Proposition: Model 1 unitary edge limit

For every fixed real $\tau ,\lambda$,

If $2\overline{C}|\tau |<1$, then on particle number at most $n$,

For a total coherent vector of mean number $\mu$, an upper bound is

which tends to zero.

Proof. On finite-particle vectors use the convergent exponential series from Lemma lem:analytic. Telescoping the $j$-th powers gives the bound

Summing with coefficients $|\tau {|}^j/j!$, and using

proves (eq:edge-unitary). Density and the norm bound two on a difference of unitaries give strong convergence for small times. For arbitrary $\tau$, divide it into finitely many sufficiently small steps and use the group law and strong convergence of uniformly bounded products.

For a coherent vector, truncate at number $n$, use (eq:edge-unitary) on the truncation, and bound the two tail images by twice the tail norm. This gives (eq:Ecoh). First choose $n$ and then let $\epsilon \to 0$.