Proximity-measurement induced random localization in quantum fluids

Proximity measurements probe whether pairs of particles are close to one another. We consider the impact of post-selected random proximity measurements on a quantum fluid of many distinguishable particles. We show that such measurements induce random spatial localization of a fraction of the particles, and yet preserve homogeneity macroscopically. Eventually, all particles localize, with a distribution of localization lengths that saturates at a scale controlled by the typical measurement rate. The steady-state distribution of these lengths is governed by a familiar scaling form.