A study of heat flow through a qubit system

My master thesis deals with the study of nonequilibrium thermodynamics for open quantum systems, in particular It focuses on heat flow through a system of qubits in order to generalize Feynman-Vernon's work to a a system of spins (qubits). In the last decades, one of the early successes of classical statistical mechanics was the consistent definition of physical quantities such as work, heat, or entropy increase, at the level of single classical trajectories. Another resounding success is the discovery of the nonequilibrium fluctuation theorems. The study of fluctuating work and heat in open quantum systems is an active interface between nonequilibrium statistical physics and quantum information. As a matter of fact, consistently defining heat production for systems that are strongly coupled to their environment is still an active open question: thermodynamic heat does not fit into standard quantum theory as heat is a property of a process, and not of a state. Quantum heat hence cannot, in general, be expressed as an expectation value of a quantum operator. Despite many efforts and recent progress, the toolbox of nonequilibrium thermodynamics for open quantum systems is still incomplete and heat in quantum systems is a contentious topic, with potential applications to quantum heat engines. In the first part of my work, I use a powerful field-theoretic approach to thermodynamics given by the notion of a trajectory, already built in the Feynman-Vernon path integral, where an open quantum system is explicitly modeled as a system of interest interacting with a bath of harmonic oscillators, with the bath variables are integrated out. The Feynman-Vernon approach opens a way to investigate, through the generating functions, quantum heat in a genuine non-equilibrium setting of heat flow between different reservoirs. Secondly, by using the Feynman-Vernon formalism, it is possible to extend the approach to systems coupled to a dissipative environment (e.g. Caldeira-Leggett, spin-boson problem) in order to study the heat exchange with the environment. At this scope, interactions between spin and bosonic environments such as phonons (in solid state systems) or photons environments (in cavity electrodynamics) have been considered. One of the main goals in this thesis project is study and evaluate the formula for the full counting statistic when the system is one qubit. However Feynman-Vernon-style results are reasonably straight-forward to apply when the state space of the quantum system is continuous. They are less straightforward to apply when the state space of the quantum system is discrete a system of qubits. The main difficulty is hence to understand what is actually the dynamics system of qubits interacting with one or several heat baths.