Dark solitons in a trapped gas of long-range interacting bosons

M. Beau, A. del Campo, D. J. Frantzeskakis, T. P. Horikis, and P. G. Kevrekidis

Phys. Rev. A 105, 023323 – Published 22 February 2022

Abstract

We consider the interplay of repulsive short-range and same-sign long-range interactions in the dynamics of dark solitons, as prototypical coherent nonlinear excitations in a trapped one-dimensional Bose gas. First, the form of the ground state is examined, and then both the existence of the solitary waves and their stability properties are explored, and corroborated by direct numerical simulations. We find that single- and multiple-dark-soliton states can exist and are generically robust in the presence of long-range interactions. We analyze the modes of vibration of such excitations and find that their respective frequencies are significantly upshifted as the strength of the long-range interactions is increased. Indeed, we find that a prefactor of the long-range interactions considered comparable to the trap strength may upshift the dark soliton oscillation frequency by an order of magnitude, in comparison to the well established one of $Ω / \sqrt{2}$ in a trap of frequency $Ω$ .

Received 2 July 2021
Revised 4 November 2021
Accepted 28 January 2022

DOI:https://doi.org/10.1103/PhysRevA.105.023323

Physics Subject Headings (PhySH)

Bose-Einstein condensates Bosons Superfluidity

1-dimensional systems Atomic gases Nonlinear waves Quantum fluids & solids Solitons Superfluids Ultracold gases

Atomic, Molecular & OpticalNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Beau^1,2, A. del Campo ^3,4,1, D. J. Frantzeskakis ⁵, T. P. Horikis⁶, and P. G. Kevrekidis ⁷

¹Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
²Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland
³Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
⁴Donostia International Physics Center, E-20018 San Sebastián, Spain
⁵Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
⁶Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
⁷Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA

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Vol. 105, Iss. 2 — February 2022

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Images

Figure 1
Ground state density profile (top row) in the limit of $μ = 1 ≫ Ω = 0.1$ . The outer, inverted parabola profile corresponds to the (local) case of $β = 0$ [28, 29], as per Eq. (10). The inner, smaller amplitude profile corresponds to the nonlocal case of $β = Ω = 0.1$ . In both cases, the solid blue line provides the numerical result, while the dashed red line corresponds to the analytical approximation. The second row presents the spectral plane ( $ω_{i}, ω_{r}$ ) of the BdG eigenfrequencies $ω = ω_{r} + i ω_{i}$ for the case of $β = 0$ (left) and $β = 0.1$ (right). The numerically obtained four lowest frequencies are shown with blue circles, while the analytical prediction of the TF limit for $β = 0$ , i.e., $ω / Ω = \sqrt{m (m + 1) / 2}$ [27, 29, 31, 32], is shown with red stars. The absence of imaginary eigenfrequencies showcases the spectral stability of the corresponding configuration. The third and fourth rows show the same features, but now for the case of $μ = 10 ≫ Ω = 0.1$ .
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Figure 2
The top panel contains the exact stationary trapped dark soliton solution in the absence (i.e., $β = 0$ , dashed line) and in the presence ( $β = Ω$ , solid line) of long-range interactions. The chemical potentials and trap parameters are directly analogous (in top and bottom panels) to those of Fig. 1. The second row panels show the BdG results (again the imaginary vs the real part of the four lowest eigenfrequencies) for $β = 0$ (left panel) and $β = Ω$ (right panel). The real nature of the eigenfrequencies indicates stability in both cases. The numerical results in both settings are indicated by blue circles. The red stars show in both cases the analytical predictions in the TF limit for $β = 0$ for comparison (see also text). The third row panel represents the solution for $β = 0$ (dashed line) and $β = 0.1$ (solid line) for the TF limit case of $μ = 10$ . The bottom panels show the corresponding BdG eigenfrequencies for $β = 0$ (left) and $β = Ω$ (right panel). Notice in the bottom left panel the coincidence of the numerical (blue circles) and analytically predicted (red stars—see also text) frequencies. However, even in the nonlocal case of $β \neq 0$ of the bottom right panel, the symmetry modes at $ω = 0$ and $ω = Ω$ and the dark soliton vibrational mode [see text around Eq. (15)] are theoretically captured.
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Figure 3
Contour plot of the dynamical space $(x) - time (t)$ evolution of a single dark soliton. The color bar indicates the modulus $| Ψ |$ of the wave function. The initial condition contains a dark soliton perturbed by the (anomalous) eigenmode associated with the dark soliton in-trap oscillation. As expected, this leads to a soliton oscillation with the frequency predicted by the BdG analysis of Eqs. (6), namely for this case of $β = Ω = 0.1, ω = 0.2089$ . The dashed (red) line shows a simple cosinusoidal curve with this frequency, illustrating excellent agreement with the BdG prediction.
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Figure 4
Comparison of the density profile of two dark solitons with and without long-range interactions, represented by a solid and dashed line, respectively. The corresponding collective frequencies are shown as in Fig. 2. Here, only the case of $μ = 1$ is shown. The red stars in both of the bottom panels reflect the analytical prediction of the lowest (background and anomalous) modes for the case of $β = 0$ .
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Figure 5
Top panel: similar to Fig. 3, but now for the in-phase dynamics of a two-soliton state. The color bar once again indicates the modulus $| Ψ |$ of the wave function. The state oscillates with a frequency $ω_{I P} = 0.1728$ identified in the BdG analysis. Indeed, as a guide to the eye for the motion of one of the solitons, the (dashed) curve which is cosinusoidal with the same frequency is also shown to illustrate the accuracy of the relevant frequency of vibration. Bottom panel: same as the top panel but now for the out-of-phase oscillation of the two dark solitons with $ω_{OP} = 0.3276$ . Once again, the dashed (red) curve represents a cosinusoidal oscillation that is superposed as a guide to the eye.
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