Other properties

We estimate the collection area of the trap as the cross-sectional region within which the potential is deeper than the typical cesium MOT energy $k_B T_{\rm MOT}$. For our example 100$\mu$K traps of Figure 8.3 this area is about 1$\mu$m$^2$. However, it is not possible to do much better than this with our design: if one tries to increase the area by increasing the overall trap depth, the $k_B T_{\rm MOT}$ contour touches the substrate, indicating that atoms at this energy can reach the substrate surface, where they will stick, limiting the effective collection area.

To investigate the lifetime of atoms transported incoherently (the multi-mode regime), we can calculate the heating rate along similar lines as Grimm and Weidemüller[85]. We start with their equation (23) which gives the rate of change of the average of the total energy of atomic motion $E = E_{\rm kin} + E_{\rm pot}$ as

$$\dot{\overline{E}} = k_B T_R \, \overline{\Gamma_{\rm scatt}} \, ,$$ (8.12)

$T_R$ being the recoil temperature, and use the assumption that in an equilibrated 3D trap $\overline{E_{\rm kin}} = {\textstyle\frac{3}{2}} k_B T$. Since there is harmonic motion in two directions but free motion in the third, the virial theorem gives us $\overline{E_{\rm pot}} = {\textstyle\frac{2}{3}} \overline{E_{\rm kin}}$. Combining this with $\overline{\Gamma_{\rm scatt}} = U_{\rm max} \Gamma / G \hbar \Delta$ from (8.8) gives the heating rate

$$\dot{T} = \frac{2}{5 G} \frac{\Gamma}{\Delta} T_R \, \frac{U_{\rm max}} {\hbar} \, ,$$ (8.13)

which is of the order of one recoil temperature per coherence time. For our 100$\mu$K depth trap at $\Delta = 15\,$nm the rate is 4.4$\mu$Ks$^{-1}$, implying that storage and transport for many seconds is possible. For simplicity, we have ignored the fact that there may be distinct longitudinal and transverse temperatures which do not equilibrate over the trapping timescales.