Quantum fluids, Josephson tunneling and gravitational waves

In this paper we study the Josephson tunneling for a quantum fluid (a Bose–Einstein condensate) in the presence of a weak gravitational wave. Starting from a Lagrangian formulation in the framework of linearized gravity, we deduce the Gross–Pitaevskii equation for the fluid in a weak gravitational background. We use such an equation to investigate the influence of a gravitational wave on the Josephson effect. Considering a double-well trap, made of two identical boxes placed orthogonally to each other and weakly coupled through a small junction, we show how the trap geometry influences the coupling between the gravitational wave and the quantum field modes in the two sides of the trap. Namely, the macroscopic wavefunction of the condensate acquires a different phase shift on the two sides, hence giving rise to a gravitationally induced ac-Josephson effect through the trap junction. Although small, such an effect is theoretically interesting, since it represents the influence of a gravitational wave on a mesoscopic quantum system. Also the effect exhibits polarimetric properties. Possible experimental detection of such an effect is briefly discussed at the end of the paper.