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We develop a functional integral formulation for binary Bose-Einsteincondensates coupled to polarized fermions. We find that spin-dependentfermion-mediated interactions have dramatic effects on the properties of thebinary condensates. The quasiparticle spectrum features two branches. The upperbranch, which is of density nature, gets modified by the induced interactions,while the lower branch, which is of spin nature, is left intact. Theground-state phase diagram consists of stable region of miscible phases andunstable region toward phase separation. In the stable region, it is furtherclassified by the damping of excitations of the upper branch. We show that itis possible to find region of well-defined, long-lived quasiparticleexcitations by tuning relevant parameters, such as boson-fermion mass ratio,boson-fermion number density ratio, and interspecies interactions betweenbosons as well. We explore the effects of quantum fluctuation due to theeffective potential on the binary condensates. It turns out that both thedensity structure factor and spin density structure factor fulfill the Feynmanrelation, except that the latter is immune to the fermion-mediatedinteractions.

Ultracold Bose Mixtures with Spin-Dependent Fermion-Mediated Interactions