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.