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

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.

10 pages, 4 figures

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• Nonclassical phase-space trajectories for the damped harmonic quantum oscillator
• Propagating wave correlations in complex systems
• Origin of the $1/f^{\alpha }$-Spectral-Noise in Chaotic and Regular Quantum Systems