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AIP conference proceedings

Application of finite difference time domain method to Gross-Pitaevskii equation

I. Wayan Sudiarta

Published Jan 1, 2019• 0 citations
Publisher

Abstract

In this paper, the finite difference time domain (FDTD) method is applied to solve the Gross-Pitaevskii (GP) equation. A simple procedure to include the non-linear term of the GP equation in the FDTD algorithm is given and discussed. For validation of the FDTD method, ground state, excited state wavefunctions and energies are computed for various trap potentials and compared to results from other numerical methods. The FDTD method is shown to be suitable and accurate for solving the GP equation.

DOI
10.1063/1.5132665

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In this paper, the finite difference time domain (FDTD) method is applied to solve the Gross-Pitaevskii (GP) equation. A simple procedure to include the non-linear term of the GP equation in the FDTD algorithm is given and discussed. For validation of the FDTD method, ground state, excited state wavefunctions and energies are computed for various trap potentials and compared to results from other numerical methods. The FDTD method is shown to be suitable and accurate for solving the GP equation.
"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."
We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our methods are as follows: (i) the fourth-order accuracy in space andrth ( r≥2 ) accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponential matrices. The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs). Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.
Many solutions have been propounded for Schrodinger’s equation. The application of these solutions is restricted to a few aspects of physics. One of the methods used for solving the Schrodinger equation is the Finite Difference Time Domain (FDTD) method [1]. It was used to solve numerically the Schrodinger equation for ground state and excited state eigenvalues, and eigenfunctions for a single particle in one-, two-, and three-dimensional potential wells. In the past, analytical methods which have been used to solve the Schrodinger equation are: supersymmetry [1], Nikiforov Uvarov method (NU) [3], asymptotic iteration method (AIM) [4,5], hyperviria perturbation [6], exact quantization rule (EQR) [7], perturbative formalism [8], wave function ansatz method [9], and path integral [10]. In all these methods highlighted above, none has been incorporated with the Bloch NMR to solve nuclear spin among other problems in physics and chemistry. The solution of the Schrodinger equation using the mathematical modeling of Bloch NMR has wide range of applications in many fields of physics and chemistry. It can be used for checking and improving models developed to study quantum mechanical systems, improve on the effective analysis of biological systems, analyzing spin transport, nuclear spin dynamics, improvement of the numerical methods and so on. In Sec. 2, the solution of the Schrodinger is given. In Sec. 3, the solution was validated by applying it to the harmonic solution. The energy
The Gross-Pitaevskii equation (GP), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GP equation consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained.This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.
We propose a numerical scheme for obtaining the stationary vortex-ring solutions of the Gross–Pitaevskii (GP) equation for an axisymmetrically trapped Bose–Einstein condensate (BEC). The effective energy functional and the associated GP equation are derived by assuming a trial phase profile for the wavefunction that is subject to the condition of circulation quantization on the rz plane. The wavefunction of the vortex ring is determined by solving the ground state of the effective GP equation numerically. Application of our method to the formation of a three-dimensional Skyrmion in a trapped two-component BEC is demonstrated.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.