Matlab s eigs computes first the lu decomposition with full pivoting and scaling of the matrix a. Recently, methods for solving the grosspitaevskii equation gpe arising in boseeinstein condensates becs 1, 20 based on riccati equation were introduced. 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, an open source matlab toolbox for the numerical simulation of grosspitaevskii equations. Boseeinstein condensate, nite element method, gross pitaevskii equation ams subject classi cations. The nls equation is a partial differential equation like the grosspitaevskii equation. The gpe equation coming from physics the aim of gpelab is to compute both stationary solutions and the dynamics of boseeinstein condensates becs 68, 70 based on grosspitaevskii equations gpes 45, 56, 57, 69. Some numerical methods have been proposed to obtain approximate solutions for fractional grosspitaevskii. Comparing three numerical solvers of the grosspitaevskii equation posted on 03 february 2016. Exact analytical solutions of threedimensional gross. Gpelab, a matlab toolbox to solve grosspitaevskii equations. The numerical simulation of boseeinstein condensates is a challenging task, even more so if we are interested in fast rotating vortices.
We solve the timeindependent grosspitaevskii gp equation which describes the dilute bosecondensed atoms in harmonic trap at zero temperature by. Solving the gpe lets us describe a dilute becs ground state. Coupled mode equations and gap solitons for the 2d gross. Gpelab gross pitaevskii equation laboratory is a matlab toolbox devoted to the numerical.
Research article an efficient compact finite difference method for the solution of the grosspitaevskii equation rongpeizhang, 1 jialiu, 2 andguozhongzhao 3. Fourthorder algorithms for solving the imaginarytime. The model equation that gpelab solves is the grosspitaevskii equation. How can solve laplacian of a function with coefficients.
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. Quantum gross pitaevskii equation jutho haegeman1, damian draxler2, vid stojevic1,3 j. The grosspitaevskii equation and boseeinstein condensates 249 of the system is the creation of vortices, which can be seen as small twisters inside the liquid that behave quite differently. Solver for the twodimensional gross pitaevskii equation. The grosspitaevskii equation describes the ground state of a quantum system of identical bosons using the hartreefock approximation and the pseudopotential interaction model. We begin with an introduction on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem.
The gross pitaevskii equation determines the shape of this wave function. This is a schrodingerlike equation, now called the grosspitaevskii equation gpe, describes the wave function of bose particles interacting coher ently and confined in an atomic trap. The solution of the twodimensional grosspitaevskii equation. Dynamics of a grosspitaevskii equation with phenomenological. However the description can only be valid provided the amplitudes can be regarded as large, and this must be the case for all modes. Relaxation algorithm to hyperbolic states in gross. This paper mainly concerns the mathematical justification of the asymptotic limit of the grosspitaevskii equation with general initial data in the natural energy space over the whole space. Then using mean eld theory, one nds that the system is. The inverse problem for the grosspitaevskii equation. Learn about these semiconductor features and more below. The square of the wave function, times the above mentioned factor, is defined as the hartree potential. Solver for the twodimensional grosspitaevskii equation.
This offers the advantage to hide all technical issues related to the implementation. The grosspitaevskii equation gpe determines the quantum wave function of a gas of bosons in the limit in which thermal and quantum fluctuations are negligibly small. This function solves the gross pitaevskii equation in a twodimensional space. Fourthorder algorithms for solving the imaginarytime gross. The toolbox provides a variety of matlab classes for simulations based on the gross pitaevskii equation, the multicon gurational hartree method for bosons, and on generic fewmode models, as well as optimization problems. Osborne5 and frank verstraete1,2 1 department of physics and astronomy, university of ghent, krijgslaan 281 s9, b9000 ghent, belgium 2 faculty of physics, university of vienna, boltzmanngasse 5, a1090 wien, austria. Comparing three numerical solvers of the gross pitaevskii equation posted on 03 february 2016. 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 4350, australia. Nonlinear quantum search using the grosspitaevskii equation.
Osborne,4 and frank verstraete1,2 1department of physics and astronomy, university of ghent, krijgslaan 281 s9, b9000 ghent, belgium. 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. 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. Complex solutions of the time fractional grosspitaevskii. Computer physics computer physics communications 00 2014 1. Asymptotic limit of the grosspitaevskii equation with.
The grosspitaevskii equation describes the properties of a dilute. This software is a cudaenabled nonlinear schrodinger gross pitaevskii equation solver. This equation is a general version of the dissipative gross pitaevskii equation including terms with firstorder derivatives in the spatial coordinates which allow for rotational contributions. In this equation only the short range part of the interaction between the. Comparing three numerical solvers of the grosspitaevskii. Pdf the grosspitaevskii equation and boseeinstein condensates. A spectral integral equation solution of the gross. Numerical solution for the grosspitaevskii equation. Grosspitaevskii equation laboratory, an advanced easytouse and flexible matlab toolbox for numerically simulating many complex physics situations related to boseeinstein condensation. The grosspitaevskii equation for in appropriate coordinates has the form.
An efficient compact finite difference method for the. Numerical solutions to the gross pitaevskii equation for boseeinstein condensates by luigi galati b. 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 gross pitaevskii equations gpes. 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. We present a new numerical system using classical finite elements with mesh adaptivity for computing stationary solutions of the grosspitaevskii equation. We present a method for numerically solving a grosspitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one and. Usually it does not have analytic solution and numerical methods, such as the splitstep cranknicolson and fourier spectral methods, are used for its solution. 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. Fourthorder algorithms for solving the imaginarytime gross pitaevskii equation in a rotating anisotropic trap siu a. 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. Hi guys, i need to solve the gross pitaevskii equation using matlab, and i need to solve something like that. The inverse problem for the grosspitaevskii equation boris a. Unlike a similar nonlinear schrodinger equation nlse, the gpe is not integrable even in the 1d case due to presence of the external potential.
Then we present a new numerical method, cfdmaif method, which combines compact finite difference method cfdm. The simulations can be controlled through a python interface. Derivation of grosspitaevskii version of nonlinear. 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.
Complex solutions of the time fractional grosspitaevskii gp. 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. This software is a cudaenabled nonlinear schrodinger grosspitaevskii equation solver. We present an efficient, unconditionally stable, and accurate numerical method for the solution of the gross pitaevskii equation. Relaxation algorithm to hyperbolic states in grosspitaevskii equation. Recently this method was used to derive the hartree equation for singular interactions 4 and the gross pitaevskii equation without positivity condition on the interaction 10 where one had to restrict the scaling behaviour of the interaction.
In this paper more general scalings shall be considered assuming positivity of the interaction. The gross pitaevskii equation is a partial differential equation in space and time variables. We demonstrate the performance of the method in numerical experiments. Computer physics computer physics communications 00. We obtain conditions for the existence of a global attractor and find bounds for its dimension. Due to the complexity and timescales needed to simulate such system, it was essential to write some accelerated code to understand the behaviour of such systems. 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. Gpelab grosspitaevskii equation laboratory is a matlab toolbox devoted to the numerical. Bifurcation analysis of stationary solutions of twodimensional coupled gross pitaevskii equations using deflated continuation. In this section we learn how to simplify the full gross pitaevskii equation in the presence of a trap. The solution of the twodimensional grosspitaevskii.
The grosspitaevskii equation, geq, describes the bosecondensates phenomena, which is a very important investigation eld in condensed matter physics, at low temperature 1. Research article an efficient compact finite difference. The grosspitaevskii equation gpe, 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 at oms. Bifurcation analysis of stationary solutions of two. Extremelycomplexphysicsfactssimplyreportedbyanapplied mathematician thesimpletruncatedwikide. Referenced in 24 articles fortran programs for the timedependent grosspitaevskii equation in a fully anisotropic trap. The fields institute is a centre for mathematical research activity a place where mathematicians from canada and abroad, from academia, business, industry and financial institutions, can come together to carry out research and formulate problems of mutual interest. 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. Quantum gross pitaevskii equation jutho haegeman,1 damian draxler,2 vid stojevic,3 j.
A timesplitting pseudospectral method for the solution of. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the grosspitaevskii equation to the strong solution of the incompressible euler equations. Numerical solutions to the grosspitaevskii equation for boseeinstein condensates by luigi galati b. Derivation of the time dependent gross pitaevskii equation. Quantum grosspitaevskii equation jutho haegeman, damian draxler, vid stojevic, j. The methods are based on pseudospectral approximation techniques and therefore provide highly accurate solutions. Fourthorder algorithms for solving the imaginarytime grosspitaevskii equation in a rotating anisotropic trap siu a. I am trying to solve the gross pitaevskii equation numerically and using ndsolve, but this seems to be a dead end. An open source virtual laboratory for the schrodinger equation. This paper mainly concerns the mathematical justification of the asymptotic limit of the gross pitaevskii equation with general initial data in the natural energy space over the whole space. A spectral integral equation solution of the grosspitaevskii.
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. Hu xiao and li biaoy nonlinear science center and department of mathematics, ningbo university, ningbo 315211, china received 23 november 2010. Relaxation algorithm to hyperbolic states in grosspitaevskii. Gpelab, an open source matlab toolbox for the numerical.
This equation is a general version of the dissipative grosspitaevskii equation including terms with firstorder derivatives in the spatial coordinates which allow for rotational contributions. This function solves the grosspitaevskii equation in a twodimensional space. The primary use of this code was for my research on rapidly rotating boseeinstein condensates. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the gross pitaevskii equation to the strong solution of the incompressible euler equations.
I am trying to solve the grosspitaevskii equation numerically and using ndsolve, but this seems to be a dead end. Introduction the riccati equation re, named after the italian mathematician jacopo fran. The dimensionless rotating grosspitaevskii equation 2. Grosspitaevskii equation gpe determines the quantum wave function of a. Quantum grosspitaevskii equation jutho haegeman1, damian draxler2, vid stojevic1,3 j. The gross pitaevskii equation, geq, describes the bosecondensates phenomena, which is a very important investigation eld in condensed matter physics, at low temperature 1. We present an efficient, unconditionally stable, and accurate numerical method for the solution of the grosspitaevskii equation. However the description can only be valid provided the amplitudes can be regarded as large, and this must be the case for. These classes can be easily combined, which has the advantage that one can adapt the simulation programs. Octbec a matlab toolbox for optimal quantum control of. Pdf gpelab, a matlab toolbox to solve grosspitaevskii. The finite element method for the timedependent gross. The problem is boundary and initial conditions that i cant find a way to determin.
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