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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.

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  • $\begingroup$ 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? $\endgroup$ – Alexey Uvarov Apr 16 at 8:07
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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).

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  • $\begingroup$ Thanks you. I will try this method. $\endgroup$ – Thapanio Apr 16 at 9:10
  • $\begingroup$ Can you provide me with a reference/textbook for this kind of interactions? $\endgroup$ – Thapanio Apr 20 at 1:10
  • $\begingroup$ 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. $\endgroup$ – DaftWullie Apr 20 at 6:29

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