Quantum poincare recurrences for a hydrogen atom in a microwave field

We study the time dependence of the ionization probability of Rydberg atoms driven by a microwave field, both in classical and in quantum mechanics. T he quantum survival probability follows the classical one up to the Heisenb erg time and then decays algebraically as P(t) proportional to 1/t. This de cay law derives from the exponentially long times required to escape from s ome region of the phase space, due to tunneling and localization effects. W e also provide parameter values which should allow one to observe such deca y in laboratory experiments.