Miscibility in a degenerate fermionic mixture induced by linear coupling

We consider a one-dimensional mean-field-hydrodynamic model of a two-component degenerate Fermi gas in an external trap, each component representing a spin state of the same atom. We demonstrate that the interconversion between them (linear coupling), imposed by a resonant electromagnetic wave, transforms the immiscible binary gas into a miscible state, if the coupling constant, $\kappa$, exceeds a critical value, $\kappa _{\mathrm{cr}}$. The effect is predicted in a variational approximation, and confirmed by numerical solutions. Unlike the recently studied model of a binary BEC with the linear coupling, the components in the immiscible phase of the binary fermion mixture never fill two separated domains with a wall between them, but rather form anti-locked ($\pi$ -phase-shifted) density waves. Another difference from the bosonic mixture is spontaneous breaking of symmetry between the two components in terms of numbers of atoms in them, $N_{1}$ and $N_{2}$. The latter effect is characterized by the parameter $\nu \equiv (N_{1}-N_{2})/(N_{1}+N_{2})$ (only $N_{1}+N_{2}$ is a conserved quantity), the onset of miscibility at $\kappa \geq \kappa_{\mathrm{cr}}$ meaning a transition to $\nu \equiv 0$. At $\kappa <\kappa_{\mathrm{cr}}$, $\nu$ features damped oscillations as a function of $\kappa$. We also briefly consider an asymmetric model, with a chemical-potential difference between the two components.