Optical Pumping of Rubidium

Ryan Frazier

Contents

1 Prelab 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Structure of Alkali Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Optical Pumping in Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Absorption of Rb resonance radiation by atomic Rb 52.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Postlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Low Field Resonances 83.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Postlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1

Chapter 1

Prelab

1.1 Introduction

Optical pumping is a process which uses photons to redistribute the states occupided by a collection ofatoms. For example, an isolated collection of atoms will occupy energy states as shown by a standardthermal distribution. However, one can apply resonance radion to alter the distribution of atoms amongthese states.The atom being optically pumped in this experiment is Rubidium, which is enclosed in a system along withNeon, for stability. Optical pumping is important due to it being the basis for all lasers. For example. aHeNe laser uses Optically pumped Helium and Neon Gas chambers to create a very narrow bandwidth laseremition at 633nm.

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1.2 Equipment

• Rubidium Discharge Lamp-The RUbidium discharge lampe consists of an RF oscillator, oven, andgas bulb. the gas bulb is filled wil a little Rubidium metal and a buffer gas. The bulb sits within thecoil of the oscillator.

• Photodiode Detector- The detector is a silicon photodiode. The active area of the diode is circularwith a diameter of 1

4 inch.

• Optics

– Plano-Convex Lens- Usedto minize spherical abbations andd focus the light from the dischargelamp

– Interference filter- used to eliminate the 780nm band emitted from the discharge lamp

– Linear Polarizers- linearly polarizes the light so that it can then be circularly polarized by thequarter wave plate. Wave plates have no effect on unpolarized light due to them creating a phaseretardation.

– Quarter Wave Plate- The quarter wave plate allows linearly polarized light to be converted intoelliptically polarized light, if the linear polarization is at a 45 degree angle to the wave plate axis,the light will be circularly polarized.

• Controller Box- The controller box is used to monitor field currents and Rubidium temperature. Italso can sweep the horizontal magnetic field and set the Main Field current value.

Figure 1.1: Experiment Equipment, Controller Box (left) and experiment set up and calibrated (right)

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1.3 Theoretical Analysis

1.3.1 Structure of Alkali Atoms

These experiments study the absorption of light by rubidium atom. During theoretical analysis the atomicstructure of Rubidium will be used.In the quantum mechanical picture, atoms are described by a central field approximateion where the nucleusis a point particle characterized by it’s charge, spin, electric moment and magnetic moment. The energylevels are described by wave functions that arrise from perturbations of the Schroedinger equation.For Alkali atoms, the angular momenta eigenstates are coupled in a Russell-Sauders coupling scheme, leadingto energy eigenstate solutions close to the observed values.Rubidium acts very similar to the hydrogen atom due to it having only a single valance electron, the electronconfiguration of Rubidium is

1s22s22p63s23p63d104s24p65s1 (1.1)

The outer electron can be described by an orbital angular momentum L, a spin angular momentum S, anda total non-nuclear angular momentum J, all of which are in units of h̄. The total angular momentum canbe written as

J = L + S (1.2)

To accurately calculate the energy levels one has to consider the dipole moments associated with the spin andangular momentum. This nuclear spin causes what is known as a hyperfine splitting, to furthere differentiatbetween energy levels. Figure 1.2 shows a pictoral diagram of hyperfine splitting.

Figure 1.2: Pictoral Diagram of Hyperfine splitting for rubidium

A weak external magnetic field applied to the energy levels will produce the Zeeman effect, and will resultin further splitting energy levels. The Hamiltonian for this effect is

H = haI · J − µjJJ ·B − µI

II ·B (1.3)

1.3.2 Optical Pumping in Rubidium

Optical Pumping is a method of driving an ensemble of atoms away from thermodynamic equilibrium bymeans of the resonant absorption of light. Rubidium resonance radiation is passed through a heated absorp-tion cell containing rubidium metal and a buffer gas. Resonance light is produced by an RF discharge lampcontaining zenon gans and a small amount of rubidium metail, which has been enriched in RB87 such thatthere are equal amount of natural Rb and Rb87.

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Chapter 2

Absorption of Rb resonance radiationby atomic Rb

In this esperiment, an approximate measurement of the cross-section for the absorption of rubidum resonanceradiation by atomic rubidium was made. The measured value is then compared with the geometric cross-section and the value calculated from theory

2.1 Procedure

1. The apparatus was arranged as shown in figure 2.1, however all polarization elements were removed.This left the Focussing lenses and the interference filter in the Optical setup.

Figure 2.1: Schematic of Optical Pumping, curtousy of umn.edu

2. The cell heater was set to 300k, after waiting approximately a half hour, thermal equilibrium wasestablished in the setup.

3. Using the controller box and high sensitivity voltmeters, the intensity of the optical signal was measuredand recorded

4. The optical signal intensity was recorded for a set of temperatures ranging from 300k-400k

5. The data series measured in step 3 was repeated five times and averaged for accuracy

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2.2 Data Analysis

2.2.1 Data

The desired nonlinear fit is modelled below.I = ae−bρ (2.1)

In order to get the voltage data to fit, an offset voltage was removed from all measurements, this is recom-mended by the lab manual, as the component on the equation that is of interest is the value of b, which isindependant of a vertical offset.The corrected voltaged are recorded in table 2.1 and plotted in figure 2.2.

Temperature Trial 1 Trial 2 Trial 3 Trial 4 Trial 5t, (K) T1 (v) T2 (v) T3 (v) T4 (v) T5 (v)300 1.37820 1.51608 1.46630 1.45622 1.55161310 1.18880 1.15152 1.13428 1.13986 1.14845320 0.84600 0.98387 0.93302 0.88245 0.99846330 0.48847 0.59720 0.48524 0.52190 0.58480340 0.30238 0.45906 0.44098 0.41486 0.47070350 0.17515 0.09503 0.02941 0.02823 0.21555360 -0.00832 -0.02441 0.01711 0.00562 -0.05398370 0.03350 -0.04123 0.08821 0.07772 0.08190380 -0.07016 -0.00658 -0.10946 0.01749 0.01013390 0.00190 0.00219 -0.05636 -0.09579 0.04650400 0.04308 0.07003 0.06050 -0.00504 -0.08455

Table 2.1:

Figure 2.2: Plot of the nonlinear fit for the five trials

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The nonlinear fitted models, as well as b values are recorded in table 2.2. These models were recovered usingMathematica’s ”NonlinearModelFit” command. using the B values, and knowing the absorption path lengthis 0.025m, one is able to calculate the value for the cross section for absorption, also plotted in table 2.2

Trial Fitted Model b-Value cross sectionσ, (m2)

1 1.37369E−0.0501849ρ 0.0501849 2.00740 ∗ 10−16

2 1.39232E−0.0367209ρ 0.0367209 1.46884 ∗ 10−16

3 1.38262E−0.0442219ρ 0.0442219 1.76888 ∗ 10−16

4 1.37131E−0.0442831ρ 0.0442831 1.77132 ∗ 10−16

5 1.39057E−0.0343622ρ 0.0343622 1.37449 ∗ 10−16

Avg 0.0419546 1.67818 ∗ 10−16

Table 2.2:

2.2.2 Calculations

Calculating the nonlinear regression

The nonlinear regression was calculated using Mathematica’s ”NonlinearModelFit” function. The fit wasonly applied to voltages where the Rubidium container was not optically thick.

Calculating Cross sectional area

The equation for the cross sectional area of the absorption is given as

lσ = b (2.2)

where l is the length of the absorption path, and be is calculated from the nonlinear fit of the data plot.This calculation was carried out using mathematica due to its repative nature.

2.2.3 Analysis

The data is in agreement with that of the lab manual, where the cross sections were theoretical and exper-imentally calculated to be 15 × 10−16 and 1.6 × 10−16 respectively. The Cross section is a function of thefrequency distribution in the absorption profile of the rubidium atom, and the intensity of the absorbed lightwill depend on the relationship of the intensity profile of the incident light to the absorption profile of theatom.

2.3 Postlab

2.3.1 Discussion

The purpose of the experiment was to calculate the cross section due to experiments. Our experimentallycalculated cross section came to 1.68 ∗ 10−16. This is in close agreement with the manual’s experimentalvalue; however, it is a whole order of magnitude different than the theoretically calculated value. This erroris examined in the next section.

2.3.2 Error Analysis

The measured cross-section is about 10 times smaller than that which is calculated in theory. Sources oferror for our measurement is the repad variation of the density of rubidium atoms in the cell as a functionof temperature. This is a source of considerable error. Also

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Chapter 3

Low Field Resonances

For the Low Field Resonance experiments, it is necesary to apply a weak magnetic field along the opticalaxis of the apparatus. One has to make sure that the residual magnetic field is as close to zero as possible.In order to observe the zero-field transition, no RF is applied and the magnetic field is swept slowly aroundzero. This is accomplished by varying the current in the sweep windings.There are two isotopes of rubidium, and they have differen nuclear spins. the goal of this experiment is tomeasure those spins by measuring the gf values fomr which the spins can be calculated. This is done bymeasuring a single resonant frequency of each isotope at a known value of the magnetic field. The magneticfield will be determined approximatedl from teh geometry of the field coils.Only the Sweep coils will be used for this experiment, their parameters are given in table 3.1.

Mean radius (m) 0.1639 B (Gauss) 8.991 ∗ 10−3IN/R̄Turns per side 11

Table 3.1:

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3.1 Procedure

1. The residual magnetic field at the location of the absorption cell was determined.

• The current in the sweep coils was adjusted to center on the zero field resonance, and the currentwas measured.

• The value of current was plugged the equation in table 3.1.

2. An RF signal was applied, it’s amplitude set to an arbritary value.

3. The frequency of the RF signal was set to 150kHz.

4. The horizontal magnetic field was swept, slowly increasing from zero, searching for zeeman resonance.zeeman resonances show up as troughs in the oscilloscope trace, see figure 3.1.

Figure 3.1: Oscilloscope image showing Zeeman Resonances

5. The characteristics of teh RF transistions were measured as a function of the amplitude of the RFmagnetic field, the value that provides optimum transition probability was derived.

6. With the main coils still disconnected, the transition frequencies of each isotope was measured as afunction of sweep coil current. The results were plotted to determine that the resonance are linear inthe magnetic field.

7. From the slope of the plots the ratio fo the gf factors was determined. This value was compared tothe ratio predicted by theory.

8. The sweep coils were calibrated using the known gf values and the previous measurements.

• From the prior measurements, the magnetic field was calculated for each isotape from the reso-nance equation

• The magnetic field was plotted vs. the current in the sweep coils.

• This date was fit to a straight line using a linear regression to obtain an equation for the magneticfield vs. current.

9. The Main field was calibrated for further experiments

• The main coils were connected so that their field is in the same direction as that of the sweepcoils.

• Both sets of coils were used to make measurements at resonance frequencies up to 1MHz

• The sweep coil calibration was used to correct the measured fields for the residual field.

• The data was plotted on a linear plot, a linear regression was used to obtain the best fit.

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3.2 Data Analysis

3.2.1 Data

Residual Field

The zero field resonance was found at a current of I=0.29807 Amps. From this and the equation in Table3.1, the residual field is calculated to be 0.179862 gauss. This value must be subracted from the field valueswhen calculating the nuclear spin.

Nuclear spins

The table below shows the isotope currents for 10 trials as well as their respective magnetic fields and thecalculated gf values.

Trial Resonance Current B-field g-factor Resonant Current B-field g-factorIres, (A) B, (G) g Ires, (A) B, (G) g

1 0.855673 0.336471 0.319207 0.628364 0.199307 0.5388872 0.780854 0.291323 0.368676 0.653968 0.214757 0.5001183 0.829979 0.320966 0.334626 0.637634 0.204901 0.5241754 0.789841 0.296746 0.361938 0.684464 0.233159 0.4606465 0.8116 0.309875 0.346603 0.630901 0.200838 0.5347796 0.836215 0.324729 0.330749 0.636433 0.204176 0.5260357 0.794457 0.299531 0.358573 0.708947 0.247933 0.4331978 0.77461 0.287555 0.373507 0.619493 0.193954 0.5537599 0.84822 0.331973 0.323531 0.617478 0.192738 0.55725210 0.886362 0.354989 0.302555 0.624883 0.197206 0.544626

Avg 0.327 0.4725

Table 3.2:

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Low Field Zeeman Effect

To confirm the low field Zeeman effect, the transition frequencies of each isotope was measured as a functionof sweep coil current. These results were plotted to determine that the resonance are in face linear in themagnetic field.

Frequency Current B-Field Current B-Fieldν (MHz) I, (A) B, (G) I, (A) B, (G)

0 0 0 0 0.1 0.34215 0.218824 0.24856 0.151523.15 0.54969 0.328236 0.36214 0.227285.2 0.67988 0.437647 0.465207 0.303047

Table 3.3:

Figure 3.2: Linearly fitted plot for Low field Zeeman Effect

Sweep Field Calibration

For this experiment, the value of the magnetic field was calculated for each isotope from the resonanceequation, the slope of the plot of the magnetic field vs. the sweep current will be used to find a formula forthe magnetic field intensity. For the data values of the magnetic field, see Table 3.3.

Figure 3.3: Linearly fitted plot for sweep field calibration

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Main Field Calibration

The data tables for the Main FIeld calibration are shown below and separated by Rubidium isotope. Ad-ditonally, the linear plot of The Coil Current vs. Magnetic field intensity has an r2 = 0.973 which is veryhigh. The slope of the plot is 8.17308 Gauss/Amp, which is close to the accepted value in the lab manual.

Frequency Total Field Sweep Current Main Current B-Sweep B-Mainν (MHz) (Gauss) (Amp) (Amp) (Gauss) (Gauss)

0.2 0.320547 0.257911 0.0340595 0.00377627 0.3167710.3003 0.452149 0.363797 0.0480685 0.00532663 0.4468220.4002 0.616946 0.496392 0.0750628 0.00726805 0.6096780.5002 0.755526 0.607893 0.0831059 0.00890061 0.746625

Table 3.4: Main Field Calibration for Rb87

Frequency Total Field Sweep Current Main Current B-Sweep B-Mainν (MHz) (Gauss) (Amp) (Amp) (Gauss) (Gauss)

0.2 0.432777 0.34821 0.0392722 0.00509841 0.4276780.3003 0.676416 0.544242 0.0732577 0.00796865 0.6684480.4002 0.826037 0.664626 0.0960853 0.00973128 0.8163050.5002 1.09192 0.878551 0.11189 0.0128635 1.07905

Table 3.5: Main Field Calibration for Rb85

Figure 3.4: Linear fit on data from Table 3.4.

3.2.2 Calculations

Calculating the g-factors

From the lab, the resonant frequencies for optical pumping are found from

ν = gFµ0B/h (3.1)

Rearranging this equation gives

gF =νh

µ0B(3.2)

This equation was used to derive the g-factors in table 3.2.

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Calculating the g-factor ratio

The Slopes of the linear fits for Figure ?? are 0.4312 and 0.28575. The ratio of these two slopes is then1.474. As the theoretical ratio is 1.5, the experimental error in this measurement is 1.734%.

3.2.3 Analysis

All Data was analyzed using Mathematica. This allowed for accurate linear regression equations as well asa very quick way to manipulate recorded data values. Since the average error in the experiments was verylow, there were no unusual measurements in the experiment.

3.3 Postlab

3.3.1 Discussion

The purpose of this lab was to use Low field resonances to measure the g-factors of R85 and R87. weconcluded that the g-factors are 0.327 and 0.472 respectively, these are very close to the accepted value of0.33 and 0.5. Our data supports the theory of quantum mechanics as well as the hyperfine fitting theory.The main difficulties with this experiment as minimizing the net magnetic field, it would take large amountsof time out of the experiment simply calibrate the setup. However, careful calibration led to accurate results.

3.3.2 Error Analysis

our g-factors were measured to be 0.327 and 0.472. These have an error of 1.9 and 5.6 percent respectively,these are very accurate measurements especially given the sensative nature of the experiment.

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