Non-Hermitian Hamiltonians and stability of pure states

TL;DR

This paper investigates how quantum fluctuations and environment-induced anti-Hermitian Hamiltonian parts can cause pure states to become unstable and evolve into mixed states, with a formalism applicable to open quantum systems.

Contribution

It introduces a quantum-statistical framework for analyzing the stability of pure states under non-Hermitian Hamiltonians, including derivation of stability equations and analysis of solvable models.

Findings

Pure states can become unstable due to quantum fluctuations.

Anti-Hermitian parts of Hamiltonians control instability.

Stability analysis applied to two-state models.

Abstract

We demonstrate that quantum fluctuations can cause, under certain conditions, the dynamical instability of pure states that can result in their evolution into mixed states. It is shown that the degree and type of such an instability are controlled by the environment-induced anti-Hermitian parts of Hamiltonians. Using the quantum-statistical approach for non-Hermitian Hamiltonians and related non-linear master equation, we derive the equations that are necessary to study the stability properties of any model described by a non-Hermitian Hamiltonian. It turns out that the instability of pure states is not preassigned in the evolution equation but arises as the emergent phenomenon in its solutions. In order to illustrate the general formalism and different types of instability that may occur, we perform the local stability analysis of some exactly solvable two-state models, which can be used in the theories of open quantum-optical and spin systems.