Depth, coherence time, and Q factor

where the trap minimum position is at , and the second equality is verified by substitution of (8.4) and (8.5), and defining and . We use this latter definition because we are interested in the coherence time of atoms spending time close to the trap minimum (which will certainly be true for the transverse ground state.) Using (8.6) to solve for and evaluate the `goodness factor', it turns out that the factor is independent of either laser power (

Combining (8.8) and (8.9) gives

fixing the product of achievable depth and coherence time as a constant multiple of the detuning. This is a remarkable result since it shows that increasing is really

where the value (taken from best-fit exponentials to the numerically-found squared electric fields for the guide dimensions of Section 8.2.2 with ) has been substituted to give the the final form. This design expression does not give the required to reach a desired balance between and , however, the total laser power is usually in the mW range, several orders of magnitude less than in most free-space trap designs. For instance, with mW, and nm we could generate a trap of 2mK depth with the relatively short coherence time of 0.9ms. The transverse oscillation frequencies in this trap would be kHz and kHz (the field shapes fix this ratio at about 1:4), giving an atomic mode spacing due to the x motion of 5.6K, roughly twice the cesium MOT temperature, and a characteristic ground-state size of 26nm by 12nm.

For coherent guiding, we can define a more physically meaningful
figure of merit,
, which tells us the typical number of
coherent transverse oscillations we can expect multiplied by
(*i.e.* it is the Q-factor of the transverse oscillations).
We should choose
since this is
the smaller of the transverse frequencies in our case.
For the transverse atomic modes will be well resolved, and our guide
can be a useful interferometric device.
Using (8.10), in conjunction with the
fact that when the trapping potential shape is fixed then
is proportional to the square root of the depth,
tells us that for a given trap and detuning,
.
For a higher we should choose smaller transverse oscillation frequencies,
that is, shallower traps.
For example, the 2mK trap discussed above has , but
if we reduce it to a 20K trap of the same (by changing or
the laser powers), the is 10 times larger.
Increasing would allow even higher to be realized.

The dependence on detuning in (8.11) is another way of expressing the advantages already known about using far off-resonant beams[2,85]. However, our single-resonance approximation will break down if the detuning is too large: we have (somewhat arbitrarily) chosen a detuning limit of 15nm, as compared to nm for cesium. At this limit, the additional dipole potential created due to the detunings from the D1 line is very significant. However, by removing the detuning symmetry (changing from +15nm to +12.07nm and from -15nm to -17.14nm), the desired D2 single-resonance approximation potential is recovered in the true physical situation of both resonances present. (These required shifts, which are of order , can easily be found using the expression for the sum of dipole potentials from the two lines.) An additional necessity for our limit is the fact that any larger detunings start to demand separate bound-mode calculations for the two colors, a treatment we reserve for the future. This detuning limit in turn limits the depths, coherence times and Q-factors we quote here, but we anticipate similar future EW atom waveguide designs which explore the region (or even ), and achieve much better coherence.