Polarization moments in quantum physics

For two levels J and J which are linked to each other by optical transition the density operator p may be represented by the matrix 

The non-diagonal submatrices j j p and j j p describe the optical coherences between the magnetic sublevels of the states J and J. The submatrices j j P and j j p describe the particles on levels J and J respectively. Their diagonal elements characterize the populations of the respective sublevels M and M, whilst the non-diagonal elements describe the Zeeman coherences. 

The expansion of p may be performed in two ways, using as bases either the operators jl j2Tq, or their conjugates j Tq)  

The difference between these expansions consists of the following. In the first case the operators jxj2Tq and the moments are contragredient 

In the second case the expansion is cogredient, and both j Tq and polarization moments j fQ transform, at the turn of coordinates, in a similar way to the form of (D.5). The form of expansion (D.3) is applied, e.g., in [46, 73, 95, 110, 187, 304, 309, 402]), and also in other papers by these authors, whilst form (D.4) was first introduced by Dyakonov in 

---

Appendix D. Various methods of introducing polarization moments D.l Polarization moments in quantum physics... 

---