Self-heterodyne detection of the in situ phase of an atomic superconducting quantum interference device

We present theoretical and experimental analysis of an interferometric measurement of the 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 in situ phase can be accurately determined. Mean-field estimates and numerical simulations show that the interatomic 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 in situ phase drop.