Electromagnetically induced transparency for a mechanical probe in a hybrid quantum circuit

We investigate electromagnetically induced transparency (EIT) in a hybrid quantum circuit composed of a three-level transmon in a ladder configuration, capacitively coupled to both a transmission line resonator (TLR) and a nanoelectromechanical system (NEMS). Within the mean-field approximation, we demonstrate that this system can exhibit a narrow transparency window for a mechanical probe, controlled by an electromagnetic excitation. This configuration enables the creation of a dark state in the transmon and allows for adiabatic population transfer between its ground and excited states. Our findings reveal a new mechanism of quantum interference between hybrid mechanical and electromagnetic modes, with potential applications in quantum control of mechanical signals.