Theories and numerical methods

On this page, I present a list of the theories and numerical methods that I have been exploring in connection with my research activities.

Highlights

• I am responsible for the first application of the Landauer-Büttiker formalism to massive bosonic systems.
  
  
• I have written my own full implementations of ...
  
  
  1. The coupled-channel approach for the numerical description of ultracold atomic collisions
     using the accumulated-phase method [Verhaar et al, PRA 79, 032711 (2009)].
  
  
  2. A numerical solver for the Gross-Pitaevskii equation in 1D, 2D, and axial 3D geometries,
     using a Crank-Nicolson ADI scheme
     (in collaboration with T. Congy).

Atomic and molecular physics

Theory

• Quantum scattering theory at low energies in 3D, in 1D, and in effective 1D geometries.
  Single-channel, two-channel, and multi-channel scattering.


• Inelastic processes.


• WKB method


• Symmetries and selection rules for atomic states and molecular terms,
  Hund's coupling cases and recoupling phenomena


• Dipole-dipole interaction (anisotropic and long-range)

Numerics

• Full implementation of the coupled-channel approach with the accumulated-phase method (cf. Highlights)


• Discrete and continuous spectrum of a linear differential operator:
  shooting and relaxation methods, matching conditions near a fitting point

Many-body physics

Theory

• Quantum harmonic lattice vibrations and dispersion relations


• Lindemann criterion for the dislocation of a crystal lattice


• Born-Oppenheimer approach, Bethe-Peierls boundary conditions


• Tight-binding model


• Quantum lattice models (extended Bose-Hubbard family)

Numerics

• Exact numerical diagonalization of small-sized lattice models


• Numerical calculation of Bloch and Wannier functions


• Lattice sums


• Full implementation of a (classical) molecular dynamics simulation, using the Velocity-Verlet algorithm

Dynamics and thermodynamics of superfluids

Theory

• Hydrodynamics and two-fluid theory for quantum fluids


• Thermodynamics of Helium 4 and Bose gases,
  thermomechanical effects


• Gross-Pitaevskii description for Bose gases,
  Thomas-Fermi limit


• Lower effective dimensionalities: quasi-2D and quasi-1D systems.


• Exact scaling solutions to the time-dependent Gross-Pitaevskii equation.

Numerics

• Full implementation of a Gross-Pitaevskii solver
  in 1D, 2D, and axial 3D geometries
  using a Crank-Nicolson ADI scheme (cf. Highlights)


• Solution of linear and non-linear PDEs:
  Spectral methods, finite-difference methods,
  approximate factorization and alternating-direction schemes

Transport properties of cold gases

Theory

• Landauer-Büttiker formalism applied to Fermi and Bose gases

Quantum optics

Theory

• Non-linear phenomena
• Two-photon interference processes