  Motivated by recent experiments on quantum mechanical charge pumping in a
Cooper pair sluice, we present a measurement scheme for observing shifts of
transition frequencies in two-level quantum systems induced by broadband
environmental fluctuations. In contrast to quantum optical and related set-ups
based on cavities, the impact of a thermal phase reservoir is considered. A
thorough analysis of Lamb and Stark shifts within weak-coupling master
equations is complemented by non-perturbative results for the model of an
exactly solvable harmonic system. The experimental protocol to measure the Lamb
shift in experimentally feasible superconducting circuits is analysed in detail
and supported by numerical simulations.
