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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    [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,” Science330, 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. Express17, 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,” Physics2, 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. E65, 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,” Nature424, 810–816 (2003).
    [CrossRef] [PubMed]
  10. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 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,” Nature478, 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,” Nature459, 550–555 (2009).
    [CrossRef] [PubMed]
  20. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 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,” Nature472, 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,” Nature456, 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. Express20, 8535–8542 (2012).
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    [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. Express19, 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,” Nature462, 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. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett.100, 173116 (2012)
[CrossRef]

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. Express20, 8535–8542 (2012).
[CrossRef] [PubMed]

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. Express19, 22316–22321 (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,” Nature478, 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,” Nature472, 69–73 (2011).
[CrossRef] [PubMed]

2010 (6)

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

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

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]

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]

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,” Nature462, 633–637 (2009).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express17, 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]

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82 (2009).
[CrossRef] [PubMed]

2008 (2)

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature456, 480–484 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (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. Express15, 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,” Nature424, 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. E65, 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,” Nature478, 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,” Science330, 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,” Nature478, 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,” Nature456, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 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,” Nature472, 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,” Nature478, 89–92 (2011).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 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,” Nature459, 550–555 (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,” Nature472, 69–73 (2011).
[CrossRef] [PubMed]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 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,” Science330, 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,” Nature472, 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,” Nature462, 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,” Nature459, 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. E65, 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,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

Girvin, S.

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–637 (2009).
[CrossRef] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature424, 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,” Nature478, 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.

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,” Nature472, 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,” Nature478, 89–92 (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,” Nature456, 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. E65, 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. E65, 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. E65, 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,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express15, 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,” Nature478, 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. Express17, 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,” Nature456, 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,” Nature472, 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,” Nature462, 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,” Physics2, 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,” Nature472, 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,” Nature478, 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,” Nature472, 69–73 (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]

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. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82 (2009).
[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]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 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]

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. Express19, 22316–22321 (2011).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express17, 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,” Nature456, 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,” Science330, 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,” Nature478, 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,” Nature472, 69–73 (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]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 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. E65, 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. Express19, 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. Express19, 22316–22321 (2011).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Optomechanical coupling in photonic crystal supported nanomechanical waveguides,” Opt. Express17, 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,” Nature456, 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.

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]

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]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82 (2009).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express15, 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,” Science330, 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. E65, 066611 (2002).
[CrossRef]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 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,” Nature472, 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. Express19, 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,” Nature456, 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)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–637 (2009).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature424, 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,” Nature478, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 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,” Nature472, 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,” Nature456, 480–484 (2008).
[CrossRef] [PubMed]

Nature Photon. (3)

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

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]

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. E65, 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]

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]

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]

Physics (1)

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

Science (2)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science330, 1520–1523 (2010).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 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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