Magnetic Dipolar Interaction between Hyperfine Clock States in a Planar Alkali Bose Gas

In atomic systems, clock states feature a zero projection of the total angular momentum and thus a low sensitivity to magnetic fields. This makes them widely used for metrological applications like atomic fountains or gravimeters. Here, we show that a mixture of two such nonmagnetic states still displays magnetic dipole-dipole interactions comparable to the one expected for the other Zeeman states of the same atomic species. Using high-resolution spectroscopy of a planar gas of ${}^{87}\mathrm{Rb}$ atoms with a controlled in plane shape, we explore the effective isotropic and extensive character of these interactions and demonstrate their tunability. Our measurements set strong constraints on the relative values of the $s$-wave scattering lengths $a_{ij}$ involving the two clock states.

Figure 1

(a) Level diagram of the hyperfine ground-level manifold showing the two states relevant to this Letter: $|1⟩\equiv |F=1,m_Z=0⟩$ and $|2⟩\equiv |F=2,m_Z=0⟩$. (b) Image of the atomic cloud obtained through absorption imaging along the direction perpendicular to the atomic plane. Atoms are confined in the $xy$ plane in a disk of radius $12\mu \mathrm{m}$. The orientation of the magnetic field $\mathit{B}$ is tuned in the $xz$ plane. (c) Schematics of atoms prepared in the state $|1⟩$, with no MDDIs in this case. MDDIs are also absent when all atoms are in $|2⟩$. (d) Significant MDDIs occur for atoms in a linear superposition of $|1⟩$ and $|2⟩$.

Figure 2

(a),(b) Normalized Ramsey oscillations measured for $\mathit{B}$ perpendicular ($\mathrm{\Theta }=0°$) or parallel ($\mathrm{\Theta }=90°$) to the atomic plane. For both cases, we show the transferred population as a function of detuning $\delta$ to the single-atom resonance. In each case the resonance is marked by a vertical dashed line. The circles (respectively, squares) correspond to a balanced (respectively, unbalanced) mixture $f=0$ (respectively, $f\approx 0.95$). Vertical error bars represent the standard deviation from the two measurements realized for each points. (c) Variation of the frequency shift $\mathrm{\Delta }\nu$ with the imbalance $f$. We restrict to positive imbalances, for which the population in $|2⟩$ remains small enough to limit the role of two-body relaxation and spin-changing collisions. For each angle, the solid line is a linear fit to the data.

Figure 3

Variation of the ratio $R(\mathrm{\Theta })$ determined from the data of Fig. 2 with the magnetic field orientation $\mathrm{\Theta }$. Blue circles (respectively, red squares) correspond to the measurement at maximum density (respectively, half density). The variation of this ratio is well fitted by a cosine variation compatible with the prediction for MDDIs. The amplitude and offset of this variation allow one to determine accurately relative values of the scattering lengths. Vertical error bars represent the uncertainty obtained from the fitting procedure of the data in Fig. 2. The uncertainty on the determination of the angles is limited by the geometrical arrangement of the coils generating the field $\mathit{B}$, estimated here at the level of 1°.

Figure 4

Interaction shift $\mathrm{\Delta }\nu$ as a function of the anisotropy parameter $\eta$. For a fixed density and an in plane magnetic field, we vary the anisotropy of the elliptically shaped 2D cloud. No dependence on the shape of the cloud is observed, in agreement with the expected isotropic character of MDDIs in 2D when $R_{x,y}\gg {\ell }_z$. Vertical error bars represent the estimated 1 Hz accuracy on the determination of the single-atom resonance frequency. Inset: interaction shift as a function of the size of the cloud, for $\mathit{B}$ normal to the atom plane.