Time-dependent condensation of bosonic dysprosium

TL;DR

The paper addresses how time-dependent Bose-Einstein condensation proceeds in strongly dipolar bosons, specifically Dy-164. It advances a nonlinear boson diffusion equation (NBDE) framework where transport coefficients reflect interaction strengths, including dipolar contributions, and derives analytic NBDE solutions for constant coefficients to predict the time evolution of the condensate fraction after a quench. A key contribution is the explicit inclusion of dipolar effects via the total elastic cross section $\sigma_{el}$ and the associated scale $a_{tot} = \sqrt{\sigma_{el}}$, enabling predictions for initiation and equilibration times and offering a testable distinction between coherence driven by contact interactions alone versus including dipolar interactions. The results guide experiments on dipolar gases and propose 2D extensions, providing a tractable approach to probing coherence buildup in time-dependent dipolar Bose gases.

Abstract

We investigate thermalization and time-dependent Bose-Einstein condensate formation in ultracold Dy-164 using a nonlinear boson diffusion equation. As compared to alkali atoms such as K-39 or Rb-87, the strong magnetic dipole interaction modifies the scattering-length dependence of the transport coefficients that govern thermalization and condensate formation. A prediction for the time-dependent condensate fraction in Dy-164 is made.

Figures (4)

• Figure 1: Thermalization of $^{164}$Dy. Nonequilibrium evolution of a quenched $^{164}$Dy gas in double-log scale for eight timesteps (increasing dash lengths) as calculated from analytical NBDE solutions. The system is taken to evolve from an initial truncated Bose--Einstein distribution with $T_\mathrm{i}=240$ nK, $\epsilon_\mathrm{i}=234$ nK (here for $\mu_\mathrm{i}=0$) to the equilibrium distribution with $T_\mathrm{i}=30$ nK and $\mu=0$. The transport coefficients are $D=45$ nK$^2$/ms, $v=-1.5$ nK/ms, the equilibration time at the cut is $\tau_\mathrm{eq}\simeq600$ ms for a scattering parameter $\sqrt{\sigma_\mathrm{el}}/a_0=637.90$.
• Figure 2: Analytical vs. numerical NBDE solution for $^{164}$Dy. The absolute errors of the numerical NBDE solutions relative to the analytical solutions shown in Fig. \ref{['fig1']} are displayed for different times. They are defined as the absolute difference between the analytical and the numerical solution $|n_\mathrm{ana}(\epsilon,t)-n_\mathrm{num}(\epsilon,t)|.$
• Figure 3: Time-dependent condensate fraction for an equilibrating 3D $^{39}$K Bose gas. The calculated condensate fraction based on the analytical solutions of the NBDE (see text) is displayed as function of time (solid curve) and compared to Cambridge data gli21 for a scattering length $a/a_0=140$ (symbols, with error bars reflecting experimental fitting errors.). In the experiment, ultracold $^{39}$K atoms with temperature $T_\mathrm{i} = 130$ nK in a box trap are subjected to a rapid quench, subsequently thermalize and -- starting at $\tau_\mathrm{ini}$=130 ms -- form a BEC with $\simeq 40\%$ particle content in the statistical equilibrium limit ($T_\mathrm{f}=32.5$ nK).
• Figure 4: Time-dependent condensate fraction for an equilibrating 3D $^{164}$Dy Bose gas. The calculations are based on the analytical solutions of the NBDE (see text) and displayed as functions of time for $a_\mathrm{tot}/a_0\equiv\sqrt{\sigma_\mathrm{el}}/a_0\simeq 637.90$, (solid curve, with $a_s/a_0=113$). Transport coefficients have then been recalculated for a reduced contact-interaction scattering length $a_\mathrm{s}/a_0=40$ (dot-dashed curve), and the correspondingly reduced total scattering length $a_\mathrm{tot}/a_0=355.30$ (dashed curve). The significant difference in scaling of the time-dependent condensate fraction with $a_s$vs.$a_\mathrm{tot}$ is expected to be measurable. It indicates the coherence mechanism (see text).