Optical Pumping Lab Report [Basic Principle]
Before discussing the optical pumping, we have to mention the energy levels of the atom and their splitting due to the interaction between the spin of electron, orbital angular momentum of electron and spin of nuclei. We denote them with S, L, and I respectively. And the total angular momentum of electron is denoted by J, (Grand) total angular momentum is denoted by F. Therefore we will have to deal with the LS interaction, IJ interaction and later the interaction between the Grand total angular momentum and external magnetic field.

A very schematic picture of the energy levels of 87Rb under a weak external field is shown in Fig. 1. After applying an weak external magnetic field on the atom, the formally degenerated energy level with same F will further split into 2F+1 sublevels, denoted by mF which is the projection of Grand total angular momentum to the direction of external field B0.

Fig. 1
Until now we haven't mention any thing about the optical pumping. Now we have the atom well prepared, the only thing we have to do is to illuminate the vapor of Rubidium with some well tuned highly polarized light. Well tuned means the spectrum of the incident light is better to be narrow, this can be achieved by using a interference filter to screen out the unwanted light and let only the D1 line pass, namely when this light illuminated on the Rb vapor, only transitions between sublevels originated from 5S1/2 and sublevels originated from 5P1/2 have high enough probabilities to happen. Furthermore, if this well tuned light goes through a linear polarizer and a quarter lambda plate successively, it will be circular polarized. If such a light been absorbed by a 87Rb atom, the only possible excitation that can happen will be the one between two levels whose mF are different by either +1 or -1, the sign is determined by the polarization of the light.

However, the atom will not stay at the exited level forever. Due to varies of mechanism, the atom will relax and de-excite back to 5S level. But this time there is no previous strong restriction on ΔmF but a weaker one which isΔmF=0, ±1. Therefore, if the absorption and relaxation happen continuously for several times, the atom (87Rb in this example)will end up been trapped in mF=2 (or -2 for opposite polarization)sublevel of 5S. And this process is the so called optical pumping.

After we pump the atom in Rb vapor into its final status, it will not be able to absorb any light from the lamp. Now we can apply another RF magnetic field to the vapor. If the frequency of the signal is resonant with the energy gap between two adjacent sublevels, mF=2 and mF=1 for example, the simulated absorption can happen. Then the vapor will be de-pumped and it can again absorb some light from the lamp. By measuring the absorption curve versus the frequency of RF signal, we can get plenty of information of the atom.

[Experiment Results]
I. Diagram of Apparatus
The diagram of apparatus is shown in Figure 1.

Fig.1
The rubidium lamp generates the light which is going to be used as the pumping light in this experiment. The light goes through a interference filter which let only the D1 line pass. Then a linear polarizer and a piece of quarter lambda plant generate the circular polarized light for the purpose of optical pumping.
The coils we used in this experiment are the Maxwell coil set A, B, C and the Helmholtz coil which is perpendicular to the Maxwell coil. Former one is generating the constant background magnetic field H0, and the Helmholtz coil is used to generate the RF field. The Maxwell coil is driven by the DC supply HP 6266B which is controlled by the computer program. Helmholtz coil is driven by RF system consists with 3330B signal generator plus 7805 PRD amplifier.
[Warming Up the Apparatus]
The warming up of lamp as well as the vapor cell take about an hour. The signal of photodiode is shown in Figure 2. The range of the multimeter was set to DC voltage 10V. Therefore the error of the measurement is:
10×(4×10-6) +measurement×(1.5×10-5)

Which is barely recognizable in the chart.

The relevant parameters apparatus are shown in Table 2. Where, Ilamp is the current of lamp; Vhearter is the voltage setup for hair dryer; Ifan is the current apply to the fan of hair dryer; Tcell is the reading form the thermometer.
Table 2 Experiment parameters | Reading | Error | ILamp/A | 0.30 | 0.05 | VHeater/V | 32.5 | 0.5 | Ifan/A | 0.34 | 0.01 | Vfan/V | 12.5 | 0.01 | Tcell/°C | 55 | Fig. 2
1 |

[Spectrum measurement]
1. Under Earth Field
After the system reach its steady state, we turned on the RF power. No additional external field was added. Only measuring the Zeeman split due to earth magnetic field. Spectrums of transmittance signal are shown in Fig 3 and Fig 4.

Fig. 3 Fig. 4
Here we rewrite the results of the fitting in table 3.
Table 3 | Center/kHz | Width/kHz(FWHM) | Asymmetricity | Peak No.1 | 205.77±4.8 | 1.48 | -23.12% | Peak No.2 | 307.79±7.2 | 2.02 | -23.81% |
The way that the asymmetricity was defined was the
1-dHdL×100%
Where dH and dL are the differences between the peak frequency and the higher and lower frequencies where signal drop to half of the peak. Namely, if the asymmetricity is positive, it means that the peak is wider in lower frequency side, if negative then vice versa.

2. Measurement of g-factors of two isotopes of Rubidium.
The geometer parameters of the Maxwell coil is shown in figure 5.

Fig. 5
Therefore, considering the thickness of the wire, we need to use the corrected radius and distances, RA=30.2cm, RB=39.7cm, D=27.0cm, the magnetic field in the center region of the coil set is
Bz≈μ02142RB+μ0RA2×110D2+RA232I+Bearth=4.14(6)×10-4 I

We scanned the RF frequency at different current of Maxwell Coil and acquired the relation between resonance frequencies and current. The result is shown in Figure 6. Where the blue and red dots are representing positive and negative current, respectively.
The linear fitting gives us the slope of the line equal to 0.684(2) MHz/Guass.
Since the resonance frequency should have the expression: ν=gfμBhBz And assuming μB=9.274×10-24J T-1, h=6.626×10-34J S, we can get that g85=0.334(1) Fig. 6
Similarly, for the second peak, which is the signal from 85Rb, we also got the relation between resonance frequencies and magnetic field as shown in Figure 7.

Fig. 7
Therefore, we can calculate g87 from this result: g87=0.503(2) 3. Measurement of Earth Field
Now that we knew the g-factors of both isotopes of Rubidium, we can calculate the earth field from either of the intersections.
Bearth=ν0hgfμB
Where ν0 is the resonance frequency when current of Maxwell coil is zero. The results are shown in table 4
Table 4. Earth Field From Peak 1 (Gauss) | From Peak 2 (Gauss) | Average (Gauss) | 0.429±0.003 | 0.420±0.005 | 0.425±0.004 |

4. Asymmetricity
We noticed that the shape of the dips in the transmission spectrum are not symmetric. And it appears that this asymmetricity is somehow related to the current in Maxwell coil, hence related to the magnetic field.
For positive current, we have the relation between the current and asymmetricity of dips from 85Rb shown in Figure 8. Where the definition of asymmetricity is given in the first section of this chapter.

Fig. 8
We can at least see that all dips are tilted toward high frequency.
A typical shape of the dip, for I=200mA, is shown in Figure 9 Fig. 9(positive current) Fig. 10 (negative current)
For the second dip which came from the 87Rb, we can get similar conclusion that under all currents the dips are tilted towards high frequency.
The tells us that the transparency signal tent to decay slower when the frequency of RF signal detunes to higher frequency than it goes to lower frequency.
However, when we reverse the current, the asymmetricity of dips also reversed. Namely the decay of signal is slower for the detuning towards lower frequency.
A typical shape of the dip, which also came from 87Rb with the reversed current I=-800mA is shown in Figure 10.
And one can also clearly see that the shape of the dip is far deviated from Lorentzian curve. I am not sure about the reason of this phenomenon. It might have something to do with changing the orientation of magnetic field relative to polarization of the pumping light.

[Summary]
In this experiment, the g-factors of ground states of 85Rb and 87Rb had been measured. The results are: g85=0.334±0.001 g87=0.503±0.002

And the Earth magnetic field is also measured, which gives the value of
Bearth=0.425±0.004 Gauss at the location of the apparatus.

Furthermore, author noticed a phenomenon that the response of transparency signal to the detuning of RF field is asymmetric with the sign of the detuning. When the current is positive, the signal decays slower when frequency is higher than resonance frequency. And when the current was reversed, the response behavior was also reversed, namely the signal decays slower when the frequency of RF signal is below resonance frequency.

[References]
1. Write-up for Optical Pumping Experiment, SUNY-SB, PHY 515.
2. Lemin Zheng, Atomic Physics, Peking University Press, first edition, Dec, 2000
3. Sakurai, Modern Quantum Mechanics

--------------------------------------------
[ 1 ]. Manual of HP 34401A multimeter, Chapter 8 DC Characteristics, page 20, 21