Non-dispersive wave packets in periodically driven quantum systems

With the exception of the harmonic oscillator, quantum wave-packets usually spread as time evolves. This is due to the non-linear character of the classical equations of motion which makes the various components of the wave-packet evolve at various frequencies. We show here that, using the nonlinear resonance between an internal frequency of a system and an external periodic driving, it is possible to overcome this spreading and build non-dispersive (or non-spreading) wave-packets which are well localized and follow a classical periodic orbit without spreading. From the quantum mechanical point of view, the non-dispersive wave-packets are time periodic eigenstates of the Floquet Hamiltonian, localized in the nonlinear resonance island. We discuss the general mechanism which produces the non-dispersive wave-packets, with emphasis on simple realization in the electronic motion of a Rydberg electron driven by a microwave field. We show the robustness of such wavepackets for a model one-dimensional as well as for realistic three dimensional atoms. We consider their essential properties such as the stability versus ionization, the characteristic energy spectrum and long lifetimes. The requirements for experiments aimed at observing such non-dispersive wave-packets are also considered. The analysis is extended to situations in which the driving frequency is a multiple of the internal atomic frequency. Such a case allows us to discuss non-dispersive states composed of several, macroscopically separated wave-packets communicating among themselves by tunneling. Similarly we briefly discuss other closely related phenomena in atomic and molecular physics as well as possible further extensions of the theory.