Fortran and C programs for the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap

Many of the static and dynamic properties of an atomic Bose-Einstein condensate (BEC) are usually studied by solving the mean-field Gross-Pitaevskii (GP) equation, which is a nonlinear partial differential equation for short-range atomic interaction. More recently, BEC of atoms with long-range dipolar atomic interaction are used in theoretical and experimental studies. For dipolar atomic interaction, the GP equation is a partial integro-differential equation, requiring complex algorithm for its numerical solution. Here we present numerical algorithms for both stationary and non-stationary solutions of the full three-dimensional (3D) GP equation for a dipolar BEC, including the contact interaction. We also consider the simplified one- (1D) and two-dimensional (2D) GP equations satisfied by cigar- and disk-shaped dipolar BECs. We employ the split-step Crank-Nicolson method with real- and imaginary-time propagations, respectively, for the numerical solution of the GP equation for dynamic and static properties of a dipolar BEC. The atoms are considered to be polarized along the z axis and we consider ten different cases, e.g., stationary and non-stationary solutions of the GP equation for a dipolar BEC in 1D (along x and z axes), 2D (in x-y and x-z planes), and 3D, and we provide working codes in Fortran 90/95 and C for these ten cases (twenty programs in all). We present numerical results for energy, chemical potential, root-mean-square sizes and density of the dipolar BECs and, where available, compare them with results of other authors and of variational and Thomas-Fermi approximations.Comment: To download the programs click other and download sourc

CUDA programs for solving the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank-Nicolson method as in the previous version (R. Kishor Kumar et al., Comput. Phys. Commun. 195, 117 (2015)), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12 to 25, depending on the program and input size.Comment: 7 pages, 2 figures; to download the programs, click other formats and download the sourc

Symmetry breaking, Josephson oscillation and self-trapping in a self-bound three-dimensional quantum ball

We study spontaneous symmetry breaking (SSB), Josephson oscillation, and self-trapping in a stable, mobile, three-dimensional matter-wave spherical quantum ball self-bound by attractive two-body and repulsive three-body interactions. The SSB is realized by a parity-symmetric (a) one-dimensional (1D) double-well potential and (b) a 1D Gaussian potential, both along the $z$ axis and no potential along the $x$ and $y$ axes. In the presence of each of these potentials, the symmetric ground state dynamically evolves into a doubly-degenerate SSB ground state. If the SSB ground state in the double well, predominantly located in the first well ($z>0$), is given a small displacement, the quantum ball oscillates with a self-trapping in the first well. For a medium displacement one encounters an asymmetric Josephson oscillation. The asymmetric oscillation is a consequence of SSB. The study is performed by a variational and numerical solution of a non-linear mean-field model with 1D parity-symmetric perturbations

OpenMP Fortran programs for solving the time-dependent dipolar Gross-Pitaevskii equation

In this paper we present Open Multi-Processing (OpenMP) Fortran 90/95 versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in one, two and three spatial dimensions. The atoms are considered to be polarized along the z axis and we consider different cases, e.g., stationary and non-stationary solutions of the GP equation for a dipolar Bose-Einstein condensate (BEC) in one dimension (along x and z axes), two dimensions (in x-y and x-z planes), and three dimensions. The algorithm used is the split-step semi-implicit Crank-Nicolson scheme for imaginary- and real-time propagation to obtain stationary states and BEC dynamics, respectively, as in the previous version [R. Kishor Kumar et al., Comput. Phys. Commun. 195, 117 (2015)]. These OpenMP versions have significantly reduced execution time in multicore processors

Faraday and Resonant Waves in Dipolar Cigar-Shaped Bose-Einstein Condensates

Faraday and resonant density waves emerge in Bose-Einstein condensates as a result of harmonic driving of the system. They represent nonlinear excitations and are generated due to the interaction-induced coupling of collective oscillation modes and the existence of parametric resonances. Using a mean-field variational and a full numerical approach, we studied density waves in dipolar condensates at zero temperature, where breaking of the symmetry due to anisotropy of the dipole-dipole interaction (DDI) plays an important role. We derived variational equations of motion for the dynamics of a driven dipolar system and identify the most unstable modes that correspond to the Faraday and resonant waves. Based on this, we derived the analytical expressions for spatial periods of both types of density waves as functions of the contact and the DDI strength. We compared the obtained variational results with the results of extensive numerical simulations that solve the dipolar Gross-Pitaevskii equation in 3D, and found a very good agreement.Comment: 18 pages, 10 figure

Characteristic features of the Shannon information entropy of dipolar Bose-Einstein condensates

Calculation of the Shannon information entropy (S) and its connection with the order-disorder transition and with inter-particle interaction provide a challenging research area in the field of quantum information. Experimental progress with cold trapped atoms has corroborated this interest. In the present work, S is calculated for the Bose-Einstein condensate (BEC) with dominant dipolar interaction for different dipole strengths, trap aspect ratios, and number of particles (N). Trapped dipolar bosons in an anisotropic trap provide an example of a system where the effective interaction is strongly determined by the trap geometry. The main conclusion of the present calculation is that the anisotropic trap reduces the number of degrees of freedom, resulting in more ordered configurations. Landsberg's order parameter exhibits quick saturation with the increase in scattering length in both prolate and oblate traps. We also define the threshold scattering length which makes the system completely disordered. Unlike non-dipolar BEC in a spherical trap, we do not find a universal linear relation between S and lnN, and we, therefore, introduce a general quintic polynomial fit rather well working for a wide range of particle numbers