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Laser Physics

W-shaped solitons in inhomogeneous cigar-shaped Bose–Einstein condensates with repulsive interatomic interactions

Houria Triki, Qin Zhou, Wenjun Liu

Published Apr 9, 2019• 12 citations
Publisher

Abstract

Abstract We analyze the formation and dynamics of W-shaped solitons in an inhomogeneous cigar-shaped Bose–Einstein condensate with repulsive interatomic interactions in the presence of spatially and temporally nonuniform nonlinearity and external potential. The condensate is assumed to have a large number of particles; hence it may be approximated by an effective one-dimensional equation, which is derived from the full three-dimensional Gross–Pitaevskii equation. New classes of W-shaped soliton solutions, which may propagate in the condensate system with the same hump value, are found for the first time by means of the similarity transformation technique. The functional form of these localized nonlinear structures is different from the usual ones reported in the setting of optical fibers. The dynamic behaviors of the W-shaped soliton pulses under different nonlinearity and external potential management that are of particular interest for applications in Bose–Einstein condensates are analyzed.

DOI
10.1088/1555-6611/ab0a69

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Abstract We analyze the formation and dynamics of W-shaped solitons in an inhomogeneous cigar-shaped Bose–Einstein condensate with repulsive interatomic interactions in the presence of spatially and temporally nonuniform nonlinearity and external potential. The condensate is assumed to have a large number of particles; hence it may be approximated by an effective one-dimensional equation, which is derived from the full three-dimensional Gross–Pitaevskii equation. New classes of W-shaped soliton solutions, which may propagate in the condensate system with the same hump value, are found for the first time by means of the similarity transformation technique. The functional form of these localized nonlinear structures is different from the usual ones reported in the setting of optical fibers. The dynamic behaviors of the W-shaped soliton pulses under different nonlinearity and external potential management that are of particular interest for applications in Bose–Einstein condensates are analyzed.
The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.
We investigate dynamical effects of a bright soliton in Bose–Einstein condensed (BEC) systems with local and smooth space variations of the two-body atomic scattering length. It includes a discussion about the possible observation of a new type of standing nonlinear atomic matter wave in cigar-type traps. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. By considering an analytical time-dependent variational approach and also full numerical simulation of one-dimensional and three-dimensional Gross–Pitaevskii equations, we study processes such as trapping, reflection and transmission of the bright matter soliton due to the impurity. We also derive conditions for the collapse of the bright solitary wave, considering a quasi-one-dimensional BEC with attractive local inhomogeneity.
We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.
Abstract In this paper, we theoretically investigate the dynamics and stability of matter‐wave solitons (MWS) in a cigar‐shaped Bose–Einstein condensate (BEC) consisting of atoms with attractive interatomic interactions trapped by Pöschl–Teller (PT) potential. In this scenario, we propose the idea of the BEC being dragged by PT potential at a constant speed. The BEC was modeled by the quasi‐one‐dimensional Gross–Piteaevskii Equation (1D‐GPE). In addition, an analytical (time‐dependent variational method) and numerical (split‐step Crank–Nicolson method) approach has been proposed to investigate the dynamic properties of solitons during the temporal evolution of this system. Both analytical and numerical results demonstrated that MWSs dragged by PT potential can be stable. From a general point of view, the results obtained in this paper can motivate both theoretical and experimental investigations on the dynamics and stability of solitons in dilute and ultracold unitary Bose gases trapped by PT potential, nonlinear optical properties in PT quantum wells, and propagation of solitons in nonlinear media.
By numerical and variational analysis of the three-dimensional Gross–Pitaevskii equation, we study the formation and dynamics of bright and vortex-bright solitons in a cigar-shaped dipolar Bose–Einstein condensate for large repulsive atomic interactions. A phase diagram showing the region of stability of the solitons is obtained. We also study the dynamics of breathing oscillation of the solitons as well as the collision dynamics of two solitons. At large velocities the frontal collision is elastic and the two three-dimensional solitons pass through each other undeformed. Two solitons placed side by side at rest coalesce to form a stable bound soliton molecule due to dipolar attraction. Movie clips illustrating collision and molecule-formation dynamics of two bright and vortex-bright solitons are included.
We investigate the profiles and stability of the localized nonlinear matter waves in a Bose–Einstein condensate with spatially inhomogeneous two- and three-body interactions in various external potentials both analytically and numerically. Several families of localized stationary solutions of the one-dimensional mean-field Gross–Pitaevskii equation with cubic–quintic nonlinearities are obtained by similarity transformation. It is shown that the whole Bose–Einstein condensate can have an infinite number of localized nonlinear matter waves determined by certain non-negative integers. The linear and dynamical stabilities of the localized stationary solutions are studied numerically, and some stable localized nonlinear matter waves are found. We demonstrate that the attractive inhomogeneous two- and three-body interactions support linearly and dynamically stable localized nonlinear matter waves, but attractive two-body interaction along with repulsive three-body interaction destroys the linear stability. Moreover, an anharmonic potential is proposed to stabilize the localized nonlinear matter waves in the Bose–Einstein condensate with spatially inhomogeneous interactions.