Two excited atoms in a resonant Tavis-Cummings mode

For two equally coupled atoms initially in $|ee;0⟩$, excitation conservation confines the interaction picture to $|ee;0⟩$, $(|eg⟩+|ge⟩)|1⟩/\sqrt{2}$ and $|gg;2⟩$. The Hamiltonian operator divided by $\hslash$ has off-diagonal entries $\sqrt{2}g$ and $2g$. Its eigenvalues are $0,\pm \sqrt{6}g$, so the amplitudes are $(2+\cos \sqrt{6}gt)/3$, $-i\sin (\sqrt{6}gt)/\sqrt{3}$ and $\sqrt{2}(\cos \sqrt{6}gt-1)/3$. An ideal one-photon number projection therefore heralds a symmetric atomic Bell state with probability ${\sin }^2(\sqrt{6}gt)/3$, at most $1/3$.