= Solution
For these three couplings, the <decoherence factor> becomes
$$
\boxed{z(t)=\cos(\pi t)\cos^2(\pi t/2)=\frac12\cos(\pi t)+\frac14\left(1+\cos(2\pi t)\right)}.
$$
It is periodic with period $2$. To locate its extrema, put $c=\cos\pi t$; then $z=c(1+c)/2$. Its maximum is $1$ at $t=0,2,4$, and its minimum is $-1/8$ where $c=-1/2$, namely $t=2/3,4/3,8/3,10/3$. The zeros on the displayed interval are $t=1/2,1,3/2,5/2,3,7/2$. The half-integer zeros are crossings; the zeros at $1$ and $3$ are double zeros, where the curve touches zero from below.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-57-spin-bath-recurrence.png]
{title=Three-spin coherence factor with exact period-two recurrences}
{height=480}
The <finite spin-bath coherence recurrence> returns the device to full coherence every two time units. Negative $z$ indicates a relative phase change and is not a negative probability. When $ab\ne0$, the device <reduced density matrix> becomes diagonal at each zero, but <quantum decoherence> is not irreversible in this finite bath. The sketch explicitly displays both loss of coherence and its revival.
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