  The Gibbs state is widely taken to be the equilibrium state of a system in
contact with an environment at temperature $T$. However, non-negligible
interactions between system and environment can give rise to an altered state.
Here we derive general expressions for this mean force Gibbs state, valid for
any system that interacts with a bosonic reservoir. First, we derive the state
in the weak coupling limit and find that, in general, it maintains coherences
with respect to the bare system Hamiltonian. Second, we develop a new expansion
method suited to investigate the ultrastrong coupling regime. This allows us to
derive the explicit form for the mean force Gibbs state, and we find that it
becomes diagonal in the basis set by the system-reservoir interaction instead
of the system Hamiltonian. Several examples are discussed including a single
qubit, a three-level V-system and two coupled qubits all interacting with
bosonic reservoirs. The results shed light on the presence of coherences in the
strong coupling regime, and provide key tools for nanoscale thermodynamics
investigations.
