# Time-dependent metric for the two dimensional, non-Hermitian coupled oscillator

## TL;DR

This paper introduces a time-dependent metric for a two-dimensional non-Hermitian harmonic oscillator, ensuring physical consistency across different symmetry regimes by explicitly incorporating time dependence into the Dyson map.

## Contribution

It presents a novel time-dependent Dyson map and metric for a non-Hermitian oscillator, addressing issues in the broken symmetry regime.

## Key findings

- Time-dependent metric maintains physicality in broken symmetry regime.
- Explicit time dependence resolves unphysical behavior in the model.
- Model exhibits a transition between unbroken and broken $ ext{PT}$-symmetry regimes.

## Abstract

We provide a time-dependent Dyson map and metric for the two dimensional harmonic oscillator with a non-Hermitian $i xy$ coupling term. This particular time-independent model exhibits spontaneously broken $\mathcal{PT}$-symmetry and becomes unphysical in the broken regime, with the spectrum becoming partially complex. By introducing an explicit time-dependence into the Dyson map, we provide a time-dependent metric that renders the model consistent across the unbroken and broken regimes.