  A hallmark of integrable systems is the purely elastic scattering of their
excitations. Such systems possess an extensive number of locally conserved
charges, leading to the conservation of the number of scattered excitations, as
well as their set of individual momenta. In this work, we show that inelastic
decay can nevertheless be observed in circuit QED realizations of integrable
boundary models. We consider the scattering of microwave photons off impurities
in superconducting circuits implementing the boundary sine-Gordon and Kondo
models, which are both integrable. We show that not only inelastic decay is
possible for the microwave photons, in spite of integrability, and thanks to a
nonlinear relation between them and the elastically-scattered excitations, but
also that integrability in fact provides powerful analytical tools allowing to
obtain exact expressions for response functions describing the inelastic decay.
Using the framework of form factors, we calculate the total inelastic decay
rate and elastic phase shift of the microwave photons, extracted from a 2-point
response function. We then go beyond linear response and obtain the exact
energy-resolved inelastic decay spectrum, using a novel method to evaluate form
factor expansions of 3-point response functions, which could prove useful in
other applications of integrable quantum field theories. We relate our results
to several recent photon splitting experiments, and in particular to recent
experimental data that provides evidence for the elusive Schmid-Bulgadaev
dissipative quantum phase transition.
