Non-equilibrium quantum dynamics of nonlinear multimode circuit QED systems

Abstract

The circuit quantum electrodynamics (cQED) architecture has been established as a highly-engineerable platform for the realization of multimode quantum nonlinear systems - necessary to harness the power of quantum computation and simulation, and for fundamental studies of many-body quantum optics. In this thesis, we provide a detailed theoretical analysis of the driven-dissipative dynamics of specific multimode quantum networks incorporating a single nonlinear element, realized in cQED by a Josephson junction. Using a stochastic description based on the positive-$P$ representation, we analyze the dynamics of such networks across a well-defined classical-to-quantum transition. In the classical limit, we identify a specific symmetry requirement of the multimode networks - intimately tied to the dispersive nature of the Kerr-type nonlinearity furnished by the Josephson junction - which enables us to semi-analytically characterize their phase diagram for arbitrary network size. In regimes where the single nonlinear mode is sufficiently strongly-coupled to the network modes, the nature of their interaction becomes non-Markovian and fundamentally modifies the classical stability properties of the network. In particular, this interaction gives rise to classically unstable phases, where in addition to the possibility of chaotic dynamics, the network can undergo stable, periodic limit cycle excursions that display a frequency comb in the power spectrum. We analyze the impact of quantum fluctuations on these limit cycle dynamics as the system is brought to a quantum regime marked by a strengthening of the nonlinearity, and find that the linewidth of stable frequency combs is fundamentally limited by the vacuum fluctuations amplified by the very nonlinearity that generates the combs. Using a linearized Floquet analysis, we obtain an approximate analytic expression for the quantum-fluctuations-induced comb linewidths for multimode networks near the threshold of comb formation, an expression also verified using exact stochastic simulations. Our results and a recent experimental realization point towards multimode Kerr networks as a powerful cQED platform to study complex multimode dynamics of quantum nonlinear systems, and to generate engineerable, quantum-coherent multifrequency light sources.