Complex symmetric Hamiltonians and exceptional points of order four and five

Abstract

In the broad context of physics ranging from classical experimental optics to quantum mechanics of unitary as well as non-unitary systems there emerge interesting phenomena related to the presence of the so called Kato's exceptional points in the space of parameters. An elementary linear-algebraic method of their localization is proposed and applied to the class of tridiagonal $N$ by $N$ complex symmetric toy-model generators of evolution $H=H(\gamma )$. The implementation of the method is shown to provide new models with the exceptional points $\gamma ={\gamma }^{(EP)}$ of higher orders. Two distinct areas of applicability are expected to lie (1) in quantum mechanics of non-Hermitian (open as well as closed) systems, and (2) in the experiments using the coupled classical optical waveguides simulating the EP-related effects in the laboratory.

25 pp., 2 figures

References in corpus (20)

• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• Non-Hermitian robust edge states in one-dimension: Anomalous localization and eigenspace condensation at exceptional points
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Light stops at exceptional points
• Time-dependent version of cryptohermitian quantum theory
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Chirality of wave functions for three coalescing levels
• Exceptional points and lasing self-termination in photonic molecules
• Light transport in PT-invariant photonic structures with hidden symmetries
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
• Gegenbauer-solvable quantum chain model
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• A return to observability near exceptional points in a schematic PT-symmetric model
• Conditional observability
• The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Three solvable matrix models of a quantum catastrophe
• Determination of the domain of the admissible matrix elements in the four-dimensional PT-symmetric anharmonic model
• Simple models of three coupled $\mathcal{P}\mathcal{T}$-symmetric wave guides allowing for third-order exceptional points

Cited by in corpus (16)

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• Anomalous-order exceptional point and non-Markovian Purcell effect at threshold in one-dimensional continuum systems
• Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT-symmetric Su-Schrieffer-Heeger model
• Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy
• Confluences of exceptional points and a systematic classification of quantum catastrophes
• Arnold's potentials and quantum catastrophes
• Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians
• Anomalous mechanisms of the loss of observability in non-Hermitian quantum models
• Broken Hermiticity phase transition in Bose-Hubbard model
• Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via $\mathcal{P}\mathcal{T}-$symmetric and exactly solvable linear-Bose-Hubbard building blocks
• Quantum phase transitions mediated by clustered non-Hermitian degeneracies
• Intrinsic exceptional point -- a challenge in quantum theory
• Avoided level crossings in quasi-exact approach
• Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation
• Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional PT-symmetric dynamics and unidirectional coupling
• Scalable higher-order exceptional surface with passive resonators