Lee-Huang-Yang effects in the ultracold mixture of ${}^{23}\mathrm{Na}$ and ${}^{87}\mathrm{Rb}$ with attractive interspecies interactions

The beyond-mean-field Lee-Huang-Yang (LHY) correction is ubiquitous in dilute ultracold quantum gases. However, its effects are often elusive due to the typically much larger influence of the mean-field (MF) energy. In this work, we study an ultracold mixture of ${}^{23}\mathrm{Na}$ and ${}^{87}\mathrm{Rb}$ with tunable attractive interspecies interactions. The LHY effects manifest in the formation of self-bound quantum liquid droplets and the expansion dynamics of the gas-phase sample. A liquid-to-gas-phase diagram is obtained by measuring the critical atom numbers below which the self-bound behavior disappears. In stark contrast to trapped gas-phase condensates, the gas-phase mixture formed following the liquid-to-gas-phase transition shows an anomalous expansion featuring a larger release energy for increasing MF attractions.

Figure 1

Creation and detection of Na-Rb mixtures for investigating LHY effects. (a) In the upper panel, the purple, yellow, and black solid lines are scattering lengths between $\mathrm{Na}\text{−}\mathrm{Na}\left(a_{11}\right)$, $\mathrm{Rb}\text{−}\mathrm{Rb}\left(a_{22}\right)$, and $\mathrm{Na}\text{−}\mathrm{Rb}\left(a_{12}\right)$, respectively. The dashed green curve is $\widetilde{\delta g}$. The dashed vertical line indicates the magnetic field for $\widetilde{\delta g}=0$. In the lower panel, the phase diagrams at different atomic numbers and magnetic fields are shown. Inset: Schematic detection method of the droplet in free space. (b) Images of the mixture during the TOF expansion in the gas phase at 350.451 G (left) and the droplet phase at 349.849 G (right). (c) Cloud sizes obtained from (b) reveal the very different behaviors of the two phases. Data points for ${}^{87}\mathrm{Rb}$ are shown in purple, while those for ${}^{23}\mathrm{Na}$ are in yellow. Inset: In the droplet phase, bimodal distributions are observed for the cloud with excess atoms. A double Gaussian fitting is used to extract the droplet size in this case.

Figure 2

Evolution of the Na-Rb droplet in free space. Panels (a), (b), and (c) show the size, atom numbers, and the ${}^{87}\mathrm{Rb}$ to ${}^{23}\mathrm{Na}$ number ratio at $\widetilde{\delta g}=-0.116$ (349.852 G) following the TOF. The droplet is created by directly releasing the sample from the trap. The vertical bar marks the approximate time of the liquid-to-gas-phase transition. Panels (d), (e), and (f) show the evolution for the droplet created with the additional magnetic-field quenching for $\widetilde{\delta g}=-0.089$ (349.880 G). The red dashed lines in (c) and (f) mark the theoretical number ratio $N_2/N_1=1.51$.

Figure 3

Critical atom numbers $N_c$ for the liquid-to-gas-phase transition as a function of $\widetilde{\delta g}$. The solid lines are calculated from coupled eGPEs. The color codes are purple for ${}^{87}\mathrm{Rb}$ and yellow for ${}^{23}\mathrm{Na}$.

Figure 4

The release energy as a function of $\widetilde{\delta g}$. The open squares represent $E_{\mathrm{rel}}$ for gas-phase mixtures released from the trap directly, while the solid circles are $E_{\mathrm{rel}}$ for the gas samples formed following the liquid-to-gas-phase transition. The red solid curve and the red dashed curve show $E_{\mathrm{rel}}$ for these two cases, calculated with eGPEs. The magenta short dashed curve is $E_{\mathrm{rel}}$ for the second case from variational theory. The blue solid curve shows the GPE calculation without the LHY term for comparison.

Figure 5

The intraspecies scattering length for ${}^{23}\mathrm{Na}$ atoms in the $|1,1\rangle$ state. The two resonances at high magnetic fields around 900 G [29] are far away from the current magnetic field of interest near 350 G. However, $a_{11}$ still varies smoothly from $54.5a_0$ at 0 G to $60.05a_0$ for 350 G.

Figure 6

Droplet formation by directly releasing the trapped sample. The blue dashed curves represent $E_{\mathrm{tot}}=E_{\mathrm{kin}}+E_{\mathrm{pot}}+E_{\mathrm{int}}$ of the trapped double BEC, and the red solid curves are $E_{\mathrm{tot}}$ in free space (i.e., $E_{\mathrm{kin}}+E_{\mathrm{int}}$). (a) For very small $|\widetilde{\delta g}|$, the free-space $E_{\mathrm{tot}}$ (the red open circle) is higher than the barrier height on the large size side (the red solid dot). In this case, the sample will keep expanding and never forms the droplet. (b) For $|\widetilde{\delta g}|$ large enough, the sample does not have enough energy and will be confined near the energy minimum to form the droplet.

Figure 7

Producing long-lived droplets by magnetic field quench. (a) For some weakly bound droplets, it is possible to find a $\widetilde{\delta g}$ value where the in-trap gas-phase sample size (blue dashed curve) is the same as the size of the free-space droplet (red solid curve). Quenching $\widetilde{\delta g}$ at the instant of releasing the sample, as depicted by the black dashed arrow, will lead to a near perfect mode matching. Panels (b) and (c) show the experimentally measured droplet signals for $\widetilde{\delta g}=-0.089$ (349.880 G) and $\widetilde{\delta g}=-0.061$ (349.910 G) created with this method. At these magnetic fields, no droplet can be observed without the quench as they fall in the situation as described in Fig. 6.

Figure 8

The release energies. (a) Origin of $E_{\mathrm{rel}}$ for the gas released from the trap. The blue dashed curve is the in-trap total energy $E_{\text{tot}}=E_{\mathrm{pot}}+E_{\text{kin}}+E_{\text{int}}$, and the red solid curve is $E_{\mathrm{tot}}$ without $E_{\mathrm{pot}}$ when the sample is suddenly released from the trap. (b) $E_{\mathrm{rel}}$ for the case following the liquid-to-gas-phase transition. The blue and orange dashed curves are $E_{\mathrm{tot}}$ of the stable and meta-stable droplets, respectively. The red solid curve is $E_{\mathrm{tot}}$ just on the phase-transition point, where the local minimum disappears. For both cases, following the size increase, the quantum pressure $E_{\mathrm{kin}}$ and the interaction energy $E_{\text{int}}$ are converted to the kinetic energy of the atoms.