  We revisit the question of how a definite phase between Bose-Einstein
condensates can spontaneously appear under the effect of measurements. We first
consider a system that is the juxtaposition of two subsystems in Fock states
with high populations, and assume that successive individual position
measurements are performed. Initially, the relative phase is totally undefined,
and no interference effect takes place in the first position measurement. But,
while successive measurements are accumulated, the relative phase becomes
better and better known, and a clear interference pattern emerges. It turns out
that all observed results can be interpreted in terms of a pre-existing, but
totally unknown, relative phase, which remains exactly constant during the
experiment. We then generalize the results to more condensates. We also
consider other initial quantum states than pure Fock states, and distinguish
between intrinsic phase of a quantum state and phase induced by measurements.
Finally, we examine the case of multiple condensates of spin states. We discuss
a curious quantum effect, where the measurement of the spin angular momentum of
a small number of particles can induce a big angular momentum in a much larger
assembly of particles, even at an arbitrary distance. This spin observable can
be macroscopic, assimilable to the pointer of a measurement apparatus, which
illustrates the non-locality of quantum mechanics with particular clarity.
