Nuclear spin readout in a cavity-coupled silicon double quantum dot

Nuclear spins show long coherence times and are well isolated from the environment, which are properties making them promising for quantum information applications. On the other hand, these same qualities make the initialization and the readout of nuclear spin qubits challenging. Here, we present a method for nuclear spin readout by probing the transmission of a microwave resonator. We consider a single electron in a silicon double quantum dot interacting with a microwave resonator via the electric dipole coupling and subjected to a homogeneous magnetic field and a transverse magnetic field gradient. In our scenario, the electron spin interacts with a $^{31}\mathrm{P}$ defect nuclear spin via the hyperfine interaction. We theoretically investigate the influence of the P nuclear spin state on the microwave transmission through the cavity and show that nuclear spin readout is feasible with current state-of-the-art devices. Moreover, we identify optimal readout points with strong signal contrast to facilitate the experimental implementation of nuclear spin readout. Furthermore, we investigate the potential for achieving coherent excitation exchange between a nuclear spin qubit and cavity photons.