Optical Pumping of Rubidium Vapor

Emily P. Wang MIT Department of Physics

The technique of optical pumping is used to measure how the characteristic pumping time τ of Rb varies according to the temperature and intensity of the beam of irradiating photons. We found that τ decreased with increased beam intensity and increased with increased temperature. This agrees with physical intuition, but our mathematical model for the effects of temperature must be reconsidered. By slowly varying B~ , we obtain the |B~ | = (294.1±21.1) mgauss for the Earth’s ambient field, which is correct to an order of magnitude. We also obtain 1.49±0.01 for the ratio of the Landé g factors for the ground states of Rb87 and Rb85, which compares well to the theoretical value of 87 1.5. For the magnitudes of the values of the Landég factors, we obtain gf (Rb ) = .520 ± .001 and 85 85 1 87 1 gf (Rb ) = .319 ± .001, while the theoretical values are gf (Rb ) = 3 and gf (Rb ) = 2 .

Hydrogenic Electron Electron- Zeeman state spin-orbit nuclear splitting 1. INTRODUCTION spin (1eV) (10e-3 eV) (10e-8 eV) Energy of (10e-6 eV) splitting 2 P3/2 mf Optical pumping is a process of raising individual +2 5P f=2 2 atoms from lower to higher states of internal energy— P -2 1/2 optical this process is used to alter the equilibrium populations f=1 +1 transition -1 +2 of atoms in different energy levels to prepare them for f=2 radio- 2 frequency 5S S 1/2 -2 spectroscopy. The word “optical” refers to the light en- transition f=1 +1 ergy that powers this atomic pump. The motivation for -1 pumping atoms is to prepare them for a particular kind of spectroscopic analysis. Atoms can emit and absorbe elec- FIG. 1: Adapted from [3] tromagnetic radiation varying from radio waves, whose wavelengths are on the order of hundreds of meters, to X-rays, which are a thousand times shorter than visible and D2 lines, respectively. In our experiment, the D2 light waves. By the 1960s, the visible and X-ray spec- line is removed from the light source by means of an nar- tra of atoms had been extensively researched, but ex- row band filter—we are only concerned with exciting the 2 2 ploration of the radio-frequency region of atomic spectra transition between the S1/2 state and the P1/2 state. had only begun [1]. Both the difficulty and the great effectiveness of using In the 1940s, microwave and radiofrequency spec- radio waves in atomic spectroscopy is due to the low en- troscopy was used to determine atomic and molecular ergy of radio waves. When atoms absorb or emit radio structure–nuclear spins and moments in particular. In waves, they experience a slight change in their magnetic 1949, Bitter raised the possibility of detecting radiofre- energy. These transitions in energy states can happen quency resonances through optical means [2]. In 1966, spontaneously in either direction–atoms at higher levels A. Kastler realized this possibility with the invention of will fall into the lower level, and those in the lower level called optical pumping, a new way of doing radio-wave will jump to the higher one, provided that the necessary spectroscopy. The g-factor and the lifetime of excited quanta of energy is available. For transitions to be de- states could be determined from such experiments. Prac- tected in a sample of matter, there must be an overall tical applications of optical pumping include the maser, excess of upward or downward jumps in energy among a the laser, and very sensitive magnetometers. In 1957, population of atoms [1]. Dehmelt introduced the transmission technique of de- At equilibrium, the atomic populations of two energy tecting magnetic resonances in the ground state of alkali levels, n1 and n2, is given by the Boltzmann Distribution: n − ∆E metal vapors [2], a method that we make use of in this 1 = e kT where ∆E is the energy difference between n2 experiment. the two levels, k is Boltzmann’s constant, and T is temperature. For the radiofrequency transitions we wish to observe, ∆E is 10−8 eV at room temperature, kT is 0.25 2. THEORY eV, and n1 differs from unity by only one part in 106, n2 and transitions between the two levels are very difficult 2.1. Optical Pumping of Rubidium Vapor to observe. Here optical pumping comes into play, altering the We investigated the radio-frequency transitions in Rb87 populations of the energy levels in such a way that the and Rb85. Rubidium is an alkali metal atom that has a atomic transitions can be detected with more ease. Fig- 2 2 2 S1/2 ground state and P1/2 and P3/2 excited states ure 2 illustrates the general principle of optical pumping. (Figure 1). The optical transitions between the ground We look at three levels: A, B, and C, two of which are state and the two excited states are the origin of the D1 closely spaced together. The transitions A↔C and B↔C 2

C Optical with orbital motion µ . µ , the combined magnetic mo- transitions After applying s j resonant e ~ radiofrequency... ment, is given by µj = −gJ 2m J, where gJ is the Landé Radiofrequency g factor of interest, which we will now seek to calculate. B transition When the strength of the external magnetic field ex- A perienced by the atom is much greater than the strength of the internal field, then the fine structure dominates, FIG. 2: Adapted from [1]. Shown are the results of optical 0 pumping and the subsequent application of a radio-frequency and Hz can be treated as a small perturbation. This signal that destroys the pumped state. is called the weak-field Zeeman effect, and is the regime of Zeeman splitting that we are concerned with in this experiment. The magnitude of splitting is given by are accompanied by the absorption or emission of an op- 0 e ~ ~ ~ Ez = 2m B < L + 2S >. When the fine structure domi- tical photon, while the transition A↔B is accompanied nates, we can use the basis |n, l, j, mj >, where n is the by the absorption or emission of a photon of radiofre- principle quantum number, l is the quantum number as- quency. Before optical pumping begins, atoms are lo- sociated withL~ ,, j is the sum of the spin (given by the cated in the lower two energy levels with an almost equal quantum number s) and l, and mj is a quantum number probability[1]. that takes on values from |l − s| to l + s. We use this ba- When a lamp emitting optical photons is shined onto sis because S~ and L~ are not separately conserved in the atoms in a vapor, the photons excite atoms in level A presence of spin-orbit coupling. Using the vector model, to level C, where they remain momentarily before falling we see that J~ = L~ + S~ remains constant, and L~ and S~ back to either B or A. If the atoms drop back to A, they precess quickly around this fixed vector sum. Using the may be excited again, but if they drop back to B, there fact that the time average value of S~ is its projection they will remain, for they may no longer absorb photons ~ ~ ~ ~ of optical wavelength. As time passes, the net effect is to along J, we can obtain the value of < L + 2S >= gJ J, “pump” atoms from level A to level B. This state can be giving us the g factor we desire: detected by observing the transmitted light that passes j(j + 1) + s(s + 1) − l(l + 1) g = 1 + (2) through the vapor of atoms, such as with a photodiode J 2j(j + 1) detector. When all of the atoms have been pumped up to the B level, they are no longer able to absorb incoming To further correct the energies of the system, we take radiation, and the vapor is relatively transparent to the into account the fact that the nucleus of the atom also ~ lamp’s light. has an angular momentum, given by the operator I and The pumped state can be destroyed by applying a ra- the quantum number i. The total angular momentum diofrequency that equals the resonant frequency of the of the atom is now described by F~ = J~ + I~, and the atom–namely, the frequency corresponding to the tran- dipole moment of the atom is now given by the equation e ~ sition between levels A and B. This change in situation, ~µ = gf 2m F , where gf is the new Landég factor we must too, can be observed by measuring the transmitted light. calculate. A sudden increase in opacity would indicate that the Using a similar procedure to the one used to calculate pumped state has been destroyed, and the atoms which gJ , we note that J~ and I~ now precess around their sum drop back down to level A are once again able to ab- F~ . Because the nucleus is so massive, however, it has a sorb incoming radiation. Optical pumping can be used much smaller angular momentum than electrons, so the to measure the relaxation times of magnetic moments of Landég factor associated with I~, gI , is negligible, and the atoms, the magnetic field experienced by the atoms, and precession of I~ around F~ can be ignored in the analysis the Landég factors of atoms. [5]. Carrying through with the algebra once more, we obtain f(f + 1) + j(j + 1) − i(i + 1) 2.2. Zeeman Effect, Derivation of Landég factor g = g [ ] (3) f J 2f(f + 1)

The Landég factor is a dimensionless factor which cor- where gJ was derived above. Evaluating these theoret- rects for the magnetic interactions in the atom that alter ical expressions for the ground state of the two differ- an atom’s energetic states. When an atom is placed in a ent rubidium isotopes used in this experiment, we find uniform external magnetic field B, the energy levels will 85 1 that both isotopes have gJ = 2, gf (Rb ) = 3 , and be shifted, and this phenomenon is called the Zeeman 87 1 gf (Rb ) = . effect. Following the analysis in [4] and [5], for a single 2 electron, the perturbation of the energy levels is given by e 2.3. Characteristic Time Constant τ for Optical H0 = −(~µ + ~µ )B~ = (L~ + 2S~)B~ = − ~µ B~ (1) Pumping z l s 2m J where L~ is the orbital angular momentum operator, S~ Our physical intuition tells us that increasing the rate is the spin operator, µl is the dipole moment associated of orientation, such as by increase the light intensity, we 3 should expect a decrease in the time it takes for atoms X Helmholtz coils Y Z to become optically pumped, and hence there should be Collimating lens Focusing lens Narrow-band filter a decrease in the time constant. If we increase the rate Rb cell of disorientation, such as by increasing the collisions of Rubidium Photodiode atoms with each other by increasing the temperature, we Lamp Linear polarizer Half wave plate Current to Voltage should increase the pumping time constant. The math- Amplifier 1 ematical model is τ = , where Wu is the rate of Wu+Wd to RF coils XYZ transitions to the pumped state and Wd is the rate of Coil current supply Oscilloscope transitions out of the pumped state. Function Generator to Trigger

FIG. 3: Setup for optical pumping, showing coordinate sys- 3. EXPERIMENT tem used—setup used to vary the B~ in order to obtain ambient field magnitude and ratio and values of Landég factors. In this experiment, we used circularly polarized light from a rubidium lamp to irradiate rubidium vapor to ex- ~ 2 2 the Helmholtz coils. The B obtained at that time was cite the optical S1/2 state to P1/2 state transition (the the B~ needed to cancel out the Earth’s B~ , and therefore D line, as mentioned above). Every photon has angu- 1 was equal in magnitude to the Earth’s B~ . The value of lar momentum ± , so transitions in the Rb atoms will ~ the resonant frequency was measured by using the cur- have ∆m = 1. Spontaneous transitions can occur with f sors on the oscilloscope screen to obtain the location of ∆m = ±1, 0. As a result, there will be a net pumping f the absorption peaks that indicated the disturbance of of atoms to the m = +1 substate of the lower electronic f the pumped vapor state. state, for once atoms land there, they are unable to ab- To produce and vary the magnetic field external to the sorb another circularly polarized photon. atom in the x-, y-, and z-directions as desired, three sets of Helmholtz coils were placed around the Rb vapor cell, and current was run through these coils, as shown in Fig- 3.1. Slowly Varying B~ ure 3. Helmholtz coils consist of pairs of identical coils which are spaced at a distance equal to their radii. They The setup used in this part of the experiment is shown produce a magnetic field at a point directly between the ~ in Figure 3. The rubidium lamp was first heated to a centers of each coil which is given by B~ = √8µ0NI , HC 125R temperature of approximately 40 degrees Celcius, and where I~ is the current in the Helmholtz coil, R is the the beam of light was focused by two lenses so that it hit radius of the coil, N is the number of turns in the coil, the photodiode squarely in the center. The photodiode −7 −1 and µ0 is an SI constant equal to 4π × 10 Wb A observed the amount of light that traversed the cell of m−1. If we substitute this into the equation for the reso- Rb vapor located in the center of the apparatus, and its nant frequency of the Rb atoms, we obtain the following output was input to the oscilloscope. A relatively high equation: transmittance was observed once the state of pumping g µ q had been achieved, and when a sudden dip in transmit- f = F B B2 + B2 + B2 tance was observed (sudden increase in vapor opacity) h x y z s (4) it could be deduced that a resonant radiofrequency had g µ 8µ N I~ N I~ N I~ been reached and the pumping state had been temporar- = F B √ 0 ( x x )2 + ( y y )2 + ( z z )2 h R R R ily destroyed. 125 x y z To measure the ambient B~ of the Earth, as well as the From Equation (4), we can see that the characteristics of ratio and the values of the g factors for Rb, a radiofre- the frequency for varying Ix and Iy will be hyperbolic. quency sweep was performed by hooking up the func- Once Bx and By are set to equal zero, however, the char- tion generator to the RF coils surrounding the Rb vapor acteristic of the frequency for the varying of Iz will simply cell, and the B~ external to the atom was slowly varied. be symmetrically linear. We note that the resonant frequency f of rubidium is related to the magnetic field experienced by the atom, ~ 3.2. Characteristic Time for Optical Pumping B~ , in the following way: f = gF µB B , and that B~ will qh ~ 2 2 2 have three components: |B| = Bx + By + Bz . Our In this part of the experiment, we first bucked out the goal was to produce a magnetic field that just canceled vertical and lateral components of the Earth’s magnetic out the Earth’s B~ and thereby minimized the resonant field using the x and y Helmholtz coils, using the values frequency observed in the Rb atoms. This was accom- obtained previously. The z-coil’s current was then turned plished by varying first the Bx until the resonant fre- on and off using the square wave on a second function quency had been minimized, re-setting Bx to this value, generator hooked up to the current controls. The wave- and repeating the procedure with the By and the Bz in forms which showed the change in opacity as a function 4

Hyperbola Fit to Varying X Current of time after turning on and oﬀ the current were captured 220 Rb87 200 Resonance frequency minimum located at: fit line on the oscilloscope and later analyzed to obtain the val- ± 180 I = (104.1 .2) mA x Rb85 2 160 χ = 1.2 fit line ues of the time constant τ related to the optical pumping ν−1 140

120 of the system. 2 100 χ =1 ν−1 The pumping rate was varied by changing the intensity 80 Frequency of Resonant Peak (kHz) 60 0 20 40 60 80 100 120 140 160 180 200 of the incoming light beam from the Rb lamp. Filters of X Current (mA) Hyperbola Fit to Varying Y Current optical density 0.1 were used to progressively block out 120 χ2 =1.6 110 ν−1 more and more light, and a screen capture of the changing 87 Rb 100 fit line Rb85 opacity was taken at each level of filtering, up to five 90 Resonance frequency minimum located at: fit line χ2 I = (−14.4±.1) mA ν =1 filters. The density of the Rb vapor was also altered by 80 y −1 70 changing the temperature of the bulb. Frequency of Resonant Peak (kHz) 60 −20 −15 −10 −5 0 5 10 15 20 Y Current (mA)

4. RESULTS AND ERROR ANALYSIS FIG. 4: Varying x- and y-current and observing change in resonant frequency of the two Rb isotopes. 4.1. Measurement of the Ambient |B~ |, Land´eg Linear Fit to Varying Z Current factors 120

110 Line intersections:

100 [Rb85]: I =(24.01±.76) mA z χ2 [Rb87]: I = (24.05±.62) mA ν =0.5 90 z −1 Average: I = (24.03±.51) mA Following Equation (4), the ﬁt line used for I and z x 80 χ2 =2.6 ν−1 b p 2 2 70 χ2 =1.1 I was hyperbolic: y = c + (x − x ) + d. The ﬁt ν−1 y a 0 60 50 χ2 =1.8 line used for Ix was linear, consisting of one line of neg- ν−1 40 ative slope and one line of postive slope. The values of 30

Frequency of Resonant Peak (kHz) 20 I and I at the minimum resonant frequency were ob- 0 10 20 30 40 50 x y Z Current (mA) tained from the fit parameter x0. The two lines were fitted separately, and the value of Iz at the point of in- FIG. 5: Varying z-current and observing change in resonant tersection was noted. Note each graph shows a plot for frequency of the two Rb isotopes. Note that the errorbars 87 85 both Rb and Rb . To obtain final values for current are larger for the lines on the left, as it was more difficult values in each direction that minimized the resonant fre- to make out the location of the resonant frequency peaks for quency (and therefore produced the correct |B~ | to cancel those points. out the Earth’s field), the current values obtained from the fits for each isotope were averaged, and errors were added in quadrature. The current values are shown on was (294.1±21.1) mgauss, compared to the ≈ 300 mgauss the plots in Figures 4 and 5. measured using a magnetometer in the lab, and the ac- In the Iz plots, the z-intercept for each isotope was tual value of the B~ at our longitude and latitude, which ar1−al1 calculated with the following equation:Iz = , al2−ar2 is ≈ 600 mgauss. The reason for this discrepency be- where ar1 and ar2 were the y-intercept and slope, re- tween measurement and accepted value is likely due to spectively, of the line on the right, while al1 and al2 the presence of a large metal pipe directly above the ex- were the y-intercept and slope of the line on the left. perimental setup, which caused the magnetic field in the The error was calculated with the following equation: room to be distorted. q σ2 +σ2 σ2 +σ2 r1 l1 l2 r2 As the resonant frequency of each isotope is directly σI = Iz[ 2 + 2 ], where the error in z (ar1−al1) (al2−ar2) proportional to its g factor, the ratio between the g fac- each fit parameter was obtained from the fitting script. 87 85 The error for each datapoint, which was input into the tors of Rb and Rb was obtained by taking all the values for the resonant frequencies of Rb87 and dividing fitting script, was obtained by noting the resolution of 85 the cursor used to measure the location of the resonant by all of the resonant frequencies of Rb . The ratio ob- peaks. Unfortunately, we were unable to fit the peaks to tained in this fashion was 1.49±0.01. Error was obtained curves due to a difficulty in the osciloscope-to-computer by adding in quadrature the standard deviations of the resonant frequencies of Rb87 to the standard deviations capture program. 85 The values for the current in each direction were of the resonant frequencies of Rb . then substituted into the equation for the B~ created To obtain the actual values of the Landég factors, by Helmholtz coils was then used to calculate the com- Eq. (4) was again used to determine how the g factors were related to the parameters obtained in the fitting ponents of the B~ necessary to cancel out Earth’s B~ . of the resonant frequency vs. current plots. Error was The error in each component of the magnetic field was propagated using the errors obtained for the parameters calculated by error propogation based on the equation through the fitting program that, again, had been input for the magnetic field of a Helmholtz coil, given the with errors for each datapoint relating to the resolution of current, number of coils, and radius characterizing the the oscilloscope cursor. The values of the Landég factors q 2 2 2 σI σ 87 85 Helmholtz coil: σB = B ( I2 + R2 ). The value ob- obtained were gf (Rb ) = .520 ± .001 and gf (Rb ) = 85 1 tained for the Earth’s magnetic field using this method .319±.001, while the theoretical values are gf (Rb ) = 3 5

Fit of Data for 1 Filter Fit of Temperature Data

0.3

4.24

0.28

4.22

0.26

2 4.2 χ = 0.35 ν−1 0.24 χ2 = 2.1 ν−1 4.18

Time Constant (s) 0.22 Intensity Signal (V)

4.16 Model: y=a+(1−exp(−x/b))*c 0.2

0.18 4.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 40 45 50 55 Time (s) Temperature (C)

FIG. 6: A sample of the data taken for the portion of this FIG. 8: Varying of temperature and observing its eﬀect on experiment involving the measurement of the time constant the time constant. for rubidium pumping. 5. CONCLUSIONS

Fit of Time Constant to Beam Intensity

2 0.3 χ = 0.68 ν−1 We found that τ decreased with increased beam intensity and increased with increased temperature, agreeing Time constant (s) 0.25 with physical intuition on both counts, but disagreeing

0.2 with our mathematical model for the effects of tempera- 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Intensity ture, suggesting we should reconsider our model. Other FIG. 7: Varying the beam intensity and observing its effect improvements would be to more accurately control the on the time constant. temperature during the temperature-varying trials. We obtained that the Earth’s ambient magnetic field has a magnitude of |B~ | = (294.1 ± 21.1) mgauss, which is cor- 87 1 and gf (Rb ) = 2 . rect to an order of magnitude, but is inaccurate due to the presence of magnetic field distortion in the room— future work could choose a site which had no interference 4.2. Characteristic Time for Optical Pumping with the Earth’s magnetic field in order to obtain a more accurate measurement. We also obtained 1.49 ± 0.01 for A portion of a typical screen capture from the oscillo- the ratio of the Landég factors for the ground states scope is shown in Figure 6. All data taken in this part of of Rb87 and Rb85, which compares well to the theoret- the experiment were fitted to exponential curves, which ical value of 1.5. For the values of the magnitudes of 87 allowed determination of the time constant that charac- the Landég factors, we obtain gf (Rb ) = .520 ± .001 85 terized the optical pumping process. To obtain error on and gf (Rb ) = .319 ± .001, while the theoretical values 85 1 87 1 the exponential fits, error for each individual datapoint are gf (Rb ) = 3 and gf (Rb ) = 2 . The actual values was estimated as a percentage of the signal, with ad- do not agree as well as the ratio, even with error bars, ditional error from deviations in temperature and deter- suggesting that there is some unaccounted-for systematic mining the time of the polarity flip. The observed depen- error which is not simple, as one value of the g factor is dence of the time constant on the temperature and beam too large, and the other value is too small. One pos- intensity were then compared to our physical intuition sibility may be that the field created by the Helmholtz and our mathematical models. We observed that the coils is not perfectly uniform, as we assumed it to be, or time constant decreased with a increase in beam inten- that the Rb cell is not perfectly located at the center of sity, as expected from both physical intuition and math- the coils, which is where the equation for the magnetic ematical models, but that the time constant increased field created by the Helmholtz coils is valid. Future work with an increased temperature, agreeing with the physi- could more closely examine the nature of the magnetic cal intuition but not the mathematical model. field produced by the Helmholtz coils.

[1] A. L. Bloom, Scientific American (1960). Acknowledgments [2] R. Bernheim, Optical pumping; an introduction (W. A. Benjamin, 1965). The author gratefully acknowledges Kelley Rivoire, her [3] J. L. Staff, Junior lab reader (2005). fellow investigator, and the junior lab staff for their as- [4] D. J. Griffiths, Introduction to Quantum Mechanics (Pear- son Prentice Hall, 2005), 2nd ed. sistance in lab. [5] R. Benumof, American Journal of Physics 33, 151 (1965).