  Atom interferometry experiments are searching for evidence of chameleon
scalar fields with ever-increasing precision. As experiments become more
precise, so too must theoretical predictions. Previous work has made numerous
approximations to simplify the calculation, which in general requires solving a
3-dimensional nonlinear partial differential equation (PDE). In this paper, we
introduce a new technique for calculating the chameleonic force, using a
numerical relaxation scheme on a uniform grid. This technique is more general
than previous work, which assumed spherical symmetry to reduce the PDE to a
1-dimensional ordinary differential equation (ODE). We examine the effects of
approximations made in previous efforts on this subject, and calculate the
chameleonic force in a set-up that closely mimics the recent experiment of
Hamilton et al. Specifically, we simulate the vacuum chamber as a cylinder with
dimensions matching those of the experiment, taking into account the
backreaction of the source mass, its offset from the center, and the effects of
the chamber walls. Remarkably, the acceleration on a test atomic particle is
found to differ by only 20% from the approximate analytical treatment. These
results allow us to place rigorous constraints on the parameter space of
chameleon field theories, although ultimately the constraint we find is the
same as the one we reported in Hamilton et al. because we had slightly
underestimated the size of the vacuum chamber. This new computational technique
will continue to be useful as experiments become even more precise, and will
also be a valuable tool in optimizing future searches for chameleon fields and
related theories.
