Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

doi:10.1016/J.CPC.2013.07.012

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Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Journal Article•DOI•

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Xavier Antoine¹, Xavier Antoine², Weizhu Bao³, Christophe Besse⁴•Institutions (4)

University of Lorraine¹, Centre national de la recherche scientifique², National University of Singapore³, university of lille⁴

01 Dec 2013-Computer Physics Communications (North-Holland)-Vol. 184, Iss: 12, pp 2621-2633

TL;DR: The nonlinear Schrodinger/Gross–Pitaevskii equation (NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well as other applications is discussed and their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, and dispersion relation to soliton solutions are discussed.

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About: This article is published in Computer Physics Communications.The article was published on 2013-12-01 and is currently open access. It has received 314 citations till now. The article focuses on the topics: Nonlinear Schrödinger equation & Gross–Pitaevskii equation.

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Dataset•DOI•

Bose-Einstein condensation

[...]

Wolfgang Ketterle

01 Jan 2014-Access Science

TL;DR: When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Einstein condensate as mentioned in this paper, which is the state of the art.

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Abstract: When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

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591 citations

Journal Article•DOI•

Mathematical theory and numerical methods for Bose-Einstein condensation

[...]

Weizhu Bao, Yongyong Cai

01 Dec 2012-Kinetic and Related Models

TL;DR: In this article, the authors mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE).

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Abstract: In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

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366 citations

Cites background or methods from "Computational methods for the dynam..."

...For simplicity of notation, here we only present the TSSP method for the GPE (2.25) in 1D truncated on a bounded interval U = (a, b) with homogeneous Dirichlet boundary conditions....
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...It is very useful in designing the most efficient and accurate numerical methods for computing ground states and dynamics, such as BESP and TSSP [5, 7, 8]), especially for the box potential....
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...d = 1, when β < 0, it admits the well-known bright soliton solution as [4, 70]...
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...25) is explicit, unconditionally stable, second-order accurate in time and spectral-order accurate in space [4, 7, 14]....
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...Various numerical methods have been proposed and studied in the literature [4, 7, 14, 20, 34, 64] for computing the dynamics of the GPE (2....
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Journal Article•DOI•

Mathematical theory and numerical methods for Bose-Einstein condensation

[...]

Weizhu Bao, Yongyong Cai

16 Mar 2014-arXiv: Mathematical Physics

TL;DR: In this article, the authors mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE).

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252 citations

Journal Article•DOI•

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions☆

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Xavier Antoine¹, Xavier Antoine², Romain Duboscq¹, Romain Duboscq²•Institutions (2)

University of Lorraine¹, French Institute for Research in Computer Science and Automation²

01 Nov 2014-Computer Physics Communications

TL;DR: GPELab (Gross–Pitaevskii Equation Laboratory), an advanced easy-to-use and flexible Matlab toolbox for numerically simulating many complex physics situations related to Bose–Einstein condensation is presented.

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122 citations

Cites background from "Computational methods for the dynam..."

...This includes the kind of computation (dynamics or ground state), the number of components, the type of scheme (BESP, BEFD, CNSP, CNFD for the ground state and Relaxation, Splitting for the dynamics [11, 12]), the semi discretization parameters and other inputs that we explain below....
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...A second paper [12] will present the numerical schemes that are included in GPELab for solving the deterministic [11] and stochastic dynamics of GPEs [1, 2, 3, 14, 36], the associated GPELab functions and a few numerical examples....
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Journal Article•DOI•

Fortran and C programs for the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap

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R. Kishor Kumar¹, Luis E. Young-S.², Dušan Vudragović³, Antun Balaž³, Paulsamy Muruganandam⁴, Sadhan K. Adhikari² - Show less +2 more•Institutions (4)

University of São Paulo¹, Spanish National Research Council², University of Belgrade³, Bharathidasan University⁴

10 Jun 2015-Computer Physics Communications

TL;DR: Numerical algorithms for both stationary and non-stationary solutions of the full three-dimensional Gross–Pitaevskii (GP) equation for a dipolar BEC, including the contact interaction are presented.

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112 citations

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References

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Journal Article•DOI•

A perfectly matched layer for the absorption of electromagnetic waves

[...]

Jean-Pierre Berenger

01 Oct 1994-Journal of Computational Physics

TL;DR: Numerical experiments and numerical comparisons show that the PML technique works better than the others in all cases; using it allows to obtain a higher accuracy in some problems and a release of computational requirements in some others.

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9,875 citations

"Computational methods for the dynam..." refers background or methods in this paper

...onal domain is chosen extremely large and/or time-dependent. Thus, in order to choose a smaller computational domain which might save memory and/or computational cost, perfectly matched layers (PMLs) [60] or high-order absorbing (or artiﬁcial) boundary conditions (ABCs) [12, 21, 23, 59, 102, 149, 154] need to be designed and/or used at the artiﬁcial boundary so that one can truncate (or approximate) t...
[...]
...solving the NLSE/GPE in the literatures. Here we simply review some of them for the completeness of this paper. In fact, PMLs were introduced by Berenger in 1994 for electromagnetic ﬁeld computations [60] and they have´ been extended to NLSE/GPE by various authors recently [153, 195]. Again, here we present the idea in 1D. Suppose that one is only interested in the solution of the NLSE/GPE (1.1) with ...
[...]

Journal Article•DOI•

Absence of Diffusion in Certain Random Lattices

[...]

Philip W. Anderson¹•Institutions (1)

Bell Labs¹

01 Mar 1958-Physical Review

TL;DR: In this article, a simple model for spin diffusion or conduction in the "impurity band" is presented, which involves transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites.

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Abstract: This paper presents a simple model for such processes as spin diffusion or conduction in the "impurity band." These processes involve transport in a lattice which is in some sense random, and in them diffusion is expected to take place via quantum jumps between localized sites. In this simple model the essential randomness is introduced by requiring the energy to vary randomly from site to site. It is shown that at low enough densities no diffusion at all can take place, and the criteria for transport to occur are given.

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9,647 citations

"Computational methods for the dynam..." refers background or methods in this paper

...Another issue is to design e fficient and accurate numerical methods and apply them for stud ying numerically NLSE/GPE with random potential [10, 98, 142, 150] or stochastic NL SE/GPE [58, 88, 89, 90, 149] with applications and NLSE/GPE in higher dimensions, i....
[...]
...with random potential [10, 98, 142, 150], then the following time-split ting finite difference (TSFD) – in which the time-splitting...
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Journal Article•DOI•

Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor

[...]

M. H. Anderson¹, J. R. Ensher¹, Michael R. Matthews¹, Carl E. Wieman¹, Eric A. Cornell² - Show less +1 more•Institutions (2)

National Institute of Standards and Technology¹, University of Colorado Boulder²

14 Jul 1995-Science

TL;DR: A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled and exhibited a nonthermal, anisotropic velocity distribution expected of the minimum-energy quantum state of the magnetic trap in contrast to the isotropic, thermal velocity distribution observed in the broad uncondensed fraction.

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Abstract: A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled. The condensate fraction first appeared near a temperature of 170 nanokelvin and a number density of 2.5 x 10 12 per cubic centimeter and could be preserved for more than 15 seconds. Three primary signatures of Bose-Einstein condensation were seen. (i) On top of a broad thermal velocity distribution, a narrow peak appeared that was centered at zero velocity. (ii) The fraction of the atoms that were in this low-velocity peak increased abruptly as the sample temperature was lowered. (iii) The peak exhibited a nonthermal, anisotropic velocity distribution expected of the minimum-energy quantum state of the magnetic trap in contrast to the isotropic, thermal velocity distribution observed in the broad uncondensed fraction.

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6,074 citations

"Computational methods for the dynam..." refers background in this paper

...own and important derivation of the NLSE is from the mean-ﬁeld approximation of many-body problems in quantum physics and chemistry [158], especially for the study of Bose-Einstein condensation (BEC) [11, 158], where it is also called the Gross-Pitaevskii equation (GPE) [29, 118, 157, 158]. Another important application of the NLSE is for laser beam propagation in nonlinear and/or quantum optics and there ...
[...]

Book•

The Principles of Quantum Mechanics

[...]

Paul Adrien Maurice Dirac

01 Jan 1930

5,991 citations

"Computational methods for the dynam..." refers background in this paper

... 81Q20, 81V45 1. Introduction The nonlinear Schrodinger equation (NLSE) is a partial di¨ erential equation (PDE) that can be met in many dierent areas of physics and chemistry as well as engineering [1, 2, 84, 92, 158, 165, 170]. The most well-known and important derivation of the NLSE is from the mean-ﬁeld approximation of many-body problems in quantum physics and chemistry [158], especially for the study of Bose-Einstein c...
[...]
...1) and its speciﬁc form depends on dierent applications [1, 2, 27, 29, 158, 170]. In fact, when f(ˆ) , with a constant, then the NLSE (1.1) collapses to the standard (linear) Schrodinger equation [92, 165]. The most popular and important nonlinearity is the cubic¨ nonlinearity [1, 2, 29, 158, 170] f(ˆ) = ˆ; 0 ˆ<1; (1.3) where (positive for repulsive or defocusing interaction and negative for attrac...
[...]

Book•

Solitons and the Inverse Scattering Transform

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Mark J. Ablowitz, Harvey Segur

01 Dec 1981

TL;DR: In this paper, the authors developed the theory of the inverse scattering transform (IST) for ocean wave evolution, which can be solved exactly by the soliton solution of the Korteweg-deVries equation.

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Abstract: : Under appropriate conditions, ocean waves may be modeled by certain nonlinear evolution equations that admit soliton solutions and can be solved exactly by the inverse scattering transform (IST). The theory of these special equations is developed in five lectures. As physical models, these equations typically govern the evolution of narrow-band packets of small amplitude waves on a long (post-linear) time scale. This is demonstrated in Lecture I, using the Korteweg-deVries equation as an example. Lectures II and III develop the theory of IST on the infinite interval. The close connection of aspects of this theory to Fourier analysis, to canonical transformations of Hamiltonian systems, and to the theory of analytic functions is established. Typical solutions, including solitons and radiation, are discussed as well. With periodic boundary conditions, the Korteweg-deVries equation exhibits recurrence, as discussed in Lecture IV. The fifth lecture emphasizes the deep connection between evolution equations solvable by IST and Painleve transcendents, with an application to the Lorenz model.

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3,415 citations