Study of phase diagrams for spin-1 Bose-Einstein Condensate under harmonic confinement

Abstract

In this thesis, we discuss a variational method that can help us to accurately determine the density profile of both single-component and multi-component ground states for a Spinor-Bose-Einstein condensate confined in a harmonic potential. Subsequently, we will apply our variational method to determine the phase boundary for a spin-1 Bose-Einstein condensate in a quasi-one-dimensional harmonic confinement . We begin by finding energy and number densities for all possible stationary states under the Thomas-Fermi approximation. Although the Thomas-Fermi approximation works better for larger condensates, it can fail even for larger condensates when the difference in energy between the competing ground states is minimal. Because of these limitations of Thomas-Fermi, researchers started exploring more accurate methods, such as numerical simulations, to determine the density profile of ground states for condensates under confinement. However, numerical simulations prove to be computationally expensive in higher dimensions. Thus, there is a growing need for analytical methods to efficiently and accurately determine the density profile of ground states for condensate under confinement and subsequently determine the phase boundary. This is where our variational method proves particularly effective. Furthermore, accurately finding the exact location of the phase boundary for a trapped system is crucial for studying critical phenomena like phase transitions, quench-induced dynamics, etc.