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arXiv:1910.07001 · Journal of Physics B Atomic Molecular and Optical Physics

Quasi-one-dimensional approximation for Bose–Einstein condensates transversely trapped by a funnel potential

Mateus C. P. dos Santos, Boris A. Malomed, Wesley Bueno Cardoso

Published Oct 21, 2019• 10 citations
arXivHTMLPDF

Abstract

Abstract Starting from the standard three-dimensional (3D) Gross–Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schrödinger equation (1D-NPSE) governing the axial dynamics of atomic Bose–Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e. an attractive 2D potential ∼−1/ r (where r is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential’s singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.

DOI
10.1088/1361-6455/ab4fb7

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Abstract Starting from the standard three-dimensional (3D) Gross–Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schrödinger equation (1D-NPSE) governing the axial dynamics of atomic Bose–Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e. an attractive 2D potential ∼−1/ r (where r is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential’s singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.
We derive an effective nonpolynomial Schr\"odinger equation (NPSE) for self-repulsive or attractive BEC in the nearly one-dimensional cigar-shaped trap, with the transverse confining frequency periodically modulated along the axial direction. In addition to the usual linear cigar-shaped trap, where the periodic modulation emulates the action of an optical lattice (OL), the model may be also relevant to toroidal traps, where an ordinary OL cannot be created. For either sign of the nonlinearity, extended and localized states are found, in the numerical form [using both the effective NPSE and the full three-dimensional (3D) Gross-Pitaevskii equation] and by means of the variational approximation (VA). The latter is applied to construct ground-state solitons and predict the collapse threshold in the case of self-attraction. It is shown that numerical solutions provided by the one-dimensional NPSE are always very close to full 3D solutions, and the VA yields quite reasonable results too. The transition from delocalized states to gap solitons, in the first finite bandgap of the linear spectrum, is examined in detail, for the repulsive and attractive nonlinearities alike.
We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective two-dimensional (2D) nonpolynomial Schr\"odinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict the density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charges in various planar trapping configurations, including the axisymmetric harmonic confinement and one-dimensional periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charges 2 or 3 in the self-attractive BEC.
Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.
This work presents a dimensional reduction of Bose-Einstein condensates confined by generalized transverse potentials, parametrized by an exponent n . Starting from the three-dimensional Gross-Pitaevskii equation, we employ a variational ansatz to derive an effective one-dimensional nonpolynomial Schrödinger equation, which self-consistently determines the transverse width dynamics. The model generalizes existing formalisms for harmonic and funnel-shaped transverse confinement potentials. We validate the approach through comprehensive numerical tests, demonstrating excellent agreement with full 3D simulations for ground-state properties across various interaction regimes. Finally, real-time simulations of matter-wave scattering at potential barriers verify the model’s dynamical robustness, successfully replicating the spatiotemporal evolution and energy-dependent transmission characteristics observed in full 3D calculations.
Abstract One‐dimensional nonlinear Schrödinger equations are derived to describe the axial effective dynamics of cigar‐shaped atomic repulsive Bose‐Einstein condensates trapped with anharmonic transverse potentials. The accuracy of these equations in the perturbative, Thomas‐Fermi, and crossover regimes were verified numerically by comparing the ground‐state profiles, transverse chemical potentials and oscillation patterns with those results obtained for the full three‐dimensional Gross‐Pitaevskii equation. This procedure allows us to derive different patterns of 1D nonlinear models by the control of the transverse confinement even in the presence of an axial vorticity.
We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.