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We obtain the quadratic-in-spin terms of the conservative Hamiltoniandescribing the interactions of a binary of spinning bodies in GeneralRelativity through $\mathcal{O}(G^2)$ and to all orders in velocity. Ourcalculation extends a recently-introduced framework based on scatteringamplitudes and effective field theory to consider non-minimal coupling of thespinning objects to gravity. At the order that we consider, we establish thevalidity of the formula proposed in \cite{Bern:2020buy} that relates theimpulse and spin kick in a scattering event to the eikonal phase.

Quadratic-in-Spin Hamiltonian at $\mathcal{O}(G^2)$ from Scattering Amplitudes