Abstract

We present an integrated optomechanical and electromechanical nanocavity, in which a common mechanical degree of freedom is coupled to an ultrahigh-Q photonic crystal defect cavity and an electrical circuit. The system allows for wide-range, fast electrical tuning of the optical nanocavity resonances, and for electrical control of optical radiation pressure back-action effects such as mechanical amplification (phonon lasing), cooling, and stiffening. These sort of integrated devices offer a new means to efficiently interconvert weak microwave and optical signals, and are expected to pave the way for a new class of micro-sensors utilizing optomechanical back-action for thermal noise reduction and low-noise optical read-out.

© 2011 OSA

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  1. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13(5), 307–320 (1901).
    [CrossRef]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15(25), 17172–17205 (2007).
    [CrossRef] [PubMed]
  3. T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
    [CrossRef] [PubMed]
  4. I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
    [CrossRef]
  5. V. B. Braginskiĭ and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (University of Chicago Press, Chicago, 1977).
  6. V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, 1992).
    [CrossRef]
  7. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
    [CrossRef]
  8. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
    [CrossRef] [PubMed]
  9. O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidman, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
    [CrossRef] [PubMed]
  10. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
    [CrossRef] [PubMed]
  11. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
    [CrossRef] [PubMed]
  12. S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science 330(6010), 1520–1523 (2010).
    [CrossRef] [PubMed]
  13. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
    [CrossRef] [PubMed]
  14. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
    [CrossRef] [PubMed]
  15. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008).
    [CrossRef]
  16. J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [CrossRef] [PubMed]
  17. C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Physics 4(7), 555–560 (2008).
    [CrossRef]
  18. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
    [CrossRef] [PubMed]
  19. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and Control of a Cavity Optoelectromechanical System,” Phys. Rev. Lett. 104(12), 123604 (2010).
    [CrossRef] [PubMed]
  20. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103, 103601 (2009).
    [CrossRef] [PubMed]
  21. A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [CrossRef]
  22. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical travelling wave phonon-photon translator,” New J. Phys. 13, 013017 (2011).
    [CrossRef]
  23. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061101 (2005).
    [CrossRef]
  24. M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
    [CrossRef] [PubMed]
  25. A. H. Safavi-Naeini, T. P. Mayer Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
    [CrossRef]
  26. E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
    [CrossRef]
  27. I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, “Programmable photonic crystal nanobeam cavities,” Opt. Express 18(8), 8705–8712 (2010).
    [CrossRef] [PubMed]
  28. R. Perahia, J. D. Cohen, S. Meenehan, T. P. Mayer Alegre, and O. Painter, “Electrostatically tunable optomechanical “zipper” cavity laser,” Appl. Phys. Lett. 97(19), 191112 (2010).
    [CrossRef]
  29. L. Midolo, P. J. van Veldhoven, M. A. Dündar, R. Nötzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavities,” Appl. Phys. Lett. 98(21), 21120 (2011).
    [CrossRef]
  30. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
    [CrossRef]
  31. See http://www.comsol.com/
  32. S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
    [CrossRef]
  33. 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(7183), 72–U5 (2008).
    [CrossRef] [PubMed]
  34. T. P. Mayer Alegre, R. Perahia, and O. Painter, “Optomechanical zipper cavity lasers: theoretical anaylysis of tuning range and stability,” Opt. Express 18(8), 7872–7885 (2010).
    [CrossRef]
  35. C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystal, and O. Painter, “An optical fiber-taper probe for wafer-scale microphotonic device characterization,” Opt. Express 15(8), 4745–4752 (2010).
    [CrossRef]
  36. P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13(3), 801–820 (2005).
    [CrossRef] [PubMed]
  37. D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006).
    [CrossRef] [PubMed]
  38. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  39. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977).
    [CrossRef]

2011 (6)

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[CrossRef] [PubMed]

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[CrossRef] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

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

L. Midolo, P. J. van Veldhoven, M. A. Dündar, R. Nötzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavities,” Appl. Phys. Lett. 98(21), 21120 (2011).
[CrossRef]

2010 (8)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
[CrossRef]

T. P. Mayer Alegre, R. Perahia, and O. Painter, “Optomechanical zipper cavity lasers: theoretical anaylysis of tuning range and stability,” Opt. Express 18(8), 7872–7885 (2010).
[CrossRef]

C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystal, and O. Painter, “An optical fiber-taper probe for wafer-scale microphotonic device characterization,” Opt. Express 15(8), 4745–4752 (2010).
[CrossRef]

S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science 330(6010), 1520–1523 (2010).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[CrossRef]

I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, “Programmable photonic crystal nanobeam cavities,” Opt. Express 18(8), 8705–8712 (2010).
[CrossRef] [PubMed]

R. Perahia, J. D. Cohen, S. Meenehan, T. P. Mayer Alegre, and O. Painter, “Electrostatically tunable optomechanical “zipper” cavity laser,” Appl. Phys. Lett. 97(19), 191112 (2010).
[CrossRef]

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and Control of a Cavity Optoelectromechanical System,” Phys. Rev. Lett. 104(12), 123604 (2010).
[CrossRef] [PubMed]

2009 (6)

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
[CrossRef]

2008 (4)

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(7183), 72–U5 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef] [PubMed]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Physics 4(7), 555–560 (2008).
[CrossRef]

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008).
[CrossRef]

2007 (1)

2006 (3)

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidman, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[CrossRef] [PubMed]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006).
[CrossRef] [PubMed]

2005 (3)

P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13(3), 801–820 (2005).
[CrossRef] [PubMed]

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76, 061101 (2005).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

2001 (1)

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

1977 (1)

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

1901 (1)

E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13(5), 307–320 (1901).
[CrossRef]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
[CrossRef]

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

Anetsberger, G.

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (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(6010), 1520–1523 (2010).
[CrossRef] [PubMed]

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008).
[CrossRef]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidman, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[CrossRef] [PubMed]

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

Asano, T.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
[CrossRef]

Ashkin, A.

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Aspelmeyer, M.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[CrossRef] [PubMed]

S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
[CrossRef]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

Barclay, P.

Bäuerle, D.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

Beveratos, A.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

Blaser, F.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

Böhm, H. R.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

Borselli, M.

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006).
[CrossRef] [PubMed]

Bowen, W. P.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and Control of a Cavity Optoelectromechanical System,” Phys. Rev. Lett. 104(12), 123604 (2010).
[CrossRef] [PubMed]

Braginskii, V. B.

V. B. Braginskiĭ and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (University of Chicago Press, Chicago, 1977).

V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, 1992).
[CrossRef]

Braive, R.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

Briant, T.

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidman, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[CrossRef] [PubMed]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

Carmon, T.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

Chan, J.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[CrossRef] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

Chang, D. E.

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A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[CrossRef]

Sagnes, I.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

Schliesser, A.

S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science 330(6010), 1520–1523 (2010).
[CrossRef] [PubMed]

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008).
[CrossRef]

Schwab, K. C.

S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
[CrossRef]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

Simmonds, R. W.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

Sirois, A. J.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Song, B.-S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
[CrossRef]

Srinivasan, K.

Teufel, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Physics 4(7), 555–560 (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(7183), 72–U5 (2008).
[CrossRef] [PubMed]

Thorne, K. S.

V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, 1992).
[CrossRef]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

van Veldhoven, P. J.

L. Midolo, P. J. van Veldhoven, M. A. Dündar, R. Nötzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavities,” Appl. Phys. Lett. 98(21), 21120 (2011).
[CrossRef]

Vanner, M. R.

S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
[CrossRef]

Weis, S.

S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science 330(6010), 1520–1523 (2010).
[CrossRef] [PubMed]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Whittaker, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[CrossRef]

Zeilinger, A.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

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(7183), 72–U5 (2008).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

A. H. Safavi-Naeini, T. P. Mayer Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[CrossRef]

R. Perahia, J. D. Cohen, S. Meenehan, T. P. Mayer Alegre, and O. Painter, “Electrostatically tunable optomechanical “zipper” cavity laser,” Appl. Phys. Lett. 97(19), 191112 (2010).
[CrossRef]

L. Midolo, P. J. van Veldhoven, M. A. Dündar, R. Nötzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavities,” Appl. Phys. Lett. 98(21), 21120 (2011).
[CrossRef]

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[CrossRef]

Nat. Photonics (2)

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[CrossRef]

Nat. Physics (2)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Physics 4(7), 555–560 (2008).
[CrossRef]

S. Gröblacher, 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. Physics 5(7), 485–488 (2009).
[CrossRef]

Nature (10)

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(7183), 72–U5 (2008).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[CrossRef] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356) (2011).
[CrossRef] [PubMed]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444(7115), 67–70 (2006).
[CrossRef] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011).
[CrossRef] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[CrossRef] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–556 (2009).
[CrossRef] [PubMed]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidman, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(7115), 71–74 (2006).
[CrossRef] [PubMed]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444(7115), 75–78 (2006).
[CrossRef] [PubMed]

Nature Mater. (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4(3), 207–210 (2010).
[CrossRef]

New J. Phys. (2)

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

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008).
[CrossRef]

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[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical Coupling in a Two-Dimensional Photonic Crystal Defect Cavity,” Phys. Rev. Lett. 106(20), 203902 (2001).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and Control of a Cavity Optoelectromechanical System,” Phys. Rev. Lett. 104(12), 123604 (2010).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103, 103601 (2009).
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[CrossRef]

Science (2)

S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science 330(6010), 1520–1523 (2010).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef] [PubMed]

Other (3)

V. B. Braginskiĭ and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (University of Chicago Press, Chicago, 1977).

V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, 1992).
[CrossRef]

See http://www.comsol.com/

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Figures (7)

Fig. 1
Fig. 1

(a) Scanning Fabry-Pérot cavity example of an electro-optomechanical system, in which cavity mirrors are attached to capacitive actuators. (b) Displacement profile of the PC implementation of an electro-optomechanical cavity. The cavity is formed as a waveguide defect in between two individual PC membrane halves (region outlined by the black rectangle), the distance between which can be adjusted using an electrostatic force generated between pairs of metal wires. (c) & (d) Electric field distribution |E|2 of the first ((c)) and second order (d)) optical cavity modes. (e) Scanning-electron micrograph of a processed device in a double-capacitor configuration. The PC membrane is suspended on struts with w1 = 250 nm and w2 = 80 nm.

Fig. 2
Fig. 2

(a) Plot of the normalized transmission spectrum of a device with zero applied voltage showing both the fundamental and the second-order optical resonance. (b) Cavity resonance wavelengths versus applied voltage V a 2, indicating quadratic wavelength-tuning of the cavity modes with tuning parameter α = 0.051 nm/V2. (c) RF power spectral density of laser light transmitted through the second order cavity mode. The resonances at 3.18, 3.28, and 3.61 MHz correspond to thermally transduced mechanical modes with hybridized in- and out-of-plane character. The insets show FEM-simulations of the eigenmodes of a single membrane half in top- and sideview.

Fig. 3
Fig. 3

Optical spring reduction of thermal noise: (a) Transmission spectrum (blue) of the second order cavity mode. The actuators are driven by a triangular signal (orange) with an amplitude of 1 V and a frequency of 50 Hz. (b) RF spectra of the pump laser transmission as function of ncav. The optical spring effect shifts the mechanical mode at 3.6 MHz to higher frequencies. (c) Normalized transmission spectrum of the fundamental cavity mode (blue) of a device with α = 0.055 nm/V2 together with a Voigt fit (red). (d) False color plots of the transmission scans of the fundamental cavity mode as function of the intracavity photon number ncav in the second order mode. The horizontal green lines indicate the intra-cavity photon numbers at which the individual scans in (e)–(g) were taken. The insets above the green lines schematically depict the detuning between the pump laser (red) and the second order cavity mode under the influence of the thermal bistability (white). (h) Linewidth of the fundamental cavity mode as function of ncav in the second order mode. Blue dots show the FWHM linewidth extracted from (d) while the red dots show the linewidth as an effective optical Q-factor.

Fig. 4
Fig. 4

Electrically controlled optomechanical back-action: (a) False-color plot of RF optical transmission spectra as a function of Va. For blue detuning we observe stiffening and amplification of mechanical motion, whereas we observe softening and damping for red detuning. (b) RF spectra for a blue-detuned pump laser below (lower) and above the lasing threshold (upper panel). (c) Time trace of the cavity optical transmission in the phonon lasing regime. (d) Waterfall plot of the RF optical transmission spectra of the mechanical modes in the cooling regime, with the pump-laser a half-linewidth red-detuned from the cavity. Curves go from red to blue as ncav is increased. (e) Plot of the higher-frequency 3.6 MHz mechanical mode linewidth (blue dots) and effective temperature (red dots) versus ncav under red-detuned pumping.

Fig. A1.
Fig. A1.

FEM-simulations of electrostatic tuning. (a) Displacement of a single cavity membrane as function of the applied voltage Va for geometry parameters similar to those realized in the device studied in the main text. Blue bullets show simulation results, the red line shows a quadratic fit to the data, and the dashed black line shows the result of Eq. (A4). (b) Frequency shift of the fundamental cavity mode as function of the displacement of a single membrane. Within numerical error, the tuning proceeds linearly with gOM = 2π × 73 GHz/nm.

Fig. A2.
Fig. A2.

Mechanical mode spectra as function of membrane stiffness. (a) Mechanical spectra for different arrangements of the fiber taper: hovering over the membranes (blue curve), touching on membrane 1 (green curve), and touching on membrane 2. (b) Mechanical mode frequencies as function of w1, the width of the strut that carries the metal wires on the. We observe a clear anticrossing between the in-plane tuning mode and a spectrally nearby flexural mode. The device studied in the main text was processed with w1 close to this anticrossing.

Fig. A3.
Fig. A3.

AC-tuning curve of the device used for Fig. 2(c) of the main text. The false-color plot shows transmission spectra of the fundamental cavity mode when applying a sinusoidal drive voltage to the cavity contacts. Mechanical resonances are clearly visible by broadening of the transmission feature, when the double drive frequency 2ν coincides with a mode frequency. The mechanical mode structure found form RF PSD measurements is clearly reproduced here.

Equations (18)

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

F cap ( x ) = F spring ( x ) ,
1 2 d d x C ( x ) V a 2 = k eff x .
C ( x ) = a ( w g x ) n ,
δ x = n a 2 k eff w g n + 1 V a 2 .
α = g OM 2 π λ c 2 c n c 2 k eff w g = 0.025 nm / V 2
ω m 2 = ω m0 2 + 2 h ¯ g OM 2 n cav m eff Δ ω Δ ω 2 + κ 2 / 4 ,
n cav = κ e 2 1 Δ ω 2 + κ 2 / 4 P in h ¯ ω c ,
V ( ω ) = + T ( ω ) G ( ω ω ) d ω
T ( ω ) = 1 κ e 4 2 κ κ e ( ω ω c ) 2 + κ 2 / 4
G ( ω ) = 1 σ 2 π e ω 2 / ( 2 σ 2 ) .
f V 0.5346 κ + 0.2166 κ 2 + f G 2 .
x rms = x 2 = k B T m eff ω m 2 .
σ = g OM 2 π λ c 2 c x rms .
F cap ( t ) = 1 2 d C d x V AC 2 cos 2 ( ω t ) = F AC [ 1 + cos ( 2 ω t ) ]
F AC = 1 2 d C d x ( V AC 2 ) 2 .
x ( t ) = F AC m eff 1 ( 4 ω 2 ω i 2 ) 2 + 4 ω 2 γ m 2 cos ( 2 ω t )
Δ ω c , AC = g OM F AC k i , eff Q m , i
γ m = γ m , 0 2 h ¯ g OM 2 n cav κ m eff Δ ω ( Δ ω 2 + κ 2 / 4 ) 2 .

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