Solution to the Landau-Zener problem via Susskind-Glogower operators

We just uploaded a short, simple manuscript showing how to deal with a time dependent Jaynes-Cummings Hamiltonian where the time dependence is linear and you end up with a quantum Landau-Zener-Majorana (LZM)-like Hamiltonian.

This LZM-like Hamiltonian, under the rotating wave approximation, is simple to diagonalize in the field basis and the solutions to the classical Landau-Zener problem holds; that is, you can express the time evolution of the state in terms of Parabolic Cylinder or Hypergeometric functions.

As it goes, the Hamiltonian without the rotating wave approximation is also simple to diagonalize in the two-level system basis. The evolution of the initial state can be reduced all the way to two-uncoupled infinite sets of first order differential equations; of course, I have no clue how to solve these or else I would not be calling this a simple manuscript *wink* hehehe.

If you want, you can learn more about this by reading the Arxiv preprint.

Edit: If someone knows a reference for equation 16, please tell me about it. I cannot believe no-one has used it before, but I haven't found anything about it in the literature so far.