OPTICAL PUMPING SERENA (DANTING) CHEN Department of Physics, University of California, Berkeley. serenadantingchen@berkeley.edu PARTNER: IWAN SMITH Department of Physics, University of California, Berkeley. mncx9is2@berkeley.edu Dated: November 13 2012

We measure the Zeeman energy levels of rubidium-85 and 87 in an inert buffer gas in a modulated magnetic field. Circularly polarised photons were aimed at the rubidium specimen, introducing an optical “pumping” to higher energy levels, and enabling the sample to become transparent under pumping light. Measurements of magnetic field and frequency values at successful pumping, paired with the Breit-Rabi equation, gave a means to calculate the nuclear spins of Rb-87 and 85 (found to be at 1.5285, and 2.538, respectively) to yield values with less than 2% error. A value for the Earth's magnetic field was also obtained to be 0.417 ±0.007 gauss, also extremely accurate for an undergraduate laboratory. 1. INTRODUCTION

2. ATOMIC STRUCTURE

Optical pumping is the production of population inversion by means of light. In this case, we use circularly polarised light to redistribute atoms in a weak magnetic field amongst their Zeeman energy levels such that they are “pumped” up to the highest sublevels of the ground state. Observation of the specific parameters needed to achieve this pumping has the notable ability to measure fine, hyperfine and Zeeman structure[1], with an astronishingly simple setup. Through similar experiments one can also observe the magnetic moments of nuclei, collisional energy perturbations[2], and extremely small magnetic fields, including the magnetic field of the Earth. This in turn leads to its application in high-precision magnetometers[3], as well as the production of atomic clocks[4] and, more recently, the probing of magnetic structures of noble gas nucleii as a means for obtaining enhanced MRI imaging[5].

Atoms bounded by potentials give rise to quantised energy levels – most simply the principal energy states from the Coulombic interaction between the positivelycharged nucleus and the negatively-charged electrons. Perturbing these levels further we obain finer and finer corrections (see FIG. 1): First, fine structure, due to relativistic corrections and the spin-orbit coupling of electrons; then, hyperfine structure, due to the interaction between the magnetic dipole of the nucleus and the electric quadrupole of the electron[6]. In this experiement, we confine our attention to the hyperfine and Zeeman energy levels (existing due to an external magnetic field). For rubidium, its electron configuraion is 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s. Note that it has only one valence electron, with its first 36 electrons in closed subshells. Hence, we only need to consider the angular momentum of the 5s electron (the rest sums to zero). Here, (1) J = L + S,

In this article we describe the measurement of nuclear magnetic momenta of rubidium-85 an 87 (calculated in §5.1), as well as the Earth's magnetic field (§5.2). This is done by bombarding rubidium vapour with rf light such that the energy carried by the photons exactly match the transitional energies needed to bump the electrons into their excited state. When this is achieved, relevant frequency and magnetic field values are measured, then inputted into the Breit-Rabi equation (a quick derivation is included in the Appendix). Perhaps the most blatant observation at first is the ease of which the setup (§3) and procedure (§4) lends to such high-precision measurements in an undergraduate lab. The validity and accuracy of these measurements, however, are not so easily inferred, and will be discussed along with the results in §5.

with J denoting total angular momentum of the electron, which is comprised of the spin (S) and orbital angular momentum (L). Similarily, the total angular momentum of the atom (F) is just the sum of the contributions from the electron (J) and the nucleus (I): F=J+I

(2)

We carry on assuming the reader is already familiar with Coulomb, fine and hyperfine structures. A light treatment of these perturbations is given in Appendix A. 2.1 ZEEMAN SPLITTING IN THE RUBIDIUM ATOM In a weak external magnetic field, B, hyperfine energy levels split further due to the Zeeman perturbation[6]:

2

FIG. 1. A quick sketch of the different levels of atomic energy level corrections.

(3) where gF is the Landé g-factor, (4) and

we were to include the D2 radiation (and above), we allow the mF=+2 excited states to further absorb D2 radiation, decreasing the polarisation and allowing, as de Zafra calls it[1], an “escape route” for the electrons, resulting in relatively short relaxation times and a mixture of atomic orientations. Thus it is favourable[7] to filter out any D2 radiation. Now let's consider the spontaneous emission of a photon from an electron in the excited state: selection rules tell us that m = 0, ±1. These m values are equally probable when a photon is emitted, which means the electron now has a 2/3 chance of having a higher mF value when it drops back down to the ground state. After a few iterations of absorption and emission, one can see that the highest possible excited level quickly gets saturated. The electrons have been “pumped” up to a higher level. For Rb-87, the saturated level is 2s1/2, F=2, mF=+2. For Rb85, it's 2s1/2, F=3, mF=+3. When the topmost energy level is saturated, the rubidium atoms cannot absorb any more photons. Recall that E = hν. Here, ν is the frequency of the circularly polarised light. When pumping is achieved, the atoms become transparent to the pumping light. We note down the pumping frequency, ν, and external magnetic field, Bext. With this data we are now able to calculate the angular momentum of the atom's nucleus, using the Breit-Rabi formula:

(5) And μB is the Bohr magneton (defined as μB = eħ/2m). Since we know that the ground state of Rubidium is 2s1/2, with s = ½ and l = 0, we can conclude that J = ½ and this calculate the Landé g-factors, shown in Table I.

(6) A quick derivation of the Breit-Rabi formula can for found in Appendix B.

TABLE I: LAND G-FACTORS FOR RB-85 and RB-87 Isotope J I F gF Rb-85 Rb-87

1/2

5/2

2

1/6 gJ

1/2

5/2

3

+1/6 gJ

1/2

3/2

1

+1/4 gJ

1/2

3/2

2

1/4 gJ

This energy splitting gives rise to 2F+1 energy levels per F state. 2.2 OPTICAL PUMPING In this experiment, we bombard the rubidium vapour with circularly polarised light – each photon carrying one unit of angular momentum. Hence, the magnetic moment, mF of the valence electron must increase by one. l = +1, taking the ground state 2s1/2 atom to 2p1/2. On top of that, m = +1 when absorbing a photon. This gives us all the possible transitions from ground states to excited states, shown for Rb-87 in FIG. 2. Note that we now are only considering the 2s1/2 and 2p1/2 states – also known as the D1 energy levels. Though the same processes occur with higher energy levels, collisions between rubidium atoms with each other and the wall of their container mixes the different mF levels. If

FIG. 2. Energy levels for the Rb-97 atom.

3

FIG. 3. Experimental setup.

3. APPARATUS The experimental setup is outlined in a schematic diagram in FIG. 3. Some quick things to note: the RF oscillator produces the pumping light, which goes through a circular polariser and D1 filter to make sure it has the needed properties as discussed above. Resonance is detected with a photodiode that registers a signal when the rubidium has been pumped into its excited state and rendered transparent. This strength of this signal is dependent on temperature, to be addressed later. 3.1 HELMHOLTZ COILS The large coils shown sandwhiching the rest of the setup in FIG. 3 produce the external magnetic field that induces the Zeeman splittings. The radius of the coil averages at R = 27.5cm (±0.2cm for measured variance from shape imperfections), and N = 135 turns of the coil. In our mathematical treatment, let's first recall the magnetic field at a distance z from the centre of a loop of current[8]: (7) Placing two coils of wire at ±d/2, by the principle of superposition, gives the sum of their individual B fields. It can be shown that the change in B at z = 0 is zero. Fir the second derivative to equal zero, we pick d = R, and hence,

(8) So in the case of many coils, we need only superimpose the fields. Substituting 0 = 4π × 10‒7 (9) This is the magnetic field inside the coils, as a function of current, which we can change. Note also from FIG. 3 that the DC power supply for the coils runs through an AC Modulation Uni – we can sweep the magnetic field to make finding the correct B field a simple task of finding resonance. The Helmholtz coils are bolted to the experiment table such that the produced magnetic field is aligned to be perfectly in parallel with the Earth's magnetic field. This is so later, we can compare two sets of data (with normal and reversed polarity) to obtain a value of the Earth's magnetic field. One may also wish to pay heed to the heating coils located inside the lamp assembly. These coils heat the rubidium vapour to operational temperatures, but also produce a magnetic field. This field will interfere with our experiment, so we must remember to switch the heating off before any meaurements are taken. 3.2 RESONANCE BULB We use a low-density specimen of rubidium atoms so

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that we may neglect the atom-atom interactions (for now). The rubidium vapour is mixed with a “buffer” gas (such as hydrogen, argon, neon) to minimise the “mixing” of states due to collisions off the side of the container or with each other[9]. Since the melting point of rubidium is at 39℃, the bulb must be kept heated in its vapour phase for pumping to occur. The bulb is shown in FIG. 3 to be surrounded by coils – these are connected to the DS345 RF function generator, which induces stimulated emission from the pumped atoms. This quick and constant depopulation of pumped states is needed to maintain an ongoing signal for the photodiode to detect – otherwise we would have no response data for when we change the frequency, or magnetic field. The intensity would stay at a relative maximum. If the frequency of this RF field is resonant with the current magnetic field, it will redistribute the atoms amongst the Zeeman levels, and pumping can once again occur. 4. PROCEDURE 4.1 SIGNAL V. TEMPERATURE For pumping to occur, we require that the rubidium be in a vapourised state. Hence, the temperature must be high enough so that it is dispersed throughout the bulb well enough that atom-atom collisions can be neglected. However, the sample cannot be too hot, else it will have too much kinetic energy and then be more likely to bump into a wall, or atom, mixing its quantum state. The signal from Rb-85 peaks and 37℃, whilst Rb-87 peaks at 42℃[10]. Henceforth we only took measurements from the specific temperature range of 34-42℃, in an attempt to produce optimal conditions for pumping. The temperature range was focused on that of Rb-85, as it is more naturally abundant (72% to 28%)[1]. Temperature should not affect the point of resonance, but it is preferable to maintain a good signal anyway. 4.2 FREQUENCY MODULATION There are two ways in which we may find matching ν and Bext values that produce pumping – modulating frequency or modulating the magnetic field. We sweep a range of values so that we may compare the signal of no pumping to the resonant signal where pumping occurs. When we reach resonance, we note the values of ν and I. The first way to do this is to modulate frequency: we set up the experiment as in the diagram in FIG. 3. The bulb is heated up to 48℃ – hot enough so that the bulb is still in the optimal temperature range for the duration of the experiment but cool enough as to not damage any of the apparatus. The sample is never to be heated above 50℃. Field modulation (i.e., modulation of the current) is turned off, and a 0.31mA current is run from the DC power supply. The function generator is set to modulate the frequency with a 100Hz ramp function at 2.2MHz. The signal from the photodiode is sent to first, a preamp, and finally to CH2 of the oscilloscope. Going to

CH1 is the signal from the RF oscillator – note here that this setup differs from the one shown in FIG. 3. Setting the scope to X-Y mode, we view the transparency of the specimen as a function of frequency. Changing the frequency, we search for resonance. To obtain a more precise measurement we may decrease the modulation span until we've “honed in” on the resonance. This method is not preferred as the function generator can only be changed in discrete steps – often making it fiddlesome and difficult to quickly and efficiently find resonance. The continuous adjustment of a knob, like the one we have for current, is preferred. 4.3 FIELD MODULATION Here the setup is exactly that of FIG. 3. The AC Modulation Unit is turned on and set to modulate the field at a value such that it was large enough to show the change from opacity to transparency, but small enough as to only include the signal from one isotope. The scope is set to X-Y mode, and we observe a Lissajou curve on the screen. We find resonance by changing the current so that the trace shown becomes symmetrical about the Y axis; this means that we have an equal change in transparency on both sides of the mid current value and thus the mid current value must therefore be at resonance. FIG. 4 shows some expected shapes on the scope.

FIG. 4. OSCILLOSCOPE TRACES OF RESONANCE (W. E. Bell and A. L. Bloom 1967)

Once again, the specimen is heated to 48℃. The heater is then turned off for the measurement phase of the experiment. We note down the current resonant current values at discrete frequencies from 0 – 6MHz, in 0.5MHz steps. We also reversed the current to find corresponding B values at “negative” frequencies. Resonance was found at each state for both isotopes. The corresponding magnetic field from current can be calculated with equation (9), derived in §3.1. It should be noted here that the Lissajou curve was extremely sensitive to changes in current – so much so that it would fluctuate with anomalous current changes, making the resonant peak unstable. A somewhat rough solution was found by securing all cables and wires to solid surfaces with tape. We now turn off the RF radiation to find a resonant current value at zero frequency. This was found to be at I = 0.0978 mA (±0.016/√3), by taking four measurements and then averaging them out. This should not exist if the Helmholtz coils were aligned perfected with the Earth's magnetic field (as we'd expect BH to cancel out all of BE),

but since such an alignment is practically impossible, we see a resonant signal at zero field. 4.4 PUMPING & RELAXATION TIME With field modulation off and a square frequency modulation at 100% depth at resonance frequency, we view the pumping and depumping of the rubidium atoms (FIG. 5). This process resembles the charging and discharging of a capacitor – when RF is on, the atoms saturate their upper levels exponentially; when RF is turned off, spontaneous emission occurs and the sample becomes exponentially opaque. By taking a photo of this trace and then entering data points we estimate that the pumping time is 9.9×10–4 s (±0.4s), and the relaxation time 1.55×10–4 s (±0.07s). The errors were estimated based on the data's spread around a fitted exponential curve.

5

weak external magnetic field in our theoretical treatment – for intermediate fields neither Zeeman nor hyperfine Hamiltonians dominate, and we get off-diagonal elements and our system becomes non-linear.[6] Using (6), we can calculate the nuclear spin from the gradient of the plots (Table II). Here we find good agreement between the calculated I values of Rb-87 an Rb-85, respectively. The K correction value was calculated by assuming that our dervied equation for the B field in the Helmholtz coil (9) was not perfect, but the I values of 3/2 and 5/2 were. We expect the K correction to be very close to 1; in recalculating the gradients with the “error-less” I values, it turned out that we were slightly overestimating the B field due to the coils. From assuming that our χ2 should be perfect, we worked backwards to find the systematic errors on the system, given by Esys in Table II. These were surprisingly low, given the sensitivity of the system to any vibrations of fluctuations in fields around it. However, by securing all screws and cables to the working surfaces, and making sure that no electronic equipment (other than the measurement apparatus that was needed) was running in the vicinity of the experiment, we managed to reduce the systematic error. 5.2 MAGNETIC FIELD OF THE EARTH

FIG. 5. Measuring pumping and relaxation times. A lot of the noise you see here is due to the 60-cycle property of the mains, and thus we could not eliminate it.

5. RESULTS AND ANALYSIS 5.1 NUCLEAR SPINS AND ZEEMAN LEVELS OF RUBIDIUM At resonance, the gradient of frequency vs. magnetic field depends on the nuclear angular moment by the Breit-Rabi equation (6). FIG. 6 plots frequency against magnetic field at resonance for both isotopes and both polarities of the field and fits the data using least-squares fitting to a line[11]. Uncertainty was taken for every single measurement during the experiment – overall they tended to increase as the magnitude of the current increased. This perhaps due to the fact that we considered a very TABLE II: NUCLEAR SPIN OF RB-85, RB-87 Gradient Icalculated Iactual

To obtain the magnetic field of the Earth, we extrapolate the lines in FIG. 6 and observe where it intercepts the x-axis. This is the zero frequency field, where the magnitude of BH equals that of BE, and they cancel to produce resonance. Combining our four sets of data and allowing “negative” frequencies, we obtain two distint lines – one for Rb-87, one Rb-85. These lines intercept at the same point on the x-axis, and gives the value of BH that completely cancels out BE. Hence we calculate the magnetic field of the Earth to be With an error calculated from the adjusted coefficients as described in §5.1 and Lyons[11]. Note well that this measurement was taken at, and is only valid for the location at the University of California, Berkeley, in the United States. Measurements at other locations on Earth will differ. Rewriting the Breit-Rabi equation (6), we find that at the resonant B value, (10) Noting that Bext = BH + BE, (11)

K correction

Esys

Zero ν field

0.6707 ± 0.0005

1.526

3/2

0.987 ± 0.001

0.0040

0.423 ± 0.014

0.6891 ± 0.0003

1.531

3/2

0.9848 ± 0.0006

0.0023

0.426 ± 0.010

0.4607 ± 0.0003

2.534

5/2

0.988 ± 0.001

0.0021

0.426 ± 0.016

0.4601 ± 0.0002

2.542

5/2

0.9863 ± 0.0009

0.0019

0.412 ± 0.020

6

And thus (12)

From this method we may obtain two averaged values of BE., one from each isotope. We would expect this value to be more accurate than the one obtained by the above method, as equation (12) eliminates the need for BH all together. As we saw, our theoretical calculations of BH was not the true magnetic field inside the coils. Unfortunately due to time contraints we have not obtained a value with this method, though it would be a simple calculation exercise â€“ perhaps one that the reader could undertake in times of extreme, unbearable boredom. 5.3 HYSTERESIS It was a shame that the experiment was somewhat rushed, and the realisation came only afterwards that we needed to check for hysteresis. This is very noticable when one pays heed to the residues of the four plots in FIG. 6: all residues seem to follow the same pattern. During the data-gathering phase of the experiment we started each measurement of current from 0, going up until we aw resonance. Sometime in the middle of the experiment it was fast and more efficient to actually start from a BH that was at the midpoint of the two resonances. This is where I suspect hysteresis played part. When one applies a magnetic field to a magnetically neutral specimen, the dipoles in this specimen will be influenced by this field and hence will eventually align themselves to become magnetised. However, if one would

like to magnetise this specimen in the opposite direction, one needs to apply double the magnetic field to first depolarise the magnetic dipoles, and then to coerce them to the reverse polarisation. This is hysteresis. Atoms of rubidium in the bulb become slightly magnetised with the application of an external magnetic field. Hence this changes their resonance position, and gives us unconsistent values depending on whether we approach resonance from the bottom up or top down. 6. CONCLUSION Measured and calculated values for nuclear spins of rubidium atoms were within 1.9% and 1.52% of their actual values for Rb-87 and Rb-85 respectively. Our calculation for the Earth's magnetic field was also within statistical, and localtional fluctuations. This highly accurate and precise agreement allows us to conclude that optical pumping, in a word, works, and is highly commended as a teaching tool in physics for its simplicity and precision. Concepts dealt with in this experiment lends to a greater conceptual understand of quantum energy levels, lasers, and other optical applications. Acknowledgements. The author would like to thank her lab partner, Iwan Smith, for his wizard-like prowess in MATLAB and frequent games of foosball, as well as GSIs David Gee, Matthew Leonard, and professors Bill Holzapfel and Hartmut Haeffner, for their endless patience in putting up with two clueless foregin exchange students doing science in America.

FIG. 6. Data plots with least-squares linear fitting for Rb-87 and 85, in both normal and reversed mgnetic fields.

7

REFERENCES [1] R. L. de Zafra, “Optical Pumping”, Amer. Journ. of Phys. 28, 646 (1970) [2] E. C. Beaty and P. L. Bender, “Narrow Hyperfine Absorption Lines of Cs-133 in Various Buffer Gases”, Phys. Rev. 112, 450 (1958) [3] W. E. Bell and A. L. Bloom, “Optical Detection of Magnetic Resonance in Alkali Metal Vapor”, Phys. Rev. 107, 1559 (1957) [4] M. Arditi and J. L. Picqué, “A cesium beam atomic clock using laser optical pumping”, J. Physique Lett. 41, 379-381 (1980) [5] M. S. Rosen, T. E. Chupp, K. P. Coulter, R. C. Welsh, and S. D. Swanson, “Polarized Xe-129 optical pumping/spin exchange and delivery system for magnetic resonance spectroscopy and imaging studies”, Rev. Sci. Instrum. 70, 1546 (1999) [6] D. J. Griffiths, An Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, New Jersey, 2005), p. 267

[7] W. Franzen and A. G. Emslie, “Atomic Orientation by Optical Pumping”, Phys. Rev. 108, 1453–1458 (1957) [8] D. J. Griffiths, An Introduction to Electrodynamics, 3rd ed. (Pearson Benjamin Cummings, San Francisco, 2008), p. 218 [9] A. L. Bloom, “Optical pumping”, Sci. Amer., 203(4), 72 (1960) [10] University of California, Berkeley, Physics 111: Advanced Lab, coursenotes. Retrieved from http://www.advancedlab.org/mediawiki/index.php/Optical_Pumping #The_Effect_of_Temperature_on_Signal_Strength [11] L. Lyons, A Practical Guide to Data Analysis for Physical Science Students, (Cambridge, 1991), p. 45

APPENDIX A. ENERGY CORRECTIONS OF ALKALI ATOMS The principal quantum number n and orbital number l characterises the Coulomb interaction, with Hamiltonian (A.1) The first correction is spin-orbit coupling, given first by the lowest-order reletavistic correction: (A.2) This, along with the effect of the magnetic dipole moment of the electron, gives

B. LIGHTNING-QUICK DERIVATION OF THE BREIT-RABI FORMULA Starting with the Hamiltonian: (B.1) we find that we can eliminate the second two terms in the brackets since BJ and Bext is comparatively small. Expanding the Hamiltonian with J and F values (Check out Griffith's Quantum Mechanics book for more details on the relationship bewteen magnetic momenta and the various quantum numbers) and then considering specifically the rubidium atom with J = ½ and m = +1 we eventually obtain

(A.3)

(B.2)

where gE is the electronic g-factor. Here, levels are split according to their J value. Next we have hyperfine splitting (also known as spinspin coupling), from the magnetic momenta of the electron and the proton. Its Hamiltonian is given by:

This, of course, is just equal to hν. And so, substituting the values F = I + J, J = ½, and rearranging, we get,

(A.4) where gN is the nuclear g-factor. In hyperfine splitting levels lift according to their F value.

(B.3) – the Breit-Rabi equation!