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

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

Phys. Rev. A 103, 043304 – Published 2 April 2021

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.

Received 22 December 2020
Accepted 18 March 2021

DOI:https://doi.org/10.1103/PhysRevA.103.043304

Physics Subject Headings (PhySH)

Bose gases Bose-Einstein condensates Decoherence in quantum gases

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Christian Baals ^1,2, Alexandre Gil Moreno ¹, Jian Jiang¹, Jens Benary ¹, and Herwig Ott ¹

¹Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany
²Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 103, Iss. 4 — April 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Iso-surface plots of the DS without dissipation after different times. We show the density at 10% of its maximum as contours and the phase as color code. (top) Initial DS at $t = 0$ ms. After 9 ms the central plane of the DS starts to bend. This is the beginning of the snaking instability. After 20 ms the VR appears as an intermediate stationary state. Being unstable itself (see the bending and shift after 40 ms), it eventually decays into a solitonic vortex (100 ms), which is dynamically stable.
Reuse & Permissions
Figure 2
(a) Number of particles $N$ and (b) loss rate $| \dot{N} |$ over time for characteristic dissipation strengths $γ_{0} = 0 kHz$ (blue dashed line), $γ_{0} = 1.2 kHz$ (orange solid line), $γ_{0} = 4.7 kHz$ (green dash-dotted line) and $γ_{0} = 7.1 kHz$ (red dotted line). The local maximum of $\dot{N}$ at 83 ms is due to a density wave that is created due to the finite extent of the dissipative potential [6]. It travels towards the edge of the BEC gets reflected there and returns towards the imaginary potential. (c) Number of particles $N$ and (d) loss rate $| \dot{N} |$ at $t = 0 ms$ (blue points) and at $t = 100 ms$ (red diamonds).
Reuse & Permissions
Figure 3
Analysis of the decay of the DS. (a) Position of the density minimum at $t = 0 ms$ (blue points) and $t = 100 ms$ (red diamonds) for different $γ$ . For large $γ$ , the dark soliton is stable and the density minimum does not move. For small $γ$ , the final structure (VR or SV) has moved away from the center. (b) Time evolution of the radially integrated density in the density minimum for characteristic dissipation strengths $γ_{0} = 0 kHz$ (blue dashed line), $γ_{0} = 1.2 kHz$ (orange solid line), $γ_{0} = 4.7 kHz$ (green dash-dotted line), and $γ_{0} = 7.1 kHz$ (red dotted line). The three solitary waves have distinct values of the density. (c) Radially integrated relative density in the density minimum after $t = 0 ms$ (blue points) and $t = 100 ms$ (red diamonds) for different $γ$ . As in panel (a) the initial and final values overlap above $γ_{crit}$ . (d) Exponential fit of the initial part of the dynamics from panel (b) to extract the decay time of the DS.
Reuse & Permissions
Figure 4
(a) Refilling time extracted from an exponential fit to the initial dynamics of $n_{rel}$ over dissipation $γ$ [compare Fig. 3] for $N = 80 \times 10^{3}$ particles in a radially isotropic trap. (b) Same data (blue points) rescaled by $γ_{crit} = 5893 Hz$ and plotted logarithmically together with a power-law fit (blue solid line). Additional simulations with $N = 80 \times 10^{3}$ particles in a radially anisotropic trap (orange diamonds and dashed line, $γ_{crit} = 5909 Hz$ ) and $N = 40 \times 10^{3}$ particles in a radially isotropic trap (green crosses and dotted line, $γ_{crit} = 4107 Hz$ ).
Reuse & Permissions
Figure 5
Attractor dynamics towards the dark soliton. (a) Number of particles $N (t)$ and (b) loss rate $| \dot{N} (t) |$ for different initial phase difference $Δ ϕ$ of the gray kink soliton, i.e., $Δ ϕ = 0$ (blue dashed line), $Δ ϕ = 0.2 π$ [orange lower solid line in panel (a) and orange upper solid line in panel (b)], $Δ ϕ = 0.5 π$ (green dash-dotted line), $Δ ϕ = 0.8 π$ (red dotted line), and $Δ ϕ = π$ [violet upper solid line in panel (a) and violet lower solid line in panel (b)]. (c) Number of particles $N (t)$ and (d) loss rate $| \dot{N} (t) |$ at $t = 0 ms$ (blue points) and $t = 70 ms$ (red diamonds).
Reuse & Permissions
Figure 6
Attractor dynamics towards the dark soliton. (a) Evolution of the position of the density minimum at different initial phase differences $Δ ϕ = 0$ (blue dashed line), $Δ ϕ = 0.2 π$ (orange upper solid line), $Δ ϕ = 0.5 π$ (green dash-dotted line), $Δ ϕ = 0.8 π$ (red dotted line), and $Δ ϕ = π$ (violet lower solid line). The initial motion of the gray soliton is damped and the DS is stabilized at the central position, where the dissipation is located for all initial phase differences as shown in (b) for the position at $t = 0 ms$ (blue points) and $t = 70 ms$ (red diamonds). (c) The density in the minimum approaches zero [color code as that of panel (a)] over time and for all initial phase differences (d) [color code is the same as that of panel (b)]. (e) The phase difference across the density minimum approaches the characteristic value of $π$ of the DS [color code is the same as that of panel (a) but with the upper solid line for $Δ ϕ = π$ and the lower solid line for $Δ ϕ = 0.2 π$ ] over time and for all initial phase differences (f) [color code is the same as that of panel (b)].
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information