Three-Dimensional Shapes of Spinning Helium Nanodroplets

Computational microscopy with coherent diffractive imaging and ptychography

Microscopy and crystallography are two essential experimental methodologies for advancing modern science. They complement one another, with microscopy typically relying on lenses to image the local structures of samples, and crystallography using diffraction to determine the global atomic structure of crystals. Over the past two decades, computational microscopy, encompassing coherent diffractive imaging (CDI) and ptychography, has advanced rapidly, unifying microscopy and crystallography to overcome their limitations. Here, I review the innovative developments in CDI and ptychography, which achieve exceptional imaging capabilities across nine orders of magnitude in length scales, from resolving atomic structures in materials at sub-ångstrom resolution to quantitative phase imaging of centimetre-sized tissues, using the same principle and similar computational algorithms. These methods have been applied to determine the 3D atomic structures of crystal defects and amorphous materials, visualize oxygen vacancies in high-temperature superconductors and capture ultrafast dynamics. They have also been used for nanoscale imaging of magnetic, quantum and energy materials, nanomaterials, integrated circuits and biological specimens. By harnessing fourth-generation synchrotron radiation, X-ray-free electron lasers, high-harmonic generation, electron microscopes, optical microscopes, cutting-edge detectors and deep learning, CDI and ptychography are poised to make even greater contributions to multidisciplinary sciences in the years to come.

Quantized vortex nucleation in collisions of superfluid nanoscopic helium droplets at zero temperature

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.

Generation of Large Vortex-Free Superfluid Helium Nanodroplets

Superfluid helium nanodroplets are an ideal environment for the formation of metastable, self-organized dopant nanostructures. However, the presence of vortices often hinders their formation. Here, we demonstrate the generation of vortex-free helium nanodroplets and explore the size range in which they can be produced. From x-ray diffraction images of xenon-doped droplets, we identify that single compact structures, assigned to vortex-free aggregation, prevail up to 10^{8} atoms per droplet. This finding builds the basis for exploring the assembly of far-from-equilibrium nanostructures at low temperatures.

Nanoscopic jets and filaments of superfluid 4He at zero temperature: A DFT study

The instability of a cryogenic ⁴He jet exiting through a small nozzle into vacuum leads to the formation of ⁴He drops, which are considered ideal matrices for spectroscopic studies of embedded atoms and molecules. Here, we present a He-density functional theory (DFT) description of droplet formation resulting from jet breaking and contraction of superfluid ⁴He filaments. Whereas the fragmentation of long jets closely follows the predictions of linear theory for inviscid fluids, leading to droplet trains interspersed with smaller satellite droplets, the contraction of filaments with an aspect ratio larger than a threshold value leads to the nucleation of vortex rings, which hinder their breakup into droplets.

Three-dimensional femtosecond snapshots of isolated faceted nanostructures

The structure and dynamics of isolated nanosamples in free flight can be directly visualized via single-shot coherent diffractive imaging using the intense and short pulses of x-ray free-electron lasers. Wide-angle scattering images encode three-dimensional (3D) morphological information of the samples, but its retrieval remains a challenge. Up to now, effective 3D morphology reconstructions from single shots were only achieved via fitting with highly constrained models, requiring a priori knowledge about possible geometries. Here, we present a much more generic imaging approach. Relying on a model that allows for any sample morphology described by a convex polyhedron, we reconstruct wide-angle diffraction patterns from individual silver nanoparticles. In addition to known structural motives with high symmetries, we retrieve imperfect shapes and agglomerates that were not previously accessible. Our results open unexplored routes toward true 3D structure determination of single nanoparticles and, ultimately, 3D movies of ultrafast nanoscale dynamics.

Finding the semantic similarity in single-particle diffraction images using self-supervised contrastive projection learning

Single-shot coherent diffraction imaging of isolated nanosized particles has seen remarkable success in recent years, yielding in-situ measurements with ultra-high spatial and temporal resolution. The progress of high-repetition-rate sources for intense X-ray pulses has further enabled recording datasets containing millions of diffraction images, which are needed for the structure determination of specimens with greater structural variety and dynamic experiments. The size of the datasets, however, represents a monumental problem for their analysis. Here, we present an automatized approach for finding semantic similarities in coherent diffraction images without relying on human expert labeling. By introducing the concept of projection learning, we extend self-supervised contrastive learning to the context of coherent diffraction imaging and achieve a dimensionality reduction producing semantically meaningful embeddings that align with physical intuition. The method yields substantial improvements compared to previous approaches, paving the way toward real-time and large-scale analysis of coherent diffraction experiments at X-ray free-electron lasers.

The Scatman: an approximate method for fast wide-angle scattering simulations

Single-shot coherent diffraction imaging (CDI) is a powerful approach to characterize the structure and dynamics of isolated nanoscale objects such as single viruses, aerosols, nanocrystals and droplets. Using X-ray wavelengths, the diffraction images in CDI experiments usually cover only small scattering angles of a few degrees. These small-angle patterns represent the magnitude of the Fourier transform of the 2D projection of the sample's electron density, which can be reconstructed efficiently but lacks any depth information. In cases where the diffracted signal can be measured up to scattering angles exceeding ∼10°, i.e. in the wide-angle regime, some 3D morphological information of the target is contained in a single-shot diffraction pattern. However, the extraction of the 3D structural information is no longer straightforward and defines the key challenge in wide-angle CDI. So far, the most convenient approach relies on iterative forward fitting of the scattering pattern using scattering simulations. Here the Scatman is presented, an approximate and fast numerical tool for the simulation and iterative fitting of wide-angle scattering images of isolated samples. Furthermore, the open-source software implementation of the Scatman algorithm, PyScatman, is published and described in detail. The Scatman approach, which has already been applied in previous work for forward-fitting-based shape retrieval, adopts the multi-slice Fourier transform method. The effects of optical properties are partially included, yielding quantitative results for small, isolated and weakly interacting samples. PyScatman is capable of computing wide-angle scattering patterns in a few milliseconds even on consumer-level computing hardware, potentially enabling new data analysis schemes for wide-angle coherent diffraction experiments.

Anisotropic Surface Broadening and Core Depletion during the Evolution of a Strong-Field Induced Nanoplasma

Strong-field ionization of nanoscale clusters provides excellent opportunities to study the complex correlated electronic and nuclear dynamics of near-solid density plasmas. Yet, monitoring ultrafast, nanoscopic dynamics in real-time is challenging, which often complicates a direct comparison between theory and experiment. Here, near-infrared laser-induced plasma dynamics in ∼600  nm diameter helium droplets are studied by femtosecond time-resolved x-ray coherent diffractive imaging. An anisotropic, ∼20  nm wide surface region, defined as the range where the density lies between 10% and 90% of the core value, is established within ∼100  fs, in qualitative agreement with theoretical predictions. At longer timescales, however, the width of this region remains largely constant while the radius of the dense plasma core shrinks at average rates of ≈71  nm/ps along and ≈33  nm/ps perpendicular to the laser polarization. These dynamics are not captured by previous plasma expansion models. The observations are phenomenologically described within a numerical simulation; details of the underlying physics, however, remain to be explored.

Noise reduction and mask removal neural network for X-ray single-particle imaging

Free-electron lasers could enable X-ray imaging of single biological macromolecules and the study of protein dynamics, paving the way for a powerful new imaging tool in structural biology, but a low signal-to-noise ratio and missing regions in the detectors, colloquially termed 'masks', affect data collection and hamper real-time evaluation of experimental data. In this article, the challenges posed by noise and masks are tackled by introducing a neural network pipeline that aims to restore diffraction intensities. For training and testing of the model, a data set of diffraction patterns was simulated from 10 900 different proteins with molecular weights within the range of 10-100 kDa and collected at a photon energy of 8 keV. The method is compared with a simple low-pass filtering algorithm based on autocorrelation constraints. The results show an improvement in the mean-squared error of roughly two orders of magnitude in the presence of masks compared with the noisy data. The algorithm was also tested at increasing mask width, leading to the conclusion that demasking can achieve good results when the mask is smaller than half of the central speckle of the pattern. The results highlight the competitiveness of this model for data processing and the feasibility of restoring diffraction intensities from unknown structures in real time using deep learning methods. Finally, an example is shown of this preprocessing making orientation recovery more reliable, especially for data sets containing very few patterns, using the expansion-maximization-compression algorithm.

Sizes of pure and doped helium droplets from single shot x-ray imaging

Advancements in x-ray free-electron lasers on producing ultrashort, ultrabright, and coherent x-ray pulses enable single-shot imaging of fragile nanostructures, such as superfluid helium droplets. This imaging technique gives unique access to the sizes and shapes of individual droplets. In the past, such droplet characteristics have only been indirectly inferred by ensemble averaging techniques. Here, we report on the size distributions of both pure and doped droplets collected from single-shot x-ray imaging and produced from the free-jet expansion of helium through a 5 μm diameter nozzle at 20 bars and nozzle temperatures ranging from 4.2 to 9 K. This work extends the measurement of large helium nanodroplets containing 10⁹-10¹¹ atoms, which are shown to follow an exponential size distribution. Additionally, we demonstrate that the size distributions of the doped droplets follow those of the pure droplets at the same stagnation condition but with smaller average sizes.

Aggregation of solutes in bosonic versus fermionic quantum fluids

Quantum fluid droplets made of helium-3 (³He) or helium-4 (⁴He) isotopes have long been considered as ideal cryogenic nanolabs, enabling unique ultracold chemistry and spectroscopy applications. The droplets were believed to provide a homogeneous environment in which dopant atoms and molecules could move and react almost as in free space but at temperatures close to absolute zero. Here, we report ultrafast x-ray diffraction experiments on xenon-doped ³He and ⁴He nanodroplets, demonstrating that the unavoidable rotational excitation of isolated droplets leads to highly anisotropic and inhomogeneous interactions between the host matrix and enclosed dopants. Superfluid ⁴He droplets are laced with quantum vortices that trap the embedded particles, leading to the formation of filament-shaped clusters. In comparison, dopants in ³He droplets gather in diffuse, ring-shaped structures along the equator. The shapes of droplets carrying filaments or rings are direct evidence that rotational excitation is the root cause for the inhomogeneous dopant distributions.

Levitodynamics: Levitation and control of microscopic objects in vacuum

[Figure: see text].

Reconstruction of nanoscale particles from single-shot wide-angle free-electron-laser diffraction patterns with physics-informed neural networks

Single-shot wide-angle diffraction imaging is a widely used method to investigate the structure of noncrystallizing objects such as nanoclusters, large proteins, or even viruses. Its main advantage is that information about the three-dimensional structure of the object is already contained in a single image. This makes it useful for the reconstruction of fragile and nonreproducible particles without the need for tomographic measurements. However, currently there is no efficient numerical inversion algorithm available that is capable of determining the object's structure in real time. Neural networks, on the other hand, excel in image processing tasks suited for such purpose. Here we show how a physics-informed deep neural network can be used to reconstruct complete three-dimensional object models of uniform, convex particles on a voxel grid from single two-dimensional wide-angle scattering patterns. We demonstrate its universal reconstruction capabilities for silver nanoclusters, where the network uncovers novel geometric structures that reproduce the experimental scattering data with very high precision.

Spontaneous Motion and Rotation of Acid Droplets on the Surface of a Liquid Metal

Self-propulsion of droplets is of great significance in many fields. The spontaneous horizontal motion and self-jumping of droplets have been well realized in various ways. However, there is still a lack of an effective method to enable a droplet to rotate spontaneously and steadily. In this paper, by employing an acid droplet and a liquid metal, the spontaneous and steady rotation of droplets is achieved. For an acid droplet, it may spontaneously move when it is deposited on the surface of the liquid metal. By adjusting experimental parameters to the proper range, the self-rotation of droplet happens. This phenomenon originates from the fluctuation of the droplet boundary and the collective movement of bubbles that are generated by the chemical reactions between the acid droplet and liquid metal. This rotation has a simpler implementation method and more steady rotation state. Its angular velocity is much higher than that driven by other mechanisms. Moreover, the movements of acid droplets on the liquid metal are classified according to experimental conditions. The internal flow fields, the movements and distribution of bubbles, and the fluctuation of the droplet boundary are also explored and discussed. The theoretical model describing the rotational droplet is given. Our work may deepen the understanding of the physical system transition affected by chemical reactions and provide a new way for the design of potential applications, e.g., micro- and nanodevices.

Fast reconstruction of single-shot wide-angle diffraction images through deep learning

Single-shot x-ray imaging of short-lived nanostructures such as clusters and nanoparticles near a phase transition or non-crystalizing objects such as large proteins and viruses is currently the most elegant method for characterizing their structure. Using hard x-ray radiation provides scattering images that encode two-dimensional projections, which can be combined to identify the full three-dimensional object structure from multiple identical samples. Wide-angle scattering using XUV or soft x-rays, despite yielding lower resolution, provides three-dimensional structural information in a single shot and has opened routes towards the characterization of non-reproducible objects in the gas phase. The retrieval of the structural information contained in wide-angle scattering images is highly non-trivial, and currently no efficient rigorous algorithm is known. Here we show that deep learning networks, trained with simulated scattering data, allow for fast and accurate reconstruction of shape and orientation of nanoparticles from experimental images. The gain in speed compared to conventional retrieval techniques opens the route for automated structure reconstruction algorithms capable of real-time discrimination and pre-identification of nanostructures in scattering experiments with high repetition rate—thus representing the enabling technology for fast femtosecond nanocrystallography.

Angular Momentum in Rotating Superfluid Droplets

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.

Alkali atoms attached to vortex-hosting helium nanodroplets

Light absorption or fluorescence excitation spectroscopy of alkali atoms attached to ⁴He droplets is investigated as a possible way for detecting the presence of vortices. To this end, we have calculated the equilibrium configuration and energetics of alkali atoms attached to a ⁴He₁₀₀₀ droplet hosting a vortex line using ⁴He density functional theory. We use them to study how the dipole absorption spectrum of the alkali atom is modified when the impurity is attached to a vortex line. Spectra are found to be blue-shifted (higher frequencies) and broadened compared to vortex-free droplets because the dimple in which the alkali atom sits at the intersection of the vortex line and the droplet surface is deeper. This effect is smaller for lighter alkali atoms and all the more so when using a quantum description since, in this case, they sit further away from the droplet surface on average due to their zero-point motion. Spectral modifications due to the presence of a vortex line are minor for np ← ns excitation and therefore insufficient for vortex detection. In the case of higher n'p ← ns or n's ← ns (n' > n) excitations, the shifts are larger as the excited state orbital is more extended and therefore more sensitive to changes in the surrounding helium density.

Imaging plasma formation in isolated nanoparticles with ultrafast resonant scattering

We have recorded the diffraction patterns from individual xenon clusters irradiated with intense extreme ultraviolet pulses to investigate the influence of light-induced electronic changes on the scattering response. The clusters were irradiated with short wavelength pulses in the wavelength regime of different 4d inner-shell resonances of neutral and ionic xenon, resulting in distinctly different optical properties from areas in the clusters with lower or higher charge states. The data show the emergence of a transient structure with a spatial extension of tens of nanometers within the otherwise homogeneous sample. Simulations indicate that ionization and nanoplasma formation result in a light-induced outer shell in the cluster with a strongly altered refractive index. The presented resonant scattering approach enables imaging of ultrafast electron dynamics on their natural timescale.

Ultrafast relaxation of photoexcited superfluid He nanodroplets

The relaxation of photoexcited nanosystems is a fundamental process of light–matter interaction. Depending on the couplings of the internal degrees of freedom, relaxation can be ultrafast, converting electronic energy in a few fs, or slow, if the energy is trapped in a metastable state that decouples from its environment. Here, we study helium nanodroplets excited resonantly by femtosecond extreme-ultraviolet (XUV) pulses from a seeded free-electron laser. Despite their superfluid nature, we find that helium nanodroplets in the lowest electronically excited states undergo ultrafast relaxation. By comparing experimental photoelectron spectra with time-dependent density functional theory simulations, we unravel the full relaxation pathway: Following an ultrafast interband transition, a void nanometer-sized bubble forms around the localized excitation (He\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{* }$$\end{document}*) within 1 ps. Subsequently, the bubble collapses and releases metastable He\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{* }$$\end{document}* at the droplet surface. This study highlights the high level of detail achievable in probing the photodynamics of nanosystems using tunable XUV pulses.

There is interest in understanding the relaxation mechanisms of photoexcitation in atoms, molecules and other complex systems. Here the authors unravel the photoexcitation and ultrafast relaxation of superfluid helium nanodroplets using a pump-probe experiment with FEL pulses.

Real-time observation of jumping and spinning nanodroplets

The manipulation of liquids at nanoscale dimensions is a central goal of the emergent nanofluidics field. Such endeavors extend to nanodroplets, which are ubiquitous objects involved in many technological applications. Here, we employ time-resolved electron microscopy to elucidate the formation of so-called jumping nanodroplets on a graphene surface. We flash-melt a thin gold nanostructure with a laser pulse and directly observe how the resulting nanodroplet contracts into a sphere and jumps off its substrate, a process that occurs in just a few nanoseconds. Our study provides the first experimental characterization of these morphological dynamics through real-time observation and reveals new aspects of the phenomenon. We observe that friction alters the trajectories of individual droplets. Surprisingly, this leads some droplets to adopt dumbbell-shaped geometries after they jump, suggesting that they spin with considerable angular momentum. Our experiments open up new avenues for studying and controlling the fast morphological dynamics of nanodroplets through their interaction with structured surfaces.

Refinement for single-nanoparticle structure determination from low-quality single-shot coherent diffraction data

With the emergence of X-ray free-electron lasers, it is possible to investigate the structure of nanoscale samples by employing coherent diffractive imaging in the X-ray spectral regime. In this work, we developed a refinement method for structure reconstruction applicable to low-quality coherent diffraction data. The method is based on the gradient search method and considers the missing region of a diffraction pattern and the small number of detected photons. We introduced an initial estimate of the structure in the method to improve the convergence. The present method is applied to an experimental diffraction pattern of an Xe cluster obtained in an X-ray scattering experiment at the SPring-8 Angstrom Compact free-electron LAser (SACLA) facility. It is found that the electron density is successfully reconstructed from the diffraction pattern with a large missing region, with a good initial estimate of the structure. The diffraction pattern calculated from the reconstructed electron density reproduced the observed diffraction pattern well, including the characteristic intensity modulation in each ring. Our refinement method enables structure reconstruction from diffraction patterns under difficulties such as missing areas and low diffraction intensity, and it is potentially applicable to the structure determination of samples that have low scattering power.

Multispectroscopic Study of Single Xe Clusters Using XFEL Pulses

X-ray free-electron lasers (XFELs) deliver ultrashort coherent laser pulses in the X-ray spectral regime, enabling novel investigations into the structure of individual nanoscale samples. In this work, we demonstrate how single-shot small-angle X-ray scattering (SAXS) measurements combined with fluorescence and ion time-of-flight (TOF) spectroscopy can be used to obtain size- and structure-selective evaluation of the light-matter interaction processes on the nanoscale. We recorded the SAXS images of single xenon clusters using XFEL pulses provided by the SPring-8 Angstrom compact free-electron laser (SACLA). The XFEL fluences and the radii of the clusters at the reaction point were evaluated and the ion TOF spectra and fluorescence spectra were sorted accordingly. We found that the XFEL fluence and cluster size extracted from the diffraction patterns showed a clear correlation with the fluorescence and ion TOF spectra. Our results demonstrate the effectiveness of the multispectroscopic approach for exploring laser–matter interaction in the X-ray regime without the influence of the size distribution of samples and the fluence distribution of the incident XFEL pulses.

Deep neural networks for classifying complex features in diffraction images

Intense short-wavelength pulses from free-electron lasers and high-harmonic-generation sources enable diffractive imaging of individual nanosized objects with a single x-ray laser shot. The enormous data sets with up to several million diffraction patterns present a severe problem for data analysis because of the high dimensionality of imaging data. Feature recognition and selection is a crucial step to reduce the dimensionality. Usually, custom-made algorithms are developed at a considerable effort to approximate the particular features connected to an individual specimen, but because they face different experimental conditions, these approaches do not generalize well. On the other hand, deep neural networks are the principal instrument for today's revolution in automated image recognition, a development that has not been adapted to its full potential for data analysis in science. We recently published [Langbehn et al., Phys. Rev. Lett. 121, 255301 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.255301] the application of a deep neural network as a feature extractor for wide-angle diffraction images of helium nanodroplets. Here we present the setup, our modifications, and the training process of the deep neural network for diffraction image classification and its systematic bench marking. We find that deep neural networks significantly outperform previous attempts for sorting and classifying complex diffraction patterns and are a significant improvement for the much-needed assistance during postprocessing of large amounts of experimental coherent diffraction imaging data.

Imaging Quantum Vortices in Superfluid Helium Droplets

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.