  We present a general and non-invasive probing scheme to perform full
momentum-resolved spectroscopy of a cold atomic gas loaded into an optical
lattice using a single quantum impurity. The protocol relies on weak
collisional interactions and subsequent population measurements of the
impurity. By tuning a few controllable external parameters the impurity-lattice
interaction can be engineered, and using two sets of measurements, performed
with the impurity in two different positions, the full dispersion relation of
the superfluid phonons can be reliably extracted.
