Research

Scientific program

Context

Since its first experimental realization in 1995 [1, 2, 3], the Bose-Einstein condensation (BEC) phenomena provides an incredible glimpse into the macroscopic quantum world and has opened a new era in atomic and molecular physics as well as in condensed matter physics. It regains vast interests and has been extensively studied both experimentally and theoretically [4, 5, 6, 7, 8]. In particular, future high-tech applications are expected like for example for high precision GPS or quantum computers. At temperatures T much smaller than the critical temperature Tc, the properties of a rotating BEC is well described by a macroscopic complex-valued wave function ψ(x, t) whose evolution is governed by the celebrating three-dimensional (3D) Gross-Pitaevskii equation (GPE). Solving the d-dimensional (d = 2 or 3) dimensionless GPE with a rotation term [M17, 9] leads to the following initial-value problem : for a given initial state ψ 0 , find the complex-valued wave function ψ(x, t) solution to

i ∂tψ(x, t) = [ − 1/2 ∇^2 + V(x, t) − ω L_z + β|ψ|^2 ] ψ(x, t), (2.1)
ψ(x, t = 0) = ψ_0(x), x ∈ R^d, t ≥ 0,

where x := (x, y, z) (:= (x, y) in 2D) and t are the space and time variables, respectively. Denoting by ∇ the gradient operator, ∇^2 is then the laplacian operator, and V(x, t) is a function corresponding to the potential. The real-valued constants β and ω respectively represent the nonlinear interaction strength and the rotating frequency. In addition, L_z = i(y ∂x − x ∂y) is the z-component of the angular momentum [M17, 9]. When ω = 0, the GPE is also often called nonlinear cubic Schrödinger equation. Being able to simulate BEC through the GPE is crucial since the experimental realization of a condensate is complex but also very fragile since the quantum behavior is destroyed almost instantaneously when the system interacts with the exterior (for example when imaging the BEC). Therefore, the numerical simulation of such complex structures is a serious experimental way to understand and manipulate BECs.

The goal of our associate team is to bring all together our high expertise on developing accurate and efficient numerical methods for solving the GPE and to build the corresponding HPC solver. Most specifically, we will consider the rotating GPE (2.1) as a first model. Indeed, solving this problem is already extremely challenging, most particularly for large nonlinearities β and fast rotations ω. In these situations, the BEC has many vortices, translating the multiscale nature of the quantum problem, which are extremely complicate to capture. For the stationary states, the problem can be recast as a constrained optimization problem of the
associated nonconvex and strongly nonlinear energy functional [M17, 9]. In particular, we are often interested in computing the ground state, which is the global minimum of the energy, or the first excited states, corresponding to higher energy local minima. Since small features must be captured, high-order discretization techniques are required. Here, we will focus on pseudospectral FFT-based methods where we have a strong expertise, preconditioned normalized gradient flows [M3, M4, M6, M12, M17, M32, M33, M34, 9] and conjugate gradients [M10,M40], accelerated by multigrid algorithms. Another problem is to compute the dynamics of the GPE. Then, one has to derive high-order time and space adaptive schemes that also conserve some physical quantities like e.g. the mass or energy [M2, 9]. We have a strong knowledge [M6, M9, M17, M24, M27, M28, M29, M30, M36, M37] in methods that can achieve this goal for 4the rotating GPE but which still need to be further investigated in a HPC environment. Even if the rotating GPE already includes some complicate problems related to quantum physics, mathematics and parallel computing, other more advanced BEC models will be further investigated, including multi-components gazes [M4, M6, M9, M17, M29, M33], nonlocal nonlinear interactions for dipolar gazes [M4, M6, M23, M26, M28], fractional models [M14, M41] to translating the multiscale behavior of the underlying physical background. In addition, most of the time, open systems must be considered. For this reason (and also computational aspects), we will develop new PML-based (Perfectly Matched Layer) algorithms [M11, M18, M19, M21] in pseudospectral techniques for simulating the unbounded characteristic of the spatial domain. Finally, pseudospectral domain decomposition techniques [M5, M7, M8, M13, M15, M16, M20] will be maybe required to accelerate the HPC efficiency of our algorithms.

Objectives

We have three main objectives for the associate team

  1. Advanced numerical methods for the GPE : The aim of this first objective is related to the design of new efficient, robust and high-order pseudospectral methods for solving GPEs for both stationary states and dynamics of BECs.
  2. HPC implementation and testing : The aim of this second objective is to implement and test in a HPC environment all the numerical methods developed during objective 1.
  3. HPC software production for the physics community to solve realistic quantum problems : The objective 3 is devoted to the design of a flexible and easy-to-use software for people in quantum physics in cold matter.

References

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    • [M12] X. Antoine, C. Besse, R. Duboscq and V. Rispoli, Acceleration of the Imaginary Time Method for Spectrally Computing Stationary States of Gross-Pitaevskii Equations, Computer Physics, 219, (2017), pp. 70-78.
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    • [M15] X. Antoine, F. Hou and E. Lorin, Asymptotic Estimates of the Convergence of Classical Schwarz Waveform Relaxation Domain Decomposition Methods for Two-Dimensional Stationary Quantum Waves, to appear in M2AN, 2018.
    • [M16] X. Antoine and E. Lorin, Multilevel Preconditioning Techniques for Schwarz Waveform Relaxation Domain Decomposition Methods Applied to Real- and Imaginary-time Nonlinear Schrödinger Equations, Applied Mathematics and Computation, 336 (1), (2018), pp. 403-417.
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    • [M18] A. Modave, J. Lambrechts, C. Geuzaine, Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations, Computers and Mathematics with Applications 73(4), (2017), pp. 684-700.
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    • [M21] A. Modave, A. Kameni, J. Lambrechts and C. Geuzaine, An optimum PML for scattering problems in the time domain, European Journal Applied Physics, 64(2) (2013), Article Number : UNSP 24502.
    • [M22] W. Bao, N. J. Mauser, H. Jian, Y. Zhang, Dimension reduction of the Schrödinger equation with Coulomb and anisotropic confining potentials, SIAM Journal on Applied Mathemathics, 73 (6) (2013), pp.2100-2123.
    • [M23] L. Exl, N. J Mauser and Y. Zhang, Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation, Journal of Computational Physics, 327 (2016), pp. 629-642.
    • [M24] Z. Ma, Y. Zhang and Z. Zhou, An improved semi-Lagrangian time splitting spectral method for the Schrödinger equation with vector potentials using NUFFT, Applied Numerical Mathematics, 111 (2017), pp. 144-159.
    • [M25] N. Mauser, H.-P. Stimming and Y. Zhang, A novel nonlocal potential solver based on nonuniform FFT for efficient simulation of the Davey-Stewartson equations, ESAIM : Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 1527–1538.
    • [M26] L. Greengard, S. Jiang and Y. Zhang, The anisotropic truncated kernel method for convolution with free-space Green’s functions, SIAM Journal on Scientific Computing, accepted.
    • [M27] W. Bao, Z. Xu and Q. Tang, Numerical methods and comparisions for computing dark and bright solitons in nonlinear Schrödinger equation, Journal of Computational Physics, 235 (2013), pp. 423–445.
    • [M28] W. Bao, D. Marahrens, Q. Tang and Y. Zhang, A simple and efficient numerical method for computing dynamics of rotating dipolar Bose–Einstein condensation via a rotating Lagrange coordinate, SIAM Journal on Scientific Computing, 35 (2013), pp. A2671–A2695.
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    • [M30] W. Bao and Q. Tang, Numerical study of quantized vortex dynamics and interaction in nonlinear Schrödinger equation on bounded domains, Multiscale Modeling and Simulation : a SIAM Interdiscipli- nary Journal, 12 (2014), pp. 411–439.
    • [M31] H. Wang and Y. Xiang, An adaptive level set method based on two-level uniform meshes and its application to dislocation dynamics, International Journal for Numerical Methods in Engineering, 94 (2013), pp. 573-597.
    • [M32] Y. Cai and H. Wang, Analysis and computation for ground state solutions of Bose-Fermi mixtures at zero temperature, SIAM Journal on Applied Mathematics, 73 (2013), pp. 757-779.
    • [M33] S. Song, Y. Zhang, L. Wen and H. Wang, Spin-orbit coupling induced displacement and hidden spin textures in spin-1 Bose-Einstein condensates, J Phys. B : At. Mol. Opt. Phys., 46 (2013), pp. 145304 (1-10).
    • [M33] H. Wang, A projection gradient method for computing ground state of spin-2 Bose-Einstein conden- sates, Journal of Computational Physics, 274 (2014), pp. 473-488.
    • [M34] H. Wang and Z. Xu, Projection gradient method for energy functional minimization with a constraint and its application to computing the ground state of spin-orbit-coupled Bose-Einstein condensates, Com- puter Physics Communications, 185 (2014), pp. 2803-2808.
    • [M35] J. Bai, H. Wang, Q. Wang, D. Zhou, K. Q. Le, and B. Wang, Coherent pulse progression of mid-infrared quantum-cascade lasers under group-velocity dispersion and Self-Phase modulation, IEEE Journal of Quantum Electronics, 52 (2016), pp. 2300106.
    • [M36] H. Wang, X. Ma, J. Lu, W. Gao, An efficient time-splitting compact finite difference method for Gross-Pitaevskii equation, Applied Mathematics and Computation, 297 (2017), pp. 131-144.
    • [M37] H. Wang, A splitting compact finite difference method for computing the dynamics of dipolar Bose- Einstein condensate, International Journal of Computer Mathematics, 94 (2017) pp. 2027-2040.
    • [M38] H. Wang, Z. Liang and, R. Liu, A splitting Chebyshev collocation method for Schrödinger-Poisson system, Computational and Applied Mathematics, 37 (2018), pp. 5034-5057.
    • [M39] J. Bai, H. Wang, J. Zhang and F. Liu, A comparative study of effects of group-velocity dispersion on mid-infrared quantum-cascade lasers with Fabry-Perot and ring cavities. Journal of Nanophotonics, 12 (2018), 026003.
    • [M40] X. Antoine, Q. Tang and Y. Zhang, A Preconditioned Conjugated Gradient method for computing ground states of rotating dipolar Bose-Einstein condensates via kernel truncation method for Dipole- Dipole Interaction evaluation, Communication in Computational Physics, 24(4) 2018, pp. 966-988.
    • [M41] X. Antoine, Q. Tang and Y. Zhang, On the ground states and dynamics of space fractional non- linear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, Journal of Computational Physics, 325 (2016), pp. 74-97.

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