  A single-particle Green's function (propagator) is introduced to study the
detection of laser-cooled Cesium atoms in an atomic fountain due to RF field
gradients in the Ramsey TE011 cavity. The detection results in a
state-dependent loss of atoms at apertures in the physics package, resulting in
a frequency bias. A model accounting only for motion in one dimension
transverse to the symmetry axis of the fountain is discussed in detail and then
generalized to two transverse dimensions. Results for fractional frequency
shifts due to transverse field gradients are computed for NIST F-1 and F-2
Cesium fountains. The shifts are found to be negligible except in cases of
higher RF power applied to the cavities.
