Dynamics of massive point vortices in a binary mixture of Bose-Einstein condensates

Andrea Richaud, Vittorio Penna, and Alexander L. Fetter

Phys. Rev. A 103, 023311 – Published 8 February 2021

Abstract

We study the massive point-vortex model introduced in Richaud et al. [A. Richaud, V. Penna, R. Mayol, and M. Guilleumas, Phys. Rev. A 101, 013630 (2020)], which describes two-dimensional point vortices of one species that have small cores of a different species. We derive the relevant Lagrangian itself, based on the time-dependent variational method with a two-component Gross-Pitaevskii (GP) trial function. The resulting Lagrangian resembles that of charged particles in a static electromagnetic field, where the canonical momentum includes an electromagnetic term. The simplest example is a single vortex with a rigid circular boundary, where a massless vortex can only precess uniformly. In contrast, the presence of a sufficiently large filled vortex core renders such precession unstable. A small core mass can also lead to small radial oscillations, which are, in turn, clear evidence of the associated inertial effect. Detailed numerical analysis of coupled two-component GP equations with a single vortex and small second-component core confirms the presence of such radial oscillations, implying that this more realistic GP vortex also acts as if it has a small massive core.

Received 26 October 2020
Accepted 22 January 2021

DOI:https://doi.org/10.1103/PhysRevA.103.023311

Physics Subject Headings (PhySH)

Vortices in superfluids

Bose-Einstein condensates

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Andrea Richaud ^1,2,*, Vittorio Penna^2,†, and Alexander L. Fetter ^3,‡

¹Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, I-34136 Trieste, Italy
²Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, I–10129 Torino, Italy
³Departments of Physics and Applied Physics, Stanford University, Stanford, California 94305-4045, USA

^*arichaud@sissa.it
^†vittorio.penna@polito.it
^‡fetter@stanford.edu

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Vol. 103, Iss. 2 — February 2021

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Images

Figure 1
Effective potential $V_{eff} (r)$ in Eq. (28) with $Φ (r) = ln (1 - r^{2})$ (as explained in the text, both quantities are dimensionless). Panel (a) is for fixed $l = 0.1$ and increasing values of the mass ratio $μ = M_{b} / M_{a} = 0.1$ (solid blue curve), $μ = 0.2$ (long-dashed ochre curve), and $μ = 0.4$ (short-dashed green curve). In contrast, panel (b) (note different vertical scale) is for fixed $μ = 0.1$ and increasing $l = 0.1$ (solid blue curve), $l =$ 0.25 (long-dashed ochre curve), and $l =$ 0.45 (short-dashed green curve). In both figures, the smallest values $μ = l = 0.1$ (blue solid curves) clearly can support stable trajectories with $r_{\min} < r < r_{\max}$ (both solid blue curves are actually the same, despite the different vertical scales). In both figures, the curves with largest parameter values (short-dashed green curves) have no stable trajectories, with the vortex moving out toward the boundary at dimensionless $r = 1$ . The curves with intermediate value (long-dashed ochre curves) are weakly stable in both figures.
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Figure 2
Three possible trajectories for a single positive massive point vortex confined in a circular rigid trap. Plots with arrows correspond to the numerical solutions of the Euler-Lagrange equation (7) specialized to the case of the $N_{v} = 1$ vortex. We follow Fig. 1 of Ref. [11] with $R = 50 μ m, N_{a} = 5 \times 10^{4} {}^{23}{Na}$ atoms and various $N_{b} {}^{39}K$ atoms. (a) Circular orbits obtained for $N_{b} = 0$ with $μ = 0$ (dashed gray curve) and for $N_{b} = 1000$ with $μ = 0.034$ (solid red curve). (b) The presence of a core mass ( $N_{b} = 1000$ with $μ = 0.034$ ) can also yield more structured trajectories for different initial conditions. (c) For a larger core mass ( $N_{b} = 1600$ with $μ = 0.054$ ), the massive vortex moves continuously to the circular boundary.
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Figure 3
Three possible dynamical regimes for a pair of corotating massive vortices confined in a circular rigid trap. Plots with arrows correspond to the solutions of the Euler-Lagrange equation (42) for vortex 1 and the similar equation for vortex 2. As in Fig. 2, we have $N_{a} = 5 \times 10^{4}$ and $N_{b} = 2000$ (there are now two vortices). In dimensionless units, we have $μ = 0.0678$ . (a) Uniform circular orbits with $r_{0} = 0.357$ and $Ω_{0} = 4.527$ [see condition (43)]. (b) Small oscillations around a uniform circular orbit arising from small inward radial displacements in the initial condition. (c) Irregular (nonperiodic) trajectories arising from asymmetric initial positions. All panels: blue and red denote the trajectories of the two vortices.
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Figure 4
Numerical result from the two-component GP numerical evolution (purple) compared with the prediction of the massive point-vortex model (orange). (a) Four complete radial oscillations shown on a linear scale, allowing detailed comparison highlighting the decreased amplitude and the altered period of GP analysis (dashed purple curve) compared to the massive point-vortex model (solid orange curve). (b) Orbital plots with arrows of both trajectories with purple from the GP analysis and orange from the massive point-vortex model [compare similar orbits in Fig. 2].
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Figure 5
Density of the $a$ component at $t = 0.04$ from the numerical evolution of the two coupled GP equations. Blue (yellow) color corresponds to zero (high) values of the density. We have employed the same model parameters used in Fig. 4.
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