Quantum thermodynamics addresses the emergence of thermodynamical laws from quantum mechanics. The III-law of thermodynamics has been mostly ignored. There are seemingly two independent formulation of the third law. The rst, known as the Nernst heat theorem, implies that the entropy ow from any substance at absolute zero temperature is zero:  and Jc is the heat current. In order to insure the fulllement of the second law when Tc >0 it is necessary that the entropy production scales The second formulation of the III-law is a dynamical one, known as the unatinability principle: No refrigerator can cool a system to absolute zero temperature at nite time. This formulation is more restrictive, imposing limitations on the on the spectral density and the dispersion dynamics of the heat bath. We quantify this formulation by evaluating the characteristic exponent  of the cooling process. We relate the III-law to a generic quantum refrigerator model which 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 related to laser cooling. When optimized, both cases lead to the same exponent , 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 cold bath composed of harmonic oscillators and a cold bath composed of ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the third law will be addressed. 

[1] Amikam Levy and Ronnie Koslo, Phys. Rev. Lett. 108, 070604 (2012).