Quantum δ-kicked rotor:: the effect of amplitude noise on the quantum resonances

We study analytically the effect of amplitude noise on the quantum resonances of an atom optics realization of the delta-kicked rotor. Noise is shown to add a time growth to the 'deterministic' energy and to induce a time increasing spreading in the momentum distribution; exact results are given for both effects. The ballistic peaks characteristic of the noiseless distribution for particular initial conditions broaden and eventually vanish, whereas the associated quadratic growth of energy persists; at large times, the survival probability decays as t(-1). Moreover, the nonexponential 'localization' linked to different initial conditions is gradually destroyed. Features specific to Gaussian noise, white and coloured, are analysed. The feasibility of experimental tests of these effects is discussed.