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

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.

• Hi! I'm not sure where exactly such a Hamiltonian comes from (the $\hat{H}$ part looks kind of like some external drive), but this is a valid model nonetheless. Could you please give more details about what you expect as "working properly" and what you get instead? – Alexey Uvarov Apr 16 at 8:07

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).

• Thanks you. I will try this method. – Thapanio Apr 16 at 9:10
• Can you provide me with a reference/textbook for this kind of interactions? – Thapanio Apr 20 at 1:10
• I'm not really sure what it is that you're after (and I'm not great on references). Many places state the form of interaction they use (e.g. arxiv.org/pdf/1512.01141.pdf) but it's just that, a statement. There's not a lot of explanation behind it. – DaftWullie Apr 20 at 6:29