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Single color photoassociation spectroscopy of ⁶Li₂ and ⁸⁵Rb₂ 2012

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Single Color Photoassociation Spectroscopy of 6Li2 and 85Rb2 by Magnus Haw A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF HONOURS BACHELORS OF SCIENCE in The Faculty of Undergraduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2012 c© Magnus Haw 2012 Abstract Single color photoassociation (PA) of 85Rb2 and 6Li2 was achieved in an optical dipole trap with the goal of measuring new vibrational levels of the 13Σg state of 6Li2. Initial benchmark tests using Rb were performed to determine the optimum conditions for observing photoassociation on the apparatus: >.5 kW/cm2 PA laser intensity, high density, low temperature (∼ 10µK), and long hold times (>1s). These tests with Rb also revealed discrepancies between dipole trap PA data and spectra taken in a magneto- optic trap on another experiment (MAT). Initial calculations for the locations of the 6Li2 lines were carried out using fitted potentials provided to us by Nikesh Dattani [5]. We observed seven 6Li2 PA lines that were within 1 cm−1 of the predicted values. Because of this good agreement between theory and measurement for all seven lines, we are confident that these lines are indeed vibrational levels of the 13Σg state of Li2. Additional studies of two of these peaks revealed five or six-fold splitting due to the magnetic field. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 What is Photoassociation? . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Molecular States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Electronic State . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Vibrational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 Rotational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Photoassociation Rate . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Photoassociation Signal . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Atom Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Magneto-Optic Trap (MOT) . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 iii Table of Contents 3 Apparatus and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 Magneto-optic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Titanium Sapphire Ring Laser (TiSapph) . . . . . . . . . . . . . . . . . 20 3.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Measurement Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Rb Photoassociation Benchmark . . . . . . . . . . . . . . . . . . . . . . 22 4.1 First Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Hold Time Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Comparison with Previous Data . . . . . . . . . . . . . . . . . . . . . . 26 5 6Li Photoassociation Results . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.1 Calculations of Li levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Measured Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Preliminary Characterization of B-Field Dependence . . . . . . . . . . . 31 5.4 Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Appendices A Two Level System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 B Additional Data/Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 C Apparatus Schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iv List of Tables 2.1 Table of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 MOT parameters using B-field gradient of 23 G/cm2 . . . . . . . . . . . 19 3.2 IPG Dipole Trap parameters . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Typical Measurement Conditions . . . . . . . . . . . . . . . . . . . . . . 21 4.1 Comparison of Rb 0+u band energy levels (-12814 cm −1) . . . . . . . . . 26 5.1 Theoretical Predictions of Li2 1 3Σg Rovibrational Levels . . . . . . . . . 29 5.2 Measurements of Li2 1 3Σg Vibrational Levels (@830 gauss) . . . . . . . 30 5.3 Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B.1 Measurements of Peak Splitting at 12546 cm−1 (See Figure 5.7) . . . . . 48 v List of Figures 1.1 A depiction of the photoassociation process for two free atoms: A + B + γ → A-B*. The ground state and excited state coulomb potentials are plotted as a function of the internuclear distance. The vertical green arrow represents the photoassociation photon which couples the free atoms with kinetic energy kBT to an excited bound state. In general, the kinetic energy of atoms used in photoassociation needs to be much less than the splitting between states to achieve appropriate resolution (kBT << E∗n − E∗n−1, where E∗n is the energy of a particular state of the excited potential). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Plot of the eight lowest lying Li electronic states. Asymptotic behavior is shown on the right hand side and standard molecular term labels are given for each potential. For scale, 1 Hartree= 219474.63 cm−1. Taken from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Plot of the triplet ground state (a3Σ+u ) and first excited state (1 3Σ+u ) of 6Li2 and the currently measured vibrational levels. Blue vibrational levels are measurements from bound-bound spectroscopy. Red levels are data from photoassociation. Taken from [5]. . . . . . . . . . . . . . . . . 8 2.3 Decay modes of photoassociated molecules. The red atoms in the excited state represent the photoassociated molecule, the black circles are free atoms, the blue circles are a ground state molecule and the green arrows are emitted photons. The measured signal is the atom loss associated with these decay processes. Although the decay mode to the ground state does not result in loss from the trap, it will result in a reduced atom number as measured by our imaging system. . . . . . . . . . . . . . . . . . . . . 12 vi List of Figures 2.4 Plot of Lithium scattering length of |1〉-|2〉 mixture versus magnetic field. |1〉, |2〉 refer to Li’s two lowest hyperfine states. Note the two Feshbach resonances at 834 G and 1.3 T. Plot taken from [8]. . . . . . . . . . . . 14 2.5 Hyperfine structure of 6Li, I=1. Selection rules prevent transitions to the singlet excited state (22P1/2) resulting in a closed optical loop (blue arrows). This structure allows 6Li to be trapped in a MOT. . . . . . . . 15 2.6 Diagram of magneto-optic trap (MOT). Red detuned laser beams are pictured as arrows and magnetic coils are pictured as loops (upper and lower currents flow in opposite directions). Note that the trapping region is in a vacuum cell (coils can be outside). . . . . . . . . . . . . . . . . . 16 2.7 Plot of dipole potential for single arm. The upper cone-shaped surface is the 3D representation of the focus of a gaussian beam. Below is plotted the dipole potential energy as a function of distance along beam axis, z, and the radius from the beam axis, r. . . . . . . . . . . . . . . . . . . . 17 4.1 Plot of two Rb photoassociation scans for a hold time of 100 ms. The data for this hold time showed no reproducible trends or evidence for photoassociation resonances. Longer hold times (>1.4 sec) were the key factor in acquiring a photoassociation signal (see Figure 4.2). . . . . . . 23 4.2 Plot of representative Rb photoassociation signal for a hold time of 10 seconds. The data agrees with the published spectra [19] within expected errors (scan resolution 156MHz), see Table 4.1 for fit comparisons. The linear decrease in baseline is indicative of decreasing steady state atom number in the MOT trap. The peak at 12814.0 cm−1 was used for lifetime measurements and estimates of photoassociation rate (see Figure 4.3). The observed structure of peaks was reproducible. . . . . . . . . . . . . 24 vii List of Figures 4.3 Plot of Rb loss due to photoassociation. Data is shown for a PA laser power of 53mW. No significant change in loss rate versus hold time was observed for different powers (26mW, 35mW) implying that the PA rate is limited by collisional frequency and not by the photon scattering rate. Note that at times ∼ 100 ms, there is no significant loss induced by photoassociation and that by 5s the loss percentage saturates at 40%. The trap lifetime without PA light is ∼ 8s. Since there is 10% scatter in the number of atoms in a full dipole trap, it was necessary to hold for > 1.4 seconds to achieve reasonable signal (i.e 25% loss). . . . . . . . . . . 25 4.4 Plot of shifted high resolution MAT PA signal overlaid on dipole trap data. We find that the MAT data is shifted by +.04 cm−1 compared to the other two signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.5 Plot of Rb photoassociation signal around 12783 cm−1 against published spectra [19]. The data matches the published spectra in most respects except that the peak at 12783.4 cm−1 is significantly narrower. . . . . . 27 4.6 Plot of Rb photoassociation signal around 12794 cm−1 against published spectra [19]. The data is substantially different from the published spectra. 28 5.1 Plot of vibrational line at 12692 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. Note that the scatter in the atom number of the full trap is ∼ 10%. The outlier to the right of the main peak suggests the existence of a narrow resonance at that location. Plots of other peaks can be found in Appendix B. . . . . . . . . . . . . . 31 5.2 Plot of peak positions versus vibrational number. Red line is a quadratic fit. Note that the residuals conceal higher order terms. This data corre- sponds to the broad features found at high magnetic field (830 G). The fit for the data is: [-2.78724682e+00, 2.74306314e+02, 7.78522728e+03]*[(ν ′+ 1/2)2, (ν ′+ 1/2), 1]. The fit for the original predicted values from the po- tential gives: [ -2.78936905e+00, 2.74570667e+02, 7.78059001e+03]*[(ν ′+ 1/2)2, (ν ′ + 1/2), 1]. These values are within 1% of the measured values. 32 viii List of Figures 5.3 Plot of peak widths (FWHM) versus vibrational number. Behavior ap- pears to be non-linear. Note that this data was taken at high magnetic field (830 G). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.4 Plot of features at 800G (1kW/cm2, 2s hold time). . . . . . . . . . . . 33 5.5 Plot of vibrational line at 12394 cm−1 for 0 and 800G magnetic fields. Note the substantial difference in peak width. The peak at 12546 cm−1 had the same structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.6 Plot of main feature splitting at low magnetic fields. Each subplot shows the ν ′ = 22 resonance at a different magnetic field. Arrows indicate peak positions. We observe that each peak splits into three daughter peaks which shift position as a function of magnetic field. It is not clear whether the left main peak (B=0) has two or three daughter peaks; this plot identifies three such daughter peaks, however, the rightmost of these peaks is not visible above 40 G. . . . . . . . . . . . . . . . . . . . . . . . 35 5.7 Plot of vibrational level splitting as a function of B field. The positions of the six daughter peaks identified in Figure 5.6 are plotted versus magnetic field. This plot shows the movement of the two satellite features as a function of magnetic field and the formation of the broad resonance at 800 G. Values are shown in Table B.1 . . . . . . . . . . . . . . . . . . . 36 5.8 Plot of satellite peak positions as a function of magnetic field for ν ′ = 21. Data was taken with 1kW/cm2 and a 5s hold time. Slopes for the right peak (blue) and the left peak (red) are, respectively, 1.003e-4±4.0e-6 and 8.874e-5±4.2e-6 cm−1/G. . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.9 Plot of satellite peak positions as a function of magnetic field for ν ′ = 22. Data was taken with 1kW/cm2 and a 5s hold time. Slopes for the right peak (blue) and the left peak (red) are, respectively, 1.099e-4±5.8e-6 and 1.075e-4±5.9e-6 cm−1/G. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 B.1 Plot of vibrational line at 13097 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. Note that the scatter in the atom number of the full trap is ∼ 10%. . . . . . . . . . . . . . . . . . . 47 ix List of Figures B.2 Plot of vibrational line at 12967 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. . . . . . . . . . . . . . . . 48 B.3 Plot of vibrational line at 12833 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. . . . . . . . . . . . . . . . 49 B.4 Plot of vibrational line at 12546 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. The outlying point to the right of the main feature suggests the existence of a narrow peak. . . . 50 B.5 Plot of vibrational line at 12394 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. . . . . . . . . . . . . . . . 51 B.6 Plot of vibrational line at 12237 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. . . . . . . . . . . . . . . . 52 B.7 Plot of leftmoving features at different magnetic fields. Data was taken with 1kW/cm2 and a 5s hold time. Our current understanding is that these three leftmost peaks coalesce into the broad feature seen at 830 G. 53 B.8 Plot of satellite features at different magnetic fields. Data was taken with 1kW/cm2 and a 5s hold time. Our current understanding is that they are a result of hyperfine splitting. . . . . . . . . . . . . . . . . . . . . . . 54 C.1 Experiment Schematic: the photoassociation laser was collinear with the dipole trap and the beam path was optimized to have the maximum intersection with the MOT. The schematic only shows a single arm of the dipole trap (setup for Rb benchmark) for clarity. The measurements for Li utilized 2 arms of the IPG. Note: this is a cartoon schematic and does not show components to scale or the Gaussian profiles of the lasers. . . . 55 C.2 Magneto Optic Trap Schematic. The trap is the intersection of three pairs of counter propagating beams along three perpendicular axes between a pair of anti-Helmholtz coils. The small white dot at the center represents the trapped atom cloud. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 x List of Figures C.3 Control rack for MOL experiment. . . . . . . . . . . . . . . . . . . . . . 57 C.4 Titanium Sapphire (TiSapph) Ring Lasers. The near laser was used in this thesis in conjunction with a control box C.5. The further laser is being set up for future two colour photoassociation experiments. . . . . 58 C.5 Titanium Sapphire (TiSapph) Ring Laser Control Box. The laser fre- quency was set by sending a DC voltage from an analog out port on the experiment control rack to this control box. . . . . . . . . . . . . . . . . 59 C.6 Fiber Coupling of TiSapph laser. Coupling the laser into the fiber elim- inated a number of alignment issues. The red line traces the beam path through several optical elements. . . . . . . . . . . . . . . . . . . . . . . 60 xi Acknowledgements I would like to acknowledge my advisor, Dr. Kirk Madison, for his support and guidance. I would also like to give special thanks to: Mariusz Semczuk, Will Gunton, Dr. Jim Booth, and Janelle Dongen, for their invaluable work and assistance on this thesis. xii Chapter 1 Introduction The fields of atomic, molecular, and optical physics have expanded significantly in the past two decades due to improvements in laser technology. Specifically, the ability to cool and confine atoms to submillikelvin temperatures has opened many new avenues of research including Bose-Einstein condensation, fermi-degenerate gases, and optical lat- tices [21]. One of the new techniques made possible by laser-cooling is photoassociation spectroscopy. Photoassociation (PA) spectroscopy is interesting for two main reasons: first, the high resolution spectra give precise measurements of atomic properties and second, it provides a controlled method of creating molecules. Both avenues of research have resulted in considerable success over the last few years. Creating ground state molecules is of particular interest because they are relatively stable and have a wealth of physics to explore. Several areas of interest regarding ground state molecules include molecular Bose-Einstein condensates, qubit implementations, and orientation-dependent collisions [6, 14]. 1.1 What is Photoassociation? Photoassociation is the formation of an excited state molecule due to the absorption of a photon during a collision between two free atoms: A+B + γ → AB∗. (1.1) The process is illustrated in Figure 1.1. In the semi-classical view, the sum of the kinetic energy (kBT ) of the colliding atoms plus the photon energy must equal a molecular bound state energy for photoassociation to occur. In general, these photoassociated 1 1.1. What is Photoassociation? Interatomic radius P o te n ti a l E n e rg y Ground StateA+B Excited StateA-B* γ kB T Figure 1.1: A depiction of the photoassociation process for two free atoms: A + B + γ → A-B*. The ground state and excited state coulomb potentials are plotted as a function of the internuclear distance. The vertical green arrow represents the photoassociation photon which couples the free atoms with kinetic energy kBT to an excited bound state. In general, the kinetic energy of atoms used in photoassociation needs to be much less than the splitting between states to achieve appropriate resolution (kBT << E ∗ n−E∗n−1, where E∗n is the energy of a particular state of the excited potential). molecules radiatively decay with a short lifetime, τ ≈ 10µs, to two free atoms [17]: AB∗ → A+B + γ. (1.2) Photoassociation spectroscopy is made possible by the negligible thermal broadening of ensembles of ultracold gases (<1 mK). To achieve high resolution through photoassoci- ation spectroscopy, the energy of the atoms has to be on the order of or smaller than the natural linewidth of a particular photoassociation transition, Γ ∼ 107 Hz. This 2 1.2. Motivation corresponds to submillikelvin temperatures: T ≤ h̄Γ/kB ≈ 0.5mK. (1.3) Consequently, photoassociation experiments are dependent on laser cooling to achieve these temperatures and usually require the use of one or more atom traps. The necessity of laser cooling restricts the possible species for photoassociation to those atoms with simple valence structure, generally alkali metals with a single valence electron. 1.2 Motivation There are two major motivations for measuring new vibrational levels of the 13Σg ex- cited state of 6Li2: additional corrections to the potential and creation of ground state molecules. Measuring new vibrational levels of the 13Σg state 6Li2 will increase our knowledge of the excited state potential and provide an important test of theoretical predictions. The particular levels we target (ν ′=20-26) are especially significant for the 13Σg state because they bisect a 5000 cm−1 gap in the currently measured levels (ν ′=8-55) (see Figure 2.2). These particular measurements also provide a test of potential fitting techniques that have made predictions for the energy states in this gap [5]. Furthermore, knowing the location of these states is a prerequisite for measuring triplet ground state (a3Σu) vibrational levels using two color photoassociation. Measuring these ground state levels will characterize the low temperature scattering properties of Li and permit creation of ground state molecules. Ground state Li2 molecules formed from laser cooled atoms are interesting for sev- eral reasons. First, since Li2 is the second simplest molecule after H2, creating these molecules and studying their behavior (scattering properties, spectroscopy, etc...) may help improve few-body molecular theory. One topic of interest is the stability of a molecule in the lowest level of the ground triplet potential (a3Σu). While such a molecule is, without external perturbation, stable, it is interesting to see whether collisions or other processes can allow relaxation to the singlet ground state X1Σg. A second ques- tion we would like to address is whether the rotational state of these molecules affects their scattering properties. 3 1.3. Overview Creating triplet ground state LiRb molecules would also be of great interest because they would be the first polar ground state molecules with a magnetic moment. In addition to the unique potential for a strongly interacting condensate, polar ground state molecules represent one of the more attractive implementations of qubits. As a result, creating and trapping such molecules would be of great significance to the field of quantum computing [6]. There are currently no measured ground or excited states of LiRb and another spectroscopic search is needed to locate them. For this additional search, it is necessary to measure Li2 vibrational levels within the range of our lasers to distinguish LiRb photoassociation signal from that of the homonuclear dimers (Li2 and Rb2). Consequently, this study will pave the way for additional spectroscopic searches for LiRb resonances. 1.3 Overview The following is a general overview of the structure of this project. The first goal was to setup the apparatus necessary for the experiment. This included fiber coupling, beam shaping, and construction of control software. The second phase optimized the conditions for photoassociation in the dipole trap using 85Rb. 85Rb was chosen as the test species because it is one of the two atoms available in the apparatus and its spectrum has been well mapped by photoassociation spectroscopy in the past. In parallel with the 85Rb tests, calculations of vibrational levels were made using potentials acquired from [5]. The final portion of the thesis was directed at finding and characterizing the relevant 6Li2 states. 4 Chapter 2 Theory The theory covers three distinct areas: the physics of molecular states, the photoasso- ciation process, and atom traps. All three areas are essential to the measurement and characterization of 6Li2. 2.1 Molecular States In our discussion of molecular states, we are primarily concerned with the energy eigen- states of the molecule. These energy eigenstates form a complete basis for the molecular wavefunction and we can express any molecular state as a linear superposition of these eigenfunctions. For the remainder of this paper, the term molecular state will be used interchangeably with energy eigenstate. Molecular states are similar in many ways to atomic wavefunctions, however, there is one key difference: molecules have more de- grees of freedom and, as a result, have a number of additional modes of quantization (vibrations and rotations). Since we are interested in the creation of 6Li2 molecules, we will restrict our discussion to the basic case of a homonuclear diatomic molecule. For this simple case we can decompose the Hamiltonian into electronic, vibrational, and rotational terms and will provide an informal discussion of the quantization of each term. Hdiatomic = He +Hv +Hr (2.1) For an in depth treatment of diatomic molecular states see [4]. 2.1.1 Electronic State We will first consider the electronic part, He, of the Hamiltonian. One common approach to describing a molecular electronic state is to assume that it can be represented as a linear combination of atomic orbitals (LCAO). This method is used because it matches 5 2.1. Molecular States the asymptotic behavior of the wavefunction at large interatomic distances. In other words, when the molecule is pulled very far apart, the molecular wavefunction should correspond to the sum of two isolated atomic wavefunctions. In general, the electronic state is the term which defines the strength of interaction between the bonding atoms and is usually visualized as a potential curve plotted as a function of interatomic distance. Figure 2.1: Plot of the eight lowest lying Li electronic states. Asymptotic behavior is shown on the right hand side and standard molecular term labels are given for each potential. For scale, 1 Hartree= 219474.63 cm−1. Taken from [3]. Molecules have many more electronic states than single atoms and consequently require a more complicated labeling system. The levels are classified by total electronic spin S, the projection of the orbital angular momentum along the internuclear axis, Λ, parity (u/g), and the excitation level, n. These four quantities comprise the molecular term symbol, a label which is equivalent to the 1s 2s 2p... orbital classification of atomic 6 2.1. Molecular States systems: n2S+1Λ (+/−) (u/g) . (2.2) Λ corresponds to the angular momentum (0 → Σ, 1 → Π) similar to the s, p, d, f atomic system. The (u/g) subscript refers to the behavior of the wavefunction when inverted through the center of mass of the molecule: if the wavefunction remains the same under inversion, it is (gerade, g) and if it changes sign under inversion then it is (ungerade, u). This (u/g) classification is only valid for homonuclear molecules. The ± exponent labels symmetry with respect to reflections of the electronic coordinates through the a plane containing the nuclei and is only relevant for Σ states. Several exceptions and addendums apply to this scheme. The first is that the excitation level of the ground state is labeled with a capital X instead of a 0. Secondly, there is another numbering scheme used occasionally for diatomic molecules: the singlet (2S + 1 = 1) excitation level integers are exchanged with capital letters (A,B,C...) and the triplet (2S + 1 = 3) state excitation levels are exchanged with lower case letters (a,b,c...). For example, Li2 has a singlet ground state, X 1Σg, a lowest lying triplet state, a 3Σu, and a first excited Σ-triplet state, 13Σg. The interatomic potentials, standard labels, and asymptotic behavior for these states are shown in Figure 2.1. This work is primarily concerned with the first excited Σ-triplet state: 13Σg. 2.1.2 Vibrational Levels For a given molecular electronic state, there are a set of quantized vibrational states similar to the quantum harmonic oscillator. Since molecular potentials have finite bind- ing energy, there will be a finite number of vibrational levels which are distributed based on the curvature of the electronic potential. Figure 2.2 shows the currently measured vibrational levels for the triplet first excited state (13Σg). The large gap in the measured vibrational levels of 13Σg is one of the main motivations for this thesis. 2.1.3 Rotational Levels Rotational quantization for homonuclear diatomic molecules is analogous to the quan- tum mechanical rigid rotor. Since the molecule is symmetric, the energy eigenvalues are 7 2.1. Molecular States Figure 2.2: Plot of the triplet ground state (a3Σ+u ) and first excited state (1 3Σ+u ) of 6Li2 and the currently measured vibrational levels. Blue vibrational levels are measurements from bound-bound spectroscopy. Red levels are data from photoassociation. Taken from [5]. degenerate for a given total angular momentum, N , and we only need a single quantum number (N) to describe the system. The eigenvalues are: EN = h̄2 2I N(N + 1), (2.3) where I is the moment of inertia perpendicular to the molecular axis and N is the total orbital angular momentum (See Table 2.1). The rotational spacing is much smaller than the vibrational spacing (∆EN  ∆E′ν). Consequently, each vibrational level will have rotational structure. However, in photoassociation experiments, only a few rotational states (N < 4) can be observed for a given vibrational level due to limited angular momentum in the formation of the molecules.[14]. However, measurements of these 8 2.2. Photoassociation levels are highly useful as one can extract information about the interatomic potential and an effective bond length from the moment of inertia. Table 2.1: Table of Angular Momenta Type of Angular Momentum Operator Total Projection Electronic orbital L L Λ Electronic spin S S Σ Rotational R R ... Total J=R+L+S J Ω = Λ + Σ Total Orbital N=R+L N Λ 2.2 Photoassociation As described in the introduction, photoassociation is the formation of an excited state molecule due to the absorption of a photon during a collision between two free atoms. For a illustrative diagram, refer to Figure 1.1. 2.2.1 Photoassociation Rate The process of photoassociation is a dynamical quantum problem and the full treatment requires quantum electrodynamics. However, we can gain significant insight into the essential dependencies of photoassociation if we only consider electromagnetic coupling between the free scattering state and the excited bound state; in other words, a two level system [21]. This is a good assumption because the spacing between adjacent Li2 vibrational levels is ∼ 106 times the natural linewidth. Given the assumption of a two level system, we can find a first order approximation for the photoassociation rate in terms of known quantities: RPA = Rc ∗NsPs→b, (2.4) where Rc is the collision rate, Ns is the number of scattered photons per collision, and Ps→b is the probability of a transition for a photon scattering event. With this formulation, the collision rate sets the maximum photoassociation rate and the product of the scattering rate and the transition probability is a scaling factor between 0 and 9 2.2. Photoassociation 1 (Ps→b  [0, 1]) where 1 represents the case where every colliding pair that scatters a photon is photoassociated. We will first calculate the transition probability for coherent light incident upon a pair of colliding atoms. Assuming the two level approximation and using first order perturbation theory, we find that this transition probability is linearly dependent on the laser intensity and inversely proportional to the detuning from resonance: Ps→b ∝ I 1 + ∆2 , (2.5) where the subscript s refers to the free scattering state, the subscript b refers to the excited bound state, I is the intensity, and ∆ is the detuning from exact resonance. This calculation is carried out explicitly in Appendix A. Another, more sophisticated, approximation of the transition probability relies on the spatial overlap between the two wavefunctions: |〈ψs|ψb〉|2. This formulation is called the Franck-Condon Principle. An in depth description of this alternate formulation is given in reference [22]. The next parameter that needs to be estimated is Ns, the number of incident photons per collision. This parameter depends on the scattering rate and the average time of collision: Ns = Rs ∗ dt. (2.6) The scattering rate for the D1 and D2 lines of 6Li is Rs/2pi =5.8724 MHz. We will assume that this rate also extends to the vibrational levels of interest. To calculate the time of collision (dt), we will assume that dt is equal to the time it would take two spheres of cross-sectional area σc to pass through one another. If we average this quantity over all offsets, we find that: dt = σc r〈v〉 = √ σcpi 〈v〉 (2.7) The last parameter is the collision rate, Rc. Since we are concerned exclusively with 2-body collisions, we take the collision rate to be: Rc = σc〈v〉 ∫ ∞ 0 n2 d3r, (2.8) where n is the number density of atoms. If we assume that the atoms are held in a 10 2.2. Photoassociation nearly spherical trap, then we can approximate the atom density as a 3D gaussian: n(x, y, z) = N pi3/2σxσyσz e−( x σx )2e −( y σy )2 e−( z σz )2 . (2.9) We can then integrate this ansatz to give us an expression for the collision rate in terms of measurable parameters: Rc = σc〈v〉 N 2 (2pi)3/2σxσyσz (2.10) Combining all of the above terms, we get an expression for the photoassociation rate: RPA ∝ ( Rsσ 3/2 c )( I 1 + ∆2 )( N2 σxσyσz ) (2.11) where I is intensity, ∆, is frequency difference from resonance (detuning), N is number of trapped atoms, (σx, σy, σz) are the trap dimensions, Rs is the photon scattering rate (≤ 2 pi 5.8724 MHz) and σc is the collisional cross section. Note that the velocity dependence cancels. For the purpose of finding resonances, the objective is to maximize the photoassocia- tion rate. Consequently, we find that we want to maximize the atom number density, n, the collisional cross section, σc, and the laser intensity while minimizing the detuning. 2.2.2 Photoassociation Signal A photoassociation signal is found by observing atom loss from a neutral atom trap. In the photoassociation process, the molecule gains significant energy from the absorbed photon (∼ 2 eV for our target states) and then quickly decays (∼ 1 µs) to two free atoms or a lower bound state (Figure 2.3). Since the atoms traps have a depth that is on the order of a million times smaller than this energy (10−6 eV), when the molecule dissociates, both atoms will escape the trap. The other decay mode to ground state molecules is extremely weak and for the purpose of single color photoassociation can be ignored [14]. Using atom loss as our signal, we find that it is the integral of the photoassociation rate over some time, thold, which we will call the hold time. 11 2.2. Photoassociation Figure 2.3: Decay modes of photoassociated molecules. The red atoms in the excited state represent the photoassociated molecule, the black circles are free atoms, the blue circles are a ground state molecule and the green arrows are emitted photons. The measured signal is the atom loss associated with these decay processes. Although the decay mode to the ground state does not result in loss from the trap, it will result in a reduced atom number as measured by our imaging system. 12 2.3. Feshbach Resonances 2.3 Feshbach Resonances Feshbach resonances occur when the energy of free colliding atoms becomes degenerate with the energy of a bound state. This can happen through a variety of coupling mecha- nisms where an attractive potential is raised in energy by a magnetic or electric field. We are concerned in particular with magnetic Feshbach resonances of 6Li and their effect on the collisional cross section. One of the most interesting effects of Feshbach resonances is that the scattering length, as diverges near resonance. The effective collisional cross section goes like: σc = 4pia2s 1 + k2a2s , (2.12) where k is the collision wave vector, k = 2pi/λ. As a result, the collision rate near resonance is greatly enhanced but still limited to the scale of the de Broglie wavelength. To provide a sense of scale, 6Li’s scattering length on resonance (834 G) is greater than 1 µm, giving an effective collisional cross section of ≈ 13µm2, over 108 times larger than the physical size of atomic 6Li (See Figure 2.4). 2.4 Atom Traps The experiment relies on two types of atom traps to confine and cool the atoms. Atoms are accumulated and cooled in a magneto-optic trap (MOT) and then transferred to a dipole trap where they will be held during the photoassociation process. This trapping sequence is necessary to create the high-density, low temperature conditions required for photoassociation. 2.4.1 Magneto-Optic Trap (MOT) MOT’s are atom traps that have two crucial trapping components: an optical molasses and a magnetic field gradient. An optical molasses is a volume where radiation pressure provides a frictional force which confines and cools atoms. This is created by employ- ing doppler cooling along 3 perpendicular axes. doppler cooling is a process where the doppler effect is used to create velocity dependent radiation pressure. This velocity de- pendence originates from detuning the trapping laser light slightly red (lower frequency) 13 2.4. Atom Traps Figure 2.4: Plot of Lithium scattering length of |1〉-|2〉 mixture versus magnetic field. |1〉, |2〉 refer to Li’s two lowest hyperfine states. Note the two Feshbach resonances at 834 G and 1.3 T. Plot taken from [8]. 14 2.4. Atom Traps 2P3/2 2S1/2 F’=5/2 F’=1/2 F’’=3/2 F’’=1/2 671 nm F’=3/2 2P1/2 F=5/2 F=1/2 2.88 MHz 1.74 MHz 1.17 MHz 8.69 MHz 17.37 MHz 76.1 MHz 152.1 MHz 10 GHz Cooling Transition Repump Figure 2.5: Hyperfine structure of 6Li, I=1. Selection rules prevent transitions to the singlet excited state (22P1/2) resulting in a closed optical loop (blue arrows). This structure allows 6Li to be trapped in a MOT. of a resonance transition. In this situation, the atoms that are moving towards the laser source see the light doppler-shifted towards resonance and atoms moving away from the source see the light shifted away from resonance. The net effect is a frictional force due to radiation pressure that opposes motion in the direction of the source. Note that this force does not affect atoms moving away from the source (i.e does not accelerate atoms away from source). An optical molasses is created when 3 opposing pairs of red-detuned laser sources along the cartesian axes (x,y,z) are overlapped in a region of space (see Figure 2.6). In this region of overlap, atoms experience a frictional force opposing any direction of motion, hence the term, optical molasses. This process limits candidate atoms to species where selection rules form an isolated two, three, or four level system (i.e a closed optical loop) such that atoms do not fall into a dark state (i.e a state where the laser light no longer induces transitions). For the case of 6Li, there are two hyper- fine ground states (F’=1/2, 3/2). Consequently, an additional laser (a repump laser) is required to cool atoms that fall into the lower hyperfine state. Figure 2.5 displays the transitions typically used to trap 6Li. The magnetic field gradient further confines atoms within the optical molasses using 15 2.4. Atom Traps Figure 2.6: Diagram of magneto-optic trap (MOT). Red detuned laser beams are pic- tured as arrows and magnetic coils are pictured as loops (upper and lower currents flow in opposite directions). Note that the trapping region is in a vacuum cell (coils can be outside). a position dependent force. The magnetic field is zero at the center of the trap and linearly increases in the radial direction. The force results from position dependent Zeeman shifts in atomic levels which increase the radiation pressure in the direction of the magnetic field zero point. Since the motive force is a result of radiation pressure, this is not a conservative potential. Since all forces are non-conservative, there is no analytical expression for the trap depth of a MOT [12]. A more in depth description of MOT physics can be found in [18]. 2.4.2 Dipole trap Dipole traps use an entirely different force to trap atoms: the dipole force. This force is generated by illuminating neutral atoms with intense laser light to create an induced dipole moment, p = αE, that oscillates at the driving frequency of the laser[10]. This interaction has an associated potential energy (Figure 2.7): Udip = −1 2 〈p ·E〉 = − 1 20c Re(α)I, (2.13) 16 2.4. Atom Traps and the corresponding dipole force is simply the negative gradient of this potential: Fdip = 1 20c Re(α)∇I(r). (2.14) Figure 2.7: Plot of dipole potential for single arm. The upper cone-shaped surface is the 3D representation of the focus of a gaussian beam. Below is plotted the dipole potential energy as a function of distance along beam axis, z, and the radius from the beam axis, r. This dipole force will attract atoms to regions of higher intensity in the dipole trap (i.e the laser’s local focus). There are additional complications for laser frequencies near resonance (ω ≈ ω0) but the traps we are concerned with are far detuned to the red of resonance (ω  ω0) so we can ignore any frequency dependent near-resonance effects [11]. The dipole trap is better suited than the MOT for most experiments because it has a greater density of atoms, is more easily modeled, and does not cycle the electronic state of the trapped species. Typical dipole trap depths are of the order of 10-100µK. See Will Gunton’s undergraduate thesis, [11], for a complete description of the theory and characterization of the dipole traps used in this experiment. 17 Chapter 3 Apparatus and Methods The measurements for this thesis were taken on the molecule experiment (MOL) at the Quantum Degenerate Gases Laboratory (QDG) at UBC. The MOL is a dual species (Li and Rb) laser cooling apparatus which consists of two overlapped magneto-optic traps (MOT’s) and two overlapped dipole traps. The general operating procedure is to trap atoms in the MOT and then transfer them to one of the dipole traps. 3.1 Magneto-optic Traps The MOT setup consists of the laser control system, the optical table, and the vacuum apparatus. A full description is given in [16]. The system uses a combination of grating- stabilized and injection-seeded diode lasers (master and slave lasers). The master lasers generate resonant light which is subsequently amplified by the slave lasers. The laser control system consists of 5 master lasers: 2 for the pump and repump transitions of each Rb isotope (85Rb, 87Rb), and a single master laser for 6Li. 6Li only requires a single master laser because the repump transition light is generated by sending part of the master light through an acousto-optic modulator (AOM) which shifts the frequency by the required ∼228 MHz; this is only possible because Li has a small hyperfine splitting in the ground state compared to other alkali species. The master lasers are “locked” to a given frequency using saturated absorption spectroscopy, a technique which passes a portion of the master light through a reference cell and measures absorption using a photodiode. Using reference cells of the target species (Rb or Li) and scanning the frequency of the master laser, we can pinpoint the desired transition in the absorption spectrum and hold the frequency at this position. This master light is then amplified to an appropriate power using seed-injected slave diodes. The optical table consists of a variety of mirrors, lenses, polarizers, shutters, and AOM’s which shape, tune, and merge the pump and repump light into the 6 perpendic- 18 3.2. Dipole trap ular MOT beams. Table 3.1: MOT parameters using B-field gradient of 23 G/cm2 6Li 85Rb/87Rb λD2,vac (nm) 670.977 780.241 Γ/2pi (MHz) 5.87 6.07 Isat (mW/cm 2) 3.8 3.58 Ipump / Isat 25 72 Irepump/Isat 14 3.0 Neq (8± 2)× 107 ≥ (7± 2)× 107 The vacuum apparatus serves two primary purposes: it provides an optically acces- sible vacuum cell and introduces gaseous species of interest into this cell. The vacuum system has a background pressure of ∼ 10−7 N/m2 (10−10 Torr), and contains Li and Rb dispensers which can be activated by passing a current through them. For comparison, atmospheric pressure is ∼ 105 N/m2. The MOT’s equilibrium atom number is approximately 108 atoms in steady state for both Li and Rb. More specific MOT parameter values are given in Table 3.1. For more details on the construction of the MOT setup, see [16]. 3.2 Dipole trap The dipole trap was produced using a IPG fiber laser. Relevant parameters are shown in Table 3.2. The IPG dipole trap was primarily used in a cross trap configuration: two nearly perpendicular overlapped dipole traps. This forms a more confined trapping region with higher atom density and results in a higher signal to noise absorption image. Table 3.2: IPG Dipole Trap parameters λ (nm) 1064 P (W) 9-12 zwaist (µm) 30 ULi (µK) 53 µK URb (µK) 130 µK 19 3.3. Titanium Sapphire Ring Laser (TiSapph) 3.3 Titanium Sapphire Ring Laser (TiSapph) The TiSapph laser is a Coherent Model 899 Titanium Sapphire Ring Laser. It has an output power of up to 500 mW and its frequency is tunable from 763-817 nm using a set of intra-cavity etalons (the intra-cavity assembly). Additional sets of internal optics can increase this range to 690-1080 nm [13]. The TiSapph light is coupled into a single mode fiber which takes the light to the optical table. The TiSapph also has an associated control box that can set the output frequency of the laser within ±100 GHz of the central frequency. It is this control box which was used to operate the fine scans in frequency. 3.4 Imaging This thesis employed absorption imaging to measure atom number in the dipole trap. Absorption imaging works by taking images of light that has passed through the atom trap. One then reconstructs a picture of the trap by differencing images with trapped atoms and images without atoms. This technique has high sensitivity and spatial resolu- tion making it ideal for studies of small dipole traps. A full explanation of this imaging system is very well described in [7]. 3.5 Measurement Procedure The general approach to the measurements of this thesis is the same as any other form of spectroscopy: scan the frequency of incident light over a range where we expect to observe resonances. For this project, the measured quantity was the number of atoms remaining in the dipole trap after a fixed hold time. If the photoassociation light was at a resonance, we would expect to see a substantial drop in the number of atoms remaining in the trap. The general procedure was to trap atoms in the dipole trap and then illuminate them for a period of time, thold, with the photoassociation light, measure the number of remaining atoms and then repeat the procedure over a range of frequencies. The average trial run consisted of a loop of six major steps: 1. Set photoassociation light frequency 20 3.5. Measurement Procedure 2. Load MOT (15-20s) 3. Load dipole trap from MOT 4. Evaporate atoms in dipole trap at a particular magnetic field 5. Illuminate atoms in dipole trap with photoassociation light (1-10s) 6. Image remaining atoms This sequence was usually repeated for 100 points in frequency steps of 300 MHz. A single run would take upwards of one hour to complete. Evaporation of atoms in the dipole trap lowered the temperature of the sample, decreased variation in final atom number, and increased number density. The Rb benchmark measurements were all taken with a single arm of the IPG dipole trap at zero magnetic field (i.e no Feshbach enhanced collision rate). The Li measurements were taken with both arms of the IPG dipole trap at high magnetic field to take advantage of Li’s increased collisional cross-section in that regime. Typical measurement conditions are shown in Table 3.3. Table 3.3: Typical Measurement Conditions n (atoms/m3) 1016 I (kW/cm2) .5-1 Utrap (µK) 50 Tatoms (µK) 5 thold (s) 3 Additional schematics and images of the apparatus are shown in Appendix C. 21 Chapter 4 Rb Photoassociation Benchmark 4.1 First Signal The primary difficulty in obtaining first signal was finding the correct hold time (the length of time to illuminate the trapped atoms with photoassociation light). In previous works [1, 2, 15, 19], hold times were reported to be on the order of 100 ms. However, we were unable to observe any significant signal with this hold time despite optimizing alignment of the traps, optimizing PA laser intensity, and increasing MOT atom number (Figure 4.1). For our system, we found it necessary to have a hold time of several seconds to achieve reasonable signal. The first signal in the dipole trap was observed with a 10s hold time, two orders of magnitude larger than the 100 ms quoted in the literature (see Figure 4.2). It should be noted that the Rb photoassociation data was taken at zero magnetic field and did not have enhanced collision rate due to Feshbach resonances. 4.2 Hold Time Dependence The next step was to determine the minimum hold time required for an adequate PA signal, so as to minimize the duration of a frequency scan. This was important in the subsequent search for Li vibrational lines. To do this, we measured the atom number for different trap hold times with and without resonant PA light. The Rb PA peak at 12814 cm−1 was used as the test resonance. Data for this measurement is shown in Figure 4.3. We observed that the PA signal was only significant after a hold time of 1.4 seconds and that the signal was insensitive to a factor of 2 change in laser intensity. This lack of sensitivity implies that the PA is limited by collisional frequency rather than the photon scattering rate. Furthermore, the length of the hold time is unexpected because it indicates a very low PA rate compared to previous experiments ([1, 2, 15, 19]). This is unexpected since since our system should have greater density, greater number of atoms, 22 4.2. Hold Time Dependence 0.00 0.05 0.10 0.15 0.20 0.25 PA Laser Frequency (+12814cm−1 ) 150 200 250 300 350 S ca le d A to m n u m b er (1 03 ) Trial 1 Trial 2 Heinzen et al. 1993 Figure 4.1: Plot of two Rb photoassociation scans for a hold time of 100 ms. The data for this hold time showed no reproducible trends or evidence for photoassociation reso- nances. Longer hold times (>1.4 sec) were the key factor in acquiring a photoassociation signal (see Figure 4.2). 23 4.2. Hold Time Dependence 0.1 0.0 0.1 0.2 0.3 0.4 PA Laser Frequency (+12814cm−1 ) 200 250 300 350 400 S ca le d A to m n u m b er (1 03 ) MOL 2012 Heinzen et al. 1993 Figure 4.2: Plot of representative Rb photoassociation signal for a hold time of 10 seconds. The data agrees with the published spectra [19] within expected errors (scan resolution 156MHz), see Table 4.1 for fit comparisons. The linear decrease in baseline is indicative of decreasing steady state atom number in the MOT trap. The peak at 12814.0 cm−1 was used for lifetime measurements and estimates of photoassociation rate (see Figure 4.3). The observed structure of peaks was reproducible. 24 4.2. Hold Time Dependence 0 100 200 300 400 500 A to m N u m b er (1 03 ) PA off PA on 0 1000 2000 3000 4000 5000 6000 7000 8000 Hold Time (ms) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 P er ce n t lo ss Figure 4.3: Plot of Rb loss due to photoassociation. Data is shown for a PA laser power of 53mW. No significant change in loss rate versus hold time was observed for different powers (26mW, 35mW) implying that the PA rate is limited by collisional frequency and not by the photon scattering rate. Note that at times ∼ 100 ms, there is no significant loss induced by photoassociation and that by 5s the loss percentage saturates at 40%. The trap lifetime without PA light is ∼ 8s. Since there is 10% scatter in the number of atoms in a full dipole trap, it was necessary to hold for > 1.4 seconds to achieve reasonable signal (i.e 25% loss). 25 4.3. Comparison with Previous Data Table 4.1: Comparison of Rb 0+u band energy levels (-12814 cm −1) Bergeman 2006 et al. [2] Dipole trap MAT -.001±.036 -.003±.012 .148±.024 .149±.016 .0108±.03 .291±.018 .291±.009 .255±.03 .4295±.015 .427±.013 and lower temperature. 4.3 Comparison with Previous Data Since Rb was only used as a benchmark test for PA, it was not intensively studied after confirming it reproduced the known spectrum around 12814 cm−1. However, there are several discrepancies worth noting. The first was the confirmation that there was a frequency shift in the measurements taken in a different type of trap (MOT) on a different apparatus (Miniature Atom Trap, MAT). The MAT data was taken using the same master lasers, the same TiSapphire laser for photoassociation, and the same wavemeter for measuring frequency. Consequently, it is surprising that the two data sets differ. The observed shift is +.03-.05 cm−1 which is 15 times the wavemeter standard error. A high resolution MAT scan is shown in Figure 4.4 and a comparison of peak fits parameters are shown in Table 4.1. The current understanding is that there are additional processes in the MOT versus the dipole trap (e.g cycling of states) which produce shifts in the measured resonances. We repeated these initial measurements in two other regions to compare with previ- ous spectra. The data is shown in Figures 4.5, 4.6. The first region around 12783 cm−1 matches well: the data follows the same trends as the published spectra. Heinzen’s data was taken with much higher intensity PA light (14kW/cm2 vs .5 kW/ cm2) and, as expected, shows broader and deeper peaks. The second region, however, does not agree with Heinzen’s data and shows different numbers of peaks in different locations. Given the good agreement seen in the 12814 cm−1 and 12783 cm−1 regions (Figures 4.2,4.5), it is surprising that the measured spectrum differs significantly from the published values in this third region. This disparity merits additional investigation but was beyond the scope of the project. 26 4.3. Comparison with Previous Data 0.10 0.15 0.20 0.25 0.30 PA Laser Frequency (+12814cm−1 ) 200 250 300 350 400 S ca le d A to m n u m b er (1 03 ) MOL 2012 Heinzen et al. 1993 MAT 2011 +.04cm-1 Figure 4.4: Plot of shifted high resolution MAT PA signal overlaid on dipole trap data. We find that the MAT data is shifted by +.04 cm−1 compared to the other two signals. 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 PA Laser Frequency (+12783cm−1 ) 10 15 20 25 30 S ca le d A to m n u m b er (1 03 ) MOL 2012 Scan 1 MOL 2012 Scan 2 Heinzen et al. 1993 Figure 4.5: Plot of Rb photoassociation signal around 12783 cm−1 against published spectra [19]. The data matches the published spectra in most respects except that the peak at 12783.4 cm−1 is significantly narrower. 27 4.3. Comparison with Previous Data 0.2 0.1 0.0 0.1 0.2 PA Laser Frequency (+12794cm−1 ) 40 60 80 100 120 S ca le d A to m n u m b er (1 03 ) MOL 2012 Scan 1 MOL 2012 Scan 2 Heinzen et al. 1993 Figure 4.6: Plot of Rb photoassociation signal around 12794 cm−1 against published spectra [19]. The data is substantially different from the published spectra. 28 Chapter 5 6Li Photoassociation Results 5.1 Calculations of Li levels 6Li2 states were calculated from a published potential through a discrete variable rep- resentation (DVR) of the Hamiltonian. The calculations were performed by adapting Roman Krem’s DVR Fortran code to run on the potential obtained from [5]. A full description of the calculations can be found in [20]. The calculated transition energy values are shown in Table 5.2. Because our single Li atoms are at a temperature of < 10µK, we expect only s-wave collisions and therefore only transitions to the N=1 rotational level. Calculations were also carried out for N=0,2 in preparation for future searches for rotational lines at higher temperatures. Table 5.1: Theoretical Predictions of Li2 1 3Σg Rovibrational Levels ν ′ N=1 (cm−1) N=0 (∆cm−1) N=2 df (∆cm−1) 20 12237.07 -0.25531296 0.765938881 21 12394.46 -0.238191683 0.714575048 22 12546.29 -0.222432344 0.667297033 23 12692.57 -0.207875321 0.623625962 24 12833.27 -0.194386443 0.583159328 25 12968.34 -0.181851513 0.545554539 26 13097.86 -0.17017335 0.510520051 These predictions compared favorably with vibrational energies calculated by Nikesh Dattani, the provider of the potential curves [5]. 29 5.2. Measured Lines 5.2 Measured Lines Armed with the above predictions and the results of the Rb PA tests, the next step was to locate the Li PA lines. The approach was to use the highest PA laser intensity (1kW/cm2), a 10s hold time, and conduct the PA at high magnetic field (800-830 G) near a Feshbach resonance (834 G). Consequently, the observed lines at these high magnetic fields are shifted and broadened. This The first observed line was for the highest vibrational state, ν ′ = 26 (within .8 cm−1 of the predicted value). The remaining lines also turned out to be very close to the predicted values and we found the locations of all 7 accessible lines within two days. Table 5.2 shows the Lorentzian fit parameters for each of the 7 lines and Figure 5.1 shows a typical fit. For our measurement range, the singlet excited state (11Σu) is also accessible and cannot be ruled out as a confounding variable. However, as all 7 measured lines agree quite well with the initial predictions, we expect that they are indeed vibrational states of the 13Σg potential. Table 5.2: Measurements of Li2 1 3Σg Vibrational Levels (@830 gauss) ν ′ Predicted (cm−1) Measured (cm−1) FWHM (cm−1) 20 12237.073 12237.178±.0006 .018±.0022 21 12394.456 12394.395±.0003 .012±.0008 22 12546.296 12546.065±.0004 .012±.0014 23 12692.573 12692.170±.0008 .022±.0020 24 12833.269 12832.698±.0005 .026±.0016 25 12968.369 12967.640±.0004 .028±.0010 26 13097.863 13096.991±.0005 .024±.0022 We also discovered that there are additional satellite features adjacent to the main feature at these high magnetic fields for at least 2 vibrational lines (ν ′ = 21, 22) and probably a third (ν ′ = 23). Figure 5.4 shows the relative spacing for these satellite peaks with respect to the main feature. The two satellite features are separated by ∼ 240 MHz and are an order of magnitude narrower than the main feature. We postulate that these satellite features are the result of splitting due to the magnetic field and that the main peak is a conglomeration of smaller split peaks. 30 5.3. Preliminary Characterization of B-Field Dependence 0.2 0.1 0.0 0.1 0.2 PA Laser Frequency (+12692cm−1 ) 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.797±0.0508 Sigma param 0.011±0.0011 Freq 12692.17±0.0008 Figure 5.1: Plot of vibrational line at 12692 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. Note that the scatter in the atom number of the full trap is ∼ 10%. The outlier to the right of the main peak suggests the existence of a narrow resonance at that location. Plots of other peaks can be found in Appendix B. 5.3 Preliminary Characterization of B-Field Dependence After locating the lines of interest, the B-field dependence of two of the peaks (ν ′ = 21, 22) was investigated. We find that at zero field, the resonance becomes a set of two narrow peaks separated by 240 MHz, slightly blue of the broad resonance at 830 G. It should be noted that 240 MHz is approximately the hyperfine ground state atomic energy level separation. A plot of the zero field behavior is overlaid on the 830 G resonance in Figure 5.5. Both vibrational levels (ν ′ = 21, 22) exhibit this same structure at zero magnetic field. Further measurements of the resonance at low magnetic field showed that the two narrow features at zero magnetic field each split into two or three daughter peaks as the field increases (Figure 5.6). If we trace the position of these daughter peaks (DP), we find that the two bluest DP’s shift to higher frequency at higher B-field becoming the satellite peaks and that the 31 5.4. Magnetic Moments 12200 12300 12400 12500 12600 12700 12800 12900 13000 13100 F re q u en cy cm −1 sq:-2.787e+00 ± 1.450e-03 slope:2.743e+02 ± 6.821e-02 incpt:7.785e+03 ± 7.974e-01 20 21 22 23 24 25 26 27 Vibrational Number (ν+1/2) 0.015 0.010 0.005 0.000 0.005 0.010 0.015 R e si d u a ls Figure 5.2: Plot of peak positions versus vibrational number. Red line is a quadratic fit. Note that the residuals conceal higher order terms. This data corresponds to the broad features found at high magnetic field (830 G). The fit for the data is: [-2.78724682e+00, 2.74306314e+02, 7.78522728e+03]*[(ν ′ + 1/2)2, (ν ′ + 1/2), 1]. The fit for the origi- nal predicted values from the potential gives: [ -2.78936905e+00, 2.74570667e+02, 7.78059001e+03]*[(ν ′ + 1/2)2, (ν ′ + 1/2), 1]. These values are within 1% of the mea- sured values. remaining peaks shift to lower frequency to form the large resonance features measured at 800-830 G. 5.4 Magnetic Moments We can also make tentative measurements of the magnetic moment for the 2 daughter peaks with strong magnetic field dependence. A plot and fit of the positions, for the ν ′ = 21, 22 levels, as a function of magnetic field is shown in Figures 5.8,5.9 respectively. Table 5.3 shows the fit values. 32 5.4. Magnetic Moments 20 21 22 23 24 25 26 27 Vibrational Number (ν+1/2) 0.010 0.015 0.020 0.025 0.030 F W H M cm −1 Figure 5.3: Plot of peak widths (FWHM) versus vibrational number. Behavior appears to be non-linear. Note that this data was taken at high magnetic field (830 G). 0.36 0.38 0.40 0.42 0.44 0.46 0.48 PA Laser Frequency (+12394 cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er Main feature Satellite Features Figure 5.4: Plot of features at 800G (1kW/cm2, 2s hold time). 33 5.4. Magnetic Moments 0.34 0.36 0.38 0.40 0.42 0.44 PA Laser Frequency (+12394 cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er 0 G 800 G Figure 5.5: Plot of vibrational line at 12394 cm−1 for 0 and 800G magnetic fields. Note the substantial difference in peak width. The peak at 12546 cm−1 had the same structure. Table 5.3: Magnetic Moments Daughter Peaks Left Peak (MHz/G) Right Peak (MHz/G) ν ′=21 2.514±100 3.018±100 ν ′=22 2.818±93 3.174±181 Atomic Li Lower state (MHz/G) Upper state (MHz/G) 2S1/2 .47 1.4 2P3/2 2.8 2.05 34 5.4. Magnetic Moments 0.066 0.0680.070 0.0720.074 0.076 0.0780.080 0.082 0.0 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er 0 G 0.066 0.0680.070 0.0720.074 0.076 0.0780.080 0.082 0.0 0.2 0.4 0.6 0.8 1.0 20 G 0.066 0.0680.070 0.0720.074 0.076 0.0780.080 0.082 PA Laser Frequency (+12546 cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er 30 G 0.066 0.0680.070 0.0720.074 0.076 0.0780.080 0.082 PA Laser Frequency (+12546 cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 40 G Figure 5.6: Plot of main feature splitting at low magnetic fields. Each subplot shows the ν ′ = 22 resonance at a different magnetic field. Arrows indicate peak positions. We observe that each peak splits into three daughter peaks which shift position as a function of magnetic field. It is not clear whether the left main peak (B=0) has two or three daughter peaks; this plot identifies three such daughter peaks, however, the rightmost of these peaks is not visible above 40 G. 35 5.4. Magnetic Moments 0 100 200 300 400 500 600 700 800 Magnetic Field (G) 0.06 0.07 0.08 0.09 0.10 0.11 0.12 P A L as er F re q u en cy ( + 12 54 6 cm −1 ) Figure 5.7: Plot of vibrational level splitting as a function of B field. The positions of the six daughter peaks identified in Figure 5.6 are plotted versus magnetic field. This plot shows the movement of the two satellite features as a function of magnetic field and the formation of the broad resonance at 800 G. Values are shown in Table B.1 36 5.4. Magnetic Moments 100 200 300 400 500 Magnetic field (G) 0.41 0.42 0.43 0.44 0.45 0.46 P ea k P os it io n ( + 12 39 4 cm −1 ) Figure 5.8: Plot of satellite peak positions as a function of magnetic field for ν ′ = 21. Data was taken with 1kW/cm2 and a 5s hold time. Slopes for the right peak (blue) and the left peak (red) are, respectively, 1.003e-4±4.0e-6 and 8.874e-5±4.2e-6 cm−1/G. 37 5.4. Magnetic Moments 100 150 200 250 300 350 400 Magnetic field (G) 0.07 0.08 0.09 0.10 0.11 P ea k P os it io n ( + 12 54 6 cm −1 ) Figure 5.9: Plot of satellite peak positions as a function of magnetic field for ν ′ = 22. Data was taken with 1kW/cm2 and a 5s hold time. Slopes for the right peak (blue) and the left peak (red) are, respectively, 1.099e-4±5.8e-6 and 1.075e-4±5.9e-6 cm−1/G. 38 Chapter 6 Future Outlook There are a variety of topics that are of interest now that the requirements for pho- toassociation have been established. The first is to further characterize the magnetic field dependence of all seven accessible lines and to corroborate that these states are vibrational levels of Li2 1 3Σg. The determination of the line origin will consist of cal- culating vibrational levels of the other local potentials as well as implementing tests for the triplet character of the measured lines. The next major topic revolves around finding and measuring rotational structure of each vibrational state. These measurements would be critical to improving the 13Σg po- tential curve. Other avenues of research include production of ground state molecules via two color photoassociation, a search for LiRb lines, and examination of the discrepancy between MAT PA lines and dipole trap measurements. 39 Chapter 7 Conclusions This project measured seven new vibrational levels of 6Li2. The conditions for mea- surement were first optimized using photoassociation of known Rb lines. This process consisted of preliminary scans to identify strong resonance lines and then optimization of the PA rate at those resonance lines Our preliminary Rb spectroscopy of two (12814, 12783 cm−1) of three sample regions agreed well with published spectra while the third (12794 cm−1) had substantial differences. We also note that our data exhibits marked differences (a +.04 cm−1 frequency shift and different peak structures) from measure- ments taken in an adjacent MOT apparatus (MAT) that uses the same master and photoassociation lasers. To optimize the conditions for photoassociation, we chose the Rb PA resonance at 12813.99 cm−1 and examined PA rate as a function of laser inten- sity and hold time (the time the atoms were illuminated with photoassociation light). This benchmark testing indicated that photoassociation on our apparatus is limited by collision rate and that we require an order of magnitude greater hold time (≥ 1s) to get an adequate PA signal. We then proceeded to use these optimized conditions to search for new 6Li2 vibra- tional levels. Calculations of the level energies were carried out by finding the energy eigenvalues of a fitted 6Li2 1 3Σg potential curve using Roman Krem’s Discrete Variable Representation (DVR) Fortran code. We then began scans around the calculated ener- gies. To enhance our low collision rate, we took our initial measurements at 830 G, near a 6Li Feshbach resonance and increased the trap number density by adding a second arm to the dipole trap. We found broad resonance features within ±.8 cm−1 of all seven calculated values. We are confident that these resonances are the expected vibrational levels of the triplet potential (13Σg, v’=20-26) because of the excellent agreement with the theoretical calculations. To better determine the behaviors of the resonances, two of the peaks (v’=21,22) were examined at different magnetic fields. We found that at zero magnetic field, both of these resonances reduce to two narrow peaks. Further scans at 40 Chapter 7. Conclusions increasing magnetic field showed that each of these narrow peaks exhibit two or three- fold splitting and that the broad features observed at 830 G are combinations of these split peaks. Additional data is needed to determine if this structure is repeated for the other measured vibrational levels. These measurements are significant because they make possible a host of new pho- toassociation experiments as well as increasing our knowledge of the 6Li2 1 3Σg state. Studies of the unexpected splitting of the newly discovered lines, rotational states, as well as ground state molecules are all made accessible by these measurements. Perhaps even more significant is that the hardware and procedures for photoassociation have been streamlined so future work can be done at an accelerated rate. 41 Bibliography [1] E. R. I. Abraham, N. W. M. Ritchie, W. I. McAlexander, and R. G. Hulet. Photoas- sociative spectroscopy of long-range states of ultracold 6li2 and 7li2. The Journal of Chemical Physics, 103(18):7773–7778, 1995. [2] T Bergeman, J Qi, D Wang, Y Huang, H K Pechkis, E E Eyler, P L Gould, W C Stwalley, R A Cline, J D Miller, and D J Heinzen. Photoassociation of 85 rb atoms into 0 u + states near the 5s+5p atomic limits. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(19):S813, 2006. [3] Boris and Minaev. Ab initio study of low-lying triplet states of the lithium dimer. Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 62(45):790 – 799, 2005. [4] John M. Brown and Alan Carrington. Rotational spectroscopy of diatomic molecules. Cambridge molecular science series. Cambridge University Press, Cam- bridge, 2003. [5] Nikesh S. Dattani and Robert J. Le Roy. A dpf data analysis yields accurate analytic potentials for and that incorporate 3-state mixing near the state asymptote. Journal of Molecular Spectroscopy, 268(1-2):199 – 210, 2011. [6] D. DeMille. Quantum computation with trapped polar molecules. Phys. Rev. Lett., 88:067901, Jan 2002. [7] M. Gao. Honours thesis, University of British Columbia, 2008. [8] M. E. Gehm. Preparation of an Optically-Trapped Degenerate Fermi Gas of 6Li: Finding the Route to Degeneracy. Phd thesis, Duke University, 2003. 42 Bibliography [9] David Griffiths. Introduction to Quantum Mechanics. Pearson Prentice Hall, 2nd edition edition, 2005. [10] Rudolf Grimm, Matthias Weidemller, and Yurii B. Ovchinnikov. Optical dipole traps for neutral atoms. volume 42 of Advances In Atomic, Molecular, and Optical Physics, pages 95 – 170. Academic Press, 2000. [11] Will Gunton. The loading and storage of li and rb in an optical dipole trap. Honours thesis, University of British Columbia, 2009. [12] Magnus Haw, Nathan Evetts, Will Gunton, Janelle Van Dongen, James L. Booth, and Kirk W. Madison. Magneto-optical trap loading rate dependence on trap depth and vapor density. J. Opt. Soc. Am. B, 29(3):475–483, March 2012. [13] Coherent Inc. Coherent 899 titanium sapphire ring laser user manual. [14] Kevin M. Jones, Eite Tiesinga, Paul D. Lett, and Paul S. Julienne. Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering. Rev. Mod. Phys., 78:483–535, May 2006. [15] S. D. Kraft, M. Mudrich, M. U. Staudt, J. Lange, O. Dulieu, R. Wester, and M. Weidemüller. Saturation of Cs2 photoassociation in an optical dipole trap. Physical Review A, 71(1):013417, January 2005. [16] Keith Ladouceur, Bruce G. Klappauf, Janelle Van Dongen, Nina Rauhut, Bastian Schuster, Arthur K. Mills, David J. Jones, and Kirk W. Madison. Compact laser cooling apparatus for simultaneous cooling of lithium and rubidium. J. Opt. Soc. Am. B, 26(2):210–217, Feb 2009. [17] Pd Lett, Ps Juilienne, and Wd Phillips. Photoassociative Spectroscopy of Laser- cooled Atoms. Annual Review of Physical Chemistry, 46:423–452, 1995. [18] H. J. Metcalf and P. van der Straten. Laser cooling and trapping of atoms. J. Opt. Soc. Am. B, 20(5):887–908, May 2003. [19] J. D. Miller, R. A. Cline, and D. J. Heinzen. Photoassociation spectrum of ultracold rb atoms. Phys. Rev. Lett., 71:2204–2207, Oct 1993. 43 [20] Eite Tiesinga, Carl J. Williams, and Paul S. Julienne. Photoassociative spec- troscopy of highly excited vibrational levels of alkali-metal dimers: Green-function approach for eigenvalue solvers. Phys. Rev. A, 57:4257–4267, Jun 1998. [21] John Weiner, Vanderlei S. Bagnato, Sergio Zilio, and Paul S. Julienne. Experiments and theory in cold and ultracold collisions. Rev. Mod. Phys., 71:1–85, Jan 1999. [22] R. Wester, S.D. Kraft, M. Mudrich, M.U. Staudt, J. Lange, N. Vanhaecke, O. Dulieu, and M. Weidemller. Photoassociation inside an optical dipole trap: absolute rate coefficients and franck-condon factors. Applied Physics B: Lasers and Optics, 79:993–999, 2004. 10.1007/s00340-004-1649-5. 44 Appendix A Two Level System The derivation given here follows Chapter 9 of Introduction to Quantum Mechanics by Griffiths [9]. We begin by assuming there are two eigenstates, ψa and ψb corresponding to the ground state and the excited state. In the absence of a perturbation, these states oscillate in time with frequencies Eah and Eb h . In this case, an arbitrary state can be expressed as: ψ(t) = caψae −iEat/h̄ + cbψbe−iEbt/h̄, (A.1) where |ca|2 + |cb|2 = 1. However, if we introduce a time dependent perturbation, H ′(t), the coefficients of each state will gain time dependence: ψ(t) = ca(t)ψae −iEat/h̄ + cb(t)ψbe−iEbt/h̄. (A.2) We then can solve for these coefficients using the time dependent wave equation: [H0 +H ′(t)]ψ = ih̄ ∂ψ ∂t . (A.3) Through careful elimination and using the orthogonality of ψa and ψb, we find the following coupled expressions for the time derivatives of ca and cb: ċa = −i h̄ H ′abe −iω0tcb, ċb = −i h̄ H ′bae iω0tca, ω0 ≡ Eb − Ea h̄ . (A.4) Since we are concerned with coupling due to resonant electromagnetic radiation, we take the following time dependent perturbation: H ′ab(t) = −Vab cos(ωt), . (A.5) 45 Appendix A. Two Level System We can then get a first order approximation for cb(t) by plugging in initial values to Eq. A.4: cb(t) = − i h̄ Vba ∫ t 0 cos(ωt′)eiωt ′ ≈ −iVba h̄ sin[(ω0 − ω)t/2] ω0 − ω e i(ω0−ω)t/2 (A.6) The square of this coefficient gives us a transition probability for a two-level system under coherent sinusoidal perturbation. Pa→b(t) = |cb(t)|2 ≈ |Vba| 2 h̄2 sin2[(ω0 − ω)t/2] (ω0 − ω)2 (A.7) For the case of coherent electromagnetic waves (i.e a laser), we have Vba = pE0 where p = q〈ψb|z|ψa〉. If we assume this case, we find that the transition probability is proportional to the intensity and inversely proportional to the square of the detuning from resonance: Pa→b ∝ I ∆2 . (A.8) 46 Appendix B Additional Data/Plots 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 PA Laser Frequency (+13097cm−1 ) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 A to m F ra ct io n R em ai n in g Amplitude 0.815±0.0366 Sigma param 0.012±0.0012 Freq 13096.991±0.0005 Figure B.1: Plot of vibrational line at 13097 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. Note that the scatter in the atom number of the full trap is ∼ 10%. 47 Appendix B. Additional Data/Plots 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 PA Laser Frequency (+12967cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.915±0.0258 Sigma param 0.014±0.0007 Freq 12967.64±0.0004 Figure B.2: Plot of vibrational line at 12967 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. Table B.1: Measurements of Peak Splitting at 12546 cm−1 (See Figure 5.7) B-field (G) 1 Peak 1 cm−1 Peak 2 Peak 3 Peak 4 Peak 5 Peak 6 0 0.0687 0.0687 0.0687 0.0763 0.0763 0.0763 20 0.0682 0.0697 0.0711 0.0756 0.077 0.0788 30 0.0675 0.0691 0.0709 0.0745 0.0764 0.0786 40 0.0682 0.0697 0.0727 0.0748 0.0778 0.0803 50 0.068 0.0701 0.0749 100 0.0674 0.0696 0.0735 0.0781 0.0816 200 0.0666 0.0696 0.0719 0.0879 0.0947 300 0.066 0.0689 0.0706 0.0961 0.104 400 0.1067 0.1138 800 0.065 0.065 0.065 48 Appendix B. Additional Data/Plots 0.8 0.7 0.6 0.5 0.4 0.3 0.2 PA Laser Frequency (+12833cm−1 ) 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.854±0.0315 Sigma param 0.013±0.0007 Freq 12832.698±0.0004 Figure B.3: Plot of vibrational line at 12833 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. 49 Appendix B. Additional Data/Plots 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 PA Laser Frequency (+12546cm−1 ) 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.843±0.071 Sigma param 0.006±0.0007 Freq 12546.065±0.0004 Figure B.4: Plot of vibrational line at 12546 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. The outlying point to the right of the main feature suggests the existence of a narrow peak. 50 Appendix B. Additional Data/Plots 0.32 0.34 0.36 0.38 0.40 0.42 0.44 PA Laser Frequency (+12394cm−1 ) 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.877±0.0391 Sigma param 0.005±0.0004 Freq 12394.396±0.0002 Figure B.5: Plot of vibrational line at 12394 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. 51 Appendix B. Additional Data/Plots 0.00 0.05 0.10 0.15 0.20 PA Laser Frequency (+12237cm−1 ) 0.2 0.4 0.6 0.8 1.0 A to m F ra ct io n R em ai n in g Amplitude 0.766±0.0558 Sigma param 0.009±0.0011 Freq 12237.178±0.0006 Figure B.6: Plot of vibrational line at 12237 cm−1. Feature is fit with a green lorentzian curve: Aσ 2 (x−f)2+σ2 , where A is the amplitude, σ is the half-width at half max (HWHM), and f is the center frequency. 52 Appendix B. Additional Data/Plots 0.415 0.420 0.425 0.430 PA Laser Frequency (+12394 cm−1 ) 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er 140 G 180 G 220 G Figure B.7: Plot of leftmoving features at different magnetic fields. Data was taken with 1kW/cm2 and a 5s hold time. Our current understanding is that these three leftmost peaks coalesce into the broad feature seen at 830 G. 53 Appendix B. Additional Data/Plots 0.085 0.090 0.095 0.100 0.105 0.110 0.115 PA Laser Frequency (+12546 cm−1 ) 0.0 0.2 0.4 0.6 0.8 1.0 S ca le d at om n u m b er 200 G 300 G 400 G Figure B.8: Plot of satellite features at different magnetic fields. Data was taken with 1kW/cm2 and a 5s hold time. Our current understanding is that they are a result of hyperfine splitting. 54 Appendix C Apparatus Schematics MOT
 Dipole
 Trap
 Photoassocia2on
 beam

 Figure C.1: Experiment Schematic: the photoassociation laser was collinear with the dipole trap and the beam path was optimized to have the maximum intersection with the MOT. The schematic only shows a single arm of the dipole trap (setup for Rb benchmark) for clarity. The measurements for Li utilized 2 arms of the IPG. Note: this is a cartoon schematic and does not show components to scale or the Gaussian profiles of the lasers. 55 Appendix C. Apparatus Schematics Figure C.2: Magneto Optic Trap Schematic. The trap is the intersection of three pairs of counter propagating beams along three perpendicular axes between a pair of anti- Helmholtz coils. The small white dot at the center represents the trapped atom cloud. 56 Appendix C. Apparatus Schematics Figure C.3: Control rack for MOL experiment. 57 Appendix C. Apparatus Schematics Figure C.4: Titanium Sapphire (TiSapph) Ring Lasers. The near laser was used in this thesis in conjunction with a control box C.5. The further laser is being set up for future two colour photoassociation experiments. 58 Appendix C. Apparatus Schematics Figure C.5: Titanium Sapphire (TiSapph) Ring Laser Control Box. The laser frequency was set by sending a DC voltage from an analog out port on the experiment control rack to this control box. 59 Appendix C. Apparatus Schematics Figure C.6: Fiber Coupling of TiSapph laser. Coupling the laser into the fiber eliminated a number of alignment issues. The red line traces the beam path through several optical elements. 60

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