The Ulam problem and the ionization of Rydberg atoms by microwave radiation

Abstract

We study the Ulam problem for long times (several million collisions) by numerical methods. We show that in the diffusion regime, which is valid for moderate times, this problem is mathematically equivalent to the problem of the diffusive ionization of atomic Rydberg states by microwave radiation. It is concluded that the diffusion regime sets in only for a very small number of initial conditions (field phases). It is theorized that the analogy between the two problems can be extrapolated to times longer than the diffusion time. We show in the Ulam problem that after the diffusional buildup of energy has finished, the quasistationary regime does not continue indefinitely: after several million particle-wall collisions the energy rapidly drops to zero. On the basis of this extrapolation we examine the possibility that an electron which has reached the continuous spectrum will not fly off to infinity (ionization), but will return to bound Rydberg states of the atoms (if the field acts for a sufficiently long time). This can make the diffusive ionization probability much lower than the value given by the known estimates.