  We discuss the emergence of non-stationarity in open quantum many-body
systems. This leads us to the definition of dissipative time crystals which
display experimentally observable, persistent, time-periodic oscillations
induced by noisy contact with an environment. We use the Loschmidt echo and
local observables to indicate the presence of a finite sized dissipative time
crystal. Starting from the closed Hubbard model we then provide examples of
dissipation mechanisms that yield experimentally observable quantum periodic
dynamics and allow analysis of the emergence of finite sized dissipative time
crystals. For a disordered Hubbard model including two-particle loss and gain
we find a dark Hamiltonian driving oscillations between GHZ states in the
long-time limit. Finally, we discuss how the presented examples could be
experimentally realized.
