We study numerically the time-independent vector Gross--Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose--Einstein condensate (BEC) at zero or a very low temperature. In preparation for the numerics, we scale the three-dimensional (3d) VGPEs, approximately reduce it to lower dimensions, present a continuous normalized gradient flow (CNGF) to compute ground states of multicomponent BEC, prove energy diminishing of the CNGF, which provides a mathematical justification, and discretize it by the backward Euler finite difference (BEFD), which is monotone in linear and nonlinear cases and preserves energy diminishing property in the linear case. Then we use a time-splitting sine-spectral (TSSP) method to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multicomponent BEC. The merits of the TSSP method for VGPEs are that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, has "good" resolution in the semiclassical regime, is of spectral-order accuracy in space and second-order accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in three dimensions for ground states and dynamics of multicomponent BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multicomponent BEC.We study numerically the time-independent vector Gross--Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose--Einstein condensate (BEC) at zero or a very low temperature. In preparation for the numerics, we scale the three-dimensional (3d) VGPEs, approximately reduce it to lower dimensions, present a continuous normalized gradient flow (CNGF) to compute ground states of multicomponent BEC, prove energy diminishing of the CNGF, which provides a mathematical justification, and discretize it by the backward Euler finite difference (BEFD), which is monotone in linear and nonlinear cases and preserves energy diminishing property in the linear case. Then we use a time-splitting sine-spectral (TSSP) method to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multicomponent BEC. The merits of the TSSP method for VGPEs are that it is explicit, unconditionally stable, time reversible and time transverse invariant if the VGPEs is, has "good" resolution in the semiclassical regime, is of spectral-order accuracy in space and second-order accuracy in time, and conserves the total particle number in the discretized level. Extensive numerical examples in three dimensions for ground states and dynamics of multicomponent BEC are presented to demonstrate the power of the numerical methods and to discuss the physics of multicomponent BEC.

arXiv:cond-mat/0305309Multiscale Modeling and SimulationJan 1, 2004160 citations

Solutions of the Gross-Pitaevskii and time-fractional Gross-Pitaevskii equations for different potentials with Homotopy Perturbation Method

Neslihan Üzar, Han, D., T. T ufekci +1 more

In this study, after we have briefly introduced the standard Gross-Pitaevskii equation, we have suggested fractional Gross-Pitaevskii equations to investigate the time-dependent ground state dynamics of the Bose-Einstein condensation of weakly interacting bosonic particle system which can includes non-Markovian processes or non-Gaussian distributions and long-range interactions. Only we focused the time-fractional Gross-Pitaevskii equation and have obtained solutions of the standard Gross-Pitaevskii and time-fractional Gross-Pitaevskii equations for attractive and repulsive interactions in the case external trap potentials $V(x)=0$ and optical lattice potential $V(x) =\pm\sin^{2}x$ by using Homotopy Perturbation Method. We have found that the Homotopy Perturbation Method solutions of the Gross-Pitaevskii equation for these potentials and interactions are the same analytical results of it. Furthermore we have also found that solutions of the time-fractional Gross-Pitaevskii equation for these potentials and interactions can be given in terms of Mittag-Leffler function. The solutions of the time-fractional Gross-Pitaevskii equation provide that the time evolution of the ground state dynamics of Bose-Einstein condensation of bosonic particles deviates exponential form, and evolutes with time as stretched exponentially.In this study, after we have briefly introduced the standard Gross-Pitaevskii equation, we have suggested fractional Gross-Pitaevskii equations to investigate the time-dependent ground state dynamics of the Bose-Einstein condensation of weakly interacting bosonic particle system which can includes non-Markovian processes or non-Gaussian distributions and long-range interactions. Only we focused the time-fractional Gross-Pitaevskii equation and have obtained solutions of the standard Gross-Pitaevskii and time-fractional Gross-Pitaevskii equations for attractive and repulsive interactions in the case external trap potentials $V(x)=0$ and optical lattice potential $V(x) =\pm\sin^{2}x$ by using Homotopy Perturbation Method. We have found that the Homotopy Perturbation Method solutions of the Gross-Pitaevskii equation for these potentials and interactions are the same analytical results of it. Furthermore we have also found that solutions of the time-fractional Gross-Pitaevskii equation for these potentials and interactions can be given in terms of Mittag-Leffler function. The solutions of the time-fractional Gross-Pitaevskii equation provide that the time evolution of the ground state dynamics of Bose-Einstein condensation of bosonic particles deviates exponential form, and evolutes with time as stretched exponentially.

arXiv:1203.3352Mar 15, 20123 citations

GROUND STATE ENERGY OF ONE-DIMENSIONAL MACROSCOPIC QUANTUM OSCILLATOR WITH GRAVITATIONAL POTENTIAL

Teguh Budi Prayitno

We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.

May 1, 20120 citations

V(x)=0

and optical lattice potential

V(x) =\pm\sin^{2}x

by using Homotopy Perturbation Method. We have found that the Homotopy Perturbation Method solutions of the Gross-Pitaevskii equation for these potentials and interactions are the same analytical results of it. Furthermore we have also found that solutions of the time-fractional Gross-Pitaevskii equation for these potentials and interactions can be given in terms of Mittag-Leffler function. The solutions of the time-fractional Gross-Pitaevskii equation provide that the time evolution of the ground state dynamics of Bose-Einstein condensation of bosonic particles deviates exponential form, and evolutes with time as stretched exponentially.