Ground state of a two-component dipolar Fermi gas in a harmonic potential

Interacting two-component Fermi gases are at the heart of our understanding of macroscopic quantum phenomena such as superconductivity. The changing nature of the interaction is expected to head to novel quantum phases. Here we study the ground state of a two-component fermionic gas in a harmonic potential with dipolar and contact interactions. Using a variational Wigner function we present the phase diagram of the system with equal but opposite values of the magnetic moment. We identify the second-order phase transition from paramagnetic to ferronematic phases. Moreover, we show the impact of the experimentally relevant magnetic field on the stability and magnetization of the system. We also investigate a two-component Fermi gas with large but almost equal values of the magnetic moment to study how the interplay between contact and dipolar interactions affects the stability properties of the mixture. To be specific we discuss experimentally relevant parameters for ultracold ${}^{161}$Dy.

Figure 1

Comparison of the deformation of the cloud in momentum ($\alpha$) and position ($\beta$) spaces between Gaussian and $\theta$ Wigner functions (W.f.) for the $g_d$ in the stable regime. Notice very good agreement for the stable range of $g_d{N}^{1/6}$.

Figure 2

Phase diagram for two-component gas. Three regimes of different magnetization are visible. For $M=0$, the size and shape of components are like those for a noninteracting gas. For $0<M<1$, the Fermi surface of the bigger component has the prolate shape (red) whereas for smaller one, the oblate (blue). Thick, dashed line presents smooth crossover to the phase with $M=1$ where the shape of the Fermi surface is prolate. The transition to the unstable regime is possible from not only the ferronematic but also the unmagnetized phase. (b-c) Deformations in momentum and position spaces (${\alpha }_i$ and ${\beta }_i$) as well as compression in the position space (${\lambda }_i$) for constant $g_d{N}^{-1/6}=1.2$ and $g_c{N}^{-1/6}=13$, respectively [dash-dotted lines in (a)]. In the paramagnetic phase, due to the contact interaction, both components are getting bigger in position space with growing $g_c$. In the ferronematic regime, the smaller component (index 2) still gets bigger; while for a constant $g_d$, parameter ${\lambda }_1$ goes to 1, which means equal occupation of the position and momentum phase spaces. For constant $g_c$, we see that the collapse is due to the higher occupation of the momentum rather than position phase space.

Figure 3

Phase diagrams showing magnetization and unstable regime for several values of $g_B{N}^{-1/3}$. For nonzero magnetic field we no longer have a paramagnetic phase. Black solid lines show the range of the unstable regime. Notice that the boundary between the unstable and fully polarized ferronematic phases does not depend on the strength of the magnetic field. Moreover, as indicated by the color, magnetization equals unity in ferronematic as well as in unstable regions. For $N={10}^6$ and $\omega =2\pi \times 100$ Hz, the diagrams correspond to $B=0.01,1,2,4$ mG.

Figure 4

(a) $\frac{N_1-N_2}{N}$ for a two-component gas of $m_1=\frac{21}{2}$ and $m_2=\frac{19}{2}$ for different numbers of atoms and $\omega =2\pi \times 100$ Hz. (b-c) Deformations in momentum and position spaces (${\alpha }_i$ and ${\beta }_i$) as well as compression in the position space (${\lambda }_i$) for constant $g_d{N}^{-1/6}=1.2$ and $g_c{N}^{-1/6}=13$, respectively [dashed lines in (a)]. In the whole range $N_1>N_2$ and both components, whenever they exist, are prolate in momentum and position spaces (more elongated is the first one). The boundary of the stability of two components against collapse is strongly changed due to the interplay between contact and dipolar interactions.

Figure 5

(a) Size of the cloud and (b) number of particles in every component for different magnetic fields. Results are for experimental parameters of Dy: $\mu =10{\mu }_B$, $\omega =2\pi \times 100$ Hz, and $N={10}^6$.