  Recently developed Josephson junction array transmission lines implement
strong-coupling circuit electrodynamics compatible with a range of
superconducting quantum devices. They provide both the high impedance which
allows for strong quantum fluctuations, and photon modes with which to probe a
quantum device, such as a small Josephson junction. In this high-impedance
environment, current through the junction is accompanied by charge Bloch
oscillations analogous to those in crystalline systems. However, the interplay
between Bloch oscillations and environmental photon resonances remains largely
unexplored. Here we describe the Bloch oscillations in a transmon-type qubit
attached to high-impedance transmission lines with discrete photon spectra.
Transmons are characterized by well-separated charge bands, favoring Bloch
oscillations over Landau-Zener tunneling. We find resonances in the
voltage--current relation and the spectrum of photons emitted by the Bloch
oscillations. The transmon also scatters photons inelastically; we find the
cross-section for a novel anti-Stokes-like process whereby photons gain a Bloch
oscillation quantum. Our results outline how Bloch oscillations leave
fingerprints for experiments across the DC, MHz, and GHz ranges.
