key: cord-0604864-ilab4nym
authors: OConnell, Sean M. O.; Tanyag, Rico Mayro P.; Verma, Deepak; Bernando, Charles; Pang, Weiwu; Bacellar, Camila; Saladrigas, Catherine A.; Mahl, Johannes; Toulson, Benjamin W.; Kumagai, Yoshiaki; Walter, Peter; Ancilotto, Francesco; Barranco, Manuel; Pi, Marti; Bostedt, Christoph; Gessner, Oliver; Vilesov, Andrey F.
title: Angular momentum in rotating superfluid droplets
date: 2019-10-28
journal: nan
DOI: nan
sha: a3c8cd10b4dcf4e39c05cb927cc5453dd0765b04
doc_id: 604864
cord_uid: ilab4nym

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 sub-micrometer superfluid 4He 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.

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 sub-micrometer 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.

A quiescent droplet is spherical, whereas it experiences pronounced shape deformations when set into rotation. Classical viscous droplets execute rigid body rotation (RBR), the kinematics of which are well understood [1] [2] [3] [4] . In contrast, a superfluid droplet is described by a quantum mechanical wave function [5, 6] and, at low temperatures, its angular momentum may be stored in quantized vortices as well as capillary waves. The rotation of superfluid droplets poses intriguing questions regarding similarities and differences between classical and quantum mechanical kinematics. Experimental studies of rotation in single, free superfluid 4 He droplets have recently become feasible with the advent of x-ray free electron lasers (XFELs) and intense extreme ultraviolet (XUV) sources [7] [8] [9] [10] . Ultrafast diffraction experiments have established the existence of oblate axisymmetric and triaxial prolate superfluid droplets with shapes that are also found in classical droplets [1] [2] [3] [4] . However, previous experiments were not able to independently probe the angular momentum and angular velocity of rotating droplets, which is imperative for understanding their underlying laws of motion.

The shapes of classical droplets undergoing RBR follow a universal stability diagram in terms of reduced angular momentum ΛRBR and reduced angular velocity ΩRBR, as shown in Fig. 1 [1] [2] [3] [4] .

The reduced quantities are linked to the angular momentum LRBR and angular velocity ωRBR 

Here, σ is the surface tension of the liquid; ρ its mass density; and R the radius of a spherical droplet with the same volume as the deformed droplet. With increasing ΛRBR, the droplets become more oblate. At ΛRBR > 1.2, however, they assume prolate shapes such as pseudo-triaxial ellipsoids, capsules and dumbbells. It is unknown whether this diagram also applies to superfluid droplets. We address this fundamental question by simultaneously determining the angular momentum, L, the angular velocity, ω, and the corresponding shapes of the rotating droplets via imaging both the shapes of 4 He droplets and the vortex configurations they contain. shapes for specific combinations of reduced angular velocity ΩRBR and reduced angular momentum ΛRBR [3, 4] . The left branch corresponds to oblate axisymmetric shapes, the right branch to prolate shapes. The unstable portion of the branch is indicated by a dashed curve. The blue curve corresponds to droplets rotating solely through capillary wave motion [11] . Black circles mark the locations of classical droplets with the same aspect ratios as the superfluid droplets studied in this work.

Rotation of prolate superfluid droplets is interesting because it can be accomplished by capillary waves, which decay rapidly in classical droplets due to viscosity-induced energy dissipation but may, in principle, be sustained indefinitely in superfluids. Capillary waves in levitating helium droplets have been investigated experimentally and theoretically [12, 13] .

Travelling capillary waves give rise to a new branch in the stability diagram as shown by the blue line in Fig. 1 [11] . Prolate superfluid droplets likely contain both capillary waves and vortices, whose combined action determines the droplet shape. The arrangement of vortices in rotating, nonaxisymmetric systems represents a general problem that has been discussed theoretically in connection with liquid helium [11, 14] and dilute Bose-Einstein condensates (BECs) [15] [16] [17] [18] . The observation, however, of vortices in rotating systems with large asymmetries remains challenging [19] .

Here, we posit that in axisymmetric, oblate superfluid droplets, angular momentum originates solely from quantized vortices that arrange in a triangular lattice [7, 20] similar to those previously observed in BECs [5, 6, 21] . In contrast, the angular momentum in prolate superfluid droplets has significant contributions from both vortices and capillary waves. We find that in capsule-shaped droplets, vortices form a triangular lattice, whereas they are arranged along elliptical contours in ellipsoidal droplets. The combined action of the vortex lattice and the capillary waves yields droplet shapes close to those in isochoric classical droplets rotating at the same angular velocity.

Experiments have been performed using the LAMP endstation at the Atomic, Molecular and Optical (AMO) beamline of the Linac Coherent Light Source (LCLS) XFEL [22, 23] . Submicrometer sized helium droplets containing NHe = 10 9 -10 11 atoms are produced via fragmentation of liquid 4 He expanded into vacuum through a 5 µm nozzle at a temperature of 4.5 K and a backing pressure of 20 bar [7, 24, 25] . The droplets evaporatively cool in vacuum to T = 0.37 K [25, 26] , becoming superfluid at T ≈ 2.17 K. In order to visualize vortices, the droplets are doped with Xe atoms, which cluster along the vortex cores [7, [27] [28] [29] [30] . The XFEL is operated at 120 Hz with a photon energy of 1.5 keV (λ = 0.826 nm) and a pulse width of < 180 fs. The ultrashort duration and high intensity of the x-ray pulses enables the instantaneous capture of the structure of individual droplets and the vortex arrays they contain. Diffraction images are recorded on a pncharge-coupled device (pn-CCD) detector that is centered along the XFEL beam axis. Since the diffraction images span only small scattering angles, they predominantly contain column density information of the scattering object in the direction perpendicular to the detector plane.

Diffraction patterns from doped droplets with slightly oblate axisymmetric, triaxial pseudoellipsoidal, and capsule shapes are presented in a logarithmic color scale in Fig. 2 A1 , B1, and C1, respectively. Figure 2 A2-C2 shows the corresponding column densities of He (blue) and Xe (red/yellow) obtained from the diffraction patterns using the droplet coherent diffractive imaging (DCDI) algorithm [28, 31] , see SM for details [32] . The diffraction pattern in Fig. 2 A1 exhibits a circular ring structure close to the center and a speckled pattern in the outer region, which is due to scattering off the He droplet and embedded Xe clusters, respectively. From the inter-ring spacing at different azimuthal angles [7, 27, 31] , the projection of the droplet contour onto the detector plane is found to be circular with a radius of 308±6 nm. As seen in Fig. 2 A2, the droplet contains 12 vortex cores, visualized by Xe filaments. The filaments are aligned along the droplet rotational axis and extend between opposite surface points. The filaments appear much shorter than the droplet diameter, indicating that the rotational axis is tilted with respect to the x-ray beam by ≈ 0.3 rad. Bragg spots each are arranged symmetrically with respect to the streak. We assign the Bragg spots to x-ray scattering off a lattice of vortex-bound Xe clusters [7] . 

in which m1 and m2 are integers, r1 = 111 nm, r2 = 100 nm, and ⃑ % and ⃑ 8 are unit vectors with ⃑ % perpendicular to the streak and ⃑ 8 oriented at a 67 o angle relative to ⃑ % as illustrated in Fig. 2 C2.

The vortex lattice is triangular, but not equilateral as in the idealized 2D case [5] and as previously observed in axisymmetric rotating He droplets [7] and BECs [5, 6, 21] . These observations suggest that the droplet shapes and vortex patterns within are closely inter-related.

We compare the angular momentum and angular velocity in superfluid droplets in Fig. 2 and (2) . The angular momentum due to vortices, LVORT, is obtained from the number of vortices, NV, and their location within the droplet [33] as detailed in the SM [32] . The angular velocity of the vortex array is obtained from the average vorticity as [34] :;"< =

where nv is the areal number density of vortices and κ = h/m4 is the quantum of circulation, derived from Planck's constant h and the mass of the 4 He atom m4.

The angular velocity of the superfluid droplet is given by ωSF = ωVORT because, at equilibrium, the vortex array must be stationary with respect to the droplet contour. The angular momentum due to capillary waves, LCAP, is obtained as the product of the effective irrotational moment of inertia of the droplets multiplied by ωSF [32, 35] . Contributions of Xe tracers to the angular momentum are neglected because the total mass of the embedded Xe atoms in Fig. 2 is approximately a factor of 100 smaller than that of the He droplet itself. Table 1 suggest that rotation of large, vortex-hosting superfluid droplets can be described in terms of a reduced angular velocity and a reduced angular momentum in a very similar fashion as the rotational motion of their classical counterparts. Details of the analysis and a comprehensive list of geometric and kinematic parameters are presented in the SM [32] , which also documents some differences between classical and superfluid shapes having the same AR.

To rationalize the similarities between the shapes of classical and superfluid droplets, we may recall that, for a given L, RBR has the smallest kinetic energy. A vortex array in the central part of a droplet creates a region of liquid that effectively moves as in RBR and minimizes the required capillary engagement. Essentially, the prolate droplet attempts to approach RBR with available means, i.e., by combining the effect of vortices and capillary waves. It appears that for each L,

there is an optimal partitioning into LVORT and LCAP that minimizes the total energy of the rotating droplet. States with larger or smaller numbers of vortices are metastable and appear in the regions in between the stability curves for capillary wave motion and axisymmetric droplets in Fig. 1 .

To ascertain how the shape of a superfluid droplet changes with the number of vortices at constant angular momentum, we have carried out density functional theory (DFT) calculations [11, 36, 37] . Due to the high computational cost, a full 3D calculation can only be performed for small droplets, which cannot support a large number of vortices [11] . Therefore, we have studied a free-standing, deformable 4 He cylinder rotating around its symmetry axis. The calculations are performed in the co-rotating frame [37] . The total angular momentum is fixed at LSF/NHe=7.83ℏ . from NV = 2 to NV = 11. The deformation of the cylinder increases for a decreasing number of vortices, as capillary waves contribute more angular momentum. We note that the energies of the configurations in Fig. 3 are rather similar; they differ less than 0.01 K per atom, with the NV = 6 configuration (Fig. 3C) representing the global minimum. Each panel in Fig. 3 shows the computed LCAP/LSF ratio and the cylinder's AR and ω. In cylinders with relatively small AR = 1.1 and 1.5, the vortices are arranged into elliptical patterns, whereas they form elongated arrays for larger ARs, in agreement with the experimental observations in Fig. 2 .

Each panel in Fig. 3 also shows the superfluid streamlines, illustrating the different irrotational velocity fields: the streamlines dominated by vortices wrap around the vortex cores, whereas those associated with capillary waves terminate at the surface. Future experiments at high-repetition rate ultrafast x-ray light sources such as the European XFEL and LCLS-II will explore a larger phase space in terms of Λ, Ω, and R. More advanced theoretical work is required to study angular velocities and angular momenta with higher accuracy and to derive a more detailed stability diagram of superfluid droplets from the measurements.

Furthermore, probing smaller droplet sizes will reveal possible finite size effects.

Previously, it was shown that densities of clusters encapsulated in helium droplets can be reconstructed from the x-ray diffraction patterns using the so-called droplet coherent diffractive imaging algorithm (DCDI) [1] . The algorithm exploits the fact that the scattering amplitude from a doped He droplet can be calculated as the sum of the known amplitude from the He droplet itself and the unknown amplitude from the dopants. Since the experiments were performed at small scattering angles [1] [2] [3] [4] , the diffraction amplitude can be well approximated by the two-dimensional Fourier Transform of the column density of the scattering object in the direction perpendicular to the detector plane. Accordingly, the DCDI algorithm requires the column density of the droplet as an input. For a spheroidal droplet, the column density can be calculated analytically from the half axes of the droplet's projection on the detector plane. The half axes are obtained from the elliptical droplet diffraction contours as described previously [1, 2, 4] . The density is then normalized to fit the intensity of the diffraction signal from the droplet [1] . However, the shape of prolate droplets cannot be obtained from the half axes because their shapes are not given by any analytic expression.

Therefore, we apply a numerical determination of the column densities for prolate droplets.

First, the contour of the droplet's projection onto the detector plane is determined. It is obtained via 2D inverse Fourier Transform (IFT) [4] . In Ref. [4] , before applying IFT, the phase was assigned to each of the diffraction contours from the droplet, which is a tedious procedure.

Therefore, in this work we employ the IFT of the square root of the diffraction intensity itself, which yields similar results in a less cumbersome manner. Because the input is positive throughout, it can be seen as a smooth positive offset function decreasing towards larger scattering angles with an oscillating component. It is easy to show that the period of oscillation is a factor of two smaller than that in the diffraction amplitude. Therefore, the obtained droplet contour is scaled down by a factor of two. Figure S1 shows the result of the application of the IFT to the square root of the diffraction intensity shown in Fig. 2 C1 of the main text. Figure S1 shows the contour of the droplet in light blue and yellow colors, the long axis of which is aligned diagonally. Intensities inside of the contour are due to the IFT of the offset function, Bragg spots and other low frequency features in the diffraction. Similar to previous work [4] , the procedure yields multiple contours because some low-frequency signals from the droplet contour are not recorded due to the central hole in the detector. The true droplet contour is expected to coincide with the center of gravity of the modulus of these contours. Figure S1 shows a set of two contours of comparable amplitude. Therefore, the droplet's contour lies in between, as indicated by the red line. Once the droplet contour is obtained, the density of the droplet is filled in axisymmetrically with respect to the long axis of the contour. This density is an approximation, because the droplet b-axis perpendicular to the rotation axis is slightly larger than the c-axis parallel to the rotation axis as noted in the main text [4] . Therefore, the actual density distribution of the prolate droplet is not strictly axisymmetric with respect to the long axis. Table S1 . Geometric and kinematic parameters for the droplets in Figure 2 A2 Table S1 shows the geometric and kinematic parameters of for the droplets in LRBR is expressed in units of ℏ per He atom, and ωRBR is expressed in radians per second. For the droplet in A2, ΛRBR and ΩRBR cannot be obtained in the same way, because its AR is unity within the experimental accuracy. Therefore, in this case, the results are based on ωSF, which is set to ωRBR from which ΛRBR and ΩRBR are deduced using eqs. S1 and S2. The angular velocity of the lattice is given as AB = = > ?

8 [5] , in which nV is the vortex density described below, and = The expression for angular momentum of NV linear vortices in spherical droplets was described by Nam et al. [6] Here, we use their expression:

in which R is the radius of the droplet, ri is the displacement of the i-vortex from the center of the droplet, ρ is the mass density of superfluid helium, and is the quantum of circulation. For this analysis, it is assumed that vortices with constant areal density fill a circular region with radius RB around the droplet center so that there is a boundary region of the droplet devoid of vortices between RB and R. In the limit of large numbers of vortices, the summation in eq. (S3) can be replaced by integration. Integration yields the angular momentum due to vortices in units of ℏ per He atom as:

"^P_" K P"`KaK

Here, we use eq. (S4) for a sphere to approximate non-spherical solutions for droplets B2 and C2. The value of RB is estimated by measuring the area filled with vortices, AV, dividing this by the area of the droplet cross section, AD, and taking the square root of the result as:

. The angular momentum of capillary waves in a triaxial ellipsoid rotating around the z-axis is given as [7] :

The estimated values of LSF contain some error due to uncertainties in determining LVORT and LCAP. One is associated with the use of eq. (S4) which is strictly only valid for spherical droplets.

No corresponding analytic expression is available for droplets of general shape. In addition, LCAP is calculated using an irrotational moment of inertia for a liquid constrained to an ellipsoid [7] , whereas the actual droplet shape is not strictly ellipsoidal. Furthermore, the values for ωSF are obtained from Feynman's equation, AB = = > ? 8 , which is valid for large 2D arrays, whereas this work considers finite 3D vortex arrays within droplets. We note, however, that the perfect match of ωSF and ωRBR in Fig. 2 C2 , where nV can be determined with high accuracy, as well as the results of the DFT calculations described in the main text, indicate that the application of Feynman's equation is likely accurate.

Upon detailed inspection, the results in Fig. 2 C of the main text indicate some deviation between the calculated RBR and measured superfluid shapes. At ΛRBR = 1.5 as in Fig. 2 C2, the classical shape has a weak depression along the b -axis, which gives rise to additional nodal structures in the diffraction signal that appear as multiple, radially emanating streaks (see SM of Bernando et al. [4] ). The diffraction pattern in Fig.2 C1 does not exhibit any such nodes, which may indicate that the presence of the vortex array and capillary waves leads to a stabilization of the straight outline of the droplet. Moreover, the diffraction maxima along the streak are not oriented perpendicular to the streak axis, as is the case for RBR shapes (see SM from Bernando et al. [4] ). Instead, the pattern is tilted counterclockwise. This tilt likely indicates that the two sides of the droplet are not exactly parallel but include a small angle on the order of a few degrees. Thus, the droplet in Fig 2 C2 is likely non-centrosymmetric, as is the case in classical droplets, which may be another indication of the interaction of the vortex lattice with the droplet shape. This wedging effect cannot be seen in the droplet contour in Fig. 2 C2 obtained from the Fourier Transform of the diffraction intensity, [4] which exclusively leads to axisymmetric shapes.

We consider a self-bound superfluid 4 He cylinder rotating around its symmetry z-axis with a constant angular velocity w. The details of the calculations are described in Ref. [8] , where vortices in axisymmetric rotating cylinders were considered. Here, we have extended that work to select configurations with the vortex-hosting cylinder allowed to deform into shapes with non-circular cross sections. We have assumed in our calculation a uniform density along the z-direction, which implies that the resulting vortices always remain linear. A complex order parameter, Y, is found to represent the superfluid helium state at zero temperature [8, 9] whose square modulus is the atomic density. The DFT equation is formulated in a rotating frame-of-reference with constant angular velocity w (co-rotating frame), which follows from the variational minimization of the energy, and is solved looking for stationary solutions. Vortex structures and positions are optimized during the functional minimization, after having been phase-imprinted, as described in

Refs. [8, 9] . Simultaneously, we allow for deviations from the circular cross section ("prolate" rotating cylinders) using the same method as previously applied to prolate spinning droplets [10] , i.e. solve the variational equation by imposing a given value for the angular momentum per atom Lz and iteratively find the associated value of the rotational frequency w. Classically, such fixed Lz calculations correspond to torque-free drops with an initially prescribed rotation ("isolated drops") and result in stable prolate configurations like the ones shown in Fig. 3 in the main text.

The Stability of a Rotating Liquid Drop

Equilibrium Configurations of Rotating Charged or Gravitating Liquid Masses with Surface-Tension .2

The Shape and Stability of Rotating Liquid-Drops

The Shape Distribution of Splash-Form Tektites Predicted by Numerical Simulations of Rotating Fluid Drops

Bose-Einstein Condensation and Superfluidity. Bose-Einstein Condensation and Superfluidity 164

Rotating Trapped Bose-Einstein Condensates

Shapes and Vorticities of Superfluid Helium Nanodroplets

Shapes of Rotating Superfluid Helium Nanodroplets

Coherent Diffractive Imaging of Single Helium Nanodroplets with a High Harmonic Generation Source

Three-Dimensional Shapes of Spinning Helium Nanodroplets

Spinning Superfluid He-4 Nanodroplets

Oscillations of Charged Helium II Drops

Cavity Optomechanics in a Levitated Helium Drop

Vortex Nucleation in Deformed Rotating Cylinders

Vortex Lattice of a Bose-Einstein Condensate in a Rotating Anisotropic Trap

Two-Dimensional Bose-Einstein Condensate Under Extreme Rotation

Vortices in a Rotating Bose-Einstein Condensate Under Extreme Elongation

Structural Change of Vortex Patterns in Anisotropic Bose-Einstein Condensates

Measurement of Equilibrium Critical Velocities for Vortex Formation in Superfluid-Helium

Vortex Arrays in Nanoscopic Superfluid Helium Droplets

Observation of Vortex Lattices in Bose-Einstein Condensates

Large-format, High-speed, X-ray pnCCDs Combined with Electron and Ion Imaging Spectrometers in a Multipurpose Chamber for Experiments at 4th Generation Light Sources

The Atomic, Molecular and Optical Science instrument at the Linac Coherent Light Source

Sizes of Large He Droplets

Cold Chemistry: Molecular Scattering and Reactivity Near Absolute Zero

Rotationally Resolved Spectroscopy of SF6 in Liquid-Helium Clusters -a Molecular Probe of Cluster Temperature

Coupled Motion of Xe Clusters and Quantum Vortices in He Nanodroplets

Imaging Quantum Vortices in Superfluid Helium Droplets

Pinning of Quantized Vortices in Helium Drops by Dopant Atoms and Molecules

Superfluid Helium -Visualization of Quantized Vortices

Communication: X-ray Coherent Diffractive Imaging by Immersion in Nanodroplets

See Supplemental Materials at //link// for additional information on the determination of the droplet parameters, some deviations between the classical and superfluid droplet shapes and details of DFT calculations

Vortex Patterns in a Freely Rotating Superfluid

Chapter II Application of Quantum Mechanics to Liquid Helium

Capture of Xe and Ar Atoms by Quantized Vortices in He-4 Nanodroplets

Density Functional Theory of Doped Superfluid Liquid Helium and Nanodroplets

Vortex Arrays in a Rotating Superfluid He-4

References

Communication: X-ray Coherent Diffractive Imaging by Immersion in Nanodroplets

Shapes and Vorticities of Superfluid Helium Nanodroplets

Coupled Motion of Xe Clusters and Quantum Vortices in He Nanodroplets

Shapes of Rotating Superfluid Helium Nanodroplets

Chapter II Application of Quantum Mechanics to Liquid Helium

Vortex Patterns in a Freely Rotating Superfluid

Capture of Xe and Ar Atoms by Quantized Vortices in He-4 Nanodroplets

Vortex Arrays in a Rotating Superfluid He-4

Density Functional Theory of Doped Superfluid Liquid Helium and Nanodroplets

Spinning Superfluid He-4 Nanodroplets