Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

TL;DR

This paper develops a full classical description of the non-Hermitian Swanson oscillator using metriplectic flow equations, revealing periodic trajectories and divergence phenomena in finite time, linking classical and quantum dynamics.

Contribution

It introduces a novel classical framework for the non-Hermitian Swanson oscillator using metriplectic flow, connecting classical and quantum Gaussian wave packet dynamics.

Findings

Classical trajectories are periodic in time.

Classical and quantum states can diverge in finite time.

Classical dynamics exactly match quantum Gaussian wave packet evolution.

Abstract

The non-Hermitian quadratic oscillator studied by Swanson is one of the popular $PT$-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the $PT$-symmetry is unbroken, i.e., the eigenvalues are purely real.