Quantum and Classical Fidelity for Singular Perturbations of the Inverted and Harmonic Oscillator

TL;DR

This paper compares quantum and classical fidelities for singular perturbations of inverted and harmonic oscillators, analyzing their behavior over time and varying coupling strength to understand stability and sensitivity in these systems.

Contribution

It introduces a definition of classical fidelity for these systems and compares it with quantum fidelity, highlighting differences and similarities in their dynamical behaviors.

Findings

Quantum fidelity exhibits specific decay patterns over time.

Classical fidelity shows different sensitivity to perturbations.

Comparison reveals insights into quantum-classical correspondence in singular systems.

Abstract

Let us consider the quantum/versus classical dynamics for Hamiltonians of the form \beq \label{0.1} H\_{g}^{\epsilon} := \frac{P^2}{2}+ \epsilon \frac{Q^2}{2}+ \frac{g^2}{Q^2} \edq where $\epsilon = \pm 1$, $g$ is a real constant. We shall in particular study the Quantum Fidelity between $H\_{g}^{\epsilon}$ and $H\_{0}^{\epsilon}$ defined as \beq \label{0.2} F\_{Q}^{\epsilon}(t,g):= < \exp(-it H\_{0}^{\epsilon})\psi, exp(-itH\_{g}^ {\epsilon})\psi > \edq for some reference state $\psi$ in the domain of the relevant operators. We shall also propose a definition of the Classical Fidelity, already present in the literature (\cite{becave1}, \cite{becave2}, \cite{ec}, \cite{prozni}, \cite{vepro}) and compare it with the behaviour of the Quantum Fidelity, as time evolves, and as the coupling constant $g$ is varied.