Precise Measurement of the Hyperfine Splittingand Isotope Shift of the 8P1/2 State in 203Tl and

205Tl Using Two-Step Laser Spectroscopy

bySauman Cheng

Professor Protik K. Majumder, Advisor

A thesis submitted in partial fulfillmentof the requirements for the

Degree of Bachelor of Arts with Honorsin Physics

WILLIAMS COLLEGEWilliamstown, Massachusetts

May 20, 2016

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Acknowledgements

First and foremost, I would like to thank my thesis advisor, Professor Tiku Ma-jumder, for all he has taught me this year and for all his support this year towardscompleting this thesis. Allison Carter, and Ariel Silbert, fellow inhabitants of theBronfman sub-basement, I thank for their unwavering presence and assistance. Iwould also like to acknowledge everyone who has worked in the lab, from ourpostdoc Milinda Rupasinghe, to Nathanial Vilas and Bingyi Wang, for their helpover the summer and Winter Study respectively. Professor Charlie Doret must alsobe thanked for being an excellent second-reader, responder to miscellaneous ques-tions, and lender of random optics. Last, but not least, I would like to thank Dr.Patrick Diehl, who first taught me in physics, when, perhaps, all of this began.

Abstract

The hyperfine splitting of the 8P1/2 excited state in 203Tl and 205Tl, as well as theisotope shift within the 7S1/2 → 8P1/2 transition has been measured using two-steplaser spectroscopy. Our measured values for the hyperfine splittings are 780.5(4)MHz in 203Tl and 788.3(7) in 205Tl, where the uncertainties in the last decimal placeare combined statistical and systematic errors. The transition isotope shift of 203Tlrelative to 205Tl was measured to be 450.3(0.9) MHz. Our value for both hyperfinesplittings is in reasonable agreement with the 1988 published values. However,our measurement of the 8P1/2 isotope shift relative to the ground state is 10 MHzless than the previous published value. In our experiment, one laser was lockedto the 6P1/2(F = 1) → 7S1/2(F ′ = 1) 378 nm transition, while a second, 672 nmred laser was scanned across the 7S1/2(F ′ = 1)→ 8P1/2(F ′′ = 0, 1) transitions. Thetwo lasers were spatially overlapped inside a heated thallium vapour cell. Radio-frequency modulation of the laser was used to create sidebands in the absorptionspectrum to introduce an method of frequency calibration internal to the absorp-tion spectra.

Executive Summary

This thesis describes the precise measurement of the hyperfine splitting of the 8P1/2

state in the two naturally occuring isotopes of thallium (203Tl and 205Tl), as well asthe isotope shift of the 7S1/2 → 8P1/2 state between the two isotopes. This mea-surement follows upon those performed over the past years by our group of thehyperfine splitting and isotope shift of the 7S1/2 state in thallium [1], and of thethallium 7P1/2 state [2]. Our interest in performing these measurements of atomicstructure stems from the need to test ab initio atomic theory calculations for GroupIIIA atoms such as thallium and indium. Our measurement of the 7P1/2 hyper-fine splitting in 2014 showed a ∼ 20 MHz discrepancy from previously publishedvalues with comparable precision [3], improved agreement with theoretical esti-mates. This motives us to remeasure other hyperfine splittings, to discover anyother possible systematic and/or calibration errors in the current literature.

In this experiment, we use a thallium vapour cell, and two external cavity diodelasers, to achieve the double-excitation of the 6P1/2 → 7S1/2 → 8P1/2 transition. Asthis transition is essentially exactly analogous to the Tl 6P1/2 → 7S1/2 → 7P1/2 tran-sition last measured in our lab [4, 5], altering the experimental apparatus used inthat measurement was a matter of making the simple and efficient move of switch-ing out the infra-red laser used then to drive the 7S1/2 → 7P1/2 transition for a 672nm (red) laser suitable for the 7S1/2 → 8P1/2 transition in which we are interested.Simultaneously, we pursued the ambition to build, and employ in the course ofthis measurement, our own external cavity diode lasers. After some trials, wehave been remarkably successful in that endeavour. In addition to the obviousfinancial incentives to master home-built diode lasers, there are also substantialpractical conveniences. For example, the 6P1/2 → 7S1/2 → 8P3/2 transition whichwe also wish to measure requires only a different laser diode 1 to be installed inthe existent Littrow configuration diode laser system to be fully accessable withthe same experimental setup. Currently, the switch has already been made, andwe are poised to collect data on that transition.

Figure 1 is a schematic of the experimental setup. A 378 nm (ultra-violet) laser,locked to the correct frequency, drives the first-step 6P1/2(F = 1) → 7P1/2(F ′ = 1)transition. A 672 nm (red) laser is scanned across the second-step 7S1/2(F ′ = 1)→8P1/2(F ′′ = 1, 0) transitions. The UV laser is split into two beams, one propagat-

1One that covers 655 nm, rather than 672 nm.

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ing with the red laser beam, the other in the opposite direction: a ‘co-propagatingbeam’, and a ‘counter-propagating beam’. The three lasers beams are physicallyoverlapped inside the thallium vapour cell, and the red laser sent to a photode-tector. The signal is very small. This is because to achieve the spectral resolutionnecessary to reveal the details of hyperfine structure, we choose to excite only afew percent of available Tl atoms. Specifically, we excite atoms of a narrow veloc-ity class, and in doing so, eliminate the vast majority of Dopper broadening fromthe resulting spectra. Thus, absorption spectra must be collected with the aid of anoptical chopper wheel and a lock-in amplifier.

Figure 1: Diagram summarising the setup of the experiment. The full locking setup maybe found in Appendix D.

Two types of spectra are collected: single-isotope spectra and dual-isotope spec-tra. For the first, the two isotopes are separately excited, and the hyperfine splittingof each may be separately and straightforwardly extracted from the resulting sixpeak spectrum. Figure 2 is an example of a collected single-isotope spectrum onwhich the actual hyperfine peaks and thus hyperfine splitting is indicated. Peak 4- peak 3 is the hyperfine splitting.

Dual-isotope spectra are collected by exciting Doppler shifted atoms of bothisotopes with one laser such that the hyperfine spectrum of both isotopes is cap-tured at the same time. This is necessary to obtain the isotope shift. Figure 3 isan example of a dual-isotope spectrum. It consists of eight peaks. Unlike in thesingle-isotope spectrum, every peak is a true hyperfine peak: two isotopes, twohyperfine levels, and two UV laser propagation directions yields eight peaks. Theanti-parallel UV beam configuration allows the Doppler shift to be accounted forby taking a simple average of corresponding peaks.

Figure 2: Sample single isotope spectrum. Red line is the best fit of a sum of six Lorentziansto the data. Peaks are number 1-6 in order of increasing frequency for ease of reference.Peaks 3 and 4 are the true hyperfine peaks. Peaks 1, 2, 5, and 6 are EOM sidebands for thepurpose of frequency calibration during the data analysis process.

Figure 3: Sample dual isotope spectrum. Red line is the best fit of a sum of eightLorentzians to the data. Peaks are labelled A - H for ease of reference. The spectrumappears complicated, but each peak can be fully specified by a combination of Tl isotope,hyperfine transition, and UV laser propagation direction.

We obtain values for the 8P1/2 hyperfine splitting for 203Tl and 205Tl, and theisotope shift of the 6P1/2 → 7S1/2 → 8P1/2 transition, with precision at or below 1MHz with the inclusion of both statistical and systematic errors. They are, in thatorder, 780.5(4) MHz, 788.3(7) MHz, and 450.3(0.9) MHz. Figure 4.24 compares ourvalues to previous measurements made by M. Grexa et al. [3] in the 1988. Ourvalues for both hyperfine splittings are in reasonable agreement with the previousvalues. However, our value for the 8P1/2 isotope shift relative to the ground stateis less than the previously measured value by 10 MHz.

Figure 4: Comparison of our results with values measured by Grexa et al. [3]. Left: 8P1/2

hyperfine splittings. Purple corresponds to 203Tl, green to 205Tl. Right: Isotope shift of8P1/2 state relative to the ground state.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Executive Summary vii

1 Introduction 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Purpose of Precise Measurements . . . . . . . . . . . . . . 11.1.2 Thallium and Indium . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Laser Spectroscopy and Atomic Structure 92.1 Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Broadening Mechanisms . . . . . . . . . . . . . . . . . . . . . . 102.2 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Isotope Effects: Isotope Shift and Hyperfine Anomaly . . . . . 18

3 Experimental Setup and Apparatus 213.1 Experimental Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 External Cavity Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . 243.3 Frequency Stabilisation of 378 nm Laser . . . . . . . . . . . . . . . . . 263.4 Diagnostic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Data Collection, Analysis Methods, and Results 314.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Single-Isotope Spectra and the Hyperfine Splitting . . . . . . . 314.1.2 Dual-Isotope Spectra and the Isotope Shift . . . . . . . . . . . 324.1.3 Summary of Collected Spectra . . . . . . . . . . . . . . . . . . 36

4.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.1 Frequency Mapping and Linearisation . . . . . . . . . . . . . . 374.2.2 Determination of Measurements . . . . . . . . . . . . . . . . . 404.2.3 Frequency Calibration . . . . . . . . . . . . . . . . . . . . . . . 434.2.4 Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 44

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4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Single-Isotope Data: Hyperfine Splitting . . . . . . . . . . . . 474.3.2 Dual-Isotope Data: Isotope Shift . . . . . . . . . . . . . . . . . 534.3.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Future Work 595.1 8P3/2 Hyperfine Splitting and Isotope Shift . . . . . . . . . . . . . . . 595.2 Moving on to Indium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A 672 nm Laser Design and Construction 63A.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A.2 Preliminary Decisions and Acquisitions . . . . . . . . . . . . . . . . . 63

A.2.1 Simple Calculations . . . . . . . . . . . . . . . . . . . . . . . . 63A.2.2 List of Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.3 Construction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.3.1 Machining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.3.2 Making Final Connections . . . . . . . . . . . . . . . . . . . . . 66

A.4 Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.5 Threshold Current Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Laser Locking 79B.1 Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.2 Temperature Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 80B.3 Manipulating the Lockpoint . . . . . . . . . . . . . . . . . . . . . . . . 82

C MATLAB Code 83C.1 Data Files to Extracted Measurements . . . . . . . . . . . . . . . . . . 83C.2 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D Direct Measurement of Fabry-Perot Cavity FSR 89D.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89D.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

List of Figures

1 Experimental Setup Diagram . . . . . . . . . . . . . . . . . . . . . . . viii2 Single-isotope spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . ix3 Dual-isotope spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . ix4 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1.1 Thallium and indium energy level diagrams . . . . . . . . . . . . . . 31.2 Tl energy level diagram, 7P1/2, 8P1/2,3/2 . . . . . . . . . . . . . . . . . 61.3 Tl 6P1/2 → 7S1/2 → 8P1/2 transition . . . . . . . . . . . . . . . . . . . . 7

2.1 Simulated 6P1/2 → 7S1/2 transition . . . . . . . . . . . . . . . . . . . . 142.2 Voigt, Lorentzian, Gaussian comparison . . . . . . . . . . . . . . . . . 162.3 Level IS and hyperfine anomaly diagram . . . . . . . . . . . . . . . . 19

3.1 Experimental setup diagram . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Littrow configuration ECDL . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Simulated error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Demonstration of laser lock quality . . . . . . . . . . . . . . . . . . . . 28

4.1 Simulated single-isotope spectrum . . . . . . . . . . . . . . . . . . . . 324.2 Dual-isotope Doppler shifted atoms . . . . . . . . . . . . . . . . . . . 334.3 Doppler shift in dual-isotope spectrum . . . . . . . . . . . . . . . . . 334.4 First-step transition, sum of Doppler shifts . . . . . . . . . . . . . . . 344.5 Simulated dual-isotope spectrum . . . . . . . . . . . . . . . . . . . . . 354.6 Sample Fabry-Perot transmission signal . . . . . . . . . . . . . . . . . 374.7 Linearisation polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 394.8 Sample single-isotope spectrum . . . . . . . . . . . . . . . . . . . . . . 404.9 Sample dual-isotope spectrum . . . . . . . . . . . . . . . . . . . . . . 414.10 Direct FSR measurement plot . . . . . . . . . . . . . . . . . . . . . . . 444.11 Sample HFS historgram . . . . . . . . . . . . . . . . . . . . . . . . . . 454.12 Fitted single-isotope data, co-propagating . . . . . . . . . . . . . . . . 474.13 Fitted single-isotope data, counter-propagating . . . . . . . . . . . . . 474.14 Scatter plot of 203Tl hyperfine splitting . . . . . . . . . . . . . . . . . . 484.15 Comparison of data subsets for 203Tl HFS . . . . . . . . . . . . . . . . 494.16 Comparison of data subsets for 205Tl HFS . . . . . . . . . . . . . . . . 494.17 Calibration factors with error bars plot . . . . . . . . . . . . . . . . . . 51

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4.18 Sample EOM sideband histograms . . . . . . . . . . . . . . . . . . . . 524.19 Fitted dual-isotope spectum . . . . . . . . . . . . . . . . . . . . . . . . 534.20 IS and Doppler shift histograms . . . . . . . . . . . . . . . . . . . . . . 544.21 Scatter plot of isotope shift . . . . . . . . . . . . . . . . . . . . . . . . . 544.22 Comparison of isotope shift data with scan direction . . . . . . . . . . 554.23 Scatter plot of IS v. HFS correlation . . . . . . . . . . . . . . . . . . . . 564.24 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Simulated 8P3/2 single-isotope spectrum . . . . . . . . . . . . . . . . . 605.2 Sample 8P3/2 fitted single-isotope spectrum . . . . . . . . . . . . . . . 605.3 Simulated 8P3/2 dual-isotope spectrum . . . . . . . . . . . . . . . . . . 615.4 Preliminary 8P3/2 dual-isotope data . . . . . . . . . . . . . . . . . . . 61

A.1 Picture of 672 nm Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2 Picture of 672 nm Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.3 Threshold Current Plot for 672 nm Laser . . . . . . . . . . . . . . . . . 77A.4 Threshold Current Plot for 655 nm Laser . . . . . . . . . . . . . . . . . 77

B.1 Locking scheme utilising a second thallium vapour cell. . . . . . . . . 79B.2 Simulated error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.3 Optimal lockpoint temperature . . . . . . . . . . . . . . . . . . . . . . 81B.4 Difference signal at high T/large α . . . . . . . . . . . . . . . . . . . . 81B.5 Peak separation at different lockpoints . . . . . . . . . . . . . . . . . . 82

C.1 Diagram of MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . 87

D.1 X2 for various assumed FSRs . . . . . . . . . . . . . . . . . . . . . . . 90D.2 Direct Measurement of FSR . . . . . . . . . . . . . . . . . . . . . . . . 90

List of Tables

1.1 Measurements of the thallium 7S1/2 HFS and IS through time. . . . . 41.2 History of Tl 7P1/2 HFS and IS . . . . . . . . . . . . . . . . . . . . . . . 41.3 History of Tl 8P1/2,3/2 HFS and IS . . . . . . . . . . . . . . . . . . . . . 4

2.1 Summary of broadening mechanisms . . . . . . . . . . . . . . . . . . 13

4.1 Parameterisation of the eight peaks in dual-isotope spectrum . . . . . 354.2 203Tl hyperfine splitting values . . . . . . . . . . . . . . . . . . . . . . 484.3 205Tl hyperfine splitting values . . . . . . . . . . . . . . . . . . . . . . 484.4 Calibration factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Weighted average results from dual-isotope spectra . . . . . . . . . . 534.6 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.7 History of Tl 8P1/2 HFS and IS . . . . . . . . . . . . . . . . . . . . . . . 57

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Chapter 1

Introduction

Atoms have long proven themselves to be eminently suitable testing grounds formodern physical theories, to shed light on the fundamental nature of the universe.But, in order to extract meaningful results from experiments exploiting atomic sys-tems it is indispensable to have atomic theory that is both accurate and precise.Despite the leaps and bounds that have been made in the century since the ven-erable plum-pudding model of the atom was in vogue, constructing accurate andprecise models of heavy atoms remains a very challenging task.

The aim of the Majumder lab is to conduct high-precision measurements ofatomic structure (hyperfine splittings, isotope shifts, and polarizabilities) such thatthey may act as tests of the approximation and calculation techniques used in state-of-the-art atomic theory. Our focus is on thallium and indium, both elements be-longing to Group IIIA. As relatively heavy atoms with three valence electrons, theystrike a balance between simple enough to be tractable and sufficiently compli-cated to be useful tests of atomic theory. Thallium in particular is also a desirablesubject because it has been historically popular in experiments studying the weakinteraction and parity nonconservation as well as atomic electric dipole moments.

1.1 Motivations

1.1.1 The Purpose of Precise Measurements

Probing Fundamental Physics

The Standard Model description of elementary particles and their interactions hasbeen repeatedly confirmed by the last forty or so years of high-energy particle col-lider experiments. The model itself nevertheless remains incomplete. Probing bothStandard Model physics and physics beyond the Standard Model through atomicsystems, being both low-energy and comparatively less demanding to constructand control, is an inviting alternative. Multiple experiments on atomic systemshave contributed to the determination of Standard Model parameters to degrees

1

of precision comparable to that achieved by high-energy particle collider experi-ments [6].

The approach typically taken by such experiments is to make a precise mea-surement of an observable dependent on some interesting fundamental interactionwithin the atomic system. However, to actually learn something useful, one mustbe able to distinguish between the effects attributable to the interesting fundamen-tal interaction and the effects that reflect the less interesting details of the atomicsystem. Consider an observable εw. We can write

εw = QwC(Z) (1.1)

whereQw is a quantification of the fundamental phenomenon of interest, andC(Z)is a description of the relationship between the fundamental phenomenon and theatomic system under study, linking a fundamental quantity to some atomic ob-servable. The uncertainty in any resulting value of Qw is then obviously limitednot only by the uncertainty in the measured observable εw but also by the uncer-tainty in the theoretical formulation of C(Z). That is, a useful measurement of QW

from an atomic system will require both high-precision experimental measurementand accurate theoretical calculations. Our current work and the subject of this the-sis is directed towards furthering the latter. Due to the properties C(Z) has beenshown to possess - in the weak interaction, for example, C(Z) scales as Z3 - it isgreatly advantageous to use heavy atoms in such atomic observations. Unfortu-nately, heavy atoms are also the atoms most difficult to accurately model throughatomic theory.

A Stringent Test of Atomic Theory

As one generally discovers in introductory quantum mechanics, it is only the light-est atom, the hydrogen atom, that has wavefunctions which are analytically solv-able. When confronted with heavy atoms, theorists must utilise various approx-imation techniques to find numerical solutions to Schrodinger’s equation. Theprimary collaborators to the Majumder lab over many years, the group of MarianaSafronova at the University of Delaware, is devoted to developing precise ab initiowavefunction calculations of heavy atomic systems. With their calculations, it isthen possible to predict the quantities characteristic of atomic structure: hyperfinesplittings, isotope shifts, polarizabilities etc. By experimentally determining thesevery quantities to a degree of precision greater than that achieved by theory, ourmeasurements serve as stringent tests of the results of theoretical wavefunctioncalculations. In this manner, we hope to push forward improved accuracy, whichcould then serve to reduce uncertainties in the quantity C(Z).

Furthermore, the range of measurements made by our group targets a varietyof theoretical predictions so as to provide complementary, rather than redundant,information. Hyperfine structure (as Chapter 2 will discuss) arises due to interac-tions between the electron and the nucleus and is thus a reflection of short-range

Figure 1.1: Partial energy level diagrams of thallium and indium. Not to scale.

wavefunction behaviour. In contrast, polarizabilities are due to perturbations ofthe electron wavefunction far away from the nucleus and are thus a reflection oflong-range wavefunction behaviour.

Ultimately, for it to be practical to test fundamental physics with heavy atomicsystems, the ‘ordinary’ quantum mechanics of the studied atom must first be suf-ficiently well understood.

1.1.2 Thallium and Indium

The basic structure of Group IIIA elements thallium and indium is well-understood.Both elements possess three valence electrons, two of which are paired in an s or-bital, hence their ns2np1 structure. Figure 1.1 shows several low-lying states ofthe valence p-electron of both elements. The parallel structure means that similaroptical systems may be used in experiments involving either element.

1.2 Previous Work

For the past two decades, a series of high-precision measurements relevant to theatomic structure of both thallium and indium has been performed in the Majumderlab using both vapour cell and atomic beam spectroscopy. These include the elec-tric quadruple amplitude in the thallium 6P1/2 → 6P3/2 transition and the hyper-fine structure, isotope shift, and Stark shift in the thallium 6P1/2 → 7S1/2 transition.Similar measurements have been conducted with indium, specifically of the hy-perfine structure of the indium 6P3/2 state, and the Stark shift in the 5P1/2 → 6S1/2

transition. However, the two previous measurements performed by the Majumder

lab most relevant to our current work are those of the hyperfine splitting (HFS) andthe isotope shifts (IS) of the thallium 7S1/2 [1], and 7P1/2 states [2].

In the late 1980s, a group from the Justus Liebig University Giessen in Hesse,Germany completed a series of higher-precision measurements of the hyperfinestructure and isotope shift of the n2PJ (n = 7−10) and n2S1/2 (n = 7, 8) in thallium-203 and thallium-205 [3]. Some of these measurements [7] were subsequently cor-rected by the same group in 1993 due to “calibration and linearization errors” [8].However, many of the values remain uncorrected in the literature, such as the7P1/2 HFS and IS measured in 2014 by the Majumder group [2]. Our HFS valueswere approximately 20 MHz larger than the preceding published values. This isan additional motivation for us to pursue the re-measurement of other hyperfinesplittings.

Tables 1.1 and 1.2 summarise the historical progression of the hyperfine struc-ture measurements of the thallium 7S1/2 and 7P1/2 states, respectively. Figure 1.2displays an energy level diagram for all the transitions discussed. Table 1.3 sum-marises the existing measurements for the 8P1/2,3/2 states that are the subject of ourcurrent work.

Year 203Tl HFS (MHz) 205Tl HFS (MHz) I6P1/2→7S1/2(MHz) Group

1990 12242.6± 1.6 12350.9± 3.9 U Giessen [7]1993 12181.6± 2.2 12297.2± 1.6 U Giessen [8]2000 12180.5± 1.8 12294.5± 1.5 1659.0± 0.6 Majumder [1]2012 12179.9± 0.7 12296.1± 0.7 1658.33± 0.70 Chen et al. [9]

Table 1.1: Measurements of the thallium 7S1/2 HFS and IS through time.

Year 203Tl HFS (MHz) 205Tl HFS (MHz) Level IS (MHz) Group1988 2134.6± 0.8 2155.5± 0.6 −134.8± 1.0 U Giessen [3]1993 −129.8± 4.7 U Giessen [8]2014 2153.3± 0.7 2173.3± 0.8 −125.4± 4.0 Majumder [2]2001 2193 (Theory) [10]

Table 1.2: Measurements of the thallium 7P1/2 HFS and IS through time.

Year 203Tl HFS (MHz) 205Tl HFS (MHz) Level IS (MHz) Group8P1/2 1988 781.7± 1.6 788.5± 0.9 −37.2± 1.2 U Giessen [3]8P3/2 1988 260.2± 0.9 260.4± 1.1 −61.8± 2.9 U Giessen [3]

1993 −56.8± 6.6 U Giessen [8]

Table 1.3: Measurements of the thallium 8P1/2,3/2 HFS and IS through time. The valueslisted here are the ones of interest in this thesis.

1.3 Current Work

This thesis reports on the measurement of the hyperfine splitting of the 8P1/2 statein both stable isotopes of thallium, as well as the isotope shift of that state us-ing absorption laser spectroscopy and a two-step excitation scheme. Two externalcavity diode lasers, a UV and a red laser, are used to drive the two steps of the6P1/2 → 7S1/2 → 8P1/2 transition. The UV laser is stabilised to the resonant fre-quency, while the red laser is scanned across the second-step transition. The lasersare overlapped and sent through a quartz cell of thallium, heated to ∼ 500 °Ctoproduce thallium vapour. The resulting signal is very small, and only detectablewith the aid of an optical chopper wheel and a lock-in amplifier.

Measuring Tl 8P1/2 hyperfine structure is a natural extension of the measure-ments already performed by the Majumder lab using this technique. The 6P1/2 →7S1/2 → 8P1/2 transition (shown in Figure 1.3) is analogous to the 6P1/2 → 7S1/2 →7P1/2 transition previously used to measure the 7P1/2 state in thallium [2, 4, 5].The two transitions feature the same first-step excitation from the 6P1/2(F = 1)hyperfine state to the 7S1/2(F ′ = 1) hyperfine state. Thus, by replacing the 1301nm (IR) laser used in the 7P1/2 measurement with a 672 nm (red) laser, we wereable to efficiently adapt the previous experimental setup for the current experi-ment. Part of this transition involved pursuing the ambition to build and use ourown diode lasers in this experiment - specifically the 672 nm laser required for the7S1/2 → 8P1/2 transition. After many trials, these efforts have ultimately provensuccessful.

Sufficient quanities of data have been collected and analysed to yield valuesfor the hyperfine splitting of the 8P1/2 state in both 203Tl and 205Tl, and the isotopeshift with precision at or below the level of 1 MHz including both statistical andsystematic errors. We report values of 780.5(5) MHz and 788.3(7) MHz for the 203Tland 205Tl hyperfine splittings, respectively, and a value of 450.3(1.0) MHz for the7S1/2 → 8P1/2 transition isotope shift.

The remainder of the thesis will proceed as follows: Chapter 2 outlines the con-cepts fundamental to absorption spectroscopy and the atomic theory that under-pins the hyperfine splitting and isotope shift. Chapter 3 describes the experimentalapparatus and the process of collecting the necessary absorption spectra. Chapter4 summarises the data analysis method and the results obtained in this experi-ment. Chapter 5 anticipates the work to be done immediately following this thesisto measure the 8P3/2 hyperfine splitting.

Figure 1.2: Thallium energy level diagram showing all of the transitions featured inTables 1.1 to 1.3. Not to scale.

Figure 1.3: Energy level diagram of 6P1/2 → 7S1/2 → 8P1/2 transition for both isotopes ofthallium showing two-step excitation from ground state, and transition isotope shift. Weelect to drive the first-step transition between the 6P1/2(F = 1) and 7S1/2(F

′ = 1) states.Not to scale.

Chapter 2

Laser Spectroscopy and AtomicStructure

2.1 Absorption Spectroscopy

Broadly speaking, our experiment seeks to reveal the structure of thallium by send-ing lasers through a vapour of hot atoms and measuring the transmitted output asa function of laser frequency. As the laser beam travels through the vapour, pho-tons with energies sufficiently close to the energy of an atomic transition will, withsome probability, be absorbed by atoms. Thus, when the frequency of the laser isclose to such resonant frequencies, the intensity of transmitted light will be lowerthan otherwise, resulting characteristic absorption dips in the observed signal. Bydetermining the centre of these peaks, we can locate energy levels of interest. Moreimportantly for this experiment, by determining the relative position of peaks, wecan extract both the hyperfine splitting and transition isotope shift from absorptionspectra.

For a laser beam of intensity I0, the transmitted intensity after interacting witha cloud of atoms with number density n, for a distance d is

It = I0e−σand (2.1)

where σa is the absorption cross section encapsulating the instrinsic ability of theatoms to absorb a photon. The variables in the exponent may be consolidated intoone paramter, the optical depth α.

It = I0e−α (2.2)

By defining a normalised absorption coefficient A(ω) that conveys the probabilityan atom will absorb a photon, and an optical depth on-resonance α0, the transmit-ted intensity may be more usefully written as

It = I0e−α0A(ω) (2.3)

9

Measuring It(ω) across an appropriate range of frequencies results in absorptionpeaks in the observed signal. By determining the centre of these peaks, we canlocate energy levels of interest. More importantly, by determining the relativeposition of peaks, we can extract both the hyperfine splitting and transition iso-tope shift from absorption spectra. The peaks found in these spectra have, due toa range of broadening mechanisms, some finite linewidth and some characteris-tic lineshape, around the true atomic resonance. The two most important to thecurrent experiment are lifetime broadening, due to the finite lifetime of an atomsexcited state, which produces a Lorentzian absorption profile, and Doppler broad-ening, due to the motion of the atoms relative to the laser beam with which theyinteract, which leads to a Gaussian absorption profile.

2.1.1 Broadening Mechanisms

Lifetime Broadening

The lifetime τ of an atoms excited state is the amount of time after which an excitedatom is expected to decay to a state of lower energy. The Heisenburg UncertaintyPrinciple in energy-time form reads

dE · dt ' ~ (2.4)

Accordingly, given that the lifetime of the excited state cannot be infinite, the en-ergy of the state correspondingly cannot be described by an infinitely narrow deltafunction at the transition resonance frequency. Rather, the absorption line mustoccupy some width around that resonance, wherein the value of that width is dic-tated by the uncertainty principle.

Letting the uncertainty in time dt ' τ , and using E = ~ ·ω, the so-called naturallinewidth of the transition is thus

Γτ =dE

~τ=dω

τ=

1

2πτ(2.5)

where Γτ is the full width at half maximum (FWHW) of the absorption featuresdue to lifetime broadening. The lifetime of the 7S1/2, 8P1/2, and 8P3/2 states inthallium are approximately 7.43 ns, 177.6 ns, and 123.5 ns respectively [11]. Usingthese values, the widths of those states due to lifetime broadening are:

Γ7S1/2= 21.4 MHz

Γ8P1/2= 0.896 MHz

Γ8P3/2= 1.29 MHz

However, if the 8P1/2,3/2 states are reached from the ground state via a two-stepexcitation, the linewidth of either transition as a whole must take into account theuncertainty in the 7S1/2 state as well. Let dE1 and dE2 be the uncertainties of the

first and second excited energy levels respectively. Then the total uncertainty dET ,and therefore the actual natural linewidth are:

dET =√dE2

1 + dE22 → Γτ =

√Γ2

1 + Γ22

For the two transitions under examination, the natural linewidths are thus:

6P1/2 → 7S1/2 → 8P1/2 : Γτ ' 21.4MHz6P1/2 → 7S1/2 → 8P3/2 : Γτ ' 21.4MHz

Evidently, the contribution from the first step of the transition dominates in bothcases.

Finally, it is to be noted that lifetime broadening is a homogenous broadeningmechanism. That is, lifetime broadening equally affects each atom in the vapourthat interacts with the probing laser. Consider the absorption line of one individualatom. If such a signal could be observed, its lineshape due to lifetime broadeningwould be Lorentzian and have the width predicted above. Consider now whatone generally observes: an absorption peak that is the product of the excitationof many atoms. The lifetime broadened shape of this signal is equally Lorentzian,and possesses the same FWHM predicted above. As will be seen in the followingsection, not all broadening mechanisms behave in this manner.

Doppler Broadening

Doppler broadening is an artifact of the range of velocities at which a cloud ofhot atoms move relative to the propagation of the laser beam used to prove theatoms. Atoms with a non-zero velocity component in the propagation direction ofthe laser perceive photons in the laser beam to not be at the output frequency ofthe laser, but at some Doppler-shifted frequency. In another manner of speaking,when these non-zero velocity class atoms encounter and absorb photons with theapparent correct transition energy, we measure absorption at frequencies shiftedfrom the resonance frequency by

df = f0(vatomc

) (2.6)

where f0 is the frequency of the laser. Now consider an ensemble of atoms. Theprobability distribution of the velocity of atomis in a vapour at some temperatureT is given by the Maxwell-Boltzmann distribution

P (v, T ) =

√( m

2πkT

)3

4πv2e−mv2

2kT (2.7)

From this, it can be seen that Doppler broadening will result in absorption fea-tures with an essentially Gaussian profile. The FWHM of the Maxwell-Boltzmanndistribution is

FWHM = 2

√2 ln 2

√kT

m(2.8)

The Doppler shift formula gives us the relation between atom velocity and laserfrequency. The FWHM of absorption peaks due to Doppler broadening is then:

ΓDoppler = 2√

2 ln 2

√kT

mc2f0 (2.9)

Where f0 is now the resonant frequency of the transition.In the first step transition from the ground state of thallium to the 7S1/2 energy

level, f0 ' c377.7 nm ' 793.7 THz. At 500 °C

ΓDoppler ' 1100 MHz

In the second step transition, where f0 ' c671.6 nm ' 336.4 THz,

ΓDoppler ' 600 MHz

In contrast to lifetime broadening, Doppler broadening is an inhomogenous broad-ening mechanism. That is, its effect on each atom is not equal, but rather depen-dant on some factor that distinguishes atom from atom; in this case, the velocity ofindividual atoms. Consider once again the absorption line of one individual atom.The shape of this line is unaffected by Doppler broadening and remains essentiallyLorentzian. However, an Doppler broadened absorption peak that reflects the ex-citation of many atoms has a Gaussian, not a Lorentzian, profile. This is becausethe Lorentzians corresponding to single atoms encounted by the probing laser willoccupy a range of frequencies centered around the true resonant frequency, as dic-tated by the velocity of the atoms and thus specified by the Maxwell-Boltzmandistribution.

Other Sources of Line Broadening

In addition to lifetime broadening and Doppler broadening, there are other broad-ening effects common to absorption spectroscopy that are worth a brief account-ing. Collisional or pressure broadening is one of these. It occurs when the densityof atomic vapour is high enough that The other three could not unreasonably begrouped together as effects dependent on the properties of the laser used. They arethe intrinsic linewidth of the laser(s), power or saturation broadening, and transit-time broadening. Although it is common to speak of the laser tuned to “somefrequency”, the output of a laser is never truly an infinitely narrow delta function,and contains some finite range of frequencies. The width of this range is the in-trinsic linewidth of the laser. Power broadening occurs when a transition is beingdriven with a laser at sufficiently high intensity such that the transition is saturatedand incident photons are more likely to result in stimulated emission, rather thanfurther population of the upper state of the transition. This effectively shortensthe lifetime of the excited state. Transit-time broadening becomes relevent whenthe interaction time of the laser with the average atom (as determined by the mean

thermal velocity of the atoms and the size of the laser beam) is short relative to thelifetimes of the excited levels, in which case, contrary to Section 2.1.1, the lifetimeof the excited states ceases to be the ‘intrinsic’ limiting variable in determining thewidth of the absorption line [12].

The effect of all of these broadening mechanisms is generally small in compar-ison to Doppler broadening and lifetime broadening in the regimes relevant to theexperiment. Table 2.1 lists typical values in order of decreasing magnitude, includ-ing the natural linewidths of our transitions calculated above for comparison:

Broadening Mechanism FWHM Lineshape TypeNatural Linewidth 21 MHz Lorentzian H

Doppler Broadening 600 MHz Gaussian IHResidual Doppler Broadening 20 MHz Gaussian IH

Power Broadening Lorentzian HIntrinsic Laser Linewidth 1 MHz [13] HTransit-Time Broadening 250 kHz [12] Gaussian IH

Collisional/Pressure Broadening 30 kHz [12] Lorentzian H

Table 2.1: Summary of all broadening mechanisms and properties mentioned relevant for6P1/2 → 7S1/2 → 8P1/2,3/2 transitions at representative experimental conditions. H and IHindicate homogenous and inhomogenous respectively.

Notice that in Table 2.1, an attempt has not been made to estimate the expectedcontribution of power broadening to linewidth. For a two-step excitation process,the degree to which absorption lines would be expected to be power broadenedis dependent not simply on the intensity of either laser, but the intensity of thesecond-step laser in the region where it is overlapped with the first-step laser.Given the difficulty of determining this, calculating the power broadening is animpractically tall order. However, power broadening for a two-level system, atleast, is not necessarily small. In fact, power broadening results in a fractional in-

crease in linewidth ΓPB

Γ0=√

1 + IIsat

where Isat = πhc3λ3τ

[14]. For example, if thefirst step 6P1/2(F = 1) → 7S1/2(F ′ = 1) transition was driven with a 1 mW laserwith a beam diameter of 2 mm 1, there would be a ∼ 27% increase in the observedlinewidth, or given Γτ from Section 2.1.1, an increase of ∼ 6 MHz.

1These are typical experimental conditions.

Two-Step Laser Spectroscopy and Doppler-free Spectra

From the previous discussion on the natural linewidth of a transition and theDoppler broadened linewidth, and their relative magnitudes, one would expecttypical absorption spectrum to be dominated by the Gaussian contribution fromDoppler broadening. However, this is not the case. It is part of the beauty ofthe two-step excitation scheme in our experiment that it produces spectra that arelargely free of Doppler broadening for second-step 7S1/2 → 8P1/2,3/2 transitions inwhich we are interested.

Figure 2.1: Simulated spectrum of the 6P1/2 → 7S1/2 transition showing the 1.6 GHz iso-tope shift as well as the 70-30 relative abundance of the 205Tl and 203Tl isotopes respec-tively.

When tuned and locked (See Section 3.3) to a single frequency, the first steplaser excites only atoms moving in relation to the propagation of the laser beamsuch that the atoms perceive the laser to be at or very close (within the homoge-nous linewidth) to the resonant frequency of the target transition. Put alternately,the first step laser serves to select only one velocity class of atoms with which topopulate the intermediate 7S1/2 state and be possibly excited by the second steplaser to the final 8P1/2,3/2 energy level, thereby eliminating the vast majority ofDoppler broadening, and producing spectra with mostly Lorentzian shapes, closeto natural linewidths. For example, if the first step laser is tuned to the transitionresonance, then the zero velocity class atoms - that is, the atoms moving only per-pendicular to the laser beam - will be selected, and there will be no Doppler shiftdue to the motion of the atoms. It is important to note that regardless of the spe-cific velocity class selected by the first step laser, the two-step excitation processwill result in the same largely Doppler-free spectra, and thus whether that laser istuned to the transition resonance or not, or how close it is tuned to the resonance,

has no impact on the expected linewidth or lineshape of the absorption peaks inour spectra. The fraction of excited atoms will be roughly,

ΓτΓDoppler

≈ 21 MHz1100 MHz

≈ 2%

Recall from Section 2.1.1 that the natural linewidths of the 8P1/2,3/2 absorption lineswere calculated to be ∼ 21 MHz. However, minimum observed linewidths aretypically approximately twice this value, about ∼ 45 MHz.This is at least in partdue to the fact that the two-step excitation process only produces largely Doppler-free spectra. Due to the non-zero divergence of the first-step laser beam, somedegree of residual Doppler broadening in the absorption peaks is to be expected.Imagine that the laser diverges by 1°. Using Equation 2.6, and a RMS velocity of∼ 300 m/s, we find that this amount of divergence in the UV laser would result in∼ 14 MHz contribution to the linewidth.

Consequently, the line shape of the spectra collected in this experiment is mostaccurately some convolution of the ‘intrinsic’ Lorentzian and the Gaussian pro-files. This is otherwise known as the Voigt profile. Figure 2.2 shows a comparisonof the Gaussian, Lorentzian, and Voigt distributions with normalised amplitudeand a common FWHM to reveal the distinctive differences between the three pro-files. Unfortunately, fitting data to the Voigt profile is a computationally intensiveprocess. From previous experience, we have found that painstakingly doing soyields neither improved quality of fits nor measurable change to peak locations[2]. Therefore, fitting our spectra to only the Lorentzian profile, which has a sim-ple analytic form, is generally sufficient. From the comparison of the observedlinewidth with the natural linewidth, we can set an upper bound on the residualDoppler broadening in Table 2.1 of ∼ 20 MHz.

Also due to the two-step transition, power broadening could occur in two dif-ferent ways, depending on whether it is the first-step laser, or the second-step laserthat is the source of the broadening. Power broadening due to the second-step(red) laser would result in a Lorentzian contribution to the linewidth, as one wouldotherwise expect from a transition in a two-level system. In contrast, if it is thefirst-step (in our case, UV) laser, that is the source of power broadening, the effectwould be to increase the range of velocity classes excited to the 7S1/2 state. Thus,power broadening from the UV laser would manifest itself as apparent ‘residual’Doppler-broadening with the associated Gaussian line profile. Given the calcu-lation in Section 2.1.1 of the power broadening in the first step transition, it is notunlikely that some of the residual Doppler broadening is due to power broadeningin the first-step transition.

Figure 2.2: Examples of Gaussian, Lorentzian, and Voigt absorption profiles with roughlyequal FWHM. Note that they diverge mainly at the tails.

2.2 Atomic Structure

The following section will give a brief overview of the hyperfine splitting, and thetwo additional measurements of interest furnished by the natural composition ofthallium: the hyperfine anomaly and the isotope shift.

2.2.1 Hyperfine Structure

In hydrogen atoms, fine structure is the result of both a relativistic correction andspin-orbit coupling: an interaction between the magnetic dipole moment of thevalence electron, and the magnetic field of the nucleus. In many-electron atoms,the latter effect is dominant. A further lifting in energy level degeneracy gener-ally labelled as hyperfine structure in turn arises from an interaction between themagnetic dipole moments of the electron and the magnetic dipole moment of thenucleus known as spin-spin coupling.

In the spin-spin interaction, the magnetic moment of the electron µe feels atorque in the presence of the magnetic field set up by the magnetic moment of thenucleus B. Therefore, the energy of this interaction is

H = −µe •B (2.10)

The term in the Hamiltonian associated with spin-spin coupling can thus be writ-ten as

Hhf = ahfI • J (2.11)

where ahf is the hyperfine coupling constant. The total angular momentum F ofthe atom is the sum of the nuclear spin I, and the total angular momentum J of theelectron.

F = I + J (2.12)

Taking the dot product of F with itself results in

(I • J) =1

2(F 2 − I2 − J2) (2.13)

If we consider Hhf as a perturbation to a zeroth order Hamiltonian H0 that alreadyencapsulates the central field, the electrostatic repulsion between electrons, andthe spin-orbit interaction, then the full Hamiltonian H would be

H = H0 +Hhf (2.14)

The first order correction to the nth energy level of our atom E1n is given by the

expectation value of the perturbation H0 in the unperturbed state ψ0n of our atom

[15]:E1n =< ψ0

n|Hso|ψ0n > (2.15)

such that the resulting correctionE1n will be the hyperfine correction to a fine struc-

ture energy level.

E1n = < ψ0

n|Hhf |ψ0n >

= ahf < ψ0n|(I • J)|ψ0

n >

E1n =

ahf2

< ψ0n|(F2 − I2 − J2)|ψ0

n > (2.16)

Applying the following,

J2ψ = ~2J(J + 1)ψ

I2ψ = ~2I(I + 1)ψ (2.17)F2ψ = ~2F (F + 1)ψ

one finds that,

E1n = ahf

~2

2(F (F + 1)− I(I + 1)− J(J + 1)) (2.18)

Unsurprisingly, the degeneracy in a given J level has been lifted. To see how ex-actly, consider the 8P1/2 state (i.e. J = 1

2) in thallium, where the nuclear spin I = 1

2.

F is evidently either 0 or 1. All together this yields,

E1n = ahf~2

{+1

4(F = 1)

−34

(F = 0).(2.19)

Furthermore, the energy difference between two consecutive hyperfine levels Fand F ′ = F + 1 for the case of J = 1

2andI = 1

2is

(E1n)F ′ − (E1

n)F = ∆EHFS = ahf (F + 1) (2.20)

Thus, not only is the amount by which the fine structure level is perturbed propor-tional to the constant ahf , in this specific case where the hyperfine splitting is thedifference between F = 0 and F ′ = 1 states, ahf is exactly the hyperfine splitting[16].

Alternately, by applying Equation 2.18 to the 8P3/2 state in thallium, it can beseen that

E1n = ahf~2

{+3

4(F = 2)

−54

(F = 1).(2.21)

The energy difference between these two hyperfine levels 8P3/2(F = 1) and 8P3/2(F =2) is, from Equation 2.20, equal to 2ahf .

2.2.2 Isotope Effects: Isotope Shift and Hyperfine Anomaly

Comparing the spectral lines of two different isotopes of the same element willlead one to notice that, though similar, the energy level structure of two atoms thatonly differ in the number of neutrons in their nuclei are not precisely the same.There are two observable effects. First, that the hyperfine splitting of any given finestructure level in one isotope is not equivalent to the same hyperfine splitting inanother isotope. This is known as the hyperfine anomaly. Second, that the distancein energy between two fine structure levels (having correctly averaged away thehyperfine splitting) in one isotope is also not equivalent to the distance between thesame two levels in another isotope. This is known as the transition isotope shift,from which the isotope shift of either one of the levels may be extracted. ConsultFigure 2.3 for a helpful illustration comparing a fine structure level in 203Tl and205Tl.

The isotope shift may in general be attributed to two distinct effects due to thepresence (or absence) of additional neutrons in the nucleus: a mass effect, and avolume effect. The mass effect is most significant when comparing isotopes of lightatoms, such as hydrogen and deuterium. In heavy atoms such as thallium, isotopeeffects can almost entirely be attributed to the volume effect.

Volume Effect

The volume effect is due to differences in the nuclear charge distribution betweenisotopes due to the presence - or lack thereof - of additional neutrons. In essense,real nuclei are not point charges but occupy real space, and different numbers ofneutrons in the nucleus of an atom changes both its size and shape. For exam-ple, the mean square charge radius of the 203Tl nucleus is larger than that of 205Tlnucleus. Valence electrons occupying both s-orbitals and p-orbitals, whose chargedensity is non-negligible in the region of the nucleus, contributes to the divergenceof the nuclear electric potential from the Coulomb potential of a point charge nu-cleus. This helps explain the reasons for which a205 > a203 as shown in Figure 2.3.A mathematical treatment of this isotope effect is beyond the scope of this thesis.

Figure 2.3: Energy level diagram of the 8P1/2 state to show the level isotope shift (the shiftin absence of hyperfine structure) and the difference in hyperfine splitting between 203Tland 205Tl. Not to scale.

The key point is that changes to both the size, and the shape of the nucleus fromthe number of neutrons present is significant to the isotope shift. For this reason itsmeasurement is very useful towards a better understanding of nuclear structure.

Chapter 3

Experimental Setup and Apparatus

Recall from Chapter 1 and Figure 1.3 that to examine the 8P1/2 state, we undergo atwo-step transition from the 6P1/2 ground state via the 7S1/2 intermediate state. Todo so and collect useful spectra such that we may illuminate the hyperfine struc-ture of thallium, there are broadly four steps: 1) frequency stabilisation of the 378nm laser, 2) incorporating requisite diagnostic tools into the optical system, 3) com-bining the two lasers in the experimental oven, 4) detecting the absorption signal.The first section of this chapter is an overview of the experimental layout and pro-cess. Sections 3.3 and 3.4 discuss steps 1 and 2 respectively. Section 3.2 is devotedto the external cavity diode lasers used in this experiment, including the 672 nmlaser built in our lab.

3.1 Experimental Layout

Figure 3.1 is a schematic of the experimental setup. Two lasers are required forthe two-step transition: a 378 nm laser and a 672 nm laser. The two are prepareddifferently, and then combined in an oven containing a quartz cell of thallium inits naturally occuring ratio of 70 % 205 Tl and 30 % 203 Tl, heated to ∼ 500 °C.

The UV laser is first passed through an AOM. The first-order beams are devotedto the locking setup (See Section 3.3), which stabilises the frequency of the UV laserto drive the 6P1/2(F = 1) → 7S1/2(F ′ = 1) transition. The undiffracted, unshifted,central beam proceeds to the rest of the experiment. It is passed through a 50/50beam splitter, and the output directed to enter the experiment oven in oppositedirections. We label these with respect to the propagation direction of the red laserthrough the oven as the co-propagating UV beam and the counter-propagating UVbeam. A shutter is placed in the path of each UV beam such that it is possible toblock and unblock each as desired. The UV beam is divided in this manner for tworeasons. First, it should make no difference to the measured hyperfine splittingwhether the two lasers were propagating in the same direction, or in opposingdirections. It would be sensible to verify this. Second, it is necessary for obtainingthe isotope shift from the dual-isotope spectra (See Section 4.1).

21

Figure 3.1: Diagram summarising the setup of the experiment. A diagram and descriptionof the full locking setup may be found in Appendix B.

Unlike the UV laser, the red laser is scanned up and down over the 7S1/2 →8P1/2 transition by triangular voltage signal produced by a function generator (SeeSection 3.2 for details). The red laser beam is split into two beams. One beam is sentto a confocal Fabry-Perot cavity, and an optical fibre that leads to a wavemeter 1 todetermine and monitor the wavelength of the red laser. The main beam proceedsthrough an electro-optic modulator (EOM), a series of collimation optics, then ontowards the experiment oven. The significance of both the Fabry-Perot cavity andthe EOM will be discussed in Section 3.4. With the help of two dichroic mirrors,the red beam, the co-propagating UV beam, and the counter-propagatin UV beamare physically overlapped and sent through the vapour cell in the oven in theirrespective directions. The importance of overlapping the red beam with both UVbeams within the vapour cell cannot be overemphasised. An atom excited by theUV laser to the 7S1/2 state will decay after moving mere microns given its RMSvelocity. Only in a place where an atom can encounter both lasers essentially atonce is it possible for the second-step transition to occur. Thus, to maximise thesignal-to-noise ratio, good alignment is indispensible.

After passing through the vapour cell and out of the oven, the red beam isisolated from the UV beams by a dichroic mirror and directed to a photodetector.

1Generously shared with us by Prof. Charlie Doret. The wavemeter is incidently located in hislaboratory next door. Thus, a portion of the optical fibre lives in the ceiling.

The Problem of Signal Detection

If by locking the UV laser to a specific frequency we have isolated one velocity classof atoms, it follows automatically that we have also simultaneously consigned thevast majority of atoms in our vapour to general irrelevance to absorption in thesecond-step transition. The magnitude of this reduction is roughly the naturallinewidth to the Doppler broadened width of the 6P1/2 → 7P1/2 transition. As wascalculated in Chapter 2,

ΓDoppler-free

ΓDoppler≈ 21 MHz

1100 MHz≈ 2%

The result is that the signal we strive to detect is very small in comparison to thenoise inherent in the experimental system. To directly detect this signal is not pos-sible. Instead, we call upon the power of the lock-in amplifier. An optical chopperwheel is configured to block and unblock the UV beam at ∼ 1000 Hz., causing thered laser absoprtion spectrum to come in and out of existence at that particular fre-quency. The chopper wheel reference signal is sent to the lock-in amplifier, whichpicks out the part of the transmitted red beam signal oscillating at the reference fre-quency. Any part of the signal that is not oscillating at the reference frequency willbe averaged away. This detection is what turns our absorption spectra that wouldlogically be small dips in some constant background (as, indeed, we illustrate spec-tra from the first-step transition to be in Figure 2.1), into well-resolved peaks on azero background. See Figure 4.8 for an example. Data was collected by sendingthe relevant signals (voltage ramp, lock-in output signal, Fabry-Perot transmissionsignal) via a data acquisition board (DAQ) to the data acquisition computer. Thesethree signals are then recorded simultaneously. All data was acquired with the aidof LabView code originally written by David Kealhofer ‘14 for the measurement ofthe 7P1/2 state in thallium, with some minor improvements for speed 2. The signalsare sampled at a rate determined by parameters input into the programme.

Collecting Data

Single-isotope and dual-isotope spectra are collected slightly differently. For single-isotope spectra, we only wish to see the spectrum due to either the co-propagatingUV beam or the counter-propagating UV beam at any one time. Thus, the shut-ters shown in Figure 3.1 are programmed to alternate between open and closedduring the course of a data run. This counter-co-counter-co pattern is used suchthat spectra originating from both UV propagation directions are recorded equallyacross the duration of a run. Dual-isotope spectra require both co-propagatingand counter-propagating UV beams to be on at once, so during their collection,both shutters are kept open at all times. In addition, during the acquisition of

2Such that spectra is recorded for every complete (up and down) sweep of the red laser, and notevery other sweep, taking twice as long as necessary.

single-isotope data, an electro-optic modulator (EOM) is turned on (See Section3.4), while for dual-isotope data, it is turned off.

In its essentials, the setup of the experiment described by this thesis is iden-tical to that used in the measurement of the 7P1/2 state in thallium, the subjectof the theses of David Kealhofer ‘13, and Gabrielle Vukasin ‘15 [2, 4, 5]. The keyalteration to the experimental setup necessary to examine the 8P1/2 instead wasto replace the 1301 nm (IR) laser with the 672 nm (red) laser appropriate to the7S1/2 → 8P1/2 transition. This 672 nm laser is the first laser to be successfully builtin our lab and used in an experiment, and is the culmination of a series of effortsin the Majumder lab to test the feasibility of building our own exernal cavity diodelasers. The final design is based on that presented by Arnold, Wilson, and Boshier[17], with modifications by Hawthorn, Weber, and Scholten [18], built from modi-fied CAD drawings originally created by Cole Meisenhelder ‘15 [19]. We requireda 672 nm laser that could scan ∼ 5 GHz. This goal was achieved with the addi-tion of a feed forward mechanism to the laser. By feeding foward, we mean thatthe voltage ramp used to scan the laser (multiplied by some suitable gain factor) issent to the laser diode controller to simultaneously modulate the operating currentof the laser diode. This substantially extends the continuous tuning range of thelaser. The UV laser used in this setup is commercial, manufactured by the Germancompany Sacher Lasertechnik.

3.2 External Cavity Diode Lasers

As the name may suggest, an external cavity diode laser (ECDL) is constructedby creating an additional external optical cavity around the internal cavity of alaser diode. The two lasers used in this experiment are both ECDLs in the Littrowconfiguration. In this arrangement, the light emitted by the laser diode is inter-cepted by a diffraction grating. In combination with the reflective back face of thediode, the grating establishes our external cavity. The grating is positioned suchthat the first-order diffracted beam is sent straight back into the laser diode, induc-ing stimulated emission in the diode gain medium at the specific wavelength ofthat first-order beam. The resulting single frequency output exits the laser in thedirection of zero-order difffraction from the grating. Figure 3.2 shows a diagramof the essential components arranged as described.

The resulting system is simple, but simply effective. In particular, two crucialimprovements have been made to the diode laser. First, the feedback mechanismestablished by the grating selects for a single frequency output of the many sup-plied by the laser diode. Second, by controlling the angular position of the gratingthe frequency output is also be controlled. As long as the diffraction grating maybe moved continuously, and the alignment of the optical components are such thata single longitudinal mode is maintained in the cavity, the laser is capable of oper-ating over some continuous frequency range.

Figure 3.2: External cavity diode laser in the Littrow configuration. The additional ofa correction mirror placed parallel to the diffraction grating is optional, but helpful, asit converts any angular deflection of the output beam due to a change in angle of thediffraction grating relative to the laser diode into a horizontal translation that does notincrease in magnitude with distance travelled.

In the usual manner, a diffraction grating hit with light of wavelength λ at θi,produces diffracted light at angles θn according to the relationship

nλ = d(sin θi + sin θn) (3.1)

where d is the distance between lines on the diffraction grating, and n is theorder of diffraction.

In the Littrow configuration, the first-order diffraction is sent back into the laserdiode, allowing the grating equation to be simplified to

nλ = 2d(sin θ) (3.2)

To summarise this equation: changing the angle at which the diffration gratingsits relative to the laser diode determines the wavelength of the first-order diffrac-tion, and thereby - given sufficiently correct alignment - the wavelength of theoutput of the laser itself.

In practice, a piezoelectric actuator (PZT) is installed behind the diffractiongrating. Varying the voltage applied to the PZT causes the small device to expandand contract, moving the diffraction grating the miniscule amount (on the order ofmicrons) necessary to change the selected frequency. By feeding a linear voltagesignal to the PZT, it is possible to scan the laser up and down a frequency rangedictated by some combination of the angle of the diffraction gravity, the length ofthe cavity, the temperature of the system, the current supplied to the diode, and

the gain curve of the diode. To see the complete single-isotope spectrum from the6P1/2 → 7S1/2 → 8P1/2 transition (See Figure 4.1 or 4.8), it is necessary to scan aminimum of ∼ 5 GHz, preferably further. Due to the properties of the PZT, thefrequency response of the laser to a linear voltage ramp is non-linear. This is oneof the challenges of extracting information from our spectra and will be addressedin Chapter 4.

3.3 Frequency Stabilisation of 378 nm Laser

To observe the 7S1/2 → 8P1/2 transition, the UV laser driving the 6P1/2 → 7S1/2

transition is kept stabilised to the correct resonance frequency in a process we referto as locking the laser. There are two reasons why this stabilisation is necessary.On one hand this is to ensure that the intermediate state is populated. On the otherhand, the degree to which the frequency of the first-step laser drifts over the timeit takes to record one spectrum (typically 10 seconds) directly contributes to un-certainty in the observed frequency differences of the second transition. Recallingthe discussion in Section 2.1.1 on Doppler broadening, the specific frequency of theUV laser determines the velocity class of atoms available for further excitation inthe intermediate state. The velocity class selected in turn determines the apparentresonant frequency of the various hyperfine transitions. As long as the velocityclass remains constant, which velocity class has been selected does not concern us.However, if this changes in the course of a frequency scan, the observed frequencydifference between the two hyperfine transitions 7S1/2(F ′ = 1) → 8P1/2(F ′′ = 0)and 7S1/2(F ′ = 1)→ 8P1/2(F ′′ = 1) will of course be affected. Drift in the UV laseris therefore a potential source of systematic error we endeavour to limit by lockingthe laser.

The technique used to lock the UV laser is a familiar one for the Majumder lab,last used in the measurement of the Tl 7P1/2 state [2, 4, 5]. In brief, it employs a sec-ond Tl vapour cell in a smaller oven, an optical system to generate an error signalshown in Figure B.2, and a proportional-integral-derivative (PID) controller. Evi-dently, there are three approximately linear sections on the error signal to which itis possible to lock, producing a 203Tl lockpoint, 205Tl lockpoint, and a dual-isotopelockpoint at which substantially Doppler shifted atoms of both isotopes are si-multaneously excited. We use the first two lockpoints to collect the single-isotopespectra from which the hyperfine splitting of each isotope is extracted. The thirdlockpoint allows for the collection of dual-isotope spectra necessary to obtain theisotope shift. There is a final advantage to locking the laser in this manner. Bychanging the lockpoint slightly, it is simple to adjust the exact frequency of the UVlaser sent to the experiment oven. We find that this ability is crucial in achievingdual-isotope spectra with eight well-resolved peaks (See Section 4.1.2).

Figure 3.3: Simulated difference signal for UV laser locking scheme showing error signalwith three quasi-linear slopes and corresponding lockpoints.

Quality of Lock

Over long time scales, the above frequency stabilisation mechanism eliminateslong-term drifts. Figure 3.4 shows the dramatically different behaviour of the laserwhen it is locked compared to when it is unlocked. Although the laser frequency isnow confined to a small region around the lockpoint, that region is still finite. Twopieces of information are necessary to determine the size of this residual drift: theRMS value of the observed voltage fluctuations in the error signal, and the slopeof the error signal at the relevent lockpoint for a voltage to frequency calibrationfactor. The latter is obtained from the comparing the known frequency separationof the extrema of the error signal in Figure B.2 to the observed slope in volts/sec-ond while the UV laser is scanned. Typical values for the slope range from 1.5MHz/mV to 2.5 MHz/mV. Given that the fluctuations of the error signal is gener-ally fractions of a mV, the RMS frequency fluctuations of the stabilised laser is wellbelow 1 MHz.

3.4 Diagnostic Tools

With the help of our chopper wheel and lock-in amplifier, spectra containing theinformation we desire (the 8P1/2 hyperfine splitting in each isotope, and the iso-tope shift) have been collected. However, our spectra are inconveniently collectedas voltage over time. Extracting meaningful information on the hyperfine splittingand isotope shift from them requires that we be able to accomplish two tasks: tomap a frequency axis onto the time axis of the spectra, and to calibrate this fre-quency axis precisely. How to do so will be defered to Chapter ??. To make it

Figure 3.4: Frequency drift of laser when it is unlocked in contrast to when it is locked.Voltage to frequency axis determined by using known loc

possible to do so, we require two diagnostic tools mentioned in the Section 3.1overview of the experimental setup and pictured in Figure 3.1: a confocal Fabry-Perot cavity and an Electro-optic Modulator (EOM).

Fabry-Perot Interferometer

We use a Fabry-Perot interferometer to help us accurately map a frequency axisonto the time axis of our raw spectra. A confocal Fabry-Perot cavity consists oftwo spherical mirrors separated by the radius of curvature R of the mirrors. In thisconfiguration, the free spectral range (FSR) of the cavity is given by c

4R, where c

is the speed of light. The red laser aligned through the cavity, then scanned overa range of frequencies, would yield periodic peaks of transmitted intensity. Infrequency space, these peaks are all equally separated by the FSR, because it is atthose frequencies that there is constructive interference inside the cavity.

The transmission T of a Fabry-Perot cavity is the well-known ‘Airy’ functiongiven by

T =1

1 + F sin2( δ2)

(3.3)

where δ is the phase gained by one round trip through the cavity, and F isthe coefficient of finesse. F is defined in terms of R, the reflectance of the cavitymirrors, as

F =4R

(1−R)2(3.4)

The finesse f of the cavity is defined as the free spectral range divided by thefull-width-half-max of the Fabry-Perot transmission peaks [20].

f =∆νFSRFWHM

≈ π√F

2=

πR12

1−R(3.5)

For the Fabry-Perot cavity used in this experiment, we have found that thecoefficent of finesse F ≈ 50, implying that the finesse f is ∼ 12. By recording thetransmission of the Fabry-Perot cavity simultaneous with the hyperfine spectra,one has essentially built a ruler into each spectrum. The ruler ends at the end ofthe spectrum, so it is not possible to know the absolute transition frequencies, butthe frequency difference between hyperfine transitions can be determined by thenumber of Fabry-Perot peaks spanned.

In the case of the cavity in this experiment,R is nominally 206.6 mm. Therefore,the FSR is approximately 363 MHz.

Electro-optic Modulator (EOM)

Yet it is only possible to say that the free spectral range is approximately 363 MHz.As we will discover in Chapter 4, to accurately calibrate the frequency axis of ourspectra, it is indispensible to know the free spectral range of the Fabry-Perot cavitymore precisely. If, for example, the cavity mirrors are not exactly separated by R(which they almost certainly are not) or the focal length of the mirrors is not exactlythe value published by its manufactureres, the FSR will diverge from the calculatedvalue. Furthermore, although the cavity is kept thermally isolated, over the courseof months, it is still possible for the cavity to drift thermally, changing the cavitylength and thus the FSR. The electro-optic modulator (EOM) exists in the opticalsystem to allow us to determine the degree to which the assumed value for the freespectral range diverges from the true free spectral range. It consists of a crystalplaced in a resonant microwave cavity. When a microwave signal at the correctresonant frequency ω is used to drive the EOM, the laser passing through gains asinusoidally modulating phase, that, at sufficiently small modulation intensity canbe approximated as a pair of sidebands±ω away from the unmodulated frequency[4]. These are what we call first-order sidebands.

Our EOM is resonant at 1000 MHz. Thus, when the EOM is turned on, eachpeak corresponding to a hyperfine transition in our spectrum gains just two addi-tional peaks that are separated from the transition peak by ±1000 MHz. Note thatsecond-order sidebands have also been visible in some of our data as evidencedby Figure 4.8. By combining these two facts: each Fabry-Perot peak is separatedby the FSR, and each sideband is separated from the central peak by 1000 MHz, itis simple to determine the correct FSR. The process by which this is done will bedescribed in detail in Chapter 4.

Chapter 4

Data Collection, Analysis Methods,and Results

4.1 Data Collection

To measure the hyperfine splitting as well as the isotope shift of the 8P1/2 state, itis necessary to collect two different types of spectra: ‘single-isotope spectra’ and‘dual-isotope spectra’. In the first type, only the atoms of one isotope are excitedat a time, while in the latter atoms of both isotopes are invited to participate in thetransition. Below, we examine the expected form of both types of spectra for thetransition of interest, drawing upon previous experimental values for the hyper-fine splitting and isotope shift (See Section 1.2) and the aspects of atomic structurediscussed in Chapter 2. We also define all the peaks in the spectrum in preparationfor our discussion of the data analysis process in Section 4.2.

4.1.1 Single-Isotope Spectra and the Hyperfine Splitting

Single-isotope spectra from 203Tl and 205Tl are qualitatively identical. Figure 4.1 isa simulation of the expectd spectrum. The six peaks in the figure are numbered 1-6in order of increasing frequency. Peaks labelled 3 and 4 represent the 7S1/2(F ′ =1) → 8P1/2(F ′′ = 0) and 7S1/2(F ′ = 1) → 8P1/2(F ′′ = 1) transitions respectively.Peaks 1 and 5 are the lower and upper first-order sidebands of peak 3, the (F = 0)peak, which are present due to the resonant EOM in the experimental setup, asdiscussed in Section 3.4. Peaks 2 and 6 are the analogous first-order sidebandsbelonging to peak 4, the (F = 1) peak.

We denote the central frequency of each peak by νi, where i = 1 . . . 6 is the num-bering of the peak. The frequency separation between two peaks is denoted as δνijwhere i and j are the numbers of the two peaks involved. The hyperfine splittingcan be extracted from in a relatively straightforward manner. In this spectrum, itis simply the position of peak 3 subtracted from the position of peak 4, or δν43. Thesplitting can also be extracted from a multitude of other peak combinations, either

31

directly (δν65) or indirectly due to the known EOM sideband separation of 1000MHZ. The hyperfine splitting from the sideband separation −δν32 is of particularinterest because peaks 2 and 3 are only separated by ∼ 200 MHz, versus the ∼ 800MHz separation of peaks 3 and 4, meaning that δν32 is less sensitive than δν43 tothe effects of residual nonlinearity in the frequency axis (Refer to Section 4.2 forfurther discussion on linearisation).

Figure 4.1: Single isotope simulated spectrum for the 6P1/2 → 7S1/2 → 8P1/2 transi-tion. Peaks 3 and 4 are the true hyperfine peaks, corresponding to the 7S1/2(F

′ = 1) →8P1/2(F

′′ = 0), and 7S1/2(F′ = 1) → 8P1/2(F

′′ = 1) transitions respectively. Peaks 1, 2, 5,and 6 are all separated from one of the hyperfine peaks by exactly 1000 MHz and exist forcalibration reasons (See Section 3.4 or Section 4.2.3).

4.1.2 Dual-Isotope Spectra and the Isotope Shift

The isotope shift is the shift in energy levels in the absence of hyperfine structure.To obtain the isotope shift, we must be able to observe the transition of interest inboth isotopes at the same time. Thus, there is a need to collect dual-isotope spectraon top of the single-isotope spectra discussed in Section 4.1.1. Unfortunately, toexcite atoms from both thallium isotopes with a single laser, at the same time, onemust excite atoms that are severely Doppler shifted. As such, dual-isotope spectraoffer certain complexities that do not arise in the case of single-isotope spectra.Examining the first-step transition absorption signal in Figure B.2 shows that atthe dual-isotope lockpoint, excited 205Tl atoms must have a component of velocityin the propogation direction of the UV laser, and excited 203Tl atoms must have acomponent of velocity opposite to the propogation direction of the UV laser.

As a result, in the dual-isotope spectrum, relative to the positions of zero-velocity class atoms, the set of two peaks corresponding to each isotope will have

Figure 4.2: Illustration of the movement of atoms of both isotopes relative to the two laserswhen the UV laser is locked to the dual-isotope lockpoint. Black arrows indicate the ve-locity component of atoms along the axis of laser propagation.

either both shifted away from each other (co-propagating) as in Figure 4.3 or to-wards each other (counter-propagating), as in Figure 4.3b, in the dual-isotope spec-trum. However, because the co-propagating and counter-propagating UV beamsoriginate from the same locked laser, regardless of the direction of the observedshift in the red spectrum, the magnitude of the shift will be the same as long asone believes that the two UV beams are truly anti-parallel. A simple average ofcorresponding co/counter-propagating peaks is then sufficient to dispose of theDoppler shift in our spectra.

Figure 4.3: a) Four co-propagating peaks ofthe eight peak dual-isotope spectrum,and b) their counter-propagating counter-parts.Observe the effect of the Doppler shiftedatoms on the spectrum. In the co-propagating spectrum the two sets of hyper-fine peaks are shifted away from the centre,while in the counter-propagating spectrumthe hyperfine peaks are shifted towards thecenter, as indicated by the black arrows.

Furthermore, regardless of what the magnitude of the red and blue Dopplershifts are - this is of course dictated by the exact position of the dual-isotope lock-point along the relevant section of the locking error signal (Refer to Figure B.2) -their sum will always equal the frequency difference between the 203Tl 6P1/2(F =1) → 7S1/2(F ′ = 1) and 205Tl 6P1/2(F = 1) → 7S1/2(F ′ = 1) transitions, as ob-servable in the 6P1/2 → 7S1/2 transition absorption signal: the familiar 1636 MHz[2]. Figure 4.4 attempts to illustrate this relationship. Notice that this is not quitethe transition isotope shift of the 6P1/2 → 7S1/2 transition, which, consulting thehandy Table 1.1, was previously measured by our group to be 1659 MHz [1]. Thisis because some of the contribution to the observed frequency difference betweenthe two isotopes in the transition is due to hyperfine structure differences.

Figure 4.4: The magnitudes of the red and blue Doppler shifts of the excited 203Tl atomsand 205Tl atoms respectively at the dual-isotope lockpoint will always sum to the samewell-known number: 1636 MHz [2]

The dual-isotope spectrum should have eight peaks, as shown in the simu-lated spectrum in Figure 4.5. We designate the peaks A to H in order of increasingfrequency. In contrast to the single-isotope spectrum, each peak here is a ‘true’hyperfine peak. There are no EOM sidebands, as adding them would result in anunresolveable mass of twenty-four peaks. Each of the eight peaks can be distin-guished from the other seven by a unique combination of Tl isotope, hyperfinetransition, and UV beam propagation direction. For example, peak D is due to the7S1/2(F ′ = 1)→ 8P1/2(F ′′ = 0) transition in 205Tl from the counter-propagating UVbeam.

Each peak can additionally be parameterised in terms of H , the 7S1/2 and 8P1/2

hyperfine splittings, I7S→8P , the transition isotope shift, and ∆f(2ndstep), the redDoppler shift for each isotope, relative to f0, the 205Tl 7S1/2 → 8P1/2 transitionfrequency in the absence of hyperfine structure. These parameterisations and theircorresponding identifications are listed in Table ??. Following the convention setabove in Section 4.1.1 in discussing the single-isotope spectrum, we denote the cen-

tral frequency of each peak as να where α = A . . . H are the peak labels in Figure4.5. The frequency separation between two peaks is denoted as δναβ where α andβ are the letters of the two peaks involved.

Figure 4.5: Dual-isotope simulated spectrum for the 6P1/2 → 7S1/2 → 8P1/2 transition.The eight peaks are designated A to H in order of increasing frequency. The orange lettersand purple letters indicate a peak from the counter-propagating, and co-propagating UVbeams respectively.

Dual Isotope Peaks Direction Isotope HF LevelνA = f0 − 1

4H7S,205 − 3

4H8P,205 −∆f205(2ndstep) CO 205 F ′′ = 0

νB = f0 − 14H7S,203 − 3

4H8P,203 −∆f203(2ndstep) + I7S→8P CTR 203 F ′′ = 0

νC = f0 − 14H7S,205 + 1

4H8P,205 −∆f205(2ndstep) CO 205 F ′′ = 1

νD = f0 − 14H7S,205 − 3

4H8P,205 + ∆f205(2ndstep) CTR 205 F ′′ = 0

νE = f0 − 14H7S,203 + 1

4H8P,203 −∆f203(2ndstep) + I7S→8P CTR 203 F ′′ = 1

νF = f0 − 14H7S,203 − 3

4H8P,203 + ∆f203(2ndstep) + I7S→8P CO 203 F ′′ = 0

νG = f0 − 14H7S,205 + 1

4H8P,205 + ∆f205(2ndstep) CTR 205 F ′′ = 1

νH = f0 − 14H7S,203 + 1

4H8P,203 + ∆f203(2ndstep) + I7S→8P CO 203 F ′′ = 1

Table 4.1: Parameterisation of the eight peaks in dual-isotope spectrum

From the parameterisation in Table 4.1, the isotope shift is

I7S→7P =3|νG+νC

2− νH+νE

2|+ |νD+νA

2− νF +νB

2|

4+H7S,203 −H7S,205

4(4.1)

where H7S,203 and H7S,205 are the hyperfine splittings of the 7S1/2 state in 203Tland 203Tl respectively, as previously measured [1].

Achieving eight well-resolved peaks in the dual-isotope spectrum requires theexperimental system to be somewhat more subtly adjusted than generally neces-sary for the single-isotope spectrum. In addition to the necessity of aligning threebeams such that they are maximally physically overlapped, from a combination ofthe 70-30 relative abundance of the two isotopes, and the larger transition prob-ability to the F ′′ = 1 triplet state, the amplitudes of the eight peaks vary widely.Peak G, for example, has unerringly had the largest relative amplitude in the spec-trum. Furthermore, in contrast to our usual indifference to where exactly the UVlaser is locked (See Section 3.3), we have found that the relative positions of theeight peaks in the dual-isotope spectrum is a non-negligible function of the chosenposition of the dual isotope lockpoint along the relevant section of the error sig-nal (See Figure B.2). That is, depending on the exact frequency at which the UVlaser is locked, the positions of the eight peaks shift sufficiently for the order of thepeaks to change. This is important to note for two reasons. First, we have dis-covered that adjusting the lockpoint to effectively shift the eight peaks around isindispensible to producing spectra with eight well-resolved peaks (See AppendixA for details). Second, the simulated spectrum in Figure 4.5 has been adjustedsuch that the order of the peaks correspond to the actual data that has been col-lected in the course of this experiment. It is, at least theoretically, by no means theonly order in which it is possible to see eight resolved peaks. Therefore, while thelabelling of A to H by increasing frequency to the corresponding identification ofeach peak in Table 4.1 is certainly correct for all the spectra we have collected, itis not necessarily transferable without alteration to any dual isotope spectrum ofthe same 6P1/2 → 7S1/2 → 8P1/2 transition, even following the same experimentalsetup described in Chapter 3.

4.1.3 Summary of Collected Spectra

Using the experimental apparatus and process described in Chapter 3, we recordeda total of approximately 2250 single-isotope scans, half 205Tl and half 203Tl, takenon 4 days interspersed between December 2015, and Mar 2016, and approximately2000 dual-isotope scans, on 8 days interspersed over the same time period. Eachset of data consisted of 60-200 scans, between which we re-optimised the alignmentof the system, varied the power of both lasers, the frequency of the scan, and thetemperature of the experimental oven. Experimental conditions are varied to checkfor consistency, and thus access accuracy in our measurements. Many scans aretaken to increase precision. After discarding 4 days of dual-isotope data, and a fewsets of single-isotope data, mostly due to Fabry-Perot instability or misalignment,we were left with approximately 1000 scans of each type of spectra, 203 single-isotope, 205 single-isotope, and dual-isotope. These we analysed as according tothe process laid out in Section 4.2 below.

4.2 Analysis Methods

Analysis of each spectrum collected, single-isotope or dual-isotope, consists in es-sentials of these three steps: 1) constructing a linear frequency axis for the spec-trum, 2) extracting the measurement(s) of interest, 3) calibration of the frequencyscale, 4) determine via statistical methods the average value of the measurement(s)of interest. This is the flow followed by the MATLAB analysis code adapted fromprevious experiments. This chapter will lay out the general methodology em-ployed.

4.2.1 Frequency Mapping and Linearisation

As discussed in Section 3.4, each hyperfine spectrum has an associated Fabry-Perotscan that allows a frequency axis to be mapped onto the raw time/point numberaxis. However, one subtlety of the process was not mentioned. Namely, as men-tioned in Section 3.2, the frequency response of an ECDL is not linear with voltage,and thus also not linear with time (or equivalently, point number) with the appli-cation of a linear voltage ramp. Examining any given Fabry-Perot scan collectedfully demonstrates this point. We expect the peaks of a Fabry-Perot transmissionsignal to be equally spaced in frequency, but as Figure 4.6 demonstrates, this isclearly not the case for our Fabry-Perot signals.

Figure 4.6: Sample Fabry-Perot signal. On the axis pictured, the Fabry-Perot peaks appearto be eveningly spaced, as one would expect them to be on a true frequency axis, but theyare definitively not! This uneven spacing is one indication of the nonlinearity buried inour spectra.

Therefore, in constructing a frequency axis for each spectrum, it is necessaryto linearize the scan by approximately capturing the nonlinear behaviour with a

higher-order polynomial. This can be achieved through one of two fitting meth-ods. The first method is to fit the Fabry-Perot signal to an adapted Airy functionallowing for the nonlinearity of the horizontal axis. The second method locates thepeaks of the Fabry-Perot signal by fitting to a sum of Lorentzians, and then explic-itly fits those points to a polynomial. We find that the Airy fitting method is lesscomputationally intensive, and thus, practically speaking, preferred. However, theLorentzian fitting method provides a better intuitive understanding of the lineari-sation process, so it will be discussed first. In theory, both fitting methods shouldyield the same linearisation polynomial.

Consider some polynomial p(j) = a0 + a1j + a2j2 + a3j

3 . . . to the desired nthorder. Such that the magnitude of the coefficients a0, a1, a2 etc. illuminates thesignificance of each corresponding term in the polynomial in capturing the non-linearity of the scan, before applying either fitting method it is a good idea to firstnormalise the horizontal axis of the Fabry-Perot signal. If the original point num-bers are integers j, we redefine an x-axis with domain −1 ≤ x ≤ 1 by scaling eachvalue j by the total number of data points N such that

xj =j − N

2N2

(4.2)

Now, |(xj)k| ≤ 1 for any k, and the relative magnitude of the coefficients out-put by the fitting programme is conveniently reflective of the significance of theirassociated terms.

Sum of Lorentzians

Proceeding with the sum of Lorentzians method, fit the Fabry-Perot signal withn peaks to a sum of n Lorentzians to find the central position of each peak. Infrequency space, each consecutive value should be separated by one free spectralrange. Labelling the first position one FSR, associate each subsequent peak posi-tion with the next integer multiples of the FSR, and fit the resulting points with anappropriate order polynomial. A plot of the result is featured in Figure 4.7.

Airy Function

Proceeding with the Airy method, two adjustments need to be made to the generalAiry function. Reproducing Equation 3.3:

T =1

1 + F sin2( δ2)

where δ is the phase gained by one round trip through the cavity, and F isthe coefficient of finesse. The finesse f of the cavity is defined as the free spectralrange divided by the full-width-half-max of the Fabry-Perot transmission peaksand is related to the coefficient F by Equation 3.5

Figure 4.7: Left: Plot of integer multiples of the FSR v. Fabry-Perot peak positions, fittedwith a 7th order polynomial. Right: Linearisation polynomial with constant and linearterms dropped to demonstrate the inherent nonlinearity embedded in our spectra. To thehuman eye, the plot on the left suspiciously resembles a line.

f ≈ π√F

2

For the Fabry-Perot cavity used in this experiment, we have found that f ≈ 12.First, replace the phase δ with a polynomial of appropriate order p(xj) = a0 +

a1xj + a2x2j + a3x

3j . . . . Second, allow the amplitude of each Fabry-Perot peak to

vary to accomodate fluctations in laser power.Making these two substitutions, we have

T =b0 + b1xj

1 + F sin2(a0 + a1xj + a2x2j + a3x3

j . . . )(4.3)

where F is again the coefficient of finesse, and b0 and b1 are constants, andan is the nth coefficient of the frequency linearisation polynomial. Fit the Fabry-Perot signal in one fell swoop with this new Airy function and use the relevent fitcoefficients to reconstruct the linearisation polynomial. In either case, by feedingall values of xj through the polynomial, we have successfully constructed a linearfrequency axis for the hyperfine spectrum using its associated Fabry-Perot signal.Depending on the number of Fabry-Perot peaks captured in a scan, we typically fitto 5th to 7th order polynomials. The necessary order is determined by fitting someportion of the data to a one higher order polynomial and seeing if the resultingextracted measurements are changed. If not, then it is sufficient to fit to the lowerorder polynomial because no additional nonlinearity was captured by increasingthe order. Thus, we default to the most efficient fitting method. S

4.2.2 Determination of Measurements

As discussed in Chapter 2, a consequence of having spectra that are largely-Doppler-free is that it is generally sufficient to fit the peaks to Lorentzians. The centre of eachfitted Lorentzian determines the central frequency of that peak, which we indicateby ν1 . . . ν6 and νA . . . νH respectively for single-isotope and dual-isotope spectra.

Single-Isotope Spectra

All single-isotope spectra were fit to a sum of six Lorentzians 1. Figure 4.8 is asample fitted single isotope spectrum labelled in exactly the same way as the sim-ulated spectrum in Figure 4.1. The fitting process locates the center of each peakas the center of its Lorentzian. Three types of values were extracted by taking thedifference of appropriate peak positions.

Figure 4.8: Sample single isotope spectrum. Red line is best fit to sum of (in this particularcase) ten Lorentzians. Peaks 3 and 4 are the actual hyperfine peaks. The rest are first andsecond order EOM sidebands.

1. The hyperfine splitting: H8P1/2= ν4 − ν3 = δν43 (Refer to Figure 4.8 for peak

numbers )

2. A second, independent method of calculating the hyperfine splitting:H8P1/2

= (1000 MHz)− δν(calibrated)32 . As the actual frequency splitting δν32

here is approximately a quarter of the hyperfine splitting δν43, thismeasurement should be much less sensitive to residual nonlinearity in thefrequency axis.

1With the exception of single-isotope spectra in which second-order EOM sidebands manifestthemselves with a sufficiently large signal-to-noise ratio to be fit to additional Lorentzians, as isclearly the case in Figure 4.8.

3. Four sideband splittings: δν31, δν42, δν53, δν64 for calibration purposes.

Dual-Isotope Spectra

Similarly, all dual-isotope spectra were fit to a sum of eight Lorentzians. Fromthese eight peaks we also extract three types of values. In accordance with Figure4.9, the central frequency of each peak will be referred to as νA . . . νH .

Figure 4.9: Sample dual isotope spectrum. Red line is best fit to sum of eight Lorentzians.

1. The hyperfine splittings: δνCA, δνEB, δνGD, δνHF .

2. The isotope shift.

3. The sum of the Doppler shifts of the two isotopes.

Hyperfine Splittings We are not ultimately interested in using the values for thehyperfine splitting obtained dual-isotope spectra. We use our single-isotope datato obtain those values. However, four frequency splittings in our spectrumshould yield values for the hyperfine splitting Hprop.direction

isotope . The possible peakcombinations are:

HCO205 = νC − νA

HCTR203 = νE − νB

HCTR205 = νG − νDHCO

203 = νH − νF(4.4)

These values were compared to those calculated from the single-isotope spectrato reassure oneself that they are not systematically different. This correspondenceis important because the dual-isotope spectrum does not contain EOM sidebands,and thus does not have an internal source of frequency calibration. To calibratethe dual-isotope spectra, we must rely on applying the calibration factors foundfrom analysing the single-isotope spectra.

Isotope Shift The transition isotope shift is calculated by accounting for theeffects of both the Doppler shift and the hyperfine anomalies. The Doppler shiftcan be removed by averaging each co-propagating/counter-propagating pair ofpeaks (by which is meant the two peaks that correspond to the same hyperfinetransition in the same isotope, but originate from different UV beams). Thehyperfine anomalies in both the 7S1/2 and 8P1/2 states can be removed byadding/subtracting the correct fraction of the hyperfine splitting. We reproduceEquation 4.1:

I7S→7P =3|νG+νC

2− νH+νE

2|+ |νD+νA

2− νF +νB

2|

4+H7S,203 −H7S,205

4

where H7S,203 and H7S,205 are the hyperfine splittings of the 7S1/2 state in 203Tl and203Tl respectively, as previously measured [1].

Doppler Shift Lastly, as discussed in Section 4.1, the difference betweenco-propagating counter-propagating peak pairs should always yield the sum ofthe blue and red Doppler shifts of the two isotopes in the spectrum of the 672 nmlaser. Thus the average Doppler shift is merely:(

|∆f205(2ndstep)|+ |∆f203(2ndstep)|)Avg

=1

4(δνBF + δνAD + δνEH + δνCG) (4.5)

Although this value is, in and of itself, not particularly interesting, it is anexcellent test of the accuracy of our other extracted values because the magnitudeof this total Doppler shift is well-known. It is merely the 1636 MHz frequencydifference between the 203Tl 6P1/2(F = 1)→ 7S1/2(F ′ = 1) and 205Tl6P1/2(F = 1)→ 7S1/2(F ′ = 1) transitions seen in the absorption signal of the fullfirst-step transition (Figure 4.4) scaled by red laser frequency

UV laser frequency , as the interval is nowobserved in the spectrum of the red laser. The Doppler shift is expected to be

|∆f205(2ndstep)|+ |∆f203(2ndstep)| = (1636 MHz)

(λUVλred

)(4.6)

= (1636 MHz)

(377.68 nm671.57 nm

)|∆f2052ndstep)|+ |∆f203(2ndstep)| ≈ 920.1 MHz

4.2.3 Frequency Calibration

EOM Sideband Calibration

Per the discussion in Section 3.4, one must assume a value for the free spectralrange of the Fabry-Perot cavity in order to carry out step one of the data analysisprocess: frequency mapping and linearisation. For example, we have assumed avalue of 363 MHz in our analysis. Having linearized the frequency axis of thespectra on this assumption, in theory, if 363 MHz is exactly the true FSR, then oneshould find that the measured sideband separation and the known sidebandseparation are the same, i.e. 1000 MHz. As it turns out, the measured sidebandvalue is not quite 1000 MHz. This suggest that our assumed FSR value is notquite correct, and that we need to scale our frequency axis such that the sidebandseparations do come out, on average, to be 1000 MHz. This calibration factorCEOM is of course merely

CEOM =1000 MHz

∆νSB(4.7)

where ∆SB is the measured average sideband separation. To exploit the power oflarge numbers, we elect to average all sideband values over a day of data, anddesignate one calibration factor to apply to that entire day of results. Multiplyingthe raw values by this correction factor yields the final measurement. Forexample,

δν(calibrated)ij = CEOMδνij (4.8)

Alternate Calibration: Direct FSR Measurement

To verify that the calibration factor deduced from the observed EOM sidebandvalue is correct, we additionally attempt to independently and directly determinethe true free spectral range of the Fabry-Perot cavity. First, we collected a range offrequencies at which a Fabry-Perot peak is located by slowly tuning the red laserby hand and watching the response of the transmission signal. Now, thedifference between any two pair of frequencies thus recorded corresponds to thedistance in frequency between two FabryPerot peaks and should be an integer

multiple of the free spectral range. The true free spectral range should thereforeminimise the quantity

N∑j=1

(∆fj − njFSRk)2 = X2(FSRk) (4.9)

where ∆fj is the jth experimental frequency difference, nj is the integer thatminimises the difference between ∆fj and a reasonable guess for the free spectralrange FSRk. Now, we compute this sum for an appropriate range of values for(FSR)k. In this case, frequencies around 363 MHz would be appropriate. FigureD.1 shows the resulting plot.

Figure 4.10: Verify that the true free spectral range is in agreement with that implied bythe EOM sideband values by conducting an independent measurement of the Fabry-PerotFSR.

By approximating the curve as a quadratic function, it is possible to fit to theregion around where X2(FSRk) is minimised. Performing this procedure twiceover the course of several months, we find that this direct measurement suggestsa true FSR = 363.77± .01 MHz. In comparison with the nominal value of 363MHz used throughout the data analysis, the calibration factor implied is

CFP =FSRtrue

FSRnominal

= 1.00212(3) (4.10)

4.2.4 Statistical Methods

After fitting all the spectra, finding the hyperfine splittings and isotope shift fromeach spectrum, and calibrating those values, we use two methods to determinewhat the average value is, and the statistical error in that value.

Histogram and Gaussian Fit

One method is finding an average value is to create a histogram from all ourvalues of (for example) the hyperfine splitting, and fit the histogram to a Gaussianfunction. Figure 4.11 shows an example of this method.

Figure 4.11: Example histogram from 205Tl containing 680 scans, showing the hyperfinesplitting from δν43 on the left, and the frequency separation δν32 on the right. The red curveis a Gaussian function fit to the historgram. Legend shows both the standard deviationand the standard error of the Gaussian fit. Note that these are raw, uncalibrated values.Histogram is unweighted.

This method of consolidating the values from some thousand scans is very goodfor visualisation, and checking whether our data distribution is reasonable in thesense that it follows, as expected, a Gaussian distribution. By fitting a Gaussianfunction to the histogram, we locate the mean value, and the standard deviationof the function σ. From σ, and the n, the number of scans in the data set, wecompute the standard error σm.

σm =σ√(n)

(4.11)

Obviously, the more scans used in the histogram, the smaller the standard errorwill become. It is for this reason that we take many, many scans. However, thishistogram approach has two disadvantages. First, although binning all of ourextracted values into one histogram a very small standard error can be obtained,we are also losing some information by doing so. Namely, we are discardinginformation about the day-to-day or data-set to data-set variations in themeasured value. That very large histogram is in some sense the aggregate of anumber of smaller histograms, each with its own mean, that are clustered aroundthe global mean displayed in the very large histogram. This day-to-day variationshould be included in the statistical error of our measured value. In short,locating the mean value of our data and the statistical error in that value from this

histogram approach underestimates the statistical error. Second, astraightforward binning of all data into a histogram and fitting to a Gaussianfunction also discards information about the relative goodness-of-fit of eachspectrum. Inevitably, some scans in a data set fit better than others, and yields adata point with a smaller associated standard deviation. Ideally, we wish toinclude that information in our measured value.

Weighted Average

A weighted average approach, while not visually illuminated, will allow us toboth take into account the error in each data point in finding the mean value.Using the results from taking the weighted averages of each data set, we can alsothen correct for the underestimation of statistical error in our value by accountingfor the data-set to data-set scatter of measured values. Consider that we have ndata sets, each containing m scans, each scan for which we have already extractedthe hyperfine splitting Hj and its standard deviation σj . The weighted averagedhyperfine splitting for each data set H is

H =

∑mj=1 Hjσ

−2j∑m

j=1 σ−2j

(4.12)

and its standard deviation σH is

σH =1√∑mj=1 σ

−2j

(4.13)

This is applied to all n data sets, yielding n weighted averaged hyperfinesplittings Hi with n standard deviations σHi

. By saying that we suspect that thestatisical error is being underestimated, we are equivalently saying that webelieve X2 is too large. To find X2 to begin with, we do a weighted fit of the list ofσHi

values to a free constant. Then the correction required is to multiply eachstandard deviation of the mean by X2 divided by the numbers of degrees offreedom, n− 1 such that

σ(corrected)

Hi= σHi

√X2

n− 1(4.14)

The value of the fitted constant is the weighted mean of all n data sets. Byapplying this method to carefully divided segments of our data, it can bedetermined whether, despite the variation of some experimental condition, theresulting values for the hyperfine splitting or isotope shift are consistent witheach other.

4.3 Results

This section presents highlights from the the data collected and analysed, as wellas searches for systematic errors. In the course of both collecting, and analysingour data, the potential for systematic error looms. To illuminate each possibility,we divide the data into subsets according to those variables (propogationdirection of the UV laser, frequency calibration, linearisation) one at a time,compute an individual weighted averaged measurement for each subset, andcompare the resulting values to one of two ends: show that the experimentalvariable has no statistically significant effect on the measurement, or quantify thedegree to which the variable is a source of systematic error.

4.3.1 Single-Isotope Data: Hyperfine Splitting

Typical fits for the single-isotope data are featured in Figures 4.12 and 4.13.

Figure 4.12: Typical fit for single-isotope data. This specific scan is from 205Tl, in the co-propagating configuration. Red line is fit to sum of ten Lorentzians for two hyperfinepeaks, four first-order sidebands, and four second-order sidebands.

Figure 4.13: Another typical fit for single-isotope data. This scan is also from 205Tl, but inthe counter-propagating configuration. Red line is fit to sum of ten Lorentzians for twohyperfine peaks, four first-order sidebands, and four second-order sidebands.

Figure 4.14 shows the scatter of the 203Tl hyperfine splitting from two data sets, of∼ 200 points each.

Figure 4.14: Scatter plot of∼ 400 values for the 203Tl hyperfine splitting from two data setscollected on the same day. Dark green and green points distinguish between the two datasets

Evaluation of Systematic Errors

As described in Chapter 3, the experiment apparatus records single-isotope datain both co-propagating and counter-propagating configurations, and scans thered laser both upwards and downwards in frequency. Together with the twoindependent methods of extracting the hyperfine splitting from a spectrum (Seesection 4.2.2), the effect of these three fundamental categorisations are separatelycompared for the two isotopes in Figures 4.15 and 4.16. The values for each subsetis compiled in Tables 4.2 and 4.3.

HCO203 HCTR

203 HUP203 HDN

203 H(43)203 H

(32)203

780.3(1) 780.6(4) 780.4(2) 780.3(5) 780.2(4) 780.7(3)

Table 4.2: 203Tl hyperfine splitting values in MHz, sorted by UV laser propagation direc-tion, red laser scan direction, and extraction method.

HCO205 HCTR

205 HUP205 HDN

205 H(43)205 H

(32)205

788.7(2) 787.9(2) 787.9(2) 788.5(3) 788.2(3) 788.3(3)

Table 4.3: 205Tl hyperfine splitting values in MHz, sorted by UV laser propagation direc-tion, red laser scan direction, and extraction method.

Figure 4.15: Comparison of data subsets for 203Tl to investigate systematic errors.CO/CTR refers to the propagation direction of the UV laser. Up/down refers to the scandirection of the red laser in frequency space. Error bars reflect one standard deviationuncertainties based on observed set-to-set scatter of H203 within the relevant subsets ofdata.

Figure 4.16: Comparison of data subsets for 205Tl to investigate systematic errors. Errorbars reflect one standard deviation uncertainties based on observed set-to-set scatter ofH205 within the relevant subsets of data.

Laser Propagation Direction and Scan Direction Examining Figure 4.16, we seethat of the pairs of data points comparing opposing UV laser propagationdirection and red laser scan direction for 205Tl, neither are in statistical agreement,although the discrepancy in both cases is under 1 MHz. We can use thediscrepancy between the pairs of values for 205Tl as a measure of the systematicerror due to laser propagation direction or scan direction. In the case of the scandirection, it is reasonable to suppose that the hyperfine splitting measurement ispulled in opposite directions because the mechanism by which the red laser isscanned up in frequency or down in frequency is exactly opposite: onecorresponds to the diffraction grating in the ECDL being pushed by theexpansion of the PZT, and the other corresponds to a contraction of the PZT (See

Section 3.2). If the actual value of the 205Tl hyperfine splitting is roughly anaverage of HUP

205 and HDN205 , then a good estimate of the systematic error would be

one-half of the difference between the values.

1

2|HUP

205 −HDN205 | = 0.3 MHz

Similarly, in the absence of any compelling reason to believe that either HCO205 or

HCTR205 is closer to the correct value for the 205Tl hyperfine splitting, it is most

reasonable for us to argue that HCO205 is an overestimate and HCTR

205 is anunderestimate of the actual value, and that, as in the case of scan direction, anaverage of the two values should be taken. This is a particularly reasonableconclusions to come to in light of the fact that for the other isotope, 203Tl (SeeFigure 4.15), the hyperfine splitting values from both the CO/CTR-propagatingsubsets and up/down scan direction subsets are in statistical agreement.Therefore, here we also estimate the systematic error to be one-half of thedifference between HCO

205 and HCTR205 .

1

2|HCO

205 −HCTR205 | = 0.4 MHz

Frequency Calibration Having elected to establish a calibration value for eachday of datasets, for each isotope, it is possible to examine the scatter of theseresulting values. From Figure 4.17 and Table 4.4, one can see that there isconsiderable variation amongst the values. C205,Dec

EOM in particular is removed fromthe weighted mean and the other three EOM calibration factors. However, thetwocalibration factors from the December data sets, and the two calibration factorsfrom the February and March data sets are in statistical agreement betweenthemselves. On the surface, this suggests that the difference in calibration factorsare only due to thernal drift in the cavity over the time scale of months.Furthermore, the calibration factor from directly measuring the FSR of the cavityis the average of two sets of data, one acquired in December, and a second inJanuary. It is therefore heartening that CFP is in statistical agreement with theDecember EOM calibration factors.

C203,MarEOM C205,F eb

EOM C203,DecEOM C205,Dec

EOM CFP1.00148(2) 1.00152(2) 1.00188(5) 1.00234(5) 1.00212(3)

Table 4.4: Calibration factors, sorted by isotope and date. Last value from direct FSR mea-surement (See Section 4.2.3.) Error bars reflect one standard deviation uncertainties basedon observed set-to-set scatter of the calibration factor, and thus averaged EOM sidebandsplitting.

To estimate the contribution of calibration to systematic error amongst thisconfluence of factors, we want to calculate the impact using either calibration

Figure 4.17: Comparison of calibration factors with error bars corrected for set to set sta-tistical variation

factor would have on final values of the hyperfine splitting and isotope shift. Wefuthermore want to capture what the difference between those final measurementswould be if one calibration factor instead of another. For the purposes of makingthis estimate, and the rest of the estimates of systematic error in this Chapter, weassume that the hyperfine splitting is 800 MHz, and the isotope shift is 450 MHz.Using C205,Dec

EOM for calibration would result in a correction of ∼ +1.9 MHz to theraw hyperfine splitting, and ∼ +1.1 MHz in the isotope shift. Using C205,F eb

EOM

would result in a change of ∼ +1.2 MHz, and ∼ +0.7 MHz in the isotope shift. Areasonable estimate of the systematic error due to frequency calibration is then

1

2|δHDec − δHFeb| = 0.35 MHz

1

2|δIDec − δIFeb| = 0.20 MHz

Scan linearisation In each single isotope spectrum there are four possiblesideband values (See Figure 4.8). We know that in a truly accurate frequency axis,all four sidebands would be located 1000 MHz from the hyperfine peak to whichit belongs. Therefore, if after averaging over the ∼ 100 scans in a data set, the foursideband values are not in statistical agreement with each other, this is anindication that there is residual nonlinearity in the frequency axis of the spectra.We quantify the possible effect of this discrepancy on the final values for thehyperfine splittings and isotope shift much in the same manner as done inconsidering the frequency calibration, by seeing what the discrepancy in finalvalues for the hyperfine splitting and isotope shift would be assuming onecalibration based on one sideband value compared to the other. Refering to themean sideband values exhibited in Figure 4.18, a sideband value of 998.9 MHzimplies a calibration factor C1 of ∼ 1.00115. Using C1 for calibration would result

Figure 4.18: Histograms of the four raw sideband values extracted from all 205Tl scans.

in a correction of ∼ +0.9 MHz in the hyperfine splitting, and ∼ +0.5 MHz in theisotope shift. Repeating the process for the sideband value from the opposite endof the spectrum (See Figures 4.18 and 4.8), C2 ' 1.00153 would result in acorrection of ∼ +1.2 MHz, and ∼ +0.7 MHz in the isotope shift. Taking half thedifference of these results as before:

1

2|δH1 − δH2| = 0.15 MHz

1

2|δI1 − δI2| = 0.1 MHz

In addition, it should be noted from Figure 4.18 that although three of thesideband splittings are all approximately 998 MHz, the mean value for the upperEOM sideband corresponding to the 7S1/2(F ′ = 1)→ 8P1/2(F ′′ = 0) transition isnoticably lower. We believe this to be due to the difficulty of fitting that sidebandaccurately given its smaller size and its position, only partially resolved from thehyperfine peak from the 7S1/2(F ′ = 1)→ 8P1/2(F ′′ = 1) transition (See Figures4.12 and 4.13). For this reason all of the calibration factors in Table 4.4 are theresult of averaging only three of the four sidebands in all single-isotope scans.

4.3.2 Dual-Isotope Data: Isotope Shift

A typical fit for the dual-isotope data is featured in Figure 4.19. Figure 4.20 showsa sample histogram for raw, uncalibrated values of the isotope shift, and averagedsum of doppler shifts, while Figure 4.21 displays the scatter of calibration isotopeshift values across four data sets. As discussed in Section 4.2.2, for dual isotopedata, we must draw upon the calibration factor determined by the EOMsidebands in the single-isotope data. Two pieces of information serve to persuadethat it is justifiable for us to do so. First, that the hyperfine splittings for bothisotopes extracted as written in Equations 4.3 are reasonably close to the valuesfrom single isotope spectra. Second, that the calibrated, average Doppler shift isclose to the expected value of 920.1 MHz. The first four values in Table 4.5 for thehyperfine splittings from dual-isotope spectra are reasonably similiar to theresults in Table 4.6. In particular, the average of the two values for the205Tlhyperfine splitting is in statistical agreement with the single-insotope value. It isunsurprising that the 203Tl values have a larger discrepancy from thesingle-isotope values. The 203Tl peaks in the dual-isotope spectrum have a smalleramplitude, and generally less well-resolved (See Figure 4.9). This is particularlythe case for peaks E and F.

Figure 4.19: Typical fit for dual-isotope spectra. Red line is fit to sum of eight Lorentzians.

HCO203 HCTR

203 HCO205 HCTR

205 I7S1/2→8P1/2DSAvg

777.8(4) 779.9(4) 788.6(3) 787.7(5) 450.3(3) 918.6(2)

Table 4.5: Weighted average hyperfine splittings, isotope shift, and averaged Doppler shiftin MHz from all dual-isotope data, calibrated by C205,F eb

EOM , as most dual-isotope data wastaken in February. Errors calculated as described in Section 4.2.4

Figure 4.20: Histogram of all isotope shift, and averaged sum of doppler shifts values,from ∼ 700 scans. Red line is Gaussian fit to histogram. Note that these are uncalibratedvalues and that histogram is unweighted.

Figure 4.21: Scatter plot of isotope shift, from four data sets and ∼ 260 scans.

Evaluation of Systematic Errors

There are two potential systematic errors applicable only to dual-isotope spectra.This is due first to the fact that unlike single-isotope spectra, dual-isotope spectradepend on optimal alignment of all three laser beams (co/counter-propagatingUV, and red) with each other. Second, the dual-isotope spectrum does not have ainternal source of frequency calibration.

Laser Scan Direction But first, it is possible to compare subsets of thedual-isotope data by red laser scan direction. Figure 4.22 shows that the weightedaverage isotope shift extracted from each subset of data is in very goodagreement.

Figure 4.22: Comparison of scan direction data subsets for isotope shift value. The dual-isotope spectrum requires both UV beams at all times, so it is only possible to compareresulting isotope shift values by their laser scan direction: up or down in frequency. Errorbars reflect one standard deviation uncertainties based on observed set-to-set scatter ofI7S1/2→8P1/2

within the relevant subsets of data.

Geometric Alignment of Beams From Table 4.5 we can see that while theexpected sum of Doppler shifts is 920.1 MHz, the measured sum of Doppler shiftsis only 918.6 MHz, 1.5 MHz smaller. This discrepancy could be attributed toimperfect alignment of the co-propagating and counter-propagating UV beams. Ifthe two beams are not perfectly anti-parallel, then of course the sum of theDoppler shifts will too not be 920.1 MHz. Such a misalignment would result in asmall systematic error in the extraction of the isotope shift. This is because theDoppler shift in the dual-isotope spectrum is accounted for in the data analysisprocess by assuming magnitude of the Doppler shift in theco/counter-propagating beams of a given peak pair will be the same, regardlessof what that Doppler shift is. Thus we taking the averaged position of theco/counter-propagating peak pairs to remove the Doppler shift. However, if theAn additional systematic error should be assosciated with this mislignment of theUV beams. We estimate this error much as we did with frequency calibration andscan linearisation above.

1

2|∆ftotal(2ndstep) −DSAvg| = 0.75 MHz

As a percentage of the total Doppler shift, this 1.5 MHz discrepancy is in line withthe analogous discrepancy in the Doppler shift found by the Majumder lab in2014 while measuring the 7P1/2 state in thallium. [2].

Correlation between HFS and IS Figure 4.23 was constructed by taking pairs ofHCO

205 , I7S→8P , from the same scan, and plotting those pairs.We can associate a systematic error with the discrepancy between the hyperfinesplittings found from the single-isotope spectra with those from the dual-isotope

Figure 4.23: Scatter plot of isotope shift I7S→8P v. hyperfine splitting HCO205 . Excluded

outliers. Red line is best linear fit through the data points.

spectra from this plot. This is done by asking what the isotope shift wouldcorrespondingly be, if we assumed H205 = 788.3(3) to be correct as in Table 4.6.The linear fit through the scatter plot in Figure 4.23 gives that the isotope shiftshould be ∼ 449.6 MHz. Therefore, following the methodology we’ve been usingthus far, we quantify the systematic error as

1

2|I7S1/2→8P1/2

− 449.2 MHz| = 0.35 MHz

4.3.3 Summary of Results

Table 4.6 summarises the systematic errors discussed above and shows thecombined total error.

(203Tl) HFS (205Tl) HFS 7S1/2 → 8P1/2 TISFinal result (MHz) 780.5 788.3 450.3

Statistical error (MHz) 0.3 0.3 0.3Systematic error sources (MHz)Beam co vs counterpropagation 0.4

Laser sweep direction 0.3Frequency calibration 0.35 0.35 0.2

Scan linearisation 0.15 0.15 0.10Geometric alignment of beams 0.75

Correlation between HFS and IS 0.35Combined total error (MHz) 0.4 0.7 0.9

Table 4.6: Summary of results and overall error for 8P1/2 HFS and isotope shift. All valuesin MHz

We can do one last calculation to find the 8P1/2 level isotope shift from thetransition isotope shift listed in Table 4.6. In particular, the value derived for thelevel isotope shift of the 7S1/2 state is +409.0(3.8) MHz [1]. From this we infer a8P1/2 level isotope shift of I8P1/2

= −41.3(3.9). In addition, in combination withthe previously measured transition isotope shift, I6P1/2→7S1/2

, it is possible to findI6P1/2→8P1/2

, as

I6P1/2→8P1/2= I6P1/2→7S1/2

− I7S1/2→8P1/2(4.15)

Refering to the value in Table 1.1, I6P1/2→7S1/2 = 1659.0± 0.3 MHz. Thus,

I6P→8P = 1208.7± 0.9 MHz

Comparison with Previous Measurements

Historical values for the 8P1/2 hyperfine splittings and the level isotope shift canbe found in Table 1.3, the relevent parts of which are produced below. Althoughwe could compare the 8P1/2 level isotope shifts with each other, the total errorquoted in the 1988 value is appreciably less than our value for the level isotopeshift, as the total error in our value is limited by the degree of precision to whichthe 7P1/2 level isotope shift was previously measured ??. Therefore, we chooseinstead to compare our value for the isotope shift of the 8P1/2 level with respect tothe ground state with the 1988 value, both of which have total errors around 1MHz.

Year 203Tl HFS (MHz) 205Tl HFS (MHz) Level IS (MHz) Group8P1/2 1988 781.7± 1.6 788.5± 0.9 −37.2± 1.2 U Giessen [3]

Table 4.7: Measurements of the thallium 8P1/2,3/2 HFS and IS through time. The valueslisted here are the ones of interest in this thesis.

Our values for both hyperfine splittings are in reasonable agreement with thevalues from 1988. However, we find the 8P1/2 isotope shift relative to the groundstate to be 10 MHz lower than the previously measured value. Figure 4.24summarises these relationships.

Figure 4.24: Comparison of our results with values measured by Grexa et al. [3]. Left:8P1/2 hyperfine splittings. Purple corresponds to 203Tl, green to 205Tl. Right: Isotope shiftof 8P1/2 state relative to the ground state.

Chapter 5

Future Work

5.1 8P3/2 Hyperfine Splitting and Isotope Shift

The next step in this experiment would be to move on to measuring the hyperfinesplitting and isotope shift of the 8P3/2 state. As shown in Figure 1.2, the onlychange to the current experimental setup that needs to be made is to switch the672 nm laser for a 655 nm laser, which can be conveniently accomplished bychanging the laser diode installed in the current laser to a similar one that coversthe desired wavelength. Simulations of the resulting single-isotope anddual-isotpe spectra are shown below in Figures 5.1 and 5.3. Figures 5.2 and 5.4 arerecently collected preliminary single-isotope and dual-isotope spectra.Continuing and finishing this measurement should be fairly straightforward.However, unlike the 8P1/2 state, the hyperfine splitting of the 8P3/2 state wasnever remeasured after the initial measurement in 1988 (See Table 1.3). Accordingto those values, the 8P3/2 HFS is only about a third the size of the 8P1/2 HFS.Furthermore, there was essentially no hyperfine anomaly detected. The 203Tl and205Tl hyperfine splitting were in statistical agreement, at around 260 MHz [3]. Itwill be interesting to see whether we observe the same result.

5.2 Moving on to Indium

After the conclusion of the 8P3/2 hyperfine splitting measurement, we mightconsider looking at the analagous transitions in indium: namely the5P1/2 → 6S1/2 → 7P1/2,3/2 transitions. Conveniently, the second step of thosetransitions fall within the range of ∼ 685− 690 nm. We would simply need toacquire a suitable laser diode, and install it into our (currently 655 nm) externalcavity diode laser. Working with Indium is an inviting prospect because incontrast to thallium’s nuclear spin of 1

2, indium has a nuclear spin of 9

2. As a

result, not only is its hyperfine structure much more complex, but higher order

59

Figure 5.1: Simulated single-isotope spectrum for 6P1/2 → 7S1/2 → 8P1/2 transition inthallium.

Figure 5.2: Sample fitted single-isotope spectrum for 6P1/2 → 7S1/2 → 8P1/2 transition inthallium. Red line is fit to sum of eight Lorentzians

electromagnetic multipole moments also become relevant to the hyperfinestructure [21].

Figure 5.3: Simulated dual-isotope spectrum for 6P1/2 → 7S1/2 → 8P1/2 transition inthallium.

Figure 5.4: Preliminary dual-isotope data for 6P1/2 → 7S1/2 → 8P1/2 transition in thallium.

Appendix A

672 nm Laser Design andConstruction

A.1 Preface

The external cavity diode laser (ECDL) design to be discussed in this appendix isthe third (and finally experimentally adequate) incarnation of the 672 nm diodelaser intended to be used in measuring the 8P1/2 hyperfine splitting that is thesubject of this thesis (See Section 3.2 for an overview of ECDLs). Sarah Peters ‘14worked on a version the summer of 2013. Talia Calnek-Sugin ‘15 and myselfreworked the design the summer of 2014, and completed a functional laser,simply not one that had a sufficiently wide scanning range, or sufficient stability,to be used in a measurement.

This final design was introduced to us by Prof. Charlie Doret, and the CADdrawings presented below are modifications on those created by his thesisstudent Cole Meisenhelder ‘15 [19]. The basic design consists of a Newportoptical mount and cylindrical adaptor, machined to accomodate a collimationtube. The laser diode is installed inside the collimation tube. Two smallaluminium pieces complete the essential parts: a ‘grating arm’ that holds thediffraction grating, and a ‘mirror arm’ that holds the correction mirror.As described in Section 3.1, the design itself is the work of Arnold, Wilson, andBoshier [17], with additions by Hawthorn, Weber and Scholten [18].

A.2 Preliminary Decisions and Acquisitions

A.2.1 Simple Calculations

Retrieving the grating equation from Section 3.2, we see that it reads

nλ = 2d(sin θ)

63

Figure A.1

where d is the line spacing of the grating, and λ is the target wavelength of thelaser. We are interested in knowing what angle the diffraction grating needs to bepositioned at to make the final diode laser capable of lasing at the wavelengthswhich one requires, so:

θ = arcsin

(λ

2d

)(A.1)

For example, if λ = 672 nm, and we had a 1800 lines/mm grating, then θ ' 37 °.

A.2.2 List of Parts

Below is a comprehensive list of parts required to assemble an ECDL along thelines of the 672 nm laser used in this thesis. There is some flexibility in choosingsome of the parts, which the list indicates to the best of its ability.

Parts To Be Machined

• Newport U100-P(2H) Ultima Platform Optical Mount, 2.0” by 2.0” (2H withactuators)

• Newport UPA-PA1 Horizontal adaptor, 1.0”, for U100-P

• Grating arm

• Mirror arm

• Base: The 672 nm laser is mounted (See Figure A.1) onto an aluminium baseplate machined to fit the bottom of the Newport U100-P platform mount.The thermoelectric coolers (TEC) are sandwiched between the bottom of thebase plate and a separate aluminium block. Nylon screws hold the baseplate to the aluminium block. The block is itself screwed into the opticaltable. The block elevates the output of the ECDL to a convenient height.

• Mount for electronic connections: In the 672 nm setup, this is merely arectangle of clear acrylic attached with screws to the base aluminium block.The acrylic is laser-cut with the appropriate sized holes to hold two 9-pinconnectors for the TEC and Laser Diode cables (that go to the laser diodecontroller (LED)), and the BNC connection to the PZT controller (See FigureA.2). This is a passable design. It’s weakness is that it is not very good atrelieving any stress from the heavy laser diode controller cables on theconnection to the laser diode itself.

• Cover for thermal isolation: Acrylic box lined with dense foam forinsulation. An alternative base/covering design can be found in the thesisof Cole Meisenhelder ‘15 [19].

• Feed forward: A feed forward system may or may not be necessary toachieve the desired tuning range. A circuit diagram is included in SectionA.4.

Parts To Be Purchased (Shopping List)

• Newport U100-P(2H) Ultima Platform Optical Mount, 2.0” by 2.0” (2Hincludes actuators) [requires machining]

• Newport UPA-PA1 Horizontal adaptor, 1.0”, for U100-P [requiresmachining]

• Collimation tube: The two collimation tubes listed requires differentmodifications to the Newport UPA-PA1 horizontal adaptor. Both drawingsare included. The adjustable collimation tube needs a shortened adaptor toaccommodate the rotating cap on the end.

– Thorlabs LTN330-B Adjustable Collimation Tube with CollimationLens, suitable for both �5.6 mm and �9 mm laser diodes, AR Coatedfor 350-700 nm

– Thorlabs LT230P-B, Collimation Tube with Collimation Lens, suitablefor both �5.6 mm and �9 mm laser diodes, AR Coated for 650-1050 nm

• An appropriate diffraction grating, such as Thorlabs GH13 -18V VisibleReflective Holographic Grating, 1800 lines/mm, 12.7 mm by 12.7 mm

• A correction mirror, such as Thorlabs BBSQ05-E02 Square BroadbandDielectric Mirror, 1/2” by 1/2”, 400-750 nm

• An appropriate laser diode

– 672 nm laser used:

* HL6714G, 670 nm, 10 mW, �9 mm, A Pin Code1, Hitachi LaserDiode

* HL6756MG, 670 nm, 15 mW, �5.6 mm, A Pin Code, Opnext LaserDiode

– 655 nm laser used:

* L658P040, 658 nm, 40 mW, �5.6 mm, A Pin Code, Laser Diode

• Laser diode socket, such as Thorlabs S7060R, or S8060

• Piezoelectric actuator (PZT), such as Thorlabs AE0203D04F

• Thermoelectric coolers (TEC)

• Thermistor

• Two 9-pin connectors

• A BNC connector

A.3 Construction Process

A.3.1 Machining

The Newport parts, the grating arm, and the mirror arm are all configured to bemachined on the CNC mill. Drawings are included in section A.4 to illuminatehow the parts fit together.

A.3.2 Making Final Connections

There are three components that require soldering to be completed.

1. Connector from laser diode to laser diode cable

1All the laser diodes currently in supply have A Pin Code, and are thus all compatible with theexisting wiring to the Laser Diode Controller.

• The two ends of this connector are the 3-pin laser diode socket (whichgoes to the laser diode), and one of the 9-pin connectors (which isplugged into a cable that eventually goes to the laser diode controller).Use the pin code in the laser diode specification sheet and the laserdiode controller manual to determine what connections need to bemade.

2. Connector from thermoelectric coolers to TEC cable

• The thermoelectric coolers need to be directly connected to the second9-pin connector. This is plugged into another cable that also goes to thelaser diode controller.

3. Connector from PZT to BNC cable

• The PZT needs to be directly connected to a BNC connector. Thiseventually goes to the PZT controller. In the current design, becausethe BNC connector can only be inserted into the acrylic mount from theoutside, small gold wire connectors have been used to ensure each partis separable without needing to unsolder anything.

Figure A.2

A.4 Drawings

• Overview of assembled laser

• Newport U100-P optical mount

• Newport UPA-PA1 adaptor

• Newport UPA-PA1 adaptor shortened for adjustable collimation tube

• Grating arm

• Mirror arm

• Base plate

• Feed forward circuit diagram

A.5 Threshold Current Plots

Figures A.3 and A.4 show threshold current plots for both the 672 nm laser andthe 655 nm laser.

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Figure A.3: Threshold current plot for 672 nm laser and diode.

Figure A.4: Threshold current plot for 655 nm laser and diode.

Appendix B

Laser Locking

B.1 Locking Scheme

Any potential locking scheme requires a suitable error signal reflecting frequencydrift in the laser. This signal is passed to a proportional-integral-derivative (PID)controller, completing a feedback mechanism that outputs a voltage correction tothe laser in accordance with the error signal such that the laser is kept at a somefrequency lockpoint. Atoms remain stable, as long as external perturbations(magnetic fields etc.) are minimised. For this reason we elect to base our lockingscheme on a second thallium vapour cell. Assuming that one would want to lockto the centre of our atomic transition, it is logical to consider using the absoprtionsignal of the transition as the error signal. However, the absorption signal (SeeFigure 2.1) is an even function around the resonance frequency. Consequently, itis not possible to distinguish betweeen a small positive drift in frequency and asmall negative drift in frequency from the absorption signal alone. What isneeded is a dispersion-type curve that is an odd function around the transitionresonance. We obtain such a signal through the set up pictured in Figure B.1.

Figure B.1: Locking scheme utilising a second thallium vapour cell.

79

The UV laser is first passed through an acousto-optic modulator (AOM) thatproduces diffracted beams at ±250 MHz intervals from the input frequency. Theundiffracted beam is allowed to pass through to the rest of the experiment. Thefirst order diffractions are deftly combined in a polarising beam splitter (PBS) andsent through a thallium vapour cell in a smaller locking oven heated to ∼ 500 °C.A second polarising beam splitter is used to separate the two beams after theoven. Finally, the two frequency shifted beams are directed to a differentialphotodetector. When the UV laser is scanned across the chosen6P1/2(F = 1)→ 7S1/2(F ′ = 1) transition, the resulting output of the differentialphotodetector is shown in Figure B.2. This signal has a dispersion-like shape thatfeature three approximately linear regions and three zero-crossings. Using a PIDcontroller, one can easily set up a feedback mechanism in which frequency drift inthe laser will result in movement either up or down the quasi-linear error signal,prompting the PID controller to send a correction signal to the laser to direct itback to the designated frequency.

Figure B.2: Simulated difference signal for UV laser locking scheme showing error signalwith three quasi-linear slopes and corresponding lockpoints.

B.2 Temperature Optimisation

It should be unsurprising that the shape of the difference signal used for ourlocking purposes is a function of temperature. There exists, therefore, an ‘optimal’temperature of the locking oven at which the shape of the difference signal is suchthat the slope at the lockpoint is maximally steep. In theory, at this point thetightest possible lock is achieved. Appendix B of the thesis of David Kealhofer ‘13describes how to simulate the difference signal to find this optimal temperature[4]. He concludes that the optimal temperature ranges from 690 K to 715 K, or 417

°Cto 442 °C. Following his instructions, one can produce the plots of the slope atthe lockpoints as a function of temperature for oneself, as in Figure B.3

Figure B.3: Simulation of error signal slope at lockpoints as a function of lock oven tem-perature.

It is also useful to recognise the way in which the difference signal 1 can appeardistored from its usual, useful, dispersion curve-like shape, when the opticaldepth of the absorption signal is too large. As Figure B.4 shows, when the opticaldepth is too large (i.e. the temperature is high), the previously linear portions ofthe error signal are no longer linear.

Figure B.4: Simulation of error signal at high values of optical depth.

It is still possible to lock to such a signal as it is still an odd function about thelockpoint. However, it is not ideal. The slope is clearly far away from its

1In this locking scheme, the terms difference signal and error signal have been used interchange-ably. It should not, however, be confused with the correction signal.

maximum value. An easy fix is simply to turn the temperature of the lock ovendown.

B.3 Manipulating the Lockpoint

As discussed in Section 4.1.2, manipulating the lockpoint and thus the exactfrequency to which the laser is locked, was crucial to getting eight well-resolvedpeaks in the dual-isotope spectrum. There are essentially two ways to change thelock frequency without unlocking the laser. One way is by moving the differencesignal up and down by changing the relative intensities of the AOM sidebands inthe locking setup. Particularly if one has half-waveplates in each UV sidebandbefore the polarising beam splitter (see Figure B.1), this is not difficult. Thecleaner method is simply to change the setpoint on the PID controller to someother voltage - not zero. Although we discuss the ‘ zero-crossings’ of thedifference signal, the point of zero voltage is not intrinsically related to thelockpoint. We may lock to any voltage crossing on the linear slope, simply bysetting the PID internal setpoint to that voltage value.For the purposes of peak resolution,by simulating the effect of moving thelockpoint on peak separation, we can predict the direction in which the lockpointneeds to be moved. Figure B.5 is one example.

Figure B.5: The separation between peaks E and F in the dual isotope spectrum (See Sec-tion 4.1.2, Figure 4.5) was found for different values of the lockpoint and then plotted.

Appendix C

MATLAB Code

The main body of MATLAB code implements steps 1-3 of the data analysisprocess as described in Section 4.2, that is, it generates a linearisation polynomialby fitting the Fabry-Perot signal, fits the hyperfine signal with a sum ofLorentzians, and then extracts the measurements of interest. The masterprogramme is called ThalliumAnalysis2016.m, and both draws together the partsof the data analysis code and allows one to control variations on the data analysisprocess. Figure C.1 displays the progression of MATLAB scripts called upon bythis master programme. Step 4 of the data analysis described in Section 4.2 usesadditional MATLAB code described in Section C.2.

C.1 Data Files to Extracted Measurements

ThalliumAnalysis2016.m takes data sets as .txt files with four columns:

1. Time in arbitrary units/data point number (consecutive integers)

2. Ramp voltage (used to scan 672 nm laser)

3. Fabry Perot transmission signal

4. Hyperfine signal from lock-in amplifier

It then loops over each scan, implementing the MATLAB code shown in FigureC.1.For each data set, three files are produced: one storing the co-efficients of thelinearisation polynomial, one storing the locations of the peaks with standarddeviations, and one storing the extracted measurements with their standarddeviations.

getdataThallium.m

Imports data sets into MATLAB.

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downsampleAndNormalizeThallium.m

• Isolates region of data where voltage ramp is linear.

• Reverses the order of up-scan data points such that frequency increaes withtime/data point number for both scan directions.

• Normalises x-axis in point numbers.

• Normalises Fabry-Perot signal, and hyperfine signal.

• Downsamples and plots both Fabry-Perot signal, and hyperfine signal.

FabryPerotFittingThallium Airy Bootstrap.m

Fits FP signal to Airy function and finds polynomial to convert horizontal axis ofspectra from point number to linearised frequency.

• Uses sympeaksThallium.m to find peaks of Fabry-Perot signal.

• Uses FindFpPeaksThallium.m to confirm all Fabry-Perot peaks have beenfound.

• Uses FPCleanUp.m to eliminate any partial peaks at beginning or end ofscan, as well as corresponding hyperfine signal data points.

• Option to eliminate additional peaks at the beginning and/or end of scan.

• Fits remaining data to Airy function with nth order polynomial frequencydependence.

• Stores fit results, and linearisation polynomial.

• Plots Fabry-Perot data and fitted function.

FabryPerotFittingThallium Lorentzian.m

Fits FP signal to sum of Lorentzians to find transmission peaks.

• Uses sympeaksThallium.m to find peaks of Fabry-Perot signal.

• Uses FindFpPeaksThallium.m to confirm all Fabry-Perot peaks have beenfound.

• Uses FPCleanUp.m to eliminate any partial peaks at beginning or end ofscan, as well as corresponding hyperfine signal data points.

• Option to eliminate additional peaks at the beginning and/or end of scan.

• Fits remaining data to sum of Lorentzians.

• Saves peak locations.

• Plots Fabry-Perot data and fitted function.

FrequencyLinearizationThallium Lorentzian.m

Finds polynomial to convert horizontal axis of spectra from point number tolinearised frequency.

• Fits nth order polynomial to j data points (xj, j(FSR)), where xj are thelocations of the FP peaks in normalised point number, and j(FSR) are thecorresponding integer multiples of the FSR.

• Plots linearisation polynomial.

LorentzianFitThallium.m

• Evaluates linearisation polynomial at each hyperfine signal point number.

• Uses sympeaksThallium.m to find peaks of hyperfine signal.

• Uses FindHFSPeaksThallium.m to confirm correct number of peaks havebeen found.

• Uses SumofLorentzians.m to generate fit function and specify fit options.

• Fits hyperfine data to resulting sum of Lorentzians.

• Stores fit results, including peak locations, amplitudes, widths, andstandard deviations.

• Calculates hyperfine splittings, and sideband separations.

• Plots hyperfine data and fitted function.

LorentzianFitThallium Dual.m

• Evaluates linearisation polynomial at each hyperfine signal point number.

• Uses sympeaksThallium.m to find peaks of hyperfine signal.

• Uses FindHFSPeaksThallium.m to confirm correct number of peaks havebeen found.

• Fits hyperfine data to sum of Lorentzians.

• Stores fit results, including peak locations, amplitudes, widths, andstandard deviations.

• Calculates hyperfine splittings, sum of doppler shifts, and isotope shift.

• Plots hyperfine data and fitted function.

WriteToFileThallium.m

Saves linearisation polynomial coefficients, all raw fit results (peak locations), andall uncalibrated extracted measurements to .txt files.

C.2 Statistical Analysis

Additional code is used to carry out statistical analysis on the uncalibratedextracted measurements.

getHistdataThallium.m

Imports measurements extracted from single-isotope data. Allows up to four datafiles to be imported at once. getHistdataThallium Dual.m performs the same taskon measurements extracted from dual-isotope data.

HistThallium.m

• Creates six histograms: for hyperfine splitting, peak 3 minus peak 2 (fromwhich the hyperfine splitting can be calculated), and four sidebandseparations.

• Fits histograms with Gaussian functions.

HistThallium Dual.m

• Creates six histograms: for hyperfine splittings, isotope shift, and averagedsum of Doppler shifts.

• Fits histograms with Gaussian functions.

WeightedAverageThallium.m

Calculates weighted averages of uncalibrated measurements.

Figure C.1

Appendix D

Direct Measurement of Fabry-PerotCavity FSR

D.1 Data Collection

To make a direct meaurement of the free spectral range of the Fabry-Perot cavity,we need to collect a range of frequencies at which a Fabry-Perot peak is located.This can be done by slowly tuning the red laser with the PZT controller andwatching the transmission signal of the cavity on the oscilliscope. Where thetransmission spikes, record the frequency of the red laser as read on thewavemeter to form a list of frequencies fj .

D.2 Analysis

By taking the difference between any pair of frequencies in that list, find a list offrequency differencies ∆fj . Now, any element of ∆fj should be an integermultiple of the free spectral range. The true free spectral range should thenminimise the quantity

N∑j=1

(∆fj − nj(FSR)k)2 = X2(FSRk)

We want to compute this X2 for a range of possible values for the free spectralrange FSRk, and then plot X2 v. FSRk to find the frequency for which X2 isminimised. For each FSRk we first find the correct integer nj for each ∆fj thatminimises the quantity

∆fj − nj(FSR)k

To do this in Mathematica, first assume some FSRk. Then, generate a list ofn(FSRk where n are integers from 1 to some reasonable maximum n for the

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values of ∆fj . Extract the closest value in that list to a given ∆fj . That value willbe nj(FSR)k. Repeat this process for the entire list of ∆fj . Now X2(FSRk) for theassumed FRSk is easily calculated. Repeat this entire process for a range ofdifferent free spectral ranges. Figure D.1 shows the resulting plot.

Figure D.1: Example plot of X2 v. FSRk.

Fit to the region around where X2(FSRk) is minimised, approximating the curveas a quadratic function. In this case, our plot of X2 v. FSRk suggests that the trueFSR ' 363.76 MHz.

Figure D.2: Fit data points around minimum X2 to a quadratic function. Red line is fittedcurve.

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