Total excitation number ($\mathcal{N}=a^{†}a+({\sigma }_z+1)/2$)

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 $|g,0⟩$ alone, and each eigenvalue $n+1\ge 1$ labels the paired sector $|e,n⟩,|g,n+1⟩$.