Kinetics of the evaporative cooling of an atomic beam

We compare two distinct models of evaporative cooling of a magnetically guided atomic beam: a continuous one, consisting in approximating the atomic distribution function by a truncated equilibrium distribution, and a discrete-step one, in which the evaporation process is described in terms of successive steps consisting in a truncation of the distribution followed by rethermalization. Calculations are performed for the semilinear potential relevant for experiments. We show that it is possible to map one model onto the other, allowing us to infer, for the discrete-step model, the rethermalization kinetics, which turns out to be strongly dependent upon the shape of the confining potential.

Figure 1

Evaporation “trajectories” in the temperature-flux plane, for evaporation with a constant $\eta$ for the continuous model (solid lines) and for the discrete one (dots, each symbol representing the effect of one evaporation step). The background gray scale (with the white labels) shows the on-axis phase-space density. The qualitative shapes of those trajectories are similar for both models. For $\eta$ high enough (e.g., $\eta =8$), the trajectories are very close in both models. The number of evaporation zones required for reaching degeneracy in the discrete-step model, for $\eta =5$ (6, 8), is 88 (152, 526).

Figure 2

Evaporation efficiency $\xi$ as a function of the evaporation parameter $\eta$, for a harmonic transverse confinement. The solid (dashed) line corresponds to the continuous (discrete-step) evaporation model.

Figure 3

Number $N_c$ of collisions necessary for rethermalization between two evaporation zones (inferred by mapping the continuous evaporation model onto the discrete one; see text) as a function of the scaled evaporation parameter ${\tilde{\eta }}_D$. The solid (open) circles correspond to a linear (harmonic) confinement.

Figure 4

Minimal evaporation length $L_{\mathrm{ev}}$ needed to reach a phase-space density $\rho =1$ with evaporation at constant $\eta$ for an atomic beam of initial flux ${\Phi }_{\mathrm{i}}$ and temperature $T_{\mathrm{i}}$, propagating at $60\mathrm{cm}∕\mathrm{s}$ in a guide with a gradient $b=800\mathrm{G}∕\mathrm{cm}$ and a longitudinal bias field $B_0=1\mathrm{G}$. The isocontours of $L_{\mathrm{ev}}$ show the large advantage of starting with a temperature on the order of $100\mu \mathrm{K}$ for reaching degeneracy in a reasonable length. The optimal evaporation parameter $\eta$ is almost constant $(\simeq 6)$ over the whole parameter space explored here.