  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 standard system Hamiltonian. Second, we derive the explicit
form of the state in the ultrastrong coupling limit, and show 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.
