Kinetic inductance coupling for circuit QED with spins

Abstract: In contrast to the commonly used qubit resonator transverse coupling via the $\sigma_{xy}$-degree of freedom, longitudinal coupling through $\sigma_z$ presents a tantalizing alternative: it does not hybridize the modes, eliminating Purcell decay, and it enables quantum-non-demolishing qubit readout independent of the qubit-resonator frequency detuning. Here, we demonstrate longitudinal coupling between a {Cr$7$Ni} molecular spin qubit ensemble and the kinetic inductance of a granular aluminum superconducting microwave resonator. The inherent frequency-independence of this coupling allows for the utilization of a 7.8 GHz readout resonator to measure the full {Cr$_7$Ni} magnetization curve spanning 0-600 mT, corresponding to a spin frequency range of $f\text{spin}=$0-15 GHz. For 2 GHz detuning from the readout resonator, we measure a $1/e$ spin relaxation time $\tau=$0.38 s, limited by phonon decay to the substrate. Based on these results, we propose a path towards longitudinal coupling of single spins to a superconducting fluxonium qubit.