Signature of exceptional point phase transition in Hermitian systems

T. T. Sergeev1,2,3, A. A. Zyablovsky1,2,3,4, E. S. Andrianov1,2,3, and Yu. E. Lozovik5,6

1Dukhov Research Institute of Automatics, 127055, 22 Sushchevskaya, Moscow Russia
2Moscow Institute of Physics and Technology, 141700, 9 Institutskiy pereulok, Dolgoprudny Moscow region, Russia
3Institute for Theoretical and Applied Electromagnetics, 125412, 13 Izhorskaya, Moscow Russia
4Kotelnikov Institute of Radioengineering and Electronics RAS, 125009, 11-7 Mokhovaya, Moscow Russia
5Institute of Spectroscopy Russian Academy of Sciences, 108840, 5 Fizicheskaya, Troitsk, Moscow, Russia
6MIEM at National Research University Higher School of Economics, 101000, 20 Myasnitskaya, Moscow, Russia

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Exceptional point (EP) is a spectral singularity in non-Hermitian systems. The passing over the EP leads to a phase transition, which endows the system with unconventional features that find a wide range of applications. However, the need of using the dissipation and amplification limits the possible applications of systems with the EP. In this work, we demonstrate an existence of signature of exceptional point phase transition in Hermitian systems that are free from dissipation and amplification. We consider a composite Hermitian system including both two coupled oscillators and their environment consisting only of several tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a transition, which occurs at the coupling strength between oscillators corresponding to the EP in the non-Hermitian system. This transition manifests itself even in the non-Markovian regime of the system dynamics in which collapses and revivals of the energy occur. Thus, we demonstrate that the phase transition occurring at the passing over the EP in the non-Hermitian system manifests itself in the Hermitian system at all time. We discuss the experimental scheme to observe the signature of EP phase transition in the non-Markovian regime.

🇺🇦 Quantum strongly condemns the 2022 invasion of Ukraine, the loss of life and war crimes inflicted by Russian forces. For more information on our policy on publishing articles by authors based in Russian institutions, see this post

An interaction of system with an environment causes an energy exchange between them. At times smaller than Poincare return time, the energy exchange leads to relaxation processes in the system. At times less than the return time, the systems interacting with the environment are often considered as non-Hermitian. The eigenstates of non-Hermitian systems are no mutually orthogonal. The point in the space of system parameters, at which some of eigenstates coalesce and their eigenvalues coincide is called an exceptional point (EP) of non-Hermitian system. The passing over the EP is accompanied by qualitative changes in the eigenstates, which is referred to as an EP phase transition. At times greater the return time, the system dynamics demonstrate collapses and revivals, which are due to finite size of the environment. In this case, the non-Hermitian consideration is not suitable and an existence of the EP phase transitions early does not discuss.
We demonstrate an existence of a signature of the EP phase transition at times greater the Poincare return time. We consider a Hermitian system including environment consisting of only a few tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a signature of the EP phase transition at times both smaller and greater than the return time. This transition occurs at the system parameters corresponding to the EP in the non-Hermitian system. We introduce an order parameter that characterizes the EP phase transition in both the Hermitian and non-Hermitian systems. We propose an experimental scheme to observe the signature of EP phase transition in the Hermitian system at time greater than the return time. Our results extend the concept of the EP phase transition to the Hermitian systems.

► BibTeX data

► References

[1] C. M. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80(24), 5243 (1998).

[2] N. Moiseyev. Non-Hermitian quantum mechanics, Cambridge University Press (2011).

[3] 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(1), 205-214 (2002).

[4] M. A. Miri, A. Alu. Exceptional points in optics and photonics, Science 363, 6422 (2019).

[5] S. K. Ozdemir, S. Rotter, F. Nori, L. Yang. Parity–time symmetry and exceptional points in photonics, Nature Mater. 18, 783 (2019).

[6] M.V. Berry. Physics of nonhermitian degeneracies, Czech. J. Phys. 54, 1039 (2004).

[7] C. M. Bender. Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947 (2007).

[8] W. D. Heiss. The physics of exceptional points, J. Phys. A 45, 444016 (2012).

[9] B. B. Wei, L. Jin. Universal critical behaviours in non-Hermitian phase transitions, Sci. Rep. 7, 7165 (2017).

[10] F. E. Öztürk, T. Lappe, G. Hellmann, et al. Observation of a non-Hermitian phase transition in an optical quantum gas, Science 372(6537), 88-91 (2021).

[11] T. T. Sergeev, A. A. Zyablovsky, E. S. Andrianov, et al. A new type of non-Hermitian phase transition in open systems far from thermal equilibrium, Sci. Rep. 11, 24054 (2021).

[12] A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, A. A. Lisyansky. PT-symmetry in optics, Phys. Usp. 57, 1063-1082 (2014).

[13] R. El-Ganainy, K. G. Makris, M. Khajavikhan, et al. Non-Hermitian physics and PT symmetry, Nat. Phys. 14(1), 11-19 (2018).

[14] S. Longhi. Parity-time symmetry meets photonics: A new twist in non-Hermitian optics, Europhys. Lett. 120, 64001 (2018).

[15] J. B. Khurgin. Exceptional points in polaritonic cavities and subthreshold Fabry–Perot lasers, Optica 7(8), 1015-1023 (2020).

[16] A. A. Zyablovsky, I. V. Doronin, E. S. Andrianov, A. A. Pukhov, Y. E. Lozovik, A. P. Vinogradov, A.A. Lisyansky. Exceptional points as lasing prethresholds, Laser Photonics Rev. 15, 2000450 (2021).

[17] T. Gao, E. Estrecho, K. Y. Bliokh, et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard, Nature 526, 554 (2015).

[18] D. Zhang, X. Q. Luo, Y. P. Wang, T. F. Li, J. Q. You. Observation of the exceptional point in cavity magnon-polaritons, Nat. Commun. 8, 1368 (2017).

[19] G. Q. Zhang, J. Q. You. Higher-order exceptional point in a cavity magnonics system, Phys. Rev. B 99(5), 054404 (2019).

[20] H. Xu, D. Mason, L. Jiang, J. G. E. Harris.Topological energy transfer in an optomechanical system with exceptional points, Nature 537(7618), 80-83 (2016).

[21] J. Zhang, B. Peng, Ş. K. Özdemir, et al. A phonon laser operating at an exceptional point, Nature Photon. 12(8), 479-484 (2018).

[22] Y. X. Wang, A. A. Clerk. Non-Hermitian dynamics without dissipation in quantum systems, Phys. Rev. A 99(6), 063834 (2019).

[23] I. V. Doronin, A. A. Zyablovsky, E. S. Andrianov, A. A. Pukhov, A. P. Vinogradov. Lasing without inversion due to parametric instability of the laser near the exceptional point, Phys. Rev. A 100, 021801(R) (2019).

[24] Y.-H. Lai, Y.-K. Lu, M.-G. Suh, Z. Yuan, K. Vahala. Observation of the exceptional-point-enhanced Sagnac effect, Nature 576, 65 (2019).

[25] H. Hodaei, A.U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D.N. Christodoulides, M. Khajavikhan. Enhanced sensitivity at higher-order exceptional points, Nature 548, 187 (2017).

[26] W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, L. Yang. Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017).

[27] 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).

[28] Z. P. Liu, J. Zhang, Ş. K. Özdemir, et al. Metrology with PT-symmetric cavities: enhanced sensitivity near the PT-phase transition, Phys. Rev. Lett. 117, 110802 (2016).

[29] A. A. Zyablovsky, E. S. Andrianov, A. A. Pukhov. Parametric instability of optical non-Hermitian systems near the exceptional point, Sci. Rep. 6, 29709 (2016).

[30] S. Longhi. Bloch Oscillations in Complex Crystals with PT Symmetry, Phys. Rev. Lett. 103(12), 123601 (2009).

[31] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides. Unidirectional invisibility induced by PT-symmetric periodic structures, Phys. Rev. Lett. 106(21), 213901 (2011).

[32] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani. Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100(10), 103904 (2008).

[33] S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, Y. S. Kivshar. Nonlinear switching and solitons in PT-symmetric photonic systems, Laser Photonics Rev. 10(2), 177-213 (2016).

[34] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip. Observation of parity–time symmetry in optics, Nat. Phys. 6(3), 192-195 (2010).

[35] A. Guo, G. J. Salamo, D. Duchesne, et al. Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103(8), 093902 (2009).

[36] H. Hodaei, M.-A. Miri, M. Heinrich, D.N. Christodoulidies, M. Khajavikan. Parity-time–symmetric microring lasers, Science 346, 975 (2014).

[37] L. Feng, Z.J. Wong, R.-M. Ma, Y. Wang, X. Zhang. Single-mode laser by parity-time symmetry breaking, Science 346, 972 (2014).

[38] B. Peng, Ş. K. Özdemir, M. Liertzer, et al. Chiral modes and directional lasing at exceptional points, Proc. Natl. Acad. Sci. 113(25), 6845-6850 (2016).

[39] M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, S. Rotter. Pump-induced exceptional points in lasers, Phys. Rev. Lett. 108(17), 173901 (2012).

[40] I. V. Doronin, A. A. Zyablovsky, E. S. Andrianov. Strong-coupling-assisted formation of coherent radiation below the lasing threshold, Opt. Express 29, 5624 (2021).

[41] J. Wiersig. Prospects and fundamental limits in exceptional point-based sensing, Nat. Commun. 11, 2454 (2020).

[42] J. Wiersig. Review of exceptional point-based sensors, Photonics Res. 8, 1457-1467 (2020).

[43] H. Wang, Y. H. Lai, Z. Yuan, M. G. Suh, K. Vahala. Petermann-factor sensitivity limit near an exceptional point in a Brillouin ring laser gyroscope, Nat. Commun. 11, 1610 (2020).

[44] W. Langbein. No exceptional precision of exceptional-point sensors, Phys. Rev. A 98(2), 023805 (2018).

[45] M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, L. Jiang. Quantum noise theory of exceptional point amplifying sensors, Phys. Rev. Lett. 123(18), 180501 (2019).

[46] C. Chen, L. Zhao. The effect of thermal-induced noise on doubly-coupled-ring optical gyroscope sensor around exceptional point, Opt. Commun. 474, 126108 (2020).

[47] H. K. Lau, A. A. Clerk. Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing, Nature Commun. 9, 4320 (2018).

[48] C. Wolff, C. Tserkezis, N. A. Mortensen. On the time evolution at a fluctuating exceptional point, Nanophotonics 8(8), 1319-1326 (2019).

[49] R. Duggan, S. A. Mann, A. Alu. Limitations of sensing at an exceptional point, ACS Photonic 9(5), 1554-1566 (2022).

[50] H.-P. Breuer, E.-M. Laine, J. Piilo, B. Vacchini. Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).

[51] I. de Vega, D. Alonso. Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).

[52] M. O. Scully, M. S. Zubairy. Quantum optics, Cambridge University Press: Cambridge (1997).

[53] H. Carmichael. An open systems approach to quantum optics, Springer-Verlag, Berlin (1991).

[54] C. W. Gardiner, P. Zoller. Quantum noise: A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics, Springer-Verlag, Berlin (2004).

[55] T. T. Sergeev, I. V. Vovchenko, A. A. Zyablovsky, E. S. Andrianov. Environment-assisted strong coupling regime, Quantum 6, 684 (2022).

[56] 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(1), 205-214 (2002).

[57] L. D. Landau, L. E. Lifshitz. Statistical Physics: Volume 5, Elsevier (1980).

[58] Y. Akahane, T. Asano, B.-S. Song, S. Noda. High-Q photonic nanocavity in a two-dimensional photonic crystal, Nature 425, 944 (2003).

[59] D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala. Ultra-high-Q toroid microcavity on a chip, Nature 421, 925 (2003).

[60] Y. Akahane, T. Asano, B.-S. Song, S. Noda. Fine-tuned high-Q photonic-crystal nanocavity, Opt. Express 13(4), 1202 (2005).

[61] T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, H. Taniyama. Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity, Nature Photon. 1, 49 (2007).

[62] X.-F. Jiang, C.-L. Zou, L. Wang, Q. Gong, Y.-F. Xiao. Whispering-gallery microcavities with unidirectional laser emission, Laser Photonics Rev. 10(1), 40-61 (2016).

[63] R. J. Schoelkopf, S. M. Gir. Wiring up quantum systems, Nature 451, 664 (2008).

[64] A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, A. Wallraff. Photon-mediated interactions between distant artificial atoms, Science 342, 1494 (2013).

[65] G. Andersson, B. Suri, L. Guo, T. Aref, P. Delsing. Non-exponential decay of a giant artificial atom, Nat. Physics 15, 1123-1127 (2019).

[66] N. M. Sundaresan, R. Lundgren, G. Zhu, A. V. Gorshkov, A. A. Houck. Interacting qubit-photon bound states with superconducting circuits, Phys. Rev. X 9, 011021 (2019).

[67] K. Lalumiere, B. C. Sanders, A. F. van Loo, A. Fedorov, A. Wallraff, A. Blais. Input-output theory for waveguide QED with an ensemble of inhomogeneous atoms, Phys. Rev. A 88, 043806 (2013).

[68] D. Vion, A. Aassime, A. Cottet, et al. Manipulating the quantum state of an electrical circuit, Science 296, 886 (2002).

[69] J. Koch, T. M. Yu, J. Gambetta, et al. Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007).

[70] V. S. Ferreira, J. Banker, A. Sipahigil, et al. Collapse and revival of an artificial atom coupled to a structured photonic reservoir, Phys. Rev. X 11(4), 041043 (2021).

[71] V. I. Tatarskii. Example of the description of dissipative processes in terms of reversible dynamic equations and some comments on the fluctuation-dissipation theorem, Sov. Phys. Usp. 30(2), 134 (1987).

Cited by

[1] Timofey T. Sergeev, Alexander A. Zyablovsky, Evgeny S. Andrianov, and Yurii E. Lozovik, "Self-consistent description of relaxation processes in systems with ultra- and deep-strong coupling", Journal of the Optical Society of America B 40 11, 2743 (2023).

[2] Bijan Bagchi and Sauvik Sen, "Artificial Hawking radiation, weak pseudo-Hermiticity, and Weyl semimetal blackhole analogy", Journal of Mathematical Physics 63 12, 122102 (2022).

[3] Artem Mukhamedyanov, Alexander A. Zyablovsky, and Evgeny S. Andrianov, "Subthreshold phonon generation in an optomechanical system with an exceptional point", Optics Letters 48 7, 1822 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-02-27 21:05:46) and SAO/NASA ADS (last updated successfully 2024-02-27 21:05:47). The list may be incomplete as not all publishers provide suitable and complete citation data.