C066 derivation

Setup

Work in the two-ion ground-qubit space spanned by $\ket{\downarrow\uparrow}$ and $\ket{\uparrow\downarrow}$, with the Bell basis $$\ket{\Psi^{\pm}} = \frac{1}{\sqrt{2}}\big(\ket{\uparrow\downarrow} \pm \ket{\downarrow\uparrow}\big).$$

The blue-sideband tone of drive (A) produces an AC Stark shift $\Delta$ on $\ket{\downarrow}$. Because the two ions see a Rabi-frequency difference $\epsilon$, the shift felt by the two ions differs. Relative to the mean, ion 1 sees $+\tfrac{1}{2}\epsilon\Delta$ and ion 2 sees $-\tfrac{1}{2}\epsilon\Delta$ on its $\ket{\downarrow}$ component (a differential $\sigma_z$-like term of total strength $\epsilon\Delta$ per ion on the $\downarrow$ level).

Relative phase

The states $\ket{\downarrow\uparrow}$ and $\ket{\uparrow\downarrow}$ differ only by which ion is in $\ket{\downarrow}$. The differential Stark Hamiltonian assigns them energies of opposite sign, $\pm \tfrac{1}{2}(\epsilon\Delta)\cdot 2 = \pm\,\epsilon\Delta$ per ion; summing the two-ion contribution the energy difference between $\ket{\downarrow\uparrow}$ and $\ket{\uparrow\downarrow}$ is $$\hbar\,\omega_{\mathrm{rel}} = 2\,\hbar\,\epsilon\,\Delta .$$ After time $t$ the relative phase is therefore $$\boxed{\phi = 2\,\epsilon\,\Delta\, t}$$ matching the paper's stated relation. (STRUCTURALLY CORRECT.)

Transfer probability

Write $\ket{\downarrow\uparrow}=\tfrac{1}{\sqrt2}(\ket{\Psi^+}-\ket{\Psi^-})$ and $\ket{\uparrow\downarrow}=\tfrac{1}{\sqrt2}(\ket{\Psi^+}+\ket{\Psi^-})$. A relative phase $\phi$ between the two product states, applied symmetrically as $U=\mathrm{diag}(e^{-i\phi/2},e^{+i\phi/2})$ in the $\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$ basis, acts on $\ket{\Psi^-}$ as $$U\ket{\Psi^-} = \tfrac{1}{\sqrt2}\big(e^{-i\phi/2}\ket{\uparrow\downarrow} - e^{+i\phi/2}\ket{\downarrow\uparrow}\big) = \cos\!\tfrac{\phi}{2}\,\ket{\Psi^-} - i\sin\!\tfrac{\phi}{2}\,\ket{\Psi^+}.$$ Hence the probability to be found in $\ket{\Psi^+}$ is $$\boxed{p = |\langle \Psi^+ | U | \Psi^- \rangle|^2 = \sin^2(\phi/2)}$$ again matching the paper's stated form. (STRUCTURALLY CORRECT.)

Numeric evaluation

With the paper's stated numbers: $\epsilon = 0.05$, $\Delta = 2\pi\times 25\,\mathrm{kHz}$, $\delta = 2\pi\times 14.7\,\mathrm{kHz}$, $t = 2\times\frac{2\pi}{\delta}$.

$$\phi = 2\epsilon\Delta t = 2(0.05)(2\pi\cdot 25\times10^3)\cdot\frac{2\cdot 2\pi}{2\pi\cdot 14.7\times10^3}.$$ Numerically (see run.py / run.log): $t \approx 136.05\,\mu s$, $\phi \approx 2.137\,\mathrm{rad}$, and $$p = \sin^2(\phi/2) \approx 0.77.$$

Conclusion

The two structural relations $\phi = 2\epsilon\Delta t$ and $p=\sin^2(\phi/2)$ are correct. The paper's specific value $p\approx 0.5$ does NOT match the recomputed $p\approx 0.77$ (and is $\approx 0.5$ only as a rough "large per-cycle error" statement). The qualitative point -- a near order-unity error that "completely destroys the protocol" -- holds for either value. Reported as partial, failure_reason: mismatch.