Quantum refrigerators and the third law of thermodynamics

The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to absolute zero. The third law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent $\zeta$ of the cooling process $\frac{dT\left(t\right)}{dt}\sim -T^{\zeta }$ when approaching absolute zero, $T\to 0$. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled two-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat-driven refrigerator (absorption refrigerator) is compared to a power-driven refrigerator. When optimized, both cases lead to the same exponent $\zeta$, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat bath models are considered: a bath composed of harmonic oscillators and a bath composed of ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the third law are discussed. In the Appendices, the theory of periodically driven open systems and its implication for thermodynamics are outlined.

Figure 1

A quantum heat pump designated by the Hamiltonian $H_s$ coupled to a work reservoir with temperature $T_w$, a hot reservoir with temperature $T_h$, and a cold reservoir with temperature $T_c$. The heat and work currents are indicated. In steady state, ${\mathcal{J}}_h+{\mathcal{J}}_c+\mathcal{P}=0$.