Density matrix, nonlinear light interaction

A formal expression for the resonant nonlinear susceptibility can be obtained by describing the light-matter interactions in a density matrix formalism (Boyd 2003 Mukamel 1995), which is beyond the scope of this chapter. A third-order perturbative expansion of the system s density matrix yields the following form for the nonlinear susceptibility ... 

In order to introduce some notation, we first recall a few of the well-known properties of the interaction of light pulses with molecules in the linear approximation. Frequently, the signals in nonlinear optical experiments are expressed in terms of the polarization induced in the medium by the incident pulses. The complex linear polarization P t) vector for a distribution of identical two-level systems is obtained from an elementary calculation of the density matrix using the Liouville equation of a system perturbed by an electric field and proceeding as follows ... 

The nonlinear interaction of light with matter is useful both as an optical method for generating new radiation fields and as a spectroscopic means for probing the quantum-mechanical structure of molecules [1-5]. Light-matter interactions can be formally classified [5,6] as either active or passive processes and for electric field based interactions with ordinary molecules (electric dipole approximation), both may be described in terms of the familiar nonlinear electrical susceptibilities. The nonlinear electrical susceptibility represents the material response to incident CW radiation and its microscopic quantum-mechanical formalism can be found directly by diagrammatic techniques based on the perturbative density matrix approach including dephasing effects in their fast-modulation limit [7]. Since time-independent (DC) fields can only induce a... 

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