Quantum ping-pong model

A dynamics of quantum particle in a triangular well in a presence of monochromatic driving field was studied in [46]. After a canonical transformation the model becomes equivalent to a ball jumping on oscillating plate in a gravitational field. The classical dynamics is described by the Chirikov standard map, which gives the ball velocity change after each elastic collision with the wall. Inspite the fact that the classical dynamics is as for the kicked rotor the quantum behaviour is different. The physical reason is due to a proportionality of the diffusion rate $D$ , measured in the number of photons, to the photon number $N$ ( $D \sim \epsilon^2 N/\omega^3$ where $\epsilon, \omega$ are the field strength and frequency, and mass $m=1$, $\hbar=1$). As the result the quantum dynamics is diffusive only for $l \sim D > N$, namely $\epsilon > \omega^{3/2}/2$. Below this delocalization border eigenstates are algebraically localized. This research together with results of [49] formed the basis of thesis of F.Benvenuto (Milano Univ., 1992). Recently this model was studied in more detail by N.Brenner and S.Fishman (1996).