A general variational approach for equilibrium phase boundaries of trapped spin-1 Bose-Einstein condensates

We develop a simple and general variational method to estimate the solutions of the Gross-Pitaevskii equations and obtain the corresponding density profiles for all spin states of a trapped spin-1 Bose-Einstein condensate. We further employ this approach to obtain the complete phase diagram of the system under quasi-one-dimensional harmonic confinement, with ferromagnetic or antiferromagnetic spin interactions. We identify a suitable scaling that collapses all phase diagrams for different system sizes (i.e., total particle number) into a universal (system size-independent) phase diagram. The complete phase diagram for a confined system shows some significant qualitative differences compared to that of a condensate with homogeneous density distribution. The phase diagrams reported here could help identify the important parameter regimes in which phase transitions in the confined system, in general, occur. This knowledge of the region of phase boundaries can enable a reliable investigation of the instabilities near the boundaries that drive phase transitions.