Stability of quantum degenerate Fermi gases of tilted polar molecules

A recent experimental realization of a quantum degenerate gas of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecules opens up prospects of exploring strong dipolar Fermi gases and many-body phenomena arising in that regime. Here, we derive a mean-field variational approach based on the Wigner function for the description of the ground-state properties of such systems. We show that the stability of dipolar fermions in a general harmonic trap is universal as it only depends on the trap aspect ratios and the dipoles' orientation. We calculate the species-independent stability diagram and the deformation of the Fermi surface (FS) for polarized molecules, whose electric dipoles are oriented along a preferential direction. Compared to atomic magnetic species, the stability of a molecular electric system turns out to strongly depend on its geometry and the FS deformation significantly increases.

Figure 1

(a) Shape of the molecular cloud and the FS of an ideal Fermi gas. (b) Schematic illustration of the ansatz for the shape of the molecular cloud and the FS of a dipolar Fermi gas. The inset shows the orientation of dipoles.

Figure 2

(a) A universal stability diagram for harmonically trapped ultracold dipolar Fermi gases at quantum degeneracy: Critical value of the relative dipole-dipole interaction strength ${\varepsilon }_{\mathrm{dd}}^{\mathrm{crit}}$ for $\theta =\phi =0$. The system has a stable ground state for ${\varepsilon }_{\mathrm{dd}}\le {\varepsilon }_{\mathrm{dd}}^{\mathrm{crit}}$. (b), (c) Critical value of the electric dipole moment $d^{\mathrm{crit}}$ for a stable ground state of $N=3\times {10}^4$ ultracold molecules of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ for $\theta =\phi =0$: (b) ${\omega }_z=2\pi \times 50\mathrm{Hz}$; (c) ${\omega }_z=2\pi \times 200\mathrm{Hz}$. White dots in (a), (c) correspond to the parameters of experiment [48], while black lines in (b), (c) correspond to the permanent dipole moment $d=0.574$ D of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecules.

Figure 3

FS deformation $\mathrm{\Delta }$ as a function of ${\omega }_x$ and ${\omega }_y$ for a ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ system with $N=3\times {10}^4, {\omega }_z=2\pi \times 200\mathrm{Hz}, \theta =\phi =0$, and electric dipole moments: (a) $d=0.22$ D; (b) $d=0.26$ D. White dots correspond to the parameters of Ref. [48]. The black region in (b) does not yield stable solutions.

Figure 4

(a) Angular stability diagram of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$: $d^{\mathrm{crit}}$ as a function of the dipoles' orientation. The black line corresponds to the permanent electric dipole moment $d=0.574$ D of ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$, and the trap frequencies are as in Ref. [48], with $N=3\times {10}^4$ molecules. (b), (c) Angular dependence of $\mathrm{\Delta }$ for a fixed value $d=0.25$ D and the trap frequencies: (b) As in Ref. [48]; (c) $({\omega }_x,{\omega }_y,{\omega }_z)=2\pi \times (50,500,900)\mathrm{Hz}$.

Figure 5

(a) Real-space aspect ratio $A_R$ of the ${}^{40}\mathrm{K}{}^{87}\mathrm{Rb}$ molecular cloud as a function of time $t$ during the TOF expansion from the ground state, after the trap is switched off. Top (red) solid and dashed lines are obtained for $d=0.5$ D and frequencies $2\pi \times (63,36,500)\mathrm{Hz}$, bottom (blue) solid and dashed lines for $d=0.22$ D and $2\pi \times (63,36,200)\mathrm{Hz}$, and inset for $d=0.35$ D and $2\pi \times (250,150,100)\mathrm{Hz}$. (b) $A_R$ (30 ms) after the TOF expansion for $t=30$ ms as a function of $d$. Trap frequencies corresponding to all line types are the same as in (a). In both plots solid lines correspond to a nonballistic expansion, where the DDI is taken into account, while the dashed lines represent calculated results for a free (ballistic) expansion, $N=3\times {10}^4, \theta =\phi =0$. $A_R$ is calculated using the imaging angle $22.5^{\circ }$ of Ref. [48], in the geometry of Ref. [23].