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arXiv:2610.35728

Critical velocity and dissipation mechanism in finite superfluid $^4$He nanodroplets

Keshav Sishodia, Ernesto García‐Alfonso, Nadine Halberstadt, Sivarama Krishnan, Marcel Carto Constantin Mudrich

Published Oct 26, 2026• 0 citations
arXivHTMLPDF

Abstract

A hallmark of superfluidity is the frictionless motion of an object below the Landau critical velocity; Whether this concept applies to nanoscale droplets remains unclear. Using time-dependent density functional theory, we investigate the motion of a sodium atom on the surface and inside a superfluid helium nanodroplet. For surface motion, we identify a critical velocity above which energy dissipation through ripplon excitation and the formation of surface-bound quantized vortices leads to rapid deceleration. Inside the droplet, the atom is slowed down to a steady-state velocity that depends on the initial speed, contrasting experiments reporting a critical velocity for dopants ejected from droplets [Phys. Rev. Lett. 111, 153002 (2013)]. Remarkably, the velocity of the ejected atom is nearly independent of its initial velocity. We demonstrate that this behavior results from the bursting of the bubble surrounding the atom at the droplet surface, challenging the previous experimental interpretation.

DOI
10.48550/arxiv.2610.35728

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Critical velocity and dissipation mechanism in finite superfluid 4^44He nanodroplets
A hallmark of superfluidity is the frictionless motion of an object below the Landau critical velocity; Whether this concept applies to nanoscale droplets remains unclear. Using time-dependent density functional theory, we investigate the motion of a sodium atom on the surface and inside a superfluid helium nanodroplet. For surface motion, we identify a critical velocity above which energy dissipation through ripplon excitation and the formation of surface-bound quantized vortices leads to rapid deceleration. Inside the droplet, the atom is slowed down to a steady-state velocity that depends on the initial speed, contrasting experiments reporting a critical velocity for dopants ejected from droplets [Phys. Rev. Lett. 111, 153002 (2013)]. Remarkably, the velocity of the ejected atom is nearly independent of its initial velocity. We demonstrate that this behavior results from the bursting of the bubble surrounding the atom at the droplet surface, challenging the previous experimental interpretation.
The best-known property of superfluid helium is the vanishing viscosity that objects experience while moving through the liquid with speeds below the so-called critical Landau velocity. This critical velocity is generally considered a macroscopic property as it is related to the collective excitations of the helium atoms in the liquid. In the present work we determine to what extent this concept can still be applied to nanometer-scale, finite size helium systems. To this end, atoms and molecules embedded in helium nanodroplets of various sizes are accelerated out of the droplets by means of optical excitation, and the speed distributions of the ejected particles are determined. The measurements reveal the existence of a critical velocity in these systems, even for nanodroplets consisting of only a thousand helium atoms. Accompanying theoretical simulations based on a time-dependent density functional description of the helium confirm and further elucidate this experimental finding.
The capture dynamics of a Ne atom by a superfluid helium nanodroplet (((4)He)N=1000; T = 0.37 K), Ne + ((4)He)N→ Ne@((4)He)N', was investigated using a quantum approach (TDDFT (helium) + quantum wave packet (Ne)) at zero angular momentum and a rather wide range of Ne atom initial mean velocities (〈v0〉: 90-1300 m s(-1)). This is probably the first quantum dynamics study focusing on the pick up process and the evolution of the dopant inside the nanodroplet and the second more detailed investigation on this topic. For 〈v0〉 = 210 m s(-1) and above the atom is always captured, but for lower velocities the probability of capture is somewhat below the unity and decreases as 〈v0〉 diminishes. The main energy exchange begins with the collision of the atom with the nanodroplet surface, and the excess of energy placed in the doped nanodroplet is progressively released by the evaporation of a small amount of (4)He atoms. Once the atom has entered into the nanodroplet its mean position follows an oscillatory trajectory, due to multiple sequential collisions with the inner surface of the nanodroplet, and its mean velocity reaches values which are below Landau's critical velocity. This probably corresponds to the general behavior of nanodroplets with a bulk-like region when moderate collision energies (i.e., similar to the ones considered here) are involved. In the future we hope to investigate the influence of angular momentum on the mechanism of the pick up process, using the same quantum dynamics method.
Helium nanodroplets are considered ideal model systems to explore quantum hydrodynamics in self-contained, isolated superfluids. However, exploring the dynamic properties of individual droplets is experimentally challenging. In this work, we used single-shot femtosecond x-ray coherent diffractive imaging to investigate the rotation of single, isolated superfluid helium-4 droplets containing ~10(8) to 10(11) atoms. The formation of quantum vortex lattices inside the droplets is confirmed by observing characteristic Bragg patterns from xenon clusters trapped in the vortex cores. The vortex densities are up to five orders of magnitude larger than those observed in bulk liquid helium. The droplets exhibit large centrifugal deformations but retain axially symmetric shapes at angular velocities well beyond the stability range of viscous classical droplets.
We have studied the merging of two identical 4He^{4}\mathrm{He}4He droplets at zero temperature, caused by their van der Waals mutual attraction. During the early stages of the merging, density structures appear which closely match the experimental observations by Vicente et al. [J. Low Temp. Phys. 121, 627 (2000)]. When the droplets are merging, quantized vortex-antivortex ring pairs nucleate at the surface and annihilate inside the merged droplet producing a roton burst. We also observe the nucleation of quantized vortex-antivortex rings that wrap the droplet surface and remain localized on the surface until they eventually decay into short-wavelength surface waves. Analysis of the kinetic energy spectrum discloses the existence of a regime where turbulence caused by vortex interaction and annihilation is characterized by a Kolmogorov power law. This is followed by another regime where roton radiation---produced by vortex-antivortex annihilation---dominates, whose hallmark is a weak, turbulent surface dynamics. We suggest that similar processes might appear in superfluid helium droplets after they capture impurities or if they are produced by hydrodynamic instability of a liquid jet. Experiments on collisions between recently discovered self-bound Bose-Einstein condensates should display a similar phenomenology.
Within density functional theory, we have investigated the coalescence dynamics of two superfluid helium nanodroplets hosting vortex lines in different relative orientations, which are drawn towards each other by the Van der Waals mutual attraction. We have found a rich phenomenology depending on how the vortex lines are oriented. In particular, when a vortex and antivortex lines are present in the merging droplets, a dark soliton develops at the droplet contact region, which eventually decays into vortex rings. Reconnection events are observed between the vortex lines or rings, leading to the creation of more vortices. Our simulations show the interplay between vortex creation and reconnections, as well as the effect of the droplet surface which pins the vortex ends and, by reflecting short-wavelength excitations produced by the interactions between vortices, strongly affects the droplet final state. Additional vorticity is nucleated in the proximity of surface indentations produced in the course of the dynamics, which in turn interact with other vortices present in the droplets. These effects, obviously absent in the case of bulk liquid helium, show that the droplet surface may act as a multiplier of vortex reconnections. The analysis of the energy spectrum shows that vortex-antivortex ring annihilation, as well as vortex-antivortex reconnections, yields roton bursts of different intensity.
We address the collision of two superfluid 4He droplets at non-zero initial relative velocities and impact parameters within the framework of liquid 4He time-dependent density functional theory at zero temperature. Despite the small size of these droplets (1000 He atoms in the merged droplet) imposed by computational limitations, we have found that quantized vortices may be readily nucleated for reasonable collision parameters. At variance with head-on collisions, where only vortex rings are produced, collisions with a non-zero impact parameter produce linear vortices that are nucleated at indentations appearing on the surface of the deformed merged droplet. Whereas for equal-size droplets, vortices are produced in pairs, an odd number of vortices can appear when the colliding droplet sizes are different. In all cases, vortices coexist with surface capillary waves. The possibility for collisions to be at the origin of vortex nucleation in experiments involving very large droplets is discussed. An additional surprising result is the observation of the drops coalescence even for grazing and distal collisions at relative velocities as high as 80 and 40 m/s, respectively, induced by the long-range van der Waals attraction between the droplets.