Kinetic inductance coupling for circuit QED with spins

While transverse coupling (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>σ</a:mi><a:mi>x</a:mi></a:msub></a:math>) is commonly used in qubit-resonator systems, longitudinal coupling via <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>σ</b:mi><b:mi>z</b:mi></b:msub></b:math> presents a tantalizing alternative: it does not hybridize the modes, eliminating Purcell decay, and it enables quantum-nondemolition qubit readout independent of the qubit-resonator frequency detuning. Here, we demonstrate longitudinal coupling between a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mo>{</c:mo><c:msub><c:mi>Cr</c:mi><c:mn>7</c:mn></c:msub><c:mi>Ni</c:mi><c:mo>}</c:mo></c:mrow></c:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>{</d:mo><d:msub><d:mi>Cr</d:mi><d:mn>7</d:mn></d:msub><d:mi>Ni</d:mi><d:mo>}</d:mo></d:mrow></d:math> magnetization curve spanning 0–600 mT, corresponding to a spin frequency range of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>f</e:mi><e:mtext>spin</e:mtext></e:msub><e:mo>=</e:mo><e:mn>0</e:mn></e:mrow></e:math>–15 GHz. For 2-GHz detuning from the readout resonator, we measure a <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mn>1</f:mn><f:mo>/</f:mo><f:mi>e</f:mi></f:mrow></f:math> spin relaxation time of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>τ</g:mi><g:mo>=</g:mo><g:mn>0.38</g:mn></g:mrow></g:math> 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.