7.8 General Interaction with Radiation

Under typical conditions, a collection of atoms is not just subjected
to a single electromagnetic wave, as described in the previous
section, but to broadband

incoherent radiation of all
frequencies moving in all directions. Also, the interactions of the
atoms with their surroundings tend to be rare compared to the
frequency of the radiation but frequent compared to the typical life
time of the various excited atomic states. In other words, the
evolution of the atomic states is collision-dominated. The question
in this subsection is what can be said about the emission and
absorption of radiation by the atoms under such conditions.

Since both the electromagnetic field and the collisions are random,
a statistical rather than a determinate treatment is needed. In it,
the probability that a randomly chosen atom can be found in a
typical atomic state

The energy of the electromagnetic radiation, per unit volume and per
unit frequency range, will be indicated by

is the nominal frequency of the photon released or absorbed in a transition between the two states.

In those terms, the fractions

In the first equation, the first term in the right hand side reflects
atoms that are excited from the low energy state to the high energy
state. That decreases the number of low energy atoms, explaining the
minus sign. The effect is of course proportional to the fraction

Similarly, the second term in the right hand side of the first
equation reflects the fraction of low energy atoms that is created
through de-excitation of excited atoms by the electromagnetic
radiation. The final term reflects the low energy atoms created by
spontaneous decay of excited atoms. The constant

The second equation can be understood similarly as the first. If
there are transitions with states other than

The constants in the equations are collectively referred to as the
“Einstein

Anyway, the

The spontaneous emission rate was found by Einstein using a dirty
trick, {D.42}. It is

One way of thinking of the mechanism of spontaneous emission is that
it is an effect of the ground state electromagnetic field. Just like
normal particle systems still have nonzero energy left in their ground
state, so does the electromagnetic field. You could therefore think
of this ground state electromagnetic field as the source of the atomic
perturbations that cause the atomic decay. If that picture is right,
then the term

It is a pretty reasonable description, but it is not quite true. In
the ground state of the electromagnetic field there is half a photon
in each mode, not one. It is just like a harmonic oscillator, which
has half an energy quantum

As a special example of the given evolution equations, consider a
closed box whose inside is at absolute zero temperature. Then there
is no ambient blackbody radiation,

where the sum is over all the lower energy states that exist. It describes the effect of all possible spontaneous emission processes that the excited state is subject to. (The above equation is a rewrite of (7.28) of section 7.5.3 in the present notations.)

The above expression assumed that the excited atoms are in a box that is at absolute zero temperature. Atoms in a box that is at room temperature are bathed in thermal blackbody radiation. In principle you would then have to use the full equations (7.45) and (7.46) to figure out what happens to the number of excited atoms. Stimulated emission will add to spontaneous emission and new excited atoms will be created by absorption. However, at room temperature blackbody radiation has negligible energy in the visible light range, chapter 6.8 (6.10). Transitions in this range will not really be affected.

Key Points

- This section described the general evolution equations for a system of atoms in an incoherent ambient electromagnetic field.

- The constants in the equations are called the Einstein
and coefficients.

- The
coefficients describe the relative response of transitions to incoherent radiation. They are given by (7.47).

- The
coefficients describe the spontaneous emission rate. They are given by (7.48).