Unveiling the BEC-droplet transition with Rayleigh superradiant scattering

TL;DR

The paper probes the BEC-to-macrodroplet transition in a dipolar $^{166}$Er gas by leveraging Rayleigh superradiant scattering as both a probe and a control knob. By tuning the dipolar-to-contact interaction ratio $\epsilon_{dd}=a_{dd}/a_s$ and varying dipole orientation via magnetic field tilt, the authors map how superradiance efficiency, depletion, and expansion dynamics signal the quantum state of the sample. The experimental results, supported by an extended Gross-Pitaevskii framework with a time-dependent Gaussian variational ansatz, reveal a non-monotonic SR response peaking near the transition and show that tilted dipoles shift the transition boundary in $a_s$ and $\epsilon_{dd}$, enabling a tunable phase diagram in $\epsilon_{dd}-N$ and $a_s-\theta$ spaces. These findings establish superradiant scattering as a powerful, fast tool to study coherence, self-bound droplets, and the interplay of contact and dipolar interactions in low-dimensional dipolar quantum gases.

Abstract

Light scattering plays an essential role in uncovering the properties of quantum states through light-matter interactions. Here, we explore the transition from Bose-Einstein condensate (BEC) to droplets in a dipolar $^{166}$Er gas by employing superradiant light scattering as both a probing and controlling tool. We observe that the efficiency of superradiant scattering exhibits a non-monotonic behavior akin to the rate of sample expansion during the transition, signaling its sensitivity to the initial quantum state, and in turn, revealing the BEC-droplet transition. Through controlled atom depletion via superradiance, we analyze the sample's expansion dynamics and aspect ratio to identify the BEC-droplet phases distinctly, supported by Gaussian variational ansatz calculations. Finally, using these two approaches, we track how the BEC-droplet transition points shift under varying magnetic field orientations. Our work opens new avenues for studying quantum states through superradiance, advancing our understanding of both the BEC-droplet crossover and its coherence properties.

Figures (4)

• Figure 1: Superradiance from a BEC to a macrodroplet state (a) An elongated condensate is exposed to the superradiance beam linearly polarized along the $z$-direction, propagating along the $y-$ axis. The magnitude ($B$) and direction ($\hat{r}(\theta)$) of the external magnetic field are independently controlled, leading to tunable $s$-wave scattering and dipolar interactions. (b) and (c) Schematic of mother cloud and recoiled atoms after 10 ms of free expansion. The scattering efficiency is enhanced near the transition point and suppressed in deep droplet and BEC regimes. Also, aspect ratio of the mother cloud shows distinct behavior in BEC and droplet phases while increasing the scattering rate, $\Gamma_R$. Qualitative change of superradiance efficiency across BEC-droplet transition (upper panel) and aspect ratio change of mother cloud due to loss of atoms (lower panel), following Rayleigh superradiant scattering.
• Figure 2: Superradiance threshold behavior from BEC to macrodroplet (a) Vertical absorption images of atomic momentum distribution after exposure to an off-resonant laser pulse of variable scattering rate. Images are taken after 10 ms time of flight. The scattering rates are 0.15, 0.35, and 0.55 ms$^{-1}$. Images on the left side represent the BEC regime ($\epsilon_{dd}=1.05$, prisms), whereas on the right side is the droplet regime ($\epsilon_{dd}=1.19$, stars). In both BEC and droplet phases, an increase in the scattering rate leads to a strong depletion of the mother cloud and thus larger daughter clouds. To clearly distinguish the aspect ratio change and increase of SR efficiency for larger $\Gamma_R$, different color scales are applied to mother and daughter clouds. (b) The portion of recoiled atoms versus the Rayleigh scattering rate, $\Gamma_R$, and dipolar strength, $\epsilon_{dd}$. The superradiance efficiency is maximum across the crossover and is progressively suppressed as the system moves away from the transition point. The absorption images corresponding to prisms and stars are depicted in (a). (c) An exponential increase in total number of recoil atoms is observed. The increase is proportional to the loss term, $L$, and is illustrated as a function of $\epsilon_{dd}$. The loss term reduces to a minimum value near the transition point and enhances again with a further increase of $\epsilon_{dd}$. Inset shows portion of recoiled atoms at $\epsilon_{dd} = 0.98$ fitted to the equation $N=A~\text{exp}\left({(\Gamma_R-L)t_{\text{pulse}}}\right)$ where $t_{\text{pulse}}$=100 $\mu$s and $A$ is arbitrary coefficient. In the fitting, $\Gamma_R$ is an independent variable (from 0.04 to 0.6 ms$^{-1}$). Lower panel: the portion of recoiled atoms for $\Gamma_R=0.55$ ms$^{-1}$ across the transition point, showing loss term and superradiant efficiency $\left(\frac{N_D}{N_D+N_M}\right)$ are inversely related.
• Figure 3: Identify BEC-droplet phase based on the aspect ratio of the depleted sample after superradiant scattering. (a) The scaled aspect ratio ($\gamma$) of the mother cloud is shown as a function of scattering rate, $\Gamma_R$ for different values of $\epsilon_{dd} = 0.98,~1.05,~1.09$ in BEC regime and $\epsilon_{dd} =~1.15,~1.19,~1.22$ in droplet regime (from top to bottom curves). The aspect ratio is rescaled by the value corresponding to the largest scattering rate for each $\epsilon_{dd}$. Ellipsoids on top and bottom depict a schematic representation of the shape of the mother cloud at different scattering rates, $\Gamma_R$ for the BEC (upper, $\epsilon_{dd}=1.05$) and droplet (lower, $\epsilon_{dd}=1.22$) phases, illustrating distinct behavior of $\gamma$ in both regimes. (b) Theoretical simulation of aspect ratio (normalized) after 10 ms of free expansion. Inset shows the loss of mother cloud atom number with an increase of scattering rate, $\Gamma_R$. (c) Experimental phase diagram for BEC-droplet crossover in $\epsilon_{dd}$-$N$ space, based on aspect ratio (normalized) change.
• Figure 4: Shift of phase transition point with different magnetic field orientation (a) The expansion velocity of the cloud is measured for different $\epsilon_{dd}$ values at magnetic field angles of 0°, 20°, and 30°. A clear shift in the point of lowest expansion rate is observed for all three angles (gray curves). The portion of recoiled atoms is measured for different $\epsilon_{dd}$ values from the BEC-droplet phase at magnetic field angles of 0°, 20°, and 30° (purple curves). (b) Experimental data for aspect ratio (normalized) of mother cloud in $\theta-$N space at $\epsilon_{dd}=1.22$. (c) Theoretical phase-diagram of BEC-droplet crossover in $a_s-\theta$ space for fixed atom number (left) and $a_s$-$N$ space for different angles (right).