  In multicomponent superfluids and superconductors, co- and counter-flows of
components have in general different properties. It was discussed in 1975 by
Andreev and Bashkin, in the context of He$^3$/He$^4$ superfluid mixtures, that
inter-particle interactions produce a dissipationless drag. The drag can be
understood as a superflow of one component induced by phase gradients of the
other component. Importantly the drag can be both positive (entrainment) and
negative (counter-flow). The effect is known to be of crucial importance for
many properties of diverse physical systems ranging from the dynamics of
neutron stars, rotational responses of Bose mixtures of ultra-cold atoms to
magnetic responses of multicomponent superconductors. Although there exists a
substantial literature that includes the drag interaction phenomenologically,
much fewer regimes are covered by quantitative studies of the microscopic
origin of the drag and its dependence on microscopic parameters. Here we study
the microscopic origin and strength of the drag interaction in a quantum system
of two-component bosons on a lattice with short-range interaction. By
performing quantum Monte-Carlo simulations of a two-component Bose-Hubbard
model we obtain dependencies of the drag strength on the boson-boson
interactions and properties of the optical lattice. Of particular interest are
the strongly-correlated regimes where the ratio of co-flow and counter-flow
superfluid stiffnesses can diverge, corresponding to the case of saturated
drag.
