Past exam of the quantum theory course of the University of Oxford / 2021 / 2 / b / ii / Solution

From part (a), $[N,a_j]=-a_j$ and $[N,a_j^{\ast }]=a_j^{\ast }$. The commutator product rule givesConsequentlyEach term in $L$ transfers one excitation between the two modes, preserving the total number operator eigenvalue. This is an instance of the oscillator bilinear commutator; every number operator eigenspace is an invariant subspace of $L$.