  We use previously obtained experimental results by neutron interferometry to
effectively constrain the parameter space of several prominent dark energy
models. This investigation encompasses the environment-dependent dilaton field,
a compelling contender for dark energy that emerges naturally within the strong
coupling limit of string theory, alongside symmetron and chameleon fields. Our
study presents substantial improvements over previous constraints of the
dilaton and symmetron fields, improving parameter constraints by several orders
of magnitude. However, the analysis does not yield any new constraints on the
chameleon field. Furthermore, we establish constraints for the projected
neutron split interferometer, which has recently concluded a decisive
proof-of-principle demonstration. Our symmetron simulations reveal that
depending on the parameter values there are multiple static solutions with
increasing number of nodes and increasing energy inside a cylindrical vacuum
chamber. This agrees with results obtained earlier in the literature for
infinitely parallel plates. Interestingly, while these multiple solutions can
correspond to domain walls forming inside the vacuum chamber, we also find
solutions that do not reach their vacuum expectation value inside the vacuum
chamber, but display multiple nodes nonetheless.
