# Mending the broken PT-regime via an explicit time-dependent Dyson map

## TL;DR

This paper introduces a method to restore meaningful quantum descriptions for non-Hermitian PT-symmetric systems with broken symmetry by employing an explicit time-dependent Dyson map, leading to real energy spectra.

## Contribution

It presents an analytical solution to the time-dependent Dyson equation with a time-dependent metric, applying it to a two-level atomic system with a nonlinear Ermakov-Pinney equation.

## Key findings

- Restores real spectra in broken PT-symmetry regimes
- Develops a time-dependent Dyson framework for non-Hermitian systems
- Solves the nonlinear Ermakov-Pinney equation explicitly

## Abstract

We demonstrate that non-Hermitian Hamiltonian systems with spontaneously broken PT-symmetry and partially complex eigenvalue spectrum can be made meaningful in a quantum mechanical sense when introducing some explicit time-dependence into their parameters. Exploiting the fact that explicitly time-dependent non-Hermitian Hamitonians are unobservable and not identical to the energy operators in such a scenario, we show that their corresponding non-Hermitian energy operators develop a different type of PT-symmetry from the Hamiltonians that ensures the reality of their energy spectra. For this purpose we analytically solve the fully time-dependent Dyson equation with all quantities involved being explicitly time-dependent giving rise to a time-dependent metric. The key auxiliary equation to be solved for the two level atomic system considered here is the nonlinear Ermakov-Pinney equation with time-dependent coefficients.