Abstract

The cross-Kerr effects between the cavity and the mechanical oscillator in a parity-time symmetric optomechanical system are investigated. It is found that in the double-passive case there appears an asymmetric optomechanically induced transparency (OMIT) spectrum which is composed of a broad absorption peak located around the resonant point and a absorption line at the frequency position mainly determined by the Kerr interaction. The distinctive asymmetry induced by the cross-Kerr coupling is precisely demonstrated by the analytic findings. In the passive-active case, the resonance peaks in the OMIT spectrum are increased with the weak tunnel coupling, which is contrary to that in the double-passive case. When the tunnel coupling is increased up in the strong coupling region, the broad absorption peak and the absorption line in the OMIT spectrum are sequentially changed into the amplification ones, and the central amplification dip is split into two parts due to the normal mode splitting induced by the strong tunnel coupling. This can be used to realize a switching from absorption to amplification by only adjusting the tunnel interaction.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95(3), 033820 (2017).
[Crossref]

Q. Yang, B. P. Hou, and D. G. Lai, “Local modulation of double optomechanically induced transparency and amplification,” Opt. Express 25(9), 9697–9711 (2017).
[Crossref] [PubMed]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752 (2017).
[Crossref]

X. Y. Zhang, Y. Q. Guo, P. Pei, and X. X. Yi, “Optomechanically induced absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 95(6), 063825 (2017).
[Crossref]

J.-S. Zhang, W. Zeng, and A.-X. Chen, “Effects of cross-Kerr coupling and parametric nonlinearity on normal mode splitting, cooling, and entanglement in optomechanical systems,” Quantum Inf. Process 16(6), 163 (2017).
[Crossref]

2016 (4)

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
[Crossref] [PubMed]

W. Xiong, D.-Y. Jin, Y. Qiu, C.-H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93(2), 023844 (2016).
[Crossref]

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

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

2015 (10)

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(3), 11278 (2015).
[Crossref] [PubMed]

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(4), 043843 (2015).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

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]

J.-M. Pirkkalainen, S.U. Cho, F. Massel, J. Tuorila, T.T. Heikkilä, P.J. Hakonen, and M.A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref] [PubMed]

R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
[Crossref]

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

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92(7), 043830 (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] [PubMed]

2014 (5)

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric phonon laser with gain and loss,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” Phys. Rev. Lett. 112(20), 203603 (2014).
[Crossref]

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]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

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

2013 (6)

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

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

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref] [PubMed]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3(42), 2943 (2013).
[Crossref] [PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2013).
[Crossref]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

2012 (5)

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

H. Xiong, L.-G. Si, A.-S. Zheng, X.-X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[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]

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]

2011 (7)

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, and M. S. Allman, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

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

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer Alegre, and J. Chan, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6064), 729 (2011).
[Crossref] [PubMed]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84(4), 040101 (2011).
[Crossref]

2010 (4)

C. E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 47 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Normal mode splitting and antibunching in stokes and anti-stokes processes in cavity optomechanics: radiation pressure induced four-wave mixing cavity optomechanics,” Phys. Rev. A 81(3), 033830 (2010).
[Crossref]

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105(10), 101101 (2010).
[Crossref] [PubMed]

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

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

2007 (3)

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

D. Vitali, S. Gigan, and A. Ferreira, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

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]

2004 (1)

M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref] [PubMed]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88(12), 120401 (2002).
[Crossref] [PubMed]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
[Crossref]

Agarwal, G. S.

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

G. S. Agarwal and S. Huang, “Normal mode splitting and antibunching in stokes and anti-stokes processes in cavity optomechanics: radiation pressure induced four-wave mixing cavity optomechanics,” Phys. Rev. A 81(3), 033830 (2010).
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J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(3), 11278 (2015).
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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]

H. Xiong, L.-G. Si, A.-S. Zheng, X.-X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Xiong, W.

W. Xiong, D.-Y. Jin, Y. Qiu, C.-H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93(2), 023844 (2016).
[Crossref]

Xu, X.-W.

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

Xu, Y.

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Xu, Y. L.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6064), 729 (2011).
[Crossref] [PubMed]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Yang, L.

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

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric phonon laser with gain and loss,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2013).
[Crossref]

Yang, Q.

Yang, X.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(3), 11278 (2015).
[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]

Yang, X.-X.

H. Xiong, L.-G. Si, A.-S. Zheng, X.-X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Yi, X. X.

X. Y. Zhang, Y. Q. Guo, P. Pei, and X. X. Yi, “Optomechanically induced absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 95(6), 063825 (2017).
[Crossref]

You, C.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(3), 11278 (2015).
[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]

You, J. Q.

W. Xiong, D.-Y. Jin, Y. Qiu, C.-H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93(2), 023844 (2016).
[Crossref]

Zeng, W.

J.-S. Zhang, W. Zeng, and A.-X. Chen, “Effects of cross-Kerr coupling and parametric nonlinearity on normal mode splitting, cooling, and entanglement in optomechanical systems,” Quantum Inf. Process 16(6), 163 (2017).
[Crossref]

Zhan, X.

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92(7), 043830 (2015).
[Crossref]

Zhang, D.

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92(7), 043830 (2015).
[Crossref]

Zhang, J.

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

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric phonon laser with gain and loss,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]

Zhang, J. Q.

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Zhang, J.-S.

J.-S. Zhang, W. Zeng, and A.-X. Chen, “Effects of cross-Kerr coupling and parametric nonlinearity on normal mode splitting, cooling, and entanglement in optomechanical systems,” Quantum Inf. Process 16(6), 163 (2017).
[Crossref]

Zhang, W. M.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3(42), 2943 (2013).
[Crossref] [PubMed]

Zhang, X. Y.

X. Y. Zhang, Y. Q. Guo, P. Pei, and X. X. Yi, “Optomechanically induced absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 95(6), 063825 (2017).
[Crossref]

Zheng, A.-S.

H. Xiong, L.-G. Si, A.-S. Zheng, X.-X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84(4), 040101 (2011).
[Crossref]

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Ann. Phys. (1)

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

Nat. Commun. (1)

J.-M. Pirkkalainen, S.U. Cho, F. Massel, J. Tuorila, T.T. Heikkilä, P.J. Hakonen, and M.A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6, 6981 (2015).
[Crossref] [PubMed]

Nat. Mater. (1)

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Nat. Photonics (1)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752 (2017).
[Crossref]

Nat. Phys. (2)

C. E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 47 (2010).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2013).
[Crossref]

Nature (3)

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, and M. S. Allman, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
[Crossref] [PubMed]

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, 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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A. H. Safavi-Naeini, T. P. Mayer Alegre, and J. Chan, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (14)

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in cavity optomechanics,” Phys. Rev. A 87(3), 031802 (2013).
[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(4), 043843 (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]

H. Xiong, L.-G. Si, A.-S. Zheng, X.-X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
[Crossref]

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95(3), 033820 (2017).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, “Enhanced nonlinear optics in coupled optical microcavities with an unbroken and broken parity-time symmetry,” Phys. Rev. A 92(7), 043830 (2015).
[Crossref]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Normal mode splitting and antibunching in stokes and anti-stokes processes in cavity optomechanics: radiation pressure induced four-wave mixing cavity optomechanics,” Phys. Rev. A 81(3), 033830 (2010).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84(4), 040101 (2011).
[Crossref]

X.-W. Xu, Y.-X. Liu, C.-P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
[Crossref]

X. Y. Zhang, Y. Q. Guo, P. Pei, and X. X. Yi, “Optomechanically induced absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 95(6), 063825 (2017).
[Crossref]

R. Khan, F. Massel, and T. T. Heikkilä, “Cross-Kerr nonlinearity in optomechanical systems,” Phys. Rev. A 91, 043822 (2015).
[Crossref]

W. Xiong, D.-Y. Jin, Y. Qiu, C.-H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93(2), 023844 (2016).
[Crossref]

Phys. Rev. Lett. (15)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

T. T. Heikkilä, F. Massel, J. Tuorila, R. Khan, and M. A. Sillanpää, “Enhancing optomechanical coupling via the Josephson effect,” Phys. Rev. Lett. 112(20), 203603 (2014).
[Crossref]

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

A. A. Geraci, S. B. Papp, and J. Kitching, “Short-range force detection using optically cooled levitated microspheres,” Phys. Rev. Lett. 105(10), 101101 (2010).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

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]

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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S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88(12), 120401 (2002).
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D. Vitali, S. Gigan, and A. Ferreira, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
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H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric phonon laser with gain and loss,” Phys. Rev. Lett. 113, 053604 (2014).
[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] [PubMed]

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT symmetry and spontaneous symmetry breaking in a microwave billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
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C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
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M. Bhattacharya and P. Meystre, “Trapping and cooling a mirror to its quantum mechanical ground state,” Phys. Rev. Lett. 99(7), 073601 (2007).
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Quantum Inf. Process (1)

J.-S. Zhang, W. Zeng, and A.-X. Chen, “Effects of cross-Kerr coupling and parametric nonlinearity on normal mode splitting, cooling, and entanglement in optomechanical systems,” Quantum Inf. Process 16(6), 163 (2017).
[Crossref]

Rev. Mod. Phys. (1)

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

Sci. China Phys. Mech. Astron. (1)

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

Sci. Rep. (4)

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3(42), 2943 (2013).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(3), 11278 (2015).
[Crossref] [PubMed]

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

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6, 31095 (2016).
[Crossref] [PubMed]

Science (3)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6064), 729 (2011).
[Crossref] [PubMed]

S. Weis, R. Rivière, 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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M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74–77 (2004).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of the two-cavity-coupled optomechanical system. In the passive cavity with a loss rate κ1, the cavity field is coupled to the mechanical oscillator not only by optomechanical interaction but by the cross-Kerr coupling mediated by a superconducting charge qubit. Additionally, the left passive cavity is directly coupled to the right active cavity with a gain rate κ2. The passive cavity is driven by a coupling field L and a probe field εP, respectively.
Fig. 2
Fig. 2 The absorption εR as a function of the redefined detuning σ = (δωm)/ωm for different values of the weak tunnel coupling (J < |κ2 + κ1|/2) (a): J = 0.5κ1(black, solid curve); J = κ1(red, dashed curve); J = 2κ1(blue, dotted curve). In the case of the strong tunnel coupling (J > |κ2 + κ1|/2), the tunnel coupling J is given by different values as (b): J = 3κ1(black, solid curve); J = 5κ1(red, dashed curve); J = 7κ1(blue, dotted curve). The other values of the parameters are given by: PL = 3mW, g = 250, ωc1/2π = 1.3GHz, κ1/2π = 0.1MHz, κ2 = −5κ1, ωm/2π = 6.3MHz, γm = 40Hz and G = 0.
Fig. 3
Fig. 3 The absorption εR as a function of the redefined detuning σ = (δωm)/ωm for different values of the cross-Kerr parameter G : G = 0.0005g (black, solid curve); G = 0.001g (red, dashed curve); G = 0.002g (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2 except for J = 0.
Fig. 4
Fig. 4 The absorption εR as a function of the redefined detuning σ = (δωm)/ωm with G = 0.001g and for different values of the weak tunnel coupling (J < |κ2 + κ1|/2) (a): J = κ1(black, solid curve); J = 1.6κ1(red, dashed curve); J = 2κ1(blue, dotted curve). In the case of the strong tunnel coupling for J > |κ2 + κ1|/2, the tunnel coupling J is given by different values as (b): J = 3κ1(black, solid curve); J = 3.6κ1(red, dashed curve); J = 5κ1(blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 5
Fig. 5 The absorption εR as a function of the redefined detuning σ = (δωm)/ωm in the passive-active case (κ1/2π = 0.1MHz, κ2 = 5κ1). The weak tunnel coupling (J < |κ2+κ1|/2) is given by different values (a): J = κ1(black, solid curve); J = 1.6κ1(red, dashed curve); J = 2κ1(blue, dotted curve). The strong tunnel coupling (J > |κ2 + κ1|/2) is given by different values (b): J = 4κ1(black, solid curve); J = 6κ1(red, dashed curve); J = 8κ1(blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 6
Fig. 6 The same settings are used as those in Fig. 3 except for the presence of tunnel coupling (J = 2κ1) and in the passive-active case(κ2 = 5κ1).
Fig. 7
Fig. 7 The same settings are used as those in Fig. 4(a) except for the presence of cross-Kerr effect (G = 0.001g).
Fig. 8
Fig. 8 The absorption εR as a function of the redefined detuning σ = (δωm)/ωm with G = 0.001g in the passive-active case. The strong tunnel coupling (J > |κ2 + κ1|/2) is given by different values: (a) J = 3κ1(black, solid curve); (b) J = 3.6κ1(red, dashed curve); (c) J = 5κ1(blue, dotted curve).The same settings are used as those in Fig. 4(b) except for the presence of cross-Kerr effect (G = 0.001g).

Equations (18)

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H = Δ 1 a ^ 1 + a ^ 1 + Δ 2 a ^ 2 + a ^ 2 + ω m b ^ + b ^ + g a ^ 1 + a ^ 1 ( b ^ + + b ^ ) + G a ^ 1 + a ^ 1 b ^ + b ^ + J ( a ^ 1 + a ^ 2 + a ^ 1 a ^ 2 + ) + i ε L ( a ^ 1 + a ^ 1 ) + i ε p ( a ^ 1 + e i δ t a ^ 1 e i δ t ) .
a ^ ˙ 1 = i Δ 1 a ^ 1 i g a ^ 1 ( b ^ + b ^ ) i G a ^ 1 b ^ + b ^ i J a ^ 2 + ε L + ε p e i δ t κ 1 a ^ 1 + 2 κ 1 a ^ in , 1 .
a ^ ˙ 2 = i Δ 2 a ^ 2 i J a ^ 1 + κ 2 a ^ 2 + 2 | κ 2 | a ^ in , 2 .
b ^ ˙ = i ω m b ^ i g a ^ 1 + a ^ 1 i G a ^ 1 + a ^ 1 b ^ γ m b ^ + ξ ^ .
a s 1 = ε L ( κ 2 i Δ 2 ) ( κ 1 + i Δ 111 ) ( κ 2 i Δ 2 ) J 2 , a s 2 = i J ε L ( κ 1 + i Δ 111 ) ( κ 2 i Δ 2 ) J 2 , b s = i g a s 1 * a s 1 γ m + i Ω m .
δ a ˙ 1 = ( i Δ 111 + κ 1 ) δ a 1 i g a s 1 ( δ b + δ b * ) i G a s 1 ( b s * δ b + b s δ b * ) i J δ a 2 + ε p e i δ t .
δ a ˙ 2 = ( i Δ 2 κ 2 ) δ a 2 i J δ a 1 .
δ b ˙ = ( i ω m + γ m ) δ b i g ( a s 1 * δ a 1 + a s 1 δ a 1 * ) i G b s ( a s 1 * δ a 1 + a s 1 δ a 1 * ) i G a s 1 * a s 1 δ b .
δ a 1 = a 1 + e i δ t + a 1 e i δ t , δ a 2 = a 2 + e i δ t + a 2 e i δ t , δ b = b + e i δ t + b e i δ t .
a 1 + = ε p ( Q 21 M + ) | Q 1 Q 2 | 2 N 12 2 + ( M + Q 12 ) ( M + Q 21 ) ,
N 12 = 1 N + 1 N + , Q 12 = | Q 1 | 2 N | Q 2 | 2 N + + J 2 H , Q 21 = | Q 2 | 2 N | Q 1 | 2 N + J 2 H + .
H ± = κ 2 ± i Δ 2 + i δ , Q 1 = i g a s 1 + i G a s 1 b s * , Q 2 = i g a s 1 + i G a s 1 b s , M ± = κ 1 ± i Δ 111 + i δ , N ± = ± i ω m ± i G a s 1 a s * γ m + i δ .
δ a 1 = ε p κ 1 i σ | g a s 1 G b s * a s 1 | 2 i ( σ G a s 1 * a s 1 ) γ m .
σ L = G a s 1 * a s 1 | g a s 1 G b s * a s 1 | 2 | G a s 1 * a s 1 | 2 + ( κ 1 γ m ) 2 , σ R = G a s 1 * a s 1 + G a s 1 * a s 1 | g a s 1 G b s * a s 1 | 2 | G a s 1 * a s 1 | 2 + ( κ 1 γ m ) 2 .
D L = κ 1 ( κ 1 γ m ) | g a s 1 G b s * a s 1 | 2 | G a s 1 * a s 1 | 2 + ( κ 1 γ m ) 2 , D R = γ m + ( κ 1 γ m ) | g a s 1 G b s * a s 1 | 2 | G a s 1 * a s 1 | 2 + ( κ 1 γ m ) 2 .
i t ( a ^ 1 a ^ 2 ) = ( Δ 111 i κ 1 J J Δ 2 + i κ 2 ) ( a ^ 1 a ^ 2 ) .
ω ± = 1 2 { Δ 111 + Δ 2 + i ( κ 2 κ 1 ) ± [ Δ 111 Δ 2 i ( κ 2 + κ 1 ) ] 2 + 4 J 2 } .
ω ± = ω m + i 2 ( κ 2 κ 1 ) ± 4 J 2 ( κ 2 + κ 1 ) 2 .

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