 Solution

For these three couplings, the decoherence factor becomesIt 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.

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 $ab\ne 0$, 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.