Optical pumping of rubidium Quinn Pratt, John Prior, Brennan Campbella) (Dated: 25 October 2015) The effects of a magnetic field incident on a sample of rubidium were examined both in the low-field Zeeman region as well as in the intermediate, quadratic region. In the Zeeman region, the Land´eg-factor (gF ) for Rubidium-85 and Rubidium-87 were calculated to be 0.3357 ± 0.010 and 0.5043 ± 0.023, respectively. These values were later used to calculate the nuclear spin for each of these isotopes. In turn, we calculate I = 1.513 and I = 2.518 compared to I = 3/2 and I = 5/2 for rubidium-87 and 85 respectively. Furthermore, in the quadratic region, we are able to resolve all of the magnetic sublevels for each isotope in accordance with quantum theory.

I. INTRODUCTION A. Quantum Mechanics

Optical pumping is the process of stimulating a sample There are a variety of quantum numbers which serve of atoms out of thermodynamic equilibrium through the to describe the various physical properties of a quantum resonant absorption of light.1In this experiment we use system. These quantum numbers and their significance optical pumping in conjunction with a magnetic field to are as follows,2 trap atoms in a specific quantum mechanical atomic sub- level. Then, we use radio frequency photons to stimulate • S: the S quantum number, or SZ corresponds to absorption. By varying the frequency of the incident rf- the z-component of the angular momentum of an photons, we can monitor the linear dependance of energy electron. with respect to the magnetic field. Furthermore, upon • L: the L quantum number corresponds to the an- increasing the magnetic field beyond the linear Zeeman gular momentum of the orbit of the electron. region, we can separate out the M-level spectrum of the atoms and account for every one of the magnetic sub- • J: the J quantum number is defined as: levels. Atomic spectroscopy is frequently the study of a se- J = L + S, (1) ries of quantum numbers. We will discuss how quantum mechanics explains effects such as Zeeman splitting and and it is known as the total angular momentum of optical pumping. Additionally we will describe the ex- the electron. perimental design including the optical and electronic as- • I: the I quantum number corresponds to the intrin- pects of the apparatus. Lastly we will discuss the results sic spin of the nucleus, this will be the subject of of both facets of this experiment: the exploration of the one of our experiments, it will be different for each linear Zeeman effect to derive the nuclear spin quantum isotope. number (I), as well as accounting for M-level absorption dips in the quadratic region. • F: this quantum number is best described as the We will begin with the theory of the experiments, then total atomic angular momentum and it is defined move onto the design of the apparatus, then the results of as: the experiments themselves, lastly we will discuss these results in the context of high resolution spectroscopy. F = J + I, (2) this is another one of the most important quantum numbers in our studies as it describes the coupling II. THEORY of the electron with the nucleus.

The theoretical underpinnings of this set of experi- • M: this quantum number is different from the oth- ments can be broken down into two main categories. ers as it is dependent on the application on an ex- First, the theory surrounding the nuclear quantum me- ternal magnetic field, M is therefore the component chanics and its treatment of magnetic effects. Secondly of F along said magnetic field (B). the theory behind optical pumping and the other optical To further demonstrate the organization of these quan- aspects of this experiment. tum numbers one should consult the vector diagram in FIG.1. The most important quantum numbers for this exper- iment are F, M and I, the first two are important be- a)also at University of San Diego: Department of Physics & Bio- cause they serve to organize the energy levels based on physics. the applied magnetic field, as shown in FIG. 2, I is also 2

FIG. 1. This simple diagram illustrates the relationship be- tween these quantum numbers. The effect of the impinging magnetic field is what we will be studying in this lab. important as our experiment serves to calculate I for each isotope. The externally applied magnetic field acts as a pertur- bation on the atomic energy levels, hamiltonian which described this interaction is:

µ µ FIG. 2. This is the principal energy diagram for this exper- H = haI · J − J J · B − I I · B (3) iment, it shows the low-field Zeeman splitting effect wherein J I the energy level separation between M-levels varies linearly µ is the total electronic magnetic dipole moment, µ with increasing magnetic field (B). It is important to know J I that this diagram is for rubidium-87, not rubidium-85. The is the nuclear magnetic dipole moment.3As we can see F-levels correspond to the hyperfine structure, whereas the this perturbation is linked to the nuclear spin as well M-levels correspond to Zeeman splitting. as the total electronic spin. The ultimate result of this perturbation is shown in FIG. 2. Upon an increasing B-field each F state gives rise to where it is clear that W scales linearly with B. 2F + 1 sublevels. Although these energy levels appear to Furthermore, split linearly with increasing B, their true nature is given by the Breit-Rabi equation: F(F + 1) + J(J + 1) − I(I + 1) g = g (7) F J 2F(F + 1) −∆W µ ∆W 4M W (F,M) = − I BM± [1+ x+x2]1/2, This relationship will be used later to calculate the nu- 2(2I + 1 I 2 2I + 1 clear spin (I) for each isotope. (4) We will also collect data in the intermediate, quadratic where, region where a simplified linear approximation cannot be made. µoB µI x = (gJ − gI ) and gI = − . (5) ∆W IµI B. Optical Pumping In terms of the dimensionless number x, three principal regions of interest exist. Next, we must address the other theoretical aspects of 1. Zeeman Region: x = 0, the energy levels split lin- the experiment regarding optical pumping and the stim- early with B. ulation of the rubidium atoms by RF-photons and light. The rubidium cell is continuously being exposed to fo- 2. Intermediate Region: 0 < x < 2, energy levels split cused light of wavelength λ = 795nm, which is consistent quadratically with B. 1/2 1/2 with the resonant wavelength for the S1/2 → P1/2 transition. This light is also circularly polarized to cause 3. Paschen-Bach Region: 2 < x, energy levels are lin- the only allowed transitions to be ∆M = +1. early separated where I and J have been decoupled. The following diagram serves best to illustrate the al- We will be investigating effects in the low-field Zeeman lowed transitions under the optical pumping conditions, region where the energy level splitting is given approxi- Note that there is no transition out of the M = 2 state. mately as: This means that the rubidium atoms will transition out of the ground state, and naturally cascade back down W = gF µoBM, (6) and transition up through the Zeeman Split states until 3

A. Investigations in Zeeman Splitting

The procedure for this experiment is as follows, 1. We begin by aligning the apparatus itself as to best cancel-out the local components of Earth’s mag- netic field. Other setup procedures include setting the temperature in the Rubidium cell to 50oC, and we must ensure the horizontal coils are set to 0. 2. Next, we monitor the current being delivered to the aligned-Helmholtz coils on the oscilloscope ver- FIG. 3. This diagram, although similar to FIG. 2 has an im- sus the absorption from the optical detector. We portant difference. Here we can see the allowed transitions adjust the range and speed of the sweep until we between energy levels. It is critical to note that there is no have a clear image of the zero-field dip. This dip transition out of the M = 2 ground state. This is the key to corresponds to the whatever magnetic field is neces- optical pumping, while an external B field is applied, the ru- sary to cancel out any residual effects from Earth’s bidium atoms will be pumped into this state. It is important magnetic field. to note that this diagram only applies to rubidium-87. 3. After obtaining a clear image of the zero-field dip we are prepared to generate RF-photons to stim- population largely inhabits the M = 2 state. This is the ulate the sample. The transition energy of these basis for optical pumping. photons is given by, E = hν. (8) These photons are used to stimulate the atoms III. EXPERIMENT DESIGN trapped in the pumped state to a more absorbent state. This de-pumping causes two more dips to appear on either side of the central zero-field dip. Using the apparatus displayed in FIG. 4, along with There are two because there is one for each isotope. other instrumentation to be explained later, we will con- duct two experiments, The first serves to investigate the 4. Starting at a frequency of about ν = 20kHz and linear Zeeman splitting at low B field values. We will use incrementing up to values near ν = 150kHz collect the data collected from the first experiment to calculate the relevant data to see the de-pumping dips drift the nuclear spin values I for each isotope. The second further and further away from the central dip. experiment takes place at higher magnetic field values to As the energy of the RF-photons increases, the Zeeman investigate the quadratic, intermediate region. level at which this energy becomes resonant occurs at a higher and higher B field value. Recall Eq. (8) where we showed in the Zeeman Region the transition energy is linear with respect to B.

B. Quadratic Zeeman Splitting

Assuming the calibration completed in step 1 of the first experiment is still sound, the second experiment is FIG. 4. This diagram represents the experimental setup. The conducted as follows: o absorption cell is maintained at 50 C. The wavelength of the 1. First we must broaden the range of our magnetic circularly polarized photons is 794.8nm. field sweep in the main coils. 2. For this experiment the horizontal magnetic field As seen in FIG. 4, we use an RF discharge lamp to will be used to cause quadratic Zeeman splitting, generate the light of λ = 780, 794.8nm. The 780nm light start by turning the horizontal field knob to cause is then filtered out. Meanwhile the 794.8nm light passes the entire 5-dip structure to shift horizontally on through a linear polarizer and a 1/4-wave plate to emerge the oscilloscope. right hand circularly polarized, carying one unit of angu- lar momentum along the central axis of the apparatus, 3. Continue to increase the horizontal magnetic field setting the ∆M selection rule. After passing through the while simultaneously increasing the frequency of gas cell the optical detector receives the light, transmit- the simulating RF-photons to keep focus on the ting any relevant absorption information. absorption structure. 4

4. At some point we must decide which of the de- This value will be subtracted from the rest of the mea- pumping dips to focus in on. This process involves surements. slowly the magnetic field and the photon frequency We began increasing the simulated de-pumping tran- until we reach the MHz range, then the sweep time sition energy at an RF frequency of ν = 20kHz, and should be increased to above 100seconds. extended up to ν = 150kHz. Then, using MatLab, we track the expansion of the dips on either side of the zero- 5. We will see a myriad of dips within the larger de- field dip. The following relationship shows the principal pumping dip. These correspond to the M-level sub- result of this experiment: levels.

IV. RESULTS

Just as the experimental design was presented in two parts, one pertaining to each experiment being con- ducted, so too will the results.

A. Low Field Effects

The figure below (FIG. 5) displays the key features of the experiment, we see the central, deepest peak, this corresponds to the zero-field de-pumping.4On either side of the central dip, we see an identical set of two dips corresponding to the stimulated de-pumping caused by the RF photons (in this case RF ν = 20kHz).

FIG. 6. Here we see the principal results of the low-field Zeeman splitting experiment. It is clear that the transition energy is linearly dependent on B. The slope of each of these lines is used to calculate the Land´eg-factors for each isotope, then the g-factors will be used to calculate the nuclear spin number I.

As noted on the graph, the slope of each line is

0.46991(R2 = 0.999) (11)

and

FIG. 5. This figure is the characteristic absorption spread 0.70579(R2 = 0.998) (12) for Rubidium subject to constant RF. For this image the fre- quency is 20kHz. The dip closest to the central, zero-field dip note: Both of these slope values have units of is the de-pumping of rubidium-87, the outermost de-pumping MHz/Gauss. dip corresponds to rubidium-85. This image also shows us Next, we use a variation of Eq. (6) given as that the zero-field transition occurs at I = 0.297amps.

Upon collecting data for current vs. absorption we be- ν = gF µoB/h, (13) gin by converting the current-to-coils data into the mag- netic field through the equation: where µo is the bohn magneton, and h is planck’s con- stant. Their combination yields, µ /h = 1.3996. Equa- B = 8.991 × 10−3IN/R (9) o tion (13) allows to compute the gF values for each isotope where I is the current data, N is the number of turns in through: the coils, given to be 11; and R is the mean radius, given to be 0.1639m. Using this information we are able to 0.46991 [MHz/G] = gF × 1.3996 [MHz/G] (14) diagnose the magnetic field needed to produce the zero- field absorption, this was found to be, and

Bo = 0.1791 Gauss. (10) 0.70579 [MHz/G] = gF × 1.3996 [MHz/G] (15) 5

Ultimately we calculate: total of 10 possible transitions. Therefore, we should see 10 dips in the Rubidium-85 Zeeman splitting. 85 ( Rb) : gF = 0.3357 ± 0.010 (16) and

87 ( Rb) : gF = 0.5043 ± 0.023 (17) We then use these values in conjunction with Eq. (7) to calculate the nuclear spin number (I) for each isotope.

(85Rb) : I = 2.518 (18) and (87Rb) : I = 1.513 (19) We can now compare these values to the known values of I = 5/2 and I = 3/2 respectively.

FIG. 8. This diagram is the other half of the results for the B. Quadratic Zeeman Effect second part of this experiment. Here we see all 10 of the M- levels for rubidium-85 accounted for. Note that this data was filtered to make the dips more prevalent. Now we will discuss the second experiment which in- vestigates the effects at much higher magnetic field val- ues. The results here are best displayed through graphi- V. DISCUSSION cal comparison between our absorption data and the en- ergy level diagram in FIG. 2. Our results for both experiments matched theory very Note that based on FIG. 2 we expect to see 6 dips in well. In the first experiment we found that the energy 87 the Zeeman splitting of Rb Corresponding to all of the levels split linearly with respect to an increase in mag- ground state M-level transitions. netic field. Part of this calculation lead us to the Land´e g-factors for the different isotopes of rubidium. We then used the g-factors to calculate the nuclear spin number I for each isotope. For Rubidium-85 we calculated I = 2.518, compared to the theoretical value of I = 5/2, this amounts to a relative error of 0.72%. For Rubidium-87 we calculated I = 1.513, whereas the theoretical value is I = 3/2. In this case, relative error is 0.867%. Upon comparing the results form our experiment with the sample results given in the Optical Pumping Man- ual, we find that they are absolutely in agreement, Al- though the diagrams under the quadratic Zeeman section in the manual were more pronounced, the same general effects are found in our figures. The values given for zero- field magnetic field are similar (although this depends on FIG. 7. This diagram is half of the principal result of the Earth’s local magnetic field), and all of their principal explorations of quadratic Zeeman splitting. Here we see all results are well within the bounds of our results. 6 of the M-transitions in rubidium-87 accounted for. This diagram exactly matches the plot given in the optical pumping manual. VI. REFERENCES As for Rubidium-85, since we do not have an energy level diagram to consult, we must first calculate how 1Teach Spin, Optical Pumping of Rubidium: Instructor’s Manual, many M-level dips we would expect to see. For Rubid- OP1-A, (Buffalo,NY, 2002), pp.2-11. 2 ium 85 we know that the F levels in the ground state are D.J. Griffiths, Introduction to Quantum Mechanics, 2nd Ed. (Pearson Education, NJ, 2005),pp.277. F = 2 and F = 3. Therefore, since we know there are 3Teach Spin, Optical Pumping of Rubidium: Instructor’s Manual, 2F + 1 M-levels per F-level, we see that F = 2 gives rise OP1-A, (Buffalo,NY, 2002), pp.2-4. to 5 M-levels, and F = 3 gives rise to 7 sublevels, for a 4B. Wolff-Reichert, A Conceptual Tour of Optical Pumping, (2009).