Frequency-resolved decoherence spectroscopy of a semiconductor charge qubit coupled to a high-impedance resonator

Superconducting resonators coupled to semiconductor quantum dots provide a powerful platform to investigate light-matter interaction and decoherence mechanisms in solid-state quantum systems. Here we study a hybrid circuit quantum electrodynamics architecture consisting of a GaAs double-quantum-dot charge qubit capacitively coupled to a high-impedance, frequency-tunable SQUID-array resonator. By tuning the qubit transition frequency over the range $\omega_\mathrm{q}/2\pi \sim 3$-$6$ GHz, we perform frequency-resolved decoherence spectroscopy of the charge qubit across a broad energy window. Time-resolved measurements enable us to disentangle relaxation and pure dephasing processes and to identify distinct decoherence regimes as a function of qubit frequency. We find that at lower frequencies ($\leq 4.5$ GHz) dephasing dominates the qubit linewidth, whereas at higher frequencies energy relaxation becomes the leading contribution. The measured frequency dependence of the relaxation rate exhibits a cubic scaling, consistent with charge-qubit decay dominated by coupling to a piezoelectric phonon bath and providing frequency-resolved access to the corresponding phonon-induced spectral density. Our results show that hybrid semiconductor--superconducting circuits can serve as sensitive spectroscopic tools to probe microscopic decoherence mechanisms relevant for a wide range of hybrid quantum devices.