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Driving an ion at a motional sideband transition induces the Jaynes--Cummings (JC) interaction.
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For the hopping Hamiltonian for two ions ( ψ = ψ Ion 1 ⊗ ψ Ion 2 ≡ | n Ion 1 , n Ion 2 ⟩ \psi=\psi_{\rm Ion\,1}\otimes\psi_{\rm Ion\,2}\equiv\ket{n_{\rm Ion\,1},n_{\rm Ion\,2}} ) H Hop = κ 12 2 ( a 1 a 2 † + a 1 † a 2 ) H_{\rm Hop}=\frac{\kappa_{12}}{2}\,(a_{1}a_{2}^{\dagger}+{a_{1}^{\dagger}}a_{2}) , the rate for each transition from | n , 0 ⟩ \ket{n,0} to | n − 1 , 1 ⟩ \ket{n-1,1} and from | 1 , 0 ⟩ \ket{1,0} to | 0 , 1 ⟩ \ket{0,1} can be calculated as ⟨ n − 1 , 1 | H Hop | n , 0 ⟩ = κ 12 2 n \bra{n-1,1}H_{\rm Hop}\ket{n,0}=\frac{\kappa_{12}}{2}\sqrt{n} and ⟨ 0 , 1 | H Hop | 1 , 0 ⟩ = κ 12 2 \bra{0,1}H_{\rm Hop}\ket{1,0}=\frac{\kappa_{12}}{2} , respectively. Therfore, the rate for | n , 0 ⟩ → | n − 1 , 1 ⟩ \ket{n,0}\rightarrow\ket{n-1,1} becomes faster than that for | 1 , 0 ⟩ → | 0 , 1 ⟩ \ket{1,0}\rightarrow\ket{0,1} by the factor n \sqrt{n}
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