## OpenMP solver for rotating spin-1 spin–orbit- and Rabi-coupled Bose–Einstein condensates

##### Abstract

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 Ω 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 rotxgamyferro or rotxgamyantiferro, where x denotes the value of the angular velocity of rotation Ω and y denotes the strength of the SO coupling γ for ferromagnetic (c0=482,c2=15) and antiferromagnetic (c0=669,c2=−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.