Photoelectron counting in quantum optics

Master equationEffective density matrix of the system ρ(t) ≡ Tr B [χ(t)].Interaction picture with respect to H 0 ,∫dtdt ˜ρ(t) = −iTr B[Ṽ (t), χ(t = 0)] −0dt ′ Tr B [Ṽ (t), [Ṽ (t′ ), ˜χ(t ′ )]].Born approximation, ˜χ(t ′ ) ≈ R 0 ⊗ ˜ρ(t ′ ), R 0 bath density matrix.System-bath interaction as V = ∑ k S k ⊗ B k ,Bath correlation functions C kl (t, t ′ ) ≡ Tr B[˜Bk (t)˜B l (t ′ )R 0],Tr B ˜B k (t)R 0 = 0.∫dtdt ˜ρ(t) = −0dt ′ ∑ kl+ C lk (t ′ − t)[}C kl (t − t ′ ){˜S k (t)˜S l (t ′ )˜ρ(t ′ ) − ˜S l (t ′ )˜ρ(t ′ )˜S k (t){˜ρ(t ′ )˜S l (t ′ )˜S k (t) − ˜S} ]k (t)˜ρ(t ′ )˜S l (t ′ ) . (4)