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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.

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.

A motional sideband is a transition that changes internal excitation and a motional quantum number together. Define the internal energy gap as $\hslash {\omega }_0>0$. A carrier drives $|g,n⟩\leftrightarrow |e,n⟩$; the first red sideband drives $|g,n⟩\leftrightarrow |e,n-1⟩$; the first blue sideband drives $|g,n⟩\leftrightarrow |e,n+1⟩$. Their resonant drive angular frequencies follow by subtracting these energy levels. For a Raman transition, the drive is a beam-difference frequency with a sign fixed by the selected internal transition.

Choose a real carrier Rabi angular frequency $\Omega$ defined so that resonant carrier excitation is ${\sin }^2(\Omega t/2)$. In the Lamb-Dicke regime, isolated first-sideband Hamiltonians in a rotating frame can be written, after choosing the pulse phase, asThese formulas use a rotating-wave approximation: neglected carrier and other-sideband terms oscillate rapidly compared with the retained coupling. The drive's effective width and sideband coupling must be small compared with $\nu$; sufficient suppression of off-resonant carrier excitation must also be checked. The red interaction is a Jaynes-Cummings model with a motional quantum replacing a cavity photon. The blue interaction creates internal excitation and a motional quantum together; the external laser supplies energy. The driven-system rotating-frame Hamiltonian alone is not the total laboratory energy.

The requested bare labels require $n\ge 1$. Their four energies areAmong these four, the pair that is fully present in one interacting sector is $|e,n-1⟩,|g,n⟩$. Its center is $C=\hslash \omega (n-1/2)$ and its dressed branches areTheir endpoint values at ${\omega }_0=0$ and $2\omega$ are the same, $C\pm \hslash \sqrt{{\omega }^2+4g^{2}n}/2$. At ${\omega }_0=\omega$ they are $C\pm \hslash |g|\sqrt{n}$. The avoided crossing therefore has resonant gapAt the three horizontal-axis landmarks, the bare energies divided by $\hslash \omega$ are respectivelywhich specifies the vertical-axis labeling without assigning an unstated numerical $g$. The original plot below uses illustrative $n=3$ and $g/\omega =0.08$ and shows this complete central doublet together with all four bare lines.

Sector qualification: the outer displayed state $|e,n⟩$ couples to $|g,n+1⟩$, outside the requested bare set. Likewise $|g,n-1⟩$ couples to $|e,n-2⟩$ if $n\ge 2$; for $n=1$ it is the isolated $|g,0⟩$. Thus the four-state set cannot be diagonalized as a closed four-dimensional block. If branches associated with these outer states are also desired, complete those neighboring sectors and use the formula in part c with indices $n$ and $n-2$.

Finally, “adjacent dressed states” within the shown resonant pair have the gap just boxed. Adjacent rungs of the dressed ladder are not equally spaced: at resonance,When neighboring doublets remain ordered, the gap between their nearest branches is $\hslash \omega -\hslash |g|(\sqrt{n+1}+\sqrt{n})$. This distinguishes the within-doublet Rabi splitting from differences between adjacent excitation sectors.

Take ${\sigma }_z|e⟩=|e⟩$ and ${\sigma }_z|g⟩=-|g⟩$. In the resonant Jaynes-Cummings model, the interaction connects only $|e,n⟩$ and $|g,n+1⟩$. Both have free energy $\hslash \omega (n+1/2)$, and their interaction matrix is $\hslash g\sqrt{n+1}{\sigma }_x$. ThereforeExponentiating this two-state block and applying it to $|e,n⟩$ yieldsThe populations undergo Rabi oscillations at angular frequency $2g\sqrt{n+1}$. The common phase in this fixed-$n$ block does not affect its probabilities, but it must be retained when blocks with different photon numbers are superposed.

Let $|g⟩,|e⟩$ be the atomic levels, ${\sigma }_{+}=|e⟩⟨g|$, and $a$ the annihilation operator of the single field mode. Choose the joint ground energy to be zero. The resonant Jaynes-Cummings model isThe first term counts field photons, the second the atomic excitation, and the interaction exchanges one photon for one atomic excitation. A phase redefinition makes $g$ real and positive. Resonance means $\omega ={\omega }_{eg}$; the total excitation $\mathcal{N}=a^{†}a+|e⟩⟨e|$ is conserved. Adding an overall constant gives the equivalent convention using ${\sigma }_z/2$ and a zero-point field term.

The model follows from the electric-dipole interaction and the rotating-wave approximation, retaining the energy-conserving terms while dropping $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$. We assume a single quantized monochromatic mode and negligible cavity leakage and atomic relaxation over the time considered. With this convention, the resonant population Rabi angular frequency in the one-excitation subspace is $2g$.

Quantize a single electromagnetic normal mode of angular frequency ${\omega }_c$ and represent one isolated atomic transition as a two-level system with raising operator ${\sigma }_{+}=|e⟩⟨g|$. The bare Hamiltonian operator isThe first main approximation is the electric-dipole approximation: the field varies negligibly across an atom, $k{a}_{\mathrm{atom}}\ll 1$, and the leading coupling is $H_d=-\mathbf{d}\cdot \mathbf{E}({\mathbf{r}}_0)$. For states of definite opposite parity, the relevant dipole part is off-diagonal. After choosing operator phases, this gives $H_d=\hslash g(a+a^{†})({\sigma }_{+}+{\sigma }_{-})$, with $g$ fixed by the transition dipole and the one-photon field wave amplitude.

In the interaction picture, $a{\sigma }_{+}$ and $a^{†}{\sigma }_{-}$ oscillate at the difference ${\omega }_a-{\omega }_c$, whereas $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$ oscillate at the sum ${\omega }_a+{\omega }_c$. The second main approximation is the rotating-wave approximation: near resonance and for $g\sqrt{n+1}\ll {\omega }_a+{\omega }_c$, the latter terms average away on the interaction time scale. The resulting Jaynes-Cummings model hasThe two-level system and single-normal mode truncations are additional assumptions: other atomic transitions and optical normal modes must be sufficiently far off resonance or weakly coupled. Dissipation is neglected over the interaction time. The retained terms exchange one atomic excitation for one photon, conserving $a^{†}a+{\sigma }_{+}{\sigma }_{-}$.

Use the resonant interaction picture and define $g>0$ by the one-atom Jaynes-Cummings model in part (a). Equal coupling to both atoms gives the Tavis-Cummings modelThe initial state has two excitations and is symmetric under atom exchange. Excitation conservation and this symmetry confine the evolution toThe antisymmetric atomic spin singlet state is uncoupled and unpopulated. Applying the annihilation operator and creation operator givesThe first off-diagonal entry combines two indistinguishable atomic emission paths; the second also includes the $\sqrt{2}$ enhancement for creating the second photon. Its eigenvalues are $0,\pm \sqrt{6}g$. Put $\Omega =\sqrt{6}g$ and $K=H_I/\hslash$, so $K^3={\Omega }^{2}K$. Its exponential isApplying this to $|A⟩$ yieldsAll three states have the same bare energy on resonance, so restoring the free Schrödinger-picture evolution adds only a common global phase. The expression starts at $|ee;0⟩$, has initial derivative $-i\sqrt{2}g|B⟩$, and its three squared probability amplitudes sum to one.

The operator $|e⟩⟨g|\otimes A$ annihilates one field quantum and excites the atom. Its adjoint $|g⟩⟨e|\otimes A^{†}$ de-excites the atom and creates one field quantum. These are the two exchange processes in the Jaynes-Cummings model.

Unitary time evolution requires the Hamiltonian to be Hermitian. Since the common coefficient $\frac{1}{2}\hslash (\Omega -\omega )$ is real, the two terms must be adjoints with equal coefficients. Therefore

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 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 $|e,n⟩,|g,n+1⟩$.