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How to rigorously define thermodynamic quantities such as heat, work, andinternal energy in open quantum systems driven far from equilibrium remains asignificant open question in quantum thermodynamics. Heat is a quantity whosefundamental definition applies only to processes in systems infinitesimallyperturbed from equilibrium, and as such, must be accounted for carefully instrongly-driven systems. A key insight from Mesoscopics is that infinitely farfrom the local driving and coupling of an open quantum system, reservoirs areindeed only infinitesimally perturbed, thereby allowing the heat dissipated tobe defined. The resulting partition of the entropy necessitates a Hilbert-spacepartition of the energetics, leading to an unambiguous operator for theinternal energy of an interacting time-dependent open quantum system. Fullygeneral expressions for the heat current and the power delivered by variousagents to the system are derived using the formalism of nonequilibrium Green'sfunctions, establishing an experimentally meaningful and quantum mechanicallyconsistent division of the energy of the system under consideration into Heatflowing out of and Work done on the system. The spatio-temporal distribution ofinternal energy in a strongly-driven open quantum system is also analyzed. Thisformalism is applied to analyze the thermodynamic performance of a modelquantum machine: a driven two-level quantum system strongly coupled to twometallic reservoirs, which can operate in several configurations--as a chemicalpump/engine or a heat pump/engine.

On the First Law of Thermodynamics in Time-Dependent Open Quantum Systems