Thermodynamic limits of dynamic cooling

We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature T-min > 0 reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum. The minimal work needed to reach T-min grows as 1/T-min. This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above T-min. Our results on T-min > 0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law. We also study cooling via a reservoir consisting of N >> 1 identical spins. Here we show that T-min alpha 1/N and find the maximal cooling compatible with the minimal work determined by the free energy. Finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature, the unattainability is recovered when taking into account imperfections in preparing the microcanonic state.