  Many different formalisms exist for computing the phase of a matter-wave
interferometer. However, it can be challenging to develop physical intuition
about what a particular interferometer is actually measuring or about whether a
given classical measurement provides equivalent information. Here we
investigate the physical content of the interferometer phase through a series
of thought experiments. In low-order potentials, a matter-wave interferometer
with a single internal state provides the same information as a sum of position
measurements of a classical test object. In high-order potentials, the
interferometer phase becomes decoupled from the motion of the interferometer
arms, and the phase contains information that cannot be obtained by any set of
position measurements on the interferometer trajectory. This phase shift in a
high-order potential fundamentally distinguishes matter-wave interferometers
from classical measuring devices.
