Transitions toward Quantum Chaos: with Supersymmetry from Poisson to Gauss

The transition from arbitrary to chaotic fluctuation properties in quantum systems is studied in a random matrix model. It is assumed that the Hamiltonian can be written as the sum of an arbitrary and a chaos producing part. The Gaussian ensembles are used to model the chaotic part. A closed integral representation for all correlations in the case of broken time reversal invariance is derived by employing supersymmetry and the graded eigenvalue method. In particular, the two level correlation function is expressed as a double integral. For a correlation index, exact diffusion equations are derived for all three universality classes which describe the transition to the chaotic regime from arbitrary initial conditions. As an application, the transition from Poisson regularity to chaos is discussed. The two level correlation function becomes a double integral for all values of the transition parameter.