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arXiv:cond-mat/0107028 · Physical Review Letters

Quantized Vortices in Mixed H3e−H4e Drops

R. Mayol, Martí Pi, M. Barranco, F. Dalfovo

Published Sep 12, 2001• 22 citations
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Abstract

Using density functional theory, we investigate the structure of mixed ${}^{3}{\mathrm{He}}_{{N}_{3}}{\ensuremath{-}}^{4}{\mathrm{He}}_{{N}_{4}}$ droplets with an embedded impurity (Xe atom or HCN molecule) which pins a quantized vortex line. We find that the $\mathrm{dopant}+\mathrm{vortex}{+}^{4}{\mathrm{He}}_{{N}_{4}}$ complex, which in a previous work [F. Dalfovo et al., Phys. Rev. Lett. 85, 1028 (2000)] was found to be energetically stable below a critical size ${N}_{\mathrm{cr}}$, is robust against the addition of ${}^{3}\mathrm{He}$. While ${}^{3}\mathrm{He}$ atoms are distributed along the vortex line and on the surface of the ${}^{4}\mathrm{He}$ drop, the impurity is mostly coated by ${}^{4}\mathrm{He}$ atoms. Results for ${N}_{4}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}500$ and a number of ${}^{3}\mathrm{He}$ atoms ranging from 0 to 100 are presented, and the binding energy of the dopant to the vortex line is determined.

DOI
10.1103/physrevlett.87.145301

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We present a diffusion Monte Carlo study of a vortex line excitation attached to the center of a 4He^{4}\mathrm{He}4He droplet at zero temperature. The vortex energy is estimated for droplets of increasing number of atoms, from N=70N=70N=70 up to 300, showing a monotonous increase with NNN. The evolution of the core radius and its associated energy, the core energy, is also studied as a function of NNN. The core radius is $\ensuremath{\sim}1\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ in the center and increases when approaching the droplet surface; the core energy per unit volume stabilizes at a value $2.8\phantom{\rule{0.3em}{0ex}}\mathrm{K}{\ensuremath{\sigma}}^{\ensuremath{-}3}$ $(\ensuremath{\sigma}=2.556\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$ for $N\ensuremath{\ge}200$.
We calculate the binding energies of a 3He{}^{3}\mathrm{He}3He impurity and a fictitious 4He{}^{4}\mathrm{He}4He impurity to a superfluid 4He{}^{4}\mathrm{He}4He vortex. Within the accuracy of the calculation, there is no difference between the mass-3 and mass-4 binding energies, suggesting that the binding is due to the displacement of atoms from the core. Comparing the results between Onsager-Feynman ( ${\ensuremath{\Psi}}_{\mathrm{OF}}$) and ``shadow'' ( ${\ensuremath{\Psi}}_{D}$) variational functions, the energy is dependent on the density for ${\ensuremath{\Psi}}_{\mathrm{OF}}$, but independent for ${\ensuremath{\Psi}}_{D}$. The binding of 2.3 K for ${\ensuremath{\Psi}}_{D}$ at zero pressure is smaller than the experimental estimate. We also estimate the binding energies of charged ions.
Using density functional theory, we have theoretically studied the formation and the stability of vortices in quantum liquid droplets composed of a mixture of hyperfine states of potassium. Following the experimental setup that produced quantum droplets for the first time, we work with squeezed drops that are compressed in one direction. By squeezing the drops even more, toward a quasi-two-dimensional geometry, we study the minimum atom number able to show a stable vortex and obtain that this number is significantly smaller than previous predictions for spherical droplets. The reduction of the critical atom number for forming a stable vortex could make their experimental observation in these droplets, which is still lacking, more feasible. Contrary to results obtained in heteronuclear mixtures, where the energetically preferred vortices are partially filled with the species not participating in the rotation, our results show a relevant stability island of fully empty vortices. Increasing the number of particles in the drop and the speed of rotation, we estimate the transition line between empty and filled vortices.
Within density functional theory, we investigate stationary many-vortex structures in a rotating 4He^{4}\mathrm{He}4He nanocylinder at zero temperature. We compute the stability diagram and compare our results with the classical model of vortical lines in an inviscid and incompressible fluid. Scaling the results to millimeter-size buckets, they can be compared with experiments on vortex arrays conducted in the past. Motivated by recent experiments that have used atomic impurities as a means of visualizing vortices in superfluid 4He^{4}\mathrm{He}4He droplets, we have also considered the formation of chains of xenon atoms along a vortex line and the interaction between xenon atoms inside the same vortex and on different neighboring vortex lines.