Detecting quadrupole interactions in ultracold Fermi gases

We propose to detect quadrupole interactions of neutral ultracold atoms via their induced mean-field shift. We consider a Mott insulator state of spin-polarized atoms in a two-dimensional optical square lattice. The quadrupole moments of the atoms are aligned by an external magnetic field. As the alignment angle is varied, the mean-field shift shows a characteristic angular dependence, which constitutes the defining signature of the quadrupole interaction. For the ${}^3{P}_2$ states of Yb and Sr atoms, we find a frequency shift of the order of tens of Hertz, which can be realistically detected in experiment with current technology. We compare our results to the mean-field shift of a spin-polarized quasi-two-dimensional Fermi gas in continuum.

Figure 1

We consider spin-polarized fermions interacting only via quadrupole-quadrupole interactions. The quadrupoles are aligned by an external field $\mathbf{B}$ that points in the direction defined by the polar angles ${\theta }_F$ and ${\varphi }_F$. We propose to measure the mean-field shift of a Mott state in a 2D optical lattice, shown in (a), as a function of these angles. For comparison, we also consider a quasi-2D system of width ${\lambda }_z$ of fermions in continuum, panel (b), for which we determine the mean-field shift as a function of ${\theta }_F$.

Figure 2

(a) We consider two quadrupoles aligned by a magnetic field $\mathbf{B}$ with a displacement $\mathbf{R}$. The resulting quadrupole-quadrupole interaction is a function of their distance, $\left|\mathbf{R}\right|$, and the angle between $\mathbf{R}$ and $\mathbf{B}$, $\theta$. In the laboratory system, $\theta$ can be expressed by the polar angles ${\varphi }_F$ and ${\theta }_F$. (b) The characteristic feature of the quadrupole-quadrupole interaction is the change from repulsive to attractive interaction and back within 0 to $\pi /2$. For comparison, we also plotted the angular dependency of dipole-dipole interaction, with only one zero-crossing, and monopole-monopole interaction, which is constant over the whole range.

Figure 3

Effective 2D interaction $U_{2\mathrm{D}}\left(\mathbf{r}\right)$, compared to the bare interaction $U\left(\mathbf{r}\right)$, for ${\lambda }_z=38.4$ nm. The spatial extent of the wave function in the $z$ direction provides a cutoff for the effective interaction. The ratio of the interactions is depicted by the solid lines, the corresponding short-range limit by the dashed lines. (a) The alignment is parallel to the $z$ axis, i.e., ${\theta }_F=0$ and ${\varphi }_F$ is arbitrary. Here the effective interaction is rescaled to a logarithmic behavior. (b) For ${\theta }_F=\pi /4$ and ${\varphi }_F=\pi /4$, the resulting effective interaction behaves as $\sim 1/r^4$ at short distances.

Figure 4

(a) Mean-field shift for spin-polarized Yb(${}^3{P}_2$) atoms in a Mott insulator state, as a function of ${\theta }_F$ and ${\varphi }_F$, for the harmonic and pointlike approximation of the Wannier states. The small plot shows the full parametric surface while the larger one shows the most divergent cases for ${\varphi }_F$. The solid line corresponds to the case of ${\lambda }_z={\lambda }_{xy}=34.8$ nm; the dashed lines correspond to ${\lambda }_z=34.8$ nm and ${\lambda }_{xy}\to 0$. (b) Ratio of the harmonic approximation of the Wannier states to pointlike orbitals in the $xy$ plane, as a function of the lattice depth $V_{xy}/E_{\mathrm{rec}}^{xy}$. The location of $V_{xy}=35E_{\mathrm{rec}}^{xy}$, corresponding to ${\lambda }_{xy}=34.8\mathrm{nm}$, is marked. (c) The difference in the mean-field frequency shift between the exact Wannier states and the harmonic approximation as a function of ${\theta }_F$ and ${\varphi }_F$.

Figure 5

Characteristic behavior of the mean-field frequency shift $2\pi \Delta \nu$ changing sign twice. ${\theta }_F$ describes the angle of the external relative to the plane. The parameters for this example are given in Sec. 4.

Figure 6

Frequency shift $2\pi \Delta \nu$ for ${\theta }_F=0$ depending on experimental parameters: (a) quadrupole moment, (b) confinement in the $z$ direction, and (c) particle density. The parameters for this example are $V_z=35E_{\mathrm{rec}}^z$, $n=14.1\mu {\mathrm{m}}^{-2}$, $T=0$, and $q=30a_B^{2}e$ for Yb(${}^3{P}_2$), or marked in the corresponding plots, respectively. Additionally, $q=16a_B^{2}e$ corresponding to Sr(${}^3{P}_2$) is specified in (a).

Figure 7

Density-density correlation function $g_{\mathrm{con}}\left(r\right)$ for different temperatures $T/T_{\mathrm{F}}=0.1,1,10$ and $n=14.1\mu {\mathrm{m}}^{-2}$ (solid lines). Analytic results for $T=0$ (dotted line) and high-temperature limit for $T/T_{\mathrm{F}}=10$ (dashed line).

Figure 8

Mean-field shift of spin-polarized Yb(${}^3{P}_2$) atoms as a function of temperature $T/T_{\mathrm{F}}$, for a quasi-2D Fermi gas. The solid line corresponds to the exact result while the dashed line corresponds to the high-temperature approximation $T/T_{\mathrm{F}}\gg 1$. The analytic limiting cases for $T=0$ and $T/T_{\mathrm{F}}\to \infty$ are marked. We use $V_z=35E_{\mathrm{rec}}^z$, $n=14.1\mu \mathrm{m}$, and $q=30a_B^{2}e$ in this example.

Figure 9

Mean-field shift of spin-polarized Yb(${}^3{P}_2$) atoms in a 2D Mott state as a function of the lattice depths $V_{xy}/E_{\text{rec}}$ and $V_z/E_{\text{rec}}$ in the $z$ and $xy$ directions for equal lattice constants $a_z=a_{xy}=266\mathrm{nm}$. The red line corresponds to $V_z=V_{xy}$, along which the mean-field shift is almost constant.