Abstract

Extraneous thermal motion can limit displacement sensitivity and radiation pressure effects, such as optical cooling, in a cavity-optomechanical system. Here we present an active noise suppression scheme and its experimental implementation. The main challenge is to selectively sense and suppress extraneous thermal noise without affecting motion of the oscillator. Our solution is to monitor two modes of the optical cavity, each with different sensitivity to the oscillator’s motion but similar sensitivity to the extraneous thermal motion. This information is used to imprint “anti-noise” onto the frequency of the incident laser field. In our system, based on a nano-mechanical membrane coupled to a Fabry-Pérot cavity, simulation and experiment demonstrate that extraneous thermal noise can be selectively suppressed and that the associated limit on optical cooling can be reduced.

© 2012 OSA

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  1. V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).
  2. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17172–17205 (2007).
    [CrossRef] [PubMed]
  3. F. Marquardt, “Optomechanics,” Physics 2, 40 (2009).
    [CrossRef]
  4. A. Cleland, “Optomechanics: Photons refrigerating phonons,” Nat. Phys. 5, 458–460 (2009).
    [CrossRef]
  5. M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, “Quantum optomechanics – throwing a glance,” J. Opt. Soc. Am. B 27, A189–A197 (2010).
    [CrossRef]
  6. S. Gröblacher, J. Hertzberg, M. Vanner, G. Cole, S. Gigan, K. Schwab, and M. Aspelmeyer, “Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity,” Nat. Phys. 5, 485–488 (2009).
    [CrossRef]
  7. O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
    [CrossRef]
  8. A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
    [CrossRef]
  9. M. Eichenfield, J. Chan, A. Safavi-Naeini, K. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17, 020078 (2009).
    [CrossRef]
  10. G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
    [CrossRef] [PubMed]
  11. B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
    [CrossRef]
  12. G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
    [CrossRef]
  13. R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
    [CrossRef]
  14. D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
    [CrossRef]
  15. J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
    [CrossRef] [PubMed]
  16. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89 (2011).
    [CrossRef] [PubMed]
  17. Prof. Jack Harris, private discussions.
  18. 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] [PubMed]
  19. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
    [CrossRef]
  20. F. Marquardt, J. Chen, A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007).
    [CrossRef]
  21. T. Carmon, T. Kippenberg, L. Yang, H. Rokhsari, S. Spillane, and K. Vahala, “Feedback control of ultra-high-q microcavities: application to micro-raman lasers and microparametric oscillators,” Opt. Express 13, 3558–3566 (2005).
    [CrossRef] [PubMed]
  22. P. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437 (1990).
    [CrossRef]
  23. L. Diósi, “Laser linewidth hazard in optomechanical cooling,” Phys. Rev. A 78, 021801 (2008).
    [CrossRef]
  24. G. Phelps and P. Meystre, “Laser phase noise effects on the dynamics of optomechanical resonators,” Phys. Rev. A 83, 063838 (2011).
    [CrossRef]
  25. 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]
  26. A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D 52, 577–585 (1995).
    [CrossRef]
  27. G. Harry, H. Armandula, E. Black, D. Crooks, G. Cagnoli, J. Hough, P. Murray, S. Reid, S. Rowan, P. Sneddon, M. M. Fejer, R. Route, and S. D. Penn, “Thermal noise from optical coatings in gravitational wave detectors,” Appl. Opt. 45, 1569–1574 (2006).
    [CrossRef] [PubMed]
  28. H. J. Butt and M. Jaschke, “Calculation of thermal noise in atomic force microscopy,” Nanotechnology 6, 1 (1995).
    [CrossRef]
  29. K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004).
    [CrossRef]
  30. A. Cleland and M. Roukes, “Noise processes in nanomechanical resonators,” J. Appl. Phys. 92, 2758–2769 (2002).
    [CrossRef]
  31. T. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron Dev. 40, 903–909 (1993).
    [CrossRef]
  32. V. Braginsky, V. Mitrofanov, V. Panov, K. Thorne, and C. Eller, Systems with Small Dissipation (Univ. of Chicago Press, 1986).
  33. S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
    [CrossRef]
  34. D. Santamore and Y. Levin, “Eliminating thermal violin spikes from ligo noise,” Phys. Rev. D 64, 042002 (2001).
    [CrossRef]
  35. P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
    [CrossRef]
  36. M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, “Feedback cooling of a cantilevere’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
    [CrossRef] [PubMed]
  37. B. Sheard, M. Gray, B. Slagmolen, J. Chow, and D. McClelland, “Experimental demonstration of in-loop intra-cavity intensity-noise suppression,” IEEE J. Quantum Electron. 41, 434–440 (2005).
    [CrossRef]
  38. C. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004).
    [CrossRef] [PubMed]
  39. V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry-perot interferometer,” Phys. Lett. A 293, 228–234 (2002).
    [CrossRef]
  40. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
    [CrossRef] [PubMed]
  41. A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
    [CrossRef]
  42. Prof. Jun Ye and Prof. Peter Zoller, private discussions.
  43. J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
    [CrossRef]
  44. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  45. E. Black, “An introduction to pounddreverhall laser frequency stabilization,” Am. J. Phys 69, 79 (2001).
    [CrossRef]
  46. For a generic time-dependent variable ζ(t), the Fourier transform is defined as ζ(f)≡∫−∞∞ζ(t)e−2πiftdt, and the one-sided power spectral density at Fourier frequency f is Sζ(f)≡2∫−∞∞〈ζ(t)ζ(t+τ)〉e2iπftdτ with unit [ζ]2/Hz. The normalization convention is that 〈ζ2(t)〉=∫0∞Sζ(f)df.
  47. Provided by Prof. Jun Ye’s group.
  48. M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).
  49. H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
    [CrossRef]
  50. www.comsol.com .
  51. D. F. Wall and G. J. Milburn, Quantum Optics (Springer, 1995), chap. 7.

2011

G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[CrossRef]

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

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

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

2010

M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, “Quantum optomechanics – throwing a glance,” J. Opt. Soc. Am. B 27, A189–A197 (2010).
[CrossRef]

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[CrossRef]

2009

M. Eichenfield, J. Chan, A. Safavi-Naeini, K. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17, 020078 (2009).
[CrossRef]

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

O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

F. Marquardt, “Optomechanics,” Physics 2, 40 (2009).
[CrossRef]

A. Cleland, “Optomechanics: Photons refrigerating phonons,” Nat. Phys. 5, 458–460 (2009).
[CrossRef]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (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]

2008

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

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

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

2007

F. Marquardt, J. Chen, A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007).
[CrossRef]

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

M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, “Feedback cooling of a cantilevere’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[CrossRef] [PubMed]

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] [PubMed]

2006

G. Harry, H. Armandula, E. Black, D. Crooks, G. Cagnoli, J. Hough, P. Murray, S. Reid, S. Rowan, P. Sneddon, M. M. Fejer, R. Route, and S. D. Penn, “Thermal noise from optical coatings in gravitational wave detectors,” Appl. Opt. 45, 1569–1574 (2006).
[CrossRef] [PubMed]

S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
[CrossRef]

2005

B. Sheard, M. Gray, B. Slagmolen, J. Chow, and D. McClelland, “Experimental demonstration of in-loop intra-cavity intensity-noise suppression,” IEEE J. Quantum Electron. 41, 434–440 (2005).
[CrossRef]

T. Carmon, T. Kippenberg, L. Yang, H. Rokhsari, S. Spillane, and K. Vahala, “Feedback control of ultra-high-q microcavities: application to micro-raman lasers and microparametric oscillators,” Opt. Express 13, 3558–3566 (2005).
[CrossRef] [PubMed]

2004

C. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004).
[CrossRef] [PubMed]

K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

2002

A. Cleland and M. Roukes, “Noise processes in nanomechanical resonators,” J. Appl. Phys. 92, 2758–2769 (2002).
[CrossRef]

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry-perot interferometer,” Phys. Lett. A 293, 228–234 (2002).
[CrossRef]

2001

D. Santamore and Y. Levin, “Eliminating thermal violin spikes from ligo noise,” Phys. Rev. D 64, 042002 (2001).
[CrossRef]

E. Black, “An introduction to pounddreverhall laser frequency stabilization,” Am. J. Phys 69, 79 (2001).
[CrossRef]

1999

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

1995

H. J. Butt and M. Jaschke, “Calculation of thermal noise in atomic force microscopy,” Nanotechnology 6, 1 (1995).
[CrossRef]

A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D 52, 577–585 (1995).
[CrossRef]

1993

T. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron Dev. 40, 903–909 (1993).
[CrossRef]

1990

P. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437 (1990).
[CrossRef]

1983

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1970

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).

1951

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Ageev, A.

S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
[CrossRef]

Alegre, T. P. M.

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

Allman, M.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

Anetsberger, G.

O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

Ansmann, M.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Arcizet, O.

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[CrossRef]

O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

Armandula, H.

Aspelmeyer, M.

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

G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, “Quantum optomechanics – throwing a glance,” J. Opt. Soc. Am. B 27, A189–A197 (2010).
[CrossRef]

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

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

Bialczak, R.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Black, E.

Braginsky, V.

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry-perot interferometer,” Phys. Lett. A 293, 228–234 (2002).
[CrossRef]

V. Braginsky, V. Mitrofanov, V. Panov, K. Thorne, and C. Eller, Systems with Small Dissipation (Univ. of Chicago Press, 1986).

Braginsky, V. B.

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).

Busby, D.

S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
[CrossRef]

Butt, H. J.

H. J. Butt and M. Jaschke, “Calculation of thermal noise in atomic force microscopy,” Nanotechnology 6, 1 (1995).
[CrossRef]

Cagnoli, G.

Callen, H. B.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Camp, J.

K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Carmon, T.

Chan, J.

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

M. Eichenfield, J. Chan, A. Safavi-Naeini, K. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17, 020078 (2009).
[CrossRef]

Chen, J.

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O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

Rokhsari, H.

Roukes, M.

A. Cleland and M. Roukes, “Noise processes in nanomechanical resonators,” J. Appl. Phys. 92, 2758–2769 (2002).
[CrossRef]

Route, R.

Rowan, S.

Rugar, D.

M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, “Feedback cooling of a cantilevere’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[CrossRef] [PubMed]

Safavi-Naeini, A.

M. Eichenfield, J. Chan, A. Safavi-Naeini, K. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17, 020078 (2009).
[CrossRef]

Safavi-Naeini, A. H.

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

Sank, D.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Sankey, J.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[CrossRef]

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Santamore, D.

D. Santamore and Y. Levin, “Eliminating thermal violin spikes from ligo noise,” Phys. Rev. D 64, 042002 (2001).
[CrossRef]

Saulson, P.

P. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437 (1990).
[CrossRef]

Schliesser, A.

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[CrossRef]

O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

Schwab, K.

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

Shanks, W.

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

Sheard, B.

B. Sheard, M. Gray, B. Slagmolen, J. Chow, and D. McClelland, “Experimental demonstration of in-loop intra-cavity intensity-noise suppression,” IEEE J. Quantum Electron. 41, 434–440 (2005).
[CrossRef]

Simmonds, R.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

Sirois, A.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

Slagmolen, B.

B. Sheard, M. Gray, B. Slagmolen, J. Chow, and D. McClelland, “Experimental demonstration of in-loop intra-cavity intensity-noise suppression,” IEEE J. Quantum Electron. 41, 434–440 (2005).
[CrossRef]

Sneddon, P.

Spillane, S.

Teufel, J.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

Thompson, J.

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Thompson, J. D.

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

Thorne, K.

V. Braginsky, V. Mitrofanov, V. Panov, K. Thorne, and C. Eller, Systems with Small Dissipation (Univ. of Chicago Press, 1986).

Tikhonov, M. Y.

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).

Tombesi, P.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

Vahala, K.

M. Eichenfield, J. Chan, A. Safavi-Naeini, K. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17, 020078 (2009).
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T. Carmon, T. Kippenberg, L. Yang, H. Rokhsari, S. Spillane, and K. Vahala, “Feedback control of ultra-high-q microcavities: application to micro-raman lasers and microparametric oscillators,” Opt. Express 13, 3558–3566 (2005).
[CrossRef] [PubMed]

Vahala, K. J.

Vanner, M.

G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

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

Vitali, D.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

Vyatchanin, S.

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry-perot interferometer,” Phys. Lett. A 293, 228–234 (2002).
[CrossRef]

Wall, D. F.

D. F. Wall and G. J. Milburn, Quantum Optics (Springer, 1995), chap. 7.

Wang, H.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Weides, M.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Weis, S.

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[CrossRef]

Welton, T. A.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Wenner, J.

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

Werbach, K.

G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

Whittaker, J.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

Willems, P.

S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
[CrossRef]

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D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
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G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

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] [PubMed]

Yang, C.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[CrossRef]

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

Yang, L.

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] [PubMed]

Zwickl, B.

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[CrossRef]

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

Zwickl, B. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
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Am. J. Phys

E. Black, “An introduction to pounddreverhall laser frequency stabilization,” Am. J. Phys 69, 79 (2001).
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Appl. Opt.

Appl. Phys. B

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Appl. Phys. Lett.

B. Zwickl, W. Shanks, A. Jayich, C. Yang, B. Jayich, J. Thompson, and J. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Appl. Phys. Lett. 92, 103125–103125 (2008).
[CrossRef]

Eur. Phys. J. D

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

IEEE J. Quantum Electron.

B. Sheard, M. Gray, B. Slagmolen, J. Chow, and D. McClelland, “Experimental demonstration of in-loop intra-cavity intensity-noise suppression,” IEEE J. Quantum Electron. 41, 434–440 (2005).
[CrossRef]

IEEE Trans. Electron Dev.

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J. Appl. Phys.

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J. Opt. Soc. Am. B

N. J. Phys.

A. Jayich, J. Sankey, B. Zwickl, C. Yang, J. Thompson, S. Girvin, A. Clerk, F. Marquardt, and J. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Nanotechnology

H. J. Butt and M. Jaschke, “Calculation of thermal noise in atomic force microscopy,” Nanotechnology 6, 1 (1995).
[CrossRef]

Nat. Commun.

G. Cole, I. Wilson-Rae, K. Werbach, M. Vanner, and M. Aspelmeyer, “Phonon-tunneling dissipation in mechanical resonators,” Nat. Commun. 2, 231 (2011).
[CrossRef] [PubMed]

Nat. Photonics

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008).
[CrossRef]

Nat. Phys.

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

A. Cleland, “Optomechanics: Photons refrigerating phonons,” Nat. Phys. 5, 458–460 (2009).
[CrossRef]

J. Sankey, C. Yang, B. Zwickl, A. Jayich, and J. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).
[CrossRef]

Nature

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

C. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432, 1002–1005 (2004).
[CrossRef] [PubMed]

A. O’Connell, M. Hofheinz, M. Ansmann, R. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[CrossRef]

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359 (2011).
[CrossRef] [PubMed]

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

Opt. Express

Phys. Lett. A

S. Penn, A. Ageev, D. Busby, G. Harry, A. Gretarsson, K. Numata, and P. Willems, “Frequency and surface dependence of the mechanical loss in fused silica,” Phys. Lett. A 352, 3–6 (2006).
[CrossRef]

V. Braginsky and S. Vyatchanin, “Low quantum noise tranquilizer for fabry-perot interferometer,” Phys. Lett. A 293, 228–234 (2002).
[CrossRef]

Phys. Rev.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Phys. Rev. A

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[CrossRef]

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

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

O. Arcizet, R. Rivière, A. Schliesser, G. Anetsberger, and T. Kippenberg, “Cryogenic properties of optomechanical silica microcavities,” Phys. Rev. A 80, 021803 (2009).
[CrossRef]

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[CrossRef]

Phys. Rev. D

A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational-wave detectors,” Phys. Rev. D 52, 577–585 (1995).
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[CrossRef]

D. Santamore and Y. Levin, “Eliminating thermal violin spikes from ligo noise,” Phys. Rev. D 64, 042002 (2001).
[CrossRef]

Phys. Rev. Lett.

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 83, 3174–3177 (1999).
[CrossRef]

M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, “Feedback cooling of a cantilevere’s fundamental mode below 5 mK,” Phys. Rev. Lett. 99, 017201 (2007).
[CrossRef] [PubMed]

K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

F. Marquardt, J. Chen, A. Clerk, and S. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 93902 (2007).
[CrossRef]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric SiN films,” Phys. Rev. Lett. 103, 207204 (2009).
[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] [PubMed]

Physics

F. Marquardt, “Optomechanics,” Physics 2, 40 (2009).
[CrossRef]

Sov. Phys. JETP

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. JETP 31, 829 (1970).

Other

Prof. Jack Harris, private discussions.

V. Braginsky, V. Mitrofanov, V. Panov, K. Thorne, and C. Eller, Systems with Small Dissipation (Univ. of Chicago Press, 1986).

www.comsol.com .

D. F. Wall and G. J. Milburn, Quantum Optics (Springer, 1995), chap. 7.

For a generic time-dependent variable ζ(t), the Fourier transform is defined as ζ(f)≡∫−∞∞ζ(t)e−2πiftdt, and the one-sided power spectral density at Fourier frequency f is Sζ(f)≡2∫−∞∞〈ζ(t)ζ(t+τ)〉e2iπftdτ with unit [ζ]2/Hz. The normalization convention is that 〈ζ2(t)〉=∫0∞Sζ(f)df.

Provided by Prof. Jun Ye’s group.

Prof. Jun Ye and Prof. Peter Zoller, private discussions.

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Figures (14)

Fig. 1
Fig. 1

Conceptual diagram of the noise suppression scheme. M1/2: cavity mirrors. DP/S: photodetector for the probe field and the science field, respectively. SW: switch. FM: electro-optic frequency modulator.

Fig. 2
Fig. 2

Measured spectrum of detuning fluctuations, S Δ ( f ) (also expressed as effective cavity length fluctuations, S L ( f ) ) for the Fabry-Pérot cavity described in Section 2. The observed noise (red trace) arises from thermal motion of the end-mirror substrates, in agreement with the finite element model (blue trace, Appendix C.1). This “substrate noise” constitutes an extraneous background for the “membrane-in-the-middle” system conceptualized in Fig. 1 and detailed in [14].

Fig. 3
Fig. 3

Model spectrum of detuning fluctuations arising from mirror substrate (blue trace) and membrane motion (red trace) for the system described in Section 2. The power and detuning of the cavity field are chosen so that the (3,3) membrane mode is optically damped to a thermal phonon occupation number of (3,3) = 50. The substrates vibrate at room temperature.

Fig. 4
Fig. 4

Location of the cavity modes relative to the membrane surface for experiments reported in Section 4.2 – 4.4. Density plots of the intra-cavity intensities of TEM00 (red) and TEM01 (blue) modes are displayed on top of a black contour plot representing the axial displacement of the (2,6) membrane mode. Averaging the displacement of the surface weighted by the intensity profile gives the “effective displacement”, δxm, for membrane motion; in this case the effective displacement of the (2,6) mode is greater for the TEM00 mode than it is for TEM01 mode.

Fig. 5
Fig. 5

Predicted suppression of substrate detuning noise (dark blue) for the science field based on a feedback with ideal differential sensing, ηps = 0, for all modes, gain G ( f ) = e π i f / f m ( 3 , 3 ), and measurement noise S ν p N ( f ) = 1 Hz 2 / Hz, where f m ( 3 , 3 ) = 2.32 MHz is the mechanical frequency of the (3,3) membrane mode. Unsuppressed substrate (light blue) and membrane noise (red) for the science field is taken from the model in Fig. 3.

Fig. 6
Fig. 6

Experimental setup: λ/2: half wave plate. λ/4: quarter wave plate. PBS: polarizing beam splitter. EOM0,P,S: electro-optical modulators for calibration and probe/science beams. CWP: split-π wave plate. OFR: optical Faraday rotator. M1,2: cavity entry/exit mirrors. DP,S: photodetectors for probe/science beams. SYN0,P,S: synthesizers for driving EOM0,P,S.

Fig. 7
Fig. 7

Substrate noise suppression implemented with the membrane removed. Gain is manually set to G(f0 = 3.8 MHz) ≈ −1 using an RF amplifier and a delay line. The ratio of the noise spectrum with (dark blue trace) and without (light blue trace) feedback is compared to the “suppression factor” |1 + G(f)|2 (red trace, right axis) with G(f) = eπif/f0.

Fig. 8
Fig. 8

Combined membrane and substrate thermal noise (blue trace) in the vicinity of f m ( 2 , 6 ) = 3.56 MHz, the frequency of the (2,6) vibrational mode of the membrane. gm has been set to ∼ 0.04g1,2 in order to emphasize the substrate noise component. For comparison, a measurement of the substrate noise with the membrane removed from the cavity is shown in red.

Fig. 9
Fig. 9

Characterizing differential sensitivity of the TEM00 (science) and TEM01 (probe) mode to membrane motion. Green and blue traces correspond to the noise spectrum of electronically added (green) and subtracted (blue) measurements of δΔp and δΔs. Red trace is a scaled measurement of the substrate noise made with the membrane removed, corresponding to the red trace in Fig. 8. Electronic gain G0(f) has been set so that subtraction coherently cancels the contribution from the (2,6) mode at f m ( 2 , 6 ) 3.568 MHz. The magnitude of the gain implies that η p ( 2 , 6 ) / η s ( 2 , 6 ) 0.59 and that η p ( 6 , 2 ) / η s ( 6 , 2 ) 0.98 for the nearby (6,2) noise peak at f(6,2) ≈ 3.572 MHz.

Fig. 10
Fig. 10

Substrate noise suppression with the membrane inside the cavity. Orange and blue traces correspond to the spectrum of science field detuning fluctuations with and without feedback, respectively. In order to suppress the substrate noise contribution near f m ( 2 , 6 ) 3.56 MHz, the feedback gain has been set to G ( f m ( 2 , 6 ) ) 1. The feedback gain is fine-tuned by suppressing the detuning noise associated with an FM tone applied to both fields at 3.565 MHz. The amplitude suppression achieved for this “Calibration peak” is 15.8 dB.

Fig. 11
Fig. 11

Lorentzian fits of the thermal noise peak near f m ( 2 , 6 ) in Fig. 10, here plotted on a linear scale. Solid blue and orange traces correspond to science field detuning noise with noise suppression off and on, respectively. Dashed traces correspond to Lorentzian fits.

Fig. 12
Fig. 12

The impact of electro-optic feedback on optical damping/cooling of the (2,6) membrane mode with a red-detuned science field, as reflected in measured ratios Rγeff (yellow circles, Eq. (9)) and RΔp (blue squares, Eq. (10)), as a function of feedback gain parameter μ. The model shown (black line) is for η s ( 2 , 6 ) / η p ( 2 , 6 ) = 0.6.

Fig. 13
Fig. 13

Visualization of “negative” differential sensing. The transverse displacement profile of the (1,5) membrane mode is shown in black (a 1D slice along the midline of the membrane is shown). Red and blue curves represent the transverse intensity profile of TEM00 and TEM01 cavity modes, both centered on the membrane. The cavity waist size is adjusted so that the displacement averaged over the intensity profile is negative for TEM01 and positive for TEM00.

Fig. 14
Fig. 14

Realization of “negative” differential sensing for the (3,2) membrane mode. The TEM00 (science) and TEM01 (probe) modes are positioned near the center of the membrane. Measurements of δΔp (blue) and δΔs (red) are combined electronically on an RF splitter with positive gain, G 0 = η p ( 3 , 2 ) / η s ( 3 , 2 ), as discussed in Section 4.3. The power spectrum of the electronic signal before (blue and red) and after (black) the splitter (here in raw units of dBm/Hz) is shown. The (2,3) noise peak in the combined signal is amplified while the (3,2) noise peak is supressed, indicating that η p ( 2 , 3 ) / η s ( 2 , 3 ) > 0 and η p ( 3 , 2 ) / η p ( 3 , 2 ) < 0.

Tables (2)

Tables Icon

Table 1 Differential sensing factor, ηps, for the (2,6) and (6,2) membrane modes, with TEM00 and TEM01 forming the science and probe modes, respectively. The values in this table are inferred from Fig. 9 and the model discussed Appendix B.

Tables Icon

Table 2 Parameters from Figs. 8 and 11. μ = 0 and μ = η p ( 2 , 6 ) / η s ( 2 , 6 ) represents the noise suppression is off and on, respectively. γeff and δ Δ s 2 are inferred from the Lorentzian fits in Fig. 11. S Δ s , e ( f m ( 2 , 6 ) ) with μ = 0 and μ = η p ( 2 , 6 ) / η s ( 2 , 6 ) are inferred from the red curve in Fig. 8 and the orange curve in Fig. 10, respectively.

Equations (49)

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δ ν c p = g 1 δ x 1 p + g 2 δ x 2 p + g m δ x m p , δ ν c s = g 1 δ x 1 s + g 2 δ x 2 s + g m δ x m s .
δ x m p , s ( f ) η p , s b m ( f ) ; η p η s δ x 1 , 2 p ( f ) δ x 1 , 2 s ( f ) δ x 1 , 2 ( f ) .
δ Δ s ( f ) = G ( f ) δ Δ p ( f ) δ ν c s ( f ) = ( g 1 δ x 1 ( f ) + g 2 δ x 2 ( f ) ) ( 1 + G ( f ) ) g m η s b m ( f ) ( 1 + ( η p / η s ) G ( f ) ) .
G ( f ) = | G ( f ) | e 2 π i f τ ( f ) ,
S Δ s ( f ) = | 1 + G ( f ) | 2 ( g 1 2 S x 1 ( f ) + g 2 2 S x 2 ( f ) ) + | 1 + ( η p / η s ) 2 G ( f ) | 2 g m 2 η p 2 S b m ( f ) + | G ( f ) | 2 S ν c N ( f ) .
γ opt ( μ ) γ opt ( μ = 0 ) 1 + μ .
b m 2 ( μ = 0 ) = γ m γ m + γ opt ( μ = 0 ) k B T b m eff ( 2 π f m ) 2 ,
b m 2 ( μ ) = γ m γ eff ( μ ) k B T b m eff ( 2 π f m ) 2 ,
R γ eff ( μ ) γ eff ( μ ) γ eff ( μ = 0 ) = 1 + γ eff ( μ = 0 ) γ m γ eff ( μ = 0 ) μ .
R Δ p ( μ ) δ Δ p 2 ( μ = 0 ) δ Δ p 2 ( μ ) = R γ eff ( μ ) ,
R Δ s ( μ ) δ Δ s 2 ( μ = 0 ) δ Δ s 2 ( μ ) = R γ eff ( μ ) ( 1 μ ) 2 ,
n ¯ min 2 Γ m S ν 0 ( f m ) g δ x zp + κ 2 f m 2 ,
g m = 2 π f m ( 2 , 6 ) δ Δ s 2 η s ( 2 , 6 ) m eff γ eff k B T b γ m = 2.1 × 10 4 MHz / μ m ,
δ ν c ( t ) ν c x ¯ 1 A 1 x ^ u 1 ( x , y , z , t ) | ψ ( x , y , z ) | 2 d A A 1 | ψ ( x , y , z ) | 2 d A | x ¯ 1 + ν c x ¯ 2 A 2 x ^ u 2 ( x , y , z , t ) | ψ ( x , y , z ) | 2 d A A 2 | ψ ( x , y , z ) | 2 d A | x ¯ 2 + ν c x ¯ m A m x ^ u m ( x , y , z , t ) | ψ ( x , y , z ) | 2 d A A m | ψ ( x , y , z ) | 2 d A | x ¯ m ,
δ ν c ( t ) = g 1 δ x 1 ( t ) + g 2 δ x 2 ( t ) + g m δ x m ( t ) ,
g 1 , 2 , m ν c x ¯ 1 , 2 , m ,
δ x 1 , 2 , m A 1 , 2 , m x ^ u 1 , 2 , m ( x , y , z , t ) | ψ ( x , y , z ) | 2 d A A 1 , 2 , m | ψ ( x , y , z ) | 2 d A | x 1 , 2 , m
g m = 2 | r m | g 0 sin ( 4 π x ¯ m / λ ) 1 r m 2 cos 2 ( 4 π x ¯ m / λ ) ,
g 1 = g 0 g m 2 , g 2 = g 0 g m 2 ,
u ( x , y , z , t ) = σ b σ ( t ) ϕ σ ( x , y , z ) ,
δ x = σ η σ b σ ( t ) ,
η σ A x ^ ϕ σ ( x , y , z ) | ψ ( x , y , z ) | 2 d A A | ψ ( x , y , z ) | 2 d A
ϕ ( i , j ) ( x ¯ m , y , z ) = sin ( i π ( y y 0 ) d m ) sin ( j π ( z z 0 ) d m ) x ^ ,
| ψ ( n , n ) ( x ¯ m , y , z ) | 2 = N ( n , n ) ( H n [ 2 y w ( x ¯ m ) ] H n [ 2 z ) w ( x ¯ m ) ] ) 2 ,
( 2 π ) 2 ( f m 2 f 2 + i f γ m ( f ) ) m eff b ( f ) χ m ( f ) 1 b ( f ) = F ext ( f ) ,
m eff = U 1 2 ( 2 π f m ) 2 b 2 = V ϕ ϕ ρ d V ,
S F ( f ) = 4 k B T b 2 π f m Im [ χ m ( f ) 1 ] = 4 k B T b γ m ( f ) m eff .
S b ( f ) = | χ m ( f ) | 2 S F ( f ) = k B T b γ m ( f ) 2 π 3 m eff 1 ( f m 2 f 2 ) 2 + f 2 γ m 2 ( f ) .
S x ( f ) = σ ( η σ ) 2 S b σ ( f ) .
S ν c ( f ) = g 1 2 S x 1 ( f ) + g 2 2 S x 2 ( f ) + g m 2 S x m ( f ) .
S b ( f ) k B T b γ m 2 π 3 m eff 1 ( ( f m + δ f opt ) 2 f 2 ) 2 + f 2 ( γ m + γ opt ) 2
b 2 = 0 S b ( f ) d f = k B T b ( 2 π f ) 2 m eff γ m γ m + γ opt ,
b ¨ ( t ) + γ m b ˙ ( t ) + ( 2 π f m ) 2 b ( t ) = F ext ( t ) + δ F rad ( t ) ,
a ˙ ( t ) + 2 π ( κ + i ( Δ δ ν c ( t ) ) ) a ( t ) = 4 π κ 1 E 0 e 2 π i ϕ ( t ) .
δ ν 0 ( t ) = ϕ ˙ ( t )
δ Δ ( t ) = ϕ ˙ ( t ) δ ν c ( t ) = ϕ ˙ ( t ) g m η s b ( t ) .
δ F rad ( t ) = g m η ν c ( a * δ a ( t ) + a δ a * ( t ) ) .
δ ˙ a ( t ) + 2 π ( κ + i ( Δ ) δ a ( t ) 2 π κ δ ν c ( t ) a = 2 π i 4 π κ 1 E 0 ϕ ( t ) ,
a = E 0 4 π κ 1 / ( κ + i Δ ) .
δ a ( f ) = i a κ + i ( Δ + f ) ( δ ν c ( f ) + ( κ + i Δ ) ϕ ( f ) ) .
δ F rad ( f ) = g m η | a | 2 ν c ( i κ + i ( Δ + f ) i κ + i ( Δ + f ) ) ( δ ν c ( f ) i ϕ ( f ) / f ) .
δ F rad ( f ) = g m η ν c | E 0 | 2 4 π κ 1 κ 2 + Δ 2 ( i κ + i ( Δ + f ) i κ + i ( Δ + f ) ) δ Δ ( f )
φ ( f ) δ Δ ( f ) = φ ( f ) g m η s b ( f ) ( 1 + μ ( f ) ) ,
χ eff ( f ) 1 = χ m ( f ) 1 δ F rad ( f ) / b ( f )
= ( 2 π ) 2 ( f m 2 f 2 + i f γ m ) m eff + φ ( f ) g m η s ( 1 + μ ( f ) ) .
χ eff ( f ) 1 ( 2 π ) 2 ( ( f m + δ f opt ) 2 f 2 + i f ( γ m + γ opt ) ) m eff
γ opt 1 ( 2 π ) 2 1 f m m eff Im ( F rad ( f m ) b ( f m ) ) = 1 ( 2 π ) 2 g m η s f m m eff Im ( ( 1 + μ ( f m ) ) φ ( f m ) ) ,
δ f opt 1 ( 2 π ) 2 1 2 f m m eff Re ( F rad ( f m ) b ( f m ) ) = 1 ( 2 π ) 2 g m η s 2 f m m eff Re ( ( 1 + μ ( f m ) ) φ ( f m ) ) .
S b ( f ) = | χ eff ( f ) | 2 4 k B T b γ m f = k B T b γ m 2 π 3 m eff 1 ( ( f m + δ f opt ) 2 f 2 ) 2 + f 2 ( γ m + γ opt ) 2 .

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