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arXiv:2012.11487 · Physical Review A

Stability analysis and attractor dynamics of three-dimensional dark solitons with localized dissipation

Christian Baals, Alexandre Gil Moreno, Jian Jiang, Jens Benary, Herwig Ott

Published Apr 2, 2021• 12 citations
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Abstract

We study the stability and the attractor dynamics of an elongated Bose-Einstein condensate (BEC) with dark or gray kink solitons in the presence of localized dissipation. To this end, the three-dimensional Gross--Pitaevskii equation with an additional imaginary potential is solved numerically. We analyze the suppression of the snaking instability in dependence of the dissipation strength and extract the threshold value for the stabilization of the dark soliton for experimentally realistic parameters. Below the threshold value, we observe the decay into a solitonic vortex. Above the stabilization threshold, we observe the attractor dynamics towards the dark soliton when initially starting from a gray soliton. We find that for all initial conditions the dark soliton is the unique steady-state of the system---even when starting from the BEC ground state.

DOI
10.1103/physreva.103.043304

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We study the stability and the attractor dynamics of an elongated Bose-Einstein condensate (BEC) with dark or gray kink solitons in the presence of localized dissipation. To this end, the three-dimensional Gross--Pitaevskii equation with an additional imaginary potential is solved numerically. We analyze the suppression of the snaking instability in dependence of the dissipation strength and extract the threshold value for the stabilization of the dark soliton for experimentally realistic parameters. Below the threshold value, we observe the decay into a solitonic vortex. Above the stabilization threshold, we observe the attractor dynamics towards the dark soliton when initially starting from a gray soliton. We find that for all initial conditions the dark soliton is the unique steady-state of the system---even when starting from the BEC ground state.
The dissipative dynamic stability is investigated of dark solitons in elongated Bose–Einstein condensates that can be described by the Gross–Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear Schrodinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.
Solitons are striking manifestations of nonlinearity, encountered in diverse physical systems such as water waves, nonlinear optics, and Bose-Einstein condensates (BECs). In BECs, dark solitons emerge as exact stationary solutions of the one-dimensional Gross-Pitaevskii equation. While they can be long-lived in elongated traps, their stability is compromised in higher dimensions due to the snake instability, which leads to the decay of the soliton into vortex structures among other excitations. We investigate the dynamics of a dark soliton in a Bose-Einstein condensate confined in an anisotropic harmonic trap. Using a variational ansatz that incorporates both the transverse bending of the soliton plane and the emergence of vortices along the nodal line, we derive equations of motion governing the soliton's evolution. This approach allows us to identify stable oscillation modes as well as the growth rates of the unstable perturbations. In particular, we determine the critical trap anisotropy required to suppress the snake instability. Our analytical predictions are in good agreement with full numerical simulations of the Gross-Pitaevskii equation.
The stability of nonlinear waves on curved surfaces is a problem of widespread interest across physics. Here, we establish the stability criteria for dark solitons on a spherical Bose-Einstein condensate. We demonstrate a sharp instability threshold in the nonlinear parameter, beyond which solitons decay into vortex dipoles via snake instabilities. Analytically and numerically, we prove this decay is dictated by a single unstable mode for each angular momentum m≥2m \geq 2m≥2, which is a universal mechanism that controls the resulting vortex state. Unlike in the full three-dimensional case, where snake instabilities lead to vortex rings, a dark soliton confined to the surface of a bubble can only decay into vortex pairs.
We report on the existence and stability of multidimensional bound solitonic states in harmonically trapped scalar Bose-Einstein condensates. Their equilibrium separation, as a measure of the strength of the soliton-soliton or the solitonic vortex-vortex interaction, is provided for varying chemical potential $\ensuremath{\mu}$. Static bound dark solitons are shown to be dynamically stable in elongated condensates within a range of intermediate (repulsive) interparticle-interaction strength. Beyond this range the snaking instability manifests during the time evolution of the planar solitons and produces the decay into nonstationary vortex states. A subsequent dynamical recurrence of solitons and vortices can be observed at low $\ensuremath{\mu}$. At equilibrium, the bifurcations of bound dark solitons are bound solitonic vortices. Among them, both two-open and two-ring vortex lines are demonstrated to exist with both counter- and co-rotating steady velocity fields. The latter flow configurations evolve, for high chemical potential, into a stationary three-dimensional (3D)-chain-shaped vortex and a three vortex-antivortex-vortex ring sequence that arrest the otherwise increasing angular or linear momentum respectively. As a feature common to the bifurcated families of vortex states, their excitation spectra present unstable modes with associated oscillatory dynamics. In spite of this, the family of two-open counter-rotating vortices support dynamically stable 3D states.
Here we revisit the topic of stationary and propagating solitonic excitations in self-repulsive three-dimensional (3D) Bose-Einstein condensates by quantitatively comparing theoretical analysis and associated numerical computations with our experimental results. Motivated by numerous experimental efforts, including our own herein, we use fully 3D numerical simulations to explore the existence, stability, and evolution dynamics of planar dark solitons. This also allows us to examine their instability-induced decay products including solitonic vortices and vortex rings. In the trapped case and with no adjustable parameters, our numerical findings are in correspondence with experimentally observed coherent structures. Without a longitudinal trap, we identify numerically exact traveling solutions and quantify how their transverse destabilization threshold changes as a function of the solitary wave speed.