Jaynes-Cummings model

The Jaynes-Cummings model couples a two-level atom to one quantized oscillator mode throughThe first term excites the atom while annihilating one field quantum; the second performs the Hermitian-conjugate process.

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 $|e,n⟩,|g,n+1⟩$.

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.

An initially excited atom interacting resonantly with a coherent state has inversion ${\sum }_n{P}_n\cos (2g\sqrt{n+1}t)$, where $P_n$ is its Poisson distribution of photon number. Different Rabi frequencies dephase on a time scale $1/g$ and approximately rephase near $2\pi \sqrt{\bar{n}}/g$. The revival estimate follows from the adjacent-component frequency difference $g/\sqrt{\bar{n}}$. Large mean photon number makes the early oscillations nearly classical without removing the discrete spectrum responsible for revivals.

The Tavis-Cummings model extends the Jaynes-Cummings model to several two-level systems exchanging excitations with a common quantized normal mode under the rotating-wave approximation. It conserves total excitation number. Equal coupling preserves permutation-symmetric atomic sectors, allowing finite-dimensional reductions for a fixed initial excitation number.

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 $1/3$.