Short Time Dynamics of Scalar Products in Hilbert Space

TL;DR

This paper uses semiclassical methods to analyze short-time dynamics of scalar products like overlap and fidelity in quantum systems, revealing classical scars and deriving quadratic decay analytically.

Contribution

It introduces a semiclassical approach to study scalar products, connecting classical periodic orbits with quantum fidelity decay and overlap behavior.

Findings

Fidelity exhibits quadratic decay at short times.

Classical scars appear naturally in the semiclassical expansion.

Overlap behavior differs between chaotic and integrable systems.

Abstract

We use the semiclassical method proposed in \cite{Adelcio2003} to study scalar products such as the overlap, Husimi functions and fidelity decay. Scars of classical periodic orbits arise naturally in this pertubative expansion. We also derive analytically a well known numerical result that fidelity has a quadratic decay for short times. We study the overlap in the chaotic regime and integrable for some simple systems.