Vorticity and quantum turbulence in the merging of superfluid Helium nanodroplets

TL;DR

This study investigates the dynamics of merging superfluid helium droplets, revealing vortex formation, turbulence regimes with Kolmogorov spectra, and roton radiation, providing insights into quantum turbulence phenomena.

Contribution

It presents a detailed simulation of vortex nucleation and turbulence in merging superfluid helium droplets, linking microscopic vortex dynamics to macroscopic turbulent behavior.

Findings

Vortex-antivortex rings nucleate and annihilate during merging.

A Kolmogorov power law regime indicates quantum turbulence.

Roton radiation dominates surface dynamics after vortex interactions.

Abstract

We have studied the merging of two $^4$He 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.

Figures

Equations

$$\frac{R_{\text{neck}}}{R}=\beta \sqrt{\frac{t}{\tau }}$$

$$E_{\text{kin}}(t)=\frac{{\hslash }^2}{2m}\int \text{d}\mathbf{r}{\left|\nabla \sqrt{\rho (\mathbf{r},t)}\right|}^2+\frac{m}{2}\int \text{d}\mathbf{r}\frac{{\mathbf{j}}^2(\mathbf{r},t)}{\rho (\mathbf{r},t)}.$$

$$E_{\text{hyd}}(t)=4\pi {\int }_0^{\infty }\text{d}k{\mathcal{E}}_{\text{hyd}}(k,t),$$

$${\mathcal{E}}_{\text{hyd}}(k,t)=\frac{m}{2}(2\pi {)}^3\frac{k^2}{4\pi }{\int }_{{\Omega }_k}\text{d}{\Omega }_k{\left|\widetilde{\frac{\mathbf{j}}{\sqrt{\rho }}}(\mathbf{k},t)\right|}^2.$$

Full text

Superfluid helium droplets are fascinating objects that have drawn the interest of both experimentalist and theoreticians Toe04 . This interest is notoriously broad and includes, e.g., the nature of superfluidity at the nanoscale, the interaction of atomic and molecular impurities with the hosting droplets, and the study of vortical states in nanodroplets, see Refs. Anc17, ; Mau18, for recent comprehensive reviews.

One of the hallmarks of superfluidity is the appearance of quantized vortices Bar01 ; Pit16 . Whereas the presence of vortices in macroscopic samples of bulk liquid helium was unambiguously proved long ago, in the case of helium droplets it remained elusive. It is only recently that vortex arrays inside droplets made of $10^{8}$–$10^{11}$ atoms were detected by coherent x-ray scattering Gom14 .

The coalescence of superfluid 4He drops of several tenths of cm radius levitating in a magnetic trap was investigated some time ago Vic00 . The experimental setup allowed to study their merging at fairly low velocity, and the temperature was kept low enough to make the helium vapor around them negligible, so that the droplets were essentially isolated.

In this work we address the coalescence of two identical 4He droplets drawn together by the mutual Van der Waals (VdW) long-range attraction. After showing that the merging yields density structures which closely match the experimental observations even in their fine details, we show that vorticity nucleates both inside and on the surface of the merged droplet. The subsequent decay of vorticity results in the appearance of two distinct turbulence regimes, which we characterize with their kinetic energy spectra. Multimedia content is provided as Supplemental Material SM .

We study the strongly-correlated superfluid 4He by using a realistic Density Functional Theory (DFT) approach which allows to reproduce complex dynamical phenomena such as vortex nucleation and vortex-density waves interactions Anc17 . One important outcome of our simulations is an accurate description of the dynamics of vortex interactions and annihilation, which is a fundamental ingredient of current studies of quantum turbulence in liquid He and in cold-gas superfluids Vin02 ; Tsu09 ; Bar14 ; Tsu17 .

The Gross-Pitaevskii (GP) approach Pit16 has been also shown to be capable to sustain these phenomena. Vortex ring emission and the possible transition to a chaotic turbulent regime due to vortex interactions and decay has been studied within time-dependent GP theory in both superfluid 4He and Bose-Einstein condensates (BEC) Bar01 ; Vil18 ; Ber01 ; Fri92 ; Nav16 ; Kob05 ; Pro09 . It is well known, however, that GP theory cannot provide an accurate description of the vortex core structure in superfluid liquid helium, and can at most reproduce the phonon part of the 4He dispersion relation. In the following, we will prove the role of roton emission in turbulence phenomena in 4He, and we will show that the roton wave-vector provides indeed a natural dividing line between the different length scales contributing to the turbulence created by the droplet collision process.

Within DFT, the total energy of a 4HeN droplet at zero temperature is written as a functional of an effective wave function $\Psi(\mathbf{r},t)$ related to its atomic density by $\rho(\mathbf{r},t)=|\Psi(\mathbf{r},t)|^{2}$. In this study, the number of helium atoms in each of the initial droplets is $N=500$, and the results we present have been obtained using the 4He-DFT BCN-TLS computing package Pi17 . The density functional used in this work is finite-range and includes non-local effects Anc05 , since both aspects are needed to describe quantitatively the response of the liquid at the scale of the vortex core radius, which is of the order of one Angstrom Gal14 . This functional reproduces quite faithfully the density modulations around the vortex core Anc17 . These modulations have been interpreted as a cloud of bound virtual rotons embodied in the phase of the vortex wave function, and they may be converted into real ones following vortex-antivortex annihilations, making vortex tangles a potential source of non-thermal rotons Ame18 . Our simulations provide a clean evidence of such conversion process.

We have first determined the equilibrium configuration of an isolated helium droplet by solving the static DFT problem Anc17 . These droplets are spherical, and their sharp density surfaces, defined as the locus where the helium density equals half the bulk liquid value $\rho_{0}=0.0218$ Å*-3*, have a radius $R=r_{0}N^{1/3}$ with $r_{0}=2.22$ Å. The energy of the 4He500 droplet is $-2474.6$ K, and that of the 4He1000 droplet is $-5400.3$ K. An energy of $451.2$ K, mainly arising from the decrease of surface energy Bar06 , is thus available for the merging.

Next, we place the droplets at rest so that their surfaces are about $6$ Å away from contact, and use this as the initial configuration for the merging dynamics within the time-dependent DFT (TDDFT) approach. This is in stark contrast with classical calculations of merging, since they have to start from two sharp surfaces with a point of contact and to assume that by some microscopic mechanism a tiny bridge appears joining the droplets Egg99 ; and with BEC calculations of confined Sun08 or self-sustained droplets, where some velocity has to be given to the droplets to trigger the collision Fer19 . In our case, it is the VdW mutual attraction that provides such a microscopic mechanism.

The TDDFT equation Anc17 ; Pi17 has been solved up to $t=600$ ps with a time step $\Delta t=1$ fs. Droplet coalescence proceeds by the development of a tiny, low density bridge connecting the droplets in about $4.5$ ps SM . Figure 1 displays the linear $t$-dependence of the squared neck radius $R_{\textrm{neck}}^{2}$ with $6.31$ Å2/ps slope, showing that $R_{\textrm{neck}}\propto t^{1/2}$. In spite of the different length and time scales, the same behavior has been observed in experiments on low viscosity fluids Wu04 . Calculations for classical droplets yielded Egg99 ; Duc03

[TABLE]

with $\tau=(\rho R^{3}/\sigma)^{1/2}$, where $\rho$ is the mass density of the fluid, $\sigma$ its surface tension, and $R$ is the droplet’s radius. The $\beta$ factor was calculated for inviscid droplets yielding a value of 1.62 Duc03 , while measurements gave values in the $1.03$–$1.29$ range Wu04 . Using for $\tau$ the liquid 4He value and the calculated slope, we find $\beta=0.93$. We have analyzed the first few frames of Fig. 2 in Ref. Vic00, , also finding $R_{\textrm{neck}}\propto t^{1/2}$ but with a significantly higher prefactor, $\beta=6.75$. This discrepancy is attributed to the different impact velocity of the droplets, which is zero in our case and a few cm/s in the experiment Vic00 .

The droplets eventually merge, producing in the process two protrusions symmetrically placed along the collision direction, as shown in Fig. 2 for $t=92$ ps. The growth of the merging neck and the appearance of the protrusions at the droplet surface displayed in Fig. 2 are remarkably similar to those found for the coalescence of 4He drops of $0.20$–$0.25$ cm radius, as shown in Fig. 2 of Ref. Vic00, , with the only obvious differences being the time and length scales involved.

As the protrusions shrink, pairs of quantized vortex-antivortex rings are nucleated at the necks that connect the protrusions with the merged droplet, with all vortex (antivortex) rings located around the left (right) neck. Once formed, the rings of each pair move symmetrically towards the droplet center SM . Two such vortex-antivortex pairs can be identified in Fig. 2 at $t=111$ ps. Additionally, lenticular, sharp density indentations appear in the regions between rings, likely shock waves produced by the sudden collapse of the protrusions; these indentations are separated by a distance similar to that between He atoms in the compressed liquid.

During their motion, the smaller radius rings catch up to the larger radius rings and pass through them, shrink and eventually disappear at $t\simeq 114$ ps. The larger radius vortex-antivortex rings last up to $t\simeq 139$ ps, when they collide against each other and annihilate, producing a roton burst (as explained in the following). We may estimate the energy $\Delta E$ released in the annihilation event by recalling that, in 4He, the energy of a vortex ring of radius $R_{v}$ is $E_{v}=(2\pi^{2}\hbar^{2}/m)\rho_{0}R_{v}[\ln(8R_{v}/a)-1.615]$ Rob71 , where $a=0.7$ Å is the vortex core size. In our case, the vortex ring radius just before annihilation is $R_{v}\sim 8$ Å, which leads to $\Delta E=2E_{v}\sim 240$ K. This corresponds to the emission of $\sim 24$ rotons Mor98 .

In the later stages of the time evolution, the droplet appears to be in a highly excited state characterized by short-wavelength nanoscopic surface waves, as shown in Fig. 2 at $t=200$ ps. Similar surface patterns have been found in the decay of multicharged vortices in trapped BEC Cid17 . Superposed to this complex surface dynamics, the merged droplet undergoes large amplitude oscillations, its shape periodically shifting from oblate to prolate SM .

The presence of quantized vortices in the droplet during the merging, their mutual interaction, and their annihilation followed by the emission of rotons, is likely a source of turbulence, which we address here with the aid of a widely used tool in studies of classical and quantum turbulence, i.e., the kinetic energy spectrum whose dependence upon the wavenumber $k$ allows to distinguish different regimes that are relevant for characterizing turbulence Tsu09 .

Writing the effective wave function as $\Psi(\mathbf{r},t)=\sqrt{\rho(\mathbf{r},t)}\exp[\imath\,\mathcal{S}(\mathbf{r},t)]$, the atom current density is ${\mathbf{j}}({\mathbf{r}},t)=\rho({\mathbf{r}},t){\mathbf{v}}({\mathbf{r}},t)$ with ${\mathbf{v}}({\mathbf{r}},t)=\hbar\,\nabla\mathcal{S}({\mathbf{r}},t)/m$. Thus, the kinetic energy of the superfluid can be written as

[TABLE]

The first term is the quantum pressure while the second is the usual hydrodynamic kinetic energy $E_{\textrm{hyd}}$. Working in Fourier space, one can rewrite $E_{\textrm{hyd}}$ as

[TABLE]

where the energy spectrum $\mathcal{E}_{\textrm{hyd}}(k,t)$ is the spherical average in $\mathbf{k}$-space Bar01 ; Nor97 ; Bra12

[TABLE]

We have decomposed it into a divergence-free (incompressible) part, related to vorticity, and a compressible part Nor97 , related to density waves, and have analyzed them separately Ree12 .

Figure 3 shows both additive components of $\mathcal{E}_{\textrm{hyd}}(k,t)$, as well as three relevant $k$ values: $k_{\textrm{vort}}=2\pi/a=6.3$ Å*-1*, $a$ being the vortex core size $\sim 1$ Å; $k_{\textrm{R}}=2\pi/R=0.3$ Å*-1*, where $R$ is the radial dimension of the droplet, and the roton wave-vector $k_{\textrm{rot}}=1.98$ Å*-1*. Both components periodically display bright regions in the small $k$ region ($k\lesssim k_{\mathrm{R}}$) in phase with the oscillations of the shape of the merged droplet. Interestingly, in the compressible part (bottom panel), a bright peak appears at $t\simeq 140$ ps corresponding to the roton burst created by the annihilation of the large vortex-antivortex rings. This peak spreads with time and is present in the rest of the simulation note1 . The fainter spot at $t\simeq 115$ ps originates from the mentioned large amplitude density waves that appear between the larger vortex and antivortex rings.

Between approximately $100$ and $140$ ps, the relevant part of the incompressible energy spectrum is dominated by the presence of vortices and their decay (top panel of Fig. 3). The top panel of Fig. 4 corresponds to this regime. It displays two powers laws, $k^{-3.00}$ and $k^{-1.64}$, which are strikingly similar, respectively, to that arising from the presence of vortices Nor97 ; Bra12 , and to the Kolmogorov classical scaling, $k^{-5/3}$, which is known to be also present in bulk superfluid turbulence Nor97 ; Sal10 .

At later times, the compressible part of the energy spectrum is dominated by rotons and their effect on the droplet surface (see below). The bottom panel of Fig. 3 corresponds to this regime, whose scaling laws are shown in the bottom panel of Fig. 4. It is the analogue of the weak-wave turbulence regime arising from acoustic radiation, characterized by a $k^{-3/2}$ law Naz06 ; Tsu09 , although other scalings, such as $k^{-7/2}$, have also been found Ree12 . In our case, the spectrum displays two powers laws with exponents $1.42$ and $-3.03$. One might expect that the scalings associated to droplet surface turbulence are not those of two-dimensional turbulence on a flat surface Naz06 ; Tsu09 ; Ree12 . Moreover, since the wave energy spectrum depends on the size and shape of the droplet surface, one should not expect a universal behavior.

Inspection of the early stages of superfluid flow passing through the neck connecting the two merging droplets shows that the circulation lines have a tendency to bend outwards at the contact region, hinting at the possible nucleation of vortex-antivortex rings on the outer droplet surfaces, which cannot be observed by the mere inspection of the droplet densities SM . To help identify the presence of vorticity in these low-density regions, we have calculated the pseudo-vorticity $\nabla\times{\mathbf{j}}({\mathbf{r}},t)$ Vil16 . Plots of $|\nabla\times{\mathbf{j}}({\mathbf{r}},t)|$ iso-surfaces allow to visualize regions of potentially non-zero vorticity Fre18 .

A video in the Supplemental Material SM shows how pseudo-vorticity spreads and slips on the outer surface of the merging droplets, and Fig. 5 displays pseudo-vorticity iso-surfaces for selected times. We have checked that, between 12 and 90 ps, the circulation around the more intense of these tiny surface structures is quantized, with charges $\pm 1$, hence signalling bona fide vortex/antivortex rings note . These vortices do not sink into the droplet but remain on the surface (shells of parallel ring-like structures appearing in Fig. 5). Eventually, they decay and fragment due to the nanoscopic indentations appearing on the surface of the merged droplet at $t\sim 160$ ps, when the surface changes its appearance from smooth to rough SM , thus causing the weak-wave turbulence discussed before. Superfluid helium droplets, after capturing impurities Anc17 or being produced by hydrodynamic instability of a liquid jet Gom14 , might experience a similar process.

Although most of the pseudo-vorticity appearing when $t>160$ ps is localized on the droplet surface region, lenticular patterns remain in its bulk. These patterns, which are remnants of the sharp density indentations discussed before, can be seen, e.g., in Fig. 5 at $t=206$ ps, where a train of such faint structures is visible along the incident direction.

Finally, we would like to mention that recent realizations of stable self-bound ultra-dilute quantum droplets, made of atoms of a binary mixture of BECs Cab18 ; Sem18 , allow to address droplet merging in a different superfluid environment Fer19 and in a likely more controllable way, opening up the possibility of studying wave turbulence on their surfaces. Self-bound BEC droplets made of dipolar gases displaying a roton minimum Kla09 might also disclose non-thermal roton emission in these systems.