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The states which have a classical limit are coherent states which, for each mode $k$, are of the form $|z\rangle =e^{-|z|^2/2}e^{z \hat a_k^\dagger}|0\rangle$. These have $$ \hat a_k|z\rangle= z|z\rangle, $$ and hence $$ \langle z |\hat a_k|z\rangle = z. $$