Universal Scaling of Spectral Fluctuation Transitions for Interacting Chaotic Systems

The statistical properties of interacting strongly chaotic systems are investigated for varying interaction strength. In order to model tunable entangling interactions between such systems, we introduce a new class of random matrix transition ensembles. The nearest-neighbor-spacing distribution shows a very sensitive transition from Poisson statistics to those of random matrix theory as the interaction increases. The transition is universal and depends on a single scaling parameter only. We derive the analytic relationship between the model parameters and those of a bipartite system, with explicit results for coupled kicked rotors, a dynamical systems paradigm for interacting chaotic systems. With this relationship the spectral fluctuations for both are in perfect agreement. An accurate approximation of the nearest-neighbor-spacing distribution as a function of the transition parameter is derived using perturbation theory.