Section: Research Program

Dynamical localization and kicked rotors

The kicked rotor is a unitary discrete time dynamics proposed in the seventies in the context of studies on quantum chaos, and used recently as a “quantum simulator” for the Anderson model. It is a quantum equivalent of the standard map and is obtained by integrating a time-dependent linear Schrödinger equation with a time-periodic, very singular (delta comb) potential. It continues to pose considerable mathematical challenges, in particular the so-called “quantum suppression of classical chaos” in the presence of a strong potential, which remains an open problem from the mathematical point of view. It can be rephrased as follows: show that the $H^1$ norm of the solution is uniformly bounded in time (see  [36] for more background). In more recent years, the question has arisen how the behavior of this system would change in the presence of a nonlinear term in the Schrödinger equation.

This problem displays both numerical and analytical challenges, in particular because of the difficulty to obtain long time simulations of the system and because of the presence of instabilities due to the nonlinearity. Preliminary theoretical results motivate some conjectures on the behavior of these systems, that we plan to validate empirically in a first step. Indeed, reliable long-time simulations of the system should allow us to get more insight into the behavior of the exact solutions in the unstable cases. One of the main difficulties for the numerical simulation is the intrinsic instability of the system, which magnifies quite rapidly the numerical error due to machine precision. This requires the use of multiprecision techniques in order to handle reasonably long times, even for moderate nonlinearities, and of the transparent boundary conditions recently introduced by members of the former SIMPAF project-team.