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, ${c}_{s}$ , includes contributions from both the two phases, implying the ``two-fluid'' feature of the system. ${c}_{s}$ can show a sharp bend at the Bose-Einstein condensation temperature.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, ${c}_{s}$, includes contributions from both the two phases, implying the ``two-fluid'' feature of the system. ${c}_{s}$ can show a sharp bend at the Bose-Einstein condensation temperature.

arXiv:cond-mat/0405094Physical Review ADec 10, 200433 citations

Spin Waves in Spin-Polarized Quantum Gases

Eugene P. Bashkin

Physical Review LettersSep 23, 198512 citations

Ferromagnetic phase transition in spinor Bose gases

Qiang Gu

The achievement in cooling alkali atomic gases, such as $^{87}$ Rb, $^{23}$ Na and $^{7}$ Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a $k^2$ -formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contributThe achievement in cooling alkali atomic gases, such as $^{87}$Rb, $^{23}$Na and $^{7}$Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a $k^2$-formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contribut

arXiv:cond-mat/0512693Dec 29, 20050 citations

{c}_{s}

, includes contributions from both the two phases, implying the ``two-fluid'' feature of the system.

{c}_{s}

can show a sharp bend at the Bose-Einstein condensation temperature.

The achievement in cooling alkali atomic gases, such as

^{87}

Rb,

^{23}

Na and

^{7}

Li, to quantum degeneracy opens up a way to study magnetism in spinor bosons, because these constituent atoms usually have a hyperfine spin degree of freedom. This article reviews several basic problems related to the ferromagnetic phase transition in spinor atomic Bose gases from a theoretical perspective. After a brief discussion on various possible origins of the ferromagnetic interaction, the phase diagram of the ferromagnetically coupled spinor bosons is investigated. It is found that the ferromagnetic transition occurs always above Bose-Einstein condensation and the Bose condensate is fully polarized. The low-lying collective excitations of the spinor condensate, including spin and density modes, are discussed. The spectrum of the density mode is of the Bogliubov form and the spin wave spectrum has a

k^2

-formed dispersion relation at long wavelengths. The spin-wave stiffness coefficient contains contribut