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 $\psi_{\rm {L}}$ of low energy will be called $P_{\rm {L}}$. Similarly, the proba­bility that an atom can be found in an atomic state $\psi_{\rm {H}}$ of higher energy will be called $P_{\rm {H}}$. More simplistic, $P_{\rm {L}}$ can be called the fraction of atoms in the low energy state and $P_{\rm {H}}$ 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 $\rho(\omega)$. The particular frequency $\omega_0$ that is relevant to transitions between two atomic states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ is related to the energy difference between the states. In particular,

$$\omega_0 = (E_{\rm {H}} - E_{\rm {L}})/\hbar$$

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

In those terms, the fractions $P_{\rm {L}}$ and $P_{\rm {H}}$ of atoms in the two states evolve in time according to the evolution equations, {D.41},

$$\fbox{$\displaystyle \frac{{\rm d}P_{\rm{L}}}{{\rm d}t} = \mbox{... ...} \rho(\omega_0)\; P_{\rm{H}} + A_{\rm{H\to{L}}}\; P_{\rm{H}} + \ldots\; $}$$ (7.45)

$$\fbox{$\displaystyle \frac{{\rm d}P_{\rm{H}}}{{\rm d}t} = \mbox{... ...rho(\omega_0)\; P_{\rm{H}} - A_{\rm{H\to{L}}}\; P_{\rm{H}} + \ldots\; $}%$$ (7.46)

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 $P_{\rm {L}}$ of low energy atoms that is available to be excited. It is also propor­tional to the energy $\rho(\omega_0)$ 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 $A_{\rm {H\to{L}}}$ is the sponta­neous emission rate. (It is really the decay rate $\lambda$ as defined earlier in section 7.5.3, but in the present context the term sponta­neous emission rate and symbol $A$ tend to be used.)

The second equation can be understood similarly as the first. If there are transitions with states other than $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$, 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 $A$ and $B$ 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 $A$ and $B$. 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 $B$ coefficients are, {D.41},

$$\fbox{$\displaystyle B_{\rm{L\to{H}}} = B_{\rm{H\to{L}}}... ...t e{\skew0\vec r}\vert\psi_{\rm{H}}\rangle\vert^2}{3} $} %$$ (7.47)

Here $\epsilon_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.854,19 10$\POW9,{-12}$ C$\POW9,{2}$/J m is the permit­tivity of space. Note from the appearance of the Planck constant that the emission and absorption of radiation is truly a quantum effect. The second ratio is the average atomic matrix element discussed in the previous section. The fact that $B_{\rm {L\to{H}}}$ equals $B_{\rm {H\to{L}}}$ reflects that the electric field is equally effective for absorption as for stimulated emission. It is a consequence of the symmetry property of two-state systems mentioned in the previous section.

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

$$\fbox{$\displaystyle A_{\rm{H\to{L}}} = B_{\rm{H\to{L}}}... ...{\rm{equiv}}(\omega) = \frac{\hbar\omega^3}{\pi^2c^3} $} %$$ (7.48)

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 $\rho_{\rm {equiv}}$ 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 $\hbar\omega$ 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, $\rho$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Now assume that initially there is a thin gas of atoms in the box in an excited state $\psi_{\rm {H}}$. These atoms will decay to whatever are the available atomic states of lower energy. In particular, according to (7.46) the fraction $P_{\rm {H}}$ of excited atoms left will evolve as

$$\frac{{\rm d}P_{\rm {H}}}{{\rm d}t} = - \left[ A_{\rm... ...o{L}}_2} + A_{\rm {H\to{L}}_3} + \ldots \right] P_{\rm {H}}$$

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

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

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The constants in the equations are called the Einstein $A$ and $B$ coefficients.

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

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