Near-Resonance-Induced Caustics and Scaling Laws in a Quantum Kicked Rotor

TL;DR

This paper explores the formation of caustic structures in the quantum kicked rotor near resonance, deriving analytical expressions for their positions, recurrence, and scaling laws, and examining classical-quantum correspondence and experimental prospects.

Contribution

It introduces a detailed analytical framework for caustic structures in the quantum kicked rotor near resonance, including scaling laws and classical-quantum relationships.

Findings

Caustic structures such as cusps and reticular patterns are observed in the near-resonant quantum kicked rotor.

A power-law scaling with an Arnold index of 1/4 relates wave amplification to kicking strength and detuning.

Chaos disrupts phase matching, eroding caustic structures and affecting classical-quantum correspondence.

Abstract

In this study, we investigate the dynamics of the quantum kicked rotor in the near-resonant regime and observe distinct caustic structures, such as recurring cusps, cusp oscillations, and reticular cusp patterns in high-order resonant cases. By deriving a path integral expression for the wave function's time evolution, we analytically determine both the positions of the caustic singularities and their recurrence periods. We further derive and validate a power-law scaling with an Arnold index of $1/4$, which establishes a quantitative relationship between the amplification of the wave amplitude, the kicking strength, and the resonant detuning parameter. We also explore the classical-quantum correspondence of these caustic singularities, demonstrating that chaos disrupts phase matching and ultimately erodes the caustic structure. Finally, we address the feasibility of experimental implementations of our findings and their broader ramifications for related research fields.