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 defined, 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, analogous to the pointer
of a measurement apparatus, which illustrates the non-locality of standard
quantum mechanics with particular clarity. The effect can be described as the
teleportation at arbitrary distances of the continuous classical result of a
local experiment. The EPR argument, transposed to this case, takes a
particularly convincing form since it does not involve incompatible
measurements and deals only with macroscopic variables.
