The smallest absorption refrigerator: the thermodynamics of a system with quantum local detailed balance

We study the thermodynamics of a quantum system interacting with different baths in the repeated interaction framework. In an appropriate limit, the evolution takes the Lindblad form and the corresponding thermodynamic quantities are determined by the state of the full system plus baths. We identify conditions under which the thermodynamics of the open system can be described only by system properties and find a quantum local detailed balance condition with respect to an equilibrium state that may not be a Gibbs state. The three-qubit refrigerator introduced in Linden et al. [Phys. Rev. Lett. 105, 130401 (2010)] and Skrzypczyk et al. [J. Phys. A: Math. Theory 44, 492002 (2011)] is an example of such a system. From a repeated interaction microscopic model we derive the Lindblad equation that describes its dynamics and discuss its thermodynamic properties for arbitrary values of the internal coupling between the qubits. We find that external power (proportional to the internal coupling strength) is required to bring the system to its steady state, but once there, it works autonomously as discussed in Linden et al. [Phys. Rev. Lett. 105, 130401 (2010)] and Skrzypczyk et al. [J. Phys. A: Math. Theory 44, 492002 (2011)].