How to construct a Hamiltonian for an ensemble of atoms interacting with each other?

Question

How to construct a Hamiltonian for an ensemble of atoms interacting with each other?

For example if the one atom hamiltonian can be written as:

$$\hat{H}=\left(\begin{matrix}0&\Omega_p(t)&0\\\Omega_p(t)&0&\Omega_r(t)\\0&\Omega_r(t)&0\end{matrix}\right)$$

so the many atoms Hamiltonian can be written as

$$\hat{\mathcal{H}}=\hat{H}\otimes\hat{H}\otimes...\otimes\hat{H}+\sum_{j,k}^{N}V_{jk}\left|R\rangle_j\langle R\right|\otimes\left|R\rangle_k\langle R\right|$$ where $\Omega_p(t)$ and $\Omega_r(t)$ are arbitrary functions, $|R\rangle$ is the Rydberg stat $V$ is the interaction energy between Rydberg states, and $N$ is the number of atoms.

I am using Wolfram Mathematica and every-time I do a simulation for 3 atom, it does not work properly.

Answer 1

I suspect that, for your single-atom Hamiltonians, you want to compose them as $$ \mathcal{H}=H\otimes I\otimes I+I\otimes H\otimes I+I\otimes I\otimes H $$ (example for the 3-qubit case).