Randomness in Dissipative and Chaotic Quantum Dynamics

This thesis studies randomness in quantum dissipative and chaotic dynamics. We first focus on a detailed study of the effect of noise in the Hamiltonian of a quantum system, going beyond the noise-average and characterizing higher moments. The main quantity we introduce is the Stochastic Operator Variance (SOV), which is an observable characterizing the spread of trajectories around the average evolution. Interestingly, this quantity fulfills different types of uncertainty relations and is related to quantum information scrambling through out-of-time-order correlators (OTOC). We illustrate the SOV-OTOC connection in a stochastic version of the Lipkin-Meshkov-Glick (LMG) model, which shows a positive Lyapunov exponent from an unstable saddle point. We find that under the action of the noise, this Lyapunov exponent can change sign, thus stabilizing the unstable phase of the model while destabilizing the stable one. We then study the interplay between noise and decay in a non-Hermitian Hamiltonian. The noise-average evolution follows an antidephasing master equation beyond Lindblad form. We characterize the purity dynamics and the steady states of this master equation, and study this new evolution in a stochastically driven version of the Dissipative Qubit. By characterizing its spectral and steady state properties, we find that there are three phases: the PT unbroken, PT broken, and a novel Noise Induced (NI) phase where the qubit converges to the lossy state. We further investigate the validity of our model to explain experimental data, such as the residual damping rate of the PT unbroken phase. In the last Chapter, we study randomness as a model for chaotic dynamics. In particular, leveraging a Wigner-like surmise for the k-th neighbor level spacing distribution, we compute analytically the k-th neighbor Spectral Form Factor (knSFF), which characterizes the contribution of the k-th neighbor spacings to the Spectral Form Factor (SFF). We study the properties of the individual knSFF and characterize their role in building the universal ramp of the SFF. Interestingly, we find that the very short-range and very long-range spacings are the ones that contribute the most to the extent of the ramp. We finish by discussing our results and possible pathways for further research.