Spontaneous emission is an inherently quantum mechanical and random effect. It is similar to inelastic collisions. This result should be expected since inelastic collisions link the upper and lower states to other levels. Spontaneous emission links the upper level to the lower level, which in turn increases the population in the lower level and decreases the population in the upper level. The normalization of the system is preserved in this case because the population of the upper state is being drained into the lower state and not into some level outside the two--state system. Thus, the dipole moment is altered, but not necessarily destroyed.
Place all the atoms in the lower state. Set Freq=1.00 and Amp=0.05 and enable spontaneous emission with . Set the analysis to vs time and uvw vs time. As the system evolves toggle between uvw vs t and dipole moment vs time.
Figure: Rabi oscillations including spontaneous emission Freq=1.00, Amp=0.05, and A=0.01
As the system reaches a steady state, in Figure , the populations of the upper and lower states are the same as in elastic collisions; however, the dipole moment is non--zero. This non-zero dipole moment occurs because the linkage in this case is specifically between two levels. The transitions are not completely random. It is the random nature of collisions which destroys the dipole moment.