= Short-time Gaussian spin-bath decoherence
{title2=$z_N(t)\simeq e^{-2Ng_*^2t^2/3}$}
For initially equatorial spins and uniform couplings on $[0,g_*]$, expand $\log\cos(2g_kt)=-2g_k^2t^2+O(g_k^4t^4)$. The <strong law of large numbers> then gives $\sum g_k^2\simeq Ng_*^2/3$. On the scale $t=O((g_*\sqrt N)^{-1})$, the total fourth-order remainder tends to zero, giving the Gaussian coherence envelope. This approximation concerns early times and does not remove <finite spin-bath coherence recurrence>.
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