High-Fidelity Control of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mmultiscripts> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mprescripts/> <mml:none/> <mml:mrow> <mml:mn>13</

Nuclear spins near group-IV defects in diamond are promising candidates for quantum memories in quantum network applications. Here, we demonstrate high-fidelity control of a single <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mmultiscripts> <a:mrow> <a:mi mathvariant="normal">C</a:mi> </a:mrow> <a:mprescripts/> <a:none/> <a:mrow> <a:mn>13</a:mn> </a:mrow> </a:mmultiscripts> </a:mrow> </a:math> nuclear spin coupled to a tin-vacancy center in diamond. We perform a combination of optical and microwave pumping to achieve initialization into a combined electronuclear spin state with a fidelity of 99.74(3)%. Harnessing a superconducting waveguide for radio-frequency driving, we demonstrate precise nuclear-spin control: Ramsey measurements reveal a coherence time of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mrow> <d:msubsup> <d:mrow> <d:mi>T</d:mi> </d:mrow> <d:mrow> <d:mn>2</d:mn> </d:mrow> <d:mrow> <d:mo>*</d:mo> </d:mrow> </d:msubsup> <d:mo>=</d:mo> <d:mn>1.5</d:mn> <d:mo stretchy="false">(</d:mo> <d:mn>1</d:mn> <d:mo stretchy="false">)</d:mo> <d:mtext> </d:mtext> <d:mtext> </d:mtext> <d:mi>ms</d:mi> </d:mrow> </d:math> , and we use dynamical decoupling to extend it to 1.35(3) s. We perform randomized benchmarking, yielding a single-qubit gate fidelity of 99.92(1)%. This demonstrates a coherent spin-photon system with promising properties for quantum network nodes.