where, H i = 1 2 ( ( Ω Ω | 1 ⟩ ⟩ i ⟨ ⟨ r | + Ω Ω ∗ ∗ | r ⟩ ⟩ i ⟨ ⟨ 1 | ) − − Δ Δ | r ⟩ ⟩ i ⟨ ⟨ r | is the Hamiltonian of i-th atom, Ω Ω is the Rabi frequency of coupling between the Rydberg states and the | 1 ⟩ ⟩ state and Δ Δ is the detuning (see figure to the right for level diagram). When | V r r | >> | Ω Ω | , | Δ Δ | , we are in the so-called Rydberg Blockade regime. In this regime, the | r r ⟩ ⟩ state is highly detuned from the rest of the system and thus is effectively decoupled. For the rest of this article, we consider only the Rydberg Blockade regime.