  We have engineered the bandgap profile of transmon qubits by combining
oxygen-doped Al for tunnel junction electrodes and clean Al as quasiparticle
traps to investigate energy relaxation due to quasiparticle tunneling. The
relaxation time $T_1$ of the qubits is shown to be insensitive to this bandgap
engineering. Operating at relatively low $E_J/E_C$ makes the transmon
transition frequency distinctly dependent on the charge parity, allowing us to
detect the quasiparticles tunneling across the qubit junction. Quasiparticle
kinetics have been studied by monitoring the frequency switching due to
even/odd parity change in real time. It shows the switching time is faster than
10 $\mu$s, indicating quasiparticle-induced relaxation has to be reduced to
achieve $T_1$ much longer than 100 $\mu$s.
