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

Phys. Lett. A 375, 3770-3774 (2011) 

In this paper, we try to show what happens when you consider the Jaynes-Cummings model in the case of a linear time-dependent detuning between the two-level system and the field.

Experimental systems: I have my heart on circuit-QED, it is the simplest thing I can think of and our result may be helpful for modular processes. A more complicated model may or may not be of relevance in BEC physics.

Our major result: By using a right unitary transform involving Susskind-Glogower operators—these are the ones introducing the right unitary characteristic—it is possible to show that the Hamiltonian is exactly solvable; specifically, the Hamiltonian is diagonalizable in the Fock state basis of the field. Moreover, the time evolution for the Hamiltonian can be written in an exact closed form given in terms of solutions to Weber Differential Equation, which are related to the exact solutions to the Landau-Zener-Majorana-Stuckelberg problem. 

With the analytical solution at hand, it is possible to calculate whatever you want. 

Minor results: It was curious for me to find out that in the case where the rotating wave approximation cannot be made it is still possible to diagonalize the Hamiltonian in the two-level system basis. Then, one can do numerics for small number of photons in the field. 

I was surprised to find that it was very simple to write a script that generates code to solve a system of some thousand coupled differential equations. 

The Physics: (Everybody always asks me about "the physics", so far I still don't have a clue what that question is really about but here's an attempt to an answer) Just for the sake of giving an example, in the article we present the physics in cases similar to those dealt by Landau, Zener and Majorana, to obtain the transition probability at the end of times for a system initially in the ground state at the beginning of time. Basically, the number of photons in the quantum field enhances the coupling between the two-level system and the field; this you can see at the level of the right unitary application.

Curious things: I learned the following,

• Majorana worked on the problem and published his results at the same time than Landau and Zenner. His formulation is closer to a full quantization of the Rabi problem. I found this while reviewing the literature on the topic.
• Stuckelberg worked in the problem, I haven't been able to get my hands on his paper so I have no clue what he said about the problem. I found this while attending the QIPC Zurich 2011 and someone gave a talk and mentionend the Landau-Zener-Stuckelberg-Mechanism.
• I should read Wikipedia at least once before finalizing a paper because in its entrance for Landau-Zener it clearly states the cites to Stuckelberg and Majorana's works.

Well, I hope you can find some use for the results we present in the paper and, as always, drop me a line, I am always glad to discuss or try to help whenever possible.