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.

💡 Research Summary

In this paper the authors investigate the emergence of caustic structures in the quantum kicked rotor (QKR) when the system is operated in the near‑resonant regime, i.e., when the kicking period deviates slightly from the primary quantum resonance condition (T=4\pi) by a small detuning (\Delta) ((\Delta/4\pi\ll1)). Using both extensive numerical simulations and an analytical framework based on Floquet theory and the Feynman path‑integral formalism, they reveal that the wave‑function amplitude develops a series of recurring cusp‑type caustics. These cusps appear at well‑defined times, oscillate in position for non‑zero initial momentum, and persist even in high‑order resonant cases where the period satisfies (T=4\pi r/s+\Delta) with coprime integers (r,s) and (s\ge2).

The authors first present the dimensionless Hamiltonian (\hat H=\hat p^{2}/2+K\cos\hat\theta\sum_{n}\delta(t-nT)) and define the effective kicking strength (K_{\rm eff}=K,T). Classical dynamics are reduced to the standard map, which becomes globally chaotic once (K_{\rm eff}>K_{c}\approx0.97). However, quantum simulations show that, despite the underlying classical chaos, the QKR exhibits regular, periodic caustic patterns when (\Delta) is small. For large detuning the quantum dynamics become chaotic, while for (\Delta\approx10^{-4}) the system displays a striking divergence between classical diffusion and quantum regularity.

To understand the origin of the caustics, the authors write the one‑period evolution operator as (\hat U(T)=\exp(-iK\cos\hat\theta)\exp(-i\hat p^{2}T/2)) and construct the propagator (G(\theta_{n},t_{n};\theta_{0},t_{0})) as a discrete path integral. By applying the Poisson summation to the Jacobi theta function that appears in the matrix elements, they isolate the dominant term in the (\Delta\to0) limit and obtain an effective discrete action
\