Abstract

Recently, the conception of PT symmetry has attracted considerable attention in various fields such as optics, acoustics, and atomic physics because of the existence of exceptional point (EP) and its importance in understanding non-Hermitian physics. Here, we propose a new scheme of investigating the mechanical-EP-induced transparency and tunable fast-to-slow light phenomena in PT-symmetric mechanical systems. We find that (i) the transmission of the probe field changes from singleto double transparency windows via the transition from a broken mechanical PT-symmetric phase to an unbroken mechanical PT-symmetric phase; (ii) the efficiency of transparency can be significantly enhanced about three orders of magnitude in the vicinity of the mechanical EP, compared to passive mechanical resonators system; and (iii) the mechanical EP can not only amplify the group delay, but also manipulate the switch from slow light to fast light, which may offer an approach to achieve the practical application of slow light and relevant to the optical switcher and communication network. Our results reveal that the exotic properties of the mechanical EP can result in enormous enhancement of the transmitted probe power and novel steering of fast and slow light.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  48. Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112, 111109 (2018).
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    [Crossref]

2019 (2)

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref] [PubMed]

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref] [PubMed]

2018 (7)

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref] [PubMed]

H. Xiong and Ying Wu, “Optomechanical Akhmediev Breathers,” Laser Photonics Rev. 12, 1700305 (2018).
[Crossref]

Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112, 111109 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

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

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

B. Wang, Z.-X. Liu, X. Jia, H. Xiong, and Y. Wu, “Polarization-based control of phonon laser action in a parity time-symmetric optomechanical system,” Commun. Phys. 1, 43 (2018).
[Crossref]

2017 (9)

W. Li, C. Li, and H. Song, “Theoretical realization and application of parity-time-symmetric oscillators in a quantum regime,” Phys. Rev. A 95, 023827 (2017).
[Crossref]

Y.-L. Liu, R. Wu, J. Zhang, Ş. K. Özdemir, L. Yang, F. Nori, and Y. 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]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref] [PubMed]

H. Lü, S. K. Özdemir, L. M. Kuang, F. Nori, and H. Jing, “Exceptional Points in Random-Defect Phonon Lasers,” Phys. Rev. Appl. 8, 044020 (2017).
[Crossref]

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

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature. 548, 23280 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 23281 (2017).
[Crossref]

G.-L. Cheng and A.-X. Chen, “Squeezing induced high-efficiency diffraction grating in two-level system,” Opt. Express 25, 4483–4492 (2017).
[Crossref] [PubMed]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma Soliton Dynamics Based on the Mechanical Effect of Light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref] [PubMed]

2016 (3)

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U.S.A. 113, 6845 (2016).
[Crossref] [PubMed]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, Coalescence, and Topological Properties of Multiple Exceptional Points and Their Experimental Realization,” Phys. Rev. X 6, 021007 (2016).

Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y. Liu, “Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

2015 (2)

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

X.-Y. Lü, H. Jing, J. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

2014 (10)

H. Wang, X. Gu, Y.-x. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90, 023817 (2014).
[Crossref]

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

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89, 013824 (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 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[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 (2014).
[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. Photonics 8, 524–529 (2014).
[Crossref]

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

2013 (1)

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

2012 (2)

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]

M.-A. Miri, P. L. Wa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref] [PubMed]

2011 (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]

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

2010 (3)

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

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon Laser Action in a Tunable Two-Level System,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755 (2010).
[Crossref] [PubMed]

2009 (2)

R. Zhang, S. R. Garner, and L. V. Hau, “Creation of Long-Term Coherent Optical Memory via Controlled Nonlinear Interactions in Bose-Einstein Condensates,” Phys. Rev. Lett. 103, 233602 (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]

2008 (3)

L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465 (2008).
[Crossref]

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical Signal Processing Using Tunable Delay Elements Based on Slow Light,” IEEE J. Sel. Top. Quantum Electron. 14, 691 (2008).
[Crossref]

2005 (1)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062 (2005).
[Crossref]

1999 (1)

A. Partovi, D. Peale, M. Wuttig, C. A. Murray, G. Zydzik, L. Hopkins, K. Baldwin, W. S. Hobson, J. Wynn, J. Lopata, L. Dhar, R. Chichester, and J. H-J Yeh, “High-power laser light source for near-field optics and its application to high-density optical data storage,” Appl. Phys. Lett. 75, 1515 (1999).
[Crossref]

1995 (1)

C. K. Law, “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation,” Phys. Rev. A 51, 2537 (1995).
[Crossref] [PubMed]

Agarwal, G. S.

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89, 013824 (2014).
[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]

Alegre, T. P. M.

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

Alù, A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref] [PubMed]

Arcizet, O.

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

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity Optomechanics,” Rev. Mod. Phys. 86, 1391 (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 (2013).
[Crossref]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465 (2008).
[Crossref]

Baldwin, K.

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B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref] [PubMed]

B. Wang, Z.-X. Liu, X. Jia, H. Xiong, and Y. Wu, “Polarization-based control of phonon laser action in a parity time-symmetric optomechanical system,” Commun. Phys. 1, 43 (2018).
[Crossref]

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112, 111109 (2018).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma Soliton Dynamics Based on the Mechanical Effect of Light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref] [PubMed]

X.-Y. Lü, H. Jing, J. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114, 253601 (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]

Wu, Ying

H. Xiong and Ying Wu, “Optomechanical Akhmediev Breathers,” Laser Photonics Rev. 12, 1700305 (2018).
[Crossref]

Wuttig, M.

A. Partovi, D. Peale, M. Wuttig, C. A. Murray, G. Zydzik, L. Hopkins, K. Baldwin, W. S. Hobson, J. Wynn, J. Lopata, L. Dhar, R. Chichester, and J. H-J Yeh, “High-power laser light source for near-field optics and its application to high-density optical data storage,” Appl. Phys. Lett. 75, 1515 (1999).
[Crossref]

Wynn, J.

A. Partovi, D. Peale, M. Wuttig, C. A. Murray, G. Zydzik, L. Hopkins, K. Baldwin, W. S. Hobson, J. Wynn, J. Lopata, L. Dhar, R. Chichester, and J. H-J Yeh, “High-power laser light source for near-field optics and its application to high-density optical data storage,” Appl. Phys. Lett. 75, 1515 (1999).
[Crossref]

Xiao, M.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, Coalescence, and Topological Properties of Multiple Exceptional Points and Their Experimental Realization,” Phys. Rev. X 6, 021007 (2016).

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. Photonics 8, 524–529 (2014).
[Crossref]

Xiao, Y.

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

Xiong, H.

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref] [PubMed]

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref] [PubMed]

H. Xiong and Ying Wu, “Optomechanical Akhmediev Breathers,” Laser Photonics Rev. 12, 1700305 (2018).
[Crossref]

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

B. Wang, Z.-X. Liu, X. Jia, H. Xiong, and Y. Wu, “Polarization-based control of phonon laser action in a parity time-symmetric optomechanical system,” Commun. Phys. 1, 43 (2018).
[Crossref]

Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112, 111109 (2018).
[Crossref]

H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma Soliton Dynamics Based on the Mechanical Effect of Light,” Phys. Rev. Lett. 119, 153901 (2017).
[Crossref] [PubMed]

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

Yan, L.

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical Signal Processing Using Tunable Delay Elements Based on Slow Light,” IEEE J. Sel. Top. Quantum Electron. 14, 691 (2008).
[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. Photonics 8, 524–529 (2014).
[Crossref]

Yang, L.

Y.-L. Liu, R. Wu, J. Zhang, Ş. K. Özdemir, L. Yang, F. Nori, and Y. 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. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 23281 (2017).
[Crossref]

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

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U.S.A. 113, 6845 (2016).
[Crossref] [PubMed]

Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y. Liu, “Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (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]

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 (2014).
[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]

Yilmaz, H.

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U.S.A. 113, 6845 (2016).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

You, C.

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
[Crossref] [PubMed]

Zhang, B.

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical Signal Processing Using Tunable Delay Elements Based on Slow Light,” IEEE J. Sel. Top. Quantum Electron. 14, 691 (2008).
[Crossref]

Zhang, J.

Y.-L. Liu, R. Wu, J. Zhang, Ş. K. Özdemir, L. Yang, F. Nori, and Y. 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, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y. Liu, “Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016).
[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]

Zhang, J.-Q.

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

Zhang, L.

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical Signal Processing Using Tunable Delay Elements Based on Slow Light,” IEEE J. Sel. Top. Quantum Electron. 14, 691 (2008).
[Crossref]

Zhang, R.

R. Zhang, S. R. Garner, and L. V. Hau, “Creation of Long-Term Coherent Optical Memory via Controlled Nonlinear Interactions in Bose-Einstein Condensates,” Phys. Rev. Lett. 103, 233602 (2009).
[Crossref]

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 (2014).
[Crossref] [PubMed]

Zhang, Z. Q.

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, Coalescence, and Topological Properties of Multiple Exceptional Points and Their Experimental Realization,” Phys. Rev. X 6, 021007 (2016).

Zhang, Z.-M.

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

Zhao, G.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 23281 (2017).
[Crossref]

Zhao, M.G.

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

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]

Zydzik, G.

A. Partovi, D. Peale, M. Wuttig, C. A. Murray, G. Zydzik, L. Hopkins, K. Baldwin, W. S. Hobson, J. Wynn, J. Lopata, L. Dhar, R. Chichester, and J. H-J Yeh, “High-power laser light source for near-field optics and its application to high-density optical data storage,” Appl. Phys. Lett. 75, 1515 (1999).
[Crossref]

Appl. Phys. Lett. (2)

A. Partovi, D. Peale, M. Wuttig, C. A. Murray, G. Zydzik, L. Hopkins, K. Baldwin, W. S. Hobson, J. Wynn, J. Lopata, L. Dhar, R. Chichester, and J. H-J Yeh, “High-power laser light source for near-field optics and its application to high-density optical data storage,” Appl. Phys. Lett. 75, 1515 (1999).
[Crossref]

Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu, “Magnetic-field-dependent slow light in strontium atom-cavity system,” Appl. Phys. Lett. 112, 111109 (2018).
[Crossref]

Appl. Phys. Rev. (1)

H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5, 031305 (2018).
[Crossref]

Commun. Phys. (1)

B. Wang, Z.-X. Liu, X. Jia, H. Xiong, and Y. Wu, “Polarization-based control of phonon laser action in a parity time-symmetric optomechanical system,” Commun. Phys. 1, 43 (2018).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. E. Willner, B. Zhang, L. Zhang, L. Yan, and I. Fazal, “Optical Signal Processing Using Tunable Delay Elements Based on Slow Light,” IEEE J. Sel. Top. Quantum Electron. 14, 691 (2008).
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H. Xiong and Ying Wu, “Optomechanical Akhmediev Breathers,” Laser Photonics Rev. 12, 1700305 (2018).
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Nat. Photonics (3)

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. Photonics 8, 524–529 (2014).
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L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474 (2008).
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T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465 (2008).
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Nat. Phys. (3)

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 (2014).
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R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
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J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712 (2013).
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Nature (4)

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 23281 (2017).
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M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, “Electromagnetically induced transparency with single atoms in a cavity,” Nature 465, 755 (2010).
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W. J. Chen, K.Ş. Özdemir, M.G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192 (2017).
[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 472, 69 (2011).
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Nature. (1)

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature. 548, 23280 (2017).
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Opt. Express (2)

G.-L. Cheng and A.-X. Chen, “Squeezing induced high-efficiency diffraction grating in two-level system,” Opt. Express 25, 4483–4492 (2017).
[Crossref] [PubMed]

B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, “Magnon-induced transparency and amplification in PT-symmetric cavity-magnon system,” Opt. Express 26, 20248–20257 (2018).
[Crossref] [PubMed]

Opt. Lett. (3)

Z.-X. Liu, C. You, B. Wang, H. Xiong, and Y. Wu, “Phase-mediated magnon chaos-order transition in cavity optomagnonics,” Opt. Lett. 44, 507–510 (2019).
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J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
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M.-A. Miri, P. L. Wa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
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Phys. Rev. A (8)

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89, 013824 (2014).
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X.-W. Xu, Y. Liu, C.-P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92, 013852 (2015).
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W. Li, C. Li, and H. Song, “Theoretical realization and application of parity-time-symmetric oscillators in a quantum regime,” Phys. Rev. A 95, 023827 (2017).
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Y.-L. Liu, R. Wu, J. Zhang, Ş. K. Özdemir, L. Yang, F. Nori, and Y. 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).
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H. Wang, X. Gu, Y.-x. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90, 023817 (2014).
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P.-C. Ma, J.-Q. Zhang, Y. Xiao, M. Feng, and Z.-M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
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H. Lü, S. K. Özdemir, L. M. Kuang, F. Nori, and H. Jing, “Exceptional Points in Random-Defect Phonon Lasers,” Phys. Rev. Appl. 8, 044020 (2017).
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T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light Stops at Exceptional Points,” Phys. Rev. Lett. 120, 013901 (2018).
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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).
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Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y. Liu, “Metrology with PT-Symmetric Cavities: Enhanced Sensitivity near the PT-Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016).
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R. Zhang, S. R. Garner, and L. V. Hau, “Creation of Long-Term Coherent Optical Memory via Controlled Nonlinear Interactions in Bose-Einstein Condensates,” Phys. Rev. Lett. 103, 233602 (2009).
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H. Xiong, J.-H. Gan, and Y. Wu, “Kuznetsov-Ma Soliton Dynamics Based on the Mechanical Effect of Light,” Phys. Rev. Lett. 119, 153901 (2017).
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Phys. Rev. X (1)

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, Coalescence, and Topological Properties of Multiple Exceptional Points and Their Experimental Realization,” Phys. Rev. X 6, 021007 (2016).

Proc. Natl. Acad. Sci. U.S.A. (1)

B. Peng, S. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. U.S.A. 113, 6845 (2016).
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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram of a three-mode coalescence system, which includes cavity mode a ^ and P T-symmetric mechanical modes b ^ 1 , b ^ 2, and μ is the coupling strength of coupled mechanical resonators, and g is the optomechanical coupling parameter. The system is pumped by a strong control field and a weak probe field. (b) The implementations of the three-mode system. The shaded area shows a P T-symmetric mechanical resonator (MR). An active mechanical resonator couples to a general optomechanical system via mechanical coupling μ. k is the cavity decay rate, γ and γ′ indicate the loss and gain of mechanical modes b ^ 1 and b ^ 2, respectively.
Fig. 2
Fig. 2 The power transmission coefficient |tp|2 varies with ω. System parameters we take here are, the wavelength of the control field is 1573 nm, Pc = 50μW, g = 2π MHz, ωm/2π = 3.68 GHz, γ = γ′ = 0.5 × 10−2 ωm, and for the cases of (a)-(d), we choose μ = 0.2 (γ + γ′), μ = 0.27 (γ + γ′), μ = 0.8 (γ + γ′), μ = 1.5 (γ + γ′) respectively.
Fig. 3
Fig. 3 Schematic of the energy-level diagram of our system, where | n , | n m 1 , and | n m 2 denote the number states of the cavity photon, passive and active mechanical resonators phonons, respectively.The control field ωl can make a single-phonon transition from | n , n m 1 + 1 , n m 2 to | n + 1 , n m 1 , n m 2 . The transition from | n , n m 1 , n m 2 to | n + 1 , n m 1 , n m 2 can be caused by the probe field ωp, and the transition of | n , n m 1 + 1 , n m 2 | n , n m 1 , n m 2 + 1 is induced by mechanical coupling. | n , n m + and | n , n m represent mechanical supermodes(dressed mechanical mode), and λ is the splitting width of the two supermodes | n , n m ± .
Fig. 4
Fig. 4 The logarithm of transmission coefficient |tp|2 varies with the mechanical coupling μ and ω. The mechanical coupling strength we chosen contains the range of P T-symmetry breaking phase and P T-symmetry phase. The parameters are the same as Fig. 2.
Fig. 5
Fig. 5 The group delay τg varies with the mechanical coupling μ under the different control power Pc (a) and cavity decay k (b), where the detuning ω = ωm. The other parameters are the same as Fig. 2.

Equations (17)

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H = Δ a ^ a ^ + ω m ( b ^ 1 b ^ 1 + b ^ 2 b ^ 2 ) μ ( b ^ 1 b ^ 2 + b ^ 2 b ^ 1 ) g a ^ a ^ ( b ^ 1 + b ^ 1 )         + i η κ ε l ( a ^ a ^ ) + i η κ ε p ( a ^ e i ω t a ^ e i ω t ) ,
d a d t = i Δ a ^ κ 2 a ^ + i g a ( b 1 + b 1 ) + η κ ε l + η κ ε p e i ω t , d b 1 d t = i ω m b 1 γ 2 b 1 + i g a a + i μ b 2 , d b 2 d t = i ω m b 2 + γ 2 b 2 + i μ b 1 ,
a ¯ = η κ ε l i Δ + κ / 2 + i g ( b ¯ 1 * + b ¯ 1 ) , b ¯ 1 = i g ( i ω m γ / 2 ) | a ¯ | 2 ( i ω m + γ / 2 ) ( i ω m γ / 2 ) + μ 2 , b ¯ 2 = i μ b ¯ 1 i ω m γ / 2 ,
d δ a d t = i Δ δ a κ 2 δ a + i g δ a ( b ¯ 1 + b ¯ 1 * ) + i g a ¯ ( δ b 1 + δ b 1 * ) + η κ ε p e i ω t , d δ b 1 d t = i ω m δ b 1 γ 2 δ b 1 + i g ( a ¯ δ a * + a ¯ * δ a ) + i μ δ b 2 , d δ b 2 d t = i ω m δ b 2 + γ 2 δ b 2 + i μ δ b 1 ,
δ a = A 1 + e i ω t + A 1 e i ω t , δ b 1 = B 1 + e i ω t + B 1 e i ω t , δ b 1 = C 1 + e i ω t + C 1 e i ω t ,
A 1 = η κ ε p Ξ ( ω ) i g a ¯ ( Λ Γ * + λ ) ,
Ξ ( ω ) = i Δ + κ / 2 i ω i g ( b ¯ 1 * + b ¯ 1 ) , α ( ω m , γ ) = i ω m i ω + γ / 2 , λ = i g a ¯ [ α ( ω m , γ ) / f 2 ( α ) α ( ω m , γ ) / f 1 ( α ) ] , λ = i g a ¯ * [ α ( ω m , γ ) / f 2 ( α ) α ( ω m , γ ) / f 3 ( α ) ] ,
f 1 ( α ) = α ( ω m , γ ) α ( ω m , γ ) + μ 2 , f 2 ( α ) = α ( ω m , γ ) α ( ω m , γ ) + μ 2 , f 3 ( α ) = α ( ω m , γ ) α ( ω m , γ ) + μ 2 ,
S o u t = S i n η κ a ^ = ε c η κ a ¯ + ( ε p η κ A 1 ) e i ω t η κ A 1 + e i ω t ,
t p = 1 η κ Ξ ( ω ) i g a ¯ ( Λ Γ * + λ ) ,
τ g = d ψ ( ω p ) d ω p = d a r g [ t p ( ω p ) ] d ω p ,
H ^ m = ( b ^ 1 b ^ 2 ) ( ω m i γ / 2 μ μ ω m + i γ / 2 ) ( b ^ 1 b ^ 2 ) ,
R = ( ω m i γ / 2 μ μ ω m + i γ / 2 ) ,
R = ( ω + i γ 0 0 ω + + i γ + ) ,
H ^ m = ( B ^ 1 B ^ 2 ) ( ω + i γ 0 0 ω + + i γ + ) ( B ^ 1 B ^ 2 ) ,
ω ± = ω m ± Re μ 2 ( γ + γ 4 ) 2 ,
γ ± = ( γ γ ) 4 ± Im μ 2 ( γ + γ 4 ) 2 ,

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