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We revisit the theoretical description of the ultrastrong light-matterinteraction in terms of exactly solvable effective Hamiltonians. A perturbativeapproach based on polaronic and spin-dependent squeezing transformationsprovides an effective Hamiltonian for the quantum Rabi model up to the secondorder in the expansion parameter. The model consistently includes both rotatingand counter-rotating terms, going therefore beyond the rotating waveapproximation. Analytical and numerical results show that the proposedHamiltonian performs better than the Bloch-Siegert model when calculatingoperator averages (e.g.\, the mean photon number and number of excitations).This improvement is due to a refined calculation of the dressed states withinthe present model. Regarding the frequency shift induced by the qubit-photoninteraction, we find a different sign from the Bloch-Siegert value. Thisinfluences the eigenstates structure in a non-trivial way and ensures thecorrect calculation of the number of excitations associated to a given dressedstate. As a consistency check, we show that the exactly solvable independentboson model is reproduced as a special limit case of the perturbativeHamiltonian.

Effective Hamiltonians in the Quantum Rabi Problem