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.

Introduction

Ultracold atomic gases have been shown to be an ideal testing ground for the experimental study of various condensed matter phenomena  [1], [2]. A particularly interesting possibility is the realization of spin–orbit interactions in cold atomic systems. Quantum particles have an internal “spin” angular momentum. Spin–orbit coupling (SOC) aims to link a particle’s spin to its motion. If successful, this can lead to many fundamental phenomena in a wide of quantum systems, which range among nuclear physics, condensed matter physics, and atomic physics. For example, in electronic condensed matter systems, SOC can produce quantum spin Hall states or topological insulators  [3], [4], which have potential applications in quantum devices. Quite recently, SOC has been induced in ultracold spinor Bose gases of ${}^{87}\mathrm{Rb}$ atoms with the so-called “synthetic non-Abelian gauge fields”[5].

The creation of this SOC in ultracold gases has rapidly attracted theoretical researchers to Bose–Einstein condensates (BEC)  [3], [4], [6], [7], [8], [9], [10], [11], [12]. In particular, it was shown recently that the ground state of an homogeneous, i.e., untrapped, two-component BEC with SOC is a single plane-wave phase or a spin stripe phase, depending on the spin-dependent interactions  [13]. By contrast, noninteracting trapped SOC BEC are expected to have a half-quantum vortex configuration [14]. It was predicted that, in addition to the plane-wave phase and the standing-wave phase, exotic lattice states called triangular-lattice states and square-lattice state also emerge as the ground state of spin-1 condensates  [15]. Previously, SOC BEC has been studied theoretically for different types of spin–orbit interactions (Rashab coupling, Dresselhaus coupling, or Rashab–Dresselhaus coupling) and different internal structures of bosons (pseudospin- 1/2 bosons, spin-1 bosons, and spin-2 bosons). It has been found that different couplings can generate non-trivial ground-state structures in spin- 1/2 BEC, spin-1 BEC, and spin-2 BEC, but an efficient numerical method for finding the ground state solution has not been studied rigorously.

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 SOC BEC at extremely low temperatures. We demonstrate that this method has the advantage of maintaining 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 the constraint only considers an energy functional with real functions as variables. We extend this method to consider complex functions as independent variables. We use this method to find the numerical ground state solution of SOC pseudo-spin 1/2 BEC. It should be noted that many other numerical methods, such as the construction of a normalized gradient flow  [16], [17], [18], [19], [20], the Sobolev gradient method  [21], [22], and the two-parameter continuation method  [23], may also be used to compute the ground state of SOC BEC. However, they require a normalization step to maintain the constraint in numerical computations. Thus, the optimal approach to the normalization step remains a numerical issue. In contrast to these three methods, our method discretizes the continuous gradient flows directly and no further normalization step procedure is required to ensure that the constraint is satisfied numerically. This may be one of the main advantages of our proposed method. In addition, our proposed method is quite general and it can easily be extended to handle many other types of functional minimization problems under constraints, such as the energy functional minimization problem with multiple constraints  [24], [25]and the isoperimetric problem  [26]. However, extending the normalized gradient flow, the Sobolev gradient method, or the two-parameter continuation method to handle the functional minimization problem with multiple constraints are not easy tasks, although they are possible.

The remainder of this paper is organized as follows. In Section  2, we review the general projection method for the functional minimization problem under a constraint. We extend this method to consider a similar functional minimization problem that allows the treatment of complex functions as independent variables. We prove that the method has the advantage of maintaining the constraint when evolving a gradient flow to find the energy functional minimization under a constraint. In Section  3, we define the ground state solutions for pseudo-spin 1/2 BEC at very low temperatures. We demonstrate how to apply the projection method to find ground state solutions for pseudo-spin 1/2 BEC. In Section  4, we report two-dimensional numerical results for the ground state of the pseudo-spin 1/2 BEC. Section  5 presents our conclusions and discussion.