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A Quantum Description of Physical Systems with Non-real Energies

Published Jan 1, 2021• 0 citations
Publisher

Abstract

While quantum systems are traditionally described by Hermitian Hamiltonians, the formalism is extendable to a non-Hermitian description for systems that are dissipative or obey parity-time symmetry.
DOI
10.7566/jpsht.1.044

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Abstract Hermiticity is the fundamental property of the physical systems, for which it should obey the laws of energy conservation and time-reversal symmetry. Parity-time symmetry is introduced to study the non-Hermitian system with real energy spectra. Considering the coupling dissipation of the practical system, there would be phase-related terms on the non-diagonal elements of the Hermitian matrix. Here in this work, we present a coupled circuit model of the system and investigate the dynamics of parity-time-anyonic Hamiltonian related to an arbitrary phase of the system. We find that parity-time symmetry can be achieved under the tunable phases in the quasi-parity-time symmetric circuit system.
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.
The exploration of non-Hermitian physics has predominantly been a frontier in quantum mechanics, revealing exotic phenomena not observed in conventional Hermitian systems. However, its extension to classical mechanics offers a rich and largely untapped avenue for understanding open and active systems characterized by dissipation and gain. This paper systematically investigates the theoretical foundations of non-Hermitian classical mechanics, focusing on the emergence and implications of dissipation and gain symmetries. We demonstrate how concepts like parity-time (PT) symmetry and pseudo-Hermiticity, originally quantum mechanical, find natural classical analogues in systems with balanced energy loss and amplification. The study delves into the modification of classical Hamiltonian and Lagrangian formalisms to accommodate non-conservative forces, leading to complex-valued potentials and phase space trajectories. We explore the role of exceptional points as spectral degeneracies in these classical systems, showing how they govern the dynamics and give rise to unique phenomena such as unidirectional transport and enhanced sensitivity. Furthermore, we discuss the practical implications for designing classical mechanical systems with tailored dissipative and amplification properties, opening new possibilities for classical sensing, energy harvesting, and wave manipulation. This work aims to establish a robust theoretical framework for non-Hermitian classical mechanics, highlighting its potential to unveil emergent symmetries and complex dynamics in macroscopic systems.