  We present path integral ground state (PIGS) quantum Monte Carlo calculations
for the ground state ($T = 0$) properties of repulsively interacting bosons in
a three-dimensional external double well potential over a range of interaction
strengths and potential parameters. We focus our calculations on ground state
number statistics and the one-body density matrix in order to understand the
level of squeezing and fragmentation that the system exhibits as a function of
interaction strength. We compare our PIGS results to both a two-mode model and
a recently-proposed eight-mode model. For weak interactions, the various models
agree with the numerically exact PIGS simulations. However, the models fail to
correctly predict the amount of squeezing and fragmentation exhibited by the
PIGS simulations for strong interactions. One novel and somewhat surprising
result from our simulations involves the relationship between squeezing and
interaction strength: rather than a monotonic relationship between these
quantities, we find that for certain barrier heights the squeezing increases as
a function of interaction strength until it reaches a maximum, after which it
decreases again. We also see a similar relationship between fragmentation and
interaction strength. We discuss the physical mechanisms that account for this
behavior and the implications for the design of atom interferometers, which can
use squeezed states to reduce measurement uncertainty.
