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The European Physical Journal Plus

Real spectra in some negative potentials: linear and nonlinear one-dimensional PT-invariant quantum systems

Biswanath Rath

Published May 1, 2021• 5 citations
Publisher
DOI
10.1140/epjp/s13360-021-01295-1

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Inspired by the discovery of real energy spectra of non-Hermitian Hamiltonians obeying parity-time (PT) symmetry, many proposals and relevant experiments have been suggested and implemented for the realization of PT-symmetric Hamiltonians in various physical systems. Among them PT-symmetric atomic gases are of great interest as they possess many unique advantages, including: (i) the possession of active PT symmetry with balanced optical gain and loss, (ii) the ability to be extended to high-dimensional ones, and (iii) the feasibility to acquire enhanced Kerr nonlinearity based on the resonance character between light and atoms. Here, we review the new progresses made on the study of PT symmetry in coherent atomic gases since our last review paper [Advances in Physics: X, 2, 737–783 (2017)]. Particularly, we focus on the newly emerged research topics in this direction, including nonlocal nonlinear effect, non-Hermitian quantum effect, exceptional point and associated phenomenon, and non-Hermitian skin effect with PT symmetry. Perspectives and potential applications of the subject are also discussed.

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On the spectrum of the system of interacting quantum nonlinear resonances

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We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.
Inspired by the discovery of real energy spectra of non-Hermitian Hamiltonians obeying parity-time (PT) symmetry, many proposals and relevant experiments have been suggested and implemented for the realization of PT-symmetric Hamiltonians in various physical systems. Among them PT-symmetric atomic gases are of great interest as they possess many unique advantages, including: (i) the possession of active PT symmetry with balanced optical gain and loss, (ii) the ability to be extended to high-dimensional ones, and (iii) the feasibility to acquire enhanced Kerr nonlinearity based on the resonance character between light and atoms. Here, we review the new progresses made on the study of PT symmetry in coherent atomic gases since our last review paper [Advances in Physics: X, 2, 737–783 (2017)]. Particularly, we focus on the newly emerged research topics in this direction, including nonlocal nonlinear effect, non-Hermitian quantum effect, exceptional point and associated phenomenon, and non-Hermitian skin effect with PT symmetry. Perspectives and potential applications of the subject are also discussed.