The relationship between real and virtual excitation mechanisms for nonlinear refraction

It is explained in physical terms why the nonlinear refraction due to real excitation is enhanced by the factor 2T1/T2 (T1 is the lifetime and T2 is the dephasing time) compared with that due to virtual excitation. In this explanation it is assumed that only changes in electron state occupation probabilities lead to changes in the refractive index, regardless of the nature, real or virtual, of the excitation. This assumption is verified for a two-level model using an exponentially growing pulse envelope, leading to the concept of generalized state filling, i.e. filling by a mixture of real and virtual excitation. To establish this concept a generalization of Fermi's Golden Rule for the transition rate is required. The required generalization is derived using the equation of motion for the density matrix. The derivation adds insight to the range of validity of the standard Fermi Golden Rule result. The concept of generalized state filling is extended to arbitrary pulse shapes. To illustrate this concept, an expression is derived for the change in the occupation probabilities of a two-level system illuminated near resonance by a pulse with exponential leading and trailing edges. The physics contained in this algebraic result is discussed and much insight into the relationship between real and virtual excitation is obtained. The algebra shows that the time constants T1, and T2/2 that govern the response for real excitation are both replaced by the same time constant for virtual excitation, namely the time constant determining the rate of change of the pulse intensity.