Abstract

The recent experimental progress in parity-time (PT) symmetry has attracted great interest. However, compared with PT symmetry, there are only a few reported results on its counterpart, anti-PT symmetry, which would lead to new insights and applications. Experimentally simulating and demonstrating the properties of anti-PT symmetry is of particular interest. Here, we present experimental research for simulating the dynamics of bosonic Bogoliubov quasi-particles with anti-PT symmetry based on single photons generated from a point defect in a gallium nitride film. The dynamical evolution under a non-unitary operator is a continuous complex Lorentz transformation in a complex Minkowski space. The evolved states are located on hyperbolic curves, depending on the values of the new defined inner product, which remain invariant during the evolution. Three types of state, space-like, light-like, and time-like, which are analogous to those predicted by the special relativity, are demonstrated. The results of our work could be helpful to investigate the dynamics of quasi-particles in diverse systems and promote deep understanding of non-Hermitian quantum mechanics in open systems.

© 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]
  33. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry II. a complete characterization of non-Hermitian Hamiltonians with a real spectrum,” J. Math. Phys. 43, 2814–2816 (2002).
    [Crossref]
  34. A. Mostafazadeh, “Pseudo-Hermiticity versus PT-symmetry III: equivalence of pseudo-Hermiticity and the presence of antilinear symmetries,” J. Math. Phys. 43, 3944–3951 (2002).
    [Crossref]
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    [Crossref]

2018 (3)

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]

Q. Zhang and B. Wu, “Lorentz quantum mechanics,” New J. Phys. 20, 013024 (2018).
[Crossref]

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

2017 (4)

F. Yang, Y.-C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96, 053845 (2017).
[Crossref]

A. M. Berhane, “Bright room-temperature single-photon emission from defects in gallium nitride,” Adv. Mater. 29, 1605092 (2017).
[Crossref]

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, 187–191 (2017).
[Crossref]

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

2016 (4)

Z.-P. Liu, J. Zhang, Ş. 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]

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[Crossref]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

2015 (2)

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref]

D. A. Antonosyan, A. S. Solntsev, and A. A. Sukhorukov, “Parity-time anti-symmetric parametric amplifier,” Opt. Lett. 40, 4575–4578 (2015).
[Crossref]

2014 (9)

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref]

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

M. Kreibich, J. Main, H. Cartarius, and G. Wunner, “Realizing PT-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials,” Phys. Rev. A 90, 033630 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

B. Peng, Ş. 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, 328–332 (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]

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]

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]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

2013 (3)

C. Hang, G. Huang, and V. V. Konotop, “PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

J.-B. Gong and Q.-H. Wang, “Time-dependent PT-symmetric quantum mechanics,” J. Phys. A 46, 485302 (2013).
[Crossref]

L. Ge and H. E. Tuereci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

2012 (1)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[Crossref]

2011 (1)

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

2010 (1)

Q.-H. Wang, S.-Z. Chia, and J.-H. Zhang, “PT symmetry as a generalization of Hermiticity,” J. Phys. A 43, 295301 (2010).
[Crossref]

2007 (1)

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[Crossref]

2002 (4)

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

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205–214 (2002).
[Crossref]

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry II. a complete characterization of non-Hermitian Hamiltonians with a real spectrum,” J. Math. Phys. 43, 2814–2816 (2002).
[Crossref]

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-symmetry III: equivalence of pseudo-Hermiticity and the presence of antilinear symmetries,” J. Math. Phys. 43, 3944–3951 (2002).
[Crossref]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1949 (1)

L. Landau, “On the theory of superfluidity,” Phys. Rev. 75, 884–885 (1949).
[Crossref]

1947 (1)

N. N. Bogoliubov, “On the theory of superfluidity,” J. Phys. USSR 11, 23–32 (1947).

Adamo, G.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Aharonovich, I.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Alù, A.

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref]

Antonosyan, D. A.

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[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]

B. Peng, Ş. 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, 328–332 (2014).
[Crossref]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[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, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Berhane, A.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Berhane, A. M.

A. M. Berhane, “Bright room-temperature single-photon emission from defects in gallium nitride,” Adv. Mater. 29, 1605092 (2017).
[Crossref]

Bodrog, Z.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bogoliubov, N. N.

N. N. Bogoliubov, “On the theory of superfluidity,” J. Phys. USSR 11, 23–32 (1947).

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[Crossref]

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

Cao, W.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Cartarius, H.

M. Kreibich, J. Main, H. Cartarius, and G. Wunner, “Realizing PT-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials,” Phys. Rev. A 90, 033630 (2014).
[Crossref]

Cerjan, A.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[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. Photonics 8, 524–529 (2014).
[Crossref]

Chen, G.

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[Crossref]

Chen, W.

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

Chia, S.-Z.

Q.-H. Wang, S.-Z. Chia, and J.-H. Zhang, “PT symmetry as a generalization of Hermiticity,” J. Phys. A 43, 295301 (2010).
[Crossref]

Christodoulides, D. N.

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]

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, 187–191 (2017).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

Deutsch, C.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

El-Ganainy, R.

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]

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, 187–191 (2017).
[Crossref]

Ellis, F. M.

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

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.

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Q.-H. Wang, S.-Z. Chia, and J.-H. Zhang, “PT symmetry as a generalization of Hermiticity,” J. Phys. A 43, 295301 (2010).
[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]

Wang, Y. T.

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[Crossref]

Wang, Z. Y.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Wen, J.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (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. Photonics 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 548, 192–196 (2017).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

Wittek, S.

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, 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]

Wu, B.

Q. Zhang and B. Wu, “Lorentz quantum mechanics,” New J. Phys. 20, 013024 (2018).
[Crossref]

Wu, J.-H.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref]

Wunner, G.

M. Kreibich, J. Main, H. Cartarius, and G. Wunner, “Realizing PT-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials,” Phys. Rev. A 90, 033630 (2014).
[Crossref]

Xiao, M.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (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. Photonics 8, 524–529 (2014).
[Crossref]

Xiao, Y.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Xu, J. S.

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[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, F.

F. Yang, Y.-C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96, 053845 (2017).
[Crossref]

Yang, L.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (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.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

B. Peng, Ş. 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, 328–332 (2014).
[Crossref]

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

Yilmaz, H.

B. Peng, Ş. 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, 328–332 (2014).
[Crossref]

You, L.

F. Yang, Y.-C. Liu, and L. You, “Anti-PT symmetry in dissipatively coupled optical systems,” Phys. Rev. A 96, 053845 (2017).
[Crossref]

Yu, S.

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[Crossref]

Zhang, J.

Z.-P. Liu, J. Zhang, Ş. 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]

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

Zhang, J.-H.

Q.-H. Wang, S.-Z. Chia, and J.-H. Zhang, “PT symmetry as a generalization of Hermiticity,” J. Phys. A 43, 295301 (2010).
[Crossref]

Zhang, Q.

Q. Zhang and B. Wu, “Lorentz quantum mechanics,” New J. Phys. 20, 013024 (2018).
[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, 972–975 (2014).
[Crossref]

Zhang, Y.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

Zhang, Z.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[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, 192–196 (2017).
[Crossref]

Zheng, M. C.

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

Zhou, Y.

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Adv. Mater. (1)

A. M. Berhane, “Bright room-temperature single-photon emission from defects in gallium nitride,” Adv. Mater. 29, 1605092 (2017).
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[Crossref]

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-symmetry III: equivalence of pseudo-Hermiticity and the presence of antilinear symmetries,” J. Math. Phys. 43, 3944–3951 (2002).
[Crossref]

J. Phys. A (2)

J.-B. Gong and Q.-H. Wang, “Time-dependent PT-symmetric quantum mechanics,” J. Phys. A 46, 485302 (2013).
[Crossref]

Q.-H. Wang, S.-Z. Chia, and J.-H. Zhang, “PT symmetry as a generalization of Hermiticity,” J. Phys. A 43, 295301 (2010).
[Crossref]

J. Phys. USSR (1)

N. N. Bogoliubov, “On the theory of superfluidity,” J. Phys. USSR 11, 23–32 (1947).

Nat. Commun. (2)

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6, 5905 (2015).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

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

J.-S. Tang, Y. T. Wang, S. Yu, D. Y. He, J. S. Xu, B. H. Liu, G. Chen, Y. N. Sun, K. Sun, Y. J. Han, C.-F. Li, and G.-G. Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system,” Nat. Photonics 10, 642–646 (2016).
[Crossref]

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, 394–398 (2014).
[Crossref]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
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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).
[Crossref]

Nature (2)

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, 187–191 (2017).
[Crossref]

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

New J. Phys. (1)

Q. Zhang and B. Wu, “Lorentz quantum mechanics,” New J. Phys. 20, 013024 (2018).
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L. Ge and H. E. Tuereci, “Antisymmetric PT-photonic structures with balanced positive- and negative-index materials,” Phys. Rev. A 88, 053810 (2013).
[Crossref]

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

M. Kreibich, J. Main, H. Cartarius, and G. Wunner, “Realizing PT-symmetric non-Hermiticity with ultracold atoms and Hermitian multiwell potentials,” Phys. Rev. A 90, 033630 (2014).
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Phys. Rev. Lett. (10)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
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J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
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J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
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H. Jing, Ş. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[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.-X. Liu, “Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition,” Phys. Rev. Lett. 117, 110802 (2016).
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Sci. Adv. (1)

Y. Zhou, Z. Y. Wang, A. Rasmita, S. Kim, A. Berhane, Z. Bodrog, G. Adamo, A. Gali, I. Aharonovich, and W.-B. Gao, “Room-temperature solid state quantum emitters in the telecom range,” Sci. Adv. 4, eaar3580 (2018).
[Crossref]

Science (2)

B. Peng, Ş. 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, 328–332 (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).
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Supplementary Material (1)

NameDescription
» Supplement 1       Detailed information about single photon sources.

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Figures (4)

Fig. 1.
Fig. 1. Experimental setup. (a) State preparation. The polarization of a single photon generated from an intrinsic defect in a gallium nitride (GaN) film is first initialized to be horizontal by a half-wave plate (H1) and a polarization beam splitter (PBS1). The polarization state is further prepared as |ψin=(m|H+n|V) with a half-wave plate (H2) and a quarter-wave plate (Q1), where |H and |V represent the horizontal and vertical polarizations, and m and n are two real amplitudes, respectively. (b) The change in photon number. The state is separated into two paths p and q with a beam displacer (BD1). The polarization in path q is rotated to be horizontal with H3, which is the same as that in path p. The transmitted photon number is controlled by H4 and PBS2. (c) The non-orthogonal state transformation. In path p, |H is transformed to |φH with H5 and Q2, whereas in path q, |V is transformed to |φV with H6 and Q3. BD2, H7, H8, and BD3 are used to combine different components denoted as Vp, Vq, Hp, and Hq. The post-selection state is chosen by H9, H10, and PBS3. The final two paths are combined by H11 and BD4, where the polarization state is rotated by H12, and the output state becomes |ψout.
Fig. 2.
Fig. 2. State evolution when υ=±0.4. The points in red circles represent the first initial states during the evolution. The arrows represent the evolution directions with the corresponding υ. The blue rhomboids and dark cyan pentagons represent evolved states with the initial state setting to be |H. The purple rhomboids and pink pentagons represent evolved states with the initial state setting to be |V. The olive hexagons and wine rhomboids represent evolved states with the initial state setting as (|H+|V)/2. The red hexagons and orange rhomboids represent evolved states with the initial state setting as (|H|V)/2. The solid lines represent corresponding theoretical predictions. The dashed lines are symmetric theoretical results with a global π phase difference. The light blue and light red regions correspond to the space-like and time-like regions, respectively. The boundary between these two regions corresponds to the light-like case. Error bars are estimated from the Poissonian counting statistics, which are small and are within the size of the symbols.
Fig. 3.
Fig. 3. State evolution with different υ. The initial states located in the red circles are set to be (|H+|V)/2 in the first row with (a) υ=0.3, (b) υ=0.4, and (c) υ=0.5, respectively, while the initial states located in the red circles are set to be |H with (d) υ=0.3, (e) υ=0.4, and (f) υ=0.5, respectively. Error bars are due to the Poissonian counting statistics, which are within the size of the symbols.
Fig. 4.
Fig. 4. New defined inner product of states as a function of n. The value is defined as (m2n2). The initial state is set to be |H with υ chosen as 0.3. The dashed line represents the theoretical prediction of 1. Error bars are estimated from counting statistics.

Equations (6)

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HBdG=(αiμ+νiμνα),
UBdG=eiHBdGt=(xy*yx*),
UBdG=11υ2(1υυ1),
Utot=|φ0s0s||0a0a|+|φ1s1s||1a0a|+|φ0s1s||0a1a|+|φ1s0s||1a1a|,
P=l|0a+|1a20a|+1a|2,
Ups=PUtot=12(|φ0s0s|+|φ1s1s|)|0a+|1a20a|,

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