  We numerically study the expansion dynamics of ultracold atoms in a
one-dimensional disordered potential in the presence of a weak position
measurement of the atoms. We specifically consider this position measurement to
be realized by a combination of an external laser and a periodic array of
optical microcavities along a waveguide. The position information is acquired
through the scattering of a near-resonant laser photon into a specific
eigenmode of one of the cavities. The time evolution of the atomic density in
the presence of this light scattering mechanism is described within a Lindblad
master equation approach, which is numerically implemented using the Monte
Carlo wave function technique. We find that an arbitrarily weak rate of photon
emission leads to a breakdown of Anderson localization of the atoms.
