Thermodynamics of open quantum systems: from a critical study to the optimization of non-equilibrium heat engines

One of the most relevant aspects of thermodynamics is its universality. Its prescriptions are ubiquitous in the characterizaton of the energy transfer between systems at equilibrium, even at the nanoscale, where quantum effects start to become important. In these models the energy balance is completely described in terms of universal quantities, like the Helmoltz free energy and the Boltzmann entropy, while the probabilistic fluctations of work, heat end particle number are tipically negligible making equilibirum thermodynamics essentially a deterministic theory. There are, however, plenty of fields in which the equilibrium approach is too limiting, for instance when dealing with steady state and driven heat engines, researching efficient quantum probes in metrology and even studying decoherence phenomena in quantum computation. In the non equilibrium scenario many specific details, usually negligible in the standard approach, become relevant and a more accurate knowledge of the dynamics is necessary to improve the capacity of controlling, measuring and manipulating energy, whose puctuations also become larger and larger making the theory intrinsecally stochastic. The characterization of out of equilibrium quantum system is the principal aim of this manuscript, which encompasses several aspects of the field. A perturbative expansion for slowly driven master equations is derived, reproducing the quasi static equilibrium trajectory for infinitely slow modulations and providing a compact formula for calculating the deviations on such a behavior. The expansion turns also to be succesful for the description of low dissipation heat engines, providing interesting connections between some celebrated efficiencies at maximum power (like the Schmiedl Seifert and Curzon Ahlborn ones [34, 37]) and the spectral density of the baths inducing thermalization.