Optical Pumping of Rubidium

Optical Pumping of Rubidium Adam Egbert,1, a) Will Wiegand,1 and Hannah Sheppard2 1)University of San Diego 2)University of San Diego (Dated: 26 September 2015) In this experiment, we explored the how optical pumping can be used to access the quantum numbers associated with Zeeman splitting inside of Rubidium-85 and Rubidium-87 atoms. Optical pumping is a process where electrons are driven into specific energy levels whose transitions can be easily detected. We were able to determine the g-factors associated with each Rb and use these factors to find values of nuclear spin (I). The quadratic Zeeman effect was also observed in this experiment.

I. INTRODUCTION L and S. Quantizing these values as multiples of ~, yields J = L ± S. (1) Understanding nuclear and electronic quantum num- bers that characterize angular momentum of an atom 2 The lone valence electron in Rb is designated by S1/2, is a fundamental part of quantum mechanics. Optical so the ground state must have L=0 and S = 1/2 giving a pumping allows us to raise electrons to specific states, value of 1/2 for J. Using the selection rule that governs but the basic states are not all there is to see in the realm transitions of angular momentum, ∆L can only shift one of optical pumping. This experiment demonstrates that whole integer up or down. Knowing that J cannot be states which appear to be degenerate are able to be split negative, the atom reaches its first excited state at L=1 further when Rb atoms are exposed to a magnetic field. and S=1/2 so J can equal 1/2 or 3/2. L, S and J describe Due to the Zeeman Effect, as an external magnetic field energy levels under no perturbation and contain degener- increases, the energy levels separate allowing for simple ate states which can be observed under a magnetic field observation. Due to the selection rules common to quan- thanks to the Zeeman Effect. tum mechanics, the allowable transitions between states of an electron are limited. Based on relative absorbance of photons passing through that atoms and strength of B. Energy Level Splitting the B-field applied, we can determine the magnitude of the splitting between the previously degenerate states. Besides the electrons momentum, the angular momen- This paper will first begin by explaining theory dis- tum of the nucleus also has a spin state, I. The combi- cussing spin states and optical pumping. The following nation of the electrons momentum as well as the nuclear section will cover the apparatus and set up used, as well angular momentum yields the total angular momentum, as the procedure involved with optical pumping. The F. F is calculated by the coupling relation paper will be concluded by analyzed data with results as well as error analysis. F = I ± J. (2)

One goal of this experiment is to empirically determine I for both isotopes of Rb. Due to difference in the nucleus II. THEORY of Rb isotopes, values of I are not the same for the dif- ferent isotopes. The difference between energy levels of A. Atomic Structure F are incredibly small relative to the difference between the ground state and first excited state leading to what Optical pumping requires a basic understanding of is first seen as degeneracy of the states. the energy level structure because of the absorption and The degeneracy just described can be eliminated when emission of photons within the atom happens at quan- the atom is exposed to a magnetic field and the F en- tized energy levels. Rubidium was used in our experiment ergy states are spilt into further sub levels and labelled because it only contains one election in its outer-most va- with a new quantum number, M, representing the Zee- lence shell, so it is a close representation of a hydrogen man states. The difference between the Zeeman states atom. Taking Rb to be similar to H, we know of three increases linearly with and increasing B-field if the B- important quantities representing its energy: the orbital field is relatively small. In accordance to perturbation angular momentum L, the spin angular momentum S, theory, the perturbation due to the magnetic field on the and the total angular momentum J, the combination of nuclear magnetic moments through the Hamiltonian

H = AI~ · J~ + gf µJ J~ · gI µN I~ · B,~ (3)

where µJ is the total electronic magnetic dipole mo- a)Physics Department, University of San Diego. ment and µI is the nuclear magnetic dipole moment. The 2

first zone is where the B-field is low and we have a linear relationship between the B-field and energy. Then a sec- ond zone appears for larger B-fields and a quadratic term comes into play. At very high B-fields all states return to normal linearity.

FIG. 1. 1 a diagram depicting the possible energy state transitions for Rb magnetic moment of the nucleus is much smaller, due to its large size, than the magnetic moment of the electron, it can be ignored. The magnetic energy can be calculated by

E = gJ µf MB, (4) where gjis the Lande g-factor and evaluated as

gj = 1+(J(J +1)+S(S +1)−L(L+1))/2J(J +1). (5) If we do consider the interaction with the nucleus we obtain a new g-factor given by FIG. 2. 1 a diagram depicting the possible energy state transitions for gf = gj((F (F +1)+J(J +1)−I(I +1))/2F (F +1)). (6) Rb in a magnetic field when spin is 3/2. The x-axis represents an increasing magnetic field and the y-axis is a Incorporating the new g-factor the interaction energy is unit-less representation of energy states given by

W = gf µ0MB, (7) C. Relevant Optics where µ0 is the Bohr Magneton.This result allows us to determine g by plotting energy levels against B-field f Photons are absorbed by electrons in the rubidium strength to obtain a slope of g µ M. Knowing g al- f 0 f atoms which raise the electrons to a particular energy lows Equation (6) to be solved for I because we already level based frequency of the photons. The electron can know F and J. This model only works under small mag- then spontaneously emit a photon of a particular wave- netic fields; the effect begins to change as magnetic field length to reach a lower energy state. The energy of the is significantly increased. photon absorbed does not have to equal the energy of What we obtain after raising the B field a significant the photon emitted, but the energy levels still follow the amount is the Quadratic Zeeman Effect where the differ- relevant selection rules. The most important one to our ences in Zeeman level change quadratically rather than experiment is ∆M = 0, ± 1. By circularly polarizing the linearly. The splitting is calculated by the Briet-Rabi photons, we limit the possible transitions of states to the equation: one previously listed. However, there is a limit in which W (F,M) = −(∆W/2(2I + 1)) − (µI /I)BM ± ∆M cannot be lower than -1, so eventually we pump (∆W/2)[1 + (4M/(2I + 1))x + x2]1/2, (8) the entire population into this higher M state. With all atoms in this higher state, no more photons can be ab- where sorbed making the atoms inaccessible to other photons disrupting the thermal equilibrium of the system. When x = (gJ − gI )(µ0B/∆W ) (9) the atoms experience no B-field (the zero-field transmis- sion), the thermal equilibrium is then reset and the Rb and atoms can absorb photons again. This results in a loss of intensity at a detector until they are fully pumped up gI = −µI /IµI. (10) once again. A plot of this equation is seen in Fig. (2.1). Upon in- Introducing a RF signal at a specific frequency allows spection we see there are 3 zones within the figure: the transitions from the M=2 state to lower states which 3 gives the electrons a break from their highly excited state. B. Procedure Again, this is observed by a decrease in photon absorp- tion at the detector. Most importantly the frequency 1. Low Field Splitting at which the phenomena occurs is proportional to the strength of the B-field allowing us to determine the en- ergy states the electrons are occupying. It is also impor- tant to note, the differences in atomic structure leads to a different energy gap between the two isotopes allowing the observation of the quantum numbers for each isotope.

III. APPARATUS AND PROCEDURE

A. Apparatus

We used a TeachSpin optical pumping apparatus for all parts of our experiment. The set up has an RF discharge lamp, interference filter, linear polarizer, quarter wave plate, rubidium absorption cell and optical detector in FIG. 4. 1 that order as seen in figure (). The RF discharge lamp Frequency vs. Magnetic Field for Rb isotopes. The slopes was the source of all our RF waves that pass through a were used to determine the I values. focusing lens to get the maximum amount of light into the absorption cell. Following the lens is the interference First we had to determine the strength of the B-field filter which only allows the wavelengths of light through that remained from the earth’s magnetic field because that we want to study. The polarizers act together to the vertical field from the Helmholtz coils leaves one axial produce circularly polarized light which limits transitions component. The voltage was measured and the strength to ∆M = ±1. Then the light enters the Rb absorption of the B-field was determined by cell which contains two naturally occurring isotopes of −3 rubidium along with xenon, which acts as a buffer. After B = 8.991 ∗ 10 (IN/R), (11) the cell, the light passes through another lens and into where B is the magnetic fiend strength, I is the current, N an optical detector. is the number of turns (11 for this set up) and R was the radius of the coils (0.1639m for this set up). The strength of the residual magnetic field could then be subtracted from all future magnetic field calculations. The next step was to introduce an RF signal to inspire the transition of the electrons from M=2 to M=1. As stated earlier, the two isotopes of Rb results in differ- ent g-factors, so we observed two distinct dips. The RF frequency was increased and readings were taken at five FIG. 3. 1 different frequencies (40 Hz, 60 Hz, 80 Hz, 100 Hz, and An overhead view of the of the optical pumping apparatus 200 Hz). The current producing the B-field was taken used at each dip and used in equation (11) to determine the strengths of the B-field at those absorptions. We then The rubidium cell is surrounded by two sets of plotted the RF frequency against the B-field strength as Helmholtz coils in order to produce magnetic fields. One seen in Fig. (4.1) to obtain our gf in accordance to equa- field should be aligned with the magnetic axis of the earth tion (7). From that we could determine I for each isotope and is used to counteract earths natural magnetic field in using equation (6). order to calibrate our measurements using the zero-field transmission. The other set is used to create a sweeping B-field to allow us to identify the strengths of the B-field 2. Quadratic Zeeman Effect in which transitions occur. The RF coil allows us to now reverse some of the effects The quadratic Zeeman Effect required a stronger B- of the pumping for both isotopes of Rb. It accomplishes field as well as a major increase in RF frequency. The this by being tuned to the difference in frequency between method we used was to center on the RB 85 peak and states driving atoms into a lower excited state. When by increasing RF frequency while increasing the main the atoms are in the lower state, the can absorb photons field we were able to observe the peak into a large B- again resulting in a drop in intensity at the detector. field where the quadratic effect could be observed. In the 4 region we could observe the transitions between the 2F+1 sublevels as six distinct dips as observed in Fig. (5.1). The strength of magnetic fields were again calculated and compared theoretical values from equation (8).

FIG. 5. 1 The observed quadratic Zeeman effect for Rb 87

V. CONCLUSION AND ERROR ANALYSIS IV. RESULTS Unfortunately we did not get the results we had hoped for, but we did manage to be a consistent with theory in some regards as well as see the quadratic Zeeman effect. WE obtained data for the low field splitting by mea- We are unsure where we went a factor of three wrong, suring the strength of the B field at dips in absorbency perhaps the B-field calibration was off, or we managed during various RF frequencies. From Fig. (4.1) we can to forget or include a resistor to the circuit. Either way I see the slop for Rb 87 is 0.0391 and for Rb 85 the slope believe optical pumping can be a useful tool to obtain in- is 0.0645. The ratio between the slopes is 1.65, close to formation about atomic structure with a high resolution theoretical value of 1.5. The gF values that were ob- which would otherwise be vastly more difficult.213 tained from equation (7) were 0.1330 ±.0005 for Rb 85 and 0.1704 ±.0005 for Rb 87. These values are a factor of 1David Griffiths. Introduction to Quantum Mechanics. Pearson 3 of from expected g values so they yield I values that Education, Upper Saddle River, NJ, 2005. F 2Adrian Melissinos. Experiments in Modern Physics. Academic are 1 what they should be, 5/6 and 1/2 respectively, but Press, San Diego, CA, 2003. 3 multiplying both of the gf values by three yields results Teach Spin. Optical Pumping Instructor Manual. Teachspin, Buf- consistent with theory. falo, NY, 2002.