A variational approach for the ground-state profile of a trapped spinor-BEC: a detailed study of phase transition in spin-1 condensate at zero magnetic field

Abstract

In this article, we introduce a multi-modal variational method to analytically estimate the full number- and corresponding energy-density profile of a spin-1 Bose-Einstein condensate (BEC) for a number of particles as low as 500 under harmonic confinement. To apply this method, we consider a system of spin-1 BEC under three-dimensional isotropic and effective one-dimensional harmonic confinement in the absence (negligible presence) of the magnetic field which has ground-state candidates of comparable energy. It should be noted that in such circumstances kinetic energy contribution to the ground state cannot be neglected which puts the applicability of Thomas–Fermi (T-F) approximation to question. For anti-ferromagnetic condensates, the T-F approximated energy difference between the competing stationary states (ground state and the first excited state) is approximately 0.5%. As T-F approximation is only good for condensates with a large number of particles, T-F approximated predictions can completely go wrong especially for small condensates. This is where comes the role of a detailed analysis using our variational method, which incorporates the kinetic energy contribution and accurately estimates the number- and energy-density profile even for condensates having a small number of particles. Results of our analytical method are supported by numerical simulation. This variational method is general and can be extended to other similar/higher-dimensional problems to get results beyond the accuracy of the T-F approximation.