  We investigate how ultracold atoms in double well potentials can be used to
study and put bounds on models describing wave function collapse. We refer in
particular to the continuous spontaneous localization (CSL) model, which is the
most well studied among dynamical reduction models. It modifies the Schrodinger
equation in order to include the collapse of the wave function in its dynamics.
We consider Bose Josephson junctions, where ultracold bosons are trapped in a
double well potential,since they can be experimentally controlled with high
accuracy and are suited and used to study macroscopic quantum phenomena on
scale of microns with a number of particles typically ranging from $\sim
10^2-10^3$ to $\sim 10^5-10^6$. We study the CSL dynamics of three atomic
states showing macroscopic quantum coherence: the atomic coherent state, the
superposition of two atomic coherent states, and the NOON state. We show that
for the last two states the suppression of quantum coherence induced by CSL
model increases exponentially with the number of atoms. We observe that, in the
case of optically trapped atoms, the spontaneous photon emission of the atoms
induce a dynamics similar to the CSL one and we conclude that magnetically
trapped atoms may be more convenient to experimentally test the CSL model. We
finally discuss decoherence effects in order to provide reasonable estimates on
the bounds that it is (or it will) possible to obtain for the parameters of the
CSL model in such class of experiments: as an example, we show that a NOON
state with $N \sim 10^3$ with a coherence time of $\sim 1$ s can constrain the
CSL parameters in a region where the other systems presently cannot.
