Spontaneous symmetry breaking of Bose-Fermi mixtures in double-well potentials

Abstract

We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF) mixture in a double-well potential (DWP). The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density functional in the unitarity limit (a similar model applies to the BCS regime, too). Straightforward SSB in the degenerate Fermi gas loaded into a DWP is impossible, as it requires an attractive self-interaction, and the intrinsic nonlinearity in the Fermi gas is repulsive. Nonetheless, we demonstrate that the symmetry breaking is possible in the mixture with attraction between fermions and bosons, like $^{40}\mathrm{K}$ and $^{87}\mathrm{Rb}$. Numerical results are represented by dependencies of asymmetry parameters for both components on particle numbers of the mixture, ${N}_{\mathrm{F}}$ and ${N}_{\mathrm{B}}$, and by phase diagrams in the $({N}_{\mathrm{F}},{N}_{\mathrm{B}})$ plane, which displays regions of symmetric and asymmetric ground states. The dynamical picture of the SSB, induced by a gradual transformation of the single-well potential into the DWP, is reported too. An analytical approximation is proposed for the case when the GPE for the boson wave function may be treated by means of the Thomas-Fermi (TF) approximation. Under a special linear relationship between ${N}_{\mathrm{F}}$ and ${N}_{\mathrm{B}}$, the TF approximation allows us to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms. The latter one directly displays the mechanism of the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture.