Abstract

When a trichromatic laser field is applied to a cavity optomechanical system within the single-photon strong-coupling regime, we find that the motion of mirror can evolve into a dark state such that the cavity field mode cannot absorb energy from the external field. Via tuning three components of the pumping field to be resonant to the carrier, red-sideband and blue-sideband transitions in the displaced representation respectively, the state of mirror motion can exhibit non-classical properties, such as that in the Lamb-Dicke limit, the state evolves into a squeezed coherent state, and beyond the limit, the state can become a squeezed non-Gaussian state.

© 2014 Optical Society of America

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2014 (3)

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
[CrossRef]

N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
[CrossRef]

2013 (7)

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 020423 (2013).
[CrossRef]

P. D. Nation, “Nonclassical mechanical states in an optomechanical micromaser analog,” Phys. Rev. A 88, 053828 (2013).
[CrossRef]

X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
[CrossRef]

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
[CrossRef]

G. F. Xu and C. K. Law, “Dark states of a moving mirror in the single-photon strong-coupling regime,” Phys. Rev. A 87, 053849 (2013).
[CrossRef]

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[CrossRef] [PubMed]

W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
[CrossRef]

2012 (7)

S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
[CrossRef]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[CrossRef] [PubMed]

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 488, 476–480 (2012).
[CrossRef] [PubMed]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012).
[CrossRef]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[CrossRef]

J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, “Quantum signatures of the optomechanical instability,” Phys. Rev. Lett. 109, 253601 (2012).
[CrossRef]

Y. Greenberg, Y. A. Pashkin, and E. II’ichev, “Nanomechanical resonators,” Phys.-Usp. 55, 382 (2012).
[CrossRef]

2011 (7)

M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (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] [PubMed]

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

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

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

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[CrossRef]

2010 (3)

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

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] [PubMed]

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

2009 (3)

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
[CrossRef]

Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
[CrossRef]

J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
[CrossRef]

2008 (4)

H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
[CrossRef] [PubMed]

K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008).
[CrossRef]

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

T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008).
[CrossRef]

2007 (3)

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]

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]

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

2005 (2)

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[CrossRef]

2004 (4)

B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
[CrossRef] [PubMed]

M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
[CrossRef] [PubMed]

M. E. A. El-Mikkawy, “On the inverse of a general tridiagonal matrix,” Appl. Math. Comput. 150, 669–679 (2004).
[CrossRef]

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
[CrossRef]

2003 (1)

J. Fiurášek, L. Mišta, and R. Filip, “Entanglement concentration of continuous-variable quantum states,” Phys. Rev. A 67, 022304 (2003).
[CrossRef]

2002 (1)

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef] [PubMed]

1997 (1)

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]

1996 (1)

R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

1990 (1)

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[CrossRef] [PubMed]

1981 (1)

C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1974 (1)

R. L. Hudson, “When is the wigner quasi-probability density non-negative?” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
[CrossRef]

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

Akram, U.

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
[CrossRef]

Alegre, T. P. M.

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

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] [PubMed]

Armour, A. D.

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
[CrossRef]

Aspelmeyer, M.

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

Barnett, S. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[CrossRef]

Blencowe, M. P.

A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
[CrossRef]

Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (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. A 56, 4175–4186 (1997).
[CrossRef]

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 488, 476–480 (2012).
[CrossRef] [PubMed]

Bowen, W. P.

U. Akram, W. P. Bowen, and G. J. Milburn, “Entangled mechanical cat states via conditional single photon optomechanics,” New J. Phys. 15, 093007 (2013).
[CrossRef]

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 488, 476–480 (2012).
[CrossRef] [PubMed]

Brennecke, F.

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

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 488, 476–480 (2012).
[CrossRef] [PubMed]

Buzek, V.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[CrossRef] [PubMed]

Caves, C.

C. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
[CrossRef] [PubMed]

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

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
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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 488, 476–480 (2012).
[CrossRef] [PubMed]

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E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

Shnirman, A.

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
[CrossRef]

Si, L. G.

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[CrossRef]

Sillanpää, M. A.

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

Son, W.

M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
[CrossRef] [PubMed]

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W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
[CrossRef]

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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 488, 476–480 (2012).
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K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, “Observation of quantum-measurement backaction with an ultracold atomic gas,” Nat. Phys. 4, 561–564 (2008).
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Tuorila, J.

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

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T. Tyc and N. Korolkova, “Highly non-Gaussian states created via cross-Kerr nonlinearity,” New. J. Phys. 10, 023041 (2008).
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Vanner, M. R.

M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).

M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” arXiv:1406.1013.

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E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
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Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
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Wang, Z. D.

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
[CrossRef]

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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, 1520–1523 (2010).
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S. Rips, M. Kiffner, I. Wilson-Rae, and M. J. Hartmann, “Steady-state negative Wigner functions of nonlinear nanomechanical oscillators,” New J. Phys. 14, 023042 (2012).
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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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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).
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M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
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Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
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H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
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H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
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G. F. Xu and C. K. Law, “Dark states of a moving mirror in the single-photon strong-coupling regime,” Phys. Rev. A 87, 053849 (2013).
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X. X. Ren, H. K. Li, M. Y. Yan, Y. C. Liu, Y. F. Xiao, and Q. H. Gong, “Single-photon transport and mechanical NOON-state generation in microcavity optomechanics,” Phys. Rev. A 87, 033807 (2013).
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H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
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Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
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J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
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H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
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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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H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
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A. J. Rimberg, M. P. Blencowe, A. D. Armour, and P. D. Nation, “A cavity-Cooper pair transistor scheme for investigating quantum optomechanics in the ultra-strong coupling regime,” New J. Phys. 16, 055008 (2014).
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G. S. Agarwal and Sumei Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[CrossRef]

H. Xiong, L. G. Si, A. S. Zheng, X. X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
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S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Phys. Rev. A 83, 043826 (2011).
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Z. Q. Yin, “Phase noise and laser-cooling limits of optomechanical oscillators,” Phys. Rev. A 80, 033821 (2009).
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J. Zhang, Y. X. Liu, and F. Nori, “Cooling and squeezing the fluctuations of a nanomechanical beam by indirect quantum feedback control,” Phys. Rev. A 79, 052102 (2009).
[CrossRef]

Y. Li, L. A. Wu, and Z. D. Wang, “Fast ground-state cooling of mechanical resonators with time-dependent optical cavities,” Phys. Rev. A 83, 043804 (2011).
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W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013).
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M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
[CrossRef]

R. L. de, M. Filho, and W. Vogel, “Nonliear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005).
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P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
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Phys. Rev. Lett. (12)

Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
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F. Khalili, S. Danilishin, H. X. Miao, H. Müller-Ebhardt, H. Yang, and Y. B. Chen, “Preparing a mechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010).
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B. Kraus and J. I. Cirac, “Discrete entanglement distribution with squeezed light,” Phys. Rev. Lett. 92, 013602 (2004).
[CrossRef] [PubMed]

M. Paternostro, W. Son, and M. S. Kim, “Complete conditions for entanglement transfer,” Phys. Rev. Lett. 92, 197901 (2004).
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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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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).
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A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108, 033602 (2012).
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P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011).
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Phys. Rev. X (2)

N. Lörch, J. Qian, A. Clerk, F. Marquardt, and K. Hammerer, “Laser theory for optomechanics: limit cycles in the quantum regime,” Phys. Rev. X 4, 011015 (2014).

M. R. Vanner, “Selective linear or quadratic optomechanical coupling via measurement,” Phys. Rev. X 1, 021011 (2011).

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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, 1520–1523 (2010).
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T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” arXiv:1311.3802.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, 1997).

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University, 1997).

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M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” arXiv:1406.1013.

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Figures (9)

Fig. 1
Fig. 1

Sketch of the optomechanical system consisting of a fixed end mirror and a movable end mirror. The cavity field couples to the movable mirror via radiation pressure, which is driven by a trichromatic laser field.

Fig. 2
Fig. 2

The transitions between the energy levels of the optomechanical system with cavity field confined within the zero- and single-photon states. Each component of trichromatic field is applied to establish a set of resonant transitions, and we denote |Pm = |(0)〉m and |m = |(1)〉m for simplicity. By choosing an appropriate g = gN, there will be no transitions to N + 1 phonons states shown by dashed lines.

Fig. 3
Fig. 3

Numerical results of phonon number distribution Pm of the dark state compared with that of squeezed coherent state in Eq. (24). The coupling strength is gN = 0.1916ωm with phonon number N = 99, laser driving strength is Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm is determined from Eq. (20) with the requirement of rank(M) = N.

Fig. 4
Fig. 4

Wigner function of the dark state with gN = 0.1916ωm corresponding to the phonon number N = 99, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm, and Ω1 = −0.0038ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape is clearly visible.

Fig. 5
Fig. 5

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. The crescent shape and negative values of the Wigner function are clearly visible.

Fig. 6
Fig. 6

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.3678ωm (N = 26), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.00963ωm determined from Eq. (20) with the requirement of rank(M) = N. Here the phonon number population is mainly distributed within N = 15.

Fig. 7
Fig. 7

Wigner function and the phonon number distribution Pm of the dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.016ωm, Ω3 = −0.01ωm and Ω1 = 0.002ωm determined from Eq. (20) with the requirement of rank(M) = N.

Fig. 8
Fig. 8

Fidelity between the system state and the dark state as a function of time. The initial system state is a dark state with the coupling strength gN = 0.5124ωm (N = 13), laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

Fig. 9
Fig. 9

Time evolution of Fidelity between the system state and the dark state. The initial system is in its ground state with the coupling strength gN = 0.5124ωm (N = 13), cavity damping rate κ/ωm = 0.08, laser driving strength Ω2 = 0.0225ωm, Ω3 = −0.015ωm and Ω1 = 0.0029ωm determined from Eq. (20) with the requirement of rank(M) = N. Laser detunings are given in Eq. (9).

Equations (30)

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H ^ = ω c a ^ a ^ + ω m b ^ b ^ g a a ^ ( b ^ + b ^ ) + [ ( Ω 1 e i ω 1 t + Ω 2 e i ω 2 t + Ω 3 e i ω 3 t ) a ^ + H . c . ] ,
H ^ = ω m b ^ b ^ g a ^ a ^ ( b ^ + b ^ ) + [ ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) a ^ + H . c . ] ,
H ^ 0 = ω m b ^ b ^ g a ^ a ^ ( b ^ + b ^ ) = n , p ε n , p | ψ n , p ψ n , p | .
| ψ n , p = | n c D ( n g / ω m ) | p m = | n c | p ˜ ( n ) m ,
a ^ = n , p n , p | ψ n , p ψ n , p | a ^ | ψ n , p ψ n , p | = n , p , p A p , p ( n ) | ψ n 1 , p ψ n , p | ,
A p , p ( n ) = { n p ! p ! e ξ 2 2 ( ξ ) p p L p p p ( ξ 2 ) , p p , n p ! p ! e ξ 2 2 ( ξ ) p p L p p p ( ξ 2 ) , p > p ,
H ^ = n , p ε n , p | ψ n , p ψ n , p | + n , p , p [ A p , p ( n ) ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) | ψ n , p ψ n 1 , p | + H . c . ] .
H ^ int = n , p , p [ A p , p ( n ) ( Ω 1 e i Δ 1 t + Ω 2 e i Δ 2 t + Ω 3 e i Δ 3 t ) × e i ( ε n , p ε n 1 , p ) t | ψ n , p ψ n 1 , p | + H . c . ] .
Δ 1 = ε 1 , p ε 0 , p = g 2 / ω m , Δ 2 = ε 1 , p ε 0 , p + 1 = ω m g 2 / ω m , Δ 3 = ε 1 , p ε 0 , p 1 = ω m g 2 / ω m ,
H ^ int = H ^ eff + V ^ ,
H ^ eff = p [ ( A p , p ( 1 ) Ω 1 | ψ 1 , p ψ 0 , p | + A p + 1 , p ( 1 ) Ω 2 | ψ 1 , p ψ 0 , p + 1 | + A p , p + 1 ( 1 ) Ω 3 | ψ 1 , p + 1 ψ 0 , p | ) + H . c . ] ,
V ^ = Σ n , p , p [ A p , p ( n ) ( Ω 1 e i δ 1 ( n , p , p ) t + Ω 2 e i δ 2 ( n , p , p ) t + Ω 3 e i δ 3 ( n , p , p ) t ) | ψ n , p ψ n 1 , p | + H . c . ] .
δ 1 ( n , p , p ) = ( p p ) ω m 2 ( n 1 ) g 2 / ω m , δ 2 ( n , p , p ) = ( p p + 1 ) ω m 2 ( n 1 ) g 2 / ω m , δ 3 ( n , p , p ) = ( p p 1 ) ω m 2 ( n 1 ) g 2 / ω m .
| D = C p = 0 β p | 0 c | p m ,
β p 1 Ω 3 A p 1 , p ( 1 ) + β p Ω 1 A p , p ( 1 ) + β p + 1 Ω 2 A p + 1 , p ( 1 ) = 0 .
A N + 1 , N ( 1 ) = 1 / ( N + 1 ) e ξ 2 / 2 ξ L N ( 1 ) ( ξ 2 ) , A N , N + 1 ( 1 ) = 1 / ( N + 1 ) e ξ 2 / 2 ( ξ ) L N ( 1 ) ( ξ 2 ) .
β N 1 Ω 3 A N 1 , N ( 1 ) + β N Ω 1 A N , N ( 1 ) = 0
M | β ] = 0 ,
| β ] = ( β 0 , β 1 , , β N 1 , β N ) T
M = [ Ω 1 A 0 , 0 ( 1 ) Ω 2 A 1 , 0 ( 1 ) Ω 3 A 0 , 1 ( 1 ) Ω 1 A 1 , 1 ( 1 ) Ω 2 A 2 , 1 ( 1 ) Ω 3 A 1 , 2 ( 1 ) Ω 2 A N , N 1 ( 1 ) Ω 3 A N , N 1 , N ( 1 ) Ω 1 A N , N ( 1 ) ] .
A p 1 , p ( 1 ) p ( ξ ) e ξ 2 / 2 , A p , p ( 1 ) e ξ 2 / 2 , A p + 1 , p ( 1 ) p + 1 ξ e ξ 2 / 2 .
β p 1 p ( Ω 3 ξ ) + β p Ω 1 + β p + 1 p + 1 ( Ω 2 ξ ) = 0 .
μ β p 1 p + ν β p + 1 p + 1 = β β p ,
β p = 1 p ! ν ( μ 2 ν ) p 2 exp ( | β | 2 2 + μ * 2 ν β 2 ) H p ( β 2 μ ν ) ,
( Δ X ^ ) 2 = X ^ 2 X ^ 2 .
W ( α ) = 2 π Tr [ D ^ ( α ) ρ ^ D ^ ( α ) ( 1 ) b ^ b ^ ] ,
A p + 1 , p ( 1 ) = 1 p + 1 e ξ 2 / 2 ξ L p ( 1 ) ( ξ 2 ) = p + 1 e ξ 2 / 2 ξ [ 1 p 2 ! ξ 2 + ( p 1 ) p 3 ! ξ 4 + ] = p + 1 e ξ 2 / 2 ξ f ( p ) ,
| D ( t ) = C p = 0 N β p e i p ω m t | 0 c | p m ,
d d t ρ ^ = i [ H ^ , ρ ^ ] + κ 2 ( 2 a ^ ρ ^ a ^ a ^ a ^ ρ ^ ρ ^ a ^ a ^ ) + γ ( n ¯ + 1 ) 2 ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) + γ n ¯ 2 ( 2 b ^ ρ ^ b ^ b ^ b ^ ρ ^ ρ ^ b ^ b ^ ) ,
F = Tr [ | D ( t ) D ( t ) | ρ ^ ( t ) ] .

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