  Relative phase is treated as a physical quantity for two mode systems in
quantum atom optics, adapting the Pegg-Barnett treatment of quantum optical
phase to define a linear Hermitian relative phase operator via first
introducing a complete orthonormal set of relative phase eigenstates. These
states are contrasted with other so-called phase states. Other approaches to
treating phase and previous attempts to find a Hermitian phase operator are
discussed. The relative phase eigenstate has maximal two mode entanglement, it
is a fragmented state with its Bloch vector lying inside the Bloch sphere and
is highly spin squeezed. The relative phase states are applied to describing
interferometry experiments with Bose-Einstein condensates (BEC), particularly
in the context of a proposed Heisenberg limited interferometry experiment. For
a relative phase eigenstate the fractional fluctuation in one spin operator
component perpendicular to the Bloch vector is essentially only of order 1/N,
so if such a highly spin squeezed state could be prepared it may be useful for
Heisenberg limited interferometry. An approach for preparing a BEC in a state
close to a relative phase state is suggested, based on adiabatically changing
parameters in the Josephson Hamiltonian starting from a suitable energy
eigenstate in the Rabi regime.
