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Relaxation terms

All the models in the literature agree on the form of transverse relaxation terms (those who apply to coherences): Q(ρ)nm = -γnm ρnm, where γnm = γmn.

On the contrary many models can be found for the longitudinal relaxation applying to populations without any comparison in the physics literature. We comapre theme from the point of view of the good modelling of spontaneous emission and trace conservation [Bid01, Bid02, BBR01]. This study shows that the best model is the Pauli master equation

Q(ρ)nn = ∑m≠n Wnmρmm - ∑m≠n Wmnρnn, where Γn = ∑m≠n Wmn,

which equilibrium states (Q(ρ)=0) are given by the relation

Wnm = Wmn eℏ(ωmn)/κT.

We want to preserve positiveness properties through the time evolution such as

  • ρnn ∈ [0,1];
  • nm|2 ≤ ρnnρmm;
  • ρ is a positive (linear) operator.

To this aim we have to impose conditions on the coefficients. For the first two properties, we find necessary and sufficient conditions that are valid in all the physical contexts:

γnm ≥ 0, Wnm ≥ 0, 2γnm ≥ Γn + Γn - (WnmWmn)1/2,

For the last property, we can make the (very restrictive but always verified in physical models) assumption that

γnm = (Γn + Γn)/2 + γncoll + γmcoll,

in which case a sufficient condition is γncoll ≥ 0.


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