Analytical description of collisional decoherence in a BEC double-well accelerometer

BEC-based quantum sensors offer a huge, yet not fully explored potential in gravimetry and ac- celerometry. In this paper, we study a possible setup for such a device, which is a weakly interacting Bose gas trapped in a double-well potential. In such a trap, the gas is known to exhibit Josephson oscillations, which rely on the coherence between the potential wells. Applying the density matrix approach, we consider transitions between the coherent, partially incoherent, and fully incoherent states of the Bose gas. We provide an analytical description of the collisional decoherence due to weak interactions, causing the Josephson oscillations to decay with time. In particular, we give the mathematical link between that decay in the density matrix approach and its interpretation in terms of phase fluctuations. To investigate the potential of the double-well setup as a quantum sensor we apply additional external acceleration to the system. The interplay of collisional interaction and ac- celeration leads to an additional shift of the oscillation frequency. We give the analytical expression for this shift and estimate the sensitivity of a hypothetical BEC double-well accelerometer based on that effect.