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.