Abstract

In this paper, a new mechanism of heat transfer is introduced. It is shown that, without emission and absorption of photons, light can operate as a channel of heat transfer between nano- or micro-mechanical oscillators. We consider the dynamics of two vibrating mirrors coupled through one optical cavity mode in an optomechanical system. It is shown that light mediates heat transfer between two micro-mirrors. When the detuning frequency of the mechanical resonators is low, fluctuations flow through the light channel from the high temperature vibrating mirror toward the low temperature one. This behavior is named the resonance heat transfer effect. The rate of heat flow between the mechanical resonators depends on the detuning frequency of mechanical resonators, heat bath temperatures, laser intensity, and optomechanical regime of operation. Heat transfer in good and bad cavity regimes of operation is investigated.

© 2014 Optical Society of America

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  1. S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
    [CrossRef]
  2. O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
    [CrossRef]
  3. M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
    [CrossRef]
  4. D. Kleckner and D. Bouwmeester, “Sub-Kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
    [CrossRef]
  5. A. Schliesser, P. DelHaye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
    [CrossRef]
  6. A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
    [CrossRef]
  7. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [CrossRef]
  8. Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5, 489–493 (2009).
    [CrossRef]
  9. F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
    [CrossRef]
  10. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
    [CrossRef]
  11. F. Farman and A. R. Bahrampour, “Effects of optical parametric amplifier pump phase noise on the cooling of optomechanical resonators,” J. Opt. Soc. Am. B 30, 1898–1904 (2013).
    [CrossRef]
  12. D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
    [CrossRef]
  13. A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
    [CrossRef]
  14. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
    [CrossRef]
  15. S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
    [CrossRef]
  16. S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
    [CrossRef]
  17. S. Huang and G. S. Agarwal, “Entangling nano-mechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
    [CrossRef]
  18. Y. Dan Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New J. Phys. 14, 105010 (2012).
    [CrossRef]
  19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
    [CrossRef]
  20. S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [CrossRef]
  21. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
    [CrossRef]
  22. D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
    [CrossRef]
  23. D. Vitali and V. Giovannetti, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
    [CrossRef]
  24. A. Hurwitz, “On the conditions under which an equation has only roots with negative real part,” in Selected Papers on Mathematical Trends in Control Theory, R. Bellman and R. Kalaba, eds. (Dover, 1964).
  25. S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009).
    [CrossRef]

2013 (1)

2012 (2)

Y. Dan Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New J. Phys. 14, 105010 (2012).
[CrossRef]

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

2010 (2)

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

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

2009 (4)

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Entangling nano-mechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[CrossRef]

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

2008 (2)

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

2007 (2)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
[CrossRef]

2006 (6)

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

D. Kleckner and D. Bouwmeester, “Sub-Kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

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

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

2003 (2)

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
[CrossRef]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

2001 (2)

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
[CrossRef]

D. Vitali and V. Giovannetti, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Entangling nano-mechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[CrossRef]

Anetsberger, G.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Arcizet, O.

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Aspelmeyer, M.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

Bahrampour, A. R.

Bauerle, D.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Blaser, F.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

Bohm, H. R.

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-Kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

Briant, T.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Buonanno, A.

A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
[CrossRef]

Chen, Y.

A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
[CrossRef]

Clerk, A. A.

Y. Dan Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New J. Phys. 14, 105010 (2012).
[CrossRef]

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

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
[CrossRef]

Cohadon, P. F.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Dan Wang, Y.

Y. Dan Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New J. Phys. 14, 105010 (2012).
[CrossRef]

Deleglise, S.

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

DelHaye, P.

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

Farman, F.

Gavartin, E.

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

Gigan, S.

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Giovannetti, V.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

D. Vitali and V. Giovannetti, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

Girvin, S. M.

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
[CrossRef]

Heidmann, A.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Hertzberg, J. B.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Huang, S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

S. Huang and G. S. Agarwal, “Entangling nano-mechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

Hurwitz, A.

A. Hurwitz, “On the conditions under which an equation has only roots with negative real part,” in Selected Papers on Mathematical Trends in Control Theory, R. Bellman and R. Kalaba, eds. (Dover, 1964).

Kim, M. S.

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

Kippenberg, T. J.

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

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

Kleckner, D.

D. Kleckner and D. Bouwmeester, “Sub-Kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

Langer, G.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Lloyd, S.

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

Mancini, S.

D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
[CrossRef]

Marquardt, F.

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
[CrossRef]

Mavalvala, N.

A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
[CrossRef]

Nooshi, N.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

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

Park, Y. S.

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

Paternostro, M.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

Pinard, M.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

Pirandola, S.

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

Riviere, R.

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Schliesser, A.

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

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

Schwab, K. C.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Tombesi, P.

D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
[CrossRef]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
[CrossRef]

Vahala, K. J.

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

Vitali, D.

D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
[CrossRef]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
[CrossRef]

D. Vitali and V. Giovannetti, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

Wang, H.

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

Wang, Y. D.

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

Weis, S.

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

Wilson-Rae, I.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

Zeilinger, A.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

J. Mod. Opt. (1)

F. Marquardt, A. A. Clerk, and S. M. Girvin, “Quantum theory of optomechanical cooling,” J. Mod. Opt. 55, 3329–3338 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

D. Vitali, S. Mancini, and P. Tombesi, “Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity,” J. Phys. A 40, 8055–8068 (2007).
[CrossRef]

Nat. Phys. (2)

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Nature (3)

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[CrossRef]

D. Kleckner and D. Bouwmeester, “Sub-Kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef]

New J. Phys. (3)

M. Paternostro, S. Gigan, M. S. Kim, F. Blaser, H. R. Bohm, and M. Aspelmeyer, “Reconstructing the dynamics of a movable mirror in a detuned optical cavity,” New J. Phys. 8, 107 (2006).
[CrossRef]

S. Huang and G. S. Agarwal, “Entangling nano-mechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[CrossRef]

Y. Dan Wang and A. A. Clerk, “Using dark modes for high-fidelity optomechanical quantum state transfer,” New J. Phys. 14, 105010 (2012).
[CrossRef]

Phys. Rev. A (6)

D. Vitali, S. Mancini, and P. Tombesi, “Optomechanical scheme for the detection of weak impulsive forces,” Phys. Rev. A 64, 051401(R) (2001).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
[CrossRef]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003).
[CrossRef]

D. Vitali and V. Giovannetti, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

S. Huang and G. S. Agarwal, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (2009).
[CrossRef]

Phys. Rev. D (1)

A. Buonanno, Y. Chen, and N. Mavalvala, “Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme,” Phys. Rev. D 67, 122005 (2003).
[CrossRef]

Phys. Rev. Lett. (5)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

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

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef]

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

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97, 150403 (2006).
[CrossRef]

Science (1)

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

Other (1)

A. Hurwitz, “On the conditions under which an equation has only roots with negative real part,” in Selected Papers on Mathematical Trends in Control Theory, R. Bellman and R. Kalaba, eds. (Dover, 1964).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of (a) an optomechanical system consisting of two vibrating mirrors. The cavity is driven by a laser from the left mirror, and (b) an optomechanical system consisting of two vibrating and two fixed mirrors, which is driven by a laser from mirror 4.

Fig. 2.
Fig. 2.

(a) Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve). (b) Effective temperature of STE (solid red curve), MTE (dashed blue curve), and CTE (green curve with marker, O) as a function of the detuning Δ0(s1) for mirrors 1 and 2. The initial temperature of both mirrors is 300 K.

Fig. 3.
Fig. 3.

(a) Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve). Effective temperature of STE (solid red curve), MTE (dashed blue curve), and CTE (green curve with marker, O) as a function of the detuning Δ0(s1) for (b) mirror 1, and (c) mirror 2. The initial temperatures of mirrors 1 and 2 are 10 K and 300 K, respectively.

Fig. 4.
Fig. 4.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve) in the bad cavity regime of operation. The initial temperatures of mirrors 1 and 2 are 10 and 300 K, respectively.

Fig. 5.
Fig. 5.

(a) Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve). Effective temperature of STE (solid red curve), MTE (dashed blue curve), and CTE (green curve with marker, O) as a function of the detuning Δ0(s1) for (b) mirror 1, and (c) mirror 2. The initial temperature of both mirrors is 300 K and ωm1=8.79×106, ωm2=8.84×106.

Fig. 6.
Fig. 6.

(a) Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve). Effective temperature of STE (solid red curve) and MTE (dashed blue curve) as a function of the detuning Δ0(s1) for (b) mirror 1, and (c) mirror 2. The initial temperature of both mirrors is 300 K, and ωm1=8.79×106, ωm2=9.06×106.

Fig. 7.
Fig. 7.

Variation of the two terms relative to the Brownian noises in DNS as a function of the frequency ω(s1). The effect of Brownian noise from (a) heat bath 1 on the DNS of mirror 1, (b) heat bath 2 on the DNS of mirror 1, (c) heat bath 1 on the DNS of mirror 2, and (d) heat bath 2 on the DNS of mirror 2. The initial temperature of both mirrors is 300 K, ωm1=8.79×106, ωm2=9.06×106, and Δ0=8.79×106s1.

Fig. 8.
Fig. 8.

(a) Effective temperature Teff(K) as a function of the detuning Δ0(s1) for mirror 1 (solid red curve) and mirror 2 (dashed–dotted blue curve). Effective temperature of STE (solid red curve) and MTE (dashed blue curve) as a function of the detuning Δ0(s1) for (b) mirror 1, and (c) mirror 2. The initial temperature of both mirrors is 300 K and ωm1=8.79×106, ωm2=17.59×106.

Fig. 9.
Fig. 9.

Variation of the two terms relative to the Brownian noises in DNS as a function of the frequency ω(s1). The effect of Brownian noise from (a) heat bath 1 on the DNS of mirror 1, (b) heat bath 2 on the DNS of mirror 1, (c) heat bath 1 on the DNS of mirror 2, and (d) heat bath 2 on the DNS of mirror 2. The initial temperature of both mirrors is 300 K, ωm1=8.79×106, ωm2=17.59×106, and Δ0=8×106s1.

Equations (22)

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(1+r2η14χ2η24)(1+r2η32χ2η31)+r2χ2η34(η1+η2)2=0,
H=(ωcωL)ccG1ccq1G2ccq2+12(p12m1+m1ωm12q12)+12(p22m2+m2ωm22q22)+iE(cc),
q˙i=pimi,i=1,2,
p˙i=miωmi2qi+Giccγipi+ζi,i=1,2,
c˙=i(Δ0G1q1G2q2)c+Eκc+2κcin.
δ¨qi=ωmi2δqi+Gimi(csδc+cs*δc)γiδ˙qi+ζimii=1,2,
δ˙c=(iΔ+κ)δc+i(G1δq1+G2δq2)cs+2κcin.
δqi(ω)=1d(ω){ζi[Xj+Dj+2Dm1]+ζj[XjGimiGj]+2κGimi(cin[cs*D2Dj+2]+cin[csD1Dj+2])}(i,j)=(1,2),(2,1),
Sqi(ω)=1|d(ω)|2[γimiωcoth(ω2KBTi)|Xj+DDj+2|2+γjmjωcoth(ω2KBTj)(XjGimiGj)2+cs2κ2Gi2mi2|Dj+2|2(|D1|2+|D2|2)](i,j)=(1,2),(2,1).
M1=(NZ1+Z2+NX1X2)2N±(NZ1+Z2+NX1X2)24NX1(NZ1+Z2)2N,
M2=X2N(X1M1),
N=2κω(ωm22ω2)+ωγ2(Δ2+κ2ω2)2κω(ωm12ω2)ωγ1(Δ2+κ2ω2),
Z1=(ωm12ω2)(Δ2+κ2ω2)2κω2γ1,
Z2=(ωm22ω2)(Δ2+κ2ω2)2κω2γ2.
E1=Einr4eikL4r(1+η32)Δ,
E2=Einr4eikL4iχ(η131)Δ,
E3=Einr4eikL4iχrη3(η1+η2)Δ,
E4=Einr4eikL4(1+r2η32χ2η31)Δ,
Δ=(1+r2η14χ2η24)(1+r2η32χ2η31)+r2χ2η34(η1+η2)2,
η2=η1,η24=1,η131.
G1=ωcr2L4+L2x2+L1r2,
G2=ωcχ2L4+L2x2+L1r2.

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