  Two-time correlations are a crucial tool to probe the dynamics of many-body
systems. We use these correlation functions to study the dynamics of
dissipative quantum systems. Extending the adiabatic elimination method, we
show that the correlations can display two distinct behaviors, depending on the
observable of interest: a fast exponential decay, with a timescale of the order
of the dissipative coupling, or a much slower dynamics. We apply this formalism
to bosons in a double well subjected to phase noise. While the single-particle
correlations decay exponentially, the density-density correlations display slow
aging dynamics. We also show that the two-time correlations of dissipatively
engineered quantum states can evolve in a drastically different manner compared
to their Hamiltonian counterparts.
