Conserved quantities in non-Hermitian systems via vectorization method

TL;DR

This paper introduces a vectorization method to identify conserved quantities in non-Hermitian systems, including time-periodic cases, expanding the tools for analyzing open quantum and classical systems with balanced gain and loss.

Contribution

It presents a novel approach to derive conserved quantities in non-Hermitian systems, applicable to static and Floquet cases, complementing existing methods.

Findings

Successfully applied to a $ ext{PT}$-symmetric dimer example

Derived operators with exponential expectation value dynamics

Extended to time-periodic (Floquet) non-Hermitian systems

Abstract

Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time $(\mathcal{PT})$ symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a $\mathcal{PT}$-symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful addition to the well-established methods for characterizing time-invariants in non-Hermitian systems.