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In the Tavis-Cummings model which is a generalization of the Jaynes-Cummings model for multi atoms, the total Hamiltonian is given by (https://arxiv.org/pdf/1201.2928.pdf):

$$ H=H_\text{q}+H_\text{f}+H_{\text{int}} $$

or $$ H=\hbar \omega _0 S_z + \hbar \omega a^+ a + \hbar \omega \beta (a^+ +a)S_x $$

Where $H_f=\hbar \omega a^+ a$ is the Hamiltonian for the light with $a$, and $a^+$ as creation and annihilation operators of the field. $\hbar \omega _0 S_z$ is the Hamiltonian for the qubit pair, with the atomic transition frequency $\omega_0$, and $S_z$ & $S_x$ are defined as addition of individual Pauli spin matrices of each qubit: $$S_z=\frac{\hbar \omega _0}{2} (\sigma _z^{(1)}+\sigma _z^{(2)})$$ and, $$S_x=\frac{\hbar \omega _0}{2} (\sigma _x^{(1)}+\sigma _x^{(2)})$$

And the last term is the interaction Hamiltonian of qubit pair with the light.

So my question is why don't they write qubit-qubit interaction term too, which should be dipole-dipole interaction of each qubit and something like $$ H_{qint}=\hbar g \sigma _x^{(1)} \sigma _x^{(2)}$$

Do you know the reason for this? Any help would be highly apprecieated!

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  • $\begingroup$ The notation must be defined in order for anyone to be able to answer this question. $\endgroup$ – DanielSank Jul 6 '17 at 6:17
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Sometimes you can extend your model through the inclusion of the term of dipole interaction between atoms. But generally, the interaction between atoms is not taken in the original Tavis-Cummings model.

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  • $\begingroup$ Wow didn't expect an answer after couple of years. Yeah, I figured that out that it isn't that necessary to include this interaction. So if I'm not wrong, it is a model that describes the system pretty well without this interaction. Still, thanks for the answer! $\endgroup$ – TheDorkSide Feb 3 at 13:24
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An important point to note is that qubit-qubit interactions can be mediated by the light field/cavity mode. The dipole-dipole interaction is a typical example thereof and can be understood in a pertubative picture as a dipole emitting a photon into the mode which is subsequently absorbed by the other dipole.

To see this formally, one can for example adiabatically eliminate the cavity mode (this works in the bad cavity regime), which produces an effective interaction Hamiltonian with terms as suggested by the OP.

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