Phase Dependence of Phonon Tunneling in Bosonic Superfluid-Insulator-Superfluid Junctions

We consider the tunneling of phonon excitations across a potential barrier spatially separating two condensates with different macroscopic phases. We analyze the relation between the phase difference $\varphi$ of the two condensates and the transmission coefficient {\it T} by solving the Bogoliubov equations. It is found that {\it T} strongly depends on $\varphi$, and that the perfect transmission of low-energy excitations disappears when the phase difference reaches the critical value which gives the maximum supercurrent of the condensate. We also discuss the feasibility of observing the phase differences in experiments.