  We develop a dissipative quantum state preparation scheme for the creation of
phase- and number-squeezed states. It utilizes ultracold atoms in a double-well
configuration immersed in a background Bose-Einstein condensate, with the
latter consisting of an atom species different from the atoms in the double
well and acting as a dissipative quantum reservoir. We derive a master equation
for this system starting from microscopic physics, and show that squeezing
develops on a time scale proportional to $1/N$, where $N$ is the number of
particles in the double well. This scaling, caused by bosonic enhancement,
allows us to make the time scale for the creation of squeezed states very
short. The lifetime of squeezed states is limited by dephasing arising from the
intrinsic structure of the setup. However, the dephasing can be avoided by
stroboscopically switching the driving off and on. We show that this approach
leads to robust stationary squeezed states. Finally, we provide the necessary
ingredients for a potential experimental implementation by specifying a
parameter regime for rubidium atoms that leads to squeezed states.
