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Model

The 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

tρnm = -iωnmρnm - 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 ρnn describe the population of level n, 1≤nN. 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 ωnmnm 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.


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