Self-heterodyne detection of the \it in-situ phase of an atomic-SQUID

We present theoretical and experimental analysis of an interferometric
measurement of the {\it in-situ} phase drop across and current flow through a
rotating barrier in a toroidal Bose-Einstein condensate (BEC). This experiment
is the atomic analog of the rf-superconducting quantum interference device
(SQUID). The phase drop is extracted from a spiral-shaped density profile
created by the spatial interference of the expanding toroidal BEC and a
reference BEC after release from all trapping potentials. We characterize the
interferometer when it contains a single particle, which is initially in a
coherent superposition of a torus and reference state, as well as when it
contains a many-body state in the mean-field approximation. The single-particle
picture is sufficient to explain the origin of the spirals, to relate the
phase-drop across the barrier to the geometry of a spiral, and to bound the
expansion times for which the {\it in-situ} phase can be accurately determined.
Mean-field estimates and numerical simulations show that the inter-atomic
interactions shorten the expansion time scales compared to the single-particle
case. Finally, we compare the mean-field simulations with our experimental data
and confirm that the interferometer indeed accurately measures the {\it
in-situ} phase drop.