Ultra-cold spinor Bose gases present rich physics due to the special combination of superfluidity and magnetism in a quantum system. This system was first realized in 1998 by confining a Bose-Einstein condensate in an all-optical trap.\nIn this thesis, we consider the fluctuations of observable quantities in a spin-1 Bose gas. Understanding how fluctuations arise due to the spin excitations of the condensate is important because it offers an insight into how measurement noise reveals the many-body physics of the system.\nTo begin, we present the mean-field and Bogoliubov theory of a uniform spin-1 Bose gas subject to a constant magnetic field, describing the condensate and its low-energy collective excitations.\nWe then develop a formalism to describe the fluctuations in a general density-like observable. We start from the two-point correlation function and cast it in the form of a generalised static structure factor determined by the three Bogoliubov quasiparticle excitation branches. We derive analytic results for the fluctuation amplitudes and the temperature-dependent static structure factors for observables of total density and the three spin densities.\nFor all four magnetic phases, we analyse the spinor order parameter and quasiparticle spectra while numerically mapping out the fluctuation amplitudes and static structure factors for the total and spin density operators.\nWe describe the fluctuations in experimental measurements made within finite cells, which is an important step to making meaningful predictions for experiments. We consider cylindrical cells and gaussian cells as two limiting cases. We apply this analysis to an experimentally realisable system of a quasi-2D spinor gas in a harmonic trap, comparing extensive numeric results with analytic limits.
We develop a general theory of spin-dependent Bragg spectroscopy for spinor Bose-Einstein condensates. This spectroscopy involves using a density and spin-coupled optical probe to excite the system. We show that within the linear-response regime the momentum or energy transferred by the probe is determined by a set of density and spin-density dynamic structure factors. We derive a set of
f-sum rules that provides rigorous constraints for the first energy moments of these structure factors. As an application we compute the dynamic structure factors for cases within all four distinct phases of a spin-1 condensate using Bogoliubov theory. Our results demonstrate that spin-dependent Bragg spectroscopy can be used to selectively investigate the various phonon and magnon excitation branches and will be a useful tool for advancing our understanding of spinor condensates.
We develop theory for fluctuations in atom number and spin within finite-sized cells of a spinor Bose-Einstein condensate. This theory provides a model of measurements that can be performed in current experiments using finite resolution in situ imaging. We develop analytic results for quantum and thermodynamic limits of the fluctuations and apply our theory to the four equilibrium phases of a spin-1 condensate. We then validate these limits and examine the behavior over a wide parameter regime using numerical calculations specialized to the case of a spinor condensate confined to be quasi-two-dimensional.
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.
Bogoliubov modes and the static structure factor for a two-species Bose–Einstein condensate were investigated within the framework of mean field approximation. We find that both quantities are strongly affected due to atomic collisions between the two species. The inter-species interaction not only changes the density distributions of the two species, but also modifies the Bogoliubov modes. As a consequence, the static structure factor exhibits a measurable difference. For a binary mixture of 87Rb and 23Na, our numerical results suggest that such an effect can be observed experimentally.