Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional PT-symmetric dynamics and unidirectional coupling

Jan Perina Jr; Kishore Thapliyal; Grzegorz Chimczak; Anna Kowalewska-Kudlaszyk; Adam Miranowicz

doi:10.22331/q-2025-12-10-1933

Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional PT-symmetric dynamics and unidirectional coupling

Jan Perina Jr¹, Kishore Thapliyal¹, Grzegorz Chimczak², Anna Kowalewska-Kudlaszyk², and Adam Miranowicz²

¹Joint Laboratory of Optics, Faculty of Science, Palacký University, Czech Republic, 17. listopadu 12, 771 46 Olomouc, Czech Republic
²Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

Published:	2025-12-10, volume 9, page 1933
Editor:	Pérola Milman
Eprint:	arXiv:2405.01667v3
Doi:	https://doi.org/10.22331/q-2025-12-10-1933
Citation:	Quantum 9, 1933 (2025).

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Updated after initial publication: This publication was updated to version v3 after the initial publication. The authors left the following comment on the arXiv:

Partial PT-symmetry like dynamics (nonconventional PT-symmetry) as well as non-Hermitian bosonic systems with unidirectional coupling are investigated. The work is in continuation of the paper "Quantum 9, 1932 (2025)". Version published in Quantum

Abstract

We analyze the existence and degeneracies of quantum exceptional, diabolical, and hybrid points in simple bosonic systems - comprising up to six modes with damping and/or amplification - under two complementary scenarios to those of Ref. [1]: (i) nonconventional PT-symmetric dynamics confined to a subspace of the full Liouville space, and (ii) systems featuring unidirectional coupling. The system dynamics described by quadratic non-Hermitian Hamiltonians is governed by the Heisenberg-Langevin equations. Conditions for the observation of inherited quantum hybrid points with up to sixth-order exceptional and second-order diabolical degeneracies are revealed, though relevant only for short-time dynamics. This raises the question of whether higher-order inherited singularities exist in bosonic systems under general conditions. Nevertheless, for short times, unidirectional coupling of various types enables the concatenation of simple bosonic systems with second- and third-order exceptional degeneracies such that arbitrarily high exceptional degeneracies are reached. Methods for numerical identifying the quantum exceptional and hybrid points together with their degeneracies are addressed. Following Ref. [1] rich dynamics of second-order field-operator moments is analyzed from the point of view of the presence of exceptional and diabolical points and their degeneracies.

Featured image: Four-mode bosonic systems in two configurations showing non-conventional higher-order hybrid singularities.

Popular summary

Parity-time (PT) symmetry describes systems in which balanced gain and loss give rise to real spectra and unconventional phase transitions. These phenomena are typically analyzed within the framework of non-Hermitian Hamiltonians, usually neglecting quantum jumps in the underlying quantum dynamics. Such transitions are governed by exceptional points (EPs)–singularities where eigenvalues and eigenvectors coalesce–distinct from diabolical points (DPs), where eigenvalues are degenerate, but eigenvectors remain linearly independent.

In open quantum systems, Liouvillian exceptional points mark the boundaries where distinct decay modes merge, giving rise to spectral and dynamical singularities that dominate the system’s evolution. Here, we investigate Liouvillian exceptional points in multimode bosonic systems governed by quadratic Hamiltonians, formulated in the Heisenberg picture. This framework provides direct access to higher-order exceptional, diabolical, and hybrid (diabolically degenerated exceptional) singularities manifested in the higher-order moments of field operators, offering clearer physical insight than the Schrödinger picture. Comprehensive analyses and classifications of such singularities are presented in Multiple Quantum Exceptional, Diabolical, and Hybrid Points in Multimode Bosonic Systems: I. Inherited and Genuine Singularities.

In the companion work, Part II: Nonconventional PT-Symmetric Dynamics and Unidirectional Coupling, we explore more unconventional configurations where PT symmetry is not globally defined but emerges only within restricted subspaces of the Liouvillian spectrum. We further analyze unidirectionally coupled systems, representing strongly non-Hermitian interactions that enable enhanced sensitivity and novel avenues for quantum sensing. While concatenation of such subsystems can engineer higher-order inherited degeneracies, we find that quantum consistency, i.e., the preservation of physical observables and commutation relations, is generally maintained only in short-time dynamics.

Our results establish a unified framework for understanding and controlling complex singular behaviors in non-Hermitian bosonic systems and open new directions for quantum sensing, mode control, and non-Hermitian quantum engineering.

► BibTeX data

@article{PerinaJr2025multiplequantum,
  doi = {10.22331/q-2025-12-10-1933},
  url = {https://doi.org/10.22331/q-2025-12-10-1933},
  title = {Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: {II}. {N}onconventional {PT}-symmetric dynamics and unidirectional coupling},
  author = {Perina Jr, Jan and Thapliyal, Kishore and Chimczak, Grzegorz and Kowalewska-Kudlaszyk, Anna and Miranowicz, Adam},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1933},
  month = dec,
  year = {2025}
}

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Cited by

[1] Javid Naikoo, Ravindra W Chhajlany, Jan Kołodyński, and Adam Miranowicz, "Precision assessment in non-Hermitian systems: a comparative study of three formalisms", New Journal of Physics 28 3, 034508 (2026).

[2] Huawei ZHAO, Xinlei LIU, Xinyao HUANG, and Guofeng ZHANG, "Research progress of non-Hermitian dynamics in quadratic bosonic systems", Acta Physica Sinica 75 4(2026).

[3] Kishore Thapliyal, Jan Perina Jr., Grzegorz Chimczak, Anna Kowalewska-Kudlaszyk, and Adam Miranowicz, "Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities", Quantum 9, 1932 (2025).

[4] Marek Kopciuch and Adam Miranowicz, "Liouvillian and Hamiltonian exceptional points of atomic vapors: The spectral signatures of quantum jumps", Physical Review Research 7 3, 033187 (2025).

[5] Jan Peřina, Karol Bartkiewicz, Grzegorz Chimczak, Anna Kowalewska-Kudlaszyk, Adam Miranowicz, Joanna K. Kalaga, and Wiesław Leoński, "Quantumness and its hierarchies in <inline-formula><mml:math><mml:mi>PT</mml:mi></mml:math></inline-formula>-symmetric down-conversion models", Physical Review A 112 4, 043545 (2025).

[6] Javid Naikoo, Ravindra W. Chhajlany, Jan Kołodyński, and Adam Miranowicz, "Precision assessment in non-Hermitian systems: a comparative study of three formalisms", arXiv:2506.22571, (2025).

[7] Huawei Zhao, Xinlei Liu, Xinyao Huang, and Guofeng Zhang, "Advances in non-Hermitian dynamics of quadratic bosonic systems", arXiv:2601.14329, (2026).

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