Abstract

We study the response of parity-time (PT)-symmetric optomechanical systems with tunable gain and loss to the weak probe field in the presence of a strong control field and a coherent phonon pump. We show that the probe transmission can exceed unity at low control power due to the optical gain of the cavity and it can be further enhanced or suppressed by tuning the amplitude and phase of the phonon pump. Furthermore, the phase dispersion of the transmitted probe field is modified by controlling the applied fields, which allows one to tune the group delay of the probe field. Based on this optomechianical system, we can realize a tunable switch between slow and fast light effect by adjusting the gain-to-loss ratio, power of the control field as well as the amplitude and phase of the phonon pump. Our work provides a platform to control the light propagation in a more flexible way.

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

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References

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  1. K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref] [PubMed]
  2. Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
    [Crossref] [PubMed]
  3. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
    [Crossref]
  4. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref] [PubMed]
  5. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
    [Crossref]
  6. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [Crossref]
  7. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [Crossref]
  8. 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, 093902 (2009).
    [Crossref] [PubMed]
  9. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref] [PubMed]
  10. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  11. C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  12. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
    [Crossref]
  13. H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
    [Crossref]
  14. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  15. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
    [Crossref]
  16. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
    [Crossref]
  17. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
    [Crossref]
  18. 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]
  19. 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 (London) 472, 69–73 (2011).
    [Crossref]
  20. 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, 12165–12173 (2013).
    [Crossref] [PubMed]
  21. V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
    [Crossref] [PubMed]
  22. C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
    [Crossref]
  23. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
    [Crossref]
  24. B. Chen, C. Jiang, and K.-D. Zhu, “Slow light in a cavity optomechanical system with a Bose-Einstein condensate,” Phys. Rev. A 83, 055803 (2011).
    [Crossref]
  25. M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (2015).
    [Crossref]
  26. X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
    [Crossref]
  27. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
    [Crossref] [PubMed]
  28. B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
    [Crossref]
  29. 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]
  30. Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
    [Crossref] [PubMed]
  31. H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
    [Crossref] [PubMed]
  32. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep.  6, 31095 (2016).
    [Crossref] [PubMed]
  33. 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, 083034 (2016).
    [Crossref]
  34. 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, 063825 (2017).
    [Crossref]
  35. D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
    [Crossref]
  36. H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
    [Crossref] [PubMed]
  37. Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
    [Crossref]
  38. 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]
  39. J. Y. Ma, C. You, L. G. Si, H. Xiong, J. H. Li, X. X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep.  5, 11278 (2015).
    [Crossref] [PubMed]
  40. X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015).
    [Crossref]
  41. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
    [Crossref]
  42. 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]
  43. L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
    [Crossref]
  44. G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
    [Crossref]
  45. J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
    [Crossref]
  46. L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
    [Crossref] [PubMed]
  47. D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
    [Crossref]
  48. C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
    [Crossref]
  49. H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (2012).
    [Crossref]
  50. 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 (1987).
    [Crossref]
  51. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
    [Crossref]
  52. 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, 63–67 (2012).
    [Crossref] [PubMed]
  53. T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
    [Crossref] [PubMed]

2018 (2)

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

2017 (7)

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[Crossref] [PubMed]

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

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]

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (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, 063825 (2017).
[Crossref]

2016 (6)

D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
[Crossref]

W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep.  6, 31095 (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, 083034 (2016).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[Crossref]

2015 (9)

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (2015).
[Crossref]

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

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]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (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, 043843 (2015).
[Crossref]

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

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
[Crossref]

2014 (6)

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

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

2013 (4)

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (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, 12165–12173 (2013).
[Crossref] [PubMed]

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

2012 (3)

H. Xiong, L.-G. Si, A.-S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86, 013815 (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, 63–67 (2012).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

2011 (5)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

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 (London) 472, 69–73 (2011).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

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

2010 (4)

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

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

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

2009 (2)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

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, 093902 (2009).
[Crossref] [PubMed]

2008 (1)

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

2001 (1)

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (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 (1987).
[Crossref]

Agarwal, G. S.

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

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Aimez, V.

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, 093902 (2009).
[Crossref] [PubMed]

Akram, M. J.

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (2015).
[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 (London) 472, 69–73 (2011).
[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]

Aspelmeyer, M.

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

Awschalom, D. D.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

Bahl, G.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

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

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Bian, X.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[Crossref]

Blanter, Y. M.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Bochmann, J.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

Boettcher, S.

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

Bosman, S. J.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Brown, E.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Castellanos-Gomez, A.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[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 (London) 472, 69–73 (2011).
[Crossref]

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 (London) 472, 69–73 (2011).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Chen, B.

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, 12165–12173 (2013).
[Crossref] [PubMed]

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

Chen, G.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[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, 12165–12173 (2013).
[Crossref] [PubMed]

Chen, W.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

Cho, S. U.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

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, 093902 (2009).
[Crossref] [PubMed]

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Cleland, A. N.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

Clerk, A. A.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Cui, Y.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[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, 12165–12173 (2013).
[Crossref] [PubMed]

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 (1987).
[Crossref]

Deléglise, S.

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, 63–67 (2012).
[Crossref] [PubMed]

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

Devoret, M. H.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Dey, T. N.

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Duchesne, D.

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, 093902 (2009).
[Crossref] [PubMed]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[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 (London) 472, 69–73 (2011).
[Crossref]

Eisfeld, A.

D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
[Crossref]

Fan, L.

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
[Crossref] [PubMed]

Fan, S.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Feng, L.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Fong, K. Y.

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
[Crossref] [PubMed]

Ganainy, R. E.

D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
[Crossref]

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Ganainy, R.E.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

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]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

Geng, Z.

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

Gianfreda, M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

Girvin, S. M.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Goldzak, T.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

Gracia, H.G.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

Gross, R.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

Guo, A.

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, 093902 (2009).
[Crossref] [PubMed]

Guo, Y. Q.

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, 063825 (2017).
[Crossref]

Hakonen, P.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Hassan, A. U.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

He, B.

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

Heikkilä, T. T.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[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 (London) 472, 69–73 (2011).
[Crossref]

Hocke, F.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

Hodaei, H.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Huang, S.

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

Huang, Y. Y.

Huebl, H.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

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, C.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[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, 12165–12173 (2013).
[Crossref] [PubMed]

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

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Jiang, Y.

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

Jiao, Y.

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, 083034 (2016).
[Crossref]

Jing, H.

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[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, 083034 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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]

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” 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,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[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 (1987).
[Crossref]

Khajavikhan, M.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

Khan, M. M.

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (2015).
[Crossref]

Kim, S.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

Kip, D.

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Kippenberg, T. J.

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

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[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, 63–67 (2012).
[Crossref] [PubMed]

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

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Lei, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, C.

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

Li, C.-W.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Li, J. H.

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

Li, W.

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

Li, X.

Li, Y.

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. 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, 083034 (2016).
[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]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015).
[Crossref]

Lin, Q.

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

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Liu, H.

Liu, Y.-L.

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

Liu, Y.-x.

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[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, 043843 (2015).
[Crossref]

Liu, Z.-P.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

Long, G. L.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Lü, H.

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[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, 083034 (2016).
[Crossref]

Lü, X. Y.

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

Lü, X.-Y.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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]

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” 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,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Ma, J. Y.

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

Ma, J.-Y.

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]

Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Mailybaev, A. A.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

Makris, K. G.

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Marquardt, F.

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

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Marx, A.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

Massel, F.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Menon, S.

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Miri, M.-A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Moiseyev, N.

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

Monifi, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Morandotti, R.

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, 093902 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Nori, F.

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[Crossref] [PubMed]

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Özdemir, S. K.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[Crossref] [PubMed]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

Painter, O.

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 (London) 472, 69–73 (2011).
[Crossref]

Pei, P.

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, 063825 (2017).
[Crossref]

Peng, B.

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Pirkkalainen, J.-M.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Poot, M.

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
[Crossref] [PubMed]

Qian, J.

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, 083034 (2016).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Regensburger, A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

Rivière, R.

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

Röter, C. E.

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Safavi-Naeini, A. H.

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

Saif, F.

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (2015).
[Crossref]

Salamo, G. J.

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, 093902 (2009).
[Crossref] [PubMed]

Saloniemi, H.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Schliesser, A.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[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, 63–67 (2012).
[Crossref] [PubMed]

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

Schneider, B. H.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Schoelkopf, R. J.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

Schönleber, D. W.

D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
[Crossref]

Segev, M.

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Si, L. G.

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

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

Si, L.-G.

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

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

Sillanpää, M. A.

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

Singh, V.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Siviloglou, G. A.

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, 093902 (2009).
[Crossref] [PubMed]

Sohn, D. B.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

Song, H.

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

Steele, G. A.

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Sterling, R.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
[Crossref]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

Suzuki, H.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
[Crossref]

Tang, H. X.

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
[Crossref] [PubMed]

Tian, L.

Vainsencher, A.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

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, 63–67 (2012).
[Crossref] [PubMed]

Volatier-Ravat, M.

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, 093902 (2009).
[Crossref] [PubMed]

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Wei, L. F.

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]

Weis, S.

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, 63–67 (2012).
[Crossref] [PubMed]

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

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Wiersig, J.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

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 (London) 472, 69–73 (2011).
[Crossref]

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

Wu, R.

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

Wu, Y.

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

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

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]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

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

Xiao, M.

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Xiong, H.

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

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

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

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

Xu, X. W.

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015).
[Crossref]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

Yang, L.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

Yang, X.

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

Yang, X. X.

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

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[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, 063825 (2017).
[Crossref]

You, C.

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

Yu, H.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[Crossref]

Zhang, J.

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[Crossref]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” 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,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref] [PubMed]

Zhang, X.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[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, 063825 (2017).
[Crossref]

Zhang, X. Z.

Zhao, G.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

Zheng, A.-S.

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

Zhou, X.

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

Zhu, K.-D.

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

Zubairy, M. S.

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

Zuo, F.

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[Crossref]

Nat. Commun (1)

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun.  6, 5850 (2015).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9, 820–824 (2014).
[Crossref] [PubMed]

Nat. Photon. (2)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

Nat. Phys. (4)

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. E. Röter, K. G. Makris, R. E. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

Nature (1)

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, 63–67 (2012).
[Crossref] [PubMed]

Nature (London) (5)

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
[Crossref]

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

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature (London) 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H.G. Gracia, R.E. Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature (London) 548, 187–191 (2017).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167171 (2012).
[Crossref]

New J. Phys (2)

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, 083034 (2016).
[Crossref]

D. W. Schönleber, A. Eisfeld, and R. E. Ganainy, “Optomechanical interactions in non-Hermitian photonic molecules,” New J. Phys.  18, 045014 (2016).
[Crossref]

Opt. Express (2)

Phys. Rev. A (15)

L.-G. Si, H. Xiong, M. S. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95, 033803 (2017).
[Crossref]

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Y.-L. Liu, R. Wu, J. Zhang, S. K. Özdemir, L. Yang, F. Nori, and Y.-x. Liu, “Controllable optical response by modifying the gain and loss of a mechanical resonator and cavity mode in an optomechanical system,” Phys. Rev. A 95, 013843 (2017).
[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]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015).
[Crossref]

C. M. Bender, M. Gianfreda, S. K. Özdemir, B. Peng, and L. Yang, “Twofold transition in PT-symmetric coupled oscillators,” Phys. Rev. A 88, 062111 (2013).
[Crossref]

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

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 (1987).
[Crossref]

C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016).
[Crossref]

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

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92, 023846 (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, 063825 (2017).
[Crossref]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A 94, 031802 (2016).
[Crossref]

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

Phys. Rev. Lett. (10)

K. G. Makris, R. E. Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

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, 093902 (2009).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

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]

Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref] [PubMed]

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

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120, 013901 (2018).
[Crossref] [PubMed]

Physics (1)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Rev. Mod. Phys. (2)

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

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

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

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]

Sci. Rep (3)

H. Jing, S. K. Özdemir, H. Lü, and F. Nori, “High-order exceptional points in optomechanics,” Sci. Rep.  7, 3386 (2017).
[Crossref] [PubMed]

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

Sci. Rep. (1)

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

Science (2)

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

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 (a) Schematic diagram of the parity-time-symmetric optomechanical system with phonon pump. An active cavity with gain rate κ2 is coupled to a passive cavity with loss rate κ1 with tunnelling strength J, and the passive cavity is also coupled to a mechanical mode of frequency ωm and damping rate γm. A strong control field and a weak probe field is simultaneously applied to the passive cavity and the mechanical mode is driven by a weak coherent phonon pump. (b) Level scheme of this optomechanical system. Here n1n2〉 and nm denote the photon number in the passive (active) cavity and the phonon number in the mechanical mode.
Fig. 2
Fig. 2 Stability diagram with respect to J/(κ1 + κ2) and κ2/κ1. We use the following parameters: the frequency of the passive cavity ω1 = 2πc/λ with λ = 1550 nm, the radius of the passive cavity R = 15 µm, g1 = ω1/R, κ1/2π = 6 MHz, m = 6.2 ng, ωm/2π = 78 MHz, γm/2π = 12 kHz, Pc = 0.1 µW, ηc = 0.4, εm = 0, and Δ 1 = Δ 2 = ω m.
Fig. 3
Fig. 3 (a) Transmission|tp|2 and (b) phase ϕt versus the detuning (Ω − ωm)/κ1 for different values of κ2/κ1 without phonon pump. The other parameters are the same as those in Fig. 2 except J = 0.2(κ1 + κ2〉.
Fig. 4
Fig. 4 Contour plot of the probe transmission|tp|2 at Ω = ωm as functions of amplitude εm and phase difference ϕ with κ2 = 0.1 κ1. We fix εp = εc /1000 throughout this work since the probe field should be much weaker than the control field. For the other parameters, see Fig. 3.
Fig. 5
Fig. 5 Transmission|tp|2 [(a) and (c)] and phase ϕt [(b) and (d)] versus the detuning between the probe and control field for different values of εm and ϕ. Figs. 5(c) and 5(d) are, respectively, the enlargement of Figs. 5(a) and 5(b) in the narrow regime around Ω = ωm. The other parameters are the same as in Fig. 4.
Fig. 6
Fig. 6 Plots of|t1|2,|t2|2 and|tp|2 at Ω = ωm as a function of the amplitude εm of the phonon pump for (a) ϕ = π/2 and (b) ϕ = 3π/2. The other parameters are the same as in Fig. 4.
Fig. 7
Fig. 7 Group delay τg as a function of the control power Pc for different values of (a) gain-to-loss ratio κ2/κ1 with εm = 0 and (b) pump amplitude εm and phase difference ϕ with κ2 = 0.1 κ1. The other parameters are the same as in Fig. 4.
Fig. 8
Fig. 8 Group delay τg as functions of (a) amplitude εm of the phonon pump with ϕ = π/2 and 3π/2 and (b) phase difference ϕ/π with εm = 2, 4, and 8 pN, respectively. The other parameters are the same as in Fig. 4.
Fig. 9
Fig. 9 Transmission|tp|2 [(a) and (c)] and group delay τg [(b) and (d)] as functions of J/(κ1 + κ2) for different values of εm and ϕ. The other parameters are the same as in Fig. 4.

Equations (25)

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H = Δ 1 a 1 a 1 + Δ 2 a 2 a 2 + p 2 2 m + 1 2 m ω m 2 x 2 J ( a 1 a 2 + a 1 a 2 ) g 1 a 1 a 1 x + H d r ,
H d r = i η c κ 1 [ ( ε c + ε p e i Ω t i ϕ p c ) a 1 H .c . ] x ε m cos ( Ω t + ϕ m ) ,
a ˙ 1 [ κ 1 / 2 + i ( Δ 1 g 1 x ) ] a 1 + i J a 2 + η c κ 1 ( ε c + ε p e i Ω t i ϕ p c ) ,
a ˙ 2 = ( κ 2 / 2 i Δ 2 ) a 2 + i J a 1 ,
x ˙ = p m ,
p ˙ = m ω m 2 x γ m p + g 1 a 1 a 1 + ε m cos ( Ω t + ϕ m ) ,
H e f f = ( Δ 1 i κ 1 2 ) a 1 a 1 + ( Δ 2 + i κ 2 2 ) a 2 a 2 J ( a 1 a 2 + a 1 a 2 ) = ( a 1 a 2 ) ( ( Δ 1 i κ 1 2 ) J J ( Δ 2 i κ 2 2 ) ) ( a 1 a 2 ) .
ω ± = Δ i κ 1 κ 2 4 ± J 2 ( κ 1 + κ 2 4 ) 2 .
a 1 s = η c κ 1 ε c ( κ 2 / 2 i Δ 2 ) ( κ 1 / 2 + i Δ 1 ) ( κ 2 / 2 i Δ 2 ) J 2 ,
a 2 s = i J η c κ 1 ε c ( κ 1 / 2 + i Δ 1 ) ( κ 2 / 2 i Δ 2 ) J 2 ,
x s = g 1 | a 1 s | 2 m ω m 2 ,
p s = 0 ,
δ a ˙ 1 = ( κ 1 / 2 + i Δ 1 ) δ a 1 + i g 1 a 1 s δ x + i J δ a 2 + η c κ 1 ε p e i Ω t i ϕ p c ,
δ a ˙ 2 = ( κ 2 / 2 i Δ 2 ) δ a 2 + i J δ a 1 ,
δ x ˙ = δ p m ,
δ p ˙ = m ω m 2 δ x γ m δ p + g 1 ( a 1 s * δ a 1 + a 1 s δ a 1 ) + ε m cos ( Ω t ϕ m ) ,
v ˙ = M v ,
M = ( κ 1 / 2 Δ 1 0 J g 0 Im [ a 1 s ] 0 Δ 1 κ 1 / 2 J 0 g 0 Re [ a 1 s ] 0 0 J κ 2 / 2 Δ 2 0 0 J 0 Δ 2 κ 2 / 2 0 0 0 0 0 0 0 ω m 4 g 0 Re [ a 1 s ] 4 g 0 Im [ a 1 s ] 0 0 ω m γ m ) .
a 1 + = η c κ 1 [ 1 + i f ( Ω ) ] ε p e i ϕ p c + i g 1 a 1 s χ m ( Ω ) ε m e i ϕ m / 2 κ 1 / 2 + i ( Δ 1 Ω ) 2 Δ 1 f ( Ω ) J 2 χ c ( ω ) ,
χ m ( Ω ) = 1 m ( ω m 2 Ω 2 i γ m Ω ) , χ 2 ( Ω ) = 1 κ 2 / 2 + i Δ 2 + i Ω , f ( Ω ) = g 1 2 | a 1 s | 2 χ m ( Ω ) κ 1 / 2 i Δ 1 i Ω J 2 χ 2 ( Ω ) , χ c ( Ω ) = χ 2 * ( Ω ) + i f ( Ω ) [ χ 2 * ( Ω ) χ 2 ( Ω ) ] .
a 1 , out ( t ) = ( ε c η c κ 1 a 1 s ) e i ω c t + ( ε p e i ϕ p c η c κ 1 a 1 + ) e i ( Ω + ω c ) t η c κ 1 a 1 e i ( ω c Ω ) t = ( ε c η c κ 1 a 1 s ) e i ω c t + ( ε p e i ϕ p c η c κ 1 a 1 + ) e i ω p t η c κ 1 a 1 e i ( 2 ω c ω p ) t .
i p = ( ε p e i ϕ p c η c κ 1 a 1 + ) / ( ε p e i ϕ p c ) = t 1 + t 2 ,
t 1 = 1 1 + i f ( Ω ) κ 1 / 2 + i ( Δ 1 Ω ) 2 Δ 1 f ( Ω ) J 2 χ c ( ω ) η c κ 1 ,
t 2 = i g 1 a 1 s χ m ( Ω ) ε m / 2 ε p κ 1 / 2 + i ( Δ 1 Ω ) 2 Δ 1 f ( Ω ) J 2 χ c ( ω ) η c κ 1 e i ϕ ,
τ g = d ϕ t ( ω p ) d ω p = d { arg [ t p ( ω p ) ] } d ω p .

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