Abstract

In this paper, the effect of parametric amplifier pump phase noise on the cooling of a micromirror in an optomechanical system with an optical parametric amplifier inside the optical cavity is investigated theoretically. It has been demonstrated that the photon number distribution of a parametric amplifier near the threshold of instability leads to improved cooling of the micromirror. But due to the presence of the parametric amplifier, there is a resonance detuning frequency for transferring noise energy to the potential and kinetic energy fluctuations of the mirror which causes the mirror mechanical oscillation mode temperature to increase. In low quality factor cavities, this effect occurs in a nonequilibrium thermodynamic process, while in high quality factor cavities, this process is a thermal equilibrium one. The effects of the Lorentzian two-time correlations of laser phase noise on the mirror mechanical mode temperature are considered in this paper.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
    [CrossRef]
  6. J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Grblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
    [CrossRef]
  7. C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
    [CrossRef]
  8. 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]
  9. L. Diosi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78, 021801(R) (2008).
    [CrossRef]
  10. P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
    [CrossRef]
  11. S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
    [CrossRef]
  12. A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
    [CrossRef]
  13. Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5, 489–493 (2009).
    [CrossRef]
  14. Z. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
    [CrossRef]
  15. G. A. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
    [CrossRef]
  16. M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
    [CrossRef]
  17. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009).
    [CrossRef]
  18. M. Suhail. Zubairy and M. Scully, Quantum Optics (Cambridge University, 1997).
  19. 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]
  20. 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), p. 7282.
  21. 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]
  22. 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]
  23. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [CrossRef]
  24. H.-J. Chen and X.-W. Mi, “Normal mode splitting and ground state cooling in a Fabry–Perot optical cavity and transmission line resonator,” Chin. Phys. B 20, 124203 (2011).
  25. S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
    [CrossRef]

2011 (4)

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

G. A. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
[CrossRef]

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[CrossRef]

H.-J. Chen and X.-W. Mi, “Normal mode splitting and ground state cooling in a Fabry–Perot optical cavity and transmission line resonator,” Chin. Phys. B 20, 124203 (2011).

2009 (8)

S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (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]

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[CrossRef]

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

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

Z. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
[CrossRef]

2008 (3)

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (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]

L. Diosi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78, 021801(R) (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]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

2006 (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]

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]

2001 (1)

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]

Abdi, M.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[CrossRef]

Agarwal, G. S.

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

Anetsberger, G.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (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]

Arcizet, O.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (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]

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.

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

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

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[CrossRef]

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[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]

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]

Barzanjeh, S.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[CrossRef]

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]

Chan, J.

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

Chen, H.-J.

H.-J. Chen and X.-W. Mi, “Normal mode splitting and ground state cooling in a Fabry–Perot optical cavity and transmission line resonator,” Chin. Phys. B 20, 124203 (2011).

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Clerk, A. A.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[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]

Cole, G. D.

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[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]

Diosi, L.

L. Diosi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78, 021801(R) (2008).
[CrossRef]

Favero, I.

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
[CrossRef]

Genes, C.

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[CrossRef]

Gigan, S.

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[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]

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.

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, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Grblacher, S.

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

Groblacher, S.

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[CrossRef]

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

Hammerer, K.

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

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[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. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[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]

Hill, J. T.

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

Huang, S.

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, “Normal-mode splitting in a coupled system of a nanomechanical oscillator and a parametric amplifier cavity,” Phys. Rev. A 80, 033807 (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), p. 7282.

Karrai, K.

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
[CrossRef]

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.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (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]

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]

Krause, A.

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

Marquardt, F.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Mayer Alegre, T. P.

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

Metzger, C.

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
[CrossRef]

Meystre, P.

G. A. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
[CrossRef]

Mi, X.-W.

H.-J. Chen and X.-W. Mi, “Normal mode splitting and ground state cooling in a Fabry–Perot optical cavity and transmission line resonator,” Chin. Phys. B 20, 124203 (2011).

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]

Ortlieb, A.

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
[CrossRef]

Painter, O.

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

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]

Phelps, G. A.

G. A. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
[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]

Rabl, P.

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[CrossRef]

Riviere, R.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (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]

Safavi-Naeini, A. H.

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

Schliesser, A.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (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]

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. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[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]

Scully, M.

M. Suhail. Zubairy and M. Scully, Quantum Optics (Cambridge University, 1997).

Suhail. Zubairy, M.

M. Suhail. Zubairy and M. Scully, Quantum Optics (Cambridge University, 1997).

Tombesi, P.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[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]

Vanner, M. R.

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

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[CrossRef]

Vitali, D.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[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]

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]

Yin, Z.

Z. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
[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]

Chin. Phys. B (1)

H.-J. Chen and X.-W. Mi, “Normal mode splitting and ground state cooling in a Fabry–Perot optical cavity and transmission line resonator,” Chin. Phys. B 20, 124203 (2011).

Nat. Phys. (4)

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]

S. Groblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

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

Nature (5)

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

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]

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

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

New J. Phys. (1)

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]

Phys. Rev. A (8)

L. Diosi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78, 021801(R) (2008).
[CrossRef]

P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, “Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems,” Phys. Rev. A 80, 063819 (2009).
[CrossRef]

Z. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
[CrossRef]

G. A. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
[CrossRef]

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84, 032325 (2011).
[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]

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. B (1)

C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, “Optical self cooling of a deformable Fabry–Perot cavity in the classical limit,” Phys. Rev. B 78, 035309 (2008).
[CrossRef]

Phys. Rev. Lett. (3)

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]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quanyum theory of cavity-assisted sideband cooling of mechamnical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[CrossRef]

Other (2)

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), p. 7282.

M. Suhail. Zubairy and M. Scully, Quantum Optics (Cambridge University, 1997).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Schematic structure of an optomechanical system that consists of two mirrors and OPA inside the cavity.

Fig. 2.
Fig. 2.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve), ΓP=1kHz, γP=100kHz (dashed green curve), ΓP=1kHz, γP=1MHz (dash-dotted red curve), and ΓP=1kHz, γP=100MHz or γP (dotted black curve). The purple curve with marker O is indicated as the parameter r. The system parameters are G=3.5×107s1, θ0=0, κ=108s1, T=300K.

Fig. 3.
Fig. 3.

(a) Effective potential temperature Tp(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve) and ΓP=1kHz, γP (dashed green curve). (b) The effective kinetic temperature Tk(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve) and ΓP=1kHz, γP (dashed green curve).

Fig. 4.
Fig. 4.

Effective temperature for three components of input noises as a function of the detuning Δ0(s1) separately. The dotted blue curve indicates the input vacuum noise. The solid green curve shows the Brownian noise and dashed red curve shows the OPA laser phase noise with linewidth ΓP=1kHz, γp. The environment temperature is T=300(K).

Fig. 5.
Fig. 5.

(a) Variation of minimum mirror temperature as a function of the laser linewidth ΓP. (b) The variation of minimum mirror temperature as a function of the laser cutoff frequency γP for ΓP=1kHz (solid green curve) and ΓP=10kHz (dashed blue curve). The parameters are G=3.5×107s1, θ0=0, κ=108s1, T=300K.

Fig. 6.
Fig. 6.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve) and ΓP=1kHz, γP (dashed green curve). The red dash-dotted curve indicates the parameter r as a function of the detuning Δ0(s1). The parameters are G=107s1, θ0=0.2467+π/2, κ=5×106s1, T=300K, F=3768s1.

Fig. 7.
Fig. 7.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve) and ΓP=1kHz, γP (dashed green curve). The red dash-dotted curve indicates the parameter r as a function of the detuning Δ0(s1). The parameters are G=5×106s1, θ0=3π/4, κ=107s1, T=300K, F=1884s1.

Fig. 8.
Fig. 8.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for ΓP=0 (solid blue curve) and ΓP=1kHz, γP (dashed green curve). The red dash-dotted curve indicates the parameter r as a function of the detuning Δ0(s1). The parameters are G=3.5×107s1, θ0=0, κ=108s1, T=1K.

Fig. 9.
Fig. 9.

Variation of minimum mirror temperature as a function of the laser cutoff frequency γP for ΓP=1kHz (solid green curve) and ΓP=10kHz (dashed blue curve). The parameters are G=3.5×107s1, θ0=0, κ=108s1, T=1K.

Fig. 10.
Fig. 10.

Effective temperature Teff(K) as a function of the detuning Δ0(s1) for (a) ΓP=0 (solid blue curve, leftmost vertical scale) and ΓP=1kHz, γP (dashed green curve, rightmost vertical scale), (b) ΓP=0 (solid blue curve), ΓP=1kHz, γP=1kHz (dotted green curve), ΓP=1kHz, γP=10kHz (dashed red curve), and γP=100kHz, γP=100kHz (dash-dotted black curve). The parameters are m=145ng, P=6.9mW, (γm/2π)=140s1, Q=104, κ=0.227×ωm, T=1K.

Equations (7)

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

H=(ωCωL)cc(χccq+12(p2m+mωm2q2)+i(E(cc)+i(G(ei(θ0+δθ(t))c2ei(θ0+δθ(t))c2),
A=[(iΔ+κ)2Geiθ00iχcs2Geiθ0(iΔ+κ)0iχcs*χcs*χcsγmmωm2001m0],
δq(ω)=1d(ω)([Δ2+(κiω)24G2]ζ(ω)iχ2κ([(ω+iκΔ)cs+2iGeiθ0cs*]δc˜in(ω)+[(ω+iκ+Δ)cs*+2iGeiθ0cs]δc˜in(ω))χ[Δ|cs|2+2Gsin(θ0)[a2b2]4abGcos(θ0)]δθ˙(ω)),
Sq(ω)=14πdΩδq(ω)δq(Ω)+δq(Ω)δq(ω).
δc˜in(ω)δc˜in(Ω)=2πδ(ω+Ω)δζ(ω)δζ(Ω)=2πγmmω[coth(ω2KBT)+1]δ(ω+Ω).
δθ˙(ω)δθ˙(Ω)=4πΓPδ(ω+Ω)γP2γP2+ω2δϕ˙(ω)δϕ˙(Ω)=4πΓLδ(ω+Ω)γL2γL2+ω2.
Sq(ω)=2χ2|d(ω)|2[ΓPγP2γP2+ω2+2ΓLγL2γL2+ω2]+(Δ|cs|2+2Gsin(θ0)[a2b2]4abGcos(θ0))2+|d(ω)|2{2κχ2[(κ2+ω2+Δ2+4G2)|cs|2+2Geiθ0cs*2(κiΔ)+2Geiθ0cs2(κ+iΔ)]mγmω[(κ2ω2+Δ24G2)2+4κ2ω2]coth(ω2KBT)}Sp(ω)=m2ω2Sq(ω).

Metrics