Time-domain scars: resolving the spectral form factor in phase space

TL;DR

This paper introduces a phase space approach to analyze the spectral form factor, revealing classical and non-classical structures that influence quantum return probabilities, supported by numerical studies.

Contribution

It defines a quantum return probability in phase space and identifies phase space manifolds contributing to the spectral form factor, including non-classical structures.

Findings

Identification of phase space manifolds affecting the form factor

Distinction between classical and non-classical contributions

Numerical validation with quantum cat map and driven oscillator

Abstract

We study the relationship of the spectral form factor with quantum as well as classical probabilities to return. Defining a quantum return probability in phase space as a trace over the propagator of the Wigner function allows us to identify and resolve manifolds in phase space that contribute to the form factor. They can be associated to classical invariant manifolds such as periodic orbits, but also to non-classical structures like sets of midpoints between periodic points. By contrast to scars in wavefunctions, these features are not subject to the uncertainty relation and therefore need not show any smearing. They constitute important exceptions from a continuous convergence in the classical limit of the Wigner towards the Liouville propagator. We support our theory with numerical results for the quantum cat map and the harmonically driven quartic oscillator.