Sound waves and modulational instabilities on continuous-wave solutions in spinor Bose-Einstein condensates

Abstract

We analyze sound waves (phonons, i.e. Bogoliubov excitations) propagating on continuous-wave (cw) solutions of repulsive $F=1$ spinor Bose-Einstein condensates (BECs) such as $^{23}\mathrm{Na}$ (which is antiferromagnetic or polar) and $^{87}\mathrm{Rb}$ (which is ferromagnetic). Zeeman splitting by a uniform magnetic field is included. All cw solutions to ferromagnetic BECs with vanishing ${M}_{F}=0$ particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields are subject to modulational instability (MI). Modulational instability increases with increasing particle density. Modulational instability also increases with differences in the components' wave numbers; this effect is larger at lower densities but becomes insignificant at higher particle densities. Continuous-wave solutions to antiferromagnetic (polar) BECs with vanishing ${M}_{F}=0$ particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields do not suffer MI if the wave numbers of the components are the same. If there is a wave-number difference, MI initially increases with increasing particle density and then peaks before dropping to zero beyond a given particle density. The cw solutions with particles in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ components and nonvanishing ${M}_{F}=0$ components do not have MI if the wave numbers of the components are the same, but do exhibit MI when the wave numbers are different. Direct numerical simulations of a continuous wave with weak white noise confirm that weak noise grows fastest at wave numbers with the largest MI and show some of the results beyond small-amplitude perturbations. Phonon dispersion curves are computed numerically; we find analytic solutions for the phonon dispersion in a variety of limiting cases.