Question about temperature dependence of heom

[Brian Rolczynski]

Hello,

I had a quick question about the heom solver's temperature dependence. I would appreciate your advice.

I am finding a temperature (T) dependence that is the opposite of what I would have expected it to be.

To illustrate my point, I am attaching a simple script of a 2-state system below. When I attempt to run this script at low and high T, I find that the higher T has longer-lived coherences. For example, when I use T=10 K, the coherence appears to halve in about 100 fs. However, when I apply T=100 K, it halves in nearly 1 ps. At higher temperature, it lasts even longer. I would have expected the opposite trend.

Am I doing something wrong here, or misunderstanding something? 

Thank you very much!

[Neill Lambert]

Dear Brian,

Sorry for the slow reply, I had a quick look at your example.

I think basically what you are seeing is a lack of convergence in the HEOM Matsubara decomposition (controlled by the N_k parameter) for the parameters you are using.

For example, T=10, and even T=100, are quite low temperatures when thinking about the bath cut-off and the system energies you are using, and its quite hard to get a converged result with the standard matsubara expansion.  

The Pade decomposition is a bit better, but you still need to increase the N_k parameter quite a bit to see a reasonable result.

With the new environment class in qutip v5.1.0 (released a few weeks ago) its quite easy to plot the effective bath your system is seeing given a particular decomposition and cut-off N_k.  I have updated your example to use this new interface, and use the Pade decomposition, and I think you should see more reasonable trends. If you dont want to update qutip you can probably still get a good result using DrudeLorentzPadeBath and a larger choice of N_k.

I also added a plot of the power_spectrum;   essentially the one for the '''real bath'' and the one for the decomposition actually used by HEOM should look kinda similar. When the temperature is low there is a hard cut-off around zero frequency which the HEOM has trouble capturing, so you can see there is still some small error.   In the new environment class @Gerardo Suarez is adding a bunch of new fitting methods which might be useful for your case, but generally speaking it requires a bit of fiddling to get a good result out of these difficult regimes.

all the best

neill

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[Brian Rolczynski]

Hi Neill,

Thank you very much! That makes sense.