Phenomenological Inclusion of Decay Phenomena

So far we have neglected the fact that the levels E and Ej are not only coupled by transitions induced by the external field but may also decay by spontaneous emission or by other relaxation processes such as collision-induced transitions. We can include these decay phenomena in our formulas by adding phenomenological decay terms to (2.98) which can be expressed by the decay constants and (see Fig.2.27). 

In the rotating-wave approximation, where the term with frequency is neglected, (2.98) then become 

The expectation value D of the dipole moment for our two-level system which interacts with the radiation field is 

When the field amplitude Aq is sufficiently small [see (2.104)] we can use the weak-signal approximation of the previous section. This means that a(t) = 1 and b(t) l and also aa - bb 1. With this approximation one obtains, after taking the second time derivative of (2.113) and using (2.111), the equation of motion for the dipole moment of the atom under the influence of the radiation field, 

The inhomogeneous equation (2.114) is the quantum mechanical equivalent of the classical equation (2.41) and shows that the induced dipole moment of the atom interacting with a monochromatic radiation field behaves like a damped harmonic oscillator with the eigen frequency = (Ej - E )/h and a 

This is a Lorentzian line profile (Fig. 2.19) with a full halfwidth yah = ya + yb- 

After taking the second-time derivative of (2.66b) and using (2.80), the equation of motion for the dipole moment D of the atom under the influence of a radiation field, becomes 

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