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Hyperfine Structure of Rubidium Phys 1560 – Brown University – August 2010Purpose: 87 To measure hyperﬁne splitting of atomic energy levels in Rb. To become familiar with optical techniques To become familiar with Michelson interferometry and measurement calibrationSafetyThe EOS laser which you will use in this experiment is a class IIIb laser with output inthe mW range; the laser won’t hurt your skin, but it could destroy your vision. Opticaltables are kept at waist level, with the idea that if all optical beams are kept in thehorizontal plane, then the probability of a laser beam inadvertently burning your eye isdramatically reduced. When doing this experiment do not bend down to ―see better‖because if you do, you may be permanently blinded. To prevent inadvertent reﬂectionof the laser beam into your eye, remove all watches, jewelry, and reflective clothing.According to 2007 ANSI standard Z13, for the range of wave lengths that include 780 nm(the approximate wave length used in this experiment) no laser safety goggles are requiredbelow 5 mW. However, since this wavelength is invisible and will not invoke the blinkreflex, ANSI requires the no goggles threshold to be at a power level lower than that requiredto damage the eye in 10 sec. - 0.5 mW (~76 mA at 780 nm). Most of the setup for thisexperiment can be done below 0.5 mW. But, to actually perform this experiment the powerwill need to be ~87 mA (~3.6 mW). If the laser power is equal to or above 0.5mW (~76mA)you must wear the appropriate laser safety goggles.Doppler-Free SpectroscopyIn this experiment you will use a technique known as saturation-absorption spectroscopyto study the hyperﬁne structure (hfs) of rubidium. This particular method is designed toovercome the limitations imposed by the Doppler-broadening of spectral lines whileavoiding the need to work at low temperatures.TheoryIn its ground state rubidium (Rb) has a single 5s electron outside a closed shell; theprincipal structure of the excited states is, therefore, similar to that of hydrogen. The ﬁrstexcited state is split in two by the spin-orbit (s-o) interaction, and each of these levels issplit further by the hyperﬁne interaction. Neglecting relativistic and reduced-masseﬀects, the Hamiltonian consists of four terms: 1
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[ ]Here J , L and S have their usual meanings, and I is the operator for nuclear magneticmoment. The Coulomb potential is assumed to be spherically symmetric, and screening −3eﬀects are accounted for via . The function f(r) is roughly proportional to r (spin-orbit coupling is discussed in Griffiths’ Introduction to Quantum Mechanics Ch 6.3.2,and the hyperfine splitting is discussed in Ch 6.5).The hyperﬁne interaction is the sum of two terms, multiplied respectively by α and β.The ﬁrst term is analogous to the spin-orbit term in that it results from considering themagnetic dipole interaction −µn · Be. In this case, however, µn refers to the nuclearmagnetic moment (isospin). Such spin-spin coupling is the dominant contribution to thehyperﬁne splitting (hfs). The constant α is called the magnetic hfs constant, and has unitsof energy. The second term is more complex, arising from the quadrupole moment of thenucleus. Not surprisingly, β is called the quadrupole hfs constant. Rubidium has nopermanent dipole moment. 87 85Rubidium has two naturally occurring isotopes, Rb (28% abundant, I =3/2) and Rb 2 2(72% abundant, I = 5/2). We will study the structure associated with the 5 P3/2 to 5 S1/2 87transitions in Rb, as shown in Figure 1. The total angular momentum of a state isdesignated by F, where F = J + I. From this you can easily verify that each of theselevels is split as shown. However, due to electric dipole selection rules, only transitionswith ΔS =0, ΔJ =0, ±1 are allowed (Griffiths Ch 9.3.3). From this we conclude that hfstransitions must have ΔF = 0, ±1.The frequency spacings between adjacent levels as shown in Figure 1 can be calculated 2 2 2by rewriting the dot product J · I = (F − J − I )/2. In an eigenstate of these operatorswe have, for example, | 〉 | 〉where k denotes the collective index. Thus we see that in such a case [ ] 2
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87Figure 1. Energy level diagram for Rb. The frequency spacing between adjacent levelsis shown for the hyperﬁne splitting.Then, from | 〉 | 〉 we ﬁndwhich gives the perturbed energy—the size of the deviation from the unperturbed state.This is often written in terms of the frequency:The accepted values of the coeﬃcients A and B are tabulated below in Table 1. From theabove equation, you should be able to convince yourself that the quadrupole term must 2vanish for 5 S1/2. Units are MHz 85Rb 87Rb A for 52S1/2 1011.91 3417.34 A for 52P3/2 25.01 84.85 B for 52S1/2 — — B for 52P3/2 25.90 12.52 Table 1 3
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Exercise: use the above equation to verify the frequency spacing for several hyperfinelevels. If you take the time to do the same for the more abundant 85Rb, you will see whywe focus our attention on the less abundant isotope.The Doppler Broadening of Absorption SpectraIf we direct a laser at a collection of Rb atoms, photons cause transitions between stateswhenever the energy is an exact match; this absorption of energy can be measured toobserve the energy transitions. At zero temperature, each energy level has a slightfuzziness given by the uncertainty principle where is the lifetime of thestate. Therefore, an atom can absorb light over a small range of frequencies, called thenatural line width of the transition. However, atoms at room temperature are in constant,random thermal motion. This means that a substantial fraction of the atoms will seeeither red-shifted or blue-shifted photons, resulting in a further broadening of the linewidth. If we consider only motion along the axis deﬁned by the beam of light, then theDoppler-shifted frequency is given byWhere 0 refers to the resonance frequency when the atom is at rest, to the lightfrequency, and V. It is possible to estimate υ from the equipartition theorem, but a moreuseful expression can be obtained by considering the Maxwell velocity distribution ofparticles in a classical ideal gas: √If we re-write the Doppler shift as ⁄and substitute this expression into the Maxwell distribution, we ﬁnd the probability thatan atom will see photons of a given frequency is √The expression is a Gaussian, given in terms of the line width parameter √ . Itis conventional to specify the width of a given transition by the full-width at halfmaximum (FWHM), which is given by √ 4
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At 300K, the 780 nm transition ( , , and . Compare this to the natural linewidth of the transition, . The Doppler broadening of the frequencies seen by the Rb at roomtemperature is two orders of magnitude larger than the natural linewidths, meaning thehyperfine structure associated with the 780 nm transition is lost in the Doppler spread.Clearly a way around this must be found in order to observe the hyperfine transitions.Saturated Absorption SpectroscopySaturated absorption spectroscopy is the name given to a very clever trick whichovercomes the limitations imposed by Doppler-broadening. The mechanism by which itworks depends critically on the rate of optical pumping and the lifetimes of the hyperﬁnestates, and is quite involved. With this in mind we will proceed from a more empiricalpoint of view.Figure 2 shows the optical apparatus. When an on-resonance laser beam passes throughthe sample, power will be absorbed by atoms in the sample and a detector picking up thebeam on the other side would register a Doppler-broadened absorption spectrum asdiscussed above. Saturation absorption spectroscopy bypasses the Doppler broadeningthrough the use of two separate beams. The higher-intensity pump beam saturates thehigher state, so as the probe beam passes through in the opposite direction, there is adecreased number of atoms in the zero-velocity ground state to absorb its photons. Thedecreased power loss resulting from reduced absorption of photons from the probe beamby atoms in the sample is measured at the detector. The beauty of this method is that because the beams travel in opposite directions, onlyatoms which are at rest relative to both beams are in resonance with both. Thus theabsorption signal detected by the probe should exhibit a small, narrow spike at thelocation of the ―zero temperature‖ line, and its width should approach the natural linewidth.Cross-Over ResonancesYou will also need to look out for ―cross-over‖ resonances, a phenomenon peculiar tosaturation absorption spectroscopy. These resonances only occur when two (or more)different transition frequencies are within the Doppler-broadened frequency distributionof the laser. Under these conditions, at a frequency halfway between two transitions,there will be some nonzero velocity with respect to the propagation of the laser for whichthe Doppler shift is just right to induce a transition. Atoms moving with this velocity willsaturate one of the two excited states—let’s say state A.However, in this scenario the other transition—to state B—is excited by the exactopposite Doppler shift. In other words, the transition to state B will be stimulated foratoms moving with the same speed but in the opposite direction with respect to thedirection of propagation of the beam. But the probe beam enters the sample in theopposite direction, so it will be these same atoms just excited by the transition to state Awhich will now see the correct Doppler shift in photons from the probe beam to transition 5
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to state B.Because of the depleted number of atoms in the ground state due to the saturation of stateA by the pump beam, there will be a decrease in absorption of photons from the probebeam. This is observed as an increase in the power reaching the detector (andcorresponding voltage spike on the oscilloscope,) just like in the case where the laser isprecisely on the transition frequency—a sort of false positive. Cross-over resonances areoften even stronger than the desired ―zero temperature‖ resonances! These cross-overresonances, however, allow for another method of measuring the ground state energy.Other NotesThere are other factors which inﬂuence line width. These interactions are given thecollective title of ―pressure broadening‖, and are represented by the expression Vn = Cn/rnwhere n = 1 and n = 2 are associated with the Stark eﬀect due to interactions withhydrogen or helium (these probably have a vanishingly small contribution in ourexperiment). Most relevant are n = 3 (dipole interactions) and n = 6 (van der Waalsinteractions). It is not expected that you will reach any comprehensive conclusions aboutthese eﬀects, though you should be able to estimate their magnitude.Now that you have considered the exercise and the questions, you should be convincedthat the F =3, 2, 1 ↔ 2 transitions should be resolvable under the conditions of thisexperiment. A central goal is to measure the frequency spacing of these levels. Note thatthe F = 2, 1, 0 ↔ 1 transitions are also resolvable, but the optical pumping (the intensityof the pump beam relative to the intensity of the probe) reduces them to marginal levels.Equipment EOS 2010 External Cavity Diode Laser with power supply IR viewing scope and IR viewing card Oscilloscope Function Generator 3 mirrors, one regular beamsplitter, 1 microscope slide beamsplitter, and 2 photodetectors assorted posts, clamps and magnetic bases assorted coaxial cablesTunable LaserThe wavelength of the EOS is tuneable thanks to a voltage-controlled piezo crystal andan external cavity feedback system, which is explained in the EOS manual. You shouldperuse this, especially pps. 2–4 (introduction and safety), pps. 13–19 (operation) and pps.27–30 (theory of external cavity operation). However, the only adjustment you shouldhave to make on the laser is the position of the tuning mirror, via the micrometer knob onthe rear of the box. Do not open the lid.Also keep in mind that it has an extensive warm-up period for optimal stability, so it isbest to leave the power supply on and the laser in sleep mode. Since only low voltage,low frequency modulations are used in this experiment, to protect the laser, keep the scan 6
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frequency below 25 Hz and the peak-to-peak voltage below 14 V. A ﬁnal point to keepin mind is that you do not want to send reflected beams back into the laser cavity, as thiswill make the frequency unstable and generate a lot of noise. Function Generator Oscilloscope Laser Detectors Rb Vapor Cell Probe Beams Beamsplitters M2 Pump Beam M1 Figure 7 Figure 2. Experimental setup. Note that one of the two probe beams crosses the pump beam inside the Rb cell, and the other does not.ProcedureFigure 2 shows the setup you will use. Notice that in this setup, there are two probebeams. If you do not allow the pump beam to interact with the second probe beam (i.e. ifthey don’t cross in the cell,) you can subtract the two signals to remove the Doppler-broadened background. This allows for direct observation and measurement of thehyperfine structure. A very comprehensive treatment of this technique is given in thereferences.Hyperfine Structure 1. Be safe. Put on your goggles, take off your watch, etc. You know the drill. 2. First you will have to tune the laser to find the ~780 nm transition frequency. Place the rubidium cell in front of the laser, turn off the lights, and use the IR goggles to observe the cell. Put the current from the laser’s power supply around 87 mA, then use the micrometer knob to slowly adjust the frequency of the laser until you hit resonance. You will know it when you see it. What changes as you turn the micrometer knob? 3. Next you will need to choose appropriate parameters for the piezo ramp which will modulate the laser frequency. Keep the Rb cell in front of the laser, and put a detector on the other side to catch the beam so you can observe the effects of the ramp. The maximum modulation input voltages at various frequencies are given on page 26 of the EOS laser manual, but these values are much higher than you will need. You should observe reasonably good results for frequencies between 3 and 15 Hz. Watch the signal and ramp on the oscilloscope, and make sure the amplitude of the ramp is sufficient to see four separate peaks on the scope (the fourth is very small.) If you see strange jumps in the signal, you may have to 7
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raise the intensity of the beam by increasing the input current. Also, use the micrometer knob on the laser to center the observed structure around the middle of the ramp (V=0.) 4. Set up the apparatus in figure 2. Be very careful to make sure that one of the probe beams crosses the pump beam inside the rubidium cell, and that the other does not. Also make sure that the lasers do not interact with the walls of the cell other than the flat ones at the ends where they enter and leave, and that no reflected beams reenter the laser cavity. 5. Once you have properly aligned the optics, you should be able to see the hyperfine structure on the oscilloscope using the Math functions. Record the data.CalibrationYou will need to calibrate the frequency shift of the laser as a function of applied voltage.You will do this by constructing a Michelson interferometer as shown if Figure 3. Afterthe beams recombine, they will have a phase differenceSince the laser wavelength is being swept continuously, we actually have Since the intensity is a maximum whenever the phase diﬀerence is an even multipleof 2π, maxima will be observed at regular frequency intervals given byWhen you observe the interference of the beams in the Michelson setup as you apply thepiezo ramp, since the peak separation will occur at a constant frequency interval given bythe above equation, you can use the average voltage difference in the ramp between peaklocations to calibrate the frequency shift of the laser to the voltage applied by the functiongenerator. 8
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Figure 3. Typical interferometer conﬁguration.You will now set up a Michelson interferometer to calibrate the amount the frequencyof the laser changes in response to voltages applied by the piezo ramp. Aligning theinterferometer can be tricky; the beams will be weak, so you will need to use the IRviewer and work in near-darkness. It is essential to realize that the recombined beamsmust be nearly parallel at the detector in order to observe interference. The followingsteps should facilitate the process.1. First, make sure all of your optical components are at approximately the same height off of the table. If they aren’t all in the same horizontal plane, it will be infinitely more difficult to get the beams to recombine parallel.2. Set up the optical apparatus depicted in figure 3—a path length diﬀerence of ~30 cm gives a nice level of sensitivity. Without the beam expander or the photodetector in place, align the endpoints of the two beams on the wall behind where the detector will be. Put the beam expander in to place, and make sure the beam enters the expander lens near its center (and perpendicular to its surface) so it is not diverted out of the plane of the optics.3. The beams should be roughly parallel as they approach the detector, and must overlap at the detector. This is the most difficult part, and it may take some time to get right. The following technique may be helpful: a. Close the adjustable aperture near the laser, then realign the dots on the wall. b. Hold a sheet of paper where the detector will be. If the dots are not aligned on the sheet of paper as well as on the wall, the beams are not parallel. (As third reference point, the beams should also hit the beam splitter at the same point, both before and after they hit mirrors 2 and 3.) 9
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Bring them as close together as possible using the knobs on mirror 1. c. Remove the sheet of paper and repeat the process, alternating between the wall and the paper. When the dots are roughly aligned on both the wall and the paper, then the beams are adequately parallel4. Open the adjustable aperture, and put the piezo ramp at ~0.1 Hz. Using the infrared goggles, you should be able to see an oscillating bulls-eye interference pattern on the wall. If you can’t, fiddle with mirrors 2 and 3 a little bit. Once you have produced the bulls-eye, place the detector in the setup and arrange it such that the bulls-eye is centered on the sensitive part of the detector. Mirrors 2 and 3 can be used for small adjustments of the location of the bulls-eye, especially vertical placement.5. Once you have accomplished this, set the ramp to the frequency used when observing the hfs and save the data; see figure 5 for a guide to what, ideally, you should see. Figure 5. Typical interferometer output. Note the behavior of the signal near the discontinuities of the ramp. 10
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