Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2310.18528 · Journal of Physics A Mathematical and Theoretical

Nonuniform Bose–Einstein condensate: I. An improvement of the Gross–Pitaevskii method

Maksim Tomchenko

Published Nov 12, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract A nonuniform condensate is usually described by the Gross–Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz Ψ ^ ( r , t ) = Ψ ( r , t ) . Proceeding from a more accurate operator ansatz Ψ ^ ( r , t ) = a ^ 0 Ψ ( r , t ) / N , where N is the number of Bose particles, we find the equation i ℏ ∂ Ψ ( r , t ) ∂ t = − ℏ 2 2 m ∂ 2 Ψ ( r , t ) ∂ r 2 + ( 1 − 1 N ) 2 c Ψ ( r , t ) | Ψ ( r , t ) | 2 , which we call the GP N equation. It differs from the GP equation by the factor ( 1 − 1 /

DOI
10.1088/1751-8121/ad9187

Related papers

Condensate dynamics at T = 0

Allan Griffin, Tetsuro Nikuni, E. Zaremba

Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.

Cambridge University Press eBooks•Feb 19, 2009•0 citations

Green function method and the Gross-Pitaevskii equation

Carlos A Trallero-Herrero, Victor Lopez‐Richard, Julio C. Drake-Pérez +1 more

"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."

Revista Mexicana de Física•Jan 1, 2007•0 citations

The Gross–Pitaevskii equation and Bose–Einstein condensates

J. Rogel-Salazar

The Gross–Pitaevskii equation (GPE) is discussed at the level of an advanced course on statistical physics. In the standard literature the GPE is usually obtained in the framework of the second quantization formalism, which in many cases goes beyond the material covered in many advanced undergraduate courses. In this paper, we motivate the derivation of the GPE in relationship to concepts from statistical physics, highlighting a number of applications from the dynamics of a Bose–Einstein condensate to the excitations of the gas cloud. This paper may be helpful for encouraging the discussion of modern developments in a statistical mechanics course, and can also be of use in other contexts such as mathematical physics and modelling. The paper is suitable for undergraduate and graduate students, as well as for general physicists.

arXiv:1301.2073•European Journal of Physics•Jan 9, 2013•132 citations

Bose-Einstein Condensation: The Odd Nonlinear Quantum Mechanics

Лев П. Питаевский

The non-linear Gross-Pitaevskii (GP) equation plays an important role in the mean-field theory of the Bose-Einstein condensation and has interesting mathematical properties. In this lecture we consider two nontrivial problems related to physical interpretation of the solutions of the GP equation: the superfluidity of a BEC gas in an optical lattice and the superfluidity of a 1D Bose-gas, with special attention to the role of grey solitons. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Kluwer Academic Publishers eBooks•Mar 1, 2006•1 citation

q-deformed Bose statistics and the Gross–Pitaevskii equation

Mahnaz Maleki, Zahra Ebadi, Hosein Mohammadzadeh

In continuation of our earlier work on the nonextensive form of the Gross–Pitaevskii equation (GPE) [M. Maleki, H. Mohammadzadeh and Z. Ebadi, Int. J. Geom. Methods Mod. Phys. 20 (2023) 2350216], we now delve into its [Formula: see text]-deformed counterpart. GPE is a type of nonlinear partial differential equation that is specifically designed to describe the behavior of a group of particles with Bose–Einstein statistics, such as atoms in a superfluid or a Bose–Einstein condensate (BEC). In some systems, the standard Bose–Einstein or Fermi–Dirac statistics may not apply, and generalized statistics may be needed to describe the behavior of the particles. Therefore in this paper, we investigate the dynamics of a system with particle obeying [Formula: see text]-deformed statistics described by the [Formula: see text]-deformed GPE. First, we use the oscillator algebra and [Formula: see text]-calculus to obtain the well-known Schrödinger equation. By selecting an appropriate Hamiltonian for the condensate phase and minimizing the free energy, we derive the [Formula: see text]-deformed time-independent GPE.

International Journal of Geometric Methods in Modern Physics•Dec 17, 2023•2 citations

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

Weizhu Bao, Dieter Jaksch, Peter A. Markowich

arXiv:cond-mat/0303239•Journal of Computational Physics•Apr 23, 2003•638 citations
Abstract A nonuniform condensate is usually described by the Gross–Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz Ψ ^ ( r , t ) = Ψ ( r , t ) . Proceeding from a more accurate operator ansatz Ψ ^ ( r , t ) = a ^ 0 Ψ ( r , t ) / N , where N is the number of Bose particles, we find the equation i ℏ ∂ Ψ ( r , t ) ∂ t = − ℏ 2 2 m ∂ 2 Ψ ( r , t ) ∂ r 2 + ( 1 − 1 N ) 2 c Ψ ( r , t ) | Ψ ( r , t ) | 2 , which we call the GP N equation. It differs from the GP equation by the factor ( 1 − 1 /
Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.
"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."
The Gross–Pitaevskii equation (GPE) is discussed at the level of an advanced course on statistical physics. In the standard literature the GPE is usually obtained in the framework of the second quantization formalism, which in many cases goes beyond the material covered in many advanced undergraduate courses. In this paper, we motivate the derivation of the GPE in relationship to concepts from statistical physics, highlighting a number of applications from the dynamics of a Bose–Einstein condensate to the excitations of the gas cloud. This paper may be helpful for encouraging the discussion of modern developments in a statistical mechanics course, and can also be of use in other contexts such as mathematical physics and modelling. The paper is suitable for undergraduate and graduate students, as well as for general physicists.
The non-linear Gross-Pitaevskii (GP) equation plays an important role in the mean-field theory of the Bose-Einstein condensation and has interesting mathematical properties. In this lecture we consider two nontrivial problems related to physical interpretation of the solutions of the GP equation: the superfluidity of a BEC gas in an optical lattice and the superfluidity of a 1D Bose-gas, with special attention to the role of grey solitons. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In continuation of our earlier work on the nonextensive form of the Gross–Pitaevskii equation (GPE) [M. Maleki, H. Mohammadzadeh and Z. Ebadi, Int. J. Geom. Methods Mod. Phys. 20 (2023) 2350216], we now delve into its [Formula: see text]-deformed counterpart. GPE is a type of nonlinear partial differential equation that is specifically designed to describe the behavior of a group of particles with Bose–Einstein statistics, such as atoms in a superfluid or a Bose–Einstein condensate (BEC). In some systems, the standard Bose–Einstein or Fermi–Dirac statistics may not apply, and generalized statistics may be needed to describe the behavior of the particles. Therefore in this paper, we investigate the dynamics of a system with particle obeying [Formula: see text]-deformed statistics described by the [Formula: see text]-deformed GPE. First, we use the oscillator algebra and [Formula: see text]-calculus to obtain the well-known Schrödinger equation. By selecting an appropriate Hamiltonian for the condensate phase and minimizing the free energy, we derive the [Formula: see text]-deformed time-independent GPE.