Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems

TL;DR

This paper investigates conserved quantities in non-Hermitian open systems, especially those with $ ext{PT}$-symmetry and other antilinear symmetries, revealing their implications for system dynamics and providing explicit methods to identify these conserved observables.

Contribution

It introduces a spectral-decomposition method to find all conserved observables in $ ext{PT}$-symmetric and other antilinear symmetric non-Hermitian systems, generalizing previous understanding.

Findings

Identifies all conserved quantities in $ ext{PT}$-symmetric systems.

Extends conservation law analysis to systems with general antilinear symmetries.

Provides explicit recursive procedures for obtaining conserved observables.

Abstract

Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general $\mathcal{PT}$-symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.