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In this work we investigate the late-time stationary states of open quantumsystems coupled to a thermal reservoir in the strong coupling regime. Ingeneral such systems do not necessarily relax to a Boltzmann distribution ifthe coupling to the thermal reservoir is non-vanishing or equivalently if therelaxation timescales are finite. Using a variety of non-equilibrium formalismsvalid for non-Markovian processes, we show that starting from a product stateof the closed system = system + environment, with the environment in itsthermal state, the open system which results from coarse graining theenvironment will evolve towards an equilibrium state at late-times. This statecan be expressed as the reduced state of the closed system thermal state at thetemperature of the environment. For a linear (harmonic) system and environment,which is exactly solvable, we are able to show in a rigorous way that allmulti-time correlations of the open system evolve towards those of the closedsystem thermal state. Multi-time correlations are especially relevant in thenon-Markovian regime, since they cannot be generated by the dynamics of thesingle-time correlations. For more general systems, which cannot be exactlysolved, we are able to provide a general proof that all single-timecorrelations of the open system evolve to those of the closed system thermalstate, to first order in the relaxation rates. For the special case of azero-temperature reservoir, we are able to explicitly construct the reducedclosed system thermal state in terms of the environmental correlations.

The equilibrium states of open quantum systems in the strong coupling regime