Measurement of the order parameter and its spatial fluctuations across Bose-Einstein condensation

We investigate the strong out-of-equilibrium dynamics occurring when a harmonically trapped ultracold bosonic gas is evaporatively cooled across the Bose-Einstein condensation transition. By imaging the cloud after free expansion, we study how the cooling rate affects the timescales for the growth of the condensate order parameter and the relaxation dynamics of its spatial fluctuations. We find evidence of a delay on the condensate formation related to the collisional properties and a universal condensate growth following the cooling rate. Finally, we measure an exponential relaxation of the spatial fluctuations of the order parameter that also shows a universal scaling. Notably, the scaling for the condensate growth and for the relaxation of its fluctuations follow different power laws.

Figure 1

Exploring the gas parameters during the evaporation ramp for a low (left column) or a high (right) cooling rate. (a) Variation of the evaporation frequency $\nu$ in time; the evaporation rate is $q=d\nu /dt$. Temperature (b) and condensed fraction (c) measured by interrupting the evaporation ramp at different times $t$ and imaging the cloud either immediately (light and dark blue circles) or keeping the gas in the trap for equilibration (orange diamonds). Temperature is extracted from a Bose distribution fit to the tails of the atomic distribution measured after time of flight and is normalized to the transition temperature $T_c$. The vertical red line represents the time $t_c$ at which $T_c$ is experimentally crossed; the horizontal blue dashed line is the final temperature after equilibration. The horizontal dashed line in (c) marks the final condensate fraction. Insets show the absorption images taken in points B, C, B', and C'. The shaded area marks the duration of the ramp. (d) Evolution of $N_0/N$ vs $T/T_c$, showing that higher evaporation rates induce strong out-of-equilibrium dynamics, as can be seen comparing the instantaneous measurements (light and dark blue circles) to the corresponding equilibrium points (orange diamonds). The arrows indicate the direction of the time evolution. The dashed and the solid lines represent the theoretical predictions for the condensate fraction for an ideal and for an interacting Bose gas, respectively.

Figure 2

Post-quench growth dynamics of the BEC order parameter for high evaporation rates. (a) Condensate fraction as a function of time for atom number $N_{\text{high}}$, for different values of $q$. The time axes are shifted according to $t_c$, see text. The dashed lines mark the equilibrium condensed fraction for each evaporation rate. (b) Same as (a), but with the vertical axis normalized to the final condensed fraction for each evaporation rate. The gray area highlights the latency time $\mathrm{\Delta }t$. (c) Final relative temperature for the various rates $q$. [(d)–(f)] Equivalent results for a smaller atom number, $N_{\text{low}}$.

Figure 3

Growth of the BEC order parameter during cooling ramps with low evaporation rate. The condensate fraction is shown for different evaporation rates on gases with $N_{\text{high}}$ atoms, as a function of $(t-t_c)$ (a) or of $(t-t_c)/{\tau }_Q$ (b). (c) shows how ${\tau }_1$ (triangles) and $t_{50\%}$ (stars) depend on ${\tau }_Q$. The lines are linear fits of ${\tau }_1\left({\tau }_Q\right)$ (solid) and $t_{50\%}\left({\tau }_Q\right)$ (dot-dashed). The inset shows a zoom of the region close to the origin, highlighting the presence of a finite offset for $t_{50\%}$, while ${\tau }_1$ tends to zero when ${\tau }_Q\to 0$. In (d), all growth curves overlap onto a universal growth curve (in black) when shifted by the $t_{50\%}$ parameter obtained from the fit and rescaled by the time ${\tau }_Q$ of each ramp. [(e)–(h)] Same as (a)–(d), but for a smaller initial atom number, $N_{\text{low}}.$ The black curve in (h) is the same as in (d), showing that the universality holds also for different atom numbers.

Figure 4

Relaxation dynamics of the order parameter fluctuations. (a) Pictures of the BEC component as a function of the universal timescale of the condensate growth. (b) The residuals $h(x,z)$ between the OD shown in (a) and the average $\overline{OD}$ at the same universal time, normalized by the peak $\overline{OD}$ at each time, are larger shortly after the BEC formation and then decay when the turbulence relaxes to the smooth equilibrium state. (a), (b) and (c), (d) correspond to evaporation rates of $q=50$ and $300\mathrm{k}\mathrm{Hz}{\mathrm{s}}^{-1}$, respectively. All images share the same spatial scale and coordinate axes indicated in (a) and (b).

Figure 5

(a) Decay of the order parameter fluctuations $H(t-t_c)$ in the case of low evaporation rates, normalized by the initial value $H\left(0\right)$. (b) Time constant ${\tau }_2$ of the exponential fit, compared with that of the condensate growth rate ${\tau }_1$. Linear fits in the log-log scale highlight the power-law scaling behavior of the two quantities with an exponent 1 (linear scaling) and 0.6, respectively for ${\tau }_1$ and ${\tau }_2$.

Figure 6

(a) Decay of the order parameter fluctuations $H(t-t_c)$ in the case of high evaporation rates, normalized by the initial value $H\left(0\right)$. (b) Time constant ${\tau }_2$ extracted from the exponential fit, plotted as a function of the quench time ${\tau }_Q$. In contrast to the low evaporation rates, there is no clear dependence on the quench time, with ${\tau }_2$ resulting fairly constant at a value of $98\pm 25\mathrm{ms}$ (gray area).