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Beliaev theory of spinor Bose-Einstein condensates

By generalizing the Green's function approach proposed by Beliaev [1, 2], we investigate the effect of quantum depletion on the energy spectra of elementary excitations in an F = 1 spinor Bose-Einstein condensate, in particular, of 87Rb atoms in an external magnetic field. We find that quantum depletion increases the effective mass of magnons in the spin-wave excitations with quadratic dispersion relations. The enhancement factor turns out to be the same for both ferromagnetic and polar phases, and also independent of the magnitude of the external magnetic field. The lifetime of these magnons in a 87Rb spinor BEC is shown to be much longer than that of phonons. We propose an experimental setup to measure the effective mass of these magnons in a spinor Bose gas by exploiting the effect of a nonlinear dispersion relation on the spatial expansion of a wave packet of transverse magnetization. This type of measurement has practical applications, for example, in precision magnetometry.