Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities

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

doi:10.22331/q-2025-12-10-1932

Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities

Kishore Thapliyal¹, Jan Perina Jr.¹, 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 1932
Editor:	Pérola Milman
Eprint:	arXiv:2405.01666v2
Doi:	https://doi.org/10.22331/q-2025-12-10-1932
Citation:	Quantum 9, 1932 (2025).

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Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:

Inherited EPs up to third-order singularities are observed in the bosonic systems with the usual (bidirectional) coupling. In the following paper "Quantum 9, 1933 (2025)", the investigation are further extended to other bosonic systems. Version published in Quantum

Abstract

The existence and degeneracies of quantum exceptional, diabolical, and hybrid (i.e., diabolically degenerated exceptional) singularities of simple bosonic systems composed of up to five modes with damping and/or amplification are analyzed. Their dynamics governed by quadratic non-Hermitian Hamiltonians is followed using the Heisenberg-Langevin equations. Their dynamical matrices generally exhibit specific structures that allow for an effective reduction of their dimension by half. This facilitates analytical treatment and enables efficient spectral analysis based on characteristic second-order diabolical degeneracies. Conditions for the observation of inherited quantum hybrid points, observed directly in the dynamics of field operators, having up to third-order exceptional and second-order diabolical degeneracies are revealed. Surprisingly, exceptional degeneracies of only second and third orders are revealed, even though the systems with up to five modes are considered. Exceptional and diabolical genuine points and their degeneracies observed in the dynamics of second-order field-operator moments are also analyzed. Each analyzed bosonic system exhibits its own unique and complex dynamical behavior.

Featured image: Four-mode bosonic system in linear configuration showing 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 the present work—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{Thapliyal2025multiplequantum,
  doi = {10.22331/q-2025-12-10-1932},
  url = {https://doi.org/10.22331/q-2025-12-10-1932},
  title = {Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: {I}. {I}nherited and genuine singularities},
  author = {Thapliyal, Kishore and Perina Jr., Jan 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 = {1932},
  month = dec,
  year = {2025}
}

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