Our latest paper: The exact solution of generalized Dicke models via Susskind–Glogower operators

Our latest paper The exact solution of generalized Dicke models via Susskind–Glogower operators  has been published a day ago in J. Phys. A: Math. and Theor.  The Dicke Hamiltonian is a workhorse of Quantum Optics, it describes the interaction of a collection of identical two-level systems with a single mode electromagnetic field under the long wave and rotating wave approximations. Surely, you will be thinking: "well, that model conserves the total number of excitations and parity; then, it's trivial to find its proper values and states." Well, it is trivial to solve the system but, as far as I know, it's not that trivial to follow the dynamics of a large ensemble under this model. 

Actually, I was looking at recent solution to a nonlinear version of this model via the Bethe ansatz method and  I got frustrated that even by using this method it was hard to follow the dynamics of a qubit ensemble of size twenty. So, Héctor and I sat down and applied a right unitary method that we had used to follow the dynamics of a quantum Landau-Zener-Majorana Hamiltonian a few months ago. It was trivial to extend the approach from a single qubit to an ensemble but the solution was not elegant enough, as you can see in the first part of the latest paper. So, we tried an alternative, instead of thinking about transformations we just thought about algebraic manipulation of the Hamiltonian at hand. After a few tries, we realized that one particular arrangement allowed us to write the evolution operator as the transform operators acting on the evolution operator of a tridiagonal matrix in the ensemble basis that depended only on the number operator. From there, it was all downhill because calculating the evolution operator of such a semi-classical-like Hamiltonian is quite simple, numerically, even for very large matrices and applying the transform operators on the initial states was easier than applying them on the time evolution operator. 

At the time when we were writting the paper, I only had my dual-core i7 laptop with 8GB of RAM, but it only took a few hours to follow the dynamics of an ensemble consisting of twenty-five qubits interacting with a coherent field with as mean photon number of twenty five. Now, I have done some simulations in my eight-core i7 desktop with 64GB RAM and I can follow the dynamics of a hundred qubits overnight with a very inefficient program. I'm hoping hat I will be able to simulate four or five hundred qubits interacting with large coherent fields as soon as I have time to sit down to think about this problem again.

Oh, I forgot to tell you. Once we obtained a result for just the Dicke model we extended the approach to include independent nonlinearities in the field and the ensemble, an approach a little bit more general than that of the guys in the Bethe ansatz method.

So, I hope you like our approach, use it and cite us in the future. You can find the Journal version at JPAMG. If you don't have access, we have prepared a manuscript with the final published version but without the journal format and uploaded it to the arXiv.