  Decoherence describes the tendency of quantum sub-systems to dynamically lose
their quantum character. This happens when the quantum sub-system of interest
interacts and becomes entangled with an environment that is traced out. For
ordinary macroscopic systems, electromagnetic and other interactions cause
rapid decoherence. However, dark matter (DM) may have the unique possibility of
exhibiting naturally prolonged macroscopic quantum properties due to its weak
coupling to its environment, particularly if it only interacts gravitationally.
In this work, we compute the rate of decoherence for light DM in the galaxy,
where a local density has its mass, size, and location in a quantum
superposition. The decoherence is via the gravitational interaction of the DM
overdensity with its environment, provided by ordinary matter. We focus on
relatively robust configurations: DM perturbations that involve an overdensity
followed by an underdensity, with no monopole, such that it is only observable
at relatively close distances. We use non-relativistic scattering theory with a
Newtonian potential generated by the overdensity to determine how a probe
particle scatters off of it and thereby becomes entangled. As an application,
we consider light scalar DM, including axions. In the galactic halo, we use
diffuse hydrogen as the environment, while near the earth, we use air as the
environment. For an overdensity whose size is the typical DM de Broglie
wavelength, we find that the decoherence rate in the halo is higher than the
present Hubble rate for DM masses $m_a \lesssim 5 \times 10^{-7}$eV and in
earth based experiments it is higher than the classical field coherence rate
for $m_a \lesssim 10^{-6}$eV. When spreading of the states occurs, the rates
can become much faster, as we quantify. Also, we establish that DM BECs
decohere very rapidly and so are very well described by classical field theory.
