Pairwise dressed states and twin photon-phonon generation in a tripartite quantum system

We present a theoretical formulation of a tripartite hybrid quantum circuit composed of a resonator, a transmon qubit, and a nanoelectromechanical oscillator coupled via effective nonlinear interactions. Starting from the full Hamiltonian, we derive an effective model that captures the essential features of the tripartite coupling, including coherent driving and dissipative mechanisms. By solving the steady-state regime of the system's quantum dynamics, we compute the reflection and transmission spectra using input-output theory. Our results reveal clear signatures of energy-level quantization in the single-excitation limit, demonstrating the formation of pairwise-dressed states emerging from the tripartite interaction. This work provides a theoretical framework for probing hybrid quantum systems in the few-quanta regime and highlights their potential for quantum information and sensing