Extended Data Fig. 6: Theoretical phonon scattering process.

a, Theoretical scattering phase \(\arg \left\langle u \vert v\right\rangle\) (blue line; right vertical axis) as a function of detuning frequency, together with single-phonon phase gate data shown in Fig. 2c (orange and aqua points; left vertical axis). The theoretical scattering phases at \(\arg \left\langle u \vert v\right\rangle =\pi /2\) and 3π/2 align in detuning frequency with the two experimental single-phonon excitation minima, corresponding to a net π phase difference. b, Theoretical pulse distortion \(1-\left\vert \left\langle u \vert v\right\rangle \right\vert\) as a function of detuning frequency. The largest pulse distortion, 3 × 10⁻², occurs at zero detuning. For a defined π-phase phonon gate implemented at detunings where \(\arg \left\langle u \vert v\right\rangle =\pi /2\) and 3π/2, the calculated pulse distortion leads to a scattering-induced phase gate infidelity of 8 × 10⁻³, as indicated by the horizontal dashed line.