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

Abstract

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 threedimensional (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, rootmean-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. Program summary Program title: (i) imag1dZ, (ii) imag1dX, (iii) imag2dXY, (iv) imag2dXZ, (v) imag3d, (vi) real1dZ, (vii) real1dX, (viii) real2dXY, (ix) real2dXZ, (x) real3d

Statics and dynamics of a self-bound dipolar matter-wave droplet

“…We solve the 3D GP equation (3) numerically by the split-step Crank-Nicolson method [28] for a dipolar BEC [27,30] using both real-and imaginary-time propagation in Cartesian coordinates employing a space step of 0.025 and a time step upto as small as 0.00001. In numerical calculation, we use the parameters of 52 Cr atoms [26], e.g., a dd = 15.3a 0 and m = 52 amu with a 0 the Bohr radius.…”

“…The stationary widths w ρ and w z of a droplet correspond to the global minimum of energy (6) [26,27] 1 (8) and (9), for different K 3 . For N < N crit and for a > a dd = 15.3a 0 (dipolar) and for a > 0 (nondipolar) no droplet can be formed.…”

Stable and mobile two-dimensional dipolar ring-dark-in-bright Bose–Einstein condensate soliton

“…This interaction leads to a fully asymmetric stable, 2D RDB soliton mobile in the x − z plane in the presence of a harmonic trap along y axis. The dimensionless GP equation in this case can be written as [27,28] i ∂φ(r, t)…”

“…The system is assumed to be in the ground state φ(y) = exp(−y 2 /2)/(π) 1/4 of the axial trap and the wave function can be written as φ(r, t) = φ(y)φ 2D ( ρ, t), where ρ ≡ {x, z} and φ 2D ( ρ, t) is an effective wave function in the x − z plane. Using this ansatz in (3), the y dependence can be integrated out to obtain the following effective 2D equation [29,28] …”

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

“…Previously published C and Fortran programs [1] for solving the dipolar GP equation are sequential in nature and do not exploit the multiple cores or CPUs found in typical modern computers. A parallel implementation exists, using Nvidia CUDA [2], and both versions are already used within the ultra-cold atoms community [3].…”