Thermal behavior of Bose-Einstein condensates of polar molecules

TL;DR

The paper addresses finite-temperature effects in a dilute Bose-Einstein condensate of polar NaCs molecules and tests the temperature-dependent extended Gross-Pitaevskii equation (TeGPE) against the recent experimental results. It uses Bogoliubov-based thermal corrections within a local density approximation to model both the condensate and depleted components, and it simulates time-of-flight expansion with a time-dependent TeGPE. The authors report quantitative agreement with measured condensate fractions, post-expansion density profiles, and peak densities, and they analyze the role of the inter-molecular interaction under double microwave shielding. The work validates TeGPE as a reliable framework for dilute dipolar molecular condensates and points to future studies of stronger dipolar regimes, droplets, and supersolidity, with quantum Monte Carlo as a possible route to go beyond mean-field and capture superfluid properties.

Abstract

We use the finite-temperature extended Gross-Pitaevskii equation (TeGPE) to study a condensate of dipolar NaCs molecules under the conditions of the very recent, breakthrough experiment [Bigagli et.al., Nature 631, 289 (2024)]. We report the condensate fraction of the system, and its density profile after a time-of flight expansion for the coldest experimental case, finding excellent agreement with the experimental measurements. We also report the peak density of the ground state and establish a comparison with the experimental estimates. Our results, derived from the TeGPE formalism, successfully describe the Bose-Einstein condensation of polar molecules at finite temperature.

Figures (6)

• Figure 1: Emergence of a molecular Bose Einstein Condensate as the temperature is lowered. The panels depict the column density $\rho(x,y) = \int dz \left( \abs{\psi({\bf r})}^2 + \rho_d({\bf r}) \right)$, with $\psi({\bf r})$ the condensate wave function and $\rho_d({\bf r})$ the density of depleted molecules. The total number of molecules is $N=298$, while the temperatures are $T = 10.5, 9.5, 6$ and $0$ nK for panels $a)$ to $d)$, respectively. The condensate fractions for each case are $f_c =0.01, 0.12, 0.58$ and $0.96$, respectively. Panel $c)$ corresponds to the coldest point of the experiment of Ref. Bigagli2024.
• Figure 2: Condensate fraction of the NaCs molecular condensate under the experimental conditions of Ref. Bigagli2024. The horizontal axis corresponds to the total number of molecules while the colorbar indicates the temperature.
• Figure 3: Ground state column density $\rho(x)$, condensate density $\rho_{\rm c}(x)$, and depleted density $\rho_{\rm d}(x)$ of the molecular condensate at three different points of the experiment of Ref. Bigagli2024: ($N=289$,$T=6$ nK) (top), ($N=394$,$T=8.2$ nK) (middle), ($N=884$,$T=12.6$ nK) (bottom). We also include the gaussian and TF contributions of the bimodal fit of Eq. \ref{['eq_bimodal']}, which are obtained by fitting the total density.
• Figure 4: Ground state (dashed line) and time-evolved density profiles (solid line) obtained, respectively, from the solution of the TeGPE and the tTeGPE. The time evolution corresponds to a 17 ms TOF expansion. Dots indicate the experimental density distribution of Ref. Bigagli2024. The number of particles is $N=289$ while the temperature is $T=6$ nK.
• Figure 5: Peak density of the molecular condensate under the experimental conditions of Ref. Bigagli2024. The horizontal axis corresponds to the total number of particles while the colorbar indicates the temperature. The "TeGPE" results correspond to the peak density of the total cloud obtained from a TeGPE simulation, while the "TeGPE cond." results correspond to the peak density of the condensate molecules alone. Note that the experimental estimation does not account for the effect of the DDI nor the thermal cloud Bigagli2024dalfovo99.