  We compare and contrast the mean-field and many-body properties of a
Bose-Einstein condensate trapped in a double well potential with a single
impurity atom. The mean-field solutions display a rich structure of
bifurcations as parameters such as the boson-impurity interaction strength and
the tilt between the two wells are varied. In particular, we study a pitchfork
bifurcation in the lowest mean-field stationary solution which occurs when the
boson-impurity interaction exceeds a critical magnitude. This bifurcation,
which is present for both repulsive and attractive boson-impurity interactions,
corresponds to the spontaneous formation of an imbalance in the number of
particles between the two wells. If the boson-impurity interaction is large,
the bifurcation is associated with the onset of a Schroedinger cat state in the
many-body ground state. We calculate the coherence and number fluctuations
between the two wells, and also the entanglement entropy between the bosons and
the impurity. We find that the coherence can be greatly enhanced at the
bifurcation.
