Light-matter decoupling and A2 term detection in superconducting circuits

When the vacuum Rabi frequency, Ω , of an electromagnetic mode is much smaller than the bare frequency of the excitation to whom it couples, ω, the simple Jaynes-Cumming or Tavis-Cumming models capture the main features of light-matter interaction and cavity QED1,2. However, already for a normalised coupling > . ω Ω 0 1, the Rotating Wave Approximation (RWA) that justifies those solvable models fails3. In this ultrastrong coupling (USC) regime, light-matter interaction must be described beyond RWA, using the Rabi4 and Hopfield-Bogoliubov5 models, that correctly describe the ground state squeezing and asymmetric splitting6–9. The USC regime, observed for the first time only few years ago10, has now been achieved in many solid-state cavity quantum electrodynamics setups11–17, with an actual coupling record of = . ω Ω 0 8718. When the normalised coupling becomes of the order one, also the aforementioned non-RWA models fail. In this regime, named deep strong coupling (DSC)19, the localised dipolar interaction dominates and a real-space description with many excited photonic modes becomes essential. Our understanding of such a deep non-perturbative regime is still incomplete20–24. A first, recent counter-intuitive result is that light and matter eventually decouple in the DSC regime: the spontaneous emission rate of the system dramatically decreases, instead of increasing, with the coupling strength25. This decoupling is associated with the diamagnetic term A2, that expels modes away from the emitter. Still, the decoupling has been rigorously proved only for linear systems —a perfect planar metallic cavity coupled to a 2D sheet of dipoles—, and the link of the decoupling effect with the A2 term remains a hypothesis. Indeed, without diamagnetic term, the model in ref. 25 becomes unstable and undergoes a superradiant phase transition26–28, impeaching a comparison of DSC physics with and without A2. In circuit QED it has been shown that a microscopic treatment of the light-matter coupling between a superconducting waveguide and a qubit gives rise to a diamagnetic term, analogous to the A2 term of the minimal coupling Hamiltonian29. Still, contrary to the usual minimal coupling case, the relative strength of the dipolar and diamagnetic terms is not fixed by the Thomas-Reiche-Kuhn sum rule. Circuit QED