Abstract

We theoretically investigate an optomechanical structure consisting of two parallel GaAs membranes with an air-slot type photonic crystal nanocavity. The optical cavity has a quality factor of 4.8 × 106 at 1.52 μm and an extremely small modal volume of 0.015 of a cubic wavelength for the fundamental mode in a vacuum. The localized electric field near the air/dielectric-object boundary provides a large optomechanical coupling factor of ~990 GHz/nm. The fundamental mechanical mode resonance is 95 MHz and a quality factor is 83,800 at room temperature, nearly seven times higher than that for a similar Si-based structure. This high mechanical quality factor of a GaAs-based structure stems from low thermoelastic loss and leads to more effective optical control of nanomechanical oscillators.

©2012 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
  24. Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12(17), 3988–3995 (2004).
    [Crossref] [PubMed]
  25. H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
    [Crossref]
  26. M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express 18(8), 8144–8150 (2010).
    [Crossref] [PubMed]
  27. C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230–235 (1937).
    [Crossref]
  28. T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
    [Crossref]

2011 (2)

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

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

2010 (7)

M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express 18(8), 8144–8150 (2010).
[Crossref] [PubMed]

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18(23), 23844–23856 (2010).
[Crossref] [PubMed]

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

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

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

D. Van Thourhout and J. Roels, “Optomechanical Device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

2009 (4)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (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), 550–555 (2009).
[Crossref] [PubMed]

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

2008 (4)

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008).
[Crossref] [PubMed]

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), 72–75 (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]

2006 (3)

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “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]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

2004 (3)

Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12(17), 3988–3995 (2004).
[Crossref] [PubMed]

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004).
[Crossref] [PubMed]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

1991 (1)

S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991).
[Crossref] [PubMed]

1990 (1)

C. N. Cohen-Tannoudji and W. D. Phillips, “New mechanisms for laser cooling,” Phys. Today 43(10), 33–40 (1990).
[Crossref]

1989 (1)

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

1937 (1)

C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230–235 (1937).
[Crossref]

Alegre, T. P. M.

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

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

Anetsberger, G.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Arakawa, Y.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express 18(8), 8144–8150 (2010).
[Crossref] [PubMed]

Aras, M. S.

Arcizet, O.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

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

Aspelmeyer, M.

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

Assefa, S.

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Bergquist, J. C.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Bouwmeester, D.

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

Briant, T.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444(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), 550–555 (2009).
[Crossref] [PubMed]

Chan, J.

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

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Chu, S.

S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991).
[Crossref] [PubMed]

Cohadon, P. F.

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

Cohen-Tannoudji, C. N.

C. N. Cohen-Tannoudji and W. D. Phillips, “New mechanisms for laser cooling,” Phys. Today 43(10), 33–40 (1990).
[Crossref]

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Diedrich, F.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Ding, L.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Ducci, S.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Ee, H. S.

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Favero, I.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Gao, J.

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18(23), 23844–23856 (2010).
[Crossref] [PubMed]

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Gibbs, H. M.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Girvin, S. 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), 72–75 (2008).
[Crossref] [PubMed]

Gotoh, H.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Gröblacher, S.

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

Harris, J. G. E.

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), 72–75 (2008).
[Crossref] [PubMed]

Heidmann, A.

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

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Hill, J. T.

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

Houston, B. H.

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

Itano, W. M.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Ito, D.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Iwamoto, S.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express 18(8), 8144–8150 (2010).
[Crossref] [PubMed]

Jayich, A. 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), 72–75 (2008).
[Crossref] [PubMed]

Jeong, K. Y.

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

Karrai, K.

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004).
[Crossref] [PubMed]

Khitrova, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Kippenberg, T. J.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

Kleckner, D.

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

Kotthaus, J. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Krause, A.

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

Kuga, T.

Kumagai, N.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

Kuramochi, E.

T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Lee, Y. H.

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

Lemaitre, A.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Leo, G.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Li, Y.

Marquardt, F.

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), 72–75 (2008).
[Crossref] [PubMed]

McMillan, J. F.

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Metcalf, T. H.

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

Metzger, C. H.

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004).
[Crossref] [PubMed]

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Nomura, M.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express 18(8), 8144–8150 (2010).
[Crossref] [PubMed]

Notomi, M.

T. Yamamoto, M. Notomi, H. Taniyama, E. Kuramochi, Y. Yoshikawa, Y. Torii, and T. Kuga, “Design of a high-Q air-slot cavity based on a width-modulated line-defect in a photonic crystal slab,” Opt. Express 16(18), 13809–13817 (2008).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Okamoto, H.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Onomitsu, K.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Ota, Y.

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

Painer, O.

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

Painter, O.

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

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Park, H. G.

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

Park, Y.-S.

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

Pate, B. B.

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

Phillips, W. D.

C. N. Cohen-Tannoudji and W. D. Phillips, “New mechanisms for laser cooling,” Phys. Today 43(10), 33–40 (1990).
[Crossref]

Photiadis, D. M.

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

Pinard, M.

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

Qiu, M.

Riviere, R.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical Device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

Rupper, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

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

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

Sanada, H.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Scherer, A.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Schliesser, A.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Senellart, P.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

Seo, M. K.

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

Shchekin, O. B.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Shinya, A.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Shu, J.

Sogawa, T.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Tanabe, K.

Tanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Taniyama, H.

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), 72–75 (2008).
[Crossref] [PubMed]

Torii, Y.

Unterreithmeier, Q. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[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), 550–555 (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]

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical Device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

Wang, H.

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

Watanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

Weig, E. M.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Wineland, D. J.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Winger, M.

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

Wong, C. W.

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18(23), 23844–23856 (2010).
[Crossref] [PubMed]

Wu, M.-C.

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Yamaguchi, H.

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

Yamamoto, T.

Yoshie, T.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

Yoshikawa, Y.

Zener, C.

C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230–235 (1937).
[Crossref]

Zhang, Z.

Zheng, J.

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express 18(23), 23844–23856 (2010).
[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), 72–75 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (5)

J. Gao, J. F. McMillan, M.-C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot modegap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96(5), 051123 (2010).
[Crossref]

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

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultra-high-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88(4), 041112 (2006).
[Crossref]

H. S. Ee, K. Y. Jeong, M. K. Seo, Y. H. Lee, and H. G. Park, “Ultrasmall square-lattice zero-cell photonic crystal laser,” Appl. Phys. Lett. 93(1), 011104 (2008).
[Crossref]

T. H. Metcalf, B. B. Pate, D. M. Photiadis, and B. H. Houston, “Thermoelastic damping in micromechanical resonators,” Appl. Phys. Lett. 95(6), 061903 (2009).
[Crossref]

Nat. Photonics (1)

D. Van Thourhout and J. Roels, “Optomechanical Device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

Nat. Phys. (3)

M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
[Crossref]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5(12), 909–914 (2009).
[Crossref]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

Nature (7)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

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

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004).
[Crossref] [PubMed]

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

C. H. Metzger and K. Karrai, “Cavity cooling of a microlever,” Nature 432(7020), 1002–1005 (2004).
[Crossref] [PubMed]

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), 72–75 (2008).
[Crossref] [PubMed]

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

Opt. Express (4)

Phys. Rev. (1)

C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230–235 (1937).
[Crossref]

Phys. Rev. Lett. (3)

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett. 105(26), 263903 (2010).
[Crossref] [PubMed]

H. Okamoto, D. Ito, K. Onomitsu, H. Sanada, H. Gotoh, T. Sogawa, and H. Yamaguchi, “Vibration amplification, damping, and self-oscillations in micromechanical resonators induced by optomechanical coupling through carrier excitation,” Phys. Rev. Lett. 106(3), 036801 (2011).
[Crossref] [PubMed]

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62(4), 403–406 (1989).
[Crossref] [PubMed]

Phys. Today (1)

C. N. Cohen-Tannoudji and W. D. Phillips, “New mechanisms for laser cooling,” Phys. Today 43(10), 33–40 (1990).
[Crossref]

Science (2)

S. Chu, “Laser manipulation of atoms and particles,” Science 253(5022), 861–866 (1991).
[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 (1)

V. Braginsky and A. Manukin, Measurement of Weak Forces in Physics Experiments (Univ. Chicago Press, 1977).

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Figures (5)

Fig. 1
Fig. 1

(a) Computed cavity electric field distribution in the air-slot PhC nanocavity structure. The long-term (Q = 4.8 × 106) and localized (V = 0.015λ3) electric field confinement around the air/semiconductor boundary results in extremely strong optomechanical interaction. (b) Color plot of the absolute displacement in y direction for the investigated fundamental mechanical oscillation mode (95 MHz). This oscillation can be optically controlled by the cavity field in the PhC nanocavity.

Fig. 2
Fig. 2

Enhanced damping rate at various laser frequency detuning with respect to the cavity resonance Δω. The lateral axis is normalized by the optical cavity decay rate κ.

Fig. 3
Fig. 3

Time evolution of the velocity of the membrane (y>0) and the cavity field amplitude (a) at the detuning of −0.35κ (red detuning, cooling case) and (b) at the detuning of 0.35κ (blue detuning, heating case).

Fig. 4
Fig. 4

(a) Mechanical frequency change induced by optical stiffening and (b) cooling factor at various detunings and values of cavity Q at an input power of 1 nW.

Fig. 5
Fig. 5

Color map of the calculated cooling factors at various detunings and values of Qm at an input power of 1 nW. Cavity Q is fixed at 5 × 105.

Tables (1)

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Table 1 Summary of the parameters used to estimate Qm and values of Qm for GaAs and Si-based structures.

Equations (6)

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Q m = C p ρ EαT ( 1+ Ω m τ th ) 2 Ω m τ th
L OM 1 = 1 2 ( q( r )· n )[ Δ ε 12 ( r ) | E ( 0 ) (r) | 2 Δ( ε 12 1 ( r ) ) | D ( 0 ) (r) | 2 ]dA ε(r) | E(r) | 2 dV .
da dt =iΔω( y )a( 1 2 τ 0 + 1 2 τ ex )a+ 1 τ ex s
d 2 y d t 2 + Ω m 2 Q m dy dt + Ω m 2 = F OM m eff = g OM ω c   | a | 2 m eff
m eff = (r r 0 )   2 ( r r 0 ) max 2 ρdV,
Γ om = ω c 2 Ω m L om 2 m eff ( 2 κ ex κ 2 +4Δ (y) 2 )[ κ/2 (Δ( y ) Ω m ) 2 + (κ/2) 2 κ/2 (Δ( y )+ Ω m ) 2 + (κ/2) 2 ]P.

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