A time-splitting pseudospectral method for the solution of the Gross–Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

Salman, Hayder

doi:10.1016/j.jcp.2013.10.009

Cited by 7 publications

(3 citation statements)

References 52 publications

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“…It should be mentioned here that the use of spherical harmonics as a basis in the angular vari-ables for states bearing radial symmetry is rather common in terms of numerical schemes for linear and nonlinear Schrödinger (NLS) equations -see, e.g., Ref. [34] for a a recent example. Earlier efforts along these lines in the context of linear Schrödinger equations can be found, e.g., in the works of Refs.…”

Section: Introductionmentioning

confidence: 99%

Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

et al. 2016

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In this work we study spherical shell dark soliton states in three-dimensional atomic Bose-Einstein condensates. Their symmetry is exploited in order to analyze their existence, as well as that of topologically charged variants of the structures, and, importantly, to identify their linear stability Bogolyubov-de Gennes spectrum. We compare our effective 1D spherical and 2D cylindrical computations with the full 3D numerics. An important conclusion is that such spherical shell solitons can be stable sufficiently close to the linear limit of the isotropic condensates considered herein. We have also identified their instabilities leading to the emergence of vortex line and vortex ring cages. In addition, we generalize effective particle pictures of lower dimensional dark solitons and ring dark solitons to the spherical shell solitons concerning their equilibrium radius and effective dynamics around it. In this case too, we favorably compare the resulting predictions such as the shell equilibrium radius, qualitatively and quantitatively, with full numerical solutions in 3D.

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Section: Introductionmentioning

confidence: 99%

Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

et al. 2016

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“…In narrow ring traps (and when g 11 = g 22 ), azimuthal phase separation is favoured (see for instance [5,20,21]), while in wider toroidal traps concentric ring configurations can occur (see for instance [20,21]). In the following we numerically solve the coupled GP equations by applying a pseudo-spectral second order Strang method with symmetric three-operator splitting [22].…”

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confidence: 99%

Emergence of classical rotation in superfluid Bose-Einstein condensates

2016

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Phase transitions can modify quantum behaviour on mesoscopic scales and give access to new and unusual quantum dynamics. Here we investigate the superfluid properties of a rotating twocomponent Bose-Einstein condensate as a function of changes in the interaction energy and in particular through the phase transition from miscibility to immiscibility. We show that the breaking of one of the hallmarks of superfluid flow, namely the quantisation condition on circulation, is continuous throughout an azimuthal phase separation process and displays intriguing density dynamics. We find that the resulting currents are stable for long times and possess a phase boundary that exhibits classical solid body rotation, despite the quantum nature of superfluid flow. To support this co-existence of classical and quantum behaviour the system develops a unique velocity flow profile, which includes unusual radial flow in regions near the phase boundary.Phase transitions in quantum systems can have a dramatic impact on the quantum mechanical behaviour on mesoscopic scales. Superfluidity in Bose-condensed gases is a mesoscopic manifestation of quantum mechanical effects and one of its hallmarks is the existence of quantised flow around phase singularities as a response to external rotation [1][2][3][4]. However, as the quantisation condition arises from the requirement of the single-valuedness of the wavefunction, an interesting, and less well investigated, generalization appears in superfluids composed of several components. In these systems, due to the interplay of intra-and inter-component interactions, the spinor order parameter can undergo a phase transition that modifies the global symmetry of the system. As the quantisation condition applies to each component independently, the path along which circulation is determined consequently depends on the presence of the other component. This has proven to be particularly striking in toroidally trapped binary mixtures of BECs, where immiscibility can drive a transition to azimuthal phase separation, breaking the requirement of quantised circulation around the toroid [5]. Below we show that this transition is continuous and leads to a phase boundary, which rotates as a classical solid body. While this might seem at first to be incompatible with the quantum nature of superfluid flow, this co-existence can be explained through the presence of a radial flow.In superfluids the circulation around a closed path p is quantised according to p v · dr = n2π /m . Here n is an integer winding number, m the atomic mass, the reduced Planck constant, and the superfluid velocity field, v = ∇θ/m, is completely determined by the gradient of the condensate phase, θ. This implies the velocity field of a vortex has a tangential 1/r velocity profile, in contrast to classical rigid-body rotation, where v = Ω × r. The creation of vortices is a response to external rotation and depends, in particular, on the confining geometry. While in simply connected trapping potentials vortices with higher winding numbers a...

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“…The numerical solutions of equation (28) for various values of α were obtained using SSFM [32]. This method has proven to be a reliable approximation for the GPE [33][34][35]. The SSFM involves discretizing space and time and then alternating between Fourier transformation and applying evolution operators.…”

Section: Numerical Analysismentioning

confidence: 99%

One dimensional Bose–Einstein condensate under the effect of the extended uncertainty principle

Benhadjira,

Benkrane,

Bentouila

et al. 2024

Phys. Scr.

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In this study, an analytical investigation was conducted to assess the effects of the extended uncertainty principle (EUP) on a Bose-Einstein condensate (BEC) described by the deformed one-dimensional Gross-Pitaevskii equation (GPE). Analytical solutions were derived for null potential, while, we used variational and numerical methods for harmonic oscillator potential. Subsequently, we analyzed the probability density, position, and momentum uncertainties as functions of the deformation parameter α

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A time-splitting pseudospectral method for the solution of the Gross–Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

Cited by 7 publications

References 52 publications

Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

Emergence of classical rotation in superfluid Bose-Einstein condensates

One dimensional Bose–Einstein condensate under the effect of the extended uncertainty principle

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