= Ensemble spin-bath coherence
{title2=$\mathbb E z_N(t)=\operatorname{sinc}(2g_*t)^N$}
For independent uniform couplings on $[0,g_*]$ and equatorial initial bath spins,
$$
\mathbb E z_N(t)=\left[\frac{\sin(2g_*t)}{2g_*t}\right]^N,\qquad
\mathbb E|z_N(t)|^2=\left[\frac12+\frac{\sin(4g_*t)}{8g_*t}\right]^N.
$$
At fixed finite $N$, the mean tends to zero and the mean square tends to $2^{-N}$ as time grows. At any fixed nonzero time, the mean-square bracket is strictly below one; the <Markov inequality> proves coherence tends to zero in probability as $N$ grows. Individual finite realizations still exhibit <finite spin-bath coherence recurrence>.
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