  We ask and answer a basic question about the length scales involved in
quantum decoherence: how far apart in space do two parts of a quantum system
have to be, before a common quantum environment decoheres them as if they were
entirely separate? We frame this question specifically in a cold atom context.
How far apart do two populations of bosons have to be, before an environment of
thermal atoms of a different species (`buffer gas') responds to their two
particle numbers separately? An initial guess for this length scale is the
thermal coherence length of the buffer gas; we show that a standard Born-Markov
treatment partially supports this guess, but predicts only inverse-square
saturation of decoherence rates with distance, and not the much more abrupt
Gaussian behavior of the buffer gas's first-order coherence. We confirm this
Born-Markov result with a more rigorous theory, based on an exact solution of a
two-scatterer scattering problem, which also extends the result beyond weak
scattering. Finally, however, we show that when interactions within the buffer
gas reservoir are taken into account, an abrupt saturation of the decoherence
rate does occur, exponentially on the length scale of the buffer gas's mean
free path.
