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