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arXiv:1111.2270 · Physical Review A

Statics and dynamics of atomic dark-bright solitons in the presence of impurities

Vassos Achilleos, P. G. Kevrekidis, Vassilis M. Rothos, D. J. Frantzeskakis

Published Nov 22, 2011• 35 citations
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Abstract

Adopting a mean-field description for a two-component atomic Bose-Einstein condensate, we study the statics and dynamics of dark-bright solitons in the presence of localized impurities. We use adiabatic perturbation theory to derive an equation of motion for the dark-bright soliton center. We show that, counterintuitively, an attractive (repulsive) delta-like impurity, acting solely on the bright-soliton component, induces an effective localized barrier (well) in the effective potential felt by the soliton; this way, dark-bright solitons are reflected from (transmitted through) attractive (repulsive) impurities. Our analytical results for the small-amplitude oscillations of solitons are found to be in good agreement with results obtained via a Bogoliubov--de Gennes analysis and direct numerical simulations.

DOI
10.1103/physreva.84.053626

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Adopting a mean-field description for a two-component atomic Bose-Einstein condensate, we study the statics and dynamics of dark-bright solitons in the presence of localized impurities. We use adiabatic perturbation theory to derive an equation of motion for the dark-bright soliton center. We show that, counterintuitively, an attractive (repulsive) delta-like impurity, acting solely on the bright-soliton component, induces an effective localized barrier (well) in the effective potential felt by the soliton; this way, dark-bright solitons are reflected from (transmitted through) attractive (repulsive) impurities. Our analytical results for the small-amplitude oscillations of solitons are found to be in good agreement with results obtained via a Bogoliubov--de Gennes analysis and direct numerical simulations.
Abstract We develop a direct perturbation theory for dark-bright solitons and derive evolution equations for the soliton parameters. In particular, first the linearization equation around the solitons is solved by expanding its solution into a set of complete eigenfunctions of the linearization operator. Then, suppression of secular growth in the linearized solution leads to the evolution equations of soliton parameters. The results are applied to a number of case examples motivated by the physics of atomic Bose-Einstein condensates, where dark-bright solitons have recently been studied both in theory and in experiments. We thus consider perturbations corresponding to (a) finite temperature-induced thermal losses, and (b) the presence of localized ( δ -function) impurities. In these cases, relevant equations of motion for the dark-bright soliton center are in agreement with ones previously obtained via alternative methods, including energy-based methods, as well as numerical linear stability analysis and direct simulations.
Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton ``molecules,'' composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark- and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. This equation provides equilibrium positions and characteristic oscillation frequencies of the solitons, which are found to be in good agreement with the eigenfrequencies of the anomalous modes of the system.
In this thesis, we have studied the behavior of two-component dark-bright solitons in multicomponent Bose-Einstein condensates (BECs) analytically and numerically in different situations. We utilized various analytical methods including the variational method and perturbation theory. By imprinting a linear phase on the bright component only, we were able to impart a velocity relative to the dark component and thereby we obtain an internal oscillation between the two components. We find that there are two modes of the oscillation of the dark-bright soliton. The first one is the famous Goldstone mode. This mode represents a moving dark-bright soliton without internal oscillation and is related to the continuous translational symmetry of the underlying equations of motion in the uniform potential. The second mode is the oscillation of the two components relative to each other. We compared the results obtained from the variational method with numerical simulations and found that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments. Also, we studied the binding energy and found a critical value for the breakup of the dark-bright soliton. Building on these results, we have studied another situation where we have the dark-bright soliton oscillate in a harmonic potential. We found for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. We found this result is robust against noise in the initial condition and should, therefore, be experimentally observable. In addition, we have studied the interaction between a moving dark-bright soliton in a uniform background with internal oscillation and a fixed impurity, modeled by a delta function potential. The interaction excites different modes in the system. Our analytical model capture two of these modes: the relative oscillation between the two components, as well as the in-sync oscillation of the widths. The numerical simulations allow further internal modes like out-of-sync oscillations of the soliton widths and even shape deformations of various kinds. We identify regions in parameter space for the transmission, reflection and inelastic scattering of the dark-bright soliton by the potential barrier. We have studied the velocity of dark-bright solitons described with an ansatz that uses one center of mass variable to represent the position of the two components. We found for a dark-bright soliton the maximum velocity is limited by the relative number of atoms in the bright component as compared to the size of the hole or density notch created by the dark component. Above this critical velocity the dark-bright soliton develops internal oscillations, and eventually unbinds and breaks apart.
Abstract We study the interaction of an atomic dark–bright soliton in a two-component Bose–Einstein condensate with an impurity modeled by a delta function. In order to capture both elastic and inelastic scattering, we allow for relative oscillations between the dark and bright soliton components as well as higher order internal modes. Using a modified perturbed dynamical variational Lagrangian approximation, we develop an analytical model to capture motion of the center of the mass (elastic scattering) and dominant relative motion (inelastic). Numerical integration further incorporates higher order inelastic effects when the kinetic energy of the dark–bright soliton approaches the impurity energy. We calculate the maximum velocity for a dark–bright soliton and find it is limited to a value below the sound speed, depending on the relative number of atoms present in the bright soliton component and excavated by the dark soliton component, respectively. Above a critical value of the maximum velocity, the two components are no longer described by one center of mass variable and spontaneously develop internal oscillations, eventually breaking apart when pushed to higher velocities. This effect limits the incident kinetic energy in scattering studies and presents a smoking gun experimental signal.
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.
By using a multiple-scale method, we analytically study the effect of a localized impurity on the soliton dynamics in the Bose-Einstein condensates. It is shown that a dark soliton can be transmitted through a repulsive (or attractive) impurity, while at the position of the localized impurity the soliton can be quasitrapped by the impurity. Additionally, we find that the strength of the localized impurity has an important effect on the dark soliton dynamics. With increasing strength of the localized impurity, the amplitude of the dark soliton becomes bigger, while its width is narrower, and the soliton propagates slower.