• non Markovian Fermi Fluids

In this thesis we have numerically studied the quantum phases of a fermionic onedimensional (1D) quantum gas of atoms in two spin levels and interacting via the photons in an optical cavity. This is especially interesting since the virtual exchange of photons from the cavity modes provides the atoms with an effective interaction that is tunable in strength, after acting on the photon-atom coupling and on the detuning, and in range by coupling the atoms to different numbers of cavity modes with the same energy.
In fact, optical cavities are progressively becoming a very versatile platform for precision measurements and as quantum resources to simulate many-body systems. In particular, the engineering of quantum technologies for ultracold atoms in optical cavities may lead to the realization of the concept of many-body entanglement and its applications to phase estimations for metrological use in atomic interferometry experiments. On the other hand, optical cavities are intrinsically open (quantum) systems, so that driving and dissipation effects are to be taken into account and carefully treated, as they can affect the persistence of selected quantum phases.
This is a complex problem, to be treated by means of suited numerical methods. In this thesis, we have implemented the HOPS algorithm, which is based on the concept of the quantum state diffusion. The work carried out for this thesis has been performed within an international collaboration, and in particular with Professor Andrew Daley (University of Strathclyde) and Professor Jonathan Keeling (University of St Andrews, UK) for the theoretical aspects, and with Professor Benjamin Lev at Stanford for the possible experimental implementations of the idea. In fact, I have spent a few months in Glasgow to learn and implement the numerical methods, using the grants made available for such purposes by the University of Pisa.
The thesis work is organized as follows. Chap. 2 is dedicated to introduce useful concepts on the physics of atoms in optical cavities and the theoretical tools needed to investigate open quantum system. Chap. 3 is devoted to discuss the model physical model under examination, and Chap. 4 to illustrate the specific numerical
methods and their implementation. We discuss the results in Chap. 5, with special attention to the possibility of realizing a superfluid phase of the driven-dissipative Fermi system in 1D sense. Finally, Chap. 6 is devoted to outline concluding remarks and future perspectives.
My personal contributions to the thesis work include: - Setting of the codes for the numerical solution using the HOPS algorithm; - Data analysis of the output obtained from the simulations; - Discussion and characterization of the physics emerging from the results.