  We study non-equilibrium dynamics of integrable and non-integrable closed
quantum systems whose unitary evolution is interrupted with stochastic resets,
characterized by a reset rate $r$, that project the system to its initial
state. We show that the steady state density matrix of a non-integrable system,
averaged over the reset distribution, retains its off-diagonal elements for any
finite $r$. Consequently a generic observable $\hat O$, whose expectation value
receives contribution from these off-diagonal elements, never thermalizes under
such dynamics for any finite $r$. We demonstrate this phenomenon by exact
numerical studies of experimentally realizable models of ultracold bosonic
atoms in a tilted optical lattice. For integrable Dirac-like fermionic models
driven periodically between such resets, the reset-averaged steady state is
found to be described by a family of generalized Gibbs ensembles (GGE s)
characterized by $r$. We also study the spread of particle density of a
non-interacting one-dimensional fermionic chain, starting from an initial state
where all fermions occupy the left half of the sample, while the right half is
empty. When driven by resetting dynamics, the density profile approaches at
long times to a nonequilibrium stationary profile that we compute exactly. We
suggest concrete experiments that can possibly test our theory.
