We present OpenMP version of a Fortran program for solving the Gross–Pitaevskii equation for a harmonically trapped three-component rotating spin-1 spinor Bose–Einstein condensate (BEC) in two spatial dimensions with or without spin–orbit (SO) and Rabi couplings. The program uses either Rashba or Dresselhaus SO coupling. We use the split-step Crank–Nicolson discretization scheme for imaginary- and real-time propagation to calculate stationary states and BEC dynamics, respectively.

New version program summary

Program title: BEC-GP-SPINOR-ROT-OMP, a program package containing programs spin-SO-rot-imre2d-omp.f90, with util.f90.

CPC Library link to program files: https://doi.org/10.17632/j3wr4wn946.2

Licensing provisions: Apache License 2.0

Programming language: OpenMP Fortran 90/95. The program is tested with the GNU, Intel, PGI, and Oracle (former Sun) compilers.

Supplementary material: File Supp.pdf gives additional details about the new program version and the underlying physical system.

Journal Reference of previous version: Comput. Phys. Commun. 259 (2021) 107657.

Does the new version supersede the previous version?: Only partially. The program spin-SO-rot-imre2d-omp.f90 supersedes spin-SO-imre2d-omp.f90, while the one-dimensional program is not part of this package.

Nature of problem: The present Open Multi-Processing (OpenMP) Fortran program solves the time-dependent nonlinear partial differential Gross–Pitaevskii (GP) equation for a trapped rotating spinor Bose–Einstein condensate (BEC) in two spatial dimensions.

Solution method: We employ the split-step Crank–Nicolson scheme to discretize the time-dependent GP equation in space and time. The discretized equation is then solved by imaginary- or real-time propagation, employing adequately small space and time steps, to yield the solution of stationary and non-stationary problems, respectively.

Reason for new version: The BEC is a special form of matter called superfluid. A hallmark of superfluidity is the formation of quantized vortices in a rotating BEC. The present program can be used to study the generation of quantized vortices in a rotating spin-1 trapped BEC and hence should be of general interest to researchers from various fields.

Summary of revisions: Previously we published Fortran [1] and C [2] programs for solving the mean-field GP equation for a BEC, which are now enjoying widespread use. Later we extended these programs to the more complex scenario of dipolar BECs [3], spin-1 spinor BECs [4], and of rotating BECs [5]. The OpenMP [6, 7] and CUDA/MPI [8, 9, 10] versions of these programs, designed to make these faster and more efficient in multi-core computers, are also available. In this paper we present Fortran 90/95 program for solving the GP equation of a two-dimensional (2D) rotating spin-1 spinor BEC with Rashba [11] and Dresselhaus [12] spin–orbit (SO) coupling and Rabi coupling, involving a modification over the same for a spin-1 spinor BEC [4]. A new input parameter OMEG, which represents the angular velocity of rotation $\Omega$ of the spin-1 spinor BEC, has been introduced in the program, following Ref. [5]. Besides this new parameter, the execution of the present program follows the same procedure as the 2D program of Ref. [4]. All other input parameters in the two programs are identical and the reader is advised to consult that reference for further details. For some values of input parameters the quantized vortices of a rotating BEC could be arranged in the form of a lattice with a certain spatial symmetry, e.g., triangular or square lattice [5]. In our numerical study, we established recently such a symmetric lattice structure for a Rashba SO-coupled rotating spin-1 BEC in the simplest case, without the Rabi coupling [13]. A Dresselhaus SO-coupled rotating spin-1 BEC should also lead to identical structure, provided the sign of the angular velocity of rotation is changed. For the sake of completeness, in the supplementary material related to this article that can be found online at URL we provide the corresponding GP equations for a rotating spin-1 BEC with some instructive numerical examples. The program package BEC-GP-SPINOR-ROT-OMP contains the programs spin-SO-rot-imre2d-omp.f90 and util.f90 in the directory src, as well as the files makefile and README.md. The makefile allows automated compilation of the program using different supported compilers (GNU, Intel, PGI, Oracle) by a simple make command, as in Ref. [4]. The file README.md contains instructions on how to compile and run the programs. The directory output contains examples of matching outputs of imaginary- and real-time propagation programs in sub-directories with a generic name rot$x$gam$y$ferro or rot$x$gam$y$antiferro, where $x$ denotes the value of the angular velocity of rotation $\Omega$ and $y$ denotes the strength of the SO coupling $\gamma$ for ferromagnetic ($c_0=482,c_2=15$) and antiferromagnetic ($c_0=669,c_2=-3.1$) cases. The results in imaginary-time sub-directories rot.3gam.5ferro and rot.3gam.5antiferro are calculated using the respective converged imaginary-time wave functions with zero angular velocity. The real-time sub-directories rot.3gam.5ferro and rot.3gam.5polar contain real-time results calculated using the respective converged imaginary-time wave functions as inputs. These sub-directories also contain gnuplot programs fig*.gnu which can be used to generate fig*.eps figure files of component densities.

References

[1] P. Muruganandam, S. K. Adhikari, Comput. Phys. Commun. 180 (2009) 1888.

[2] D. Vudragović, I. Vidanović, A. Balaž, P. Muruganandam, S. K. Adhikari, Comput. Phys. Commun. 183 (2012) 2021.

[3] R. Kishor Kumar, L.E. Young-S., D. Vudragović, A. Balaž, P. Muruganandam, S.K. Adhikari, Comput. Phys. Commun. 195 (2015) 117.

[4] R. Ravisankar, D. Vudragović, P. Muruganandam, A. Balaž, S. K. Adhikari, Comput. Phys. Commun. 259 (2021) 107657.

[5] R. K. Kumar, V. Lončar, P. Muruganandam, S. K. Adhikari, A. Balaž, Comput. Phys. Commun. 240 (2019) 74.

[6] L.E. Young-S., D. Vudragović, P. Muruganandam, S.K. Adhikari, A. Balaž, Comput. Phys. Commun. 204 (2016) 209.

[7] L. E. Young-S., P. Muruganandam, S. K. Adhikari, V. Lončar, D. Vudragović, A. Balaž, Comput. Phys. Commun. 220 (2017) 503.

[8] V. Lončar, A. Balaž, A. Bogojević, S. Škrbić, P. Muruganandam, S.K. Adhikari, Comput. Phys. Commun. 200 (2016) 406.

[9] V. Lončar, L.E. Young-S., S. Škrbić, P. Muruganandam, S.K. Adhikari, A. Balaž, Comput. Phys. Commun. 209 (2016) 190.

[10] B. Satarić, V. Slavnić, A. Belić, A. Balaž, P. Muruganandam, S.K. Adhikari, Comput. Phys. Commun. 200 (2016) 411.

[11] E. I. Rashba, Fiz. Tverd. Tela 2 (1960) 1224; English Transla.: Sov. Phys. Solid State 2 (1960) 1109.

[12] G. Dresselhaus, Phys. Rev. 100 (1955) 580.

[13] S. K. Adhikari, J. Phys.: Condens. Matter 33 (2021) 065404.

中文翻译：

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OpenMP解算器，用于旋转自旋1自旋轨道和Rabi耦合的Bose-Einstein冷凝物

我们提供了Fortran程序的OpenMP版本，用于求解二维捕获的谐波捕获的三分量旋转自旋1旋轴玻色-爱因斯坦凝聚物（BEC）的Gross–Pitaevskii方程，无论是否具有自旋轨道（SO）和Rabi耦合。该程序使用Rashba或Dresselhaus SO耦合。我们使用虚步和实时传播的分步式Crank-Nicolson离散化方案分别计算稳态和BEC动力学。

新版本程序摘要

程序标题： BEC-GP-SPINOR-ROT-OMP，一个程序包，其中包含程序spin-SO-rot-imre2d-omp.f90和util.f90。

CPC库链接到程序文件： https : //doi.org/10.17632/j3wr4wn946.2

许可条款： Apache License 2.0

编程语言： OpenMP Fortran 90/95。该程序已经过GNU，Intel，PGI和Oracle（以前的Sun）编译器的测试。

补充材料：文件Supp.pdf提供了有关新程序版本和基础物理系统的更多详细信息。

先前版本的期刊参考：计算。物理 公社 259（2021）107657。

新版本会取代旧版本吗？：仅部分。程序spin-SO-rot-imre2d-omp.f90取代了spin-SO-imre2d-omp.f90，而一维程序则不是此软件包的一部分。

问题的本质：目前的Open Multi-Processing（OpenMP）Fortran程序解决了二维空间中旋转自旋的Bose-Einstein凝结水（BEC）的时变非线性偏微分Gross-Pitaevskii（GP）方程。

解决方法：我们采用分步式Crank-Nicolson方案离散时空GP方程的时空分布。然后通过采用足够小的空间和时间步长的虚或实时传播来求解离散方程，分别得出固定和非固定问题的解。

新版本的原因：BEC是一种特殊形式的物质，称为超流体。超流动性的标志是在旋转的BEC中形成量化的涡流。本程序可用于研究旋转的spin-1捕集的BEC中量化涡旋的产生，因此，各个领域的研究人员应该对此感兴趣。

修订摘要：以前，我们发布了Fortran [1]和C [2]程序，用于求解BEC的平均场GP方程，这些程序现在正得到广泛使用。后来，我们将这些程序扩展到了偶极BEC [3]，spin-1旋转BEC [4]和旋转BEC [5]的更为复杂的场景。这些程序的OpenMP [6，7]和CUDA / MPI [8，9，10]版本也旨在使这些程序在多核计算机中更快，更高效。在本文中，我们提出了Fortran 90/95程序，该程序用于求解具有Rashba [11]和Dresselhaus [12]自旋轨道（SO）耦合和Rabi耦合的二维（2D）旋转spin-1 spinor BEC的GP方程，涉及对spin-1 spinor BEC的相同修改[4]。一个新的输入参数OMEG，它代表旋转的角速度$\Omega$自旋1 spinor BEC的“参考”已在程序中引入，参考文献。[5]。除此新参数外，本程序的执行遵循与Ref。2D程序相同的过程。[4]。两个程序中的所有其他输入参数都相同，建议读者参考该参考资料以获取更多详细信息。对于某些输入参数值，旋转BEC的量化涡流可以安排成具有一定空间对称性的格子形式，例如三角形或正方形格子[5]。在我们的数值研究中，我们最近在最简单的情况下为没有Rabi耦合的Rashba SO耦合旋转spin-1 BEC建立了这种对称晶格结构[13]。Dresselhaus SO耦合的旋转spin-1 BEC也应导致相同的结构，只要旋转角速度的符号改变。为了完整起见，在可在URL上在线找到的与本文相关的补充材料中，我们提供了旋转spin-1 BEC的相应GP方程，并提供了一些说明性的数值示例。程序包BEC-GP-SPINOR-ROT-OMP包含目录src中的程序spin-SO-rot-imre2d-omp.f90和util.f90，以及文件makefile和README.md。makefile允许通过简单的make命令使用不同的受支持的编译器（GNU，Intel，PGI，Oracle）自动编译程序，如Ref.1中所述。[4]。文件README.md包含有关如何编译和运行程序的说明。目录输出包含具有通用名称rot的子目录中的虚构和实时传播程序的匹配输出的示例$X$加姆$ÿ$铁或腐烂$X$加姆$ÿ$反铁，哪里 $X$ 表示旋转角速度的值 $\Omega$ 和 $ÿ$ 表示SO耦合的强度 $\gamma$ 用于铁磁（$C_0=482，C_{\mathrm{2个}}=15$）和反铁磁（$C_0=669，C_{\mathrm{2个}}=-3。\mathrm{1个}$）的情况。虚时子目录rot.3gam.5ferro和rot.3gam.5antiferro中的结果是使用各自的零角速度的收敛虚时波函数计算的。实时子目录rot.3gam.5ferro和rot.3gam.5polar包含实时结果，这些结果是使用各自的收敛虚时波函数作为输入来计算的。这些子目录还包含gnuplot程序fig * .gnu，可用于生成组件密度的fig * .eps图形文件。

参考

[1] P. Muruganandam，SK Adhikari，Comput。物理 公社 180（2009）1888。

[2] D.Vudragović，I。Vidanović，A。Balaž，P。Muruganandam，SK Adhikari，Comput。物理 公社 183（2012）2021。

[3] R. Kishor Kumar，LE Young-S。，D。Vudragović，A。Balaž，P。Muruganandam，SK Adhikari，Comput。物理 公社 195（2015）117。

[4] R. Ravisankar，D。Vudragović，P。Muruganandam，A。Balaž，SK Adhikari，Comput。物理 公社 259（2021）107657。

[5] RK Kumar，V。Lončar，P。Muruganandam，SK Adhikari，A。Balaž，Comput。物理 公社 240（2019）74。

[6] LE Young-S。，D。Vudragović，P。Muruganandam，SK Adhikari，A。Balaž，Comput。物理 公社 204（2016）209。

[7] LE Young-S。，P。Muruganandam，SK Adhikari，V。Lončar，D。Vudragović，A。Balaž，计算机。物理 公社 220（2017）503。

[8] V.Lončar，A。Balaž，A。Bogojević，S。Škrbić，P。Muruganandam，SK Adhikari，Comput。物理 公社 200（2016）406。

[9] V.Lončar，LE Young-S。，S。Škrbić，P。Muruganandam，SK Adhikari，A。Balaž，Comput。物理 公社 209（2016）190。

[10] B.Satarić，V。Slavnić，A。Belić，A。Balaž，P。Muruganandam，SK Adhikari，计算机。物理 公社 200（2016）411。

[11] EI Rashba，菲兹。特维尔 Tela 2（1960）1224；英文翻译：Sov。物理 固态2（1960）1109。

[12] G. Dresselhaus，物理学。Rev.100（1955）580。

[13] SK Adhikari，《物理学报》：Condens。问题33（2021）06540​​4。