#253 Add a Demo for non-Markovian dynamics

[Ulf Lorenz]

The goal is to recalculate some of the results of Meier and Tannor, JCP 111:3365.

A first attempt can be found under the branch non-markovian-demo, and requires debugging, because I could not reproduce the results of the paper. Open problems:

• Is it possible to reproduce the correlation function, fig. 2? I got the correct qualitative behavior, but the imaginary part is much too small
• Are eqs. (16) and (17) correct or is there maybe a typo?
• The initial state in VI (b) is not what the figures show (e.g., fig. 3), we rather start with a pure state (1, 0).
• And of course, there are lots of equations to be set up, there can be arbitrary errors in the code. Or in the equations in the paper.

Once we manage to recreate one of the calculations of fig. 3, we could add initial correlations and time-dependent effects as additional demos. Ideally, the code is rewritten in such a way that it can deal with arbitrary systems and spectral densities, then we would have actual added value for users. Note: This requires the addition of the renormalization term that plays no role for the example two-level system.

Issues to report back to David Tannor:

• typo in spectral density definition below (28), omega_e should be omega_c
• initial state 1/2 (sigma_x - sigma_z) unphysical, trace would be zero and <sigma_z> = 0. The reference uses the initial state (1, 0) x (1,0) with <sigma_z> = 1</sigma_z></sigma_z>