Abstract

We investigate the optomechanical interaction in two coupled microresonators. Compared to the single resonator optomechanical system where the input light is required to detune from the cavity resonance to generate two asymmetrical sidebands and thus large mechanical damping/amplification, the coupled resonator system can allow both the input light and its frequency sideband to be on resonance. In this configuration, we find that the optomechanical interaction can be enhanced and optically induced energy transfer between different mechanical oscillators is possible.

© 2012 OSA

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  1. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [Crossref] [PubMed]
  2. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
    [Crossref]
  3. 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]
  4. W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express 17, 12424–12432 (2009).
    [Crossref] [PubMed]
  5. D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nature Photon. 4, 211–217 (2010).
    [Crossref]
  6. F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
    [Crossref]
  7. S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
    [Crossref]
  8. M. L. Povinelli, M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005).
    [Crossref] [PubMed]
  9. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [Crossref] [PubMed]
  10. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [Crossref] [PubMed]
  11. A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
    [Crossref] [PubMed]
  12. J. Chan, T. P. M. 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, 89–92 (2011).
    [Crossref] [PubMed]
  13. Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
    [Crossref] [PubMed]
  14. Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nature Phys. 5, 489–493 (2009).
    [Crossref]
  15. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
    [Crossref] [PubMed]
  16. A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
    [Crossref]
  17. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
    [Crossref]
  18. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phononphoton translator,” New J. Phys. 13, 013017 (2011).
    [Crossref]
  19. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
    [Crossref] [PubMed]
  20. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
    [Crossref] [PubMed]
  21. 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, 69–73 (2011).
    [Crossref] [PubMed]
  22. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
    [Crossref] [PubMed]
  23. X. Zhao, J. M. Tsai, H. Cai, X. M. Ji, J. Zhou, M. H. Bao, Y. P. Huang, D. L. Kwong, and A. Q. Liu, “A nano-opto-mechanical pressure sensor via ring resonator,” Opt. Express 20, 8535–8542 (2012).
    [Crossref] [PubMed]
  24. M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Topics Quantum Electron. 16, 276–287 (2010).
    [Crossref]
  25. X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (2011).
    [Crossref] [PubMed]
  26. X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
    [Crossref]
  27. Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
    [Crossref]
  28. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
    [Crossref] [PubMed]
  29. S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” OSA/CLEO (2011).

2012 (2)

X. Zhao, J. M. Tsai, H. Cai, X. M. Ji, J. Zhou, M. H. Bao, Y. P. Huang, D. L. Kwong, and A. Q. Liu, “A nano-opto-mechanical pressure sensor via ring resonator,” Opt. Express 20, 8535–8542 (2012).
[Crossref] [PubMed]

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
[Crossref]

2011 (4)

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (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, 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

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

2010 (6)

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[Crossref] [PubMed]

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]

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nature Photon. 4, 211–217 (2010).
[Crossref]

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Topics Quantum Electron. 16, 276–287 (2010).
[Crossref]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

2009 (8)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[Crossref] [PubMed]

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express 17, 12424–12432 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
[Crossref]

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nature Phys. 5, 489–493 (2009).
[Crossref]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

2008 (2)

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

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

2007 (2)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

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

2006 (1)

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

2005 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. 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, 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, 1520–1523 (2010).
[Crossref] [PubMed]

Aspelmeyer, M.

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

Bao, M. H.

Cai, H.

Camacho, R.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Capasso, F.

Chan, J.

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, 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

Chang, D.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[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, 69–73 (2011).
[Crossref] [PubMed]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[Crossref] [PubMed]

Deléglise, S.

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]

DelHaye, P.

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Domokos, P.

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[Crossref] [PubMed]

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, 69–73 (2011).
[Crossref] [PubMed]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

Fink, Y.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Fong, K. Y.

Freegarde, T.

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[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, 1520–1523 (2010).
[Crossref] [PubMed]

Girvin, S.

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[Crossref] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Gröblacher, S.

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

Grudinin, I. S.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

Hill, J. T.

J. Chan, T. P. M. 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, 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, 69–73 (2011).
[Crossref] [PubMed]

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

Horak, P.

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[Crossref] [PubMed]

Hossein-Zadeh, M.

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Topics Quantum Electron. 16, 276–287 (2010).
[Crossref]

Huang, Y. P.

Ibanescu, M.

M. L. Povinelli, M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005).
[Crossref] [PubMed]

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Ji, X. M.

Jiang, X.

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

Joannopoulos, J.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Joannopoulos, J. D.

Johnson, S.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Johnson, S. G.

Kippenberg, T.

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Kippenberg, T. J.

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]

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

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

Krause, A.

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

Kwong, D. L.

Lee, H.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

Li, M.

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express 17, 12424–12432 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

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, 69–73 (2011).
[Crossref] [PubMed]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

Lipson, M.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[Crossref] [PubMed]

S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” OSA/CLEO (2011).

Liu, A. Q.

Loncar, M.

Manipatruni, S.

S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” OSA/CLEO (2011).

Marquardt, F.

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[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, 69–73 (2011).
[Crossref] [PubMed]

Michael, C. P.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

Nooshi, N.

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Painter, O.

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phononphoton translator,” New J. Phys. 13, 013017 (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, 69–73 (2011).
[Crossref] [PubMed]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

Park, Y.-S.

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nature Phys. 5, 489–493 (2009).
[Crossref]

Perahia, R.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

Pernice, W. H. P.

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (2011).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express 17, 12424–12432 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

Povinelli, M. L.

Rivière, R.

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]

Roels, J.

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nature Photon. 4, 211–217 (2010).
[Crossref]

Rosenberg, J.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phononphoton translator,” New J. Phys. 13, 013017 (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, 69–73 (2011).
[Crossref] [PubMed]

Schliesser, A.

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]

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Skorobogatiy, M.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Smythe, E. J.

Sun, X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
[Crossref]

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (2011).
[Crossref] [PubMed]

Tang, H. X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
[Crossref]

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (2011).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express 17, 12424–12432 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

Thourhout, D. V.

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nature Photon. 4, 211–217 (2010).
[Crossref]

Tsai, J. M.

Vahala, K.

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Vahala, K. J.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Topics Quantum Electron. 16, 276–287 (2010).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

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

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

Wang, H.

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nature Phys. 5, 489–493 (2009).
[Crossref]

Weiderhecker, G.

S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” OSA/CLEO (2011).

Weis, S.

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]

Weisberg, O.

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[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, 69–73 (2011).
[Crossref] [PubMed]

Xiong, C.

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, “GHz optomechanical resonators with high mechanical Q factor in air,” Opt. Express 19, 22316–22321 (2011).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

Xuereb, A.

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[Crossref] [PubMed]

Zhang, X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
[Crossref]

Zhao, X.

Zhou, J.

Appl. Phys. Lett. (1)

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012)
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Topics Quantum Electron. 16, 276–287 (2010).
[Crossref]

Nature (7)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[Crossref] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (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, 69–73 (2011).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[Crossref] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462, 633–637 (2009).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

J. Chan, T. P. M. 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, 89–92 (2011).
[Crossref] [PubMed]

Nature Photon. (3)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nature Photon. 1, 416–422 (2007).
[Crossref]

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nature Photon. 4, 211–217 (2010).
[Crossref]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nature Photon. 4, 236–242 (2010).
[Crossref]

Nature Phys. (1)

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nature Phys. 5, 489–493 (2009).
[Crossref]

New J. Phys. (1)

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

Opt. Express (4)

Opt. Lett. (1)

Phys. Lett. (1)

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Lett. 103, 223901 (2009).
[Crossref]

Phys. Rev. E (1)

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

Phys. Rev. Lett. (4)

A. Xuereb, T. Freegarde, P. Horak, and P. Domokos, “Optomechanical cooling with generalized interferometers,” Phys. Rev. Lett. 105, 013602 (2010).
[Crossref] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[Crossref] [PubMed]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

A. Schliesser, P. DelHaye, N. Nooshi, K. Vahala, and T. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Physics (1)

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Science (2)

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]

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

Other (1)

S. Manipatruni, G. Weiderhecker, and M. Lipson, “Long-range synchronization of optomechanical structures,” OSA/CLEO (2011).

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Figures (6)

Fig. 1
Fig. 1

The schematics and sideband picture of the optomechanical system based on (a) a single resonator; (b) two coupled resonators. The mechanical motion modifies the optical resonance frequency.

Fig. 2
Fig. 2

Optical field energies of the two resonators in the: (a) weak coupling regime, κ = Γ2; (b) strong coupling regime, κ = 10Γ2. The intrinsic Q-factors are Qi1 = Qi2 = 106, and the external Q-factor of the first resonator is Qe1 = 105. The input optical power is 1mW. For simplicity, the two resonators are assumed to have the same intrinsic resonance, Δ = Δ1 = Δ2.

Fig. 3
Fig. 3

Calculated (a)(b) optomechanical amplification (Γ′m1 < Γm1) or cooling (Γ′m1 > Γm1) and (c)(d) optical spring effect for (1) single optical resonator with Qe = Qi = 106, Ωm = 2π × 500MHz; (2) coupled resonator with Qe1 = Qi1 = Qi2 = 106, κ = 0.5Ωm. The blue curves denote the normalized optical energy inside the resonators. The input power is 1mW. The optomechanical coupling coefficient is gi = 10GHz/nm. The effective mass and Q-factor of the mechanical oscillator are set at mi = 10pg and Qm=1000, respectively.

Fig. 4
Fig. 4

(a) Normalized power transmission in the coupled resonator system for OMIT effect. The control light is on resonance with one of the optical resonances with smaller frequency. The frequency detuning of the probe light from the control light is scanned to obtain its transmission spectrum. The input power of control and probe light are 1mW and 10μW, respectively. Qi1 = Qi2 = Qe1 = 106, m1 = 10pg, g1 = 20GHz/nm, Qm = 1000, Ωm = 2π × 500MHz, κ = 0.5Ωm. (b) Comparison of the normalized peak transmission for OMIT effect in a single resonator and coupled resonator system. The single resonator system has the same optical decay rates and properties of mechanical oscillation as the first resonator in the coupled resonator system.

Fig. 5
Fig. 5

(a) The optical energy inside the first (red) and second (blue) resonator. (b) The calculated quadrature force component generated by optical field in the first (red) and second (blue) resonator. The solid lines are calculated when mechanical oscillation exists in both resonators. The dotted lines are calculated when only one resonator has mechanical motion. The Q-factors of the two resonators are Qi1 = Qi2 = 106, Qe = 105, Qκ = 106.

Fig. 6
Fig. 6

(a) The optical energy and (b) the calculated quadrature force component generated by optical field in the first (red) and second (blue) resonator. The Q-factors of the two resonators are: Qi1 = Qκ = 106, Qi2 = Qe = 5 × 105,.

Equations (18)

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

d a 1 d t = ( i Δ 1 Γ 1 / 2 i g 1 x 1 ) a 1 + i Γ e A in + i κ a 2
d a 2 d t = ( i Δ 2 Γ 2 / 2 i g 2 x 2 ) a 2 + i κ a 1
d 2 x i d t 2 + Γ m i d x i d t + Ω m i 2 x i = F i m i + F th m i
a ¯ 1 = i Γ e A in i Δ ¯ 1 Γ 1 + κ 2 i Δ ¯ 2 Γ 2 / 2
a ¯ 2 = i κ a ¯ 1 i Δ ¯ 2 Γ 2 / 2
x ¯ i = g i | a ¯ i | 2 m i Ω m i 2 ω
d δ a 1 d t = ( i Δ ¯ Γ 1 2 ) δ a 1 i g 1 a ¯ 1 δ x 1 + i κ δ a 2 + i Γ e δ A in d δ a 2 d t = ( i Δ ¯ Γ 2 2 ) δ a 2 i g 2 a ¯ 2 δ x 2 + i κ δ a 1 d 2 δ x i d t 2 + Γ m i d δ x i d t + Ω m i 2 δ x i = g i m i ω ( a ¯ i δ a i * + a ¯ i * δ a i ) + F th m i
[ ( Ω m 1 2 Ω 2 i Γ m 1 Ω ) + i g 1 2 | a ¯ 1 | 2 m 1 ω [ f + ( Δ ¯ , Ω ) + f ( Δ ¯ , Ω ) ] ] δ x ˜ 1 ( Ω ) = F ˜ th ( Ω ) m 1
f + ( Δ ¯ , Ω ) = [ i ( Δ ¯ + Ω ) Γ 1 / 2 + κ 2 i ( Δ ¯ + Ω ) Γ 2 / 2 ] 1 f ( Δ ¯ , Ω ) = [ i ( Δ ¯ Ω ) + Γ 1 / 2 + κ 2 i ( Δ ¯ Ω ) + Γ 2 / 2 ] 1
( Ω m 1 ' ) 2 = Ω m 1 2 + { i g 1 2 | a ¯ 1 | 2 m 1 ω [ f + ( Δ ¯ , Ω ) + f ( Δ ¯ , Ω ) ] }
Γ m 1 ' = Γ m 1 { i g 1 2 | a ¯ 1 | 2 m 1 ω Ω [ f + ( Δ ¯ , Ω ) + f ( Δ ¯ , Ω ) ] }
δ a ˜ 1 ( Ω ) = i Γ e f + ( Ω ) { δ A ˜ in ( Ω ) + i g 1 2 ω 1 χ ( Ω ) a ¯ 1 f ( Ω ) [ a ¯ 1 * δ A ˜ in ( Ω ) a ¯ 1 δ A ˜ in * ( Ω ) ] } 1 + i g 1 2 ω 1 χ ( Ω ) | a ¯ 1 | 2 [ f + ( Ω ) + f ( Ω ) ]
d δ a 1 d t = ( i Δ 1 Γ 1 ) δ a 1 + i κ δ a 2 i g 1 x 1 cos ( Ω m t ) a ¯ 1
d δ a 2 d t = ( i Δ 2 Γ 2 ) δ a 2 + i κ δ a 1 i g 2 x 2 cos ( Ω m t ) a ¯ 2
δ a 1 ± = 1 2 i g 1 x 1 a ¯ 1 [ i ( Δ 2 Ω m ) Γ 2 / 2 ] + g 2 x 2 a ¯ 2 κ [ i ( Δ 1 Ω m ) Γ 1 / 2 ] [ i ( Δ 2 Ω m ) Γ 2 / 2 ] + κ 2
δ a 2 ± = 1 2 i g 2 x 2 a ¯ 2 [ i ( Δ 1 Ω m ) Γ 1 / 2 ] + g 1 x 1 a ¯ 1 κ [ i ( Δ 1 Ω m ) Γ 1 / 2 ] [ i ( Δ 2 Ω m ) Γ 2 / 2 ] + κ 2
F I i = 2 g i ω i [ a ¯ i * ( δ a i + + δ a i ) ]
F Q i = 2 g i ω i [ a ¯ i * ( δ a i + δ a i ) ]

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