Fortran programs for the timedependent GrossPitaevskii equation in a fully anisotropic trap
Abstract
Here we develop simple numerical algorithms for both stationary and nonstationary solutions of the timedependent GrossPitaevskii (GP) equation describing the properties of BoseEinstein condensates at ultra low temperatures. In particular, we consider algorithms involving real and imaginarytime propagation based on a splitstep CrankNicolson method. In a onespacevariable form of the GP equation we consider the onedimensional, twodimensional circularlysymmetric, and the threedimensional sphericallysymmetric harmonicoscillator traps. In the twospacevariable form we consider the GP equation in twodimensional anisotropic and threedimensional axiallysymmetric traps. The fullyanisotropic threedimensional GP equation is also considered. Numerical results for the chemical potential and rootmeansquare size of stationary states are reported using imaginarytime propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of nonstationary oscillation for different trap symmetries using realtime propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, Fortran 90/95 versions provide some simplification over the Fortran 77 programs, and these programs are also included (six programs in all). Program summaryProgram title: (i) imagetime1d, (ii) imagetime2d, (iii) imagetime3d, (iv) imagetimecir, (v) imagetimesph, (vi) imagetimeaxial, (vii) realtime1d, (viii) realtime2d, (ix) realtime3d, (x) realtimecir, (xi) realtimesph, (xii) realtimeaxial Catalogue identifier: AEDU_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDU_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 122 907 No. of bytes in distributed program, including test data, etc.: 609 662 Distribution format: tar.gz Programming language: FORTRAN 77 and Fortran 90/95 Computer: PC Operating system: Linux, Unix RAM: 1 GByte (i, iv, v), 2 GByte (ii, vi, vii, x, xi), 4 GByte (iii, viii, xii), 8 GByte (ix) Classification: 2.9, 4.3, 4.12 Nature of problem: These programs are designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in one, two or threespace dimensions with a harmonic, circularlysymmetric, sphericallysymmetric, axiallysymmetric or anisotropic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Solution method: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation, in either imaginary or real time, over small time steps. The method yields the solution of stationary and/or nonstationary problems. Additional comments: This package consists of 12 programs, see "Program title", above. FORTRAN77 versions are provided for each of the 12 and, in addition, Fortran 90/95 versions are included for ii, iii, vi, viii, ix, xii. For the particular purpose of each program please see the below. Running time: Minutes on a medium PC (i, iv, v, vii, x, xi), a few hours on a medium PC (ii, vi, viii, xii), days on a medium PC (iii, ix). Program summary (1)Title of program: imagtime1d.F Title of electronic file: imagtime1d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 1 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in onespace dimension with a harmonic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (2)Title of program: imagtimecir.F Title of electronic file: imagtimecir.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 1 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in twospace dimensions with a circularlysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (3)Title of program: imagtimesph.F Title of electronic file: imagtimesph.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 1 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with a sphericallysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (4)Title of program: realtime1d.F Title of electronic file: realtime1d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 2 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in onespace dimension with a harmonic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems. Program summary (5)Title of program: realtimecir.F Title of electronic file: realtimecir.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 2 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in twospace dimensions with a circularlysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems. Program summary (6)Title of program: realtimesph.F Title of electronic file: realtimesph.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 2 GByte Programming language used: Fortran 77 Typical running time: Minutes on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with a sphericallysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems. Program summary (7)Title of programs: imagtimeaxial.F and imagtimeaxial.f90 Title of electronic file: imagtimeaxial.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 2 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time: Few hours on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with an axiallysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (8)Title of program: imagtime2d.F and imagtime2d.f90 Title of electronic file: imagtime2d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 2 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time: Few hours on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in twospace dimensions with an anisotropic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (9)Title of program: realtimeaxial.F and realtimeaxial.f90 Title of electronic file: realtimeaxial.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 4 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time Hours on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with an axiallysymmetric trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems. Program summary (10)Title of program: realtime2d.F and realtime2d.f90 Title of electronic file: realtime2d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 4 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time: Hours on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in twospace dimensions with an anisotropic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems. Program summary (11)Title of program: imagtime3d.F and imagtime3d.f90 Title of electronic file: imagtime3d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum RAM memory: 4 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time: Few days on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with an anisotropic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems. Program summary (12)Title of program: realtime3d.F and realtime3d.f90 Title of electronic file: realtime3d.tar.gz Catalogue identifier: Program summary URL: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computers: PC/Linux, workstation/UNIX Maximum Ram Memory: 8 GByte Programming language used: Fortran 77 and Fortran 90 Typical running time: Days on a medium PC Unusual features: None Nature of physical problem: This program is designed to solve the timedependent GrossPitaevskii nonlinear partial differential equation in threespace dimensions with an anisotropic trap. The GrossPitaevskii equation describes the properties of a dilute trapped BoseEinstein condensate. Method of solution: The timedependent GrossPitaevskii equation is solved by the splitstep CrankNicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and nonstationary problems.
 Publication:

Computer Physics Communications
 Pub Date:
 October 2009
 DOI:
 10.1016/j.cpc.2009.04.015
 arXiv:
 arXiv:0904.3131
 Bibcode:
 2009CoPhC.180.1888M
 Keywords:

 02.60.Lj;
 02.60.Jh;
 02.60.Cb;
 03.75.b;
 Ordinary and partial differential equations;
 boundary value problems;
 Numerical differentiation and integration;
 Numerical simulation;
 solution of equations;
 Matter waves;
 Condensed Matter  Quantum Gases;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Physics  Computational Physics
 EPrint:
 34 pages, 11 figures, 18 Fortran programs included (to download the programs click other and download source), output files (not included) available in Comput. Phys. Commun. Library