Isolated ground sector of the Jaynes-Cummings model ($\mathcal{H}|g,0⟩=-\hslash {\omega }_0|g,0⟩/2$)

Conservation of excitation number separates the Jaynes-Cummings model into two-dimensional sectors $\{|e,n⟩,|g,n+1⟩\}$ for $n\ge 0$, together with the isolated one-dimensional sector $|g,0⟩$. The latter is annihilated by both interaction terms. Omitting it makes the often-stated paired-sector decomposition incomplete.