Radiation tensors, molecular photonics

The latter result (82) yields a quantum probability amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate downconversion. To see this, we note that the matrix element for SHG invokes the tensor product Py (—2co co, ) p([/lC., where the brackets embracing two of the subscripts (jk) in the radiation tensor denote index symmetry, reflecting the equivalence of the two input photons. As shown previously [1], this allows the tensor product to be written without loss of generality as ( 2co co, co), entailing an index-symmetrized form of the molecular response tensor,... 

In these equations the position of the molecule is described by the vector R the wavevectors of the two beams of modes r2 and are k2 and k3 respectively, with ( 2) and (q3) the corresponding mean photon numbers (mode occupancies) and is a unit vector describing the polarization state of mode rn. In deriving Eqs. (120) and (121), the state vectors describing the radiation fields have been assumed to be coherent laser states, and so, for example, (<72) = (oc n a(2 ), where a ) is the coherent state representing mode 2 and h is the number operator a similar expression may be written for (<73). Also, the molecular parameters apparent in Eqs. (120) and (121) are the components of the transition dipole, p °, and the index-symmetric second-order molecular transition tensor,... 

Rayleigh and Raman scattering are two-photon processes in which absorption of a photon is followed coherently by emission of the same and of a different photon, respectively. They are thus second-order processes if we use a description in which the interaction between matter and radiation is used as a perturbation. The complete second-order treatment (Sushchinskii, 1972 Behringer, 1974) is outside the scope of this contribution. We shall start with the simplified description of the scattering tensor that is familiar from many previous treatments. The scattering tensor describing the Raman transition from the molecular eigenstate / > to /> is represented by a 3 X 3 matrix with (molecule-fixed) Cartesian components (in atomic units, ft = 1)... 

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