  Designing the spatial profile of the superconducting gap -- gap engineering
-- has long been recognized as an effective way of controlling quasiparticles
in superconducting devices. In aluminum films, their thickness modulates the
gap; therefore, standard fabrication of Al/AlOx/Al Josephson junctions, which
relies on overlapping a thicker film on top of a thinner one, always results in
gap-engineered devices. Here we reconsider quasiparticle effects in
superconducting qubits to explicitly account for the unavoidable asymmetry in
the gap on the two sides of a Josephson junction. We find that different
regimes can be encountered in which the quasiparticles have either similar
densities in the two junction leads, or are largely confined to the lower-gap
lead. Qualitatively, for similar densities the qubit's excited state population
is lower but its relaxation rate higher than when the quasiparticles are
confined; therefore, there is a potential trade-off between two desirable
properties in a qubit.
