Propagating the Time Dependent Schroedinger Equation

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Presentation transcript:

Propagating the Time Dependent Schroedinger Equation B. I. Schneider Division of Advanced Cyberinfrastructure National Science Foundation September 6, 2013

What Motivates Our Interest Novel light sources: ultrashort, intense pulses  Nonlinear (multiphoton) laser-matter interaction Attosecond pulses probe and control electron dynamics XUV + IR pump-probe Free electron lasers (FELs) Extreme intensities  Multiple XUV photons

Basic Equation Possibly Non-Local or Non-Linear Where

Properties of Classical Orthogonal Functions

More Properties

Matrix Elements

Properties of Discrete Variable Representation

Its Actually Trivial

Multidimensional Problems Tensor Product Basis Consequences

Multidimensional Problems Two Electron matrix elements also ‘diagonal” using Poisson equation

Finite Element Discrete Variable Representation Properties Space Divided into Elements – Arbitrary size “Low-Order” Lobatto DVR used in each element: first and last DVR point shared by adjoining elements Elements joined at boundary – Functions continuous but not derivatives Matrix elements requires NO Quadrature – Constructed from renormalized, single element, matrix elements Sparse Representations – N Scaling Close to Spectral Accuracy

Finite Element Discrete Variable Representation Structure of Matrix

Time Propagation Method Diagonalize Hamiltonian in Krylov basis Few recursions needed for short time- Typically 10 to 20 via adaptive time stepping Unconditionally stable Major step - matrix vector multiply, a few scalar products and diagonalization of tri-diagonal matrix

Putting it together for the He Code NR1 NR2 Angular Linear scaling with number of CPUs Limiting factor: Memory bandwidth XSEDE Lonestar and VSC Cluster have identical Westmere processors

Extensive convergence tests: Comparison of He Theoretical and Available Experimental Results NSDI -Total X-Sect Considerable discrepancies! Rise at sequential threshold Extensive convergence tests: angular momenta, radial grid, pulse duration (up to 20 fs), time after pulse (propagate electrons to asymptotic region)  error below 1%

Two-Photon Double Ionization in The spectral Characteristics of the Pulse can be Critical

Can We Do Better ? How to efficiently approximate the integral is the key issue