Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this paper, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of self-consistency methods. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase in temperature induces a traceable shift of the ferromagnetic-polar transition boundary, behavior previously not noted by self-consistent mean-field calculations.
We investigate condensation and finite-temperature magnetic phase transitions in a coherently (Rabi) coupled Bose gas confined in a three-dimensional harmonic trap. For the noninteracting system, we derive analytical expressions for the critical temperature and condensate fraction using a semiclassical description of the single-particle spectrum and density of states. At fixed particle number, coherent coupling enhances the critical temperature relative to the uncoupled system, and this enhancement decreases with increasing particle number
N. The finite-size correction, in contrast, lowers the transition temperature, with its effect diminishing for larger
N. We then incorporate repulsive interactions within the Hartree--Fock--Bogoliubov--Popov framework to investigate the finite-temperature phase diagram in the temperature--Rabi coupling plane. The system undergoes successive transitions from a ferromagnetic to a paramagnetic condensate and, at higher temperature, to a thermal gas. The transition boundaries obtained from the vanishing of the condensate fraction show excellent agreement with the analytical Hartree-Fock predictions. We find that the interactions substantially suppress the variation in the condensation temperature with an increase in the coherent coupling.
In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field
B. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field
Bc, asymptotic values of
n0 (relative atom number in
mF=0 state) at large
B and the phase diagram of
n0. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of
n0. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature"
Ts and the ``kinetic temperature"
Tk with
Ts≪Tk, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.
We investigate the ferromagnetic-paramagnetic phase transition in coherently (Rabi) coupled Bose-Einstein condensates at zero and finite temperatures, exploring different routes to the transition by tuning the Rabi coupling or increasing the temperature at a fixed coupling. Using the Hartree-Fock-Bogoliubov theory within the Popov approximation, we map out the finite-temperature phase diagram of a three-dimensional homogeneous condensate and identify the critical line through the softening of the spin gap. Magnetization and the spin dispersion branch reveal the progressive suppression of the ferromagnetic order with increasing temperature. In quasi-one-dimensional harmonic traps, the transition, driven by Rabi coupling, is inferred through the softening of the spin breathing mode with its minimum shifting to lower coupling values with increasing temperature. Notably, the thermally driven transition causes monotonic hardening of all the spin modes. For both coupling and temperature-driven transition, the hybridized density modes in the ferromagnetic phase acquire more density character while approaching the critical point.
The finite temperature Green's function technique is used to calculate the energies and damping rates of the elementary excitations of homogeneous, dilute, spin-1 Bose gases below the Bose-Einstein condensation temperature in both the density and spin channels. For this purpose a self-consistent dynamical Hartree-Fock model is formulated, which takes into account the direct and exchange processes on equal footing by summing up certain classes of Feynman diagrams. The model is shown to satisfy the Goldstone theorem and to exhibit the hybridization of one-particle and collective excitations correctly. The results are applied to gases of
23Na and
87Rb atoms.