  We experimentally and theoretically study phase coherence in two-component
Bose-Einstein condensates of $^{87}{\rm Rb}$ atoms on an atom chip. Using
Ramsey interferometry we measure the temporal decay of coherence between the
$|F=1,m_{F}=-1\rangle$ and $|F=2,m_{F}=+1\rangle$ hyperfine ground states. We
observe that the coherence is limited by random collisional phase shifts due to
the stochastic nature of atom loss. The mechanism is confirmed quantitatively
by a quantum trajectory method based on a master equation which takes into
account collisional interactions, atom number fluctuations, and losses in the
system. This decoherence process can be slowed down by reducing the density of
the condensate. Our findings are relevant for experiments on quantum metrology
and many-particle entanglement with Bose-Einstein condensates and the
development of chip-based atomic clocks.
