Solvable dilation model of $\cal PT$-symmetric systems

TL;DR

This paper introduces an exactly solvable model for a two-dimensional, time-dependent $ ext{PT}$-symmetric quantum system that transitions through exceptional points, providing analytical insights into its dilation and evolution.

Contribution

It presents the first explicit analytical solution for a time-dependent $ ext{PT}$-symmetric dilation problem crossing exceptional points.

Findings

Exceptional points are less physically relevant in time-dependent systems.

Analytical solutions for the dilated Hamiltonian and system evolution are obtained.

System transitions from unbroken to broken $ ext{PT}$-symmetry phase.

Abstract

The dilation method is a practical way to experimentally simulate non-Hermitian, especially $\cal PT$-symmetric quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet non-trivial exactly solvable dilation problem with two dimensional time-dependent $\cal PT$-symmetric Hamiltonian. Our system is initially set in the unbroken $\cal PT$-symmetric phase and later goes across the so-called exceptional point and enters the broken $\cal PT$-symmetric phase. For this system, the dilated Hamiltonian and the evolution of $\cal PT$-symmetric system are analytically worked out. Our result clearly showed that the exceptional points do not have much physical relevance in a \textit{time-dependent} system.