In recent years, progress in quantum engineering have provided new tools for simulating the dynamics of isolated many body quantum systems, e.g. with trapped ions or cold atoms . Interestingly, despite the ideal unitary evolution of their quantum state, these systems can display signatures of a local equilibration. More strikingly, this equilibrium state can be in strong disagreement with the predictions of statistical physics, e.g. when a Many Body Localization phase transition takes place . These experimental facts are clearly questioning what kind of statistical description is relevant for isolated many body quantum systems.
In this talk, I will present our recent results on a theoretical model dedicated to this question and involving an arbitrary quantum system coupled to a large arbitrary quantum environment. By introducing randomness in the interaction Hamiltonian, we showed that the system exhibits typical dynamics, in other words its state is self-averaging .
This property sets the ground for an averaging procedure over random interactions which can be used for analytical non-perturbative calculations performed with full generality, i.e. for arbitrary system, environment, and initial state. We applied this technique to calculate analytically the stationary state of the system at long times and found a new thermodynamical ensemble and a new partition function of purely quantum origin . This partition function predicts for the first time unconventional stationary states and recovers the microcanonical and canonical ensembles as particular cases.
 Gross et al, 357 6355, pp. 995-1001, Science (2017)
 Schreiber et al, 349, 6250, pp. 842-845, Science, (2015)
 Ithier et al, Phys. Rev. A 96, 012108 (2017)
 Ithier et al, Phys. Rev. E (Rapid Com.) 96, 060102(R) (2017)