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The concept of a quantum heat engine (QHEN) has been discussed in the literature, not only due to its intrinsic scientific interest, but also as an alternative to efficiently recover, on a nanoscale device, thermal energy in the form of useful work. The quantum character of a QHEN relies, for instance, on the fact that any of its intermediate states is determined by a density matrix operator. In particular, this matrix can represent a mixed state. For a classical heat engine, a theoretical upper bound for its efficiency is obtained by analyzing its quasi-static operation along a cycle drawn by a sequence of quasi-equilibrium states. A similar analysis can be carried out for a quantum engine, where quasi-static processes are driven by the evolution of ensemble-averaged observables, via variation of the corresponding operators or of the density matrix itself on a tunable physical parameter. We recently proposed two new conceptual designs for a magnetically-driven quantum engine, where the tunable parameter is the intensity of an external magnetic field. Along this article, we shall present the general quantum thermodynamics formalism developed in order to analyze this type of QHEN, and moreover, we shall apply it to describe the theoretical efficiency of two different practical implementations of this concept: an array of semiconductor quantum dots and an ensemble of graphene flakes submitted to mechanical tension

Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency