We present a simple derivation of the interference pattern in matter-wave
interferometry as predicted by a class of master equations, by using the
density matrix formalism. We apply the obtained formulae to the most relevant
collapse models, namely the Ghirardi-Rimini-Weber (GRW) model, the continuous
spontaneous localization (CSL) model together with its dissipative (dCSL) and
non-markovian generalizations (cCSL), the quantum mechanics with universal
position localization (QMUPL) and the Di\'{o}si-Penrose (DP) model. We discuss
the separability of the collapse models dynamics along the 3 spatial
directions, the validity of the paraxial approximation and the amplification
mechanism. We obtain analytical expressions both in the far field and near
field limits. These results agree with those already derived in the Wigner
function formalism.
We compare the theoretical predictions with the experimental data from two
relevant matter-wave experiments: the 2012 far-field experiment and the 2013
Kapitza Dirac Talbot Lau (KDTL) near-field experiment of Arndt's group. We show
the region of the parameter space for each collapse model, which is excluded by
these experiments. We show that matter-wave experiments provide model
insensitive bounds, valid for a wide family of dissipative and non-markovian
generalizations.Comment: 49 pages,16 figure
