 Normalization of photon-added and photon-subtracted states

Raw vectors $a^{†}|\psi ⟩$ and $a|\psi ⟩$ have squared norms $\mu +1$ and $\mu$. Matrix elements of the number operator in these vectors must be divided by their squared norms before calling them means. For an optical squeezed vacuum with mean $s$, the raw moments are $(3s+1)(s+1)$ and $s(3s+1)$, while both normalized means equal $3s+1$ when subtraction is possible.