Zakari Denis —  Università Paris-Sud e Università di Parma #
Quantum transport of ultracold atoms #
We would want to find a description for ultracold atoms (bosons, possibly extendable to fermions) that are injected and transported along a lattice. Starting from the Bose-Hubbard model coupled to one lead (or eventually more than one), equations of motion should be derived for correlation functions of mode creation and annihilation operators. With appropriate approximations these equation may be brought into a closed form. While this first part is purely analytic, the obtained equations of motions can be solved only in special cases (e.g. without atom-atom interactions) and numerical integration methods must be applied in general.


A first step is the understanding of a method already introduced and applied to open quantum systems. This BBGKY hierarchy expansion consists of dynamical equations for two, four, etc. point correlations functions, i.e. the expectation values of the two-, four-, etc. body reduced density matrices. The interaction term in the Hubbard-Hamiltonian then induces the coupling between these dynamical equations to higher orders. Typically, we truncate at the second order, approximating the six-point correlator by products of two- and four-point correlation functions. This allows us to arrive at a closed system of coupled equations, which we can subsequently solve. The complication arises from the additional expectation values containing the reservoir operators (the boundary terms), which have to be taken into account in addition. A second step may be the extension of the BBGKY hierarchy expansion from time-equal expectation values to different-time ones for the one-, two-, etc. particle reduced density matrix elements. With appropriate approximations (e.g. the truncation of the hierarchy equations for higher-order expectation values) this should again lead to a closed but much larger system of equations for C-numbers, which can be solved numerically in principle.