Abstract

We proposed a scheme for detecting the atom-field coupling constant in the Dicke superradiation regime based on a hybrid cavity optomechanical system assisted by an atomic gas. The critical behavior of the Dicke model was obtained analytically using the spin-coherent-state representation. Without regard to the dynamics of cavity field an analytical formula of one-to-one correspondence between movable mirror’s steady position and atom-field coupling constant for a given number of atoms is obtained. Thus the atom-field coupling constant can be probed by measuring the movable mirror’s steady position, which is another effect of the cavity optomechanics.

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  1. P. Lebedew, “Experimental examination of light pressure,” Ann. Phys. (Leipzig)6, 433–458 (1901).
  2. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev.13, 307 (1901).
  3. T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt6, S675–S683 (2004).
    [CrossRef]
  4. T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett.98, 150802 (2007).
    [CrossRef] [PubMed]
  5. S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
    [CrossRef] [PubMed]
  6. P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett.833174 (1999).
    [CrossRef]
  7. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett.99, 093902 (2007).
    [CrossRef] [PubMed]
  8. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett.99, 093901 (2007).
    [CrossRef] [PubMed]
  9. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008).
    [CrossRef] [PubMed]
  10. 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,” Nature478, 89–92 (2011).
    [CrossRef] [PubMed]
  11. D. Teufe, 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,” Nature475, 359–363 (2011).
    [CrossRef]
  12. K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today58, 36–42 (2005).
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  13. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A56, 4175 (1997).
    [CrossRef]
  14. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett.91, 130401 (2003).
    [CrossRef] [PubMed]
  15. F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett.105, 070403 (2010).
    [CrossRef] [PubMed]
  16. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett.98, 030405 (2007).
    [CrossRef] [PubMed]
  17. C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A77, 050307 (2008).
    [CrossRef]
  18. L. Zhou, Y. Han, J. Jing, and W. Zhang, “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A83, 052117 (2011).
    [CrossRef]
  19. M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A82, 012333 (2010).
    [CrossRef]
  20. K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett.107, 123601 (2011).
    [CrossRef] [PubMed]
  21. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett.88, 120401 (2002).
    [CrossRef] [PubMed]
  22. L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett.93, 266403 (2004).
    [CrossRef]
  23. K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong coupling of a mechanical oscillator and a single atom,” Phys. Rev. Lett.103, 063005 (2009).
    [CrossRef] [PubMed]
  24. P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensate coupled to a nanomechanical resonator on an atom chip,” Phys. Rev. Lett.99, 140403 (2007).
    [CrossRef] [PubMed]
  25. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322, 235–238 (2008).
    [CrossRef] [PubMed]
  26. A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A80, 043607 (2009).
    [CrossRef]
  27. D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett.104, 143002 (2010).
    [CrossRef] [PubMed]
  28. S. K. Steinke, S. Singh, M. E. Tasgin, P. Meystre, K. C. Schwab, and M. Vengalattore, “Quantum-measurement backaction from a Bose-Einstein condensate coupled to a mechanical oscillator,” Phys. Rev. A84, 023841 (2011).
    [CrossRef]
  29. H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A78, 013824 (2008).
    [CrossRef]
  30. Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A83, 053834 (2011).
    [CrossRef]
  31. G. Chen, Y. Zhang, L. Xiao, J.-Q. Liang, and S. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express18, 23016–23023 (2010).
    [CrossRef] [PubMed]
  32. K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett.102, 020501 (2009).
    [CrossRef] [PubMed]
  33. Q. Sun, X.-H. Hu, W. M. Liu, X. C. Xie, and A.-C. Ji, “Effect on cavity optomechanics of the interaction between a cavity field and a one-dimensional interacting bosonic gas,” Phys. Rev. A84, 023822 (2011).
    [CrossRef]
  34. Q. Sun, X.-H. Hu, A.-C. Ji, and W. M. Liu, “Dynamics of a degenerate Fermi gas in a one-dimensional optical lattice coupled to a cavity,” Phys. Rev. A83, 043606 (2011).
    [CrossRef]
  35. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93, 99–110 (1954).
    [CrossRef]
  36. K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Annals Phys.(N.Y.)76, 360–404 (1973).
    [CrossRef]
  37. Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A7, 831–836 (1973).
    [CrossRef]
  38. F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A8, 1440–1445 (1973).
    [CrossRef]
  39. C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett.90, 044101 (2003).
    [CrossRef] [PubMed]
  40. C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E67, 066203 (2003).
    [CrossRef]
  41. N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A71, 053804 (2005).
    [CrossRef]
  42. G. D. Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A83, 052324–052329 (2011).
    [CrossRef]
  43. G. Chen, J. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A74, 054101 (2006).
    [CrossRef]
  44. T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev.58, 1098–1113 (1940).
    [CrossRef]
  45. J. P. Santos, F. L. Semião, and K. Furuya, “Probing the quantum phase transition in the Dicke model through mechanical vibrations,” Phys. Rev. A82, 063801 (2010).
    [CrossRef]
  46. E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett.99, 173601 (2007).
    [CrossRef] [PubMed]
  47. S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A81, 042311 (2010).
    [CrossRef]
  48. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature454, 310–314 (2008).
    [CrossRef] [PubMed]
  49. F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A75, 013804 (2007).
    [CrossRef]

2011

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,” Nature478, 89–92 (2011).
[CrossRef] [PubMed]

D. Teufe, 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,” Nature475, 359–363 (2011).
[CrossRef]

L. Zhou, Y. Han, J. Jing, and W. Zhang, “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A83, 052117 (2011).
[CrossRef]

K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett.107, 123601 (2011).
[CrossRef] [PubMed]

S. K. Steinke, S. Singh, M. E. Tasgin, P. Meystre, K. C. Schwab, and M. Vengalattore, “Quantum-measurement backaction from a Bose-Einstein condensate coupled to a mechanical oscillator,” Phys. Rev. A84, 023841 (2011).
[CrossRef]

Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A83, 053834 (2011).
[CrossRef]

Q. Sun, X.-H. Hu, W. M. Liu, X. C. Xie, and A.-C. Ji, “Effect on cavity optomechanics of the interaction between a cavity field and a one-dimensional interacting bosonic gas,” Phys. Rev. A84, 023822 (2011).
[CrossRef]

Q. Sun, X.-H. Hu, A.-C. Ji, and W. M. Liu, “Dynamics of a degenerate Fermi gas in a one-dimensional optical lattice coupled to a cavity,” Phys. Rev. A83, 043606 (2011).
[CrossRef]

G. D. Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A83, 052324–052329 (2011).
[CrossRef]

2010

J. P. Santos, F. L. Semião, and K. Furuya, “Probing the quantum phase transition in the Dicke model through mechanical vibrations,” Phys. Rev. A82, 063801 (2010).
[CrossRef]

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A81, 042311 (2010).
[CrossRef]

G. Chen, Y. Zhang, L. Xiao, J.-Q. Liang, and S. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express18, 23016–23023 (2010).
[CrossRef] [PubMed]

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett.104, 143002 (2010).
[CrossRef] [PubMed]

M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A82, 012333 (2010).
[CrossRef]

F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett.105, 070403 (2010).
[CrossRef] [PubMed]

2009

K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong coupling of a mechanical oscillator and a single atom,” Phys. Rev. Lett.103, 063005 (2009).
[CrossRef] [PubMed]

A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A80, 043607 (2009).
[CrossRef]

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett.102, 020501 (2009).
[CrossRef] [PubMed]

2008

H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A78, 013824 (2008).
[CrossRef]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science322, 235–238 (2008).
[CrossRef] [PubMed]

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A77, 050307 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008).
[CrossRef] [PubMed]

M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature454, 310–314 (2008).
[CrossRef] [PubMed]

2007

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A75, 013804 (2007).
[CrossRef]

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett.99, 173601 (2007).
[CrossRef] [PubMed]

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

T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett.98, 150802 (2007).
[CrossRef] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett.98, 030405 (2007).
[CrossRef] [PubMed]

P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensate coupled to a nanomechanical resonator on an atom chip,” Phys. Rev. Lett.99, 140403 (2007).
[CrossRef] [PubMed]

2006

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
[CrossRef] [PubMed]

G. Chen, J. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A74, 054101 (2006).
[CrossRef]

2005

N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A71, 053804 (2005).
[CrossRef]

K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today58, 36–42 (2005).
[CrossRef]

2004

T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt6, S675–S683 (2004).
[CrossRef]

L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett.93, 266403 (2004).
[CrossRef]

2003

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett.91, 130401 (2003).
[CrossRef] [PubMed]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett.90, 044101 (2003).
[CrossRef] [PubMed]

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E67, 066203 (2003).
[CrossRef]

2002

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

1999

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett.833174 (1999).
[CrossRef]

1997

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A56, 4175 (1997).
[CrossRef]

1973

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Annals Phys.(N.Y.)76, 360–404 (1973).
[CrossRef]

Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A7, 831–836 (1973).
[CrossRef]

F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A8, 1440–1445 (1973).
[CrossRef]

1954

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93, 99–110 (1954).
[CrossRef]

1940

T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev.58, 1098–1113 (1940).
[CrossRef]

1901

P. Lebedew, “Experimental examination of light pressure,” Ann. Phys. (Leipzig)6, 433–458 (1901).

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

Allman, M. S.

D. Teufe, 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,” Nature475, 359–363 (2011).
[CrossRef]

Ansmann, M.

M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature454, 310–314 (2008).
[CrossRef] [PubMed]

Ashhab, S.

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A81, 042311 (2010).
[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,” Nature478, 89–92 (2011).
[CrossRef] [PubMed]

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett.102, 020501 (2009).
[CrossRef] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett.98, 030405 (2007).
[CrossRef] [PubMed]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
[CrossRef] [PubMed]

Bäuerle, D.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
[CrossRef] [PubMed]

Bhattacherjee, A. B.

A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A80, 043607 (2009).
[CrossRef]

Bialczak, R. C.

M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature454, 310–314 (2008).
[CrossRef] [PubMed]

Blaser, F.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
[CrossRef] [PubMed]

Bohm, H. R.

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature444, 67–70 (2006).
[CrossRef] [PubMed]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett.98, 030405 (2007).
[CrossRef] [PubMed]

Børkje, K.

K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett.107, 123601 (2011).
[CrossRef] [PubMed]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A56, 4175 (1997).
[CrossRef]

Bouwmeester, D.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett.91, 130401 (2003).
[CrossRef] [PubMed]

Brandes, T.

N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A71, 053804 (2005).
[CrossRef]

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E67, 066203 (2003).
[CrossRef]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett.90, 044101 (2003).
[CrossRef] [PubMed]

Brennecke, F.

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Figures (3)

Fig. 1
Fig. 1

A collective of N two-level atoms interact with a single-mode quantized cavity field. The movable cavity mirror is in a harmonic motion due to a linear restoring force from the spring.

Fig. 2
Fig. 2

The expectation values of the number of photons and the number of atoms in excited states per atom as a function of atom-field coupling constant λ with ω = ω0 = 2λc.

Fig. 3
Fig. 3

The movable mirror’s steady position qs as a function of atom-field coupling constant λ for a given number of atoms. qs is in unit of xzp/L. The solid line (red), dashed line (blue) and dotted line (black) represents N =5, 10 and 20 respectively. The system parameters are ω = ω0 = 10ωm and λc = 1.

Equations (22)

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H D M = ω a a + ω 0 J z + λ N ( J + + J ) ( a + a )
[ J + , J ] = 2 J z , [ J z , J ± ] = ± J ±
J u = 1 2 i = 1 N σ u ( i ) ( u = x , y , z ) J ± = 1 2 i = 1 N ( σ x ( i ) ± i σ y ( i ) )
a | α = α | α
| θ , φ = R θ , φ | j , j
J n | θ , φ = j | θ , φ
E ± ( α ) = ω ( μ 2 + v 2 ) ± N 2 ω 0 2 + ( 4 λ μ N ) 2
E ( α ) μ = 0 , E ( α ) v = 0.
α = { 0 , λ < λ c N ω 0 2 ( λ 4 / λ c 4 1 ) 16 λ 2 , λ > λ c
J z = θ , φ | R J z R | θ , φ = { N 2 , λ < λ c N λ 2 2 λ c 2 , λ > λ c
H O P M = ω m c c g a a ( c + c )
H O P M = P 2 2 M + M ω m 2 2 Q 2 ω L a a Q
q = M ω m 2 Q p = 1 2 M ω m P
H O P M = ω m 2 ( p 2 + q 2 ) 2 g a a q
q ˙ = ω m p Γ 2 q + Γ q in p ˙ = ω m q + 2 g a a Γ 2 p + Γ p in
q ˙ = ω m p Γ 2 q p ˙ = ω m q + 2 g | α | 2 Γ 2 p
q s = 2 g | α | 2 / ω m 1 + Γ 2 4 ω m 2 p s = Γ q s 2 ω m = g | α | 2 Γ / ω m 2 1 + Γ 2 4 ω m 2
δ q ( t ) = q ( t ) q s δ p ( t ) = p ( t ) p s
d d t ( δ q δ p ) = ( Γ 2 ω m ω m Γ 2 ) ( δ q δ p )
q ( t ) = e Γ 2 t [ cos ω m t 2 g | α | 2 Γ 2 4 + ω m 2 cos ( ω m t ϕ ) ] + 2 g | α | 2 / ω m 1 + Γ 2 4 ω m 2 = e Γ 2 t [ cos ω m t 2 g | α | 2 / ω m 1 Q m 2 + 1 cos ( ω m t tan ( 1 Q m ) ) ] + 2 g | α | 2 / ω m 1 Q m 2 + 1
q ( t ) = ( 1 2 g | α | 2 ω m ) e Γ 2 t cos ω m t + 2 g | α | 2 ω m
q s = 2 g | α | 2 ω m = N ω ω 0 2 x z p 8 ω m λ 2 L ( λ 4 λ c 4 1 )

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