Highlights

## Dynamics of condensed particles in a few site system ()

The physics of $N$ bosons in an $M$ site system is described by the Bose-Hubbard Hamiltonian. The $M=2$ problem is integrable and equivalent to $j=N/2$ spin problem, while the $M=3$ problem has both non-trivial topology and mixed phase space. Our studies concerns the analysis of fluctuations, occupation statistics and quantum stirring in such systems.

The time dependent dynamics in a Bosonic Josephson junction with no driving can be regarded as a quantum version of a pendulum. Adding driving to this system, we have studied the adiabatic-diabatic-sudden crossover of the many-body Landau-Zener dynamics; the Kapitza pendulum dynamics; and the Quantum Zeno dynamics due to erratic driving. In particular we have studied the stabilization and the suppression of collision-induced dephasing by periodic, erratic, or noisy driving.

The $M=4$ problem is the minimal configuration for the study of quantum thermalization and many-body localization. Specifically we consider the equilibration of the occupation in a weakly coupled subsystems.

Fig1: From left to right: (a) The Wigner function of a prepared Fock state. The red dashed line is the separatrix. (b) The same, viewed from the right, after a finite time evolution.

Fig2: From left to right: (a) The time evolution of the Bloch vector for various coherent preparations. (b) The quantum Zeno effect for Pi preparation due to the presence of noise.

Fig3: Phase space tomography for a kicked-top: the participation number for all coherent preparations is imaged for an integrable phase-space (left), for a mixed phase-space (middle) and for a chaotic phase-space (right). In the latter case one observes the effect of scarring.