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arXiv:2106.14115 · Physical Review A

Dark solitons in a trapped gas of long-range interacting bosons

Mathieu Beau, Adolfo del Campo, D. J. Frantzeskakis, Theodoros P. Horikis, P. G. Kevrekidis

Published Feb 22, 2022• 2 citations
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Abstract

We consider the interplay of repulsive short-range and same-sign long-range interactions in the dynamics of dark solitons, as prototypical coherent nonlinear excitations in a trapped one-dimensional Bose gas. First, the form of the ground state is examined, and then both the existence of the solitary waves and their stability properties are explored, and corroborated by direct numerical simulations. We find that single- and multiple-dark-soliton states can exist and are generically robust in the presence of long-range interactions. We analyze the modes of vibration of such excitations and find that their respective frequencies are significantly upshifted as the strength of the long-range interactions is increased. Indeed, we find that a prefactor of the long-range interactions considered comparable to the trap strength may upshift the dark soliton oscillation frequency by an order of magnitude, in comparison to the well established one of $\mathrm{\ensuremath{\Omega}}/\sqrt{2}$ in a trap of frequency $\mathrm{\ensuremath{\Omega}}$.

DOI
10.1103/physreva.105.023323

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We consider the interplay of repulsive short-range and same-sign long-range interactions in the dynamics of dark solitons, as prototypical coherent nonlinear excitations in a trapped one-dimensional Bose gas. First, the form of the ground state is examined, and then both the existence of the solitary waves and their stability properties are explored, and corroborated by direct numerical simulations. We find that single- and multiple-dark-soliton states can exist and are generically robust in the presence of long-range interactions. We analyze the modes of vibration of such excitations and find that their respective frequencies are significantly upshifted as the strength of the long-range interactions is increased. Indeed, we find that a prefactor of the long-range interactions considered comparable to the trap strength may upshift the dark soliton oscillation frequency by an order of magnitude, in comparison to the well established one of \ensuremathΩ/2\mathrm{\ensuremath{\Omega}}/\sqrt{2}\ensuremathΩ/2​ in a trap of frequency \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ.
We study the one-dimensional dynamics of dark-dark solitons in the miscible regime of two density-coupled Bose-Einstein condensates having repulsive interparticle interactions within each condensate (g>0g>0g>0). By using an adiabatic perturbation theory in the parameter g12/g{g}_{12}/gg12​/g, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive g12>0{g}_{12}>0g12​>0. As a result, the relative motion of dark solitons with equal chemical potential $\ensuremath{\mu}$ is well approximated by harmonic oscillations of angular frequency ${w}_{r}=(\ensuremath{\mu}/\ensuremath{\hbar})\sqrt{(8/15){g}_{12}/g}$. We also show that, in finite systems, the resonance of this anomalous excitation mode with the spin-density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter. In the presence of harmonic trapping (with angular frequency $\mathrm{\ensuremath{\Omega}}$) the solitons are driven by the superposition of two harmonic motions at a frequency given by ${w}^{2}={(\mathrm{\ensuremath{\Omega}}/\sqrt{2})}^{2}+{w}_{r}^{2}$. When g12<0{g}_{12}<0g12​<0, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them.
We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective high-order Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg--de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.
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Abstract: In this thesis we experimentally observe the dynamics of dark solitons in elongated Bose-Einstein condensates (BEC) in the crossover regime between 1D and 3D. Dark solitons are dynamically stable in this regime. This allows the observation of the collisional and oscillational dynamics of two dark solitons in a harmonically confined BEC over several oscillation periods. We can detect the repulsive intersolitonic interaction as an increase of the oscillation frequency of the two solitons strongly depending on their mean distance. Thereby we give the first experimental evidence of the theoretically predicted repulsive interaction between the solitons. We compare our experimental findings to numerical simulations of the Gross-Pitaevskii equation and effective 1D equations. Additionally we develop a model in which the solitons are approximated by repulsively interacting classical particles. This model is based on a soliton interaction potential suggested by [Kiv1995] and can describe the essentials of the soliton dynamics. We observe well agreement between the soliton oscillation frequencies measured in the experiment, the numerical simulations and the results of the particle model. Besides we verify the theoretical prediction that the oscillation frequency of dark solitons in a harmonically confined BEC should be slower than the trap frequency.
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.