An important statement and two definitions
▶ Important Statement If a photon has the same energy as the energy difference between the ground state and an excited state, the atom will absorb that photon and move an electron to the excited state. If the photon does not have the same energy as that energy difference, the atom will ignore the photon completely.
• Resonance: When an atom gets excited by a photon from one state to another, we say the atom “goes through resonance.” This is similar to playing the trumpet. When you blow into a trumpet, you excite a standing wave in the pipe to create a note. The same thing happens with an atom. When you excite an atom, the electron goes from one standing wave mode to another. Atomic physicists use the word “excitation” and “resonance” interchangeably.
• Resonance frequency: When the frequency of the laser is just right to excite the atom from the ground state to an excited state, we call that the resonance frequency. We use the variable fr for resonance frequency. Since resonance frequency is a frequency, it uses units of Hz, MHz, GHz, or THz.
Imagine we have a bunch of atoms in a vapor cell, which is a sealed glass tube containing only the atoms we are interested in. We are going to make a few assumptions:
• The atoms are in their gaseous form.
• All of the atoms are frozen in place. In other words, the speed of every atom is 0 m/s (at rest).
• None of the atoms are interacting with anything, including each other. This means that all the atoms are neutral (each atom has an equal number of electrons and protons)
• Every atom has only 2 energy states. The state with lower energy is called the ground state. The state with higher energy is called the excited state. This simplified atom has a very descriptive label: the two-level atom.
Now, we do spectroscopy. We start by sending a laser through the vapor cell and detecting how much light makes it through, see Fig. 3.1. At the beginning of the thought experiment, the photons in the laser do not have enough energy to excite the atoms from the ground state to the excited state. In other words, the frequency of the laser is lower than the resonance frequency. As such, we don’t expect the
Fig. 1 The experimental setup for the thought experiment
atoms to absorb photons from the laser beam. Next, we will smoothly increase the frequency of the laser until the photons in the laser beam have more energy than the energy difference between the ground and excited states. In going from too small to too big, the laser frequency will, at some point, be just right so that the photons have the right amount of energy to excite the atom, which we call resonance. In the lab, smoothly changing the laser frequency over time is referred to as “scanning the laser.”
A transmission plot is a plot of the fraction of photons that make it through the vapor cell as a function of laser frequency. The question is, what does our transmission plot look like for this thought experiment? Based on what we’ve learned so far, a completely reasonable guess would be that the atoms completely ignore the photons unless the photons have the perfect energy to excite the atom from the ground state to the excited state. So, you might guess that our transmission plot looks like the sharp dip in Fig. 3.2. The plot has a single, sharp dip that occurs at the resonance frequency. If the laser has any other frequency the photons do not have the correct energy to excite the atom. However, this is not quite right. A more realistic transmission plot can be seen in Fig. 3.3a. The transmission plot does indeed have a dip at the resonance frequency, but the dip has a width. This width is
Fig. 2 A completely reasonable, but incorrect, guess for the transmission plot. For a transmission plot, 100% means that no photons are absorbed by the atoms, 50% means that half of the photons are absorbed, and 0% means that all of the photons are absorbed by the atoms.
Fig. 3 (a) A more accurate transmission plot for the thought experiment. (b) A picture of a laser beam whose frequency matches a resonance frequency near 459 nm (blue light) passing through a vapor cell of cesium atoms. As we scan the laser frequency from below the resonance frequency to above the resonance frequency, we visually see no glowing cylinder, followed by a glowing cylinder (resonance), followed by no glowing cylinder. On resonance, the transmission decreases as shown in (a) because the atoms take photons from the laser and re-emit them in all directions, and almost none of these re-emitted photons continue along their original path to the detector. Note that the transmission plot in (a) is a theory plot assuming that the atoms are at rest. In the vapor cell the atoms are not frozen in place. Chapter 4 will explore a transmission plot for atoms moving around .
called the natural linewidth of the transition and is represented by the lowercase Greek letter gamma, γ . The natural linewidth is a frequency, so it has units of hertz. Spectroscopists often just say “linewidth” instead of “natural linewidth.” In an experiment, we would see our atoms start to glow as the laser frequency passes through resonance, see Fig. 3.3b.
Often times spectroscopists would rather look at an absorption plot instead of a transmission plot, see Fig. 3.4. An absorption plot is the fraction of photons lost as a function of laser frequency. It looks very similar to a transmission plot, but it has a bump instead of a dip. Both an absorption plot and a transmission plot tell us the same information: atoms absorb (and re-emit) photons from the laser around the resonance frequency. If you add absorption to transmission, you should get 100% for all laser frequencies.
Fig. 4 An illustrative example of an absorption plot (left) and a transmission plot (right). If you add these two plots together, you would get 100% for all laser frequencies
Fig. 5 A zoom in of a spectral feature. The natural linewidth is the full width half maximum of this feature .
The natural linewidth is the full width at half the maximum (FWHM) of the absorption bump, see Fig. 3.5. It is a property of the transition that we cannot change. As an analogy, think about the charge or mass of an electron. The charge of the electron is simply the charge of the electron, which is 1.602×10−19 coulombs. The mass of the electron is simply the mass of the electron, which is 9.11 × 10−31 kg. These are intrinsic properties of the electron that we cannot change. The natural linewidth of a transition is inherent for that transition, and we cannot change it.
The shape of the bump (or dip in the transmission plot) is often referred to as a spectral feature or spectral profile. For completeness, the width of the spectral feature we measure in the lab is always larger than the natural linewidth because of various “broadening” mechanisms. One of these broadening mechanisms is laser power, which we will discuss in Sect. 3.5. If the laser power was the only broadening mechanism, we would find that as the laser power gets smaller and smaller, the width gets narrower and narrower until it reaches the natural linewidth. The natural linewidth is the minimum possible FWHM of a spectral feature.
The mathematical shape of the spectral feature is a Lorentzian function. In the absence of any broadening mechanism, the mathematical form is:
(1)
where γ is the natural linewidth and A is the maximum absorption, which describes the amount of laser light lost when traveling through an atomic sample when the laser light is perfectly on resonance. A is a number between 0 (no light absorbed) and 1 (all light absorbed, which is also 100%). You may be familiar with a similarly shaped Gaussian function. A Lorentzian function looks very similar, but it is slightly different in shape.
Important Spectroscopists being spectroscopists call the spectral feature many different things. Other common descriptions of the spectral feature include the Lorentzian profile, the absorption profile, the absorption lineshape, the spectral profile, the spectral lineshape, or the Doppler-free spectrum. The description “Doppler-free spectrum”... Some atomic physicists use the descriptor “Lorentzian function”, but that should be avoided because that phrase is a mathematical definition of that function itself. As an analogy, think about how many times you have used a sine wave or a cosine wave in math. These are mathematical functions that are used to describe lots of different physical phenomena like an oscillating spring or a swinging pendulum. The same thing is true with the Lorentzian function. There are other physical phenomena described by a Lorentzian function, so we instead use more specific language. For the rest of the book, we will use the term spectral feature or spectral profile....
▶ Important Statement A photon has a probability of being absorbed by an atom depending upon the photon’s energy. It is most likely to be absorbed if the photon’s energy exactly matches the energy difference between the ground and excited state, but there is a non-zero probability of absorption off resonance.
Specifically, in the absence of any broadening mechanism (like laser power), if the photon’s energy is off by hγ/ 2 from that resonance energy, the photon is half as likely to be absorbed compared to a photon that has the energy equal to the energy difference between the ground and excited states.
