Dominant mechanism of spin-spin relaxation at clock transition in [Ho_xY_1-x(W₅O₁₈)₂]^9-

Spin qubits at clock transition exhibit remarkable insensitivity to the surrounding magnetic fluctuations. Recent experimental studies on the single crystals of Na₉[Ho_xY_1-x(W₅O₁₈)₂].nH₂O, which host the molecular spin qubits in the [Ho(W₅O₁₈)₂]^9- units, have unveiled a peculiar correlation between the Hahn-echo decoherence time at clock transition and the density x of spin qubits, where the decoherence time increases with decreasing x and remains constant when x is below 0.01. In this work, we theoretically study the electron spin-electron spin relaxation time in this single crystal as a function of both x and the inhomogeneity in qubit frequency and study the decoherence of single Ho spin due to surrounding nuclear spins with the method of cluster-correlation expansion for clock-transition qubits. In both cases, spin Hamiltonians with ab initio parameters are used. We find that the calculated electron spin-electron spin and electron spin-nuclear spin decoherence times agree with the experimentally measure ones at high and low Ho density, respectively, providing an explanation of the peculiar correlation. This work elucidates some possible dominant mechanisms of molecular spin decoherence at clock transition and provides insights on how we can further increase coherences in molecular clock spin qubits.