This tutorial guides you in the first steps for using matlab. The model equation that gpelab solves is the grosspitaevskii equation. The models are motivated from spinor boseeinstein condensate bec. Numerical solution of the grosspitaevskii equation for bose. I am trying to solve the gross pitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Sep 19, 2006 we solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by symplectic shooting method ssm. Notes on the ginzburglandau theory icmr summer school on novel superconductors university of california, santa barbara. Solver for the twodimensional gross pitaevskii equation. Numerical methods for the dynamics of the nonlinear schrodinger gross pitaevskii equations weizhu bao department of mathematics. Part i is dedicated to exponential operator splitting methods 11,30 32 34 35 41 50 for ordinary differential equations. Gpelab1 gross pitaevskii equation laboratory is an open access matlab toolbox 8 developed for computing the stationary states and the nonlinear deterministic and stochastic dynamics of 1d2d3d gross pitaevskii equations gpes 24, 32, 33, 40, 41. The problem is boundary and initial conditions that i cant find a way to determin. Pdf gpelab, a matlab toolbox to solve grosspitaevskii.
Algorithm 31, 38, newtonlike method based on sequential quadratic programming. Gpelab a matlab toolbox for computing stationary solutions and. Osborne,4 and frank verstraete1,2 1department of physics and astronomy, university of ghent, krijgslaan 281 s9, b9000 ghent, belgium. A computation with the standard besp scheme see gpelab. Gpelab1 grosspitaevskii equation laboratory is a flexible matlab toolbox devoted to the numerical computation. The coefficients of terms of ggpe studied are arbitrary functions of time t.
A finiteelement toolbox for the stationary grosspitaevskii equation. Bec is the minimization of the grosspitaevskii gp energy functional with. We solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by symplectic shooting method ssm. But under certain conditions, one can assume that only twobody contact interactions contribute and the swave scattering length ais much less than the interparticle spacing. Abstract gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of gross pitaevskii equations that arise in the physics of boseeinstein condensates. From the programming point of view, the toolbox presents the following advantages. This is a stiff system because the limit cycle has portions where the solution components change slowly alternating with. It can compute the ground state in the defocusing case, for any number of equations with harmonic or quasiharmonic trapping potentials, in spatial dimension one, two or three. In this paper, we discuss the different splitting approaches to numerically solve the grosspitaevskii equation gpe. This program is particularly well designed for the computation of ground states of bose. Gpelab1 grosspitaevskii equation laboratory is an open access matlab toolbox 8 developed for computing the stationary. Stepanyants2 1department of physical electronics, school of electrical engineering, faculty of engineering, tel aviv university, tel aviv 69978, israel 2department of mathematics and computing, faculty of sciences, university of southern queensland, toowoomba, queensland. Computer physics computer physics communications 00 2014. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in the physics of boseeinstein condensates.
This function solves the gross pitaevskii equation in a twodimensional space. Function v is the external confining potential for example. Gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of gross pitaevskii equations that arise in the physics of boseeinstein condensates. Matlab tutorial francesco franco matlab is a software package that makes it easier for you to enter matrices and vectors, and manipulate them. The gross pitaevskii equation for in appropriate coordinates has the form. Solver for the twodimensional grosspitaevskii equation. Numerical solutions to the grosspitaevskii equation for bose. Numerical methods for the dynamics of the nonlinear.
Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to. Lecture notes on classical mechanics a work in progress. Nonlinear quantum search using the gross pitaevskii equation 2 interaction terms. Grosspitaevskii pitaevskii gp equation macroscopic quantum wavefunction a quantum state with large number of bosons occupying the same oneparticle states can be described by a macroscopic wavefunction 0 r t, 0 r t n r t, remark. Matlab, unfortunately, doesnt support block comments and so any comment blocks must use a % in front of each line. Notes on the ginzburglandau theory icmr summer school on novel superconductors university of california, santa barbara august 2 august 15, 2009 m. Gpelab 1 g ross p itaevskii e quation lab oratory is an open access matlab toolbox developed for computing the stationary states and the nonlinear deterministic and stochastic dynamics of 1d2d3d grosspitaevskii equations gpes. Then using mean eld theory, one nds that the system is. Computation of ground states of the grosspitaevskii. Numerical solution for the grosspitaevskii equation.
We investigate the 1d generalized gross pitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. The inverse problem for the grosspitaevskii equation boris a. Provided programs are written in fortran 25, 27, c 29, 30 or matlab. Department of computational science national university of singapore, singapore 117543. Computing stationary solutions for a class of systems multicomponents of gross pitaevskii equations in 1d, 2d and 3d. I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Numerical simulation of boseeinstein condensates based on. The grosspitaevskii description is not valid in this regime and new. Permission is granted to make and distribute verbatim copies of this manual.
The aim of this second paper, which follows \\citegroundgpelab, is to first present the various pseudospectral schemesavailable in gpelab for computing the deterministic and stochastic nonlinear dynamics of gross. Quantum gross pitaevskii equation jutho haegeman,1 damian draxler,2 vid stojevic,3 j. We consider the generalized purestate density matrix, which depends on different time moments, and obtain the evolution equation for this density matrix for the case where the density matrix corresponds to solutions of the grosspitaevskii equation. This system is formed of coupled meanfield equations, which are based on coupled grosspitaevskii equations. This paper presents gpelab gross pitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. The input and output for solving this problem in matlab is given below. The ground state of a system of grosspitaevskii equations is. Though matlab is primarily a numerics package, it can certainly solve straightforward di. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to the numerical. The present manuscript shall provide an introduction to advanced integration methods for nonlinear schrodinger equations that rely on highorder timesplitting hermite and fourier spectral methods. Freezing relative equilibria in equivariant firstorder evolution equations with comsol multiphysics 5. The gpe is widely used for modeling boseeinstein condensates becs 6, 11, 17, 18. Numerical solution of the grosspitaevskii equation for.
Grosspitaevskii equation for the density matrix in the. Gpelab1 grosspitaevskii equation laboratory is an open access matlab. Gpelab, an open source matlab toolbox for the numerical simulation of grosspitaevskii equations. The model equation that gpelab solves is the gross pitaevskii equation. Gsgpes is a matlab gnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. For users of the semiconductor module, comsol multiphysics version 5. Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to the numerical. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions.
The aim of this second paper, which follows \citegroundgpelab, is to first present the various pseudospectral schemesavailable in gpelab for computing the deterministic and stochastic nonlinear. Referenced in 27 articles fortran programs for the timedependent grosspitaevskii equation in a fully anisotropic trap. Nonlinear schrodinger equation multidimensional matlab. These solvers can be used with the following syntax. Grosspitaevskii equation gpe determines the quantum wave function of a. Pdf gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in. A system of grosspitaevskii equations gpes is used to mathematically model a.
The problem is boundary and initial conditions that i. Numerical solution of the grosspitaevskii equation for boseeinstein condensation weizhu bao. The grosspitaevskii equation describes the properties of a dilute. The inverse problem for the grosspitaevskii equation. Timesplitting spectral methods for nonlinear schrodinger. Gpelab, an open source matlab toolbox for the numerical. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to. It may be exploited to simulate the evolution of boseeinstein condensate in a static external potential, or to calculate the groundstate using the imaginary time evolution. Exact soliton solutions of the 1d generalized gross. We investigate the 1d generalized grosspitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. The grosspitaevskii equation gpe in one, two, and three dimensions 1d, 2d, and 3d is a fundamental model of the dynamics of boseeinstein condensates becs in ultracold gases. Grosspitaevskii equation in the present section, we state a normalisation of the ddimensional gpe 1 that is obtained by a linear transformation of the spatial variable and a rescaling of the wave function, see also caliari et al. Pdf the threedimensional 3d grosspitaevskii equation with strongly anisotropic confining potential is analyzed.
Gpelab1 grosspitaevskii equation laboratory is a flexible matlab toolbox. Pdf this paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and. This function solves the grosspitaevskii equation in a twodimensional space. Matlab will stop processing anything on that line to the right of the % symbol and will resume with the next line. The simulations can be controlled through a python interface. Referenced in 27 articles fortran programs for the timedependent gross pitaevskii equation in a fully anisotropic trap. Computer physics communications 00 2014 8 3 cpc library classi. The nls equation is a partial differential equation like the grosspitaevskii equation. Unlike a similar nonlinear schrodinger equation nlse, the gpe is not integrable even in the 1d case due to presence of the external potential. Nonlinear quantum search using the grosspitaevskii equation 2 interaction terms.
Grosspitaevskii equation gpe for boseeinstein condensates. Numerical solutions to the grosspitaevskii equation for boseeinstein condensates by luigi galati b. Miet 36486065 meng telecommunication systems upc etsetb barcelona. In matlab, the percentage symbol, %, is used to signify a comment. Numerical solutions to the gross pitaevskii equation for boseeinstein condensates by luigi galati b. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. Usually it does not have analytic solution and the same numerical methods used to solve the grosspitaevskii equation, such as the splitstep cranknicolson and fourier spectral methods, are used for its solution. Nonlinear quantum search using the grosspitaevskii equation. An open source virtual laboratory for the schrodinger equation. This paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. Gpelab, a matlab toolbox to solve grosspitaevskii equations i. Chapter 3 superconductivity as a macroscopic quantum phenomenon.
The interface follows a language that is designed to look like the notation used in linear algebra. Gpelab, a matlab toolbox to solve grosspitaevskii equations. Quantitative derivation of the grosspitaevskii equation. Unfortunately, the energy is not exactly conserved for example when. The exact solutions of the ggpe are obtained via expansion method with particular soliton features highlighted. Matlab has several different functions builtins for the numerical solution of odes. Ed, university of florida a thesis submitted to the graduate faculty of georgia southern university in partial ful llment of the requirement for the degree master of science statesboro. Gpelab, an open source matlab toolbox for the numerical simulation of gross pitaevskii equations.
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