Engineering quantum states, nonlinear measurements and anomalous diffusion by imaging

We show that well-separated quantum superposition states, measurements of strongly nonlinear observables and quantum dynamics driven by anomalous diffusion can all be achieved for single atoms or molecules by imaging spontaneous photons that they emit via resonance florescence. To generate anomalous diffusion we introduce continuous measurements driven by Levy processes and prove a number of results regarding their properties. In particular, we present strong evidence that the only stable Levy density that can realize a strictly continuous measurement is the Gaussian.