  We present a technique for atomic density measurements by the off-resonant
phase-shift induced on a two-frequency, coherently-synthesised light beam. We
have used this scheme to measure the column density of a magnetically trapped
atom cloud and to monitor oscillations of the cloud in real time by making over
a hundred non-destructive local density measurments. For measurements using
pulses of 10,000-100,000 photons lasting ~10 microsecond, the precision is
limited by statistics of the photons and the photodiode avalanche. We explore
the relationship between measurement precision and the unwanted loss of atoms
from the trap and introduce a figure of merit that characterises it. This
method can be used to probe the density of a BEC with minimal disturbance of
its phase.
