# Qubit-qubit interaction Hamiltonian term

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!

• The notation must be defined in order for anyone to be able to answer this question. – DanielSank Jul 6 '17 at 6:17