  A scheme is discussed that allows one for performing homodyne detection of
the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe
lasers setup, that both illuminates a Bose-Einstein condensate, acting as
reference system, and a second ultracold gas, composed by the same atoms but in
a quantum phase to determine. Photon scattering outcouples atoms from both
systems, which then propagate freely. Under appropriate conditions, when the
same photon can either be scattered by the Bose-Einstein condensate or by the
other quantum gas, both flux of outcoupled atoms and scattered photons exhibit
oscillations, whose amplitude is proportional to the condensate fraction of the
quantum gas. The setup can be extended to measure the first-order correlation
function of a quantum gas. The dynamics here discussed make use of the
entanglement between atoms and photons which is established by the scattering
process in order to access detailed information on the quantum state of matter.

Quantum Gases;Quantum Physics