Scattering distributions in the presence of measurement backaction

Scattering probe particles from a quantum system can provide experimental access to information about the system's state. However, measurement backaction and momentum transfer during scattering changes the state of the system, potentially destroying the state of the system we wish to probe. Here we investigate how to probe the system's initial state even in the presence of backaction and momentum transfer. We show that summing the scattering distributions of an ensemble of measurements reveals the initial state scattering pattern even when each of the individual measurements completely destroys the initial state. This procedure is effective provided the scattering takes place on a timescale that is short compared with the free evolution of the system.