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arXiv:1910.12926 · Physical Review Letters

Angular Momentum in Rotating Superfluid Droplets

Sean M. O. O’Connell, Rico Mayro P. Tanyag, Deepak Verma, Charles Bernando, Weiwu Pang, Camila Bacellar, Catherine A. Saladrigas, Johannes Mahl, Benjamin W. Toulson, Yoshiaki Kumagai, Peter Walter, Francesco Ancilotto, M. Barranco, Martí Pi, Christoph Bostedt, Oliver Geßner, Andrey F. Vilesov

Published May 26, 2020• 42 citations
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Abstract

The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay between which remains to be uncovered. Here, the rotation of isolated submicrometer superfluid ^{4}He droplets is studied by ultrafast x-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder.

DOI
10.1103/physrevlett.124.215301

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The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay between which remains to be uncovered. Here, the rotation of isolated submicrometer superfluid ^{4}He droplets is studied by ultrafast x-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder.
Helium nanodroplets are considered ideal model systems to explore quantum hydrodynamics in self-contained, isolated superfluids. Rotation of a superfluid droplet is manifested as a collection of quantum vortices. Here, the authors use single-shot femtosecond X-ray coherent diffractive imaging to investigate the shapes and vorticity of rotating superfluid 4{}^{4}4He droplets. They find that most of the diffraction contours exhibit noticeable ellipticity. A smaller but still significant number of distorted diffraction contours are marked by regions of high intensity concentrated along the direction of the long axis in the diffraction image, indicating the presence of extremely deformed 4{}^{4}4He droplets. Forward modeling of the diffraction images shows that the shapes of superfluid 4{}^{4}4He droplets are very similar to their classical counterparts. In particular, the existence of both axially symmetric (oblate) and, for higher angular momenta, two-lobed (triaxial prolate) droplets is revealed. The latter shapes are strongly deformed compared to the spheroidal bodies that are expected for smaller rotational speeds. The observed shapes provide direct access to the droplet angular momenta, angular velocities, and estimated number of quantum vortices inside the droplets.
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.
Within density functional theory, we have studied the interplay between vortex arrays and capillary waves in spinning prolate 4He^{4}\mathrm{He}4He droplets made of several thousand helium atoms. Surface capillary waves are ubiquitous in prolate superfluid 4He^{4}\mathrm{He}4He droplets, and depending on the size and angular momentum of the droplet, they may coexist with vortex arrays. We have found that the equilibrium configuration of small prolate droplets is vortex free, evolving towards vortex hosting as the droplet size increases. This result is in agreement with a recent experiment [O'Connell et al., Phys. Rev. Lett. 124, 215301 (2020)] that disclosed that vortex arrays and capillary waves coexist in the equilibrium configuration of very large drops. In contrast to viscous droplets executing rigid-body rotation, the stability phase diagram of spinning 4He^{4}\mathrm{He}4He droplets cannot be universally described in terms of dimensionless angular momentum and angular velocity variables: Instead, the rotational properties of superfluid helium droplets display a clear dependence on the droplet size and the number of vortices they host.
Previous single-pulse extreme ultraviolet and x-ray coherent diffraction studies revealed that superfluid 4He^{4}\mathrm{He}4He droplets obtained in a free jet expansion acquire sizable angular momentum, resulting in significant centrifugal distortion. Similar experiments with normal fluid 3He^{3}\mathrm{He}3He droplets may help elucidate the origin of the large degree of rotational excitation and highlight similarities and differences of dynamics in normal and superfluid droplets. Here, we present a comparison of the shapes of isolated 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He droplets following expansion of the corresponding fluids in vacuum at temperatures as low as \ensuremath{\sim}2 K. Large 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He droplets with average radii of \ensuremath{\sim}160 and \ensuremath{\sim}350 nm, respectively, were produced. We find that the majority of the shapes of 3He^{3}\mathrm{He}3He droplets in the beam correspond to rotating oblate spheroids, in agreement with previous observations for 4He^{4}\mathrm{He}4He droplets. The aspect ratio of the droplets is related to the degree of their rotational excitation, which is discussed in terms of reduced angular momenta (\ensuremath{\Lambda}) and reduced angular velocities (\ensuremath{\Omega}), the average values of which are found to be similar in both isotopes. This similarity suggests that comparable mechanisms induce rotation regardless of the isotope. We hypothesize that the observed distribution of droplet sizes and angular momenta originates from processes in the dense region close to the nozzle, where a significant velocity spread and frequent collisions between droplets induces excessive rotation followed by droplet fission.
Free superfluid helium droplets constitute a versatile medium for a diverse range of experiments in physics and chemistry that extend from studies of the fundamental laws of superfluid motion to the synthesis of novel nanomaterials. In particular, the emergence of quantum vortices in rotating helium droplets is one of the most dramatic hallmarks of superfluidity and gives detailed access to the wave function describing the quantum liquid. This review provides an introduction to quantum vorticity in helium droplets, followed by a historical account of experiments on vortex visualization in bulk superfluid helium and a more detailed discussion of recent advances in the study of the rotational motion of isolated, nano- to micrometer-scale superfluid helium droplets. Ultrafast X-ray and extreme ultraviolet scattering techniques enabled by X-ray free-electron lasers and high-order harmonic generation in particular have facilitated the in situ detection of droplet shapes and the imaging of vortex structures inside individual, isolated droplets. New applications of helium droplets ranging from studies of quantum phase separations to mechanisms of low-temperature aggregation are discussed.
A significant fraction of superfluid helium nanodroplets produced in a free-jet expansion has been observed to gain high angular momentum resulting in large centrifugal deformation. We measured single-shot diffraction patterns of individual rotating helium nanodroplets up to large scattering angles using intense extreme ultraviolet light pulses from the FERMI free-electron laser. Distinct asymmetric features in the wide-angle diffraction patterns enable the unique and systematic identification of the three-dimensional droplet shapes. The analysis of a large data set allows us to follow the evolution from axisymmetric oblate to triaxial prolate and two-lobed droplets. We find that the shapes of spinning superfluid helium droplets exhibit the same stages as classical rotating droplets while the previously reported metastable, oblate shapes of quantum droplets are not observed. Our three-dimensional analysis represents a valuable landmark for clarifying the interrelation between morphology and superfluidity on the nanometer scale.