To get the prediction from LLT for comparison to t-dep simulations, run GetXi.m, followed by Fit_Delande_den.m (see comments therein).

To reproduce the numerical results stated in the paper regarding time dependent simulations, follow these instructions.

First of all, I'm providing you with some precomputed workspaces (below you'll see how to generate them). For the results on page 20 of the paper (in section 5.2), they are in the subfolder t-dep/TinT_SCff0_1. For the results on page 25 (in section 6), they are in the subfolder t-dep/TinT_SCff0_2k0_5. Each of these folders contains workspaces numbered 1-20 (or 1-30), corresponding to different noise realisations.

These are generated by running the script tdepsims.m and saving the entire workspace. See comments within for details.

Then we have to average over the total number of realisations to get the mean density profile as a function of time. This is achieved by running the script mean1Dden.m (see comments for details). The important variables generated by this have been saved for you in the main folder as:

For the results on page 20 of the paper (in section 5.2), using data from the subfolder t-dep/TinT_SCff0_1: meany_TinT_SCff0_1.mat
For the results on page 25 (in section 6), using data from the subfolder t-dep/TinT_SCff0_2k0_5: meany_TinT_SCff0_2k0_5.mat

Then you have to inspect the density profile as a function of time (movies are plotted in mean1Dden) and decide two things: 
(1) at what point in time to start the fitting. Pick a time at which the density more or less settles down, becomes exponential, and only changes slowly afterwards. This will usually happen at t ~ 50t_0.
(2) what segment of the density profile in the channel to fit with an exp. Look for linear behaviour on the log plots, obviously. Almost always you'll have to discard the initial bit (close to R_1) as the density there will not be exponential, and often (but not always), a segment at the boundary with R_2 (coupling to the noise-free reservoir changes the wavefunction).

Armed with this info, run fit_1Dden.m (see comments therein). This will produce a plot of \xi in \ell as a function of time in t_0. The paper simply describes these plots in words.