Effective Hamiltonians in the Quantum Rabi Problem

We revisit the theoretical description of the ultrastrong light-matter interaction in terms of exactly solvable effective Hamiltonians. A perturbative approach based on polaronic and spin-dependent squeezing transformations provides an effective Hamiltonian for the quantum Rabi model up to the second order in the expansion parameter. The model consistently includes both rotating and counter-rotating terms, going therefore beyond the rotating wave approximation. Analytical and numerical results show that the proposed Hamiltonian performs better than the Bloch-Siegert model when calculating operator averages (e.g.\, the mean photon number and number of excitations). This improvement is due to a refined calculation of the dressed states within the present model. Regarding the frequency shift induced by the qubit-photon interaction, we find a different sign from the Bloch-Siegert value. This influences the eigenstates structure in a non-trivial way and ensures the correct calculation of the number of excitations associated to a given dressed state. As a consistency check, we show that the exactly solvable independent boson model is reproduced as a special limit case of the perturbative Hamiltonian.