  Active interferometers use amplifying elements for beam splitting and
recombination. We experimentally implement such a device by using spin exchange
in a Bose-Einstein condensate. The two interferometry modes are initially empty
spin states that get spontaneously populated in the process of parametric
amplification. This nonlinear mechanism scatters atoms into both modes in a
pairwise fashion and generates a nonclassical state. Finally, a matched second
period of spin exchange is performed that nonlinearly amplifies the output
signal and maps the phase onto readily detectable first moments. Depending on
the accumulated phase this nonlinear readout can reverse the initial dynamics
and deamplify the entangled state back to empty spin states. This sequence is
described in the framework of SU(1,1) mode transformations and compared to the
SU(2) angular momentum description of passive interferometers.
