Abstract

This paper proposes a scheme for wide-range precision measurement of the environmental temperature in a double-cavity optomechanical system. This system consists of an optomechanical cavity coupling to the other cavity via photon tunneling interaction. Bycontrolling the tunnelling strength between the two cavities, double optomechanically induced transparency (double OMIT) effect is observed in the homodyne spetra of the outfield. It is shown that the central peak value depends linearly on the environmental temperature. Based on this linear relationship, the environmental temperature can be inferred from the central peak value of the output homodyne spectrum. This scheme is robust against mechanical decay and it shows high sensivity over a wide temperature range.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

Q. Wang, “Precision temperature measurement with optomechanically induced transparency in an optomechanical system,” Laser Physics 28,075201 (2018).
[Crossref]

2017 (4)

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Applied Physics Letters 110, 171102 (2017).
[Crossref]

Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
[Crossref]

Q. Wang and W. Li, “Precision mass sensing by tunable double optomechanically induced transparency with squeezed field in a coupled optomechanical system,” International Journal of Theoretical Physics 56, 1346–1354 (2017).
[Crossref]

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

2016 (1)

Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
[Crossref]

2015 (7)

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Physical Review A 92, 033829 (2015).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

Q. Wu, “Tunable optomechanically induced absorption with quantum fields in an optomechanical system,” JOSA B 32, 1712–1717 (2015).
[Crossref]

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

2014 (5)

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

W. Gu and Z. Yi, “Double optomechanically induced transparency in coupled-resonator system,” Optics Communications 333, 261–264 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanic,” Reviews of Modern Physics 86, 1391 (2014).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Physical Review A 90, 043848 (2014).
[Crossref]

2013 (5)

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Physical Review Letters 110, 071105 (2013).
[Crossref] [PubMed]

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

D. Tarhan, S. Huang, and Ö. E. Müstecaplioǧlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Physical Review A 87, 013824 (2013).
[Crossref]

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

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

2012 (4)

J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nature Nanotechnology 7, 509 (2012).
[Crossref] [PubMed]

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Physical Review A 85, 021801 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

2011 (8)

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
[Crossref]

C. Jiang, C. Bin, and K. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhysics Letters 94, 38002 (2011).
[Crossref]

B. Chen, C. Jiang, and K. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Physical Review A 83, 055803 (2011).
[Crossref]

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

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

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” Journal of Physics B: Atomic, Molecular and Optical Physics 44, 165505 (2011).
[Crossref]

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

2010 (2)

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

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

2009 (1)

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

2008 (2)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Physics 4, 555 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

2007 (1)

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
[Crossref]

2006 (1)

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

2004 (1)

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

2002 (1)

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry–Perot interferometer,” Physics Letters A 293, 228–234 (2002).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Physical Review A 85, 021801 (2012).
[Crossref]

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

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

Alegre, T. P. M.

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

Allman, M. S.

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

Arcizet, O.

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

Arvanitaki, A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Physical Review Letters 110, 071105 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanic,” Reviews of Modern Physics 86, 1391 (2014).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
[Crossref]

Bin, C.

C. Jiang, C. Bin, and K. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhysics Letters 94, 38002 (2011).
[Crossref]

Bowen, W. P.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Braginsky, V. B.

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry–Perot interferometer,” Physics Letters A 293, 228–234 (2002).
[Crossref]

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,” Physical Review Letters 99, 250401 (2007).
[Crossref]

Budakian, R.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

Castellanos-Beltran, M. A.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

Chan, J.

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

Chang, D. E.

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

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
[Crossref]

Chen, B.

B. Chen, C. Jiang, and K. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Physical Review A 83, 055803 (2011).
[Crossref]

Cheng, J.

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” Journal of Physics B: Atomic, Molecular and Optical Physics 44, 165505 (2011).
[Crossref]

Cho, S. U.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

Chui, B. W.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

Cicak, K.

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

Deléglise, S.

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

Donner, T.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

Du, C.

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
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A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69 (2011).
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M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
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Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
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Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
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P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
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J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
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S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
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Gao, M.

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
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E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nature Nanotechnology 7, 509 (2012).
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S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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Geng, Z.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
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A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Physical Review Letters 110, 071105 (2013).
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M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
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X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
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W. Gu and Z. Yi, “Double optomechanically induced transparency in coupled-resonator system,” Optics Communications 333, 261–264 (2014).
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H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
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D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
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F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
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Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” Journal of Physics B: Atomic, Molecular and Optical Physics 44, 165505 (2011).
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J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
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S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
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Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
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Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
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Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
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F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
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Hill, J. T.

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

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
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Hou, B. P.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Physical Review A 92, 033829 (2015).
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Huang, S.

D. Tarhan, S. Huang, and Ö. E. Müstecaplioǧlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Physical Review A 87, 013824 (2013).
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G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Physical Review A 85, 021801 (2012).
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S. Huang and G. S. Agarwal, “Electromagnetically induced transparency with quantized fields in optocavity mechanics,” Physical Review A 83, 043826 (2011).
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G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Physical Review A 81, 041803 (2010).
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X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
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Jiang, C.

C. Jiang, C. Bin, and K. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhysics Letters 94, 38002 (2011).
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B. Chen, C. Jiang, and K. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Physical Review A 83, 055803 (2011).
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Jing, H.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Jing, Q. L.

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

Kim, M. 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,” Physical Review Letters 99, 250401 (2007).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanic,” Reviews of Modern Physics 86, 1391 (2014).
[Crossref]

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nature Nanotechnology 7, 509 (2012).
[Crossref] [PubMed]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Knittel, J.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Kong, C.

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

Lehnert, K. W.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Physics 4, 555 (2008).
[Crossref]

Lei, F.

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

Li, D.

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

Li, W.

Q. Wang and W. Li, “Precision mass sensing by tunable double optomechanically induced transparency with squeezed field in a coupled optomechanical system,” International Journal of Theoretical Physics 56, 1346–1354 (2017).
[Crossref]

Li, W. J.

Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
[Crossref]

Li, Y.

J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
[Crossref]

Lin, Q.

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

Liu, Y.

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

Liu, Z.

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

Long, G. L.

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

Lü, X. Y.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Ma, P.

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

Ma, P. C.

Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
[Crossref]

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

Mamin, H. J.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanic,” Reviews of Modern Physics 86, 1391 (2014).
[Crossref]

Marx, A.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

Massel, F.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

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D. Meiser and P. Meystre, “Coupled dynamics of atoms and radiation-pressure-driven interferometers,” Physical Review A 73, 033417 (2006).
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Meystre, P.

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

Miranowicz, A.

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

Müstecaplioglu, Ö. E.

D. Tarhan, S. Huang, and Ö. E. Müstecaplioǧlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Physical Review A 87, 013824 (2013).
[Crossref]

Naderi, M. H.

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

Nori, F.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

Özdemir, S. K.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Painter, O.

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
[Crossref]

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

Paternostro, M.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
[Crossref]

Peng, B.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Pirkkalainen, J. M.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

Prams, S.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Regal, C. A.

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Physics 4, 555 (2008).
[Crossref]

Rivière, R.

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

Rubinsztein-Dunlop, H.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Rugar, D.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

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

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
[Crossref]

Saloniemi, H.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

Schliesser, A.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

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

Shahidani, S.

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

Si, L. G.

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

Si, L.-G.

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Applied Physics Letters 110, 171102 (2017).
[Crossref]

Sillanpää, M. A.

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

Simmonds, R. W.

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

Sirois, A. J.

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

Soltanolkotabi, M.

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

Swaim, J. D.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Szorkovszky, A.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Tarhan, D.

D. Tarhan, S. Huang, and Ö. E. Müstecaplioǧlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Physical Review A 87, 013824 (2013).
[Crossref]

Taylor, J. M.

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Physical Review A 90, 043848 (2014).
[Crossref]

Teufel, J. D.

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

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Physics 4, 555 (2008).
[Crossref]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

van Ooijen, E. D.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Verlot, P.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nature Nanotechnology 7, 509 (2012).
[Crossref] [PubMed]

Vitali, D.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
[Crossref]

Vyatchanin, S. P.

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry–Perot interferometer,” Physics Letters A 293, 228–234 (2002).
[Crossref]

Wang, B.

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

Wang, H.

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

Wang, Q.

Q. Wang, “Precision temperature measurement with optomechanically induced transparency in an optomechanical system,” Laser Physics 28,075201 (2018).
[Crossref]

Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
[Crossref]

Q. Wang and W. Li, “Precision mass sensing by tunable double optomechanically induced transparency with squeezed field in a coupled optomechanical system,” International Journal of Theoretical Physics 56, 1346–1354 (2017).
[Crossref]

Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

Wang, S. J.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Physical Review A 92, 033829 (2015).
[Crossref]

Wei, L. F.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Physical Review A 92, 033829 (2015).
[Crossref]

Weis, S.

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

Whittaker, J. D.

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

Winger, M.

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

Wu, J. H.

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

Wu, Q.

Q. Wu, “Tunable optomechanically induced absorption with quantum fields in an optomechanical system,” JOSA B 32, 1712–1717 (2015).
[Crossref]

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

Wu, Q. Q.

Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
[Crossref]

Wu, Y.

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Applied Physics Letters 110, 171102 (2017).
[Crossref]

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

Xiao, Y.

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

Xiong, H.

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Applied Physics Letters 110, 171102 (2017).
[Crossref]

Xu, X.

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Physical Review A 90, 043848 (2014).
[Crossref]

Xu, Y.

J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
[Crossref]

Yan, L.

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

Yang, L.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Yang, X.

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

Yao, C. M.

Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
[Crossref]

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

Yao, Y. C.

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

Yi, Z.

W. Gu and Z. Yi, “Double optomechanically induced transparency in coupled-resonator system,” Optics Communications 333, 261–264 (2014).
[Crossref]

Zhan, X.

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

Zhang, J.

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
[Crossref]

Zhang, J. Q.

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

Zhang, Z. M.

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

Zheng, A. S.

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

Zhou, L.

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” Journal of Physics B: Atomic, Molecular and Optical Physics 44, 165505 (2011).
[Crossref]

Zhou, X.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

Zhu, K.

B. Chen, C. Jiang, and K. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Physical Review A 83, 055803 (2011).
[Crossref]

C. Jiang, C. Bin, and K. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhysics Letters 94, 38002 (2011).
[Crossref]

Applied Physics Letters (1)

H. Xiong, L.-G. Si, and Y. Wu, “Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics,” Applied Physics Letters 110, 171102 (2017).
[Crossref]

Europhysics Letters (1)

C. Jiang, C. Bin, and K. Zhu, “Tunable pulse delay and advancement device based on a cavity electromechanical system,” Europhysics Letters 94, 38002 (2011).
[Crossref]

IEEE Photonics Journal (1)

Z. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Highly sensitive optical detector for precision measurement of coulomb coupling strength based on a double-oscillator optomechanical system,” IEEE Photonics Journal 10, 1–11 (2018).

International Journal of Theoretical Physics (3)

Q. Wang, W. J. Li, P. C. Ma, and Z. He, “Precision mass measurement with optomechanically induced transparency in an optomechanical system,” International Journal of Theoretical Physics 56, 2212–2220 (2017).
[Crossref]

Q. Wang, C. M. Yao, Q. Q. Wu, and Z. He, “Tunable Multiple Optomechanically Induced Transparency with Squeezed Fields in an Optomechanical System,” International Journal of Theoretical Physics 55, 5385–5392 (2016).
[Crossref]

Q. Wang and W. Li, “Precision mass sensing by tunable double optomechanically induced transparency with squeezed field in a coupled optomechanical system,” International Journal of Theoretical Physics 56, 1346–1354 (2017).
[Crossref]

JOSA B (1)

Q. Wu, “Tunable optomechanically induced absorption with quantum fields in an optomechanical system,” JOSA B 32, 1712–1717 (2015).
[Crossref]

Journal of Physics B: Atomic, Molecular and Optical Physics (2)

Y. Han, J. Cheng, and L. Zhou, “Electromagnetically induced transparency in a cavity optomechanical system with an atomic medium,” Journal of Physics B: Atomic, Molecular and Optical Physics 44, 165505 (2011).
[Crossref]

X. Zhan, L. G. Si, A. S. Zheng, and X. Yang, “Tunable slow light in a quadratically coupled optomechanical system,” Journal of Physics B: Atomic, Molecular and Optical Physics 46, 025501 (2013).
[Crossref]

Laser Physics (1)

Q. Wang, “Precision temperature measurement with optomechanically induced transparency in an optomechanical system,” Laser Physics 28,075201 (2018).
[Crossref]

Nature (4)

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

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

F. Massel, T. T. Heikkilä, J. M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351 (2011).
[Crossref] [PubMed]

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329 (2004).
[Crossref] [PubMed]

Nature Nanotechnology (2)

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nature Nanotechnology 4, 820 (2009).
[Crossref] [PubMed]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nature Nanotechnology 7, 509 (2012).
[Crossref] [PubMed]

Nature Physics (2)

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nature Physics 4, 555 (2008).
[Crossref]

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nature Physics 9, 179 (2013).
[Crossref]

New Journal of Physics (1)

D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” New Journal of Physics 13, 013824 (2011).
[Crossref]

Optics Communications (1)

W. Gu and Z. Yi, “Double optomechanically induced transparency in coupled-resonator system,” Optics Communications 333, 261–264 (2014).
[Crossref]

Optics Express (2)

F. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Optics Express 23, 11508–11517 (2015).
[Crossref] [PubMed]

Q. Wu, J. Q. Zhang, J. H. Wu, M. Feng, and Z. M. Zhang, “Tunable multi-channel inverse optomechanically induced transparency and its applications,” Optics Express 23, 18534–18547 (2015).
[Crossref] [PubMed]

Physical Review A (13)

G. S. Agarwal and S. Huang, “Optomechanical systems as single-photon routers,” Physical Review A 85, 021801 (2012).
[Crossref]

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

P. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Physical Review A 90, 043825 (2014).
[Crossref]

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

J. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Physical Review A 86, 053806 (2012).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, Y. C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Physical Review A 91, 063827 (2015).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Physical Review A 92, 033829 (2015).
[Crossref]

H. Wang, X. Gu, Y. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Physical Review A 90, 023817 (2014).
[Crossref]

B. Chen, C. Jiang, and K. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Physical Review A 83, 055803 (2011).
[Crossref]

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

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

D. Tarhan, S. Huang, and Ö. E. Müstecaplioǧlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Physical Review A 87, 013824 (2013).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Physical Review A 90, 043848 (2014).
[Crossref]

Physical Review Letters (3)

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Physical Review Letters 99, 250401 (2007).
[Crossref]

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Physical Review Letters 110, 071105 (2013).
[Crossref] [PubMed]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Physical Review Letters 108, 120801 (2012).
[Crossref] [PubMed]

Physics Letters A (1)

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry–Perot interferometer,” Physics Letters A 293, 228–234 (2002).
[Crossref]

Reviews of Modern Physics (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanic,” Reviews of Modern Physics 86, 1391 (2014).
[Crossref]

Science (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

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

Scientific Reports (2)

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X. Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Scientific Reports 5, 9663 (2015).
[Crossref] [PubMed]

Z. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Scientific Reports 7, 12521 (2017).
[Crossref] [PubMed]

The European Physical Journal D (1)

Q. Wang, L. Yan, P. C. Ma, Z. He, and C. M. Yao, “Tunable double optomechanically induced transparency with quantized fields in an optomechanical system,” The European Physical Journal D 69,213 (2015).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic of the system and the measurement. Two cavities are coupled via photon tunneling effect with strength J. The field a1 in the left cavity is driven by a strong classical field with frequency ωc and a squeezed vacuum at frequency ωp. In combination with a 1 , i n, the output field a 1 , o u t turns out to be a o u t. Then a o u t is mixed with a strong local field alo at a 50 : 50 beam splitter (BS). The strong local field is centered around the probe frequency and can be depicted as a l o ( t ) = a l o e i ω m t. Based on the difference between the output signals of the two photodetectors (PD), the homodyne spectrum can be obtained from the spectrum analyzer (SA).
Fig. 2
Fig. 2 Homodyne spectrum X ( ω ) versus the normalized frequency ω / ω m (a) for different coupling strength J with T = 100mK, N = 5, (b)for different environmental temperature T with J = κ , N = 5, (c) for different photon number N with J = κ , T = 100mK.
Fig. 3
Fig. 3 (a) Homodyne spectrum X ( ω ) as a function of ω / ω m for J = κ 1, T = 100 m K,N = 5. (b) The thermal-related homodyne spectrum X ( ω ) = X ( ω ) g 0 X ( ω ) g = 0 versus ω / ω m. Other parameters are chosen as in Fig. 2(a).
Fig. 4
Fig. 4 (a) Homodyne spectrum X ( ω ) as a function of ω / ω m for different temperature. (b) The rescaled central peak value versus the temperature T. Other parameters are chosen as in Fig. 2(a).
Fig. 5
Fig. 5 The quantum signal visibility VQS as a function of the photon number N and the temperature T with J = κ 1. Other parameters are chosen as in Fig. 2(a).
Fig. 6
Fig. 6 (a) Homodyne spectrum X ( ω ) as a function of ω / ω m for J = κ 1, T = 100 m K,N = 5. (b) The rescaled central peak value Xp (in unit of X 0 = 22.11) as a function of T. Other parameters are chosen as in Fig. 2(a).
Fig. 7
Fig. 7 Homodyne spectrum X ( ω ) versus ω / ω m (a) for different Δ2 with Δ 2 = ω m, (b) for different Δ1 with Δ 1 = ω m.(c) The rescaled central peak value X p as a function of T for different detunings.

Equations (34)

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

H = i = 1 , 2 Δ i a i a i + m ω m 2 2 q 2 + p 2 2 m g a 2 a 2 q + J ( a 1 a 2 + a 1 a 2 ) + i ε c ( a 1 a 1 ) .
d d t q = p m ,
d d t p = γ m p m ω m 2 q + g a 2 a 2 + ξ ,
d d t a 1 = ( κ 1 + i Δ 1 ) a 1 i J a 2 + 2 κ 1 a 1 , i n ,
d d t a 2 = ( κ 2 + i Δ 2 i g q ) a 2 i J a 1 + 2 κ 2 a 2 , i n ,
< ξ ( ω ) ξ ( Ω ) > = 4 π γ m ω ω m [ 1 + coth  ( ω 2 k B T ) ] δ ( ω + Ω ) ,
< a 1 , i n ( ω ) a 1 , i n ( Ω ) > = 2 π M Γ 2 Γ 2 + ( ω ω m ) 2 δ ( ω + Ω 2 ω m ) ,
< a 1 , i n ( ω ) a 1 , i n ( Ω ) > = 2 π N Γ 2 Γ 2 + ( ω ω m ) 2 δ ( ω + Ω ) ,
< a 2 , i n ( ω ) a 2 , i n ( Ω ) > = δ ( ω + Ω ) ,
q s = g a 2 s * a 2 s ω m ,
p s = 0 ,
a 1 s = ε c [ κ 2 + i ( Δ 2 + g q s ) ] ( κ 1 + i Δ 1 ) ( κ 2 + i Δ 2 ' ) + J 2 ,
a 2 s = i J ε c ( κ 1 + i Δ 1 ) ( κ 2 + i Δ 2 ' ) + J 2 ,
d d t δ q = δ p m ,
d d t δ p = γ m δ p m ω m 2 δ q + g a 2 s * δ a 2 + g a 2 s δ a 2 + ξ ,
d d t δ a 1 = ( κ 1 + i Δ 1 ) δ a 1 i J δ a 2 + 2 κ 1 a 1 , i n ,
d d t δ a 2 = ( κ 2 + i Δ 2 i g q s ) δ a 2 i J δ a 1 + i g a 2 s δ q + 2 κ 2 a 2 , i n .
A = ( i m ω 1 0 0 0 0 m ω m 2 γ m i ω 0 0 g a 2 s g a 2 s * 0 0 κ 1 + i ( Δ 1 ω ) 0 i J 0 0 0 0 κ 1 i ( Δ 1 + ω ) 0 i J i g a 2 s 0 i J 0 κ 2 + i ( Δ 2 g q s ω ) 0 i g a 2 s 0 0 i J 0 κ 2 i ( Δ 2 g q s + ω ) ) ,
X = ( δ q ( ω ) , δ p ( ω ) , δ a 1 ( ω ) , δ a 1 ( ω ) , δ a 2 ( ω ) , δ a 2 ( ω ) ) T ,
B = ( 0 , ξ ( ω ) , 2 κ 1 a 1 , i n ( ω ) , 2 κ 1 a 2 , i n ( ω ) , 2 κ 2 a 2 , i n ( ω ) , 2 κ 1 a 1 , i n ( ω ) ) T .
δ a o u t ( ω ) = V ( ω ) ξ + E 1 ( ω ) a 1 , i n ( ω ) + E 2 ( ω ) a 2 , i n ( ω ) + F 1 ( ω ) a 1 , i n ( ω ) + F 2 ( ω ) a 2 , i n ( ω ) ,
V = 2 κ 1 g a 2 s M 2 J / R ,
E 1 = 2 κ 1 / N 1 J 2 2 κ 1 M 1 N 1 i ( g a 2 s J ) 2 M 2 2 κ 1 M 1 R ,
E 2 = 2 i κ 1 κ 2 J M 1 + 2 κ 1 κ 2 ( g a 2 s ) 2 N 1 J M 2 M 1 R ,
F 1 = i 2 κ 1 ( g a 2 s J ) 2 R ,
F 2 = 2 ( g a 2 s ) 2 κ 1 κ 2 N 2 J N 1 R ,
Δ ¯ = ω m 2 ω 2 i γ m ω ,
N 1 = κ 1 + i ( Δ 1 ω ) ,
N 2 = κ 1 i ( Δ 1 + ω ) ,
M 1 = [ κ 2 + i ( Δ 2 g q s ω ) ] [ κ 1 + i ( Δ 1 ω ) ] + J 2 ,
M 2 = [ κ 2 i ( Δ 2 g q s + ω ) ] [ κ 1 i ( Δ 1 + ω ) ] + J 2 ,
R = i g 2 a 2 s 2 N 1 M 2 + i g 2 a 2 s 2 N 2 M 1 + m Δ ¯ M 1 M 2 .
[ a l o * ( t ) a o u t ( t ) + c . c . ] [ a l o * ( t ) a o u t ( t ) + c . c . ] = a l o 2 2 π d ω e i ω ( t t ) X ( ω ) .
X ( ω ) = E 1 ( ω + ω m ) E 1 ( ω + ω m ) M Γ 2 Γ 2 + ω 2 + | E 1 ( ω + ω m ) | 2 ( N Γ 2 Γ 2 + ω 2 + 1 ) + | E 1 ( ω + ω m ) | 2 N Γ 2 Γ 2 + ω 2 + E 1 * ( ω + ω m ) E 1 * ( ω + ω m ) M Γ 2 Γ 2 + ω 2 + | F 1 ( ω + ω m ) | 2 ( 1 + N Γ 2 Γ 2 + ( ω 2 ω m ) 2 )   + | F 1 ( ω + ω m ) | 2 N Γ 2 Γ 2 + ( ω + ω m ) 2 + | E 2 ( ω + ω m ) | 2 ( 1 + N Γ 2 Γ 2 + ω 2 ) + | E 2 ( ω + ω m ) | 2 N Γ 2 Γ 2 + ω 2   + | F 2 ( ω + ω m ) | 2 ( 1 + N Γ 2 Γ 2 + ( ω 2 ω m ) 2 ) + | F 2 ( ω + ω m ) | 2 N Γ 2 Γ 2 + ( ω + 2 ω m ) 2   + 2 γ m π m ( ω + ω m ) [ 1 + coth   ( ( ω + ω m ) 2 κ B T | V ( ω + ω m ) | 2 ]   + 2 γ m π m ( ω ω m ) [ 1 + coth   ( ( ω ω m ) 2 κ B T | V ( ω m ω ) | 2 ] .

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