Abstract

We study the controllable optical response in a three-mode optomechanical system comprised of two indirectly coupled cavity modes and an intermediate mechanical mode. The two cavity modes are assumed to have different frequencies and driven by two control fields on the red and blue sidebands, respectively. When the system is perturbed by two probe fields satisfying the specific matching condition, a series of intriguing phenomena can be observed by adjusting phases and amplitudes of the control fields, such as absorption-amplification switching, ultra-narrow response windows, frequency-independent perfect reflection, and ultralong optical group delay. We also compare our system with conventional optomechanical systems to highlight its distinct features. Our results may have potential applications in optical communication and quantum information processing.

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

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2019 (3)

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

2018 (7)

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

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

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

2017 (7)

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
[Crossref]

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

2016 (3)

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

2015 (5)

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

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

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

2014 (4)

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

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

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

X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

2013 (7)

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

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

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

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

2012 (2)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
[Crossref] [PubMed]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

2011 (3)

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, 69–73 (2011).
[Crossref] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

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

2010 (3)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

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

2008 (4)

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

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

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

2007 (2)

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

2005 (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

2003 (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

2001 (1)

2000 (1)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

1999 (3)

E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999).
[Crossref]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

1998 (1)

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

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

Akulshin, A. M.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[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, 69–73 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Amitai, E.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

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,” Phys. Rev. Lett. 110, 071105 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

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

Bai, C.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Barreiro, S.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

Barzanjeh, S.

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

Bernier, N. R.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Bin, Q.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Bruder, C.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[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]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[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, 69–73 (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, 69–73 (2011).
[Crossref] [PubMed]

Chen, A.-X.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Chen, R.-X.

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

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Clerk, A. A.

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

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

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

Cui, C.-L.

DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

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]

Didier, N.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Du, L.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Eichenfield, 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, 69–73 (2011).
[Crossref] [PubMed]

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]

Fan, C.-H.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Farace, A.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Fazio, R.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Feofanov, A. K.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

Fu, C.-B.

Gao, J.-Y.

Gavartin, E.

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]

Geraci, A. A.

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

Gigan, S.

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

Giovannetti, V.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Girvin, S. M.

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

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Gross, R.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Gu, K.-H.

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Hafezi, M.

Han, Y.

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

Harlow, J. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Harris, J. G. E.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

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

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, 69–73 (2011).
[Crossref] [PubMed]

Hocke, F.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Hofheinz, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Hou, B. P.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

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

Huang, S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

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

Huang, Y. Y.

Huebl, H.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Jacobs, K.

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

Jayich, A. M.

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

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Jia, W. Z.

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

Jiang, B.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

Jiang, C.

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

Jing, H.

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Kaufman, C.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Kessler, S.

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.

Kippenberg, T. J.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

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

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[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]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Lai, D. G.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Lehnert, K. W.

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J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (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, 69–73 (2011).
[Crossref] [PubMed]

Wong, C. W.

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

Wu, D.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Wu, J.-H.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
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X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
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Wu, Y.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

Xiao, Y.-F.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
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Xie, H.

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

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W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

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

Xu, X.-W.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

Yan, X.-B.

Yang, C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
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Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
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Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Yu, J.

Yuan, P.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Yuan, S.

Zhang, H.-X.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
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Zhang, X. Z.

Zhang, Y.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
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Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
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Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
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[Crossref]

Zheng, L.-L.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
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Zhou, X.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
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Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
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Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Europhys. Lett. (1)

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
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J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
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Nat. Photonics (1)

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Optica (1)

Phys. Lett. A (1)

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
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Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
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Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
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X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
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C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
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X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
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X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
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L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
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Sci. Rep. (1)

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
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M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).

D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).

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Figures (8)

Fig. 1
Fig. 1 (a) Fabry-Pérot implementation of our three-mode optomechanical system. The intermediate membrane oscillator with damping rate γm is linearly coupled with two optical cavities simultaneously. The left (right) cavity mode with decay rate κ is driven by a control field with amplitude εc1 (εc2), frequency ωc1 (ωc2), and phase ϑ1 (ϑ2). We also employ a probe field with amplitude εp1 (εp2) and frequency ωp1 (ωp2) to perturb the left (right) cavity mode. Phases of the two probe fields, however, have been assumed to be vanishing without the loss of generality. (b) Effective energy level configuration exhibiting the closed-loop transition for our three-mode optomechanical system. Relevant states of the left cavity, intermediate mechanical, and right cavity modes are denoted by excitation numbers na1, nb, and na2 in order.
Fig. 2
Fig. 2 Stability diagrams attained with G1 = κ (a) and θ = 0 (b) in the case of Δ′1 = −Δ′2 = ωm and γm = 10−4κ. The yellow and blue regions denote the stable and unstable regimes, respectively.
Fig. 3
Fig. 3 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = 0.
Fig. 4
Fig. 4 Profiles of reflectivity RL with η = 0.4 (a), G = κ (b), and η = 0 (c). (d) Effective mechanical damping rates against G with η = 0.9 (red) or against η with G = κ (blue) in the case of σ = 0, where the solid, dashed and dot-dashed lines correspond to Γopt,1, Γopt,2 and Γopt,3, respectively. Other parameters are γm = 10−4κ and θ = 0 if not given in relevant panels.
Fig. 5
Fig. 5 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = π.
Fig. 6
Fig. 6 Profiles of reflectivity RL for different values of θ. Relevant parameters are γm = 10−4κ, G = κ, and η = 0.9 except θ specified in the figure.
Fig. 7
Fig. 7 Profiles of optical group delay τL (a, c) and τ′L (b, d). Relevant parameters are the same as in Fig. 5 except G specified for panels (a, b) and η specified for panels (c, d) in the figure.
Fig. 8
Fig. 8 Profiles of optical group delay τL (a) and left output phase ϕ (b). Relevant parameters are γm = 10−4κ, G = κ, and η = 0.98 except θ specified for panels (a, b) in the figure.

Equations (21)

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= ω m b b + j = 1 , 2 [ ω j a j a j + ( 1 ) j g j a j a j ( b + b ) + i ( ε c j a j e i ϑ j e i ω c j t + ε p a j e i ω p j t H . c . ) ] ,
= ω m b b + j = 1 , 2 [ Δ j a j a j + ( 1 ) j g j a j a j ( b + b ) + i ( ε c j a j e i ϑ j + ε p a j e i δ j t H . c . ) ] ,
a ˙ 1 = ( κ + i Δ 1 ) a 1 + i g 1 a 1 ( b + b ) + ε c 1 e i ϑ 1 + ε p e i δ 1 t + 2 κ a 1 in , a ˙ 2 = ( κ + i Δ 2 ) a 2 + i g 2 a 2 ( b + b ) + ε c 2 e i ϑ 2 + ε p e i δ 2 t + 2 κ a 2 in , b ˙ = ( γ m + i ω m ) b + i ( g 1 a 1 a 1 g 2 a 2 a 2 ) + 2 γ m b in ,
α 1 s = ε c 1 e i ϑ 1 κ + i Δ 1 , α 2 s = ε c 2 e i ϑ 2 κ + i Δ 2 , β s = i ( g 1 | α 1 s | 2 g 2 | α 2 s | 2 ) γ m + i ω m
δ a 1 . = ( κ + i Δ 1 ) δ a 1 + i G 1 ( δ b + δ b ) + ε p e i δ 1 t + 2 κ a 1 in , δ a 2 . = ( κ + i Δ 2 ) δ a 2 i G 2 ( δ b + δ b ) + ε p e i δ 2 t + 2 κ a 2 in , δ b . = ( γ m + i ω m ) δ b + i ( G 1 * δ a 1 + G 1 δ a 1 G 2 * δ a 2 G 2 δ a 2 ) + 2 γ m b in ,
δ a 1 . = ( κ i Δ 1 ) δ a 1 + i G 1 * ( δ b + δ b ) + ε p e i δ 1 t + 2 κ a 1 in , δ a 2 . = ( κ i Δ 2 ) δ a 2 + i G 2 * ( δ b + δ b ) + ε p e i δ 2 t + 2 κ a 2 in , δ b . = ( γ m i ω m ) δ b i ( G 1 * δ a 1 + G 1 δ a 1 G 2 * δ a 2 G 2 δ a 2 ) + 2 γ m b in .
δ a 1 δ a 1 e i Δ 1 t , δ a 1 in δ a 1 in e i Δ 1 t , δ a 2 δ a 2 e i Δ 2 t , δ a 2 in δ a 2 in e i Δ 2 t , δ b δ b e i ω m t , δ b in δ b in e i ω m t ,
δ a 1 . = κ δ a 1 + i G 1 δ b + ε p e i σ 1 t + 2 κ a 1 in , δ a 2 . = κ δ a 2 + i G 2 * δ b + ε p e i σ 2 t + 2 κ a 2 in , δ b . = γ m δ b + i ( G 1 * δ a 1 G 2 δ a 2 ) + 2 γ m b in ,
δ a 1 . = κ δ a 1 + i G δ b + ε p e i σ 1 t + 2 κ a 1 in , δ a 2 . = κ δ a 2 + i η G e i θ δ b + ε p e i σ 2 t + 2 κ a 2 in , δ b . = γ m δ b + i G ( δ a 1 η e i θ δ a 2 ) + 2 γ m b in
u ˙ = M u + f + n ,
M = ( κ 0 i G 0 κ i η G e i θ i G i η G e i θ γ m ) .
A 1 + = f 1 f 2 + η G 2 ( e i θ η ) f 1 [ f 1 f 2 + G 2 ( 1 η 2 ) ] ε p , A 2 + = 0
ε outL = 2 κ δ a 1 ε p e i σ t ,
ε L + = 2 κ A 1 + ε p ,
A 1 + = f 1 f 2 + η G 2 ( e i θ + η ) f 1 [ f 1 f 2 + G 2 ( 1 + η 2 ) ] ε p ,
( γ m i σ + | G 1 | 2 | G 2 | 2 κ i σ ) B + = i ( G 1 * G 2 ) κ i σ ε p
Γ opt , 1 = γ m + κ G 2 ( 1 η 2 ) κ 2 + σ 2 .
Γ opt , 2 = γ m + κ G 2 κ 2 + σ 2 ,
Γ opt , 3 = γ m + κ G 2 ( 1 + η 2 ) κ 2 + σ 2 ,
τ L = d ϕ d ω p 1 ,
τ L = d ϕ d ω p

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