Some non-algebraic forms of $\exp(A+B)$

Abstract: We present examples where expressions for $\exp(\hat{A}+\hat{B})$ can be derived even though the operators (or superoperators) $\hat{A}$ and $\hat{B}$ do not commute in a manner that leads to known factorizations. We apply our factorization to the case of a Lindblad operator modeling single photon decay and to a binary Glauber-Fock photonic lattice.