Evaporative cooling at low trap depth

A quantitative, analytic model of evaporative cooling covering both the small- $(<4)$ and large- $(>4)$ $\eta$ regimes is presented. $\eta$ is the dimensionless parameter defined as the trap depth divided by the temperature of the trapped sample. Although some of the same general properties present at large $\eta$ are also present at small $\eta$, there are significant quantitative differences. These differences must be taken into account in order to accurately extract from the trapping data quantitative measurements of, for example, collisional atomic cross sections.

Figure 1

Effective trapping potentials for atoms with angular momentum taking on values of $\pounds =0,0.1,\dots ,1$. The $x$ axis of this plot is the radial coordinate of the trap in units of the maximum trap radius. The $y$ axis is the effective potential (including angular momentum barrier) in units of the potential energy at the trap wall.

Figure 2

(a) These five traces are the radial probability distributions for energies of $Q=0.3$, 0.6, 0.9, 1.2, 1.45. The distributions have been scaled to have a peak value of unity. (b) These four traces are the energy probability distributions for atoms found at radii $\rho =0$, 0.3, 0.6, 0.9. They have also been scaled to have peak values of unity.

Figure 3

The solid lines show the trapped density distributions obtained by numerically integrating Eq. (47) for $\eta =2$, 4, 6, 8 10. The large-$\eta$ approximation for the density, $n=n_0{e}^{-\eta \rho }$, is shown by the dotted traces for the same values of $\eta$.

Figure 4

The solid line is the result of numerically integrating Eq. (54) to obtain the fraction of elastic collisions that result in an evaporating atom. The dashed line is a plot of the large-$\eta$ approximation for the evaporation fraction given in Eq. (11).

Figure 5

The solid line shows the $\eta$ dependence of the average energy of a trapped atom, $\overline{E}∕kT$, as given by numerically integrating Eq. (55). The dashed line is a plot of the large-$\eta$ approximation as given by Eq. (22).

Figure 6

The solid curve shows the $\eta$ dependence of the average energy of atoms lost due to inelastic collisions, ${\overline{E}}_d∕kT$, as given by numerically integrating Eq. (56). The dashed curve is a plot of the large-$\eta$ approximation as given by Eq. (27).

Figure 7

The solid line shows the $\eta$ dependence of the average energy of an evaporated atom, ${\overline{E}}_v∕kT$, as given by numerically integrating Eq. (57). The dashed line is the large-$\eta$ approximation given by Eq. (29).

Figure 8

The ratio between the elastic and inelastic scattering rates causes the temperature of the trapped atoms to approach a fixed fraction of the trap depth. This figure shows the cross section ratio that must exist to produce a given “equilibrium $\eta$.” The solid curve is the result of the exact treatment. The dashed curve is the result of the large-$\eta$ approximation.