Real eigenvalues in non-Hermitian Hamilton quantum physics

TL;DR

This paper explores how non-Hermitian Hamiltonians in quantum and optical systems exhibit real eigenvalues under certain conditions, with phase transitions occurring at eigenvalue crossings, enabling the design of systems with tailored properties.

Contribution

It demonstrates the formal equivalence between quantum and optical systems and analyzes the conditions under which eigenvalues remain real or become complex, revealing insights into dynamical phase transitions.

Findings

Eigenvalues are real in PT symmetric optical lattices when environmental influence is small.

Avoided level crossings lead to symmetry breaking and complex eigenvalues.

The study enables designing systems with specific dynamical properties across parameter ranges.

Abstract

The dynamics of open quantum systems is determined by avoided and true crossings of eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases of the eigenfunctions are not rigid so that environmentally induced spectroscopic redistribution processes may take place and a dynamical phase transition may occur. Due to the formal equivalence between the quantum mechanical Schr\"odinger equation and the optical wave equation in PT symmetric lattices, the dynamics of the system is determined also in this case by avoided and true crossings of eigenvalue trajectories of the non-Hermitian Hamiltonian. In contrast to the eigenvalues characterizing an open quantum system, the eigenvalues describing the PT symmetric optical lattice are real as long as the influence of the environment (lattice) onto the optical wave equation is small. In the regime of avoided level crossings, the symmetry is destroyed, the eigenvalues become complex and a dynamical phase transition occurs similar as in the open quantum system. The redistribution processes in the regime of avoided level crossings allow to design systems with desired properties in a broad parameter range.