Topology and Nonlinearity in Driven-Dissipative Photonic Lattices: Semiclassical and Quantum Approaches

This Thesis aims to advance the understanding of photonic systems along two parallel but complementary tracks: the semiclassical analysis of lattices involving a macroscopic number of photons and the quantum analysis of strongly-correlated photonic lattices. The semiclassical analysis is particularly suited to investigate systems, among others, that emerge in the context of topological photonics. In particular, this Thesis focuses on the analysis of so-called topological lasers, devices that aim to improve on traditional lasers by operating in special lasing modes, i.e. topological modes, that are more resilient against disruptions. Via a semiclassical description of the laser physics, we are able to determine the most efficient gain configurations, uncover new effects and new instability regimes, investigate the resilience to disorder, and design subtle mechanisms that allow the laser to select the desired topological mode even under unfavorable conditions. The quantum analysis of strongly-correlated photonic lattices is instead complicated by the sheer size of the Hilbert space and by the inherent many-body correlations, which do not allow for an effective single-particle scenario. In absence of a coherent pump, though, small enough systems can still be studied via exact diagonalization by exploiting their phase rotation symmetry; we do so in the analysis of a driven-dissipative Bose-Hubbard dimer, where two nonlinear cavities are coupled. In this system, we reveal signatures of a localization-delocalization transition in steady-state observables and in steady-state response functions, as well as in the time dynamics of the observables. For the more ambitious goal of studying a full lattice of highly nonlinear cavities, though, this is no longer enough. Therefore, we employ a technique called dynamical mean-field theory, borrowed from the treatment of strongly-correlated electron systems, to reduce the size of the effective Hilbert space by replacing the interaction generated by the lattice on any one of its sites with an effective time-dependent field. As a case study for the effectiveness of our specific implementation of this technique in the context of driven-dissipative photonic lattices, we demonstrate the ability to reproduce the so-called quantum Zeno effect in a Bose-Hubbard lattice with strong two-particle dissipation.