AUTHOR=Gutiérrez-Jáuregui R. , Jáuregui R. 

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

JOURNAL=Frontiers in Physics

VOLUME=Volume 10 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.896174

DOI=10.3389/fphy.2022.896174

ISSN=2296-424X

ABSTRACT=Atomic gases tightly trapped near the focus of an electromagnetic wave interact with photons that exhibit a complex structure: with strong gradients of the local polarization and field amplitude due to focusing and, eventually, to topological phase singularities. We illustrate the consequences of this structure on a paradigmatic nonlinear optical process: three-wave mixing. A proper selection of the pump light structure yield 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 high effectiveness in the generation of nonlinear optics events conditions the properties of the  down converted photons to those of the pump field through phase matching conditions. It is emphasized that the expression of the latter  must incorporate both the structure of the vectorial EM modes, and the spatial configuration of the atomic trap. Three dimensional focusing prevents the applicability of the slowly varying envelope approximation on the description of the scattered EM field. An alternative is discussed using a Green function formalism  valid for any configuration of the field that also allows the identification of phase matching conditions. Spherical vectorial waves exemplify most concepts here discussed, including the possibility of observing nonlinear quantum phenomena at the single photon level.