  The interference of two Bose-Einstein condensates, initially in Fock states,
can be described in terms of their relative phase, treated as a random unknown
variable. This phase can be understood, either as emerging from the
measurements, or preexisting to them; in the latter case, the originating
states could be phase states with unknown phases, so that an average over all
their possible values is taken. Both points of view lead to a description of
probabilities of results of experiments in terms of a phase angle, which plays
the role of a classical variable. Nevertheless, in some situations, this
description is not sufficient: another variable, which we call the "quantum
angle", emerges from the theory. This article studies various manifestations of
the quantum angle. We first introduce the quantum angle by expressing two Fock
states crossing a beam splitter in terms of phase states, and relate the
quantum angle to off-diagonal matrix elements in the phase representation. Then
we consider an experiment with two beam splitters, where two experimenters make
dichotomic measurements with two interferometers and detectors that are far
apart; the results lead to violations of the Bell-Clauser-Horne-Shimony-Holt
inequality (valid for local-realistic theories, including classical
descriptions of the phase). Finally, we discuss an experiment where particles
from each of two sources are either deviated via a beam splitter to a side
collector or proceed to the point of interference. For a given interference
result, we find "population oscillations" in the distributions of the deviated
particles, which are entirely controlled by the quantum angle. Various versions
of population oscillation experiments are discussed, with two or three
independent condensates.
