What's the physical interpretation of ρ for dipole forbidden channel?

[zhdx]

Consider we have a $\Lambda$ type three level system, the one upper level is denoted by 3, and the two lower levels are denoted by 1 and 2. Assuming that the transitions between 1$\leftrightarrow$3 and 2$\leftrightarrow$3 are dipole alowed, while the transition between two lower states 1 and 2 is dipole forbidden. As we know the real and imaginary part of ρ₃₁ describe the dispersion and absorption of the external light in 1$\leftrightarrow$3 channel. But what does the ρ₁₂ mean?

 

[M Quack]

The same thing.

Dipole-forbidden means that the electric dipole moment vanishes. Higher order multipole transitions (E.g. quadrupole) can still occur, but the matrix elements and thus the transition probabilities are much lower. Therefore the electric dipole is usually very dominant, and if the dipole is allowed, then all other transitions can be neglected. The net effect is that the 1-2 transitions are very weak (but still not completely forbidden)= the 2 state has a very long lifetime.

 

[zhdx]

Does the ρ₁₂ has relation to the population distributions?

The net effect is that the 1-2 transitions are very weak (but still not completely forbidden)= the 2 state has a very long lifetime.

Thanks!
Does the ρ₁₂ has relation to the population distributions? Say consider two cases:
Case 1: ρ₁₁=0.9, ρ₂₂=0.1
Case 2: ρ₁₁=0.1, ρ₂₂=0.2
What I've found is that the ρ₁₂ in Case 2 is much less than in Case 1. (I've add a dephasing constant γ_2d for level 2 compared to level 1.)