Rabi model exactly solved

Rabi model looks very simple: it describes a two-level atom coupled to a monochromatic beam of light through dipole interactions. However, an exact solution is not found until recently. A paper [the preprint of which is here] in PRL reports such a solution. Exactly solvable models are always interesting, because they can offer insights in many areas that may seem irrelevant at first glance.

Braak’s unexpected full analytical solutions of the quantum Rabi model are, however, worthy of celebration [2]. In mathematics and physics there are many, not necessarily compatible, criteria for a model to be both integrable and solvable. Examples include Frobenius’ condition for integrability in differential systems and Liouville’s condition for integrability in dynamical systems [6]. In the realm of quantum physics, it has been assumed that the existence of invariant subspaces associated with conserved quantities, other than energy, might be a necessary condition, as is the case in the Jaynes-Cummings model. The quantum Rabi model possesses naturally an additional discrete symmetry, the parity, which was assumed for some time to be insufficient for yielding a solvable model. Braak has proved that this is not the case and has presented exact analytical solutions of the quantum Rabi model for all parameter regimes. This is a remarkable achievement that adds the model to the short list of integrable quantum systems. Furthermore, Braak is able to take advantage of this key result to propose an operational criterion of integrability, inspired by the case of the hydrogen atom.

Integrability is, following Braak, equivalent to the existence of quantum numbers that classify eigenstates uniquely. It does not presuppose the existence of a family of commuting operators. Surprisingly, he has been able to apply this novel integrability criterion to a more elaborate case, the generalized quantum Rabi model, where a term that allows tunneling between the two atomic states is added. For the latter, he was able to prove that the model is not integrable, because it has an additional symmetry that is broken. However, the model is exactly solvable, and Braak presents the solutions. These are important results advancing the mathematical aspects of the quantum Rabi model in terms of integrability and solvability. Moreover, we should not overlook that the quantum Rabi model is a key physical model describing the interaction of quantum light and matter. [http://physics.aps.org/viewpoint-for/10.1103/PhysRevLett.107.100401]