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

TL;DR

This paper explores novel non-algebraic methods for expressing the exponential of sums of non-commuting operators, with applications to quantum decay processes and photonic lattices.

Contribution

It introduces new examples of exponential operator expressions for non-commuting operators, extending beyond traditional algebraic factorizations.

Findings

Derived new operator exponential expressions for non-commuting cases

Applied factorization to Lindblad operators in quantum decay

Extended methods to Glauber-Fock photonic lattices

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.