Derivation — C013 (loop-closing times)

From C012, $\alpha(t)=-i\frac{\eta\Omega}{\delta}e^{-i\delta t/2}\sin(\delta t/2)$ and $\Phi(t)=\frac{\eta^2\Omega^2}{4\delta^2}(\delta t-\sin\delta t)$.

Substitute $t=2n\pi/\delta$, $n\in\mathbb Z$. Then $\delta t=2n\pi$ and $\delta t/2=n\pi$.

Displacement closes

$$\alpha(2n\pi/\delta)=-i\frac{\eta\Omega}{\delta}e^{-in\pi}\sin(n\pi) =-i\frac{\eta\Omega}{\delta}(\pm1)\cdot0=0.$$ So the displacement generator $(\alpha a^\dagger-\alpha^* a)S_{x,e}$ vanishes and $U_A=\exp(i\Phi S_{x,e}^2)$ — a pure $S_{x,e}^2$ coupling, decoupled from the oscillator (the phase-space loop closes).

Geometric phase

$$\Phi(2n\pi/\delta)=\frac{\eta^2\Omega^2}{4\delta^2}\big(2n\pi-\sin(2n\pi)\big) =\frac{\eta^2\Omega^2}{4\delta^2}\cdot2n\pi =\frac{n\pi\,\eta^2\Omega^2}{2\delta^2}.$$

Exactly the claimed value. Confirmed symbolically in sympy_check.py with $n$ an integer symbol (sympy evaluates $\sin(n\pi)=0$, $\cos(n\pi)=(-1)^n$).