After the workshop, we will gather the pdf presentation for each talk.
- BAO Weizhu
(University of Singapore).
Modeling, analysis and simulation for degenerate dipolar quantum gas.
Abstract,
Slides
In this talk, I will present our recent work on mathematical
models, asymptotic analysis and numerical simulation for degenerate
dipolar quantum gas. As preparatory steps, I begin with the
three-dimensional Gross-Pitaevskii equation with a long-range
dipolar interaction potential which is used to model the degenerate
dipolar quantum gas and reformulate it as a Gross-Pitaevskii-Poisson
type system by decoupling the two-body dipolar interaction potential
which is highly singular into short-range (or local) and long-range
interactions (or repulsive and attractive interactions).
Based on this new mathematical formulation, we prove rigorously existence
and uniqueness as well as nonexistence of the ground states, and
discuss the existence of global weak solution and finite time blowup
of the dynamics in different parameter regimes of dipolar quantum
gas. In addition, a backward Euler sine pseudospectral method is
presented for computing the ground states and a time-splitting sine
pseudospectral method is proposed for computing the dynamics of
dipolar BECs. Due to the adoption of new mathematical formulation,
our new numerical methods avoid evaluating integrals with high
singularity and thus they are more efficient and accurate than those
numerical methods currently used in the literatures for solving the
problem. In addition, new mathematical formulations in two-dimensions
and one dimension for dipolar quantum gas are obtained when the external
trapping potential is highly confined in one or two directions.
Numerical results are presented to confirm
our analytical results and demonstrate the efficiency and accuracy
of our numerical methods. Some interesting physical phenomena are discussed too.
- BARENGHI Carlo
(University of Newcastle, UK).
Superfluid turbulence near a boundary.
Abstract,
Slides
In recent years, a number of experimental, numerical and theoretical
investigations have revealed an interesting similarity between
homogeneous isotropic turbulence in ordinary fluids and turbulence in
quantum fluids (superfluid helium and atomic Bose-Einstein
condensates). This similarity is striking because quantum fluids
differ from ordinary fluids in two important respects: the absence of
viscosity (superfluidity) and the nature of the vorticity (which is
concentrated along thin filaments of fixed circulation). In this talk
I shall describe a recent investigation of superfluid flow near a
boundary performed by numerically solving the Gross-Pitaevskii
equation. The surprising result is the emergence of a turbulent
boundary layer, despite the absence of viscosity.
- BRACHET Marc
(ENS Paris, France).
Dual cascade and dissipation mechanisms in helical quantum turbulence
.
Abstract,
Slides
While in classical turbulence helicity depletes nonlinearity and can
alter the evolution of turbulent flows, in quantum turbulence its
role is not fully understood. We present numerical simulations of
the free decay of a helical quantum turbulent flow using the
Gross-Pitaevskii equation at high spatial resolution. The evolution
has remarkable similarities with classical flows, which go as far
as displaying a dual transfer of incompressible kinetic energy and
helicity to small scales. Spatiotemporal analysis indicates that
both quantities are dissipated at small scales through nonlinear
excitation of Kelvin waves and the subsequent emission of
phonons. At the onset of the decay, the resulting turbulent flow
displays polarized large scale structures and unpolarized patches
of quiescence reminiscent of those observed in simulations of
classical turbulence at very large Reynolds numbers.
- CAI Yongyong
(Beijing Computational Science Center, China).
Ground state problem in two component Bose-Einstein condensates.
Abstract,
Slides
We study analytically and asymptotically as well as numerically the
nonlinear eigenvaule problem (ground state) of two-component
spin-orbit-coupled Bose-Einstein condensates (BECs) modeled by the
coupled Gross-Pitaevskii equations (CGPEs). In fact, due to the
appearance of the spin-orbit (SO) coupling in the two-component BEC
with a Raman coupling, the ground state structures become very rich
and complicated. For the ground states, we establish the existence
and non-existence results under different parameter regimes, and
obtain their limiting behaviors and/or structures with different
combinations of the SO and Raman coupling strengths. We will start
with two-component BEC without SO coupling and investigate its
limiting behavior and phase separation.
- CANCÈS Eric
(École des Ponts ParisTech, CERMICS).
Numerical analysis of linear and nonlinear eigenvalue problems.
Abstract,
Slides
In this talk, I will review some recent advances on the numerical
analysis of linear and nonlinear elliptic eigenvalue problems. I
will first present the derivation of accurate, fully computable,
lower and upper bounds of the exact eigenvalues of the Laplace
operator on a polyhedral bounded domain with Dirichlet boundary
conditions, for both conforming and non-conforming discretization
methods. I will then focus on the nonlinear Gross-Pitaevskii
equation, and show that a priori error analysis results can be used
to construct more efficient numerical schemes for computing the
steady states of Bose-Einstein condensates.
- CARLES Rémi
(Univ Montpellier).
Universal dynamics for the logarithmic Schrodinger equation.
Abstract,
Slides
We consider the nonlinear Schrodinger equation with a logarithmic
nonlinearity, whose sign is such that no non-trivial stationary
solution exists. Explicit computations show that in the case of
Gaussian initial data, the presence of the nonlinearity affects the
large time behaviour of the solution, on at least three aspects. The
dispersion is faster than usual by a logarithmic factor in time. The
positive Sobolev norms of the solution grow logarithmically in
time. Finally, after rescaling in space by the dispersion rate, the
modulus of the solution converges to a universal Gaussian profile
(whose variance is independent of the initial variance). In the case
of general initial data, we show that these properties remain, up to
weakening the third point (weak convergence instead of strong
convergence). One of the key steps of the proof for the last point
consists in using the Madelung transform. It reduces the equation to a
variant of the isothermal compressible Euler equation, whose large
time behaviour turns out to be governed by a parabolic equation
involving a Fokker-Planck operator.
- CARRETERO-GONZÁLEZ Ricardo
(San Diego State University, USA).
Vortex Rings in Bose-Einstein Condensates.
Abstract,
Slides
We review recent results for the emergence, existence, dynamics and
interactions of vortex rings in Bose-Einstein condensates (BECs)
modelled by the Gross-Pitaevskii equation (GPE). We focus our
attention on the two opposite regimes of low and high atomic density
limits in the BEC as well as in the intermediate transition between
these two limits. In the low density limit, corresponding to the
linear limit, we study the emergence of single and multiple vortex
rings emanating from planar 3D dark solitons through bifurcations. We
characterize such bifurcations quantitatively using a Galerkin-type
approach, and find good qualitative and quantitative agreement with
our Bogoliubov-de Gennes (BdG) numerical analysis. Under appropriate
conditions for the trapping strengths, we find that vortex rings might
be stabilized for large enough atomic densities (large chemical
potentials). On the other hand, in the large density limit, the
vortex rings acquire stability and are effectively robust coherent
structures. We study different single and multi-vortex-ring
configurations together with their (normal) modes of vibration. Exotic
structures such as Hopfions, the one-component counterpart to
Skyrmions, are also constructed and tested for stability. Finally, we
discuss some interactions dynamics between vortex rings such as
periodic leapfrogging of co-axial vortex rings and the scattering
behavior for co-planar collisions between vortex rings.
- CHEVY Frédéric
(ENS Paris).
Revisiting Landau's critical velocity.
Abstract,
Slides
The celebrated Landau's argument predicts that superfluid flows can
only exist below a certain critical velocity above which dissipation
reappears. In this talk I will review the experimental effort to
demonstrate Landau's argument in ultracold atoms . I will show that in
most experiments the critical velocity is lower than expected and I
will show that this discrepancy can be attributed to the breakdown of
Landau's argument underlying assumptions due to the external trap
confining the atoms.
- DANAILA Ionut
(University of Rouen Normandy).
A finite-element toolbox for the simulation of Bose-Einstein condensates.
Abstract,
Slides
We present several numerical tools using classical finite elements with mesh adaptivity for solving
different models used for the study of Bose-Einstein condensates. The programs are written as a toolbox
for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms [1].
For solving the stationary (imaginary-time) Gross-Pitaevskii equation, we use two robust and optimised numerical methods:
a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization
library IPOPT. A very recent conjugate-gradient method using concepts of Riemannian optimization is also presented [2].
For the Bogoliubov-de Gennes system, representing a linearisation of the Gross-Pitaevskii equation, a Newton method
and a fast algorithm based on ARPACK for the calculation of eigenvalues are available. For the real-time Gross-Pitaevskii
equation, classical splitting and relaxation methods were implemented and intensively tested.
Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments:
single-line vortex, Abrikosov lattice, giant vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates [3].
This is a joint work with P.E. Emeriau, F. Hecht, P. Kevrekidis, B. Protas, G. Vergez.
[1] G. Vergez, I. Danaila, S. Auliac, F.Hecht, A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation,
Computer Physics Communications, 209, p. 144-162, 2016.
[2] I. Danaila, B. Protas, Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization, arXiv:1703.07693v1, 2017.
[3] I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent
Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016.
- IGNAT Radu
(Institut de Mathématiques de Toulouse).
Interaction energy between vortices of vector fields on surfaces.
Abstract,
Slides
We study a variational Ginzburg-Landau type model depending on a small
parameter ε>0 for (tangent) vector fields on a 2-dimensional
Riemannian surface.
As ε tends to zero, the vector
fields tend to be of unit length and will have singular points of a
(non-zero) index, called vortices.
Our main result determines
the interaction energy between these vortices as a Gamma-limit (at the
second order) as ε tends to zero.
In particular, this
points out how the Gauss curvature of the surface influences the
location of vortices. We also prove similar results for problems
involving vector fields on compact surfaces embedded in R³.
This is joint work with Robert Jerrard (Toronto).
- KOBAYASHI Michikazu
(Univ Kyoto, Japan).
Energy and helicity cascade in non-Abelian quantum turbulence.
Abstract,
Slides
Non-Abelian quantum turbulence is defined as quantum turbulence
comprised of non-Abelian vortices having topological charges
classified by the non-Abelian fundamental group.
Sinse reconnection dynamics breaking the helicity conservation
is prohibited for non-Abelian vortices, helicity cascades to small
scales with conserved in non-Abelian quantum turbulence, giving
the energy spectrum with the power of -7/3.
Energy is also conserved and we can also observe the cascade of the
energy giving the energy spectrum with the power of -5/3.
The direction of the energy cascade is opposite to the helicity cascade:
the energy cascade to larger scales. Our prediction can be tested
for the cyclic phase of spin-2 spinor Bose-Einstein condensates
- KOMINEAS Stavros
(University of Crete, Greece).
Continuous and discontinuous solitons in polariton condensates.
Abstract,
Slides
Bose-Einstein condensates of exciton-polaritons are described by a
Schrodinger system of two equations for the wavefunctions of the
excitons and the photons. The system is nonlinear due to exciton
interactions. We have derived the non-traveling soliton solution for
the one-dimensional lossless system. We will present in detail the
frequency bands of dark soliton solutions. For positive detuning
(photon frequency higher than exciton frequency), there is a frequency
band for which the exciton wavefunction becomes discontinuous when the
operating frequency exceeds the exciton frequency. The exciton
wavefunction is discontinuous at its symmetry point, where it
undergoes a phase jump of pi. A band of ordinary (continuous) dark
solitons merges with the band of discontinuous dark solitons, forming
a larger band over which the soliton far-field amplitude varies from
zero to infinity. This phenomenon lies outside the parameter regime of
validity of the Gross-Pitaevskii (GP) model.
Within its regime of validity, we give a derivation of a single-mode
GP model from the initial Schrodinger system and compare the
continuous polariton solitons and GP solitons using the healing length
notion. Finally, we will discuss a second band of dark solitons, and
the bands of bright soliton solutions.
- KROPIELNICKA Karolina
(University of Gdańsk & Polish Academy of Sciences Poland).
Decomposition methods for the time dependent linear Schrödinger equation.
Abstract,
Slides
Precise discretization in time of the linear, time dependent Schrödinger equation has been
a subject of interest for many recent years. This task becomes particularly difficult if
potential also depends on time. In this talk I will present a commutator-free (thus computationally
cheap) Magnus based methods that can be applied in this case. I will also present asymptotic
symmetric splitting, which is especially useful in semiclassical regime. Based on this approach
and on the 4^{th} order compact method for Schrödinger equation, some new approach towards
6^{th} order splitting methods will be presented, which seems to be very efficient not only
in semiclassical regime.
- LEVEQUE Emmanuel
(ENS Lyon).
Lattice Boltzmann modeling of superfluid counterflow in a pipe.
Abstract,
Slides
In collaboration with P-E Roche (Institut Néel Grenoble)
Simulations of the two-fluid model of superfluid helium (He-II)
encompassing the HVBK mutual coupling are developed based on the
lattice Boltzmann method. A Boussinesq-like hypothesis is introduced
to omit temperature variations along the pipe. The thermo-mechanical
forcings of the normal and superfuid components are taken into account
by a pressure term related to their mass-density variations under an
approximation of weak compressibility. This modeling framework allows
us to easily reproduce the essential features of a thermally-driven
counterflow of superfluid helium.
- MADAY Yvon
(Université Pierre et Marie Curie, LJLL).
Exercises in Style about post processing methods for getting more from the discretization to nonlinear eigenvalue problems.
Abstract,
Slides
A priori, a posteriori, post processing, adaptation approaches are classical techniques for improving the discretization of partial differential equations by going one step further and get a more accurate approximation with roughly the same computational cost. These approaches have been imported and extended to problems in quantum chemistry and variations of those have been proposed from simple toy problems like one dimensional GP problem to more involved KS problems.
We present a comprehensive survey of some of these approaches that allow to compare them both at the level of the techniques, the implementation, and the accuracy they lead to.
- OSTERMANN Alexander
(University of Innsbruck, Austria).
A splitting approach for the magnetic Schrödinger equation.
Abstract,
Slides
The Schrödinger equation in the presence of an external electromagnetic field is an important problem in computational
quantum mechanics. It also provides a nice example of a differential equation where it is advantageous to employ operator
splitting and to split the flow into three physically different parts.
The treatment of the advection part requires special care in order not to lose the conservation properties of the scheme.
We discuss several options. Numerical examples in one, two and three space dimensions show that the method of
characteristics coupled with a non-equispaced fast Fourier transform (NFFT) provides a fast and reliable technique
for achieving mass conservation at the discrete level.
This is joint work with M. Caliari (Verona) and C. Piazzola (Innsbruck).
- PARNAUDEAU Philippe
(University of Poitiers, France).
High performance computing of the 3D Bose Einstein condensates.
This talk has been given by I. Danaila
Abstract,
Slides
We present a new robust and scalable parallel code called GPS (Gross-Pitaevskii Simulator). GPS is designed
to efficiently run on various computing systems, from personal computers to massive parallel high-performance
architectures. The code can be used to numerically solve different forms of the Gross-Pitaevskii (GP)
mean-field equation: time-independent (stationary) or time-dependent (real-time), with or without rotation.
GPS can be thus used to investigate various phenomena related to Bose-Einstein condensates (BEC) or other
quantum systems (superfluid Helium): vortex arrangements in ground or meta-stable states of rotating systems,
dynamics of quantized vortices, quantum turbulence, etc.
Two high-order space discretisations are implemented within GPS: Fourier spectral and 6th order compact
finite-difference schemes (with spectral accuracy). This allows to impose either periodic boundary conditions
(Fourier spectral) or homogeneous Dirichlet (compact schemes). Several recent numerical methods are used in
GPS to solve the stationary GP equation (a semi-implicit backward-Euler scheme and a Sobolev gradient descent
method) and the time-dependent GP equation (relaxation, splitting and Crank-Nicolson methods).
Capabilities of GPS are highlighted in this paper by simulating several challenging 2D and 3D configurations
of rotating BEC and two-component BEC.
[Joint work with Jean-Marc Sac-Epée, Atsushi Suzuki and Ionut Danaila]
- PROTAS Bartosz
(McMaster, Canada).
Computation of Ground States of the Gross-Pitaevskii Functional
via Riemannian Optimization.
Abstract,
Slides
This presentation concerns a novel approach to the
computation of ground states in Bose-Einstein condensates where 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 in the system
constraints the minimizers to lie on a Riemannian manifold
corresponding to the unit L2 norm. The idea developed in our study is
to transform the original constrained optimization problem to an
unconstrained problem on this (spherical) Riemannian manifold, so that
faster minimization algorithms can be applied. We first obtain Sobolev
gradients using an equivalent definition of an H^1 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. Then, we use the concept of the Riemannian vector transport
to propose a new 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. Numerical tests carried out in the finite-element
setting based on Lagrangian piecewise quadratic space discretization
demonstrate that the proposed RCG method outperforms the simple
gradient descent RG method in terms of rate of convergence. The RCG
method is extensively tested by computing complicated vortex
configurations in rotating Bose-Einstein condensates, a task made
challenging by large values of the non-linear interaction constant and
the rotation rate. Finally, we will also discuss our on-going work on
the design and implementation of the Riemannian Newton method for the
minimization of the Gross-Pitaevskii energy functional.
[Joint work with Ionut Danaila from Université de Rouen]
- PU Han
(Rice University).
Synthetic spin-orbit coupling in mixtures of spinor superfluids.
Abstract,
Slides
Laser field can induce an effective coupling between the (pseudo-)spin
degrees of the atom and its external motion. This synthetic
spin-orbit coupling (SOC) has attracted much attention over the past
few years. Another new development in cold atoms is the creation of
mixtures of superfluids, particularly mixtures of Bose-Fermi
superfluids. This offers the opportunity of exploring SOC in a
superfluid mixtures. In this talk, I will discuss a mixture of
spin-1/2 Bose superfluids, with one of the species subject to
laser-induced SOC. This system exhibits very rich phase diagram, some
of which are unique to the mixture.
- SAITO Hiroki
(Tokyo).
Matter-wave droplets in a dipolar Bose-Einstein condensate.
Abstract,
Slides
Recently, Bose-Einstein condensates of dysprosium atoms have been
realized, which have a large magnetic dipole-dipole interaction (DDI).
In the experiments, anisotropic self-trapped condensates were
observed, which however cannot be reproduced by the standard
mean-field theory with DDI. I will show you various theoretical
approaches to explain the observed phenomena in this system with
strong DDI.
- SCHMELCHER Peter
(Hamburg, Germany).
Applications of the Multi-Layer Multi-Configuration Time-Dependent
Hartree Method for Bosons to the Nonequilibrium Quantum Dynamics
of Ultracold Systems.
Abstract,
Slides
The multi-layer multi-configuration time-dependent Hartree method for
bosons represents a powerful ab initio method for the investigation of
the non-equilibrium quantum dynamics of single and multi-species
bosonic systems in traps and optical lattices [1,2]. After briefly
introducing the method we discuss a number of different applications
ranging from the correlated quantum dynamics in optical lattices to
beyond mean-field behaviour of solitons and collisionally coupled
correlated species. Firstly we demonstrate [3] in a `bottom-up
approach' the correlated many-particle effects in the collective
breathing dynamics for few- to many-boson systems in a harmonic
trap. Many-body processes in black and grey matter-wave solitons are
explored thereby demonstrating that quantum fluctuations limit the
lifetime of the soliton contrast, which increases with increasing
soliton velocity [4]. For atomic ensembles in optical lattices we
explore the interaction quench induced multimode dynamics leading to
the emergence of density-wave tunneling, breathing and cradle-like
processes. An avoided-crossing in the respective frequency spectrum
provides to a beating dynamics for selective modes [5,6].
A particular far from equilibrium system is then studied at hand of
the correlated quantum dynamics of a single atom collisionally coupled
to a finite bosonic reservoir [7].
In the last part of the presentation we provide some selective aspects
of our recent investigations on atom-ion hybrid systems [8-9] using
the same methodology. First the ground state properties of ultracold
trapped bosons with an immersed ionic impurity are discussed.
Subsequently the capture dynamics of ultracold atoms in the presence
of the impurity ion is explored.
References
[1] L. Cao, S. Krönke, O. Vendrell and P. Schmelcher, NJP 15, 063018 (2013)
[2] L. Cao, S. Krönke, O. Vendrell and P. Schmelcher, JCP 139, 134103 (2013)
[3] R. Schmitz, S. Krönke, L. Cao and P. Schmelcher, PRA 88, 043601 (2013)
[4] S. Krönke and P. Schmelcher, PRA 91, 053614 (2015)
[5] S. Mistakidis, L. Cao and P. Schmelcher, JPB 47, 225303 (2014)
[6] S.I. Mistakidis, L. Cao and P. Schmelcher, PRA 91, 033611 (2015)
[7] S. Krönke, J. Knörzer and P. Schmelcher, NJP 17, 053001 (2015)
[8] J.M. Schurer, P. Schmelcher and A. Negretti, PRA 90, 033601 (2014)
[9] J.M. Schurer, P. Schmelcher and A. Negretti, NJP 17, 083024 (2015)
- THALHAMMER Mechthild
(University of Innsbruck, Austria).
Local error control for operator splitting methods
.
Abstract,
Slides
Exponential operator splitting methods are known to be favourable
integration methods for time-dependent nonlinear Schrödinger
equations. In this talk, I will review different approaches for the
design and theoretical analysis of local error estimators: embedded
splitting methods, defect-based a posteriori local error estimators,
and a recently developed strategy with the advantage that the
additional computational cost is negligible. Numerical examples
illustrate the capability of a local error control to enhance
reliability and efficiency of the time integration.
- WEN Zaiwen
(Peking University, Beijing, China).
Adaptive Regularized Newton Method for Riemannian Optimization.
Abstract,
Slides
Optimization on Riemannian manifold widely arises in eigenvalue
computation, density functional theory, Bose-Einstein condensates, low
rank nearest correlation, image registration and signal processing,
etc. We propose an adaptive regularized Newton method which
approximates the original objective function by the second-order
Taylor expansion in Euclidean space but keeps the Riemannian manifold
constraints. Both global convergence and superlinear local convergence
rate are guaranteed under mild conditions. Our algorithm is very
promising in extensive experiments compared with a few state-of-art
methods.