Nonlinear Quantum Optics With Structured Light: Tightly Trapped Atoms in the 3D Focus of Vectorial Waves

Abstract

Atomic gases tightly trapped near the focus of an electromagnetic wave interact with photons that exhibit a complex structure, displaying strong gradients of field amplitude and local polarization that can lead to topological phase singularities. We illustrate the consequences of this structure on a paradigmatic nonlinear optical process: three-wave mixing. The process begins by proper selection of the pump field, whose spatial structure is tailored to present huge gradients of the EM field that enhance atomic excitations through forbidden transitions. Atoms can then be depopulated via two electric dipole decays in a cascade configuration, thus providing the three necessary waves. The properties of the down-converted photons are conditioned to those of the pump field through phase matching conditions. It is emphasized that the expression of the photons must incorporate both the structure of the vectorial EM modes and the spatial configuration of the atomic trap. Due to the three-dimensional focusing, the slowly varying envelope approximation becomes inadequate when describing the scattered EM field. We discuss an alternative using a Green function formalism valid for any configuration of the field that also allows to identify the phase matching conditions. Spherical vectorial waves exemplify most concepts here discussed, including the possibility of observing nonlinear quantum phenomena at the single photon level.