Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid

In a joint experimental and theoretical effort, we report on the formation of a macrodroplet state in an ultracold bosonic gas of erbium atoms with strong dipolar interactions. By precise tuning of the $s$-wave scattering length below the so-called dipolar length, we observe a smooth crossover of the ground state from a dilute Bose-Einstein condensate to a dense macrodroplet state of more than $2\times {10}^4\mathrm{atoms}$. Based on the study of collective excitations and loss features, we prove that quantum fluctuations stabilize the ultracold gas far beyond the instability threshold imposed by mean-field interactions. Finally, we perform expansion measurements, showing that although self-bound solutions are prevented by losses, the interplay between quantum stabilization and losses results in a minimal time-of-flight expansion velocity at a finite scattering length.

Popular Summary

Recent experiments of magnetic atoms at extremely low temperatures have revealed a surprising phase of matter: The atoms, behaving like tiny magnets, form a qualitatively new type of quantum liquid that is much more dilute than standard liquids. These so-called “quantum droplets” can preserve their form in the absence of external confinement, because of quantum effects, paving the way for completely novel research in the active field of ultracold gases. Here, we have, for the first time, realized a controlled crossover from an ordinary Bose-Einstein condensate, which behaves as a superfluid gas, into a single giant quantum droplet. Our joint experimental and theoretical results conclusively demonstrate the key role that quantum many-particle physics plays in the droplet regime.

Typical experiments relating to Bose-Einstein condensates can be well described within the so-called “mean-field approximation.” Our combined analysis, performed with a condensate of roughly 100,000 erbium atoms, focuses on the density distribution, elementary excitations, atom losses, and expansion dynamics after a precise tuning of the scattering length of the gas. Our results prove precisely and quantitatively that the mere existence of the droplet regime and its properties are crucially determined by quantum fluctuations beyond the mean-field scenario. These fluctuations result in an effective repulsion that provides the necessary surface tension to stabilize the quantum droplet against collapse. Our work shows convincingly that quantum droplets, far from being specific to any given species, constitute instead a general feature of strongly dipolar gases.

Our work, which reveals an effect common to dipolar gases, paves the way for complementary follow-up studies characterizing the new superfluid droplet state.

Figure 1

Scattering length in ${}^{166}\mathrm{Er}$. $a_s$ as a function of $B$. The data points (circles) are extracted from spectroscopic measurements in a lattice-confined gas and the solid line is a fit to the data with its statistical uncertainty (gray shaded region [30]). Upper inset: Zoom-in of ${ϵ}_{\mathrm{dd}}$ as a function of $B$. The gray dashed line marks ${ϵ}_{\mathrm{dd}}=1$; see also the other figures. The lower inset illustrates the geometry of our experimental setup, the relevant axes ($x$, $y$, $z$), the optical-dipole-trap beam (shaded region), the magnetic field orientation (green arrow) along which the dipoles are aligned, and the $\parallel$- and $\perp$-imaging view axes (blue arrows). The dashed lines picture the small angles of these axes to $y$ and $z$ [30].

Figure 2

Density profiles in the BEC-to-droplet crossover. (a)–(c) 2D column density distributions probed with $\parallel$ imaging and (d) corresponding central cuts along the $x=0$ line (dots) for $t_r=10\mathrm{ms}$, $t_h=6\mathrm{ms}$ ($>1/{\nu }_{\perp }$), and different $a_s$ (see legend). Each distribution is obtained by averaging four absorption images taken after $t_{\mathrm{TOF}}=27\mathrm{ms}$. In (d), the lines show the central cuts of the 2D bimodal fit results, the solid (dashed) lines showing the two-Gaussian (MF-TF plus Gaussian) distributions and the dotted lines the corresponding broad thermal Gaussian part.

Figure 3

Axial mode. (a) Illustration of the axial mode in our experimental setup. The black arrows sketch the oscillations of the widths of the coherent gas along the characteristic axes of the trap, with weights indicating their relative amplitudes. (b),(c) Measured ${\nu }_{\text{axial}}/{\nu }_{\parallel }$ (squares) as a function of $a_s$ together with the theoretical predictions, including (solid line) or not (dashed lines) the LHY term for $t_r=100\mathrm{ms}$ (b) and $t_r=10\mathrm{ms}$ (c). Theoretical predictions are obtained from RTE (see text) for $a_s$ varied from $50a_0$ to $95a_0$. In the MF case, predictions fail for $a_s\le a_c$ (orange area) due to the occurrence of the collapsing dynamics which rules out the collective excitation picture. $a_c=57a_0$ [$a_c=64a_0$] in (b) [(c)]. In (c), ${\nu }_{\text{axial}}$ cannot be reliably extracted for quenches to $a_s\le 56a_0$, nor from the experiment (open squares) or from the LHY theory (open circles, thin line). The inset in (b) exemplifies a measurement of $R_{\parallel }$ (triangles) and its fit to a damped sine (solid line) for $a_s=80a_0$. We typically fit 4–5 oscillations for all our $a_s$.

Figure 4

Lifetime and in situ density of the high-density core. (a),(b) Measured $N_{\mathrm{core}}$ (squares) and $N_{\mathrm{th}}$ (circles) versus $a_s$ after (a) a nonadiabatic ($t_r=10\mathrm{ms}$, $t_h=8\mathrm{ms}$) and (b) an adiabatic ($t_r=45\mathrm{ms}$, $t_h=0\mathrm{ms}$) ramp. The data show a better agreement with the theory with the LHY term (solid line) as compared to the MF theory (dashed line). (c) Time decay of $N_{\mathrm{core}}$ for $a_s=65a_0$ (triangles), $57a_0$ (circles), and $50a_0$ (squares) after quenching $a_s$ ($t_r=10\mathrm{ms}$). We fit a double exponential function to the data (solid lines) [30]. (d) From the fit, we deduct the mean in situ density of the core $\overline{n}$ (see text) for $t_h=4\mathrm{ms}$ (triangles) and 16 ms (squares) and as a function of $a_s$. The error bars include the statistical errors on the fit and on $L_3$. The solid lines show results of the RTE including the LHY correction for $t_h=0\mathrm{ms}$ (red), $t_h=25\mathrm{ms}$ (blue).

Figure 5

Expansion dynamics across the BEC-to-droplet crossover. (a) TOF evolution of the width ${\sigma }_x$ of the high-density component for $a_s=93a_0$ (squares), $64a_0$ (circles), $55.5a_0$ (triangles). The lines are fit to the data using ${\sigma }_x(t_{\mathrm{TOF}})=\sqrt{{\sigma }_{x,0}^2+v_x^2{t}_{\mathrm{TOF}}^2}$, from which we extract $v_x$. (b) $v_x$ as a function of $a_s$ (squares). For comparison, the $a_s$-independent expansion velocities of the thermal component are also shown (circles). The experimental data are in very good agreement with our parameter-free theory from RTE simulations including the LHY term (solid line) and rule out the MF scenario (dotted line). For clarity, we show only $v_x$; similar results are found for $v_z$.