It is customary to associate a superconducting phase with zero resistance and an insulating phase with infinite resistance. However, near a quantum phase transition, the experimentally measured resistance does not attain either value, and hence the physics becomes hidden in the scaling of resistance with temperature, system size, magnetic field, etc. Instead, we consider a finite frequency spectroscopy of the collective phase-mode of the Cooper pair condensate, excited with less than a single quantum, on average. This approach is particularly well suited for 1D chains of Josephson junctions, where the phase-mode corresponds to a guided electromagnetic wave, and spectroscopy can be done using superconducting qubit techniques [1,2]. Notably, the insulating phase manifests itself in the damping of the phase-mode by the quantum phase-slip fluctuations (instantons) [3,4]. This damping is a many-body effect, related to relaxation of plasmons in non-linear Luttinger liquids, or decay of instantons in quantum field theory. The dependence of the damping rate on frequency (at a nearly zero temperature) serves as a novel experimental and theoretical tool for characterizing quantum phase transitions.

[1] R. Kuzmin et al. “Quantum electrodynamics of a superconductor–insulator phase transition” Nature Physics 15, 930 (2019).

[2] I. V. Pechenezhskiy et al. “The superconducting quasicharge qubit” Nature 585, 368 (2020).

[3] A. Burshtein et al. "Photon-instanton collider implemented by a superconducting circuit" PRL 126, 137701 (2021).

[4] R. Kuzmin et al. “Inelastic scattering of a photon by a quantum phase-slip” PRL 126, 197701 (2021)