Measurement of the Hyperfine Structure of the 7P 1/2 state and 8P 1/2 state in 205 Tl and 203 Tl

Transcription

1 Measurement of the Hyperfine Structure of the 7P 1/2 state and 8P 1/2 state in 205 Tl and 203 Tl by Gabrielle Vukasin Professor Protik Majumder, Advisor A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in Astrophysics WILLIAMS COLLEGE Williamstown, Massachusetts May 18, 2014

2 Abstract We report a final value of the hyperfine splitting of the 7P 1/2 state of 205 Tl and 203 Tl made using a two-step excitation. Our final values are (8) MHz and (7) MHz respectively. We also measured the isotope shift of the 7S 1/2 7P 1/2 transition to be 534.4(9) MHz. These experimental hyperfine splitting values are 20 MHz larger than those measured by another group in 1988 [1]. Our values bring the experimental values closer to the theoretical values published in 2001 [2]. Our data consists of spectra taken by scanning the second-step laser 6 GHz. For precise measurement of these spectra, we stabilize the first-step excitation using a method called laser locking. Using the same experimental layout, we are now working to measure the the hyperfine splitting of the 8P 1/2 state of both isotopes.

3 Acknowledgments I would first like to thank my advisor, Tiku Majumder, for his guidance and generous assistance throughout every step of this thesis. I would like to thank Gambhir Ranjit for his diligent work in the lab and for acquainting me with the experimental processes of this lab; Nathan Bricault, for his help in the lab and general cheerfulness; Michael Taylor, for his expertise; Ward Lopes, for his helpful comments and advice; Sarah Peters, for her help constructing the external cavity diode laser over the summer; Ben Augenbraun for his work during Winter Study and the spring; and my friends and family, for all of the support they provided.

4 Executive Summary This thesis describes the final steps of the precise measurement of the F = 0 F = 1 hyperfine splitting of the 7P 1/2 state in the two naturally occurring isotopes of thallium: 205 Tl and 203 Tl. Additionally, this thesis describes the preliminary steps of measuring the analogous hyperfine splitting of the 8P 1/2 state in the same two isotopes. We also precisely measure the hyperfine anomaly and isotope shift between the two isotopes. Hyperfine anomaly and isotope shift illuminate the isotopic differences in the nuclear structure between 205 Tl and 203 Tl. Ultimately we would like to use our precise measurements of hyperfine structure to test the accuracy and guide the refinement of atomic theory surrounding modeling short-range wavefunctions of the valence electron. Due to thallium s very heavy nature (Z = 81), accurate atomic theory, coupled with existing and future precision measurements can result in important atomic-physics-based tests of the Standard Model and physics beyond it. Atomic theory models benefit from the single valence p electron structure in thallium, allowing a semi-hydrogenic starting point for calculation. We study hyperfine structure of thallium through a double-excitation of thallium vapor. The Majumder group has used this method to measure the hyperfine structure through hyperfine splitting and Stark shift in both thallium and a similar element, indium, with great precision [3] [4] [5]. When studying the 7P 1/2 state, we use a UV (378 nm) external cavity diode laser (ECDL) to excite the first transition from 6P 1/2 7S 1/2 and an IR (1301 nm) ECDL to excite the second transition from 7S 1/2 7P 1/2. Figure 1 shows this doubleexcitation scheme. In the future, to study the hyperfine structure of the 8P 1/2 state, we will use the same UV laser to excite the first transition, but with the substitution of a 671 nm ECDL to excite the second transition of 7S 1/2 8P 1/2.

5 ii Figure 1: 7P 1/2 Energy Level Diagram An energy level diagram of the double excitation of the 7S 1/2 7P 1/2 transition. The hyperfine splittings (HFS) of each isotope and the isotope shift are labeled for the 7P 1/2 state. F'' = 1 F'' = 1 7P 1/2 { 7P 1/2 HFS 1301 nm F'' = 0 7P 1/2 Isotope Shift F'' = 0 7P 1/2 HFS 7S 1/2 F' = nm 6P 1/2 F = 1 Thallium 205 Thallium 203 The experimental setup (see figure 2) needed to execute a double excitation of thallium atoms requires the two lasers to spacially overlap in a heated cell of thallium vapor. In the 7P 1/2 experiment, we lock the UV laser to the 6P 1/2 (F = 1) 7P 1/2 (F = 1) hyperfine transition and sweep the IR laser 5 GHz across the 7P 1/2 resonant frequency. The locking technique that was developed in this lab previously is based on the thallium atoms themselves. We then look at the absorption spectra of the IR laser (see figure 3) using a photodiode and lock-in amplifier because the absorption is very small. Using the same vapor cell, we can excite either a single isotope or both isotopes by locking the frequency of the first-step UV laser to a particular frequency point in the UV transition spectrum. Before the IR laser beam reaches the thallium vapor cell, it is split and half is sent through an electro-optic modulator (EOM) to the vapor cell and the other half if sent to a Fabry-Perot (FP) cavity to monitor the frequency scan. Both the EOM and the FP cavity are important to linearize and calibrate the frequency axis of the absorption spectra. In our analysis of the resulting spectra, the first step is to fit the FP transmission spectra to remove the the scan non-linearity caused by the piezoelectric transducer (PZT) used to tune the laser frequency during the scan. Having removed the non-linearity, we use the

6 iii frequency modulated (FM) sidebands produced by the EOM to perform absolute frequency calibration. As seen in figure 3b, these FM sidebands produce copies of the hyperfine peaks at a precisely known ( MHz) frequency separation. Thallium Cell in Oven 1301 nm laser PD Shutters Half-Wave Plates Lock-In Amplifier 50/50 Beam Splitter Dichroic Optical Chopper PBS EOM FP Cavity 600 MHz Synthesizer PD 378 nm laser Isolator Locking Setup AOM To Computer Figure 2: Experimental Setup As mentioned before, we can choose to excite either one or both isotopes by choosing a certain frequency with which to lock the UV laser. We can determine the hyperfine splitting of the 7P 1/2 state from the single isotope spectra of each isotopes. To determine the isotope shift, however, we need to have the spectra of both isotopes on the same frequency axis, which means we must excite both isotopes at the same time. We can excite Doppler shifted atoms of both isotopes by locking our UV laser at a frequency between the resonance frequencies of the two isotopes. By studying spectra for both the co and counter-propagating laser beam configurations, we can remove the relative Doppler shift and reveal the true isotope shift. The measurement of the hyperfine splitting and isotope shift for the 7P 1/2 state of both isotopes have been completed. We analyzed both single and dual isotope spectra accounting for many sources of systematic and statistical error. The resulting HFS value of 205 Tl is (8) MHz and of 203 Tl is (7) MHz. We found the isotope shift for the 7S 1/2 7P 1/2 transition to be 534.4(9) MHz. The error associated with each of these three values is less than 1 MHz, allowing a careful comparison of our results to previous work. These results and the corresponding data analysis were published in January 2014 [10]. Our results show distinct disagreement with the experimental results of the Grexa et al. group [1] and show improved agreement with theoretical calculations [2]. The rest of this thesis focuses on working towards the measurement of the hyperfine struc-

7 iv ture of the 8P 1/2 state. The laser controlling the second-step excitation is a 671 nm ECDL that we have created in our lab. We are working to get the optical setup ready to take spectra for this state. Figure 3: 7P 1/2 Dual and Single Isotope Spectra Normalized Intensity Frequency (MHz) (a) Dual Isotope Spectrum Normalized Intensity Frequency (MHz) (b) Single Isotope Spectrum

8 Contents 1 Introduction The Standard Model and Parity Non-Conservation Purpose for Hyperfine Structure Study of Thallium Measurement of the 7P 1/2 Hyperfine Structure Measurement of the 8P 1/2 Hyperfine Structure Atomic Structure Details Fine Structure Hyperfine Structure Hyperfine Difference and Hyperfine Anomaly Thallium Atomic Structure Isotope Shift Hole Burning and Doppler Broadening Approximating Atomic Transition Lineshapes in Spectra Experimental Setup External Cavity Diode Lasers nm Laser nm Laser Laser Locking Importance of Laser Locking Laser Locking Set-Up nm Signal Experimental Layout Detection of the Second Step Signal Data Analysis and Results for the 7P 1/2 Experiment Data Acquisition Linearization and Calibration of the Frequency Axis v

9 CONTENTS vi Fabry-Perot Calibration and Linearization EOM Calibration Interpreting Spectra Single Isotope Spectra Single Isotope Error Analysis Systematic Error Search Subdividing Data Dual Isotope Spectra Eliminating Doppler Shift with Dual Beam Configuration Interpreting Dual Isotope Spectra Dual Isotope Error Analysis Conclusions The 8P 1/2 Experiment Introduction P 1/2 Simulated Spectra Peak Approximation Single Isotope Spectra Dual Isotope Spectra Current State of Experiment and Future Work A Matlab Code 55 A.1 ThalliumFitting.m A.1.1 getdatathallium.m A.1.2 downsampleandnormalizethallium.m A.1.3 FabryPerotFittingThallium.m A.1.4 FrequencyLinearizationThallium.m A.1.5 VoigtFitThallium.m A.1.6 VoigtFitThallium235.m A.2 DataAnalysisThallium.m B 7P1/2 Data Tables 60 C Detailed Description and Usage of the 671 nm Laser 64

10 List of Figures 1 7P 1/2 Energy Level Diagram ii 2 Experimental Setup iii 3 7P 1/2 Dual and Single Isotope Spectra iv 2.1 Hyperfine Differences P 1/2 Energy Level Diagram P 1/2 Energy Level Diagram Hole Burning Comparison of Three Distributions for Comparable Widths The 6P 1/2 7S 1/2 Transition Internal Diode Laser Cavity nm External Cavity Diode Laser Laser Locking Setup Stability of a Locked Laser versus an Unlocked Laser [6] Simulated Absorption Signal of the 6P 1/2 (F=1) 7S 1/2 (F =1) Transition for Both Isotopes Experimental Setup Red Laser Transmission Spectrum due to Optical Chopper Simultaneous Lockin Amplifier and Fabry-Perot Data Polynomial of Fabry-Perot Peak Positions Second Step Transition Single-Isotope Data Run with Residuals Comparison of HFS Values by Data Subsets Dual Beam Configurations P 1/2 Dual Isotope Spectrum Beam Misalignment Correlation of Transition Isotope Shift and Doppler Shift Simulated 205 Spectrum vii

11 LIST OF FIGURES viii 5.2 Simulated Dual Isotope Spectrum C.1 External Laser Cavity C.2 Mount for the Laser Diode and Collimating Lens C.3 Sliding Mount for the Mirror Mount C.4 Wedge Diffraction Grating Mount C.5 PZT Tube C.6 PZT Cap

12 Chapter 1 Introduction 1.1 The Standard Model and Parity Non-Conservation The study of atomic structure is of great importance, for the knowledge of subatomic interactions remains incomplete. Experiments of all varieties are essential in testing the Standard Model of Elementary Particles and physics beyond what we already know. The Standard Model describes the way particles interact based on the electromagnetic force, the strong force, and the weak force. The focus of this introduction is on the electroweak interaction, a force normally studied in accelerator-based experiments. The weak interaction presents itself in the decay of charged particles, mediated by the W+ and W- bosons. We now know that the weak interaction is also manifested in the exchange of the neutral Z 0 boson by neutral particles. In 1957, an unusual property of the weak force was discovered called parity non-conservation (PNC), or violation of parity symmetry [7]. Of the three forces that govern interactions of elementary particles, only the weak force has been proven to violate parity. PNC of the weak force means that weak force interactions of elementary particles, including their exchange of W+, W-, and Z 0 bosons, show a fundamental handedness in nature. 1.2 Purpose for Hyperfine Structure Study of Thallium The exchange of W+ and W- bosons cause beta-decay of atoms. We are interested in stable atoms, so we study weak interactions involving the Z 0 boson. The Z 0 bosons are exchanged between electrons and nucleons just as photons are exchanged between electrons and protons in the electromagnetic interaction. Unlike the photon, however, the Z 0 boson has a very short range of influence. The short lifespan of the Z 0 boson is about seconds due to its very large mass [6]. This means the maximum distance traveled during its lifetime is sec/10 8 m/s = meters, assuming its speed is on the order of the speed of light. 1

13 CHAPTER 1. INTRODUCTION 2 Because this is a very short distance, only electrons traveling through the nucleus, with an average diameter of meters, will have an effect on the nucleus via the Z 0 boson exchange [6]. Although the probability seems low, Z 0 boson exchange has been observed because of the characteristic parity violating signature of the weak force. In an atom, the observed PNC effect, ɛ P NC, is governed by the equation [8]: ɛ P NC = Q C(Z) (1.1) where Q is the weak charge of the weak interaction, C(Z) is a quantity determined by the wavefunction of the valence electron, and Z is the atomic number of the atom. Q generally increases as Z increases. It is also the quantity that we wish to measure, which can only be measured indirectly if we have C(Z) and ɛ P NC. Furthermore, we know that C(Z) Z 3 f(z), where f(z) is an increasing first order function [6]. Thus, an atom with a large atomic number will will show a greater effect of the PNC force from both C(Z) and Q. Because we cannot calculate the exact wavefunctions for the orbiting electrons for Z 1, we must approximate C(Z). The type of study that incorporates such approximations is called ab initio atomic theory. As a consequence, the accuracy of Q is limited by the accuracy of C(Z) in heavy complicated atoms. Due to the constraints of C(Z), we chose thallium to study because both of its natural isotopes, 203 Tl and 205 Tl, are heavy hydrogenic atoms, meaning they each have essentially one valence electron in the 6P orbital at the ground state. There are actually three valence electrons, but the first two fill the 6S orbital, so we can treat the 6P electron as the singular valence electron. Because thallium is a hydrogenic atom, the wavefunction of the 6P electron looks like the wavefunction of the electron of the hydrogen atom to at least first order. This is a starting point for theoretical approximations of C(Z) and other atomic structure calculations. Although the atomic number of thallium (Z = 81) is large, ɛ P NC is 9 orders of magnitude smaller than the electromagnetic force. However, it has been measured before. In our lab, we study a wide variety of atomic properties in thallium and indium. Some of these experiments involve long-range and short-range atomic interactions. Short-range atomic interactions, such as the hyperfine interaction, are useful to study because they give us insight on the true nature of the electrons wave functions close to the nucleus, something essential for the calculation of parity violation. These atomic structure phenomena are calculated using equations analogous to equation 1.1 but for physical properties other than the weak charge. For example, the approximated valence electron wavefunction associated with another physical property would be referred to as D(Z). Since we can approximate this D(Z) in hydrogenic atoms using hydrogen as an initial model and we observe the effect of the phenomena, ɛ, we can then test and adjust the models for this D(Z). The atomic theory used to approximate the wavefunctions in order to approximate D(Z) can then be

14 CHAPTER 1. INTRODUCTION 3 applied to C(Z) to make it more accurate, thus making Q more accurate. It is important to test these theoretical ab initio atomic calculations with experimental results. If atomic ab initio theory is well understood, then atomic experiments can probe physics of a more fundamental nature, thus testing the Standard Model. The specific atomic property that is the focus of this thesis is called the hyperfine structure. Hyperfine structure is caused by the interaction between the nuclear magnetic moment and the magnetic field created by the moving electrons. This creates a splitting in the energy level structure in absence of spin-orbit coupling. This splitting is called hyperfine splitting (HFS). The purpose of these experiments is to accurately and precisely measure the hyperfine structure of both thallium 205 Tl and 203 Tl using methods developed in this lab previously. More specifically, we would like to test the theoretical calculation of hyperfine structure of thallium done by the group Kozlov et al. [2] to test and improve their approximations of quantities determined by wavefunctions of the electrons, which is just a D(Z) relating to HFS. Our results will test the accuracy of and guide refinement of the atomic theory used to approximate this D(Z), which is the ultimate goal of this lab. A secondary reason for the specific experiment to be explained in the rest of this document is to find an accurate value for the measurement of the 7P 1/2 hyperfine splitting. The purpose of this is to get a precise and accurate value to compare to the experimental value of the group Grexa et al. [1]. In 1988, Grexa et al. published a paper with experimental results for the HFS and isotope shift, to be explained later, of many hyperfine levels in thallium [1]. In 1993, this same group published another paper with corrected values for some of the hyperfine splitting values they published in their previous paper [9]. The 7P 1/2 values were not among those values corrected in the second paper. Therefore, our goal is to remeasure the 7P 1/2 HFS. To ensure calibration accuracy, we employ two complementary methods of calibration. With the knowledge of success of the measurement of the hyperfine splitting in the 7P 1/2 state, we will use this method for higher levels. 1.3 Measurement of the 7P 1/2 Hyperfine Structure In this first experiment, we measure the 7P 1/2 hyperfine structure and transition isotope shift in both isotopes of thallium. In order to excite the valence electron to the 7P 1/2 state, we use a double excitation. Because of selection rules, we cannot excite the thallium atoms from 6P 1/2 to 7P 1/2 with one single wavelength. Electric dipole selection rules of transitions do not allow for a transition between two levels with the same angular momentum quantum number, L. Because 6P 1/2 and 7P 1/2 have the same L = 1, this transition is not possible. Thus the need for a double excitation of the atoms arises. We use a 378 nm laser to excite

15 CHAPTER 1. INTRODUCTION 4 the atoms from their ground state, 6P 1/2 (L = 1), to 7S 1/2 (L = 0). Then we use a 1301 nm laser to excite these atoms from 7S 1/2 to 7P 1/2 (L = 1). Both of these transitions are allowed by the selection rules. The work done by David Kealhofer [8] of setting up the experiment and taking the initial data was finished this year with more data taken and a complete data analysis. The result of David Kealhofer s work was an initial measurement of the hyperfine splitting of 205 Tl of 2177 ± 1 MHz [8]. With additional trials and data analysis, we find a hyperfine splitting value for 205 Tl and 203 Tl, and we find a value for the transition isotope shift between the two isotopes. This data analysis and error analysis that will be discussed in chapter 4 expands upon what is in our manuscript published in January of 2014 [10]. 1.4 Measurement of the 8P 1/2 Hyperfine Structure Our new experiment focuses on measuring the 8P 1/2 hyperfine splitting between the F =0 and F =1 hyperfine energy levels. This also requires a double-excitation of 205 Tl and 203 Tl vapor using two diode lasers for the same reason as for the 7P 1/2. Except for section 4, the rest of this research focuses on the 8P 1/2 hyperfine structure. Section 3 describes the setup of the entire experiment as well as individual pieces of equipment such as the 671 nm external cavity diode laser and the locking procedure of the 378 nm laser. Section 5.2 includes the simulated spectra that we expect to see in the future for this experiment.

16 Chapter 2 Atomic Structure Details 2.1 Fine Structure Fine structure is the first deviation from the energy level structure in absence of angular momentum coupling. In our research we measure the effects of a further deviation, hyperfine structure, but a certain understanding of fine structure is needed to research additional deviations. Fine structure is caused by the interactions between the spin and orbital moments of an electron; it is called spin-orbit coupling [11, p. 187]. Fine structure can be described by three quantum numbers: J, S, and L. The total angular momentum, J, is the sum of the angular momentum due to the spin moment of the electron, S, and the angular momentum due to the orbital moment of the electron, L, which are related by the vector addition of J = S + L. S equals ± 1 for all electrons. L has a range of 0 to n 1, where n is 2 the principle quantum number. The following equation relates these terms to produce the spin-orbit coupling energy, v L,S [11, p. 190]: v l,s = a f [J(J + 1) l(l + 1) S(S + 1)] (2.1) 2 where a f is the spin-orbit coupling constant generated by a radial integral. We can also express how the spin-orbit coupling energy affects the wavefunction of the valence electron by appending the following Hamiltonian to the Hamiltonian of the total wavefunction [12, p. 125]: H fine = a f L S (2.2) The quantum numbers S, L and J are used to describe the fine states created by spinorbit coupling as follows. Every orbital has two fine structure state; one with J = L and one with J = L 1. Each level is denoted, by n, J, and the letter corresponding to 2 L, producing the notation nl J. For example, the fine states 6P 1/2, 7S 1/2, and 8P 1/2 are 5

17 CHAPTER 2. ATOMIC STRUCTURE DETAILS 6 due the spin-orbit coupling of the s = 1 6P electron, s = + 1 7S, and s = 1 8P electron respectively. We will now explore how these states are further split into finer quantized levels. 2.2 Hyperfine Structure The magnetic field created by the moving valence electron interacts with the magnetic dipole moment of the nucleus. The result is a coupling between the angular momentum of the electron, J, and the angular momentum of the nucleus, I. This coupling is called the hyperfine interaction. It creates splittings of the fine energy states in wavelength on the order of 10 4 to 10 7 ev [12, p. 168]. Hyperfine splitting can only occur if the spin of the nucleus is not zero and hence has a nonzero magnetic moment [13, p. 9]. The total angular moment, F, due to the coupling of the angular momenta of the nucleus and electron is summed by the vector addition of [11, p. 342]: F = J + I (2.3) where F ranges from J I, J I + 1,..., J + I. Therefore, the number of hyperfine energy states (possible values of F ) is: # of HF states = { 2I + 1 2J + 1 if I < J if I > J (2.4) The reason our lab studies the hyperfine structure in Thallium is because it gives us insight on the wavefunction of the valence electron. Just as with fine structure and the spin-orbit coupling addition to the Hamiltonian of the wavefunction of the valence electron, the addition to the Hamiltonian due to the hyperfine splitting is [12, p. 170] : H hf = ai J (2.5) a is the magnetic dipole hyperfine coupling constant, which is slightly different for each isotope due to the slight change in mass and charge density. a also depends on the details of the wavefunction of the electron in the vicinity of the nucleus. There can be other multipole terms in the multipole expansion of the gradient of an electric or magnetic field, due to interactions such as the electrostatic interaction between the electrons and protons of the nucleus, for nuclei with I > 1. The interactions will contribute more terms to the 2 Hamiltonian in addition to the two dipole terms we have described thus far: H fine and H hf. However, thallium s nucleus has a property that I = 1 2, so we do not have to take multipole terms beyond dipoles into consideration [13, p. 10]. The other element we study

18 CHAPTER 2. ATOMIC STRUCTURE DETAILS 7 in our lab, indium, has a nucleus with I = 9 and consequently has dipole, quadrupole, and 2 octupole terms in the multipole expansion of the magnetic interaction between its electrons and nucleus. Now we must find the energy, E, corresponding to the hyperfine Hamiltonian through H hf ψ = Eψ, which must be a true statement for eigenfunction ψ because H hf is additive to the original Hamiltonian. The first step is to find I J. A simple trick is to take the dot product of equation 2.3 with itself. This results in: F F = I I + 2I J + J J (2.6) F 2 = I 2 + 2I J + J 2 (2.7) I J = 1 2 (F 2 I 2 J 2 ) (2.8) Because I J is actually the dot product of two operators, we should include the eigenfunctions in equations 2.6 through 2.8. J 2 ψ is the operator of the electronic angular moment which can be described by [14, p ]: J 2 ψ = h 2 J(J + 1)ψ (2.9) Similarly, the angular momentum operators F 2 and I 2 have energy eigenvalues of h 2 F (F + 1) and h 2 I(I + 1) respectively. Very closely related to the fine structure spin-orbit coupling energy, the resulting eigenvalue equation, equivalent to equation 2.8, for each hyperfine energy level is: E HF S = a [F (F + 1) I(I + 1) J(J + 1)] (2.10) 2 where a is again the hyperfine constant. To find the energy between two consecutive hyperfine energy levels, since I and J are the same for both hyperfine levels, we subtract equation 2.10 for F + 1 from that for F and end up with: E F +1 E F = a(f + 1) (2.11) This is the energy that we want to find in our experiments, for the 7P 1/2 and 8P 1/2 states, between the hyperfine energy levels F = 0 and F = 1, so a is exactly the hyperfine splitting. We call this energy the hyperfine splitting of a fine structure state. Since the goal of our experiment is to measure the hyperfine splitting, we will be experimentally finding a value for a. We will be measuring the effect a has on the addition to the Hamiltonian of equation 2.5.

19 CHAPTER 2. ATOMIC STRUCTURE DETAILS Hyperfine Difference and Hyperfine Anomaly Figure 2.1: Hyperfine Differences This is an exaggerated comparison of the hyperfine splittings of both isotopes of thallium. The level isotope shift and hyperfine difference, which leads to the calculation of hyperfine anomaly, are both illustrated for a generic fine structure energy level. F = 1 FS Level a 205 /4 a 205 3a 205 /4 Isotope{ Shift a 203 /4 205 Tl 3a 203 /4 203 Tl a 203 F = 0 Hyperfine Difference = a a 203 The spectra of both isotopes differ in two ways. The first is that the relative sizes of the hyperfine splittings are different. The second is that the relative sizes of the second transition in the double excitation differ as well. The former effect is called the hyperfine anomaly and the latter is called the transition isotope shift, which will be discussed in section 2.4. Both affects are exaggerated in figure 2.1. It shows the relative sizes of the spacing between F = 0 and F = 1 hyperfine states of this arbitrary fine structure level for both isotopes. Hyperfine anomaly can be calculated by: = [ H 7P,205 H 7P,203 g 203 g 205 1] (2.12) where H 7P,205 and H 7P,203 are the hyperfine splittings of 7P 1/2 of 205 Tl and 203 Tl respectively. g 203 and g 205 are the nuclear g-factors of 203 Tl and 205 Tl. Hyperfine anomaly is caused by the change in charge distribution due to the additional neutrons because the nuclear magnetic dipole moment is an average over the total volume of the nucleus and thus so must be the hyperfine splitting phenomenon [12, p. 178].

20 CHAPTER 2. ATOMIC STRUCTURE DETAILS Thallium Atomic Structure As mentioned in the introduction, of the ground state configuration, [Xe] 4f 14 5d 10 6s 2 6p 1, we can treat the singular 6P electron as a valence electron because the two 6S electrons are paired. We are interested in the multiple excited states of this valence electron. In both figures 2.2 and 2.3, the fine and hyperfine structure of the excited states in which we are interested in our experiments is expanded. One thing to notice about these diagrams is that each fine structure state only has two corresponding hyperfine states. Thallium s nuclear angular moment is 1. This means that there can only be two hyperfine levels at any given 2 fine structure level, using equation 2.4. The spacing between the fine states of the 7S 1/2 8P 1/2 transition in figure 2.3 is larger than that of the 7S 1/2 7P 1/2 transition in figure 2.2 as an exaggeration of the actual relative sizes of these two transitions in energy. The electron in the 8P 1/2 state has higher energy than in the 7P 1/2 state, so the transition wavelength from 7S 1/2 is shorter. Another point of interest is the spacing of the F = 0 and F = 1 levels relative to the fine energy levels. Substituting our values for I = 1, J = 12, and F = 0, 1, we find the hyperfine 2 levels have a shift relative to the fine structure of: F = 0 : E HF S = a 2 [0(0 + 1) 1 2 ( ) 1 2 ( )] = a 2 [ ] = 3a 4 F = 1 : E HF S = a 2 [1(1 + 1) 1 2 ( ) 1 2 ( )] = a 2 [ ] = +a 4 (2.13) (2.14) These HFS shifts are indicated in figure 2.1. Note that this insures that the weighted average of the two levels is 0 because F = 1 has three times the number of sublevels as F = 0.

21 CHAPTER 2. ATOMIC STRUCTURE DETAILS 10 F'' = 1 7P 1/2 { 7P 1/2 HFS 1301 nm F'' = 0 7P 1/2 Isotope Shift F'' = 1 F'' = 0 7P 1/2 HFS 7S 1/2 F' = GHz { F' = GHz F' = 0 F' = nm F = 1 F = 1 6P 1/ GHz 21.1 GHz F = 0 Thallium 205 Thallium 203 F = 0 Figure 2.2: 7P 1/2 Energy Level Diagram A partial diagram of the hyperfine energy structure of both isotopes for the 6P valence electron. It is not to scale. Each nl label represents the fine structure energy levels and the levels labeled with various degrees of F represent the hyperfine energy levels. The double excitation experiment from 6P 1/2 to 7P 1/2 is represented. The first excitation from 6P 1/2 to 7S 1/2 is depicted by the blue arrow at 378 nm. In reality, the second laser will be scanning a few GHz to reach both of the 7P 1/2 F =0,1 hyperfine levels. The absorption peaks on the individual IR scans will be the transitions indicated in red. The isotope shift is highlighted in green.

22 CHAPTER 2. ATOMIC STRUCTURE DETAILS 11 F'' = 1 F'' = 1 8P 1/2 8P 1/2 HFS F'' = 0 { F'' = 0 8P 1/2 HFS 671 nm 8P 1/2 Isotope Shift 7S 1/2 F' = GHz { F' = GHz F' = 0 F' = nm F = 1 F = 1 6P 1/ GHz 21.1 GHz F = 0 Thallium 205 Thallium 203 F = 0 Figure 2.3: 8P 1/2 Energy Level Diagram A partial diagram of the hyperfine energy structure of both isotopes of the 6P valence electron, much like figure 2.2, which is again not to scale. The double excitation experiment from 6P 1/2 to 8P 1/2 is represented here. The 6P 1/2 to 7S 1/2 transition is again depicted by the blue arrow. The 7S 1/2 to 8P 1/2 transitions from F = 1 to F = 0,1 are highlighted in red, at a frequencies near 1301 nm. The red arrows indicate the two transitions that will be visible on the red laser spectra. The isotope shift is highlighted in green.

23 CHAPTER 2. ATOMIC STRUCTURE DETAILS Isotope Shift The study of the transition isotope shift is the study of two phenomena called the field effect, also known as the volume effect, and the mass effect. The mass effect is proportional 1 to a change of for atoms of large Z [12, p. 194]. Therefore, the mass effect is so small in M 2 thallium that it is negligible compared to the volume effect and we do not need to account for it in our experiment. The volume effect gets its name because the increased number of neutrons in different isotopic nuclei increases the volume of the nucleus. This effect manifests in changing the energy of the atom because the charge density of the nucleus changes when the number of neutrons changes, which will ultimately alter the spectra of the different isotopes in our experiment [11, p. 339]. Thus, this effect is an important measurement to our experiment because it illuminates differences in atomic level phenomena due to additional neutrons. Unlike the hyperfine splitting, the isotope shift does not rely on a nonzero nuclear spin. Therefore, we see an isotope shift between isotopes of any kind of nucleus. The transition isotope shift presents itself as the difference of the frequency of a certain transition for one isotope and the frequency of the same transition for another isotope. In the first experiment we find the isotope shift for the 7S 1/2 7P 1/2 transition and in the second experiment we will find an isotope shift for the 7S 1/2 8P 1/2 transition. Measuring the transition isotope shift cannot be done using the single isotope data we use to measure the hyperfine splitting. In order to measure differences between the spectra of the two isotopes, we need to have their spectra on the same frequency axis. The frequency axis of each spectrum we take represents relative and not absolute frequency. Therefore, we take dual isotope data with both isotopes F = 0 and F = 1 hyperfine energy states on one spectrum. Dual isotope spectra will be discussed in section Hole Burning and Doppler Broadening A gradient of velocity classes of thallium atoms in the ground state arises because atoms in the gas state within the quartz cell move at different velocities. The only velocity component of the atoms that creates different velocity classes is the velocity component with respect to the axis of propagation of the laser. The different velocity classes see the frequency of the laser light as: f = f o 1 + v (2.15) c according to their respective velocity components, v, and the frequency seen by atoms with a velocity of zero, f o. In double-excitation laser spectroscopy, the second laser excites a

24 CHAPTER 2. ATOMIC STRUCTURE DETAILS 13 single velocity class of atoms. By selecting the frequency of the first laser, we excite only one velocity class of atoms to the second step, in which each doubly-excited atom has relatively the same Doppler shift. The method of exciting a single velocity class in the first transition in a two-step excitation is called hole burning. Hole burning, or saturated absorption, gets its name because by exciting a very specific velocity class of atoms, the atoms left at the ground state now have a Maxwell velocity distribution with a hole cut out due to the missing velocity class, as seen in figure 2.4 [11, p. 381]. Figure 2.4: Hole Burning The bottom plot represents the absorption profile of the ground state atoms in the presence of hole burning. The top plot represents the absorption profile of the excited atoms due to hole burning. Excited State Natural Line Width Hole Burning Doppler Width Ground State Frequency The line width of the absorption profile of the excited atoms is approximately the natural line width, with a small contribution from Doppler broadening, if the laser beam is diverging, for example. This line width is much smaller than the line width of the ground state absorption profile, which is due to Doppler broadening. Therefore, it is clear that hole burning reduces the effects of Doppler broadening, although there will always be a small component of the line width of the absorption profile of the excited atoms due to Doppler

25 CHAPTER 2. ATOMIC STRUCTURE DETAILS 14 broadening. Additionally, the excited state absorption profile may be further broadened by power broadening if the power is high enough. However, compared to Doppler broadening, power broadening is negligible for the most part. 2.6 Approximating Atomic Transition Lineshapes in Spectra Gaussian Lorentzian Voigt Figure 2.5: Comparison of Three Distributions for Comparable Widths We can model the absorption curve using a Voigt profile, which is a convolution of the Gaussian and Lorentzian profiles. The Gaussian, Lorentzian, and Voigt distributions are compared in figure 2.5. These distributions are scaled to have the same full width at half maximum (FWHM). The biggest difference between the three profiles occurs in the tails of the distributions where they begin to deviate from each other noticeably. The Gaussian nature of the peak is due to the inhomogeneous Doppler broadening caused by the Maxwell- Bolzmann distribution of the kinetic energy of the thallium atoms at T = 400 C to 450 C. The Doppler effect causes different velocity (kinetic energy) classes to arise and thus is an inhomogeneous form of broadening because it affects each atom or groups of atoms differently [6]. Table 2.1: Lifetimes of Excited Thallium Atoms Fine State Lifetime (ns) [15] Natural Line Width (MHz) 7S 1/ P 1/ P 1/

26 CHAPTER 2. ATOMIC STRUCTURE DETAILS 15 The Lorentzian nature of the absorption peaks is due to the homogeneous lifetime broadening of the transitions. The term lifetime broadening means that the finite lifetime, τ, of an electron in an excited state produces uncertainty in energy [8]. This is explained by the Heisenberg Uncertainty Principle. Uncertainty in lifetime presents itself as the FWHM, or γ, of the Lorentzian. We call this the natural line width of the transition. Table 2.1 contains the average lifetimes an electron has in the 7S 1/2, 7P 1/2, and 8P 1/2 states and their corresponding natural line widths from the equation: γ = 1 2πτ (2.16) Because both the 7P 1/2 and 8P 1/2 experiments have electrons excited from the 7S 1/2 state to their respective final states, the γ of both experiments must include the total γ of the 7S 1/2 state. Therefore, the actual γ s for the 6P 1/2 7S 1/2, 7S 1/2 7P 1/2 and 7S 1/2 8P 1/2 transitions are: 6P 1/2 7S 1/2 : γ = γ 6P + γ 7S = 21.4MHz (2.17) 7S 1/2 7P 1/2 : γ = γ 7S + γ 7P = 24.0MHz (2.18) 7S 1/2 8P 1/2 : γ = γ 7S + γ 8P = 22.3MHz (2.19) (2.20) In practice, given the laser power levels we use, power broadening also occurs. Power broadening decreases the lifetimes of the atoms in the excited state because stimulated emission begins to occur at higher laser powers. The true Lorentzian width is about 2 times greater as a result, giving us actual γ s of 50 MHz. Now that we know the causes of the shape of the transition peaks we should theoretically be getting, we must compare the width contribution of each. The Doppler width (FWHM) of the Gaussian is calculated using: f F W HM = 8kT ln 2 mc 2 f o (2.21) where f o is the frequency of the transition. The estimate of the relative sizes of the Lorentzian width and the Doppler width as a measure of the relative number of atoms excited by the second-step laser in the 7P 1/2 experiment is: γ Γ Doppler 50MHz Γ 7P,Doppler 5% (2.22)

27 CHAPTER 2. ATOMIC STRUCTURE DETAILS 16 Figure 2.6: The 6P 1/2 7S 1/2 Transition This is the spectrum of the 6P 1/2 7S 1/2 transition including the hyperfine transitions of both isotopes. 203Tl 205Tl Transmission Signal F = 1 to F = 0 F = 1 to F = 1 F = 0 to F = Frequency (GHz) Figure 2.6 shows the total spectrum of the hyperfine transitions in the 6P 1/2 7S 1/2 fine structure transition. Using equation 2.21, the Doppler widths for 203 Tl and 205 Tl of the F=1 F =1 transition are MHz and MHz respectively. For this transition, the Doppler width is about 50 times greater than the natural line width. Therefore, we only need to worry about the Gaussian contribution of the Voigt profile for this transition. On the other hand, the hole burning of the second step excitation largely gets rid of Doppler broadening. Thus, in the second excitation, we can approximate the transitions in the spectra by Lorentzians. The natural line width dominates, as seen in figure 2.4. This will be more important when we fit the spectra from the red laser in chapter 4. Consider the absorption spectrum of the 6P 1/2 7S 1/2 transition for both 203 Tl and 205 Tl in figure 2.6. Of the three hyperfine transitions for both isotopes, we are interested in the F=1 F =1 transition, the middle two peaks. These two peaks span only a few GHz of the entire hyperfine structure of the 6P 1/2 7S 1/2 transition spanning about 35 GHz. We know that the wavelength of this transition for 205 Tl is nm and for 203 Tl is nm.

28 CHAPTER 2. ATOMIC STRUCTURE DETAILS 17 If we scanned the UV laser over only 3 GHz above and below the nm 6P 1/2 (F=1) 7S 1/2 (F =1) transition, we would see the entire F=1 F =1 transition, as seen in the spectrum in figure 3.5. We know that we will be using a Gaussian line shape to approximate this transition because Doppler broadening is a factor in this transition. To simulate the locking spectrum with the two modulated acousto-optic modulator (AOM) frequencies, to be described in the next chapter, we first take the Gaussian equation with the Doppler widths just calculated. Each Gaussian will have some scaling coefficient that relates to the height of the absorption peaks. The total absorption spectrum in the sum (ignoring isotopic differences in line width): A(f) = A 203 e (f f o,203) 2 /2Γ 2 + A 205 e (f f o,205) 2 /2Γ 2 (2.23) where A 203 is the relative abundance of 203 Tl and A 205 is that of 205 Tl. Since the relative abundances of 203 Tl and 205 Tl are 30% and 70%, we can normalize the signal using the values 3 and 1 for A and A 205 respectively. Finally, the transmission signal is given by: T (f) = I o e αa(f) (2.24) where α is the optical depth and I o is the intensity at 100% transmission. Since A(f) is normalized to be unity on resonance of 205 Tl, α is the optical depth on resonance. We choose a vapor cell temperature which produces α 1 because this ensures that the transmission dip, due to absorption of both isotopes, is easily measurable and not saturated.

29 Chapter 3 Experimental Setup Both the 7P 1/2 and 8P 1/2 experiments utilize the same setup except the second laser in the 7P 1/2 experiment has a 1301 nm laser in place of the 671 nm laser of the 8P 1/2 experiment. Other minimal differences include the exact Fabry-Perot (FP) cavity length and numerous focusing and beam-shaping optics. However, the concepts are the same, so I will only describe the set up of the 8P 1/2 experiment in this chapter. 3.1 External Cavity Diode Lasers Both lasers that we use are external cavity diode lasers (ECDL). This means that the laser diode is surrounded by a cavity formed by reflective optical elements. A diode laser uses the p-n junction of a diode to produce a coherent beam of light. Figure 3.1 depicts a simple version of an internal diode laser cavity. By sending current from the +Electrode to the -Electrode, electrons and holes are brought from the valence band to the conduction band. Holes and electrons then annihilate in the active region of the p-n junction. When a hole from the conduction band of the p-type semiconductor and an electron from the conduction band of the n-type semiconductor annihilate, a photon is emitted. The photons travel through the active region and reflect off of the flat faces, which act like mirrors. They travel back through the active region and if they collide with a free hole and electron, they induce stimulated emission. The amplified light is emitted through the flat faces of the active region. An external cavity diode laser works differently than a cavity of a laser diode. One of the flat faces of the diode is coated with an anti-reflection coating so that no stimulated emission persists, but spontaneous emission still occurs (becoming a light-emitting diode). The light then travels to a diffraction grating and the first order diffracted light is reflected back to the diode, as seen in figure 3.2. The grating selects one wavelength from the spontaneously 18

30 CHAPTER 3. EXPERIMENTAL SETUP 19 pn Junction p-type semiconductor + Electrode Active Region n- type semiconductor Flat Faces n-substrate Current - Electrode Figure 3.1: Internal Diode Laser Cavity Laser Output Diffraction Grating 1200 lines/mm First Order Diffraction Spontaneous Emission from Diode 671 nm Diode Figure 3.2: 671 nm External Cavity Diode Laser emitted photons to send back to the diode as the first order diffraction beam because the grating makes the external cavity a Fabry-Perot cavity with discrete wavelengths with which the cavity will lase. The first order diffraction beam induces stimulated emission in the

31 CHAPTER 3. EXPERIMENTAL SETUP 20 diode at its specific wavelength. The amplified light reflects off of the back face of the diode and returns to the diffraction grating. To change the wavelength of the laser, we apply a voltage to the PZT that changes the angle of diffraction grating ever so slightly resulting in a large change in wavelength. This works because the stimulated emission amplifies the first-order diffracted beam from the grating. So, changing the angle of the grating changes the wavelength of the first-order diffracted beam thereby changing the wavelength of the amplified light (diode output). Using the expression, nλ = dsin(θ) (3.1) we see that λ and Sin(θ) are directly proportional, so we can change λ accordingly. In this experiment we want to scan the laser about 7 GHz, which corresponds to a tiny change in λ of about , so we drive the PZT by a ramping the voltage of sufficient size to move the PZT by the required small distance. The result is continuous up and down frequency scans that can be used to acquire spectra. There are three benefits of the ECDL over the laser diode for our experiment. Laser diodes emit a large spread of frequencies above and below the center frequency. Therefore, the first benefit of the ECDL is to decrease the spread of frequencies around the center frequency emitted. Because the first-order diffracted beam is the frequency amplified, there is less of a range of other frequencies amplified at the same time. The frequency that the diode laser outputs itself can be changed by changing the temperature and current of the diode. However, we cannot take a spectrum by changing the temperature and current of the laser diode because these changes are not smooth, quick enough, nor directly proportional to the output frequency of the laser diode. Thus the second benefit of the external cavity is to provide a way to rapidly and proportionally change the frequency of the laser. The ECDL also allows us to induce lasing everywhere in the gain profile of the laser. One unfortunate consequence of using the PZT to scan the red laser is that the response of the PZT to a linear increase in voltage is nonlinear. This creates an issue when we look at spectra from the scanning red laser because the frequency axis will not be linear. This is one of the main reasons that we must calibrate the frequency axis nm Laser The gain medium of the 378 nm laser is a GaN semiconductor diode. However, it is different from the red laser because it does not have an anti-reflective coated diode. The wavelength of 378 nm is short enough that any coating on the diode wears rapidly over time. This is not the case for the red laser. Therefore, the UV laser is much more susceptible to mode hops

32 CHAPTER 3. EXPERIMENTAL SETUP 21 and is not as easily controlled by changing the angle of the grating as is the red laser. There is a constant battle between the internal laser cavity and the external cavity. If the right temperature, cavity length, current, etc. are chosen, the influence of the external cavity will prevail. Because we do not need to take spectra with the UV laser as we do with the red laser, it is adequate for our experiment that the tuning ability of the UV laser is low. It has a scanning range of about 2 GHz [8]. Despite its limited tuning ability, we can tune and lock the UV laser as necessary for hours at a time. It emits between 2 6 mw depending on its frequency. This laser system is manufactured by the German company Sacher Lasertechnik nm Laser In appendix C, we included an overhead view of the current 671 nm ECDL laser system that we built ourselves. Initially, we used our own 670 nm laser diode and coated it with antireflection coating ourselves using an evaporation chamber. Unfortunately, the evaporation chamber broke during the coating process and we did not get to coat the diode fully. When we tested the capabilities of the semi-ar-coated diode, we found that it was able to lase in the external cavity, but not at a stable enough frequency. Currently, the 671 nm ECDL has an AR-coated diode, confocal lens, and laser mount from Toptica Photonics. The laser diode is an AlGaInP semiconductor that outputs up to 30 mw of power. The specifications of the laser diode and parts we built can be found in appendix C. We use a Stanford Research Systems (SRS) LDC501 to control the laser diode s current and temperature. It sends current to the laser diode controller and controls the external cavity temperature via a thermoelectric device mounted below the laser. This ECDL has exactly the same functions as the 378 nm ECDL except that we can change the laser cavity length. Although, this is not an extra tuning feature, we have chosen a length of 2 cm and fixed the cavity there. One piece of this laser that is important to note is a wedge to tilt the diffraction grating closer to the θ we want in figure 3.2. From the equation 3.1, the angle that will provide us with a first order diffraction of 671 nm is: Sin 1 ( m ) 54 (3.2) m where m is the inverse of the 1200 lines per millimeter, which is characteristic of our grating. We cut the actual wedge to be 45, so we have tilted the diffraction grating an extra 9 to get to the desired 54.

33 CHAPTER 3. EXPERIMENTAL SETUP 22 Differential Photodiode and Amplifier Figure 3.3: Laser Locking Setup Thallium Cell in Oven PBS PID Servo Controller PBS Half-Wave Plates 378 nm Laser AOM Experiment 3.2 Laser Locking Importance of Laser Locking Laser locking is very important to minimize the the drift and instability of the resonance frequency of the second step transition. By locking the first-step frequency, this fixes the resonance frequency of the second transition. Without a fixed frequency of the first laser, the velocity class excited would change over time. Consequently, the resonance frequency of the second laser would drift many MHz because of the changing Doppler shift due to the change in which velocity class is excited. This would compromise the reliability of the second-step spectra which require 10 seconds to complete Laser Locking Set-Up Figure 3.3 is a diagram of the locking setup. To choose the correct frequency for the first excitation of both isotopes of thallium, we employ a supplementary smaller oven containing a vapor cell of thallium. We direct a 378 nm beam through an acousto-optic modulator (AOM), a device which uses a crystal to create two first-order, diffracted beams 260 MHz above and below the laser s frequency. The undiffracted beam proceeds to the main experiment as described later. The two diffracted beams are combined in a polarizing beam splitter (PBS). They then travel through the supplementary vapor cell and oven setup and go through another polarizing beam splitter to separate the beams before entering a special differential photodiode and amplifier system that outputs their difference signal, as seen in

34 Frequency (uncallibrated units) CHAPTER 3. EXPERIMENTAL SETUP 23 figure 3.5. We use this difference signal as the basis of our locking scheme. This signal is sent to a commercial servo proportional-integral-differential (PID) controller, which is designed to construct an appropriate correction signal that can be fed back to the laser. The servo PID controller works by taking the photodiode signal and translating it to [16]: P ɛ(t) + I t t τ ɛ(t)dt + D dɛ dt + N (3.3) where P, I, and D are the respective proportional gain, integral gain, and derivative gain. ɛ(t) is the input signal to the PID. τ is a characteristic time constant and N is the offset adjustable voltage. We use a method called the Ziegler-Nichols tuning method to optimize the output signal from the PID [8]. After the signal goes through the PID, it applies a correcting voltage to the piezoelectric transducer (PZT) of the UV laser, which steers the UV laser frequency to keep it locked. If the values for P and I are too large or too small, the PID will constantly be overshooting the lock point or reach it too slowly [6]. Therefore, it is important that these two parameters are chosen carefully. Figure 3.4: Stability of a Locked Laser versus an Unlocked Laser [6] A comparison of the frequency of the UV laser when it is and is not locked. The output of the locked UV laser is in red and the output of the unlocked UV laser is in black. We convert voltage into frequency using the slope of the linear portion of the difference signal near. 3.0 unlocked signal 2.0 ~10 MHz 1.0 locked signal 0.0 RMS ~ 0.7 MHz time (minutes)

35 CHAPTER 3. EXPERIMENTAL SETUP 24 Figure 3.4 is a comparison of the behavior of the UV laser when it is locked (black) and when it is unlocked (red). Over an 80 minute period, the unlocked UV laser both jumps and slowly drifts in frequency by 10 s of MHz from the original frequency. This is not adequate for our experiment. However, using this method of locking, we see that a noisy drifting laser output can be made stable at the < 1 MHz level. We calibrate the plot of figure 3.4 so that we can determine the residual RMS frequency of the locked laser. Initially, the amplitude of the residual RMS frequency is measured in voltage because it is a measure of the change in voltage of the difference signal. We can approximate the portion of the difference signal, of figure 3.5, relatively near the locking point as having a linear slope. The linear slope around the locking point is v, so we can use this ratio to convert voltage into frequency. However, f the initial x-axis of the difference signal is voltage scaled by the PZT ramp [6]. To convert the x-axis of the differential signal into frequency, we use simultaneous FP scans that have the same voltage x-axis. Since we know the free spectral range (FSR) of the FP cavity, we can use this to calibrate the x-axis of the difference signal into frequency and then calibrate the residual RMS voltage into RMS frequency nm Signal We use the 6S 1/2 (F=1) 7S 1/2 (F =1) transition spectrum to lock the laser. The simulated effects of the AOM are shown on figure 3.5 by shifting eq by +260 MHz and 260 MHz. The dashed line indicates the unshifted frequency component which is sent to the experiment. It is then straightforward to take the difference of the two diffracted signals and plot the result. In figure 3.5, the difference signal below has three corresponding areas where there is a linear portion of the signal. To take single isotope data, we want to lock the UV laser at the frequency of the center of the resonance peaks (dashed line) for either of the two isotopes. As mentioned before, the job of the servo PID is to send a correction signal to the laser to bring it back to the desired frequency. In the absence of the difference signal (that is looking at the absorption directly), at the center of the resonant frequency peaks of either isotope of the 6S 1/2 (F=1) 7S 1/2 (F =1) transition, a decrease in absorption will accompany any change in the frequency of the laser. Consequently, the PID will not be able to send a correction signal because it does not know in which direction the frequency of the laser has drifted. The difference signal allows us to easily tell if the laser is drifting above or below the desired resonance frequency. This is because the slope of the difference signal at the resonant frequencies of both isotopes is linear. Besides the resonant frequency of each isotope, there is a third linear location of the difference signal to which we can lock. Fortunately, this point is directly in between the iso-

36 CHAPTER 3. EXPERIMENTAL SETUP Normalized Transmission Signal Downshifted Signal 203 Lock Dual Isotope Lock 205 Lock Upshifted Signal 0.2 Difference Signal Frequency (MHz) Figure 3.5: Simulated Absorption Signal of the 6P 1/2 (F=1) 7S 1/2 (F =1) Transition for Both Isotopes The dotted blue line is the original signal of that would be transmitted if the UV laser scanned across the 6P 1/2 (F=1) 7S 1/2 (F =1) transition of both isotopes. The red and green lines represent the up and down shifted versions of the original signal. The solid blue line represents the difference of the up and down shifted versions of the original signal. topic resonances. Some atoms of each isotope will be excited because the absorption of each isotope is not zero at this frequency. To determine the transitional isotope shift, we need to excite both isotopes simultaneously, so this lock point is crucial. Because the dual isotope lock point is not at the frequency of either isotopic resonance, we are exciting a blue shifted velocity class of 203 Tl atoms and a red shifted velocity class of 205 Tl atoms. Luckily, the Doppler widths of the isotopic resonances are sufficient that we are able to excite substantial amounts of both isotopes. 3.4 Experimental Layout A sketch of the experimental setup is shown in figure 3.4. The double excitation of thallium requires first and foremost two lasers. The first is a 378 nm external cavity diode laser (ECDL) and the second is a 671 nm ECDL that we built ourselves. In our earlier 6S 1/2 7P 1/2 experiment, this laser was an infrared laser tuned to 1301 nm. There is an oven

37 CHAPTER 3. EXPERIMENTAL SETUP 26 Thallium Cell in Oven 671 nm laser PD Shutters Half-Wave Plates Lock-In Amplifier 50/50 Beam Splitter Dichroic Optical Chopper PBS EOM FP Cavity 200 MHz Synthesizer PD 378 nm laser Isolator Locking Setup AOM To Computer Figure 3.6: Experimental Setup containing a quartz cell of naturally occurring thallium (70% 205 Tl and 30% 203 Tl). The undiffracted component of the now-stabilized 378 nm laser beam passes through a 50/50 beam splitter. The two outputs send light through our thallium interaction region in opposite directions. After the beam splitter, there is one shutter in front of each UV beam that blocks and unblocks each beam before it reaches the interaction region. The 671 nm laser beam path is more involved. First, the beam is separated in a polarizing beam splitter. Half of the beam goes into a confocal Fabry-Perot cavity and the other half of the beam goes through an electro-optic modulator (EOM). The EOM is a device that uses electro-optic elements and a radio frequency signal (200 MHz) to create frequency modulated (FM) sidebands above and below the center frequency of the 671 nm laser beam. This is used for one of the frequency calibration methods, which is described in chapter 4. The frequency modulated 671 nm beam is then combined via a dichroic mirror with one of the UV beam components and sent to the interaction region. There are now three laser beams intersecting in the interaction region: two UV and one red. The shutters can block and unblock the two UV beams separately before they reach the interaction region. Using these shutters we can arrange the the two beams copropagating (CO) through the cell or counterpropagating (CTR) through the cell, as seen in figure 3.6. We align the three beams so that they spacially overlap in the interaction region. It is important that the two beams, in both configurations, maximally overlap in the interaction region so that the most UV excited atoms are excited by the red laser. The decay time for a UV excited atom is 7.43 ns [15]. In this time, the atom can travel a distance of 250m/s s 2µm given the approximate velocity of the hot atoms. Therefore, to have the greatest absorption signal possible from the red laser, the UV and red beams can be misaligned no more than 2µm, so

38 CHAPTER 3. EXPERIMENTAL SETUP 27 that the least amount of the UV excited atoms escape the red laser. After passing through the interaction region, the red beam then travels to a photodiode, which is connected to a lock-in amplifier using the signal from the optical chopper as its reference signal. The lock-in amplifier output is sent to a computer to collect and analyze the final spectra. 3.5 Detection of the Second Step Signal The first issue that we must deal with is the small scale of the resulting absorption signal. The diameter of both beams going through the cell is about 2 mm. The second laser can only excite the already excited 7S 1/2 atoms in this small volume constrained by the diameter of the beam and length of the cell. Therefore, as discussed, it must be very well aligned with the first laser beam to maximize the amount of excited atoms. On top of this, even if the beams are perfectly aligned, there will still be a very small absorption signal because the hole burning, as discussed in chaper 2, only excites a small portion of the UV excited atoms. We estimated the fraction of ground state atoms excited by the UV laser was 5%. So direct detection of the red beam transmission would show a tiny absorption dip with large background as suggested in figure 3.7a. To decrease the noise and thereby increase the signal-to-noise ratio, we use an optical chopper and a lock-in amplifier. We set the motor of the optical chopper to some speed and place it in front of the 378 nm laser so that the beam is modulated at 1000 Hz. It provides a signal that looks like a Sine function. When the wheel blocks the UV beam, there is 100% transmission of the red laser beam as seen in figure 3.7b. This is because the atoms are not excited to the 7S 1/2 level enabling them to absorb the red laser beam. Therefore, the frequency of the modulation of the UV laser beam is the frequency with which an absorption signal of the red laser is present. The lock-in amplifier takes the reference signal from the chopper wheel controller and multiplies it by the input signal. The input signal is the output of the photodiode reading the power of the scanning red laser. Multiplying the input signal by the reference signal after some uninteresting math will allow the lock-in amplifier to record the differences between the red spectra when the UV beam is blocked and unblocked. The lock-in performs S 1 (f) S 2 (f) for S 1 (f) as the spectrum of figure 3.7b and S 2 (f) as the spectrum of figure 3.7a. The lock-in amplifier will reduce the noise in the S 2 (f) spectrum because it will only be sensitive to noise at a frequency near 1000 Hz. In essence, the lock-in will take the Fourier component of the difference signal at 1000 Hz. The noise from lower frequencies will be ignored by the lock-in amplifier. The second job of the lock-in amplifier is to amplify the signal as well. The output signal of the lock-in amplifier will look like figure 3.7c.

39 CHAPTER 3. EXPERIMENTAL SETUP 28 Figure 3.7: Red Laser Transmission Spectrum due to Optical Chopper 100% S 2 (f) Transmission 50% 0% Frequency (MHz) (a) UV Beam Unblocked by Optical Chopper 100% S 1 (f) Transmission 50% 0% Frequency (MHz) (b) UV Laser Beam Blocked by Optical Chopper

40 CHAPTER 3. EXPERIMENTAL SETUP 29 Arbitrary Units S 1 (f) S 2 (f) Frequency (MHz) (c) Output of the Lock-in Amplifier of the Spectrum from figure a

41 Chapter 4 Data Analysis and Results for the 7P 1/2 Experiment This chapter contains the data analysis of the 7P 1/2 experiment. Once we begin acquiring spectra for the 8P 1/2 experiment, our analysis procedure will be identical. The discussion below expands upon the results and error analysis in our Physical Review manuscript [10]. 4.1 Data Acquisition As a first step in data-taking, we lock the UV laser to the appropriate lock point making sure that both configurations of laser beams through the cell are spacially overlapped to a maximum by visual alignment of the beam via peak asymmetry minimization. A Labview program, created by David Kealhofer [8], programs the DAQ to control the shutters, the FP photodiode, the output of the lock-in amplifier, and the sweep of the piezo. Depending on the type of spectrum we wish to take, single or dual isotope, the program sets the UV shutters to the desired configuration. Manually, we turn the EOM on only if we desire single isotope spectra. The program then sends a voltage signal to the PZT to sweep the IR laser 5-7 GHz up and then back down to the starting frequency of the IR laser. As the sweep occurs, we sample the output signal of the FP photodiode and lock-in amplifier several hundred times during both the upward and downward sweep of the IR laser frequency. The two sweeps represent a pair of data runs; one for each sweeping direction. We store each data run in a separate text file including the point number, along with the sampled FP photodiode and lock-in amplifier signals. If we are taking single isotope data, the configuration of the UV shutters switch to set the other propagation configuration. If we are taking dual isotope data, the program continues to take pairs of data runs with the same UV shutter configuration. This process is repeated a few hundred times before we stop acquisition. We 30

42 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 31 call the collection of the hundreds of pairs (up and down scan) of data runs a data set. Throughout the data acquisition of a data set, the configuration of the beams in both the CO and CTR configurations remains constant. Other conditions of the experiment remain constant during the acquisition of a data set. Each data set takes of order one hour. Between data sets, we tweak the beam alignment of both configurations and change other parameters of the set up such as oven temperature, IR laser power, IR and UV laser polarization, etc. We take between 2 and 5 data sets in one day. 4.2 Linearization and Calibration of the Frequency Axis Fabry-Perot Calibration and Linearization The Fabry-Perot cavity we use is a parallel plate FP cavity with a finesse of 50 and an FSR of approximately 500 MHz. To maintain thermal stability within the cavity, it is made out of low-expansion material and we cover it with a thermally-insulated box. The purpose of the parallel confocal Fabry-Perot cavity is to calibrate and linearize the frequency axis of the final spectra from the 671 nm laser. As mentioned before, when we sweep the laser by applying a voltage to the PZT, its response is hysteretic and nonlinear. Because we simultaneously extract the signal data and the FP data, while the IR laser sweeps, both are subject to the same nonlinearities in the frequency axis. To illustrate this, figure 4.1 contains the signal spectrum of a 205 Tl data run with its corresponding nonlinearized FP data fit with an Airy function. We convert the raw point number, i, of our scans to a normalized x-axis scale using the equation: x i = i N/2 N/2 (4.1)

43 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 32 1 rmse = Normalized Amplitude rmse = Airy Function Normalized Ampitude Normalized Point Number, x i Figure 4.1: Simultaneous Lockin Amplifier and Fabry-Perot Data A plot of the simultaneously taken samples of FP and lock-in amplifier signal output for 205 Tl. The red line on the FP data represents the Airy function fit to this data. The x-axis is the normalized point number associated with each FP and lock-in sample taken. The point number is proportional to the ramp-step of the voltage applied to the PZT, so we can treat the x-axis as normalized applied PZT voltage. where N is the total number of points. We see that 1 x i 1. When we fit the FP data to our Airy function, we express the frequency argument as a polynomial function of x i, and find the best fit coefficients. The first step to calibrate the frequency axis consists of fitting the FP data versus the normalized point to an Airy function. Because the ramp-step of the voltage applied to the PZT is constant, the point numbers are proportional to the applied PZT voltage. Therefore, we can treat the x-axis as of the plot in figure 4.1 as normalized PZT voltage. We took the positions of the FP peaks from the Airy function fit. To begin with, we take the frequency spacing between each successive FP peak to be 500 MHz. We then plotted these frequency spacings of the FP peaks by their x i positions, see figure 4.2. We fit a fourth-degree polyno-

44 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 33 mial to these data points. We found that the fourth degree polynomial was suitable because higher-order polynomials did not have fits with any better statistical quality. The position of the peaks do not fit on a straight line as they would if the PZT scanned linearly. This is illuminated by the nonzero polynomial coefficients of the fitted line in figure 4.2. It is also evident in the straight red line that joins the first two peak positions and extends linearly. We can now use this fourth-order polynomial as a function of normalized point number to convert the x-axis of figure 4.1 to frequency. In effect, this linearizes the x-axis of the spectra. We are now in a position to fit the atomic spectra with a linearized frequency scale. Note absolute calibration requires precise knowledge of the true FP FSR. In fact, the true FSR is slightly larger than MHz , , , , Frequency (MHz) Normalized Point Number, x i Figure 4.2: Polynomial of Fabry-Perot Peak Positions The blue line through all of the FP peak positions represents the forth-degree polynomial from the Airy function. The red line represents the line fitted to the FP peak positions if the fourth-degree polynomial were a linear fit instead. This is a plot for FP peak positions for a data run with an upsweeping PZT. The down sweeping plots have forth-degree polynomials that have opposite concavity. The true FSR is above the MHz is because the FSR of the FP cavity is determined by the radius of curvature of each of the confocal mirrors. Thus, the precision of the FSR is determined by the precision of their radii. However, the radius of curvature of the mirrors typically differs from the nominal quoted value. After determining the FSR precisely as

45 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 34 described below, it is a simple matter to scale all frequency intervals in our fitted spectra by: C F P = T rue F SR 500 MHz (4.2) To find the actual FSR of the FP cavity, we sent a portion of the IR laser beam to a Burleigh WA-1500 wavemeter, which reads absolute frequencies to a precision of 30 MHz, while the FP cavity transmission was sent to an oscilloscope to help us determine where the transmission peaks of the FP cavity are. We tuned the IR laser to a transmission peak of the FP and recorded the frequency from the wavemeter. Then we would tune the laser to a different transmission peak and record that frequency, repeating this process until we had at least 100 frequencies of FP transmission peaks measured out of a range of 100 GHz. To find the FSR, we took pairs of transmission peaks and subtracted their frequencies. The frequency difference of any pair of these 100 transmission peaks is an integer multiple, n, of the true FSR. To find the FSR of two distant transmission peaks, we can guess an n such that the distance between the two frequencies is n times our nominal estimate of the FSR, F SR o. We chose n to minimize the residual between δf and n F SR o. In this way, we can find an FSR of improved precision. We took the FSR between every possible pair of frequencies we recorded using this method, which increases the precision of the average FSR value. The average FSR we found after several iterations of this process was 501.2(3) MHz MHz The corresponding correction factor is therefore = (5). This is in agreement with the FP calibration factor of 1.002(1) that David Kealhofer came up with 500 MHz using the same IR laser and FP cavity [8]. We can thus calibrate the HFS and IS raw data that we get from our spectra by simply multiplying the raw frequency interval values by (5) EOM Calibration The second method of calibration of the linearized frequency axis is to use an electro-optic modulation device (EOM), which provides FM sidebands about the center frequency of the IR laser at intervals of exactly MHz. In our experiment, only the first-order sidebands are visible in the IR spectra. The resulting IR spectra contain one spectral peak for each of the two transition frequencies, each surrounded by two sideband copies, see the top of figure 4.1. We measure the sideband splitting as the difference in frequency between the resonant spectral peak and its two surrounding sidebands. All linearized spectra start with the nominal assumption that the FP FSR is 500 MHz. Because we now know that the FP cavity does not have an FSR of exactly MHz, we expect the apparent sideband splitting to be slightly below MHz. We create a correction factor for the linearized frequency axis by simply dividing the average experimental value of the sideband size into

46 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 35 F'' = 1 { { F'' = 0 7P 1/2 F'' = 1 F'' = 0 7P 1/2 7S 1/2 F' = 1 7S 1/2 F' = 1 F' = 0 Thallium 205 Thallium 203 F' = 0 Figure 4.3: Second Step Transition The red arrows represent the IR laser scanning 6-7 MHz across the 7P 1/2 -state hyperfine structure MHz. The average correction factor, C EOM, from this method was (2) for all 203 Tl data and (2) for all 205 Tl data [10]. We found that both methods of calibration were reasonably close and in good statistical agreement with C F P. Furthermore, when we looked at the spread of calibration factors from each individual data set, they did not vary more than from their respective average correction factors [10]. Again, once we have the raw HFS and IS values, we can simply multiply them by the C EOM for the appropriate isotope. 4.3 Interpreting Spectra Figure 4.3 represents the hyperfine transitions, 7S 1/2 (F = 1) 7P 1/2 (F = 0, 1), that the IR laser scans across. To extract information from these scans, we use the ThalliumFitting.m code summarized in appendix A. For simplicity, we will call the data taken from the lock-in amplifier the signal data. The ThalliumFitting.m code first linearizes and converts the x i axis to a frequency axis of the signal data with the FP data for each data run, using the FP linearization method previously explained. This code then fits a sum of Lorentzians, six for single isotope spectra and eight for dual isotope spectra, to each linearized data run. As mentioned in section 2.6, we approximate the shape of the IR spectral peaks with Lorentzians because the Gaussian contribution caused by Doppler broadening is negligible

47 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 36 because of our hole-burning technique. The result of this fit is the solid red line in the spectra in figures 4.4 and 4.7. Each of the blue points represents a data point. Each peak s Lorentzian includes a variable for position, common width, and amplitude. We are interested in the the position variable for each peak. Along with these spectral figures, ThalliumFitting.m also produces four text files for every data set containing these peak positions and their uncertainty. There is one text file for each combination of beam configuration and sweeping direction: copropagating upscan, counterpropagating upscan, copropagating down scan, and counterpropagating downscan. HFS value and four sideband splittings are recorded for each data run. A second program, DataAnalysisThallim.m (appendix A), takes these four files of peak positions and produces tables with measurements of HFS values, EOM sideband values, IS values and their corresponding variances. The DataAnalysisThallium.m also creates histograms and comparison figures to assess the quality of the data of a particular set, which are useful to seek out potential sources of error. 4.4 Single Isotope Spectra Figure 4.4: Single-Isotope Data Run with Residuals An upscan 203 Tl spectrum [10]. The fitted Lorentzian and the data points from the data run are visibly in agreement, as seen in corresponding residual plot below. Hyperfine Splitting E B Sideband A C D F

48 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 37 From the single isotope spectra such as figure 4.4 for each data run, we extract HFS value and four sideband splittings from each plot. The frequencies of each of the labeled peaks in figure 4.4 are denoted by ν N for peak label N. For the 203 Tl isotope, each of the six peaks in the single isotope spectrum represent: ν A = ν B f AOM (4.3) ν B = f o 3 4 H 7p, H 7s,203 (4.4) ν C = ν B + f AOM (4.5) ν D = ν E f AOM (4.6) ν E = f o H 7p, H 7s,203 (4.7) ν F = ν E + f AOM (4.8) where f o is the transition of 7S 1/2 7P 1/2 in absence of hyperfine structure, f AOM is the 600 MHz sideband frequency, H 7p,203 is the hyperfine splitting of 7P 1/2 for 203 Tl, and H 7s,203 is the hyperfine splitting of the 7S 1/2 level for 203 Tl, which subtracts out in our HFS determination of ν E ν B. Because we use f o with no HFS as our reference point, we use the prefactors 3 and + 1 for the F =0 and F =1 hyperfine levels, as calculated in equations and 2.14 respectively. Figure 4.4 is the resulting spectrum from an upscan of the PZT. A downscan of the PZT results in a mirror image of the upscan, with A through F labeled right to left. Experimental spectra for single isotope scans of 205 Tl look exactly the same with roughly 20 MHz greater hyperfine splittings. The measured splittings of the sidebands going from peak A to peak F are: S 1, S 2, S 3, and S 4. We can calculate the hyperfine splitting values and sideband sizes using the following equations: HF S 1 = ν E ν B (4.9) HF S 2 = ν D ν C + 2 f AOM (4.10) S 1 = ν B ν A (4.11) S 2 = ν C ν B (4.12) S 3 = ν E ν D (4.13) S 4 = ν F ν E (4.14) Later we will discuss the importance of the comparison of HF S 1 and HF S 2 as a measure of how well we linearized and calibrated the frequency axis. It is important to note that if the lock point is not quite at the center of isotopic resonance, for the desired isotope, the entire spectra is slightly Doppler-shifted. However, the hyperfine splitting is not affected.

49 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT Single Isotope Error Analysis From the initial spread of raw hyperfine splitting values, we want to get rid of the data with bad frequency linearization. One way to do this is to assess the quality of the fitted data. Bad Lorentzian fits of the scans lead to more error in the position of the peaks and thus more error in the HFS values. Bad fits occur because the frequency linearization of the scan did not sufficiently correct for the nonlinearity in the frequency axis. This results in asymmetric fits. Another cause of poor fits is if the beams are not properly spacially overlapped, leading to asymmetry in the spectral peaks. Because we change optimize the overlap of the UV and IR beams between each data set, we effectively randomize the error due to bad spacial overlap of the beams and does not affect the final average values of our hyperfine splitting. The only cause of asymmetry that remains as a problem is therefore due to frequency axes with bad linearization. We can find inconsistent asymmetric fits by measuring the residual between the data points and fitted Lorentzian of each scan. Figure 4.4 is a scan with a good residual value. The residual is magnified so that we can better study it. A small fraction of our fits were rejected based on significantly larger residuals, as measured by the statistical quantity χ 2 (the sum of squares of the deviations of the fit data). We ultimately want to get a spread of HFS values for both isotopes that resembles a Gaussian because this will mean we have a normal distribution of our experimental results. A normal distribution of our results is important because it means we have a resulting HFS value that is not perturbed by any additional random variables or error. Tables of the raw results of each data set, by day, exist for each isotope in appendix B. Initially the spread of this raw data was not in a sufficient normal distribution. Since the calibration of the frequency axis is one of the main factors that sets our experiment apart from that of Grexa et al. [1], it should be a major factor used to cut down the raw single isotope data. This is done by restricting the range of fitted sideband values. Because we expect the average sideband value to be slightly below MHz, the maximum and minimum sideband values are not centered around MHz, but are centered around the estimated average sideband value. For each isotope, we restrict our data to those scans whose sideband values fall within the specified range. For 205 Tl this range was between 591 MHz and 605 MHz and for 203 Tl this range was between 592 MHz and 604 MHz. It is important to note that adjusting the exact cutoff range did not change our final mean value for the HFS. Therefore, the resulting average HFS value is only from data that we have some confirmation of a good frequency calibration. We apply this approach to all set of data and arrive at a final statistical error of 0.20 MHz for the 205 Tl HFS and 0.25 MHz for the 203 Tl HFS.

50 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT Systematic Error Search Equally important to the statistical error, the search for systematic errors is crucial to our data analysis. First, we considered a number of experimental parameters that ought not affect our final results. We used five different powers of the second laser in the double excitation [10]. There was no statistically significant difference in the peak positions in the spectra and thus the HFS values for both isotopes. Second, the relative polarization was changed, resulting in a relative change in the relative heights of the peaks of the spectra, but we saw no change in the splitting at the level of statistical error. The effects of Doppler broadening due to different temperatures of the thallium atoms should not be significant. We varied the temperature of the main oven that contained the thallium cell for the final spectra. This also alters the thallium vapor density. Simultaneously, these temperatures are not high enough for Doppler broadening effects to change the profile of the absorption peak shape because we found that only temperatures above 550 C would create Doppler broadened effects in the spectra 1 [10]. We saw no effects of this at lower temperatures between 400 C and 450 C. We also explored the effect of the speed and width of the laser sweep. The laser sweep is caused by sending voltage to the piezoelectric device to sweep the laser frequency 6-7 GHz up and down the frequency scale. There was no difference found from changing the laser sweep between 5-8 GHz Subdividing Data Table 4.1 shows our results for the 7P 1/2 hyperfine splitting for both isotopes. For several important potential sources of error, we subdivided the data according to categories: the scanning direction of the PZT, the configuration of the UV and IR lasers, and the 4-3 vs. 5-2 HFS measurement. Differences in the sweep direction could lead be a result of two issues. The first issue is that the response of the PZT is different between increasing and decreasing applied voltages, meaning the PZT s hysteresis is not the same in both scanning directions. The second issue could be that the time constant of the lock-in amplifier creates a lag in response time of the lock-in. In figure 4.5 the hyperfine splitting values for each scan direction differ by a little bit less than 0.5 MHz in both isotopes. The upscan HFS value is in reasonable statistical agreement with the downscan HFS value. Since the two scanning directions produce differences in different parts of the spectra due to the two issues mentioned, the agreement of the HFS values from both directions shows the validity our linearization method. Any residual 1 This effect is called radiation trapping, which is due to the relatively immediate absorption of a photon after it has been emitted by another atom. It happens in high densities. This effect presents itself as Doppler broadened pedestals.

51 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 40 discrepancy was at the 0.3 MHz level, which was included in the final value. The CO and CTR beam configurations of the UV and IR laser beams excite different velocity classes of the thallium vapor depending on the locking point of the UV laser. When we lock to the dual isotope locking point, we excite certain velocity classes of both isotopes. It must follow that when we lock the UV laser to a single isotope resonant frequency, we excite not only the zero velocity class of the desired isotope, but also a small amount of highly Doppler-shifted atoms of the other isotope. The non-resonant isotope could potentially contribute to the IR spectra. This is a problem because this could cause error in fitting the resonant isotope s data, and finding the correct frequency splittings of that isotope. Because the Doppler shift has an opposite sign in the CO configuration versus the CTR configuration, the contribution of the other isotope to the IR spectra will be different in both configurations. Therefore, if we see no difference in frequency splittings of CO and CTR single isotope spectra, then we know that the non-resonant isotope s contribution to the spectra is negligible. We switch the beam configuration hundreds of times automatically during the experiment with two shutters hooked up to the DAQ. Looking at figure 4.5, the CO and CTR HFS values for both isotopes are in statistical agreement. Lastly, we compared the HFS values resulting from subtracting the peak positions of the third peak from the fourth peak (δν D δν C ) and the second peak from the fifth peak (δν E δν B ). The HFS resulting from subtracting the third peak position from the fourth peak position and adding 1200 MHz, two sideband lengths, is less affected by a potential residual scan nonlinearity and calibration error in the frequency scale. Comparing the 4-3 and 5-2 HFS values is therefore a good measure of how well-calibrated our frequency scale is. In figure 4.5, of both isotopes the greatest discrepancy in comparison parameters is between the 4-3 and 5-2 HFS values. However, the two values are still within their combines standard deviation of 1.5 MHz. In our final error budget, we add a small systematic error component based on these comparisons. Finally, we want to add a systematic error from the overall level of disagreement between the correction factors of the two methods of calibration. Due to the discrepancy between the frequency modulation correction factors and the Fabry-Perot correction factor, we associate an error of MHz 0.4MHz. The error of each source of systematic error is added in quadrature in conjunction with the statistical error to get our final resulting error, listed in parentheses of the first line of table 4.1.

52 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 41 Figure 4.5: Comparison of HFS Values by Data Subsets The upper graph is a comparison of 203 Tl HFS values, and the lower graph is a comparison of the 205 Tl HFS values. Both graphs are representations of the hyperfine splitting values based on six parameters: copropagating (CO) laser configuration, counterpropagating (CTR) laser configuration, upscan (UP) of the piezo, downscan (DN) of the piezo, 3rd peak position subtracted from the 4th peak position, and 2nd peak position subtracted from the 5th peak position. All of these comparisons come from the same set of data, so CO and CTR, UP and DN, and 4-3 and 5-2 each represent the entire final data set. Error bars represent one standard deviation of the HFS mean value determined from the uncertainty associated with the peak positions. 203Tl Hyperfine Splitting (MHz) Hyperfine Splitting Values by Data Subsets CO CTR UP DN Tl Hyperfine Splitting (MHz) CO CTR UP DN Table 4.1: 7P 1/2 Results 7P 1/2 HFS 205 Tl 7P 1/2 HFS 203 Tl 7S 1/2-7P 1/2 Isotope Shift Final Result (MHz) (8) (7) 534.4(9) Statistical Error (MHz) Systematic Error (MHz) Laser Sweep Beam Propagation (co vs. counter) N/A Frequency Calibration Scan Linearization Thallium Cell Temp Alignment of Beams N/A N/A 0.25 Correl. with HFS and Doppler Shift N/A N/A 0.30

53 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT Dual Isotope Spectra Eliminating Doppler Shift with Dual Beam Configuration In the single isotope trials, we purposefully choose to excite the zero velocity class atoms by choosing the locking point to be on resonance of the 6P 1/2 7S 1/2 transition. Even if the class of atoms excited by locking is just off resonance, and they do have nonzero velocity, we do not have to worry about the Doppler shift caused by the nonzero velocity because both hyperfine transitions will be affected by the same Doppler shift. Since we only care about the relative difference in the hyperfine transitions, we can ignore the Doppler shift. Thus, there is no issue with Doppler shift in the single isotope spectra. However, we do have to account for the Doppler shift in the dual isotope spectra. The locking point we choose to take the red dual isotope spectra is in the middle of the resonances of the 6P 1/2 7S 1/2 transition of both isotopes at nm. A non-zero velocity class of atoms of each isotopes will then be excited. This produces a problem because the velocity classes of atoms of both isotopes are different. Therefore, we cannot ignore the relative Doppler shift if we want to measure a quantity such as the isotope shift. Since we must take differences in of transitions from both isotopes to find the isotope shift in the dual isotope spectra, we need to find a way to eliminate the Doppler shift of both isotopes completely in order to calculate the isotope shift. This is still not a problem for calculating HFS of each isotope because each isotope s peak is Doppler shifted by the same amount. The solution emerges in the simultaneous use of the CO and CTR configurations. Both of these configurations and the velocity classes of atoms they excite are depicted in figure 4.6. The velocity class of 205 Tl atoms excited by the UV laser will be moving towards the direction of propagation of the UV beam because with a resonance of nm, these atoms will be blue shifted. The velocity class of 203 Tl atoms excited by the UV laser will be moving away from the UV beam because they are redshifted atoms of the resonant wavelength

54 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 43 Figure 4.6: Dual Beam Configurations The UV beam excites moving velocity classes of thallium atoms along the direction of propagation as shown. In the CO configuration, the 205 Tl atoms are blue shifted by the UV and IR beams. The 203 Tl atoms are traveling in the opposite direction, so they are red shifted by both beams. In the CTR configuration, the 205 Tl atoms are blue shifted by the UV beam and red shifted by the IR beam. The 203 Tl atoms are red shifted by the UV beam and blue shifted by the IR beam. Copropagating Configuration IR Beam 205 Tl Atoms 203 Tl Atoms UV Beam Counterpropagating Configuration IR Beam 203 Tl Atoms 205 Tl Atoms UV Beam In the CO configuration, the 205 Tl atoms are blue shifted and the 203 Tl atoms are redshifted in the final spectra because the red beam propagates in the same direction as the UV beam. In the CTR configuration, however, the 205 Tl atoms will be moving away from the IR beam (redshifted) and the 203 Tl atoms will be moving towards the red beam (blue shifted). We can calculate the velocity and Doppler shift of the two isotopes in both configurations using the equations: f = v c f o (4.15) f 2 = f o,2 f o,1 f 1 (4.16) Equation 4.16 is found by setting the velocity found from equation 4.15 for f 1 equal to the velocity of equation 4.15 for f 2, another frequency. 205 Tl is found to have a Doppler shift of about 880 MHz and 203 Tl is found to have a Doppler shift of about 755 MHz with the UV laser [8]. Regardless of the exact Doppler shifts of the isotopes, their sum is related to the

55 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 44 isotopic peak splitting in the UV spectrum. Using equation 4.15 the velocities of the excited 205 Tl and 203 Tl atoms are approximately 330 m/s and 285 m/s respectively. The Doppler shifts of the 205 Tl and 203 Tl atoms in the IR beam are approximately 219 MHz and 256 MHz respectively, where f o,1 = c and f 378nm o,2 = c. 1301nm The only difference in the frequencies of one isotope s Doppler shifted transition peaks in the CO configuration and the CTR configuration is the sign of the Doppler shift. we average the CO and CTR frequencies for a specific hyperfine transition of a single isotope, we will get the non-doppler shifted value of this transition. Therefore, we eliminate the IR Doppler shift using the average of the CO and CTR frequency of each transition peak. If Interpreting Dual Isotope Spectra F G 0.8 Normalized Intensity A B C D E H Frequency (MHz) Figure 4.7: 7P 1/2 Dual Isotope Spectrum The dual isotope spectra have eight peaks. Four peaks correspond to the two hyperfine transitions of each isotope in the CO configuration and four peaks correspond to the two

56 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 45 hyperfine transitions of each isotope in the CTR configuration. As discussed, these sets of peaks are Doppler shifted form one another. The frequency parametrization of each peak in figure 4.7 is described by the following equations: 7S 1/2 7P 1/2 Peak Equation Prop. Config. Hyperfine Transition Isotope ν A = f o 3 4 H 7p, H 7s,205 δf 205 CO F = 1 F = ν B = f o 3 4 H 7p, H 7s,205 + δf 205 CT R F = 1 F = ν C = f o H 7p, H 7s,203 + I 7s 8p δf 203 CT R F = 1 F = ν D = f o 1 4 H 7p, H 7s,203 + I 7s 8p + δf 203 CO F = 1 F = ν E = f o H 7p, H 7s,205 δf 205 CO F = 1 F = ν F = f o 3 4 H 7p, H 7s,205 + δf 205 CT R F = 1 F = ν G = f o H 7p, H 7s,203 + I 7s 8p δf 203 CT R F = 1 F = ν H = f o H 7p, H 7s,203 + I 7s 8p + δf 203 CO F = 1 F = where f o is the frequency of the 205 Tl 7S 1/2 7P 1/2 transition in the absence of HFS. I 7s 7p is the isotope shift between the two isotopes for the transition 7S 1/2 7P 1/2. δf 203 and δf 205 are the Doppler shifts due to the IR excitation only. From these 8 peaks, labeling their frequency splittings by δν ij, we can isolate the HFS values for each isotope as well as the transition isotope shift: HF S 205,1 = ν E ν A = ν EA (4.17) HF S 205,2 = ν F ν B = ν F B (4.18) HF S 203,1 = ν G ν C = ν GC (4.19) HF S 203,2 = ν H ν D = ν HD (4.20) I 7S 7P = 3 ν G + ν H 2 ν E + ν F 2 + ν C + ν D 2 4 ν A + ν B 2 + HF S 7S,203 HF S 7S,205 4 (4.21) The transitional isotope shift, equation 4.21, is calculated by taking averages of the CO and CTR peaks to remove the Doppler shift leaving the frequency due to the transitional isotope shift and the frequency due to the hyperfine anomaly. The second term in this equation removes the affects of hyperfine anomaly in the 7S 1/2 level. The previous 7S 1/2 HFS work done in this lab provides precise values for these terms. This second term is important because isotope shift is the difference between the the 7P 1/2 states in the absence of hyperfine states. Using this equation, the value of the transition isotope shift for the 7S 1/2 7P 1/2 that we found is 534.4(9) MHz. We can take this transition isotope shift and convert it into a 7P 1/2 level isotope shift because this transition isotope shift is the difference of the 7S 1/2 and 7P 1/2 level isotope shifts. The 7S 1/2 level isotope shift was previously determined to be (3.8) MHz [4]. We simply take the difference between the 7S 1/2 level isotope

57 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 46 shift and the 7S 1/2 7P 1/2 transition isotope shift, 409.0(3.8) MHz 534.4(9) MHz, to get the 7P 1/2 level isotope shift of (4.0) MHz. The total Doppler shift is the sum of the Doppler shift due to the IR excitation of the 205 Tl atoms and the Doppler shift due to the IR excitation of the 205 Tl atoms. We can estimate the total Doppler shift by taking the difference between the four corresponding peaks in the CO and CTR configurations: ftotal D = δf δf 205 = 1 4 [δν BA + δν DC + δν F E + δν HG ] (4.22) Each difference accounts for twice the Doppler shift by which both peaks are shifted. The expected value of equation 4.22 is 475 MHz, by estimating the effects of the Doppler shift due to the two velocity classes excited by the CO and CTR configurations. This is estimated using the isotopic separation in figure 3.5 of 1636 MHz: f D total = f IR f UV 1636 MHz = 378 nm 1639 MHz 475 MHz (4.23) 1301nm Dual Isotope Error Analysis Many of the systematic errors that plague the single isotope data also affects the dual isotope data. Therefore, similar systematic error values are associated with the dual isotope data. Because we do not use the frequency modulation (FM) calibration method with the dual isotope scans, we must estimate a frequency-scale correction factor based on the results of single isotope scans. We chose to the average of the single isotope correction factors from the FM method of There are three additional sources of systematic error to consider in the dual isotope scans. The first is the geometric alignment of the beams. We rely on the exact counterpropagation since our isotope shift values come from taking the average of CO and CTR. Because there are simultaneous CO and CTR configurations of the UV and IR beams, three beams must be perfectly aligned inside of the thallium cell. We noticed that there was a greater scatter and more asymmetries in isotope shift data from the dual isotope spectra than there was of HFS values from the single isotope spectra. Due to the increase in asymmetries present, we attributed the scatter to the greater difficulty of aligning three laser beams in the dual isotope experiments than of aligning the two laser beams in single isotope experiments. Again, realigning the beams between sets of data taken effectively randomizes this error and we can associate an error for misalignment of all of the sets. We take as our final error as the scatter between data sets. However, we should consider any biases misalignment could produce.

58 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 47 Figure 4.8: Beam Misalignment Unaligned UV Beam Red Beam Aligned UV Beam Using both the CO and CTR configurations to eradicate the Doppler shift can fail if the Doppler shift of a certain transition peak is not the same in both configurations. This can happen if the geometrical alignment of the beams is not perfectly overlapping in both configurations. For example, if the red beam and the CO configuration UV beam overlap perfectly, but the CTR configuration UV beam comes into the cell at an angle, θ, with the red beam, different velocity classes of atoms will be excited and thus the Doppler shifts will be different in each configuration. The Doppler shift is at a maximum when the beams are aligned perfectly, so the Doppler shift of misaligned beams will result in a smaller Doppler shift. This reduces the total Doppler shift by: Sin(θ) δf(ir) m 1 m 475 MHz 0.5 MHz (4.24) where the dimensions of the collimator are 1 m 1 mm, which allow us to approximate Sin(θ) θ. Using the approximate Doppler shift of the IR excited atoms in both isotopes calculated earlier, 0.5 MHz. Figure 4.9 is a plot of the isotope shift versus Doppler shift. A fitted line shows a slight trend between isotope shift and Doppler shift. The mean isotope shift of approximately 534 MHz correlates to a Doppler shift of about MHz, which is slightly below the expected value. This suggests that the CO and CTR configurations may not have eradicated the Doppler shift fully due to misalignment of the three beams. The second additional source of systematic error is the correlation between the HFS values and the transitional isotope shift (IS). We can calculate IS and HFS values from dual isotope spectra, we restricted our data to those runs for which the corrected HFS values agreed with the single isotope scans within ±5 MHz. The last additional source of error is the correlation of the total Doppler shift and the

59 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 48 Figure 4.9: Correlation of Transition Isotope Shift and Doppler Shift Scatterplot of transition isotope shift versus the total Doppler shift for one set of data.

60 CHAPTER 4. DATA ANALYSIS AND RESULTS FOR THE 7P 1/2 EXPERIMENT 49 transitional isotope shift. The total Doppler shift is the sum of the Doppler shift due to both excitations. We made scatterplots of IS vs. Doppler shift and used them to look for any correlation between the IS and the total Doppler shift. From these plots of every data set, such as figure 4.9, we some times saw that the average value of the total Doppler shift is 0.5 MHz lower than expected (475 MHz). This could be due to a small beam misalignment as we have already discussed. For both the total Doppler shift and HFS correlations with IS, we considered the extra error associated with the correlation slope and then combined the two sources of error in the last entry of table Conclusions Table 4.2: Comparing Results of 7P 1/2 HFS and IS Values Group 203 Tl HFS (MHz) 205 Tl HFS (MHz) 7P 1/2 IS Grexa et al.(1988) [1] ± ± 0.6 N/A Hermann et al. (1993) [9] N/A N/A -130 ± 5 Theoretical Results[2] N/A 2193 N/A Our Results ± ± ± 4.0 Combining all sources of statistical and systematic error we see that both of our results exceed those of Grexa et al. [1] by about 20 MHz. However, both of our experiment and the experiment of Grexa et al. [1] quote errors below 1 MHz. Therefore, it seems that the calibration errors that plagued the earlier experiments of Grexa et al. [1] may have also been present in their quoted HFS values of 7P 1/2 for both isotopes. The most recent ab initio theory calculation of the 205 Tl hfs is 2193 MHz, with an error of about 2-3% [2]. Thus, our results improve agreement with the atomic theory, with a 20 MHz closer HFS value than the 1988 results. The isotope shift of the 7P 1/2 level is in very good agreement with the previously determined value of -130(5)MHz [4]. Because the isotope shift is a measure of the effects of an increased number of neutrons in the nucleus, it gives us insight into the difference in mean square isotopic charge radius. We can access this using the equation 2.12, getting a hyperfine anomaly, which is related to the ratio of 203 Tl and 205 Tl HFS values, of 7P 1/2 = 5(4) 10 4 [10]. This barely resolved value is in agreement with the hyperfine anomaly calculated in a previous experiment in this lab for 7S1/2 [4].

61 Chapter 5 The 8P 1/2 Experiment 5.1 Introduction Now that we have completed the 7P 1/2 experiment and published the results, our focus is now to find the hyperfine splitting and isotope shift for the 8P 1/2 level. The experimental layout of this experiment, as described in chapter 3, is depicted in figure 3.4. The equipment changes this experiment requires are the second step laser, the FP cavity, and the EOM synthesizer. Instead of a 600 MHz synthesizer, the EOM of this experiment would use a 200 MHz synthesizer. The greatest change made between these two experiments is the choice of the second step laser. Instead of the 1301 nm ECDL laser, we use a 671 nm ECDL laser built in our lab, see section 3.1. Because the second step laser has been changed, we must also change some specifications of the FP cavity. As mentioned before, in 1993 the Hermann et al. group [9] republished the values of HFS and IS after finding a calibration error in their previous results published in 1988 [1]. Unlike the 7P 1/2 state, this group quotes republished HFS values for the 8P 1/2 state. For the 203 Tl isotope they found an HFS of ±0.7 MHz and for the 205 Tl they found an HFS of ± 0.6 MHz [9]. Just as the 7P 1/2, the 8P 1/2 results will also help test the accuracy and guide the refinement of the atomic theory used to approximate quantities that depend on the wavefunctions of electrons P 1/2 Simulated Spectra Peak Approximation The spectra are simulated using Lorentzians of equal natural line width to approximate the shape of the peaks corresponding to the absorption transitions for the same reason that we fit Lorentzians to the IR spectra. These HFS values are much smaller than the HFS of the 50

62 CHAPTER 5. THE 8P 1/2 EXPERIMENT 51 7P 1/2 state. For consistent precision with the 7P 1/2 experiment, we need to have a FP cavity with a different FSR and smaller frequency modulated sidebands from the EOM Single Isotope Spectra Figure 5.1: Simulated 205 Spectrum E 0.8 Normalized Peak Heights A B C D F Frequency (MHz) Figure 5.1 is a simulated spectrum of 205 Tl taken by scanning the red laser 3 GHz. The simulated spectrum of 203 Tl is exactly the same except the space between peaks B and E are smaller by roughly a few tens of MHz. Except for the smaller frequency splittings, this spectrum is identical to the 7P 1/2 single isotope spectra. Peak B is the resonance peak of the 7S 1/2 (F = 1) 8P 1/2 (F = 0) transition. Peak E is the resonance peak of the 7S 1/2 (F = 1) 8P 1/2 (F = 1) transition. Peaks A, C, D, and F represent the upshifted and downshifted peaks due to the EOM. The equations for the peak positions are as follows: ν A = ν B f EOM (5.1) ν B = f o 3 4 H 8p, H 7s,205 (5.2) ν C = ν B + f EOM (5.3) ν D = ν E f EOM (5.4) ν E = f o H 8p, H 7s,205 (5.5) ν F = ν E + f EOM (5.6)

63 CHAPTER 5. THE 8P 1/2 EXPERIMENT 52 where f EOM is the sideband frequency of 200 MHz. H 8p,205 is the hyperfine splitting for the 8P 1/2 state of 205 Tl and H 7s,205 is the hyperfine splitting for the 7S 1/2 state for 205 Tl. The heights of the frequency modulated (FM) sideband peaks relative to their corresponding resonant frequency peaks are defined by Bessel functions. Here I have assumed a so-called modulation depth of Dual Isotope Spectra Figure 5.2: Simulated Dual Isotope Spectrum The solid lines represent the part of the spectrum due to the CO configuration of the UV and red laser beams. The dashed lines represent the part of the spectrum due to the CTR configuration C G 0.8 Normalized Peak Heights A B D E F H Frequency (MHz) The dual isotope spectrum, figure 5.2, was constructed by taking the single isotope equation for each isotope, without the sidebands, and offsetting each by the doppler shift each isotope encounters in both CO and CTR configurations. Because the frequency splittings in the hyperfine structure of the 8P 1/2 state are smaller than those of the 7P 1/2 state, the order of the peaks in the dual isotope 8P 1/2 spectrum is not the same as the order of the peaks in the 7P 1/2 spectrum. The equations for the frequencies of these peaks are: Peak Equation Prop. Configuration Hyperfine Transition Isotope ν A = f o 3 4 H 8p, H 7s,205 δf 205 CO F = 1 F = ν B = f o 3 4 H 8p, H 7s,203 I 7s 8p δf 203 CTR F = 1 F = ν C = f o H 8p, H 7s,205 δf 205 CO F = 1 F = ν D = f o 3 4 H 8p, H 7s,205 + δf 205 CTR F = 1 F = ν E = f o H 8p, H 7s,203 + I 7s 8p δf 203 CTR F = 1 F = ν F = f o 3 4 H 8p, H 7s,203 + I 7s 8p + δf 203 CO F = 1 F = ν G = f o H 8p, H 7s,205 + δf 205 CTR F = 1 F = ν H = f o H 8p, H 7s,203 + I 7s 8p + δf 203 CO F = 1 F = 1 203

64 CHAPTER 5. THE 8P 1/2 EXPERIMENT 53 where f o is the transition frequency of 7S 1/2 7P 1/2 for 205 Tl. I 7s 8p is the isotope shift between the two isotopes for the transition 7S 1/2 8P 1/2. When we fit the actual data for this experiment in the future, there may be some error associated in the determination of the positions of peaks E and F because they appear very close together in this simulated spectra. The calculations of the hyperfine splittings, the isotope shift and the Doppler shifts are also different: HF S 205,1 = ν C ν A = ν CA (5.7) HF S 205,2 = ν G ν D = ν GD (5.8) HF S 203,1 = ν E ν B = ν EB (5.9) HF S 203,2 = ν H ν F = ν HF (5.10) I 7S 7P = 3 ν E + ν H 2 ν G + ν C 2 + ν F + ν B 2 4 ν A + ν D 2 + HF S 7S,203 HF S 7S,205 4 (5.11) 5.3 Current State of Experiment and Future Work The first thing to do is to get the 671 nm laser lasing and tuning in a single mode fashion. This will be done by focusing the spontaneous emission that comes from the laser diode so that the beam does not diverge much at least a meter from the source. The diffraction grating must be angled 54 and then carefully aligned so that they first-order diffraction beam and the output of the laser diode spacially overlap well. Once the 671 nm laser is lasing, we will check its wavelength using the wavemeter and then tune it to 671 nm. We will also check that the laser is lasing in single mode using the spare FP cavity that is coated for this wavelength from the previous indium experiment. One will keep adjusting and tweaking the temperature and current of the laser until it can scan at least 5 GHz across the 671 nm 7S 1/2 8P 1/2 resonance. Next we will have to get the properly sized 200 MHz synthesizer for the EOM for 671nm laser. Once this is ordered, it will be simple to align the beam through the EOM to get the maximum power through the device. The FSR of the FP cavity from the 7P 1/2 experiment will not be sufficient for the 8P 1/2 experiment. We would like to have an FSR of 300 MHz. The HFS of both isotopes is smaller, so we need to have a smaller FSR. We have the longer rods cut to increase the size of the FP cavity. However, we do not have the correct mirrors for this length. Our original plan was to get plano-concave lens blanks with a focal length of 25 to 30 cm when coated. Because the evaporator broke, we do not have the capability to coat lens blanks. Therefore, we will need

65 CHAPTER 5. THE 8P 1/2 EXPERIMENT 54 to purchase two mirrors of this specification. They can have a diameter of either an inch or half and inch because we currently have mirror mounts for one inch diameter mirrors with shop-made mounts for half inch diameter mirrors. Before we can take data for the 8P 1/2 experiment, we need a Labview program for the data acquisition. Our computer crashed and we lost the program used for the data acquisition for the 7P 1/2 experiment. Fortunately, there is a copy of a similar program that can be modified to fit the needs of our experiment, so we do not have to start from scratch. The Matlab data analysis code described in appendix A is ready to analyze the 8P 1/2 data once it is taken. As long as the Labview program creates a data text file for each run containing the count number as the first column, the FP signal voltage as the second column, and the lock-in amplifier voltage (from photodiode of red laser) as the third column. Each file name must specify the run number, the propagation configuration and the laser sweep direction in its title, such as 88Copupscan.txt. All of the text files for each data run for the same set should be automatically put in a folder labeled with the date, isotope, and data set number, such as Thu, Jul 18, Following these guidelines will ensure easy use of the Matlab data analysis code.

66 Appendix A Matlab Code This code was a collaboration of efforts from Nathan Schine, David Kealhofer, Dr. Gambhir Ranjit, and myself. The following code was used to take the data from the 7P 1/2 experiment and find the hyperfine splitting for both isotopes, as well as the isotope shift. The results of this code were used in the analysis of the 7P 1/2 experiment described in chapter 4. This code will be easily modifiable to analyze the 8P 1/2 data that will be taken in the future. A.1 ThalliumFitting.m All of the code in the following section can be found in the MATLAB folder of the middle computer. This code takes in a matrix of data from the experiment as three columns of numbers. The first column numbers each data point taken. The second column is the data from the lock-in amplifier. The third column is the data from the Fabry-Perot of the 671 nm laser. Lastly, if there is a fourth column, it holds the frequency modulation signal data. Each data set must have the following filename exactly including the spaces: Z : \Data T hallium\dayofw eek, Month Day, Y ear T ypeofdata T rialnumber\. It is a file that contains all of the scans from a certain trial. DayofWeek is a three letter abbreviation for the day of the week with the first letter capitalized. Month is the first three letters of the month also with the first letter capitalized. Day is the number of the day. Year is the year in four numbers. TypeofData is 203 for 203 Tl single isotope data, 205 for 205 Tl single isotope data, and 235 for dual isotope data. The output of this code is a matrix, Z, with the peak positions and their standard deviations. Z also has a goodness of fit parameter to determine how well the polynomial fits the data taken. This code can be run for the single isotope 205 or 203 data as well as the dual isotope data. The only change with the dual isotope data is to use VoigtFitThallium235.m instead of VoigtFitThallium.m. The rest of this file names FName corresponding to the configuration of the lasers. 55

67 APPENDIX A. MATLAB CODE 56 n1 is the number of data scans in the data trial folder, so you must change the second number for each data trial file you run through this code. The even n1 s mean the data from a scan in the copropagating laser configuration and the odd ones means the data is from a scan in the counterpropagating laser configuration. For each n1 in the data trial folder, there are two files: the first is for the down scan of the piezo and the second is for the up scan of the piezo. n2 differentiates down scan, 1, from up scan, 2. The rest of this code calls the other files to be mentioned in the following subsections. This code is executed for each of the subfiles in the main data trial file by looping through n1 and n2. A.1.1 getdatathallium.m This code opens each file and reads the three columns of data. If your data has more or less than three columns you must add of take away %f for each column in your data file. If you have more than 2 headerlines, change the 2 in the data declaration accordingly. A.1.2 downsampleandnormalizethallium.m This code decreases the number of data points in each data file, because in order to get good fits later in the ThalliumFitting.m code, you must decrease the number of data points. It also makes the fitting part of the program run faster. If you have to edit this part of the code, you must be weary of getting rid of too many points for a good fit. First, the x-axis is normalized, resulting in the column vector x-norm from -1 to 1. Then we choose a factor by which to down-sample the data. This is manifested in the variable fmds factor for the frequency modulation (FM) data if it exists, fpds factor for the Fabry-Perot (FP) data, and sads factor for the spectrum analyzer (SA) data. What ever number you choose for each down-sample factor, n, the downsample method will keep every nth data point starting from the first point. The down-sampled data is then plotted versus the normalized x-axis, down-sampled by the corresponding factor for the FM data, the FP data, and the SA data. A.1.3 FabryPerotFittingThallium.m This program uses sympeaksthallium.m to get the peak heights and locations (on the frequency axis) of the down-sampled FP data (x fp DS and fp DS). In lines 12 through 25, the loop is executed if the first peak is in a position greater than or equal to npoints and the last peak is in a position less than or equal to a position npoints from the end of the

68 APPENDIX A. MATLAB CODE 57 graph. ind11 is the position npoints before the first peak, so xbegin is just the position npoints units before the first peak. Similarly, xend is defined as the position of the last peak plus npoints. indx fp is then defined as the portion of x fp DS, the x-axis values, between xbegin and xend. Then fp is defined as the FP data corresponding to the x-axis values of indx fp and xfp is defined as the values of x fp DS equivalent to the positions indx fp. Finally, Uin is defined as the matrix made of the two column vectors: peak positions and peak heights (the same as UinFP). Lastly, N is defined as NFP, which is the number of Fabry-Perot peaks in the scan. The if condition of lines 28 through 42 is similar to the last one except it is executed if the first peak occurs in a position before npoints. ind11 is defined as the position npoints before the second peak and xbegin is defined as the position ind11. xend is the same as it was in the previous if statement. Lines 36 through 40 are the same as the previous if statement. Then Uin is defined as matrix of peak position and heights of the second through last peak. The third similar if condition is executed if the first peak position is greater than or equal to npoints and the last peak position is greater than or equal to npoints before the end of the scan. ind11 and xbegin are the same as in the first if statement. However, this if statement contains another if statement. The first part of the statement is executed if the xfppeak value, peak position, for the last peak minus that of the second to last peak is less than or equal to half of the difference of the x values of the middle peak and the peak in front of it. xend is defined as npoints after the third to last peak. Uin is defined as the matrix of peak positions and heights of all of the peaks except the last two. If the second part of the if statement is executed, it is similar to the first part except Uin is the matrix of peak positions and heights of all the peaks but the last one. Next, this program plots xfp vs. fp and then fits the FP data to a sum of Lorentzians based on how many FP peaks there are, N. The bounds for this fit are set in lines 89 thorough 94, where I is defined as the vector of ones with length N. Lines 96 through 178 have the actual parameters to fit the FP peaks to sums of Lorentzians and then these fits are plotted with their corresponding RMSE values. These plots are only saved if the RMSE value, gdfp, is less than goodnessthreshold defined at the beginning of ThalliumFitting.m. The b coefficients are also saved as vector FPoints. If the RMSE value is not greater than goodnessthreshold, then the whole program is called again. sympeaksthallium.m This method takes the frequency (x value) and Fabry-Perot height data as well as and averaging number and minimum peak height for each FP peak. It outputs the peak heights

69 APPENDIX A. MATLAB CODE 58 (pks) and their frequency locations (locs). A.1.4 FrequencyLinearizationThallium.m This script first plots the b Lorentzian coefficients versus a frequency axis spacing the points 500 MHz apart, which is the FSR of the Fabry-Perot. The code then fits a fourth order polynomial to the data and plots this, saving it in a certain file. A.1.5 VoigtFitThallium.m This program first uses sympeaksthallium.m to get the positions and heights of each of the six peaks of the single isotope spectra. Lines 8 through 19 run change the avg and rerun sympeaksthallium.m until exactly 6 peaks are found. Then it plots these peaks, where freqsig are the corresponding x values for the down-sampled signal data, sig DS. Before this plot is created, the signal data is fit to a sum of Lorentzians. The output of this fit is the fit results, cf SIG, and the goodness of fit, gd SIG. The fit results are then plotted in the same frame as the down-sampled signal data. The a and b coefficients from the Lorentzian model are declared. Finally, an if statement is executed and will print the matrix Z of position and standard deviation of each of the six peaks if the goodness of fit parameter is less than goodnessthreshold defined in ThalliumFitting.m. The standard deviation of each peak position is calculated by subtracting the confidence bounds of the coefficients of the fit from each of the b coefficients. If the goodness of fit parameter is not sufficient, the goodnessthreshold is increased by 0.01 and the program is called again to refit the peaks. A.1.6 VoigtFitThallium235.m The same as VoigtFitThallium.m, except it fits the 8 peaks of the dual isotope spectra. Finally, the data and plots are stored in files in writetofile.m and writetofile235.m, where the vector Z containing the peak positions and standard deviation of these positions. A.2 DataAnalysisThallium.m This next program takes the matrix Z containing the peak positions and the error associated with each peak and takes the difference of certain peaks to find the hyperfine splitting and

70 APPENDIX A. MATLAB CODE 59 isotope shift values.

71 Appendix B 7P1/2 Data Tables This is a collection of data tables for every day data was taken that averages certain parameters for each set of data. Every thing is measured in MHz. The length of each data set is the number of scans averaged in this set. Config. stands for beam configuration and scan direction. The elements of this consist of copropagation upscan (CO UP), copropagation downscan (CO DN), counterpropagating upscan (CTR UP), and counterpropagating downscan (CTR DN). Avg. Sideband is the average size of the four sidebands in each scan. SB SD is the standard deviation of this average. CF is the correction factor calculated by the average sideband size and CF SD is its standard deviation. Raw HFS is the raw average of HFS values and Raw SD is its standard deviation. Corr. HFS is the corrected HFS values by the CF and Corr. SD is its standard deviation. 05/13/2013 Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD Set 1: length = 247 CO UP CO DN CTR UP CTR DN Set 2: length = 94 CO UP CO DN CTR UP CTR DN /14/2013 Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD Set1 has bad difference calculations for CO UP an CO DN Set 2: length = 292 CO UP CO DN CTR UP

72 APPENDIX B. 7P1/2 DATA TABLES 61 CTR DN /01/2013 Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD Set 1: length = 64 CO UP CO DN CTR UP CTR DN Set 2: length = 120 CO UP CO DN CTR UP CTR DN Set 3: length = 328 CO UP CO DN CTR UP CTR DN /2/2013 Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD Set 1: length = 84 CO UP CO DN CTR UP CTR DN Set 2: length = 202 CO UP CO DN CTR UP CTR DN /03/2013 Config. Avg. Sideband SB SD CF CF SD Raw HFS Raw SD Corr. HFS Corr. SD Set 1: length = 30 CO UP CO DN CTR UP CTR DN Set 2: length = 150 CO UP CO DN CTR UP CTR DN Set 3: length = 270 CO UP CO DN

73 APPENDIX B. 7P1/2 DATA TABLES 62 CTR UP CTR DN Table B.1: 203 Tl Raw Data Sets by Date 03/15/2013 Config. Avg. Sideband CF Raw HFS Raw SD Corr. HFS Corr. SD Set 1 CTR UP CO UP CTR DO CO DO /15/2013 Config. Avg. Sideband CF Raw HFS Raw SD Corr. HFS Corr. SD Set 1 CTR UP CO UP CTR DO CO DO Set 2 CTR UP CO UP CTR DO CO DO Set 3 CTR UP CO UP CTR DO CO DO Set 5 CTR UP CO UP CTR DO CO DO Set 7 CTR UP CO UP CTR DO CO DO Table B.2: 205 Tl Raw Data Sets by Date

74 APPENDIX B. 7P1/2 DATA TABLES 63 Table B.3: Dual Isotope Data Date Raw Isotope Shift (MHz) Std. Dev. (MHz) Corr. Factor Corrected Isotope Shift (MHz) 5/16/ /17/ /20/ /23/ /27/ /28/

75 Appendix C Detailed Description and Usage of the 671 nm Laser The AR coated laser diode, collimating lens, and diode mount all came from the company Toptica Photonics. The following papers are some specifications about the AR coated diode. Following these drawings is a labeled picture of the laser in its current form. 64

76 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 65

77 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 66

78 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 67

79 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 68 Figure C.1: External Laser Cavity This is an overhead picture of the current red laser. The coated diode and surrounding confocal lens (1) lie inside a mount (2) sending the beam at the center of the diffraction grating (3) that has 1200 lines per mm. We mount the diffraction grating on a 45 metal wedge (4) attached to a mirror mount (5). The metal wedge is near the angle for the first order diffracted beam to leave the laser cavity perpendicular to the cavity. One of the tuning nuts is replaced by a piezoelectric driver (6) on the mirror mount (5). The piezoelectric transducer (PZT) is hooked up to a signal generator (7). The SRS LDC501 current hookup (9) and temperature control hookup (8) are both female 9 pin connectors. A thermoelectric device underneath the laser cavity changes the temperature of the laser. The thermometer (10) for this device is in the laser bed closest to the diode mount (2), because the temperature that we really care about is the temperature of the diode. We can angle the diffraction grating vertically using the knob(12).

80 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 69 The next five figures are the CAD drawings of the laser pieces. These CAD drawings were created by Sarah Peters 14. The first is the original diode and collimating lens mount. We did not use this in the final laser cavity because we bought a coated laser diode from Toptica Photonics that came with a collimating lens and mount. If these drawings are used to build another ECDL laser in the Littrow configuration, make sure you make mount for the diode and collimating lens such as this. All of the pieces, except as noted, are built out of aluminum. The following details about each piece of the laser are adopted from the work of Sarah Peters 14. The Base We used a scrap piece of aluminum bar about 3 wide for the base, or thermal reservoir. This does not have to have a precise size, it just needs to be wider than the diffraction grating mount. It needs to have tapped holes to secure the track and to secure it to the table. The holes that attach it to the optical table will need to be spaced by integer multiples of an inch, due to the spacing of holes on standard optical tables. Additionally, two tapped holes are needed to attach the plastic plate that holds the connections to the electric circuitry. The Track The track is an aluminum rectangle with a 1 notch cut at one end to allow the laser 4 mount to attach and slide horizontally. We ended up not even using this notch because we got a new diode later in the year that had its own laser mount. We attached this laser mount by cutting a 45 notch to fit the bottom of the mount and tapped two holes for the screws it came with. We then chamfered the edges to allow for sliding fit of the mirror mount. Holes to attach the track to the base were drilled and tapped. Lastly, we drilled a small hole in the side, near the laser mount, to fit the temperature sensor with thermal-electric grease/epoxy. The Laser Mount: Figure C.2 The central hole for the tube containing the diode and collimating lens was created using an adjustable reamer to ensure a precise fit of the odd size. In order to secure the tube in place, we cut a slit at the top and drilled and tapped two holes so that two setscrews can tighten the drilled hole for the tube. On the bottom leg of the mount, we drilled and tapped a hole so that we can secure the mount with a setscrew once a suitable position on the track is chosen. The laser mount is 1 thick, which may be excessive can be cut down to 0.5 and still maintain stability when attached to the track. Sliding Mount for the Mirror Mount: Figure C.3 The piece is based on the dimensions of the Newport U100-P2K mirror mount. The

81 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 70 lengths of the arms do not need to be exact. However, the holes drilled to match the clearance holes in the mirror mount must be precise. We used a vertical bandsaw to cut the L-shape of the mount where the mirror mount will attach. We drilled and tapped two holes, one on each leg, and put in setscrews to secure the mount on the track. The Wedge: Figure C.4 The wedge attaches the diffraction grating to the mirror mount. It holds the diffraction grating at a 45 angle. You can adjust the mirror mount using the set screws and the PZT to get the diffraction grating at the correct angle depending on the wavelength of the laser diode. The four holes are drilled based on the holes in the mirror mount. The PZT System: Figures C.5 and C.6 The barrel and piston system that holds the PZT consists of three pieces: the barrel, the cap, the sliding cylinder. This system ensures reproducible translation of the mirror mount by the PZT. The cap remains in place. The PZT pushes the sliding cylinder and then the sliding cylinder will push on the mirror mount. We put a ball bearing inside the end of the sliding cylinder to eliminate possible horizontal translation. The barrel is made of brass and has an inner and outer part of the tapped hole. The outer part of the tapped hole has a setscrew. The inner part is wider and will contain the PZT and the cylinder. The tapped portion of the hole needs to be done very carefully with a drill and a?-100 tap. It is important to note that the exterior and interior of the barrel both have two sections with different diameters. These sections are not the same length in the exterior versus the interior. The cap and sliding cylinder are made of steel and hold the PZT in place. Each piece has a slit made to fit the PZT, so to encapsulate it. We feed the electrical leads of the PZT out of the barrel. The dimensions are based on the mm piezoelectric actuator from Thorlabs (Part Number AE0203D04F). However, if one decides to reproduce this laser, one must measure the exact dimensions of the PZT and decide upon a depth for the slits on the cap and sliding cylinder. We made sure that the pieces had a snug fit, so that no other axial translation is possible. The other end of the cap is flat. The other end of the sliding cylinder has a conical hole the ball bearing is placed. To assemble this system, we drilled holes a bit larger than 0.04 to fit the wires through the mirror mount and barrel. It is important to choose a place for this hole on the mirror mount that will not damage it. We chose to put the hole 0.30 from the outermost edge of the mirror mount. We then had to disassemble the Thorlabs mirror mount by removing one end of the springs that holds the two faces together. We removed the commercial barrel and setscrew and replaced them with barrel. The setscrew we used was a 100TPI setscrew. We

82 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 71 placed the flat face of the cap against the setscrew. Then we gently pulled the PZT electric leads through the barrel holes and fit the PZT head into the slit of the cap. We fit the sliding cylinders slit side against the other side of the PZT head. Finally, we put a ball bearing on the other side of the cylinder and reassembled the mirror mount.

83 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 72 Figure C.2: Mount for the Laser Diode and Collimating Lens

84 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 73 Figure C.3: Sliding Mount for the Mirror Mount

85 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 74 Figure C.4: Wedge Diffraction Grating Mount

86 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 75 Figure C.5: PZT Tube

87 APPENDIX C. DETAILED DESCRIPTION AND USAGE OF THE 671 NM LASER 76 Figure C.6: PZT Cap