Spatially inhomogeneous phase evolution of a two-component Bose-Einstein condensate

We investigate the spatially dependent relative phase evolution of an elongated two-component Bose-Einstein condensate. The pseudospin-$\frac{1}{2}$ system is comprised of the $|F=1,{m}_{F}=\ensuremath{-}1⟩$ and $|F=2,{m}_{F}=+1⟩$ hyperfine ground states of $^{87}\text{R}\text{b}$, which we magnetically trap and interrogate with radio-frequency and microwave fields. We probe the relative phase evolution with Ramsey interferometry and observe a temporal decay of the interferometric contrast well described by a mean-field formalism. Inhomogeneity of the collective relative phase dominates the loss of interferometric contrast, rather than decoherence or phase diffusion. We demonstrate a technique to simultaneously image each state, yielding subpercent variations in the measured relative number while preserving the spatial mode of each component. In addition, we propose a spatially sensitive interferometric technique to image the relative phase.