Spin waves in ferromagnetically coupled spinor Bose gases

It is shown that the ferromagnetic transition takes place always above Bose-Einstein condensation in ferromagnetically coupled spinor Bose gases. We describe the Bose ferromagnet within Ginzburg-Landau theory by a ``two-fluid'' model below Bose-Einstein condensation. Both the Bose condensate and the normal component are spontaneously magnetized. As a main result we show that spin waves in the two fluids are coupled together so as to produce only one mixed spin-wave mode in the coexisting state. The spin-wave stiffness coefficient of the mixed mode, cₛ, includes contributions from both the two phases, implying the ``two-fluid'' feature of the system. cₛ can show a sharp bend at the Bose-Einstein condensation temperature.