- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantumness and its hierarchies in PT-symmetric down-conversion models

Jan Peřina, Jr.*

Karol Bartkiewicz, Grzegorz Chimczak, Anna Kowalewska-Kudlaszyk, and Adam Miranowicz

Joanna K. Kalaga and Wiesław Leoński

  • Quantum Optics and Engineering Division, Faculty of Physics and Astronomy, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

  • *Contact author: jan.perina.jr@upol.cz
PDF

Phys. Rev. A 112, 043545 – Published 29 October, 2025

DOI: https://doi.org/10.1103/9vty-ctf7

Abstract

We investigate the hierarchy of quantum correlations in a quadratic bosonic parity-time-symmetric system (PTSS) featuring distinct dissipation and amplification channels. The hierarchy includes global nonclassicality, entanglement, asymmetric quantum steering, and Bell nonlocality. We elucidate the interplay between the system physical nonlinearity (which serves as a source of quantumness) and the specific dynamics of bosonic PTSSs, which are qualitatively influenced by their damping and amplification characteristics. Using a set of quantifiers (including local and global nonclassicality depths, negativity, steering parameters, and the Bell parameter) we demonstrate that the standard PTSS typically exhibits weaker quantumness than its counterparts affected solely by damping or solely by amplification. Both the maximum values attained by these quantifiers and the speed and duration of their generation are generally lower in the standard PTSS. A comparative analysis of three two-mode PTSSs (standard, passive, and active) with identical eigenvectors and real parts of eigenfrequencies, but differing in their damping and amplification strengths, reveals the crucial role of quantum fluctuations associated with gain and loss. Among them, the passive PTSS yields the most strongly nonclassical states. Nevertheless, under suitable conditions, the standard PTSS can also generate highly nonclassical states. The supremacy of the passive PTSS is further supported by its fundamental advantages in practical realizations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  2. C. M. Bender, S. Boettcher, and P. N. Meisinger, PT-symmetric quantum mechanics, J. Math. Phys. 40, 2201 (1999).
  3. C. M. Bender, D. C. Brody, and H. F. Jones, Must a Hamiltonian be Hermitian? Am. J. Phys. 71, 1095 (2003).
  4. S. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity-time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  5. M. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  6. 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).
  7. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature (London) 548, 192 (2017).
  8. H. Hodaei, U. H. Absar, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  9. B. He, S.-B. Yan, J. Wang, and M. Xiao, Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides, Phys. Rev. A 91, 053832 (2015).
  10. S. Vashahri-Ghamsari, B. He, and M. Xiao, Continuous-variable entanglement generation using a hybrid PT-symmetric system, Phys. Rev. A 96, 033806 (2017).
  11. J. Peřina, Jr. and A. Lukš, Quantum behavior of a PT-symmetric two-mode system with cross-Kerr nonlinearity, Symmetry 11, 1020 (2019).
  12. J. Peřina, Jr., A. Lukš, J. K. Kalaga, W. Leoński, and A. Miranowicz, Nonclassical light at exceptional points of a quantum PT-symmetric two-mode system, Phys. Rev. A 100, 053820 (2019).
  13. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. Bender, and L. Yang, Parity–time-symmetric whispering-gallery microcavities, Nat. Phys. 10, 394 (2014).
  14. 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 (2014).
  15. 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).
  16. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Parity-time synthetic photonic lattices, Nature (London) 488, 167 (2012).
  17. S. Scheel and A. Szameit, PT-symmetric photonic quantum systems with gain and loss do not exist, Europhys. Lett. 122, 34001 (2018).
  18. R. W. Boyd, Nonlinear Optics (Academic, New York, 2003).
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, England, 1995).
  20. J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer, Dordrecht, The Netherlands, 1991).
  21. J. Peřina, Jr. and J. Peřina, Quantum statistics of nonlinear optical couplers, in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 2000), Vol. 41, pp. 361–419.
  22. Parity-Time Symmetry and its Applications, edited by D. N. Christodoulides and J. Yang (Springer, Singapore, 2018).
  23. C. M. Bender, PT symmetry, in Quantum and Classical Physics (World Scientific, Singapore, 2020).
  24. V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88, 035002 (2016).
  25. L. Feng, R. El-Ganainy, and L. Ge, Non-Hermitian photonics based on parity-time symmetry, Nat. Photon. 11, 752 (2017).
  26. S. K. Gupta, Y. Zou, X.-Y. Zhu, M.-H. Lu, L.-J. Zhang, X.-P. Liu, and Y.-F. Chen, Parity-time symmetry in non-Hermitian complex optical media, Adv. Mater. 32, 1903639 (2020).
  27. 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. 16, 15 (2020).
  28. R. J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131, 2766 (1963).
  29. E. C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Rev. Lett. 10, 277 (1963).
  30. J. Peřina, Jr., A. Miranowicz, J. K. Kalaga, and W. Leoński, Unavoidability of nonclassicality loss in PT-symmetric systems, Phys. Rev. A 108, 033512 (2023).
  31. G. Chimczak, A. Kowalewska-Kudlaszyk, E. Lange, K. Bartkiewicz, and J. Peřina, Jr., The effect of thermal photons on exceptional points in coupled resonators, Sci. Rep. 13, 5859 (2023).
  32. J. Kadlec, K. Bartkiewicz, A. Černoch, K. Lemr, and A. Miranowicz, Experimental hierarchy of the nonclassicality of single-qubit states via potentials for entanglement, steering, and Bell nonlocality, Opt. Express 32, 2333 (2024).
  33. C. T. Lee, Measure of the nonclassicality of nonclassical states, Phys. Rev. A 44, R2775 (1991).
  34. S. A. Hill and W. K. Wootters, Computable entanglement, Phys. Rev. Lett. 78, 5022 (1997).
  35. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  36. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Phys. Rev. A 80, 032112 (2009).
  37. J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  38. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  39. J. Peřina, Jr., A. Miranowicz, G. Chimczak, and A. Kowalewska-Kudlaszyk, Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach, Quantum 6, 883 (2022).
  40. K. Thapliyal, J. Peřina, Jr., A. K. G. Chimczak, and A. Miranowicz, Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities, arXiv:2405.01666.
  41. J. Peřina, Jr., K. Thapliyal, A. K. G. Chimczak, and A. Miranowicz, Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional pt-symmetric dynamics and unidirectional coupling, arXiv:2405.01667.
  42. W. Vogel, D. G. Welsch, and S. Walentowicz, Quantum Optics (Wiley-VCH, Weinheim, Germany, 2001).
  43. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer, Berlin, 2007).
  44. S. B. Jaäger, T. Schmit, G. Morigi, M. J. Holland, and R. Betzholz, Lindblad master equations for quantum systems coupled to dissipative bosonic modes, Phys. Rev. Lett. 129, 063601 (2022).
  45. G. Adesso and F. Illuminati, Entanglement in continuous variable systems: Recent advances and current perspectives, J. Phys. A: Math. Theor. 40, 7821 (2007).
  46. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1989).
  47. G. S. Agarwal and K. Qu, Spontaneous generation of photons in transmission of quantum fields in PT-symmetric optical systems, Phys. Rev. A 85, 031802(R) (2012).
  48. V. Peřinová, A. Lukš, and J. Křepelka, Quantum description of a PT-symmetric nonlinear directional coupler, J. Opt. Soc. Am. B 36, 855 (2019).
  49. P.-C. K. J.-D. Lin, N. Lambert, A. Miranowicz, F. Nori, and Y.-N. Chen, Non-Markovian quantum exceptional points, Nat. Commun. 16, 1289 (2025).
  50. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  51. P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232, 333 (1997).
  52. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  53. K. Banaszek and K. Wódkiewicz, Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation, Phys. Rev. A 58, 4345 (1998).
  54. K. Banaszek and K. Wódkiewicz, Testing quantum nonlocality in phase space, Phys. Rev. Lett. 82, 2009 (1999).
  55. S. Olivares and M. G. A. Paris, Enhancement of nonlocality in phase space, Phys. Rev. A 70, 032112 (2004).
  56. K. Thapliyal and J. Peřina, Jr., Ideal pairing of the Stokes and anti-Stokes photons in the Raman process, Phys. Rev. A 103, 033708 (2021).
  57. J. Kadlec, K. Bartkiewicz, A. Černoch, K. Lemr, and A. Miranowicz, Experimental relative entanglement potentials of single-photon states, Phys. Rev. A 110, 023720 (2024).
  58. M. Mehmet, S. Ast, T. Eberle, S. Steinlechner, H. Vahlbruch, and R. Schnabel, Squeezed light at 1550 nm with a quantum noise reduction of 12.3 db, Opt. Express 19, 25763 (2011).
  59. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum state tomography, Rev. Mod. Phys. 81, 299 (2009).
  60. M. Drong, T. Fordos, H. Y. Jaffres, J. Peřina, Jr., K. Postava, J. Pištora, and H. J. Drouhin, Local and mean-field approaches for modeling semiconductor spin-lasers, J. Opt. 22, 055001 (2020).
  61. M. Drong, T. Fordos, H. Y. Jaffres, J. Peřina, Jr., K. Postava, P. Ciompa, J. Pištora, and H. J. Drouhin, Spin-VCSELs with local optical anisotropies: Toward terahertz polarization modulation, Phys. Rev. Appl. 15, 014041 (2021).
  62. M. Drong, M. Dems, J. Peřina, Jr., T. Fordos, H. Y. Jaffres, K. Postava, and H. J. Drouhin, Time-dependent laser cavity perturbation theory: Exploring future nano-structured photonic devices in semi-analytic way, J. Lightwave Technol. 40, 4735 (2022).
  63. M. Drong, J. Peřina Jr., T. Fördös, H. Y. Jaffrès, K. Postava, and H.-J. Drouhin, Spin vertical-cavity surface-emitting lasers with linear gain anisotropy: Prediction of exceptional points and nontrivial dynamical regimes, Phys. Rev. A 107, 033509 (2023).
  64. J. Perina, K. Bartkiewicz, G. Chimczak, A. Kowalewska-Kudłaszyk, A. Miranowicz, J. Kalaga, and W. Leonski, Quantumness and its hierarchies in PT-symmetric down-conversion models [Data set], In Physical Review A (1.0), Zenodo (2025), https://doi.org/10.5281/zenodo.17243232.

Outline

Information

Phys. Rev. A 112, 043545– Published 29 October, 2025
  • Received 14 May 2025
  • Accepted 23 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation