  We derive a phase-entanglement criterion for two bosonic modes which is
immune to number fluc- tuations, using the generalized Moore-Penrose inverse to
normalize the phase-quadrature operator. We also obtain a phase-squeezing
criterion that is immune to number fluctuations using similar techniques. These
are utilized to obtain an operational definition of relative phase-measurement
sensitivity, via analysis of phase measurement in interferometry. We show that
these measures are proportional to enhanced phase-measurement sensitivity. The
phase-entanglement criterion is a hallmark for a new type of quantum squeezing,
namely planar quantum squeezing. This has the property that it squeezes two
orthogonal spin directions simultaneously, which is possible owing to the fact
that the SU(2) group that describes spin symmetry has a three-dimensional
parameter space, of higher dimension than the group for photonic quadratures.
The practical advantage of planar quantum squeezing is that, unlike
conventional spin-squeezing, it allows noise reduction over all phase-angles
simultaneously. The application of this type of squeezing is to quantum
measure- ment of an unknown phase. We show that a completely unknown phase
requires two orthogonal measurements, and that with planar quantum squeezing it
is possible to reduce the measurement uncertainty independently of the unknown
phase value. This is a different type of squeezing to the usual spin-squeezing
interferometric criterion, which is only applicable when the measured phase is
already known to a good approximation, or can be measured iteratively. As an
example, we calcu- late the phase-entanglement of the ground state of a
two-well, coupled Bose-Einstein condensate, similar to recent experiments. This
system demonstrates planar squeezing in both the attractive and repulsive
interaction regimes.
