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The Journal of Open Source Software

GPUE: Graphics Processing Unit Gross--Pitaevskii Equation solver

James Schloss, L. ORiordan

Published Dec 11, 2018• 17 citations
Publisher

Abstract

consisting of bosonic atoms that have been cooled and condensed into a single, macroscopic ground state (Fetter, 2009;Pethick & Smith, 2008).These systems can be created in an experimental laboratory and allow for the the exploration of physical phenomenon such as superfluid turbulence (Navon, Gaunt, Smith, & Hadzibabic, 2016;Roche & Barenghi, 2008;White, Anderson, & Bagnato, 2014), chaotic dynamics (Gardiner, 2002;Kyriakopoulos, Koukouloyannis, Skokos, & Kevrekidis, 2014;Zhang, 2017), and analogues of other quantum systems (Dalibard, Gerbier, Juzeliūnas, & Öhberg, 2011).Numerical simulations of BECs that directly mimic experiments are valuable to fundamental research in these areas and allow for theoretical advances before experimental validation.The dynamics of BEC systems can be found by solving the non-linear Schrödinger equation known as the Gross-Pitaevskii Equation (GPE),

DOI
10.21105/joss.01037

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consisting of bosonic atoms that have been cooled and condensed into a single, macroscopic ground state (Fetter, 2009;Pethick & Smith, 2008).These systems can be created in an experimental laboratory and allow for the the exploration of physical phenomenon such as superfluid turbulence (Navon, Gaunt, Smith, & Hadzibabic, 2016;Roche & Barenghi, 2008;White, Anderson, & Bagnato, 2014), chaotic dynamics (Gardiner, 2002;Kyriakopoulos, Koukouloyannis, Skokos, & Kevrekidis, 2014;Zhang, 2017), and analogues of other quantum systems (Dalibard, Gerbier, Juzeliūnas, & Öhberg, 2011).Numerical simulations of BECs that directly mimic experiments are valuable to fundamental research in these areas and allow for theoretical advances before experimental validation.The dynamics of BEC systems can be found by solving the non-linear Schrödinger equation known as the Gross-Pitaevskii Equation (GPE),
The Gross–Pitaevskii equation (GPE) describes the dynamics of superflows and Bose–Einstein Condensates (BEC) at very low temperature. When a truncation of Fourier modes is performed, the resulting truncated GPE (TGPE) can also describe the correct thermal behavior of a Bose gas, as long as all relevant modes are highly occupied [M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature, Phys. Rev. Lett. 87 (16) (2001) 160402]. We review some of our groupʼs recent GPE- and TGPE-based numerical studies of superfluid dynamics and BEC stability. The relations with experiments are discussed.
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.
Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.