  Scattering scanning near-field optical microscopy (s-SNOM) is a promising
technique for overcoming Abbe diffraction limit and substantially enhancing the
spatial resolution in spectroscopic imaging. The s-SNOM works by exposing an
atomic force microscope (AFM) tip to an optical electromagnetic (EM) field,
while the tip is so close to a dielectric sample that the incident beam lies
within the near-field regime and displays nonlinear behaviour. We replace the
incident EM field by photons generated by a single photon emitter, and propose
a quantum model for the suggested system by employing electric-dipole
approximation, image theory, and perturbation theory. Quantum state of
scattered photons from the AFM tip is extracted from the proposed model, which
contains information about electrical permittivity of the dielectric material
beneath the tip. The permittivity of the sample can be extracted through
spectroscopic setups. Our proposed scheme has potential applications for
high-resolution quantum sensing and metrology, especially for quantum imaging
and quantum spectroscopy.

Quantum Physics