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We present a finite element toolbox for the computation of Bogoliubov-de Gennes modes used to assess the linear stability of stationary solutions of the Gross-Pitaevskii (GP) equation. Applications concern one (single GP equation) or two-component (a system of coupled GP equations) Bose-Einstein condensates in one, two and three dimensions of space. An implementation using the free software FreeFem++ is distributed with this paper. For the computation of the GP stationary (complex or real) solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicated libraries for the associated eigenvalue problem. Mesh adaptivity is proved to considerably reduce the computational time for cases implying complex vortex states. Programs are validated through comparisons with known theoretical results for simple cases and numerical results reported in the literature., Comment: 7 figures

We present finite-element numerical algorithms for the identification of vortices in quantum fluids described by a macroscopic complex wave function. Their implementation using the free software FreeFem++ is distributed with this paper as a post-processing toolbox that can be used to analyse numerical or experimental data. Applications for Bose-Einstein condensates (BEC) and superfluid helium flows are presented. Programs are tested and validated using either numerical data obtained by solving the Gross-Pitaevskii equation or experimental images of rotating BEC. Vortex positions are computed as topological defects (zeros) of the wave function when numerical data are used. For experimental images, we compute vortex positions as local minima of the atomic density, extracted after a simple image processing. Once vortex centers are identified, we use a fit with a Gaussian to precisely estimate vortex radius. For vortex lattices, the lattice parameter (inter-vortex distance) is also computed. The post-processing toolbox offers a complete description of vortex configurations in superfluids. Tests for two-dimensional (giant vortex in rotating BEC, Abrikosov vortex lattice in experimental BEC) and three-dimensional (vortex rings, Kelvin waves and quantum turbulence fields in superfluid helium) configurations show the robustness of the software. The communication with programs providing the numerical or experimental wave function field is simple and intuitive. The post-processing toolbox can be also applied for the identification of vortices in superconductors.

Numerical methods for solving the Navier-Stokes equations for classical (or normal) viscous fluids are well established. This is also the case for the Gross-Pitaevskii equation, governing quantum inviscid flows (or superfluids) in the zero temperature limit. In quantum flows, like liquid helium II at intermediate temperatures between zero and 2.17 K, a normal fluid and a superfluid coexist with independent velocity fields. The most advanced existing models for such systems use the Navier-Stokes equations for the normal fluid and a simplified description of the superfluid, based on the dynamics of quantized vortex filaments, with ad hoc reconnection rules. There was a single attempt (C. Coste, The European Physical Journal B - Condensed Matter and Complex Systems, 1998) to couple Navier-Stokes and Gross-Pitaevskii equations in a global model intended to describe the compressible two-fluid liquid helium II. We present in this contribution a new numerical model to couple a Navier-Stokes incompressible fluid with a Gross-Pitaevskii superfluid. Coupling terms in the global system of equations involve new definitions of the following concepts: the regularized superfluid vorticity and velocity fields, the friction force exerted by quantized vortices to the normal fluid, the covariant gradient operator in the Gross-Pitaevskii based on a slip velocity respecting the dynamics of vortex lines in the normal fluid. A numerical algorithm based on pseudo-spectral Fourier methods is presented for solving the coupled system of equations.Finally, we numerically test and validate the new numerical system against well-known benchmarks for the evolution in a normal fluid of different types or arrangements of quantized vortices (vortex crystal, vortex dipole and vortex rings). The new coupling model opens new possibilities to revisit and enrich existing numerical results for complex quantum fluids., Comment: 32 pages, 7 figures

Self-organized silicide nanowires are considered as main building blocks of future nanoelectronics and have been intensively investigated. In nanostructures, the lattice vibrational waves (phonons) deviate drastically from those in bulk crystals, which gives rise to anomalies in thermodynamic, elastic, electronic, and magnetic properties. Hence, a thorough understanding of the physical properties of these materials requires a comprehensive investigation of the lattice dynamics as a function of the nanowire size. We performed a systematic lattice dynamics study of endotaxial FeSi$_2$ nanowires, forming the metastable, surface-stabilized $\alpha$-phase, which are in-plane embedded into the Si(110) surface. The average widths of the nanowires ranged from 24 to 3 nm, their lengths ranged from several $\mu$m to about 100 nm. The Fe-partial phonon density of states, obtained by nuclear inelastic scattering, exhibits a broadening of the spectral features with decreasing nanowire width. The experimental data obtained along and across the nanowires unveiled a pronounced vibrational anisotropy that originates from the specific orientation of the tetragonal $\alpha$-FeSi$_2$ unit cell on the Si(110) surface. The results from first-principles calculations are fully consistent with the experimental data and allow for a comprehensive understanding of the lattice dynamics of endotaxial silicide nanowires., Comment: 9 pages, 7 figures, 3 tables

We determined the lattice dynamics of metastable, surface-stabilized $\alpha$-phase FeSi$_2$ nanoislands epitaxially grown on the Si(111) surface with average heights and widths ranging from 1.5 to 20 nm and 18 to 72 nm, respectively. The crystallographic orientation, surface morphology and local crystal structure of the nanoislands were investigated by reflection high-energy electron diffraction, atomic force microscopy and X-ray absorption spectroscopy. The Fe-partial phonon density of states (PDOS), obtained by nuclear inelastic scattering, exhibits a pronounced damping and broadening of the spectral features with decreasing average island height. First-principles calculations of the polarization-projected Si- and Fe-partial phonon dispersions and PDOS enable the disentanglement of the contribution of the $xy$- and $z$-polarized phonons to the experimental PDOS. Modeling of the experimental data with the theoretical results unveils an enhanced damping of the $z$-polarized phonons for islands with average sizes below 10 nm. This phenomenon is attributed to the fact that the low-energy $z$-polarized phonons couple to the low-energy surface/interface vibrational modes. The thermodynamic and elastic properties obtained from the experimental data show a pronounced size-dependent behavior., Comment: 13 pages, 9 figures, 3 tables

Measurement modalities in Bragg coherent diffraction imaging (BCDI) rely on finding signal from a single nanoscale crystal object, which satisfies the Bragg condition among a large number of arbitrarily oriented nanocrystals. However, even when the signal from a single Bragg reflection with (hkl) Miller indices is found, the crystallographic axes on the retrieved three-dimensional (3D) image of the crystal remain unknown, and thus, localizing in reciprocal space other Bragg reflections becomes in reality impossible or requires good knowledge of the orientation of the crystal. We report the commissioning of a movable double-bounce Si (111) monochromator at the 34-ID-C end station of the Advanced Photon Source, which aims at delivering multi-reflection BCDI as a standard tool in a single beamline instrument. The new instrument enables this through rapid switching from monochromatic to broadband (pink) beam permitting the use of Laue diffraction to determine crystal orientation. With a proper orientation matrix determined for the lattice, one can measure coherent diffraction near multiple Bragg peaks, thus providing sufficient information to image the full strain tensor in 3D. We discuss the design, concept of operation, the developed procedures for indexing Laue patterns, and automated measuring of Bragg coherent diffraction data from multiple reflections of the same nanocrystal.