We show that sizeable Abelian and non-Abelian gauge fields arise in the relative quantum motion of two dipole-dipole interacting Rydberg atoms. Our system exhibits two magnetic monopoles for adiabatic motion in one internal two-atom state. These monopoles occur at a characteristic distance between the atoms that is of the order of one micron. The deflection of the relative motion due to the Lorentz force gives rise to a clear signature of the broken symmetry in our system. In addition, we consider non-adiabatic transitions between two near-degenerate internal states and show that the associated gauge fields are non-Abelian. We present quantum mechanical calculations of this synthetic spin-orbit coupling and show that it realizes a velocity-dependent beamsplitter.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

The study of impurities immersed in a Bose-Einstein condensate (BEC) has become an active field of research during the past few years both on the theoretical and experimental side. In my talk I will present theoretical results on the behaviour of impurities obtained within the framework of coupled Gross-Pitaevskii-Schrödinger (GPS) equations. This approach describes effects on the impurity such as self-trapping, breathing oscillations and induced impurity-impurity interactions. I will show that variational and numerical solutions of the coupled GPS equations provide an intuitive physical picture of the statics and dynamics of the impurity and the BEC.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Realizing building blocks of photon quantum optics for matter waves is a long-standing goal. We present an efficient source for twin-atom beams, in analogy to parametric down-conversion in non-linear optics. The source shows strong non-classical correlations in the population of the two beams, - 10dB below the classical limit. We also realized an integrated Mach-Zehnder interferometer for matter waves by combining a spatial beam splitter for BEC, a gravity-dependent phase-shifter and a recombined based on a pulsed Josephson tunnel junction. The intrinsic non-linearity of the matter waves leads to number squeezing in the splitting process and to fundamental phase diffusion in the interferometer sequence. We will discuss performance limits towards matter wave metrology.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

I will present results (obtained with Jerry Bona and Christof Sparber) on the dispersive blow-up phenomenum for various Schrödinder type equations (both linear and nonlinear). Those results might be one explanation to optical rogue waves formation.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

In this talk, I will present an averaging procedure, – namely Stroboscopic averaging –, for highly-oscillatory evolution equations posed in a Hilbert space, typically partial differential equations (PDEs) in a high-frequency regime where only one frequency is present. A high order averaged system is constructed, whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, . . . ) as compared to the original system, and is non-oscillatory. The results will illustrated numerically in the case of the Gross-Pitaesvkii equation. Joint works with: F. Castella, P. Chartier, A. Murua, Y. Zhang and N. Mauser

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

In this work, I will discuss the extension of stroboscopic averaging to quasi-periodic highly-oscillatory differential equations and envisage their application to partial differential equations (PDEs) in a high-frequency regime where only a finite number of frequencies are present. The application of these resuts to Gross-Pitaesvkii equation will be envisaged.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Lower dimensional physical systems often exhibit stronger quantum behavior in comparison with high dimensional ones. Quantum gases of cold atoms can be confined in traps with effectively low spatial dimension. In this talk, I will discuss the properties of a 1D gas of two-component fermions. When the population in the two components are unequal, such a system supports a ground state that is the analog of the so-called Fulde-Ferrel-Larkin-Ovchinnikov state, an exotic superfluid state with finite-momentum Cooper pairs. Through both a mean-field Bogoliubov-de Gennes study and an essentially exact numerical investigation (TEBD), we show how FFLO can manifest itself in the expansion dynamics of the gas.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Click here to see the abstract

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

We present a rigorous dimension reduction analysis for the three dimensional (3D) Sch"{o}dinger equation with the Coulombic interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in two or one space dimensions and obtain rigorously the surface adiabatic model (SAM) or surface density model (SDM) in two dimensions (2D) and the line adiabatic model (LAM) in one dimension (1D). Efficient and accurate numerical methods for computing ground states and dynamics of the SAM, SDM and LAM models are presented based o n efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models. They are applied to find numerically convergence and convergence rates for the dimension reduction from 3D to 2D and 3D to 1D in terms of ground state and dynamics, which confirm some existing analytical results for the dimension reduction in the literatures. In particular, we explain and demonstrate that the standard Sch-Poisson system in 2D is not appropriate to simulate a ``2D electron gas" of point particles confined into a plane (or more general a 2D manifold), whereas SDM should be the correct model to be used for describing the Coulomb interaction in 2D in which the square root of Laplacian operator is used instead of the Laplacian operator. Finally, we report ground states and dynamics of the SAM and SDM in 2D and LAM in 1D under different setups.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

The aim of this talk is to develop some robust and accurate computational methods for solving Bose-Einstein condensates. Most particularly, we are interested in the case where fast rotations arise as well as strong nonlinear interactions. We consider single and multi components BEC. Furthermore, we will give some numerical examples computed by GPELab which is a freely available Matlab toolbox currently developed in collaboration with Xavier Antoine and Jean-Marc Sac-Epée (IECL).

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

In this talk, I shall primarily address the issue of efficient numerical methods for the space and time discretisation of nonlinear Schrödinger equations such as systems of coupled time-dependent Gross–Pitaevskii equations arising in quantum physics for the description of multi-component Bose–Einstein condensates. For the considered class of problems, a variety of contributions confirms the favourable behaviour of pseudo-spectral and exponential operator splitting methods regarding efficiency and accuracy. However, in the absence of an adaptive local error control in space and time, the reliability of the numerical solution and the performance of the space and time discretisation strongly depends on the experienced scientist selecting the space and time grid in advance. I will illustrate the reliable time integration of Gross–Pitaevskii equations on the basis of a local error control for splitting methods and indicate the main tools for a stability and error analysis justifying the use of the employed space and time discretisations.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Click here to see the abstract

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

In this talk, we give a review of our Bloch-decomposition based time-splitting spectral method to conduct numerical simulations of the dynamics of (non)linear Schroedinger equations subject to periodic and confining potentials. We consider this system as a two-scale asymptotic problem with different scalings of the nonlinearity. Moreover we demonstrate the superiority of our method over the classical pseudo-spectral method in many physically relevant situations. We also extended/coupled with other methods to the simulation of other wave type equations with periodic coefficients.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Fourier series serve as a powerful tool for finding approximate numerical solutions to partial differential equations. This talk will discuss the use of collocation methods to investigate solutions to partial differential equations, including the Kohn-Muller, Aviles-Giga and Klein-Gordon equations. Of particular interest is asymptotic behavior when there is a large or small coefficient. The implementation of these methods on parallel computers will also be addressed.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

We discuss the time integration of nonlinear Schrödinger equations by high-order splitting methods. The convergence is analyzed first for the semi-discretization in a general Banach space framework. For the Gross-Pitaevskii equation with rotation term, a generalized Laguerre-Fourier-Hermite method is employed for the full discretization. The convergence of this method is established theoretically and illustrated by numerical examples. To obtain efficient integrators, adaptive time-stepping is introduced. As a basis, two classes of local error estimators based on embedded pairs of splitting formulae and the defect correction principle are put forward and their asymptotical correctness is demonstrated. Numerical examples illustrate the success of the solution approach.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

Exterior Complex Scaling and Perfectly Matched Layers are two related methods for artificial absorption on bounded computational domains. The differences in the theory of both methods are discussed and the consequences of these for applicability, stability and accuracy of both methods.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

This work is devoted to the numerical simulation of nonlinear Schrödinger equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency. This is a stronger future than the usual so called "Asymptotic preserving" property, the last being therefore satisfied by our scheme in the highly oscillatory limit. Our strategy enables to simulate the oscillatory problem without using any mesh or time step refinement, and the order of the scheme is preserved in all regimes. In other words, since our numerical method is not based on the derivation and the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, in the highly oscillatory limit regime, and in the intermediate regime as well. The method is based on a "double-scale" reformulation of the original equation, with the introduction of an additional variable. Then a link is made with classical strategies based on Chapman-Enskog expansions in kinetic theory despite of the dispersive context of the Schrödinger equation.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

We introduce a new class of geometric numerical integrators for the time integration of highly oscillatory systems of differential equations using large time steps. These methods are based on composition methods and can be considered as numerical homogenization integrators. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results and the versatility of the methods. This is joint work with P. Chartier, J. Makazaga, and A. Murua.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

We analyse a concrete example to show how to uniformize a two-phase WKB function by applying an appropriate "asymptotic surgery" of its Wigner transform

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

This work is motivated by the description of the classical limit of the Schrodinger equation in terms of Wigner measures. Specifically, we study the structure of the Wigner measure at time t corresponding to a WKB ansatz for the initial wave function. We also provide information on the number of folds in the Lagrangian manifold associated to the propagated measure. Our theory applies to situations where the momentum profile is continuous with derivatives in some appropriate Lorenz space. Finally we give information on the evolution under the dynamics of the Schrodinger equation in classical scaling of a WKB ansatz that cannot be attained with the usual WKB theory for lack of regularity on the initial phase function. (Work in collaboration with C. Bardos, P. Markowich and T. Paul).

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)

This variant of the Vlasov equation is dubbed Vlasov-Dirac-Benney because the original potential is replaced by a Dirac mass and because it is very similar to the Benney equation used in water waves. Beside shaving a broad range of applications it is really at the cross road of different techniques in partial differential equations ranging from non linear hyprbolic problems to spectral theory, integrability and semi classical limit.

• Thematic program: Confined Quantum Systems (2012, Prolongation from 2011)
• Event: Workshop on "Confined Quantum Systems: Modeling, Analysis and Computation" (2013)