Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap

Dion, Claude M.; Cancès, Éric

doi:10.1103/physreve.67.046706

Cited by 55 publications

(63 citation statements)

References 38 publications

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“…In fact, it is easy to show that the GPE (1.1) conserves the total mass wheref denotes the conjugate of f . Along the numerical front, different efficient and accurate numerical methods including the time-splitting pseudospectral method [7,25,36,37], finite difference method [2,3], and Runge-Kutta or CrankNicolson pseudospectral method [14,20] have been developed for the GPE without and with [6,9,11] the angular momentum rotation term. Of course, each method has its advantages and disadvantages.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

2012

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Abstract. We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h 2 + τ 2 ) in the l 2 -norm and discrete H 1 -norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l ∞ -norm by using the inverse inequality and the l 2 -norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.

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Section: Introductionmentioning

confidence: 99%

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

2012

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“…Integrals involved in Eq. (10) can be computed numerically using the methods of Gaussian quadrature [11,16], or analytically [8] using the formulas derived by Busbridge [19]. In the present work, we have used this analytical expression-derived in the appendix for the sake of completeness-to compute the values of J integrals.…”

Section: Theorymentioning

confidence: 99%

“…We will now discuss the basis-set expansion technique, used for solving the GPE [8,11,16]. In this approach, one expandsψ(r) as a linear combination of basis functions of three-dimensional anisotropic simple harmonic oscillator…”

Section: Theorymentioning

confidence: 99%

“…Recently, Bao and Tang developed a novel scheme for obtaining the ground state of the GPE, by directly minimizing the corresponding energy functional [15]. Additionally, utilizing harmonic oscillator basis set, and Gauss-Hermite quadrature integration scheme, Dion and Cancès have proposed a scheme for solving both the time-dependent and -independent GPE [16]. Earlier, we had also proposed an alternative scheme for dealing with condensates with a large number of particles, and high number densities [17].…”

Section: Introductionmentioning

confidence: 99%

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A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Tiwari¹,

Shukla²

2006

Computer Physics Communications

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Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simpleharmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way. Program summary Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Unusual features of the program: None

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“…On the theoretical side various diverse methods for the solution of the GPE have been developed. Amongst them, some based on mathematical theorems [12,13], others based on spectral expansions [14], others using extensive numerical methods [15], and others that also include the interaction of the BEC atoms with the surrounding non-condensed atomic medium [16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

Rawitscher¹

2013

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The Gross-Pitaevskii 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 atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.

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Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap

Cited by 55 publications

References 38 publications

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation

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