We develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional linear, two-dimensional circularly symmetric, and the three-dimensional spherically-symmetric 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. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time 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).
Deep Dive into Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap.
We develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional linear, two-dimensional circularly symmetric, and the three-dimensional spherically-symmetric 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. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-st
After a successful experimental detection of Bose-Einstein condensates (BEC) of dilute trapped bosonic alkali-metal atoms 7 Li, 23 Na, and 87 Rb [1,2] at ultra-low temperatures, there have been intense theoretical activities in studying properties of the condensate using the time-dependent mean-field Gross-Pitaevskii (GP) equation under different trap symmetries. Among many possibilities, the following traps have been used in various studies: three-dimensional (3D) spherically-symmetric, axially-symmetric and anisotropic harmonic traps, two-dimensional (2D) circularly-symmetric and anisotropic harmonic traps, and one-dimensional (1D) harmonic trap. The inter-atomic interaction leads to a nonlinear term in the GP equation, which complicates its accurate numerical solution, specially for a large nonlinearity. The nonlinearity is large for a fixed harmonic trap when either the number of atoms in the condensate or the atomic scattering length is large and this is indeed so under many experimental conditions. Special care is needed for the solution of the time-dependent GP equation with large nonlinearity and there has been an extensive literature on this topic .
The time-dependent GP equation is a partial differential equation in space and time variables involving first-order time and second-order space derivatives together with a harmonic and a nonlinear potential term, and has the structure of a nonlinear Schrödinger equation with a harmonic trap. One commonly used procedure for the solution of the time-dependent GP equation makes use of a discretization of this equation in space and time and subsequent integration and time propagation of the discretized equation. From a knowledge of the solution of this equation at a specific time, this procedure finds the solution after a small time step by solving the discretized equation. A commonly used discretization scheme for the GP equation is the semi-implicit Crank-Nicolson discretization scheme [49][50][51] which has certain advantages and will be used in this work.
In the simplest one-space-variable form of the GP equation, the solution algorithm is executed in two steps. In the first step, using a known initial solution, an intermediate solution after a small interval of time ∆ is found neglecting the harmonic and nonlinear potential terms. The effect of the potential terms is then included by a first-order time integration to obtain the final solution after time ∆. In case of two or three spatial variables, the space derivatives are dealt with in two or three steps and the effect of the potential terms are included next. As the time evolution is executed in different steps it is called a split-step real-time propagation method. This method is equally applicable to stationary ground and excited states as well as non-stationary states, although in this paper we do not consider stationary excited states. The virtues of the semi-implicit Crank-Nicolson scheme [49][50][51] are that it is unconditionally stable and preserves the normalization of the solution under real-time propagation. A simpler and efficient variant of the scheme called the split-step imaginary-time propagation method obtained by replacing the time variable by an imaginary time is also considered. (The GP equation involves complex variables. However, after replacing the time variable by an imaginary time the resultant partial differential equation is real, and hence the imaginarytime propagation method involves real variables only. This trick leads to an imaginary-time operator which results in exponential decay of all states relative to the ground state and can then be applied to any initial trial wave function to compute an approximation to the actual ground state rather accurately. We shall use imaginary-time propagation to compute the ground state in this paper.) The split-step imaginary-time propagation method involving real variables yields very precise result at low computational cost (CPU time) and is very appropriate for the solution of stationary problems involving the ground state. The split-step real-time propagation method uses complex quantities and yields less precise results for stationary problems; however, they are appropriate for the study of non-equilibrium dynamics in addition to stationary problems involving excited states also.
Most of the previous studies [3][4][5]8,9,12,18,20,21,24,25,[33][34][35]39,45,46] on the numerical solution of the GP equation are confined to a consideration of stationary states only. Some used specifically the imaginary-time propagation method [6,31,44,45]. There are few studies [30,37,38,41] for the numerical solution of the timedependent GP equation using the Crank-Nicolson method [49][50][51]. Other methods for numerical solution of the time-dependent GP equation have also appeared in the literature [7,14,16,23,[26][27][28][29]32,42,43,47,48]. These time-dependent methods can be used for studying non-equilibrium dynamics of the condensate invo
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