The microhydration of rock salt (NaCl) molecules was investigated theoretically by density-functional theory and force field calculations and experimentally by high-resolution Penning ionization electron spectroscopy (PIES) in helium nanodroplets. The calculations reveal a transition from contact ion pair structures to solvent-separated ion pairs at -15. However, it takes water molecules to form a complete solvation shell around the Cl- anion and as many as to fully hydrate the Na+ cation, thus the entire NaCl molecule. Although NaCl molecules are predicted to be fully submerged inside the droplets, the PIES of NaCl are highly resolved, in stark contrast to other molecular species. Codoping the droplets with a controlled number of -10 water molecules leads to efficient quenching of the NaCl Penning ionization signal and to its full suppression for n greater than or similar to 30, in line with the calculations.

The microhydration of rock salt (NaCl) molecules was investigated using high-resolution Penning ionization electron spectroscopy (PIES) in helium nanodroplets. Although model calculations predict that NaCl molecules are fully submerged inside the droplets, PIES of NaCl are highly resolved, in stark contrast to other molecular species. Co-doping the droplets with a controlled number of n=5–10 water molecules leads to efficient quenching of the NaCl Penning ionization signal and to its full suppression for n≳ 30. Accompanying density-functional theory (DFT) and force field calculations reveal a transition from contact ion pair structures to solvent-separated ion pairs at n=12–15. However, it takes n≈ 17 water molecules to form a complete solvation shell around the Cl^- anion and as many as n≈ 34 to fully hydrate the Na^+ cation, thus the entire NaCl molecule, which rationalizes the experimental findings.

The solvation process of an alkali ion (Na+, K+, Rb+, and Cs+) inside a superfluid 4He2000 nanodroplet is investigated theoretically using liquid 4He time-dependent density functional theory at zero temperature. We simulate both steps of the pump-probe experiment conducted on Na+ [Albrechtsen et al., Nature 623, 319 (2023)], where the alkali atom residing at the droplet surface is ionized by the pump pulse and its solvation is probed by ionizing a central xenon atom and detecting the expulsed Na+Hen ions. Our results confirm the Poissonian model for the binding of the first five He atoms for the lighter Na+ and K+ alkalis, with a rate in good agreement with the more recent experimental results on Na+ [Albrechtsen et al., J. Chem. Phys. 162, 174309 (2025)]. For the probe step, we show that the ion takes several picoseconds to get out of the droplet. During this rather long time, the solvation structure around it is very hot and far from equilibrium, and it can gain or lose more He atoms. Surprisingly, analyzing the Na+ solvation structure energy reveals that it is not stable by itself during the first few picoseconds of the solvation process. After that, energy relaxation follows a Newton behavior, as found experimentally, but with a longer time delay, 5.0 ≤ t0 ≤ 6.5 ps vs 0.23 ± 0.06 ps, and characteristic time, 7.3 ≤ τ ≤ 16.5 ps vs 2.6 ± 0.4 ps. We conclude that the first instants of the solvation process are highly turbulent and that the solvation structure is stabilized only by the surrounding helium "solvent."

We study superfluid helium droplets multiply charged with Na+ or Ca+ ions. When stable, the charges are found to reside in equilibrium close to the droplet surface, thus representing a physical realization of Thomson's model. We find the minimum radius of the helium droplet that can host a given number of ions using a model whose physical ingredients are the solvation energy of the cations, calculated within the helium density functional theory approach, and their mutual Coulomb repulsion energy. Our model goes beyond the often used liquid drop model, where charges are smeared out either within the droplet or on its surface, and which neglects the solid-like helium shell around the ions. We find that below a threshold droplet radius R0, the total energy of the system becomes higher than that of the separated system of the pristine helium droplet and the charges embedded in their solvation microcluster ("snowball"). However, the ions are still kept within the droplet by the presence of energy barriers, which hinder Coulomb explosion. A further reduction of the droplet radius below a value Rexpl eventually results in the disappearance of such barrier, leading to Coulomb explosion. Surprisingly, our results are rather insensitive to the ion atomic species. This makes room to discuss them in the context of intrinsic multicharged helium droplets, where the charges are triatomic He3+ ions. Our calculated values for Rexpl display the correct scaling with the number of cations compared to the available experimental results, at variance with other estimates for the critical radii.

The sinking of alkali cations in superfluid 4He nanodroplets is investigated theoretically using liquid 4He time-dependent density functional theory at zero temperature. The simulations illustrate the dynamics of the buildup of the first solvation shell around the ions. The number of helium atoms in this shell is found to linearly increase with time during the first stages of the dynamics. This points to a Poissonian capture process, as concluded in the work of Albrechtsen et al. on the primary steps of Na+ solvation in helium droplets [Albrechtsen et al., Nature 623, 319 (2023)]. The energy dissipation rate by helium atom ejection is found to be quite similar between all alkalis, the main difference being a larger energy dissipated per atom for the lighter alkalis at the beginning of the dynamics. In addition, the number of helium atoms in the first solvation shell is found to be lower at the end of the dynamics than at equilibrium for both Li+ and Na+, pointing to a kinetic rather than thermodynamical control of the snowball size for small and strongly attractive ions.

Solvation is a ubiquitous phenomenon in the natural sciences. At the macroscopic level, it is well understood through thermodynamics and chemical reaction kinetics 1 , 2 . At the atomic level, the primary steps of solvation are the attraction and binding of individual molecules or atoms of a solvent to molecules or ions of a solute 1 . These steps have, however, never been observed in real time. Here we instantly create a single sodium ion at the surface of a liquid helium nanodroplet 3 , 4 , and measure the number of solvent atoms that successively attach to the ion as a function of time. We found that the binding dynamics of the first five helium atoms is well described by a Poissonian process with a binding rate of 2.0 atoms per picosecond. This rate is consistent with time-dependent density-functional-theory simulations of the solvation process. Furthermore, our measurements enable an estimate of the energy removed from the region around the sodium ion as a function of time, revealing that half of the total solvation energy is dissipated after four picoseconds. Our experimental method opens possibilities for benchmarking theoretical models of ion solvation and for time-resolved measurements of cation–molecule complex formation.

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.

Due to an error in the assigned value to the k = 0 component of the FFT of the screened Lennard-Jones potential, the DFT results shown in this article 1 are affected to some extent.Here Fig. 1 presents the correct appearance of Fig. 5

We have studied self-sustained, deformable, rotating liquid He cylinders of infinite length. In the normal fluid $^3$He case, we have employed a classical model where only surface tension and centrifugal forces are taken into account, as well as the Density Functional Theory (DFT) approach in conjunction with a semi-classical Thomas-Fermi approximation for the kinetic energy. In both approaches, if the angular velocity is sufficiently large, it is energetically favorable for the $^3$He cylinder to undergo a shape transition, acquiring an elliptic-like cross section which eventually becomes two-lobed. In the $^4$He case, we have employed a DFT approach that takes into account its superfluid character, limiting the description to vortex-free configurations where angular momentum is exclusively stored in capillary waves on a deformed cross section cylinder. The calculations allow us to carry out a comparison between the rotational behavior of a normal, rotational fluid ($^3$He) and a superfluid, irrotational fluid ($^4$He).

We have investigated how the Rayleigh-Plateau instability of a filament made of a 41K-87Rb self-bound mixture may lead to an array of identical quantum droplets, with typical breaking times which are shorter than the lifetime of the mixture. If the filament is laterally confined -- as it happens in a toroidal trap -- and atoms of one species are in excess with respect to the optimal, equilibrium ratio, the droplets are immersed into a superfluid background made by the excess species which provides global phase coherence to the system, suggesting that the droplets array in the unbalanced system may display supersolid character. This possibility has been investigated by computing the non-classical translational inertia coefficient. The filament may be a reasonable representation of a self-bound mixture subject to toroidal confinement when the bigger circle radius of the torus is much larger than the filament radius.

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.

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 anti-vortex 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.

The clustering, collision, and relaxation dynamics of pristine and doped helium nanodroplets is theoretically investigated in cases of pickup and clustering of heliophilic argon, collision of heliophobic cesium atoms, and coalescence of two droplets brought into contact by their mutual long-range van der Waals interaction. Three approaches are used and compared with each other. The He time-dependent density functional theory method considers the droplet as a continuous medium and accounts for its superfluid character. The ring-polymer molecular dynamics method uses a path-integral description of nuclear motion and incorporates zero-point delocalization while bosonic exchange effects are ignored. Finally, the zero-point averaged dynamics approach is a mixed quantum-classical method in which quantum delocalization is described by attaching a frozen wavefunction to each He atom, equivalent to classical dynamics with effective interaction potentials. All three methods predict that the growth of argon clusters is significantly hindered by the helium host droplet due to the impeding shell structure around the dopants and kinematic effects freezing the growing cluster in metastable configurations. The effects of superfluidity are qualitatively manifested by different collision dynamics of the heliophilic atom at high velocities, as well as quadrupole oscillations that are not seen with particle-based methods, for droplets experiencing a collision with cesium atoms or merging with each other.

Within density functional theory, we have studied the interplay between vortex arrays and capillary waves in spinning prolate He-4 droplets made of several thousand helium atoms. Surface capillary waves are ubiquitous in prolate superfluid He-4 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 He-4 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.

The relaxation dynamics of superexcited superfluid He nanodroplets is thoroughly investigated by means of extreme-ultraviolet (XUV) femtosecond electron and ion spectroscopy complemented by time-dependent density functional theory (TDDFT). Three main paths leading to the emission of electrons and ions are identified: droplet autoionization, pump-probe photoionization, and autoionization induced by re-excitation of droplets relaxing into levels below the droplet ionization threshold. The most abundant product ions are He2+, generated by droplet autoionization and by photoionization of droplet-bound excited He atoms. He+ appear with some pump-probe delay as a result of the ejection He atoms in their lowest excited states from the droplets. The state-resolved time-dependent photoelectron spectra reveal that intermediate excited states of the droplets are populated in the course of the relaxation, terminating in the lowest-lying metastable singlet and triplet He atomic states. The slightly faster relaxation of the triplet state compared to the singlet state is in agreement with the simulation showing faster formation of a bubble around a He atom in the triplet state.

We present an experimental study of the dynamics following the photoexcitation and subsequent photoionization of single Cs atoms on the surface of helium nanodroplets. The dynamics of excited Cs atom desorption and readsorption as well as CsHe exciplex formation are measured by using femtosecond pump-probe velocity map imaging spectroscopy and ion time-of-flight spectrometry. The time scales for the desorption of excited Cs atoms off helium nanodroplets as well as the time scales for CsHe exciplex formation are experimentally determined for the 6p states of Cs. For the 6p 2Π1/2 state, our results confirm that the excited Cs atoms only desorb from the nanodroplet when the excitation wavenumber is blue-shifted from the 6p 2Π1/2 ← 6s 2Σ1/2 resonance. Our results suggest that the dynamics following excitation to the 6p 2Π3/2 state can be described by an evaporation-like desorption mechanism, whereas the dynamics arising from excitation to the 6p 2Σ1/2 state is indicative for a more impulsive desorption process. Furthermore, our results suggest a helium-induced spin-orbit relaxation from the 6p 2Σ1/2 state to the 6p 2Π1/2 state. Our findings largely agree with the results of time-dependent 4He density functional theory (DFT) simulations published earlier [Eur. Phys. J. D 2019, 73, 94].

We study the properties of singly-quantized linear vortices in the supersolid phase of a dipolar Bose-Einstein condensate at zero temperature modeling $^{164}$Dy atoms. The system is extended in the $x-y$ plane and confined by a harmonic trap in the the polarization direction $z$. Our study is based on a generalized Gross-Pitaevskii equation. We characterize the ground state of the system in terms of spatial order and superfluid fraction and compare the properties of a single vortex and of a vortex dipole in the superfluid phase (SFP) and in the supersolid phase (SSP). At variance with a vortex in the SFP, which is free to move in the superfluid, a vortex in the SSP is localized at the interstitial sites and does not move freely. We have computed the energy barrier for motion from an equilibrium site to another. The fact that the vortex is submitted to a periodic potential has a dramatic effect on the dynamics of a vortex dipole made of two counter rotating parallel vortices; instead of rigidly translating as in the SFP, the vortex and anti-vortex approach each other by a series of jumps from one site to another until they annihilate in a very short time and their energy is transferred to bulk excitations.

Vortices are expected to exist in a supersolid but experimentally their detection can be difficult because the vortex cores are localized at positions where the local density is very low. We address here this problem by performing numerical simulations of a dipolar Bose-Einstein Condensate (BEC) in a pancake confinement at $T=0$ K and study the effect of quantized vorticity on the phases that can be realized depending upon the ratio between dipolar and short-range interaction. By increasing this ratio the system undergoes a spontaneous density modulation in the form of an ordered arrangement of multi-atom "droplets". This modulated phase can be either a "supersolid" (SS) or a "normal solid" (NS). In the SS state droplets are immersed in a background of low-density superfluid and the system has a finite global superfluid fraction resulting in non-classical rotational inertia. In the NS state no such superfluid background is present and the global superfluid fraction vanishes. We propose here a protocol to create vortices in modulated phases of dipolar BEC by "freezing" into such phases a vortex-hosting superfluid (SF) state. The resulting system, depending upon the interactions strengths, can be either a SS or a NS To discriminate between these two possible outcome of a "freezing" experiment, we show that upon releasing of the radial harmonic confinement, the expanding vortex-hosting SS shows tell-tale quantum interference effects which display the symmetry of the vortex lattice of the originating SF, as opposed to the behavior of the NS which shows instead a ballistic radial expansion of the individual droplets. Such markedly different behavior might be used to prove the supersolid character of rotating dipolar condensates.

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.