Time invariant $\mathcal{PT}$-product and phase locking in $\mathcal{PT}$-symmetric lattice models

TL;DR

This paper explores the phase dynamics and phase locking phenomena in $ ext{PT}$-symmetric lattice models, revealing universal phase locking behavior in the broken symmetry region, with implications for non-unitary quantum dynamics.

Contribution

It demonstrates the invariance of the $ ext{PT}$ inner product over time and uncovers universal phase locking behavior in $ ext{PT}$-symmetric lattice models.

Findings

Universal phase locking occurs in the $ ext{PT}$-symmetry broken region.

Phase locking depends on gain-site location.

Time invariance of the $ ext{PT}$ inner product influences wave-function dynamics.

Abstract

Over the past decade, non-Hermitian, $\mathcal{PT}$-symmetric Hamiltonians have been investigated as candidates for both, a fundamental, unitary, quantum theory, and open systems with a non-unitary time evolution. In this paper, we investigate the implications of the former approach in the context of the latter. Motivated by the invariance of the $\mathcal{PT}$ (inner) product under time evolution, we discuss the dynamics of wave-function phases in a wide range of $\mathcal{PT}$-symmetric lattice models. In particular, we numerically show that, starting with a random initial state, a universal, gain-site location dependent locking between wave function phases at adjacent sites occurs in the $\mathcal{PT}$-symmetry broken region. Our results pave the way towards understanding the physically observable implications of time-invariants in the non-unitary dynamics produced by $\mathcal{PT}$-symmetric Hamiltonians.