An ultracold Bose gas of two-level atoms can be thought of as a spin-1/2 Bose gas. It supports spin-wave collective modes due to the exchange mean field. Such collective spin oscillations have been observed in recent experiments at JILA with
87Rb atoms confined in a harmonic trap. We present a theory of the spin-wave collective modes based on the moment method for trapped gases. In the collisionless and hydrodynamic limits, we derive analytic expressions for the frequencies and damping rates of modes with dipole and quadrupole symmetry. We find that the frequency for a given mode is given by a temperature-independent function of the peak density n, and falls off as
1/n. We also find that, to a very good approximation, excitations in the radial and axial directions are decoupled. We compare our model to the numerical integration of a one-dimensional version of the kinetic equation and find very good qualitative agreement. The damping rates, however, show the largest deviation for intermediate densities, where one expects Landau damping---which is unaccounted for in our moment approach---to play a significant role.
We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.
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 present experimental observations of coherent spin-population oscillations in a cold thermal, Bose gas of spin-1 23Na atoms. The population oscillations in a multi-spatial-mode thermal gas have the same behavior as those observed in a single-spatial-mode antiferromagnetic spinor Bose-Einstein condensate. We demonstrate this by showing that the two situations are described by the same dynamical equations, with a factor of 2 change in the spin-dependent interaction coefficient, which results from the change to particles with distinguishable momentum states in the thermal gas. We compare this theory to the measured spin population evolution after times up to a few hundreds of ms, finding quantitative agreement with the amplitude and period. We also measure the damping time of the oscillations as a function of magnetic field.
Bose-Einstein condensation (BEC) is a phenomenon in which identical bosons occupy the same quantum state below a certain critical temperature. A hallmark of BEC is the coherence between particles every particle shares the same quantum wavefunction and phase. This coherence has been demonstrated for the external (motional) degrees of freedom of the atomic condensates by interfering two condensates. In this thesis, the coherence is shown to extend to the internal spin degrees of freedom of a spin-1 Bose gas evidenced by the observed coherent and reversible spin-changing collisions. The observed coherent dynamics are analogous to Josephson oscillations in weakly connected superconductors and represent a type of matter-wave four-wave mixing. Control of the coherent evolution of the system using magnetic fields is also demonstrated. The studies on spinor condensates begin by creating spinor condensates directly using all-optical approaches that were first developed in our laboratory. All-optical formation of Bose-Einstein condensates (BEC) in 1D optical lattice and single focus trap geometries are developed and presented. These techniques offer considerable flexibility and speed compared to magnetic trap approaches, and the trapping potential can be essentially spin-independent and are ideally suited for studying spinor condensates. Using condensates with well-defined initial non-equilibrium spin configuration, spin mixing of F = 1 and F = 2 spinor condensates of rubidium-87 atoms confined in an optical trap is observed. The equilibrium spin configuration in the F = 1 manifold confirms that 87Rb is ferromagnetic. The coherent spinor dynamics are demonstrated by initiating spin mixing deterministically with a non-stationary spin population configuration. Finally, the interplay between the coherent spin mixing and spatial dynamics in spin-1 condensates with ferromagnetic interactions is investigated.
Abstract We study the structure of quasi-one-dimensional Bogoliubov–de Genes collective excitations of inhomogeneous F = 1 spinor Bose–Einstein condensates confined by a harmonic trap potential and loaded into an optical lattice. Employing a perturbative method, we report general analytical expressions for the polar and ferromagnetic confined collective oscillations. In both cases, the excited eigenfrequencies are given as functions of the effective 1D coupling constants, the trap frequency and the optical lattice parameters. While the presence of the trap yields quantized longitudinal collective modes, whose spectra depends on the collision scattering lengths, the optical lattice shifts their frequencies and, more importantly, yields tunable oscillations of the spin populations, for the even modes, as experimentally observed.