We study a highly efficient, matter-wave amplification mechanism in a longitudinally-excited, Bose-Einstein condensate and reveal a very large enhancement due to nonlinear gain from a sixmatter- optical, wave-mixing process involving four photons. Under suitable conditions this opticallydegenerate, four-photon process can be stronger than the usual two-photon inelastic light scattering mechanism, leading to nonlinear growth of the observed matter-wave scattering independent of any enhancement from bosonic stimulation. Our theoretical framework can be extended to encompass even higher-order, nonlinear superradiant processes that result in higher-order momentum transfer.
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