Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: Application to two-component Bose-Einstein condensates (original) (raw)

Dark soliton dynamics in confined Bose-Einstein condensates

Dilute atomic Bose-Einstein condensates are inherently nonlinear systems and support solitary wave solutions. An important distinction from optical systems is the inhomogeneous background density, which results from the traps used to confine the atoms. As in optical systems, dark solitary waves in three dimensional geometries are unstable to transverse excitations, which lead to a bending of the dark soliton plane and decay into vortex rings. Highly elongated geometries can now be achieved experimentally, in which the condensate dynamics are effectively one-dimensional, and the motion of the dark soliton is governed by the inhomogeneous longitudinal density. We show that a dark soliton is fundamentally unstable to such a changing background density, by means of numerical simulations of the soliton under various potentials (e.g. steps, ramps, harmonic traps, and optical lattices). This leads to the emission of radiation in the form of sound waves. The power emitted is found to be proportional to the square of the soliton acceleration. The latter quantity is shown to be proportional to the deformation of the apparent soliton profile, arising from the sound field in the region of the soliton. We demonstrate that the ensuing interactions between the soliton and sound field, and therefore the dynamics of the soliton, can be controlled experimentally via manipulation of the emitted sound, achieved by modifying the trap geometry. In this manner, it is possible to induce a rapid decay of the soliton, stabilise the soliton, or even pump energy into the soliton by means of parametric driving.

On creation and evolution of dark solitons in Bose-Einstein condensates

2003

Generation of dark solitons from large initial excitations and their evolution in a quasi-onedimensional Bose-Einstein condensate trapped by a harmonic potential is studied analytically and numerically. In the case of a single deep soliton main characteristics of its motion such as a frequency and amplitude of oscillations are calculated by means of the perturbation theory which in the leading order results in a Newtonian dynamics, corrections to which are computed as well. It is shown that long-time dynamics of a dark soliton in a generic situation deviates substantially from outcomes of the naive application of the Ehrenfest theorem. We also consider three different techniques of controllable creation of multi-soliton structures (soliton trains) from large initial excitations and calculate their initial parameters (depths and velocities) with the use of a generalized Bohr-Sommerfeld quantization rule. Multi-soliton effects are discussed.

Generation of dark solitons in oscillating Bose–Einstein condensates

We propose an experimentally tractable setting for observing an "instability" of a repulsive oscillating Bose-Einstein condensate that leads to the generation of dark solitons. We illustrate that when the trap of the condensate (which incorporates a localized impurity) is displaced so that the condensate flow is characterized by an atomic velocity larger than the local speed of sound, dark solitons are generated. The subcritical, near critical and supercritical are analyzed in detail. (D.J. Frantzeskakis). and the generation of bright solitons , the transverse instability of dark solitons [6] and the generation of vortices [7] and robust vortex clusters [8], the generation of Faraday patterns [9], and so on. Additionally, much attention has been paid, both on the theoretical and the experimental sides (see also the discussion and references therein), to the study of dynamical instabilities and their relation to the Landau instability (see also ). The latter instability is known to be a (effectively) "dissipative" mechanism, responsible for the breakup of superfluidity, which also results in the emission of phonon radiation from a superfluid moving with a speed larger 0375-9601/$ -see front matter 

Dynamics of shallow dark solitons in a trapped gas of impenetrable bosons

Physical Review A, 2004

The dynamics of linear and nonlinear excitations in a Bose gas in the Tonks-Girardeau (TG) regime with longitudinal confinement are studied within a mean field theory of quintic nonlinearity. A reductive perturbation method is used to demonstrate that the dynamics of shallow dark solitons, in the presence of an external potential, can effectively be described by a variable-coefficient Korteweg-de Vries equation. The soliton oscillation frequency is analytically obtained to be equal to the axial trap frequency, in agreement with numerical predictions obtained by Busch et al. [J. Phys. B 36, 2553] via the Bose-Fermi mapping. We obtain analytical expressions for the evolution of both soliton and emitted radiation (sound) profiles.

Dynamics of ring dark solitons in Bose–Einstein condensates and nonlinear optics

Physics Letters A, 2010

Quasiparticle approach to dynamics of dark solitons is applied to the case of ring solitons. It is shown that the energy conservation law provides the effective equations of motion of ring dark solitons for general form of the nonlinear term in the generalized nonlinear Schrödinger or Gross-Pitaevskii equation. Analytical theory is illustrated by examples of dynamics of ring solitons in light beams propagating through a photorefractive medium and in non-uniform condensates confined in axially symmetric traps. Analytical results agree very well with the results of our numerical simulations.

Two-dimensional solitons in Bose-Einstein condensates with a disk-shaped trap

Physical Review A, 2003

We consider, both analytically and numerically, the evolution of two-dimensional ͑2D͒ nonlinear matterwave pulses in a Bose-Einstein condensate with a disk-shaped trap and repulsive atom-atom interactions. Due to the strong confinement in the axial direction the sound speed of the system is cϭ(1/2 1/4)c 0 , where c 0 is the corresponding value without the trap. From the 3D order-parameter equation of the condensate, we derive a soliton-bearing Kadomtsev-Petriashvili equation with positive dispersion. When the trapping potential is weak in two transverse directions, a low-depth plane dark soliton can propagate in the condensate with a changing profile but preserving its structure down to the boundary of the condensate. We show that high-depth plane dark solitons are unstable to long-wavelength transverse disturbances. The instability appears as a longitudinal modulation of the soliton amplitude decaying into vortices. We also show how a dark lumplike 2D nonlinear excitation can be excited in the system. Furthermore, a dark lump decaying algebraically in two spatial directions can propagate rather stable in the condensate, but disappears near the boundary of the condensate where two vortices are nucleated. The vortices move in opposite directions along the boundary and when meeting merge creating a new lump. Finally, we also provide results for head-on and oblique collisions of two lumps in the system.

Dark soliton dynamics in spatially inhomogeneous media: Application to Bose–Einstein condensates

Mathematics and Computers in Simulation, 2005

We study the dynamics of dark solitons in spatially inhomogeneous media with applications to cigar-shaped Bose-Einstein condensates trapped in a harmonic magnetic potential and a periodic potential representing an optical lattice. We distinguish and systematically investigate the cases with the optical lattice period being smaller, larger, or comparable to the width of the dark soliton. Analytical results, based on perturbation techniques, for the motion of the dark soliton are obtained and compared to direct numerical simulations. Radiation effects are also considered. Finally, we demonstrate that a moving optical lattice may capture and drag a dark soliton.

Motion of Dark Solitons in Trapped Bose-Einstein Condensates

Physical Review Letters, 2000

We use a multiple time scale boundary layer theory to derive the equation of motion for a dark (or 'grey') soliton propagating through an effectively one-dimensional cloud of Bose-Einstein condensate, assuming only that the background density and velocity vary slowly on the soliton scale. We show that solitons can exhibit viscous or radiative acceleration (anti-damping), which we estimate as slow but observable on experimental time scales.

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Nonlinearity, 2008

The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools. Recall that the underlying model, namely the completely integrable NLS equation, has infinitely many symmetries, including translational and Galilean invariances.