# Why is there no atom-atom interaction term in the Jaynes-Cummings Hamiltonian?

The Jaynes-Cummings Hamiltonian for a single mode interacting with multiple atoms is $$H/\hbar = \sum_{j=1}^N \omega_{\text{atom},j}\sigma_{+,j}\sigma_{-,j} + \omega_{\text{field}} a^\dagger a + g/\hbar (S_+ a + S_- a^\dagger)$$ where $$S = \sum_{j=1}^N \sigma_j$$

I'm wondering why there is no term for atom-atom interaction. I'm imagining that the interactions between two atoms be treated as dipole interactions. And if it is the case, is it justified to say that this interaction is too small to be included in the JC Hamiltonian? And if it too small, why don't we use perturbation theory to take it into effect, instead of neglecting it completely?

Because the (standard) Jaynes-Cunning model focusses on the interaction of a single atom with a single mode of quantum electromagnetic field.

You can extend it to e.g. a non-linear model to deal with spatially varying interactions.

You can also extend it to be more realistic by including spontaneous emission of atoms, cavity modes' intensity decay, drive terms etc. like in section 10.4 of Steck's quantum optics notes. I would imagine one could also add a similar term for atomic interactions, though this will probably only affect the $$\sigma$$ operators.

I'm imagining that the interactions between two atoms be treated as dipole interactions

It really depends on the atom and on the internal state, and on the energy regime (temperature, hence available energy to excite scattering channels). They could have a large magnetic moment and hence long range interactions, or they could ionised and hence have a non-vanishing electric dipole moment (and hence even larger long range interactions), or they could be near $$T=0$$ where usually you just deal with contact interactions.

The Jaynes-Cummings Hamiltonian, if we are being historical purists, refers specifically to the interaction between a single two level system and a single harmonic oscillator mode. This is a toy model for the interaction between a single atom and an optical cavity.

One can generalize this Hamiltonian in many ways. As you point out, we can include many atoms. We can also allow the atoms to have more complicated level structures than just two levels. We can allow the atoms to move through the cavity mode (so that $$g$$ is a function of atomic position $$g(\hat{x})$$). We can also include many longitudinal and transverse cavity modes.

The generalization you suggest of allowing the spins to interact with each other is a valid extension which has not been wholly unexplored. Why is it not typically included?

It should only be included if it has a big impact on the physics. In many atomic physics experiments the direct interaction between neutral atoms is very very minor. So minor that, even at a perturbation theory level, it just doesn't affect the physics that is observed.

One case where direct interactions between the spins of neutral atoms is when the atoms are Bose condensed. In this case the density is very high that atom-atom collisions can affect the physics. In the case of spinor BECs this physics can be spin-dependent.

Another case is in the case of highly magnetic atoms whos magnetic moments can interact at a longer distance.

Another case in which direct spin spin interactions are significant is the case of Rydberg atoms.

It's also worth mentioning that the model of a cavity interacting with multiple atoms can, in certain regimes, give rise to effective spin-spin interactions between different atoms. Sometimes it is said that the "direct" interaction you are curious about is mediated by real photons being exchanged between the atoms while the effective spin-spin atoms to which I am referring are mediated by virtual cavity photons.

edit: Short version: One pedantic answer is that there's no spin-spin interaction because then it wouldn't be the Jaynes-Cummings Hamiltonian! The Jaynes-Cummings Hamiltonian should only have 1 atom anyways! The more serious answer is that, at least in neutral atom experiments where this model is studied, the spin-spin interactions are typically so small that they don't affect the physics at all. But, that's not to say it's impossible to develop experiments in which direct spin-spin interactions between neutral atoms DO affect the physics.