Abstract

In this work we investigate an optomechanical system consisting of two cavities coupled to the same mechanical resonator. We consider each cavity being weakly pumped as well as a small tunneling rate between the cavities. In such conditions, the system can be studied via quantum Langevin equations and the steady-state solution can be found perturbatively. In order to ensure that the approximations and methods used to study the system are suitable, the analytical results were compared to numerical results. We study the statistical properties of the cavity radiation fields and we show that depending on the values of the parameters of the system, it is possible to modify the spectrum of the cavities and enhance significantly the sub-Poissonian character of the cavity field.

© 2014 Optical Society of America

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    [CrossRef]
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  9. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
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    [CrossRef]
  36. L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
    [CrossRef]
  37. L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108, 153604 (2012).
    [CrossRef]
  38. Y.-D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108, 153603 (2012).
    [CrossRef]
  39. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
    [CrossRef]
  40. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
    [CrossRef]
  41. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196–1206 (2012).
    [CrossRef]
  42. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon photon translator,” New J. Phys. 13, 013017 (2011).
    [CrossRef]
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  44. S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B 1, 424–432 (1999).
    [CrossRef]

2013 (4)

A. Kronwald, M. Ludwig, and F. Marquardt, “Full photon statistics of a light beam transmitted through an optomechanical system,” Phys. Rev. A 87, 013847 (2013).
[CrossRef]

X.-W. Xu, Y.-J. Li, and Y.-X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
[CrossRef]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[CrossRef]

L. Qiu, L. Gan, W. Ding, and Z.-Y. Li, “Single-photon generation by pulsed laser in optomechanical system via photon blockade effect,” J. Opt. Soc. Am. B 30, 1683–1687 (2013).
[CrossRef]

2012 (6)

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108, 153604 (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]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[CrossRef]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196–1206 (2012).
[CrossRef]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
[CrossRef]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

2011 (8)

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

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

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[CrossRef]

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

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (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 471, 204–208 (2011).
[CrossRef]

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

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[CrossRef]

2010 (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

2009 (4)

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

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2009).
[CrossRef]

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

2008 (3)

F. Brennecke, S. Ritter, T. Donner, and T. Esslinge, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

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

J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[CrossRef]

2007 (3)

T. Carmon and K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1  GHz frequency,” Phys. Rev. Lett. 98, 123901 (2007).
[CrossRef]

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

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, 71–74 (2006).
[CrossRef]

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

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

2002 (1)

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

1999 (1)

S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B 1, 424–432 (1999).
[CrossRef]

1997 (2)

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[CrossRef]

S. Mancini, V. I. Man’ko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042–3050 (1997).
[CrossRef]

1995 (1)

M. Pinard, C. Fabre, and A. Heidmann, “Quantum-nondemolition measurement of light by a piezoelectric crystal,” Phys. Rev. A 51, 2443–2449 (1995).
[CrossRef]

1994 (3)

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, 1337–1343 (1994).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef]

1970 (1)

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 31, 829–830 (1970).

1967 (1)

V. B. Braginsky and A. B. Manukin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 25, 653–655 (1967).

Allman, M. S.

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

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (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, 1520–1523 (2010).
[CrossRef]

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

Aspelmeyer, M.

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

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

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

Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[CrossRef]

Bäuerle, D.

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

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Blaser, F.

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

Bohm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

Böhm, H. R.

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

Borkje, K.

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[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, 1337–1343 (1994).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 31, 829–830 (1970).

V. B. Braginsky and A. B. Manukin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 25, 653–655 (1967).

V. B. Braginsky, Measurement of Weak Forces in Physics Experiments (University of Chicago, 1977).

Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinge, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

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, 71–74 (2006).
[CrossRef]

Camacho, R. M.

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

Carmon, T.

T. Carmon and K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1  GHz frequency,” Phys. Rev. Lett. 98, 123901 (2007).
[CrossRef]

Chan, J.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196–1206 (2012).
[CrossRef]

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

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

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

Chang, D. E.

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

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

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O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
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K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
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J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
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T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2009).
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M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
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A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon photon translator,” New J. Phys. 13, 013017 (2011).
[CrossRef]

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

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

Schwab, K. C.

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2009).
[CrossRef]

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

Senellart, P.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[CrossRef]

Simmonds, R. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (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 471, 204–208 (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 471, 204–208 (2011).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

Stamper-Kurn, D. M.

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

Stannigel, K.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

Tan, S. M.

S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B 1, 424–432 (1999).
[CrossRef]

Teufel, J. D.

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

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[CrossRef]

Thompson, J. D.

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

Tian, L.

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[CrossRef]

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108, 153604 (2012).
[CrossRef]

Tikhonov, M. Y.

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 31, 829–830 (1970).

Tombesi, P.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

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

S. Mancini, V. I. Man’ko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042–3050 (1997).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef]

Vahala, K. J.

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

T. Carmon and K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1  GHz frequency,” Phys. Rev. Lett. 98, 123901 (2007).
[CrossRef]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

Vanner, M. R.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Vedral, V.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

Vitali, D.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

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

Walls, D. F.

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef]

Wang, H.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[CrossRef]

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]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Whittaker, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (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 471, 204–208 (2011).
[CrossRef]

Winger, M.

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

Xu, X.-W.

X.-W. Xu, Y.-J. Li, and Y.-X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
[CrossRef]

Zeilinger, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

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

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Zwickl, B. M.

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

Appl. Phys. Lett. (1)

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett. 98, 113108 (2011).
[CrossRef]

J. Opt. B (1)

S. M. Tan, “A computational toolbox for quantum and atomic optics,” J. Opt. B 1, 424–432 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196–1206 (2012).
[CrossRef]

Nature (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,” Nature 478, 89–92 (2011).
[CrossRef]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (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, 71–74 (2006).
[CrossRef]

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

T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, “Preparation and detection of a mechanical resonator near the ground state of motion,” Nature 463, 72–75 (2009).
[CrossRef]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[CrossRef]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[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 471, 204–208 (2011).
[CrossRef]

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

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

New J. Phys. (1)

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

Phys. Rev. A (8)

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, 1337–1343 (1994).
[CrossRef]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[CrossRef]

K. Jacobs, P. Tombesi, M. J. Collett, and D. F. Walls, “Quantum-nondemolition measurement of photon number using radiation pressure,” Phys. Rev. A 49, 1961–1966 (1994).
[CrossRef]

M. Pinard, C. Fabre, and A. Heidmann, “Quantum-nondemolition measurement of light by a piezoelectric crystal,” Phys. Rev. A 51, 2443–2449 (1995).
[CrossRef]

A. Kronwald, M. Ludwig, and F. Marquardt, “Full photon statistics of a light beam transmitted through an optomechanical system,” Phys. Rev. A 87, 013847 (2013).
[CrossRef]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175–4186 (1997).
[CrossRef]

S. Mancini, V. I. Man’ko, and P. Tombesi, “Ponderomotive control of quantum macroscopic coherence,” Phys. Rev. A 55, 3042–3050 (1997).
[CrossRef]

X.-W. Xu, Y.-J. Li, and Y.-X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87, 025803 (2013).
[CrossRef]

Phys. Rev. Lett. (13)

A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011).
[CrossRef]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109, 063601 (2012).
[CrossRef]

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
[CrossRef]

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

D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, 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]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[CrossRef]

T. Carmon and K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1  GHz frequency,” Phys. Rev. Lett. 98, 123901 (2007).
[CrossRef]

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108, 153604 (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]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[CrossRef]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett. 101, 197203 (2008).
[CrossRef]

Physics (1)

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

Science (3)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[CrossRef]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinge, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[CrossRef]

Sov. Phys. J. Exp. Theor. Phys. (2)

V. B. Braginsky and A. B. Manukin, “Ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 25, 653–655 (1967).

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Sov. Phys. J. Exp. Theor. Phys. 31, 829–830 (1970).

Other (2)

V. B. Braginsky, Measurement of Weak Forces in Physics Experiments (University of Chicago, 1977).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

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Figures (6)

Fig. 1.
Fig. 1.

Energy level diagram of Hamiltonian (1) for E=0. If the cavity is in the |nc state, the radiation pressure force displaces the mechanical oscillator by ncg/ωm and the eigenenergies of the system are lowered by Δgnc2.

Fig. 2.
Fig. 2.

(a) In the optomechanical system considered here we have two optical cavities optomechanically coupled to the same mechanical oscillator with the same optomechanical coupling parameter g. The cavities are also tunnel-coupled, with tunneling amplitude J, and are weakly pumped by lasers with the same frequency ωL. The colors of the cavities do not have any relationship with the detuning of the cavities. (b) Energy level diagram of Hamiltonian (7). We shall call the states for which n1+n2=n the n’th group of states.

Fig. 3.
Fig. 3.

(a) S1 against Δ1. In the blue curve we have J=0, in the green curve J=0.05ωm, Δ2=0.4ωm, and E2=E1, and in the red curve J=0.05ωm, Δ2=0.4ωm, and E2=2E1. The dashed curves were obtained numerically. (b) g1(2)(0) against Δ1. In the blue curve we have J=0, in the green curve J=0.05ωm, Δ2=0.4ωm, and E2=E1, and in the red curve J=0.05ωm, Δ2=0.1ωm, and E2=E1. The dashed curves were obtained numerically. The others parameters, which are the same in both graphics and in all curves, are g=0.5ωm, E1=0.001ωm, κ1=κ2=0.3ωm, and γ=0.005ωm.

Fig. 4.
Fig. 4.

(a) g1(2)(0) against Δ1 for E2=E1 and for different values of κ2. The black curve is a reference curve with J=0, in the blue curve J=0.05ωm and κ2=0.3ωm, in the red curve J=0.05ωm and κ2=0.2ωm, and in the green curve J=0.05ωm and κ2=0.4ωm. (b) g1(2)(0) against Δ1 for κ1=κ2 and for different values of E2. The black curve is a reference curve with J=0, in the blue curve J=0.05ωm and E2=E1, in the red curve J=0.05ωm and E2=E1/2, and in the green curve J=0.05ωm and E2=2E1. In both graphics and in all curves g=0.5ωm, κ1=0.3ωm, E1=0.001ωm, and Δ2=0.4ωm.

Fig. 5.
Fig. 5.

Minimum value of g1(2)(0) as a function of E2 and Δ2 for different values of κ1 and κ2. (a) κ1=κ2=0.15ωm. (b) κ1=κ2=0.3ωm. (c) κ1=κ2=0.6ωm. (d) Legend. In all graphics g=0.5ωm, J=0.05ωm, and E1=0.001ωm. All the graphics were obtained using the analytical approach developed here.

Fig. 6.
Fig. 6.

(a) Minimum value of g1(2)(0) against g for different values of J and E1=E2=0.001ωm. In the black curve J=0, in the blue curve J=0.02ωm, in the red curve J=0.05ωm, and in the green curve J=0.08ωm. (b) The minimum value of g1(2)(0) against g for different values of E2. The continuous curves were obtained using the analytical approach presented in this paper (n¯m=0) and the dashed curves were obtained numerically with n¯m=1. In the black curve and in the black dashed curve we have J=0, in the blue curve and in the blue dashed curve we have J=0.05ωm and E2=E1=0.001ωm, and in the red curve and in the red dashed curve we have J=0.05ωm and E2=5E1=0.005ωm. In the numerical simulations γ=0.001ωm. In both graphics and in all curves κ1=κ2=0.3ωm.

Equations (43)

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

H^=Δa^a^+ga^a^(b^+b^)+ωmb^b^+iE(a^a^),
U^=exp[ga^a^(b^b^)/ωm].
H=U^H^U^=Δa^a^Δg(a^a^)2+ωmb^b^+iE(a^eiP^a^eiP^),
H^S=j=1,2Δja^ja^j+iEj(a^ja^j)+ga^ja^j(b^+b^)J(a^1a^2+a^1a^2)+ωmb^b^,
U^=exp[g(a^1a^1+a^2a^2)(b^b^)/ωm].
HS=j=1,2Δja^ja^jΔg(a^ja^j)2+iEj(a^jeiP^a^jeiP^)2Δga^1a^1a^2a^2J(a^1a^2+a^1a^2)+ωmb^b^.
H^=j=1,2ωja^ja^jΔg(a^ja^j)22Δga^1a^1a^2a^2.
|1,±1J=(|1,0±|0,1)/2,E1,±1=ωΔg±J,
|2,ϵJ=(|2,0+2ϵ|1,1+|0,2),E2,ϵ=2ω4Δg2εJ,
H^=H^S+0dωωd^(ω)d^(ω)+iγ2π0dω[d^(ω)+d^(ω)][b^b^]+j=1,20dωωc^j(ω)c^j(ω)+iκj2π0dω[c^j(ω)+c^j(ω)][a^ja^j],
H=HS+0dωωd^(ω)d^(ω)+iγ2π0dω[d^(ω)+d^(ω)][b^b^]+j=1,20dωωc^j(ω)c^j(ω)+iκj2π0dω[c^j(ω)+c^j(ω)][a^jeiP^a^jeiP^],
H=HS+dωωd^(ω)d^(ω)+iγ2πdω[d^(ω)b^d^(ω)b^]+j=1,2dωωc^j(ω)c^j(ω)+iκj2πdω[c^j(ω)a^jeiP^c^j(ω)a^jeiP^],
ddtO^=i[HS,O^][O^,b^][γ2b^+γb^in]+[γ2b^+γb^in][O^,b^]j=1,2[O^,a^j][κj2a^j+eiP^κja^j,in]+[κj2a^j+eiP^κja^j,in][O^,a^j],
a^j,in(t)a^j,in(t)=n¯jδ(tt),
a^j,in(t)a^j,in(t)=(n¯j+1)δ(tt),
b^in(t)b^in(t)=n¯mδ(tt),
b^in(t)b^in(t)=(n¯m+1)δ(tt).
ddta˜j=(κj/2+iΔj)a˜+iΔg(a˜j+2a˜ja˜j2+2a˜ja˜pa˜p)+iJa˜p+eiP˜(Ejκja˜j,in),
ddtb˜=(γ+iωm)b˜+j=1,2Ejgωm(a˜jeiP˜+a˜jeiP˜)+γb˜in.
ddtb˜=(γ+iωm)b˜+γb˜in.
P˜(t)=ig[b˜(0)eiωmtb˜(0)eiωmt]/ωm.
ρ^i=|010||020|ρ^th,
a˜j(t)=tdτexp[(κj2+iΔ¯j)(tτ)][iJa˜p(τ)+eiP˜(τ)ξ˜j(τ)],
a^j(t)=tdτexp[(κj2+iΔ¯j)(tτ)]eiP^(t)×eiP^(τ)[iJa^p(τ)+ξ^j(τ)],
a^j(t)=tdτ0exp[(κj2+iΔ¯j)(tτ0)]eiP^(t)×{eiP^(τ0)ξ^j(τ0)+τ0dτ1exp[(κp2+iΔ¯p)×(τ0τ1)]eiP^(τ1)[iJξ^p(τ1)J2a^j(τ1)]}.
a^j(t)=n=0a^j,n(t)Jn,
a^j,n(t)=tdτ0τ0dτ1τn1dτninexp[(κj2+iΔ¯j)×(tτ0)(κp2+iΔ¯p)(τ0τ1)(κr2+iΔ¯r)×(τn1τn)]eiP^(t)eiP^(τn)ξ^r(τn),
Sj=κj24Ej2limta^j(t)a^j(t),
ddta˜j2=(κj+2iΔj)a˜j2+4iΔg(a˜j2+a˜ja˜j3+a˜j2a˜pa˜p)+2iJa˜1a˜2+2eiP˜a˜jξ˜j.
a˜j2(t)=tdτexp[(κj+2iΔ¯j2iΔg)(tτ)]×[2eiP˜(τ)a˜j(τ)ξ˜j(τ)+2iJa˜1(τ)a˜2(τ)].
a^j2(t)=tdτexp[(κj+2iΔ¯j2iΔg)(tτ)]e2iP^(t)×e2iP^(τ)[2a^j(τ)ξ^j(τ)+2iJa^1(τ)a^2(τ)].
ddt(a˜1a˜2)=(κ12+κ22+iΔ1+iΔ2)a˜1a˜2+4iΔg(a˜1a˜2+a˜1a˜12a˜2+a˜2a˜22a˜1)+iJ(a˜12+a˜22)+eiP˜a˜2ξ˜1+eiP˜a˜1ξ˜2.
a˜1(t)a˜2(t)=tdτexp[(κ12+κ22+iΔ¯1+iΔ¯22iΔg)(tτ)]{iJ[a˜12(τ)+a˜22(τ)]+eiP˜(τ)ξ˜2(τ)a˜1(τ)+eiP˜(τ)ξ˜1(τ)a˜2(τ)}.
a^1(t)a^2(t)=tdτexp[(κ12+κ22+iΔ¯1+iΔ¯22iΔg)(tτ)]e2iP^(t)e2iP^(τ){iJ[a^12(τ)+a^22(τ)]+ξ^2(τ)a^1(τ)+ξ^1(τ)a^2(τ)}.
gj(2)(0)=limta^j2(t)a^j2(t)/a^j(t)a^j(t)2,
ddtρ^=i[ρ^,H^]+κ1D[a^1]+κ2D[a^2]+γ(n¯m+1)D[b^]+γn¯mD[b^],
a^1(t)a^2(t)=tdτ0exp[(κ12+κ22+iΔ¯1+iΔ¯22iΔg)(tτ0)]e2iP^(t){e2iP^(τ0)ξ^2(τ0)a^1(τ0)+e2iP^(τ0)ξ^1(τ0)a^2(τ0)+j=1,2τ0dτ1exp[(κj+2iΔ¯j2iΔg)(τ0τ1)]e2iP^(τ1)[2iJa^j(τ1)ξ^j(τ1)2J2a^1(τ1)a^2(τ1)]}.
a^1(t)a^2(t)=n=0(a^1a^2)n(t)Jn,
(a^1a^2)n(t)=tdτ0τ0dτ1τn1dτnR0(tτ0)××Rn1(τn2τn1)e2iP^(t)e2iP^(τn)Sn(τn1τn),
Rn(t)={exp[(κ12+κ22+iΔ¯1+iΔ¯22iΔg)t],ifniseven,j=1,2exp[(κj+2iΔ¯j2iΔg2)t],ifnisodd,
Sn(t)={Rn(t)(2)n/2[ξ^1(t)a^2(t)+ξ^2(t)a^1(t)],ifniseven,2i(2)(n1)/2{exp[(κ1+2iΔ¯12iΔg)t]×ξ^1(t)a^2(t)+exp[(κ2+2iΔ¯22iΔg)t]×ξ^2(t)a^1(t)},ifnisodd.
a^j2(t)=n=0(a^j2)n(t)Jn,
(a^j2)n(t)=tdτexp[(κj+2iΔ¯j2iΔg)(tτ)]×e2iP^(t)e2iP^(τ)[2a^j,n(τ)ξ^j(τ)+2i(a^1a^2)n1(τ)].

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