Spontaneous symmetry breaking in a mixed superfluid of fermions and bosons trapped in double-well potentials

We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF) mixture loaded into a double-well potential (DWP), in the effectively onedimensional setting. The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, which is coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density-functional model in the unitarity limit (a similar model may apply to the Bardeen-Cooper-Schriefer (BCS) regime too). Straightforward SSB in the quantum Fermi gas loaded into a DWP is impossible, as it requires an attractive self-interaction acting in the medium, while the intrinsic nonlinearity in the Fermi gas may only be repulsive. However, we demonstrate that the symmetry breaking can be made possible in the mixture, provided that interaction between the fermions and bosons is attractive (a real example is the mixture of potassium and rubidium atoms, which represent fermions and bosons, respectively). Numerical results for the SSB are represented by dependencies of asymmetry parameters for both components on numbers of particles in the mixture, and by phase diagrams in the plane of these two numbers. The diagrams display regions of symmetric and asymmetric ground states of the mixture. Dynamical pictures of the SSB, induced by a gradual transition from the single-well potential into the DWP, are reported too. In addition to the systematic numerically generated results, an analytical approximation is elaborated for the case when the GPE for the boson wave function is amenable to the application of the Thomas-Fermi (TF) approximation. Under a special linear relation between the numbers of fermions and bosons, the TF approximation makes it possible to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms, of powers 7/3 and 3, respectively. The latter terms directly illustrates the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture, whose density is "enslaved" to the fermion density, in that case.