Supersolidity of dipolar Bose-Einstein condensates induced by coupling to fermions

We study a mixture of a repulsive dipolar condensate and a degenerate Fermi gas in a quasi-one-dimensional geometry. We demonstrate that the presence of fermions, which attract bosons, drastically changes the behavior of the dipolar condensate. For a strong enough boson-fermion attraction, a dipolar Bose-Fermi droplet appears in the mixture, and as the attraction becomes stronger, a roton excitation develops in the Bogoliubov spectrum, leading to the formation of a supersolid, and eventually a crystal of isolated droplets. We describe the system by coupled extended Gross-Pitaevskii (bosons) and Hartree-Fock (fermions) equations. We study the excitation spectrum of the system and identify a number of Goldstone and Higgs modes in the supersolid regime.