Projection gradient method for energy functional minimization with a constraint and its application to computing the ground state of spin–orbit-coupled Bose–Einstein condensates

Abstract

In this study, we propose a projection gradient method for energy functional minimization with a constraint, which we use to compute the ground state of spin–orbit-coupled Bose–Einstein condensates at extremely low temperatures. The method has the advantage that it maintains the constraint when evolving a gradient flow to find the energy functional minimization under a constraint. The original gradient projection method for energy functional minimization under a constraint only considers an energy functional with real functions as variables. Thus, we extend it to consider complex functions as independent variables. We apply the newly proposed method to study the ground state solution of spin–orbit-coupled pseudo-spin 1/2 Bose–Einstein condensates. Detailed numerical results demonstrate the effectiveness of our method. Using this method, we found various types of ground state structures of spin–orbit coupled Bose–Einstein condensates.