  Despite the fact that atom interferometry has been a successful application
of quantum sensing, a major topic of interest is the further improvement of the
sensitivity of these devices. In particular, the area enclosed by the
interferometer (which controls the sensitivity) can be increased by providing a
larger momentum kick to the atom cloud, increasing the extent of the momentum
axis. One such atom optics technique involves increasing the number of central
$\pi-$Raman pulses. This technique, while providing the prerequisite additional
momentum boost, also causes the atom to remain in the intermediate high energy
state for longer periods of time. This additional length of time is often
neglected in many treatments due to the adiabatic elimination of the higher
energy state enabled by the large optical detuning. The increased time in the
intermediate high energy state results in a higher probability of undesired
spontaneous decay and a loss of quantum information, thereby adding error to
the atom interferometer. In this work, we consider an open quantum system using
the Lindblad master equation to devise a model for the atomic state dynamics
that includes the undesired spontaneous decay from the intermediate high energy
state. We formulate an error figure of merit to analyze limitations of an atom
interferometer configured for acceleration measurements. Our theoretical
results show the error figure of merit will be dominated by a $N_{R}^{-2}$
scaling factor for low numbers of $\pi-$Raman pulses, but will be dominated by
a monotonic increase in error for high number of $\pi-$Raman pulses. We
determined the number of $\pi$-Raman pulses that accomplishes maximal momentum
transfer with a the minimal error, depending on major system parameters.
