From here on we consider vertical optical dipole transition, and thus have transition probability based on time-dependent perturbation theory that Γ Γ i f = 2 π π ℏ ℏ | ⟨ ⟨ f | H ′ | i ⟩ ⟩ | 2 δ δ ( E f − − E i ± ± ℏ ℏ ω ω ) , with H ′ ≈ ≈ e A 0 m 0 ε ε ⋅ ⋅ p , where ε ε is the light polarization vector. | i ⟩ ⟩ and | f ⟩ ⟩ are the Bloch wavefunction of the initial and final states. Here the transition probability needs to satisfy the energy conservation given by δ δ ( E f − − E i ± ± ℏ ℏ ω ω ) . From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the bracket.