Abstract

Optomechanics describes the interaction between the optical field and mechanics, and the optomechanical system provides an ideal interface between photons and phonons. The role of the electromagnetic field during optomechanical interaction is studied in this paper as it is regarded as a phonon transmission medium. An analytical model is built to study the phononic mode resonance and reveals the transmission properties of the phonons, which are related to the variance of the frequency of the electromagnetic field. Moreover, when one mechanical mode is driven, different mode resonant properties could be achieved on the transmission spectrum of phonons between the two mechanical modes. We believe that the current work provides significant results for the research of phononic devices.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  7. T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett. 98, 150802 (2007).
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  9. F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y.-x. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399–405 (2016).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  22. C. Wang, W.-W. Shen, S.-C. Mi, Y. Zhang, and T.-J. Wang, “Concentration and distribution of entanglement based on valley qubits system in graphene,” Sci. Bulletin 60, 2016–2021 (2015).
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  23. F.-G. Deng, B.-C. Ren, and X.-H. Li, “Quantum hyperentanglement and its applications in quantum information processing,” Sci. Bulletin 62, 46–68 (2017).
    [Crossref]
  24. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  25. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).
    [Crossref] [PubMed]
  26. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
    [Crossref]
  27. B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
    [Crossref] [PubMed]
  28. M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758 (2010).
    [Crossref] [PubMed]
  29. K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
    [Crossref] [PubMed]
  30. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998).
    [Crossref]
  31. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  32. A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, “Optically mediated hybridization between two mechanical modes,” Phys. Rev. Lett. 112, 013602 (2014).
    [Crossref] [PubMed]
  33. H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
    [Crossref] [PubMed]
  34. S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
    [Crossref] [PubMed]
  35. E. Gil-Santos, M. Labousse, C. Baker, A. Goetschy, W. Hease, C. Gomez, A. Lemaître, G. Leo, C. Ciuti, and I. Favero, “Light-mediated cascaded locking of multiple nano-optomechanical oscillators,” Phys. Rev. Lett. 118, 063605 (2017).
    [Crossref] [PubMed]
  36. M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
    [Crossref] [PubMed]
  37. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
    [Crossref] [PubMed]
  38. Y.-C. Liu, Y.-F. Xiao, Y.-L. Chen, X.-C. Yu, and Q. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
    [Crossref] [PubMed]
  39. X.-X. Ren, H.-K. Li, M.-Y. Yan, Y.-C. Liu, Y.-F. Xiao, and Q. Gong, “Single-photon transport and mechanical noon-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
    [Crossref]
  40. M. Ludwig, B. Kubala, and F. Marquardt, “The optomechanical instability in the quantum regime,” New J. Phys. 10, 095013 (2008).
    [Crossref]
  41. A. A. Gangat, T. M. Stace, and G. J. Milburn, “Phonon number quantum jumps in an optomechanical system,” New J. Phys. 13, 043024 (2011).
    [Crossref]
  42. Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
    [Crossref] [PubMed]
  43. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon–photon translator,” New J. Phys. 13, 013017 (2011).
    [Crossref]
  44. M. Aspelmeyer and K. Schwab, “Focus on mechanical systems at the quantum limit,” New J. Phys. 10, 095001 (2008).
    [Crossref]
  45. S. Singh, G. A. Phelps, D. S. Goldbaum, E. M. Wright, and P. Meystre, “All-optical optomechanics: An optical spring mirror,” Phys. Rev. Lett. 105, 213602 (2010).
    [Crossref]
  46. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  47. C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  48. S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
    [Crossref]
  49. M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).
    [Crossref]
  50. M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).
    [Crossref] [PubMed]

2017 (3)

F.-G. Deng, B.-C. Ren, and X.-H. Li, “Quantum hyperentanglement and its applications in quantum information processing,” Sci. Bulletin 62, 46–68 (2017).
[Crossref]

E. Gil-Santos, M. Labousse, C. Baker, A. Goetschy, W. Hease, C. Gomez, A. Lemaître, G. Leo, C. Ciuti, and I. Favero, “Light-mediated cascaded locking of multiple nano-optomechanical oscillators,” Phys. Rev. Lett. 118, 063605 (2017).
[Crossref] [PubMed]

Y.-P Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonon and magnon,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

2016 (7)

X.-F. Liu, F. Lei, M. Gao, X. Yang, C. Wang, Ş. K. Özdemir, L. Yang, and G.-L. Long, “Gain competition induced mode evolution and resonance control in erbium-doped whispering-gallery microresonators,” Opt. Express 24, 9550–9560 (2016).
[Crossref] [PubMed]

X.-F. Liu, F. Lei, M. Gao, X. Yang, G.-Q. Qin, and G.-L. Long, “Fabrication of a microtoroidal resonator with picometer precise resonant wavelength,” Opt. Lett. 41, 3603–3606 (2016).
[Crossref] [PubMed]

H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
[Crossref] [PubMed]

Y.-F. Xiao and Q. Gong, “Optical microcavity: from fundamental physics to functional photonics devices,” Sci. Bulletin 61, 185–186 (2016).
[Crossref]

C. Cao, X. Chen, Y. Duan, L. Fan, R. Zhang, T. Wang, and C. Wang, “Concentrating partially entangled w-class states on nonlocal atoms using low-q optical cavity and linear optical elements,” Sci. Chin. Phys. Mech. Astronomy 59, 100315 (2016).
[Crossref]

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

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y.-x. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399–405 (2016).
[Crossref]

2015 (5)

C. Wang, W.-W. Shen, S.-C. Mi, Y. Zhang, and T.-J. Wang, “Concentration and distribution of entanglement based on valley qubits system in graphene,” Sci. Bulletin 60, 2016–2021 (2015).
[Crossref]

S. Y. Shah, M. Zhang, R. Rand, and M. Lipson, “Master-slave locking of optomechanical oscillators over a long distance,” Phys. Rev. Lett. 114, 113602 (2015).
[Crossref] [PubMed]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

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, 11508–11517 (2015).
[Crossref] [PubMed]

X. Yang, Ş. K. Özdemir, B. Peng, H. Yilmaz, F.-C. Lei, G.-L. Long, and L. Yang, “Raman gain induced mode evolution and on-demand coupling control in whispering-gallery-mode microcavities,” Opt. Express 23, 29573–29583 (2015).
[Crossref] [PubMed]

2014 (5)

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

B. Peng, Ş. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

M. H. Matheny, M. Grau, L. G. Villanueva, R. B. Karabalin, M. Cross, and M. L. Roukes, “Phase synchronization of two anharmonic nanomechanical oscillators,” Phys. Rev. Lett. 112, 014101 (2014).
[Crossref] [PubMed]

A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, “Optically mediated hybridization between two mechanical modes,” Phys. Rev. Lett. 112, 013602 (2014).
[Crossref] [PubMed]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

2013 (3)

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref] [PubMed]

Y.-C. Liu, Y.-F. Xiao, Y.-L. Chen, X.-C. Yu, and Q. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111, 083601 (2013).
[Crossref] [PubMed]

X.-X. Ren, H.-K. Li, M.-Y. Yan, Y.-C. Liu, Y.-F. Xiao, and Q. Gong, “Single-photon transport and mechanical noon-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
[Crossref]

2012 (4)

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109, 233906 (2012).
[Crossref]

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. M. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[Crossref] [PubMed]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).
[Crossref] [PubMed]

2011 (3)

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

A. A. Gangat, T. M. Stace, and G. J. Milburn, “Phonon number quantum jumps in an optomechanical system,” New J. Phys. 13, 043024 (2011).
[Crossref]

A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon–photon translator,” New J. Phys. 13, 013017 (2011).
[Crossref]

2010 (4)

S. Singh, G. A. Phelps, D. S. Goldbaum, E. M. Wright, and P. Meystre, “All-optical optomechanics: An optical spring mirror,” Phys. Rev. Lett. 105, 213602 (2010).
[Crossref]

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

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755–758 (2010).
[Crossref] [PubMed]

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12, 065038 (2010).
[Crossref]

2009 (1)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

2008 (4)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

M. Aspelmeyer and K. Schwab, “Focus on mechanical systems at the quantum limit,” New J. Phys. 10, 095001 (2008).
[Crossref]

M. Ludwig, B. Kubala, and F. Marquardt, “The optomechanical instability in the quantum regime,” New J. Phys. 10, 095013 (2008).
[Crossref]

2007 (2)

T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett. 98, 150802 (2007).
[Crossref] [PubMed]

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

2005 (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

2003 (2)

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

1998 (2)

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1993 (1)

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high- q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327 (1993).
[Crossref]

1991 (1)

K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref] [PubMed]

1955 (1)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[Crossref]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[Crossref]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Alegre, T. P. M.

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

Anetsberger, G.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Arcizet, O.

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

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

Armani, D.

D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Aspelmeyer, M.

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

M. Aspelmeyer and K. Schwab, “Focus on mechanical systems at the quantum limit,” New J. Phys. 10, 095001 (2008).
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Figures (5)

Fig. 1
Fig. 1

The structure of the system. Here an input optical field works as a medium to connect the detached mechanical resonators. The CW pumping denote a continuous laser pump. The extra driven force acts on the mechanical resonator is symboled as EF.

Fig. 2
Fig. 2

The real and imaginary part of the optical material parameter Ξ. Fig. (a) shows the real part of this transmission parameter. While Fig. (b) corresponds to the imaginary part. In Fig. (c) we zoom the extremum value part of the imaginary part.

Fig. 3
Fig. 3

The module of the transmission rate from right hand excition to the postive and negative model of the left side resonator.(a) shows the absolute value of positive transmission coefficient | t 12 |, (b) shows the absolute value of positive transmission coefficient | t 12 + |.

Fig. 4
Fig. 4

The absolute value of the transmission coefficient for both positive and negative mode. (a) The transmission coefficient for the phonon with frequency 1GHz under different electromagnetic frequency, (b) The transmission coefficient for the phonon with frequency 0.5GHz under different electromagnetic frequency, (c) The transmission coefficient for the phonon with frequency 0.1GHz under different electromagnetic frequency.

Fig. 5
Fig. 5

The transmission coefficient for both positive and negative part under unbalcanced driven. (a) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −1GHz, (b) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −0.99GHz, (c) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −0.8GHz,.

Equations (53)

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H ^ = H ^ m e c h + H ^ o p t + H ^ i n t + H ^ d r i v e ,
H ^ m e c h = p ^ 1 2 2 m e f f + 1 2 m e f f Ω m , 1 x ^ 1 2 + p ^ 2 2 2 m e f f + 1 2 m e f f Ω m , 2 x ^ 2 2 ,
H ^ o p t = ω c a v ( a ^ a ^ + 1 2 ) ,
H ^ i n t = ( g 1 x ^ 1 + g 2 x ^ 2 ) a ^ a ^ ,
H ^ d r i v e = i η o κ o ( s i n , o ( t ) a ^ s i n , o * ( t ) a ^ ) + i η r κ r ( s i n , r ( t ) b ^ 1 s i n , r * ( t ) b ^ 1 ) .
δ a ˙ = ( i ( Ω δ Ω ) κ 2 ) δ a i α ( g 1 x 1 , z p f ( δ b 1 * + δ b 1 ) + g 2 x 2 , z p f ( δ b 2 * + δ b 2 ) ) ,
δ b ˙ 1 = ( i Ω κ r , 1 2 ) δ b 1 i g 1 x 1 , z p f ( α δ a + α δ a ) + η r κ r , i s r e i Ω p t ,
δ b ˙ 2 ( i Ω κ r , 2 2 ) δ b 2 i g 2 x 2 , z p f ( α δ a + α a ) .
δ a = A e i Ω p t + A + e i Ω p t ,
δ b 1 = B 1 e i Ω p t + B 1 + e i Ω p t ,
δ b 2 = B 2 e i Ω p t + B 2 + e i Ω p t .
χ r 1 ( Ω p ) B 1 = Ξ ( Ω p , δ Ω ) B 1 + s r ,
χ r 1 ( Ω p ) B 2 = Ξ ( Ω p , δ Ω ) B 1 ,
χ r 1 ( Ω p ) B 2 + = Ξ ( Ω p , δ Ω ) B 1 .
χ O ( Ω P , δ Ω ) = ( i Ω p i ( Ω δ Ω ) κ / 2 ) 1
χ r ( Ω 1 p ) = ( i Ω p i Ω κ r , i ) 1 ,
Ξ ( Ω p , δ Ω ) = | α | 2 G 2 [ χ O ( Ω P , δ Ω ) + χ O ( Ω P , δ Ω ) ] .
t 12 = Ξ ( Ω p , δ Ω ) κ r , 1 κ r , 2 χ r 2 ( Ω p ) χ r 1 ( Ω p ) Ξ ( Ω p , δ Ω ) ,
t 12 + = Ξ ( Ω p , δ Ω ) κ r , 1 κ r , 2 χ r 1 ( Ω p ) χ r 1 ( Ω p ) χ r 1 ( Ω p ) Ξ ( Ω p , δ Ω ) .
H ^ = H ^ m e c h + h ^ o p t + H ^ i n t + H ^ d r i v e
H ^ m e c h = p ^ 1 2 2 m e f f + 1 2 m e f f Ω m , 1 x ^ 1 2 + p ^ 2 2 2 m e f f + 1 2 m e f f Ω m , 2 x ^ 2 2
H ^ o p t = ( a ^ a ^ + 1 2 )
H ^ i n t = ( g 1 x ^ + g 2 x ^ 2 ) a ^ a ^
H ^ d r i v e = i η r κ r ( s i n , o ( t ) a ^ s i n , o * ( t ) a ^ ) + i η r κ r ( s i n , r ( t ) b ^ 1 s i n , r * ( t ) b ^ 1 )
d a ^ d t = ( i Δ κ 2 ) a ^ i ( g 1 2 m e f f Ω m , 1 ( b ^ 1 + b ^ 1 ) + g 2 2 m e f f Ω m , 1 ( b ^ 2 + b ^ 2 ) ) a ^ + η r κ r s i n , o ( t )
d b ^ 1 d t = ( i Ω m , 1 κ r , 1 2 ) b ^ 1 i g 1 2 m e f f Ω m , 1 a ^ a ^ + η r κ r , i s i n , r ( t )
d b ^ 2 d t = ( i Ω m , 2 κ r , 2 2 ) b ^ 2 i g 1 2 m e f f Ω m , 2 a ^ a ^
0 = ( i Δ κ 2 ) a ^ i ( g 1 ( b ^ 1 + b ^ 1 ) + g 2 ( b ^ 2 + b ^ 2 ) ) a ^ a ^ + η o κ o s i n , o ( t )
0 = ( i Ω i κ r , i 2 ) b ^ i i g i a ^ a ^ i = 1 , 2 .
α = η o κ o i Δ ¯ κ / 2 s i n , o
x ¯ i = a ¯ 2 m e f f , i Ω i 2 i = 1 , 2 .
δ a ^ ˙ = ( i Δ ¯ κ 2 ) δ a ^ i α ( g 1 ( δ b ^ 1 + δ b ^ 1 ) + g 2 ( δ b ^ 2 + δ b ^ 2 ) )
δ b ^ ˙ 1 = ( i Ω 1 κ r , 1 2 ) δ b ^ 1 i g 1 ( α δ a ^ + α δ a ^ ) + η r κ r , i s i n , r ( t )
δ b ^ ˙ 2 = ( i Ω 2 κ r , 2 2 ) δ b ^ 2 i g 2 ( α δ a ^ + α δ a ^ )
δ a ˙ = ( i ( Ω δ Ω ) κ 2 ) δ a i α ( g 1 ( δ b 1 + δ b 1 ) + g 2 ( δ b 2 + δ b 2 ) )
δ b ˙ 1 = ( i Ω κ r , 1 2 ) δ b 1 i g 1 ( α δ a + α δ a ) + η r κ r , i s r e i Ω p t
δ b ˙ 2 = ( i Ω κ r , 2 2 ) δ b 2 i g 2 ( α δ a + α δ a )
δ a = A e i Ω p t + A + e i Ω p t
δ b 1 = B 1 e i Ω p t + B 1 + e i Ω p t
δ b 2 = B 2 e i Ω p t + B 2 + e i Ω p t
χ O 1 ( Ω P , δ Ω ) A = i α { g 1 [ B 1 + + B 1 ] + g 2 [ B 2 + + B 2 ] }
χ O 1 ( Ω P , δ Ω ) A + = i α { g 1 [ B 1 + B 1 + ] + g 2 [ B 2 + B 2 + ] }
χ r 1 ( Ω p ) B 1 = i g 1 [ α A + + α A ] + s r
χ r 1 ( Ω p ) B 1 + = i g 1 [ α A + α A ]
χ r 1 ( Ω p ) B 2 = i g 2 [ α A + + α A ]
χ r 1 ( Ω p ) B 2 + = i g 2 [ α A + α A ]
χ O ( Ω P , δ Ω ) = ( i Ω p i ( Ω δ Ω ) κ / 2 ) 1
χ r ( Ω 1 p ) = ( i Ω p i Ω κ i , i ) 1
χ r 1 ( Ω p ) B 1 = | α | a g a [ χ O ( Ω P , δ Ω ) + χ O ( Ω P , δ Ω ) ] B 1 + s r
χ r 1 ( Ω p ) B 2 = | α | a g a [ χ O ( Ω P , δ Ω ) + χ O ( Ω P , δ Ω ) ] B 1
Ξ ( Ω p , δ Ω ) = | α | 2 g 2 [ χ O ( Ω P , δ Ω ) + χ O ( Ω P , δ Ω ) ]
t 12 = Ξ ( Ω p , δ Ω ) κ r , 1 κ r , 2 χ r 2 ( Ω p ) χ r 1 ( Ω p ) Ξ ( Ω p , δ Ω )
t 12 + = Ξ ( Ω p , δ Ω ) κ r , 1 κ r , 2 χ r 1 ( Ω p ) χ r 1 ( Ω p ) χ r 1 ( Ω p ) Ξ ( Ω p , δ Ω )

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