Abstract

Optical bistability has been studied theoretically in a multi-mode optomechanical system with two mechanical oscillators independently coupled to two coupled cavities. It is found that the multi-mode optomechanical system allows one to control the optical bistability in a much more flexible way. Specifically, the bistable behavior of the mean intracavity photon number in one cavity can be tuned by the strength and frequency of the pump laser beam driving another cavity. Meanwhile, it is also found that the coupling between the two cavities and the coupling between mechanical oscillators and cavities can effectively affect the optical bistability behavior in a sensitive manner. Moreover, the mechanical steady-state position exhibits clear bistability in the situation of relatively lower phonon number, and it exhibits more controllability as well. This investigation on optical bistability in multi-mode optomechanical systems will have promising applications in optical quantum computing and quantum information processing.

© 2020 Optical Society of America

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References

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2019 (3)

2017 (2)

Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25, 9697–9711 (2017).
[Crossref]

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

2016 (3)

2015 (3)

2014 (2)

2013 (1)

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

2012 (2)

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

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

2011 (3)

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

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

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

2010 (2)

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

S. M. Sadeghi, “Tunable nanoswitches based on nanoparticle meta-molecules,” Nanotechnology 21, 355501 (2010).
[Crossref]

2009 (2)

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

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

2008 (2)

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (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. A 77, 033804 (2008).
[Crossref]

2007 (4)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

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

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

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

2003 (3)

A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83, 1301–1303 (2003).
[Crossref]

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

2002 (1)

H. Wang, D. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2002).
[Crossref]

1996 (1)

H. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref]

Agarwal, G. S.

H. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref]

Akimov, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[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. A 77, 033804 (2008).
[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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Bai, C. H.

Bennett, S. D.

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

Berman, P. R.

P. R. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

Bian, X. T.

Bouwmeester, D.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

Brown, A.

A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83, 1301–1303 (2003).
[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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Camacho, R. M.

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

Casteels, W.

W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, “Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators,” Phys. Rev. A 93, 033824 (2016).
[Crossref]

Chan, J.

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

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

Chang, D. E.

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Chang, Y.

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

Chen, A. X.

Chen, G. B.

Chen, H. J.

Chen, Y. T.

Ciuti, C.

W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, “Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators,” Phys. Rev. A 93, 033824 (2016).
[Crossref]

Cui, C. L.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Cui, Y. S.

Demler, E. A.

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Dijkhuis, J. I.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Du, L.

Eichenfield, M.

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

Eisert, J.

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

Fu, C. B.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Fu, T. D.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Gao, X. Y.

Genes, C.

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. A 77, 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. A 77, 033804 (2008).
[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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Golubev, V. G.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2002).
[Crossref]

Gu, K. H.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Guan, S. Y.

Habraken, S. J. M.

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

Haghighi, I. M.

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[Crossref]

Harris, J. G. E.

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Harshawardhan, H.

H. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref]

Hartmann, M. J.

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
[Crossref]

Hill, J. T.

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

Hou, B. P.

Huang, J. F.

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

Jiang, C.

Joshi, A.

A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83, 1301–1303 (2003).
[Crossref]

Kerst, R.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

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

Kolkowitz, S. J.

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Komar, P.

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

Koppens, F. H. L.

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Kuang, L. M.

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

Kurdyukov, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Lai, D. G.

Lan, Y. H.

Le Boité, A.

W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, “Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators,” Phys. Rev. A 93, 033824 (2016).
[Crossref]

Li, G.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

Li, G. X.

Li, J.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[Crossref]

Li, Y.

L. Du, Y. T. Chen, Y. Li, and J. H. Wu, “Controllable optical response in a three-mode optomechanical system by driving the cavities on different sidebands,” Opt. Express 27, 21843–21855 (2019).
[Crossref]

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

Liao, J. Q.

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

Liu, Y. X.

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

Lukin, M. D.

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

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Malossi, N.

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Marshall, W.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

Mazurenko, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Nie, W. J.

Nooshi, N.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Nori, F.

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

Painter, O.

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

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

Pan, G. X.

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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Penrose, R.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

Pevtsov, A. B.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Plenio, M. B.

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
[Crossref]

Rabl, P.

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

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Sadeghi, S. M.

S. M. Sadeghi, “Tunable nanoswitches based on nanoparticle meta-molecules,” Nanotechnology 21, 355501 (2010).
[Crossref]

Safavi-Naeini, A. H.

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

Sarma, A. K.

Sarma, B.

Sel’kin, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Shi, T.

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

Simon, C.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

Soensen, A. S.

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

Sorensen, A. S.

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

Stannigel, K.

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

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

Storme, F.

W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, “Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators,” Phys. Rev. A 93, 033824 (2016).
[Crossref]

Sun, C. P.

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (2009).
[Crossref]

Tombesi, P.

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. A 77, 033804 (2008).
[Crossref]

Vahala, K. J.

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

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

Vitali, D.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[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. A 77, 033804 (2008).
[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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Wang, D. Y.

Wang, H.

H. Wang, D. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2002).
[Crossref]

Wang, H. F.

Wang, Z. D.

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

Wilson-Rae, I.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Wu, H. W.

Wu, J. H.

L. Du, Y. T. Chen, Y. Li, and J. H. Wu, “Controllable optical response in a three-mode optomechanical system by driving the cavities on different sidebands,” Opt. Express 27, 21843–21855 (2019).
[Crossref]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Wu, L. A.

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

Wu, S. P.

Xiao, M.

A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83, 1301–1303 (2003).
[Crossref]

H. Wang, D. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2002).
[Crossref]

Xiao, R. J.

Yan, X. B.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

Yang, J. Y.

Yang, Q.

Yang, Y. P.

Yi, Z.

Zhang, T.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

Zhao, D. M.

Zhou, L.

Zhu, A. D.

Zippilli, S.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[Crossref]

Zoller, P.

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

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Brown, A. Joshi, and M. Xiao, “Controlled steady-state switching in optical bistability,” Appl. Phys. Lett. 83, 1301–1303 (2003).
[Crossref]

J. Opt. Soc. Am. B (3)

Nanotechnology (1)

S. M. Sadeghi, “Tunable nanoswitches based on nanoparticle meta-molecules,” Nanotechnology 21, 355501 (2010).
[Crossref]

Nat. Commun. (1)

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

Nat. Phys. (2)

D. E. Chang, A. S. Soensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007).
[Crossref]

P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, “A quantum spin transducer based on nanoelectromechanical resonator arrays,” Nat. Phys. 6, 602–608 (2010).
[Crossref]

Nature (London) (1)

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

New J. Phys. (1)

J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17, 103037 (2015).
[Crossref]

Opt. Express (6)

Phys. Rev. A (10)

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95, 043819 (2017).
[Crossref]

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

J. Q. Liao, J. F. Huang, Y. X. Liu, L. M. Kuang, and C. P. Sun, “Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array,” Phys. Rev. A 80, 044301 (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. A 77, 033804 (2008).
[Crossref]

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

Y. Chang, T. Shi, Y. X. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83, 063826 (2011).
[Crossref]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: beyond the weak-cavity-field approximation,” Phys. Rev. A 87, 053841 (2013).
[Crossref]

H. Harshawardhan and G. S. Agarwal, “Controlling optical bistability using electromagnetic-field-induced transparency and quantum interferences,” Phys. Rev. A 53, 1812–1817 (1996).
[Crossref]

H. Wang, D. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A 65, 011801 (2002).
[Crossref]

W. Casteels, F. Storme, A. Le Boité, and C. Ciuti, “Power laws in the dynamic hysteresis of quantum nonlinear photonic resonators,” Phys. Rev. A 93, 033824 (2016).
[Crossref]

Phys. Rev. Lett. (6)

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

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

M. J. Hartmann and M. B. Plenio, “Steady state entanglement in the mechanical vibrations of two dielectric membranes,” Phys. Rev. Lett. 101, 200503 (2008).
[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,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref]

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91, 213903 (2003).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Other (1)

P. R. Berman, Cavity Quantum Electrodynamics (Academic, 1994).

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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of multi-mode optomechanical system where the two cavity modes, $ {a_1} $ and $ {a_2} $, are coupled by two mechanical modes $ {b_1} $ with frequency $ {\omega _{m,1}} $ and $ {b_2} $ with frequency $ {\omega _{m,2}} $, respectively. The two cavities are coupled with each other via a common waveguide with coupling intensity $ J $. The left cavity is driven by a strong pump beam $ {E_{\rm pu}} $ in the simultaneous presence of a weak probe beam $ {E_{\rm pr}} $, and the right cavity is driven by a control beam $ {E_{\rm co}} $.
Fig. 2.
Fig. 2. Mean intracavity photon number of the left cavity $ {n_{p,1}} $ as a function of $ {\Delta _1} $ with $ J = 2\pi \times 0.14\,\,{\rm GHz} $, $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ {E_{\rm pu}} $: $ {E_{\rm pu}} = 2\pi \times 0.0\,\,{\rm GHz} $, $ 2\pi \times 0.4\,\,{\rm GHz} $, and $ 2\pi \times 0.7\,\,{\rm GHz} $, respectively.
Fig. 3.
Fig. 3. Mean intracavity photon number of the right cavity $ {n_{p,2}} $ as a function of $ {\Delta _1} $ with $ J = 2\pi \times 0.14\,\,{\rm GHz} $, $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ {E_{\rm pu}} $: $ {E_{\rm pu}} = 2\pi \times 0.0\,\,{\rm GHz} $, $ 2\pi \times 0.4\,\,{\rm GHz} $, and $ 2\pi \times 0.7\,\,{\rm GHz} $, respectively.
Fig. 4.
Fig. 4. Mean intracavity photon number of the right cavity $ {n_{p,2}} $ as a function of $ {\Delta _1} $ with $ {E_{\rm pu}} = 2\pi \times 0.8\,\,{\rm GHz} $, $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $, $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $ for different values of $ J $: $ J = 2\pi \times 0.00\,\,{\rm GHz} $, $ 2\pi \times 0.10\,\,{\rm GHz} $, $ 2\pi \times 0.15\,\,{\rm GHz} $, and $ 2\pi \times 0.20\,\,{\rm GHz} $, respectively.
Fig. 5.
Fig. 5. Mean intracavity photon number of the left cavity $ {n_{p,1}} $ as a function of $ {E_{\rm pu}} $ with $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ J $: $ J = 2\pi \times 0.00\,\,{\rm GHz} $, $ 2\pi \times 0.10\,\,{\rm GHz} $, and $ 2\pi \times 0.15\,\,{\rm GHz} $, respectively.
Fig. 6.
Fig. 6. Mean intracavity photon number of the right cavity $ {n_{p,2}} $ as a function of $ {E_{\rm pu}} $ with $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ J $: $ J = 2\pi \times 0.00\,\,{\rm GHz} $, $ 2\pi \times 0.5\,\,{\rm GHz} $, $ 2\pi \times 0.10\,\,{\rm GHz} $, and $ 2\pi \times 0.15\,\,{\rm GHz} $, respectively.
Fig. 7.
Fig. 7. (a) Mechanical steady-state position in the left cavity $ {X_{1,s}} $ as a function of $ {E_{\rm pu}} $ with $ J = 2\pi \times 0.14\,\,{\rm GHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ {g_1} $: $ {g_1} = 2\pi \times 365\,\,{\rm kHz} $, $ 2\pi \times 465\,\,{\rm kHz} $, and $ 2\pi \times 565\,\,{\rm kHz} $, and $ 2\pi \times 965\,\,{\rm kHz} $, respectively. (b) Mechanical steady-state positions in the right cavity $ {X_{2,s}} $ as a function of $ {E_{\rm pu}} $ with $ J = 2\pi \times 0.14\,\,{\rm GHz} $, $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ {g_1} $: $ {g_1} = 2\pi \times 365\,\,{\rm kHz} $, $ 2\pi \times 465\,\,{\rm kHz} $, $ 2\pi \times 565\,\,{\rm kHz} $, and $ 2\pi \times 965\,\,{\rm kHz} $, respectively.
Fig. 8.
Fig. 8. Mechanical steady-state position in the right cavity $ {X_{2,s}} $ as a function of $ {E_{\rm pu}} $ with $ J = 2\pi \times 0.14\,\,{\rm GHz} $, $ {g_1} = 2\pi \times 965\,\,{\rm kHz} $, $ {\Delta _1} = 2\pi \times 2.1\,\,{\rm GHz} $, $ {\Delta _2} = 2\pi \times 2.1\,\,{\rm GHz} $, and $ {E_{\rm co}} = 2\pi \times 0.5\,\,{\rm GHz} $ for different values of $ {g_2} $: $ {g_2} = 2\pi \times 450\,\,{\rm kHz} $, $ 2\pi \times 451\,\,{\rm kHz} $, and $ 2\pi \times 452\,\,{\rm kHz} $, respectively.

Equations (6)

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H = l = 1 2 [ ω l a l a l + ω m , l b l b l g l a l a l ( b l + b l ) ] J ( a 1 a 2 + a 1 a 2 ) + H d r ,
H d r = i κ e 1 ( E p u e i ω p u t + E p r e i ω p r t ) a 1 + i κ e 2 E c o e i ω c o t a 2 + H . c . ,
H = l = 1 2 [ Δ l a l a l + ω m , l b l b l g l a l a l ( b l + b l ) ] J ( a 1 a 2 + a 1 a 2 ) + i κ e 1 ( E p u + E p r e i δ t ) a 1 + i κ e 2 E c o a 2 + H.c. ,
d a 1 d t = ( i Δ 1 + κ 1 2 ) a 1 + i g 1 a 1 X 1 + i J a 2 + κ e 1 ( E p u + E p r e i δ t ) + 2 κ 1 a i n , 1 , d a 2 d t = ( i Δ 2 + κ 2 2 ) a 2 + i g 2 a 2 X 2 + i J a 1 + κ e 2 E c o + 2 κ 2 a i n , 2 , d 2 X 1 d t 2 = γ 1 d X 1 d t ( ω m , 1 ) 2 X 1 + 2 ω m , 1 g 1 a 1 a 1 + ξ 1 , d 2 X 2 d t 2 = γ 2 d X 2 d t ( ω m , 2 ) 2 X 2 + 2 ω m , 2 g 2 a 2 a 2 + ξ 2 ,
n p , 1 = κ e 1 ( E p u ) 2 + J 2 n p , 2 ( κ 1 2 ) 2 + [ Δ 1 2 ( g 1 ) 2 ω m , 1 n p , 1 ] 2 , n p , 2 = κ e 2 ( E c o ) 2 + J 2 n p , 1 ( κ 2 2 ) 2 + [ Δ 2 2 ( g 2 ) 2 ω m , 2 n p , 2 ] 2 .
X 1 , s = 2 g 1 ω m , 1 | a 1 , s | 2 , X 2 , s = 2 g 2 ω m , 2 | a 2 , s | 2 .

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