The heat and work of quantum thermodynamic processes with quantum coherence

We study quantum thermodynamics for a time-dependent system in a non-adiabatic evolution. The definitions of heat and work are clarified for the general time evolution with quantum coherence characterized by the off-diagonal elements of the density matrix. By integrating the coherence effects in the exactly-solvable dynamics of quantum-spin precession, the internal energy is transferred as work rigorously in the thermodynamic adiabatic process. The present study demonstrates that quantum adiabatic process certainly results in a thermodynamic adiabatic process, but not all thermodynamic adiabatic processes are due to the quantum-mechanical adiabatic processes.