Theory of a Quantum Scanning Microscope for Cold Atoms

We propose and analyze a scanning microscope to monitor `live' the quantum dynamics of cold atoms in a Cavity QED setup. The microscope measures the atomic density with subwavelength resolution via dispersive couplings to a cavity and homodyne detection within the framework of continuous measurement theory. We analyze two modes of operation. First, for a fixed focal point the microscope records the wave packet dynamics of atoms with time resolution set by the cavity lifetime. Second, a spatial scan of the microscope acts to map out the spatial density of stationary quantum states. Remarkably, in the latter case, for a good cavity limit, the microscope becomes an effective quantum non-demolition (QND) device, such that the spatial distribution of motional eigenstates can be measured back-action free in single scans, as an emergent QND measurement.