Abstract

We propose a system for realizing controllable optomechanically induced transparency (OMIT) and ponderomotive squeezing. In this system, an atomic ensemble driven by an external optical field couples with the cavity field in a typical optomechanical cavity. When the cavity is driven by a coupling laser and a probe laser, we can produce a switch for the probe field and adjust the width of the transparency window flexibly by manipulating the coupling strength between the atomic ensemble and the external optical field. We also investigate the ponderomotive squeezing properties of the transmitted field by analyzing its spectrum. Interestingly, the coupling strength between the atomic ensemble and the cavity field plays an important role in controlling the squeezing properties and the squeezing spectrum presents distinct features at red-detuned and blue-detuned frequencies by adjusting the coupling strength.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. 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 (London) 471, 204 (2011).
    [CrossRef]
  4. A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
    [CrossRef]
  5. Devrim Tarhan, Sumei Huang, Ö zgür, and E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
    [CrossRef]
  6. Bin Chen, Cheng Jiang, and Ka-Di Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Phys. Rev. A 83, 055803 (2011).
    [CrossRef]
  7. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009).
    [CrossRef]
  8. Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
    [CrossRef] [PubMed]
  9. E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
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  10. G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801(R) (2012).
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  11. C. Jiang, B. Chen, and K.-D. Zhu, “Demonstration of a single-photon router with a cavity electromechanical system,” J. Appl. Phys. 112, 033113 (2012).
    [CrossRef]
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    [CrossRef]
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  18. L. Tian and H. L. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82, 053806 (2010).
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  19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
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  22. M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
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  23. A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
    [CrossRef]
  24. Jie-Qiao Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
    [CrossRef]
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    [CrossRef] [PubMed]
  26. N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
    [CrossRef] [PubMed]
  27. E. Alebachew and K. Fessah, “Interaction of a two-level atom with squeezed light,” Opt. Commun. 271, 154 (2007).
    [CrossRef]
  28. S. Mancini and P. Tombesi, “Quantum noise reduction by radiation yressure,” Phys. Rev. A 49, 4055 (1994).
    [CrossRef] [PubMed]
  29. D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
    [CrossRef]
  30. A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
    [CrossRef]
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  32. D. Meiser and P. Meystre, “Coupled dynamics of atoms and radiation-pressure-driven interferometers,” Phys. Rev. A 73, 033417 (2006).
    [CrossRef]
  33. K. Hammerer, K. Stannigel, C. Genes, and P. Zoller, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803(R) (2010).
    [CrossRef]
  34. C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91, 147903 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  38. S. Gröblacher, K. Hammerer, M. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature (London) 460, 724 (2009).
    [CrossRef]
  39. C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
    [CrossRef]
  40. Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44165505 (2011).
    [CrossRef]
  41. M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887 (1985).
    [CrossRef] [PubMed]
  42. E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288 (1987).
    [CrossRef] [PubMed]

2013 (5)

Devrim Tarhan, Sumei Huang, Ö zgür, and E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
[CrossRef]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[CrossRef]

T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013).
[CrossRef]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
[CrossRef]

T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X,  3, 031012 (2013).

2012 (5)

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[CrossRef] [PubMed]

E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
[CrossRef]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801(R) (2012).
[CrossRef]

C. Jiang, B. Chen, and K.-D. Zhu, “Demonstration of a single-photon router with a cavity electromechanical system,” J. Appl. Phys. 112, 033113 (2012).
[CrossRef]

2011 (6)

Bin Chen, Cheng Jiang, and Ka-Di Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Phys. Rev. A 83, 055803 (2011).
[CrossRef]

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 (London) 471, 204 (2011).
[CrossRef]

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[CrossRef] [PubMed]

Jie-Qiao Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[CrossRef]

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44165505 (2011).
[CrossRef]

2010 (4)

K. Hammerer, K. Stannigel, C. Genes, and P. Zoller, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803(R) (2010).
[CrossRef]

L. Tian and H. L. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82, 053806 (2010).
[CrossRef]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

2009 (3)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009).
[CrossRef]

F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
[CrossRef]

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

2008 (5)

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[CrossRef]

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
[CrossRef] [PubMed]

A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[CrossRef]

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

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

2007 (4)

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]

M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (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]

E. Alebachew and K. Fessah, “Interaction of a two-level atom with squeezed light,” Opt. Commun. 271, 154 (2007).
[CrossRef]

2006 (1)

D. Meiser and P. Meystre, “Coupled dynamics of atoms and radiation-pressure-driven interferometers,” Phys. Rev. A 73, 033417 (2006).
[CrossRef]

2003 (1)

C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91, 147903 (2003).
[CrossRef] [PubMed]

2001 (1)

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

1995 (1)

N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
[CrossRef] [PubMed]

1994 (2)

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation yressure,” Phys. Rev. A 49, 4055 (1994).
[CrossRef] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
[CrossRef] [PubMed]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288 (1987).
[CrossRef] [PubMed]

1985 (1)

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

1958 (1)

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555 (1958).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Phys. Rev. A 85, 021801(R) (2012).
[CrossRef]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[CrossRef]

Alebachew, E.

E. Alebachew and K. Fessah, “Interaction of a two-level atom with squeezed light,” Opt. Commun. 271, 154 (2007).
[CrossRef]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[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 (London) 471, 204 (2011).
[CrossRef]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

Aspelmeyer, M.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
[CrossRef]

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

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

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]

Barbour, Russell

Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[CrossRef] [PubMed]

Bhattacharya, M.

M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99, 073601 (2007).
[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]

Botter, T.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Bourzeix, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
[CrossRef] [PubMed]

Brahms, N.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Brooks, D. W. C.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Chan, J.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
[CrossRef]

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Chen, B.

C. Jiang, B. Chen, and K.-D. Zhu, “Demonstration of a single-photon router with a cavity electromechanical system,” J. Appl. Phys. 112, 033113 (2012).
[CrossRef]

Chen, Bin

Bin Chen, Cheng Jiang, and Ka-Di Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Phys. Rev. A 83, 055803 (2011).
[CrossRef]

Cheng, J.

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” J. Phys. B: At. Mol. Opt. Phys. 44165505 (2011).
[CrossRef]

Cicak, K.

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 (London) 471, 204 (2011).
[CrossRef]

Clerk, A A

A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[CrossRef]

Clerk, A. A.

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288 (1987).
[CrossRef] [PubMed]

Delalise, S.

E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
[CrossRef]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

Edamatsu, K.

N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
[CrossRef] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Fabre, C.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
[CrossRef] [PubMed]

Ferreira, A.

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]

Fessah, K.

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C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
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N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
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C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
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A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
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A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
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A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
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T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X,  3, 031012 (2013).

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N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
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E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
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S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
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T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172 (2008).
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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).
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Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
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Jie-Qiao Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013).
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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 (London) 471, 204 (2011).
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Jie-Qiao Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
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A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009).
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S. Mancini and P. Tombesi, “Quantum noise reduction by radiation yressure,” Phys. Rev. A 49, 4055 (1994).
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A A Clerk, F Marquardt, and K Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
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F. Marquardt and S. M. Girvin, “Trend: optomechanics,” Physics 2, 40 (2009).
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Devrim Tarhan, Sumei Huang, Ö zgür, and E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
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Naderi, M. H.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
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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]

Painter, O.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
[CrossRef]

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
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T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013).
[CrossRef]

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N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
[CrossRef] [PubMed]

Peterson, R.W.

T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X,  3, 031012 (2013).

Pinard, M.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
[CrossRef] [PubMed]

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M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
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N. Ph. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426 (1995).
[CrossRef] [PubMed]

Purdy, T. P.

T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X,  3, 031012 (2013).

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Regal, C. A.

T. P. Purdy, P.-L. Yu, R.W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X,  3, 031012 (2013).

Reynaud, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337 (1994).
[CrossRef] [PubMed]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature (London) 500, 185 (2013).
[CrossRef]

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Sanders, B. C.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009).
[CrossRef]

Schliesser, A.

E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

Schreppler, S.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Shahidani, S.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[CrossRef]

Simmonds, R. W.

T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013).
[CrossRef]

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 (London) 471, 204 (2011).
[CrossRef]

Sirois, A. J.

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 (London) 471, 204 (2011).
[CrossRef]

Soltanolkotabi, M.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[CrossRef]

Stamper-Kurn, D. M.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum-noise-driven cavity optomechanics,” Nature (London) 448, 476 (2012).
[CrossRef]

Stannigel, K.

K. Hammerer, K. Stannigel, C. Genes, and P. Zoller, “Optical lattices with micromechanical mirrors,” Phys. Rev. A 82, 021803(R) (2010).
[CrossRef]

Sun, C. P.

C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91, 147903 (2003).
[CrossRef] [PubMed]

Tarhan, Devrim

Devrim Tarhan, Sumei Huang, Ö zgür, and E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87, 013824 (2013).
[CrossRef]

Teufel, J. D.

T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanical oscillator,” Nature (London) 495, 210–214 (2013).
[CrossRef]

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 (London) 471, 204 (2011).
[CrossRef]

Tian, L.

L. Tian and H. L. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82, 053806 (2010).
[CrossRef]

Tian, Lin

Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[CrossRef] [PubMed]

Tittel, W.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3, 706 (2009).
[CrossRef]

Tombesi, P.

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

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[CrossRef]

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. Mancini and P. Tombesi, “Quantum noise reduction by radiation yressure,” Phys. Rev. A 49, 4055 (1994).
[CrossRef] [PubMed]

Vahala, K. J.

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

Vanner, M.

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

Vedral, V.

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]

Verhagen, E.

E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
[CrossRef]

Vitali, D.

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[CrossRef]

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

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]

V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion,” Phys. Rev. A 63, 023812 (2001).
[CrossRef]

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M. J. Collett and D. F. Walls, “Squeezing spectra for nonlinear optical systems,” Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

Wang, H. L.

L. Tian and H. L. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82, 053806 (2010).
[CrossRef]

Wang, Hailin

Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[CrossRef] [PubMed]

Wang, Y. D.

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
[CrossRef] [PubMed]

Weis, S.

E. Verhagen, S. Delalise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature (London) 482, 63 (2012).
[CrossRef]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[CrossRef] [PubMed]

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 (London) 471, 204 (2011).
[CrossRef]

Wilson-Rae, I.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[CrossRef] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. M. 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 (London) 472, 69 (2011).
[CrossRef]

Yang, Yong

Victor Fiore, Yong Yang, Mark C. Kuzyk, Russell Barbour, Lin Tian, and Hailin Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[CrossRef] [PubMed]

Yu, P.-L.

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Figures (3)

Fig. 1
Fig. 1

Sketch of the system. A two-level atomic ensemble driven by an external optical field couples with the cavity field in a typical optomechanical cavity. The cavity is driven by a coupling laser and a probe laser.

Fig. 2
Fig. 2

(a) The transparency window with the laser power Pc as a parameter. (b) The transparency window with the coupling strength χ as a parameter. (c) The optical switch for the probe field. (d) The behaviour of the dispersion.

Fig. 3
Fig. 3

Squeezing with different coupling strength GA

Equations (43)

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H = h ¯ ω 0 c c + h ¯ ω a 2 i = 1 N σ z i + ( h ¯ G c i = 1 N σ + i + H . c ) + ( h ¯ Ω e i ω c t i = 1 N σ + i + H . c ) + ( p 2 2 m + 1 2 m ω m 2 q 2 ) h ¯ g c c q + i h ¯ ε c ( c e i ω c t H . c ) + i h ¯ ( ε p c e i ω p t H . c ) ,
A = 1 N i = 1 N σ + i , A = ( A ) .
[ A , A ] 1 .
i = 1 N σ z i = 2 A A N .
H = h ¯ ω 0 c c + h ¯ ω a A A + ( h ¯ G A c A + h ¯ χ A e i ω c t + H . c ) + p 2 2 m + 1 2 m ω m 2 q 2 h ¯ g c c q + i h ¯ ε c ( c e i ω c t H . c ) + i h ¯ ( ε p c e i ω p t H . c ) ,
H = h ¯ Δ c c c + h ¯ Δ a A A + ( h ¯ G A c A + h ¯ χ A + H . c ) + p 2 2 m + 1 2 m ω m 2 q 2 h ¯ g c c q + i h ¯ ε c ( c H . c ) + i h ¯ ( ε p c e i δ t H . c ) ,
c ˙ = i Δ c c i G A A + i g c q κ c + ε c + ε p e i δ t + 2 κ c in ,
A ˙ = i Δ a A i G A c i χ γ a A + 2 κ A in ,
p ˙ = m ω m 2 q + h ¯ g c c γ m p + ξ ,
q ˙ = p m ,
A s = i f c s i χ i Δ a + γ a , p s = 0 , q s = h ¯ g | c s | 2 m ω m 2 , c s = ε c f χ i ( Δ c | f | 2 Δ a g q s ) + ( κ + | f | 2 γ a ) ,
δ c ˙ = ( κ + i Δ ) δ c i G A δ A + i λ δ q + ε p e i δ t + 2 κ c in ,
δ A ˙ = ( γ a + i Δ a ) δ A i G A δ c + 2 κ A in ,
δ p ˙ = γ m δ p m ω m 2 δ q + h ¯ λ ( δ c + δ c ) + ξ ,
δ q ˙ = δ p m ,
c + = m [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 m [ κ + i ( Δ + δ ) + β ] [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 ( 2 i Δ α + β ) ε p ,
c out ( t ) = ( ε c 2 κ c s ) e i ω c t + ( ε p 2 κ c + ) e i ( δ + ω c ) t 2 κ c e i ( δ ω c ) t ,
t p = ε p 2 κ c + ε p = 1 2 κ m [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 m [ κ + i ( Δ + δ ) + β ] [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 ( 2 i Δ α + β ) .
T p = t p t r 1 t r ,
t r = t p ( δ = ω m , λ = 0 ) .
| T p | 2 = | 1 κ m [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 m [ κ + i ( Δ + δ ) + β ] [ κ i ( Δ + δ ) + α ] ( ω m 2 δ 2 i δ γ m ) + i h ¯ λ 2 ( 2 i Δ α + β ) | 2 .
γ eff = γ m ( 1 + C ) ,
| c s | 2 | ε c f χ | 2 .
S θ ( ω ) = δ X θ out ( t + τ ) δ X θ out ( t ) e i ω τ d τ = δ X θ out ( ω ) δ X θ out ( ω ) ,
S θ ( ω ) = 1 + 2 B c c + e 2 i θ B c c + e 2 i θ B c c ,
e 2 i θ opt = ± B c c ( ω ) | B c c ( ω ) | ,
S opt ( ω ) = 1 + 2 B c c 2 | B c c | .
u ( t ) = 1 2 π e i ω t u ( ω ) d ω , u ( t ) = 1 2 π e i ω t u ( ω ) d ω ,
c in ( Ω ) c in ( ω ) = δ ( Ω ω ) ,
A in ( Ω ) A in ( ω ) = δ ( Ω ω ) ,
ξ ( Ω ) ξ ( ω ) = h ¯ γ m m ω ( 1 + coth h ¯ ω 2 k B T ) δ ( Ω + ω ) .
M ( ω ) Z ( ω ) = Y ( ω ) ,
M = ( Λ 1 0 i G A 0 0 i λ 0 Λ 2 0 i G A 0 i λ i G A 0 Θ 1 0 0 0 0 i G A 0 Θ 2 0 0 h ¯ λ h ¯ λ 0 0 γ m i ω m ω m 2 0 0 0 0 1 i m ω ) ,
δ c ( ω ) = E 1 ( ω ) c in E 2 ( ω ) c in + E 3 ( ω ) A in + E 4 ( ω ) A in + E 5 ( ω ) ( ξ ) ,
δ c ( ω ) = E 1 * ( ω ) c in + E 2 * ( ω ) c in + E 3 * ( ω ) A in + E 4 * ( ω ) A in + E 1 * ( ω ) ξ ,
E 1 ( ω ) = 2 κ Θ 1 [ i h ¯ λ 2 Θ 2 + m ( G A 2 + Θ 2 Λ 2 ) ( ω m 2 ω 2 i ω γ m ) ] d ( ω ) ,
E 2 ( ω ) = i 2 κ h ¯ λ 2 Θ 1 Θ 2 d ( ω ) ,
E 3 ( ω ) = i 2 γ a G A [ i h ¯ λ 2 Θ 2 + m ( G A 2 + Θ 2 Λ 2 ) ( ω m 2 ω 2 i ω γ m ) ] d ( ω ) ,
E 4 ( ω ) = 2 γ a h ¯ λ 2 G A Θ 1 d ( ω ) ,
E 5 ( ω ) = i λ Θ 1 ( G A 2 + Θ 2 Λ 2 ) d ( ω ) ,
d ( ω ) = i h ¯ λ 2 [ Θ 1 ( G A 2 + Θ 2 Λ 2 ) Θ 2 ( G A 2 + Θ 1 Λ 1 ) ] m ( G A 2 + Θ 2 Λ 2 ) ( G A 2 + Θ 1 Λ 1 ) ( ω m 2 ω 2 i ω γ m ) .
B c c = 2 κ [ | E 2 ( ω ) | 2 + | E 4 ( ω ) | 2 + h ¯ γ m m ω ( 1 + coth [ h ¯ ω 2 k B T ] ) | E 5 ( ω ) | 2 ] ,
B c c = 2 κ [ E 1 ( ω ) E 2 ( ω ) + E 3 ( ω ) E 4 ( ω ) E 2 ( ω ) 2 κ + E 5 ( ω ) E 5 ( ω ) h ¯ γ m m ω ( 1 + coth [ h ¯ ω 2 k B T ] ) ] .

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