotabi, M.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

Stannigel, K.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Sterling, R.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Suzuki, H.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Takahashi, Y.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Tanaka, Y.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Teufel, J. D.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

Tian, L.

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

Tombesi, P.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Upham, J.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Vahala, K. J.

Vanner, M. R.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Vedral, V.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Vitali, D.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Wan, L. L.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

Wang, H.

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

Wang, L. L.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Wang, Q.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

Wang, S. J.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

Wang, Y. D.

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

Wei, L. F.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Wong, C. W.

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Wu, D.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

Wu, Q. Q.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

Wu, X. Y.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Wu, Y.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[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(4), 043825 (2014).
[Crossref]

Xiao, Y. F.

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Xiong, H.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Xu, X. W.

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

Xu, Y.

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Yamaguchi, H.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Yang, X.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Yao, C. M.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

Yelin, S. F.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

You, C.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

Yu, H. L.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Zeilinger, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Zhang, J. Q.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[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(4), 043825 (2014).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Zhang, W. M.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

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(4), 043825 (2014).
[Crossref]

Zheng, A. S.

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Zhou, Z. H.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Zhuang, X. G.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Zuo, F.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Ann. Phys. (1)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

Appl. Phys. Express (1)

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Nat. Commun. (1)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

Nat. Photonics (2)

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Nat. Phys. (1)

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Nature (3)

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (20)

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[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(4), 043825 (2014).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93(4), 043804 (2016).
[Crossref]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Phys. Rev. Lett. (11)

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

Sci. China Phys. Mech. Astron. (3)

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Sci. China Technol. Sci. (1)

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Sci. Rep. (2)

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

Science (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of a mechanically coupled optomechanical system, in which the two coupled nanomechanical resonators are driven by the time-dependent forces, respectively.
Fig. 2
Fig. 2 The absorption Re[εT] as a function of σ/ωm1 for different values of the mechanical interaction: V = 0 (black, solid curve); V = 1.5 × 8 × 105 Hz/m2 (red, dashed curve); V = 3 × 8 × 105Hz/m2 (blue, dotted curve). The other values of the parameters are given by: PL = 2μW, m1 = m2 = 145ng, κ = 2π × 215 × 103Hz, ωm1 = ωm2 = 2π × 947 × 103Hz, γ1 = γ2 = 2π × 140Hz, L = 25mm, D1 = 0, D2 = 0, ϕ1 = 0 and ϕ2 = 0.
Fig. 3
Fig. 3 The level-diagram picture constructed by the optical and mechanical modes (a), and the dressed mode picture with double Λ configuration (b).
Fig. 4
Fig. 4 The absorption Re[εT ] as a function of σ/ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D1: D1 = 0 (black, solid curve); D1 = 1 × 109N (red, dashed curve); D1 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 5
Fig. 5 The absorption Re[εT] as a function of D1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 4.
Fig. 6
Fig. 6 The absorption Re[εT] as a function of σ /ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D2: D2 = 0 (black, solid curve); D2 = 1 × 109N (red, dashed curve); D2 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 7
Fig. 7 The absorptions Re[εT ] as a function of D2 at σ = −0.2ωm1 (black, solid curve) and σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 6.
Fig. 8
Fig. 8 The absorption Re[εT ] as a function of σ /ωm1 with D1 = 1 × 109N for different values of the phase ϕ1: ϕ1 = 0 (black, solid curve); ϕ1 = π/2 (red, dashed curve); ϕ1 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 3.
Fig. 9
Fig. 9 The absorption Re[εT] as a function of ϕ1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 8. The dotted line denotes the X-axis.
Fig. 10
Fig. 10 The absorption Re[εT] as a function of σ /ωm1 for different values of the phase ϕ2 with D2 = 1 × 109N: ϕ2 = 0 (black, solid curve); ϕ2 = π/2 (red, dashed curve); ϕ2 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 6.
Fig. 11
Fig. 11 The absorption Re[εT] as a function of ϕ2: σ = −0.2ωm1 (black, solid curve); σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 10. The dotted line denotes the X-axis.

Equations (22)

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H= Δ c a ^ + a ^ + ω m1 b ^ 1 + b ^ 1 + ω m2 b ^ 2 + b ^ 2 +g a ^ + a ^ ( b ^ 1 + + b ^ 1 )+V( b ^ 1 + b ^ 2 + b ^ 1 b ^ 2 + ) + i D 1 ( b ^ 1 + e i ϕ 1 e i ω D 1 t b ^ 1 e i ϕ 1 e i ω D 1 t )+i D 2 ( b ^ 2 + e i ϕ 2 e i ω D 2 t b ^ 2 e i ϕ 2 e i ω D 2 t ) + i E L ( a ^ + a ^ )+i E P ( a ^ + e iδt H.c.).
d a ^ dt ={ κ+i[ Δ c +g( b ^ 1 + + b ^ 1 ) ] } a ^ + E P e -iδt + E L + 2κ a ^ in ,
d b ^ 1 dt =( γ 1 +i ω m1 ) b ^ 1 ig a ^ + a ^ iV b ^ 2 + D 1 e i ϕ 1 e i ω D 1 t + 2 γ 1 b ^ 1,in ,
d b ^ 2 dt =( γ 2 +i ω m2 ) b ^ 2 iV b ^ 1 + D 2 e i ϕ 2 e i ω D 2 t + 2 γ 2 b ^ 2,in .
a s = E L κ+iΔ ,
b 1s = ig E L 2 ( γ 1 +i ω m1 + V 2 γ 2 +i ω m2 )( κ 2 + Δ 2 ) ,
b 2s = Vg E L 2 [ ( γ 1 +i ω m1 )( γ 2 +i ω m2 )+ V 2 ]( κ 2 + Δ 2 ) ,
δ a ˙ =( κ+iΔ )δaig( δ b 1 + +δ b 1 ) a s + E P e iδt ,
δ b ˙ 1 =( γ 1 +i ω m1 )δ b 1 ig( a s δa+δ a a s )iVδ b 2 + D 1 e i ϕ 1 e i ω D 1 t ,
δ b ˙ 2 =( γ 2 +i ω m2 )δ b 2 iVδ b 1 + D 2 e i ϕ 2 e i ω D 2 t .
δa= a 1+ e iδt + a 1 e iδt + a 11+ e i ω D 1 t + a 11 e i ω D 1 t + a 12+ e i ω D 2 t + a 12 e i ω D 2 t ,
δ b 1 = b 1+ e iδt + b 1 e iδt + b 11+ e i ω D 1 t + b 11 e i ω D 1 t + b 12+ e i ω D 2 t + b 12 e i ω D 2 t ,
δ b 2 = b 2+ e iδt + b 2 e iδt + b 21+ e i ω D 1 t + b 21 e i ω D 1 t + b 22+ e i ω D 2 t + b 22 e i ω D 2 t .
a 1+ = E P ( 1 M 2 M 1 M 3 1 M 4 ) Γ 1 ,
a 11+ = igD 1 e i ϕ 1 [ a s M 5 ( 1 M 8 ) G 3 Γ 21+ + a s G 3 Γ 21 ] 1 M 6 M 5 M 7 1 M 8 ,
a 12+ = gV D 2 e i ϕ 2 Γ 62 [ a s M 9 ( 1 M 12 ) G 5 Γ 22+ + a s G 5 Γ 22 ] 1 M 10 M 9 M 11 1 M 12 .
M 1 = g 2 a s 2 Γ 1 ( 1 G 1+ 1 G 1 ), M 2 = g 2 | a s | 2 Γ 1 ( 1 G 1+ 1 G 1 ), M 3 = g 2 a s 2 Γ 1+ ( 1 G 1 1 G 1+ ), M 4 = g 2 | a s | 2 Γ 1+ ( 1 G 1 1 G 1+ ), M 5 = g 2 a s 2 Γ 21 ( 1 G 21+ 1 G 21 ), M 6 = g 2 | a s | 2 Γ 21 ( 1 G 21+ 1 G 21 ), M 7 = g 2 a s 2 Γ 21+ ( 1 G 21 1 G 21+ ), M 8 = g 2 | a s | 2 Γ 21+ ( 1 G 21 1 G 21+ ), M 9 = g 2 a s 2 Γ 22 ( 1 G 22+ 1 G 22 ), M 10 = g 2 | a s | 2 Γ 22 ( 1 G 22+ 1 G 22 ), M 11 = g 2 a s 2 Γ 22+ ( 1 G 22 1 G 22+ ), M 12 = g 2 | a s | 2 Γ 22+ ( 1 G 22 1 G 22+ ),
G 1± = Γ 3± + V 2 Γ 5± , G 2j± = Γ 4j± + V 2 Γ 6j± , (j=1,2)
Γ 1± =κ±i( δ±Δ ), Γ 2j± =κ±i( ω D j ±Δ ), Γ 3± = γ 1 +i( ω m 1 ±δ ), Γ 4j± = γ 1 +i( ω m 1 ± ω D j ), Γ 5± = γ 2 +i( ω m 2 ±Δ ), Γ 6j± = γ 2 +i( ω m 2 ± ω D j ).
ε T = 2κ( a 1+ + a 11+ + a 12+ ) E p .
H= Δ c a ^ + a ^ + ω m b ^ + b ^ + ω m+ b ^ + + b ^ + + 1 2 g a ^ + a ^ ( b ^ + + + b ^ + )+ 1 2 g a ^ + a ^ ( b ^ + + b ^ ) + i E L ( a ^ + a ^ )+i E P ( a ^ + e iδt H.c.),
H= [ Δ c g ω m+ ( D 1 sin( ϕ 1 + ω D 1 t ) D 2 sin( ϕ 2 + ω D 2 t ) ) g ω m ( D 1 sin( ϕ 1 + ω D 1 t )+ D 2 sin( ϕ 2 + ω D 2 t ) ) ] a ^ + a ^ + j=+, ω mj B ^ j + B ^ j + 1 2 g a ^ + a ^ j=+, ( B ^ j + + B ^ j ) + i( E L a ^ + + E P a ^ + e iδt H.c.),

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