Short-Time Loschmidt Gap in Dynamical Systems with Critical Chaos

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.