Interspecies interactions in an ultracold dipolar mixture

We experimentally and theoretically investigate the influence of the dipole-dipole interactions (DDIs) on the total interspecies interaction in an erbium-dysprosium mixture. By rotating the dipole orientation we are able to tune the effect of the long-range and anisotropic DDI, and therefore the in-trap displacements of the erbium and dysprosium clouds. We present a theoretical description for our binary system based on an extended Gross-Pitaevskii theory, including the single-species beyond mean-field terms, and we predict a lower and an upper bound for the interspecies scattering length $a_{12}=105[-65,+162]a_0$. Our work is a step towards the investigation of the experimentally unexplored dipolar miscibility-immiscibility phase diagram and the realization of quantum droplets and supersolid states with heteronuclear dipolar mixtures.

Figure 1

Trap geometry, ground-state column density, and dipole potential. (a) Illustration of the geometry of our ${}^{164}\mathrm{Dy}$ (red ellipse) and ${}^{166}\mathrm{Er}$ (blue ellipse) mixture. The orientation of the magnetic field is defined by the angles $\varphi$ and $\theta$. The imaging beam propagates in the horizontal plane, at an angle of ${45}^{\circ }$ with respect to the $y$ axis (not shown). (b) Ground-state column density for an imbalanced mixture with $N_{\text{Dy}}=1.3\times {10}^4$, $N_{\text{Er}}=4.9\times {10}^4$, $a_{12}=100a_0$. Dashed and solid lines show the isodensity contour levels for Dy and Er, respectively. For comparison, the in-trap displacement due to the gravitational sag for a noninteracting mixture is also shown (black dashed lines). We set $z=0$ at the center of the gravitational sag. (c) Heat map of the dipole potential produced by the Dy condensate (parameters below), ${\tilde{V}}_{Dy}\left(\mathit{r}\right)\equiv \int V_{\text{dd}}(\mathit{r}-{\mathit{r}}^{\prime })n_{Dy}\left({\mathit{r}}^{\prime }\right)d{\mathit{r}}^{\prime }$, in the $x=0$ plane. Here, the magnetic field points along the $z$ axis. The dotted-dashed lines represent the isodensity contour levels of the Er component, indicating that in this regime the interspecies dipolar interaction is predominantly attractive.

Figure 2

Experimental protocol. After preparing our Er-Dy mixture with $\mathit{B}\parallel \mathit{z}$, the magnetic field is rotated to an arbitrary direction, defined by $\theta$ and $\varphi$, in 120 ms. The atomic clouds are held in the trap for 50 ms to reach equilibrium, before either of the species is removed with resonant light. The remaining cloud is held for a variable hold time $t_h$. The cloud is then released from the trap and imaged with standard absorption imaging after a TOF expansion of $t_{\text{TOF}}=26$ ms. We prepare imbalanced mixtures with condensed atom numbers $N_{\text{C}}$ in the range $[1–3]\times {10}^4$ and $[4–6]\times {10}^4$ for Dy and Er, respectively (see Appendix pp1).

Figure 3

COM oscillations after removal of either one of the species. (a) Vertical COM position of Dy after removing Er and (b) vice versa. The vertical position $Z_i$ is recorded after a TOF expansion of 26 ms, as a function of the holding time. The offset $z_{\text{off}}$ has been subtracted to facilitate comparison. The measurements are repeated for two magnetic-field orientations: $\mathit{B}\parallel \mathit{z}$, $\theta =0^{\circ }$, $\varphi =0^{\circ }$ (circles) and $\mathit{B}\in \mathit{xy}$, $\theta ={90}^{\circ }$, $\varphi ={15}^{\circ }$ (diamonds). The atom numbers are $N_{\text{Dy}}=1.3\left(2\right)\times {10}^4$, $N_{\text{Er}}=4.9\left(7\right)\times {10}^4$ and $N_{\text{Dy}}=3.1\left(5\right)\times {10}^4$, $N_{\text{Er}}=4.7\left(5\right)\times {10}^4$ for $\mathit{B}\parallel \mathit{z}$ and $\mathit{B}\in \mathit{xy}$, respectively. The error bars reported represent the standard error on the mean over three experimental trials, and are mostly smaller than the markers. We fit Eq. (4) to the data for $\mathit{B}\parallel \mathit{z}$ (solid lines) and $\mathit{B}\in \mathit{xy}$ (dotted lines).

Figure 4

Mean-field displacement and theory prediction. Experimental estimation of the mean-field displacement $\mathrm{\Delta }{z}_{\text{MF},i}$ for Dy (red points) and Er (blue points), as a function of the magnetic-field orientation. (a) $\theta =[0^{\circ },{90}^{\circ }]$, $\varphi =0^{\circ }$. (b) $\varphi =[0^{\circ },{90}^{\circ }]$, $\theta ={90}^{\circ }$. (c) $\theta =[0^{\circ },{90}^{\circ }]$, $\varphi ={90}^{\circ }$. Theory prediction for an interspecies scattering length $a_{12}=100a_0$ (gray lines). The gray shaded area takes into account the experimental uncertainty on the estimation of the atom number. The error bars in $\mathrm{\Delta }{z}_{\text{MF},i}$ correspond to the statistical uncertainty from the fit. The mismatch between the data points at $\theta =0^{\circ }$ in (a) and (c) is due to different atom numbers (see Appendix pp1).

Figure 5

Calculated mean-field displacement as a function of $a_{12}$. Calculated mean-field displacement for (a) Dy and (b) Er as a function of the interspecies scattering length $a_{12}$. The red dashed line and the red shaded area in (a) represent the Dy experimental mean-field displacement and its error, respectively. The magnetic field is oriented along the $z$ axis. (c) In-trap density cut along $y=0$ for Dy (red) and Er (blue), for $a_{12}=30a_0$ (solid lines), $a_{12}=100a_0$ (dashed lines), and $a_{12}=200a_0$ (dotted lines). Here, $N_{\text{Dy}}=0.8\times {10}^4$ and $N_{\text{Er}}=5.9\times {10}^4$.

Figure 6

Calculation of an interaction-induced supersolidity. Ground-state configurations for an imbalanced dipolar mixture with the magnetic field pointing along the $x$ axis, for two different values of the interspecies scattering length: (a) $a_{12}=100a_0$, (b)$a_{12}=150a_0$. Dashed and solid lines show the isodensity contour levels for Dy and Er, respectively. Here, $N_{\text{Dy}}=1.2\times {10}^4$, $N_{\text{Er}}=6\times {10}^4$, $a_{11}=83a_0$, and $a_{22}=95a_0$.

Figure 7

Condensed atom numbers as a function of the magnetic-field orientation. Measured atom numbers in the BEC for Dy (red points) and Er (blue points) related to the measurement shown in Fig. 4 of the main text. The magnetic field is oriented (a) in the $XZ$ plane, (b) in the $XY$ plane, and (c) in the $YZ$ plane. The error bars reported represent the standard error on the mean over three experimental trials.

Figure 8

${\chi }^2$ distribution of the mean-field shift. ${\chi }^2$ distribution for the Dy mean-field shift (black points). We estimate the interspecies scattering length to be $a_{12}=105[-65,+162]a_0$, by locally doing a Gaussian fit around the minimum of the distribution (gray line) and by defining the lower and upper bounds as the values at which ${\chi }^2=1$ (black dashed lines).