Wave mixing of hybrid Bogoliubov modes in a Bose-Einstein condensate

Abstract

Mode-mixing of coherent excitations of a trapped Bose-Einstein condensate is modelled using the Bogoliubov approximation. Hybridization of the modes of the breather ($l=0$) and surface ($l=4$) states leads to the formation of a Bogoliubov dark state. Calculations are presented for second-harmonic generation between the two lowest-lying even-parity $m=0$ modes in an oblate spheroidal trap. Two hybrid modes are strongly excited near second-harmonic resonance, and the coupling strength, and hence conversion rate, to the breather mode is half that given by an equivalent hydrodynamic estimate.