$\mathcal{PT}$-symmetry in compact phase space for a linear Hamiltonian

TL;DR

This paper investigates the dynamics of a PT-symmetric non-Hermitian quantum system with a compact phase space, using a linear Hamiltonian in angular momentum, and analyzes state evolution via phase space distributions.

Contribution

It introduces a disentangling method for evolution operators that remains accurate near Exceptional Points and explores state evolution in a compact phase space for the first time.

Findings

Disentangling method remains accurate near Exceptional Points.

Coherent states follow classical trajectories in phase space.

Dicke states do not exhibit classical trajectory correspondence.

Abstract

We study the time evolution of a PT-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We focus on the simplest non-trivial example of such a Hamiltonian, which is linear in the angular momentum operators. In order to describe the evolution of the system, we use a particular disentangling decomposition of the evolution operator, which remains numerically accurate even in the vicinity of the Exceptional Point. We then analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states. For that purpose we calculate the Husimi distribution or Q function and study its evolution in phase space. For coherent states, the characteristics of the evolution equation of the Husimi function agree with the trajectories of the corresponding angular momentum expectation values. This allows to consider these curves as the trajectories of a classical system. For other types of quantum states, e.g. Dicke states, the equivalence of characteristics and trajectories of expectation values is lost.