Fingerprints of the Schmid transition in a DC-biased transmon shunted by a high-impedance transmission line

The energy spectrum of an isolated Josephson junction, i.e. a superconducting qubit, is 2e-periodic in the displacement charge. In a transmon qubit, the corresponding energy bands are widely separated, and the lowest band captures the low-energy physics of the transmon. A galvanic connection to a transmission line enables quantum fluctuations of the displacement charge across the transmon. It also allows one to pass a direct current through the circuit. We calculate the voltage--current relation in the high-current limit and find that the corresponding resistance V(I)/I ~ I^{2*R_Q/R - 2}. Here R is the low-frequency impedance of the transmission line and R_Q = h/4e^2 is the resistance quantum. That the exponent changes signs at R/R_Q = 1 signals the Schmid insulator-to-superconductor transition. To better connect our theory to experiments with transmission lines realized by Josephson junction arrays, we also consider an array of finite length contacted by an external circuit. The impedance mismatch at that interface induces standing-wave resonances in V(I), whose positions and widths we compute in the weak and strong mismatch limits.