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arXiv:0809.2216 · Physical Review A

Matter-wave dark solitons in a double-well potential

Ryoko Ichihara, Ippei Danshita, Tetsuro Nikuni

Published Dec 3, 2008• 15 citations
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Abstract

We study stability of the first excited state of quasi-one-dimensional Bose-Einstein condensates in a double-well potential, which is called ``$\ensuremath{\pi}$ state.'' The density notch in the $\ensuremath{\pi}$ state can be regarded as a standing dark soliton. From the excitation spectrum, we determine the critical barrier height, above which the $\ensuremath{\pi}$ state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the $\ensuremath{\pi}$ state by solving the time-dependent Gross-Pitaevskii equation. We show that due to the dynamical instability the dark soliton starts to move away from the trap center and exhibits a large-amplitude oscillation.

DOI
10.1103/physreva.78.063604

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We study stability of the first excited state of quasi-one-dimensional Bose-Einstein condensates in a double-well potential, which is called ``\ensuremathπ\ensuremath{\pi}\ensuremathπ state.'' The density notch in the \ensuremathπ\ensuremath{\pi}\ensuremathπ state can be regarded as a standing dark soliton. From the excitation spectrum, we determine the critical barrier height, above which the \ensuremathπ\ensuremath{\pi}\ensuremathπ state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the \ensuremathπ\ensuremath{\pi}\ensuremathπ state by solving the time-dependent Gross-Pitaevskii equation. We show that due to the dynamical instability the dark soliton starts to move away from the trap center and exhibits a large-amplitude oscillation.
We investigate the stability of the first excited state, the so-called “π-state,” of Bose-Einstein condensates in a double-well potential. From the condition of complex excitation energies, we determine the critical barrier height, above which the π-state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the π-state by solving the time-dependent Gross-Pitaevskii equation. Our simulation results show that the π-state in the dynamically unstable region exhibits distinctively different behavior from that in the dynamically stable region.
We study the statics and dynamics of dark solitons in a cigar-shaped Bose-Einstein condensate confined in a double-well potential. Using a mean-field model with a noncubic nonlinearity, appropriate to describe the dimensionality crossover regime from one- to three-dimensional, we obtain branches of solutions in the form of single and multiple dark soliton states, and study their bifurcations and stability. It is demonstrated that there exist dark soliton states which do not have a linear counterpart and we highlight the role of anomalous modes in the excitation spectra. Particularly, we show that anomalous mode eigenfrequencies are closely connected to the characteristic soliton frequencies as found from the solitons' equations of motion, and how anomalous modes are related to the emergence of instabilities. We also analyze in detail the role of the height of the barrier in the double-well setting, which may lead to instabilities or decouple multiple dark soliton states.
We study the stability, form, and interaction of single and multiple dark solitons in quasi-one-dimensional dipolar Bose-Einstein condensates. The solitons are found numerically as stationary solutions in the moving frame of a nonlocal Gross Pitaevskii equation and characterized as a function of the key experimental parameters, namely the ratio of the dipolar atomic interactions to the van der Waals interactions, the polarization angle, and the condensate width. The solutions and their integrals of motion are strongly affected by the phonon and roton instabilities of the system. Dipolar matter-wave dark solitons propagate without dispersion and collide elastically away from these instabilities, with the dipolar interactions contributing an additional repulsion or attraction to the soliton-soliton interaction. However, close to the instabilities, the collisions are weakly dissipative.
Dark soliton states of Bose-Einstein condensates in harmonic traps are studied both analytically and computationally by the direct solution of the Gross-Pitaevskii equation in three dimensions. The ground and self-consistent excited states are found numerically by relaxation in imaginary time. The energy of a stationary soliton in a harmonic trap is shown to be independent of density and geometry for large numbers of atoms. Large-amplitude field modulation at a frequency resonant with the energy of a dark soliton is found to give rise to a state with multiple vortices. The Bogoliubov excitation spectrum of the soliton state contains complex frequencies, which disappear for sufficiently small numbers of atoms or large transverse confinement. The relationship between these complex modes and the snake instability is investigated numerically by propagation in real time.
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 study the dynamics of matter waves in an effectively one-dimensional Bose-Einstein condensate in a double well potential. We consider in particular the case when one of the double wells confines excited states. Similarly to the known ground state oscillations, the states can tunnel between the wells experiencing the physics known for electrons in a Josephson junction, or be self-trapped. As the existence of dark solitons in a harmonic trap are continuations of such non-ground state excitations, one can view the Josephson-like oscillations as tunnelings of dark solitons. Numerical existence and stability analysis based on the full equation is performed, where it is shown that such tunneling can be stable. Through a numerical path following method, unstable tunneling is also obtained in different parameter regions. A coupled-mode system is derived and compared to the numerical observations. Regions of (in)stability of Josephson tunneling are discussed and highlighted. Finally, we outline an experimental scheme designed to explore such dark soliton dynamics in the laboratory.