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

TL;DR

The paper studies near-resonant quantum kicked rotor dynamics and uncovers recurring cusp caustics, cusp oscillations, and reticular patterns arising from near-primary resonances. It develops a path-integral, stationary-phase framework to derive caustic positions and recurrence times, and establishes a catastrophe-theory–driven scaling law for cusp amplitudes with Arnold index $1/4$, predicting $|\psi| \propto (K/\Delta)^{1/8}$. A detailed classical-quantum correspondence links caustic formation to the behavior of the fluctuation determinant, while chaos is shown to erode phase matching and gradually destroy caustics. The results have implications for quantum control in Floquet systems and are testable in optical, atomic, and related platforms, with potential extensions to high-order resonances and robustness to noise.

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.

Figures (7)

• Figure 1: Dynamical evolution of the QKR (top) and its classical counterpart (bottom). The quantum system is initialized in the zero-momentum plane-wave state $\psi(\theta_0,t_0)=1/\sqrt{2\pi}$, while classical trajectories are uniformly sampled with initial momentum $p_{0}=0$. Left and right panels correspond to $\Delta=\sqrt{2}$ and $\Delta=0.0001$, respectively. Other parameters are fixed at $K=5$ and $T=4\pi+\Delta$. The wave amplitude $|\psi(\theta_{n},t_{n})|$ is displayed in panels (a) and (b).
• Figure 2: Evolution of the wave amplitude $|\psi(\theta_{n},t_{n})|$ for the QKR with $K=5$ and $\Delta=0.0001$. The system is initialized in: (a) a plane-wave state; (b) a Gaussian wave packet in angular space with a standard deviation of 1 and centered at $\pi/2$. The initial momentum of all states is $p_0=100$.
• Figure 3: Evolution of the wave amplitude $|\psi(\theta_{n},t_{n})|$ for the QKR near high-order resonance with detuning parameter $\Delta=0.0001$, initialized in a plane-wave state $\psi(\theta_{0},t_{0})=1/\sqrt{2\pi}$. Parameters: (a) $K=100,~T=2\pi+\Delta$; (b) $K=1,~T=4\pi/3+\Delta$.
• Figure 4: Dynamical evolution of the QKR and its corresponding classical mapping in Eq. (\ref{['Equivalent Mapping']}). Surface plots depict the evolution of $|\psi(\theta_n,t_n)|$ with the initial plane-wave state $\psi(\theta_0,t_0)=(1/\sqrt{2\pi})e^{ip_{0}\theta_{0}}$. The black solid lines represent numerical simulations of the classical mapping with initial momentum $p_{0}^{\text{cl}}=p_{0}\Delta$. The magenta solid lines correspond to the caustic curve of the first caustic. Other parameters are $K=5$ and $\Delta=0.0001$.
• Figure 5: Comparison between classical discrete mapping and continuum analytical solutions. The black solid and green dashed lines correspond to Eq. (\ref{['Equivalent Mapping']}) and Eq. (\ref{['Analytical Solution']}), respectively. The red crosses mark the cusp points from Eqs. (\ref{['Caustic Time']}) and (\ref{['Cusp Points']}). Parameters are identical to those in Fig. \ref{['Fig-Classical Paths']}.