9 research outputs found
Efficient Manipulation of Bose-Einstein Condensates in a Double-Well Potential
We pose the problem of transferring a Bose-Einstein Condensate (BEC) from one side of a double-well potential to the other as an optimal control problem for determining the time-dependent form of the potential. We derive a reduced dynamical system using a Galerkin truncation onto a finite set of eigenfunctions and find that including three modes suffices to effectively control the full dynamics, described by the Gross-Pitaevskii model of BEC. The functional form of the control is reduced to finite dimensions by using a Galerkin-type method called the chopped random basis (CRAB) method, which is then optimized by a genetic algorithm called differential evolution (DE). Finally, we discuss the the extent to which the reduction-based optimal control strategy can be refined by means of including more modes in the Galerkin reduction
Identification of vortices in quantum fluids: finite element algorithms and programs
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
A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation
We present a new numerical system using classical finite elements with mesh
adaptivity for computing stationary solutions of the Gross-Pitaevskii equation.
The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free
finite-element software available for all existing operating systems. This
offers the advantage to hide all technical issues related to the implementation
of the finite element method, allowing to easily implement various numerical
algorithms.Two robust and optimised numerical methods were implemented to
minimize the Gross-Pitaevskii energy: a steepest descent method based on
Sobolev gradients and a minimization algorithm based on the state-of-the-art
optimization library Ipopt. For both methods, mesh adaptivity strategies are
implemented to reduce the computational time and increase the local spatial
accuracy when vortices are present. Different run cases are made available for
2D and 3D configurations of Bose-Einstein condensates in rotation. An optional
graphical user interface is also provided, allowing to easily run predefined
cases or with user-defined parameter files. We also provide several
post-processing tools (like the identification of quantized vortices) that
could help in extracting physical features from the simulations. The toolbox is
extremely versatile and can be easily adapted to deal with different physical
models
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
BEC2HPC: a HPC spectral solver for nonlinear Schrödinger and Gross-Pitaevskii equations. Stationary states computation
International audienceWe present BEC2HPC which is a parallel HPC spectral solver for computing the ground states of the nonlinear Schrödinger equation and the Gross-Pitaevskii equation (GPE) modeling rotating Bose-Einstein condensates (BEC). Considering a standard pseudo-spectral discretization based on Fast Fourier Transforms (FFTs), the method consists in finding the numerical solution of the energy functional minimization problem under normalization constraint by using a preconditioned nonlinear conjugate gradient method. We present some numerical simulations and scalability results for the 2D and 3D problems to obtain the stationary states of BEC with fast rotation and large nonlinearities. The code takes advantage of existing HPC libraries and can itself be leveraged to implement other numerical methods like e.g. for the dynamics of BECs
Raman measurements on plasmon-phonon coupled systems
In this thesis, Raman spectroscopy is used to characterize the interaction between a plasmon and the lattice vibration of a solid state material. Two systems have been analyzed: the first is composed of a metallic nanostrucure and a carbon material (carbon nanotubes and graphene), the second consists of beryllium-doped gallium arsenide nanowires. In the first system, additionally to the electromagnetic enhancement, a cooperative process (dynamical back-action) between the localized surface plasmon-polariton and the lattice vibration can occur. This process leads to a non-linear response of the Raman signal in dependence on the laser power. In this work the occurrence of this non-linearity is experimentally observed and compared with the theoretical prediction. In the second system, the charge-carriers provided by the dopant act as a plasma, interacting with the electric field related to the longitudinal phonon mode of the crystal lattice. This interaction causes a change in the position and width of the Raman peak, which can consequently been used to extrapolate the change carriers concentration and mobility. The appearance of surface phonons, typical of nanostructures, is also observed and discussed