I = ∑_{ext spins} |⟨out |T̂| in⟩|^{2} = ∑_{ext spins} ⟨out |T̂| in⟩ ⟨in |T̂^{∗}| out⟩

Define the spin density matrix ρ such that:

|in ⟩⟨ in| = ∑_{i,j} ρ_{i,j} = ∑_{i,j} |i ⟩⟨ j|

Now

I = ∑_{ext spins} ∑_{i,j} ⟨out |^{i}T̂ρ_{i,j}^{j}T̂^{∗}| out⟩

Let T̂ = T̂_{d}T̂_{p}, where subscript d ≡ decay and p ≡ production

I = ∑_{ext spins} ∑_{i,j} ⟨out |T̂_{d}^{i}T̂_{p}ρ_{i,j}^{j}T̂_{p}^{∗}T̂_{d}^{∗}| out⟩

Insert

1 = ∑_{X} |X ⟩⟨ X|

and

1 = ∑_{X′} |X′ ⟩⟨ X′|

I = ∑_{ext spins} ∑_{i,j} ∑_{X,X′} ⟨out| T̂_{d} |X⟩ ⟨X|^{i} T̂_{p}ρ_{i,j}^{j}T̂_{p}^{∗} |X′⟩ ⟨X′| T̂_{d}^{∗} |out⟩

I(Ω, Φ) = 2κ ∑_{k} {(1 − P_{γ}) |∑_{l,m}[l]_{m;k}^{(−)}Re[Z_{l}^{m}(Ω, Φ)]|^{2} + (1 − P_{γ}) |∑_{l,m}[l]_{m;k}^{(+)}Im[Z_{l}^{m}(Ω, Φ)]|^{2} +

(1 + P_{γ}) |∑_{l,m}[l]_{m;k}^{(+)}Re[Z_{l}^{m}(Ω, Φ)]|^{2} + (1 + P_{γ}) |∑_{l,m}[l]_{m;k}^{(−)}Im[Z_{l}^{m}(Ω, Φ)]|^{2}} (1)

If P = 0

I(Ω, Φ) = 2κ ∑_{k} {|∑_{l,m}[l]_{m;k}^{(−)}Z_{l}^{m}(Ω, Φ)|^{2} + |∑_{l,m}[l]_{m;k}^{(+)}Z_{l}^{m}(Ω, Φ)|^{2}}  (2)

Let r = (−), (+) then

I(Ω, Φ) = 2κ ∑_{k,r} {|∑_{l,m}[l]_{m;k}^{r}Z_{l}^{m}(Ω, Φ)|^{2}}  (3)