We report on the on-chip detection of microwaves in the frequency range around 100 GHz. For the purpose of detection, we employ a discrete transport channel triggered in a superconducting single-electron transistor by photon-assisted tunneling of quasiparticles. The technique is successfully applied to observe the spectrum of the dressed states of a model cQED system consisting of a superconducting coplanar resonator coupled to a quantum Josephson oscillator. The dressed states appear as typical resonance anticrossing exhibiting, in our case, an expectedly wide frequency splitting corresponding to the Jaynes-Cummings coupling strength, g/π ∼ 10 GHz. Due to the high decay rate, γ ∼ 20÷40 GHz, in the very transparent Josephson junctions used, the strong coupling limit, g γ, which is required for qubit operation, is not achieved, and the photon population in the resonator is low, n < 1. Remarkably, the continuous readout of the low population states demonstrates the high microwave sensitivity of the detector.