Hamiltonian formalism of state-dependent diffusion and application to laser-cooling atoms

A consistent treatment of inhomogeneous diffusion requires a fully Hamiltonian formalism to account for the underlying couplings between various degrees of freedom. We do so by adopting a well-established approach, whereby the cold atom can be modeled as an open system nonlinearly coupled to a heat bath which mimics the light fields. By analyzing the explicit equation of motion for the system obtained, we demonstrate that there is not a microscopic correspondence to non-Gaussian thermal noise, as well as the spatial and momentum-dependent frictions giving play to the novel effects on anomalous diffusion. For application to cold atoms diffusing in dissipative optical lattices, we provide that the Hamiltonian-Langevin dynamics needed to be used to study this issue, because the drift coefficient and the partial diffusion coefficient are the momentum dependence and satisfy the fluctuation-dissipation theorem, however, the current phenomenological models lack a few momentum and coordinate coupling terms. Our results reveal that the mechanism of the spatial Lévy-like distribution of cold atoms originates from an intrinsic stochastic metastable potential.