Anomalous Doppler effect in two-component Bose-Einstein condensates

We show that two-component Bose-Einstein condensed mixtures, in presence of a persistent current, exhibit a nontrivial Doppler shift of the sound velocities. The peculiarity is due to the interspecies interaction and the possibility of generating a counterflow persistent current. Analytical predictions are derived using superfluid hydrodynamics. While the existence of anomalous Doppler shifts at finite temperature has been discussed a long time ago in the case of superfluid $^{4}\mathrm{He}$, an experimental verification of the effect is still missing. For this reason, we also propose a protocol for the measurement of the Doppler shifts, based on the density-density response function. The dynamical protocol is simulated by means of coupled Gross-Pitaevskii equations.