Abstract

The absorption properties in double optomechanical cavities coupled by the photonic and phononic paths are investigated. The sideband absorption peaks and the transparency dips, located symmetrically around the resonant point, characterize the respective effects of the photonic and phononic interactions on the transparency spectra: the sideband absorption peaks move outward with increase of the photonic interaction due to the normal-mode splitting, and the distance between the two sideband transparency dips is determined by the phononic interaction strength. Meanwhile, the study reveals the competition and the harmonization between the photonic and phononic interactions. When simultaneously switching on the photonic and phononic paths, it is shown from the absorption spectra that the features of the interacting path with stronger strength dominate over the other one with weaker strength. Additionally, the transparency around the resonant point manifests the destructive interference between the photonic and phononic coupling paths when the strength of two paths are comparable. These findings display the flexible tunability of the transparency, and can be used to distinguish between the two quantum paths in the absorption spectrum.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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  5. S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
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  6. J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  27. Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
    [Crossref]
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    [Crossref]
  29. H. Li and M. Li, “Nano-optomechanical systems,” Nat. Nano. 9(11), 913–919 (2014).
    [Crossref]
  30. J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
    [Crossref] [PubMed]
  31. M. Bhattacharya and P. Meystre, “Multiple membrane cavity optomechanics,” Phys. Rev. A 78(4), 041801 (2008).
    [Crossref]
  32. A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
    [Crossref]
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    [Crossref]
  36. M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
    [Crossref]
  37. K. Fang, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photon. 10(7), 489–496 (2016).
    [Crossref]
  38. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
    [Crossref]
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    [Crossref]
  40. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
    [Crossref] [PubMed]
  41. Y. Q. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51(4), R2703–R2706 (1995).
    [Crossref] [PubMed]
  42. N. Mulchan, D. G. Ducreay, R. Pina, M. Yan, and Y. Zhu, “Nonlinear excitation by quantum interference in a doppler-broadened rubidium atomic system,” J. Opt. Soc. Am. B 17(5), 820–826 (2000).
    [Crossref]
  43. Q. Wang, J. Q. Zhang, P. Ma, C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
    [Crossref]
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    [Crossref] [PubMed]
  45. P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013)
    [Crossref]
  46. J. C. Howell and J. A. Yeazell, “Quantum computation through entangling single photons in multipath interferometers,” Phys. Rev. Lett. 85(1), 198–201 (2000).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2017 (3)

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
[Crossref]

Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25(9), 009697 (2017).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

2016 (8)

K. Fang, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photon. 10(7), 489–496 (2016).
[Crossref]

M. Pang, W. He, X. Jiang, and P. Russell, “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photon. 10(7), 454–458 (2016).
[Crossref]

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(4), 043804 (2016).
[Crossref]

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
[Crossref] [PubMed]

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
[Crossref]

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

Y. Yao, G. H. Dong, L. Ge, M. Li, and C. P. Sun, “Maximal coherence in a generic basis,” Phys. Rev. A 94(6), 062339 (2016).
[Crossref]

2015 (3)

T. Huan, R. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92(2), 022301 (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(3), 033829 (2015).
[Crossref]

Q. Wang, J. Q. Zhang, P. Ma, C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

2014 (5)

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

H. Li and M. Li, “Nano-optomechanical systems,” Nat. Nano. 9(11), 913–919 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

2013 (3)

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013)
[Crossref]

2012 (8)

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
[Crossref] [PubMed]

C. Xiong, X. Sun, K. Y. Fong, and H. X. Tang, “Integrated high frequency aluminum nitride optomechanical resonators,” Appl. Phys. Lett. 100(17), 171111 (2012).
[Crossref]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

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(1), 013603 (2012).
[Crossref] [PubMed]

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14(12), 125005 (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,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

C. A. Holmes, C. P. Meaney, and G. J. Milburn, “Synchronization of many nanomechanical resonators coupled via a common cavity field,” Phys. Rev. E 85(6), 066203 (2012).
[Crossref]

2011 (6)

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[Crossref]

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

A. Safavi-Naeini and O. Painter, “Proposal for an optomechanical travelling wave phonon-photon translator,” New J. Phys. 13(1), 013017 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref] [PubMed]

2010 (3)

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Riviére, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

M. Li, W. Pernice, and H. X. Tang, “Ultra-high-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97(18), 183110 (2010).
[Crossref]

2009 (1)

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

2008 (2)

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77(3), 033819 (2008).
[Crossref]

M. Bhattacharya and P. Meystre, “Multiple membrane cavity optomechanics,” Phys. Rev. A 78(4), 041801 (2008).
[Crossref]

2007 (2)

M. Bhattacharya and P. Meystre, “Trapping and Cooling a Mirror to Its Quantum Mechanical Ground State,” Phys. Rev. Lett. 99(7), 073601 (2007).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref] [PubMed]

2000 (2)

J. C. Howell and J. A. Yeazell, “Quantum computation through entangling single photons in multipath interferometers,” Phys. Rev. Lett. 85(1), 198–201 (2000).
[Crossref] [PubMed]

N. Mulchan, D. G. Ducreay, R. Pina, M. Yan, and Y. Zhu, “Nonlinear excitation by quantum interference in a doppler-broadened rubidium atomic system,” J. Opt. Soc. Am. B 17(5), 820–826 (2000).
[Crossref]

1995 (1)

Y. Q. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51(4), R2703–R2706 (1995).
[Crossref] [PubMed]

Agarwal, G. S.

K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Aksyuk, V.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Allman, M.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

Arcizet, O.

S. Weis, R. Riviére, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
[Crossref] [PubMed]

J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Bagheri, M.

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[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(4), 043804 (2016).
[Crossref]

Bennett, S. D.

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F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

Nori, Franco

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
[Crossref] [PubMed]

Özdemir, S. K.

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

Özdemir, Sahin Kaya

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
[Crossref] [PubMed]

Painter, O.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

K. Fang, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photon. 10(7), 489–496 (2016).
[Crossref]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

A. Safavi-Naeini and O. Painter, “Proposal for an optomechanical travelling wave phonon-photon translator,” New J. Phys. 13(1), 013017 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref] [PubMed]

Pang, M.

M. Pang, W. He, X. Jiang, and P. Russell, “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photon. 10(7), 454–458 (2016).
[Crossref]

Peng, B.

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
[Crossref] [PubMed]

Pernice, W.

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[Crossref]

M. Li, W. Pernice, and H. X. Tang, “Ultra-high-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97(18), 183110 (2010).
[Crossref]

Pina, R.

Plenio, M. B.

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
[Crossref] [PubMed]

Poot, M.

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[Crossref]

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,” Phys. Rev. Lett. 108(12), 120801 (2012).
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Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
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K. Qu and G. S. Agarwal, “Phonon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
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Rabl, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013)
[Crossref]

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(1), 013603 (2012).
[Crossref] [PubMed]

Rakher, M. T.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Retzker, A.

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
[Crossref] [PubMed]

Riviére, R.

S. Weis, R. Riviére, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 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,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Russell, P.

M. Pang, W. He, X. Jiang, and P. Russell, “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photon. 10(7), 454–458 (2016).
[Crossref]

Safavi Naeini, A. H.

J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Safavi-Naeini, A.

A. Safavi-Naeini and O. Painter, “Proposal for an optomechanical travelling wave phonon-photon translator,” New J. Phys. 13(1), 013017 (2011).
[Crossref]

Safavi-Naeini, A. H.

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref] [PubMed]

Schliesser, A.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

S. Weis, R. Riviére, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
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Schmidt, M.

M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14(12), 125005 (2012).
[Crossref]

Si, L. G.

Simmonds, R.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

Sirois, A.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

Sohail, A.

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
[Crossref]

Srinivasan, K.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Stannigel, K.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013)
[Crossref]

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(1), 013603 (2012).
[Crossref] [PubMed]

Sun, C. P.

Y. Yao, G. H. Dong, L. Ge, M. Li, and C. P. Sun, “Maximal coherence in a generic basis,” Phys. Rev. A 94(6), 062339 (2016).
[Crossref]

Sun, X.

C. Xiong, X. Sun, K. Y. Fong, and H. X. Tang, “Integrated high frequency aluminum nitride optomechanical resonators,” Appl. Phys. Lett. 100(17), 171111 (2012).
[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,” Phys. Rev. Lett. 108(12), 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,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Tang, H. X.

C. Xiong, X. Sun, K. Y. Fong, and H. X. Tang, “Integrated high frequency aluminum nitride optomechanical resonators,” Appl. Phys. Lett. 100(17), 171111 (2012).
[Crossref]

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[Crossref]

M. Li, W. Pernice, and H. X. Tang, “Ultra-high-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97(18), 183110 (2010).
[Crossref]

Teufel, J.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
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Tian, L.

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
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Usman, M.

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
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Uys, H.

M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77(3), 033819 (2008).
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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,” Phys. Rev. Lett. 108(12), 120801 (2012).
[Crossref] [PubMed]

Vanner, M. R.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Verhagen, E.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

Wang, C.

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, J. Q. Zhang, P. Ma, C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
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Wang, S. J.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
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Wang, T. J.

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
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Wei, L. F.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
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E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

S. Weis, R. Riviére, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Whittaker, J.

J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

Wieczorek, W.

S. Machnes, J. Cerrillo, M. Aspelmeyer, W. Wieczorek, M. B. Plenio, and A. Retzker, “Pulsed laser cooling for cavity optomechanical resonators,” Phys. Rev. Lett. 108(15), 153601 (2012).
[Crossref] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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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(4), 043804 (2016).
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Xiao, M.

Y. Q. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51(4), R2703–R2706 (1995).
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P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
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Xiong, C.

C. Xiong, X. Sun, K. Y. Fong, and H. X. Tang, “Integrated high frequency aluminum nitride optomechanical resonators,” Appl. Phys. Lett. 100(17), 171111 (2012).
[Crossref]

Xiong, H.

Yan, M.

Yang, D.

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
[Crossref] [PubMed]

Yang, L.

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

Yang, Lan

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
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Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25(9), 009697 (2017).
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Yang, X.

Yao, C.

Q. Wang, J. Q. Zhang, P. Ma, C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
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Yao, Y.

Y. Yao, G. H. Dong, L. Ge, M. Li, and C. P. Sun, “Maximal coherence in a generic basis,” Phys. Rev. A 94(6), 062339 (2016).
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J. C. Howell and J. A. Yeazell, “Quantum computation through entangling single photons in multipath interferometers,” Phys. Rev. Lett. 85(1), 198–201 (2000).
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Yu, C.

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
[Crossref]

Zhang, J.

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

Zhang, J. Q.

Q. Wang, J. Q. Zhang, P. Ma, C. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

Zhang, R.

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
[Crossref] [PubMed]

Zhang, Y.

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
[Crossref]

Zhang, Z. M.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
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T. Huan, R. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92(2), 022301 (2015).
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Zhu, Y.

Zoller, P.

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013)
[Crossref]

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(1), 013603 (2012).
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Ann. Phys. (1)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
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Appl. Phys. Lett. (2)

C. Xiong, X. Sun, K. Y. Fong, and H. X. Tang, “Integrated high frequency aluminum nitride optomechanical resonators,” Appl. Phys. Lett. 100(17), 171111 (2012).
[Crossref]

M. Li, W. Pernice, and H. X. Tang, “Ultra-high-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97(18), 183110 (2010).
[Crossref]

Eur. Phys. J. D. (1)

A. Sohail, Y. Zhang, M. Usman, and C. Yu, “Controllable optomechanically induced transparency in coupled optomechanical systems,” Eur. Phys. J. D. 71(4), 103 (2017).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Nat. Commun. (1)

B. Peng, Şahin Kaya Özdemir, W. J. Chen, Franco Nori, and Lan Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082–6082 (2014).
[Crossref] [PubMed]

Nat. Nano. (2)

M. Bagheri, M. Poot, M. Li, W. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nano. 6(11), 726–732 (2011).
[Crossref]

H. Li and M. Li, “Nano-optomechanical systems,” Nat. Nano. 9(11), 913–919 (2014).
[Crossref]

Nat. Photon. (3)

F. Monifi, J. Zhang, Ş. K. Özdemir, B. Peng, Y. X. Liu, F. Bo, F. Nori, and L. Yang, “Optomechanically induced stochastic resonance and chaos transfer between optical fields,” Nat. Photon. 10(6), 399–405 (2016).
[Crossref]

M. Pang, W. He, X. Jiang, and P. Russell, “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photon. 10(7), 454–458 (2016).
[Crossref]

K. Fang, M. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photon. 10(7), 489–496 (2016).
[Crossref]

Nat. Phys. (1)

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Nature (5)

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
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E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
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J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
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J. Chan, T. P. M. Alegre, A. H. Safavi Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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New J. Phys. (3)

A. Safavi-Naeini and O. Painter, “Proposal for an optomechanical travelling wave phonon-photon translator,” New J. Phys. 13(1), 013017 (2011).
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M. Schmidt, M. Ludwig, and F. Marquardt, “Optomechanical circuits for nanomechanical continuous variable quantum state processing,” New J. Phys. 14(12), 125005 (2012).
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Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
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Opt. Express (2)

Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25(9), 009697 (2017).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
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Opt. Lett. (1)

Phys. Rev. A (12)

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P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
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B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
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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(4), 043804 (2016).
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T. Huan, R. Zhou, and H. Ian, “Dynamic entanglement transfer in a double-cavity optomechanical system,” Phys. Rev. A 92(2), 022301 (2015).
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G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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M. Bhattacharya, H. Uys, and P. Meystre, “Optomechanical trapping and cooling of partially reflective mirrors,” Phys. Rev. A 77(3), 033819 (2008).
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Phys. Rev. E (1)

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Phys. Rev. Lett. (7)

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,” Phys. Rev. Lett. 108(12), 120801 (2012).
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Rev. Mod. Phys. (1)

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Sci. Rep. (1)

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6, 22920 (2016).
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Science (1)

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of the optomechanical systems composed of two cavities (â1, â2) and two mechanical resonators (1, 2). The coupling between both cavities (J) are due to the quantum tunnelling, and the interaction between two resonators (V) is mediated by the phononic crystal waveguides. Each cavity, driven by a strong control field (EL1, EL2) separately, is optomechanically coupled to the corresponding resonator by g1 or g2. γm1 and γm2 represent the damping rates of resonators, while κ1 and κ2 denote the noise imposed on cavities. The first cavity is probed by a weak field EP, and εout is the output field.
Fig. 2
Fig. 2 Absorption of the output field εR as a function of the normalized frequency σ/ωm1. The first optomechanical cavity is driven by a field of 3 mW, while the optomechanical interaction of the coupled (second) cavity is switched off. In part (a), the photonic tunnel coupling J between two cavities is set to 0, and the mechanical interactions are chosen to be V = 0.5V0, V0 and 2.0V0 where the parameter V0 is equal to 2π × 105 Hz, while in part (b), the mechanical interaction keeps at zero, and the photonic couplings are chosen to be J = 0.5κ, κ, and 2.0κ, as shown in the red solid, blue dashed dot, and green dashed curves, respectively.
Fig. 3
Fig. 3 Absorption of the output field εR as a function of the normalized frequency σ/ωm1, where the mechanical interaction is switched off. The red solid curve shows that both cavities are driven by a control field of 3 mW and decoupled with each other. The blue dash-dotted curve shows that the first cavity is bare optical cavity and the second cavity is driven by a control field of 3 mW, where two cavities are coupled by J = 0.5κ. The green dashed curve shows that two cavities are coupled by J = 0.5κ and driven by a control field of 3 mW.
Fig. 4
Fig. 4 Absorption of the output field εR as a function of the normalized frequency σ/ωm1 with driving field of both cavities taking the value of 3 mW. In part (a), the tunnel coupling is switched off and the mechanical interaction is chosen as V = 0.5V0, V0, and 2.0V0, and in part (b), the mechanical interaction is switched off and the photonic coupling is chosen as J = 0.5κ, κ, and 2.0κ, as shown in the red solid, blue dashed dot, and green dash curves, respectively.
Fig. 5
Fig. 5 Level diagram of a double N-type for a mechanically coupled optomechanical cavities, where a double Λ-type is shown in the dashed square.
Fig. 6
Fig. 6 Absorption of the output field εR as a function of the normalized frequency σ/ωm1 with the control fields driving the cavities set by the value of 3 mW, where the mechanical interactions are taken as V = 0.5V0, 1.5V0, and 2.0V0, as shown in the red solid, blue dashed-dotted and green dashed curves, respectively. The photonic coupling J is taken to be 0.5κ in (a), and 2.0κ in (b). The parameter V0 takes the same value as Fig. 2 (a).
Fig. 7
Fig. 7 Absorption of the output field εR as a function of the normalized frequency σ/ωm1, where both optomechanical cavities are driven by a control field of 3 mW, and the mechanical interaction is taken as a constant of V0. The red solid, blue dashed dot, and green dashed curves show those with the photonic interaction taken as J = 0.5κ, 1.0κ and 2.0κ.
Fig. 8
Fig. 8 Absorption of the output field εR as a function of the normalized frequency σ/ωm1 with the control field of the probing cavity switched off and that of the coupled cavity taking the value of 3 mW. The solid red curve shows the absorption with the mechanical coupling V = 0 and the tunnel coupling J = 2κ. The blue dashed-dotted curve gives that with V = 2V0 and J = 0, while the green dashed curve gives that with V = 2V0 and J = 2κ. The other parameters are the same as Fig. 2(a).

Equations (26)

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H ^ = i = 1 , 2 Δ ci a ^ i + a ^ i + J ( a ^ 1 + a ^ 2 + a ^ 1 a ^ 2 + ) + i = 1 , 2 ω mi b ^ i + b ^ i + V ( b ^ 1 + b ^ 2 + b ^ 1 b ^ 2 + ) + j = 1 , 2 g j ( b ^ j + + b ^ j ) a ^ j + a ^ j + i E L 1 ( a ^ 1 + a ^ 1 ) + i E L 2 ( a ^ 2 + a ^ 2 ) + i E P ( a ^ 1 + e i δ t a ^ 1 e i δ t ) ,
a ^ ˙ 1 = i Δ c 1 a ^ 1 i J a ^ 2 i g 1 ( b ^ 1 + + b ^ 1 ) a ^ 1 + E P e i δ t + E L 1 κ 1 a ^ 1 + 2 κ 1 a ^ in , 1 ,
a ^ ˙ 2 = i Δ c 2 a ^ 2 i J a ^ 1 i g 2 ( b ^ 2 + + b ^ 2 ) a ^ 2 + E L 2 κ 2 a ^ 2 + 2 κ 2 a ^ in , 2 ,
b ^ ˙ 1 = i ω m 1 b ^ 1 i V b ^ 2 i g 1 a ^ 1 + a ^ 1 γ m 1 b ^ 1 + ξ 1 ,
b ^ ˙ 2 = i ω m 2 b ^ 2 i V b ^ 1 i g 2 a ^ 2 + a ^ 2 γ m 2 b ^ 2 + ξ 2 ,
a s α = E L α ( i Δ β + κ β ) i J E L β ( i Δ α + κ α ) ( i Δ β + κ β ) + J 2 ,
b s α = V g β a s β * a s β i g α a s α * a s α ( i ω m β + γ m β ) ( i ω m α + γ m α ) ( i ω m β + γ m β ) + V 2 , α , β = 1 , 2 , α β ,
δ a ˙ 1 = ( i Δ c 1 + κ 1 ) δ a 1 i J δ a 2 + E P e i δ t i g 1 ( b s 1 * δ a 1 + b s 1 δ a 1 + δ b 1 + a s 1 + δ b 1 a s 1 ) ,
δ a ˙ 2 = ( i Δ c 2 + κ 2 ) δ a 2 i J δ a 1 i g 2 ( b s 2 * δ a 2 + b s 2 δ a 2 + δ b 2 + a s 2 + δ b 2 a s 2 ) ,
δ b ˙ 1 = i ω m 1 δ b 1 i V δ b 2 i g 1 ( a s 1 * δ a 1 + δ a 1 + a s 1 ) γ m 1 δ b 1 ,
δ b ˙ 2 = i ω m 2 δ b 2 i V δ b 1 i g 2 ( a s 2 * δ a 2 + δ a 2 + a s 2 ) γ m 2 δ b 2 ,
δ a i = a i + e i δ t + a i e i δ t , δ b i = b i + e i δ t + b i e i δ t .
ε out = 2 κ 1 a 1 + E P ,
a 1 + = Λ 1 + Λ 5 + Λ 6 + η 1 1 Λ 2 Λ 3 Λ 4 ,
Λ 1 = Ω 2 η 2 ( Ω 8 + Ω 9 Ω 11 ) Φ , Λ 2 = Ω 2 Ψ Φ , Λ 3 = Ω 1 [ Φ ( Ω 4 + Ω 6 Ω 10 ) + Ψ ( Ω 5 + Ω 6 Ω 12 ) ] Φ ( 1 Ω 6 Ω 11 ) , Λ 4 = Ω 3 [ Φ ( Ω 10 + Ω 4 Ω 11 ) + Ψ ( Ω 12 + Ω 5 Ω 11 ) ] Φ ( 1 Ω 6 Ω 11 ) , Λ 5 = Ω 1 η 2 [ ( Ω 8 + Ω 9 Ω 11 ) ( Ω 5 + Ω 6 Ω 12 ) + Φ ] Φ ( 1 Ω 6 Ω 11 ) , Λ 6 = Ω 3 η 2 [ ( Ω 8 + Ω 9 Ω 11 ) ( Ω 12 + Ω 5 Ω 11 ) + Ω 11 Φ ] Φ ( 1 Ω 6 Ω 11 ) ,
Ψ = ( 1 Ω 6 Ω 11 ) ( Ω 7 + Ω 9 Ω 10 ) + ( Ω 6 Ω 10 + Ω 4 ) ( Ω 8 + Ω 9 Ω 11 ) , Φ = ( 1 Ω 6 Ω 11 ) ( 1 Ω 9 Ω 11 ) ( Ω 8 + Ω 9 Ω 10 ) ( Ω 5 + Ω 6 Ω 12 ) .
Ω 1 = Γ 2 a s 2 * O 1 Y 1 Y 2 Γ 1 a s 1 * , Ω 2 = Γ 2 a s 1 O 1 Y 1 Y 2 Γ 1 a s 1 * , Ω 3 = Γ 2 a s 2 O 1 Y 1 Y 2 Γ 1 a s 1 * , Ω 4 = Γ 3 a s 1 * O 1 Y 1 Y 2 Γ 4 a s 2 * , Ω 5 = Γ 3 a s 1 O 1 Y 1 Y 2 Γ 4 a s 2 * , Ω 6 = Γ 4 a s 2 O 1 Y 1 Y 2 Γ 4 a s 2 * , Ω 7 = Γ 5 a s 1 * O 2 Y 1 Y 2 Γ 5 a s 1 , Ω 8 = Γ 6 a s 2 * O 2 Y 1 Y 2 Γ 5 a s 1 , Ω 9 = Γ 6 a s 2 O 2 Y 1 Y 2 Γ 5 a s 1 , Ω 10 = Γ 7 a s 1 * O 2 Y 1 Y 2 Γ 8 a s 2 , Ω 11 = Γ 8 a s 2 * O 2 Y 1 Y 2 Γ 8 a s 2 , Ω 12 = Γ 7 a s 1 O 2 Y 1 Y 2 Γ 8 a s 2 .
η 1 = Y 1 Y 2 L 2 E P ( O 1 Y 1 Y 2 Γ 1 a s 1 * ) , η 2 = i J E p Y 1 Y 2 ( O 1 Y 1 Y 2 Γ 4 a s 2 * ) .
Γ 1 = g 1 2 a s 1 ( Y 2 L 2 R 2 + Y 1 L 2 R 4 ) + J V g 1 g 2 a s 2 ( Y 2 + Y 1 ) , Γ 2 = i V g 1 g 2 a s 1 ( Y 2 L 2 + Y 1 L 2 ) i J g 2 2 a s 2 ( Y 2 R 1 + Y 1 R 3 ) , Γ 3 = i J g 1 2 a s 1 ( Y 2 R 2 + Y 1 R 4 ) + i V g 1 g 2 a s 2 ( Y 2 L 1 + Y 1 L 1 ) , Γ 4 = J V g 1 g 2 a s 1 ( Y 2 + Y 1 ) + g 2 2 a s 2 ( Y 2 L 1 R 1 + Y 1 L 1 R 3 ) , Γ 5 = g 1 2 a s 1 * ( Y 2 L 4 R 2 + Y 1 R 4 L 4 ) + J V g 1 g 2 a s 2 * ( Y 2 + Y 1 ) , Γ 6 = i V g 1 g 2 a s 1 * ( Y 2 L 4 + Y 1 L 4 ) i J g 2 2 a s 2 * ( Y 2 R 1 + Y 1 R 3 ) , Γ 7 = i J g 1 2 a s 1 * ( Y 2 R 2 + Y 1 R 4 ) i V g 1 g 2 a s 2 * ( Y 2 L 3 + Y 1 L 3 ) , Γ 8 = J V g 1 g 2 a s 1 * ( Y 2 + Y 1 ) g 2 2 a s 2 * ( Y 2 L 3 R 1 + Y 1 L 3 R 3 ) .
Y 1 = R 1 R 2 + V 2 , Y 2 = R 3 R 4 + V 2 .
O 1 = L 1 + L 2 + J 2 , O 2 = L 3 L 4 + J 2 .
L 1 = i Δ 1 + κ 1 i δ , L 2 = i Δ 2 + κ 2 i δ , L 3 = κ 1 i δ i Δ 1 , L 4 = κ 2 i δ i Δ 2 .
R 1 = i ω m 1 i δ + γ m 1 , R 2 = i ω m 2 i δ + γ m 2 , R 3 = γ m 1 i δ + i ω m 1 , R 4 = γ m 2 i δ i ω m 2 .
a 1 + = E p κ 1 i σ + G 1 2 γ m 1 2 i σ + J 2 κ 2 i σ + G 2 2 γ m 2 2 i σ ,
b ^ ± = 1 2 ( b ^ 1 b ^ 2 ) ,
H ^ = Δ c 1 a ^ 1 + a ^ 1 + Δ c 2 a ^ 2 + a ^ 2 + ( ω m + V ) b ^ + b ^ + ( ω m V ) b ^ + + b ^ + + 2 ( b ^ + b ^ + ) ( g 1 a ^ 1 + a ^ 1 + g 2 a ^ 2 + a ^ 2 ) + 2 ( b ^ + + b ^ + + ) ( g 1 a ^ 1 + a ^ 1 g 2 a ^ 2 + a ^ 2 ) + i E L 1 ( a ^ 1 + a ^ 1 ) + i E L 2 ( a ^ 2 + a ^ 2 ) ,

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