  The Andreev-Bashkin effect, or superfluid drag, is predicted in a system of
Bose-condensed excitonic polaritons in optical microcavity coupled by
electron-exciton interaction with a superconducting layer. Two possible setups
with spatially indirect dipole excitons or direct excitons are considered. The
drag density characterizing a magnitude of this effect is found by many-body
calculations with taking into account dynamical screening of electron-exciton
interaction. For the superconducting electronic layer, we assume the recently
proposed polaritonic mechanism of Cooper pairing, although the preexisting
thin-film superconductor should also demonstrate the effect. According to our
calculations, the drag density can reach considerable values in realistic
conditions, with excitonic and electronic layers made from GaAs-based quantum
wells or two-dimensional transition metal dichalcogenides. The predicted
nondissipative drag could be strong enough to be observable as induction of a
supercurrent in the electronic layer by a flow of polariton Bose condensate.
