  We report observations of discrete charge states of a coherent dielectric
two-level system (TLS) that is strongly coupled to an offset-charge-sensitive
superconducting transmon qubit. We measure an offset charge of 0.072$e$
associated with the two TLS eigenstates, which have a transition frequency of
2.9 GHz and a relaxation time exceeding 3 ms. Combining measurements in the
strong dispersive and resonant regime, we quantify both transverse and
longitudinal couplings of the TLS-qubit interaction. We further perform joint
tracking of TLS transitions and quasiparticle tunneling dynamics but find no
intrinsic correlations. This study demonstrates microwave-frequency TLS as a
source of low-frequency charge noise.
