Abstract

We investigate the generation of squeezed state of the mirror motion in a dissipative optomechanical system driven with a strong laser field accompanied with two periodically-modulated lights. Using the density operator approach we calculate the variances of quantum fluctuations around the classical orbits. Both the numerical and analytical results predict that the squeezed state of the mirror motion around its ground state is achievable. Moreover, the obtained squeezed state is robust against the thermal noise because of the strong cooling effect outside the resolved-sideband regime, which arises from the destructive interference of quantum noise.

© 2013 OSA

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    [CrossRef]

2013

T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett.110, 193602 (2013).
[CrossRef] [PubMed]

G. Z. Cohen and M. Di Ventra, “Reading, writing, and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID,” Phys. Rev. B87, 014513 (2013).
[CrossRef]

W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A88, 013835 (2013).
[CrossRef]

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys.15, 045017 (2013).
[CrossRef]

2012

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A85, 022327 (2012).
[CrossRef]

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A86, 042323 (2012).
[CrossRef]

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys.8393–397 (2012).
[CrossRef]

B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A85, 063820 (2012).
[CrossRef]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol.7, 509514 (2012).
[CrossRef]

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A86, 013820 (2012).
[CrossRef]

2011

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
[CrossRef] [PubMed]

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

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

P. Doria, T. Calarco, and S. Montangero, “Optimal control technique for many-body quantum dynamics,” Phys. Rev. Lett.106, 190501 (2011).
[CrossRef] [PubMed]

A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a Michelson–Sagnac interferometer,” Phys. Rev. Lett.107, 213604 (2011).
[CrossRef]

D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
[CrossRef]

2010

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
[CrossRef]

K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett.105, 220501 (2010).
[CrossRef]

2009

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009).
[CrossRef]

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

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A79, 013821 (2009).
[CrossRef]

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A79, 063819 (2009).
[CrossRef]

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett.102, 207209 (2009).
[CrossRef] [PubMed]

2008

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A78, 062303 (2008).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

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

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

2007

G. X. Li, H. T. Tan, and M. Macovei, “Enhancement of entanglement for two-mode fields generated from four-wave mixing with the help of the auxiliary atomic transition,” Phys. Rev. A76, 053827 (2007).
[CrossRef]

2006

G. S. Agarwal, “Interferences in parametric interactions driven by quantized fields,” Phys. Rev. Lett.97, 023601 (2006).
[CrossRef] [PubMed]

2004

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B70, 205304 (2004).
[CrossRef]

2003

X. G. Wang and B. C. Sanders, “Relations between bosonic quadrature squeezing and atomic spin squeezing,” Phys. Rev. A68, 033821 (2003).
[CrossRef]

2002

Y. Wu and C. Côté, “Quadrature-dependent bogoliubov transformations and multiphoton squeezed states,” Phys. Rev. A66, 025801 (2002).
[CrossRef]

2000

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B280, 555–556 (2000).
[CrossRef]

1999

Y. Wu, M. C. Chu, and P. T. Leung, “Dynamics of the quantized radiation field in a cavity vibrating at the fundamental frequency,” Phys. Rev. A59, 3032–3037 (1999).
[CrossRef]

1996

M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,” Rev. Mod. Phys.68, 755–799 (1996).
[CrossRef]

1990

M. T. Jaekel and S. Reynaud, “Quantum limits in interferometric measurements,” Europhys. Lett.13, 301–306 (1990).
[CrossRef]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A79, 013821 (2009).
[CrossRef]

G. S. Agarwal, “Interferences in parametric interactions driven by quantized fields,” Phys. Rev. Lett.97, 023601 (2006).
[CrossRef] [PubMed]

Anetsberger, G.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009).
[CrossRef]

Arcizet, O.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009).
[CrossRef]

Aspelmeyer, M.

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys.8393–397 (2012).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via refection from a silicon micromechanical resonator,” arXiv: 1302.6179v2 (2013).

Barbour, R.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
[CrossRef] [PubMed]

Blencowe, M. P.

M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B280, 555–556 (2000).
[CrossRef]

Bocko, M. F.

M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,” Rev. Mod. Phys.68, 755–799 (1996).
[CrossRef]

Borkje, K.

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

Bruder, C.

T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys.15, 045017 (2013).
[CrossRef]

Brukner, C.

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys.8393–397 (2012).
[CrossRef]

Calarco, T.

P. Doria, T. Calarco, and S. Montangero, “Optimal control technique for many-body quantum dynamics,” Phys. Rev. Lett.106, 190501 (2011).
[CrossRef] [PubMed]

Chan, J.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via refection from a silicon micromechanical resonator,” arXiv: 1302.6179v2 (2013).

Chu, M. C.

Y. Wu, M. C. Chu, and P. T. Leung, “Dynamics of the quantized radiation field in a cavity vibrating at the fundamental frequency,” Phys. Rev. A59, 3032–3037 (1999).
[CrossRef]

Clerk, A. A.

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett.102, 207209 (2009).
[CrossRef] [PubMed]

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

Cohen, G. Z.

G. Z. Cohen and M. Di Ventra, “Reading, writing, and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID,” Phys. Rev. B87, 014513 (2013).
[CrossRef]

Côté, C.

Y. Wu and C. Côté, “Quadrature-dependent bogoliubov transformations and multiphoton squeezed states,” Phys. Rev. A66, 025801 (2002).
[CrossRef]

Danilishin, S. L.

D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
[CrossRef]

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
[CrossRef]

Danzmann, K.

D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
[CrossRef]

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
[CrossRef]

De Chiara, G.

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A86, 042323 (2012).
[CrossRef]

Di Ventra, M.

G. Z. Cohen and M. Di Ventra, “Reading, writing, and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID,” Phys. Rev. B87, 014513 (2013).
[CrossRef]

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. A78, 062303 (2008).
[CrossRef]

Doria, P.

P. Doria, T. Calarco, and S. Montangero, “Optimal control technique for many-body quantum dynamics,” Phys. Rev. Lett.106, 190501 (2011).
[CrossRef] [PubMed]

Eisert, J.

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

Elste, F.

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett.102, 207209 (2009).
[CrossRef] [PubMed]

Farace, A.

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A86, 013820 (2012).
[CrossRef]

Ficek, Z.

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A85, 022327 (2012).
[CrossRef]

Fiore, V.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
[CrossRef] [PubMed]

Friedrich, D.

D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
[CrossRef]

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
[CrossRef]

Gardiner, C. W.

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

Gavartin, E.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol.7, 509514 (2012).
[CrossRef]

Genes, C.

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A79, 063819 (2009).
[CrossRef]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

Gigan, S.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

Giovannetti, V.

A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A86, 013820 (2012).
[CrossRef]

Girvin, S. M.

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

F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett.102, 207209 (2009).
[CrossRef] [PubMed]

Gobler, S.

D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
[CrossRef]

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
[CrossRef]

Gong, Z. R.

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

Gröblacher, S.

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T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett.110, 193602 (2013).
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D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
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D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
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K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
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K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
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K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett.105, 220501 (2010).
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T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett.110, 193602 (2013).
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K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett.105, 220501 (2010).
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T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett.110, 193602 (2013).
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H. Ian, Z. R. Gong, Yu-xi Liu, C. P. Sun, and Franco Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A78, 013824 (2008).
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L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A85, 022327 (2012).
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G. X. Li, H. T. Tan, and M. Macovei, “Enhancement of entanglement for two-mode fields generated from four-wave mixing with the help of the auxiliary atomic transition,” Phys. Rev. A76, 053827 (2007).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
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E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol.7, 509514 (2012).
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K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A79, 063819 (2009).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
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G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009).
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D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
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K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett.107, 133601 (2011).
[CrossRef] [PubMed]

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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. A88, 013835 (2013).
[CrossRef]

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T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” arXiv: 1306.1268v2 (2013).

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T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett.110, 193602 (2013).
[CrossRef] [PubMed]

K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett.105, 220501 (2010).
[CrossRef]

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A79, 063819 (2009).
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M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
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Europhys. Lett.

M. T. Jaekel and S. Reynaud, “Quantum limits in interferometric measurements,” Europhys. Lett.13, 301–306 (1990).
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Nat. Nanotechnol.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol.7, 509514 (2012).
[CrossRef]

Nat. Phys.

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys.8393–397 (2012).
[CrossRef]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009).
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New J. Phys.

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys.10, 095010 (2008).
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D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys.13, 093017 (2011).
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T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys.15, 045017 (2013).
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Phys. Rev. A

B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A86, 042323 (2012).
[CrossRef]

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A85, 022327 (2012).
[CrossRef]

X. G. Wang and B. C. Sanders, “Relations between bosonic quadrature squeezing and atomic spin squeezing,” Phys. Rev. A68, 033821 (2003).
[CrossRef]

M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A78, 062303 (2008).
[CrossRef]

W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A88, 013835 (2013).
[CrossRef]

K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A81, 033849 (2010).
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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A77, 033804 (2008).
[CrossRef]

S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A79, 013821 (2009).
[CrossRef]

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

K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A79, 063819 (2009).
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Figures (6)

Fig. 1
Fig. 1

Sketch of the effective Fabry-Pérot interferometer coupled to the cavity mode via the dispersive and dissipative couplings. The cavity field is pumped by a periodically-modulated laser.

Fig. 2
Fig. 2

Phase space trajectories of the first moments of the movable mirror and cavity light modes. Numerical simulations for t ∈ [0, 100τ] (thin blue solid lines) and analytical approximations of the asymptotic orbits (green dashed lines) from t = 99τ to t = 100τ. The plots are obtained with the parameters shown in the text.

Fig. 3
Fig. 3

Variances of the mirror position and momentum operators with the initial mean thermal phonon number th = 103. The numerical results in the periodically-modulated model are shown by blue solid lines (with the well-chosen modulation frequency Ω around twice the frequency ωm because of the cavity-induced energy shift of the oscillator), and the numerical results in the nonmodulated model are shown by the red dashed lines for t ∈ [100τ, 102τ].

Fig. 4
Fig. 4

Comparison between the analytical approximations of phase orbits for cavity variables of the zeroth order in λ (red dashed line) and the numerical asymptotic orbit (blue solid line) which is also equal to the corrections up to λ3, for t ∈ [99τ, 100τ].

Fig. 5
Fig. 5

Analytical approximation of the variance of the mirror momentum operator (red dashed line) compared with the numerical result (blue solid line) for t ∈ [100τ, 102τ].

Fig. 6
Fig. 6

Numerical calculations of the variance of the mirror momentum operator with the following set of input powers PR = 10 mW (blue solid line), PR = 20 mW (red dashed line), PR = 40 mW (green dash-dotted line), PR = 80 mW (black dotted line) for t ∈ [100τ, 102τ].

Equations (49)

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H ^ = H ^ c + H ^ m + H ^ R + H ^ c R + H ^ int , H ^ c = ω a a ^ a ^ , H ^ m = ω m b ^ b ^ , H ^ R = d ω ω a ^ ω a ^ ω , H ^ c R = i κ c / π d ω ( a ^ ω a ^ a ^ a ^ ω ) , H ^ int = g 0 [ α a ^ a ^ + i β L / ( 2 π c ) d ω ( a ^ ω a ^ a ^ a ^ ω ) ] ( b ^ + b ^ ) .
a ^ ω = 2 π a ¯ in [ 1 + ε cos ( Ω t ) ] exp ( i ω R t ) .
ρ ^ ˙ s = i Tr R [ 𝒱 ^ ( t ) , ρ ^ s ( t ) ρ ^ R ( t i ) ] Tr R t i t [ 𝒱 ^ ( t ) , [ 𝒱 ^ ( t ) , ρ ^ s ( t ) , ρ ^ R ( t i ) ] ] d t .
ρ ^ ˙ s = i [ H ^ 0 , ρ ^ s ] + ^ 1 ρ ^ s + ^ 2 ρ ^ s ,
H ^ 0 = Δ a ^ a ^ + ω m b ^ b ^ ,
λ = g 0 β L / κ c c
^ 1 ρ ^ s = i [ i 2 ( 1 + ε cos ( Ω t ) ( a in * ¯ C ^ a ¯ in C ^ ) , ρ ^ s ) ] , ^ 2 ρ ^ s = 2 C ^ ρ ^ s C ^ C ^ C ^ ρ ^ s ρ ^ s C ^ C ^ ,
x ^ = ( a ^ + a ^ ) / 2 , y ^ = ( a ^ a ^ ) / i 2 , q ^ = ( b ^ + b ^ ) / 2 , p ^ = ( b ^ b ^ ) / i 2 .
^ m ρ ^ s = i γ m 2 [ q ^ , { p ^ , ρ ^ s } ] + γ m [ n ¯ th + 1 2 ] [ q ^ , [ q ^ , ρ ^ s ] ] .
q ˙ ¯ = ω m p ¯ , p ˙ ¯ = ω m q ¯ γ m p ¯ + λ 2 E ( 1 + ε cos Ω t ) y ¯ , x ˙ ¯ = Δ y ¯ κ c x ¯ ( 1 + λ q ¯ ) 2 2 E ( 1 + ε cos Ω t ) ( 1 + λ q ¯ ) , y ˙ ¯ = Δ x ¯ κ c y ¯ ( 1 + λ q ¯ ) 2 ,
O ¯ ( t ) = j = 0 O ¯ j ( t ) λ j .
O ¯ j ( t ) = n = O ¯ n , j exp ( in Ω t ) .
a ^ = a ¯ ( t ) + d ^ , b ^ = b ¯ ( t ) + f ^ ,
d d t ρ ^ s = ^ d ρ ^ s + ^ f ρ ^ s + ^ d f ρ ^ s .
^ d ρ ^ s = i [ Δ d ^ d ^ , ρ ^ s ] + κ c ( 1 + λ q ¯ ( t ) ) 2 ( 2 d ^ ρ ^ s d ^ d ^ d ^ ρ ^ s ρ ^ s d ^ d ^ ) ,
^ f ρ ^ s = i [ ω m 2 ( P ^ 2 + Q ^ 2 ) , ρ ^ s ] + 𝒟 ^ f ρ ^ s + ^ m ρ ^ s ,
𝒟 ^ f ρ ^ s = κ c | a ¯ ( t ) | 2 λ 2 ( 2 Q ^ ρ ^ s Q ^ Q ^ 2 ρ ^ s ρ ^ s Q ^ 2 )
^ d f ρ ^ s = [ λ E ( 1 + ε cos Ω t ) ( d ^ d ^ ) Q ^ , ρ ^ s ] + { 2 λ κ c a ¯ ( t ) ( 1 + λ q ¯ ( t ) ) ( Q ^ ρ ^ s d ^ d ^ Q ^ ρ ^ s ) + H . c . } .
X ( t ) = ( d ^ d ^ , d ^ 2 , d ^ 2 , d ^ Q ^ , d ^ Q ^ , d ^ P ^ , d ^ P ^ , Q ^ 2 , P ^ 2 , Q ^ P ^ ) T ,
d d t X ( t ) = M ¯ ( t ) X ( t ) + C ( t ) .
M ¯ ( t ) = n = 2 2 M n exp ( in Ω t ) , C ( t ) = n = 2 2 C n exp ( in Ω t ) .
X ( t ) = n = X n exp ( in Ω t ) .
𝒱 ^ ( t ) = i κ c π { 1 + λ 2 [ b ^ exp ( i ω m t ) + b ^ exp ( i ω m t ) ] + λ ( 2 ) 2 [ b ^ exp ( i ω m t ) + b ^ exp ( i ω m t ) ] 2 } × d ω { a ^ ω a ^ exp [ i ( ω a ω ) t ] a ^ a ^ ω exp [ i ( ω a ω ) t ] } ,
C ^ = κ c [ 1 + λ / 2 ( b ^ + b ^ ) + λ ( 2 ) / 2 ( b ^ + b ^ ) 2 ] a ^ .
^ ext ρ ^ s = λ ( 2 ) E ( 1 + ε cos Ω t ) [ ( a ¯ a * ¯ ) Q ^ 2 + 2 q ¯ ( d ^ d ^ ) Q ^ , ρ ^ s ] + 2 κ c λ ( 2 ) q ¯ 2 ( 2 d ^ ρ ^ s d ^ d ^ d ^ ρ ^ s ρ ^ s d ^ d ^ ) + { 4 κ c λ ( 2 ) a q ¯ ( Q ^ ρ s d ^ d ^ Q ^ ρ ^ s ) + H . c . } ,
a ¯ ( t ) = E i Δ κ c + ε E / 2 exp ( i Ω t ) i ( Δ Ω ) κ c + ε E / 2 exp ( i Ω t ) i ( Δ + Ω ) κ c .
d d t ρ ^ f = Tr d t 0 t ^ d f ( t ) ^ d f ( t ) ρ ^ d ( t 0 ) ρ ^ f ( t ) d t + 𝒟 ^ f ρ ^ f + ^ m ρ ^ f ,
d d t ρ ^ f = i [ ^ f , ρ ^ f ] + 𝒦 ^ f ρ ^ f .
^ f = ω f s f ^ f ^ ,
ω f s = 1 2 λ 2 E 2 { j = 1 , 2 ( 1 ) j + 1 [ θ j ( Δ , ω m ) + θ j ( Δ , ω m ) + ε 2 4 ( θ j ( Δ Ω , ω m ) + θ j ( Δ + Ω , ω m ) + θ j ( Δ Ω , ω m ) + θ j ( Δ + Ω , ω m ) ) ] } ,
𝒦 ^ f ρ ^ f = [ A ( ω m ) + γ m 2 ( n ¯ th + 1 ) ] ( f ^ ρ ^ f f ^ f ^ f ^ ρ ^ f ) + [ A ( ω m ) + γ m 2 n ¯ th ] ( f ^ ρ ^ f f ^ f ^ f ^ ρ ^ f ) + B 1 exp ( i δ t ) ( 2 f ^ ρ ^ f f ^ f ^ 2 ρ ^ f ρ ^ f f ^ 2 ) + C 1 exp ( i δ t ) [ ρ ^ f , f ^ 2 ] + H . c ..
A ( ω m ) = 1 2 κ c λ 2 E 2 F ( Δ , ω m ) , B 1 = 1 4 λ 2 E 2 ε 2 [ h ( Δ , ω m ) + h * ( Δ , ω m ) + h ( Δ Ω , ω m ) + h * ( Δ + Ω , ω m ) 2 Δ ˜ ] , C 1 = 1 4 λ 2 E 2 ε 2 [ h ( Δ , ω m ) h * ( Δ , ω m ) + h ( Δ Ω , ω m ) h * ( Δ + Ω , ω m ) ] ,
F ( Δ , ω m ) = f ( Δ , ω m ) + ε 2 4 [ f ( Δ Ω , ω m ) + f ( Δ + Ω , ω m ) ] , f ( Δ , ω m ) = ( 2 Δ + ω m ) 2 ( Δ 2 + κ c 2 ) [ ( Δ + ω m ) 2 + κ c 2 ] , h ( Δ , ω m ) = ( Δ i κ c ) 2 Δ 2 + κ c 2 i ( Δ + ω m ) + κ c ( Δ + ω m ) 2 + κ c 2 , Δ ˜ = κ c ( i Δ κ c ) [ i ( Δ + Ω ) + κ c ] + κ c ( i Δ + κ c ) [ i ( Δ Ω ) κ c ] .
d d t ( f ^ f ^ ) = 2 [ A ( ω m ) A ( ω m ) + γ m / 2 ] f ^ f ^ + 2 [ C 1 f ^ 2 + c . c . ] + 2 A ( ω m ) + γ m n ¯ th , d d t f ^ 2 = 2 [ A ( ω m ) A ( ω m ) + γ m / 2 ] f ^ 2 + 4 C 1 f ^ f ^ 2 ( B 1 C 1 ) .
A ( ω m ) = κ c λ 2 E 2 ε 2 8 [ f ( Δ Ω , ω m ) + f ( Δ + Ω , ω m ) ] ,
κ m = A ( ω m ) A ( ω m ) + γ m / 2 ,
f ^ f ^ 1 2 [ f ^ 2 exp ( i 2 ω m t ) + f ^ 2 exp ( i 2 ω m t ) ] < 0 .
f ^ f ^ 1 2 | f ^ 2 | { exp [ i ( 2 ω m t + θ ) ] + exp [ i ( 2 ω m t + θ ) ] } < 0 .
n ( q ˙ ¯ n , 0 + in Ω q ¯ n , 0 ) exp ( in Ω t ) = ω m n p ¯ n , 0 exp ( in Ω t ) , n ( p ˙ ¯ n , 0 + in Ω p ¯ n , 0 + γ m p ¯ n , 0 ) exp ( in Ω t ) = ω m n q ¯ n , 0 exp ( in Ω t ) , n ( x ˙ ¯ n , 0 + in Ω x ¯ n , 0 + κ c x ¯ n , 0 ) exp ( in Ω t ) = Δ n y ¯ n , 0 exp ( in Ω t ) 2 E ( 1 + ε cos Ω t ) , n ( y ˙ ¯ n , 0 + in Ω y ¯ n , 0 + κ c y ¯ n , 0 ) exp ( in Ω t ) = Δ n x ¯ n , 0 exp ( in Ω t ) .
n ( q ˙ ¯ n , 1 + in Ω q ¯ n , 1 ) exp ( in Ω t ) = ω m n p ¯ n , 1 exp ( in Ω t ) , n ( p ˙ ¯ n , 1 + in Ω p ¯ n , 1 + γ m p ¯ n , 1 ) exp ( in Ω t ) = ω m n q ¯ n , 1 exp ( in Ω t ) + 2 E ( 1 + ε cos Ω t ) y ¯ 0 , n ( x ˙ ¯ n , 1 + in Ω x ¯ n , 1 + κ c x ¯ n , 1 ) exp ( in Ω t ) = Δ n y ¯ n , 1 exp ( in Ω t ) , n ( y ˙ ¯ n , 1 + in Ω y ¯ n , 1 + κ c y ¯ n , 1 ) exp ( in Ω t ) = Δ n x ¯ n , 1 exp ( in Ω t ) .
n ( q ˙ ¯ n , 2 + in Ω q ¯ n , 2 ) exp ( in Ω t ) = ω m n p ¯ n , 2 exp ( in Ω t ) , n ( p ˙ ¯ n , 2 + in Ω p ¯ n , 2 + γ m p ¯ n , 2 ) exp ( in Ω t ) = ω m n q ¯ n , 2 exp ( in Ω t ) , n ( x ˙ ¯ n , 2 + in Ω x ¯ n , 2 + κ c x ¯ n , 2 ) exp ( in Ω t ) = Δ n y ¯ n , 2 exp ( in Ω t ) 2 E ( 1 + ε cos Ω t ) q ¯ 1 2 κ c x ¯ 0 q ¯ 1 , n ( y ˙ ¯ n , 2 + in Ω y ¯ n , 2 + κ c y ¯ n , 2 ) exp ( in Ω t ) = Δ n x ¯ n , 2 exp ( in Ω t ) 2 κ c y ¯ 0 q ¯ 1 .
n ( q ˙ ¯ n , 3 + in Ω q ¯ n , 3 ) exp ( in Ω t ) = ω m n p ¯ n , 3 exp ( in Ω t ) n ( p ˙ ¯ n , 3 + in Ω p ¯ n , 3 + γ m p ¯ n , 3 ) exp ( in Ω t ) = ω m n q ¯ n , 3 exp ( in Ω t ) + 2 E ( 1 + ε cos Ω t ) y ¯ 2 , n ( x ˙ ¯ n , 3 + in Ω x ¯ n , 3 + κ c x ¯ n , 3 ) exp ( in Ω t ) = Δ n y ¯ n , 3 exp ( in Ω t ) 2 E ( 1 + ε cos Ω t ) q ¯ 2 2 κ c x ¯ 0 q ¯ 2 , n ( y ˙ ¯ n , 3 + in Ω y ¯ n , 3 + κ c y ¯ n , 3 ) exp ( in Ω t ) = Δ n x ¯ n , 3 exp ( in Ω t ) 2 κ c y ¯ 0 q ¯ 2 .
O ¯ ( t ) = j = 0 3 n = 1 1 O ¯ n , j ( t ) λ j exp ( in Ω t )
M ¯ ( t ) = ( 2 ξ 1 0 0 ξ 3 * ξ 3 0 0 0 0 0 0 2 ξ ˜ 1 0 2 ξ 3 0 0 0 0 0 0 0 0 2 ξ ˜ 1 * 0 2 ξ 3 * 0 0 0 0 0 0 0 0 ξ ˜ 1 0 ω m 0 ξ 3 0 0 0 0 0 0 ξ ˜ 1 * 0 ω m ξ 3 * 0 0 ξ 2 ξ 2 0 ω m 0 ξ ˜ 1 γ m 0 0 0 ξ 3 ξ 2 0 ξ 2 0 ω m 0 ξ ˜ 1 * γ m 0 0 ξ 3 * 0 0 0 0 0 0 0 0 0 2 ω m 0 0 0 0 0 2 ξ 2 2 ξ 2 0 2 γ m 2 ω m 0 0 0 ξ 2 ξ 2 0 0 ω m ω m γ m ) ,
C ( t ) = ( 0 , 0 , 0 , 0 , 0 , i ξ 3 , 0 , i ω m , i ω m + 2 κ c λ 2 | a ¯ ( t ) | 2 + 2 γ m ( n ¯ th + 1 / 2 ) , i γ m / 2 ) T ,
X ˙ 0 = M 0 X 0 + M 1 X 1 + M 1 X 1 + M 2 X 2 + M 2 X 2 + C 0 , X ˙ 1 + i Ω X 1 = M 0 X 1 + M 1 X 0 + M 2 X 1 + M 1 X 2 + C 1 , X ˙ 1 + i Ω X 1 = M 0 X 1 + M 1 X 0 + M 2 X 1 + M 1 X 2 + C 1 , X ˙ 2 + i 2 Ω X 2 = M 0 X 2 + M 2 X 0 + M 1 X 1 + C 2 , X ˙ 2 + i 2 Ω X 2 = M 0 X 2 + M 2 X 0 + M 1 X 1 + C 2 .
X ± 2 = M ± 1 X ± 1 + M ± 2 X 0 + C ± 2 M 2 i Ω I ,
( M 2 M 1 M 1 M 2 M 0 2 i Ω I M 0 i Ω I M 1 M 1 M 0 2 i Ω I M 0 + i Ω I M 1 M 1 M 0 + 2 i Ω I M 1 M 1 M 2 M 0 + 2 i Ω I M 2 M 1 M 2 M 1 M 0 + 2 i Ω I M 0 M 2 M 2 M 0 + 2 i Ω I M 2 M 2 M 0 2 i Ω I M 1 M 2 M 1 M 0 2 i Ω I ) × ( X 1 X 0 X 1 ) = ( C 1 + M 1 C 2 M 0 2 i Ω I C 1 + M 1 C 2 M 0 + 2 i Ω I C 0 + M 2 C 2 M 0 + 2 i Ω I + M 2 C 2 M 0 2 i Ω I ) .
X ( t ) = n = 2 2 X n exp ( in Ω t ) ,

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