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 of low energy will be called . Similarly, the probability that an atom can be found in an atomic state of higher energy will be called . More simplistic, can be called the fraction of atoms in the low energy state and the fraction in the high energy state.

The energy of the electromagnetic radiation, per unit volume and per
unit frequency range, will be indicated by . The
particular frequency that is relevant to transitions
between two atomic states and is
related to the energy difference between the states. In particular,

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

In those terms, the fractions and of atoms
in the two states evolve in time according to the evolution equations,
{D.41},

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 of low energy atoms that is available to be excited. It is also proportional to the energy of the electromagnetic waves that do the actual exciting.

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 is the spontaneous emission rate. (It is really the decay rate as defined earlier in section 7.5.3, but in the present context the term spontaneous emission rate and symbol tend to be used.)

The second equation can be understood similarly as the first. If there are transitions with states other than and , all their effects should be summed together; that is indicated by the dots in (7.45) and (7.46).

The constants in the equations are collectively referred to as the “Einstein and coefficients.” Imagine that some big shot in engineering was too lazy to select appropriate symbols for the quantities used in a paper and just called them and . Referees and standards committees would be on his/her back, big shot or not. However, in physics they still stick with the stupid symbols almost a century later. At least in this context.

Anyway, the coefficients are, {D.41},

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 in the expression above should be the energy of the field in the ground state. In terms of the analysis of chapter 6.8, that would mean that in the ground state, there is exactly one photon left in each radiation mode. Just drop the factor (6.10) from (6.11).

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 left in its ground state, chapter 4.1. Also, a ground state energy should not make a difference for the evolution of a system. Instead, because of a twilight effect, the photon that the excited atom interacts with is the one that it will emit, addendum {A.24}.

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, 0. Now assume that
initially there is a thin gas of atoms in the box in an excited state
. These atoms will decay to whatever are the
available atomic states of lower energy. In particular, according to
(7.46) the fraction of excited atoms left will
evolve as

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).