Towards a Precise Measurement of the Electric Quadrupole Amplitude within the 6p 23 P 0!

Transcription

1 Towards a Precise Measurement of the Electric Quadrupole Amplitude within the 6p 23 P 0! 3 P 2 Transition in Atomic Lead by Eli Hoenig Professor Majumder, Advisor A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in Physics WILLIAMS COLLEGE Williamstown, Massachusetts May 22, 2017

2 Abstract We present a measurement of the 6s 2 6p 2 3 P 0! 3 P 2 electric quadrupole (E2) transition amplitude relative to the 3 P 0! 3 P 1 magnetic dipole (M1) transition amplitude in 208 Pb. The motivation for this measurement stems from recent advancements in wavefunction calculations that have renewed lead s viability to test the electroweak interaction in an atomic physics based experiment. Our measurement aims to guide further refinement of the theory that will enable a precise calculation of the nuclear weak charge. Our preliminary value for the relative size of the E2 reduced matrix element compared to the known M1 matrix element is E2/M1 = (03)(10) where the first error is statistical and the second systematic. We employ Faraday optical rotation techniques with two external cavity diode lasers, one operating at 939 nm and one at 1279 nm, to measure the line-strengths for the E2 and M1 transitions, respectively. We find that this technique is capable of detecting optical rotations below the the level of 1µrad, easily resolving the E2 transition. We also present a measurement of the hyperfine structure splitting of the 6s 2 6p 2 3 P 1 state and the 6s 2 6p 2 3 P 0! 3 P 1 isotope shift between 208 Pb and 207 Pb. i

3 Acknowledgments First I would like to thank Professor Tiku Majumder, who has not only made this thesis possible, but who has guided me with great patience and knowledge throughout the year. I also extend my gratitude to Dr. Milinda Rupasinghe, who was eminently helpful in lab on a day-to-day basis; and to Professor Catherine Kealhofer for being my second reader. I would also like to thank Michael Taylor and Jason Mativi for their expertise and assistance in the construction of the experiment apparatus, Hallee Wong for building the 939 nm laser, and Ashay Patel for his help over the month of January. My experience at Williams has been defined by the coaches of, and my friends on the Nordic Ski Team. Every morning Ildiko Bodor and Karen Marchegiani provide the day s lab fuel consisting of smiles, yogurt and co ee. Nathaniel Vilas has been an excellent labmate throughout the year, and I could not have asked for a better partner-in-physics than Daniel Wong. Grace Flaherty, thank you for always being there and for lending an ear to my confused physics rambles. Finally, thank you to my parents for showing up to every important thing I do, and of course for allowing me to come to Williams. ii

4 Executive Summary We present a preliminary measurement of the electric quadrupole transition amplitude relative to the magnetic dipole transition amplitude in the 6s 2 6p 2 triplet state of 208 Pb. This measurement is conducted in response to recent wavefunction calculations for Pb given in [1] that have revived the element s relevance as an atomic testing ground of the Standard Model. The experiment detailed here marks the first time lead has been studied in the Majumder lab, although similar experimental techniques were employed in 1999 in a transition amplitude ratio measurement in thallium. To even detect the electric quadrupole (E2) transition, whose strength is two orders of magnitude less than that of the magnetic dipole (M1) transition, we employ a highly sensitive spectroscopic method that detects the Faraday rotation induced by lead atoms in the path of the laser. More common techniques, such as direct absorption spectroscopy, that record the attenuation of the laser beam after it passes through the lead vapor fail to observe the E2 signal. As shown in figure 1, not only does the Faraday rotation technique resolve the E2 transition, it produces spectra with a high signal to noise ratio. (a) A typical Faraday rotation spectrum for the E2 transition with fit residuals. (b) A typical Faraday spectrum for the M1 transition with fit residuals. Figure 1: Representative Faraday rotation spectra for the E2 and M1 transitions. Note the y-axis scales are di erent for the two spectra. iii

5 EXECUTIVE SUMMARY iv Fundamentally, the Faraday rotation apparatus is comprised of two crossed prisms (the polarizer and the analyzer), a heated vapor cell containing the lead atoms, and a Faraday modulator made up of a solenoid and a crystal with a high Verdet constant. With no optically active material in between the two prisms, virtually no light can pass through the analyzer. When the lead cell is heated and a longitudinal magnetic field is applied to the lead vapor, the atoms produce a frequency dependent rotation of the light polarization with an amplitude at the level of several hundred µrad for the E2 transition. Figure 2: A simplified schematic of the polarimetry setup. The atomically induced rotation is augmented by the Faraday modulator, which rocks the polarization at a known frequency. With this additional modulation, the amplitude of the spectral peak is linearly instead of quadratically related to the lead-induced rotation; furthermore, we employ two lock-in amplifiers for each laser in order to record the signal s first and second harmonic components. These components are proportional to the rotation and transmission lineshapes, respectively a comparison of the amplitudes of the two types of spectra serves as a cross check to the final measurement. If the transition amplitude ratio were given by the ratio of the peak heights in figure 1, our measurement would be immediately precise and straightforward to implement. The ratio of the transition amplitudes, however, depends not only on the peak heights, but also on the parameters of the lineshape describing the spectra. These parameters include the Zeeman splitting, the Gaussian width and the Lorentzian width, the last of which is particularly di cult to determine. Progress is currently being made to establish tight constraints on these parameters, but they are still the major source of uncertainty in our preliminary measurement.

6 EXECUTIVE SUMMARY v (a) E2/M1 values collected at 945 C (b) E2/M1 values collected at 913 C Figure 3: Transition amplitude measurements collected at two di erent temperatures: 913 and 945 C. The temperature dependence of the final value is one of the sources of uncertainty we quantify. The error quoted for these histograms is the standard deviation the uncertainty for any single data point not the standard error which is a factor of ten smaller. Our preliminary value for the transition amplitude ratio E2/M1 is 0.148(2)(8), where the first error is statistical and the second systematic. In reduced form, the ratio is a factor of p 5/3 larger. Themagnitudeofthesystematicerror,whichisafull5%ofthefinalvalue, is a conservative estimate, but still shows that more must be done to mitigate systematic error in this experiment. One such source of systematic uncertainty is the dependence of the transition amplitude ratio on the temperature of the vapor cell, as depicted in figure 3.

7 Contents Abstract Acknowledgments Executive Summary i ii iii 1 Introduction Motivation Previous and Related Work Present Work Outline of this Thesis Transition Amplitudes and the Spectral Lineshape Transition Matrix Elements The Electric Dipole Approximation The Multipole Expansion Relative Strengths of the Higher Order Transitions The Incident EM Wave Conversion from the Linear to Circular Basis The Form of the Refractive Index The Zeeman E ect and Di erential Absorption The Weak Field Approximation Zeeman Energy Levels Faraday Rotation The Idealized Transmission Lineshape Line Broadening and the Convolution Profile Doppler Broadening Homogeneous Broadening Mechanisms Convolution Connecting Measurement and Matrix Element Extracting the MEs from the Faraday Rotation Spectra Extracting the MEs from the Transmission Spectra The Reduced Matrix Element vi

8 CONTENTS vii 3 HFS and TIS Measurements in the 6p 2 3 P 1 State Atomic Structure of Lead Fine Structure Hyperfine Structure The Isotope Shift Experiment Setup and Data Acquisition Apparatus Details Frequency Calibration Data Acquisition and Analysis Fitting the Spectra Results and Evaluation of Errors Statistical Error Systematic Error Scan Direction and Speed HFS and TIS Results Faraday Rotation Experiment and Apparatus The Faraday Rotation Method Apparatus Details Sensitivity and Noise Analysis The Polarimetry System The Oven System The Lasers and Their Operation The Fabry-Pérots and Frequency Calibration Data Acquisition Signal Detection Data Collection Calibration Data Summary of Collected Data Data Analysis and Preliminary Results Data Analysis Summary of Collected Spectra Calibration Scheme The Fitting Process Constraining the Fitting Parameters Preliminary Result Systematic Error Analysis Uncertainty in the Lorentzian Width Further Systematic Error in Transition Amplitudes E ect of Magnetic Field Strength

9 CONTENTS viii 6 Future Work Refinement of the Faraday Technique Future Directions with Lead A HFS and IS Measurements of Thallium 8P 3/2 62 A.1 Experiment Apparatus A.1.1 Apparatus Overview A.1.2 Locking the 655 nm Laser A.1.3 Setup for Isotope Shift Measurements A.1.4 Signal Detection A.1.5 Frequency Calibration A.2 Data Processing A.2.1 Data Acquisition A.2.2 Acquiring and Fitting the Fabry-Perot Spectrum A.2.3 Acquiring and Fitting the Vapor Cell Signal A.3 Results and Errors B The LS Coupling Scheme 75

10 List of Figures 1.1 Pb Energy Levels Energy Levels with Zeeman Splitting Lorentzian and Dispersion Lineshapes The Faraday Lineshapes Energy Levels with HFS Splitting Pb HFS and TIS Experiment Layout Pb Spectrum with Sidebands Pb and 207 HFS and TIS Spectrum Histogram of HFS splitting of 207 Pb and TIS in 208 Pb and 207 Pb HFS TIS Scan Direction Error HFS TIS Comparison With Old Values Faraday Rotation Experimental Setup Polarimeter Apparatus Simulated Rotation Lineshape Sensitivity of Faraday Apparatus Oven Schematic FP spectrum with sidebands Fitted 1279 nm Faraday Spectra Fitted 939 nm Faraday Spectra Faraday Fit Schematic Lorentz Width Error Propagation Lead vapor pressure as a function of temperature Temperature Dependence of Result E2:M1 ratio as a function of magnetic field strength Pb Energy Levels for Atomic Beam Experiment A.1 Experimental Setup A.2 LockedvsUnlocked A.3 Laser Locking Signal ix

11 LIST OF FIGURES x A.4 Spectrum with FP A.5 A sample fit and linearization polynomial for a single Fabry-Perot spectrum 71 A.6 Dual-isotope spectrum with fit residuals

12 Chapter 1 Introduction The properties that make heavy atoms, which are the objects of study in the Majumder lab, useful testing grounds of the Standard Model and its possible extensions are the same that make these atoms particularly complicated to model. The Majumder lab conducts measurements that erode that barrier, providing a base of experimental knowledge to better the quantum mechanical description of atoms that serve as tests beds of the Standard Model. For twenty years, it was understood that the complex atomic structure of lead with atomic number Z = 82 and with four valence electrons precluded an accurate quantum mechanical description of the element; even with precise experimental measurements that showed evidence of weak interactions in this system, extracting useful information from lead was considered too di cult. Tests in cesium and thallium with Z = 55 and 81, and with one and two valence electrons respectively were considered more practical due to their simpler atomic structures. In the past year, new wavefunctions have been calculated for lead, reviving its importance in tests of the Standard Model. In this thesis, we describe a new lead experiment conducted in the Majumder lab in order to guide the further refinement of these recent calculations. 1.1 Motivation The search for physics of beyond the Standard Model (SM) is ongoing, both in particle colliders and in experiments probing atomic systems on optical tables. We assist the latter, providing an experimental framework to test the electroweak interaction, one of the four fundamental forces predicted by the SM. The weak force interaction is particularly important as certain universal symmetries, such as charge conjugation and parity are not conserved under its influence. Parity symmetry means there is no fundamental di erence between left handedness and right handedness; if the coordinates describing the universe flipped from (x, y, z)! ( x, y, z) the universe would be una ected. The weak force is the only one of the four fundamental forces to violate this symmetry. Parity non-conservation (PNC) tests in lead were made in 1995 by the Seattle group [2] and by the Oxford group [3]. The groups predicted that PNC would mix an electric dipole 1

13 CHAPTER 1. INTRODUCTION 2 element (E1 PNC )intothe6s 2 6p 2 3 P 0! 3 P 1 transition, which would otherwise be exclusively magnetic dipole driven. The presence of this E1 PNC element leads to optical rotation of the incident light. The amplitude of this optical rotation, whose size is of order 1 µrad and is proportional to E1 PNC, was detected and measured with 1% precision by the Seattle group. 1 The E1 PNC amplitude is related to the nuclear weak charge, Q W,afundamentalquantity predicted for atoms by the SM, according to the relation, E1 PNC / Q W C(N,Z) (1.1) where C(N,Z) contains information regarding the atomic structure of the atom. The measurement described in [2] has great potential to provide an accurate value for Q W, as C(N,Z) scales as Z 3 ;E1 PNC is amplified by atomic lead s large nuclear mass. However, the size of the nucleus and the corresponding number of electrons make wavefunction calculations di cult. At the time of the experiment, the atomic structure factor, C(N,Z), had yet to be calculated with su cient accuracy to be of fundamental interest, preventing an accurate measurement of the weak charge. In 2016, the theory group Safranova et al. based in the University of Delaware formulated new approximation methods to calculate the wavefunctions of lead. They in turn used these wavefunctions to calculate various atomic properties, including the E1 PNC amplitude, improving the theoretical precision of the weak charge by a factor of two [1]. Still, their calculations are three times less precise than experimental measurements. It is the aim of this thesis to catalyze the further refinement of this theory by providing experimental parameters of quantum mechanical properties in Pb against which they can check their calculations. 1.2 Previous and Related Work For the past decade and a half, the Majumder lab has focused on measuring Stark Shift amplitudes (i.e. scalar polarizabilities) of Group IIIA atoms and using two-step doppler free spectroscopy to measure the hyperfine structure. These experiments work in concert, as measurements of the polarizabilities test wavefunction behavior far away from the nucleus, and measurements of the hyperfine structure test short range wavefunction behavior. We have conducted a measurement of the hyperfine structure and isotope shifts in thallium and indium over the summer of The measurements of thallium are recorded in appendix A; for a record of the indium measurements see Nathaniel Vilas s bachelor s thesis [4]. In 1999, the Majumder lab conducted a similar experiment to that outlined in this thesis. The group measured the electric quadrupole (E2) to magnetic dipole (M1) amplitude ratio within the 1283 nm 6P 1/2! 6P 3/2 transition in thallium [5]. They measured this ratio, = with a precision of 1.6%, directly leading to a newly calibrated value for the E1 PNC amplitude in thallium. To perform this experiment and measure the very small 1 the Oxford group s measurement is in good agreement, although its quoted uncertainty is slightly larger at 3%.

14 CHAPTER 1. INTRODUCTION 3 E2 transition, they used Faraday rotation methods in which changes to the polarization of light, related to meaningful physics in the atom, are detected and recorded. We have reassembled much of the apparatus that was used in that work, and employed and adapted it to measure the E2/M1 ratio in the 6p 2 triplet state of lead. Although the experiment conducted in 1999 resulted in an analogous measurement of the E2/M1 ratio, the experimental and analysis demands of the previous work greatly di er from that of the current experiment. The ground and first excited states of thallium are split into two hyperfine structure energy levels, as the Tl nucleus has finite spin. The transitions from the F = 1 hyperfine ground state to the F = 1 and F = 2 hyperfine excited states (which are not resolved from each other due to Doppler broadening) are of mixed M1 and E2 character. The Faraday rotation lineshape given by these transitions has a term linear in the ratio E2/M1, so a well designed fitting program can extract this parameter precisely and directly. As explained below, such a direct extraction is not possible in the present experiment. 1.3 Present Work We work towards a measurement of the ratio between the 6s 2 6p 2 3 P 0! 3 P 1 M1 transition amplitude and the 3 P 0! 3 P 2 E2 transition amplitude in 208 Pb. In a similar fashion to the 1999 experiment, we employ Faraday rotation methods to resolve the E2 transition, which cannot be observed with less sensitive techniques such as direct absorption spectroscopy. Our experiment di ers from the 1999 experiment in that the the 1999 group measured this ratio within a single atomic transition, whereas we measure the amplitude ratio between two separate transitions; the E2 transition is driven by light at 939 nm and the M1 transition is driven by light at 1279 nm. Thus our apparatus is extended to accommodate two lasers operating simultaneously, and we analyze two separate spectra taken in rapid succession then take the appropriate ratios to extract the quantity of interest. The low lying energy levels of lead are diagrammed in figure 1.1. The states are labelled according to the LS coupling scheme, which is described in appendix B. The transitions studied in this thesis occur within the triplet 6s 2 6p 2 state. As the azimuthal quantum number does not change in these transitions, they cannot be driven by the otherwise strong electric dipole moment and therefore, if studied closely, reveal the characteristics of the higher order multipole moments, E2 and M Outline of this Thesis The following chapters present a comprehensive summary of the theory underlying our measurement and experimental techniques, the details of the experimental apparatus, and our data analysis methods. I begin by defining the electric quadrupole and magnetic dipole matrix elements, and give an estimate for their sizes as compared to the more familiar electric dipole matrix element. I proceed to explain the fundamentals of laser spectroscopy, optical

15 CHAPTER 1. INTRODUCTION 4 Figure 1.1: Low-lying states for Pb are displayed. Details of the energy levels and their notation scheme are given in appendix B. Hyperfine structure which does not occur in the even isotopes of lead is omitted. rotation and the source of optical activity in our experiment. I conclude the theory portion of this thesis by connecting the measurement we directly make to the transition amplitude ratio we seek to calculate. In the subsequent chapter, I detail a preliminary experiment in which we measure the hyperfine-structure splitting of the 6s 2 6p 2 3 P 1 state and the isotope shift between 207 Pb and 208 Pb in the 6s 2 6p 2 3 P 0! 3 P 1 transition. I then delve into the heart of the experiment, detailing the apparatus and details of Faraday rotation spectroscopy. I continue with a

16 CHAPTER 1. INTRODUCTION 5 description of our data analysis methods, and with a presentation and discussion of our results. I conclude with a discussion of the immediate steps that must be taken in order to finalize the preliminary result presented here, in addition to the long term directions the Majumder lab may take that are related to this experiment.

17 Chapter 2 Transition Amplitudes and the Spectral Lineshape On the most basic level, our experiment uses the same tool that spectroscopic laboratories have employed for decades: a controlled electromagnetic wave. When this EM wave oscillates at the resonance frequency of an atom it interacts with, both the atom and the field are perturbed. The atom, whose properties we care about but cannot directly see, is excited to ahigherenergystate. Ifwewishtolearnabouttheatom,welooktotheEMwave,whose changes mirror those of the atom precisely. Yet we study two forbidden transitions in Pb: the magnetic dipole transition to the first excited state, and the electric quadrupole to the second. These transitions are two to four orders of magnitude weaker than their electric dipole counter-part, so we must employ specialized techniques to even see signs of absorption. In this chapter, I begin by describing the matrix elements (MEs) that we measure. I express these MEs first without consideration of the finite lifetime of the excited state in order to reveal how the higher order multipole moments arise, to detail their basic characteristics and determine their relative sizes. By then relating the MEs to the refractive index of the lead vapor and including the finite lifetime of the excited state omitted previously I derive the lineshape of the recorded spectra. In doing so, I also provide the basic theory underlying Faraday rotation spectroscopy. I conclude the chapter by outlining the connection between the MEs and the lineshape parameters. 2.1 Transition Matrix Elements We measure the ratio of matrix elements governing the intrinsic strength of the 6s 2 6p 23 P 0! 3 P 1,2 transitions, expressed in bra-ket form as: hj 0 =2 q e j =0i hj 0 =1 µ m j =0i, (2.1) where q e and µ m are the electric quadrupole and magnetic dipole moments, respectively. 6

18 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 7 In order to derive the origin of these MEs, I first give the general result for the hamiltonian denoting the interaction between an EM wave and an atom. I subsequently derive the interaction terms for the specific cases of electric dipole, magnetic dipole and electric quadrupole transitions. As in [6] and [7], I take a semi-classical approach wherein the incident EM wave is described according to Maxwell s equations, but the atom is described entirely with quantum mechanics. The derivation must include the interaction of the laser s magnetic field with the atom in order to describe M1, so we write Ĥ in terms of the scalar and vector potentials, V and A. The vector potential represents the incoming plane wave, but the scalar potential of the system is simply the energy holding the atom together, and is therefore part of the unperturbed Hamiltonian, Ĥ 0. The total Hamiltonian is: Ĥ = 1 2m [ˆp ea(r, t)]2 + ev (r) = e Ĥ 0 (ˆp A + A ˆp )= 2m e Ĥ 0 ˆp A, m ˆp 2 2m (2.2) where Ĥ0 is the familiar Hamiltonian for an atom, + ev (r). In the second step, we have followed [7] and let the wave be transverse, which leads to [A, ˆp ]=0. Inthefirst step, we have dropped the term e 2 A 2, as it is a small correction on top of an already small perturbation. Now, the vector potential represents an incoming EM plane wave, with its polarization perpendicular to its propagation direction. We can simplify the following analysis by assuming the wave propagates in the ẑ direction, so the vector potential is, A = A 0 cos(!t kz)e, (2.3) where k is the wavevector,! is the frequency of oscillation and e is the transverse direction in which the electric or magnetic field points. In absorption processes, under the well-known rotating wave approximation see [8] or [7] only the e i!t term of the two exponentials composing the plane wave contributes to the transition probability. The approximation is valid when! is close to the resonance frequencies of the transitions, which are the only regions we are interested in anyway. Thus the MEs of the perturbing hamiltonian are (doing away with the time-dependence, as we assume that the atom reaches a steady state): e Ĥ ij = A 0 hi ˆp e exp( ikz) ji (2.4) m Without considering the decay of the excited state, these matrix elements determine the rate of absorption for a given transition, and therefore the transition strength The Electric Dipole Approximation At this stage, it is common to assume that the spatial variation of the field over the length of the atom is negligible. After all, the atomic diameter of lead is 3.60 Å[9]andthe

19 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 8 wavelengths of light from our lasers are of the order 1 µm 10 4 atomic lead diameters. Under the assumption that the spatial variation of the EM wave is small, it can be expressed as a standing wave so e ikz is replaced with unity, which simplifies things dramatically. Following [7], we use the commutation relation, im [Ĥ0, r] =ˆp,tofactorouttheenergy ~ eigenvalues and replace the momentum operator with the position. Eq. 2.4 reduces to the more familiar: Ĥ ij = A 0 i! 0 hi er ji. (2.5) The matrix element in Eq. 2.5 is simply the electric dipole moment, and as it depends on only the position and the electric charge instead of on the electron spin the total matrix element has no dependence on the magnetic field. The electric dipole ME vanishes, however, when ` =0 seee.g. [8]. Thetransitionswestudytakeplacewithin the 6p orbital so cannot be driven by the electric dipole moment. The spatial variation of the EM wave may produce a small e ect, but for our transition it is the only perturbation the atom experiences The Multipole Expansion Instead of ignoring kz entirely, we assume that it is small and keep the first term in the Taylor expansion, e ikz =1 ikz +... (2.6) The derivations of the M1 and E2 contributions to the transition matrix element are similar in character, only more involved, than that of the electric dipole moment and can be found in [7] and [6]. Drawing from these texts directly, the contribution to the total ME of the perturbing hamiltonian are: Ĥm 2 = A 2!0 2 0 hi µm ji 2 c! Ĥq 2 = A (2.7) 0 0 hi e(xz + yz) ji 2 2c The matrix element in the first expression is the magnetic dipole, M1, the ME in the second is the electric quadrupole, E2. µ m in the expression for M1 is the magnetic dipole operator i.e. the spin operator. Due to the di erent symmetries of M1 and E2 as compared to E1, transitions with ` = 0 are allowed for both. However, only M1 can drive the 3 P 0! 3 P 1 transition and only E2 can excite the atom to the 3 P 2 state; we consider pure magnetic dipole and electric quadrupole transitions, unlike in previous work done by the Majumder lab, e.g. in [5], where mixed M1 and E2 transitions were seen in thallium 6p 1/2! 6p 3/2.

20 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE Relative Strengths of the Higher Order Transitions We can now estimate the magnitude of transitions driven by higher order matrix elements compared to those driven by the electric dipole. For M1, we can even plug in the precisely calculated value given in [1]. We find, unsurprisingly, that the higher order transitions are several orders of magnitude weaker than the electric dipole transitions. The strength 1 of any transition is proportional to the magnitude of the relevant squared matrix element (this is Fermi s Golden Rule, outlined in e.g. [10]); therefore we obtain relative line-strengths by taking the ratio of the expressions given in Eq. s 2.7 and 2.5. h Ĥm i 2 1 h hi µ ji i 2 h µb /c i 2 = (Z ) 2 Ĥ e1 c 2 hi er ji ea 0 /Z (2.8) Where is the fine structure constant, and I have approximated the magnetic dipole moment as the Bohr magneton, and the electric dipole moment as the product of the electric charge and the radius of the atom, using dimensional analysis as in [7]. Evaluated for neutral lead (Z = 1), the ratio of the transition amplitudes is approximately We can make this ratio more exact by plugging in numbers from [1]. Comparing the magnetic dipole transition to the nearest electric dipole transition, 6p 2 3 P 1 6p7s 3 P 0,the 1 ratio of the transition amplitudes is thesameorderofmagnitudethateq.2.8gives, 400 which is all we expect given the nature of the approximations used above. The ratio of the electric quadrupole to the electric dipole is necessarily a more approximate calculation, as it is highly dependent on the charge distribution of the atom. According to [7], the electric quadrupole ME should be the same order of magnitude as the magnetic dipole ME. We have seen in our experiments, however, the electric quadrupole transition amplitude in the 6P state of Pb is an order of magnetic smaller than the magnetic dipole transition amplitude. The following sections in this chapter provide the base of knowledge necessary to understand how we not only detect but measure these minuscule matrix elements. We go over, in the following sections of this chapter, why the experiment works, not how we implement it. We begin, naturally, with the perturbing EM wave. 2.2 The Incident EM Wave The complex EM wave passing through our apparatus has both electric and magnetic field components: E = E 0 e i(!t B = E (2.9) 0 c ei(!t kz) ŷ, where, following the geometry of our experiment, the EM wave is propagating in the ẑ direction and is polarized in the transverse plane. We represent E and B in complex form with 1 strength is defined as the square of the ME, whereas the amplitude of a transition linear in the ME kz)ˆx

21 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 10 the understanding that any measurement is the real component of the resulting expression. The oscillating frequency of the EM wave in Eq. 2.9 is given by!, andk is the complex wavenumber. k can also be written as: k =!ñ c, (2.10) where ñ is the complex index of refraction. The refractive index, it turns out, depends on the laser frequency, atomic resonance frequencies and the transition strengths, thereby governing the lineshape and amplitude of the recorded spectra Conversion from the Linear to Circular Basis As explained in section 2.3 and shown in Eq. 2.19, under the application of a magnetic field the index of refraction for left-handed circularly polarized light ( )di ersfromthatofrighthanded circularly polarized light ( +.)Inanticipation,webreakE and B in Eq. 2.9 into their + and components. We can express the ˆx and ŷ components of the polarization direction as a linear combination of ˆ+ and ˆ,whereˆis the unit vector denoting the direction of : ˆx = 1 p 2 (ˆ ŷ = ˆ+) i p 2 (ˆ + ˆ+). (2.11) We have defined the EM wave s polarization to be ˆx, thereforewecanwritee and B in terms of its and +componentsbysubstitutingthefirstexpressionofeq.2.11intoeq The resulting expression describes the EM wave prior to its interaction with the vapor cell. As the EM wave propagates through the cell, its refractive index for a given polarization is n ± where ± denotes + and,andñis as in Eq Writing Eq. 2.9 with respect to the left-handed and right-handed polarization components, and allowing ñ + 6=ñ, i!t i!ñc z E = E h 0 p exp ˆ exp 2 B = ie h 0 c p exp i!t i!ñc 2 z ˆ +exp i!t i!ñ + c i!t i!ñ + c z z ˆ+ ˆ+ i i. (2.12) The real part of ñ ± clearly determines the phase shifts in + and,whereastheimaginary part of ñ ± is an absorption term. In our experiment, we record both transmission and rotation signals, so in the derivation of our lineshapes we consider both real and imaginary components of ñ ±. In the following section I derive the form of the refractive index, which in turn determines the spectral shape we observe and its connection to the transition amplitude. Subsequently, I describe the Zeeman e ect, which determines how the refractive index for left-handed and right-handed circularly polarized light di er.

22 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE The Form of the Refractive Index In the following, I provide expressions for the refractive index relevant for magnetic dipole and electric dipole asopposedtoelectricquadrupole transitions. Thetheorybehindelectric quadrupole transitions is complicated by the second order tensor relating the quadrupole moments of the atomic vapor to the gradient of the magnetic field, the form of the result, however, is the same as that for the electric dipole after replacing the dipole ME with the quadrupole ME, as shown in [11]. The expression for ñ is derived by first relating the macroscopic density of the induced electron dipole moment to the magnitude of the applied electric field, hdi = o (1 ) E. (2.13) Or, for the case of an applied magnetic field, the relation is between the induced magnetic dipole moment and the applied magnetic field, hµ m i = 1 µ o (1 µ) B, (2.14) where the electric permittivity is and the magnetic permeability is µ. Itisnotdi cult,as shown in [12], to use first order, time dependent perturbation theory to find an expression for the induced dipole (electric or magnetic) amplitude in terms of the dipole matrix element, h2 d e 0i or h2 µ m 0i. The derivation in [12] includes a phenomenological damping parameter,, to account for decay of the excited state. The inclusion of this term leads to the lineshapes observed in spectroscopic experiments, where is the Lorentzian width. The result in [12] is for electric dipole transitions, but the same basic method applies for magnetic dipole transitions as well: hdi = 1 h2 d e 0i 2 ~ (!! o) (! 2! o + i ) E hµ m i = 1 h1 µ 0i 2 ~ (!! o) (! 2! o + i ) B (2.15) Turning back to Eq. s 2.13 and 2.14, the refractive index can be expressed in terms of the electric permittivity and magnetic permeability: ñ = p µ. For an electric dipole or quadrupole transition µ 1, so in Eq can be replaced with p ñ;andforamagnetic dipole transition 1soµ in Eq can be replaced with p ñ. Substitution of Eq into Eq. s 2.13 and 2.14, and replacing and µ as ñ 2,leadstoanexpressionfortherefractive index in terms of the transition matrix elements: ñ =1+!! 0 + i (!! 0 ) 2 + 2, (2.16) where, for the electric dipole and magnetic dipole terms respectively,

23 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 12 = h2 d e 0i ~ = h1 µ m 0i 2 µ (2.17) 0 2~. in Eq is the number density of atoms and h2 d e 0i or h1 µ m 0i is the ME driving the transition. The factor of 1/2 comes from a taylor expansion of the square root over the right side of the expression. The lineshape of our transmission spectra is determined by the imaginary part of Eq. 2.16, however the more useful Faraday spectra are more complicated. In the following, I derive the polarization dependance of the Faraday lineshape. 2.3 The Zeeman E ect and Di erential Absorption The line shape of our spectra fundamentally depends on the Zeeman e ect: the shifting of the previously degenerate m j sub-levels under the application of an external magnetic field. In this section, I show how Zeeman splitting leads to di erential absorption for + and, thus laying the groundwork for our understanding of Faraday rotation The Weak Field Approximation As shown in e.g. [8], the Zeeman splitting in the weak field approximation is given by,! j = µ Bg J B ext m j, (2.18) ~ where B ext is the strength of the external magnetic field, µ B is the Bohr magneton and g J is the Landé g-factor for a given fine-structure state. The Landé g-factors for the two states we study have been measured to be and for the 6s 2 6p 23 P 1 and 3 P 2 states, respectively [1]. The weak field approximation requires that the Zeeman splitting be small compared to all other transitions to or from that state. The 208 Pb isotope has nuclear spin I =0and therefore has no hyperfine structure, thus we need only consider fine structure splitting. The frequency of the 6s 2 6p 23 P 0! 3 P 1,2 transitions are approximately 10 7 times greater than the Zeeman splitting, so the small field approximation holds. Future experiments may involve spectroscopy of the 207 Pb isotope, which has nuclear spin I =1/2, and therefore does have hyperfine structure, in which case the weak field approximation would be less accurate.

24 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE Zeeman Energy Levels The Zeeman e ect splits the degenerate m j energy levels of the 3 P 1 state into three nondegenerate energy levels, and those of the 3 P 2 state into five, as shown in figure 2.1. Although there are five non-degenerate energy levels in the 3 P 2 state, the m j = ±2 states cannot be excited from the ground state, as transitions with m j 6=0, ±1 areforbidden. This condition is one of the selection rules for photon absorption, but is equivalent to the constraint that angular momentum be conserved a photon with angular moment ~ cannot impart 2~ on the system. Polarization Dependent Excitation Angular momentum conservation applied to photon absorption not only excludes two of the m j states in the 3 P 2 level, but it directly forms the basis for Faraday rotation spectroscopy. Right-circularly polarized light, +, iscomposedexclusivelyofphotons with angular momentum ~, whereas leftcircularly polarized light,, is composed exclusively of photons with angular momentum ~. Therefore + light excites transitions to m j =1statesand polarized light excites transitions to m j = 1states. The Figure 2.1: Energy levels of the 6s 2 6p 2 triplet state of 208 Pb. Zeeman splitting energies are given for a roughly 7 gauss magnetic field. Note that the Zeeman splitting would be imperceptibly small if shown to scale with the fine structure splitting, transition to the m 0 state can be driven by light polarized in the ẑ direction,. Tomaximize the rotation signal, however, we align the magnetic field parallel to the laser propagation direction, thereby e ectively eliminating light. The resonant frequency of + light and light di ers by the Zeeman splitting, 2 j.thispolarization-dependentresonanceleads, as I show below, to an overall rotation of the linearly polarized light passing through the lead vapor in our experiment this frequency-dependent rotation is our signal. 2.4 Faraday Rotation Now, we bring together the concepts outlined in the last two sections: the polarizationdependent absorption of the vapor cell due to Zeeman splitting, the frequency dependance of the refractive index, and the + and components of the EM wave. Taken together,

25 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 14 they form the basis for Faraday rotation. We now know how ñ + and ñ di er in E.q. 2.12: the 0 term is shifted by the total Zeeman splitting between the two reachable m j levels, 2! j.therealandimaginaryparts of ñ become: Rñ =1+ (!! 0 ) (!! 0 ) 2 +( 2 ) 2! Rñ ± =1+ (!! 0 ±! j ) (!! 0 ±! j ) 2 +( 2 ) 2 (2.19) Iñ = 2 (!! 0 ) 2 +( 2 ) 2! Iñ ± = 2 (!! 0 ±! j ) 2 +( 2 ) 2. Figure 2.2: The Lorentizan and dispersion functions. Note that although the amplitude of the functions are arbitrarily scaled, their relative maxima are fixed. The shape of Rñ ± is the dispersion curve, D(!), and the imaginary part is simply a Lorentzian, L(!); both are shown in figure 2.2. After passing through a cell of length `, + polarized light experiences a phase shift!`rñ +,whereas light undergoes a phase shift c!`rñ. We go about understanding the consequences of this relative phase shift by working c backwards from what we expect: an overall rotation in the linear polarization direction, which is equivalent to applying the general rotation matrix to ˆx: cos sin 1 =(e i + e i )ˆx i(e i e i )ŷ (2.20) sin cos 0 Replacing ˆx and ŷ with the linear combination of + and given in Eq the expression simplifies to:

26 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 15 1 p 2 (e i ˆ e i ˆ+) (2.21) Eq maps directly onto the phase terms of Eq. 2.12, which reveals that the overall rotation angle of linearly polarized light passing through the vapor cell is given by f =! 0` c (Rñ +(!) Rñ (!)), (2.22) where I approximated! as! 0, as the ratio between the full-width at half maximum (FWHM) of the lineshape and! 0 is Finally, we plug in Rñ ± and arrive at the equation for the Faraday rotation lineshape, plotted in figure 2.3: F (!) =! 0` c [D(!! 0! j ) D(!! 0 +! j )] (2.23) In this and previous Faraday rotation spectroscopy experiments, e.g. [13] and [5], the D(!) form of equation 2.23 is approximated by 2! j,whichcanbefittodatamoree ciently, d! as! j becomes simply a coe cient in front of the lineshape. The approximation necessitates that the Zeeman splitting be much less than the total width, which allows us to approximate! j as an infinitesimal shift in frequency. The Lorentzian FWHM (in linear frequency units) in our experiment is of the order 100 MHz, and the doppler width is of 450 MHz and, as shown in figure 2.1, 2 j (also in linear frequency units) is characteristically 15 MHz. Therefore 2 j 0.03 total and the approximation holds. 2.5 The Idealized Transmission Lineshape In direct absorption measurements, we record the overall intensity of light after it passes through the vapor cell; we therefore consider the imaginary part of the index of refraction given in Eq Iñ still di ers, however, for + and light, so we keep those terms separate for now. The overall intensity of light after it has passed through the vapor cell is: h I / E E = E2 0 exp 2!Iñ ` c ˆ exp!iñ +` c ˆ+i 2 = (2.24) E0 2 h 2!Iñ ` 2!Iñ i +` exp +exp 2 c c It is not immediately clear whether the di erence in the refractive indices is relevant for the transmission profile. If we assume that the two exponentials give approximately the same lineshape, only with one shifted to the left by j and one shifted to the right by j, then the intensity takes the familiar form for spectroscopic transmission, I = E0 2 exp 2!I(ñ)`. (2.25) c

27 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 16 Figure 2.3: The Faraday rotation lineshape is shown along with its convolution with a Gaussian function (described in section 2.6.3). Note that the amplitude of the convolution lineshape has been scaled up by a factor of 10. As with the Lorentzian and dispersion shapes, the absolute scaling is arbitrary, but the relative maxima are dependent; in this case, their ratio is a function of the Lorentzian and Gaussian widths Such an assumption only leads to a.06% di erence in the peak of the observed lineshape, thus is negligible given our final uncertainty goal of 1%. The lineshape given by Eq is governed by the form of Iñ which in our experiment is a Lorentzian, plotted in figure 2.2. From here on out, we denote the idealized transmission lineshape by, T (!) =Eo 2 exp 2!ñ` c L(!) 2.6 Line Broadening and the Convolution Profile (2.26) The lineshapes derived above do not consider any of the experimental realities of the interaction region. The gaseous lead atoms in our vapor cell have a distribution of velocities and consequently will according to the Doppler e ect experience a range of laser frequencies. Doppler broadening is an in-homogeneous mechanism, as the Doppler width is a function of the distribution of atomic resonant frequencies among all the distinct atoms in the cell. Homogeneous broadening mechanisms, on the other hand, a ect each atom identically; the distribution of frequencies which could excite a given atom is the same for any other atom. The measure of the frequencies over which an atom may be excited, due to homogeneous broadening, is the Lorentzian width. Each component of the homogeneous

28 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 17 broadening simply adds to in equation The Doppler width, however, does not simply add to the Lorentzian width it requires that we convolve the faraday rotation lineshape with a Gaussian distribution Doppler Broadening The innocent-looking quartz cell placed at the heart of our experiment is one of the hardiest pieces of equipment in the laboratory. It must not only withstand temperatures up to 1000 C, it must also keep contained approximately atoms, all moving at 300 m/s in every direction. Of course, the quartz has no trouble dealing with these atoms, but the high velocity and random motion of the atoms introduces a major challenge to our experiment. Doppler broadening, the result of the velocity distribution of the atoms, is the outstanding component of the spectral linewidth it increases the uncertainty of our measurement, and adds complexity to the functions describing our spectra. Conveniently, it is straightforward to calculate the Doppler width for it is caused by the motion of the atoms viewed as a macroscopic system, thus can be treated classically. The distribution of atomic velocities in a gas is given by the Maxwell-Boltzmann distribution: m(v)dv = r M 2 k B T exp Mv 2 Due to the Doppler e ect, which to first order is simply 2k B T dv (2.27) v = ( 0)c, (2.28) 0 2 this distribution of velocities translates to a distribution of resonant frequencies. Substituting q Eq into Eq making sure to replace dv with d c k 0 anddefining g = B T 0 Mc 2 the distribution of frequencies reduces to a Gaussian, centered at 0 with standard deviation g: G( ) = r 1 2 ( exp 0 ) g (2.29) The FWHM of our spectra will thereby be largely defined by 2 p 2ln2 g,which ata temperature of 950 C for the E2 transition is 544 MHz, and for the M1 transition is 400 MHz. As we will see, this is the dominant broadening mechanism Homogeneous Broadening Mechanisms Homogeneous broadening accounts for a small portion of the overall width of our spectra see figure 2.3 but its contribution must be known precisely if we are to extract an accurate value for the matrix elements. We must know the Lorentzian width accurately to calculate peak of the line-shape, a parameter in the conversion from the measured value to the matrix element. The components of the homogeneous width, along with their approximate values, are given in table 2.1.

29 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 18 Broadening Mechanism FWHM Natural width 200 Hz Pressure broadening 75 MHz Transit-time broadening 70kHz Saturation broadening 0 Laser line width 2 MHz Table 2.1: Approximations for the components of the homogeneous width The Natural Width The distribution of frequencies that can drive an atom to an excited state is not a simple delta function even ignoring the experimental broadening mechanisms. The so-called natural width can be understood and calculated by starting at the beginning with the time-energy form of the Heisenberg uncertainty principle. E t = ~ (2.30) Eq is an approximation given in [14] of the true uncertainty principle, which is of course an inequality. t in this context is the time it takes before a state changes considerably for an atom t is given by the lifetime,. Thustheuncertaintyofthetransition energy, and thereby the transition frequency, to the first excited state is simply: 0 = 1 2. (2.31) We can related the lifetime of the excited state to the rate of spontaneous emission, i.e. the Einstein A coe cient, which in turn is proportional to the square of the relevant matrix element. But M1, as shown in section 2.8, is 400 times smaller than electric dipole MEs. As calculated in many theses from the Majumder lab, e.g. [15], electric dipole natural widths are 25 MHz. Therefore the natural widths of our spectra will be 160, 000 times smaller, so Hz, which is completely negligible. Pressure Broadening When two atoms collide, their wavefunctions interfere and their energy levels are perturbed. This shift depends on the distance between the two atoms and their respective electron configurations. As shown in [10] and [16], pressure broadening results in another Lorentzian distribution with a FWHM of r 2 p =2N v b =4p v b (2.32) 2 µk B T where p v is the vapor pressure inside of the cell µ is the reduced mass of the two interacting atoms, and b is the collisional cross section. At 950 C(theaverageoperatingtemperature

30 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 19 of the furnace) the vapor pressure of the lead cell is approximately 226 Pa (of order 1 Torr). [17] The collisional cross section is di cult to calculate, but is very roughly 500 Å 2 [18]. Using these values in Eq. 2.32, we arrive at the result in table 2.1 by far the dominating homogeneous broadening mechanism. Transit-time and Power Broadening Transit-time broadening is function of the duration an atom interacts with the beam of the laser. Following the explanation provided in [15], the finite time an atom interacts with the incident EM wave can be modeled by a rectangular pulse, multiplied by the light s oscillations. The rectangular pulse transforms the delta function representing the oscillation in frequency space to a distribution with a finite width. The derivation in [15] and [14] results in the simple expression t = d v rms,wheredisthediameterofthelaser. Thediameter of our beam is approximately 2 mm, and v rms is 357 m/s, leading to the value given in table 2.1. Power broadening occurs when the excited state is saturated. Because both the M1 and E2 transitions are so weak, the excited state will never be highly populated. Therefore we do not need to consider power broadening as a potential source of the total width Convolution To incorporate both Lorentzian and Doppler broadening in our lineshape, we perform a so called convolution of the two functions. It is possible to imagine a convolution physically as follows: imagine that the light propagating through the vapor cell is oscillating at the exact resonance frequency of the transition. If there were no Lorentzian broadening, only the atoms that stood completely still would be excited. Of course, there is Lorentzian broadening, so we imagine placing our homogeneous distribution given by L( ) orf( ), if we are recording a rotation spectrum centered at the resonance frequency; thus some atoms to the left and to the right of the center frequency have a chance of being excited too. The number of atoms to the left and to the right of this center frequency drop o according to G( ), however, and the chance of those atoms being excited drop o according L( ). Now, the center frequency does not have to correspond to the zero-velocity class of atoms, we could choose any velocity class, then place L( ) attheappropriatepointinfrequency space. The number of atoms excited when the laser is operating a given frequency 0 is therefore proportional to the product of L( ) andg( ), where L( ) isshiftedfromthe center of G( ) by 0.Wecalculatethisquantitybyintegratingovertheproductofthe two functions. The function of produced by this integral is called a convolution: C( 0 )= Z 1 1 G( )H( 0 )d (2.33) G( )inequation2.6.3isthegaussiandistributiongivenbyequation2.29andh( 0 )could be the Faraday function given by 2.23 or the Lorentzian lineshape, L(!). The convolution,

31 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 20 as one would expect, gives the relevant lineshape a Gaussian character, and as the Doppler width is by far the dominant source of line-broadening, increases the width of the function by several factors see figure 2.3. The convolution function is not analytically solvable, but due to its ubiquity in spectroscopy experiments, the convolution of the Lorentzian and the Gaussian shapes called a Voigt profile has been well studied and an accurate and computationally cheap analytic approximation to the Voigt has been developed[19]. We have implemented this approximation to fit our transmission spectra. The convolution of the Faraday shape and the Gaussian, however, is not well studied, thus we manually convolve the functions using Eq , in addition to employing the discrete convolution function, implemented in MatLab. 2.7 Connecting Measurement and Matrix Element The spectra we acquire do not reveal the MEs right away. The peak of the Faraday shape and the trough of the absorption dip are proportional to not only h2 q e 0i 2 or h1 µ m 0i 2 but also the length of the interaction region `, the density of atoms, the frequency of the laser, 0,andthepeakoftheconvolvedlineshapestrippedofitsprefactors Extracting the MEs from the Faraday Rotation Spectra Eq describes the lineshape of the Faraday rotation spectra before it is convolved with a Gaussian function. Recall that this lineshape describes the angle through which the beam s polarization rotates when passed through magnetized atomic lead vapor. If we plug in Eq for the constant in Eq. 2.23, and express h2 q e 0i 2 / o or h1 µ m 0i 2 µ o as a generic matrix element hj 0 T k ji 2,wherekindicatesthemultipoleorder,theun-convolvedFaraday shape is 2 : F (!) =! 0 hj 0 T k ji 2 ` h i D(!! 0! j ) D(!! 0 +! j ), (2.34) 2~c where! j is the frequency shift due to the Zeeman e ect in a given m j state. Now if there were no Doppler broadening, and we did not have to convolve F (!) withthegaussian function, G(!), extracting the matrix elements would only require a little rearranging. It is easiest to measure the maximum height of the Faraday rotation spectrum, so we evaluate Eq at resonance, and from there solve for hj 0 T k ji 2.Thematrixelementistherefore: hj 0 T k ji 2 = 2~c F (! 0 ) expmt, (2.35)! 0 ` F peak 2 Although I have only explicitly derived the amplitude of the spectra for the E1 case, the E2 case is equivalent with the substitution of the quadrupole ME. In its full form, the quadropole ME has a factor of p4 3, but it is standard practice to absorb it into the definition of the ME.

32 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 21 where F peak = Recall that the dispersion shape is given by, apple D(!! 0! j ) D(!! 0 +! j )!=! 0. (2.36) D(!! 0! j )=!! 0! j (!! 0 ±! j ) Evaluated at resonance, Eq reduces to a simple constant dependent on the Lorentzian FWHM and the Zeeman shift; expressed in angular frequency units: F peak = 2! j! j (2.37) 2 F (! 0 ) expmt and F peak are easily confused, but they represent distinct quantities. F (! 0 )isthe maximum height of the recorded spectrum; we measure this quantity directly by determining the peak of the spectra. F peak,incontrast,istheheightofthelineshapebyitself. Itisa parameter that we can either extract from the coe cients of the fitting function, or from independent measurements of the Lorentzian width and the Zeeman splitting. The experimental parameters and l in Eq. 2.35, which we do not know with any great precision, require that we measure the ratio of the transition strengths instead of each transition independently. Eq applies for both transitions, so long as we calculate F peak,! 0 and F (! 0 ) exmpt for each transition. Solving for the MEs and taking their ratio gives (labelling the quantities referring to the M1 transition with an m subscript and those referring to the E2 transition with an e subscript), h2 q e 0i 2 / o h1 µ m 0i 2 µ o = F (! 0) expmt,e F (! 0 ) expmt,m F peak,m F peak,e! 0,m! 0,m. (2.38) This result takes the same form under the non-ideal case where we must convolve the Faraday shape with a Gaussian function. Referring back to figure 2.3, it is clear that the convolution brings the height of the spectrum down. The Gaussian function is normalized such that it always integrates to unity, instead of the maximum peak height always equalling one; the total amount of absorption by the lead atoms remains constant, whether these lead atoms are distributed over a wide range of velocities or not. The decrease in peak height does not, however, correspond to a decrease in the transition strength this quantity should never change. Going back to Eq. 2.34, we now convolve F (!) withagaussianfunction: C (!)= Z 1 1 G( )F (!! 0 )d =! 0 hj 0 T k ji 2 ` Z 1 h i 2~c G( ) D(!! 0! j ) D(!! 0 +! j ) d 1 (2.39)

33 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE 22 We now solve for the matrix element in the same fashion as we did above, only now the peak of the lineshape, which we will call C peak and is given by the integral in the second expression, cannot be solved for analytically. C peak is not only a function of the Lorentzian width and Zeeman splitting, it also depends on the Gaussian width. Although C(!) is more involved than its idealized equivalent, F (!), we can solve for the ME in the same exact fashion. Thus substituting the F 0 s for C 0 s in equation 2.38, the ME extraction formula is: h2 q e 0i 2 / o h1 µ m 0i 2 µ o = C(! 0) expmt,e C(! 0 ) expmt,m C peak,m C peak,e! 0,m! 0,e. (2.40) In the approximation form of the Faraday function, the Zeeman splitting term is a scaling factor in front of the lineshape; therefore by taking E2/M1, the ratio of Zeeman splittings becomes the ratio of Landé g-factors. By factoring out the g-factors, the term C peak,m C peak,e depends on only the Lorentzian and Gaussian widths Extracting the MEs from the Transmission Spectra As the E2 ME is small, it is impossible to extract the ME element from direct absorption spectroscopy on the 6s 2 6p 2 3 P 0! 3 P 2 transition. We can, however, extract the M1 ME from the direct absorption spectra of the 3 P 0! 3 P 1 transition. As this value is taken simultaneously with the rotation spectra, we can use the value as a check for systematic errors. Using Eq and substituting the constant with Eq we note that the optical depth (the natural log of the peak absorption) of the non-convolved transmission spectra is, =! 0 hj 0 T k ji 2 ` ~c L(! 0). (2.41) We can extract the ME ratio by rearranging terms in the same fashion as Eq. 2.38, but first we note that in our experiment L(!) isreplacedbytheconvolutionofthelorentzian with a Gaussian function the Voigt function, which we write as V (!). We denote the nonidealized optical depth as v,andthepeakofthevoigtfunctionbyitself(thecounterpart to C peak )asv peak.byanalogywiththefaradayrotationmeextractionprocess, hi T k ji 2 = ~c v. (2.42)! 0 ` V peak It is important to keep in mind that V peak in the above expression is the peak of the convolution function appearing inside the exponential. We take the ratio of the E2 to M1 MEs again, only this time the M1 ME is extracted using the transmission spectra, h2 q e 0i 2 / o h1 µ m 0i 2 =2 C (! 0 ) e V peak,m! 0,m. (2.43) µ o v,m C peak,e! 0,e The value for M1 has been calculated with great precision, thus Eqs and 2.43 allow us to calculate the value for the E2 ME.

34 CHAPTER 2. TRANSITION AMPLITUDES AND THE SPECTRAL LINESHAPE The Reduced Matrix Element The matrix elements expressed above are dependent on the polarization of the incident light, and are therefore not equivalent to a more general ME a theorist will calculate. The generic ME hi T k ji is defined for a given circular polarization direction, therefore is more precisely written as hi T k ± ji, where± denote right and left handed circularly polarized light. When comparing our measurement to theoretical calculations, it is necessary to express the ME without such experimental dependencies. Therefore we define the reduced matrix element as the ME above, independent of the polarization direction. The conversion from the ME element we measure to the reduced ME is given by the Wigner-Eckart theorem, whichionlystatetheresultofhere: hj 0 m 0 T k q jmi = hj0 m 0 kq; jmihj 0 T k ji p 2j0 +1 (2.44) where q denotes the polarization. For the transitions we study, Eq becomes, for the M1 transition, hj 0 =1m 0 = ±1 T 1 ± j =0m =0i = hj 0 =1m 0 ± 1 k =1q = ±1; j =0m =0ihj 0 =1 T k j =0i p 3 = (2.45) hj 0 =1 T k j =0i p 3. In the last step we have noted that the magnetic level of the excited transition always has the same quantum number as the polarization, and the excited fine structure state has the same quantum number as the order of the ME, thereby equating the bra-ket with unity. The same steps apply to the E2 transition, only now k = j = 2, leading to the relation: hj 0 =2m 0 = ±1 T 1 ± j =0m =0i = hj0 =2 T k j =0i p 5. (2.46) To connect the matrix elements we measure to theory, therefore, we use the relationship: r h2 q e 0i 5 h1 µ m 0i = h2 q e 0i 3 h1 µ m 0i. (2.47) In the rest of this thesis, I express the transition matrix elements (not in the reduced form) as he2i and hm1i and in reduced form as he2 red i and hm1 red i

35 Chapter 3 HFS and TIS Measurements in the 6p 23 P 1 State As a preliminary experiment, we have completed measurements of the hyperfine structure (HFS) of the 6s 2 6p 2 3 P 1 state in Pb, in addition to the 6s 2 6p 2 3 P 0! 3 P 1 transition isotope shift (TIS) 1 between two isotopes in 208 Pb and 207 Pb. We use direct absorption spectroscopy on two vapor cells simultaneously, each containing a sample of a single Pb isotope. We have found that the HFS splitting is (9) MHz, and that the TIS shift is 138.7(15) MHz. These measurements serve as checks on previous measurements made in [20] and [21] as they have equivalent uncertainties; but their main purpose is to provide a foundation of experimental knowledge on which the Faraday rotation experiment may rely. We also hoped to observe the E2 transition and measure the HFS in the 6p 2 3 P 2 state but found that the transition was too weak to be observed with direct absorption or even frequency modulation spectroscopy. Before commencing our measurement of the MEs in the 6p 2 3 Pmanifold,weperform the relatively standard measurement of the HFS and TIS, using techniques familiar in the Majumder lab, to establish that the many novel aspects of the central experiment will function properly and if they do not, determine how best to fix them. To finalize our preliminary result, we must collect more data and explore the systematic errors dominating the uncertainty of our measurement. 3.1 Atomic Structure of Lead In the simplest model of atomic structure, orbitals can be described by the principal quantum number alone. With closer examination, the states within each principle quantum number break their degeneracy, first into fine structure (FS) then under closer scrutiny hyperfine 1 The transition isotope shift (TIS) is the di erence in transition energies for two isotopes. The isotope shift (IS), on the other hand, compares the energies for a given state relative to the same ground state energy in two di erent isotopes. 24

36 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 25 structure (HFS). We study the latter in this experiment, but before describing HFS it is important to understand FS, whose splittings are one step larger Fine Structure FS is a consequence of the interaction between the electron s orbital angular momentum and its spin. The orbital motion of the electron about the nucleus sets up a magnetic field, which then exerts a torque on the electron s spin. The FS hamiltonian is therefore, Ĥ fs = µ e B = fs L S. (3.1) The last step in Eq. 3.1 follows from the proportionality of the magnetic field, B, with the electron s angular momentum, L; and the electron s magnetic moment, µ e,withthe electron s spin, S. FS breaks the degeneracy of the azimuthal quantum number into states that depend on the total angular momentum of the electron, J = L + S. Using the fact that L S = 1 2 (J 2 L 2 S 2 ), the energy eigenvalues of Ĥfs are (see e.g. [8]): E fs = fs (j(j +1) `(` +1) s(s +1)), (3.2) 2 where fs is a constant governing the overall size of the splitting. For a single electron, ` can take the integral values 0! n 1(nistheprinciplequantumnumber),sis± 1 and 2 jisthesumof` and s. Lead has, for the purpose of describing most energy levels, two valence electrons, however, thus s can be 0 or ±1, leading to the FS shown in figure 1.1. The energy splittings given by the fine structure relative to the principle energy level splittings are hh fs i / hh o i (Z ) 2,whereZistheatomicnumberandalphaisthefinestructure constant. As Z is 82 for lead, hh fs i is 1/3 thesizeofhh o i Hyperfine Structure In our experiment, we consider the HFS of lead, which provides a significantly smaller (two orders of magnitude) correction to the energy levels. HFS is determined by the interaction of the nuclear spin with the magnetic field produced by the angular momentum of the electrons; it is therefore called IJ coupling I being the spin of the nucleus. The perturbing hamiltonian depends on the induced magnetic field (a function of the electrons dipole moments and positions) and its interaction with the nuclear dipole moment: Ĥ hfs = µ I B el = hfs I J, (3.3) where I have used the fact that B el is proportional to J; andµ I is proportional to I. Following the same process used to determine the FS, and defining F = J + I, wedeterminetheenergy eigenvalues of Ĥhfs, E fs = hfs (F (F +1) J(J +1) I(I +1)), (3.4) 2

37 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 26 where hfs is the hyperfine constant, determined by experiments like this one. The nuclear spin, I in Eq. 3.4, can take on di erent values depending on the element and isotope in question. For the even isotopes of lead (for example 208 Pb) I =0,thuseliminatingthe possibility for HFS. The odd isotopes of lead have finite nuclear spins, for example 207 Pb has I =1/2. For a given fine structure level, F can take on the vector sum of I and J, leading to the HFS levels shown in figure 3.1. Figure 3.1: Energy level diagram of the 6p 2 3 Ptripletstateinlead. 207 Pb di ers from 208 Pb in that it has HFS and that the weighted average of the HFS states are shifted slightly compared to the FS states of 208 Pb. Although the ground state for 207 Pb is at depicted as having the same energy as the ground state of 208 Pb, in reality they di er by some small, unknown amount. Thus the 138 MHz TIS is the shift relative to the isotopes respective ground states The Isotope Shift In the standard treatment of an atom, the nucleus is considered an infinitely heavy (relative to the electron) point charge. Such a treatment is of course an approximation, and the mass and charge distribution of the nucleus have produce measurable energy level perturbations.

38 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 27 One way to measure these e ects (called the mass and field e ects) isbyconsideringtwo di erent isotopes of the same element; the slight shift in the energy levels in one isotope relative to the other casts light on the magnitude of the nuclear perturbations. The Mass E ect The magnitude of the mass e ect is determined by the di erence between the mass of the electron and the reduced mass of the electron with the nucleus. For two di erent isotopes, the mass e ect scales by 1,whereMisthemassofthenucleus. ThusforPb,aheavy M 2 element, it contributes to only a small fraction of the total isotope shift. Specifically, as given by [7], the first order frequency di erence between two isotopes due to the mass e ect is proportional to the mass of the electron, m e,thedi erenceinthetwoisotopicmasses, and the frequency of the given transition. For 207 Pb and 208 Pb it is, is m e M 207,208 M 208 M 207 3MHz. (3.5) Given that the total magnitude of the TIS is 138 MHz, the mass e ect s contribution is relatively small (although not completely negligible). The Field E ects The field e ect is governed by the shape and size of the nucleus. This e ect is both richer in atomic theory and far more di cult to calculate. The first order field e ect is the volume e ect, whichdependssolelyonthesizeofthenucleus. Calculationsconsideringhigherorder contributions, must include the shape and variable charge density of the nucleus. Even estimations of this e ect in as complicated an element as lead are beyond this thesis. We can, however, measure the magnitude of the isotope shift and compare to theoretical models that account for isotopic changes in the charge distribution. In the following section, I outline the experimental methods used to do so and to measure the HFS splitting. 3.2 Experiment Setup and Data Acquisition We employ direct absorption spectroscopy with a 1279 nm external cavity diode laser in the Littrow configuration on the two principal isotopes of lead. In order to record the two absorption signals corresponding to the two isotopes on the same frequency axis, we split the laser beam and pass it through two heated vapor cells simultaneously. In this section, I detail the experimental apparatus, highlighting where it di ers from the apparatus described in appendix A Apparatus Details The experiment layout is shown in figure 3.2. It accommodates both the 939nm and 1279 nm lasers to enable measurement of the HFS in both the 3 P 0! 3 P 1 and 3 P 0! 3 P 2 transitions.

39 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 28 As mentioned above, we were not able to resolve the 939 nm transition, thus experimental components related to the 939 nm laser are omitted in the following. The 1279 nm laser is scanned through the appropriate range of frequencies by a piezoelectric actuator (PZT), described in section 4.2. A small fraction of the laser beam is directed into a Fabry-Pérot (FP) cavity to facilitate frequency calibration, while the majority of the beam is directed first through an electro-optic modulator (EOM), detailed in Appendix A, then is split and passed through the two ovens holding the two Pb vapor cells. We use two commercial ovens of di erent make: a Lindberg/Blue, 120 Volt, and an AVS 3210, 115 Volt. The Lindberg is temperature controlled with an internal regulation system; the AVS is connected to a PID, (proportional-integral-derivative) controller to keep the temperature constant. Finally, the transmitted intensities of both beams are recorded by New Focus 2053 photodetectors on the far side of the ovens. The details of the 1279 and 939 nm lasers are given in section 4.2. We note that a major motivation for this preliminary experiment was to test the home-built 939 nm laser. We are able to scan the laser 8GHz, which is enough to not only observe the 208 Pb peak but also the two HFS peaks separated by 6GHz in 207 Pb. Figure 3.2: Layout for the Pb HFS and TIS experiment.

40 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE Frequency Calibration As in many Majumder lab experiments, establishing accurate frequency axes for the collected spectra is the heart of our measurement. In its raw form, the absorption signal is a function of time; we must translate time to frequency to extract any physically meaningful information from the spectra. To do so we employ two optical components: a Fabry-Pérot cavity and an EOM. The application of these components is described in detail in Appendix A, in addition to [22] and [23]. Briefly, the intensity of light passing through Fabry-Pérot peaks every integer multiple of the FP s free spectral range; aswescanthelaseroverarangeof frequencies, the FP e ectively provides tick marks on the spectra see figure 3.3. The EOM, whose mechanics are detailed in Appendix A, provides sideband peaks at ±1000 MHz from the center frequency. The peaks of the absorption signal are therefore mirrored at precisely ± 1000 MHz. By demanding that these peaks are actually separated from the central peak by the correct frequency, we check the calibration provided by the FP. Note that in figure 3.3 not only are the first order sidebands apparent, but the 2nd order SBs are also just visible above the noise. These SBs overlap slightly with the 1st order SBs drawing the apparent peaks of the 1st order SBs outward. This e ect would not be an issue if it were possible to fit the 2nd order sidebands, as the resulting function would be parameterized with the Figure 3.3: Example spectrum for 208 Pb, with EOM sidebands and corresponding FP spectrum. correct peak locations. It is di cult to fit the second order SBs given their size relative to surrounding noise; thus the SB peaks can no longer be considered to provide an absolute frequency calibration. In the Faraday rotation experiment, we avoid this issue by applying SBs to the FP spectrum itself, as shown in the following chapter. FP Calibration The FP is an excellent mechanism for x-axis calibration if one knows the FSR precisely. The FSR of a confocal cavity is, FSR = c (3.6) 4L

41 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 30 where L is the length of the cavity and c is the speed of light in air. From this relation, one could hypothetically calculate the FSR by measuring the length of the cavity. Such a measurement will be imprecise, however, so we employ a more rigorous technique developed in the Majumder lab and explained in detail in e.g. [22]. In essence, to calibrate the FP more accurately (and independently of the EOM), we tune the laser to a wide range of frequencies, and map the intensity peaks to the corresponding laser frequency (using a WA 1500 wavemeter). The di erences between these frequencies should all be integral multiples of the the FSR, so we find the frequency value which, multiplied by some integer, best fits into every di erence. This value is the free spectral range of our Fabry-Pérot Data Acquisition and Analysis We collected a total of 2000 spectra for both 208 Pb and 207 Pb from January 3rd to January 9th. We collected preliminary scans, which we discarded due to fitting and FP irregularities, starting in November. During this data collection period, we varied experimental conditions such as scan speed, laser power, oven temperature and scan direction to establish that the HFS and TIS values remain constant. We recorded the spectra using a LabView program and analyzed the data using MatLab. Experiment Parameters The HFS and TIS for a given state do not change when the laser probing the atom changes power, when the frequency of the laser is increasing as opposed to decreasing, or when the oven temperature changes. We make sure that our experiment does not rely on these conditions by varying them, and making sure the final answer does not vary too. Table 3.1 outlines the parameters we change over the course of the experiment. Exp. Parameter Range Temperature C Scan period seconds Laser power mw EOM waveplate Table 3.1: Experimental conditions for data collection EOM Frequency Calibration The EOM provides an excellent check to the FP calibration, especially in doppler free spectroscopy, e.g. in the experiment detailed in Appendix A. For doppler broadened peaks, however, whose total width is approximately 500 MHz, the EOM is inherently less accurate. Compare, for example, the simple 208 Pb lineshape in figure 3.4 to the lineshape with the

42 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 31 EOM sidebands in figure 3.3. Furthermore, the second order sidebands, which are so small as to elude fitting, are nearly convolved with the 1st order sidebands, contributing to the overall uncertainty of the sideband locations. Due to this added complexity, we recorded spectra both with and without sidebands, and used the scans with sidebands only to provide an upper bound for the frequency calibration error. All of the final center values from this experiment are derived from spectra without sidebands Fitting the Spectra Doppler broadened absorption spectra are, as explained in section 2.6, described by a socalled Voigt function the convolution of the Lorentzian and Gaussian functions. We fit our spectra (which, when digitized, are represented by discrete data points) with the Voigt function to calculate the frequencies corresponding to the absorption peaks. Arepresentativescan,withthe 208 Pb and 207 Pb spectra overlaid, is shown in figure 3.4. Not only must we fit the absorption peaks of the spectra, we must also fit the frequency axes. As the PZT scans the laser through a range of frequencies, the scan speed changes slightly. The point number separation between FP peaks therefore varies with time, although the separation in frequency space stays constant. To account for scanning hysteresis, we fit the FP peaks to asixthorderpolynomial,anduse this new polynomial to linearize the x-axis. We find that higher order polynomials do not change the value of the HFS or TIS shift significantly. Amoredetailedexplanationisgiven in Appendix A and [22]. Figure 3.4: Example spectrum for 208 Pb and 207 Pb. Residual Isotopic Impurities When analyzing these spectra, we evaluate the potential e ects of other lead isotopes. The 208 Pb lead cell, purified by Oak Ridge National Labs, is quoted to be % pure and we see no evidence of other isotopes in our spectra. The 207 Pb lead cell, on the other hand ( 207 Pb has a natural abundance of 22%, compared to 208 Pb which has a natural abundance of 52%) is only 92% pure, thus the presence of other isotopes must be considered when analyzing the 207 Pb spectra figure 3.4 clearly shows the presence of 208 Pb in the 207 Pb cell. We are able to resolve the 208 Pb peak and include it in our

43 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 32 fitting function, so it does not directly add uncertainty in our measurement. If one examines figure 3.4 closely, however, it is apparent that the 208 Pb peak in the 207 Pb spectrum does not align with the 208 Pb peak in the 208 Pb spectrum. Furthermore, the fit does not exactly capture the residual 208 Pb peak curve. This e ect is due to 206 Pb which has a natural abundance of 24%, so we expect it has half the concentration in the 207 Pb cell. Its isotope shift relative to 208 Pb has been measured in [20] as 220.4(15) MHz. If we were to provide a final measurement, we would use this value to explore fits including this contribution, even though it is never resolved. 3.3 Results and Evaluation of Errors The fitting process and x-axis calibration provides a HFS splitting and TIS value for every spectrum; we then bin these data into a histogram to find the mean value from each set of data, encapsulating 200 spectra. By exploring the variation within each set, then compared to other sets, we can determine the final uncertainty in our measurement. The method by which we evaluate the systematic error of our final value has been developed in the Majumder lab over several years. We follow this process, in which we isolate and methodically vary each component of the experiment that could possibly bias the final value. If we find a correlation between the value of the experimental component and the measured quantity, we assign an appropriate systematic error to the final result Statistical Error Sample histograms of the HFS splitting and TIS values are shown in figure 3.5. We can easily determine the standard deviation of each data set, as the values are distributed approximately normally. Adding these standard deviations in quadrature, we derive the first measure of the statistical error over all the data. This process does not account for the variation between sets, however. If each mean value from each data set had a very small standard deviation, but each data set produced a dramatically di erent value, we would expect the total error to be very high, which the quadrature sum of the standard errors would not show. To account for such inter-set variation, we scale the standard error by the square root of the reduced chi-squared of the mean values from each data set: 2 red. = 2 N points N d.o.f., where N d.o.f. is the number of degrees of freedom in our case N d.o.f. =1. Thefinalstatistical errors for the HFS splitting and the TIS are listed in table 3.2. The statistical error, while important, is small compared to the systematic errors of this experiment. The evaluation of such systematics is given in the following.

44 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 33 Figure 3.5: Histogram of values from a single data run for both the HFS splitting and the TIS. The standard deviation indicates the rough precision of each binned value, whereas the standard error indicates the precision of the overall mean Systematic Error We evaluate the systematic errors by comparing data recorded under various experimental parameters. The results of the error analysis for this experiment are given in table 3.2. Besides the sources listed, we have also considered the potential e ects of laser power and oven temperature, but find no significant systematic from these sources. To present a final result, more analysis must be done to understand and mitigate the dominant errors listed below (Calibration for the HFS and scan direction for the TIS). Source Error for HFS splitting in MHz Error for the TIS in MHz Statistical Scan direction Scan period Calibration 0.85 Table 3.2: Systematic errors for the HFS splitting and the TIS. Calibration Error The dominating source of error for the HFS splitting measurement is the calibration error. We derive this error through a concert of experimental techniques and ad hoc approximations.

45 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 34 (a) (b) Figure 3.6: Scan direction systematic errors We performed the independent calculation of the FSR twice, several months apart. The first calculation resulted in an FSR of MHz, the second resulted in an FSR of MHz. We cannot derive a mathematically precise uncertainty from two measurements, so to be conservative we take the di erence between the two measurements and double it. The resulting uncertainty of the HFS splitting is 0.3 MHz. We have also calibrated the x-axis using the EOM, which gives a FSR of , but with an uncertainty in the final HFS measurement of 2.61 MHz. To account for this third calibration factor, we take the weighted mean of the HFS values using all three. The final uncertainty is calculated using the ad hoc method of taking the upper and lower limits of the HFS splitting, taking the di erence of the two and (since we weight the mean towards the value with the lower uncertainty) multiply that di erence by the ratio of uncertainties. The final error calculated this way is 0.85 MHz for the HFS splitting, and is negligible for the TIS. Such a large systematic error would need to be examined in greater detail, with a larger volume of data to be resolved finally Scan Direction and Speed To calculate the systematic error resulting from the scan direction, we divide up our data into two categories according to whether the spectra were collected while the frequency was increasing or decreasing. The mean values of the HFS and TIS, divided this way, are plotted in figure 3.6. As is shown, the up-scans and down-scans for the TIS result in significantly di erent mean values, whereas for the HFS they are in good agreement. This discrepancy is consequence of the requirement that, for the TIS, we compare two di erent spectra collected simultaneously. If there is any overall asymmetry in the fit which shifts the absolute position of only one of the spectra, the comparison between the two will be shifted, whereas the relative peak positions within each spectra will remain constant. The last significant systematic error in our experiment is the scan speed. Organizing the scans by period, we find that the longer HFS scans have a lower overall value, whereas the shorter TIS scans have a higher overall value. We fit a line to the HFS splitting and TIS

46 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 35 (a) HFS values given by our experiment as com-(bpared to those given by [21] in pared to those given by [20] in TIS values given by our experiment as com- Figure 3.7: Comparison of our HFS and TIS measurements, with those made previously. shift plotted against scan period, then find the value of the linear function at the highest and lowest scan speeds. We estimate that the final scan speed error is simply half the di erence between the two. The scan speed errors for the TIS and HFS splitting are given in table HFS and TIS Results Our results are in good agreement with previous measurements, made by [20] and [21]. Our final values with their errors are given in table 3.3, and are compared to the old measurements in figure 3.7. The results we produce have similar, if slightly smaller, error bars than the old results, thus serve only to corroborate these measurements. As discussed, to achieve a final measurement, we would need to collect more data and explore the calibration and scan direction systematic errors in greater depth. Measurement Center value Error HFS MHz 0.9 MHz TIS MHz Table 3.3: Final measurement and uncertainties of the HFS splitting within the 3 P 1 state and the 3 P 0! 3 P 1 isotope shift. The consistency of our results, however does provide confidence in our experiment moving forward. We know that the home-build 939 nm laser scans over many GHz, the optics work for both wavelengths, and the lead cells are pure as claimed. More importantly, however, we have learned that a few things are di cult to implement in the Faraday rotation experiment. We know that applying EOM sidebands to the principle spectra overly complicates the lineshape; thus in the Faraday rotation experiment we apply sidebands to the FP spectra instead. Furthermore, we know that we cannot expect to even see the 939 nm transition

47 CHAPTER 3. HFS AND TIS MEASUREMENTS IN THE 6P 2 3 P 1 STATE 36 without employing more sensitive techniques; I describe our implementation of one such technique in the following chapter.

48 Chapter 4 Faraday Rotation Experiment and Apparatus The fundamental idea behind Faraday Rotation spectroscopy is that two linear polarizers in succession, crossed with respect to each other, will let no light through; if we rock the polarization just a bit before the second polarizer, however, some light will be able to escape. As I show in Eq. 2.23, a gas of lead atoms rotates the plane of polarization according to the real part of the index of refraction, which in turn is proportional the transition strengths. Thus by measuring how much light escapes the apparatus, we determine the transition MEs thegoalofourexperiment. This technique is sensitive to rotations of light polarization down to the level of a few to µrad, not only resolving the elusive E2 transition, but also providing a high signal to noise ratio provided we augment the polarimetry setup with additional modulation. In its simplest form, the intensity, and therefore the amplitude of our signal, recorded by the PD is proportional to the square of the (very small) rotation angle. We apply additional modulation to produce a linear relationship. Another method, in which we operate the polarizers at 45,ispossiblebutwouldresultinenormousbackground. In the following, I detail the experimental apparatus shown in figure 4.1 we build and methods we employ to simultaneously detect changes to the polarization and intensity of the light as it passes through the interaction region. I first lay out the basic concepts of the Faraday rotation method, explaining how it works, its key apparatus elements and the practical limits to its sensitivity. I proceed to detail the most important components of the experiment apparatus in general. I conclude by outlining the data collection and calibration schema. 4.1 The Faraday Rotation Method When coherent light leaves a laser diode, it is nominally linearly polarized but its absolute polarization direction is not well defined. Our experiment necessitates that the light be linearly polarized in a consistent direction. We therefore pass the light through two crossed 37

49 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 38 Figure 4.1: A graphical overview of the experiment layout. Note that the apparatus accommodates both lasers a necessity as we study two transitions. The heart of the experiment is the polarimetry setup, comprised by the calcite prisms and Faraday modulator. The oven components are displayed in detail in figure 4.5. Glan-Thompson prisms; the first picks the component of the incident light polarized in the ˆx direction, while the other picks out the light polarized in the ŷ direction. Ideally, with the polarizers crossed in this manner and without any optically active material between the two, no light would pass through the second prism. In practice, two Glan-Thompson prisms in direct succession transmit <10 6 of the incident light this is the system s finite extinction. In our apparatus, with optical elements installed between the two prisms, we measure a finite extinction ratio of <10 5. This finite extinction is e ectively an uncrossing of the two 2 polarizers, which we call o in anticipation of the result given by Eq When we insert an optically active material in between the two prisms, the polarization of the light rotates slightly. This rotation projects some of the light onto the ŷ direction; the ŷ components can then can pass through the second prism. In our experiment, the optically active material is atomic lead vapor under an applied magnetic field. When the laser is resonant with the lead atoms, its polarization changes according to Eq The total intensity after the analyzer is to first order, I = I o sin 2 ( ) I o 2, (4.1)

50 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 39 where is the angle through which the polarization rotates. The first order approximation is valid given that never exceeds 10 2 radians. Theoretically, we could directly record the intensity as a function of frequency, and extract the MEs from the resulting lineshape. Such a technique, however, would not distinguish high-frequency fluctuations in the laser power from an E2 transition-induced rotation. Furthermore, we would optimally be able to record both rotation and transmission signals in order to check that the results derived from the two agree. To do so, we employ a clever method in which we introduce a Faraday modulator between the two prisms, as in figure 4.2. The Faraday modulator, whose details are described in section 4.2.2, rotates the polarization of the light according to the strength of a magnetic field applied to a crystal through which the laser travels. The magnetic field is controlled with an AC current run through a solenoid wrapping the crystal. Figure 4.2: Schematic of the polarimeter apparatus, with the intensities after each component. The light passing through the modulator undergoes a rotation of m cos(!t), where! is the frequency of the AC current and m is its amplitude. This applied rotation simply adds to the lead-induced rotation, leading to a total e ective rotation angle of = pb + m cos(!t). Typically m 10 2 to rad; m 2 o,soitdoublesthefiniteextinctionatthepeakof its oscillation. Plugging into Eq. 4.1, the total intensity of light after the analyzer becomes, h i 2 I( ) =I 0 ( ) pb + m cos(!t) + 2 o. (4.2) It is valid at this point to drop the finite extinction term, which is not part of the coherant angle rotation, as it is a constant background, and will be zero after the lock-in filter. It is, therefore, just added noise. We note that I 0 is in fact proportional to the transmission lineshape given in Making this substitution and expanding the square in Eq. 4.2, the light intensity after the Faraday apparatus is, h I( ) =I o T ( ) 2 m pb cos(!t) m cos(2!t)+ 2 pb i 2 m. (4.3) The intensity, I( ), is recorded by a photo detector and sent into two lock-in amplifiers referenced to the first and second harmonics of the modulation frequency. The lock-ins ignore the two DC terms in Eq. 4.3 but can pick up the oscillating terms. The lock-ins output the amplitude of these two terms:

51 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 40 1f :2I o T ( ) m pb 2f : 1 2 I ot ( ) 2 m (4.4) The first term, oscillating at the first harmonic, is proportional to both the transmission and rotation signals, whereas the second term, oscillating at the second harmonic, is proportional to just the transmission. To extract the rotation lineshape by itself, we simply divide the first harmonic signal by the second, as in figure 4.3. From Eq. 4.4, it s clear that the frequency dependent transmission term cancels, and only the atomically induced rotation angle and constant scaling factors are left. We have found that the absorption term for the E2 transition is nearly imperceptible, thus it does not provide information on its own. We still record the transmission term for the 939 nm laser, however, in order to eliminate the residual slope Figure 4.3: Simulation of the the transmission rotation and the pure rotation shapes. resulting from changes to the laser power as we scan, and to remove any minor e ect the transmission may have on the rotation lineshape. 4.2 Apparatus Details In this section, I provide details on the experiment components, and discuss the practical limitations of the apparatus Sensitivity and Noise Analysis In practice, there is a non-zero laser intensity after the analyzer, even without the magnetic field producing the optical activity in the lead turned on. For every optical element between the two prisms, the polarization of the light is slightly distorted, and the convection currents inside the heated oven distort the polarization even further. We minimize this e ect by sending the laser straight from the polarizer to the analyzer, passing it through only the unavoidable optics: the windows of the furnace, the quartz vapor cell and the Faraday modulator. It is important to quantify the residual intensity after the second prism, as it establishes an upper bound on the sensitivity of the apparatus; any signal smaller than the baseline photodetector input will be lost in the noise.

52 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 41 (a) The resolution of the Faraday apparatus without the oven or vapor cell. (b) The resolution of the Faraday apparatus with heated oven and vapor cell, with the laser tuned o resonance. Figure 4.4: Sensitivity of the Faraday apparatus, with and without the heated furnace and vapor cell. The baseline noise level without oven or vapor cell is.05 µrad/ p Hz. With the additional noise from these components, the baseline noise level is still only 0.3 µrad/ p Hz. Note the presence of both high-frequency noise and a slow drift in figure b. We find that without the furnace or quartz vapor cell in between the polarizer and analyzer, the apparatus can easily resolve rotations down to 1µrad. With the inclusion of the heated furnace and vapor cell the conditions under which we run the experiment the noise level increases ten-fold, as shown in figure 4.4. Such resolution is easily su cient to resolve the E2 transition, which produces rotations that are several hundred µrad, as we will see. When the heated oven and vapor cell are installed, we notice not only an increase in highfrequency noise, but also the presence of a slow drift, presumably due to gradual changes in the modulator temperature, or mechanical shift of the apparatus. If such a drift were significant over the time-scale of our scan, we would worry as it would cause asymmetry in the Faraday shape. As shown in figure 4.4b, the laser intensity drifts 0.2 µrad over 10 seconds. Such a drift is only a fraction of a percent of the signal, thus is not of great concern. Still, we keep track of it as a potential systematic error The Polarimetry System The Faraday Modulator Our Faraday modulator contains a 5cm long Hoya FR5 glass cylinder which, when subject to a magnetic field, becomes optically active via the Faraday e ect. The glass is designed to have a large Verdet constant, so that these rotations are large in proportion to the applied magnetic field. We apply the magnetic field via a solenoid wrapped around the glass, through which we run an AC current. The resulting AC magnetic field rocks the polarization by roughly 1-10 mrad. The magnetic field coils have a resistance of 7.2, and we run 2 amps of

53 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 42 current through them at 1 khz, controlled by a frequency generator amplified by a Pyle PTA 1000 stereo amplifier. With 2 amps of current running through 7 ohms, we expect the modulator to heat, which is unacceptable as that would change the refractive properties of the crystal. To keep the temperature of the modulator constant as we apply the AC current, we run water cooling tubes inside the solenoid, and let the temperature equilibrate before collecting data. The Polarimeters The two Glan-Thompson prisms are mounted on thick steel stages (built by Leo Tsai in 1998) to minimize vibration, and set at the appropriate height so the prisms are aligned with the oven windows. The angle of a given prism with respect to the polarization of the incoming light is controlled using a dual micrometer attached to a 10 cm lever arm. The angle of the prism can be controlled down to 5µR. There is a small uncertainty in the length of the lever arm and therefore the calibration of the micrometer. This uncertainty proves inconsequential, as our final measurement is proportional to the ratio of the two Faraday rotation amplitudes, and any extra scaling factor will cancel when we take that ratio The Oven System The oven is the dominant structure on our optics table: it is approximately two meters long, a meter wide and requires four people to move. It is purposefully so hefty, as its large thermal mass increases the stability and uniformity of the temperature inside. We take several additional measures to minimize the oven s contribution to overall noise of the signal. Aschematicoftheovenisprovidedinfigure4.5. Heating Elements We heat the oven with four clamshell heaters, two placed on both ends of the furnace. To bring the oven up to the necessary temperature 915 to 1000 C werun5ampsof current through the heaters. In order to maintain a constant temperature, we must run the heaters constantly. Such a large wall current in close proximity to the rest of the apparatus is unacceptable, as it causes large unwanted 60 Hz oscillations in the Faraday rotation, adding 60 Hz noise in the resulting Faraday signal. We overcome this obstacle by connecting a function generator operating at 10 khz to four stereo amplifiers (Gemini XGA 5000), and heating the ovens with the amplifier output. This frequency is high enough that it interferes minimally with the electronics nearby, and is far enough removed from the 1 khz modulation frequency to avoid mixing with that signal. Water Cooling and Evacuation The fibre glass insulation in between the inner and outer cylinders is not enough by itself to keep the outer surface of the oven cool enough to touch. With temperatures as high as 1000

54 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 43 (a) Overview of the oven system, from above. (b) The front end of the oven looking head on. Figure 4.5: Schematic of the oven system.

55 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 44 C, the insulation itself begins to singe. To prevent these unhealthy fumes from entering the room and to cool down the outer surface of the oven, we run water cooling tubes around the outer cylinder. The end caps sealing the inner cylinder are made of metal, and thus can also get very hot; this is especially bad as the light must pass through the windows on the end caps, and when the windows heat, their refractive index changes, thus distorting the polarization of the light. To cool down the end-caps, we extend the water cooling system from the inside the oven to copper tubes circling the end-caps themselves. We also worry about the e ects that convection currents inside the oven may have on the refractive index of the air inside. To mitigate these e ects, we evacuate the inner cylinder down to 2-5 torr, then backfill the inner cylinder with argon gas to keep the pressure constant. The incident light will not be refracted by argon, and thus can pass through the oven chamber with minimal distortion. Transverse and Applied Magnetic Fields As we see in section 5.1, the precise value for the applied longitudinal magnetic field is important to know so we can calculate the Zeeman splitting for a given spectrum. We use a 3-axis magnetometer to record the longitudinal magnetic field in the center of the oven where the cell sits as a function of DC current through the solenoid. We performed this calibration three times over the course of two days and found that for every amp of current through the solenoid, gauss of longitudinal field is produced in the center of the oven. The oven outer shell of the oven is made of Mu-metal shielding, which prevents the vapor cell from experiencing stray, unwanted magnetic fields. The longitudinal shielding factor of the Mu-metal is roughly 100, thereby decreasing stray fields to a low level. In future iterations of this experiment, we must ensure that the magnetic field and the light beam are truly co-axial and that the transverse magnetic fields are minimized The Lasers and Their Operation We use two lasers, one operating at 1279 nm and one operating at 939 nm. The former, a Sacher Lasertechnik TEC 150, is commercially made. The latter is homebuilt, constructed by Hallee Wong in the summer of They are both external cavity diode lasers in the Littrow configuration. The diode itself operates at approximately the right wavelength, but we require even further precision, and the ability to scan the laser through many GHz. We meet these requirements by installing a di raction grating external to the diode, and sending the first order reflection back into the internal cavity; only the reflected frequencies will continue to be emitted. By gradually changing the angle of the di raction grating we change the wavelength of the first order reflection, thus scanning the output frequency of the laser. The angle of the di raction grating must be scanned continuously through very small angles. We accomplish this by placing a piezoelectric inducer (PZT) a small device which expands when a voltage is applied across it up against the grating. We scan the laser by using a function generator

56 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 45 to send a triangle wave to the PZT, thus expanding and contracting it in regular intervals at a consistent speed. We increase the range over which the laser can scan in a single mode by the employing a feed-forward mechanism. As the voltage into the piezo increases, the laser current is reduced. With the feed-forward mechanism, we are able to scan the 939 nm laser 8 GHz. It does, however, add an overall slope to the absorption spectra. The typical short-term noise for the lasers, equipped with the external cavity but without laser-locking, is negligible, as determined by previous work e.g. [15]. The intrinsic noise of the laser, therefore, has a negligable contribution to the observed lineshape The Fabry-Pérots and Frequency Calibration Frequency calibration does not play as crucial a role in this experiment as it has in in the Majumder lab previously, as we measure the peak amplitudes instead of peak positions. Still, defining a frequency axis enables us to accurately fit our lineshapes to the appropriate functions. The lineshape amplitude a crucial factor in determining the transition amplitude depends on the Zeeman splitting and the Gaussian and Lorentzian widths. In order to provide strict constraints for these parameters in the fitting process, and to check the fitted values to their theoretical predictions, the frequency axes must be accurate. The Fabry-Pérot (FP) also simply helps us scan the laser, as it tells us the frequency range we cover, and if the laser operates in a single mode. Figure 4.6: FP spectrum with sidebands at 600 MHz from the center peak. although the fit is not perfect, it accurately captures the position of each peak. Note that,

57 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 46 Prior to merging the beams, we send each laser through its own FP. The 1279 nm laser FP has a measured free spectral range (FSR) of nm, and the 939 nm laser FP (built in the Bronfman machine shop) has an FSR of nm. As shown in figure 4.6, we apply 600 Mhz sidebands with an electro-optic modulator (whose mechanisms are described in Appendix A) to the light entering the 939 nm FP in order to provide an extra calibration check. The sidebands are opportunely placed 122 MHz away from the closest FP peak, so are easily resolved. 4.3 Data Acquisition In this section, I outline the process by which we translate the variations in beam intensity after the analyzer to a properly normalized, physically meaningful spectrum Signal Detection The extinction ratio of the polarimetry setup is on the order 10 5, and the incoming light originally has a power of 10 mw for the 940 nm laser and 1 mw for the 1280 nm laser. The photodetectors must therefore respond to signals that are only nw. To accurately detect such small signals, we use high-sensitivity photodetectors for the two di erent lasers. For the 1279 nm laser, we employ a photodetector of the same design as that used to record parity-violating optical rotation in Pb and Tl, twenty years ago see Appendix A of [13] for a circuit diagram. To detect the 939 nm signal, we use a ThorLabs PDA 100A Si photodetector, sensitive to signals down to a nw. Both of these photodetectors provide ample response to the incident beam. The tiny laser signal that we are interested in recording is easily drowned out by ambient room lights and the blackbody radiation emanating from the oven. The room lights are easily blocked by a box installed around the photodetector; eliminating the light from the oven takes a little more thought. A component of the blackbody radiation travels co-linearly with the beam, thus the box blocking the room lights does not e ectively eliminate the radiation. Furthermore, thermal radiation peaks at infrared wavelengths the same region in which our lasers operate. To di erentiate the beam from this extraneous light, we install a di raction grating downstream of the oven system. The grating reflects incident light at an angle determined by the beam s wavelength. The vast majority of thermal radiation does not have the exact same wavelength as the laser beam, thus reflects at di erent angles than the beam. We then use an iris to pick out only the light traveling along the path of the laser, essentially eliminating the e ects of thermal radiation on our signal. The di raction grating has the added bonus that it separates the 1279 nm and 939 nm lasers. Our di raction grating reflects the two laser beams at an angle relative to each other of 11,givingusenoughroomtoredirecteachbeamintoitsappropriatedetector.

58 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS Data Collection We record four principal signals when we conduct our experiment: the 1f and 2f harmonics for both the 1279 and 939 nm lasers. As auxiliary data, we record the FP spectra and ramping voltage for both lasers. Because we take the ratio of the two transition strengths to cancel out di cult-to-measure experimental parameters, it is necessary to record spectra for both transitions in rapid succession. We therefore use a shutter system, built by Nathaniel Vilas in the winter of 2015, to alternately block the 1279 or 939 nm laser after every laser scan. We link this shutter system to a LabVIEW data acquisition program that collects and stores the 1f and 2f spectra along with the relevant ramp voltage and FP spectrum. We require that the laser s frequency scans are synchronized, so we use the same function generator to scan both lasers. To provide the freedom to scan the lasers over di erent ranges, we install a variable attenuator in between the function generator and the 1279 nm laser. We collect data with a variety of di erent parameters to ensure that these are not a ecting the end result significantly. The important experimental parameters that we vary for each data set are listed in table 4.1. Parameter Faraday magnetic field Oven temperature Beam overlap Scan period Modulating current Manual calibration turn Incident laser power Typical Range 1-10gauss C NA 5-15seconds 1-2amps mrad 1-5mW Table 4.1: Parameters for data collection, and their typical ranges Calibration Data In its raw form, the y-axis of a recorded spectrum is arbitrary. The amplitude of a given spectrum changes according to the gain of the photodetector, the sensitivity of the lock-in amplifier, the amplitude of the modulation and the power of the laser beam. We calibrate the spectra by periodically interrupting the data collection process and measuring how the signal changes according to a manual uncrossing of the polarizers. The calibration process is as of yet unrefined: optimally we would automate the calibration method by periodically applying a DC o set current to the Faraday modulator, thereby adding a known rotation to the polarization of the light. This way, we could calibrate the spectra frequently and without having to interrupt data collection. Currently, with no automated system in place, we take six scans for both lasers with the polarizers completely crossed; then we cross the polarizers by a known amount a good

59 CHAPTER 4. FARADAY ROTATION EXPERIMENT AND APPARATUS 48 number is 100 µrad and record six more scans. We can then record the overall DC shift of the signal before and after the uncrossing. We know that this shift corresponds to 100 µrad, thus providing our calibration factor. In section 5.1.2, I detail how we apply this calibration factor to the Faraday spectra Summary of Collected Data We have collected data over two periods: from March 31 to April 2nd and from May 4th to May 5th, The data in the first period are used to test the apparatus; specifically, we collected these data under various applied magnetic fields to test that changes to the Zeeman splitting do not a ect the ratio of the transition strengths. We have a total of 100 usable spectra for each transition from these data. The final result from this first period is in good agreement with that from the second period, but we only use it for testing purposes as there is unusually large background noise in the 1279 nm scans. From May 4th to May 5th we collected a total of 600 scans for the 939 and 1279 nm transitions. We collected these data while varying scan direction, oven temperature and scan speed. These data bring us closer to a final result and allow us to determine an initial uncertainty in the Lorentzian width a particularly di cult parameter to extract. Yet our measurement is not yet complete, for we must conduct more systematic error analysis, further constrain the fitting parameters and continue to improve the apparatus.

60 Chapter 5 Data Analysis and Preliminary Results We extract physically meaningful values from the spectra through a sequence of normalization, calibration and numerical fitting. These final aspects of the experiment are allimportant, for our final results depend on more than the amplitude of the peak as measured directly from the spectrum itself. Instead, to calculate the transition strengths we must not only measure the amplitude of the raw lineshape, but we must also convert the y-axis of the spectrum to radians, calibrate the frequency axes of our spectra and divide the absorption signal from the transmission signal appropriately. Furthermore, we must derive the height of the line-shape, without the pre-factors scaling its amplitude. The fitting function must be accurate, so the parameters we use to describe the function must be as well. In this section, I describe the calibration process, the fitting process, and our preliminary result with its corresponding uncertainty. 5.1 Data Analysis Summary of Collected Spectra Representative spectra of both Faraday rotation and absorption are shown in figures 5.2 and 5.1. These y-axes of these spectra are already calibrated to denote the rotation angle, the frequency calibration process has been completed, and the spectra have been fit to the appropriate functions. Note that the Faraday spectrum for the M1 transition has an amplitude of approximately 30 mrad, whereas the amplitude of the Faraday spectrum for the E2 transition has an amplitude of 0.6 mrad. The ratio between the peaks does not exactly equal the ratio of the transition strengths, but gives us some idea for what to expect. The frequency axes of these spectra are calibrated according to the same method employed in the Majumder lab over many years, detailed in this thesis in appendix A, in addition to [15] and [22]. 49

61 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 50 (a) Faraday spectrum for the M1 transition with fit residuals. (b) Absorption spectrum for the M1 transition with fit residuals. Figure 5.1: Representative spectra from the M1 transition. Note that the y-axis scales are di erent for each spectrum; they must be, as one depicts absorption, the other rotation. (a) Faraday spectrum for the E2 transition, with fit residuals. (b) Absorption spectrum for the E2 transition three scans overlapped. Figure 5.2: Representative spectra from the E2 transition. As in figure 5.1, the y-axis scales are di erent for each spectrum. Figure a. shows only one scan, whereas Figure b. shows the signal from three, in order to distinguish noise from overall trends. Figure 5.2a depicts the (formerly) elusive electric quadrupole transition. Despite the small transition strength, the signal-to-noise ratio is still of order 100. Figure 5.2b depicts representative transmission signals for the 939 nm laser. Several di erent scans are shown, in order to di erentiate random fluctuations from overall trends. We expect, without any absorption, the laser power to change steadily with frequency as a consequence of the feed forward mechanism explained in section 4.2.4, leading to the slope in figure 5.2b.

62 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS Calibration Scheme The process by which we normalize the transmission spectrum, divide the Faraday rotation shape by the absorption shape, then calibrate the rotation shape is outlined below. First, recall that the rotation and transmission signals, corresponding to the first and second harmonics respectively are, F ( ) = 1 2 I( )T ( ) m pb T ( ) =2BI( )T ( ) 2 m. (5.1) IhavesubstitutedI( ) for the constant I o,toaccountforoverallchangesinlaserintensity as we scan. I have also inserted the constant B in the transmission expression to account for di erences in lock-in model and sensitivity between the two signals. We do not know B, but if we follow the correct process, outlined as follows, its value is inconsequential. 1. We normalize the transmission spectra such that the mean o -resonance value (defined arbitrarily as 2 FWHM away from the center peak) is unity. This choice of normalization may seem arbitrary, and to a certain degree it is, but it is important that this process remains consistent. 2. We divide the Faraday shape by the normalized transmission spectrum. This cancels out the frequency dependent terms in the first expression of Eq Combining all constant terms into a single constant C, we are left with, F ( ) =C pb. (5.2) 3. We perform steps 1. and 2. again, only now with the calibration scans. It is crucial at this stage that we normalize the transmission spectra (step 1) using the same points in frequency space as we did for the Faraday spectra. We require that the constant factor C in Eq. 5.2 be equivalent for both the calibration spectra and the principle spectra; otherwise, the scaling on the two y-axes will di er and the voltage to rotation angle factor we derive will not be equal. We average over as many o -resonance points as possible in step 1 to reduce statistical uncertainty in the calibration factor. 4. The manual uncrossing amounts to adding a constant term to pb,let scallit uncr. So the manual uncrossing amounts to a vertical shift in F ( ) ofc uncr. If the uncrossing corresponds to a DC o set of C uncr =,thenc = / uncr and the calibration factor is simply 1/C. WeextractCusingthenormalizedrotationscansfromstep3bysimplytaking the mean value of a scan before and after the uncrossing. 5. Finally, we apply the calibration factor 1/C to the normalized rotation scans from step 2 to obtain the calibrated rotation spectra The Fitting Process The fitting and analysis process is similar to that described in [22] and in Appendix A, with a few key di erences. I will therefore omit description of the frequency linearization

63 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 52 and calibration process, and focus on the fitting of the Faraday spectra. We assume in the following description that the spectra are normalized and calibrated, however in practice the whole process is performed in one large MatLab script. Agivenscanisrepresentedbyavectorofdiscretevectorpoints,pairedwithanothervector of frequency values. Recall that the rotation spectra are represented by the convolution of a Gaussian and the Faraday shape, C( ). As C( ) isnotanalyticallysolvable,wecannot directly use the MatLab fitting function employed in [22]. Instead, we must implement our own fitting function. First, we make educated guesses for the function parameters and set upper and lower bounds to these parameters. The tighter we can make these bounds the more reliable and faster the fitting process will be. Next, we calculate the convolution function using these initial guesses. We calculate this function one of two ways: using the discrete conv() function implemented by MatLab, or by manually convolving the two functions by numerically integrating Eq The di erence between the two functions is negligible with any vector with more than 500 elements; we employ the faster conv() function. Figure 5.3: Schematic for fitting the Faraday lineshape. The box size shows the hierarchy of functions: fmincon() calls nfaradayresiduals iteratively, and nfaradayresiduals calls the convolution function. We wrap the whole process in a function whose dependent variable is the sum of the residuals of the fit, and whose independent variables are the fit parameters see 5.3. We then minimize this function using the built-in MatLab program fmincon(). The resulting fit parameters are those that minimize the residuals. These parameters comprehensively describe our function. Finally, we can use the relations given in Eq to extract the MEs from the fitting function Constraining the Fitting Parameters The Faraday lineshape is exhaustively described by five parameters: the resonance frequency, the Zeeman splitting, the Gaussian width and the Lorentzian width. As the fitting function is a complicated numerical convolution, we constrain the parameters as much as possible to ensure its accuracy. The bounds of the peak position are set tightly, as the resonance frequency should correspond closely with the center of the lineshape (this would not be the case if we had to worry about other overlapping peaks). This parameter is found simply by finding the location of

64 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 53 Parameter Peak Position Zeeman splitting Gaussian FWHM Lorentzian FWHM Constraint ± 10 MHz ± 0.1 MHz ± 10 MHz ± 60 MHz Table 5.1: Constraints on the fitting parameters, implemented in the MatLab fitting function. the maximum data point. A typical scan consists of 500 data points and spans 8000 MHz. We therefore let the center vary by > 8000/500 = 16 MHz. The Zeeman splitting is determined by the magnetic field applied parallel to the laser beam. We have established the linear relationship between the magnetic field and input current to within 0.05 Gauss, thus we can determine the Zeeman splitting to a level of less than 0.1 MHz. The Gaussian width is known through the the simple relation derived in section 2.6, g =2 p 2ln2 0 r kb T Mc 2. The only uncertainty in the Gaussian width is therefore / p T. Given the large thermal mass of the oven corresponding to a constant and uniform temperature distribution we can estimate the Gaussian width to within at least 20 MHz. When we do let the width vary, the theoretical values for the Gaussian widths are in good agreement with the fit parameters, as shown in table 5.2. Extraction Method Gaussian FWHM (MHz) 1279 nm Faraday fit 404(4) 1279 nm absorption fit 404(6) 1279 nm calculation 407(7) 939 nm Faraday fit 542(5) 939 nm calculation 555(10) Table 5.2: Gaussian widths from fitting and theory for data collected at 945 C The Lorentzian width is left to vary widely, as its absolute value cannot be externally determined. We make an initial estimate by considering the calculation given in 2.6 and by iteratively fitting our spectra. The values obtained for the Lorentzian width from fitting both the Faraday and the transmission spectra are given in section5.3.1.

65 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS Preliminary Result We present the following result to provide a preliminary value for the E2/M1 transition amplitude ratio, however more work must be done to arrive at a final value. Our preliminary findings are given in Eq. 5.3 with the corresponding final error including both statistical and systematic uncertainty (statistical)(systematic). We extract this value from the calibrated lineshape using the formula developed in chapter 2: he2i hm1i = h C(!0 ) expmt,e C(! 0 ) expmt,m C peak,m C peak,e! 0,m! 0,e i 1/2 =0.148(2)(8). (5.3) Converting this result to the reduced ME form by scaling the ratio by he2 red. i hm1 red. i q 5 3, =.191(03)(10). (5.4) This measurement of E2 in the 6p M1 2 3 P state in lead is similar to the same ratio, only measured within the 6P 1/2! 6P 3/2 transition in Tl. The Majumder lab measured this ratio in 1999 [5] as E2 = We do not expect that the two ratios be the same; but as they M1 are both heavy metals, we are reassured by the similarity. 5.3 Systematic Error Analysis As is clear from Eq. 5.3, systematic errors are the dominant source of uncertainty in our measurement. The majority of the systematic error is a consequence of the uncertainty in the fitting parameters. If we could simply measure the peak of the lineshape, this would not be the case. The transition amplitudes, however, are related to the the peak of a recorded spectrum by the height of the lineshape this is the result shown in Eq. 5.3 by the term C fpeak,m C fpeak,e.theheightofthelineshapeisnotadirectlyobservablequantity itmustbederived using the Gaussian width, the Zeeman splitting and most importantly the Lorentzian width. We find that the Gaussian width and the Zeeman splitting contribute negligible systematic uncertainties compared to the Lorentzian width contribution. In the following sections, I outline the uncertainties derived from this problematic fitting parameter Uncertainty in the Lorentzian Width The Lorentzian width is the most imprecisely known, independent parameter of the fitting functions. It is not possible to provide an accurate, a priori calculation of the Lorentzian width, as is feasible for Doppler broadening and the Zeeman splitting. Instead, the Lorentzian width must be derived through fitting, and by clever comparison of the absorption and rotation spectra.

66 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 55 Lorentzian Width Error Propagation To determine the dependence of the transition amplitude on the Lorentzian width, simulated data with known widths and the appropriate amount of noise are fit with various fixed, erroneous values of the Lorentzian width. The other parameters, such as the Gaussian width and Zeeman splitting, are allowed to vary within their typical constraints. For a wide range of Lorentzian widths, the peak of the lineshape is linearly dependent on the Lorentzian width; a conservative estimate of the slope is 0.09 for the E2 transition and 0.13 for the M1 transition, as shown for the E2 case in figure 5.4. Therefore, the Lorentzian width uncertainty propagates to the uncertainty in the final E2 and M1 transition amplitudes according to: E2 =0.05 E2 (5.5) M1 =0.07 M1 The proportionality constants in Eq. 5.5 are 0.05 and 0.07 instead of 0.09 and 0.13, as the transition amplitude is proportional to the square root of the lineshape peak. The di erence between the two transitions is a function of the typical Gaussian widths for each (550 MHz for E2 and 405 MHz for M1). The result from this method is roughly half of that obtained by simply numerically calculating the lineshape peak for a variety of Lorentzian widths under a range of Gaussian widths and Zeeman splitting values. This latter method overestimates the Lorentzian error propagation, for the other fitting parameters are not allowed to vary in order to compensate for changes in the Lorentz width. We note that the statistical uncertainty in the Lorentzian width already propogates through to the final statistical uncertainty in the extracted transition strengths. Eq. 5.5 is applicable to systematic errors of the Lorentzian width, which are calculated in the following. Temperature Dependence of the Lorentzian Width The Lorentzian width is primarily a function of collisional broadening, as explained in section 2.6. From the results of that section, we infer that the collisional width is proportional to the vapor pressure of the lead cell. The proportionality constants are not accurately known, but an expression for the vapor pressure of lead (along with most other metals) as a function of temperature is tabulated in [17], and plotted over typical operating temperatures for this experiment in figure 5.5. The ratios of the Lorentzian widths determined through the fitting process, should follow this curve. At present, a limited amount of data has been collected therefore we can only compare the Lorentzian widths extracted from fitted data at two di erent temperatures: 913 and 945 C. As shown in table 5.3, the ratio of the mean Lorentzian widths derived from the rotation spectra at these two temperatures is of order unity for the M1 transition, whereas the vapor pressure curve predicts a ratio of 945/ 913 =1.6. TheratioofLorentzianwidthsfortheE2

67 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 56 Figure 5.4: The fractional change in peak amplitude as a function of the fractional change in Lorentzian width, for the E2 transition. As the Lorentzian width decreases, its contribution to the error correspondingly decreases. As we expect the Lorentzian width is no larger than 80 MHz, the proportionality constant is 0.1. Figure 5.5: Lead vapor pressure as a function of temperature, with points indicating the temperatures at which data was collected. This experimentally derived curve is given by: P/pa = /T.

68 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 57 transition for the two temperatures is 1.6(2), and is therefore in excellent agreement with the theoretically predicted ratio. Spectral Type FWHM at 945 C FWHM at 914 C E2 rotation 61(2) MHz 37 (4) MHz M1 rotation 70 (1) 71 (1) M1 transmission 74 (1) 98 (1) Table 5.3: Lorentzian widths collected at 945 and 914 C. The 60% error in the M1 Lorentzian widths propagates to a 3% uncertainty in the final uncertainty of our measurement. This systematic error comprises the majority of the uncertainty of our measurement, thus must be investigated further in order to present a final result. Additionally, the inconsistencies between the M1 transmission and rotation scans at 914 Cmustalsobeinvestigated Further Systematic Error in Transition Amplitudes There is more variation between the final results derived from the data taken at the two di erent temperatures than can be accounted for by the Lorentzian width alone. We find, as shown in figure 5.6, an 10% discrepancy between data collected at the two temperatures, of which only 3% can be accounted for by the Lorentzian width variation. It is also highly possible that this discrepancy is a result of the two di erent calibration factors used for these data. Since we do not have enough data to isolate temperature dependence from calibration error, we simply assign an overall 3.5% error to the final uncertainty. More work must be done to further quantify this additional source of error E ect of Magnetic Field Strength Our first set of preliminary data were collected at systematically increasing magnetic field strengths, in order to establish that the size of the Zeeman splitting does not a ect our final result. As we increase the strength of the magnetic field, the size of the 1279nm and 939nm signals should increase, and their shapes should very gradually change. Their peak height ratios, however, must remain constant. If they do not remain constant, we infer that either the beam overlap is imperfect, or the calibration process is flawed. As shown in figure 5.7, we find that there is no significant overall trend in the ratio of the two transitions corresponding to the strength of the magnetic field, although more data must be taken to explore this potential e ect further.

69 CHAPTER 5. DATA ANALYSIS AND PRELIMINARY RESULTS 58 (a) E2/M1 values collected at 945 C (b) E2/M1 values collected at 913 C Figure 5.6: Transition amplitude measurements collected at two di erent temperatures: 913 and 945 C. The error quoted for these histograms is the standard deviation the uncertainty for any single data point not the standard error which is a factor of ten smaller. Figure 5.7: he2i : hm1i ratio as a function of magnetic field strength