Quantum dynamics and Random Matrix theory

We compute the survival probability of an initial state, with an energy in a certain window, by means of random matrix theory. We determine its probability distribution and show that is is universal, i.e. caracterised only by the symmetry class of the hamiltonian and independent of the initial state. In classical mechanics, temporal chaos is caracterised by the extreme sensibility of a trajectory to variation of initial conditions. No direct analog of this phenomenon has been found in quantum mechanics so far. On the other hand, numerical evidence has been accumulated [1], showing that energy levels of a quantum system, whose classical counterpart is chaotic, have a statistical behavior described by Wigner’s random matrix theory (RMT), on the mean level spacing scale. The question we want to adress is the following: are there specific predictions of RMT for quantum dynamics, which would caracterise the temporal behavior of ”chaotic” quantum systems. We consider the following situation: The system is prepared in an initial state φ at time 0, with an energy in a certain window, centered at e and of width 2sl(e), where l(e) is the mean level spacing, and we want to compute the probability to find our system again in the state φ, at a later time t. This quantity that we call the survival probability R is given by R = ∣