  Most interferometers operate with photons or dilute, non-condensed cold atom
clouds in which collisions are strongly suppressed. Spinor Bose-Einstein
condensates (BECs) provide an alternative route toward realizing three-mode
interferometers; in this realization, spin-changing collisions provide a
resource that generates mode entanglement. Working in the regime where the pump
mode, i.e., the m=0 hyperfine state, has a much larger population than the side
or probe modes (m=+1 and m=-1 hyperfine states), f=1 spinor BECs approximate
SU(1,1) interferometers. We derive analytical expressions within the undepleted
pump approximation for the phase sensitivity of such an SU(1,1) interferometer
for two classes of initial states: pure Fock states and coherent spin states.
The interferometer performance is analyzed for initial states without seeding,
with single-sided seeding, and with double-sided seeding. The validity regime
of the undepleted pump approximation is assessed by performing quantum
calculations for the full spin Hamiltonian. Our analytical results and the
associated dynamics are expected to guide experiments as well as numerical
studies that explore regimes where the undepleted pump approximation makes
quantitatively or qualitatively incorrect predictions.
