Abstract

We systematically study the influence of amplitude modulation on the steady-state bosonic squeezing and entanglement in a dissipative three-mode optomechanical system, where a vibrational mode of the membrane is coupled to the left and right cavity modes via the radiation pressure. Numerical simulation results show that the steady-state bosonic squeezing and entanglement can be significantly enhanced by periodically modulated external laser driving either or both ends of the cavity. Remarkably, the fact that as long as one periodically modulated external laser driving either end of the cavities is sufficient to enhance the squeezing and entanglement is convenient for actual experiment, whose cost is that required modulation period number for achieving system stability is more. In addition, we numerically confirm the analytical prediction for optimal modulation frequency and discuss the corresponding physical mechanism.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

S. Chakraborty and A. K. Sarma, “Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics,” Phys. Rev. A 97, 022336 (2018).

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).

2017 (3)

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

R. X. Chen, C. G. Liao, and X. M. Lin, “Dissipative generation of significant amount of mechanical entanglement in a coupled optomechanical system,” Sci. Rep. 7, 14497 (2017).

Z. X. Chen, Q. Lin, B. He, and Z. Y. Lin, “Entanglement dynamics in double-cavity optomechanical systems,” Opt. Express 25, 17237 (2017).

2016 (2)

F. Monifi, J. Zhang, K. Özdemir, Ş B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399 (2016).

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

2015 (10)

M. Abdi and M. J. Hartmann, “Entangling the motion of two optically trapped objects via time-modulated driving fields,” New J. Phys. 17, 013056 (2015).

Z. Li, S. L. Ma, and F. L. Li, “Generation of broadband two-mode squeezed light in cascaded double-cavity optomechanical systems,” Phys. Rev. A 92, 023856 (2015).

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).

X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to Chaos in Optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 1 (2015).

T. P. Purdy, P. L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).

J. Q. Liao, C. K. Law, L. M. Kuang, and F. Nori, “Enhancement of mechanical effects of single photons in modulated two-mode optomechanics,” Phys. Rev. A 92, 013822 (2015).

T. T. Huan, R. G. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92, 022301 (2015).

2014 (6)

K. Qu and G. S. Agarwal, “Strong squeezing via phonon mediated spontaneous generation of photon pairs,” New J. Phys. 16, 113004 (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).

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321 (2014).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Dissipative optomechanical squeezing of light,” New J. Phys. 16, 063058 (2014).

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).

2013 (9)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525, 215 (2013).

W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21, 20423 (2013).

L. Tian, “Robust Photon Entanglement via Quantum Interference in Optomechanical Interfaces,” Phys. Rev. Lett. 110, 233602 (2013).

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88, 062341 (2013).

Y. D. Wang and A. A. Clerk, “Reservoir-Engineered Entanglement in Optomechanical Systems,” Phys. Rev. Lett. 110, 253601 (2013).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

S. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88, 042331 (2013).

2012 (14)

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si 3 N 4 membranes at cryogenic temperatures,” New J. Phys. 14115021 (2012).

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).

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).

J. H. Teng, S. L. Wu, B. Cui, and X. X. Yi, “Quantum optomechanics with quadratic cavity-membrane couplings,” J. Phys. B: At. Mol. Phys. 45, 185506 (2012).

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).

A. Mari and J. Eisert, “Opto- and electro-mechanical entanglement improved by modulation,” New J. Phys. 14, 075014 (2012).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14, 125005 (2012).

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

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,”Phys. Rev. A 86, 042323 (2012).

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

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).

F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

2011 (5)

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).

P. Doria, T. Calarco, and S. Montangero, “Optimal Control Technique for Many-Body Quantum Dynamics,” Phys. Rev. Lett. 106, 190501 (2011).

H. K. Cheung and C. K. Law, “Nonadiabatic optomechanical Hamiltonian of a moving dielectric membrane in a cavity,” Phys. Rev. A 84, 023812 (2011).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

2010 (2)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).

2009 (5)

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Phys. 2, 40 (2009).

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Chapter 2 Quantum Effects in Optomechanical Systems,” Adv. Atom. Mol. Opt. Phy. 57, 33 (2009).

A. Mari and J. Eisert, “Gently Modulating Optomechanical Systems,” Phys. Rev. Lett. 103, 213603 (2009).

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard Quantum Limit for Probing Mechanical Energy Quantization,” Phys. Rev. Lett. 103, 100402 (2009).

Z. Q. Yin and Y. J. Han, “Generating EPR beams in a cavity optomechanical system,” Phys. Rev. A 79, 024301 (2009).

2008 (6)

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

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 (2008).

M. Ludwig, B. Kubala, and F. Marquardt, “The optomechanical instability in the quantum regime,” New J. Phys. 10, 095013 (2008).

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).

L. Tian, M. S. Allman, and R. W. Simmonds, “Parametric coupling between macroscopic quantum resonators,” New J. Phys. 10, 115001 (2008).

2007 (4)

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).

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

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).

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).

2004 (1)

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).

2002 (2)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002).

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).

Abdi, M.

M. Abdi and M. J. Hartmann, “Entangling the motion of two optically trapped objects via time-modulated driving fields,” New J. Phys. 17, 013056 (2015).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

Adesso, G.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).

Agarwal, G. S.

K. Qu and G. S. Agarwal, “Strong squeezing via phonon mediated spontaneous generation of photon pairs,” New J. Phys. 16, 113004 (2014).

Allman, M. S.

L. Tian, M. S. Allman, and R. W. Simmonds, “Parametric coupling between macroscopic quantum resonators,” New J. Phys. 10, 115001 (2008).

Alvermann, A.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to Chaos in Optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).

Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321 (2014).

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).

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).

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

Bakemeier, L.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to Chaos in Optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).

Barzanjeh, S.

S. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88, 042331 (2013).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

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).

Bhattacharya, M.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

Bo, F.

F. Monifi, J. Zhang, K. Özdemir, Ş B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399 (2016).

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).

Børkje, K.

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).

Brukner, C.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

Calarco, T.

P. Doria, T. Calarco, and S. Montangero, “Optimal Control Technique for Many-Body Quantum Dynamics,” Phys. Rev. Lett. 106, 190501 (2011).

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).

Chakraborty, S.

S. Chakraborty and A. K. Sarma, “Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics,” Phys. Rev. A 97, 022336 (2018).

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 (2012).

Chen, J. P.

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).

Chen, R. X.

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).

R. X. Chen, C. G. Liao, and X. M. Lin, “Dissipative generation of significant amount of mechanical entanglement in a coupled optomechanical system,” Sci. Rep. 7, 14497 (2017).

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

Chen, Y.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard Quantum Limit for Probing Mechanical Energy Quantization,” Phys. Rev. Lett. 103, 100402 (2009).

Chen, Z. X.

Chesi, S.

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).

Cheung, H. K.

H. K. Cheung and C. K. Law, “Nonadiabatic optomechanical Hamiltonian of a moving dielectric membrane in a cavity,” Phys. Rev. A 84, 023812 (2011).

Cho, S. U.

F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Cicak, K.

T. P. Purdy, P. L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321 (2014).

Clerk, A. A.

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).

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).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Dissipative optomechanical squeezing of light,” New J. Phys. 16, 063058 (2014).

Y. D. Wang and A. A. Clerk, “Reservoir-Engineered Entanglement in Optomechanical Systems,” Phys. Rev. Lett. 110, 253601 (2013).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

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).

Corbitt, T.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard Quantum Limit for Probing Mechanical Energy Quantization,” Phys. Rev. Lett. 103, 100402 (2009).

Cui, B.

J. H. Teng, S. L. Wu, B. Cui, and X. X. Yi, “Quantum optomechanics with quadratic cavity-membrane couplings,” J. Phys. B: At. Mol. Phys. 45, 185506 (2012).

Danilishin, S.

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard Quantum Limit for Probing Mechanical Energy Quantization,” Phys. Rev. Lett. 103, 100402 (2009).

De Chiara, G.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,”Phys. Rev. A 86, 042323 (2012).

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).

Doherty, A. C.

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).

Dong, C. H.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).

Doria, P.

P. Doria, T. Calarco, and S. Montangero, “Optimal Control Technique for Many-Body Quantum Dynamics,” Phys. Rev. Lett. 106, 190501 (2011).

Eisert, J.

A. Mari and J. Eisert, “Opto- and electro-mechanical entanglement improved by modulation,” New J. Phys. 14, 075014 (2012).

A. Mari and J. Eisert, “Gently Modulating Optomechanical Systems,” Phys. Rev. Lett. 103, 213603 (2009).

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

Farace, A.

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

Fehske, H.

L. Bakemeier, A. Alvermann, and H. Fehske, “Route to Chaos in Optomechanics,” Phys. Rev. Lett. 114, 013601 (2015).

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).

Fiore, V.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).

Galassi, M.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

García-Patrón, R.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).

Gardiner, C.

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, vol. 56 (Springer Science & Business Media) (2004).

Genes, C.

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Chapter 2 Quantum Effects in Optomechanical Systems,” Adv. Atom. Mol. Opt. Phy. 57, 33 (2009).

Gigan, S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

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).

Giovannetti, V.

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

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002).

Girvin, S. M.

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Phys. 2, 40 (2009).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

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 (2008).

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).

Giuseppe, G. D.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

Gu, W. J.

Guerreiro, 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).

Habraken, S. J. M.

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).

Hakonen, P. J.

F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Han, Y. J.

Z. Q. Yin and Y. J. Han, “Generating EPR beams in a cavity optomechanical system,” Phys. Rev. A 79, 024301 (2009).

Harris, J. G.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

Harris, J. G. E.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 1 (2015).

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).

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 (2008).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

Hartmann, M. J.

M. Abdi and M. J. Hartmann, “Entangling the motion of two optically trapped objects via time-modulated driving fields,” New J. Phys. 17, 013056 (2015).

He, B.

Heikkilä, T. T.

F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Hill, J. T.

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

Hoch, S. W.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 1 (2015).

Huan, T. T.

T. T. Huan, R. G. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92, 022301 (2015).

Ian, H.

T. T. Huan, R. G. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92, 022301 (2015).

Illuminati, F.

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

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 (2008).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

Jing, H.

X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).

Kampel, N. S.

T. P. Purdy, P. L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).

Karuza, M.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

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M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Chapter 2 Quantum Effects in Optomechanical Systems,” Adv. Atom. Mol. Opt. Phy. 57, 33 (2009).

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).

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002).

Underwood, M.

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 1 (2015).

Uys, H.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).

van Enk, S. J.

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88, 062341 (2013).

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).

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).

Vitali, D.

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Chapter 2 Quantum Effects in Optomechanical Systems,” Adv. Atom. Mol. Opt. Phy. 57, 33 (2009).

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

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).

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002).

Wang, H.

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88, 062341 (2013).

Wang, H. L.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).

Wang, M.

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

Wang, Y. D.

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).

Y. D. Wang and A. A. Clerk, “Reservoir-Engineered Entanglement in Optomechanical Systems,” Phys. Rev. Lett. 110, 253601 (2013).

Weedbrook, C.

S. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88, 042331 (2013).

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).

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).

Woolley, M. J.

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).

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).

Wu, H. Z.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

Wu, S. L.

J. H. Teng, S. L. Wu, B. Cui, and X. X. Yi, “Quantum optomechanics with quadratic cavity-membrane couplings,” J. Phys. B: At. Mol. Phys. 45, 185506 (2012).

Wu, Y.

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).

Xie, H.

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).

Yang, C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

Yang, L.

F. Monifi, J. Zhang, K. Özdemir, Ş B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399 (2016).

Yang, Z. B.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

Yi, X. X.

J. H. Teng, S. L. Wu, B. Cui, and X. X. Yi, “Quantum optomechanics with quadratic cavity-membrane couplings,” J. Phys. B: At. Mol. Phys. 45, 185506 (2012).

Yin, T. S.

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

Yin, Z. Q.

Z. Q. Yin and Y. J. Han, “Generating EPR beams in a cavity optomechanical system,” Phys. Rev. A 79, 024301 (2009).

You, J. Q.

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

Yu, P. L.

T. P. Purdy, P. L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si 3 N 4 membranes at cryogenic temperatures,” New J. Phys. 14115021 (2012).

Zeilinger, 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).

Zhang, J.

F. Monifi, J. Zhang, K. Özdemir, Ş B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399 (2016).

Zheng, L. L.

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

Zheng, S. B.

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

Zhou, R. G.

T. T. Huan, R. G. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92, 022301 (2015).

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).

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, vol. 56 (Springer Science & Business Media) (2004).

Zwerger, W.

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).

Zwickl, B. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

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 (2008).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

Adv. Atom. Mol. Opt. Phy. (1)

C. Genes, A. Mari, D. Vitali, and P. Tombesi, “Chapter 2 Quantum Effects in Optomechanical Systems,” Adv. Atom. Mol. Opt. Phy. 57, 33 (2009).

Ann. Phys. (1)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525, 215 (2013).

J. Opt. (1)

M. Karuza, M. Galassi, C. Biancofiore, C. Molinelli, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Tunable linear and quadratic optomechanical coupling for a tilted membrane within an optical cavity: theory and experiment,” J. Opt. 15, 025704 (2013).

J. Phys. B: At. Mol. Phys. (1)

J. H. Teng, S. L. Wu, B. Cui, and X. X. Yi, “Quantum optomechanics with quadratic cavity-membrane couplings,” J. Phys. B: At. Mol. Phys. 45, 185506 (2012).

Nat. Commun. (3)

D. Lee, M. Underwood, D. Mason, A. B. Shkarin, S. W. Hoch, and J. G. E. Harris, “Multimode optomechanical dynamics in a cavity with avoided crossings,” Nat. Commun. 6, 1 (2015).

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

F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3, 987 (2012).

Nat. Photonics (1)

F. Monifi, J. Zhang, K. Özdemir, Ş B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photonics 10, 399 (2016).

Nat. Phys. (2)

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321 (2014).

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707 (2010).

Nature (1)

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 (2008).

New J. Phys. (10)

M. Karuza, C. Molinelli, M. Galassi, C. Biancofiore, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanical sideband cooling of a thin membrane within a cavity,”New J. Phys. 14, 095015 (2012).

T. P. Purdy, R. W. Peterson, P. L. Yu, and C. A. Regal, “Cavity optomechanics with Si 3 N 4 membranes at cryogenic temperatures,” New J. Phys. 14115021 (2012).

M. Ludwig, B. Kubala, and F. Marquardt, “The optomechanical instability in the quantum regime,” New J. Phys. 10, 095013 (2008).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Dissipative optomechanical squeezing of light,” New J. Phys. 16, 063058 (2014).

K. Qu and G. S. Agarwal, “Strong squeezing via phonon mediated spontaneous generation of photon pairs,” New J. Phys. 16, 113004 (2014).

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).

L. Tian, M. S. Allman, and R. W. Simmonds, “Parametric coupling between macroscopic quantum resonators,” New J. Phys. 10, 115001 (2008).

A. Mari and J. Eisert, “Opto- and electro-mechanical entanglement improved by modulation,” New J. Phys. 14, 075014 (2012).

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14, 125005 (2012).

M. Abdi and M. J. Hartmann, “Entangling the motion of two optically trapped objects via time-modulated driving fields,” New J. Phys. 17, 013056 (2015).

Opt. Express (2)

Phys. (1)

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Phys. 2, 40 (2009).

Phys. Rev. A (29)

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008).

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82, 021806 (2010).

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94, 053807 (2016).

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,”Phys. Rev. A 86, 042323 (2012).

Z. Li, S. L. Ma, and F. L. Li, “Generation of broadband two-mode squeezed light in cascaded double-cavity optomechanical systems,” Phys. Rev. A 92, 023856 (2015).

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89, 023843 (2014).

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

S. Chakraborty and A. K. Sarma, “Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics,” Phys. Rev. A 97, 022336 (2018).

C. G. Liao, R. X. Chen, H. Xie, and X. M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).

T. S. Yin, X. Y. Lü, L. L. Zheng, M. Wang, S. Li, and Y. Wu,“Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95, 053861 (2017).

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).

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).

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).

R. X. Chen, L. T. Shen, and S. B. Zheng, “Dissipation-induced optomechanical entanglement with the assistance of Coulomb interaction,” Phys. Rev. A 91, 022326 (2015).

J. Q. Liao, C. K. Law, L. M. Kuang, and F. Nori, “Enhancement of mechanical effects of single photons in modulated two-mode optomechanics,” Phys. Rev. A 92, 013822 (2015).

T. T. Huan, R. G. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92, 022301 (2015).

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77, 033819 (2008).

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88, 062341 (2013).

Z. Q. Yin and Y. J. Han, “Generating EPR beams in a cavity optomechanical system,” Phys. Rev. A 79, 024301 (2009).

H. K. Cheung and C. K. Law, “Nonadiabatic optomechanical Hamiltonian of a moving dielectric membrane in a cavity,” Phys. Rev. A 84, 023812 (2011).

C. Biancofiore, M. Karuza, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Quantum dynamics of an optical cavity coupled to a thin semitransparent membrane: Effect of membrane absorption,” Phys. Rev. A 84, 033814 (2011).

T. P. Purdy, P. L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, “Optomechanical Raman-ratio thermometry,” Phys. Rev. A 92, 031802 (2015).

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).

G. Adesso, A. Serafini, and F. Illuminati, “Extremal entanglement and mixedness in continuous variable systems,” Phys. Rev. A 70, 022318 (2004).

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).

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. D. Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).

S. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88, 042331 (2013).

Phys. Rev. Lett. (15)

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).

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).

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).

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).

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, “Standard Quantum Limit for Probing Mechanical Energy Quantization,” Phys. Rev. Lett. 103, 100402 (2009).

L. Tian, “Robust Photon Entanglement via Quantum Interference in Optomechanical Interfaces,” Phys. Rev. Lett. 110, 233602 (2013).

Y. D. Wang and A. A. Clerk, “Reservoir-Engineered Entanglement in Optomechanical Systems,” Phys. Rev. Lett. 110, 253601 (2013).

P. Doria, T. Calarco, and S. Montangero, “Optimal Control Technique for Many-Body Quantum Dynamics,” Phys. Rev. Lett. 106, 190501 (2011).

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible Optical-to-Microwave Quantum Interface,” Phys. Rev. Lett. 109, 130503 (2012).

A. Mari and J. Eisert, “Gently Modulating Optomechanical Systems,” Phys. Rev. Lett. 103, 213603 (2009).

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Oscillators Exploiting Radiation Pressure,” Phys. Rev. Lett. 88, 120401 (2002).

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).

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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).

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Sci. Rep. (1)

R. X. Chen, C. G. Liao, and X. M. Lin, “Dissipative generation of significant amount of mechanical entanglement in a coupled optomechanical system,” Sci. Rep. 7, 14497 (2017).

Science (1)

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609 (2012).

Other (2)

G. Teschl, Ordinary Differential Equations and Dynamical Systems vol. 140 (American Mathematical Society Providence) (2012).

C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, vol. 56 (Springer Science & Business Media) (2004).

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of the optomechanical system.
Fig. 2
Fig. 2 Time evolution of the real and imaginary parts of the mean values in the case of identical cavity detuning for three different modulation driving lasers. (a) symmetric modulation with EL(t) = ER(t) = 7 × 104 + 3.5 × 104 × eiΩt + 3.5 × 104 × eiΩt; (b) single cavity driving with EL(t) = 7 × 104 + 3.5 × 104 × eiΩt + 3.5 × 104 × eiΩt, ER(t) = 0; (c) single cavity modulation with EL(t) = 7 × 104 + 7 × 104 × eiΩt + 7 × 104 × eiΩt, ER(t) = 7 × 104. The chosen parameters in units of ωm are: Ω = 2, κ = 0.1, γm = 0.001, J = 2, Δ = 3, and g = 4 × 10−6.
Fig. 3
Fig. 3 Phase space trajectories of the classical c-number mean values. (a) Phase space trajectories of 〈AL(t)〉 from t = 0 to t = 30τ for symmetric modulation; (b) Phase space trajectories of cavity field mean values and the dimensionless mechanical position and momentum mean values for asymmetric modulation. The left and right columns are results of single cavity driving and single cavity modulation, respectively. All the chosen parameters are identical to those in Fig. 2.
Fig. 4
Fig. 4 Real and imaginary parts of cavity mode mean value 〈Aj(t)〉 as a function of time in the long time limit. The chosen parameters in units of ωm are: Ω = 2, κ = 0.1, γm = 0.001, J = 2, Δ = 3, g = 4 × 10−6, A L 0 = 0.1 / 2 g, A L 1 = 0.04 / 2 g, A R 0 = 0.08 / 2 g, and A R 1 = 0.02 / 2 g.
Fig. 5
Fig. 5 Variance of the mechanical oscillator position operator σ11(t) as a function of time in the long time limit from t = 598τ to t = 600τ. (a) Ω = 2 for symmetric modulation; (b) Ω = 1.97 for single cavity driving; (c) Ω = 1.97 for single cavity modulation. In all figures, the solid (red) and dashed (blue) lines correspond to the cases of a = 0, m = 0 and a = 0, m = 1 respectively and are plotted with logarithmic coordinates. The other parameters are the same as those in Fig. 2.
Fig. 6
Fig. 6 Asymptotic evolution of cavity-cavity entanglement EN as a function of time in the long time limit from t = 0 to t = 1500τ. We take a = 0, m = 0, and Ω = 2. (a) symmetric modulation; (b) single cavity driving; (c) single cavity modulation. The other parameters are the same as those in Fig. 2 except κ = 0.001 and γm = 0.1.
Fig. 7
Fig. 7 Mimimum variance σ11,min of the mechanical oscillator position operator versus the modulation frequency Ω. The chosen parameters in units of ωm are κ = 0.1, γm = 0.001, J = 2, Δ = 3, g = 4 × 10−6, GL0 = 0.13, GL1 = 0.12, GR0 = 0.07, and GR1 = 0.06.
Fig. 8
Fig. 8 Maximum cavity-cavity entanglement EN,max versus the modulation frequency Ω. All the other parameters are the same as those in Fig. 7 except κ = 0.001, γm = 0.1.

Equations (78)

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H = j = L , R [ Δ j A j A j + i E j ( t ) A j i E j * ( t ) A j ] + ω m 2 ( P 2 + Q 2 ) + g ( A L A L A R A R ) Q + J ( A L A R + A L A R ) .
Q ˙ = ω m P ,
P ˙ = ω m Q g ( A L A L A R A R ) γ m P + ξ ( t ) ,
A ˙ L = ( κ + i Δ L ) A L i g A L Q i J A R + E L ( t ) + 2 κ a L in ( t ) ,
A ˙ R = ( κ + i Δ R ) A R + i g A R Q i J A L + E R ( t ) + 2 κ a R in ( t ) ,
a j in ( t ) a j in ( t ) = ( n ¯ a + 1 ) δ ( t t ) ,
a j in ( t ) a j in ( t ) = n ¯ a δ ( t t ) ,
ξ ( t ) ξ ( t ) + ξ ( t ) ξ ( t ) / 2 = γ m ( 2 n ¯ m + 1 ) δ ( t t ) ,
Q ˙ = ω m P ,
P ˙ = ω m Q γ m P g ( A L * A L A R * A R ) ,
A ˙ L = ( κ + i Δ L ) A L i g A L Q i J A R + E L ( t ) ,
A ˙ R = ( κ + i Δ R ) A R + i g A R Q i J A L + E R ( t ) ,
q ˙ = ω m p ,
p ˙ = ω m q γ m p g ( A L * a L A R * a R + h . c . ) + ξ ( t ) ,
a ˙ L = ( κ + i Δ L ) a L i g ( A L q + Q a L ) i J a R + 2 κ a L in ( t ) ,
a ˙ R = ( κ + i Δ R ) a R + i g ( A R q + Q a R ) i J a L + 2 κ a R in ( t ) ,
H lin = ( Δ L + g Q ) a L a L + ( Δ R g Q ) a R a R + ω m 2 × ( p 2 + q 2 ) + J ( a L a R + a R a L ) + g ( A L * a L + A L a L A R * a R A R a R ) q .
O ( t ) = l = 0 n = O n , l e in Ω t g l ,
E L ( t ) = n = E n L e in Ω t ,
E R ( t ) = n = E n R e in Ω t .
P n , 0 = Q n , 0 = 0 ,
A n , 0 L = i J E n R ( κ + i Δ R + in Ω ) E n L J 2 ( κ + i Δ R + in Ω ) ( κ + i Δ L + in Ω ) ,
A n , 0 R = i J E n L ( κ + i Δ L + in Ω ) E n R J 2 ( κ + i Δ R + in Ω ) ( κ + i Δ L + in Ω )
P n , l = in Ω ω m Q n , l ,
Q n , l = ω m ( k = 0 l 1 m = A m , k L * A n + m , l k 1 L ω m 2 + i γ m n Ω ( n Ω ) 2 k = 0 l 1 m = A m , k R * A n + m , l k 1 R ω m 2 + i γ m n Ω ( n Ω ) 2 ) ,
A n , l L = i k = 0 l 1 m = A m , k L Q n m , l k + 1 + J A n , l R κ + i Δ L + i n Ω ,
A n , l R = i k = 0 l 1 m = A m , k R Q n m , l k + 1 J A n , l L κ + i Δ R + i n Ω
P n , 0 = Q n , 0 = 0 ,
A n , 0 L = A n , 0 R = E n L [ κ + i ( Δ + n Ω + J ) ] = E n R [ κ + i ( Δ + n Ω + J ) ] ,
P n , l = Q n , l = A n , l L = A n , l R = 0 .
A L ( t ) = A L 0 + A L 1 e i Ω t ,
A R ( t ) = A R 0 + A R 1 e i Ω t ,
P ( t ) i g Ω ( A R 0 A R 1 A L 0 A L 1 ) ( Ω 2 ω m 2 ) ( e i Ω t e i Ω t ) ,
Q ( t ) g ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m + g ( A R 0 A R 1 A L 0 A L 1 ) ω m × ( 1 Ω 2 Ω 2 ω m 2 ) ( e i Ω t + e i Ω t ) ,
E L ( t ) E 0 L + E 1 L e i Ω t + E 1 L e i Ω t + E 2 L e 2 i Ω t ,
E R ( t ) E 0 R + E 1 R e i Ω t + E 1 R e i Ω t + E 2 R e 2 i Ω t ,
E 0 L = ( κ + i Δ L ) A L 0 + i J A R 0 + i g 2 A L 0 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m + i g 2 A L 1 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 1 L = [ κ + i ( Δ L Ω ) ] A L 1 + i J A R 1 + i g 2 A L 1 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m + i g 2 A L 0 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 1 L = i g 2 A L 0 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 2 L = i g 2 A L 1 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 0 R = ( κ + i Δ R ) A R 0 + i J A L 0 i g 2 A R 0 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m i g 2 A R 1 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 1 R = [ κ + i ( Δ R Ω ) ] A R 1 + i J A L 1 i g 2 A R 1 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m i g 2 A R 0 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 1 R = i g 2 A R 0 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) ,
E 2 R = i g 2 A R 1 ( A R 0 A R 1 A L 0 A L 1 ) ω m ( 1 Ω 2 Ω 2 ω m 2 ) .
x j = a j + a j 2 ,
y j = a j a j i 2 ,
x j in ( t ) = a j in ( t ) + a j in ( t ) 2 ,
y j in ( t ) = a j in ( t ) a j in ( t ) i 2 ,
U = ( q , p , x L , y L , x R , y R ) T ,
N ( t ) = ( 0 , ξ ( t ) , 2 κ x L in ( t ) , 2 κ y L in ( t ) , 2 κ x R in ( t ) , 2 κ y R in ( t ) ) T ,
U ˙ = R ( t ) U + N ( t )
R ( t ) = ( 0 ω m 0 0 0 0 ω m γ m G Lr ( t ) G Li ( t ) G Rr ( t ) G Ri ( t ) G Li ( t ) 0 κ Δ 1 ( t ) 0 J G Lr ( t ) 0 Δ 1 ( t ) κ J 0 G Ri ( t ) 0 0 J κ Δ 2 ( t ) G Rr ( t ) 0 J 0 Δ 2 ( t ) κ ) ,
Δ 1 ( t ) = Δ L + g Q ,
Δ 2 ( t ) = Δ R g Q ,
G j ( t ) = 2 g A j ( t ) = 2 g ( A j 0 + A j 1 e i Ω t ) = G j 0 + G j 1 e i Ω t .
σ k , l = < U k ( t ) U l ( t ) + U l ( t ) U k ( t ) > / 2 .
σ ˙ ( t ) = R ( t ) σ ( t ) + σ ( t ) R ( t ) T + D ,
δ ( t t ) D k , l = N k ( t ) N l ( t ) + N l ( t ) N k ( t ) / 2 .
D = diag ( 0 , γ m ( 2 n ¯ m + 1 ) , κ ( 2 n ¯ a + 1 ) , κ ( 2 n ¯ a + 1 ) , κ ( 2 n ¯ a + 1 ) , κ ( 2 n ¯ a + 1 ) ) .
σ ( t ) = σ ( t + τ ) .
σ ( t ) = ( σ M σ ML σ MR σ ML T σ L σ LR σ MR T σ LR T σ R ) ,
σ r ( t ) = ( σ 1 σ c σ c T σ 2 ) .
E N = max [ 0 , ln ( 2 η ) ]
η 2 1 / 2 { [ 2 4 det σ r ] 1 / 2 } 1 / 2 ,
det σ 1 + det σ 1 2 det σ c .
b = ( q + i p ) / 2 , b = ( q i p ) / 2
c 1 = ( a L + a R ) / 2 , c 2 = ( a L a R ) / 2 .
H lin = Δ 3 c 1 c 1 + Δ 4 c 2 c 2 + ω m b b + 1 2 2 { [ G L * ( t ) G R * ( t ) ] c 1 + [ G L * ( t ) + G R * ( t ) ] c 2 + h . c . } ( b + b ) ,
H ˜ = g 2 { [ ( A L 0 A R 0 ) e i ( Δ 3 + ω m ) t + ( A L 1 A R 1 ) e i ( Δ 3 + ω m Ω ) t ] c 1 b + [ ( A L 0 + A R 0 ) e i ( Δ 4 + ω m ) t + ( A L 1 + A R 1 ) e i ( Δ 4 + ω m Ω ) t ] c 2 b + [ ( A L 0 A R 0 ) e i ( Δ 3 ω m ) t + ( A L 1 A R 1 ) e i ( Δ 3 ω m Ω ) t ] c 1 b + [ ( A L 0 + A R 0 ) e i ( Δ 4 ω m ) t + ( A L 1 + A R 1 ) e i ( Δ 4 ω m Ω ) t ] c 2 b + h . c . } .
Δ 3 = Δ 1 ( t ) + J 5 ω m + g 2 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m ,
Δ 4 = Δ 2 ( t ) J ω m g 2 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m ,
Ω = 2 ω m g 2 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) ω m ,
H ˜ g 2 [ ( A L 1 + A R 1 ) c 2 b + ( A L 0 + A R 0 ) × e i g 2 ( A R 0 2 + A R 1 2 A L 0 2 A L 1 2 ) t ω m c 2 b + h . c . ] .
β 1 = b cosh r + b sinh r ,
β 2 = c 2 cosh r + c 2 sinh r ,
H ˜ χ c 2 β 1 + h . c .
H ˜ χ b β 2 + h . c .
χ = g ( A L 0 + A R 0 ) 2 ( A L 1 + A R 1 ) 2 / 2 .

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