Bloch equations Bloch equations Relaxation terms ModelThe derivation of Bloch equations can be found in numerous textbooks (e.g. [Boy92, LDK94, Lou83, PP73, She84]). They describe the time-evolution of an observable called density matrix ∂_{t}ρ_{nm} = -iω_{nm}ρ_{nm} - iE(t)·[μ,ρ]_{nm} + Q(ρ)_{nm}. In this description, we assume that matter is well described by its N first eigenstates. The matrix diagonal entries ρ_{nn} describe the population of level n, 1≤n≤N. Off-diagonal entries model the coherence between levels. The first term of the right handside stems from the unperturbed hamiltonian of the system which is characterized by the transition frequency ω_{nm}=ω_{n}-ω_{m} between two levels. The second term is due to the action of the electromagnetic field on the system. We consider here that the field E(t) is given. The dipolar moment μ=p/ℏ is a vector valued matrix which describes the ability of a transition to generate a polarization in each space direction. Last, [μ,ρ] = μρ-ρμ is a commutator and Q(ρ) a phenomenological term which stems from numerous sources (statistical mixing, collisions, vibrations,…) and is the subject of the next paragraph. |