Referenced in 27 articles fortran programs for the timedependent grosspitaevskii equation in a fully anisotropic trap. In matlab, the percentage symbol, %, is used to signify a comment. The simulations can be controlled through a python interface. Pdf this paper presents gpelab grosspitaevskii equation laboratory, an advanced easytouse and. 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. We investigate the 1d generalized grosspitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to. The grosspitaevskii equation describes the properties of a dilute. The inverse problem for the grosspitaevskii equation boris a. Gpelab, a matlab toolbox to solve grosspitaevskii equations i. Solver for the twodimensional gross pitaevskii equation. Solver for the twodimensional grosspitaevskii equation. Nonlinear quantum search using the gross pitaevskii equation 2 interaction terms.
Exact soliton solutions of the 1d generalized gross. The gpe is widely used for modeling boseeinstein condensates becs 6, 11, 17, 18. Grosspitaevskii equation gpe determines the quantum wave function of a. Numerical solution of the grosspitaevskii equation for boseeinstein condensation weizhu bao. Numerical solutions to the gross pitaevskii equation for boseeinstein condensates by luigi galati b. Nonlinear schrodinger equation multidimensional matlab. 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. This function solves the grosspitaevskii equation in a twodimensional space. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to the numerical. Freezing relative equilibria in equivariant firstorder evolution equations with comsol multiphysics 5.
Gpelab1 grosspitaevskii equation laboratory is an open access matlab. The nls equation is a partial differential equation like the grosspitaevskii equation. 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. 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. 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. Numerical simulation of boseeinstein condensates based on. 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. The model equation that gpelab solves is the grosspitaevskii equation. The exact solutions of the ggpe are obtained via expansion method with particular soliton features highlighted. Numerical solution of the grosspitaevskii equation for bose. 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. Computer physics computer physics communications 00 2014. Notes on the ginzburglandau theory icmr summer school on novel superconductors university of california, santa barbara august 2 august 15, 2009 m.
Lecture notes on classical mechanics a work in progress. The inverse problem for the grosspitaevskii equation. Computing stationary solutions for a class of systems multicomponents of gross pitaevskii equations in 1d, 2d and 3d. Quantum gross pitaevskii equation jutho haegeman,1 damian draxler,2 vid stojevic,3 j. Referenced in 27 articles fortran programs for the timedependent gross pitaevskii equation in a fully anisotropic trap. Numerical solution of the grosspitaevskii equation for. A system of grosspitaevskii equations gpes is used to mathematically model a. Miet 36486065 meng telecommunication systems upc etsetb barcelona. Numerical methods for the dynamics of the nonlinear schrodinger gross pitaevskii equations weizhu bao department of mathematics.
Gpelab a matlab toolbox for computing stationary solutions and. For users of the semiconductor module, comsol multiphysics version 5. 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. Bec is the minimization of the grosspitaevskii gp energy functional with. Gpelab, an open source matlab toolbox for the numerical. Gpelab, an open source matlab toolbox for the numerical simulation of grosspitaevskii equations. Computer physics communications 00 2014 8 3 cpc library classi. Numerical solution for the grosspitaevskii equation. Computation of ground states of the grosspitaevskii. Timesplitting spectral methods for nonlinear schrodinger. Notes on the ginzburglandau theory icmr summer school on novel superconductors university of california, santa barbara. 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. 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. The coefficients of terms of ggpe studied are arbitrary functions of time t.
Permission is granted to make and distribute verbatim copies of this manual. 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. The gross pitaevskii equation for in appropriate coordinates has the form. Function v is the external confining potential for example. Matlab will stop processing anything on that line to the right of the % symbol and will resume with the next line. Nonlinear quantum search using the grosspitaevskii equation 2 interaction terms. 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. Introduction the purpose of this paper is to introduce the principles of gps theory, and to. Gpelab, an open source matlab toolbox for the numerical simulation of gross pitaevskii equations. Gsgpes is a matlab gnu octave suite of programs for the computation of the ground state of systems of grosspitaevskii equations. The ground state of a system of grosspitaevskii equations is. Matlab tutorial francesco franco matlab is a software package that makes it easier for you to enter matrices and vectors, and manipulate them. Pdf gpelab, a matlab toolbox to solve grosspitaevskii.
Unlike a similar nonlinear schrodinger equation nlse, the gpe is not integrable even in the 1d case due to presence of the external potential. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions. I am trying to solve the gross pitaevskii equation numerically and using ndsolve, but this seems to be a dead end. In this paper, we discuss the different splitting approaches to numerically solve the grosspitaevskii equation gpe. The problem is boundary and initial conditions that i cant find a way to determin. 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. Gpelab1 grosspitaevskii equation laboratory is a flexible matlab toolbox. Gpelab, a matlab toolbox to solve grosspitaevskii equations. Gpelab1 grosspitaevskii equation laboratory is an open access matlab toolbox 8 developed for computing the stationary. Matlab, unfortunately, doesnt support block comments and so any comment blocks must use a % in front of each line. Unfortunately, the energy is not exactly conserved for example when. This is a stiff system because the limit cycle has portions where the solution components change slowly alternating 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. 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.
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. These solvers can be used with the following syntax. The grosspitaevskii description is not valid in this regime and new. An open source virtual laboratory for the schrodinger equation. Gpelab1 grosspitaevskii equation laboratory is a flexible matlab toolbox devoted to the numerical computation. 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. Grosspitaevskii equation for the density matrix in the. Numerical solutions to the grosspitaevskii equation for boseeinstein condensates by luigi galati b. Nonlinear quantum search using the grosspitaevskii equation. Though matlab is primarily a numerics package, it can certainly solve straightforward di. Grosspitaevskii equation gpe for boseeinstein condensates. 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. The models are motivated from spinor boseeinstein condensate bec. The model equation that gpelab solves is the gross pitaevskii equation.
Chapter 3 superconductivity as a macroscopic quantum phenomenon. 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. This system is formed of coupled meanfield equations, which are based on coupled grosspitaevskii equations. Matlab has several different functions builtins for the numerical solution of odes. Then using mean eld theory, one nds that the system is. Quantitative derivation of the grosspitaevskii equation. Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to. This program is particularly well designed for the computation of ground states of bose. Pdf gpelab is a free matlab toolbox for modeling and numerically solving large classes of systems of grosspitaevskii equations that arise in. From the programming point of view, the toolbox presents the following advantages. 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. The problem is boundary and initial conditions that i. Osborne,4 and frank verstraete1,2 1department of physics and astronomy, university of ghent, krijgslaan 281 s9, b9000 ghent, belgium.
Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to the numerical. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. 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. The interface follows a language that is designed to look like the notation used in linear algebra. Numerical solutions to the grosspitaevskii equation for bose. Provided programs are written in fortran 25, 27, c 29, 30 or matlab. This tutorial guides you in the first steps for using matlab. The input and output for solving this problem in matlab is given below. I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Numerical methods for the dynamics of the nonlinear. We investigate the 1d generalized gross pitaevskii equation ggpe with quadratic potential and parameterized nonlinearity. 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.
370 935 1316 1017 958 1044 29 761 84 1457 850 97 654 1433 1073 929 188 811 501 455 833 1194 1254 676 362 1276 166 1018 172 475 367 1357 1288 599 359 42 1005 168 490 1266 1281 235 292 850