Optical pumping of rubidium

Transcription

1 Optical pumping of rubidium Quinn Pratt, John Prior, Brennan Campbell a) (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 Lé g-factor (g F ) for Rubidium-85 Rubidium-87 were calculated to be ± ± 0.023, respectively. These values were later used to calculate the nuclear spin for each of these isotopes. In turn, we calculate I = I = compared to I = 3/2 I = 5/2 for rubidium 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 Optical pumping is the process of stimulating a sample of atoms out of thermodynamic equilibrium through the resonant absorption of light. 1 In this experiment we use optical pumping in conjunction with a magnetic field to trap atoms in a specific quantum mechanical atomic sublevel. Then, we use radio frequency photons to stimulate absorption. By varying the frequency of the incident rfphotons, we can monitor the linear dependance of energy with respect to the magnetic field. Furthermore, upon increasing the magnetic field beyond the linear Zeeman region, we can separate out the M-level spectrum of the atoms account for every one of the magnetic sublevels. Atomic spectroscopy is frequently the study of a series of quantum numbers. We will discuss how quantum mechanics explains effects such as Zeeman splitting optical pumping. Additionally we will describe the experimental design including the optical electronic aspects of the apparatus. Lastly we will discuss the results of both facets of this experiment: the exploration of the linear Zeeman effect to derive the nuclear spin quantum number (I), as well as accounting for M-level absorption dips in the quadratic region. We will begin with the theory of the experiments, then move onto the design of the apparatus, then the results of the experiments themselves, lastly we will discuss these results in the context of high resolution spectroscopy. A. Quantum Mechanics There are a variety of quantum numbers which serve to describe the various physical properties of a quantum system. These quantum numbers their significance are as follows, 2 S: the S quantum number, or S Z corresponds to the z-component of the angular momentum of an electron. L: the L quantum number corresponds to the angular momentum of the orbit of the electron. J: the J quantum number is defined as: J = L + S, (1) it is known as the total angular momentum of the electron. I: the I quantum number corresponds to the intrinsic spin of the nucleus, this will be the subject of one of our experiments, it will be different for each isotope. F: this quantum number is best described as the total atomic angular momentum it is defined as: F = J + I, (2) II. THEORY this is another one of the most important quantum numbers in our studies as it describes the coupling of the electron with the nucleus. The theoretical underpinnings of this set of experiments can be broken down into two main categories. First, the theory surrounding the nuclear quantum mechanics its treatment of magnetic effects. Secondly the theory behind optical pumping the other optical aspects of this experiment. a) also at University of San Diego: Department of Physics & Biophysics. M: this quantum number is different from the others as it is dependent on the application on an external magnetic field, M is therefore the component of F along said magnetic field (B). To further demonstrate the organization of these quantum numbers one should consult the vector diagram in FIG.1. The most important quantum numbers for this experiment are F, M I, the first two are important because they serve to organize the energy levels based on the applied magnetic field, as shown in FIG. 2, I is also

2 2 FIG. 1. This simple diagram illustrates the relationship between 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 perturbation on the atomic energy levels, hamiltonian which described this interaction is: H = hai J µ J J J B µ I I I B (3) µ J is the total electronic magnetic dipole moment, µ I is the nuclear magnetic dipole moment. 3 As we can see this perturbation is linked to the nuclear spin as well 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 2F + 1 sublevels. Although these energy levels appear to split linearly with increasing B, their true nature is given by the Breit-Rabi equation: W (F, M) = where, W 2(2I + 1 µ I I BM± W 2 4M [1+ 2I + 1 x+x2 ] 1/2, (4) x = (g J g I ) µ ob W g I = µ I Iµ I. (5) In terms of the dimensionless number x, three principal regions of interest exist. 1. Zeeman Region: x = 0, the energy levels split linearly with B. 2. Intermediate Region: 0 < x < 2, energy levels split quadratically with B. 3. Paschen-Bach Region: 2 < x, energy levels are linearly separated where I J have been decoupled. We will be investigating effects in the low-field Zeeman region where the energy level splitting is given approximately as: W = g F µ o BM, (6) FIG. 2. This is the principal energy diagram for this experiment, it shows the low-field Zeeman splitting effect wherein the energy level separation between M-levels varies linearly with increasing magnetic field (B). It is important to know that this diagram is for rubidium-87, not rubidium-85. The F-levels correspond to the hyperfine structure, whereas the M-levels correspond to Zeeman splitting. where it is clear that W scales linearly with B. Furthermore, g F = g J F(F + 1) + J(J + 1) I(I + 1) 2F(F + 1) (7) This relationship will be used later to calculate the nuclear spin (I) for each isotope. We will also collect data in the intermediate, quadratic region where a simplified linear approximation cannot be made. B. Optical Pumping Next, we must address the other theoretical aspects of the experiment regarding optical pumping the stimulation of the rubidium atoms by RF-photons light. The rubidium cell is continuously being exposed to focused light of wavelength λ = 795nm, which is consistent with the resonant wavelength for the 1/2 S 1/2 1/2 P 1/2 transition. This light is also circularly polarized to cause the only allowed transitions to be M = +1. The following diagram serves best to illustrate the allowed transitions under the optical pumping conditions, Note that there is no transition out of the M = 2 state. This means that the rubidium atoms will transition out of the ground state, naturally cascade back down transition up through the Zeeman Split states until

3 3 A. Investigations in Zeeman Splitting FIG. 3. This diagram, although similar to FIG. 2 has an important difference. Here we can see the allowed transitions between energy levels. It is critical to note that there is no transition out of the M = 2 ground state. This is the key to optical pumping, while an external B field is applied, the rubidium atoms will be pumped into this state. It is important to note that this diagram only applies to rubidium-87. population largely inhabits the M = 2 state. This is the basis for optical pumping. III. EXPERIMENT DESIGN Using the apparatus displayed in FIG. 4, along with other instrumentation to be explained later, we will conduct two experiments, The first serves to investigate the linear Zeeman splitting at low B field values. We will use the data collected from the first experiment to calculate the nuclear spin values I for each isotope. The second experiment takes place at higher magnetic field values to investigate the quadratic, intermediate region. 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 magnetic field. Other setup procedures include setting the temperature in the Rubidium cell to 50 o C, we must ensure the horizontal coils are set to Next, we monitor the current being delivered to the aligned-helmholtz coils on the oscilloscope versus the absorption from the optical detector. We adjust the range speed of the sweep until we have a clear image of the zero-field dip. This dip corresponds to the whatever magnetic field is necessary to cancel out any residual effects from Earth s magnetic field. 3. After obtaining a clear image of the zero-field dip we are prepared to generate RF-photons to stimulate the sample. The transition energy of these photons is given by, E = hν. (8) These photons are used to stimulate the atoms 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. There are two because there is one for each isotope. 4. Starting at a frequency of about ν = 20kHz incrementing up to values near ν = 150kHz collect the relevant data to see the de-pumping dips drift further further away from the central dip. As the energy of the RF-photons increases, the Zeeman level at which this energy becomes resonant occurs at a higher 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 FIG. 4. This diagram represents the experimental setup. The absorption cell is maintained at 50 o C. The wavelength of the circularly polarized photons is 794.8nm. As seen in FIG. 4, we use an RF discharge lamp to generate the light of λ = 780, 794.8nm. The 780nm light is then filtered out. Meanwhile the 794.8nm light passes through a linear polarizer a 1/4-wave plate to emerge right h circularly polarized, carying one unit of angular momentum along the central axis of the apparatus, setting the M selection rule. After passing through the gas cell the optical detector receives the light, transmitting any relevant absorption information. Assuming the calibration completed in step 1 of the first experiment is still sound, the second experiment is conducted as follows: 1. First we must broaden the range of our magnetic field sweep in the main coils. 2. For this experiment the horizontal magnetic field will be used to cause quadratic Zeeman splitting, start by turning the horizontal field knob to cause the entire 5-dip structure to shift horizontally on the oscilloscope. 3. Continue to increase the horizontal magnetic field while simultaneously increasing the frequency of the simulating RF-photons to keep focus on the absorption structure.

4 4 4. At some point we must decide which of the depumping dips to focus in on. This process involves slowly the magnetic field the photon frequency until we reach the MHz range, then the sweep time should be increased to above 100seconds. 5. We will see a myriad of dips within the larger depumping dip. These correspond to the M-level sublevels. This value will be subtracted from the rest of the measurements. We began increasing the simulated de-pumping transition energy at an RF frequency of ν = 20kHz, extended up to ν = 150kHz. Then, using MatLab, we track the expansion of the dips on either side of the zerofield dip. The following relationship shows the principal result of this experiment: IV. RESULTS Just as the experimental design was presented in two parts, one pertaining to each experiment being conducted, 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. 4 On 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 Lé g-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 (R 2 = 0.999) (11) FIG. 5. This figure is the characteristic absorption spread for Rubidium subject to constant RF. For this image the frequency is 20kHz. The dip closest to the central, zero-field dip is the de-pumping of rubidium-87, the outermost de-pumping dip corresponds to rubidium-85. This image also shows us that the zero-field transition occurs at I = 0.297amps. Upon collecting data for current vs. absorption we begin by converting the current-to-coils data into the magnetic field through the equation: B = IN/R (9) where I is the current data, N is the number of turns in the coils, given to be 11; R is the mean radius, given to be m. Using this information we are able to diagnose the magnetic field needed to produce the zerofield absorption, this was found to be, B o = Gauss. (10) (R 2 = 0.998) (12) note: Both of these slope values have units of MHz/Gauss. Next, we use a variation of Eq. (6) given as ν = g F µ o B/h, (13) where µ o is the bohn magneton, h is planck s constant. Their combination yields, µ o /h = Equation (13) allows to compute the g F values for each isotope through: [MHz/G] = g F [MHz/G] (14) [MHz/G] = g F [MHz/G] (15)

5 5 Ultimately we calculate: ( 85 Rb) : g F = ± (16) total of 10 possible transitions. Therefore, we should see 10 dips in the Rubidium-85 Zeeman splitting. ( 87 Rb) : g F = ± (17) We then use these values in conjunction with Eq. (7) to calculate the nuclear spin number (I) for each isotope. ( 85 Rb) : I = (18) ( 87 Rb) : I = (19) We can now compare these values to the known values of I = 5/2 I = 3/2 respectively. B. Quadratic Zeeman Effect Now we will discuss the second experiment which investigates the effects at much higher magnetic field values. The results here are best displayed through graphical comparison between our absorption data the energy level diagram in FIG. 2. Note that based on FIG. 2 we expect to see 6 dips in the Zeeman splitting of 87 Rb Corresponding to all of the ground state M-level transitions. FIG. 7. This diagram is half of the principal result of the explorations of quadratic Zeeman splitting. Here we see all 6 of the M-transitions in rubidium-87 accounted for. This diagram exactly matches the plot given in the optical pumping manual. As for Rubidium-85, since we do not have an energy level diagram to consult, we must first calculate how many M-level dips we would expect to see. For Rubidium 85 we know that the F levels in the ground state are F = 2 F = 3. Therefore, since we know there are 2F + 1 M-levels per F-level, we see that F = 2 gives rise to 5 M-levels, F = 3 gives rise to 7 sublevels, for a FIG. 8. This diagram is the other half of the results for the 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. V. DISCUSSION Our results for both experiments matched theory very well. In the first experiment we found that the energy levels split linearly with respect to an increase in magnetic field. Part of this calculation lead us to the Lé 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 Manual, we find that they are absolutely in agreement, Although 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 zerofield magnetic field are similar (although this depends on Earth s local magnetic field), all of their principal results are well within the bounds of our results. VI. REFERENCES 1 Teach Spin, Optical Pumping of Rubidium: Instructor s Manual, OP1-A, (Buffalo,NY, 2002), pp D.J. Griffiths, Introduction to Quantum Mechanics, 2nd Ed. (Pearson Education, NJ, 2005),pp Teach Spin, Optical Pumping of Rubidium: Instructor s Manual, OP1-A, (Buffalo,NY, 2002), pp B. Wolff-Reichert, A Conceptual Tour of Optical Pumping, (2009).