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Polarization and two-photon spectroscopy of xenon for optical magnetometry Miller, Eric Robert 2018

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Polarization and Two-Photon Spectroscopy of Xenon for OpticalMagnetometrybyEric Robert MillerB. Science, University of Northern British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Chemistry)The University Of British Columbia(Vancouver)October 2018c© Eric Robert Miller, 2018The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Polarization and Two-Photon Spectroscopy of Xenon for Optical Magnetometrysubmitted by Eric Miller in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin ChemistryExamining Committee:Takamasa Momose, ChemistrySupervisorEd Grant, ChemistrySupervisory Committee MemberAndrew MacFarlane, ChemistryUniversity ExaminerFei Zhou, PhysicsUniversity ExaminerAdditional Supervisory Committee Members:Akira Konaka, ChemistrySupervisory Committee MemberAlex Wang, ChemistrySupervisory Committee MemberiiAbstractThis dissertation presents work in the hyperpolarization of 129Xe and in precision Xe spectroscopyusing two-photon absorption. Both projects contribute to the development of optical magnetometryusing 129Xe. Motivating this work is the proposal for a new 129Xe-based comagnetometer at TRI-UMF for experiments searching for a permanent electric dipole moment of the neutron. In theseproposed experiments 129Xe will occupy the same experimental volume as ultracold neutrons andbe used to measure drifts in an applied magnetic field. A scheme is described for optical magnetom-etry which involves the production of polarized 129Xe followed by measurements using two-photonlaser induced fluorescence as a probe.Spin exchange optical pumping is used to produce polarized 129Xe, which is a necessary pre-cursor for optical magnetometry. The first part of this dissertation presents the implementationand operation of a polarizer using a diode laser and a Xe-Rb-N2-He mixture. We have achievedhyperpolarization of 129Xe up to PXe ≈ 5%, which is many times greater than the thermal equilib-rium polarization. We measure the nuclear magnetic resonance signal from polarized 129Xe usinga low-field detection apparatus, and compare the signal with predictions based on a rate equationmodel. Efforts to optimize the degree of polarization in the present apparatus are described, as wellas purification of Xe gas through freezeout.The second part of this dissertation presents precision spectroscopy on natural abundance Xe us-ing a narrow linewidth laser. The two-photon transition studied here ( 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0))is suitable to probe the ground state 129Xe polarization for optical magnetometry. We present theimplementation of a CW laser source with narrow linewidth followed by the excitation of Xe two-photon absorption in a resonant cavity. We measure and report hyperfine constants and isotopeshifts from the observed laser induced fluorescence spectra. From the observed signal to noise ratiowe estimate a magnetometric sensitivity based on this detection scheme over a range of Xe pres-sures. Results from this two-photon absorption measurement are essential in the determination ofparameters for final implementation in the nEDM experiment.iiiLay SummaryThe effects of magnetism can change the way in which certain atoms absorb light. Some of themost precise magnetic sensors, used in both medical sciences and fundamental physics inquiries,are based on measuring these effects. Typically the atoms need to be prepared by aligning eachatom’s internal magnetic field to point in the same direction, producing a polarized state, before theeffects of an external magnetic field can be detected. Only a few elements are suitable for use insensors.This dissertation studies the methods of producing polarized states in xenon atoms. It alsostudies the interaction between xenon atoms and light through a sensitive process called two-photonabsorption, which only occurs when using powerful ultraviolet lasers. This work suggests a newtechnique for detecting the effects of magnetism on xenon atoms based on this process. The impactof this work is to enable the development of xenon-based precision magnetic sensors for specialtyapplications, such as measuring the magnetic field in a shared volume with ultracold neutrons, wheresensors based on other atoms are not suitable.ivPrefaceThis dissertation is based on the construction and operation of apparatus for the polarization andspectroscopy of 129Xe, which was performed at UBC under the supervision of Dr. Takamasa Mo-mose, with additional guidance from Dr. David Jones and Dr. Kirk Madison. The work performedwas a joint effort in support of the international ultracold neutron collaboration at TRIUMF. Back-ground information presented in Chapters 1 and 2 is the work of the collaboration ([1, 3, 126]),but none of the text of this dissertation is taken directly from previous work. The discussion ofprevious nEDM experiments at ILL and the 199Hg comagnetometer are the work of the ILL collab-oration, and figures are used with permission. The proposal to use two-photon excitation of 129Xecomes from T. Chupp and A. Leanhardt [4]. The dual comagnetometer proposal and derivation ofSection 2.1.4 are the unpublished work of Dr. C.A. Miller and Dr. Momose.The 129Xe polarizer presented in Chapter 3 was implemented by me with contributions andadvice from Dr. Jeff Martin, Dr. Chris Bidinosti, and Mike Lang at U. Winnipeg and Dr. Jeff Sonierat Simon Fraser U. The Rb-filled glass cell used for spin exchange optical pumping (SEOP), andspecifications for the gas mixture, were provided by Mike Lang. The NMR detector was providedby Dr. Sonier, and modified by me. Tomo Hayamizu did the COMSOL simulation which I used tocalculate the field gradient induced relaxation. Dr. Jeff Martin provided the Python Bloch equationssimulation used to optimize conditions for adiabatic fast passage (AFP). The results of the Blochsimulation and the rate equation model are my own work. Experimental work was mainly performedby myself with the help of two undergraduate students: Bill Wong and Aaron Ngai. Data collectionand some of the data analysis on temperature dependence, batch mode polarization, and relaxationwere performed by Aaron Ngai under my direction. Tomo Hayamizu installed SEOP cell heatersfor optimizing heating conditions. The remaining experimental work and analysis was performedby me.Chapter 4 presents two-photon spectroscopy with pulsed and continuous laser sources. Thepulsed laser setup was implemented by Dr. Momose. I configured and installed the detection opticsand implemented the data acquisition in LabView, and performed most of the data acquisition. Someof the data collection for polarization dependence was performed by Bill Wong under my direction.The analysis is my own work. The narrow-linewidth continuous laser was developed in Dr. Jonesvlab by Emily Altiere and Joshua Wienands, who assembled the doubling stages and optimized modematching and cavity locks. I helped locate and eliminate electronic noise sources to improve lockstability. Tomo Hayamizu offered suggestions based on parallel development of a laser for Hgspectroscopy. The doubling cavities and vacuum box were machined by Technical Services in thePhysics and Chemistry Departments. I designed and assembled components for the vacuum box.The frequency comb used to obtain a beatnote reference was provided by Dr. Kirk Madison andDr. Jones. Data collection for the high-resolution spectroscopy was performed by Emily Altiereand I. The LabView code used to frequency-sweep the laser and obtain the beatnote reference alongwith data acquisition is my work. The Matlab code used to stitch together frequency and signalmeasurements to calibrate the frequency axis was written by Emily Altiere and I. The multipeakimplementation of the ODR fitting routine in Python and corresponding Lorentzian fit analysiswas performed by me. The isotope shift analysis, Zeemax simulation of detector acceptance, andmagnetometer sensitivity estimates are all my own work. The work in Chapter 4 has led to apublication “High-resolution two-photon spectroscopy of a 5p56p← 5p6 transition of xenon” inPhysical Review A (2017) [8].viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Ultracold Neutrons and the Neutron Electric Dipole Moment measurement . . . . . 21.1.1 The search for Charge Parity (CP) violation . . . . . . . . . . . . . . . . . 21.1.2 Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Neutron Electric Dipole Moment measurement in UCN . . . . . . . . . . . 41.2 Motivation for developing a Xe comagnetometer . . . . . . . . . . . . . . . . . . 71.2.1 Advantages of using 129Xe . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Techniques planned for the Xe comagnetometer . . . . . . . . . . . . . . . . . . . 81.4 Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Literature and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Comagnetometry and the Neutron Electric Dipole Moment experiment . . . . . . . 112.1.1 Ramsey Resonance technique for nEDM measurements . . . . . . . . . . 112.1.2 Need for comagnetometry and principle of operation . . . . . . . . . . . . 132.1.3 Limitations of present comagnetometry and residual effects . . . . . . . . 16vii2.1.4 Proposal to implement a dual magnetometer . . . . . . . . . . . . . . . . . 182.2 Optical Pumping and SEOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Optical Pumping Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Spin Exchange Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 SEOP Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.4 Spin relaxation mechanisms for polarized Xe . . . . . . . . . . . . . . . . 252.2.5 NMR techniques used to measure optical pumping . . . . . . . . . . . . . 262.3 Two Photon Transitions and Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Derivation of two photon transition probability . . . . . . . . . . . . . . . 302.3.2 Doppler-Free Two Photon Spectroscopy . . . . . . . . . . . . . . . . . . . 332.3.3 Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.1 Angular momentum coupling schemes in Xe . . . . . . . . . . . . . . . . 412.4.2 History of Xenon Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 412.4.3 Detection scheme for polarized 129Xe . . . . . . . . . . . . . . . . . . . . 423 Production of polarized 129Xe by Spin Exchange Optical Pumping . . . . . . . . . . 443.1 Experimental Apparatus and Technique . . . . . . . . . . . . . . . . . . . . . . . 443.1.1 Spin Exchange Optical Pumping Apparatus . . . . . . . . . . . . . . . . . 443.1.2 NMR Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Rate equation model for SEOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.1 Determining and Optimizing Adiabatic limits . . . . . . . . . . . . . . . . 583.3.2 Estimation of the degree of 129Xe polarization . . . . . . . . . . . . . . . . 613.3.3 Improvement of Rb absorption by pressure broadening . . . . . . . . . . . 633.3.4 SEOP cell temperature dependence . . . . . . . . . . . . . . . . . . . . . 643.3.5 Continuous vs. stopped flow effects on 129Xe polarization . . . . . . . . . 663.3.6 Measurements of polarization relaxation time . . . . . . . . . . . . . . . . 693.3.7 Xe purification by freezeout and initial attempt at polarization recovery . . 723.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Two-Photon Spectroscopy of Xe with Pulsed UV laser . . . . . . . . . . . . . . . 754.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.2 Observation of Laser-Induced Fluorescence . . . . . . . . . . . . . . . . . 764.1.3 Efforts to reduce bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Two-Photon Spectroscopy with Narrow Linewidth CW Laser . . . . . . . . . . . . 824.2.1 CW Narrow Laser Development . . . . . . . . . . . . . . . . . . . . . . . 82viii4.2.2 Vacuum chamber for two photon excitation . . . . . . . . . . . . . . . . . 864.2.3 Experimental Setup for two-photon detection . . . . . . . . . . . . . . . . 884.2.4 Results: Detection, Hyperfine splitting, and Isotope shifts . . . . . . . . . 884.2.5 Pressure broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2.6 Hyperfine constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2.7 Isotope shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.8 Comparison of signal amplitude and natural abundance . . . . . . . . . . . 1004.2.9 Determination of two-photon transition probability from LIF signal . . . . 1024.3 Estimate of Magnetometer Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.1 Mercury Comagnetometer uncertainty . . . . . . . . . . . . . . . . . . . . 1064.3.2 Sensitivity estimate based on Xe two photon SNR . . . . . . . . . . . . . . 1064.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2.1 Improvement of 129Xe Polarization and Freezeout . . . . . . . . . . . . . . 1105.2.2 Towards optical polarization detection at low pressure . . . . . . . . . . . 1115.2.3 Determination of absolute two-photon transition frequencies . . . . . . . . 1115.2.4 Measurement of Xenon Electric Dipole Moment . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.1 Magnetometry Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.1.1 Optically Pumped Atomic Magnetometers . . . . . . . . . . . . . . . . . . 129A.2 NMR Q-Factor Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.3 Frequency comb method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133ixList of TablesTable 1.1 Behaviour of measurables under the operators C,P,T. . . . . . . . . . . . . . . . 2Table 1.2 Experimental results of EDM searches in neutrons, and in atomic and molecularsystems (adapted and updated from [47]). Each result is represented either as thetotal atomic or molecular EDM, or is interpreted as an EDM limit on the electronde. The neutron EDM is dn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Table 2.1 Nuclear magnetic moment, g-factor, and gyromagnetic ratios of relevant iso-topes. Magnetic moments reported are from [155] . . . . . . . . . . . . . . . . 21Table 2.2 Common notations for two photon cross section in the literature. . . . . . . . . 37Table 2.3 Natural abundances and nuclear spin I of the stable xenon isotopes. . . . . . . . 40Table 3.1 Typical SEOP parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 3.2 Coil Parameters for NMR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 3.3 Typical NMR parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Table 3.4 Estimate of optical pumping and relaxation rates inside the SEOP cell. Rateconstants ki come from Reference [63]. ρi represents the (per-atom) rate of spinpolarization production or loss for the relevant species. d[X]/dt represents thesame overall rate including number density, i.e. [X] = [Rb] or [Xe]. Each rate’scontribution to the overall spin polarization production or loss is evaluated as apercent in the final column. The rates in each section sum to 100%. . . . . . . . 53Table 3.5 Parameters used for the estimation of xenon polarization based on observedNMR signal. All 129Xe signals were obtained at ωXe/2pi = 15.555kHz reso-nance frequency. Relative measurements were made against H2O signals ob-tained at either 15.555 kHz or 58.82 kHz, and denoted as “fixed frequency” or“fixed field”, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 4.1 Pulsed laser excitation results. The initial laser used had a laser bandwidth over30 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78xTable 4.2 Comparison of absorption lineshape fitting using either a Voigt profile or a Lorentzianprofile. Peak widths are in MHz. Also shown is the difference in center fre-quency fitting parameter between the two profiles in MHz. . . . . . . . . . . . 90Table 4.3 Hyperfine splitting constants (in MHz) for the 5p5(2P3/2)6p 2[3/2]2 excited stateof 129Xe and 131Xe. Values obtained by previous works are listed in the lastcolumn for comparison. Values in parentheses are the 1σ standard deviation ofthe last digit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Table 4.4 Isotope shifts δνi,136 = ν136−νi of the transition 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0).We follow the sign convention for isotope shift outlined in [14]. Shifts for the oddisotopes were determined using the center of gravity from the hyperfine splitting. 93Table 4.5 Nuclear charge radii values δ 〈r2〉i,136 = 〈r2〉136− 〈r2〉i relative to 136Xe, usedin the calculation of absolute K and F . Some of the data has been rearrangedfrom ladder-type pairs which entails some propagation of error. In the actual fits,our data was rearranged to match the published format, to avoid propagation oferror. Also shown are our calculations for K and F for our transition based onthe respective source. [14],[29],[67] . . . . . . . . . . . . . . . . . . . . . . . 98Table 4.6 Observed peak heights and natural abundance for each isotope relative to 132Xe. 100Table 4.7 Experimental values used in the determination of two photon α for the isotope132Xe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Table 4.8 Extrapolation of magnetometer uncertainty at low Xe pressure. Calculated usingEquation 4.21 with T ′ = 150 s, n = 1500, τ=160 s. We consider the followingcases: (i) Current experimental conditions, with 1.6 Torr of mixed 50-50 Xe(nat. abund.) and O2, (ii) 1.6 Torr isopure 129Xe. (iii-vi) 10 mTorr isopure Xeat current laser power (iii) and maximum achievable to date (iv), 1mTorr isopureXe at current laser power (v) and maximum achievable to date(vi) . . . . . . . . 107Table A.1 Q-factor measurements for tuning boxes “Xe” (resonance 15.555kHz) and “Wa-ter” (resonance 58.82 kHz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133xiList of FiguresFigure 1.1 Beamline and superfluid He-II cryostat for producing UCN by spallation of atungsten target. [126] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.2 Proposed configuration for UCN guides and neutron EDM experimental cell.[126] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.3 Scheme for 129Xe-based comagnetometry: polarization of 129Xe, (blue) via spinexchange optical pumping with polarized Rb (green), followed by freezeout,precession, and detection using a two-photon transition. . . . . . . . . . . . . 8Figure 2.1 The Ramsey Resonance technique used for high precision measurements ofneutron precession frequency. See text for description. ~B0 is vertical in thefigure while ~B1 is horizontal. During the free precession time, neutrons com-plete hundreds of precession periods and may acquire a phase shift from the(gated off) stable oscillator ω1 (red). (Reprinted from [16]pg.187 with permis-sion from Elsevier) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Ramsey Resonance fringes used to precisely measure the precession frequency.(Reprinted from [16]pg.187 with permission from Elsevier) . . . . . . . . . . 14Figure 2.3 Comagnetometer performance in the ILL experiment. Magnetic field drifts (a)on the order of 70 ppm cause a corresponding shift in neutron precession fre-quency (b), which is corrected for by simultaneous monitoring of the precessionfrequency of 199Hg. (Reprinted from [16]pg.193 with permission from Elsevier) 15Figure 2.4 Zeeman splitting of the 129Xe ground state due to nuclear spin. . . . . . . . . . 21Figure 2.5 Zeeman splitting of the 129Xe 6p[3/2]2 hyperfine components F = 3/2 (solid)and F = 5/2 (dotted), respective to their line zero-field energies. The hyperfinesplitting (not pictured) between F = 3/2 and F = 5/2 is 2 GHz, with F = 3/2having higher energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.6 Optical pumping in a spin-1/2 system. . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.7 Applied field and magnetization response during the AFP technique, in the sim-plified case with no relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . 28xiiFigure 2.8 Applied field and magnetization response during the FID technique, in the sim-plified case with no relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.9 General representation of a two-photon transition (in this case, absorption) be-tween two states |i〉 and | f 〉. Excitation is by two photons ω and ω ′, which sat-isfy the conservation of energy h¯(ω+ω ′) = h¯ω0. The transition occurs througha virtual state (dotted line), given by a sum over off-resonant intermediate states|k〉 which are dipole-allowed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.10 Example plot of the combined lineshape for Doppler-free excitation (blue) com-pared with the Doppler-free background (red) as a function of detuning. The on-resonant lineshape value can be many times larger than the Doppler broadenedvalue for a sufficiently small homogeneous linewidth. The Doppler-broadenedwidth is typically wider than shown, but is exaggerated here to make the am-plitude visible. The area of the Lorenztian Doppler-free profile is twice that ofthe retroreflected Doppler-broadened profile, and four times that of a Doppler-broadened profile for absorption of a travelling wave. . . . . . . . . . . . . . 36Figure 2.11 Excited state energy levels of Xe I relative to the ground state energy, shownwith their respective electron configurations. Here a prime on the configura-tion indicates the state with a core term symbol 2P1/2, and the unprimed stateindicates the 2P3/2 core. Spin-orbit coupling is determined by the jl or Racahcoupling scheme. The dotted line indicates the ionization threshold for the 2P3/2core. Created using the values for energy levels cited in [147] . . . . . . . . . 39Figure 2.12 Allowed two-photon transitions to the Xe 6p states 2[1/2]0, 2[3/2]2, and 2[5/2]2.The transitions occur between states of the same parity (here both ground andexcited state are even parity). . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.13 Detection scheme for polarized 129Xe using two photon transitions. . . . . . . 43Figure 3.1 Gas flow and optics of the SEOP polarizer. BS: beam splitter, PBS: polarizingbeam splitter, PM: power meter, λ /4: quarter wave plate. . . . . . . . . . . . . 45Figure 3.2 SEOP cell with heater tape and solenoid removed, showing the Rb reservoir(left) and optical pumping volume (center). . . . . . . . . . . . . . . . . . . . 45Figure 3.3 Beam recombination optics “upgrade” used to circular polarize and overlap bothlinear polarized outputs (the primary p-pol and secondary s-pol) from the PBS.BS: beam splitter, PBS: polarizing beam splitter, PM: power meter, λ /4: quarterwave plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.4 SEOP cell during operation. Scattered D1 pumping light appears in false-coloras purple on the camera’s sensor. . . . . . . . . . . . . . . . . . . . . . . . . . 47xiiiFigure 3.5 Optical pumping of Rb is observed when an applied B field breaks the degener-acy of Zeeman levels. The level probed by circular polarized light only absorbswith the field OFF; with the field ON, atoms are rapidly pumped into the darkstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.6 Coils and detection electronics for the free-standing NMR spectrometer . . . . 48Figure 3.7 Worm gear used for coarse adjustment to minimize cross-talk between NMRcoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.8 Bucking coil (right) used to eliminate residual cross talk in the RLC circuittuning box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.9 Longitudinal (a) and transverse (b) field components of the SEOP solenoid mag-netic field (in units of 10−3 T), modelled in COMSOL for the calculation ofgradient relaxation ρgrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.10 Simulation results for dependence of xenon polarization on Rb number density,using a resonant laser power of 1.2 W (red) and 2.4 W (blue). The range ofnumber densities shown corresponds to Rb saturation vapour pressure for thetemperature range 300-423 K. . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.11 A typical water AFP signal during continuous ramping. The change in peakdirection is due to rapid relaxation between ramps, which creates a signal out ofphase with the Lock-in reference. (Conditions: B0 coil current = 4.63 A, ramp= 400 mV, 200 mHz.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.12 A typical xenon AFP signal during continuous ramping. (Conditions: B0 coilcurrent = 4.43 A, ramp = 250 mV, 100 mHz.) . . . . . . . . . . . . . . . . . . 59Figure 3.13 Bloch equation simulation shows H2O AFP signal increase for simultaneousincrease of B1 and dB0dt . (a) low field conditions with B1 = 0.628µT anddB0dt =3 µTs . (b) high field conditions with B1 = 2.52µT anddB0dt = 429µTs . Red:longitudinal magnetization Mz. Green: transverse magnetization Mx. . . . . . . 60Figure 3.14 The absorption linewidth of Rb increases as the Rb vapour is pressure broad-ened by collisions with He gas. Red: measured signal linewidth (convolvedwith 0.1 nm resolution bandwidth of spectrum analyzer). Blue: linewidth afterdeconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.15 Rb absorbance as a function of cell temperature compared with theoretical num-ber density temperature dependence based on vapor pressure. . . . . . . . . . . 65Figure 3.16 AFP signal dependence on SEOP cell average temperature during continuousflow at 0.3 slm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 3.17 AFP signal dependence on flow rate under continuous flow-conditions. . . . . 67xivFigure 3.18 Blue: AFP signal from batch mode operation with 8 min. buildup time, fol-lowed by transfer to NMR at 0.200 slm. The peak at t = 50s represents thebatch polarization, while signal at t > 150s is the continuous-flow (steady-state)polarization at 0.200 slm. Red: AFP ramp dB0/dt. . . . . . . . . . . . . . . . 67Figure 3.19 Batch mode conditions for buildup times up to 60 min, followed by transfer toNMR at 0.200 slm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 3.20 The Xe AFP signal decreases as polarized Xe travels through longer lengths oftubing. Signal peaks from consecutive runs in (a) appear closer together dueto a slow B0 drift. Error bars in (b) are from detection noise only and do notaccount for the slowly-changing experimental conditions. . . . . . . . . . . . . 69Figure 3.21 Xe relaxation under repeated AFP. The decay of signal here is dominated by T2effects and the incomplete inversion that occurs near the bottom of the B0 ramp. 70Figure 3.22 The ramp technique for a T1 measurement. We implement a variable time delayTD between successive up/down ramps and measure the decrease in AFP signaldue to T1 relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.23 Exponential decay of AFP signal using the ramp technique above. The T1 life-time is inferred from the empirical fit τ = 878±26 s. . . . . . . . . . . . . . . 71Figure 3.24 Schematic of Xe cold trap located between polarizer and NMR detector, withB0 = 8mT provided by a solenoid. . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.25 The downstream AFP signal disappears while performing freezeout during t =200−500s, with all the Xe condensing in the cold trap; signal recovery occurspost-thaw at t = 500s, but without the anticipated signal increase (see text). . . 73Figure 4.1 Schematic of the setup used for pulsed laser spectroscopy. Two-photon absorp-tion in the Xe cell creates laser-induced fluorescence, which may be detectedparallel or transverse to the pump beam. . . . . . . . . . . . . . . . . . . . . . 77Figure 4.2 Laser-induced fluorescence from 252.5 nm two-photon excitation of Xe. . . . . 77Figure 4.3 Frequency scan of the fluorescence following 252.5 nm two-photon excitation.The LIF signal has been corrected for PMT sensitivity and monochromator ef-ficiency at the LIF wavelengths 823 nm and 895 nm. . . . . . . . . . . . . . . 78Figure 4.4 (a) Forward and (b) transverse LIF at 828nm vs UV pulsed laser intensity at249 nm. (a) demonstrates the threshold at 50 a.u. for the onset of bidirectionalemission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.5 Pressure dependence of threshold energy (and power density) for observationof strong bidirectional emission at 828 nm. . . . . . . . . . . . . . . . . . . . 79xvFigure 4.6 Observed LIF dependence on QWP rotation angle for the 249 nm transition.For this transition only ∆M = 0 is allowed. This simple demonstration of theselection rules for unpolarized Xe vapour in the J=0 state is indicative of theexpected behavior for polarized Xe vapour in the J=2 states. . . . . . . . . . . 81Figure 4.7 Layout of the CW laser system. Schematic made by E.Altiere [7] . . . . . . . 82Figure 4.8 MATLAB simulation of frequency modes supported by the OPSL cavity, show-ing the effect of intracavity elements to force single mode lasing. The x-axisshows the OPSL frequency, offset by 296841 GHz. (a) Free spectral range ofthe intracavity etalon. (b) Transmittance function of intracavity birefringent fil-ter mounted near the Brewster’s angle. (c) Cavity mode structure modified bythe intracavity elements. (d) Magnified view of (c) showing preferential gainfor one frequency mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 4.9 Trace of signals used for PDH lock on the LBO cavity, obtained by applyinga monotonic voltage ramp to the cavity PZT. The signals shown are the photo-diode DC signal (yellow), photodiode AC signal (pink), and error signal (blue)derived from the phase of the AC signal. The slope of the error signal deter-mines which direction to drive the cavity. The oscillation that appears on theright-hand side of the photodiode DC and error signals may result from me-chanical vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.10 Schematic of the vacuum chamber and detection optics used for two-photonLIF spectroscopy. L1, L2: modematching lenses. L3: f = 19mm collectionlens. L4: f = 50mm collection lens. PBS: polarizing beam splitter. PZT:piezoelectric transducer. λ/4: quarter wave plate. IC: input coupler. HR: highreflectivity mirror. APD: avalanche photodiode. . . . . . . . . . . . . . . . . . 87Figure 4.11 Photo inside the vacuum box, showing turning mirror from brewster windowUV input; holder for QWP and lens; input coupler and PZT mounted to hollowcopper block; LIF collection lens; HR mirror; turning mirror to UV cavity monitor. 87Figure 4.12 Excitation spectrum 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0) transition in naturalabundance Xe. The total pressure was 1.6 Torr, with a 50-50 ratio of Xe and O2.The x-axis corresponds to the Xe transition frequency, eight times larger thanthe OPSL frequency. The y-axis is the observed LIF intensity of the combined895 nm and 823 nm emission. The peaks are shown with the fitted Lorentzianlineshape as described in the text. Each peak is labeled with its mass number;additionally, odd isotopes are labeled with their excited state hyperfine levelF in parentheses. The stick diagram shows the calculated peak positions andintensities obtained from the Lorentzian fit. . . . . . . . . . . . . . . . . . . . 89xviFigure 4.13 Pressure broadening for a 50-50 mix of Xe and O2. The x-axis reports totalpressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 4.14 Pressure broadening for Xe with 1 Torr of O2. The x-axis reports total pressure. 91Figure 4.15 Isotope shift relative to mass 136. A line of best fit for the even isotopes showsthe odd-even staggering observed by King. Error bars are smaller than the datapoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.16 King Plot of isotope shifts for two different two-photon transitions at 252.5 nm(present work) and 249 nm (as measured by Plimmer et al. [128]) . . . . . . . 97Figure 4.17 Linear fits to the calculated field shift vs. charge radii parameter δ 〈r2〉, plottedfor each of the three data sets in Table 4.5. The field shift is determined usingthe respective values for mass shift parameter K to subtract off the mass shiftfrom the total isotope shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.18 Residuals (∆ν f ield,calc−∆ν f ield, f it) of the fits plotted in Fig. 4.17 for field shiftvs. charge radii parameter δ 〈r2〉. We calculate the mean squared error of theseresiduals to determine the goodness of fit.) . . . . . . . . . . . . . . . . . . . . 99Figure 4.19 Signal S (at resonant frequency) normalized to power P squared for a mixtureof natural abundance Xe and 1 Torr O2, as a function of total pressure. . . . . . 101Figure 4.20 Signal S (at resonant frequency) normalized to power P squared for a 50-50mixture of natural abundance Xe and O2 . . . . . . . . . . . . . . . . . . . . . 102Figure 4.21 Zeemax simulation showing the path of rays from a point 3mm from the UV fo-cus, to illustrate the vignetting. The detection configuration is strongly sensitiveto emission less than 1 mm from the UV beam focus. . . . . . . . . . . . . . . 103Figure 4.22 Vignetting fraction (over 4pi) of rays emitted isotropically from a point on thebeam axis. The detection configuration is most sensitive to fluorescence emittedwithin 1 mm from the UV beam focus. . . . . . . . . . . . . . . . . . . . . . 104Figure A.1 Tuning box resonance at 15kHz . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure A.2 Tuning box resonance at 58kHz . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure A.3 Lowest-frequency beatnotes νA and νB generated by mixing OPSL light withthe self referenced frequency comb. The detuning on the x-axis is relative toνcomb(n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure A.4 Beatnote (blue) and wavemeter (red) readings during a monotonic increasingsweep of the OPSL frequency, while recording the second-nearest neighbourbeatnote frequency νB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135xviiGlossaryAFP Adiabatic Fast PassageBBO Beta Barium BorateCPT Charge Parity Time reversal symmetryDUV Deep UltravioletEMF Electromotive ForceFID Free Induction DecayFSR Free Spectral RangeLBO Lithium TriborateLIF Laser Induced FluorescenceNEDM Neutron Electric Dipole MomentNIR Near InfraredNMR Nuclear Magnetic ResonanceOPSL Optically Pumped Semiconductor LaserPDH Pound Drever HallPMT Photomultiplier TubeREMPI Resonance-Enhanced Multiphoton IonizationRF RadiofrequencyRLC Resistor-Inductor-CapacitorSEOP Spin Exchange Optical PumpingxviiiSHG Second Harmonic GenerationSQUID Superconducting Quantum Interference DeviceTALIF Two Photon Absorption Laser Induced FluorescenceTPA Two Photon AbsorptionTPE Two Photon EmissionUCN Ultracold NeutronsxixAcknowledgmentsI’m immensely grateful for the leadership, support, and guidance I’ve received from professors,coworkers, family and friends that led to the success of this project. To my supervisor, Dr. TakamasaMomose, thank you for taking me on as a student and patiently providing support and guidance.Thank you also to Dr. David Jones and Dr. Kirk Madison for making a collaborative team, forlending support, equipment, ideas and instruction. To my partners in the lab, Emily Altiere, JoshuaWienands, Dr. Tomo Hayamizu: thank you for teaching me from your own expertise, workingalongside to get results, and sharing the frustrations and ultimately successes. Thanks to PavleDjuricanin for your technical support with each and every thingamajig that just wouldn’t work.Our collaborators at the University of Winnipeg, Mike Lang, Dr. Jeff Martin and Dr. ChrisBidinosti, shared both equipment and expertise on the SEOP polarizer. Thank you for your supportand frequent Skype meetings to compare results and discuss new ideas.Thanks to the many past and present labmates who shared experience and banter, includingManish Vashishta, Brendan Moore, Thomas Prescott, Cindy Toh, Angel Wong, Mario Michan,Omid Nourbakhsh, Sida Zhou, Bill Wong, Aaron Ngai, and Danial Yazdan.Thanks to my family and friends, for love and support, including the previous housemates whobecame like family in Vancouver. Thanks to my parents for a lifetime of support. Thank you to myamazing wife Joanna, who fed me, spurred me on to finish and gave me lots of hugs.xxChapter 1IntroductionAtomic magnetometry has found application in the investigation of fundamental symmetries insubatomic systems. In particular 199Hg was used as a magnetometer to great success in the latestprecision measurement looking for a non-zero permanent Neutron Electric Dipole Moment (NEDM)[15, 124]. Other diamagnetic atoms with nuclear spin have been proposed as superior candidatesto 199Hg (I = 1/2), subject to demonstration of suitable detection schemes. Among these is 129Xe(I = 1/2).The goal of this work is to perform precision spectroscopy on atomic Xe using two-photontransitions, and to develop techniques and apparatus for optical magnetometry using two-photontransitions. The longterm goal of the project is to build an atomic magnetometer based on the spinprecession of 129Xe, which will share a chamber with trapped Ultracold Neutrons in an experimentat TRIUMF measuring the neutron electric dipole moment. Such a magnetometer must detect mag-netic fields in the experiment and be free of dielectric breakdown in the high voltage electric fieldsalso being applied. The device is based on polarization of Xe via spin exchange optical pumping,and detection of the degree of polarization during spin precession using two photon absorption.Spin exchange optical pumping is a technique to polarize Xe through collisions with much moreeasily polarized alkali metal atoms. Two photon absorption is a technique using intense laser beamsto cause atomic transitions into excited states through the simultaneous absorption of two photonsof the requisite energy. The theory of these are covered in Chapter 2.This introduction presents the motivation for development of a Xe-based comagnetometer. Sec-tion 1.1 describes the experimental goals of the neutron EDM measurement performed using ultra-cold neutrons, which tests time-reversal symmetry. Section 1.2 describes the need for magnetometryand some properties of 129Xe that make it suitable as a comagnetometer. Section 1.3 summarizesthe component techniques of polarization, precession, and two-photon detection necessary to utilize129Xe in a comagnetometer. Section 1.4 is the road map of the theory and research work for the restof the thesis.11.1 Ultracold Neutrons and the Neutron Electric Dipole Momentmeasurement1.1.1 The search for Charge Parity (CP) violationTable 1.1: Behaviour of measurables under the operators C,P,T.C P T~s Angular momentum + + -~µ Magnetic dipole moment - + -~d Electric dipole moment - + -~E Electric field - - +~B Magnetic field - + -There are three transformations under which the laws of physics, in the past, were thought tobe invariant[76]. These are charge-conjugation (C), referring to the replacement of every particlewith its antiparticle; parity (P), referring to the coordinate transformation (x,y,z)→ (−x,−y,−z);and time-reversal (T), referring to running a process backwards in time. Table 1.1 shows how spinand fields transform under the application of these operators. To date no experimental evidence hasbeen observed for breaking of the combined operations (product) of Charge Parity Time reversalsymmetry (CPT); and therefore CPT conservation is assumed in the present work. Nonetheless,certain violations of these symmetries have been observed to occur in specific combinations innature, as described below.CP-violating processes are implicated as a reason why we observe predominantly matter andvery little antimatter in the present universe. The observed universe contains, almost exclusively,“normal” matter, e.g., positively charged protons and negatively charged electrons. One way toquantify the prevalence of normal matter over antimatter is by calculating the “baryon asymmetry”of the universe, defined as η = (nB−nB¯)/nγ , with nB, nB¯, and nγ the number densities of normalbaryons, antibaryons, and cosmic background radiation photons, respectively. The present observedbaryon asymmetry is η = (6.21±0.16)×10−10 [54]. In the early universe there should have beenequal amounts of matter and antimatter, and at some time an interaction arose which favoured pro-duction of matter. CP-violation is one of the three “Sakharov conditions” necessary for interactionsthat favour production of matter, along with ii) baryon number violation and iii) non-equilibriumthermal conditions[146].Strong and electromagnetic interactions are observed to conserve C, P, and T symmetries, butthe weak interaction maximally violates C and P. The first experimental proof of parity violationwas observed by Wu [167], who showed that there was an asymmetry in the beta decay of polarizedcobalt-60, with electrons emitted exclusively in the direction of the nuclear spin. CP violation was2observed soon after by Fitch and Cronin [46], who observed a long lived beam of neutral kaonsto occasionally decay into two pions, when CP invariance required it decay exclusively into threepions. CP violation has more recently been observed in the decays of neutral B mesons[13]. Theobserved CP violation comes from the complex phase of the quark mixing (Cabbibo-Kobayashi-Maskawa) matrix [40, 97].The predominance of normal matter over antimatter, as defined by the baryon asymmetry, im-plies that CP violation occured at higher rates than the Standard Model currently accounts for bythe above processes. Therefore there must be other sources of CP violation that we are missing.New experiments are looking for evidence of symmetry violation in atomic systems and in freeneutrons. The existence of a permanent electric dipole moment in a subatomic particle violatestime-reversal (T) symmetry; both the electric and magnetic dipole moment will be parallel to spin,i.e. ~dn||~µ . Under time-reversal the magnetic dipole moment reverses orientation, but an EDM doesnot. Thus there is a contradiction: the two dipole moments will be parallel in the normal particle andantiparallel in the time-reversed particle. Under the assumption that CPT symmetry is conservedthen a permanent EDM is also a source of CP-violation. Placing a limit on the value of the EDMhelps to constrain theoretical models of CP violation. The most recent measurements looking for apermanent neutron EDM have been performed using Ultracold Neutrons.1.1.2 Ultracold NeutronsMeasurements of the nEDM require confinement of free neutrons for a suitable measurement timeon the order of hundreds of seconds. Previous experiments used neutron beams in vacuum; however,the modern implementation of nEDM experiments use confined ultracold neutrons.Ultracold Neutrons (UCN) are free neutrons (quark composition udd, charge = 0, mass = 1.67×10−27 kg [113]) with kinetic energies lower than 10−7 eV [71]. This is orders of magnitude lowerenergy than typical for subatomic particles, and corresponds to a velocity around 7ms−1. UCN areso slow-moving that they can be confined in closed bottles, without magnetic or electric trapping.This is because UCN have long de Broglie wavelengths and can interact repulsively with nuclei inthe walls of the bottle via the strong nuclear force. The exact kinetic energy of the UCN corre-spond to their gravitational potential energy of 102neVm−1. This can be used advantageously inexperiments; for example, the velocity distribution may be modified by placing a neutron absorberat a specified height in a cell to absorb all neutrons reaching that height; similarly, it is possibleto provide enough kinetic energy to transmit UCN through a potential barrier such as a metal foil,merely by dropping them from an altitude of several meters. UCN may be produced by coolingthe neutrons produced by nuclear fission or by spallation of nuclei. As with all free neutrons, thelifetime of UCN via β -decay is 877.7(8)s [121].TRIUMF has installed a UCN source which aims to produce world-record UCN densities forfundamental physics experiments. The source produces highly energetic MeV neutrons through3Figure 1.1: Beamline and superfluid He-II cryostat for producing UCN by spallation of atungsten target. [126]proton spallation on a tungsten target. A multi-layer cryostat cools these neutrons: first they aremoderated in nested volumes of liquid and solid (cryogenic) D2O to cold neutron energies; second,they downscatter in a superfluid He – II filled volume to become UCN and are trapped by the stronginteraction within the walls of the volume. From there UCN can be released through a valve andpolarized via a strong magnetic field, and delivered to experiments. The flagship experiment forthe new facility is the neutron electric dipole moment (EDM) measurement described below. Theexisting source was developed at the Research Center for Nuclear Physics (RCNP) in Osaka, Japanand produced a UCN density of 26UCNcm−3 [106] when irradiated with a 400 MeV, 1µA protonbeam. The target density at TRIUMF is 7000UCNcm−3 in the source and 200UCNcm−3 in theEDM cell [3] This is attainable by irradiation with the higher intensity 500 MeV, 40µA proton beamavailable at TRIUMF.1.1.3 Neutron Electric Dipole Moment measurement in UCNSearching for the permanent electric dipole moment of a subatomic particle is a test of symmetry inthe Standard Model. The existence of such a permanent charge separation in a nucleon or electronviolates P and T; this can be derived from Table 1.1. The Hamiltonian of a fermion with magneticdipole moment µ and electric dipole moment dn in combined magnetic and electric fields is givenby:H =−~µ ·~B− ~dn ·~E (1.1)The result of the parity operator acting on the Hamiltonian above is P(H) =−~µ ·~B−~dn ·( ~−E) 6=H. Because the Hamiltonian is not invariant under the transformation, it violates parity. It is specifi-4Figure 1.2: Proposed configuration for UCN guides and neutron EDM experimental cell.[126]cally the term ~dn ·~E which is the source of the parity violation; the magnetic dipole term is invariant.Likewise, under time reversal T (H) = −( ~−µ) · ( ~−B)− ( ~−dn) ·~E 6= H. As CPT is still assumed tobe conserved, T-violation implies CP-violation also.Electric Dipole Moment searches have been carried out on neutrons as well as in a numberof atomic and molecular systems. The results of EDM experiments on some of these systems arepresented in Table 1.2. The experimental EDM under study in each atom or molecule may arisefrom a combination of either an electron EDM de, a neutron EDM dn, or certain electron-nucleonand pion-nucleon interactions, as described in Reference [47]; each system varies in its sensitivityto the above interactions, making the study of EDM in many systems a complementary effort. Manyof the experiments listed in the table are ongoing; there are additional systems currently under studyor proposed, including atomic Fr [88], and protons in a storage ring [9]. There has been no non-zero measurement of a permanent EDM to date; in the case of neutrons the EDM predicted by theStandard Model is ≈ 10−31 e · cm [69]. For comparison, this is twenty-two orders of magnitudesmaller than the dipole moment of a water molecule (1.85 debye = 3.85×10−9 e · cm). Extensions(Beyond Standard Model theory) predict larger values, thus nEDM measurements also serve as atest of Beyond Standard Model theories.The typical procedure in an EDM measurement is the controlled application of parallel and5antiparallel electric and magnetic fields. Although the neutron is electrically neutral, it has a nuclearmagnetic moment due to its spin I = 1/2 and will undergo precession at the Larmor frequency ω =γB. The interaction between electric dipole and electric field ~dn ·~E imposes a slight frequency shiftwhich can be detected as a difference δω between runs with B and E either parallel or antiparallel.Table 1.2: Experimental results of EDM searches in neutrons, and in atomic and molecularsystems (adapted and updated from [47]). Each result is represented either as the totalatomic or molecular EDM, or is interpreted as an EDM limit on the electron de. Theneutron EDM is dn.System Result (e.cm) Year & Sourceneutron dn =−0.21(1.82)×10−26 e · cm 2015 [124]Cs dCs =−1.8(6.9)×10−24 e · cm 1989 [115]Tl dTl =−4.0(4.3)×10−25 e · cm 2002 [139]YbF de =−2.4(5.9)×10−28 e · cm 2011 [84]ThO de =−2.1(4.5)×10−29 e · cm 2014 [17]199Hg dHg =−2.2(3.1)×10−30 e · cm 2016 [74]129Xe dXe = 0.7(3.0)×10−27 e · cm 2001 [142]TlF d =−1.7(2.9)×10−23 e · cm 1991 [45]225Ra dRa = 4(6)×10−24 e · cm 2016 [23]The first nEDM measurement was performed by Smith, Purcell, and Ramsey on a neutron beamin vacuum using separated oscillatory fields, a technique which has come to be known as RamseyResonance. This technique is described in detail in 2.1.1. This early measurement yielded the value(−0.1± 2.4)× 10−20 e · cm [151]. Experiments since the 1980s used UCN confined in a bottle,starting with the research group at Leningrad Nuclear Physics Institute (LNPI) [6], to improvesystematic uncertainties. This makes it possible to reduce the width of the magnetic resonanceline for a fast-moving neutron beam. Additionally the long lifetime of UCN makes it possible toincrease the interaction time with the applied fields. The best result to date [124] sets an upper limitof 3.0× 10−26 e · cm (90% C.L.). The TRIUMF experiment wants to lower the nEDM upper limitto < 10−27 e · cm [3], representing an improvement over an order of magnitude. The target will beacheivable in large part due to an increase in neutron density attainable at TRIUMF. Fig. 1.2 showsthe experimental configuration planned at TRIUMF for the neutron EDM measurement, which isdescribed further in Reference [126]. UCN will be confined in a vacuum bottle (diameter ≈ 20−30cm) between high voltage electrodes separated by a cylindrical insulator ring. The experimentalapparatus is decoupled from stray environmental magnetic fields by layers of mu-metal shieldingand active field compensation; inside the shielding, additional field coils will produce a very uniformmagnetic field (|B0| ≈ 1µT). Precise magnetometry is required in the technique to reduce systematicuncertainties; this motivates development of the Xe comagnetometer described below.61.2 Motivation for developing a Xe comagnetometerThe search for a neutron EDM is limited by many systematic effects. The experiment requires avery stable magnetic field, in order to track changes in the neutron precession frequency correlatedonly with a reversal in electric field. Drifts in magnetic field (on the order of a few tens of pT)due to changing ambient conditions can completely mask the true EDM signal and are a limitingfactor to sensitivity. The most significant innovation of previous generation nEDM experimentsreaching 10−26 e · cm was the installation of a cohabiting 199Hg optically pumped atomic magne-tometer to measure and account for magnetic field fluctuations [15, 75]. This “comagnetometer”shares the same experimental volume as UCN and samples the same time- and volume-averagedmagnetic field with sub-pT accuracy. Optically pumped magnetometers themselves belong to alarger family of magnetic sensors technologies, including flux gates, Hall probes, proton-precessionmagnetometers, SQUID, nonlinear magneto-optical rotation magnetometers and magnetoresistivedevices [104]. Each of these fulfill different requirements and have different limitations[77] [103].A short description of some of the techniques is given in A.1. In the nEDM experiment, prior effortsto track the drifting magnetic field using external Rb vapour magnetometers mounted outside theexperiment proved ineffective due to the separation distance between magnetometer and neutrons[75]. Only a comagnetometer occupying the same volume could suitably measure the field. Detaileddescription of the comagnetometer operation is given in Section 2.1.2.1.2.1 Advantages of using 129XeOptical magnetometry can in principle be performed using any atomic species possessing a non-zero total spin F in the ground state. This includes both paramagnetic (Rb, Cs) and diamagneticatoms (3He, 199Hg, 129Xe). 129Xe possesses certain advantages particular to the nEDM experiment.One major advantage is that xenon has a cross section for neutron absorption which is two ordersof magnitude smaller than that of mercury. The absorption cross sections for thermal neutrons are2.1×101 barns (1 barn = 10−28 m2) for 129Xe and 2.15×103 barns for 199Hg [99]. One of the orig-inal motivations for choosing 129Xe at TRIUMF was the possibility to reduce the frequency shiftsfrom the so-called geometric phase effect (described in Section 2.1.3) by increasing the atomic den-sity enough to have a very short mean free path, although it remains to be determined if such highdensity is compatible with the high voltage (15 kVcm−1) electrodes used in nEDM measurements.Additional advantages are that Xe is a noble gas, non toxic and inert in the nEDM cell, in constrastto alkali metal atoms. 129Xe has the same I = 1/2 nuclear spin configuration as 199Hg which resultsin a simple electronic and Zeeman level structure. Table 2.1 shows that the gyromagnetic ratio of129Xe is the same sign as that of the neutron and opposite to that of 199Hg, which helps to identifyeffects of diurnal rotation in the (non-inertial) laboratory frame. 129Xe can be easily polarized, ashas been previously demonstrated by Spin Exchange Optical Pumping (SEOP) (see Section 2.2.27Figure 1.3: Scheme for 129Xe-based comagnetometry: polarization of 129Xe, (blue) via spinexchange optical pumping with polarized Rb (green), followed by freezeout, precession,and detection using a two-photon transition.and Chapter 3); SEOP cross sections for Xe are the highest of the noble gases. [162]. Like 199Hg, asuitable optical transition for 129Xe has been identified; however, at convenient laser wavelengths,the excited electronic states of 129Xe are accessible only by two photon transitions. Demonstratingoptical detection of 129Xe using two photon transitions is the focus of this current work. Finally, byproviding a second nEDM-compatible comagnetometer species, we open the possiblity of measur-ing not only the magnetic field but also the vertical gradient dB0/dz by simultaneous measurementson two comagnetometer species. This is discussed in Section 2.1.4.1.3 Techniques planned for the Xe comagnetometerThe future xenon-based comagnetometer will make use of very similar techniques to that success-fully used in previous nEDM measurements with mercury. The optical detection scheme proposedfor 129Xe is shown in Fig. 1.3. Samples of polarized 129Xe will be prepared by spin exchangeoptical pumping, starting with (likely isopure) 129Xe mixed with Rb vapour, He, and N2. The po-larization will be calibrated using Nuclear Magnetic Resonance (NMR) techniques. The polarized129Xe will be separated from Rb and purified via freezeout in a cryogenic trap under strong appliedmagnetic field, and the residual gas pumped away. Controlled heating of the trap will deliver acontrolled vapour pressure of polarized 129Xe to the experimental nEDM cell. The transport lineswill be arranged to allow adiabatic transport into the cell without loss of polarization. Samples ofpolarized UCN, 129Xe, and 199Hg will be injected into the cell. Three separate Radiofrequency (RF)8pulses will initiate free spin precession of the three species. During the free spin precession time of≈ 40s, each species undergoes hundreds of periods of precession; the UCN spin precession will bemeasured using Ramsey Resonance while spin precession of 129Xe, and 199Hg will be monitoredoptically. For 199Hg, monitoring involves excitation with a weak 253.7 nm probe beam and detec-tion of the transmitted light. For 129Xe, monitoring involves two-photon excitation with 252.5 nmlight and detection of the Laser Induced Fluorescence (LIF) at 823 nm. Both signals (129Xe, and199Hg) will display modulation at their respective Larmor frequencies, and will take the form of adecaying sinusoid. For each species this signal will be digitized and fit to extract an average ωXe(Hg).The fitting routine will likely involve counting zero crossings of the several hundred detected spinprecession periods to estimate the frequency and adding a small correction based on fitting to theinitial and final phase.The precession frequencies of 129Xe and 199Hg will be used to correct the neutron precessionfrequency for any time-dependant drift in the magnetic field. Additionally, the combined measure-ments of both precession frequencies can be used to correct for frequency shifts due to a verticalfield gradient, as described in Section 2.1.41.4 Dissertation overviewThis dissertation describes work developing a 129Xe polarizer based on spin exchange optical pump-ing and a optical detection scheme based on two photon transitions. This introductory chapter de-scribed the motivation to develop a comagnetometer, namely, detection of magnetic field drifts toimprove the sensitivity of neutron EDM experiments which search for new sources of CP violation.The first section of Chapter 2 describes comagnetometry in greater detail, with an overview ofexperimental techniques used in the neutron EDM measurement. The remaining sections describethe theory behind two important techniques necessary for Xe comagnetometry: spin exchange op-tical pumping and two-photon absorption. The electronic structure of Xe is described along withthe states accessible by two-photon absorption. Chapter 3 describes the implementation of a SEOP-based Xe polarizer and its operation. A modest polarization is generated as proof of polarizationand detected by performing Adibatic Fast Passage, and compared with predictions based on a rateequation model. SEOP has been well studied and even commercialized elsewhere; the value of thework done in this lab lies in developing the ability to provide polarized Xe for testing of two-photoncomagnetometer signals. Chapter 4 demonstrates a breakthrough in excitation of the two-photontransition with narrow linewidth light suitable to selectively probe the appropriate hyperfine levelsin the scheme described above. The chapter describes initial experiments in excitation of the rele-vant two-photon transitions with a pulsed laser, followed by high-resolution spectroscopy enabledby development of a narrow linewidth CW laser. From this spectroscopic work we extract improvedmeasurement of the hyperfine constants and isotope shifts. Measurements of pressure broadeningand signal intensity are performed, motivated by the need to determine parameters for implemen-9tation in the nEDM experiment. Specifically, we consider the signal amplitude and correspondingestimates of magnetometric sensitivity at low Xe partial pressures (approaching 1 mTorr). Such lowpressures may be required when Xe is introduced into the high-voltage nEDM cell, in order to avoidany electrical breakdown. The concluding chapter describes ongoing and future work focused onestablishing the suitability of 129Xe as a comagnetometer species.10Chapter 2Literature and Theory2.1 Comagnetometry and the Neutron Electric Dipole MomentexperimentComagnetometers are optical magnetometers which rely on the detection of spin precession. Theterm comagnetometer refers to a separate spin-polarized gas that shares the same volume as thespecies under study in an EDM measurement, and provides a time- and volume- averaged measure-ment of the magnetic field. Comagnetometry was discussed by Lamoreaux [102], who proposed199Hg as a volume comagnetometer for the ILL nEDM experiment. Description of the techniquesused for previous experiments utilizing 199Hg are given in this section.Other uses of comagnetometryComagnetometry was also used in a 129Xe EDM measurement [142], with 3He as the comagne-tometer. The EDM of 3He was assumed to be smaller than that of 129Xe because the effects ofCP-violation in the atom scale as Z2 or more with atomic number, making 3He a suitable comag-netometer. The 3He resonance was detected using coils rather than optically. A new search for the129Xe EDM uses a 3He comagnetometer with resonance detected via a Superconducting QuantumInterference Device (SQUID) magnetometer [101]. These applications will be discussed in Sec-tion 5.2.4.2.1.1 Ramsey Resonance technique for nEDM measurementsIt is necessary to describe the experimental technique used to measure the precession frequencyof the neutrons, as the comagnetometer is subjected to the same fields, and in fact the techniquesshare many similarities. The TRIUMF nEDM experiment utilizes a technique known as Ramseyresonance, first described by Ramsey in 1950 [136]. The key to this technique is the use of separatedoscillating fields to make a precise measurement of the precession frequency based on comparison11Figure 2.1: The Ramsey Resonance technique used for high precision measurements of neu-tron precession frequency. See text for description. ~B0 is vertical in the figure while ~B1is horizontal. During the free precession time, neutrons complete hundreds of preces-sion periods and may acquire a phase shift from the (gated off) stable oscillator ω1 (red).(Reprinted from [16]pg.187 with permission from Elsevier)of the phase difference with a stable oscillator. In initial neutron beam experiments the oscillatingfields were separated in distance; in today’s experiments with UCN the fields are separated in time.The technique proceeds as follows (see Fig. 2.1):1. neutrons are prepared in a “spin up” MI = 1/2 state in the presence of an applied homoge-neous magnetic field ~B0 on the order of µT.122. A rotating magnetic field B1 perpendicular to ~B0 at frequency ω1 ≈ γB0 is applied to rotatethe spins away from ~B0. The magnitude and duration of ~B1 are controlled to ensure a spin flipof exactly 90 degrees (pi/2) and then gated off.3. The spins are subjected to free spin precession about ~B0 for hundreds of precession cyclesduring time t. During this time which they acquire a phase ((ω1− γB0)t).4. A second pi/2 pulse is applied to the spins. This rotates the spins either up or down dependingon the accumulated phase, compared against the stable oscillator at ω1. If the phase is aninteger multiple of 180 degrees, the spins rotate back parallel to ~B0. If the phase is zero or aninteger multiple of 360 degrees, the spins rotate downwards and become antiparallel to ~B0.The population of one or both spin states is measured following the cycle and plotted againstfrequency ω1. This is shown in Fig. 2.2 for the spin up population. Fringes appear for everyaccumulated phase shift of pi between ω1 and γB0. The resonant Larmor frequency is determinedby the value of ω1 at the central fringe. As plotting the entire curve would require hundreds ofdata-taking cycles, a precise measurement of the resonant frequency is typically determined bytaking data at four points, two on either side of the resonance, for which the change in counts issteepest. A fit to these points determines the resonant frequency. This technique is often used inEDM experiments to allow a long interaction time with an applied electric field ~E ‖ ~B0. The phaseshift acquired in step 3) can come from either term of the Hamiltonian in Eq. 1.1, which means thatinteraction of an EDM with ~E shifts the resonant frequency. Comagnetometry is used during step3) to monitor ~B0 for drifts.2.1.2 Need for comagnetometry and principle of operationComagnetometry is necessary to make a precise measurement of the EDM in the presence of fielddrifts and other systematic effects. It is readily apparent that some form of magnetometry is needed,because the effect of ~B0 is so much larger than the effect of ~E. For example, taking ~B0 = 1µT and~E = 15kVcm−1 as typical parameters for an EDM search, the ratio between the effect of ~B0 andthat of ~E for an EDM at the current 10−26 ecm level of sensitivity is µBdnE ≈ 109. This means that ppbdrifts in the magnetic field, left uncorrected, will completely mask the presence of an EDM. Driftsfrom external environmental sources, such as that of moving an overhead crane, have been identifiedand corrected for in past experiments using atomic magnetometers mounted outside the nEDM cell.However, certain local magnetic field changes affect the EDM cell in a way that such externalmagnetometers cannot readily detect. One of these effects is the local magnetic field generated byleakage currents flowing between HV electrodes through the EDM cell wall, which doubles as theelectrode spacer [152]. Another possibility is the residual magnetization of an internal component.A comagnetometer occupies the same volume as the EDM species, and is precisely sensitive to theeffects of leakage currents and other localized magnetic fields.13Figure 2.2: Ramsey Resonance fringes used to precisely measure the precession frequency.(Reprinted from [16]pg.187 with permission from Elsevier)Principle of operationAs a type of optical magnetometer, the comagnetometer uses techniques of magnetic resonanceand spin precession similar to the EDM species. This section describes the operation followingthe 199Hg comagnetometer successfully operated in the nEDM measurement at ILL [16, 75]. Therequirements on the species used as comagnetometer are: it must not itself have an EDM, at leastnot one large enough to introduce uncertainty in the interpretation of any measured frequency shift;it must occupy the same volume, and must be compatible with the requirements of the host experi-ment. 199Hg satisfies these requirements. The intrinsic EDM of 199Hg was measured to be smallerthan |dHg| < 7.4× 10−30e · cm [74], in an optical experiment conducted on a stack of four cellsfilled with 199Hg exposed to ~B0 and ~E fields and externally shielded. The cells are optically pumpedtransverse to ~B0 by synchronous optical pumping.The operation of the mercury comagnetometer essentially follows steps 1-3) of Fig. 2.1. First,199Hg atoms in an external cell are polarized parallel to ~B0 using optical pumping (see Section 2.2) ofthe transition 6s6p(3P1) ← 6s2(1S0) with circular polarized light at 253.7 nm. Atoms are opticallypumped for a period of approximately four minutes and injected into the nEDM cell at the start ofan experimental cycle. The cell is filled with 199Hg and UCN, and two pi/2 pulses are applied at14(a) (b)Figure 2.3: Comagnetometer performance in the ILL experiment. Magnetic field drifts (a) onthe order of 70 ppm cause a corresponding shift in neutron precession frequency (b),which is corrected for by simultaneous monitoring of the precession frequency of 199Hg.(Reprinted from [16]pg.193 with permission from Elsevier)the respective Larmor frequencies for 199Hg and UCN. Both species exhibit free spin precessionabout ~B0 in step 3) until the end of the cycle. During this free spin precession, the light absorptionof 199Hg is probed with circular polarized light propagating through the cell transverse to ~B0. Theabsorption of this light varies sinusoidally at the Larmor frequency ωHg. The transmitted light isdetected and converted to a voltage signal with the shape of a sinusoidal decay curve. The time-averaged resonance frequency ωHg is determined by counting zero crossings and fitting waveformsto the first and last few seconds of data. A precise measurement of ~B0 was effectively determinedfrom the Larmor frequency ωHg(= 2piνHg) = γB0; typically, however, ωHg is used directly in afrequency ratio (with∣∣γn/γHg∣∣≈ 3.77) to calculate the nEDM:νnνHg=∣∣∣∣ γnγHg∣∣∣∣+ dn+ |γn/γHg|dHgνHg E (2.1)The performance of the comagnetometer in the presence of external magnetic field drifts is shown inFig. 2.3. Over the course of a 25 hour run, a cycle-by-cycle field shift induced changes on the orderof 70 ppm to the resonant frequency for both 199Hg and UCN. That the change occurred in bothchannels indicates that it is due to magnetic field drift and not to an EDM. The resonant frequencycan be corrected appropriately as shown in Fig. 2.3b; the ratio in Equation 2.1 is insensitive to suchdrifts.152.1.3 Limitations of present comagnetometry and residual effectsThe comagnetometry techniques described above can correct for magnetic field drifts smaller thanppm, but still suffer from systematic effects. One of these is the so-called light shift. This is a shiftin the mercury precession frequency caused by a component of the probe light which is parallelto ~B0 and shifts the ground and excited energy levels. This depends on the intensity of the probelight, and can mimic an EDM if there happens to be any correlation between |~E| and light intensity.Another systematic effect is that the UCN are slow enough that their center of mass is below thecenter of the EDM cell, so they sample a slightly different field than the thermal comagnetometeratoms. Finally, the geometric phase effect described below causes a frequency shift that can mimican EDM in the presence of magnetic field gradients.Geometric Phase EffectThe EDM experiment measures a shift in the neutron frequency, associated with a reversal in appliedelectric field ~E. The underlying assumption is that the neutron magnetic moment is unaffected bysuch reversals. However, there is an effect which can shift the neutron frequency in the same wayand can therefore be mistaken for an EDM[123]. This effect is caused by the motional magneticfield experienced by a particle moving through an electric field (Equations 2.2 through 2.9 are takenfrom [123]):Bv =E× vc2, (2.2)combined with the horizontal component of a slightly inhomogenous magnetic field, which resultsfrom Maxwell’s equations ∇ ·~B = 0:B0xy =−dB0zdzr2(2.3)Both of these fields are small compared to ~B0, and orthogonal to ~B0. Unfortunately, for particlesin certain orbits, the effects of these fields don’t average out with particle motion in the trap butinstead impart a measurable frequency shift. The shift is known as a geometric phase effect, sincethe particle’s wavefunction obtains a so-called geometric phase1 resulting from transport througha closed loop in its parameter space. For particles whose orbits have a distinct rotation in the xyplane, these horizontal fields induce a Ramsey-Bloch-Siegert shift in the precession frequency:∆ω =ω2xy2(ω0−ωr) (2.4)1For slow moving transport that can be considered adiabatic, this geometric phase is directly related to the quantummechanical Berry’s phase[123].16where ω0 is the natural Larmor precession, ωr is the rotational frequency of the particle’s trajectory,and whereω2xy = γ2B2xy = γ2 (Bv+B0xy)2 = γ2((dB0zdzr2)2+(E× vc2)2+2(dB0zdzr2)(E× vc2))(2.5)All three terms shift the precession frequency of the particle; however, only the latter term which islinear in E changes sign with electric field reversal. Since the neutron EDM experiment measuresthe difference in precession frequency under opposite direction electric fields, the last term hasthe potential to be interpreted falsely as an EDM. The shift has the potential to affect both EDMspecies and comagnetometer species alike, although the form is different for fast- and slow-movingparticles. Slow moving particles such as UCN experience a shift which generates a false EDM givenby:da f =Jh¯2(dB0z/dzB20z)v2xyc2[1− ω2rω20]−1(2.6)Faster moving particles such as the comagnetometer atoms will experience a shift which generatesa false atomic EDM da f given by:da f =Jh¯2(dB0z/dz)γ2R2c2[1− ω2oω2r]−1(2.7)The false atomic EDM will also be interpreted as a false neutron EDM given by:da f n =|γn||γa|da f , (2.8)and may be larger in magnitude. In previous generation experiments at ILL[123], the intrinsic falseEDM of the neutron was calculated to be −1.1×10−27 e · cm, while the false EDM stemming fromthe false atomic EDM was a much larger 5.0× 10−26 e · cm. One solution to reduce the geometricphase effect is to decrease the mean free path λ of comagnetometer atoms in the trap by increasingtheir partial pressure. This has the effect of introducing a suppression factor into the measured falseEDM:da fλ =[pivxyλ2R2ω0]2da f , (2.9)where R ≈ 0.25m is the cell dimension. Such an approach has been suggested for 129Xe comag-netometry; however, it is still unknown whether the partial pressure of 129Xe can be increasedsufficiently while avoiding high voltage discharge.172.1.4 Proposal to implement a dual magnetometerThe geometric phase effect described above generates a false EDM proportional to the magneticfield gradient, which is difficult to measure experimentally. However, it can potentially be solvedfor, to first order, by a technique involving two comagnetometer species. This has been proposedfor the nEDM experiment at TRIUMF [111]. Essentially two polarized spin-1/2 species, 199Hg and129Xe, would fill the EDM cell and be subjected to free spin precession. The sinusoidal decay ofeach would be fit independently to extract the precession frequency, given for mercury and xenon,respectively, by:ωHg↑↑ =−γHgB0z−γ2HgR22c2∂B0z∂ zE +γ3HgR22c4B0zE2+3γ3HgR416〈v2Hg〉B0z(∂B0z∂ z)2(2.10)ωXe↑↑ =−γXeB0z− γ2XeR22c2∂B0z∂ zE +γ3XeR22c4B0zE2+3γ3XeR416〈v2Xe〉B0z(∂B0z∂ z)2(2.11)This results in a system of two equations and two unknowns, B0z and∂B0z∂ z . The second term is thecross term between electric field E and magnetic field gradient ∂B0z∂ z , and is linear in both. This termis the source of the false EDM. The last term is quadratic in ∂B0z∂ z , and is therefore small enough tobe neglected. Then we can solve for B0z and∂B0z∂ z according to:B0z =γ2XeωHg− γ2HgωXeγXeγHg (γXe− γHg)( 12c4 γXeγHgR2E2+1) (2.12)∂B0z∂ z=2c2[γXe(γ2XeR2E2−2c4)ωHg− γHg(γ2HgR2E2−2c4)ωXe]γXeγHgR2[γXe(γ2HgR2E2−2c4)− γHg(γ2XeR2E2−2c4)] (2.13)Such knowledge of the magnetic field gradient can be used in the analysis of the neutron precessionfrequency to calculate a result free, to first order, of geometric phase effects. This motivates thedevelopment of optical magnetometry using polarized 129Xe.2.2 Optical Pumping and SEOPIn this section we describe the theory of optical pumping, and spin exchange optical pumping inparticular. Optical pumping is a necessary step for magnetometer development because it providesthe atoms with an alignment that makes them sensitive to magnetic fields. Spin exchange opticalpumping (Section 2.2.2) is the specific technique by which polarized 129Xe is produced.182.2.1 Optical Pumping TheoryOptical pumping refers to the use of light to drive atoms into specific quantum states, or changethe population distribution between states. In particular we study Zeeman optical pumping in theground state of our atoms.Zeeman splittingOptical pumping relies on the energy difference between populated states. In particular, a quantumstate with angular momentum number J has 2J + 1 degenerate sublevels, with projection valuesMJ = J,J− 1...− J. Applying a weak magnetic field ~B breaks the degeneracy of the sublevels bycausing a Zeeman shift governed by the Hamiltonian [166]H =−~µ ·~B =〈~µ · ~J〉J(J+1)~J ·~B, (2.14)where ~µ is the magnetic dipole moment of the atom, given by ~µ =−µB~L−gSµB~S =−gJµB~J. Theenergy shift of each Zeeman sublevel MJ is ∆E = gJµBBMJ . gJ is the Lande g-factor, and under LScoupling conditions gJ is given by:gJ = gLJ(J+1)+L(L+1)−S(S+1)2J(J+1)+gSJ(J+1)−S(S+1)+L(L+1)2J(J+1)(2.15)The sign conventions above are given such that gJ is a positive number. The value of gJ is differentfor each state J and indicates the strength of the magnetic dipole for that state. Experimental valuesof gJ have been tabulated for many atomic states; for examples in Xe see Ref. [147]. The jl couplingscheme described in Section 2.4.1 is a more accurate model for Xe. The g-factor for the jl couplingscheme is calculated by the equation:gJ =2J+12K+1+2K(K+1)+ j( j+1)− l(l+1)(2K+1)(2J+1)(gJc−1), (2.16)where J,K, j,, l, and gJc follow the definitions in Section 2.4.1. In particular, gJc is the g-factor ofthe parent core with angular momentum Jc as described in that section and can be calculated fromEquation 2.15.For atoms posessing nuclear spin I, the magnetic dipole is modified by an additional term cor-responding to the nuclear dipole ~µI = gIµN~I [166]. The sign convention for ~µI is opposite to ~µJ; theg-factor gI is often given in nuclear magnetons µN , and the sign of gI may be positive or negativedepending on the nucleus in question. The magnetic dipole including nuclear spin is given by:~µ =−gJµB~J+gIµN~I =−gFµB~F (2.17)19and has corresponding Zeeman shifts:∆E = gFµBBMF (2.18)where F is the total angular momentum having projection MF , with g-factor gF given by:gF =F(F +1)+ J(J+1)− I(I+1)2F(F +1)gJ− F(F +1)− J(J+1)+ I(I+1)2F(F +1)µNµBgI (2.19)The second term, relating to the nuclear magnetic moment, is roughly three orders of magnitudesmaller ( µNµB ≈ 11836 ) than the first term and is often neglected in calculations for paramagnetic atoms.In that case, the main effect of the nuclear spin is its contribution to the total angular momentum F ,leading to an increase in splitting into 2F +1 sublevels. In 129Xe and other diamagnetic atoms, theground state has J = 0 and the ground state Zeeman splitting is due solely to the second term (nuclearmagnetic moment). The nuclear magnetic moments for relevant paramagnetic and diamagneticatoms are given in Table 2.1. The g-factor gI and gyromagnetic ratio γ = µI = gIµNh¯ there are bothcalculated based on data from [155]. The values in the table are those of the bare nuclei, and differfrom reported NMR values in atoms (e.g. γ/2pi = 11.77 MHz in [125]) due to diamagnetic shieldingof the magnetic dipole moment. The Zeeman splitting of the 129Xe 1S0 ground state and 129Xe6p[3/2]2 excited state are shown in Fig. 2.4 and Fig. 2.5. A state with a negative-slope Zeemanshift is called high-field seeking, and that with a positive Zeeman shift is called low-field seeking.In a 1µT field, the ground state Zeeman splitting of 129Xe is 11.86Hz. The excited state splittingis larger due to the electron spin but still below MHz. The form of the Hamiltonian changes whenthe atom is subjected to strong fields where the interaction is greater than the hyperfine couplingA~I · ~J. The strong field case is not calculated here, because such strong fields are not necessaryfor optical pumping. The role of Zeeman splitting in optical pumping is to break the degeneracybetween magnetic sublevels and keep them sufficiently well-separated in energy to prevent rapidrelaxation between levels; practically, this is satisfied for room temperature atoms by a 1µT fieldin EDM experiments. We take advantage of optical pumping to populate and depopulate particularsublevels.Optical Pumping and PolarizationOptical pumping was first proposed and demonstrated by A. Kastler in the 1950s. A descriptionof the method is given in [49]. Atoms possessing nuclear spin or electronic spin have degener-ate ground state sublevels which experience Zeeman splitting in a weak magnetic field. Opticalpumping is a method to change the population distribution between these sublevels using (typicallycircular) polarized light. An example for a spin-1/2 system is shown in Fig. 2.6. Absorption ofcircular-polarized light imparts angular momentum to the atom causing a change in the total angu-20Figure 2.4: Zeeman splitting of the 129Xe ground state due to nuclear spin.Table 2.1: Nuclear magnetic moment, g-factor, and gyromagnetic ratios of relevant isotopes.Magnetic moments reported are from [155]Nuclei Spin magnetic moment gI γ/2pi[µN] [J/T] [MHz/T]1n 1/2 -1.91304(5) -9.7×10−27 -3.82609 -29.16471H 1/2 2.792847(3) 1.41×10−26 5.585695 42.577483He 1/2 -2.1275(3) -1.1×10−26 -4.255 -32.434185Rb 5/2 1.35298(10) 6.83×10−27 0.541192 4.12528687Rb 3/2 2.75131(12) 1.39×10−26 1.834207 13.98141129Xe 1/2 -0.777976(8) -3.9×10−27 -1.55595 -11.8604131Xe 3/2 0.6915(2) 3.49×10−27 0.461 3.514015199Hg 1/2 0.505886(9) 2.56×10−27 1.011771 7.712319201Hg 3/2 -0.5602257(14) -2.8×10−27 -0.37348 -2.8469221Figure 2.5: Zeeman splitting of the 129Xe 6p[3/2]2 hyperfine components F = 3/2 (solid) andF = 5/2 (dotted), respective to their line zero-field energies. The hyperfine splitting (notpictured) between F = 3/2 and F = 5/2 is 2 GHz, with F = 3/2 having higher energylar momentum projection. In the case of one-photon transitions, the selection rule ∆M =+1 appliesfor excitation with σ+ light, and ∆M =−1 for σ− light. In the example shown, only the M =−1/2state will absorb σ+ light due to the selection rule; the nonabsorbing state is called a “dark state”.Relaxation occurs from the excited state back to both sublevels, with the ratio determined by theClebsch-Gordan coefficients. The result is that after many cycles of absorption and emission, theM =−1/2 state population will be depleted while the M =+1/2 state population will be increased.The degree of polarization of the ground state for such a spin-1/2 system is defined by therelative population of the Zeeman sublevels as [107]:P =N+−N−N++N−(2.20)where N+ and N− are the respective populations in the M = 1/2 and M =−1/2 states. In systemswith more than two levels, polarization is defined by comparing the total magnetization per unit vol-22Figure 2.6: Optical pumping in a spin-1/2 system.ume Mz to the theoretical maximum magnetization at absolute zero temperature, i.e. P =Mz(T )Mz(T=0K).Under a weak magnetic field, the Zeeman splitting is sufficiently small that all ground state Zee-man sublevels are nearly equally populated at ambient temperatures (those satisfying the conditionkT  h¯ω0) according to a Boltzmann distribution. At absolute zero, only the lowest energy Zee-man sublevel is populated. At finite temperature kT  h¯ω0, the thermal equilibrium polarization ina magnetic field B0 is given approximately by [107]:P0 =|γ|h¯B03kBT(I+1) (2.21)where γ is the gyromagnetic ratio, I the nuclear spin, and kB is Boltzmann’s constant. In opticalpumping, the term hyperpolarized describes a species with polarization larger than the thermalequilibrium polarization. Optical pumping can produce polarization which is orders of magnitudelarger than the thermal polarization. For example, the thermal polarization for 129Xe in a 1mT fieldis P = 9×10−10, compared to hyperpolarized 129Xe which can have a polarization reaching severaltens of percent.Optical pumping is used as a precursor to spin exchange optical pumping, which is describedbelow. Near-total polarization of rubidium and potassium is acheivable by diode lasers. In someoptical magnetometers, precession is detected by monitoring the transmission of circular-polarizedlight through the ensemble of precessing atoms, making use of the same selection rules which giverise to optical pumping.232.2.2 Spin Exchange Optical PumpingSpin exchange optical pumping (SEOP) refers to the polarization of noble gas atoms through col-lisions with optically pumped alkali metal atoms that result in spin exchange. These collisions canbe simple binary collisions, or can be three-body collisions leading to the formation of short livedalkali-metal-noble-gas molecules bound by van der Waals forces. Spin exchange is the transferof angular momentum from one atom to the other, accompanied by a transition between Zeemansublevels. During the interaction, spin exchange as well as relaxation can occur through the inter-actions between polarized spins and the magnetic field generated by moving electric charges. TheHamiltonian for spin exchange is [43, 162]H = A ~IAM ·~S+α ~ING ·~S+ γa~N ·~S+gsµB~B ·~S (2.22)where IAM is the alkali metal nuclear spin, ING is the noble gas nuclear spin, ~S is the alkali metalelectron spin, ~N is the molecular rotational angular momentum, and A,α,γa are respectively thecoupling constants for the alkali metal hyperfine interaction, the isotropic hyperfine interaction be-tween noble gas nuclei and alkali metal electron spin, and spin-rotation interaction. In particular thehyperfine interaction α ~ING ·~S is responsible for spin exchange. The strength, α , of the interactiondepends on the overlap of the electron wavefunction of the spin-polarized electron with the nucleusof the noble gas atom; the wavefunction is perturbed by the presence of the noble gas atom resultingin an enhancement in the overlap and in spin exchange for atomic separations on the order of anangstrom (A˚)[162]. The spin-rotation interaction ~N ·~S between the alkali metal electron spin andmolecular rotational angular momentum ~N causes spin relaxation.2.2.3 SEOP Literature ReviewA thorough review of spin exchange optical pumping was presented by Walker and Happer [162],and builds on the earlier theory published by Happer [82]. It presents the basic experimental tech-nique, and theory of the dominant spin interactions of the Hamiltonian which govern polarizationand relaxation via spin exchange. Rb is often used as the alkali metal of choice for spin exchangedue to its easily-driven optical resonance at the D1 line (795 nm), for which high-power diode lasersare readily available. Typical SEOP starts with optical pumping of a small amount of Rb vapour ina heated reservoir, followed by addition of a gas mixture containing the noble gas atoms. N2 andHe are often present in the gas mixture; N2 helps to quench the Rb excited state by collisional de-excitation and thereby prevent reabsorption of unpolarized Rb emission (radiation trapping), whileHe is a buffer gas which pressure broadens the absorption lineshape, in order to better take ad-vantage of the typical pump laser power spectrum. These gases also participate in van der Waalsmolecule-forming collisions.Bouchiat et al [31] first observed spin exchange optical pumping in 3He used as a buffer gas24in Rb optical pumping cells. It was stated that the technique should be applicable to any noblegas having a nuclear moment. Indeed, SEOP has been demonstrated in He, Ar, Ne, Kr and Xe;Calaprice et at [41] demonstrated SEOP on metastable nuclei and radioactive 133Xe [41]. SEOP on129Xe was first observed by Kanegsberg and later published by Grover [78]. Bouchiat [30] identifiedthe role of “sticking collisions” between Rb and noble gas atoms due to van der Waals forces as afactor in the relaxation of optically pumped Rb.Much of the work on SEOP has been to identify the relative contributions to the relaxation ofeither the alkali metal or noble gas polarization. Nelson et al [116] found that under high-pressureconditions with He buffer gas, Rb relaxation rates are dominated by Rb-Xe binary collisions, forwhich the rate is linearly proportional to the gas densities. Meanwhile, relaxation by formation ofvan der Waals molecules was responsible for less than 25% of the Rb relaxation (as the molecularlifetime becomes short at high pressure), followed by contributions from binary collisions of Rb orformation of Rb2 dimers, and diffusion to cell walls. Similar results were found by Jau et al [93]for SEOP cells with higher Xe density. Cates et al [43] found that the spin exchange rates for Xeand Rb depend on the Xe pressure. At low Xe pressure, van der Waals molecules are the dominantsource of spin exchange, while at high Xe pressure binary collisions dominate. Simulations carriedout by Fink et al [63] studied the dependence of Xe polarization on temperature and partial pressureof each gas component, finding good agreement with experiment. Anger et al [11] studied spin-relaxation mechanisms of 129Xe in the gas phase, confirming intrinsic spin relaxation at low fielddue to spin rotation interactions between 129Xe nuclei. They found the dominant contribution wasdue to the formation of Xe2 dimers bound by van der Waals forces. They also found significantextrinsic relaxation from collisons with cell walls.Inexpensive diode lasers have increased SEOP rates. Wagshul and Chupp [161] demonstratedup to 70% polarization for 3He pumped with a GaAlAs laser. Driehuys et al [59] used He topressure broaden the Rb transition and better match the spectral profile of a 140 W diode laser.They demonstrated 5% polarization and predicted up to 60% was attainable for volumes up toone liter by choosing suitable flow rates. Double-digit polarizations have been acheived by manygroups, including those from Princeton[42], New Hampshire [144], Harvard[119], Vanderbilt[118],Berlin[100], and Illinois[163]. A collaboration based at Vanderbilt demonstrated polarization up to90.9% using a homebuilt open-source polarizer with a 200 W frequency-narrowed laser [117].Spin-polarized noble gases have been used for studies of surface interactions [134], magneticresonance imaging of the lungs [117], neutron polarizers [51], and fundamental symmetry tests ofthe Xe EDM [142].2.2.4 Spin relaxation mechanisms for polarized XeThe hyperpolarization of Xe is temporary; there are many processes by which the atoms can relax totheir naturally unpolarized state. Relaxation is measured by the T1 lifetime which describes the loss25of longitudinal polarization by various processes in gas, liquid, or solid phase. The most commonprocess is the nuclear spin coupling to the angular momentum of a nearby particle [145]. In the caseof gas-phase Xe, relaxation is dominated by “self-relaxation” caused by collisions with other nearbyXe atoms through the spin-rotation Hamiltonian ~N ·~I identified by Carr et al. [86]. The collidingXe atoms form either a transient (in the case of binary collisions) or persistent dimer. Here ~N is theorbital angular momentum of the dimer and ~I is the spin of the polarized Xe atom. The resultinglifetime for binary collisions is inversely proportional to pressure, but is measured to be tens of hoursat atmospheric pressure. Faster relaxations from persistent dimers can be reduced by introducinga buffer gas to breakup molecules. Another relaxation mechanism is through wall collisions. Theatomic spin can couple with dipoles on the wall. It has been found that applying silane-basedcoatings to the walls of the gas cell help reduce the effect of wall collisions and can extend Xe T1lifetimes to 20-40 min [34], still limited by coupling to protons in the coating itself. Others chooseto use specially selected glass with high purity (e.g GE180), and produce some cells with lifetimesof hours even without coatings. Field gradients in the cell contribute also to relaxation. Rapidlymoving atoms can violate the adiabatic spin condition and rapidly flip when transiting small fieldgradients. This sort of gradient relaxation gives lifetimes on the order of minutes in inhomogeneous(eg. 10%) regions. Finally, any oxygen impurities in the gas will cause relaxation from coupling of129Xe to oxygen’s large molecular magnetic dipole moment.Lifetimes in the solid state can be up to hundreds of hours. Higher field strength and lowertemperatures contribute to long lifetimes. Since Xe freezes at 161.4 K, a typical continuous flowSEOP polarizer uses a LN2 cold trap at 77K to separate polarized Xe from the He/N2 mixture. Thesame spin rotation interaction occurs, except in the solid phase it is lattice vibrations and phononscattering that briefly couple the spins. At lower fields there is also a (weak) nuclear dipole-dipoleinteraction, and the potential of cross-polarization between 131Xe and 129Xe. A strong dependenceon temperature makes it important to freezeout all Xe at as low temperature as possible. Previousdesigns [144] made efforts to maximize the cold trap surface area, because Xe atoms that stickdirectly to the wall are likely colder than Xe that stick atop an already-frozen layer. Sublimation ofthe frozen Xe helps to avoid the rapid relaxation which occurs in the liquid phase.2.2.5 NMR techniques used to measure optical pumpingNMR techniques are used on the polarized nuclei to quantify the polarization as a percentage. Thetechniques of most practical use are Adiabatic Fast Passage (AFP) and Free Induction Decay (FID),which are described below. AFP is the simplest technique for measuring the polarization betweenspecies. FID serves as a testbed for applying pi/2-pulses and testing the effects of transverse relax-ation.26Adiabatic Fast PassageAFP is an NMR technique used to assess the degree of polarization of a species by measuring themagnetic induction from a precessing sample during a pulse sequence that exactly inverts the nuclearspin. It is adiabatic in the sense that the magnetic moment maintains the same magnetic quantumnumber with respect to the field in a specially-defined rotating reference frame. The techniquewas proposed by Bloch [24] in 1946 while formulating the well-known Bloch equations, and wasdemonstrated the same year [25] in samples of water and paraffin.The technique is illustrated in Fig. 2.7. In the technique, polarized atoms are subjected to twofields: first, a strong field B0kˆ which defines a Larmor frequency for the nucleiωL = γB0; (2.23)second, a weaker oscillating RF field 2×B1 cos(ωt)iˆ applied transverse to B0 with nearly-resonantωRF ≈ ωL. The polarization, which may be the result of thermal Boltzmann polarization or opticalpumping, results in a net magnetization ~M in the direction of B0. The magnetization follows theBloch equations and has components (Mx,My,Mz). In the presence of relaxation the componentshave the following time dependence [24]:dMxdt= γ(~M×~B)x−MxT2(2.24)dMydt= γ(~M×~B)y−MyT2(2.25)dMzdt= γ(~M×~B)z−Mz−M0T1(2.26)The relaxation times T1 and T2 refer to longitudinal and transverse relaxation, respectively; M0 isthe (thermal) equilibrium magnetization under B0.Ramping B0 field strength changes the Larmor frequency, bringing the atoms into and out ofresonance with the fixed B1 frequency. Under suitable ramp conditions (described below), the atomsadiabatically follow the effective field caused by the combination of B0 and B1. This effective fieldis most easily calculated in a rotating reference frame rotating at ω = ωRF . The linear B1 can bedecomposed into a sum of two fields rotating in different directions, (B1)(cos(ωt)iˆ± sin(ωt) jˆ), oneof which appears stationary in the rotating frame, the other of which is rapidly counter-rotating andcan be neglected (more precisely, it contributes a small Ramsey-Bloch-Siegert shift). The effectivefield in the rotating frame is given by vector addition of B0 and B1 [132]:~Be f f =(B0− ωγ)kˆ+B1 iˆ (2.27)27Figure 2.7: Applied field and magnetization response during the AFP technique, in the sim-plified case with no relaxation.The additional term ωγ is a result of the coordinate transformation. As B0 is ramped from B0 >ωγto B0 < ωγ , the direction of ~Be f f changes from kˆ to −kˆ. Under adiabatic conditions ~M follows ~Be f fand completely inverts. The adiabatic conditions are [24]:B1T1,B1T2 dB0dt γB21 (2.28)The left hand side indicates that the magnetization must pass through resonance before significantrelaxation occurs; the right hand side indicates that precession around | ~Be f f | (having a minimumvalue B1) must be sufficiently fast for the magnetization to remain “locked” to ~Be f f . The detectedsignal is the Faraday induction produced by the precessing ~M in a pickup coil mounted along they-axis. After a spin flip, the magnetization slowly relaxes to equilibrium value ~M0 with decay timeT1.28Figure 2.8: Applied field and magnetization response during the FID technique, in the sim-plified case with no relaxation.Free Induction DecayFID is another useful technique, which can measure the transverse lifetime T2 of the spins. Anillustration of the technique is shown in Fig. 2.8. The applied B0 field is fixed and highly uni-form. Polarized species are subjected to a gated oscillating RF field with rotating components(B1)(cos(ωt)iˆ± sin(ωt) jˆ). The gate time and amplitude of this RF field are carefully chosen toinitiate a rotation of the magnetization vector through an angle pi/2 into the xy-plane. After this theRF field is gated off and the spins undergo free spin precession about B0 according to the Blochequations. The magnetization dephases in the xy-plane with relaxation time T2 and undergoes lon-gitudinal relaxation with time T1. Typically transverse relaxation dominates, a result of spin-spinrelaxation and inhomogeneities in B0. The detected signal again comes from Faraday induction pro-duced by ~M precessing in the xy-plane. An identical pulse sequence is used in free spin precessionmagnetometers, but detection for magnetometry is performed using optical methods or sensitiveSQUIDs.292.3 Two Photon Transitions and SpectroscopyTwo photon transitions are those involving either the simultaneous absorption or simultaneous emis-sion of two photons with total energy equal to the difference in initial and final states of the atomor molecule. The two photons can be of the same or different wavelength so long as the combinedenergy of the two satisfies conservation of energy. An example is shown in Fig. 2.9. The transitionis often said to occur through a virtual intermediate state, which is in fact a sum over all off-resonantdipole-allowed states. Since there is no actual resonant intermediate state, two photon transitionsdiffer from stepwise excitation obtained by two sequential electric dipole E1 transitions. Nor arethey higher-order multipole (e.g., electric quadrupole E2) transitions. The transition probabilitycomes from second-order perturbation theory following the dipole approximation. The selectionrules for two photon transitions are different than those for one photon transitions; in particular, twophoton transitions occur between states of the same parity.Two Photon Absorption (TPA) is a process where two photons are absorbed “simultaneously”(meaning within the lifetime of the virtual state). The atom can reside in the virtual intermediatestate only for a short time limited by the uncertainty principle. TPA requires absorption of thesecond photon within the virtual state lifetime; therefore high intensity excitation light must be usedto excite TPA at detectable rates. The transition rate (see below) is proportional to the squaredintensity of the incoming radiation. There is a corresponding emission process called Two PhotonEmission (TPE), which can occur out of excited states. There is a finite, nonzero probability fortwo photon emission over a continuous spectrum of wavelengths. Two photon emission transitionscan, however, be most readily observed in metastable states with long lifetimes, as single photonemission from such levels is dipole-forbidden.A typical scheme involves excitation to a dipole-forbidden state by two-photon absorption,followed by stepwise relaxation by two dipole-allowed transitions. The observed fluorescence istermed Two Photon Absorption Laser Induced Fluorescence (TALIF). These transitions are partic-ularly useful, as the dipole allowed LIF transitions typically occur at wavelengths far from that ofthe excitation light, and so are easy to resolve. A common use of two-photon absorption [66] is theuse of a single high-power laser to investigate transitions between states of same parity, in partic-ular those transitions that would otherwise require VUV radiation. Another is the ability to exciteDoppler-free transitions (see section below).2.3.1 Derivation of two photon transition probabilityThe theory of two photon transitions was first developed in 1931 by Maria Goeppert Mayer [72],who showed the possibility of two photon absorption and emission, as well as Raman transitions,by performing second order perturbation theory in the second quantization (where both the energylevels and the number of photons are quantized).30Figure 2.9: General representation of a two-photon transition (in this case, absorption) be-tween two states |i〉 and | f 〉. Excitation is by two photons ω and ω ′, which satisfy theconservation of energy h¯(ω +ω ′) = h¯ω0. The transition occurs through a virtual state(dotted line), given by a sum over off-resonant intermediate states |k〉 which are dipole-allowed.We derive here the transition probability for two photon absorption based on the above work.We consider a stationary atom with initial state |i〉, intermediate state |k〉, and final state | f 〉, excitedby two laser beams of electric field strength E1,E2 (intensity I1, I2) and of the same polarization,with respective frequencies ω1,ω2. The intermediate state is not resonant with either of ω1,ω2.Under the dipole approximation, the perturbing Hamiltonian between photon and atom is:H =∑rE1e~r cos(ω1t)+E2e~r cos(ω2t) (2.29)where e~r is the usual electric dipole operator. Expressed in terms of matrix elements Hki,n =En 〈k|∑r e~r |i〉, where n = 1,2 refers to the two beams, the perturbation is (neglecting counter-31rotating terms):Hki =Hki12e−iω1t +Hki22e−iω2t (2.30)First order perturbation theory describes transitions between the initial and intermediate state, withprobability amplitude a(1)k :a(1)k =12h¯[Hki1(ei(ωki−ω1)t −1)ωki−ω1 +Hki2(ei(ωki−ω2)t −1)ωki−ω2](2.31)Evaluating the perturbation to second order with perturbing Hamiltonian H f k describes transitionsto the final state. This yields four terms of the following form (each with different H f k1Hki2):H f k1Hki14h¯(ωki−ω1)[ei(ω0−2ω1)t −1h¯(ω0−2ω1) −ei(ωk f−ω1)t −1h¯(ω f k−ω1)](2.32)where ω0 = ω f k +ωki. The second term in the expression can be neglected [72], as it is an artifactof turning on the perturbation abruptly at t = 0. The resulting probability amplitude a(2)f summedover all intermediate states |k〉 is therefore:a(2)f =14h¯2∑k(H f k1Hki1(ωki−ω1)ei(ω0−2ω1)t −1(ω0−2ω1) +H f k2Hki2(ωki−ω2)ei(ω0−2ω2)t −1(ω0−2ω2)+H f k2Hki1(ωki−ω1)ei(ω0−ω1−ω2)t −1(ω0−ω1−ω2) +H f k1Hki2(ωki−ω2)ei(ω0−ω1−ω2)t −1(ω0−ω1−ω2))(2.33)The transition probability is proportional to |a(2)f |2. The first denominator of each term describesthe off resonant interaction of photon ω1,2 with the intermediate state |k〉, which has energy defecth¯(ωki−ω1,2). The second denominator is the energy conservation term which yields the absorptionlineshape described in the following section. The sum over all (off-resonant) one-photon allowedintermediate states |k〉 is often described as a transition to a short lived “virtual” state. Since thetransition dipole for each intermediate state is weighted by its respective energy defect, only thestates with energy levels close to the energy of the incident photon contribute significantly.Selection rules for two photon transitionsThe levels accessible by two photon transition are the same as those accessible by two sequentialelectric dipole transitions [66]. A consequence of this is that two photon transitions only occurbetween states of the same parity. The selection rules for two photon absorption are given in [28],and come from angular momentum conservation. The rules can be determined by evaluating theClebsch-Gordan coefficients. The general rule is ∆F ≤ 2 for an atom with total angular momentum32F and projection MF . The values of allowed ∆F and ∆MF depend upon the orientation and polariza-tion of the incident light. For two photons from a single beam with linear polarization parallel to thequantization axis (sometimes denoted pi) we have ∆F = 0,±2 and ∆MF = 0. For two photons froma single beam with circular polarization σ+, we have ∆F = 0,±2 and ∆MF = +2. A result of theselection rules is that one can selectively probe certain levels, e.g.) circular polarized light can beused in the detection of polarized Xe to selectively probe the ground state MF = −1/2 populationas described in Section 2.4.3.2.3.2 Doppler-Free Two Photon SpectroscopyIn Section 2.3.1, we considered only a stationary atom. If we expand the treatment to include anatom moving with velocity ~υ and consider two laser beams with wave vectors ~k1,~k2, Equation 2.33becomes:a(2)f =14h¯2∑k(H f k1Hki1(ωki−ω1−~υ ·~k1)ei(ω0−2ω1−2~υ ·~k1)t −1(ω0−2ω1−2~υ ·~k1)+H f k2Hki2(ωki−ω2−~υ ·~k2)ei(ω0−2ω2−2~υ ·~k2)t −1(ω0−2ω2−2~υ ·~k2)+H f k2Hki1(ωki−ω1−~υ ·~k1)ei(ω0−ω1−ω2−~υ ·(~k1+~k2))t −1(ω0−ω1−ω2−~υ · (~k1+~k2))+H f k1Hki2(ωki−ω2−~υ ·~k2)ei(ω0−ω1−ω2−~υ ·(~k1+~k2))t −1(ω0−ω1−ω2−~υ · (~k1+~k2)))(2.34)As a result of the interaction between ~υ and~k, lineshapes of atomic transitions at room temperatureare typically Doppler-broadened on the order of GHz, requiring techniques such as saturated absorp-tion spectroscopy to achieve better resolution. Each photon’s frequency in the atom’s rest frame isDoppler-shifted by ~υ ·~k. Typically, only one velocity class of atoms can absorb light from ω1 or ω2for a given detuning, equal to its Doppler shift. With two-photon absorption it is possible to choose~k1 and ~k2 such that the sum of the respective Doppler shifts is zero. This can be achieved by usingcounterpropagating ~k1 and ~k2, either by splitting or retroreflecting a single laser source, or by usingcounterpropagating light from two laser sources. If the Doppler shifts sum to zero, then atoms ofall velocity classes can participate simultaneously in the on-resonance absorption by absorbing onephoton from each beam. The result is a large Doppler-free absorption signal on-resonance with anarrow Lorentzian lineshape, superimposed on a much broader Doppler-broadened Gaussian peak.The amplitude of the on-resonance signal is inversely proportional to the linewidth via the lineshapefunction g(ω).Vasilenko was the first to calculate the Doppler-free lineshape for two counterpropagating beams[159]. For the case of equal intensities I1 = I2 = I and equal frequencies ω1 = ω2 = ω , Grynberg33et. al [79] showed that the area of the Doppler-free Lorentzian curve for counterpropagating beamsis two times the area of the Doppler-broadened Gaussian curve. To demonstrate this the termsof Equation 2.34 must be considered independently. The Gaussian curve comes from the sum ofthe two independant transition probabilities for the first two terms, integrated over the velocitydistribution; the area is two times larger than the area for a single travelling wave. The Lorentziancurve comes from the transition probability of the indistinguishable third and fourth terms, for whichthe amplitudes must be squared; the corresponding area is four times larger than the area for a singletravelling wave. The sum of the Lorentzian and Gaussian curves therefore has a total area six timeslarger than the single travelling wave case. The two-photon absorption transition probability is givenby [57, 159]:W = |a(2)f |2 =∑k∣∣H f k∣∣2 |Hki|216h¯4(ωki−ω)2{1(Ω−2kυ)2+ γ2 +1(Ω+2kυ)2+ γ2+4(Ω)2+ γ2}(2.35)where Ω = ω0−ω1−ω2 = ω0− 2ω is the detuning and γ is the homogeneous linewidth. Afteraveraging over the velocity distribution this yields the transition probability W :W =∣∣H f k∣∣2 |Hki|216h¯4(ωki−ω)26g(ω) (2.36)with lineshape g(ω):g(ω) =16(2× 22√pikυe−Ω2/(2kυ)2 +4× 2piΓL/2Ω2+(ΓL/2)2). (2.37)The Doppler-broadened Gaussian FWHM is ΓG = 4√ln(2)kυ ,while ΓL is the homogeneous (e.g.,pressure-broadened) Lorentzian FWHM of the Doppler-free peak. The lineshape is normalized bythe factor 16 such that∫g(ω)dω = 1. The combined lineshape profile is shown in Fig. 2.10. Onexact resonance (Ω= 0), we have:g(0) =16(2√pikυ+16piΓL). (2.38)The Gaussian FWHM is typically orders of magnitude larger than ΓL under room temperature andat pressures of a few Torr. Therefore the first term of Equation 2.38 is typically less than 1% andcan be neglected.High-resolution spectroscopy is made possible by Doppler-free two photon excitation. The firstexperimental Doppler-free two photon absorption spectrum was obtained by Biraben et. al. [22] forthe sodium 3s−5s transition, with a signal roughly an order of magnitude larger than the Doppler-broadened background. They also showed complete elimination of the Doppler background by34circular polarizing the light in such a way that the selection rules for the particular transition forbidabsorption of two co-propagating photons. Simultaneous observations were made by [105, 130].Next, Hansch et al [80] measured the Doppler free spectrum in hydrogen of the metastable 1s−2stransition, and measured the Lamb shift very precisely. In xenon, Raymond et al [138] measured thetwo photon absorption coefficients of three dipole-allowed 6p← 5p Xe transitions using a pulsedlaser and high Xe pressures to measure collisional excitation between excited states for pressures100 Torr or greater. Later, Plimmer et al [128] used a CW dye laser to perform high-resolutionDoppler-free two-photon spectroscopy on the 249 nm transition in Xe, which showed no hyperfinestructure as the ground and excited states are both J = 0. This made for ease of identifying isotopeshifts, on the order of 130 MHz between neighbouring even isotopes as well as between 129Xe and131Xe. The measurement was enabled in part by the use of an optical cavity enclosed in a cell,which enhanced the intensity of excitation light by an order of magnitude, and enabled two-photonabsorption measurements at Xe pressures as low as 100 mTorr. At higher pressures they measureda Lorentzian line shape and pressure broadening of FWHM of 28.8 MHz/Torr, and a shift of 9.5MHz/Torr towards lower frequency. Seiler et al [150] perform high-resolution Doppler-free twophoton spectroscopy on the 249 nm transition of xenon in a gas jet, and use subsequent ionizationby a third photon for detection. They were able to obtain isotope shifts with resolution limited to 10MHz by their pulsed laser linewidth.Cross section and transition rateA review article by Rumi and Perry defines the two-photon cross section δ (with units of GM =10−50cm4 s after Maria Goeppert-Mayer) analogous to the single-photon cross section σ in terms ofthe change in photon flux φ = I/h¯ω per path length z, for excitation with a single beam [143, 156]:dφdz=−σnφ −δnφ 2, (2.39)where n is the number density of atoms, and goes on to show the two-photon excitation rate (numberof atoms per unit time and volume) is:dndt=12δnφ 2, (2.40)More common in the literature is the definition σ (2) = 12δ such thatdndt= σ (2)nφ 2, (2.41)Literature often reports the transition rate W = 1ndndt per atom:W = σ (2)(I/h¯ω)2, (2.42)35Figure 2.10: Example plot of the combined lineshape for Doppler-free excitation (blue) com-pared with the Doppler-free background (red) as a function of detuning. The on-resonant lineshape value can be many times larger than the Doppler broadened valuefor a sufficiently small homogeneous linewidth. The Doppler-broadened width is typ-ically wider than shown, but is exaggerated here to make the amplitude visible. Thearea of the Lorenztian Doppler-free profile is twice that of the retroreflected Doppler-broadened profile, and four times that of a Doppler-broadened profile for absorption ofa travelling wave.It must be noted that the cross section σ (2) is dependant on lineshape g(ω) and frequency ω . Onecan alternately define [148] a lineshape-independant cross section σ (2)0 with units of cm4:σ (2) = σ (2)0 G(2)g(ω). (2.43)where G(2) is a dimensionless photon statistical factor that accounts for photon coherence in ex-citation with multimode pulsed lasers, and g(ω) is the absorption lineshape function. In our CWmeasurements we assume G(2) = 1. The lineshape function g(ω) used depends on the experimentalconditions, such as the beam configuration, laser linewidth, and atomic linewidth due to homo-36geneous (lifetime, pressure) and inhomogeneous (Doppler) broadening. In general it should benormalized such that∫g(ω)dω = 1. The cross section σ (2)0 is then directly related to the quantumstate by Equation2.35:σ (2)0 ∝∑k∣∣H f k∣∣2 |Hki|216h¯4(ωki−ω)2(h¯ωI)2(2.44)Often experimental reports in the literature define a two-photon coefficient α in cm4 J−2 whichis also lineshape-independant. For excitation with a single laser of intensity I, the two photonexcitation rate W may be written as [127, 138]:W = αI2g(ω) (2.45)where the lineshape g(ω) is typically Doppler-broadened and resembles only the first term of Equa-tion 2.37. The total number of atoms excited per second is given by:dNdt=WnV = αI2g(ω)nV (2.46)where N = nV is the number of Xe atoms in excitation volume V . By comparison of Eqns. 2.42 and2.45, one can infer the relation:α =σ (2)0(h¯ω)2(2.47)(One can never have too many conventions. There is yet another definition α∗ = σ (2)/h¯ω , which isstill lineshape dependent and has units of cm4 W−1. Table 2.2 lists the various expressions.)Table 2.2: Common notations for two photon cross section in the literature.Symbol Formula Unitsδ - cm4 sσ (2) δ/2 cm4 sσ (2)0 σ(2)/(g(ω)G(2)) cm4α σ (2)0 /(h¯ω)2 cm4 J−2α∗ σ (2)/h¯ω cm4 W−1Earlier in this section we discussed the enhancement in transition rate when performing Doppler-free two photon absorption. We make this enhancement explicit by multiplying Equation 2.45 by afactor of six following Raymond [138]:W = 6αI2g(ω) (2.48)with the lineshape g(ω) normalized to unity by Equation 2.37. This accounts for the combined37Doppler-broadened and Doppler-free profiles, and ensures that the value of α is consistent with thatobserved from Doppler-broadened excitation. This allows us to compare our results with those inthe literature from previous experiments, the majority of which are Doppler-broadened.2.3.3 SuperradianceIn 1954 Dicke predicted spontaneous coherent emission coming from excited atoms [58], which hassince been observed. In his highly mathematical paper, Dicke modeled a system of two-level atomswith a state that had a positional part and an internal energy part. While in general the emissionintensity of spectroscopy experiments is proportional to the number of emitters N, in superradianceit is the electric field emission which is proportional to N, making the emission intensity propor-tional to N2. The lifetime of an excited state under superradiance decreases (according to N). Thelifetime of a superradiant state also decreases with higher excitation laser power. An effect whichdisplays some of the properties of superradiance has been observed in Xe [137] under two photonexcitation. Section 4.1.2 presents our experimental observation of this effect using a pulsed laser.If an enhancement in LIF emission intensity can be observed under the excitation conditions in theplanned Xe comagnetometer, it means lower Xe pressures can be used in the experimental cell.2.4 XenonXenon was discovered in 1898 by Ramsay and Travers [135] through fractional distillation of air.It found early use in flashlamps [60] and as an anesthetic [52]. Neil Bartlett discovered the firstnoble gas compound at UBC in 1962; xenon formed an ionic compound with the strongly-oxidizingplatinum hexafluoride PtF6, due to similarity of xenon’s ionization energy (12.13 eV) and that ofdioxygen (12.2 eV) which had been previously shown to undergo oxidation [18]. This discoverysparked the renaming of the inert gases as noble gases.Xenon is a noble gas with atomic number 54. Natural abundance xenon consists of eight stableisotopes between masses 124-134 and one slightly radioactive isotope 136 (half-life≈ 1021 yr), listedin Table 2.3. The two stable odd isotopes possess non-zero nuclear spin. Xenon has a ground stateelectron configuration of [147]1s22s22p63s23p64s23d104p65s24d105p6 (2.49)and corresponding angular momentum term symbol 1S0, and is therefore diamagnetic. The lowest-lying electronic states of Xe are shown in Fig. 2.11. As a noble gas, xenon has a closed valenceshell meaning electrons are tightly bound. Xe has a first ionization energy of 97833.79 cm−1 or12.1 eV [147]. Excitation to the first excited electronic states 6s or 6s′ requires either discharge orVUV photon. The next excited state 6p, however, (around 78000 cm−1) is accessible by two-photontransitions at Deep Ultraviolet (DUV) wavelengths and is the focus of this work.38Figure 2.11: Excited state energy levels of Xe I relative to the ground state energy, shown withtheir respective electron configurations. Here a prime on the configuration indicates thestate with a core term symbol 2P1/2, and the unprimed state indicates the 2P3/2 core.Spin-orbit coupling is determined by the jl or Racah coupling scheme. The dotted lineindicates the ionization threshold for the 2P3/2 core. Created using the values for energylevels cited in [147]There are two-photon allowed transitions to 6p states (see Fig. 2.12) at 249.6, 252.5 and 255.9nm,labelled in Racah’s notation (see Section 2.4.1) as 2[1/2]0, 2[3/2]2, and 2[5/2]2, respectively. Addi-tionally, based on the ionization energy it is possible for a (2+1) Resonance-Enhanced MultiphotonIonization (REMPI) process to ionize Xe following TPA upon absorption of a third UV photon. Thisis not studied in the present work. The TPA excited 6p levels above can spontaneously decay to 6slevels 2[3/2]1 and 2[3/2]2 by emitting a Near Infrared (NIR) photon, and subsequently decay to theground state by emitting 147.0 nm light [147]. One 6s state in particular (2[3/2]2) is metastable.Spectroscopic study of the above 6p states by two-photon absorption is covered in Chapter 4.39Figure 2.12: Allowed two-photon transitions to the Xe 6p states 2[1/2]0, 2[3/2]2, and 2[5/2]2.The transitions occur between states of the same parity (here both ground and excitedstate are even parity).Table 2.3: Natural abundances and nuclear spin I of the stable xenon isotopes.Isotope Natural Abundance I124Xe 0.095% 0126Xe 0.089% 0128Xe 1.91% 0129Xe 26.40% 1/2130Xe 4.10% 0131Xe 21.20% 3/2132Xe 26.90% 0134Xe 10.40% 0136Xe 8.90% 0402.4.1 Angular momentum coupling schemes in XeLS couplingThe splitting of energy levels within each electron configuration is determined by electrostatic in-teractions and spin-orbit interactions. The most commonly encountered scheme is called LS (orRussell-Saunders) coupling [153], and is used when the electrostatic interactions dominate. Thetotal orbital angular momentum L and total electron spin S are individually determined and sum toJ = L+ S. The opposite extreme where spin-orbit interactions dominate is called j j coupling. Inthat case ji = li+ si is determined for each electron i before summing to a total J = ∑ i ji.These schemes hold approximately for light and heavy atoms, respectively. Various intermediatecoupling schemes exist including the jl coupling observed in xenon and described below. In singly-excited xenon, LS coupling is used to describe the Xe (2P1/2 or 2P3/2) core.jl coupling and Racah notationFor xenon and other noble gases, the filled outermost p− shell is tightly bound. Any excited electronhas much lower binding energy than the tightly bound core. For these atoms the suitable couplingscheme is the jl coupling described by Racah [133], which describes coupling between a parent corewith total angular momentum Jc (described well by LS coupling) and a valence electron with orbitalangular momentum l. The two momenta add to form an intermediate angular momentum, K = Jc+ lwhich then couples with the spin s of the excited electron to form a total angular momentum J =K+s. The commonly-used notation (found in tables published by Moore [114] and Saloman [147])is that suggested by Racah (where nclc is the electron core configuration):nclc(2S+1LJc)nl(2s+1)[K]J (2.50)The result of the spin-orbit coupling to the single valence electron means every term is a doubletwith two values, J = K±1/2. Thus, the term 2[1/2]0 describes a doublet with intermediate angularmomentum K = 1/2 and total angular momentum J = 0.Some authors (see, e.g. [168]) label the excited states of xenon using Paschen notation, inwhich 1s1,1s2, ... are labels for singly excited states with the lowest electron configuration (5p56s),followed by 2p1,2p2, ... for states of configuration (5p56p). The subscripts indicate terms of succes-sively lower energies. For xenon, the 5p5(2P3/2)6s 2[3/2]2 state is labelled 1s5 in Paschen notation.2.4.2 History of Xenon SpectroscopyThe first systematic spectra were obtained in discharge cells by Meggers [108] and Humphreys[109], motivated by a search for new wavelength standards. By analysis of the complicated dis-charge spectra, Jones [95] was able to confirm the nuclear spin I = 1/2 for 129Xe and recommend41I = 3/2 for 131Xe. This was followed by many studies of hyperfine structure and isotope shifts.In particular, Jackson et al. [50, 89–92] undertook a systematic study of the hyperfine structureand isotope shifts of emission spectra. In time more detailed observations were made possible byDoppler free methods of saturated absorption [21] and two photon absorption [128]. Measurementswere made also of excited level natural lifetimes and collisional deexcitation rates [35, 61].Optical pumping was demonstrated in Xe as early as 1969 in metastable Xe [149]. Later effortsshowed the ground state could be polarized through spin exchange with Rb [78]. The theory of spinexchange was presented in detail by Happer [81, 82]. There are recent efforts producing large vol-umes of hyperpolarized Xe with polarization approaching 100% [117, 144]. These find applicationin clinical studies, in particular lung imaging.Starting in the 1980s two photon excitation and REMPI were studied with an aim of single atomdetection using powerful new pulsed lasers [44, 73]. Plimmer [128] demonstrated high precisionDoppler-free spectroscopy using retroreflected narrow-linewidth CW light. As resolution improvedefforts were made to determine absolute transition frequencies [33, 122, 150]. Goehlich et al [70]use two-photon absorption in xenon as a calibration to determine the number density of atomicoxygen via laser-induced fluorescence at similar wavelengths.2.4.3 Detection scheme for polarized 129XeTwo photon excitation enables a scheme for detection of the Xe ground state spin polarizationwhich is analogous to that of one-photon optical pumping in alkali atoms. The scheme is depictedin Fig. 2.13. Using circular polarized light at 252.5 nm, one can selectively excite transitions with∆M =+2. The ground state of 129Xe is twofold degenerate with total angular momentum F = 1/2;therefore only the MF = −1/2 sublevel can be excited following the ∆M = +2 rule. The othersublevel MF = 1/2 becomes a dark state. We target the level 2[3/2]2(F = 3/2); an equivalentscheme is also possible for 2[5/2]2(F = 3/2). The rate of two-photon transitions in a coherentensemble of Xe atoms precessing in the plane of~k will therefore oscillate at the Larmor frequency,which will in turn generate an oscillating rate of laser induced fluorescence. We detect emission ofNIR light at 895.5 nm and 823.4 nm from the 6p−6s transition, although it is in principle possibleto also detect VUV emission from 6s→ 5p.42Figure 2.13: Detection scheme for polarized 129Xe using two photon transitions.43Chapter 3Production of polarized 129Xe by SpinExchange Optical PumpingBefore Xe can be used in an optical magnetometer it must be polarized. The purpose of this chapteris to describe the methods used to produce polarized 129Xe, namely spin exchange optical pumping(SEOP) with Rb vapor. We describe the apparatus used for SEOP and for NMR detection; a rateequation model used to estimate polarization and relaxation rates for comparison with experimentalresults; and measurement of some of the factors affecting the polarization.3.1 Experimental Apparatus and TechniqueWe constructed a SEOP polarizer and low-field NMR spectrometer for the Xe polarization pro-cess, with components contributed by J. Martin (U. Winnipeg) and J. Sonier (SFU). These are bothdescribed under their own headings below:3.1.1 Spin Exchange Optical Pumping ApparatusSpin exchange optical pumping in our apparatus occurs in a mixture of Xe, Rb, N2, and He in apressurized Pyrex cell irradiated with polarized NIR 795nm photons. The N2 facilitates the spin-exchange process and provides the collisional de-excitation of the Rb vapour commonly known as“quenching” to prevent radiation trapping. Helium is used as a buffer gas to pressure broaden theD1 transition and maximize laser absorption. This section describes the SEOP cell, optical andgas-handling setups. A schematic of our polarizer is shown in Fig. 3.1.The polarizer cell provided by our collaborators is designed by the Steacie Institute for Molec-ular Sciences (National Research Council, Ottawa), [34] and manufactured by U. Winnipeg glass-blowing. It is a Pyrex-constructed flow through cell consisting of i) a 2.54 cm diam. reservoirvolume filled with 1 gram Rb metal connected by a 1 cm diam. stem to ii) a 2.54 cm diam. cylindri-44Figure 3.1: Gas flow and optics of the SEOP polarizer. BS: beam splitter, PBS: polarizingbeam splitter, PM: power meter, λ /4: quarter wave plate.Figure 3.2: SEOP cell with heater tape and solenoid removed, showing the Rb reservoir (left)and optical pumping volume (center).cal volume, 10 cm in length, with IR-transparent optical flats at both ends for Rb optical pumping.The pumping volume is coated with Surfasil to reduce depolarization on the cell walls. Fig. 3.2shows a picture of the cell.Typical SEOP parameters are listed in Table 3.1. We use a gas sample of of 1% natural abun-dance Xe, 3% N2, and 96% He (Praxair, premixed), at total pressure 60 psi(g) controlled by a pres-sure regulator. The premixed gas passes through two gas chromatography filters via stainless steel1/4-inch lines: a Model 1000 oxygen purifier (Chromatography Research Supplies) and a combinedoxygen/moisture trap (Agilent, OT3-4-SS). A purge valve before each trap helps remove impurities45Table 3.1: Typical SEOP parametersParameter ValueTypical Rb reservoir temp 200-300 CTypical SEOP temp 80-100 CSEOP solenoid field 10 mTDensity of gases at 50 PSIg:He 8×1019cm−3N2 2.4×1018cm−3Xe 8×1017cm−3Rb (at 400K) 6×1012cm−3Rb (at 500K) 4×1015cm−3Resonant laser flux of 1.2 W 5×1018photons−1and extend the trap lifetime. The purified gas passes through a mass flow meter (Kofloc 8100) tothe Rb reservoir, mixes with Rb vapour (nRb ≈ 1012−1015 cm−3), and enters the pumping volumewhere spin exchange occurs. The SEOP cell is centered in a 6.5” long solenoid (750 turns of 20-AWG magnet wire, magnet constant 5 mT/A), which produces a 10 mT magnetic field for Zeemansplitting, aligned antiparallel to the direction of gas flow. The cell and reservoir are both individuallywrapped with heater tape controlled by separate Variac transformers. Three K-type thermocouplesare fixed to the cell body, near the center and at each window, to aid in uniform heating. Downstreamof the cell, a small piece of glass wool condenses out the remaining Rb vapour. A needle valve witha custom brass needle (Swagelok B-4MG, modified by UBC Chemistry Mechanical Engineering)controls the flow rate of the polarized gas. The mixture is pushed along by the pressure difference(slightly greater than 1 atm), through 1/4-inch tubing to the NMR apparatus.The pumping laser is a Coherent fiber array packaged diode laser producing up to 30 W of795nm (Rb D1) light with a 2nm linewidth. The unpolarized multimode output is circularly polar-ized by using a polarizing beam splitter (Casix, BPS0402) and quarter wave plate (Casix, WPZ1425-800). An additional lens mounted before the beamsplitter helps collimate the multimode light exit-ing the laser head to achieve a roughly 2cm beam diameter. We later installed beam recombinationoptics (Fig. 3.3) to “recycle” the s-polarized light which is otherwise wasted. An additional quarterwave plate converts the secondary beam to the same circular polarization as the primary beam, andthen it is injected into the cell at a slight beam path angle to the primary beam.The polarizer is shown operating in Fig. 3.4. Due to the multimode nature of the pump laser,the output cannot be well collimated. This is apparent by the significant divergence and scatter ofthe light as it appears on the downstream power meter. Detection of the transmitted power throughthe SEOP cell is by a fiber-coupled optical spectrum analyzer (HP 86142A) with 0.1 nm resolutionbandwidth and a sweep rate ≈ 1Hz. A spectrum of the optically pumped Rb transmission is shownin Fig. 3.5. The evidence for optical pumping is the increased transmission of resonant light that46Figure 3.3: Beam recombination optics “upgrade” used to circular polarize and overlap bothlinear polarized outputs (the primary p-pol and secondary s-pol) from the PBS. BS: beamsplitter, PBS: polarizing beam splitter, PM: power meter, λ /4: quarter wave plate.Figure 3.4: SEOP cell during operation. Scattered D1 pumping light appears in false-color aspurple on the camera’s sensor.occurs rapidly (within 1s) when the SEOP solenoid magnetic field is switched on. There is a dra-matic difference in light transmission between field ON/OFF conditions, such that the cell becomesalmost completely transparent to the pumping light. Light in the spectral wings is not absorbedbecause our laser linewidth is significantly broader than the pressure-broadened atomic linewidth.The “useful” power of our beam therefore depends on the pressure broadening determined in Sec-tion 3.3.3. Based on integration of the spectral power density in Fig. 3.5, we estimate that less than12 % of photons in the laser beam are resonant with the optical pumping transition.3.1.2 NMR ApparatusDetection of polarized Xe is by the NMR technique of Adiabatic Fast Passage (AFP) described inSec. 2.2.5. The NMR detector used was modified from an existing system constructed in the SonierLab at SFU, itself modeled on the polarizer design by Ruset [144]. It was previously conceived as47Figure 3.5: Optical pumping of Rb is observed when an applied B field breaks the degeneracyof Zeeman levels. The level probed by circular polarized light only absorbs with the fieldOFF; with the field ON, atoms are rapidly pumped into the dark state.Figure 3.6: Coils and detection electronics for the free-standing NMR spectrometer .48a stacked spin-exchange polarizer and NMR detector, with the NMR coils mounted atop a verticaloven and polarizer cell. We decommissioned the polarizer portion and reconfigured this detectorfor high field uniformity in the NMR region. A separate design, with even higher field uniformitysuitable for Free Induction Decay, is described in Ref. [165] and will be used in future work.The free-standing spectrometer consists of three orthogonal pairs of coils (B0, B1, pickup) shownschematically in Fig. 3.6. The B0 coil is a Helmholtz pair (96 turns/coil), with 55 cm diameter. Thecoils are driven by a DC power supply (KEPCO BOP-50-8D) which can be ramped by means ofa control voltage input from a function generator. The B0 field inhomogeneity from calculation isless than 0.02% change over the 10 cm sample region for the Helmholtz configuration. The B1 coilis a 5-turn saddle pair orthogonal to B0, driven by RF radiation (usually provided by the lock-inamplifier described below). B1 is typically driven at fixed amplitude for AFP experiments, withfrequency corresponding to the Larmor frequency imposed by B0. It can also be pulsed in the caseof FID. The pickup coil is another saddle pair concentric with B1 and oriented orthogonal to it. A 3Dprinted worm gear (shown in Fig. 3.7) permits fine tuning of the angle between the B1 and pickupcoil axes to minimize cross-talk, described below. The pickup coil is sensitive to precession of thenet magnetization through Faraday induction, which converts it to an Electromotive Force (EMF).The pickup is wired to a capacitor in a small tuning box outside the coil system; this forms atuned Resistor-Inductor-Capacitor (RLC) resonance circuit which increases the signal strength atthe desired resonance frequency by the quality factor Q (see Appendix A.2). The voltage acrossthis capacitor is amplified by a preamp (Stanford SRS552) and measured by a lock-in amplifier(Stanford SRS830) locked to the B1 frequency.Coil constants are listed in Table 3.2. Typically we obtain an AFP signal by ramping the B0 fieldslowly through resonance and recording the lock-in output. Table 3.3 lists typical parameters usedduring a scan.Table 3.2: Coil Parameters for NMR.B0 B1 pickup SEOPRadius (m) 0.28 0.05 0.025 0.038Length (m) - 0.2 0.1 0.165Turns 96 5 100 750Resistance (Ω) 1.4 9.85 6Inductance (mH) 4.6 .034 4.7 19Magnet constant (mT/A) 0.30729 0.05426 2.2848 5.0908Voltage, typ. (V) 36 1 - 20Current, typ. (A) 4 0.714286 - 2Field, typ. (mT) 1.22916 0.038757 - 10.181649Figure 3.7: Worm gear used for coarse adjustment to minimize cross-talk between NMR coils.Adiabatic Fast Passage TechniqueThe theory of AFP is presented in Section 2.2.5. Here we present the technical details of our im-plementation of AFP. We apply a B1 field using the lock-in reference output, at the nominal Larmorfrequency ω1 of either water or Xe. After thermal stabilization, we align the B1 and pickup coilsorthogonal by rotating the worm gear to minimize the background signal. We call this backgroundcross-talk; it results from mutual induction (or possibly capacitive coupling) between B1 and thepickup owing to imperfect alignment. Any residual cross-talk is eliminated by a different method:A small bucking coil, pictured in Fig. 3.8, is mounted in the tuning box overlapping the area of thetuned RLC circuit and driven also at ω1 by a function generator (with a suitable phase shift from thelock-in amplifier output). The amplitude of the bucking coil is chosen to match the experimentallymeasured cross-talk, and then phase shifted 180 degrees to completely cancel it. Noise from exter-nal RF sources is cancelled using a 2m long cylindrical aluminum shield. This cancellation yieldssignal sensitivity better than 1 mV.The sample volume is a 147 mL Nalgene bottle. We measure AFP signals from both H2O and50Figure 3.8: Bucking coil (right) used to eliminate residual cross talk in the RLC circuit tuningbox.Xe samples at two nominal Larmor frequencies (58.82 kHz and 15.555 kHz, respectively). Manyof the H2O samples are doped with CuSO4. The CuSO4 is known to shorten the T1 lifetime , e.g. to0.116 s at 16 mM concentration[131, 158] so we can rapidly acquire scans. The H2O was thermallypolarized in a field nominally set at 1.4 mT and a ramp applied through resonance. Ramp rates andlock-in time constants were chosen to minimize noise while preserving the fast rise times of AFPsignals. The on-resonance precession signal is demodulated by the lock-in amplifier and recordedusing an oscilloscope or computer data acquisition (DAQ).Xe AFP signal is obtained in a similar fashion. An empty 147 mL Nalgene bottle is connectedby tubing to the SEOP system. The mixture of Xe, N2 and He, containing hyperpolarized Xe,flows into the measurement region. Care is taken to avoid any rapid field changes or zero-fieldregions along the tubing path, to prevent loss of polarization. B1 is driven at a lower RF frequency(≈15 kHz) for the same nominal B0 field, as determined by the smaller Xe gyromagnetic ratio.3.2 Rate equation model for SEOPWe present here a simple rate-equation model of the processes happening during spin exchangeoptical pumping, in order to identify their relative contributions and look for ways to increase thepolarization. The phenomena are grouped as follows: polarization of Rb by optical pumping, loss ofRb polarization by collisions and radiation trapping, polarization of 129Xe through spin exchange,51Table 3.3: Typical NMR parametersParameter ValueB0 current 3.95-4.25 AB0 voltage 36 VB0 nominal field 1.4 mTB0 ramp frequency 200 mHz (5s period)B0 ramp amplitude 100mVB1 voltage from lock-in 1.00 VB1 field strength (rotating, 1.0V driving) 0.628 µTB1 frequency 15.555 kHz (H2O)58.82 kHz (H2O)lock-in sensitivity 1mVpreamp setting 100xon screen signal amplitude 1 Vsignal amplitude 10 uVnoise amplitude 5 uVand subsequent depolarization of Xe. The rate equation model is based primarily on a descriptionsummarized by Brunner [36], and uses our experimental values given in Table 3.1. Estimates of thepumping and spin exchange rates are given in Table 3.4.Briefly, resonant 795 nm light drives the optical pumping of Rb. Collisions of the polarizedRb with other species lead in general to a loss of Rb polarization (non-spin-conserving collisions),while a fraction of the Rb-Xe collisions lead to the spin-exchange reactions which transfer po-larization from the Rb electron spin to the Xe nuclear spin. Polarized Xe, likewise, undergoespolarization-destroying collisions with other Xe, other species and with the cell walls. The equilib-rium polarization of Xe in the cell comes from a balance of these rates.Rb optical pumpingCircular-polarized radiation is used to selectively pump on the MJ = −1/2 sublevel (the “brightstate”) and deplete its population. Atoms relax either by emission to the ground state sublevelswith rates determined by the corresponding dipole matrix elements, or by collisional deexcitationby a buffer gas used to prevent radiation trapping (see below). Collisional mixing of the excitedstate sublevels is assumed. The Rb polarization builds up in the MJ = +1/2 sublevel. The opticalpumping rate, meaning the absorption rate per Rb atom, is given byρopt =∫Φopt(λ )σopt(λ )dλ (3.1)52Table 3.4: Estimate of optical pumping and relaxation rates inside the SEOP cell. Rate con-stants ki come from Reference [63]. ρi represents the (per-atom) rate of spin polarizationproduction or loss for the relevant species. d[X]/dt represents the same overall rate includ-ing number density, i.e. [X] = [Rb] or [Xe]. Each rate’s contribution to the overall spinpolarization production or loss is evaluated as a percent in the final column. The rates ineach section sum to 100%.ki rate ρi d[X]/dt (Percentage)(m3s−1) (s−1) (m−3s−1)Rb spin production rateOptical pumping: 1.16E+03 1.2E+23Rb spin polarization loss rateBinary collisions:Rb-Rb kRb 4.00E-20 4.14E+00 4.29E+20 0.01%Rb-Xe kXe 6.48E-20 5.82E+04 6.02E+24 92.86%Rb-N2 kN2 9.00E-24 2.42E+01 2.51E+21 0.04%Rb-He kHe 2.00E-24 1.72E+02 1.78E+22 0.28%Radiation Trapping 957.7476 9.92E+22 1.53%Spin Exchange collisionsBinary spin exchange collisions kse,binary 3.00E-22 2.69E+02 2.79E+22 0.43%He-mediated spin exchange collisions kse,He 1.70E+04 1.77E+02 1.83E+22 0.28%Xe-mediated spin exchange collisions kse,Xe 5.23E+03 2.87E+03 2.97E+23 4.58%Xe spin polarization production rateBinary collisions kse,binary 3.00E-22 3.11E-02 2.79E+22 8.13%He-mediated collisions kse,He (s−1) 1.70E+04 2.04E-02 1.83E+22 5.35%Xe-mediated collisions kse,Xe (s−1) 5.23E+03 3.31E-01 2.97E+23 86.52%Xe spin polarization loss rateSelf-relaxation/ “transient dimers” kXe−Xe 1.86E-31 1.67E-07 1.50E+17 0.02%Persistent dimers/vanderWaals 5.86E-06 5.26E+18 0.60%Wall relaxation 0.000417 3.74E+20 42.59%Gradient relaxation 0.000556 4.99E+20 56.79%Rb equilibrium polarization PRb 1.82%Xe equilbrium polarization PXe 1.82%whereΦopt(λ ) and σopt(λ ) are the photon flux and the absorption cross section of Rb at wavelengthλ , respectively. Based on the roughly 12% absorption shown in Fig. 3.5, we calculate the totalabsorbed photon flux to be 5× 1018 photonss−1. The per-atom rate ρopt is difficult to estimate asthe cell is optically thick; the optical pumping rate is nonuniform throughout the cell and is likelyhighest near the entrance window. As a rough estimate we assume uniform absorption over the full10 cm cell length and calculate ρopt ≈ 103 s−1.53Rb spin lossesRubidium spin relaxation occurs through binary collisions with other species, through radiationtrapping, and through successful spin-exchange reactions during the lifetime of short-lived van derWaals molecules. Quoted here are per-alkali atom loss rates, as opposed to integrated rates d[Rb]/dt.The net polarization of Rb is determined by a balance of optical pumping ρopt and spin-destructionρsd rates:PRb =(ρoptρopt +ρsd)×100% (3.2)where the spin destruction is given by:ρsd =∑XρX +ρtrap+∑ρSE . (3.3)The rates for each relaxation mechanism are listed in Table 3.4. Binary collisions are of theform Rbpol + X −−→ Rbunpol + X. The relaxation rate for binary collisions with each species X isgiven by :d[Rb]dt= ρX [Rb] = kX [X ][Rb] (3.4)For [Xe] mixing ratios of a few percent, Rb-Xe binary collisions are a dominant source of loss,accounting for over 92% of Rb relaxation in the current model as shown in Table 3.4.Radiation trapping describes the reabsorption by one Rb atom of previously emitted Rb fluores-cence from a nearby atom. Due to collisional mixing of the upper state, the emission is assumedto have a random direction and orientation, and thereby depolarizes any atom that reabsorbs it.Therefore a buffer gas such as N2 is usually added to collisionally de-excitate atoms and preventfluorescence. The rate of loss due to radiation trapping in the presence of a buffer gas is given by[36, 63]:ρtrap =3A3+7.5PN2(3.5)where A= 40000s−1 from literature [63] and PN2 is the N2 partial pressure given in kPa. As a resultof including 3% N2 in our gas mixture, relaxation due to radiation trapping is predicted to be smallin comparison with other loss mechanisms.Even successful spin-exchange naturally results in a loss of Rb spin polarization. In our currentmodel this represents about 5% of all Rb loss, and is covered in the section below.Xe spin polarization productionXe spin exchange occurs via the Hamiltonian ~IXe ·~S both through binary collisions and throughpolarization transfer during the formation and breakup of short-lived van der Waals molecules. Thecross section for spin exchange in spin-conserving binary collisions leads to a per-atom rate for54xenon polarization given by [59]:ρSE,binary = kse,binary[Rb] (3.6)The probability of a successful spin-exchange collision in a van der Waals molecule depends onthe lifetime of the molecule. Collisions with a third body facilitate the formation and breakup ofthe vdW molecule and control the lifetime. Increasing the background pressure increases the rateof these collisions. The spin-exchange rate via vdW molecules is given by [63]:ρSE,Xe,He = [Rb](kse,He[He]+kse,Xe[Xe]+0.275[N2])(3.7)It is apparent from the formula that molecule-mediated spin exchange plays an important role atlower SEOP pressures. In our current setup, the model indicates that spin-exchange in van der Waalsmolecules where Xe is the third body account for more than 80% of the total rate of spin-exchange.Xe spin relaxationPolarized Xe and polarized Rb in the same cell will eventually reach an equilibrium, with spinexchange reactions happening in both directions. The Rb number density is orders of magnitudesmaller that the Xe number density, so loss of Xe polarization due to reverse spin exchange withRb is negligible. Another loss of Xe polarization occurs through the formation of Xe-Xe dimers.These can be either “transient”, equivalent to a binary collision, or “persistent,” corresponding tothe formation of a Xe2 van der Waals molecule. Two other mechanisms for relaxation are wall col-lisions and depolarization near a magnetic gradient. The glass cells used can contain paramagneticimpurities near the surface which can depolarize the Xe. A common solution is to coat the wallswith a wax or sillicon coating to increase the separation between impurities and polarized Xe; inour model we use the relaxation rate of our wall coating Surfasil for which the relaxation lifetimeis 1/ρwall = 40min as reported by [145]. The lifetime is still limited by collisions with protons inthe coating, which are still capable to a lesser extent of depolarizing the Xe. Finally, large magneticgradients can depolarize Xe if the atoms have a large enough velocity. We calculated the magneticfield gradients using a model of the SEOP solenoid in COMSOL, shown in Fig. 3.9. From the modelwe estimated the depolarization rate ρgrad based on the formula[145]:ρgrad = D|∇Bx|2+ |∇By|2B20, (3.8)where D is the diffusion coefficient in cm2 s−1. Wall collisions and gradient relaxation yield com-parable loss rates in the model, both approximately 5×10−1 s−1.55(a) Longitudinal magnetic field Bz.(b) Transverse magnetic field Bx.Figure 3.9: Longitudinal (a) and transverse (b) field components of the SEOP solenoid mag-netic field (in units of 10−3 T), modelled in COMSOL for the calculation of gradientrelaxation ρgrad .Results of rate equation modelThe steady-state Xe polarization PXe for which spin exchange rates ρSE and loss rates ρloss are equalis given by:PXe = PRb(ρSEρSE +ρloss). (3.9)The model predicts PXe = 1.8% based on a Rb number density corresponding to a reservoir temper-ature of 423 K (150 C). We suspect the polarization is correct within an order of magnitude. Thelargest uncertainty in the prediction comes from the estimates of the Rb number density and thecorresponding optical pumping rate, which as we previously indicated is not uniform through theSEOP cell. Fig. 3.10 shows the polarization dependence on Rb number density from the model.56Figure 3.10: Simulation results for dependence of xenon polarization on Rb number density,using a resonant laser power of 1.2 W (red) and 2.4 W (blue). The range of numberdensities shown corresponds to Rb saturation vapour pressure for the temperature range300-423 K.One observes the predicted polarization to peak and then fall; this is due to an increase in SEOPefficiency at low Rb number density, and decrease in PXe related to the overall decrease in PRb athigher number density. Section 3.3.4 compares this rate equation model result with experimentalobservations from heating the SEOP cell.3.3 Experimental ResultsIn this section we present results from both the SEOP polarizer and NMR detection. Many of theexperiments studied the relationship between SEOP parameters (changing temperature, pressure,etc) and Xe polarization using AFP, and are presented here qualitatively, due to day-to-day variationin operational parameters. A quantitative estimate of the Xe polarization based on calibration ofour Xe AFP signals against water AFP is also presented. Finally, first efforts are shown to purifypolarized 129Xe in a freezeout cell.A typical H2O AFP signal is shown in Fig. 3.11. Signals from H2O display successive upand down peaks which indicate precession in phase and out of phase, respectively, with the lock-in reference signal. This is a result of near-complete longitudinal relaxation between scans, since57Figure 3.11: A typical water AFP signal during continuous ramping. The change in peakdirection is due to rapid relaxation between ramps, which creates a signal out of phasewith the Lock-in reference. (Conditions: B0 coil current = 4.63 A, ramp = 400 mV, 200mHz.)the time between ramps is longer than the T1 lifetime of a few hundred milliseconds; however,the asymmetry in amplitudes for up and down peaks may indicate that equilibrium has not beenreached. An AFP signal for Xe is shown in Fig. 3.12. Xe has a gas phase T1 lifetime on the orderof minutes or even hours; therefore successive peaks may be seen in the same direction, dependingon the ramp rate and flow rate. If the Xe peak changes sign between ramps under continuous flowconditions, it indicates that the sample volume is being replenished with fresh polarized Xe fromthe SEOP polarizer at a rate faster than the rate of spin flips.3.3.1 Determining and Optimizing Adiabatic limitsAdiabatic Fast Passage requires a ramp rate well within the limits established by Formula 2.28.Approaching too close to either the upper or lower limit causes a decrease in signal due to eitherloss of adiabaticity or relaxation. Optimization of the adiabatic conditions is possible by changingeither B1 or dB0/dt. Xe and H2O possess different T1 lifetimes and different gyromagnetic ratios,which means that the adiabatic limits will be different for both species. Most of the AFP signalswere obtained by driving the B1 coil at 1.0VRMS, which yields B1 = 0.628µT. From Equation 2.28we find for Xe (taking T1 = 878s as measured in Section 3.3.6):0.00072µTs(dB0dt)Xe 29.2 µTs(3.10)58Figure 3.12: A typical xenon AFP signal during continuous ramping. (Conditions: B0 coilcurrent = 4.43 A, ramp = 250 mV, 100 mHz.)For water with 2mM CuSO4, the longitudinal relaxation lifetime is T1 = 0.274s and we have:2.3µTs(dB0dt)p 105.5 µTs(3.11)In the experiments with B1 = 0.628µT, the optimal H2O signal was found for a ramp rateapproximately dB0/dt = 3µTs . However, it is clear from inspection of Equation 2.28 that increasingB1 strength also increases the range between upper and lower limits. The upper limit is proportionalto B21 and grows faster than the lower limit which is proportional only to B1. Increasing the rangebetween limits could result in signal increase if we can ramp B0 at a rate which better satisfies theadiabatic conditions.We simulated evolution of the Bloch equations to test the AFP limits, using an ordinary differ-ential equation (ODE) solver written in Python by Jeff Martin at U. Winnipeg. The code modelsthe net magnetization M under the effects of parameters B0,B1,T1 and T2. We modified the scriptto accept arbitrary functions for the B0 field as input. The new B0 function ramps once throughresonance, holds B0 constant for time T  T1 to allow time for relaxation, then ramps through res-onance a second time. We vary dB0/dt and B1 using this code to determine conditions for AFP thatyield the highest signal.The simulation shows that at higher B1, increasing the ramp rate yields a larger AFP signal. Anexample of this is given in Fig. 3.13, for water AFP at 56 kHz resonance driven by either low orhigh B1 fields. (It should be noted that higher B1 also slightly increases the off-resonant horizontal59(a) (b)Figure 3.13: Bloch equation simulation shows H2O AFP signal increase for simultaneous in-crease of B1 and dB0dt . (a) low field conditions with B1 = 0.628µT anddB0dt = 3µTs . (b)high field conditions with B1 = 2.52µT and dB0dt = 429µTs . Red: longitudinal magneti-zation Mz. Green: transverse magnetization Mx.components.) The simulation indicates that increasing B1 to 2.52µT and ramp rate to 429µT/swould increase the AFP signal by approximately 2x.We found experimentally that we could increase B1 up to 3.14µT, corresponding to the maxi-mum 5.0VRMS output of the lock-in amplifier reference, and still adequately cancel cross-talk. Thismade it possible to increase the ramp rate dB0/dt while still satisfying the adiabatic limits. At highfield (B1 = 3.14µT), the adiabatic limits for Xe are given by:0.0036µTs dB0dt 729 µTs(3.12)and the corresponding limits for H2O are:11.5µTs dB0dt 419 µTs. (3.13)We found a 2x increase in water AFP signal at high B1 field by increasing the ramp rate from 3µT/sto 40µT/s, and a corresponding 3x increase in 129Xe signal obtained under continuous Xe flowconditions. Inspection of Equations 3.12 and 3.13 suggest that we are well within the adiabaticlimits; however, it may still be possible to increase B1 and optimize the AFP signals further. Non-adiabatic conditions for either Xe or H2O cause a decrease in the AFP signal (from the theoreticalmaximum) that can result in either an underestimation or overestimation, respectively, of the 129Xepolarization.603.3.2 Estimation of the degree of 129Xe polarizationThere are two techniques to measure polarization, which we label relative and absolute. An absolutemeasurement determines the polarization of one species directly from the NMR signal it produces.Absolute measurements require calibration of the gain parameters in the electronics. Relative mea-surements determine the polarization from a ratio of signal strengths for different species. We usewater NMR to make relative measurements because its thermal Boltzmann polarization is known.Results for calculating xenon polarization by relative and absolute methods are given in Table 3.5and described below.Absolute Polarization MeasurementsWe can estimate the expected NMR signal directly by considering the Faraday induction causedby a precessing magnetization in the pickup coil. An order-of-magnitude estimate for the inducedEMF isε =−dΦdt= NcAωµ0M = NcAωµ0NVµiPi (3.14)where Nc and A are the number of turns and area of the pickup coil, ω is the angular precessionfrequency, µ0 is the permeability of free space, and M = NV µiPi is the sample magnetization per unitvolume. µi and Pi are the magnetic dipole moment and polarization, respectively, of the precessingnuclei i. We employ the resonant RLC circuit described in Section 3.1.2 to increase the detectedsignal by a factor Q = ω∆ω , where ω and ∆ω are the center frequency and full width at half power,respectively, of the RLC circuit transfer function. In Appendix A.2 we calculate the value Q = 26.0for the 15.555 kHz circuit used for Xe. The resonance-enhanced signal is then further amplified bya preamp with gain Gpreamp = 100 before detection at the lock-in amplifier. The signal expected atthe lock-in amplifier input is therefore:Sp = GpreampQε (3.15)Applying the derivation to xenon and solving for PXe, we arrive at:PXe = SXe[GpreampQXeNcAωXeµ0NXeVµXe]−1(3.16)The largest AFP signal measured to date is SXe = 520µV for 15.555 kHz resonance when polarizingin batch mode (see Section 3.3.5). Based on this result and the parameters in Table 3.5, we estimatePXe = 0.5%, which will be discussed further in the section below.61Relative Polarization MeasurementsOur relative measurements compare the 129Xe signal strength to that of thermally polarized protonsin a H2O sample. The protons occupy both spin states almost equally at room temperature, withexact populations determined by a Boltzmann distribution. The thermal polarization Pp of protonsin the B0 field is given by Equation 2.21. We calculate Pp = 1.24× 10−9 at ω/2pi =15.555 kHzresonance and Pp = 4.70×10−9 at ω/2pi =58.82 kHz resonance.Spin exchange creates hyperpolarized populations of Xe atoms, with polarization orders of mag-nitude larger than the thermal polarization. Hence Equation 2.21 applies in this case only to water.We measure the relative NMR signal strengths of the two species, taking into account differentnumber densities and magnetic moments, and calculate the Xe polarization from the ratio SXeSp usingEquation 3.16 [55, 164]:PXe = PpSXeSp∣∣∣∣ µpµXe∣∣∣∣ NpNXe QpQXe ωpωXe (3.17)Here SXe is the signal strength, µXe the magnetic moment and NXe the number of precessingnuclei. A gain factor defined by GpGXe =QpQXeωpωXe represents the potentially different experimental gainfor the two samples. We can perform measurements at fixed frequency or fixed fields (differentfrequency). When comparing NMR signal obtained at different frequencies, we need to account forboth (i) the gain due to the different Q factors of the resonant pickup RLC circuits, and (ii) the largerinduced EMF at faster frequency ωp due to the changing flux dφdt . When comparing signals from Xeand H2O at fixed frequency, we use the same RLC circuit for both samples and therefore the gainis by definition unity, i.e. GpGXe = 1. Under fixed frequency conditions at 15.555 kHz, the best valuesmeasured to date are SXe = 520µV and Sp = 40µV. The number of proton spins and 129Xe spinsare 9.81×1024 and 1.04×1019, respectively. From these values we calculate PXe = 5.4%. (It mustbe noted that a higher polarization was estimated under fixed field conditions at 58.82 kHz, but thisis attributed to underestimation of Sp caused by an accidental detuning of the lock in amplifier awayfrom the RLC resonance.)Estimates of PXe by the absolute method are an order of magnitude lower than predictions fromrelative measurements. One source of discrepancy is overestimation of the coil flux and thereforethe induced EMF, due to the angle between sample and pickup coil; another source is uncertainty inthe measurement of the Q-factor. Both of these strongly affect the absolute measurement but not therelative measurements. Finally, it should be emphasized that our best measurement of SXe = 520µVwas obtained under low B1 field conditions, before the AFP optimization of Section 3.3.1, and wasnot repeated after optimization due to deterioration of the SEOP cell. We expect that estimates ofPXe based on the unoptimized conditions therefore underestimate the true polarization by roughly afactor of three.62Table 3.5: Parameters used for the estimation of xenon polarization based on observed NMRsignal. All 129Xe signals were obtained at ωXe/2pi = 15.555kHz resonance frequency.Relative measurements were made against H2O signals obtained at either 15.555 kHz or58.82 kHz, and denoted as “fixed frequency” or “fixed field”, respectively.Fixed frequency Fixed fieldωP/2pi (kHz) 15.555 58.82ωXe/2pi (kHz) 15.555 15.56T (K) 300 300Pp - 1.24×10−9 4.70×10−9SXe (µV) 520 520Sp (µV) 40 600Gpreamp - 100 100Ncoil - 200 200A (m2) 5.80×10−3 5.80×10−3N129Xe - 1.05×1019 1.05×1019Np - 9.81×1024 9.81×1024Volume V (cm3) 147 147µp (J/T) 1.41×10−26 1.41×10−26µ129Xe (J/T) 3.93×10−27 3.93×10−27µp/µ129Xe - 3.59 3.59Q129Xe - 26 26Qp - 26 38.9Gp/G129Xe - 1 5.66PXe (Relative) (%) 5.4 7.8PXe (Absolute) (%) 0.5 0.53.3.3 Improvement of Rb absorption by pressure broadeningThe NIR laser used for pumping is broadband, with a linewidth of roughly 2 nm. This is much widerthan the Rb transition Doppler-broadened linewidth; therefore much of the light is nonresonantand does not contribute to optical pumping. Pressure broadening the resonance with He allowsus to deliver more power to the Rb and increase the optical pumping rate ρopt . Our SEOP gasmixture contains 96% Helium by volume, the primary purpose of which is to pressure broadenthe Rb D1 absorption line. We measured pressure broadening by operating the SEOP cell at arange of pressures between 20 and 60 psi(g), controlled by a regulator on the gas cylinder. Werecorded the transmission spectrum of circular polarized light through the Rb cell using the opticalspectrum analyzer with the solenoid field switched “ON” and “OFF” and while flowing gas throughthe polarizer at flow rates 0.1 slm1 and below, and measured the width of the absorption feature inthe spectrum analyzer signal (see Fig. 3.5), calculated from a fit to the lineshape of the transmissionratio I/I0. Fig. 3.14 shows the increase in absorption linewidth as we increase the pressure; the1standard litre per minute63Figure 3.14: The absorption linewidth of Rb increases as the Rb vapour is pressure broadenedby collisions with He gas. Red: measured signal linewidth (convolved with 0.1 nmresolution bandwidth of spectrum analyzer). Blue: linewidth after deconvolution.expected qualitative behaviour (linear increase with pressure) is observed. However, the observedlinewidth is dominated by the instrument resolution bandwidth of 0.1 nm (50 GHz). We approximatethe observed linewidth (red) to be the Voigt profile convolution of the true pressure broadenedLorentzian (blue) with a Gaussian resolution bandwidth, and extract the Lorentzian linewidth viathe approximation [120] ΓV ≈ 0.5346ΓL +√0.2166Γ2L+Γ2G. We find 9± 2 GHzamagat−1 fromour data2, which is not in agreement with the ∆ν = 18.0± 0.2 GHzamagat−1 linewidth measuredby Romalis [141]. Quantitative agreement requires analysis of the spectrum with higher resolutionhardware.3.3.4 SEOP cell temperature dependenceRb melts at 312 K and boils at 961 K. Heating the body of the SEOP cell increases the Rb numberdensity (and absorbance) as shown in Fig. 3.15 by causing Rb to desorb from the cell walls. Italso therefore increases the rate of spin-exchange collisions with Xe. We measured the effect oftemperature by heating the SEOP cell with heater tape wrapped at the entrance and exit windowswhile flowing gas at a fixed rate. We limited the heating range to below 150 ◦C to avoid damage tothe SurfaSil coating. By careful adjustment we maintained a temperature uniformity of 2 K during21 amagat = 2.69×1025 m−364Figure 3.15: Rb absorbance as a function of cell temperature compared with theoretical num-ber density temperature dependence based on vapor pressure.Figure 3.16: AFP signal dependence on SEOP cell average temperature during continuousflow at 0.3 slm.65continuous flow as measured by three K-type thermocouples mounted along the cell body. Fig. 3.16shows the AFP signal plotted against average cell body temperature. The cell body temperature is,however, a poor proxy for the internal temperature; the temperature of the SEOP gas mixture couldbe even higher. Using the empirical formula for vapor pressure [36]log( pRbPa)= 9.550− 4132T/K, (3.18)predicts [Rb] = 4× 1018 m−3 based on cell body temperature and [Rb] = 1× 1020 m−3 based onreservoir temperature. However, from comparison with the model results in Fig. 3.10, we note thatthe experimentally observed increase in AFP signal with increasing cell temperature suggests lowerthan expected Rb number density. The first reason for this discrepancy is that the optical pumpingrate is nonuniform across the pumping cell and becomes increasingly nonuniform as the Rb vapourbecomes more optically thick with heating. This nonuniformity is not well accounted for in themodel. One factor which supports lower than expected Rb number density is that during continuousflow the cell will not reach equilibrium with the Rb saturation vapour pressure; the number densitycould also be lower than expected due to reactions between Rb and contaminants on the cell walls,or due to Rb condensation on localized cold spots between the Rb reservoir and cell body.3.3.5 Continuous vs. stopped flow effects on 129Xe polarizationThe polarizer can either be operated with a continuous stream of SEOP gas or with a stopped flowby adjusting the downstream needle valve. Both are described below:Flow rateAdjusting the flow rate has two effects. First, it directly controls the time Xe experience spin-exchange collisions with optically pumped Rb. Second, it controls the transport time between SEOPpolarizer and NMR cell (which are nominally separated by 4 m of tubing), during which T1 relax-ation can occur. One expects an optimized flow rate that gives Xe a long interaction time followedby quick delivery to the NMR. We adjust the needle valve downstream of the SEOP cell, and mea-sure the flow rate upstream in the pressurized gas line before the polarizer (the flowmeter is locatedupstream to prevent it from depolarizing the Xe), keeping cell temperatures constant. Flow rates inthe low pressure downstream lines were confirmed with an additional flow meter prior to startingthe experiment. Results are shown in Fig. 3.17 with flow rate measured in standard litres per minute(slm). The best continuous-flow signal is observed at 0.1 slm flow rate. This corresponds to a Xeinteraction time t ≈ 30s in the SEOP cell.66Figure 3.17: AFP signal dependence on flow rate under continuous flow-conditions.Figure 3.18: Blue: AFP signal from batch mode operation with 8 min. buildup time, followedby transfer to NMR at 0.200 slm. The peak at t = 50s represents the batch polarization,while signal at t > 150s is the continuous-flow (steady-state) polarization at 0.200 slm.Red: AFP ramp dB0/dt.67Figure 3.19: Batch mode conditions for buildup times up to 60 min, followed by transfer toNMR at 0.200 slm.Stopped-flow “batch” modeUnder continuous flow conditions often the Rb polarization does not reach equilibrium along theentire length of the cell. Fig. 3.19 shows the AFP signal amplitude from operation under stopped-flow “batch-mode” conditions. The needle valve is closed for a buildup time up to 60 min to allowthe polarization to reach an equilibrium value, after which the valve is opened and the sample istransferred through tubing to the NMR region. Equilibrium polarization is observed after 8-10 minin the SEOP cell. The stopped-flow polarization can be much larger than that under continuousflow conditions. This can be seen from Fig. 3.18, where the maximum signal from stopped-flowpolarization at t = 50s is nearly three times larger than the continuous-flow signal observed for t >150s. Stopped-flow polarization has an added technical advantage: after polarization is complete,the polarized samples can be rapidly transferred to the NMR cell at arbitrarily fast flow rates. Thiswill help minimize relaxation during transfer; in contrast to continuous flow conditions in which theflow rate through the SEOP cell also determines the flow rate to the NMR experiment.The largest Xe AFP signal measured to date was obtained by stopped-flow polarization, witha signal magnitude SXe = 520µV at the lock-in amplifier input. This signal however represents anunderestimate of the true Xe polarization for two reasons. First, the signal was obtained beforethe optimization of AFP abiabatic conditions discussed in Section 3.3.1. Second, the B0 ramp wason during the transfer of polarized Xe into the NMR cell, which causes some mixing of gas inopposite spin states. (We were able to minimize the mixing by adjusting B0 such that resonanceoccurs near the bottom of the ramp, and thereby minimize the time Xe spends in the spin down68(a) (b)Figure 3.20: The Xe AFP signal decreases as polarized Xe travels through longer lengths oftubing. Signal peaks from consecutive runs in (a) appear closer together due to a slowB0 drift. Error bars in (b) are from detection noise only and do not account for theslowly-changing experimental conditions.state). Larger Xe AFP signals are expected following optimization of the adiabatic conditions andfollowing improvements to the polarization transfer method (i.e., without any B0 ramp).Tubing Length DependenceIdeally we should locate the SEOP polarizer as close as possible to the detection chamber; however,to meet the requirements of nEDM measurements at TRIUMF it may be necessary to locate thepolarizer several metres from the EDM cell. We measured AFP signals while changing the pathlength from SEOP cell to NMR bottle, in order to observe how quickly the Xe becomes depolarized.This was achieved by flowing at 0.200 slm through lengths of nylon tubing (between 4 m and 34m) coiled up between the SEOP polarizer and NMR. The coiled tubing was supported at least 10cm above the optics table to prevent relaxation, because we previously observed complete loss ofpolarization when the tubing was allowed to contact the magnetized table. Fig. 3.20 shows the signaldependence on path length. The extra path length increases the transit time (up to 115 s), allowingmore T1 relaxation and wall collisions. Therefore the decrease in AFP signal with path length isexpected. An exponential fit of signal against transit time yields a rough estimate of τ = 190±40sfor decay time during transport.3.3.6 Measurements of polarization relaxation timeHyperpolarized Xe has a surprisingly long relaxation lifetime in the gas phase. Efforts to measurethis relaxation are described below.69Figure 3.21: Xe relaxation under repeated AFP. The decay of signal here is dominated by T2effects and the incomplete inversion that occurs near the bottom of the B0 ramp.Relaxation during repeated AFPThe first measurements of Xe spin relaxation in the lab come from a Xe sample subjected to thecontinuous AFP ramping sequence shown in Fig. 3.21. The cell is first filled with polarized Xe undera fixed B0 field, flowing gas at a fixed flow rate for long enough (> 30s) to completely replenish theNMR sample volume. The sample volume was sealed off immediately prior to turning on the B0ramp. The observed signal rapidly decays while repeatedly undergoing AFP. We expect that underrepeated AFP ramping, both T1 and T2 decay contribute to the observed relaxation. Additionally,Fig. 3.21 reveals that there is incomplete polarization inversion from the negative-going ramp beforethe positive-going ramp begins, such that two overlapping peaks occur near the bottom of each B0ramp. We observe rapid signal relaxation between the overlapping peaks, for which the decaytime constant (averaged over the first five pairs) is τ = 12(3)s. The relaxation is slower from onepair of overlapping peaks to the next, and we determine τ = 67(4)s, based on fitting only the firstpeak of each overlapping pair. This lifetime is short compared to that obtained by more directmeasurements of T1 (presented in the following section), which indicates that T2 relaxation andincomplete polarization inversion dominate under the continuous AFP ramping conditions presentedhere.70Figure 3.22: The ramp technique for a T1 measurement. We implement a variable time delayTD between successive up/down ramps and measure the decrease in AFP signal due toT1 relaxation.Figure 3.23: Exponential decay of AFP signal using the ramp technique above. The T1 life-time is inferred from the empirical fit τ = 878±26 s.71Figure 3.24: Schematic of Xe cold trap located between polarizer and NMR detector, withB0 = 8mT provided by a solenoid.T1 Lifetime MeasurementT1 is longitudinal relaxation towards the thermal equilibrium polarization given by the Boltzmanndistribution, and occurs even when there is no transverse polarization component. We measure T1by inducing two consecutive AFP ramps separated by a time delay TD, as shown in Fig. 3.22. Thecell is first filled with polarized Xe under a fixed B0 field, flowing gas at a fixed flow rate for longenough (> 30s) to completely replenish the NMR sample volume. After shutting off flow, the B0ramp sequence shown in green is applied. The sample polarization decays through wall collisions,gradient-induced relaxation, and dipole-induced relaxations, resulting in the smaller signal whichappears after delay time TD. We plot the logarithmic ratio of the peak height ln(SXe(TD)/SXe(0))while varying delay time TD to measure the decay rate. We measured for up to 1000s, shown inFig. 3.23, and found τ = 878±26 s from the fit. Because additional (e.g. T2) relaxation can occurduring the resonance part of the AFP ramp, this measurement τ represents only a lower bound forT1.3.3.7 Xe purification by freezeout and initial attempt at polarization recoveryTo perform high-resolution spectroscopy free of Doppler broadening, and ultimately to use Xe asa comagnetometer, the Xe must be separated out from the Rb, He, and N2. Separation of Rboccurs naturally through condensation of Rb on the walls downstream of the SEOP cell. We sep-arate Xe from the N2 and He by using a cold trap immersed in liquid N2. Xenon possesses ahigher freezing point (161.4 K) than either N2 or He, so can be effectively purified using a standardfreeze/pump/thaw technique.Apparatus for proof-of-principle tests of freezeout are shown in Fig. 3.24. We flow mixed gasthrough a 1-inch diameter cold trap while pumping on the trap and gas line with a scroll pump, andmonitor the pressure with a Baratron gauge. The trap and gas lines have a combined volume of 104mL. After 45s of flow at 0.14 slm flow rate, the total volume of gas through the trap equals 10572Figure 3.25: The downstream AFP signal disappears while performing freezeout during t =200− 500s, with all the Xe condensing in the cold trap; signal recovery occurs post-thaw at t = 500s, but without the anticipated signal increase (see text).mL, of which 1% by volume is Xe. The flow of gas is stopped and the trap is isolated by shutoffvalve. The trap contains visible solid Xe which forms a frost on the walls. Thawing the trap yieldsa pressure increase of 6.7 torr, which indicates that 87% of the Xe was captured by the freezeout.It has been shown [42, 64, 68] that the Xe polarization can be preserved in the solid state undera strong magnetic field, with a T1 lifetime of many hours. Fig. 3.25 shows an attempt to purifypolarized Xe in a cold trap. The trap is cooled with LN2 while flowing polarized Xe from the SEOPpolarizer in continuous flow mode. Another solenoid applies an 8mT field surrounding the trap,which corresponds to a relaxation rate 1/T1 = 10−2 ∼ 10−3 s−1 in solid state Xe at 77 K accordingto Ref. [64]. The loss of AFP signal between t = 200−500s, while maintaining a continuous flowof polarized Xe throughout, again indicates successful, near-complete freezout of Xe in the trap.We estimate the collection of pure Xe to be 7 mL at STP. At t = 500s we thaw the cold trap whilemaintaining gas flow. A continuous flow of (freshly-polarized) mixed gas from the SEOP polarizerpushes the thawed gas sample into the NMR cell for detection. Technical constraints preventedoperation of the NMR cell under vacuum, which would allow the free expansion of Xe into thecell post-thaw; a new rigid cell is under construction. The sample which gives rise to the signalappearing after t = 500s is likely a mix of thawed purified Xe and freshly polarized mixed gas;the signal amplitude shows no increase over pre-freezeout conditions, and likely indicates loss ofpolarization in the purified Xe sample. We anticipate that the post-freezeout signal for a pure Xegas sample with polarization maintained will be larger than the continuous flow signal by at least an73order of magnitude, due to the increase in Xe number density.3.4 DiscussionIn this section we have described the design and operation of a flow-through polarizer. We havedemonstrated polarization of 129Xe using spin exchange optical pumping with polarizations of a fewpercent. The rate equation model used to estimate the polarization indicates that the Rb polarizationis limited by binary collisions with Xe (the dominant source of Rb polarization loss). Measurementsof the pressure broadening up to 60 psi indicates resonant absorption of at most 12% of the opticalpumping light. An increase in polarization of Rb, and therefore of 129Xe, could be acheived eitherby increased broadening in a high-pressure cell, or by increasing the power spectral density witha narrowed pump laser. We find the best polarizations are acheived in batch-mode operation forat least 10 minutes buildup time. The long T1 lifetimes observed are sufficiently longer than theneutron EDM experimental cycle; even longer T1 is expected in an experimental cell with improveduniformity. The freezeout and purification of 129Xe is necessary for future integration with the neu-tron EDM cell and for experiments probing the polarization by two photon transitions. Experimentsat TRIUMF including the Xe comagnetometer are planned to begin in late 2020. Development offreezeout apparatus will continue at UBC and U. Winnipeg.74Chapter 4SpectroscopyAs discussed in Section 2.4.3, optical magnetometry of 129Xe requires a suitable electronic tran-sition and optical probe. This chapter covers spectroscopic results from two photon absorption inXe 6p← 5p transitions1. Section 4.1 presents early experiments with a pulsed laser: we observedLIF from all two-photon allowed 6p levels, tested the light polarization dependence of the 249 nmtransition rate, and observed an on-axis LIF enhancement due to coherent emission. Section 4.2describes the experimental setup and results of high-resolution spectroscopy with a CW laser on the5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0) transition. Hyperfine constants and isotope shifts for the tran-sition were determined with high accuracy and compared with literature, as well as the two-photontransition probability. The results of this section, including hyperfine constants and isotope shifts,have been published in Physical Review A [8]. Section 4.3 calculates the theoretical comagnetome-ter sensitivity based on the observed SNR for two-photon detection at the Xe pressures studied, andextrapolates to predict the sensitivity down to 1 mTorr pressure.4.1 Two-Photon Spectroscopy of Xe with Pulsed UV laserThe first experiments in our lab were done using a pulsed laser to a) find the peaks, b) test thepolarization dependence of the selection rules and c) investigate previously-reported bidirectionalemission.4.1.1 Experimental SetupThe schematic of the pulsed setup is shown in Fig. 4.1. We generated the UV light for thesepulsed experiments using a grating-tunable dye laser (LambdaPhysik Scanmate) with Coumarin307/MeOH dye. The gain medium was pumped by 355 nm third harmonic light from a Q-switchedNd:YAG laser (Coherent Surelight, 10Hz, 650mJ/pulse) (This particular dye efficiently absorbs UV1We use the spectroscopic convention where a left arrow indicates absorption and a right arrow indicates emission.75pump light in the 360-420 nm range, and fluoresces around 485-546 nm with a peak at 508 nmand 15% efficiency[32]). The dye laser grating was tuned to 499.2 nm. We used a non linear(phase-matched) BBO crystal to produce the UV 249.6 nm light using Second Harmonic Genera-tion (SHG). Phase matching was obtained by manually rotating the BBO crystal. After separating thefundamental and harmonic using a Pellin-Broca prism, the doubled radiation was loosely focused( f =+500 mm) into a 20 cm long vacuum cell constructed from a KF40 tee with 25 mm diameterMgF windows (Fig. 4.1). The cell was evacuated and backfilled with a partial pressure of research-grace natural abundance xenon (Praxair) between 12 mTorr and 1 Torr. Using computer controlwe stepped the dye laser output frequency through the two photon resonance. The gas absorbed atthree wavelengths corresponding to 5p← 6p (J = 0,2) transitions and emitted fluorescence lightin the NIR corresponding to 6p→ 6s transitions. We measure fluorescence both orthogonal andparallel to the excitation light. Off axis, emission was filtered by a monochromator (SPEX 1680)and recorded by a Photomultiplier Tube (PMT) (Hamamatsu R943-02) with a quantum efficiency ofapproximately 10% at the fluorescence wavelengths and operated with an approximate gain of 106.The monochromator was operated with a wide 3.6 nm bandpass to obtain maximum light collection,by setting the entrance and exit slits to 2.0 mm (the dispersion of the gratings is 1.9 nm/mm), asit was only necessary to separate spectral lines more than 60 nm apart. Smaller spectral featureswere on the order of 2 GHz and not resolvable by the monochromator. We used a gated integratorand boxcar averager (Stanford SR250) set to integrate signals over the 5 ns pulse duration of thedetected emission. The averaged DC signal was recorded using an oscilloscope or DAQ system.4.1.2 Observation of Laser-Induced FluorescenceTo first identify the lines we set the monochromator to detect 823 nm emission and scanned the dyelaser grating across the 252.5 nm two-photon resonance in 0.0001 nm wavelength increments untilwe observed fluorescence. Figure 4.2 shows the observed TALIF signal, with a FWHM of 0.002 nmcorresponding to a laser bandwidth of approximately 30 GHz. With such a broadband laser we areunable to resolve the hyperfine levels of the transition (F = 3/2 and F = 5/2), which are separatedby only 2 GHz, or isotope shifts. Similar spectra are obtained at 249.6 nm and 255.9 nm.Once the excitation wavelengths were determined we fixed the dye laser frequency at the centerof a resonance and stepped the monochromator by 0.1 nm increments to resolve the emission spectraas in Figure 4.3. The resolution of the monochromator is sufficient to resolve the fine splitting ofspin-orbit coupled levels but not that of hyperfine levels. The results are summarized in Table 4.1.Bidirectional EmissionA paper by Rankin[137] on two-photon excitation of xenon reported detection of an enhanced on-axis 828 nm LIF signal following excitation at 249 nm with a pulsed UV laser beam. This enhance-ment was labelled bidirectional emission as it was observed in both the forward and backwards76Figure 4.1: Schematic of the setup used for pulsed laser spectroscopy. Two-photon absorp-tion in the Xe cell creates laser-induced fluorescence, which may be detected parallel ortransverse to the pump beam.Figure 4.2: Laser-induced fluorescence from 252.5 nm two-photon excitation of Xe.77Figure 4.3: Frequency scan of the fluorescence following 252.5 nm two-photon excitation.The LIF signal has been corrected for PMT sensitivity and monochromator efficiency atthe LIF wavelengths 823 nm and 895 nm.Table 4.1: Pulsed laser excitation results. The initial laser used had a laser bandwidth over 30GHz.TPA Transition 5p5(2P3/2)6p 2[1/2]0 5p5(2P3/2)6p 2[3/2]2 5p5(2P3/2)6p 2[5/2]2← 5p6 (1S0) ← 5p6 (1S0) ← 5p6 (1S0)λT PA (nm) 249.6 252.5 256.0FWHM (nm) 0.0125 0.0127 0.0157LIF Transition → 2[3/2]1 → 2[3/2]2 → 2[3/2]1 → 2[3/2]2 → 2[3/2]1λLIF (nm) 828.0 823.2 895.2 904.5 992.3FWHM (nm) 2.28 2.36 2.26 2.19 -direction collinear with the axis of the excitation light. This phenomenon has been investigated insubsequent papers and is now thought to result from a mixture of amplified spontaneous emission(ASE), occurring exactly at the two-photon resonance, and nonlinear parametric emission processessuch as four-wave mixing or stimulated electronic Raman scattering, which can occur even for near-resonant light [5]. To test for bidirectional emission we set up photomultiplier tubes on-axis and off-axis of the excitation beam to compare the bidirectional and spontaneous fluorescence signals. Theforward LIF signal was separated from the intense UV laser light using a UV-reflective mirror as alow pass filter. The backwards-emitted LIF signal was observed by using a beamsplitter and repo-sitioning the spectrometer and photomultiplier tube. The optics collected light from a solid angleof approximately 0.01 steradian for the on-axis backwards emission. We varied the excitation light78(a) (b)Figure 4.4: (a) Forward and (b) transverse LIF at 828nm vs UV pulsed laser intensity at 249nm. (a) demonstrates the threshold at 50 a.u. for the onset of bidirectional emission.Figure 4.5: Pressure dependence of threshold energy (and power density) for observation ofstrong bidirectional emission at 828 nm.79intensity with a neutral-density filter and recorded LIF intensity. A clear threshold was observedfor the onset of bidirectional emission as can be seen in Fig. 4.4, which agrees qualitatively withthe literature observations. We observed bidirectional emission following both 249 nm and 252.5nm excitation. The detected power in the forward signal is several orders of magnitude higher thanfor transverse, without any correction for solid angle collection or monochromator efficiency. Thethreshold power density for this enhanced emission depends upon pressure as shown in Fig. 4.5.Calibrating the absolute threshold power density for a given Xe pressure requires better knowledgeof the focusing parameters of the UV excitation beam. Assuming a Gaussian beam profile, we yielda beam waist w0 = 20µm and a power density threshold of roughly 109 W/cm2 at 10 mTorr. Thisis three orders of magnitude higher than previously reported [137]. The disagreement indicates thatthe Gaussian beam model is likely not accurate for our laser. We find the threshold power depen-dence for 828nm LIF to be I0 ∝ p(−0.6±0.3) which agrees with the dependence I0 ∝ p−0.65 observedby Rankin [137]. The extrapolated threshold at 5 mTorr Xe from previously reported values is ap-proximately 108 W/cm2. This indicates tight focusing is required if we want to utilize this effect tosee enhancement of LIF at low Xe pressures.Polarization DependenceDifferent excited levels are accessible by different polarizations of UV light. For example, referringto Fig. 2.13 for 129Xe, the 249 nm transition 5p5(2P3/2)6p 2[1/2]0← 5p6 (1S0) is an F = 1/2←F = 1/2 transition for which the change in angular momentum can only be ∆M = 0,1. Two equal-frequency photons with linear polarization parallel to the quantization axis (and wavevector~k⊥ ~B0)can only impart an angular momentum change of ∆M = 0. Two circular polarized photons with~k ‖ ~B0 change the angular momentum by ∆M = ±2. Therefore only linear polarized light (or acombination of both circular polarizations) can excite the 249 nm transition. To test this we probedthe Xe cell with 249 nm light and monitored the absorption via 828 nm fluorescence as we changedthe UV light polarization. The linearly polarized dye laser pulses generate a UV second harmonic inthe BBO crystal, orthogonal to the incident light. We pass this UV light through a polarizing beamsplitter, followed by a quarter wave plate, with the QWP slow axis and beam splitter axis initiallyparallel. We then rotate the QWP fast axis in 5-degree increments while recording fluorescencesignals at 828 nm. An alignment of 0 degrees or 90 degrees corresponds to linear polarized light,while a 45 degree alignment produces σ+ circular polarized light; in between these orientations thepolarization is elliptical. As observed in Fig. 4.6, emission reached a maximum for the 0 degreeand 90 degree linear polarizations, while no signal was observed from circular polarized light at 45degrees. No excitation can occur due to the selection rules; thus both sublevels of the 129Xe groundstate form a dark state with respect to circular polarized 249 nm radiation.The same selection rules which prevent σ+ excitation at 249nm also make optical pumping at252.5 nm possible. Specifically, the transition 5p5(2P3/2)6p 2[3/2]2(F = 3/2)← 5p6 (1S0) to the80Figure 4.6: Observed LIF dependence on QWP rotation angle for the 249 nm transition. Forthis transition only ∆M = 0 is allowed. This simple demonstration of the selection rulesfor unpolarized Xe vapour in the J=0 state is indicative of the expected behavior forpolarized Xe vapour in the J=2 states.excited state F=3/2 hyperfine level has magnetic sublevels M=±3/2, ±1/2; when we pump with σ+light the ground state M=-1/2 sublevel is absorbing while the M=+1/2 sublevel is dark. Continuouspumping will populate the dark state; however with the broadband pulsed laser we used to makethese measurements we cannot resolve the excited state F=3/2 from the F=5/2 hyperfine level, forwhich both ground state sublevels are absorbing. A narrower-linewidth laser is necessary for opticalpumping.4.1.3 Efforts to reduce bandwidthWith a broadband pulsed laser, it is not possible to resolve the hyperfine transitions in any of ourtwo-photon processes. The hyperfine levels of 6p 2[3/2]2 for example, are separated by 2 GHz [53].We attempted to narrow the pulsed laser linewidth using a single plate etalon, and also an etalonformed by a pair of high-reflectivity mirrors. Neither was successful; even with an etalon installed,our pulsed laser linewidth is greater than 8 GHz. The linewidth of the pulsed laser was a limitingfactor that motivated excitation with the narrow linewidth laser described in the following section.81Figure 4.7: Layout of the CW laser system. Schematic made by E.Altiere [7]4.2 Two-Photon Spectroscopy with Narrow Linewidth CW LaserWe developed a tunable, narrow (sub-MHz) linewidth CW laser at 252.5 nm in collaboration withDavid Jones lab. The design follows that of ([7]) for Lyman-alpha generation and is based on har-monic generation from a 1010 nm Optically Pumped Semiconductor Laser (OPSL) in two successivecavities. The design was for 200 mW of CW UV output at 252.5 nm.4.2.1 CW Narrow Laser DevelopmentA tunable UV CW laser using an OPSL and doubling crystals was recently developed at UBC togenerate narrow linewidth DUV light at 243 nm, and presented in the Master’s thesis of E. Altiere[7]. The design of the present system shown in Fig. 4.7 is a replica of the system in that work, usinga new custom OPSL chip. For full details we refer the reader to the description in her thesis. Thislaser is used to excite two photon Xe transitions. The laser uses an 808nm diode laser (pump powerapproximately 20 W) to pump a custom-made semiconductor stack which emits multimode around821010 nm. A birefringent filter mounted at Brewster’s angle and etalon mounted internally to theOPSL cavity force single mode operation (see section below). The fundamental light is doubledto 505 nm and quadrupled to 252.5 nm by nonlinear Lithium Triborate (LBO) and Beta BariumBorate (BBO) crystals, respectively. The doubling crystals are each placed in a resonant cavity toincrease the intensity of SHG generated. Each doubling cavity has an input coupler with a reflec-tivity specially chosen to satisfy impedance matching conditions for maximum intensity buildup.Doubling cavities are made resonant by changing the length of the cavity using a piezoelectrictransducer, and are locked to resonance using a Pound Drever Hall (PDH) scheme. A ring cavitywas chosen so that light only travels one direction in the cavity; there are no standing waves and thisresults in one SHG beam instead of two. Optics were used to match the near-Gaussian beam profileto match the TEM00 mode accepted by the cavity. The previously demonstrated coupling in the firstcavity was 74%, and 83.6% in the second cavity. The buildup in the cavity is approximately 85×for the LBO and 55× for the BBO cavity. The typical conversion efficiency from IR to UV is 11%.The similar laser developed by our collaborator in [7] was shown in that work to have a 87 kHzlinewidth in the fundamental OPSL frequency when locked. The linewidth increases by a factorof√2 in each second harmonic conversion stage; hence we expect a 174 kHz linewidth for thefrequency-quadrupled UV light. Without frequency stabilization of the OPSL, the free-runninglaser drifts slowly by up to 200 MHz on a timescale of 25 minutes. The OPSL is tunable over arange of roughly 1-2 GHz before a mode hop occurs (see section below); this corresponds to a UVtuning range of up to 8 GHz, which covers 16 GHz in the (two-photon) Xe spectrum.Single-mode operation of the OPSLThe gain bandwidth of the OPSL chip is large, so this design uses an intracavity birefringent filterand etalon to force single-mode operation. The OPSL cavity itself is 60 mm long with a FreeSpectral Range (FSR) of roughly 2.5 GHz, and sets the narrow linewidth described above. Thecavity is tunable by means of a piezoelectric mounted to the output coupler. Mounted intracavityis an etalon 0.8 mm thick made of fused silica (with a FSR of 129 GHz) which serves to modifythe transmittance of the various modes as in Fig. 4.8a. The birefringent filter has a similar effect;by rotating the polarization of light it introduces a frequency-dependent loss to the various modesdepicted in Fig. 4.8b. Figs. 4.8c and 4.8d show the combined effect of the two intracavity elements:the cavity acquires a frequency-dependent transmission function which prefers one single frequencymode over all others. Changing the cavity length by PZT fine-tunes the exact frequency over a rangeup to the cavity FSR; any further tuning causes a mode-hop, where a neighbouring mode becomesmore favoured.83(a) Etalon (b) Birefringent filter(c) Supported mode structure (d) Magnified view of (c) to show individual modes.Figure 4.8: MATLAB simulation of frequency modes supported by the OPSL cavity, showingthe effect of intracavity elements to force single mode lasing. The x-axis shows theOPSL frequency, offset by 296841 GHz. (a) Free spectral range of the intracavity etalon.(b) Transmittance function of intracavity birefringent filter mounted near the Brewster’sangle. (c) Cavity mode structure modified by the intracavity elements. (d) Magnifiedview of (c) showing preferential gain for one frequency mode.Pound-Drever Hall scheme for locking doubling cavitiesLight generated at 1010 nm for the Xe laser is frequency quadrupled using two SHG doublingcavities. A crystal of LBO or BBO provided the phase matching conditions to generate SHG. Theprocess of conversion is nonlinear: the efficiency is greater for higher intensity radiation. Thus, tomaximize generation of SHG we used a buildup cavity to enhance the fundamental. The buildupused a bowtie configuration and was locked using a piezo-controlled mirror in a Pound-Drever-Hallscheme.This scheme adds frequency sidebands to the light and uses them to generate a signal propor-tional to the deviation from resonance. In this scheme, the incoming fundamental light is phase-modulated at about 11 MHz by an EOM (see Fig. 4.7). This generates two sidebands on the1010 nm light, each 11 MHz away from the fundamental. The two sidebands are out of phase,84Figure 4.9: Trace of signals used for PDH lock on the LBO cavity, obtained by applying amonotonic voltage ramp to the cavity PZT. The signals shown are the photodiode DCsignal (yellow), photodiode AC signal (pink), and error signal (blue) derived from thephase of the AC signal. The slope of the error signal determines which direction to drivethe cavity. The oscillation that appears on the right-hand side of the photodiode DC anderror signals may result from mechanical vibrations.and as such the total intensity of the beam is constant. This light is coupled into the buildup cav-ity, which has a narrow linewidth (4 MHz) and a large FSR (1.29 GHz). We use a photodiode torecord the feedback beam, which is the superposition of (i) the initial reflected intensity with (ii) acomponent of the enhanced circulating intensity in the cavity, transmitted through the input coupler.On resonance, the feedback signal is close to zero. The fundamental reflection and its transmittedcomponent destructively interfere, and so do the two out-of phase sidebands. Slightly above res-onance, the positive (high-frequency) sideband is more strongly enhanced than the fundamental.The built-up sideband reacts with the reflected (and incompletely cancelled) fundamental to createan 11 MHz beat frequency on the PD. Slightly below resonance, the same is true for the negative(low-frequency) sideband, with a beat frequency of opposite phase. The signal from the PD is thenmixed with the original 11 MHz oscillator and a low-pass filter: the result is a DC-signal called the85error signal, which is proportional to the phase - negative for one sideband and positive for the other.The phase dependence is vital, because it generates the slope of the error signal which determineswhich direction to drive the cavity. The error signal is used in a feedback loop to adjust the posi-tion of the piezo and lock to the fundamental frequency. The cavity lock follows the fundamentalfrequency, and remains locked during ramping of the fundamental frequency. An oscilloscope traceof the signals from the feedback beam PD while ramping the cavity PZT is shown in Fig. 4.9. Asdescribed above, the feedback signal DC component (yellow) decreases almost to zero, while an11 MHz AC component (pink) can be observed on either side of resonance. Noise on these signalsis an important factor limiting the stability of the cavity locks.4.2.2 Vacuum chamber for two photon excitationWe coupled the UV light from the laser described above into a linear Fabry-Perot enhancementcavity enclosed in a vacuum chamber. A schematic of the cavity inside the vacuum chamber isshown in Fig. 4.10 and a photo in Fig. 4.11. The reasons for an additional cavity are twofold:first, since the probability of two-photon absorption depends quadratically on light power, we wantto maximize the UV power available. Second, the cavity provides the retroreflection of the UVbeam which is necessary to perform Doppler-free two-photon spectroscopy with counterpropagatingphotons. The cavity was designed with a length of 260 mm, intended to simulate the dimensionsof the d ≈ 200 mm UCN test cell used at RCNP for neutron EDM tests, as reported in Ref. [1].Two mirrors with radius of curvature = 150 mm were used, an input coupler and high reflector.The reflectivities of the two mirrors chosen were based on the initial plan to install an intracavityspectroscopy cell with AR-coated UV windows (ATFilms). The high reflector mirror was chosen forthe highest available reflectivity, while the input coupler reflectivity was chosen to impedance matchwith the losses from the cell windows. In practice, however, we found that even the best availableAR-coated UV windows with reflectivities up to 98.2% were still too lossy; they reduced the cavityfinesse to unacceptable levels such that the circulating UV power was less than the input power.Therefore we constructed the vacuum chamber to enclose the cavity, as an alternative which doesnot require intracavity windows. For future integration with the nEDM experiment, it is necessaryto source suitable AR-coated UV windows with higher transmission.We enclosed our Fabry-Perot cavity in vacuum and backfilled the entire volume with a par-tial pressure of Xe. The entire volume is pumped with a dry scroll pump (Agilent SH-110). Thebase pressure achieved with this pump is 0.1Torr, and the leak rate with the chamber sealed is0.01Torr/min. The cavity was stabilized by a dither lock, monitored by a photodiode on the cavityreflection which dips at resonance. Turning mirrors inside the vacuum were mounted with picomo-tor steppers to accurately couple the UV light to the cavity. At the time of data collection, the UVlaser output was 40 mW; buildup in the Fabry-Perot cavity increased the power by five times, yield-ing a circulating power of 200mW. We filled the box with a partial pressure of 0.8 - 10 Torr xenon.86Figure 4.10: Schematic of the vacuum chamber and detection optics used for two-photon LIFspectroscopy. L1, L2: modematching lenses. L3: f = 19mm collection lens. L4:f = 50mm collection lens. PBS: polarizing beam splitter. PZT: piezoelectric trans-ducer. λ/4: quarter wave plate. IC: input coupler. HR: high reflectivity mirror. APD:avalanche photodiode.Figure 4.11: Photo inside the vacuum box, showing turning mirror from brewster windowUV input; holder for QWP and lens; input coupler and PZT mounted to hollow copperblock; LIF collection lens; HR mirror; turning mirror to UV cavity monitor.87It was found that in a pure Xe environment at such low pressures, the cavity finesse deteriorated inless than a minute. A small partial pressure of oxygen (around 1 Torr) prevented this loss of cavityfinesse. We suspect that the oxygen undergoes UV photolysis and generates O3 which helps removedeposits of hydrocarbon or other residual gases from the dielectric mirrors.4.2.3 Experimental Setup for two-photon detectionA schematic of the detection optics in the vacuum box is shown in Fig. 4.10. We couple UV lightin to the cavity to excite two photon transitions at the beam waist. An avalanche photodiode (APD,Hammamatsu C5460-01 with onboard preamplifier) is set up outside vacuum, orthogonal to thebeam axis and centered on the beam waist to detect LIF at 823 nm and 895 nm. Collection opticsL3 and L4 collimate the LIF to pass through an AR-coated window in the vacuum box, and focuson the APD. For the sake of maximum LIF collection, we make no attempt to separate the emissionwavelengths; instead we calibrate our LIF signal using the previously observed branching ratios of823 nm and 895 nm LIF multiplied by the different APD sensitivities.The OPSL is tunable across a maximum 1 GHz range, with the locked doubling stages andFabry-Perot cavity automatically adjusting to track any wavelength change. The OPSL laser wave-length is sampled by a Bristol wavemeter which determines absolute frequency up to a 30 MHzcalibration uncertainty. A more accurate measure of relative frequency change is obtained by beat-ing the OPSL light against a fiber-based self-referenced frequency comb available in our lab andrecording the beat spectrum. The comb frequency is locked around 125 MHz; knowledge of thecomb frequency and offset allows for absolute frequency determination. The wavemeter alloweddetermination of the frequency up to the nearest comb tooth, and the addition of the beatnote fre-quency potentially allows determination of the absolute frequency. Details of the comb are presentedin Section A.3.4.2.4 Results: Detection, Hyperfine splitting, and Isotope shiftsWe observed LIF from two-photon excitation of the transition 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0)using 0.8-10 Torr of natural abundance Xe mixed with a small partial pressure of oxygen. The data isshown in Fig. 4.12. We scanned the OPSL over a 300 MHz IR range, which corresponds to 2.5 GHzin the Xe energy spectrum, as follows from two successive SHG stages and a two-photon transition.LIF peaks were observed from all stable isotopes with mass 129-136. The natural abundance of124Xe, 126Xe, and 128Xe were too small to observe peaks from these isotopes. At 1.5 Torr andbelow, the absorption linewidth was sufficiently narrow to resolve each of the observed peaks. Wefit these peaks to a Lorentzian lineshape; as the Doppler-free nature negated the need for a Voigtprofile, as described below. In the low-pressure data, all fit parameters were left free to vary. In 5and 10 Torr data, it became necessary to lock the amplitude ratios to reasonably fit the profile of theunresolved peaks. From the difference in peak positions we are able to extract hyperfine constants88Figure 4.12: Excitation spectrum 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0) transition in naturalabundance Xe. The total pressure was 1.6 Torr, with a 50-50 ratio of Xe and O2. Thex-axis corresponds to the Xe transition frequency, eight times larger than the OPSLfrequency. The y-axis is the observed LIF intensity of the combined 895 nm and 823 nmemission. The peaks are shown with the fitted Lorentzian lineshape as described in thetext. Each peak is labeled with its mass number; additionally, odd isotopes are labeledwith their excited state hyperfine level F in parentheses. The stick diagram shows thecalculated peak positions and intensities obtained from the Lorentzian fit.A and B as well as isotope shifts.Fitting of Doppler-free lineshapeWe expect from Section 2.3.2 that the Doppler-free spectra should be well described by a Lorentzianprofile. To confirm this, we test fit a data sample using the built-in Gaussian, Lorentzian, and Voigtprofiles of the IGOR software multipeak fit package. Using the Voigt fit function returns estimatesof the FWHM for the convoluted Gaussian and Lorentzian components of each peak. We comparedthese to the FWHM given by a Lorentzian function alone. Table 4.2 lists the Lorentzian and Voigtfits. With the exception of two peaks, the analysis using a Voigt profile supports a completelyLorentzian lineshape. The difference in peak center between the two fits is also less than 0.5 MHz.We use only Lorentzian fits in the remaining data analysis. The residual contribution of the Gaussianto the on-resonance signal less than 1%, and is accounted for in the fit by the same constant thataccounts for the APD stray light background.To determine the exact resonance frequencies of the Xe spectral lines, we fit Lorentizan line-shapes to a plot of LIF signal vs. laser frequency. The OPSL frequency measurement was made by89a spectrum analyzer (Advantest) with a resolution bandwidth of RBW = 125 kHz. This means uponconversion to Xe energy, our resolution is 1 MHz. Therefore 1 MHz is is the uncertainty in the fre-quency axis. Since Ordinary Least Squares fitting routines assume no error in the independant axis,we had to resort to more sophisticated fitting techniques. In particular, we used Orthogonal DistanceRegression [27], so named because it measures the orthogonal distance between each sample pointand the estimated fit function. There is a readily available public domain software package calledODRPACK, now implemented with a Python interface, which we used for our fitting. We createda user-defined multipeak fit function comprised of a superposition (sum) of independant Lorentzianpeaks each with a free position and amplitude parameter. A single FWHM parameter described allpeaks in the dataset. In the case of the odd isotopes, each peak position was fit according to theformula for hyperfine splitting described in Section 4.2.6 below.Table 4.2: Comparison of absorption lineshape fitting using either a Voigt profile or aLorentzian profile. Peak widths are in MHz. Also shown is the difference in center fre-quency fitting parameter between the two profiles in MHz.Peak Voigt fit Lorentzian fit DifferenceGaussian LorentzianFWHM σ FWHM σ FWHM σ ν0,Voigt −ν0,Lor1 52.7 4.3 2.7 6.4 42.4 1.6 -0.32 0.0 0.0 51.1 0.5 50.3 0.4 -0.33 0.0 0.0 57.1 0.5 57.9 0.8 0.14 0.0 0.0 62.7 0.4 61.3 0.6 -0.45 0.0 0.0 54.8 1.2 54.5 1.1 0.36 8.7 2.7 51.4 0.7 52.4 0.4 0.07 0.0 0.0 56.8 0.4 57.4 0.6 0.08 0.0 0.0 63.0 0.7 63.3 0.7 0.09 0.0 0.0 60.6 0.8 63.2 0.6 -0.210 0.0 0.0 57.4 0.5 57.8 0.5 0.04.2.5 Pressure broadeningWe obtained data for a 50-50 mix of Xe and O2 at pressures ranging 1-10 Torr. The results areshown in Fig. 4.13. The slope has a value of 31.3(0.3)MHz/Torr, which is comparable to Plimmer’svalue of 28.8(2.6) MHz/Torr for the 249 nm line [128].We also measured the pressure broadening over the same range of total pressure, with a fixed1 Torr partial pressure of O2, as shown in Fig. 4.14. For this data we measure 32.6(0.3) MHz/Torr.90Figure 4.13: Pressure broadening for a 50-50 mix of Xe and O2. The x-axis reports totalpressure.Figure 4.14: Pressure broadening for Xe with 1 Torr of O2. The x-axis reports total pressure.91Table 4.3: Hyperfine splitting constants (in MHz) for the 5p5(2P3/2)6p 2[3/2]2 excited stateof 129Xe and 131Xe. Values obtained by previous works are listed in the last column forcomparison. Values in parentheses are the 1σ standard deviation of the last digit.Hyperfine SplittingConstant This Work Previous WorksA129 −886.3(2) −886.1(8)[29], −889.6(4)[53],−886.2(28)[157]A131 262.6(10) 263.1(6)[29], 262.7(4)[53],263.2(13)[157]B131 34.8(5) 29(2)[29], 21.3(6)[53],26.8(60)[157]4.2.6 Hyperfine constantsOdd isotopes exhibit hyperfine splitting due to the interaction of their nuclear spin I with electronangular momentum J. A common model used to derive the Hamiltonian is the potential of a mag-netic dipole in the field caused by the orbiting electrons. The vector sum of these angular momentaleads to the total angular momentum F = J + I. The hyperfine splitting is well approximated (tosecond order) by magnetic-dipole and electric-quadrupole interactions. The ground state of Xe is1S0 with J = 0 so it has no hyperfine splitting; only the excited state exhibits splitting. The transitionfrequency of the peak νi(I,J,F) corresponding to hyperfine level F is shifted away from its centerof gravity ν0,i by an amount: [154]νi(I,J,F) = ν0,i+AiK2+Bi32 K(K+1)−2I(I+1)J(J+1)4I(2I−1)J(2J−1) , (4.1)whereK = F(F +1)− I(I+1)− J(J+1), (4.2)ν0,i is the center of gravity frequency for isotope i (i = 129 or 131), and Ai and Bi are the magnetic-dipole and electric-quadrupole hyperfine constants, respectively. The next order term is magnetic-octopole and is typically of order kHz. This formula was used in the multipeak Python fit byspecifying I, J and F , with ν0,i, Ai and Bi left as free parameters for their respective peaks. In 129Xethe quadrupole term vanishes because I = 1/2.We report values for hyperfine constants Ai and Bi of the 5p5(2P3/2)6p 2[3/2]2 state in Ta-ble 4.3, alongside previously reported results. The magnetic dipole constants Ai agree well with thepreviously reported values for both 129Xe and 131Xe shown in Table 4.3. On the other hand, theelectric-quadrupole term B131 is 20 % - 40 % larger than previous measured values. Our determinedB131 value is expected to be more accurate than the previous values since, unlike previous measure-ments, it was determined by a transition directly from the ground state which is free of hyperfine92Figure 4.15: Isotope shift relative to mass 136. A line of best fit for the even isotopes showsthe odd-even staggering observed by King. Error bars are smaller than the data points.splitting. Previous measurements for the splitting of the 5p56p state are based on transtions to the6s(J = 2) state, which has its own hyperfine splitting and respective uncertainty.Since Ai is proportional to the nuclear g-factor gI = µI/(µNI) [154], we can compare di-rectly with values determined via nuclear magnetic resonance. We calculate the ratio A129/A131 =−3.38(1), which agrees with the value g129/g131 = −3.375(1) calculated from the magnetic mo-ments and spins given in reference [155].4.2.7 Isotope shiftsTable 4.4: Isotope shifts δνi,136 = ν136 − νi of the transition5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0). We follow the sign convention for isotopeshift outlined in [14]. Shifts for the odd isotopes were determined using the center ofgravity from the hyperfine splitting.i Isotope Shift (MHz)129 586.8(4)130 467.7(5)131 461.2(4)132 326.0(3)134 187.0(4)The addition or removal of neutrons from an atom’s nucleus will slightly shift the electronic93energy levels, causing each isotope to have a slightly different transition frequency. This effect isknown as isotope shift. Both the ground and excited state energy levels of a transition may be shiftedfor each isotope, with a net result of a shift of the transition frequency between those levels. Thereare two contributions to the isotope shift:• A mass shift, where the extra nuclei shift the reduced mass of the atom, and• A field (or volume) shift, where the extra nuclei change the nuclear charge distribution andresulting electric field.The isotope shift δνi,i′ is conventionally defined as the difference in transition frequency be-tween a heavy and light isotope (with corresponding energy gap hδνi,i′) [96]:IS = δνi,i′ = (νi′−νi) = ∆νmass+∆ν f ield (4.3)where νi′ and νi are the transition frequencies of the heavy and light isotope, respectively. A positiveshift indicates the heavy isotope has the larger energy difference. Typically a single isotope (in ourcase, 136Xe) is chosen as a point of reference for all other isotope shifts in an element.The observed isotope shifts are tabulated in Table 4.4 and plotted against atomic mass in Fig. 4.15.To first order the isotope shifts are roughly linear in atomic mass. Odd isotopes display a staggeringeffect, where the odd isotope lies closer in frequency to the lighter even isotope; King explained thisas due to a smaller change in mean square radius of the nuclear charge distribution when adding aneutron to an even-N isotope than for an odd-N isotope [96]. Similar staggering was found in Xefor transitions to 8d and 7s by [33].Mass shiftOne can calculate the mass and field shift to varying levels of detail. The mass shift is commonlysplit into “normal” and “specific” mass terms. Normal mass shift is the mass shift that would beobserved for a single-electron atom, and has the form [66, 96]:δνNMSi,i′ = ν0(meu)( i′− iii′)(4.4)where ν0 is the nominal transition frequency, me is the electron mass, and u is the unified atomicmass unit. The formula predicts that heavier isotopes i′ have the higher wavenumber (more energetictransition).In multielectron atoms, the mass shift is a sum of normal and specific mass shifts; this sum isoften expressed by a dimensionless parameter K.∆νmass = δνNMSi,i′ +δνSMSi,i′ = KδνNMSi,i′ (4.5)94Often the specific mass shifts (of terms) are of opposite sign to the normal mass shift, such thatK < 1. This is a result of electron momenta.Field shiftThe field shift measures the interaction of electrons with the nucleus. It affects most strongly thoseelectrons which have a probability density function that overlaps the nuclear charge radius, i.e.s-orbital electrons. Electrons in higher orbitals can have a small but finite charge density at thenucleus; additionally, non-s electrons can screen s-electrons, which changes the charge density ofthe s orbital.In particular, the shift for an s-orbital is: [96]δE = h∆ν f ield = pi|ψ(0)|2 a30Zf (Z)δ 〈r2〉i,i′ (4.6)where |ψ(0)|2 is the electron probability density at the nucleus, a0 is the Bohr radius, and Z is theatomic number. f (Z) is a function of Z which accounts for distortion of the electronic wavefunctionby the nucleus. δ 〈r2〉i,i′ is the change in the mean square radius of the nuclear charge distributionbetween isotopes. For an s− p transition: the energy level of the s level is raised for the heavierisotope (i.e. the term value is smaller); while the effect on the p orbital is smaller. Thus for theheavier isotope there is a smaller transition energy and an overall negative field shift as observedin [157]. In contrast, we find for a p− p transition, that the heavier isotope has a larger transitionenergy. This is consistent with observations by Plimmer for the 249 nm transition [128]. The termsin Equation 4.6 which don’t vary between isotopes can be collected together as a field shift constantF , such that ∆ν f ield = Fδ 〈r2〉i,i′ . The resulting expression for isotope shift is:δνi,i′ = KδνNMSi,i′ +Fδ 〈r2〉i,i′ . (4.7)King PlotSince the isotope shift differs for every electronic transition, it has become standard to plot the(modified) isotope shifts of one transition at νA against those of another transition at νB in a waythat sheds light on their respective mass shift and field shift components. This is done by solvingboth isotope shift equations for their shared value δ 〈r2〉,δ 〈r2〉i,i′ =δνi,i′−KδνNMSi,i′F, (4.8)95substituting together against a data point from the other set:δνAi,i′−KAδνA,NMSi,i′FA=δνBi,i′−KBδνB,NMSi,i′FB, (4.9)(where subscripts A and B indicate the isotopes shifts of the two respective transitions), multiplyingeach pair by a factor(ii′i′−i)that accounts for the different masses, and finally rearranging to solvefor one of the modified shifts in terms of the other:δνAi,i′(ii′i′− i)=FAFBδνBi,i′(ii′i′− i)+(KAδνA,NMSi,i′(ii′i′− i)− FAFBKBδνB,NMSi,i′(ii′i′− i)). (4.10)The advantage of such a method is that it produces a linear plot; by plotting δνAi,i′(ii′i′−i)againstδνBi,i′(ii′i′−i), the slope and intercept are both independant of the particular choice of isotopes pairs(i, i′). One can use pairs of neighbouring isotopes or pairs of widely different masses. The resultingplot is known as a King plot. The slope yields the ratio of field shifts FAFB for the two levels. Theintecept yields a relation between the mass shifts. The limitation of this method is that it does notyield absolute values for either K or F .A slight variation is to define the modified mass shift MA relative to a standard pair of isotopes,in this case 136Xe and 134Xe, by dividing by the mass shift by the factor g(i, i′) =(i′−iii′)(136×134136−134)such that:MA =KAδνA,NMSi,i′g(i, i′)= KAν0(meu)(136−134136×134). (4.11)as defined by King [96]. The advantage is that the modified shifts are of similar size to the measuredshifts and have dimensions of frequency. The resulting fit line simplifies to:δν∗Ai,i′ =FAFBδν∗Bi,i′ +(MA− FAFBMB). (4.12)where δν∗i,i′ =δνi,i′g(i,i′) is the modified isotope shift.An example of a King plot for our data is shown in Fig. 4.16. We plot our modified isotope shifts(using the second definition) against those of the two photon transition to the 2p5 state, measuredat 249 nm by Plimmer et. al [128]. The two levels in the comparison differ only in their angularmomentum J, due to different orbital angular momentum terms. Plimmer’s data is from the 6p(J =0) state. The slope of the resulting King plot yields F249F252 = 1.00(.01), and the intercept yieldsM249− F249F252 M252 = −0.8(1.8)MHz. The respective mass shifts and field shifts are equal withinerror. This shows to first order, both mass shift and field shift depend on quantum number n but areindependant of J. Further analysis of the isotope shifts are presented below. While the fit is verygood, one observed that the data point at upper right is more than 1σ off the fit line. In fact King96Figure 4.16: King Plot of isotope shifts for two different two-photon transitions at 252.5 nm(present work) and 249 nm (as measured by Plimmer et al. [128])plots are linear to first order only; there is a quadratic dependence on atomic number Z, and somenuclear polarizability dependence. Recent papers [20, 65, 110] have studied the next to leadingorder terms which cause the nonlinearity. After these corrections are accounted for, any remainingnonlinearity could indicate new physics such as an interaction between a new light boson and thenucleus. Transitions which do not involve s-orbital electrons are especially suitable to look for newphysics as both the field shift and quadratic field shift nonlinearity are typically smaller than fortransitions involving s-orbital electrons. The field shift is calculated in the following section; in ourcase the magnitude is similar to that of certain s− p transitions [53], possibly due to the effects ofp electrons in screening inner s electrons [128]. The nonlinearity observed in Fig. 4.16 is likelydue to measurement error. The frequencies of some atomic clock transitions are known with sub-Hzaccuracy. As the accuracy of isotope shift measurements improves, these measurements can be usedto constrain the search for new physics interactions.Determining charge radii based on isotope shiftsEq. 4.7 above indicates that the field shift is dependent in part on the nuclear charge radii. Todate these values have been measured by various techniques, including by x-ray, muonic atom,and optical spectroscopy. Values of δ 〈r2〉i,i′ from several measurement techniques are shown inTable 4.5. We can use these values to determine directly the mass shift and field shifts. Becauseour data is high resolution, we can use it to compare the different literature values. We used datafrom three sources for δ 〈r2〉: Aufmuth [14] is a nuclear data table derived from optical spectroscopy97Table 4.5: Nuclear charge radii values δ 〈r2〉i,136 = 〈r2〉136−〈r2〉i relative to 136Xe, used in thecalculation of absolute K and F . Some of the data has been rearranged from ladder-typepairs which entails some propagation of error. In the actual fits, our data was rearranged tomatch the published format, to avoid propagation of error. Also shown are our calculationsfor K and F for our transition based on the respective source. [14],[29],[67]Aufmuth[14] Borchers[67] Fricke[29]δ 〈r2〉i,136 129 0.133(30) 0.152(40) 0.220130 0.097(24) 0.117(40) 0.153131 0.115(26) 0.124(30) 0.172132 0.075(18) 0.0844(200) 0.115134 0.046(9) 0.0518(120) 0.067K 0.54(8) 0.36(2) 0.47(16)F (MHzfm−2) 2391(292) 2640(80) 1631(384)Figure 4.17: Linear fits to the calculated field shift vs. charge radii parameter δ 〈r2〉, plottedfor each of the three data sets in Table 4.5. The field shift is determined using therespective values for mass shift parameter K to subtract off the mass shift from the totalisotope shift.98Figure 4.18: Residuals (∆ν f ield,calc−∆ν f ield, f it) of the fits plotted in Fig. 4.17 for field shiftvs. charge radii parameter δ 〈r2〉. We calculate the mean squared error of these residualsto determine the goodness of fit.)of many isotopes, with a stated uncertainty of 10%. The values for Xe come from observing theNIR discharge spectra from isotopically enriched samples. Fricke [67] publishes nuclei data formuonic atom transition energies, electron scattering, and x-ray isotope shifts. Muonic atoms havean electron replaced by a muon, yielding smaller Bohr orbits. The Xe values published here comefrom muonic atoms. In particular, muonic values should be viewed with some uncertainty due tothe proton radius puzzle.2 Borchers et. al. [29] published charge radii measurements in Xe basedon optical detection of isotopes produced by the ISOLDE collaboration. In their analysis they makethe assumption of specific mass shift equal to zero plus or minus the normal mass shift. (i.e., anuncancelled normal mass shift), allowing them to directly estimate δ 〈r2〉.The charge radii results of all three sources are listed in Table 4.5, along with our calculatedvalues for K and F resulting from a fit to the published δ 〈r2〉i,i′ . Charge radii data in the table areshown as differences relative to 136Xe. Both Borchers and Fricke present data in that format. Inthe case of Aufmuth, data were tabulated as differences in charge radii for closest neighbouringisotopes with corresponding uncertainty. To avoid propagation of errors, our isotope shift data wasrearranged to match that particular dataset, rather than rearrange the published data. Using the2Measurements of the proton radius from laser spectroscopy on muonic hydrogen [129] yield a precise value for theproton radius that disagrees with the previous accepted value by almost five standard deviations.99data reported by Borchers et. al [29] yields the fit with the smallest uncertainties and the valuesK = 0.36(2) and F = 2640(80) MHzfm−2.A clearer comparison was facilitated by subtracting off the mass shift from the isotope shift,using the respective fit parameters for K. This leaves only the field shift term which is linear inδ 〈r2〉. We performed a linear fit of field shift vs. δ 〈r2〉 as shown in Fig. 4.17, and calculated themean squared error (MSE) of the residuals shown in Fig. 4.18:MSE =1nn∑i(∆ν f ield,calc−∆ν f ield, f it)2 (4.13)We calculate the MSE values 1.7MHz2 for Borchers’ data, and 38.8MHz2 and 57.9MHz2 for Auf-muths’ and Fricke’s data, respectively. The much smaller error on Borchers data indicates that ourobserved isotope shifts are consistent with Borchers’ values of δ 〈r2〉. We find similar agreementwith Borchers’ data for the isotope shifts measured by Plimmer et. al [128] for two-photon ex-citation to the nearby 5p5(2P3/2)6p 2[1/2]0 state. The most recent data tables [10] cite Borchers’data.4.2.8 Comparison of signal amplitude and natural abundanceThe peak height we observe in the LIF signal is proportional to the transition probability, whichdepends among other parameters on both the number density and transition dipole moment. Forthese experiments we used natural abundance Xe, so we expect the signal for each isotope to beproportional to its respective natural abundance. These are listed in Table 4.6, where the observedsignals have been normalized to the largest peak 132Xe. For the odd isotopes 129Xe and 131Xe, wesum the amplitude of the two or four hyperfine peaks, respectively. We find agreement between thepeak heights and natural abundance to within 4%.Table 4.6: Observed peak heights and natural abundance for each isotope relative to 132Xe.Peak height Natural Abundance(relative to 132) (relative to 132)129Xe (sum of 2 peaks) 0.94 0.98130Xe 0.16 0.15131Xe (sum of 4 peaks) 0.82 0.79132Xe 1 1134Xe 0.41 0.39136Xe 0.35 0.33100Figure 4.19: Signal S (at resonant frequency) normalized to power P squared for a mixture ofnatural abundance Xe and 1 Torr O2, as a function of total pressure.Signal amplitude pressure dependenceThe number density of xenon in the cell n, under the ideal gas law, is proportional to the Xe partialpressure. The lineshape parameter g(ω) discussed in Section 2.3.2 is inversely proportional to thelinewidth ∆ω . Under a certain range of pressures the lineshape will be dominated by collisionalbroadening (pressure broadening), for which the linewidth will be linearly proportional to the totalpressure. Under those conditions, a sample of pure Xe would have a two-photon transition ratethat is independent of pressure, emitting a constant LIF signal at any pressure. The same does nothold for a fixed partial pressure of Xe collisionally broadened by other gases, or for a linewidthdetermined by another broadening regime.To demonstrate this pressure independence we collected data at 1, 5 and 10 Torr for Xe and O2mixtures, shown in Fig. 4.20 and Fig. 4.19. The fixed O2 data shows that the signal is approximatelyconstant. The data for the 50-50 mixture is less convincing, suggesting a calibration error; however,it is at least apparent that the signal does not decrease with pressure, as might be expected due tothe decrease in nXe.Horiguchi et al [83] reports a lifetime of 38±2 ns for the 5p5(2P3/2)6p 2[3/2]2 level, whichsuggests a natural linewidth of 4 MHz . In that case we can reduce the pressure to about 130 mTorrand see the same signal; beyond this we expect the signal to decrease linearly with pressure.101Figure 4.20: Signal S (at resonant frequency) normalized to power P squared for a 50-50mixture of natural abundance Xe and O24.2.9 Determination of two-photon transition probability from LIF signalBased on the observed LIF signal intensity, we calculate the transition probability for two-photonabsorption.Derivation of transition rateExcitation of two photon transitions occurs along the UV beam axis, with the largest transition rateat the beam waist. As a starting point, we approximate our excitation region as a long cylindricalvolume V with beam waist wo and Rayleigh range zR. Taking into account the spatial variation ofthe UV beam profile, Equations 2.46 and 2.48 give the total number of excitations N per second involume V for Doppler-free excitation as:(dNdt)tot= 6αg(ω)n∫I(r,z)2dV (4.14)One should distinguish between the total transition rate, at the detected photon rate at the APD.Assuming isotropic emission of one NIR photon for every two-photon absorption, and neglectingnonradiative decay, the detected photon count rate is:(dNdt)det= 6αg(ω)n∫I(r,z)2Adet(r,z)dV , (4.15)102Figure 4.21: Zeemax simulation showing the path of rays from a point 3mm from the UVfocus, to illustrate the vignetting. The detection configuration is strongly sensitive toemission less than 1 mm from the UV beam focus.where Adet is the dimensionless acceptance fraction of the APD. This integral can be evaluatedexplicitly. Since the beam waist is over an order of magnitude smaller than the sensitive radius ofthe detector acceptance, we integrate over r = (0,∞) and approximate the acceptance to be constantover the beam radius such that Adet(r,z)≈ Adet(z) is a function of z only. The integral becomes:∫I2(r,z)Adet(r,z)dV =∫I20 (w0w(z))4(piw2(z)4)A(z)dz (4.16)Where I0 is the (circulating) laser intensity at the beam waist. The beam radius w(z) only increasesby√2 along the Rayleigh range. For integration of length l  zR along the beam axis, we canapproximate w(z)≈ w0 to get:∫I2(r,z)Adet(r,z)dV =P20piw20∫Adet(z)dz =P20piw20Aavgl (4.17)where P0 is the light power and Aavg is the acceptance fraction averaged over length l. Estimationof the acceptance fraction by a ray tracing simulation is shown in the following section.Detector acceptanceDetection in these experiments was by an avalanche photodiode (APD) transverse to the UV beamaxis as shown in Fig. 4.10 and described in Section 4.2.3. Two convex lenses were used to collectthe light: one to collimate the (assumed) isotropic emission through the vacuum box exit window,and another to focus the now-collimated light onto the APD. Due to this configuration, it is clear thatthe APD is most sensitive to emission from the focal point. Therefore detection of LIF is limited103Figure 4.22: Vignetting fraction (over 4pi) of rays emitted isotropically from a point on thebeam axis. The detection configuration is most sensitive to fluorescence emitted within1 mm from the UV beam focus.to emission near the beam waist, and not along the whole Raleigh range as initially assumed. Inaddition, the lenses are sufficiently thick that the thin lens approximation does not hold.We used ray tracing software Zeemax OpticStudio to simulate the propagation of light throughthe detection optics. The UV beam axis defines the object plane, and the APD detector definesthe image plane. Rays of LIF emission produced within 3 mm of the UV focus are traced throughlenses and windows, to the APD, as shown in Fig. 4.21. The simulation shows that the chromaticdispersion for 823 nm vs. 895 nm emission is negligible. The vignetting feature is used to measurethe fraction of rays that reach the APD window in the image plane, assuming isotropic emission,in contrast to rays that are vignetted by the physical edges of the detection optics and the box. Wefind that 3.3% of the rays emitted from the focus into 4pi reach the detector. The acceptance rapidlydrops for points more than z= 1mm from the focus as shown in Fig. 4.22. In contrast, the UV beamin these experiments had a waist w0 = .064mm and zR = 51mm. Thus the detected LIF is a smallfraction of the total emission.We averaged the simulation results for points on the UV beam axis up to±2mm from the focus,finding an average collection fraction of Aavg = 0.010(2).Photon count rate and transition probabilityIn the section above we showed that detection of LIF is limited to emission which occurs within±2mm of the UV beam waist. The excitation region used for the following calculations is a cylinder104Table 4.7: Experimental values used in the determination of two photon α for the isotope132Xe.Signal (V) S -0.0247(5)FWHM (Hz) ΓL/2pi 55.8(6)×106Lineshape (s) g(0) 2.42(2)×10−9Circulating Power (W) P0 0.228(11)Beam Waist(m) w0 0.0000640(3)Total pressure (Torr) Ptot 1.61(4)132Xe Partial Pressure(Torr) P129Xe 0.216(5)132Xe Number Density (m−3) n129Xe 6.97(16)×1021Branching Ratio F823 0.7F895 0.3Photon Energy (J) E823 2.41×10−19E895 2.22×10−19APD Sensitivity (V/W) Q823 -1.5×108Q895 -1.2×108Peak Detector Acceptance A(0) 0.033Volume-integrated Intensity (W2m−1)∫I2(r,z)A(r,z)dV 164.09Detector Acceptance, averaged over l = 4mm Aavg 0.010(2)Excitation length (m) l 0.004Two-photon coefficient (cm4/J2) α 4.5(1.1)with beam waist w0 and length 4mm, centered at the beam waist. Our detector is sensitive to 6p−6sfluorescence at both wavelengths 823 nm and 895 nm. The observed APD signal voltage S is:S(V ) =(dNdt)det(E823Q823F823+E895Q895F895) (4.18)where E895 is the photon energy in J, Q895 is the detector sensitivity in V/W, and F895 is the fluores-cence branching fraction from the upper level. From [83] we find the branching fractions F895 = 0.30and F823 = 0.70. Solving for excitation rate, we find:(dNdt)det=S(E823Q823F823+E895Q895F895)(4.19)From Equation 4.19, our 132Xe peak signal corresponds to a photon count rate 7.4×108 photons−1at the APD, and an estimated total emission of 1010 photons−1 in the excitation region near the UVbeam waist.Comparing Equations 4.19 and 4.15, we can use the observed photon count rate to solve for αby the following substitution:α =16G(ω)n(E823Q823F823+E895Q895F895)S∫I2(r,z)Adet(r,z)dV(4.20)105Table 4.7 lists the relevant parameters for the largest isotope LIF peak, that of 132Xe. Basedon these values we calculate α = 4.5(1.1)cm4J−2. This finds agreement with the result α =4.5(2)cm4J−2 reported by Raymond et al [138].4.3 Estimate of Magnetometer Sensitivity4.3.1 Mercury Comagnetometer uncertaintyComagnetometry performed in the ILL nEDM experiment determined the 199Hg precession fre-quency by fitting to an exponentially decaying modulation of optical transmission. The accuracy ofthe magnetometer depends in part on the signal to noise ratio (SNR) achieved in the 199Hg spec-troscopy. The uncertainty in precession frequency is parametrized as follows [75]:σ f ≈ 14T ′anas1√n(1+ e2T′/τ)1/2 (4.21)Here an is the RMS noise in the ring-down signal (decaying sinusoid), as is the signal amplitudeat the start of precession, n is the number of points fit, T ′ is the precession time and τ is the decaytime. The fraction asan is the RMS signal to noise ratio. This frequency fitting uncertainty translatesto a field uncertainty σB according to2piσ f = γσB. (4.22)The typical performance of the magnetometer in the 1 µT field had an uncertainty of σB ≈ 50 fT,or 50 ppb [75].4.3.2 Sensitivity estimate based on Xe two photon SNRBased on our current signal to noise ratio, we can extrapolate our current results to estimate an ef-fective magnetometer sensitivity using Equation 4.21, with as/an determined from the (modulated)LIF signal following two photon excitation. We focus here on estimation of the signal size at lowpressure, such as is required to avoid dielectric breakdown under the high voltage conditions of thenEDM experiment. We consider the following:1. partial pressure of isopure vs. natural abundance Xe: other isotopes contribute to pressurebroadening and a lineshape-related loss in signal.2. partial pressure of O2: it is expected that without a buildup cavity in vacuum, O2 will not benecessary.3. polarization of 129Xe: 50% polarization has been demonstrated by commercial SEOP polar-izers.106Table 4.8: Extrapolation of magnetometer uncertainty at low Xe pressure. Calculated usingEquation 4.21 with T ′ = 150 s, n = 1500, τ=160 s. We consider the following cases:(i) Current experimental conditions, with 1.6 Torr of mixed 50-50 Xe (nat. abund.) andO2, (ii) 1.6 Torr isopure 129Xe. (iii-vi) 10 mTorr isopure Xe at current laser power (iii)and maximum achievable to date (iv), 1mTorr isopure Xe at current laser power (v) andmaximum achievable to date(vi)Case Detector Ptot Plaser as an SNR σ f σB(Torr) (W) (V) (V) (µHz) (fT)(i) APD 1.6 0.228 0.00531 0.0021 2.5 46.7 3965(ii) APD 1.6 0.228 0.040846 0.0021 19.5 6.07 515(iii) PMT 0.01 0.228 4.54×10−6 0.00000005 90.8 1.3 110(iv) PMT 0.01 0.4 1.4×10−5 0.00000005 279 0.42 36(v) PMT 0.001 0.228 4.54×10−7 0.00000005 9.1 13 1105(vi) PMT 0.001 0.4 1.4×10−6 0.00000005 27.9 4.22 3594. detector: detection of weak LIF signals requires a detector with high SNR.TRIUMF has a stated goal in the 2015 CDR of 30fT uncertainty per cycle.[1]We evaluate a few cases in Table 4.8. Based on our current LIF signals, we anticipate 6 µHz(500 fT) uncertainty for 1.6 Torr of isopure 129Xe, assuming a polarization of 50%. Extrapolatingto 1 mTorr, the signal decreases due to a change of broadening regime: the lineshape factor g(ω)reaches its maximum, while number density continues to decrease. It may be possible to mitigatethe loss by increasing the laser intensity, and replacing the APD with a low-noise cooled PMT,which can have orders of magnitude better SNR when cooled, and a low noise current preamp. Weestimate σB = 36fT may be attainable at 10 mTorr pressure for a 400 mW laser power.4.4 DiscussionIn this section we described the observation of LIF signals from two-photon transitions in Xe usingboth a broadband pulsed laser and narrow linewidth CW laser. Using the pulsed laser, we inves-tigated all two-photon allowed transitions to the 6p level, namely, to 2[1/2]0, 2[3/2]2, and 2[5/2]2states, and measured the LIF wavelengths. While the pulsed laser linewidth is too broad to performoptical pumping, we did demonstrate the selection rules which forbid excitation to 2[1/2]0 by cir-cular polarized light. We compared LIF emission on- and off-axis and measured the threshold forbidirectional emission.We performed high-resolution spectroscopy with the CW laser and successfully measured hy-perfine constants of the 5p5(2P3/2)6p 2[3/2]2 level and isotope shifts for the 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0)transition at 252.5 nm. The uncertainties in the isotope shifts are small enough to assign mass shiftand field shift parameters with uncertainty of only a few percent. We compare our isotope shifts tomeasured charge radii values reported in the literature and verify the values reported by Borchers107et al. We calculate the two-photon absorption coefficient (i.e. cross-section) for the 252.5 nm tran-sition and find agreement with literature. We find some evidence that LIF signal is constant downto 1.6 Torr based on the pressure broadened absorption lineshape, and predict a continuation of thistrend with decreasing pressure down to the natural linewidth at 130 mTorr. Pressure broadeningmeasurements at lower Xe pressure are currently underway in this lab. Finally, we extrapolate theobserved signal to low pressures to estimate the SNR of polarized 129Xe for comagnetometry, andestimate the acheivable magnetic field measurement uncertainty. We predict detection with 36 fTsensitivity is possible using very low-noise PMTs.108Chapter 5Conclusions5.1 ConclusionThis chapter outlines work done to date and the current status of experiments working towardspolarization of 129Xe and high resolution spectroscopy at low pressures. This is an exploratorywork investigating experimental parameters necessary for the implementation of 129Xe as a comag-netometer in the neuton EDM experiment at TRIUMF’s ultra cold neutron facility, which is themotivation described in Chapters 1 and 2. We outlined the nEDM measurement technique and theproperties of 129Xe which make it a suitable magnetometer candidate. Chapter 2 outlined a schemecomprising hyperpolarization of 129Xe through spin exchange optical pumping, purification throughfreezeout, and detection of the polarization by two-photon transition using circular polarized light.Transitions to excited states 2[3/2]2(F = 3/2) and 2[5/2]2(F = 3/2) were identified as having suit-able selection rules to facilitate polarization detection.A review of spin exchange optical pumping literature shows that many of the factors govern-ing the production and relaxation of hyperpolarized noble gases have been identified and suitablycontrolled. This has lead to near-unity polarizations and relaxation lifetimes of many hours, espe-cially in the solid state. Chapter 3 covers the development of SEOP capabilities in our lab at UBC,motivated by the need to have a source of polarized 129Xe to measure changes in the two-photontransition spectra. Using a broadband diode laser, we demonstrated SEOP by detecting AdiabaticFast Passage NMR signals from polarized 129Xe samples. We observed that high SEOP cell temper-atures and pressures correspond with higher polarization. We compared batch mode operation withcontinuous flow operation, finding the highest polarization for 10 min batch mode preparation. Wecalibrated our 129Xe AFP signals against those of a H2O sample and find PXe = 5.5% based on therelative signals. In comparison with the literature, we find that higher polarization can be achievedusing frequency-narrowed pump lasers.Chapter 4 described measurements on the properties of two-photon transitions to 6p states1092[1/2]0, 2[3/2]2 and 2[5/2]2. We measured LIF signals with a pulsed laser and verified the two-photon selection rules prohibiting excitation to the 6p2[1/2]0 state by circular polarized light. Wealso made qualitative measurements of the on-axis LIF enhancement by bidirectional emission. Thiswas followed by high-resolution CW laser spectroscopy. We generated CW UV light at 252.5 nm,using an OPSL and nonlinear doubling, to excite Xe to the 6p2[3/2]2 state inside a vacuum chamberat 1.6 Torr total pressure. The hyperfine constants and isotope shift we found improve the precisionof these measurements over previous literature values. The high resolution permits selective excita-tion of the transition to the 2[3/2]2(F = 3/2) hyperfine level. Therefore this transition can be usedas a probe of optical pumping with no background from excitation of even isotopes, or even of the2[3/2]2(F = 5/2) state. We measured α = 4.5(1.1)cm4J−2 for the two-photon coefficient, in goodagreement with literature. Based on observed signals, we extrapolated to low pressures to predictσB = 36fT magnetometer uncertainty at 10 mTorr.5.2 Future Work5.2.1 Improvement of 129Xe Polarization and FreezeoutThe polarization of 129Xe affects the visibility of the Larmor precession, and therefore directlyaffects the signal to noise ratio used to determine the magnetometer uncertainty. For this reason,the polarization should be made as high as possible. Values over PXe = 90% have been reportedin the literature using frequency narrowed lasers. There is a tradeoff between high-volume andhigh-polarization production. For the purposes of nEDM measurements at TRIUMF, the 129Xeconsumption can be loosely estimated as:(0.001Torr)(2.59×1019 atomscm−3760Torr)(20×103 cm3cell)( 24hr/day4min/cycle)(5.1)which yields 2× 1020 atoms/day, or less than 10 mL at STP. The amount used by the experimentwill be likely small compared to amounts lost through the filling and thawing process. These volumerequirements are small enough to be satisfied by methods that favour high polarization production,typically in batch mode operation. The T1 lifetimes for 129Xe in the solid state are such that theSEOP polarizer should be operated several times per day. Per experimental cycle, a controlledpressure of the gas can be released by slightly warming solid Xe in a cold trap. Continued work onfreezeout at UBC is necessary to perform two-photon spectroscopy on polarized 129Xe free fromthe pressure broadening effects of the mixed He and N2.1105.2.2 Towards optical polarization detection at low pressureOptical polarization detection for comagnetometry requires polarized 129Xe and low-pressure spec-troscopy. In this thesis we performed high-resolution Xe spectroscopy at 1.6 Torr total pressuredriving two-photon transitions with approximately 200 mW of UV light in an optical cavity. Exper-iments are currently underway to measure the pressure dependence of the two-photon absorptionand LIF signal down to a few tens of mTorr. This will be accomplished by replacing the vacuumchamber and optical cavity with a short gas cell and retroreflection mirror. An increase in laserpower and tighter UV focus can be used to compensate for the loss of the optical cavity. Removalof the cavity removes the need for a partial pressure of O2, as transmission loss at the windows isnot compounded by multiple reflections. Additionally, the gas cell and lines can be baked whilepumping to reduce the offgassing rate which was previously 0.01Torr/min. We will test the pre-diction from Section 4.2.8 that as the pressure-broadened linewidth is reduced, down to the limitestablished by the excited state lifetime, the excitation rate remains constant.Freezeout of polarized 129Xe is necessary to perform optical detection at low pressure withsufficient signal to noise ratio. The SEOP gas mixture is only 1% Xe by volume. As noted above,the buffer gas He and N2 must be removed to prevent pressure broadening of the linewidth and acorresponding 100x reduction in LIF signal. It is necessary also to preserve the polarization duringfreezeout. The currently observed LIF signal corresponds to unpolarized 129Xe with cross sectiongiven by the two photon coefficient α . For polarized 129Xe, there is an effective increase (1+PXe)αin the two photon absorption coefficient, which linearly affects the LIF signal. Detection of a changein the LIF signal upon polarization is a current priority. This requires performing spectroscopy onthe atoms while under magnetic field, and in part motivates the replacement of the vacuum chamber,and its magnetic parts, with a glass vapour cell.At pressures below 130 mTorr the LIF signal is expected to decrease linearly with pressure/num-ber density. The enhancement of bidirectional emission observed in pulsed experiments could po-tentially compensate for this decrease in signal. This has been proposed as a way to achieve themaximum sensitivity to the two-photon transition rate. Ongoing experiments will compare the LIFemission detected on- and off-axis of the UV excitation, and determine if the same phenomena isobserved for excitation with CW light. The estimated threshold at 5 mTorr, 108 W/cm2, is severalorders of magnitude larger than the CW intensity used in the current experiments. However, evenif enhancement from bidirectional emission is not observed, on-axis detection is still anticipated toyield larger LIF signals, as an on-axis detector can collect emission from along the entire Rayleighrange and not only at the beam waist.5.2.3 Determination of absolute two-photon transition frequenciesExperiments in Chapter 4 used a self-referenced frequency comb. The absolute frequency resolutionattainable by the frequency comb could potentially allow us to set a new reference standard for the111absolute energies of excited Xe electronic states. The NIST reported value of the absolute transitionfrequency for 5p5(2P3/2)6p 2[3/2]2 ← 5p6 (1S0) is 79 212.465 cm−1 or 2 374 729. 958 GHz, withan absolute uncertainty of 0.0035 cm−1 or 100 MHz[147]. The values cited were determined bytaking values from [85] which measured excited state transitions in 136Xe, and applying a shift ofabout 0.5 cm−1 to match the absolute frequencies found by [33] for certain VUV transitions to 5d,8d, and 7s. These absolute frequencies had been calibrated by scanning the excitation laser throughnearby iodine resonance lines and counting the number of interferometric fringes produced by areference etalon while scanning. Using our technique one could set a reference standard traceableto a GPS reference oscillator.Using the frequency comb described in A.3, we measured the isotope weighted average for theabsolute transition frequency to be 2374730. 490 GHz. This is 0.532 GHz higher than the NISTreported value. Factors affecting the uncertainty of this measurement are as follows: (i) frequentloss of connection with the 10 MHz GPS reference oscillator (which has since been resolved),and (ii) drift in the unlocked comb offset frequency due to technical limitations (namely, the lackof an additional branch to simultaneous lock the comb offset and obtain our beat note). If boththese limitations are overcome, the frequency comb method can provide an accurate measure of theabsolute transition frequency with sub-MHz precision. While this is of great theoretical interest,current measurements of the transition frequency are sufficient for neutron EDM experiments. Inparticular, it is the uncertainty in precession frequency which determines magnetometer accuracy,rather than the uncertainty in absolute transition frequency.5.2.4 Measurement of Xenon Electric Dipole MomentXenon can only be used as a comagnetometer if its own permanent electric dipole moment is knownto high accuracy, and is substantially smaller than that of the neutron. From equation 2.1 we ob-served that the uncertainty of the nEDM measurement dn is limited by the uncertainty in the EDMof the comagnetometer species, e.g. dXe. Additionally, since the origin of an EDM in Xe has adifferent theoretical basis than that of the neutron (i.e, the Xe EDM may originate from the electronEDM, or also from T- and P-violating electron-nucleon or pion-nucleon interactions), the search fora Xe EDM is complementary to the nEDM experiments [47]. Development work on 129Xe comag-netometry using two-photon transitions simultaneously offers new tools to be used in the search forthe 129Xe EDM.Vold et al. [160] measured the 129Xe EDM in a Rb-Xe mix, using SEOP on a stack of threecells under applied ~B and ~E fields. They optically pumped in an applied 10 mG field followed byfree spin precession in a 0.1 mG transverse field. Pumping and detection were both performed usingRb D1 light at 795nm. While 129Xe precesses, the Rb polarization maintains a large polarizationcomponent parallel to the pump light, due to the effect of optical pumping. This component is drivenwith a controlled oscillation, and becomes sensitive to the 129Xe precession which is detected using112a lock-in amplifier. The measured EDM was (−0.3±1.1)×10−26 e · cm.Rosenberry and Chupp [48, 142] measured the 129Xe EDM with 3He added as a comagnetome-ter. Their apparatus implemented a spin-maser configuration with separate pumping and masercells. In the pumping cell, both noble gas species were spin polarized through collisions with Rboptically pumped at the D1 line. N2 buffer gas was used to prevent radiation trapping. Polar-ized atoms diffuse down to the maser cell where precession is monitored by a separate pickupcoil for each gas. The pickup coil forms part of a resonant tank circuit, which uses feedbackon the spontaneous relaxation to drive Rabi oscillations. The 129Xe signal in one coil is mixeddown with a reference frequency standard using phase-sensitive detection and used to lock the129Xe precession frequency by active correction of the static B field. This also maps any ~E-correlated frequency shifts onto the 3He precession frequency where they can be detected. Theyreport (−0.7±3.3(stat)±0.1(syst)×10−26 e · cm for the Xe EDM limit.Other 129Xe EDM experiments are underway. Asahi et al [12, 87, 169] optically probe thesteady state precession of 129Xe in a single maser cell containing Rb using D1 light, and generatea feedback magnetic field which locks the precession frequency. Kuchler et al [101] measure thesimultaneous free spin precession of polarized 129Xe and 3He using SQUID detection to measurethe magnetization of the precessing nuclei. In this experiment, 3He is used as a comagnetometer toaccount for static field drifts. They measure precession for over 4000 s, with relaxation lifetimes inexcess of 2500 s, and predict 10−28 e · cm sensitivity per cycle to the 129Xe EDM.Direct optical detection of the two-photon transition can be performed without disrupting thespin ensemble significantly. For example, the current two photon excitation rate in the cell is only1010 s−1. In contrast, using spin exchange with Rb for detection introduces additional relaxationwhich can shorten the free precession lifetime. 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Scalar magnetometers include proton precession andoptically pumped atomic magnetometers.Induction CoilThe most conceptually simple magnetic sensors are induction coils. At their simplest these arecompletely passive wire loops in which a voltage will be induced by Faraday induction; that is, bya changing magnetic flux V = −dφ/dt. Sensitivity to the signal can be increased with an op-amp(active electronics) or by inserting a ferromagnetic core to increase the flux density (passive). Theirsensitivity can be as good at 20 fT [103], and is primarily to AC fields in the range 1 Hz to 1 MHz.Induction coils are a vector magnetometer because they sense only the magnetic field componentnormal to the coil plane.FluxgateA fluxgate works by applying a driving magnetic field HE to saturate a ferromagnetic core suchthat B reaches a maximum, and sensing the voltage V = Hdµ/dt corresponding to the changingpermeability µ = B/H. Measurement of the desired (background) field is possible because it shiftsthe saturation point of the magnetic core, causing a hysteresis effect. A typical fluxgate consistsof a pair of solenoidal coils wrapped around one or two saturable magnetic cores. One coil drives127an AC field to saturate the magnetic core twice per cycle with reversing direction while the othermeasures the flux from B. This technique requires one sensor per component of the magnetic fieldvector. The sensitivity is from 10−2−107nT and can measure DC or AC up to 10 kHz, limited bythe need to sweep the saturating field.SQUIDSuperconducting Quantum Interference Device (SQUID) is a vector magnetometer device consist-ing of a superconducting coil cooled below its critical temperature Tc. Below Tc any field passingthrough the coil area would induce current which flows with zero resistance; furthermore, the fluxthrough the coil will be quantized. The coil is split at one or two points by a non-superconducting“Josephson Junction,” through which the measured current becomes an oscillating function of mag-netic field, due to an interference effect of the quantized flux. The small area of the SQUID loopis often coupled to a larger coil via a flux transformer [140]. A null-field technique with feedbackallows measurement of larger DC fields. SQUIDs are sensitive from 10 fT up to 1 nT but requirecryogenic cooling. Specific orientations of two SQUIDs can detect magnetic field vector gradients.Hall EffectThe Hall probe measures the effect of the Lorentz force F = q~v×~B on a current travelling through athin conductor. A current travelling along~x will generate a detectable voltage across the~y directionin response to a field applied along~z. Typically a semiconductor is used to see a larger Hall effect.The Hall probe is a vector magnetometer and is sensitive from approximately 10−5− 10−1T overDC-1MHz.Magneto-resistanceCertain materials such as permalloy increase their resistance in response to an applied magneticfield. A current passed through a thin film of magnetized metal will measure less resistance if thefilm magnetization is aligned with an applied field. These sense 10−6− 10−3T over DC - 1GHz.Layered structures of ferromagnet and conductor can exhibit so-called Giant Magneto-resistancewith sensitivity 10−7−10−1T.Faraday rotationThe Faraday effect is rotation of the plane of polarized light as it travels through a magneto-opticcrystal with polarization parallel to both the crystal axis and to an applied magnetic field [103].There is a different index of refraction for each circular component of the light, creating a phasedifference resulting in rotation. The strength of the effect depends on the crystal’s Verdet constant.Sensitivity of 10−11T has been obtained with response DC-GHz.128Proton precession magnetometersProton magnetometers align the spins in a hydrogen-rich fluid such as benzene by briefly applying astrong B field, generating a Boltzmann polarization. When this field is turned off, the protons precessaround the ambient field at the Larmor frequency ω = γB and induce a voltage in a sensing coil. Itcan be sensitive from 10−10−10−4T A variation on the technique uses RF to polarize the electronicspins of nearby free radicals, which then polarize the proton nuclear spins by the Overhauser Effect.The detection is analogous but an order of magnitude more sensitive. Any nuclear spin can be usedin a similar manner to the proton magnetometer, although typically this requires the optical pumpingdescribed below.A.1.1 Optically Pumped Atomic MagnetometersOptically pumped atomic magnetometers measure the Zeeman splitting of atomic levels in a mag-netic field, by detecting the precession frequency of an optically pumped sample. A thorough reviewof optical magnetometry is Ref. [37]. Optical magnetometry has been performed on paramagneticalkali metal vapours of Rb, Cs, K and on metastable 4He, in addition to diamagnetic 3He (throughspin exchange optical pumping) and 199Hg. Typical configurations measure either the longitudi-nal (Mz) or transverse (Mx) component of the precessing magnetization. Many optically pumpedmagnetometers are classified as being double resonant [77]: they use resonant light to couple elec-tronic states for optical pumping and detection, along with resonant RF radiation to couple Zeemansublevels and initiate precession. Other variations use modulated transverse (or synchronous) op-tical pumping instead of RF. Detection of the magnetization (Mz or Mx) is typically through theabsorption of resonant probe light, but detection may also be performed by NMOR or even SQUID.In general, atomic magnetometers are scalar magnetometers, measuring only the magnitude |B0|.However, measurement of the vector components of |B0| may be achieved by applying a strongoffset field ~Bo f f set along each component, one at a time [77]; then to first order, the magnitude|~B0+~Bo f f set | depends only on the component of |B0| parallel to ~Bo f f set .Optical pumping of atomic vapours has been demonstrated starting with sodium by AlfredKastler in 1949 [81]. The idea for using lamp-based optical pumping to measure magnetic fieldswas proposed in the 1950s and 60s, by Dehmelt [56] and Bloom [26], and enabled sensing of fieldsas low as 10−12 T. Recent laser advances have expanded the usability of the technique to differentgases and lower fields [38].Mz and Mx typesThe Mz magnetometer type [37, 77] uses circular polarized light with~k||~B0 (i.e, ~k||zˆ) as a pumpand probe. The atomic ensemble is continuously pumped by the light, while a resonant orthogonalRF field couples the Zeeman-split sublevels, driving the atoms away from zˆ. One sweeps the RF129frequency through resonance while looking for changes in absorption. The signal has slow responsebut high accuracy.In contrast, the Mx magnetometer type [37, 77] uses a beam of circular polarizedlight where~k has a component perpendicular to B0 in order to measure the precession frequency.Pump and probe may be achieved using two orthogonal beams or a single combined beam. Thereis again a persistent RF field driving the precession. Measurements of the precession frequency aresampled from the detected oscillating signal and used as feedback to the RF coils. The responseis fast but somewhat less accurate. Both types suffer from dead zones; that is, relative orientationsof~k and ~B0 where the desired magnetization component cannot be sampled. The sensitivity at lowfields is typically limited by broadening of the resonance due to spin exchange relaxation; however,spin exchange relaxation free (SERF) magnetometers have been shown to remove this broadeningby making collisions more frequent than the rate of precession [98].Bell-Bloom typeThe Bell-Bloom magnetometer operates with a circular polarized light beam~k ⊥ ~B0 which is mod-ulated at the nominal Larmor precession frequency. Modulation of either the amplitude, frequency,or polarization is possible. This removes the need for resonant RF radiation, meaning it is not adouble-resonance type. Bell and Bloom were the first to demonstrate such a technique and called itoptically driven spin precession [19]; it is also known as transverse or synchronous optical pumping.Precise determination of the precession frequency may come from feedback of a detected photodi-ode signal to the oscillator providing the modulation; another method is to monitor the free spinprecession via NMOR. An example of a Bell-Bloom type is the Univ. Washington 199Hg EDMmeasurement [74].Free Spin PrecessionFree spin precession refers to a magnetometer in which there is no persistent RF field driving theatoms. After being polarized either by longitudinal optical pumping followed by an RF pi/2 pulse,or by transverse optical pumping, the population is allowed to precess freely about B0. Detection ofthe precession frequency is by measuring the modulated transmission of a weak circular polarizedprobe beam~k⊥ ~B0 or by the rotation of a linearly polarized probe beam by NMOR. An example ofthis type is the 199Hg comagnetometer used in previous nEDM experiments [16, 62]. The techniquehas also been demonstrated for non-cohabiting magnetometers [2].Nonlinear magneto-optical rotationNonlinear magneto-optical rotation (NMOR) magnetometers measure a rotation of the polarizationvector of linear polarized light transmitted through a gas which has been polarized through opticalpumping [39]. The rotation is caused by the difference in refractive index for right- and left-circular130polarized beams in the polarized medium. It is nonlinear since the signal depends on the populationdifference of Zeeman sublevels in the ground state, which in turn depends on the degree of polariza-tion by optical pumping. The 199Hg EDM measurement at Univ. Washington uses NMOR to detectthe free spin precession of the polarized nuclei [74].A.2 NMR Q-Factor MeasurementTo increase the gain of our NMR pickup coil, we use a “tuning box” which forms together with thepickup a tuned RLC circuit. The current in the circuit is given by Kirchhoff’s law [94]:I =V(R+iωC− iωL)(A.1)The resonance frequency is:ωr = 1/√LC (A.2)In particular, at resonance the voltage across the capacitor will beVc =VR√LC=V Q, (A.3)where Q = 1R√LC is the quality factor or Q-factor of the circuit. The Q-factor is given empiricallybyQ =ωr∆ω, (A.4)where ∆ω is the half-power (-3dB) full width of the resonance. Therefore Q represents a signalgain. In our pickup circuit the measured voltage will by Q times the induced emf.Due to their different respective gyromagnetic ratios, Xe and H2O have different Larmor pre-cession frequencies in the same field B0. We often operate near the maximum acheivable field.The corresponding resonance frequencies for Xe and H2O differ by approximately a factor of four.We built tuning boxes resonant at 15.555 kHz and 58.82 kHz for Xe and H2O respectively. Theexact frequency was determined by a suitable capacitor to tune the circuit. Each circuit has its ownQ-factor.We determine the Q-factor of each circuit theoretically and experimentally. The results aregiven in Table A.1. We calculate the theoretical Q-factor by first principles using Equation A.3and by simulation in software package LTspice using an AC voltage source. In the simulation Q iscalculated by two methods: first by Q = ω0∆ω at -3dB points, and second by the ratio of peak powerto input.We measure the Q-factor of the circuit experimentally by sweeping an input voltage in fre-quency, recording the resonance ωr and full-width ∆ω at -3dB points of the frequency spectrum.131Figure A.1: Tuning box resonance at 15kHzFigure A.2: Tuning box resonance at 58kHz132Table A.1: Q-factor measurements for tuning boxes “Xe” (resonance 15.555kHz) and “Water”(resonance 58.82 kHz)“Xe” “Water”R (Ω) 9.3 9.3L (mH) 4.13 4.13C (nF) 25.9 1.67ωr/2pi (kHz) 15.555 58.82QXe QH2OFirst principles LRC series 42.6 167.7LTspice (0 Ω input) 42.7 172.9LTspice (2.2 Ω input) 33.4 137.6Network Signal Analyzer (< 5Ω input), ω0∆ω 23.0 46.6Lock-in amplifier, ω0∆ω 26.0 38.9The first test was performed driving LRC with the output of a network signal analyzer and recordingthe voltage across C; comparison of the measured Q-factors agreed within 10% with the observedpeak voltage on resonance (Q = Vc/Vin). The second test was performed with the lock-in ampli-fier by inducing a real EMF, by driving a magnetic dipole (either a small 3-turn dipole coil, or theB1 saddle coil pair) with the reference output and recording the mutual induction power spectrumwhile sweeping the reference frequency. The power spectrum is shown in Fig. A.2 and Fig. A.1.The measured Q-factor is much smaller than predicted from first principles; this discrepancy may bedue to capacitive coupling of the RF coils, as well as lossy electronic components. In the case of thenetwork signal analyzer, part of the discrepancy is due to the small output impedance of the mea-surement device. Similar loss in Q-factor was observed in simulations using LTspice software whenan additional series resistor representing input impedance was added to the model. Experimentalresults using both methods yielded agreement within 20% for each Q-factor. We take the valuesQ = 26.0 for the 15.555 kHz resonant RLC circuit (“Xe”) when calculating polarization estimates,and Q = 38.9 for the 58.82 kHz RLC circuit (“Water”).A.3 Frequency comb methodFor greater laser frequency resolution, we beat our OPSL light against a fiber-based self-referencedfrequency comb described in [112]. The method of obtaining the beatnote follows the descriptionin [66], and a brief description follows here. The comb is a pulsed femtosecond fiber laser with arepetition rate of frep = 125 MHz. It generates light at evenly spaced “teeth”, 125 MHz apart, overthe entire wavelength range of 1− 2µm, each with near 10 kHz linewidth. The formula for thecenter frequency of the nth comb tooth is given by:νcomb(n) = fceo+n frep (A.5)133Figure A.3: Lowest-frequency beatnotes νA and νB generated by mixing OPSL light with theself referenced frequency comb. The detuning on the x-axis is relative to νcomb(n).where fceo is the offset frequency where the first tooth would be located if the comb was extendeddown to zero frequency. The offset can be determined by beating the comb against its own SHG toobtain a difference frequency. The doubled comb also has equally spaced teeth 2 frep apart:νdoubled(i) = 2 fceo+2i frep, (A.6)The difference frequency generated by beating the nth comb tooth of the fundamental against itsnearest-frequency SHG neighbour allows determination of the offset fceo:νdoubled(i = n/2)−νcomb(n) = 2 fceo+2(n/2) frep− fceo−n frep = fceo (A.7)In practice fceo can be determined by the difference frequency between any two teeth of the respec-tive combs. In the lab we measure a beatnote aroud 340 MHz to determine the offset. Both frep andfceo can be frequency stabilized to RF synthesizers locked to a 10 MHz GPS reference. This yieldsabsolute frequency determination of the comb teeth.Our OPSL light at 1009 nm is always within 125 MHz of a comb tooth. We pick off some of theOPSL light and couple it into a fiber where it is mixed with comb light and detected by a photodiode.This generates a difference frequency at νbeat = νOPSL− νcomb against each nearest comb tooth,which tells us how far detuned light is away from that respective comb tooth. Since the comb teethare evenly spaced frep apart, we see the lowest frequency beatnotes at νA = νOPSL− νcomb(n) andat νB = νcomb(n+ 1)− νOPSL, as shown in Fig. A.3. If we already know the OPSL frequency towithin 62.5 MHz by another source such as a wavemeter, then we can determine the position ofthe nearest comb tooth by Equation A.5, and use our beatnote to determine our absolute frequencybased on the detuning from that nearest tooth. In practice, we know the OPSL frequency only with100 MHz uncertainty, and cannot assign the nearest comb tooth with absolute certainty based onwavemeter readings alone. We overcome this limitation by recording the frequency of a beatnote134Figure A.4: Beatnote (blue) and wavemeter (red) readings during a monotonic increasingsweep of the OPSL frequency, while recording the second-nearest neighbour beatnotefrequency νB.in the range 62.5-125 MHz (the second-nearest neighbour frequency) using an RFSA (AdvantestU3700 series) as a function of time, while applying a monotonic voltage ramp to the OPSL PZT toincrease the OPSL frequency, as shown in Fig. A.4. We observe the trend in the beatnote - whetherit is increasing or decreasing - and use this knowledge to determine whether the particular beatnoteunder observation is beating against a higher or lower frequency comb tooth, so we can unwrap thedata accordingly.The comb described above is capable of absolute frequency measurements with sub-MHz ac-curacy. However, the comb had no free port at the time of Xe experiments, so it was necessary toredirect light from the fceo stabilization loop to obtain the comb light used to generate our beatnote.This introduced an uncertainty that precludes absolute frequency measurements. We measured theunlocked fceo before and after experiments and observed a 2 MHz drift over two hours. Addition-ally, we experienced frequent loss of the 10 MHz clock signal used to lock frep, which was obtainedfrom a GPS reference frequency generator with satellite receiver. Without the clock signal, the un-certainty in frep is on the order of tens or hundreds of Hz, which multiplies by n ≈ 2000000 combteeth to generate absolute frequency uncertainty of hundreds of MHz. With continued technicalefforts it should be possible to obtain absolute frequency measurements in Xe spectroscopy.135

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