In this talk I will review some of our recent work on relaxation after a quantum quench. For dynamics generated by bosonic quadratic many-body Hamiltonians, I will show under which conditions the state is locally described by a Gaussian state in the long time limit. That is, subsystems converge to Gaussian states in trace norm for large times. These Gaussian states can be given explicitly and there is no need for time averaging. In the second part of the talk, I will introduce a setting in general spin systems for which thermalization and thermalization time can be addressed: Fixing the spectrum of the Hamiltonian and picking its basis from the Haar measure, the time-dependent state may locally be described by the maximally mixed state for almost all times in [0,T]. Here, T is algebraically small in the system size.
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