Optical Pumping of Natural Rubidium

Erick Arabia University of San Diego (Jonathon Garamilla and Enrique Lance) (Dated: March 26, 2011) A gas of natural Rubidium, which consists of Rubidium-87 and Rubidium-85 was optically pumped in order to determine the nuclear spin number (I) for each isotope experimentally, as well as to confirm the Breit-Rabi equation for large magnetic fields. The g-factor was determined to be 0.47 for Rb-87 and 0.33 for Rb-85, and the results of the large field calculations can be found in Table (II).

I. INTRODUCTION ¯h, and are related by

J = L + S. (1) In 1950, French physicist Alfred Kastler introduced the concept of modern optical pumping, for which he was Each electron is described in spectroscopic notation by 2S+1 awarded the Nobel Prize in 1966. The process of optical LJ ; the valence electron in Rubidium is therefore 2 pumping involves using photons of a desired wavelength designated by S1/2. This corresponds to J = 1/2, L = 0, and polarization to change the distribution of a gas of and S = 1/2. atoms from thermal equilibrium. Today, optical pump- It is also important to consider the angular momen- ing is a critical tool in modern Physics experiments, and tum of the nucleus. The nuclear spin I combines with is the basis of lasers. This process utilizes quantum me- J to form the total angular momentum number, F. This chanical ideas, and can therefore be used to gain a better coupling has a few important properties. First of all, understanding of quantum and atomic theory. because of the interaction of the nucleus and the elec- In this experiment optical pumping processes were per- tron, the quantized energy levels of the atom are split formed on natural Rubidium gas, which is a mixture of into 2S + 1 sublevels for each energy state of the electron Rb-87 and Rb-85. Because Rubidium only has a sin- (known as Hyperfine Splitting). Second, this splitting is gle valence electron, it can be approximated as a single- degenerate, meaning that atoms with slightly different electron atom (i.e. Hydrogen). Optically pumping the quantum numbers will occupy the same Hyperfine split Rubidium sample achieved two main goals, which was states, which are designated by F. It should be noted that the determination of the nuclear spin angular momenta the difference in energy between the F = 1 and F = 2 (I) for each isotope, as well as the confirmation of Zee- states is insignificant compared to the difference between man splitting due to a magnetic field according to the the energies of the ground state and first excited state. Breit-Rabi equation. In Section (II), the theory behind the optical pumping is discussed. Section (III) explores the experimental B. Zeeman Splitting setup and procedure. The results and their analyzations are in Section (IV), and the conclusions can be found in The degeneracy in the Hyperfine splitting described Section (V). earlier becomes important when a magnetic field is introduced to the atom in question. When this occurs, the F energy states are split further into 2F + 1 sub- II. THEORY levels, designated by the new quantum number M. At very small magnetic field levels, the difference between the Zeeman states is approximately equal for each value A. Atomic Structure of F, and as the magnetic field is increased the difference between the states also increases linearly. As is the case In order to understand the concept of optical pumping, when comparing the energy levels of the Hyperfine split- it is first necessary to understand a few basic principles ting and the electron energy states, the Zeeman splitting about atomic structure. Because Rubidium is an alkali is significantly smaller than the Hyperfine splitting. atom, it only has a single valence electron. That is to However, as the magnetic field is further increased be- say that all the electrons are paired except for one. It yond the low-field levels, the Zeeman split energy states follows that it can be approximated as a hydrogen-like begin to change in a non-linear fashion. These changes atom, with a single free electron. are designated by the Breit-Rabi equation, This electron has three important quantities associated 1/2 with it: the orbital angular momentum L, the spin angu- ∆W µI ∆W 4M W (F,M) = − − BM± 1+ x+x2 , lar momentum S, and the total electron angular momen- 2(2I + 1) I 2 2I + 1 tum J. These numbers are quantized, are all in values of (2) 2

FIG. 2: Allowed transitions due to photon absorption during the experiment. Note that while electrons in the M=2 state cannot absorb photons, excited electrons can relax back into this state.

D. Optical Pumping

FIG. 1: Plot of the Breit-Rabi equation. The horizontal In this experiment the light introduced is of a very axis shows the magnetic field strength as the particular wavelength and therefore energy. The light dimensionless unit x, and the vertical axis shows the introduced to the sample of natural Rubidium will only energy as the dimensionless unit W/∆W. Individual induce changes between the ground state and the first lines are designated by their quantum numbers as excited state. Because this energy is so much larger than |F M > the Hyperfine and magnetic splitting energies, the selection rules for F and M still apply (and in fact are the only important selection rules as other transitions are not where W is the interaction energy and ∆W is the Hyper- possible). The light introduced will also be circularly po- fine energy splitting. Furthermore, larized, which means that it will have a very particular angular momentum. It follows that for the atoms ab- sorbing these photons, only transitions of ∆M=+1 are µ0B x = (gJ − gI ) , (3) possible (however the normal selection rules for M still ∆W apply when the electrons relax and re-emit photons). µI gI = − . (4) The allowed transitions are shown in Figure (2). Be- Iµ 0 cause the incident photons are circularly polarized, only transitions of ∆M = +1 are possible, and therefore any A plot of the Breit-Rabi equation, using the dimen- electron in the M=2 state will be unable to absorb a pho- sionless axis W/∆W (energy) versus x (magnetic field ton. Any electron in this state will be unable to leave. strength) can be seen in Figure (1). However, the excited electrons do not have this constraint when they re-emit a photon during their relaxation process, which means that transitions into the M=2 state are allowed. It follows that eventually all the electrons C. Photon Absorption will occupy the M=2 state, and the incident light will not be absorbed. In a sense, the sample of natural Ru- As photons are absorbed by the atoms, the electrons bidium gas becomes “invisible” to the incident light! This are excited to higher energy states designated by the en- is process is referred to as optical pumping. ergy level of the particular photon absorbed. When the The optical pumping will break down in certain situa- electrons relax, they re-emit a photon corresponding to tions. First of all, if the Rubidium atoms collide with the the drop in energy they undergo. It is important to un- walls of the container, it is possible to force electrons from derstand that the re-emitted photon does not have to the M=2 state (more on this later). Second, if the mag- be of the same energy of the absorbed photon. Further- netic field applied to the sample is just strong enough to more, because the possible energy levels of the electron cancel out the magnetic field of the Earth, then there will are quantized, and because of the interaction of the quan- be no net magnetic field at the sample and the pumping tum numbers, there are certain selection rules that des- will also break down (if there is no magnetic field then ignate how an atom can absorb and emit a photon. The there is no Zeeman splitting). Because the strength of rules for an atom in a magnetic field are ∆S=0, ∆J=0, the magnetic field applied in this experiment is actually ±1, ∆L=0, ±1 (but not L=0 to L=0), ∆F=0, ±1, and swept over a wide range, this breakdown is observed and ∆M=0, ±1. is referred to as the “zero-field transition.” 3

bidium gas, two perpendicular pairs of Helmholtz coils, a set of RF pulse coils, a Silicon photodiode detector, and several lenses and optical ﬁlters. These components were all arranged along a straight track. Additionally, there was a base unit for controlling the apparatus (the strength of the magnetic ﬁelds, the temperature of the Rubidium lamp, etc.), an oscilloscope for reading the photodetector output, a computed connected to the oscilloscope capable of printing the oscilloscope data, and

Transmitted Light Intensity Light Transmitted a wave function generator. The experimental setup is shown in Figure (4).

Magnetic Field 1. Natural Rubidium

FIG. 3: Observed resonances during the low field The lamp was filled with natural Rubidium gas and optical pumping trials. The large dip to the left a small amount of gaseous Xenon as a buffer. This was represents the zero-field transition, and the two dips to done so that the collisions between the Rubidium atoms the right represent the Rb-87 and Rb-85 Zeeman and the walls of the container were minimized. The gases resonances, respectively. Note that the magnetic field in the lamp were heated and thus excited, and as they strength increases to the right and the transmitted light relaxed photons relating to the relaxation energy were intensity increases upward released. It should be noted that this results in a broad spectrum of emitted light, but this was filtered by the optical setup described later. There is another way in which the optical pumping can The natural Rubidium in the sample also contained a be broken down, and that is by introducing an RF signal buffer gas, which in this case was Neon. Again, the buffer perpendicular to the path of the incident light. By doing gas was used to minimize the collisions with the walls of this at the resonant frequency for the M=2 to M=1 tran- the container. If this were not done, collisions with the sition, it is possible to induce transitions from the M=2 container would result in photon emission, which would state and thus to temporarily break the optical pumping. destroy the optical pumping. This has a few important implications. Because the frequency needed to cause this transition is proportional to the applied magnetic field with relation to the particular 2. Optics atomic structure of the atom in question, it is possible to experimentally determine the quantum numbers of the The optical setup consisted of two plano-convex lenses, atom. And because the atomic structure of Rb-87 and an interference filter, two linear polarizers, and a quarter Rb-85 are different, the energy gap corresponding to the wavelength plate. As the Rubidium lamp was heated, M=2 to M=1 transition is different. It is therefore pos- the gas emitted two main lines at 780nm and 795 nm sible to observe both dips corresponding to each isotope’s (due to the two isotopes found in natural Rubidium). resonant breakdown of the optical pumping, as shown in The interference filter removed the 780nm line, and the Figure (3). remaining light was linearly polarized. Next, the quarter Furthermore, when a large magnetic field is applied to wavelength plate turned the linearly polarized light into the sample, it is possible to break down the pumping even circularly polarized light before it was introduced to the further and elicit transitions between all the split Zeeman natural Rubidium sample in the cell. This was all done states. In this way the full nature of the Zeeman split- to ensure that the electrons were excited by photons of a ting can be observed (however it should be noted that precise energy and angular momentum. The two plano- the transitions that occur from ∆F = ±1 take place at convex lenses were used to focus the light. several GHz and will not be observed in this experiment).

3. Helmholtz Coils III. EXPERIMENT AND APPARATUS There were three main sets of magnetic fields applied A. Apparatus to the Rubidium sample. These were calibrated to counteract the effect of the Earth’s magnetic field along the The TeachSpin optical pumping apparatus was used x, y, and z axis of the sample. The entire apparatus for all experiments performed. The apparatus consisted (i.e. the track with the sample, lamp, etc.) was alined of a Rubidium discharge lamp, a sample of natural Ru- along one of the Earth’s magnetic axis. The vertical 4

FIG. 4: Overhead view of the experimental setup.

Helmholtz coil was adjusted until it produced a mag- B. Procedure netic field just strong enough to counteract the vertical field of the Earth. Both of these processes were done by 1. Low Field Resonance Determination of I making adjustments until the observed zero-field dip was as deep and narrow as possible. It should be noted that First, the value of the residual magnetic field was deter- these adjustments did not have any effect on the numer- mined. Because the orientation of the apparatus and the ical values of the data, but rather they simply made the strength of the vertical field only counteracted two axial dips easier to find and read. components of the Earth’s magnetic field, the zero-field The other fields generated were the main horizontal transition was actually not observed at B= 0. The volt- field and the horizontal sweep field. Both fields were age across the sense resistor of the sweep coil (and thus perpendicular to the vertical field axis and parallel to the current through the sweep coils) was measured as the the track (and thus both the main and sweep fields were field was slowly swept through the zero field transition. coaxial). The sweep field started from a predetermined The strength of the sweep field was roughly determined value and increased linearly for a predetermined range by (the maximum current possible through the coils was −3 ¯ 1A). The time the sweep coil took to complete a full B = 8.991 × 10 IN/R, (5) sweep was also adjustable. Therefore, by controlling the where B is the magnetic field strength in gauss, I is the value of the start field, the range over which the field was current through the coils in Amperes, N is the number of swept, and the time it took to complete a full sweep, it turns of coil on each side (11 for this setup), and R¯ is the was possible to observe many of the characteristics of the mean radius of the coils (which was 0.1639 m). The value optically pumped Rubidium. determined to be the strength of the residual magnetic The main field was simply a more powerful horizontal field was then subtracted from all future magnetic field field coaxial with the sweep field. This was applied in values determined by Equation (5). order to observe the quadratic Zeeman splitting at higher Next, an RF signal was introduced in order to generate fields, which the sweep field was not able to find on its transitions from the M= 2 to M= 1 state for each iso- own. Both horizontal fields had a sense resistor on the tope. Because the two isotopes have different g-factors base unit, across which it was possible to read the voltage (and thus different I values), this resulted in two sepa- difference and thus current through the coils by use of rate observed dips. The RF signal was applied at six V = IR. The sense resistor for the main field was 0.5 Ω, separate frequencies ranging from 40 KHz to 65 KHz; and the voltage in the sweep coils was read across a 1 Ω and for each frequency the current through the sweep resistor. coils was measured at the two observed dips. From this data, plots were created of transition frequency versus the magnetic field strength, and the data was fit to two separate straight lines.

4. RF generator ν = gF µ0B/h (6) was used to determine the Lande g-factor for each iso- The RF generator was applied perpendicular to the tope, where ν is the frequency in MHz, µ0 is the Bohr horizontal and vertical magnetic ﬁelds. It was also con- Magneton, h is Plank’s constant, and B is the magnetic nected to the wave function generator. Therefore it was ﬁeld in gauss. Once the g-factors were determined, possible to apply an oscillating RF signal of a known F (F + 1) + J(J + 1) − I(I + 1) frequency and amplitude. This was done to allow tran- g = g (7) F J 2F (F + 1) sitions from the M= 2 state, which made it possible to observe the transitions between the Zeeman states. was used to determine I, as gJ is known for both isotopes.

Transmitted Light Intensity Light Transmitted Intensity Light Transmitted

Magnetic Field Magnetic Field

(a) The ﬁrst 5 dips (b) The second 5 dips

FIG. 5: Observed resonance at high magnetic fields. The magnetic field strength increases to the right, and the transmitted light intensity increases upward. Note that there are only 6 total dips, and therefore a 4 dip overlap between figures.

2. Quadratic Zeeman eﬀect IV. RESULTS AND ANALYSIS

A. Low Field Determination of I For the next experiment the accepted g-factor values were used, along with the data already taken, to generate The zero field dip was observed to occur at 314 mV, a more precise equation for the magnetic fields with re- which corresponds to a current through the coils of 314 spect to current. The resonance equation (Equation (6) mA and a magnetic field of .189 gauss. This value was from above) was used on all the low field data to create then subtracted from the values of the magnetic field a plot of the magnetic field strength versus the current calculated from Equation (5); the final calculated values through the sweep coils. Next, the main horizontal coils can be found in Table (I). The data in the fourth and fifth were turned on in a direction opposite the sweep coils. column were plotted against the frequency at which they The main field was then increased slightly, and the sweep were observed to form the straight lines described earlier. coil was also increased in the opposite direction so as to From this, the Lande g-factor of Rb-87 was determined center on the zero-field transition. Because the strength to be 0.48, and the g-factor of Rb-85 was determined to of the magnetic sweep field with respect to the current be 0.33. The accepted values are 1/2 and 1/3, which is through the sweep coils was known, the main coils were in good agreement with the data. Equation (7) yielded therefore generating a magnetic field of equal (though op- nuclear spin values of 3/2 for Rb-87 and 5/2 for Rb-85. posite) strength. By repeating this procedure over a few points, it was possible to generate another plot for the main magnetic field strength versus the current through B. RF Spectroscopy the main coils. By doing this, two equations were created (one for the sweep field and one for the main field) that gave a precise measurement of the magnetic field The sweep field was calibrated as described in Sec- strength. tion (III), which resulted in the calibrated equation B = −0.187 + 0.595I. Note that the residual magnetic field from the Earth is taken into account when I = 0. Next, the sweep field was centered on the Rubidium- Likewise, the calibrated equation for the main field was 87 dip, and the main field was turned to zero. By slowly B = 6.829I. increasing both the main field and RF pulse frequency, The quadratic splitting was observed at a frequency it was possible to “follow” the dip out to a high field of 4.064 MHZ, and a main field voltage of 506.8 mV. region. In this region transitions between the 2F+1 sub- This corresponded to a 6.921 gauss magnetic field. The levels were observed as six separate dips. The strength data from the spectroscopy measurements can be found of the combined magnetic fields was determined at each in Table (II). The calculated values are actually not in dip, and the results compared to the Breit-Rabi equation. good agreement with those predicted by the Breit-Rabi The resonant dips can be observed in Figure (5). equation, which were all around 5.8 gauss. 6

TABLE I: Low ﬁeld data used to determine the g-factor for each isotope. There is an uncertainty of ±0.5 mA for each current measurement, which corresponds to a ±3 × 10−4 uncertainty in the magnetic ﬁeld calculations.

Frequency (KHz) Current at Rb-87 Current at Rb-85 Magnetic ﬁeld at Magnetic Field at dip (mA) dip (mA) Rb-87 dip (gauss) Rb-85 dip (gauss) 40 408 455 0.057 0.086 45 422 476 0.066 0.098 50 434 496 0.073 0.110 55 446 511 0.080 0.119 60 459 529 0.088 0.130 65 469 546 0.094 0.140

TABLE II: High field resonant dips. All data taken at a V. CONCLUSIONS main field strength of 6.921 gauss. The Breit-Rabi equation was used for the values in the last column. It appears that there may be something wrong with the values predicted by the Briet-Rabi equation. For in- Sweep Coil Sweep Field Total Field Predicted stance, the fact that there is repetition in four of the Current (A) (gauss) (gauss) Value values is unsettling, as six dips were clearly observed. (gauss) That is not to say that the measurements were perfect, 0.319 0.003 6.918 5.805 as the calibration equations could be wrong. If this ex- 0.358 0.027 6.895 5.813 periment were to be performed again, more data would 0.397 0.050 6.871 5.820 be taken for the calibration measurements to ensure a more accurate line fitting. 0.435 0.072 6.849 5.813 However, it should be noted that the theory behind 0.472 0.094 6.827 5.805 the experiment was shown to be sound. The sample of 0.509 0.116 6.805 5.798 Rubidium was observed to have been Optically Pumped, and the values obtained for the nuclear spin numbers were very close to the accepted values. Furthermore, the dips observed confirm that Zeeman splitting of the nuclear energy states did in fact take place, and that the optical pumping was disrupted by the applied RF signal. The process of optical pumping remains a valuable tool for any Physicist. Quantum mechanical theories were explored and confirmed throughout the experiment.

[1] Melissinos, A, & Napolitano, J. (2003). Experiments in [3] Wolﬀ-Reichert, B. (2009). Conceptual tour of op- modern physics. San Diego: Academic press. tical pumping. TeachSpin Newsletters, Retrieved [2] Optical pumping of Rubidium: Instructor’s manual. Buf- from http://teachspin.com/newsletters/OP% falo: TeachSpin 20ConceptualIntroduction.pdf.