Dropping cold quantum gases on Earth over long times and large distances

We analyze the evolution of a degenerate quantum gas (bosons and fermions) falling in Earth's gravity during long times $(10\phantom{\rule{0.3em}{0ex}}\mathrm{s})$ and over large distances $(100\phantom{\rule{0.3em}{0ex}}\mathrm{m})$. This models an experiment that is currently performed by the QUANTUS Collaboration at ZARM drop tower in Bremen, Germany. Starting from the classical mechanics of the drop capsule and a single particle trapped within, we develop a quantum field theoretical description for this experimental situation in an inertial frame, the corotating frame of the Earth, as well as the comoving frame of the drop capsule. Suitable transformations eliminate noninertial forces, provided all external potentials (trap, gravity) can be approximated with a second-order Taylor expansion around the instantaneous trap center. This is an excellent assumption, and the harmonic potential theorem applies. As a first application, we study the quantum dynamics of a cigar-shaped Bose-Einstein condensate in the Gross-Pitaevskii mean-field approximation. Due to the instantaneous transformation to the rest frame of the superfluid wave packet, the long-distance drop can be studied easily on a numerical grid.