Optical Pumping of Rubidium

A. J. Sternbach, and M. Pourmand Department of Physics, Boston University, Boston, Massachusetts 02215, USA

We report on the observation of zero field-induced transition, weak field and quadratic Zeeman splitting in Rubidium. We begin with a theoretical description of the expected atomic spectra in the case of hydrogen. Noting the key differences we present, without derivation, the Breit-Rabi equation. From the weak-field Zeeman split spectral lines, we extract a measurement of the g! factor. In the hyperfine data, those key features from the Breit-Rabi equation, which were expected, were confirmed. This lab report was written with a dual purpose – to report on my findings in this experiment as a conclusion to this semesters work, and to serve as a starting point for future students who will utilize this apparatus in their own projects. As such, suggestions for experiments that were not tried, or not completed are included in the Appendix.

I. INTRODUCTION unaffected by stray magnetic fields to first order. By some fortune of nature, emission The possessions of one valence electron in 2 2 from the 5 P1/2 to F=2 5 S1/2 transition in Alkali Metal atoms like Rubidium allow for 87 2 2 Rb is absorbed by the 5 P1/2 to F=3 5 S1/2 description similar to Hydrogen. This terrific transition in 85Rb. Although transitions back advantage allows us to employ the fully 2 to 5 S1/2 F=2 and F=1 states will occur at solved wave function of hydrogen to nearly equal rates, the filter provided by calculate the atomic spectra, without 85Rb thus leads, after many cycles, to a non- difficulties stemming from treatment of thermal population imbalance. This leads to many body coulomb interactions, which a relatively high transmitted intensity render, for many other atoms, the between the F=2 and F=1 states. The well- Schrödinger equation unsolvable. Utilizing defined transition frequency between time independent perturbation theory, we hyperfine levels is then phase sensitively can thus describe exactly the expected detected and a feedback loop is used to response of Rubidium, with the small stabilize the frequency of a crystal oscillator. addition of inclusion of nuclear structure. By this means, propagation times across Rubidium has found wide usage in 20,000 km from satellites are detected on the modern technology, mainly in its application order of nanoseconds and position is thus to GPS devices1. The Rubidium atom serves calibrated with extreme precision. as the foundation for this device, by creating Additionally, the population inversion found 87 an atomic clock. The 0-0 transition from the in filtered Rb vapor, gives it the key ingredient for which it is exploited in lasers. F=2, mF=0 to F=1, mF=0 hyperfine levels is The far-reaching application of GPS, which the fine structure corrections. First, there is a is widely employed, is but a demonstration relativistic correction to the energy levels, of the power behind a mastery of the subtle which can be shown to be: interworking of quantum mechanics. ! ! !(!!) !" E! = − 3 (2) II. THEORY !"!! !!!/!

Within a Hydrogen atom, the relative Already, a dependence of energy on the l motion between electron and the orbital value has emerged. Secondly we must gives rise to a magnetic field. When this is consider the presence of spin orbit coupling. accounted for, time dependent perturbation In this case, there is a magnetic field due to theory predicts splitting of the formally the proton, and taking a classical approach, this field should be B = !!! where I = ! with degenerate energy levels. We will discuss !" ! this effect separated into three categories, T representing the period. Noting that the the fine structure that allows for zero field !"!!! angular momentum L = mvr = we transitions between states of different ! have: quantum number l, hyperfine structure, and the effect of Zeeman splitting. Since the ! ! B = ! ! L (3) mathematics is simplified by neglecting the !"!! !! ! nuclear structure of Rubidium, we will Classically, the magnetic dipole moment of neglect this in our discussion and note the an electron is easily shown to be: key differences in our concluding remarks. µ = ! S (4) 1) Zero field transition: !"

In an unperturbed hydrogen atom, the wave Noting the presence of internal magnetic function may be exactly solved and the full field, the perturbing Hamiltonian becomes energy spectrum may be found. Borrowing H! = −µ • B (5) the solution from Griffiths2, one finds that in !" this case, where the potential is the coulomb Inserting the classically derived equations 3 repulsion, the energy levels are given by: and 6 into this Hamiltonian neglects the g-

! factor of the electron that gives a factor of 2, E = ! !! ; where E = −13.6 eV (1) ! !! ! and Thomas precession that gives a factor of ½. These terms conveniently cancel and thus This result, which depends only on the our classically derived equations will luckily quantum number n, predicts that states with give the correct answer! One can now solve different values of the orbital quantum for fine structure corrections. In time number l are degenerate. Due to the independent perturbation theory, the presence of internal magnetic field, we will problem can be simplified to the non- shortly find this to be false. To understand degenerate predictions if we can find an why there is an energy difference between operator A such that [H, A] = [H’, A] = 0. In the 5S and 5P states at all, we must discuss this case, spin-orbit coupling between the proton and electron requires that we use the further neglect the important nuclear coupled representation, as A = J! will work. components for reasons that will become In this representation it can easily be shown clear in the final section of theory. Still, we that: may derive a descriptive picture of hyperfine atomic structure that will still yield a !(! )! ![! !!! !! !!! !!/! E! = ! (6) qualitative agreement to observations in !" !"!! !(!!!/!)(!!!) Rubidium. Employing our assumptions the Thus hyperfine Hamiltonian is:

! ! ! !!! ! ! ! ! E + E = 1 + − (7) H!" = −µ! • B! (8) !" !! !! !!!/! !

! This equation has broken the degeneracy in where B! = ∇xA = ∇×µ!×r/r can easily l. Now, in order to obtain a bit of useful be derived. A little algebra shows that: information, we would like to calculate the ! ! !!! !! order of magnitude of the energy difference H!" = − ! S! • S! (9) !"!!!!! between relevant states. Rubidium has an atomic structure of Finally, noting that S • S = ! (S! − S! − ! ! ! ! 2 2 6 2 6 10 2 6 ! 1s 2s 2p 3s 3p 3d 4s 4p 5s S!), and that the electron and proton have spin of ½ we find a difference in energy With a single valence electron in the 5S between the singlet and triplet states of: state. Gearing our interest towards 2 2 !! ℏ! transitions between the 5 S1/2 and 5 P1/2 ! ∆E = ! ! ≈ O(µeV) (10) states of rubidium, these should have an !!!!!! energy difference of : Indeed, the F=2 to F=1 transition in 87Rb has 1 ! an energy difference of 28 µeV. Now that ∆E ≈ !!! ≈ O(meV) (10) !! we have obtained the hyperfine corrections we have found the expected energies of an 2) Hyperfine Structure unperturbed hydrogen system to an The Hyperfine structure comes from the extremely good approximation. interaction between the electron spin and 3) Weak Zeeman splitting magnetic moment of the nucleus. Although both nuclear and electronic structure play an For a constant magnetic field, we must be important role of the physics involved in aware of the relative magnitude of the Rubidium at this energy scale, we will applied field vs. the protons magnetic field employ a simplified where we consider before we begin a discussion of how to solve splitting of the l=0 state in the presence of for corrections. Utilizing the simplifying an applied field. This assumption simplifies assumption that L ≈ ℏ we find Bint ≈ 10 T, the problem greatly as we are able to neglect and thus we rightfully consider our applied the effect of spin-orbit coupling, and thus field, which is on the order of 10 Gauss, as a use spin coupled representation. We will weak perturbation. In this case, the coupled moments of the given by Landau and Lifshitz3. We will proton and electron will define the magnetic simply identify the difference between the moment in our Hamiltonian: Hamiltonian that leads to this equation, and that which we employed. Finally we will ! H! = −(µ! + µ!) • B!"# (11) then present the Breit-Rabi equation. The Hamiltonian, which considers both Again, this is solved in the coupled hyperfine structure and Zeeman splitting, representation. Omitting the details, one and allows for nuclear structure is given by: finds the energy level splitting from this 2 Hamiltonian to be : ! � = �!! + �! = �!!J • I − µ! • B − ! E! = µ!g!B!"#m! (12) µ! • B (13)

The complete energy spectrum as a function Where, I is the Nuclear spin, J = L + S, and of corrections to the energy spectrum can be though it is not present in the above equation seen below in figure 1. F = J + I is the total atomic angular momentum. Solving this in the intermediate field region by diagonalization, and simplifying with a bit of algebra yields:

!(!±,!!) = − ! ± ! 1 + !!! x + x! ∆!!" ! !"!! ! !"!! (14)

4 where x = g!µ!B ∆E!". We note that in this equation, the necessity of circularly polarized light becomes transparent if we note that in response to circularly polarized Figure 1: Energy level diagram for 87Rb. One can see the qualitative features discussed, where formally light, only transitions with Δm! = ±1 can unperturbed energy levels with orbital motion are split be induced. To see this, one only needs to by the fine structure corrections, each level is further consider n!l!m! [L , x] nlm and note that split by spin-spin corrections, and finally each of these ! 2 levels is further split by the presence of an external field. L!, x = iℏy. Equation 14 is known as the Taken from reference [6] Breit-Rabi equation. 4) Breit-Rabi Equation III. Instrumentation The Breit-Rabi equation, which includes In order to observe the atomic physics nuclear structure, is valid for all Zeeman- discussed in the previous theory section, we hyperfine problems in which one spin is ½. employed an optical pumping apparatus This equation will not be derived in this from TeachSpin. This can be seen in the presentation, as it requires diagonalization of figure on the following page: the Hamiltonian matrix in the intermediate field region. A full description is however, equipment before making or removing connections.

V. Data taking

Figure 2: Optical Pumping Apparatus from Teachspin. The key to our data taking was to ensure that Light exiting an RF discharge lamp is focused through an interference filter into a polarizer. This polarized light all equipment was properly set up. Firstly, enters a quarter wave plate where it becomes circularly the quarter wave-plate needed to be set to polarized. This then enters the heat-controlled cell, which is housed in a region experiencing magnetic field from produce circularly polarized light. In order three Helmholtz coils. The horizontal, and vertical fields to do this, the sample was kept near room are arranged to cancel the ambient magnetic field. The third Helmholtz coil carries an AC current and thus temperature to yield maximum transmission. produces an RF filed. The exiting light is then focused into The detection sensitivity and offset were a detector. Taken from reference (5) then set to yield a clear signal, which was In order to use this apparatus we used a not saturated when the analyzer was either controller from Teachspin for heat and crossed or aligned with the polarizer. The magnetic fields. This supplied the static field analyzer was then crossed with the analyzer. used to cancel the earth’s field. The RF field A quarter wave-plate was inserted into the was driven by a 2005B signal generator position shown in figure 1 and rotated until from B&K Precision. The Tektronx 220 oscilloscope was used to read the output transmission through crossed polarizers was signals. maximized. In order to effectively cancel the earth’s To directly retrieve transmission vs. magnetic field we positioned this apparatus magnetic field data, our oscilloscope was with the aid of a compass. After this was put into XY mode with infinite persistence. carefully placed, we swept the static field As mentioned above, the apparatus was slowly, and simultaneously observed the set- positioned to have its horizontal field in the in of the zero-field absorption line as the compass needle flipped. The detection opposing direction to the earth’s field. After control was carefully controlled to ensure ensuring this was accurate to within +/- 2 that our signal was not saturated at any point degrees, the cell was heated, and the during the measurement. When the sensitivity and offset of detection were again horizontal field was confirmed to be zero, set. The magnetic field was then swept we tuned the vertical field to yield the through zero-field, which was tracked using sharpest possible line width of our zero-field a compass. Although this signal was, at first, absorption signal. not observable, appropriately adjusting our IV. Safety Issues detector made the absorption lines quite plain. This process was repeated with a The key dangers of the Optical pumping slowly varying magnetic field and the zero experiment were protection of the apparatus. field dip was found at the same time as our Since the Rubidium vapor was fully compass changed orientation. Then, to enclosed, and no high voltages or currents arrive at zero vertical field, we tuned our were used there was little danger to the user. signal to yield the narrowest possible line We were extremely careful to turn off all width of the zero-field dip. The inclusion of RF field was relatively RF field is applied, straightforward. The output of our signal ! ℏ generator, after frequency was verified on = (15) ! !!!! our oscilloscope, was directed to the RF 2 85 input of the TeachSpin controller. To obtain For the S1/2, F=3 state of Rb, this is 0.466 3 precise measurements needed, we utilized MHz/Gauss, with a corresponding gF=1/3. the DC horizontal offset and swept the field The observed results are shown in figure 4. slowly across a narrow range of interest. Noting that the observed slope of the transition υ B = 0.55 V/div, and using the VI. Data Analysis conversion factor 7.4 div = 1 gauss, we were able to estimate the nuclear spin (detailed Our first observation was of zero-field plot of 87Rb line not shown). transition in Rubidium. The observed line corresponds to the pumped transition in the unperturbed Rubidium gas. This transition is from the 5S  5P state, and occurs due to the spin-orbit and relativistic fine splitting corrections discussed in the theory section. The energy separation of meV is characteristic of orbital modes. The observed signal can be seen below in figure 3. This data may be compared to that shown in reference 5.

Figure 3: Zero field transition. Figure 4: Weak Zeeman Splitting of the Hyperfine energy levels. Distinct absorption lines can be seen for 85Rb We then wished to split and retrieve the (closer to zero field) and 87Rb (further from zero field). separate absorption lines of 85Ru and 87Ru. These lines are the manifestation of This calculation yielded υ B = 0.22 MHz/ matching the ratio of magnetic field to RF Gauss. We note that, as can be observed in frequency to the g! value of rubidium. As an the above figure, while the frequency was incident photon has energy hv, absorption decreased linearly the magnetic field will occur when this is equal to the Zeeman decrease was not linear. A close inspection splitting in the weak limit. Thus, when an of this figure reveals that as the frequency is lowered, the difference in magnetic field study of the atomic spectra in Rubidium. We points, where the 85Ru spectral line is found, observed phenomena stemming from the decreases. This large error can likely be fine, hyperfine, and Zeeman splitting of attributed to an improper zeroing of the energy levels. Our theoretical description earth’s vertical magnetic field, which is finds quantitative agreement with our supported by the apparent, albeit small experimental results. Although no bumps near the zero field resonance, and qualitative predictions were accurately relatively broad transmission. This data set made, we have collected data regarding the can be compared to figure 3, which is shown working order of the TeachSpin system, on approximately the same scale, but with which could prove valuable to students clear features that are much closer to what is seeking to perform further experiments expected in the presence of zero magnetic using this apparatus. Finally, in our field. This data may be compared to that concluding remarks, we would like to shown in reference 7. suggest experiments that are observable with this apparatus.

VIII. Appendix

One can directly measure the cross section of for the absorption of resonant radiation by Rubidium atoms. To summarize briefly, from only knowledge of temperature, one can theoretically find the density of rubidium. By taking data of temperature vs. transmission the expected exponential decrease for cross section vs. transmitted

intensity may be found via a standard fitting Figure 5: Hyperfine Splitting of 87Rb spectral lines in the procedure. Further details may be found in intermediate field region. Four distinct spectral lines can reference [8]. be seen corresponding to transitions between different values of m . F Additionally, although discussed in many Finally we observed hyperfine splitting in places, the observation of Rabi Oscillations 85Rb. The data nicely reproduces the features with this apparatus can be a bit tricky. This, expected from the Breit-Rabi equation, (14). I found not to be so much because of the 87 2 We note that for Rb, I=3/2 and in the 5 S1/2 theoretical description behind such state, F=2. Therefore after splitting, there are Oscillations, but the proper set up of the 2F+1=5 total lines of different energy which equipment. After some number of hours of 2 makeup 5 S1/2 state. We observe an trial and error, I was able to gate the RF absorption dip when we are matched, by the signal, by inserting a separate 0-5V TTL Breit-Rabi equation to the energy difference square wave input into the RF modulation, of two states. The four total spectral lines while simultaneously feeding the RF signal. with increasing amplitude that were Although I was able to see an expected dip expected can be seen in figure 5. in transmission, tuning this proved to be more challenging than I had imagined, and I VII. Conclusion was not able to complete this in time. The results I obtained can be seen below for In this report, we have provided a qualitative comparison. IX. References

[1] J. Camparo, Physics Today 60, 33 (2007)

[2] D. J. Griffiths “Introduction to Quantum Mechanics”

[3] Landau and Lifshitz “Mechanics”

[4] Wolfram “http://demonstrations.wolfram.com/BreitRa biDiagram/” The correct result would include a far sharper decrease in transmission and [5] Teachspin manual exponential rise, as can be seen in [ref 9] [6] Optical pumping of rubidium vapor, J. H. Taylor, Steve Smith, Bob Austin, Mike Romalis

[7] Teachspin newsletter II “Beyond the Manual – but Latent in the Apparatus”

[8] Optical Pumping http://internal.physics.uwa.edu.au/~stamps/2

006Y3Lab/SteveAndBlake/absorption- experimental.html [9] TeachSpin newsletter II “Even More Experiments with Optical Pumping”