  We present a detailed theoretical description of an atomic scanning
microscope in a cavity QED setup proposed in Phys. Rev. Lett. 120, 133601
(2018). The microscope continuously observes atomic densities with optical
subwavelength resolution in a nondestructive way. The super-resolution is
achieved by engineering an internal atomic dark state with a sharp spatial
variation of population of a ground level dispersively coupled to the cavity
field. Thus, the atomic position encoded in the internal state is revealed as a
phase shift of the light reflected from the cavity in a homodyne experiment.
Our theoretical description of the microscope operation is based on the
stochastic master equation describing the conditional time evolution of the
atomic system under continuous observation as a competition between dynamics
induced by the Hamiltonian of the system, decoherence effects due to atomic
spontaneous decay, and the measurement backaction. Within our approach we
relate the observed homodyne current with a local atomic density, and discuss
the emergence of a quantum nondemolition measurement regime allowing continuous
observation of spatial densities of quantum motional eigenstates without
measurement backaction in a single experimental run.
