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In a Bose-Einstein condensate (BEC), if the hyperfine spins of the constituent atoms are available degrees of freedom, it is referred to as a spinor BEC. In this talk, we focus on spin-1 BEC, where the constituent atoms have hyperfine spin-1. In the mean-field regime, the dynamics of the system are captured in the Gross-Pitaevskii (GP) equations, which is a set of three, coupled non-linear Schrodinger equations (for a spin-1 system). One can analytically solve the GP equations, by using the well-known Thomas-Fermi (T-F) approximation, which gives all possible stationary state structures in presence of the trapping. Cumulatively, there are eight possible stationary states, all competing to become the ground state of the system. The T-F approximation is typically valid for large condensates where the density is high enough for the kinetic energy contributions to be neglected in comparison to the interaction energy. But this is not always true, for example, in the case of condensates with a smaller number of particles (so that the density is lower) or even near the tail (near the Thomas-Fermi radius) of a large condensate. For such situations, we propose an analytical method involving variational approximation which works really well in estimating all the relevant physical parameters and provides a full profile of the condensate backed by numerical simulation. We will focus on the simplest case of harmonically trapped condensate having the anti-ferromagnetic type of spin-spin interaction (Na-23), in the absence of a magnetic field. The stationary states competing to become the ground state have a very small energy difference as found from T-F. The energy difference is so small that neglecting the kinetic contribution can have severe consequences. This being an ideal scenario to apply the variational method, we will discuss the effect of including the kinetic energy contribution [1].
We will also focus on the multi-component stationary states that become the ground state in presence of the magnetic field. The T-F approximation fails to give a good description of the multi-component states and provides a wrong physical conclusion even for large condensates where typically T-F should be valid. We extend the variational method which works in presence of a magnetic field. We will show that the method is indispensable to explain the ground state structure involving the multi-component stationary states, which is an obvious pre-requisite if one intends to draw a phase diagram of a trapped condensate [2]. |