Understanding curvature (quantum capacitance) couplings of spin qubits to a superconducting cavity as a low-energy limit

We describe in a general perturbation theory approach the origin of the couplings of a spin qubit to a superconducting (electromagnetic) cavity. Special attention is paid to the so-called curvature couplings related to qubit’s quantum capacitance, which is derived as a low-energy limit of a perturbation expansion to second order. We discuss the applicability of these couplings to current experiments with multi-dot spin-qubits, including single-, double, and triple-quantum dot qubits, as well as with superconducting qubits.