C programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

Abstract

We present C programming language versions of earlier published Fortran programs (Muruganandam and Adhikari (2009) [1]) for calculating both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation. The GP equation describes the properties of dilute Bose-Einstein condensates at ultra-cold temperatures. C versions of programs use the same algorithms as the Fortran ones, involving real-and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-spacevariable form of the GP equation, we consider the one-dimensional, two-dimensional, circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form, we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. In addition to these twelve programs, for six algorithms that involve two and three space variables, we have also developed threaded (OpenMP parallelized) programs, which allow numerical simulations to use all available CPU cores on a computer. All 18 programs are optimized and accompanied by makefiles for several popular C compilers. We present typical results for scalability of threaded codes and demonstrate almost linear speedup obtained with the new programs, allowing a decrease in execution times by an order of magnitude on modern multi-core computers. [14][15][16]. This new version represents translation of all programs to the C programming language, which will make it accessible to the wider parts of the corresponding communities. It is well known that numerical simulations of the GP equation in highly experimentally relevant geometries with two or three space variables are computationally very demanding, which presents an obstacle in detailed numerical studies of such systems. For this reason, we have developed threaded (OpenMP parallelized) versions of programs imagtime2d, imagtime3d, imagtimeaxial, realtime2d, realtime3d, realtimeaxial, which are named imagtime2d-th, imagtime3d-th, imagtimeaxial-th, realtime2d-th, realtime3d-th, realtimeaxial-th, respectively. Figure 1 shows the scalability results obtained for OpenMP versions of programs realtime2d and realtime3d. As we can see, the speedup is almost linear, and on a computer with the total of 8 CPU cores we observe in Fig. 1(a) a maximal speedup of around 7, or roughly 90% of the ideal speedup, while on a computer with 12 CPU cores we find in Fig. 1(b) that the maximal speedup is around 9.6, or 80% of the ideal speedup. Such a speedup represents significant improvement in the performance.

Evolution and dynamical properties of Bose-Einstein condensate dark matter stars

“…Our present study is based on a previously developed numerical solution of the GPE [26][27][28][29], which was obtained using the Crank-Nicholson method in combination with Cayley's formula [34], in the presence of an isotropic trapping potential (for a numerical investigation of BECs in the presence of anisotropic traps see [35,36].) In particular, the use of Cayley's formula ensures that the numerical solution remains stable, and the unitarity of the wavefunction is maintained.…”

Quantification of the memory effect of steady-state currents from interaction-induced transport in quantum systems

“…The zero-temperature Bose-Einstein condensate and its dynamics may be studied by the mean-field Gross-Pitaevskii equation (GPE) [69,70]. It can be generalized to describe timedependent systems [71][72][73] and has been previously implemented in modeling coherent transport [74,75]. In one dimension, the time-dependent GPE can be written as…”

“…In our simulation, the external potential V ext (x) corresponds to a box potential which confines the atoms. Here we solve the GPE with algorithms involving real-and imaginary-time propagation based on a splitstep Crank-Nicolson method [71,76] and follow Ref. [77] to normalize the wavefunction with dx|Φ(x)| 2 = 1.…”

OpenMP, OpenMP/MPI, and CUDA/MPI C programs for solving the time-dependent dipolar Gross–Pitaevskii equation

“…To parallelize the programs, we have used OpenMP with the same approach as described in [4], and extended the parallelization routines to include the computation of the dipolar term. The FFT, used in computation of the dipolar term, was also parallelized in a straightforward manner, by using the built-in support for OpenMP in FFTW3 library [5].…”

“…In this case the result of the transform has Hermitian symmetry, where one half of the values are complex conjugates of the other half. The fast Fourier transformation (FFT) libraries we use can exploit this to compute the result faster, using half the memory.To parallelize the programs, we have used OpenMP with the same approach as described in [4], and extended the parallelization routines to include the computation of the dipolar term. The FFT, used in computation of the dipolar term, was also parallelized in a straightforward manner, by using the built-in support for OpenMP in FFTW3 library [5].…”