Effects of surface interactions

The EW trap has the benefit of creating high field gradients near a surface, but along with this comes the disadvantage that interactions with that surface that can alter the trap potential and even cause heating and loss of trapped atoms.

An atom's change in potential near a surface is known as the van der Waals interaction ($l \ll \lambda$) or the Casimir interaction ($l \gg \lambda$), depending on the distance $l$ from the surface compared to $\lambda$, the dominant wavelength responsible for the polarizability of the atom (in our case of Cs this is the D line resonance, the same as our trapping resonance). There is a smooth cross-over from van der Waals ($U \sim l^{-3}$, which can be viewed as the atom's electrostatic interaction with the image of its own fluctuating dipole) to Casimir ($U \sim l^{-4}$, which can be viewed as a retarded van der Waals attraction or equally well as an atomic level shift due to a cavity QED effect) at $l \approx \lambda/10$[96]. In the case of a perfect mirror surface, the full form is known for any $l$, but for a dielectric surface, the expression becomes much more complicated to evaluate[184].

Since our trapping distances are larger than this cross-over point, we will use the Casimir form, which is correct for asymptotically large $l$, and is always an overestimate of the true potential[96]. The dependence of the coefficient with dielectric constant is complicated [68,207], but we will use the simpler approximate form given by Spruch and Tikochinsky[184], to give

$$U_{\rm Cas}(l) = -\frac{3}{8 \pi} \frac{\hbar c \,\alpha(0... ..., \frac{\epsilon - 1}{\epsilon+ (30/23)\epsilon^{1/2} + 7/23}$$ (8.14)

(in the MKSA system). This approximate form is known to be within 6% of the exact expression for any dielectric constant $\epsilon$ [207]. Substituting the recently calculated[60] static polarizability of cesium, $\alpha(0) = 399.9$ a.u., gives a Casimir interaction coefficient of 4.9nK$\mu$m$^4$ for $n_g = 1.56$.

Figure 8.1b shows the effect of this potential on a typical trap of depth 150$\mu$K and distance $y_0 = 270$nm. It is clear that the change is negligible further than 100nm from the surface, and a WKB tunneling calculation along this straight-line path (at $x=0$) shows that even if all atoms that reach the surface stick, the loss rate from the first few transverse modes is entirely negligible. However, care should be taken with the multi-mode regime, or in the case of high-$p$ traps, since the tunneling via the corners of the ``bean'' shape may dominate for $p>0.4$ (Figure 8.3).

The issue of energy transfer to trapped atoms due to a finite (and possibly room) temperature nearby surface is far less well understood, and may be a problem with many surface-based particle traps, as discussed by Henkel and Wilkens [92]. However, since we are trapping neutral particles and the conductivity of our surface is low, we expect a decoherence rate negligible compared to that already present from spontaneous absorption and emission cycles.