In this thesis, we study nonlinear dissipative dynamics of two different systems: (i) synthetic dynamical gauge fields for photons in optomechanical arrays and (ii) laser arrays with a topologically nontrivial structure. The chosen models are relevant to state-of-the-art optomechanical experiments as well as to recent implementations of topological lasers based on semiconductor ring-resonator arrays. We present our results with a focus on applications.

First, we investigate nonlinear dynamics of synthetic gauge fields in optomechanical arrays in the classical regime. We demonstrate that synthetic electric fields for photons are generated in open one-dimensional arrays, leading to the suppression of light transport. Importantly, the generation of synthetic electric fields depends on the direction of light propagation, giving rise to unidirectional light transport. In a second step, we investigate the quantum dynamics of synthetic gauge fields in the minimal setup composed of two optical modes with photon tunneling assisted by a mechanical self-oscillator. Employing the quantum van-der-Pol oscillator as the simplest dynamical model for a mechanical self-oscillator enables us to take quantum fluctuations into account using the quantum master equation formalism. We show that the generation of synthetic electric fields is robust against fluctuations and unidirectional light transport can be achieved also in the quantum regime.

In the second part of the thesis, we study topological lasing, which arises from the combination of topologically-protected chiral light transport and laser amplification, leading to laser operation that is robust against local disorder in system parameters and defects. We study a topological laser arrays based on the photonic Haldane model with selective pumping of chiral edge modes described by saturable gain. We investigate elementary excitations around the mean-field steady state and their consequences for the coherence properties. In particular, we show that the hybridization of chiral edge modes gives rise to long-lived elementary excitations, leading to large phase fluctuations in the emitted light field and a decrease of light coherence. In contrast to topologically trivial lasers, the lifetime of elementary excitations is robust against disorder in topological lasers. The lifetime depends strongly on the edge-mode dispersion around the lasing frequency. As a result, the lifetime can be reduced by orders of magnitude for lasing of different edge modes, leading to a suppression of phase fluctuations and, consequently, larger coherence of the emitted light. On the other hand, amplitude fluctuations and the second-order autocorrelation function are moderately increased at the same time.