  Matter-wave interferometer of ultracold atoms with different linear momenta
has been extensively studied in theory and experiment. The vortex matter-wave
interferometer with different angular momenta is applicable as a quantum sensor
for measuring the rotation, interatomic interaction, geometric phase, etc. Here
we report the first experimental realization of a vortex matter-wave
interferometer by coherently transferring the optical angular momentum to an
ultracold Bose condensate. After producing a lossless interferometer with atoms
only populating the two spin states, we demonstrate that the phase difference
between the interferences in the two spin states is locked on $\pi$. We also
demonstrate the robustness of this out-of-phase relation, which is independent
of the angular-momentum difference between the two interfering vortex states,
constituent of Raman optical fields and expansion of the condensate. The
experimental results agree well with the calculation from the unitary evolution
of wave packet in quantum mechanics. This work opens a new way to build a
quantum sensor and measure the atomic correlation in quantum gases.
