  Optical Bloch Equations (OBE) describe the coherent exchange of energy
between a quantum bit (qubit) and a quasi-resonant driving field in the
presence of a thermal bath. Despite it being an ubiquitous process in quantum
technologies, a sound thermodynamic analysis is still missing. We hereby
provide such an analysis, by deriving the relevant framework from first
principles. Building on a microscopic model of the bath, we first express heat,
work and entropy production for the closed qubit-bath system where these
definitions are unambiguous. We coarse-grain the obtained expressions, using a
methodology similar to the derivation of the dynamical master equation and
giving rise to consistent expressions of the First and Second Law. We verify
that long coarse graining times yield the Floquet Master Equation and its
already known thermodynamic description. Conversely, short coarse-graining
times yield the OBE and its consistent thermodynamic framework, whose variables
explicitly depend on the quantum coherences in the qubit's energy basis. These
quantum signatures in the heat and in the entropy production flows allow us to
characterize a genuinely quantum non-equilibrium situation, where the
coherences created by the driving field are continuously erased by the bath.
Our findings can be readily extended to larger open quantum systems. They carry
the seeds for future thermodynamic analyses of quantum gates and the design of
quantum engines in the strong coherent driving regime.
