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Propagating the Time Dependent Schroedinger Equation

This document explores the propagation of the time-dependent Schrödinger equation, emphasizing its significance in probing and controlling electron dynamics through advanced techniques such as attosecond pulses, XUV + IR pump-probe methods, and free electron lasers (FELs). It delves into the mathematical foundations, such as tensor product bases and finite element discrete variable representation, and addresses challenges, convergence tests, and stability issues in numerical implementations. The findings indicate discrepancies in two-photon double ionization, opening discussions for further research and improved methodologies.

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Propagating the Time Dependent Schroedinger Equation

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  1. Propagating the Time DependentSchroedinger Equation • B. I. Schneider • Division of Advanced Cyberinfrastructure • National Science Foundation • National Science Foundation • September 6, 2013

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

  3. Basic Equation Possibly Non-Local or Non-Linear Where

  4. Properties of Classical Orthogonal Functions

  5. More Properties

  6. Matrix Elements

  7. Properties of Discrete Variable Representation

  8. Its Actually Trivial

  9. Multidimensional Problems • Tensor Product Basis • Consequences

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

  11. 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

  12. Finite Element Discrete Variable Representation • Structure of Matrix

  13. 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

  14. 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

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

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

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

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