A hydrodynamic description is used to study the zero-temperature properties of a trapped spinor Bose-Einstein condensate in the presence of a uniform magnetic field. We show that, in the case of antiferromagnetic spin-spin interaction, the polar and ferromagnetic configurations of the ground state can coexist in the trap. These two phases are spatially segregated in such a way that the polar state occupies the inner part while the ferromagnetic state occupies the outer part of the atomic cloud. We also derive a set of coupled hydrodynamic equations for the number density and spin density excitations of the system. It is shown that these equations can be analytically solved for the system in an isotropic harmonic trap and a constant magnetic field. Remarkably, the related low lying excitation spectra are completely determined by the solutions in the region occupied by the polar state. We find that, within the Thomas-Fermi approximation, the presence of a constant magnetic field does not change the excitation spectra which still possess the similar form of that obtained by Stringari.
We consider a spin-1 Bose-Einstein condensate trapped in a harmonic potential under the influence of a homogeneous magnetic field. We investigate spatial and spin structure of the mean-field ground states under constraints on the number of atoms and the total magnetization. We show that the trapping potential can make the antiferromagnetic condensate separate into three distinct phases and ferromagnetic condensate into two distinct phases. In the ferromagnetic case, the magnetization is located in the center of the harmonic trap, while in the antiferromagnetic case magnetized phases appear in the outer regions. We describe how the transition from the Thomas-Fermi regime to the single-mode approximation regime with decreasing number of atoms results in the disappearance of the domains. We suggest that the ground states can be created in experiment by adiabatically changing the magnetic-field strength.
We report results for the Thomas-Fermi ground state of a spin-polarized dipolar interacting Bose-Einstein condensate for the case when the external magnetic field
B is not orientated parallel to a principal axis but is aligned parallel to a symmetry plane of a harmonic anisotropic trap. For a dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ the release energy of the condensate depends on the trap orientation angle ${\ensuremath{\vartheta}}_{T}$ between the principal axis
ez,T of the trap and the field
B. From the quasiclassical Josephson equation of macroscopic quantum physics we determine the low-lying eigenfrequencies of small-amplitude collective modes of the condensate density for various trap frequencies ${\ensuremath{\omega}}_{a}$ and trap orientation angles ${\ensuremath{\vartheta}}_{T}$. For the special case of a spherical harmonic trap with trap frequency $\ensuremath{\omega}$ it is rigorously shown for $\ensuremath{-}\frac{1}{2}<{\ensuremath{\varepsilon}}_{D}<1$ that a pure
s-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency ${\ensuremath{\Omega}}_{s}=\sqrt{5}\ensuremath{\omega}$ around the ground-state cloud, despite the well-known fact that the shape of the ground-state cloud of a spin-polarized dipolar condensate is for ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ not isotropic. For ${\ensuremath{\vartheta}}_{T}\ensuremath{\ne}0$ the small-amplitude modes of the particle density with isotropic and quadrupolar symmetry consist of two groups. There exist four modes that are combinations of basis functions, with
s-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and
dz2-wave, and
dxz-wave symmetry, and two modes that are combinations of basis functions with
dyz- and
dxy-wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}$ for prolate and oblate harmonic triaxial traps, respectively, is suggested to be used as an experimental method to measure the
s-wave scattering length
as of the atoms.
We analyze the structure of spin-one Bose-Einstein condensates in the presence of a homogeneous magnetic field. We classify the homogeneous stationary states and study their existence, bifurcations, and energy spectra. We reveal that phase separation can occur in the ground state of antiferromagnetic (polar) condensates while the spin components of the ferromagnetic condensates are always miscible and no phase separation occurs. Our theoretical model, confirmed by numerical simulations, explains that this phenomenon takes place when the energy of the lowest homogeneous state is a concave function of the magnetization. In particular, we predict that phase separation can be observed in a
23Na condensate confined in a highly elongated harmonic trap. Finally, we discuss the phenomena of dynamical instability and spin domain formation.
We calculate the spectrum of low-lying collective excitations in a mesoscopic cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas as a function of the boson-fermion repulsions. The cloud is under isotropic harmonic confinement and its dynamics is treated in the collisional regime by using the equations of generalized hydrodynamics with inclusion of surface effects. For large numbers of bosons, we find that as the cloud moves towards spatial separation (demixing) with increasing boson-fermion coupling, the frequencies of a set of collective modes show a softening followed by a sharp upturn. This behavior permits a clear identification of the quantum phase transition. We propose a physical interpretation for the dynamical transition point in a confined mixture, leading to a simple analytical expression for its location.
A finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the many-particle quantum hydrodynamic method. It is presented as the two fluid model of the Bose–Einstein condensate (BEC) and normal fluid. The continuity, Euler, spin evolution, and nematic tensor evolution equations are derived for each fluid. The linear and quadratic Zeeman effects are included. Scalar and spin–spin like short-range interactions are considered in the first order by the interaction radius. Obtained hydrodynamic equations are also represented as the set of two nonlinear Pauli equations. The spectrum of the bulk collective excitations is considered for the ferromagnetic phase in the small temperature limit. The spin wave is not affected by the presence of the small temperature in the described minimal coupling model, where the thermal part of the spin-current of the normal fluid is neglected. The two sound waves are affected by the spin evolution in the same way as the change of spectrum of the single sound wave in BEC, where speed of sound is proportional to g1+g2 with gi as the interaction constants.
It is investigated theoretically that magnetized Bose-Einstein condensation (BEC) with the internal (spin) degrees of freedom exhibits a rich variety of phase transitions, depending on the sign of the interaction in the spin channel. In the antiferromagnetic interaction case there exist always double BEC transitions from single component BEC to multiple component BEC. In the ferromagnetic case BEC becomes always unstable at a lower temperature, leading to a phase separation. The detailed phase diagram for the temperature vs the polarization, the spatial spin structure, the distribution of non-condensates and the excitation spectrum are examined for the harmonically trapped systems.