Resources

• Inguscio, M., & Fallani, L. (2013). Atomic Physics: Precise Measurements and Ultracold Matter. https://doi.org/10.1093/acprof:oso/9780198525844.001.0001
• Budker, Dmitry, Derek F. Kimball, and David P. DeMille. Atomic physics: an exploration through problems and solutions. Oxford University Press, USA, 2004.
• Spectroscopy data

Questions

Alkali atom level structure

1. List alkali atoms that you know.
2. What are similarities and differences between hydrogen and alkali atom structures?
3. Draw D1 and D2 lines including fine and hyperfine structures for alkali atoms
4. What does this notation stand for \(5^2S_{1/2}\)?
5. What is a cause of fine and hyperfine structures?
6. Typical wavelengths, splittings, and lifetimes for all alkali atoms?
7. What are the specific values for Rb87?

Rubidium isotopes

1. What an isotope of an atom?
2. What are magic numbers?
3. What are two commonly used isotopes of Rb?
4. What are 85 and 87 numbers mean? What is difference between their structures?
5. What is ratio of these two isotopes in a natural abundance?
6. What is a King plot?

Atomic linewidth

1. What is homogeneous and inhomogeneous broadening of atoms?
2. Which lineshape has natural and Doppler line?
3. What is a typical linewidth for a room temperature gas and for a cold gas? Which formula would you use?

Atomic spectrum measurement

1. Draw a simplified experimental set-up for atomic spectrum measurement?
2. How to scan through different frequencies and see it on an oscilloscope?
3. Draw an expected trace on an oscilloscope, what is on the y axis? What is a span of your scanning?

Doppler-free saturated absorption

1. What is Doppler free saturated absorption?
2. Draw a scheme?
3. How many peaks do we expect to see? What is a crossover peak?
4. How to select a specific velocity class?

Working with a cell with hot gas

1. How to heat up a cell?
2. What is a cell with buffer gas?
3. What is a coherence preserving coating?

Two valence electrons

1. What is para- and ortho-helium?

Answers

Alkali atom level structure: answers

1. List Alkali atoms that you know. Alkali atoms include lithium (Li), sodium (Na), potassium (K), rubidium (Rb), cesium (Cs), and francium (Fr).
2. What are similarities and differences between hydrogen and alkali atom structures? Similarities between hydrogen and alkali atom structures include having a single valence electron and similar electronic configurations. However, alkali atoms have additional inner electrons, leading to a larger atomic size and different energy levels. Shielding results in D1 and D2 lines.
3. Fine and hyperfine splitting of atomic levels of alkali atoms

   

4. What does this notation stand for \(5^2S_{1/2}\)? The notation \(5^2S_{1/2}\) represents the electronic configuration of an alkali atom. The “5” indicates the principal quantum number. The “2” is the multiplicity, as such that 2S+1=2, so the spin is 1/2. The letter “S” represents the total angular momentum electron spin, so L=0. The 1/2 is total electron angular momentum value J=L+S =1/2, where L=0 and S=1/2.
5. What is a cause of fine and hyperfine structures? Fine structure arises from interactions between the electron’s spin and electron’s orbital momentum. The hyperfine structure arises from interactions between the electron’s total angular momentum and the magnetic field created by the atomic nucleus. These interactions cause energy level splittings in the electronic transitions, resulting in the observed spectral line patterns. Look into Budker
6. The typical wavelengths, splittings, and lifetimes for alkali atoms vary depending on the specific element. Below there is a table with specific values and typical ranges.
   • Wavelengths: Visible to near-infrared range (600-900 nm)
   • Fine structure splittings: in the range of 10 GHz – 20 THz,
   • Hyperfine structure splittings: in the range of 200 MHz – 9.2 GHz.
   • Lifetimes of optically excited states are interestingly similar.
   • They have linewidths of 5–10 MHz, which corresponds to tens of nanoseconds in terms of lifetimes.

     

7. Specific values for Rubidium 87 are
   • central wavelengths: D1 line 794.9 nm, D2 line at around 780.0 nm
   • Hyperfine splitting for D1 line is around 3.03 GHz, D2 line is around 6.83 GHz
   • Lifetimes: D1 line lifetime of about 27.7 ns, D2 line lifetime of approximately 26.2 ns
8. Spectroscopy data can be found here:
   • database Safronova
   • NIST database
   • Steck

Rubidium isotopes: answers

1. What is an isotope? An isotope of an atom has the same number of protons but different number of neutrons. Thus isotopes have different mass numbers, which are the total number of protons and neutrons in an atom’s nucleus. The restrictions on isotopes are primarily based on the stability and natural occurrence of different neutron configurations within an element.
2. Magic numbers
3. Two isotopes of rubidium are rubidium-85 (85Rb) and rubidium-87 (87Rb).
4. What are 85 and 87 numbers mean? What is difference between their structures? In 85Rb and 87Rb, the numbers 85 and 87 refer to their respective mass numbers. Rubidium atomic number is 37, 85Rb has 48 neutrons and 87Rb has 50 neutrons.
5. The natural abundance of rubidium. The ratio of two isotopes of 85Rb to 87Rb is approximately 2:1. The exact percentages are 85Rb (72.2%) to 87Rb (27.8%)
6. Kings plot

Atomic linewidth: answers

1. What is homogeneous and inhomogeneous broadening. Homogeneous broadening refers to the broadening of spectral lines when all atoms have identical conditions. It occurs when all the atoms experience the same environment and exhibit the same energy transitions. Inhomogeneous broadening, on the other hand, arises from variations in the local environment of atoms, leading to a spread of transition energies and resulting in broader line shape. An example of inhomogeneous broadening is Doppler broadening.
2. Different lineshapes The natural line shape, also known as the Lorentzian line shape, is associated with natural broadening due to the finite lifetime of excited states in atoms. The Doppler line shape, or Gaussian line shape, is caused by Doppler broadening resulting from the thermal motion of atoms or molecules.
3. Doppler broadening as a function of temperature The linewidth of a spectral line is typically characterized by its full-width at half maximum (FWHM). For a room temperature gas, the linewidth is generally on the order of several gigahertz. In contrast, a cold gas or a gas at low temperatures can have linewidths on the order of kilohertz or even lower. Let’s put some simple estimations together: $$\Delta \nu = \frac{2\nu_0}{c}\sqrt{2\ln(2)\frac{k T}{m}}$$ Let’s estimate for a few temperatures for Rb87 $$\Delta \nu = \frac{2\cdot384.230484468 \cdot10^{12} \text{Hz}}{2.99792\cdot 10^8\text{m/s}}\sqrt{2\ln(2)\frac{1.38\cdot 10^{-23}\text{J/K} \cdot T}{87\cdot1.66\cdot 10^{-27}\text{kg}}}$$ $$\Delta \nu =29.5023 \sqrt{T} \text{MHz}$$ Let’s consider three possible situations:
   • Room temperature gas T=400K, \(\Delta \nu=510\) MHz
   • MOT atoms T=100 uK, \(\Delta \nu=0.3\) MHz, which is below natural linewidth
   • BEC T=400 nK, \(\Delta \nu\)=18 kHz
4. What is Voigt profile?

Atomic spectrum measurement: answers

1. A simplified experimental setup for atomic spectrum would look like this:

   

2. How to scan through different frequencies and see it on an oscilloscope? To see absorption spectrum you need to change central frequency of your laser. Typically you would have a scanning knob on your laser controller, which modulates laser’s piezo or current. Modulation signal is typically a ramp signal.
3. Draw an expected trace on an oscilloscope, what is on the y axis? What is a span of your scanning? Let’s assume that we have a cell with a natural Rubidium abundance. This means that it has both Rb85 and Rb87 atoms.

   

   Image is taken from here. If se scan laser frequency over thatn 6.8 GHz we see four dips, two for each isotope. Two inner one correspond to Rb85, which are split by 3 GHz, two outer ones are RB87. Each dip is Doppler broadened hiding hyperfine structure. In the next section we discuss how to resolve down to natural linewidth.

Doppler-free saturated absorption: answers

1. What is Doppler-free saturated absorption? Doppler-free saturated absorption is a spectroscopic technique used to achieve high-resolution spectroscopy by eliminating the broadening effect of Doppler shifts. It involves using two counter-propagating laser beams: a pump beam and a probe beam, which are tuned to a specific atomic transition. The pump beam saturates the transition, while the probe beam measures the absorption. This results in elimination of the Doppler broadening, and being able to resolve natural linewidth. Next we break it into a step-by-step description.
2. Draw a scheme? The scheme would look like this.

   

3. When counter-propagating pump and probe will be resonant with the same atoms? Due to Doppler effect moving atom will see the laser frequency differently, depending if it is co-propagating or counter-propagating. See image.

   

   We had initially a laser beam red-detuned (black arrow), then beam co-propagating with an atom is further off resonant. The counter-propagating beam will be seen as closer to resonance. We asked ourselves, when probe and pump fields will be simultaneously resonant with atoms. From our simple diagram we can conclude that they will be addressing the same atoms, only if atoms have zero velocity.
4. How will absorption spectrum modify in presence of the pump? If initially we had a broad dip in a probe spectrum. If we switch on a pump field it starts competing with probe for atoms. For those atoms, which are in resonance together with both probe and pump, the pump, having higher power, will take them out of interaction with probe. Thus probe won’t be absorbed by the zero class atoms, the spectrum will have a peak with a natural linewidth.

   

5. How many peaks do we expect to see? Why new peaks appear? As we discussed earlier we expect to resolve natural linewidth peaks. If before the internal structure of F=1->F’ was washed out by different velocity classes; now we will see each transition. As an example, the D2 line of rubidium 87 we see peaks for next transitions F=1 <-> F’=0,1,2 and F=2 <-> F’=1,2,3. Don’t forget that selection rules will allow only three transitions with \(\Delta F=0,\pm 1\).
6. What is a crossover peak? Let’s remember that upper level consists of many levels. So there is an interesting condition making an additional peaks appear. Let’s put our laser frequency exactly in the middle between two resonances. Let’s call splitting between two upper levels \(\Delta\), see picture.

   

   If Doppler broadening covers both resonances, then there exists a class of atoms with velocity \(v=c\Delta/2\). These atoms will be simultaneously resonant with both probe and pump, but on different transitions. The signal will be twice bigger, crossover peaks are ususally higher.
7. How to select a specific velocity class? If you have a spare AOM, you can shift frequency of probe with respect to pump field. Thus they will be resonant with the non-zero velocity class. This makes it possible to lock your frequency to be off-resonant. Let’s write simple equations to have them handy. They would apply for the previous descriptions, if you put AOM shift to zero.

   

   Before interaction with atoms frequencies of two fields are \(\begin{cases}\omega_\text{probe} = \omega_0\newline \omega_\text{pump} = \omega_0 + \omega_\text{AOM}\end{cases}\) Atoms will see modified frequencies by the Doppler effect \(\begin{cases}\omega_\text{probe}’ = \omega_0+kv\newline \omega_\text{pump}’ = \omega_0 + \omega_\text{AOM}-kv\end{cases}\) We find that velocity class that is resonant together with both pumps is \(kv=\omega_\text{AOM}/2\)

Working with a cell with hot gas: answers

1. Atomic vapor cells can be heated using various methods such as resistive heating or oven heating. Resistive heating involves passing an electric current through a resistive wire or element wrapped around the cell. Oven heating involves placing the cell in a temperature-controlled oven or furnace. In our lab we wind a wire around a cell in both directions with the same amount of turns and send constant current through. In other lab we used machined ovens, which were heated with these resitive elements. If you have money Thorlabs has a prebuilt solution.
2. Tips and tricks about heating. Monitor that you evenly heat up the cell, sometimes atoms can get absorbed by walls in the coldest spot, which could very easily be the input/output coupler. Don’t overheat the cell, I noticed that after heating it too much atoms tend to get absorbed by the walls at room temperature.
3. A cell with a buffer gas refers to an atomic vapor cell that contains an additional gas, called a buffer gas, along with the atomic vapor. The buffer gas can increase coherence time, as atoms collide with the same species rarer (Look into Budker’s paper).
4. A coherence preserving coating, also known as an anti-relaxation paraffin coating, is a special coating applied to the inner surface of an atomic vapor cell. It is designed to reduce the relaxation rate of the atomic spins caused by collisions with the cell walls. (Budker, there Budker everywhere!)

Two valence electrons: answers

It is interesting to see how spectrum modifies if we add one more electron to an outer shell.