Emergence of Gaussian fields in noisy quantum chaotic dynamics

Abstract: We study the long time Schr\"odinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schr\"odinger operators $P_h^\alpha=-h^2\Delta_g +h^\alpha Q_\omega$, $0<\alpha\leq 1$, where the semiclassical Laplace-Beltrami operator $-h^2\Delta_g$ on $X$ is subject to a small random perturbation $h^\alpha Q_\omega$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_\omega$ is bounded, but oscillates and decorrelates at scale $h^{\beta}$, $0< \beta < \frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $\alpha,\beta$, in probability with respect to $\omega$ the long time propagation $$\mathrm{e}^{\frac{i}{h}t_h P_h^\alpha } f_h, \quad o(|\log h|)=t_h\to\infty, ~~h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $\omega$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$.