Atomic Physics for Everyone
Will Raven
p-57
3.6 Example
... two of the important concepts are that (1) the scattering rate tells us how many photons per second an atom absorbs (and emits) from the laser and (2) the cumulative effect of all the atoms taking photons results in the absorption profile. Let’s solidify these concepts with an example. Throughout this example, we will also introduce some commonly used language in atomic physics. A lot of the language will be explored in more detail as we work our way through this book.
A particular type of barium atom called barium-135 has one ground state and three closely spaced excited states, see Fig.1. The spacing between the excited states is called hyperfine splitting, . The reason there are three excited hyperfine states is because the nucleus has angular momentum. If the nucleus did not have angular momentum, there would be only one excited state. Again. Before we do an example, we need a few more definitions.
Fig. 1 A simplified Grotrian diagram for a transition in the atom known as barium-135. The energy spacings are not to scale. The energy spacings between the excited hyperfine levels are calculated from the results of Baird et al.[1]. The center of gravity frequency is extracted using numbers from both Baird et al. [1] and Karlsson et al. [2]. The uncertainty in the center of gravity frequency is about 30 MHz. [2] While that isn’t a terrible uncertainty, modern day spectroscopic methods can do better!
As seen in Fig. 1, we are going to label the ground state 
and the three excited states 
, and 
. 
is called a quantum number, and it is always a positive integer, a half integer, or zero; quantum numbers ... For now, these quantum numbers are just being used to label our states. Notice there is a center of gravity frequency, 
, that tells us the energy difference between the center of gravity for the two states. This tells us that our transitions are all around 541.4 THz, or 553.7 nm. Each hyperfine level is shifted from their center of gravity state by a small amount. I want to emphasize that if a nucleus has angular momentum, the center of gravity states do not exist in real life! The hyperfine states are the actual states. However, we can learn a lot of physics by determining how the hyperfine levels shift from the center of gravity.
With that nice long intro, let’s get into the actual example. We have four states: one ground state and three excited states. We are going to assume that our atoms are all at rest and send a laser beam through the sample, see Fig. 2. All of the atoms will start in the ground state 
An electron can be excited to a higher energy level as long as the following rule, derived from quantum mechanics, is satisfied: A transition can change F by −1, 0, or 1. Memorize this rule! For completeness, there is one exception to this rule. If the ground state has F = 0, that electron cannot be excited to an F = 0 excited state.
Fig. 2 The experimental setup for the thought experiment
(3.15)
In this example, an electron in the barium ground state can be excited to any of the excited states. However, if there were an excited state with quantum number F = 7/2, an electron in the ground state could not be excited to this state because ΔF = 2.
The natural linewidth of all three transitions is about γ = 19 MHz with a saturation intensity of Is = 14.6 mW/cm2. Let’s set our laser intensity to I = 21.9 mW/cm2, corresponding to a saturation parameter of s = 1.5, and predict an absorption plot. In this example, we are interested in the frequency of each spectral feature, so we are, for now, going to ignore the amplitudes of the spectral features. .
First, we calculate the resonance frequency for each transition:
These are the center frequencies of each spectral feature, and each feature has a Lorentzian lineshape with a width of 
Figure 3 shows the simulated results. The horizontal axis represents the laser frequency relative to the center of gravity frequency, = 541,432,745 MHz.
Fig. 3A simulated absorption plot for a transition in barium-135. In this example, we are assuming that all the atoms are at rest and the experimental setup is shown in Fig. 2 The horizontal axis is the laser frequency with respect to the center of gravity frequency
