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In this work we develop a semi-analytical variational ansatz to study theproperties of few photon excitations interacting with a collection of quantumemitters in regimes that go beyond the rotating wave approximation. This methodcan be used to approximate both the static and dynamical properties of asuperconducting qubit in an open transmission line, including the spontaneousemission spectrum and the resonances in scattering experiments. Theapproximations are quantitatively accurate for rather strong couplings, asshown by a direct comparison to Matrix-Product-State numerical methods, andprovide also a good qualitative description for stronger couplings well beyondthe Markovian regime.

Dynamical polaron ansatz: a theoretical tool for the ultra-strong coupling regime of circuit QED