  We consider two dilute gas Bose-Einstein condensates with opposite velocities
from which a monochromatic light field detuned far from the resonance of the
optical transition is coherently scattered. In the thermodynamic limit, when
the relative fluctuations of the atom number difference between the two
condensates vanish, the relative phase between the Bose-Einstein condensates
may be established in a superposition state by detections of spontaneously
scattered photons, even though the condensates have initially well-defined atom
numbers. For a finite system, stochastic simulations show that the measurements
of the scattered photons lead to a randomly drifting relative phase and drive
the condensates into entangled superpositions of number states. This is because
according to Bose-Einstein statistics the scattering to an already occupied
state is enhanced.
