Four-wave mixing with multiple spin states

James P. Burke, Jr., Carl J. Williams, and Paul S. Julienne
Atomic Physics Division
National Institute of Standards and Technology
Gaithersburg, MD. 20899
Y. B. Band Department of Chemistry and Physics
Ben-Gurion University of the Negev,
Beer-Sheva, Israel 84105
Marek Trippenbach Institute of Experimental Physics,
Optics Division Warsaw University,
Warsaw, Poland

Four-wave mixing is a common process in nonlinear optics wherein the nonlinear interaction between light and matter allows three light waves to interact or "mix" and produce a fourth wave with a different central wavevector (and/or frequency). The realization of Bose-Einstein condensation (BEC) in dilute gases of alkali atoms has created an entirely new field of nonlinear atom optics. In a BEC, the nonlinear interaction is a result of atomic interactions whose strength is characterized by the two-body scattering length. Four-wave mixing of matter waves was first demonstrated in experiments conducted here at NIST. In these first experiments, the internal hyperfine spin states of the four BEC wavepackets were identical. In this work, we explore the consequences of different hyperfine spin states of the BEC wavepackets on the four-wave mixing process. We show that the three or four spin state case is phenomenologically different from the one or two spin state case and in particular, the nonlinear coupling strength of the former depends on differences in scattering lengths greatly reducing the population of the fourth wave. Three-dimensional numerical examples are presented.