Sensitivity of Quantum Motion to Perturbation in Triangle Map

TL;DR

This paper investigates how quantum fidelity decays in the triangle map with weak chaos, revealing three distinct regimes of decay depending on perturbation strength and providing numerical and theoretical insights.

Contribution

It identifies and characterizes three regimes of quantum fidelity decay in a weakly chaotic system, extending understanding of quantum-classical correspondence in such regimes.

Findings

Fidelity decays as exp(-cε² t^γ) with γ ≈ 1.7 in weak perturbation regime

Fidelity approximately depends on ε t^{2.5} in strong perturbation regime, decaying slower than power-law

Intermediate regime shows approximately exponential decay of fidelity with rate proportional to ε t

Abstract

We study quantum Loschmidt echo, or fidelity, in the triangle map whose classical counterpart has linear instability and weak chaos. Numerically, three regimes of fidelity decay have been found with respect to the perturbation strength $\epsilon$. In the regime of weak perturbation, the fidelity decays as $\exp (-c\epsilon^2 t^{\gamma})$ with $\gamma \simeq 1.7$. In the regime of strong perturbation, the fidelity is approximately a function of $\epsilon t^{2.5}$, which is predicted for the classical fidelity [G. Casati, {\it et al}, Phys.Rev.Lett.{\bf 94}, 114101 (2005)], and decays slower than power-law decay for long times. In an intermediate regime, the fidelity has approximately an exponential decay $\exp (-c' \epsilon t)$.