The semiclassical equations describe how the kth atom in the system will respond to a radiation field. This set of equations is incomplete. It does not take into account how the atoms will interact with each other. Two types of atomic interactions will be added: collisions and spontaneous emission. There are two types of collisions that will be examined: elastic and inelastic.
Figure: Collision and Spontaneous Emission Constants
An elastic collision does not affect the energy of the total system because the atoms are indistinguishable particles. A symmetric energy change does not affect the system's total energy. However, it will alter the distribution of energy in the system, which will change the quantum phase of the system. This quantum phase is described by the off--diagonal elements of the density matrix and is related to the total dipole moment of the system. On the average, the rate of change due to collisions in the off--diagonal terms is directly proportional to the off--diagonal terms themselves. will be used as the constant of proportionality. Physically, elastic collisions modify the direction of the individual atom's dipole moment. The effect is completely random and eventually destroys the system's total dipole moment. The lifetime of the dipole moment is therefore related to the constant, .
An inelastic collision will alter the energy of the total system and transfers an atom out of the two--state system. Visualize the system as an atom with numerous states. An inelastic collision will send an atom into a state other than |1> or |2>. This result will affect the level probabilities and thus the diagonal elements; indeed, it will unnormalize the distribution. On the average, this effect is proportional to the diagonal elements. The upper and lower states will be allowed to have different loss rates via the proportionality constants and .
There is a certain probability that an atom will spontaneously emit a photon while in the upper state and then drop back into the lower state. Both the probability of being in the lower and in the upper state are affected. On the average, this effect is proportional to . Spontaneous emission makes it harder to find the atom in the upper state; so, the constant of proportionality is positive in the lower state and negative in the upper state. This constant is called the Einstein A Coefficient. Spontaneous emission has an effect on the off--diagonal elements as well via the coupling of the differential equations.
Collisions and spontaneous emission alter the system homogeneously, i.e., they alter the entire system in the same way. There are also inhomogeneous effects which affect individual atoms in different ways. For example, the doppler effect varies the resonant frequency, , of a given atom. Both inhomogeneous and homogeneous results are observed in nature. This paper will focus on the homogeneous effects only, and hereafter it will be assumed that all atoms have the same transition frequency, .