Tracking Diabatic States in a numerically produced energy spectrum

[yoavpol]

I have diagonalized a Hamiltonian matrix many times with a varying parameter (varying magnetic field).
This gives me the eigenstates and eigenvalues of the matrix for the different field values.
I now need to track the diabatic states through (avoided) level crossings of the eigenvalues (energies/adiabatic states). In other words, if the system has an initial populated eigenstate and I start increasing the field very rapidly, I would like to know which eigenstate would be populated at the end

Does anyone know an easy way to do that?
Is there a readymade code I can use?

Thank you

Yoav.

 

[DrDu]

Generally, you will also need the non-adiabatic coupling to calculate the transition probability. However if - as you write - the variation of the magnetic field is very rapid, then you could try sudden approximation, i.e. decompose the initial eigenstate in the eigenstates pertaining to the large field value.

 

[yoavpol]

Thanks DrDu, I am indeed interested in the sudden approximation.

If I understand you correctly than I've already tried this.

Perhaps the graphic results would illustrate my problem better.
The figure shows the energy levels as a function of applied field. let's say I start at the top blue level at zero field. If I ramp up the field very rapidly, I will stay at the blue level when I go past the red level at Bx=0.2T for example (the diabatic states are coupled and if you looked at a higher resolution you could theoretically see an avoided crossing).
However I do not know where will I end up when I get to Bx=2T. The blue level? a red one? or a purple one?

I've tried taking the dot product of the initial state (Bx=0) with the various states at a large field value (say Bx=2T) and looking for the largest product. This should be similar to what DrDu suggested if I'm not mistaken.
This gave the correct results for the crossings at low field values, but wrong results at higher field values. This is because the diabatic states themselves change with field.
A better method is to compare the eigenstate just before each crossing (lower field) to the eigenstate just after. However the question arises as to where does a crossing start and how to recognize it.

I've thought of a very lengthy solution (which I'm still working on) but I'm sure this kind of thing was tackled many times before. I have a feeling if I could name the problem properly, I'll find a well known algorithm for this.

Has anyone heard of something like this?

 

[DrDu]

If the field changes at infinite speed, the sudden approximation becomes exact. On the other hand at very slow changes of the field, the states will follow the adiabatic surfaces.
Inbetween you have to solve in principle the total time dependent Schroedinger equation. This should not be too demanding in your problem. There are approximate formulas like the Landau-Zener formula.
Maybe the book: Molecular Collision Theory by M. S. Child is helpful as he discusses all known approximation schemes at length, especially Chapter 8.

What's the exact problem you are looking at?

 

[yoavpol]

I'm interested in the sudden approximation as I've said.

The problem is that here I have multiple levels interacting and the width of the avoided crossings varies considerably. If I just had two levels, I could study them in detail and compare the eigenstates before the crossings to those after the crossings.

Here however because of the complexity of the problem (I have to do this for many levels in each plot and I have many such plots), I need an algorithm that would automatically correlate between eigenstates before and after the crossings.


The system is that of two ions, where each of them can be either up or down with various nuclear spin values (this is not a classical spin half problem though). As the field grows, an energy difference appears (and grows) between sates for which both ions are down and states for which both ions are up (at low fields this would correspond to the difference between the two blue levels). I need to find this energy difference.



Thanks

 

[DrDu]

What is this saturation due too, which occurs at field strength > 1.4, say?

 

[yoavpol]

I still haven't verified it, but it should be due to the states ceasing to have a clear up or down character for each ion. Instead ions should have an up+-down character at these high fields

 

[DrDu]

This gave the correct results for the crossings at low field values, but wrong results at higher field values.

In which sense are the results incorrect, exactly?

 

[yoavpol]

well like I said, I can do this task manually for a specific crossing. I decide "by sight" where the crossings begins and where it ends, and I take the dot product of the initial eigenstate (just before the crossings) with eigenstates just after the crossing. The "end eigenstate" should be the one with the largest such dot product.

The problem is producing an algorithm that knows where a crossing begins and where it ends since they vary in width (and to some extent even identifying a crossing at all because of degeneracies in the energy levels).

My current algorithm deals well only with crossings which are narrow (these are found at low field values). I can tell it gives the wrong answer for wide crossings because for some of these crossings it produces different results from the manual check (it doesn't identify the beginning of a crossing correctly).

shouldn't this problem, of predicting the end state in an instantaneous approximation in a many level spectrum, have a known solution?

 

[DrDu]

shouldn't this problem, of predicting the end state in an instantaneous approximation in a many level spectrum, have a known solution?

The solution are the projections of the initial state on the final states at the very end.
The problem you have is that you don't seem to assume that the magnetic field switches fast enough to make the sudden approximation for transitions between all the states.

 

[yoavpol]

Since the actual magnetic field would change at some finite (though fast) rate, I cannot use the projection of the initial state on the states at the end. That means I cannot assume the instantaneous aprroximation for the entire 2T range. I can only assume the instantaneous approximation for narrow regions such as the width of a crossing.

This means I have to identify crossings and project the relevant state just before the crossing to the states just after. However as I've said, recognizing the beginning and end of a crossing is the problem.

 

[DrDu]

As long as you don't give us a fully specified hamiltonian I fear I can't help you.