Investigation of ground states of spin-1 Bose-Einstein condensate in a harmonic trap

Abstract

In this thesis, we develop general methods to analytically obtain the ground state profiles and associated ground state properties of a spin-1 Bose-Einstein condensate under harmonic confinement with contact interaction. Firstly, from the Gross-Pitaevskii equation, we obtain the number density and energy density profiles of all possible stationary states using the Thomas-Fermi approximation for generic confinement. These stationary states compete to become the ground state in different parameter regimes. We show that a general method exists that can capture a lot of domain structures in a unified way. We show that by comparing the Thomas-Fermi approximated energy densities of different stationary states of a trapped system, under an essential constraint of the same chemical potential of the neighboring domains, one can actually capture the full spectrum of possible domain structures. While it is generally accepted that the Thomas-Fermi approximation is accurate for large condensates where the density is high enough to neglect the kinetic energy contribution compared to the interaction energies, we encounter situations, where even for large condensates, the Thomas-Fermi approximated predictions become inconclusive. We show that in the absence of the magnetic field, the Thomas-Fermi approximation predicts a ground state with which the other competing stationary states have a small energy difference, and the difference is of the order of the error introduced by the Thomas-Fermi approximation itself. Also in the presence of the magnetic field, for multi-component stationary states, the Thomas-Fermi approximation indicates domain structures in the ground state. In contrast, numerical simulations do not predict the same. The single-mode approximation, on the other hand, is also inaccurate in producing the sub-component profiles of the multi-component ground states. In such situations, one needs a multi-modal method to explain the ground state profiles and related properties analytically. We introduce a multi-modal variational method that smoothly incorporates the kinetic energy contribution and analytically estimates the complete profile of the number density. In the absence of the magnetic field, this variational method not only produces a more accurate prediction for large condensates but also is accurate for condensates with particles as low as 500. For multi-component stationary states, the variational method accurately estimates the tail part of each sub-components ruling out the domain formation possibility in the ground state. We employ the variational method to analytically estimate the phase transition boundaries between the phase-matched and polar states for a condensate with a ferromagnetic type of spin-spin interaction under 3-D isotropic harmonic trapping. This helps to draw a detailed comparison with the same phase transition in a homogeneous system, i.e., in the absence of trapping. There exists universality of these phase boundaries with respect to varying numbers of condensate particles under a scaling of the coordinates that comes out from the analytical calculation of the variational method. The multi-modal variational method introduced in this thesis opens up the route for a range of analytical studies that requires the ground state profiles to start with.