Gaussian dynamics in the Lindblad equation and its stochastic unravellings

In this study, we analyze the evolution of Gaussian states in open quantum systems governed by Lindblad equations, which can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians. These equations demonstrate the familiar phenomena of dissipation and decoherence and can be understood as an average over the stochastic pure-state dynamics of individual experimental realizations. We examine the quantum-jump and stochastic Schrödinger dynamics for initially Gaussian states and find that while both approaches converge to the Lindblad dynamics when averaged, the individual dynamics can differ significantly. For the stochastic Schr\"odinger equation, Gaussian states remain Gaussian, with the evolution of the phase-space center governed by stochastic differential equations and the covariance matrix evolving deterministically. In contrast, pure-state dynamics arising from quantum-jump evolution do not generally remain Gaussian. We develop a method for generating quantum-jump and stochastic Schr\"odinger trajectories for arbitrary initial states based on the evolution of an underlying Gaussian state. Using several quadratic examples, we compare the behavior of the different unravellings to the Lindblad dynamics before studying the non-quadratic systems in which the earlier derived Gaussian dynamics are not an exact representation of the full quantum dynamics, yet they may be thought of as their semiclassical limit. As an example, we study a thermally driven particle in a double well, investigating the effect of temperature on tunneling properties and timescales. In this system, we also examine the transition rates of the quantum dynamics and its semiclassical and classical limits.Open Acces