  We present a comprehensive numerical simulation of an echo-type atom
interferometer. The simulation confirms a new theoretical description of this
interferometer that includes effects due to spontaneous emission and magnetic
sub-levels. Both the simulation and the theoretical model agree with the
results of experiments. These developments provide an improved understanding of
several observable effects. The evolution of state populations due to
stimulated emission and absorption during the standing wave interaction imparts
a time-dependent phase on each atomic momentum state. This manifests itself as
an asymmetry in the signal shape that depends on the strength of the
interaction as well as spontaneous emission due to a non-zero population in the
excited states. The degree of asymmetry is a measure of a non-zero relative
phase between interfering momentum states.
