Solution of the independent boson model with quadratic coupling

We revisit the model of a two-level system coupled to phonons, and provide analytical solutions including a quadratic coupling, which was until now unsolved. Indeed, current techniques to analytically solve this important problem, like the polaron transformation which eliminates what is commonly called system-phonon couplings, hide the fundamental physics. These couplings can be eliminated using a diagonal basis, in which phonons depend on the electronic level. Doing so, we obtain analytical results through a simple algebra, by switching back and forth from ground-phonons to excited-phonons. We easily recover standard results for linear coupling, like state dynamics and correlation functions for absorption/emission lineshapes, in a way far simpler than previous procedures. More importantly, we find solution for the Hamiltonian including quadratic coupling, for which no analytical results have been reported yet. This approach opens a conceptually new route to more complicated matter-boson systems.