Short-Time Loschmidt Gap in Dynamical Systems with Critical Chaos

TL;DR

This paper investigates the short-time Loschmidt echo gap in systems with critical chaos, revealing a classical dependence on potential singularity and a quantum insensitivity explained by diffraction effects and Random Matrix Theory modeling.

Contribution

It introduces a novel analysis of the Loschmidt echo gap in critical chaotic systems, highlighting the quantum-classical discrepancy and providing a scaling law for quantum echoes.

Findings

Classical echo gap scales with potential singularity and initial phase space spread.

Quantum echo gap is insensitive to potential singularity, following a different scaling law.

Diffraction effects explain the quantum-classical discrepancy in the Loschmidt echo.

Abstract

We study the Loschmidt echo F(t) for a class of dynamical systems showing critical chaos. Using a kicked rotor with singular potential as a prototype model, we found that the classical echo shows a gap (initial drop) 1-F_g where F_g scales as F_g(\alpha, \epsilon, \eta)= f_cl(\chi_cl equiv\eta^{3-\alpha}/\epsilon); \alpha is the order of singularity of the potential, \eta is the spread of the initial phase space density and \epsilon is the perturbation strength. Instead, the quantum echo gap is insensitive to \alpha, described by a scaling law F_g = f_q(\chi_q = \eta^2/\epsilon) which can be captured by a Random Matrix Theory modeling of critical systems. We trace this quantum-classical discrepancy to strong diffraction effects that dominate the dynamics.