Total excitation number
= Total excitation number
{title2=$\mathcal N=a^\dagger a+(\sigma_z+1)/2$}
= Excitation number
{synonym}
For the rotating-wave coupling in the <Jaynes-Cummings model>, one atomic excitation is exchanged for one <photon>, so the displayed operator commutes with the <Hamiltonian>. Its <eigenvalue> zero labels $|g,0\rangle$ alone, and each <eigenvalue> $n+1\geq1$ labels the paired sector $|e,n\rangle,|g,n+1\rangle$.