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Photoassociation and Feshbach resonance studies in ultra-cold gases of ⁶Li and Rb atoms Gunton, Will 2016

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Photoassociation and FeshbachResonance Studies in Ultra-ColdGases of 6Li and Rb AtomsbyWill GuntonB.Sc., The University of British Columbia, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2016c© Will Gunton 2016AbstractThis thesis presents an experimental apparatus capable of producing and studying ultra-cold mixtures of 6Li and Rb, and progress towards the creation of ultra-cold ground state6Li2 and LiRb molecules. Ultra-cold experiments with molecules have applications in manyquantum computation and simulation experiments. We discuss elements of the apparatuswhich are important to these experiments, including electric field plates in air capable ofproducing fields up to 18 kV/cm, and a laser system for photoassociation spectroscopybased on two Ti:Sapphire lasers phase locked to the same optical frequency comb. Withrespect to 6Li and 85Rb mixtures, we report on the observation of six Feshbach resonances,which represent an important step towards molecule production and future experiments inthis heteronuclear mixture. In addition, we demonstrate the production of a BEC of 6Li2Feshbach molecules, a degenerate Fermi gas of 6Li and the formation of BCS pairs.In the 6Li system, we report on the high resolution spectroscopy of the v′ = 20 − 26vibrational levels of the c(13Σ+g ) potential and the v′ = 29 − 35 vibrational levels of theA(11Σ+u ) potential. In the A(11Σ+u ) potential, we find that the v′ = 31 and v′ = 35 levelshave the largest transition strength and are therefore good candidates to use as intermediatestates in molecule formation. We demonstrate atom-molecule dark states in the BEC-BCS crossover regime and additionally use dark-state spectroscopy to make extremely highresolution measurements of the least bound N ′′ = 0 ro-vibrational levels in the X(11Σ+g ) anda(13Σ+u ) potential. In addition, we present spectroscopy of all ten N′′ = 0 and N ′′ = 2 ro-vibrational levels in the a(13Σ+u ) potential and furnish a preliminary interpretation of theobserved energy structure.Finally, we report on the observation of anomalous Autler-Townes and dark-state spec-trum. Using an extension to the standard three level model, we show that these anomalousprofiles are due to degeneracies that exist in the bound molecular states, and to the choice ofpolarization of the coupling fields. These results have a direct impact on molecule formation,and provide a clear guide to future experiments.iiPrefaceAll the work presented in this thesis was conducted on an apparatus in the QuantumDegenerate Gas laboratory at the University of British Columbia, Point Grey campus. Iwas heavily involved in the design and construction of the apparatus with the exception ofthe home-built Rb and Li master lasers discussed in Section 2.1. I was also heavily involvedin the data taking and analysis of all of the results presented in this thesis.The ratchet lock technique presented in Section 2.3 has been published in W. Gunton etal., Optics Letters 40 pp. 4372-4375 (2015) [78]. Figure 2.16 was made by Kirk Madison,Fig. 2.17 was made by me, and both appear in [78]. I was responsible for the rebuild of thefrequency comb discussed in the section, with assistance from Julien Witz and Art Mills.The design of the Zeeman Slower presented in Section 3.1 and the vacuum systemdiscussed in Section 2.2 was led by William Bowden as a Master’s thesis project [99]. Theconstruction of this iteration of the apparatus was built in collaboration with WilliamBowden, Steven Novakov, Kahan Dare and Mariusz Semczuk. The results presented inSection 3.1 have been accepted for publication in Review of Scientific Instruments [125], andI am the second author on the paper. Figures 3.5, 3.6, and 3.7 appear in the manuscriptand were made by me.Chapter 4 is heavily based on a manuscript that has been published in W. Gunton etal., Review of Scientific Instruments 87(3) (2016) [142]. The construction of the electricfields plates and associated high-voltage electronics was carried out with the help of GenePolovy, and the majority of the data was taken in collaboration with Mariusz Semczuk.Figures 4.8, 4.9, 4.10, 4.12, 4.13, and 4.14 appear in the manuscript and were made by me.In Chapter 5, Fig. 5.3 was made by C. Chin et al., and appears in [10]. Figure 5.4 wasmade by S. Simonucci et al., and appears in [147]. The results of Section 5.1.3 have beenpublished in W. Gunton et al., Phys. Rev. A. 88, 062510 (2013) [165]. The data in thissection was taken in collaboration with Mariusz Semczuk. Figures 5.6 and 5.7 were madeby me, Mariusz Semczuk, and Kirk Madison and appear in [165].iiiPrefaceThe results of Section 6.1.1 have been published in M. Semczuk et al., Phys. Rev. A87, 052505 (2013) [84] on which I am the second author. Figure 6.2 was made by Nikesh S.Dattani and appears in [84]. The results of Section 6.1.2 have been published in W. Guntonet al., Phys. Rev. A 88, 062510 (2013) [85]. The theoretical analysis of the molecularpotentials presented in these two papers was done by Nikesh S. Dattani and is not includedin this thesis.The analysis of the electromagnetically induced transparency (EIT) spectra (includingthat of the dark state spectroscopy data) presented in Chapter 6 was done by me using asimulation that I originally wrote and was modified and updated (and cleaned up) by Tedvan der Weerden. The analysis of the eigenstates of the N ′′ = 2 level in Section 6.3.2 wasdone by me, using code that was written by Fernando Luna. The high resolution dark statespectroscopy and the atom-molecule dark state data presented in Sections 6.1.3 and 6.3.1has been published in M. Semczuk et al., Phys. Rev. Lett. 113, 055302 (2014) [159]. Thedata in this section was taken in collaboration with Mariusz Semczuk. Figure 6.5 was madeby me and appears in [159]. Figure 6.11 was made by Mariusz Semczuk and appears in [98].Chapter 7 presents results that have been published in B. Deh et al., Phys Rev. A 82,020701(R) (2010) [52], on which I am the second author. The data in this section was takenin collaboration with Ben Deh, Bruce Klappauf, and Mariusz Semczuk. The theoreticalanalysis was done by Zuan Li. Figures 7.1 and 7.2 were made by Ben Deh and the latterappears in [52]. Figure 7.3 was made by Zuan Li.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Laser Cooling Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Lithium Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Rubidium Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Vacuum System and Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Photoassociation Laser System . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 Femtosecond Optical Frequency Comb . . . . . . . . . . . . . . . . 362.3.2 Ti:Sapphire Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Preparation and Detection of Cold Atoms . . . . . . . . . . . . . . . . . . 583.1 MOT and Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vTable of Contents3.2 Optical Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3 Optical Pumping and State Selection . . . . . . . . . . . . . . . . . . . . . 863.4 Imaging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.4.1 Fluorescence Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 973.4.2 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Electric Field Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1 Design of Electric Plates and HV Components . . . . . . . . . . . . . . . . 1154.2 Testing and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.1 Timing of Charging and Discharging . . . . . . . . . . . . . . . . . 1214.2.2 Electric Field Measurement . . . . . . . . . . . . . . . . . . . . . . . 1234.3 Field Shielding and Trap Loss . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3.1 Electric Field Shielding . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3.2 Residual Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3.3 Trap Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3.4 Tests with Small Prototype Plates . . . . . . . . . . . . . . . . . . . 1365 Overview of 6Li2 System and Photoassociation . . . . . . . . . . . . . . . 1385.1 Introduction to 6Li and 6Li2 System . . . . . . . . . . . . . . . . . . . . . . 1385.1.1 Two-Atom Scattering State . . . . . . . . . . . . . . . . . . . . . . . 1415.1.2 s-wave Feshbach Resonances in the |12〉 Hyperfine Mixture . . . . . 1445.1.3 Degenerate Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.1.4 6Li2 Bound Molecular States . . . . . . . . . . . . . . . . . . . . . . 1575.2 Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.2.1 Spectroscopy Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1715.2.2 Dark States and Stimulated Raman Adiabatic Passage . . . . . . . 1746 6Li2 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.1 Summary of Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.1.1 Spectroscopy of the c(13Σ+g ) Potential . . . . . . . . . . . . . . . . . 1896.1.2 Spectroscopy of the A(11Σ+u ) Potential . . . . . . . . . . . . . . . . 1936.1.3 Spectroscopy of the X(11Σ+g ) Potential . . . . . . . . . . . . . . . . 1976.2 Degenerate Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203viTable of Contents6.2.1 Singlet States: Five Level Model . . . . . . . . . . . . . . . . . . . . 2036.2.2 Effect in Triplet States . . . . . . . . . . . . . . . . . . . . . . . . . 2156.3 Characterization of a(13Σ+u ) Potential . . . . . . . . . . . . . . . . . . . . . 2226.3.1 High Resolution Spectroscopy of |v′′ = 9, N ′′ = 0〉 Manifold . . . . . 2246.3.2 Low Resolution Spectrscopy of N ′′ = 0 and N ′′ = 2 Levels . . . . . 2287 Heteronuclear Mixtures: 6Li+85Rb . . . . . . . . . . . . . . . . . . . . . . . 2417.1 Production of Ultra-Cold Bose-Fermi Mixtures . . . . . . . . . . . . . . . . 2437.2 LiRb Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256A List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272viiList of Tables2.1 Hyperfine constants for 6Li . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Hydrogen conductance for various vacuum components . . . . . . . . . . . . 282.3 Magnetic field produced by the MOT and compensation coils . . . . . . . . 323.1 Optimal loading parameters for the 6Li and 85Rb MOT . . . . . . . . . . . 663.2 Currents used in each coil segment for the 6Li and 85Rb Zeeman slower . . 673.3 Properties of the high and low power dipole traps . . . . . . . . . . . . . . . 773.4 Pixelink camera calibration factors for 6Li and Rb . . . . . . . . . . . . . . 985.1 State labels for the six Zeeman sublevels in the ground 2S1/2 manifold of6Li 1395.2 Properties of Feshbach resonances in 6Li . . . . . . . . . . . . . . . . . . . . 1495.3 Allowed rotational levels and nuclear spin configurations for 6Li2 . . . . . . 1655.4 Accessible excited state potentials in 6Li . . . . . . . . . . . . . . . . . . . . 1675.5 Selection rules for Σ→ Σ electric dipole transitions in 6Li2 . . . . . . . . . 1686.1 Location of measured PA resonances for the c(13Σ+g ) potential. . . . . . . . 1916.2 Shift rate and systematic shift for PA resonances in the c(13Σ+g ) potential . 1916.3 Measued spin-spin and the spin-rotation constants for the c(13Σ+g ) potential 1926.4 Experimentally measured PA resonances in the A(11Σ+u ) potential . . . . . 1946.5 Location of the v′′ = 37 and v′′ = 38 levels in the X(11Σ+g ) potential . . . . 1996.6 Location of the three v′′ = 9 hyperfine states in the a(13Σ+u ) potential . . . 2276.7 Shift rate of the three v′′ = 9 hyperfine states in the a(13Σ+u ) potential . . . 2276.8 Predicted and measured hyperfine splitting in the a(13Σ+u ) potential . . . . 2286.9 Location of all ten N ′′ = 0 ro-vibrational levels in the a(13Σ+u ) potential . . 2306.10 Location of all ten N ′′ = 2 ro-vibrational levels in the a(13Σ+u ) potential . . 2407.1 Experimentally measured s-wave FRs for 6Li–85Rb . . . . . . . . . . . . . . 248viiiList of Figures2.1 Spectrum from the Sacher TAs at different seed powers. . . . . . . . . . . . 192.2 Layout of amplifier setup for the generation of 6Li light . . . . . . . . . . . 202.3 Layout of updated amplifier setup for the generation of 6Li light . . . . . . 202.4 Layout of the amplifier setup for Rb light generation . . . . . . . . . . . . . 242.5 Solidworks drawing of the experiment apparatus . . . . . . . . . . . . . . . 262.6 Atomic shutter, cold finger and differential pumping tube. . . . . . . . . . . 272.7 Solidworks drawing of the 6Li and Rb effusive source . . . . . . . . . . . . . 302.8 MOT coil base design showing damns . . . . . . . . . . . . . . . . . . . . . 342.9 Cross sectional view of the MOT coil design . . . . . . . . . . . . . . . . . . 342.10 Location of the compensation coils relative to MOT region . . . . . . . . . . 352.11 Schematic of the frequency comb and associated locking electronics . . . . . 372.12 Output of the frequency comb in the time and frequency domain . . . . . . 382.13 Comb spectrum before and after supercontinuum generation . . . . . . . . . 432.14 Transmission of the frequency selective elements in the Ti:Sapphire lasers . 482.15 Layout for stabilizing and referencing the Ti:Sapphire lasers to the FFC . . 502.16 Illustration of the ratchet scanning method . . . . . . . . . . . . . . . . . . 542.17 Demonstration of ratchet lock via heterodyne measurement . . . . . . . . . 563.1 Magnetic field profile used the 6Li and 85Rb Zeeman slower . . . . . . . . . 633.2 Slowing transition used in the 6Li Zeeman slower . . . . . . . . . . . . . . . 643.3 Slowing transition used in the Rb Zeeman slower . . . . . . . . . . . . . . . 653.4 Effect of slowing beam power on the 6Li and 85Rb MOT loading rate . . . . 673.5 Effect of hyperfine repumping light on 85Rb MOT loading rate . . . . . . . 683.6 6Li MOT performance versus oven temperature . . . . . . . . . . . . . . . . 703.7 85Rb MOT performance versus oven temperature . . . . . . . . . . . . . . . 703.8 CDT trap frequencies as a function of the crossing angle . . . . . . . . . . . 75ixList of Figures3.9 Schematic of the beam paths for ODT and PA laser . . . . . . . . . . . . . 783.10 SPI beam radius through Pyrex and Quartz cell as a function of power . . . 803.11 Timing of the transfer from the SPI to IPG and evaporation in the IPG . . 843.12 85Rb Hyperfine changing collisions in the SPI . . . . . . . . . . . . . . . . . 873.13 ac Stark shift of the 85Rb optical pumping beam . . . . . . . . . . . . . . . 923.14 Influence of magnetic field direction on 85Rb optical pumping efficiency . . 933.15 85Rb state population with and without optical pumping . . . . . . . . . . 943.16 RF frequencies of the |1〉 → |2〉 and the |2〉 → |3〉 transition in 6Li . . . . . 963.17 Example of a fluorescence image of a 6Li MOT . . . . . . . . . . . . . . . . 993.18 Example of an absorption image of a 6Li ODT . . . . . . . . . . . . . . . . 1023.19 Transitions used in high field imaging of 6Li . . . . . . . . . . . . . . . . . . 1063.20 RF transitions in the ground state of 85Rb in a small magnetic field . . . . 1094.1 Illustration of the effects of Townsend discharge and ion avalanche . . . . . 1144.2 Expected electric field strength as a function of plate size . . . . . . . . . . 1164.3 Picture and schematic of electric field plate assembly . . . . . . . . . . . . . 1174.4 High voltage switching network schematic . . . . . . . . . . . . . . . . . . . 1194.5 Example of a high voltage connection . . . . . . . . . . . . . . . . . . . . . 1204.6 Placement of the field plates in the experimental setup . . . . . . . . . . . . 1214.7 Illustration of the dc Stark shift in 85Rb . . . . . . . . . . . . . . . . . . . . 1264.8 Expected and measured dc Stark shift as a function of applied voltage . . . 1274.9 influence of the dc Stark effect the 85Rb MOT and Imaging . . . . . . . . . 1284.10 Electric field strength versus time for single plate bias . . . . . . . . . . . . 1304.11 Electric field strength versus time when both plates are biased . . . . . . . 1324.12 Effects of shielding on the electric field strength over sequential runs . . . . 1334.13 Loss of 6Li atoms from the ODT as a function of plate voltage . . . . . . . 1354.14 Loss of 6Li atoms from the ODT and MOT with both plates biased . . . . 1375.1 Zeeman states for the two ground hyperfine states of 6Li . . . . . . . . . . . 1405.2 Magnetic Field dependence of the |1〉 and |2〉 states . . . . . . . . . . . . . . 1455.3 Two channel model of a Feshbach resonance . . . . . . . . . . . . . . . . . . 1465.4 Origin of s-wave Feshbach resonance in 6Li . . . . . . . . . . . . . . . . . . 1515.5 Closed channel fraction of Feshbach molecule state . . . . . . . . . . . . . . 153xList of Figures5.6 Evaporation ramp to form, and evidence of, a mBEC . . . . . . . . . . . . . 1565.7 Emergence of BCS Pairing signal in RF spectroscopy . . . . . . . . . . . . . 1585.8 The first seven potentials of Li2 . . . . . . . . . . . . . . . . . . . . . . . . . 1665.9 Summary of the accessible levels used in photoassociation . . . . . . . . . . 1685.10 Cartoon of three levels system used in photoassociation . . . . . . . . . . . 1705.11 Example of single and two color photoassocation . . . . . . . . . . . . . . . 1735.12 Pulse sequence and population evolution for the STIRAP process . . . . . . 1775.13 Atom-molecule dark state formed in the X(11Σ+g ) potential . . . . . . . . . 1805.14 Model dark state spectrum predicted from one-body and two-body loss . . 1866.1 Photoassociation spectra for the v′ = 21 level of the c(13Σ+g ) potential. . . . 1906.2 Comparison of experimentally determined and ab initio spin-spin constant . 1936.3 Ω1 as a function of vibrational level in the A(11Σ+u ) potential at 754 G . . . 1966.4 Ω1 as a function of B field for v′ = 31 in the A(11Σ+u ) potential . . . . . . . 1976.5 Revival height of dark state in the BEC-BCS crossover regime . . . . . . . . 2016.6 States and couplings in the five level model . . . . . . . . . . . . . . . . . . 2046.7 Polarization angles of probe and control fields . . . . . . . . . . . . . . . . . 2066.8 Dependence of five-level EIT spectrum on polarization . . . . . . . . . . . . 2096.9 Theoretical Dark State revival height in a five level system . . . . . . . . . 2106.10 Autler-Townes splitting in a five level system . . . . . . . . . . . . . . . . . 2126.11 Schematic of the previous setup of ODT and PA beams . . . . . . . . . . . 2136.12 Anomalous Autler-Townes with recycled photoassociation beams . . . . . . 2146.13 Summary of energy structure for the a(13Σ+u ) and c(13Σ+g )6Li2 potentials . 2166.14 Relevant states and couplings for coupling to the F ′′ = 1 ground state . . . 2186.15 F ′′ = 1 dark state spectrum with coupling to a decaying excited state. . . . 2196.16 Effect of couplings to the |e′〉 state on the dark state revival height . . . . . 2226.17 Dark state spectra for v′′ = 9, N ′′ = 0 in the a(13Σ+u ) potential . . . . . . . 2266.18 Two-color spectroscopy of v′′ = 8, N ′′ = 0 ro-vibrational level . . . . . . . . 2316.19 Two-color spectroscopy of v′′ = 7, N ′′ = 0 ro-vibrational level . . . . . . . . 2326.20 Two-color spectroscopy of v′′ = 2, N ′′ = 0 ro-vibrational level . . . . . . . . 2336.21 Two-color spectrum for the v′′ = 9, N ′′ = 2 ro-vibrational level . . . . . . . 2356.22 |J ′′〉 components of the N ′′ = 2 eigenstates in the a(13Σ+u ) potential . . . . 2376.23 Two-color spectroscopy of v′′ = 1, N ′′ = 2 ro-vibrational level . . . . . . . . 238xiList of Figures7.1 Schematic of setup used for LiRb Feshbach resonance search . . . . . . . . . 2437.2 Feshbach resonances in the LiRb mixture . . . . . . . . . . . . . . . . . . . 2477.3 Predicted s-wave elastic cross section for 6Li and 85Rb mixtures . . . . . . . 250xiiList of AbbreviationsABM asymptotic bound state modelAOM acousto-optic modulatorBCS Bardeen-Cooper-SchriefferBEC Bose-Einstein condensateCDT crossed optical dipole trapECDL extended cavity diode laserEDFA erbium-doped fiber amplifierEIT electromagnetically induced transparencyFC Franck-CondonFR Feshbach resonanceHNLF highly non-linear fiberITO indium tin oxideMOT magneto-optical trapmBEC molecular Bose-Einstein condensateNEG non-evaporable getterODT optical dipole trapOFC femtosecond optical frequency combPA photoassociationPZT piezoelectric transducerRGA residual gas analyzerSTIRAP stimulated raman adiabatic passageTA tapered amplifierWDM wavelength division multiplexerxiiiAcknowledgementsI was lucky enough to have Kirk Madison teach one of my undergraduate physics courses,which lead to an undergraduate Honours thesis project in his lab. The decision to stay onas a graduate student was made much easier by his presence and leadership. His optimism,enthusiasm, and intuition is inspiring, and it is clear that he cares deeply about his studentsand their development, both academically and personally. Thank you for this opportunity.When I started in our lab, I was fortunate enough to work with Bruce Klappauf, BenDeh, and Janelle Van Dongen. I could not have had a better introduction to research thanworking with Ben and Bruce. Janelle is one of the most patient and kind people I have met,and her willingness to help with any and all problems in the lab is invaluable. In addition,I owe a great deal of thanks to David Jones and Art Mills for their help and advice, andfor providing me with an excellent crash course in optical frequency combs.None of the work in this thesis would have been possible without Mariusz Semczuk.Not only was Mariusz crucial in ensuring that the experiment worked (and that it wasaesthetically pleasing), he was always thinking about what else our experiment could doand was willing to give almost anything a try. It was a pleasure to work with him and couldnot imagine my time as a graduate student without his help and support.I worked with many other students during my tenure in the lab, all of whom I would tothank for their help. Thank you to William Bowden, without whom the new iteration of theexperiment would not exist. I owe a great deal of thanks to the current MOL team of GenePolovy, Kahan Dare, and Koko Yu both for their excellent work on the MOL experimentand their support and help during the last year. Our experiment (and wherever they maybe in the future) is in good hands. Thank you to Jim Booth, Kais Jooya, and Janelle fortaking care of the Rb laser system and for always being up for an interesting discussion.I would like to thank my family for their support, and humoring me as I was the firstof my brother and sister to leave for University, and then the last person to finish. Finally,thank you to Shira Daltrop for sticking with me through all these years of graduate school.I look forward to whatever is next.xivChapter 1IntroductionThe field of Atomic, Molecular, and Optical (AMO) Physics encompasses the study ofinteractions between matter and matter and/or matter and light which occur on a smallscale, where quantum effects play a role in the interactions. Rapid developments and newmethods for cooling and controlling atomic and molecular systems brought with them theability to study gases in an extremely “clean” (high precision) setting. These develepomentswere highlighted by the formation of the first Bose-Einstein Condensate (BEC) in 1995with systems of 87Rb [1], Na [2] and 7Li [3]. Several years later degeneracy was reached inFermionic systems using 40K [4] and 6Li [5]. These degenerate Fermi gases differ from theirbosonic counterparts in that the Pauli exclusion principle forbids identical particles fromoccupying the same quantum state. This unique property plays a key role in many physicaleffects, from electrical conductivity in metals to the stability of white dwarf and neutronstars, and motivates the interest in, and study of, fermionic systems.The ability to reach these quantum regimes, where the quantum nature of the particlesand the underlying Bose or Fermi statistics play a role, requires that the thermal de Brogliewavelengthλdb =√2pi~2mkBT, (1.1)is on the same order as the interparticle spacing, given by n−1/3 where n is the density. Thistransition is often characterized by the phase space density of the gas, where PSD = nλ3db,where the transition from a classical gas to a quantum gas occurs when the PSD is of orderone. Given the scaling of the thermal de Broglie wavelength with temperature (see Eq. 1.1),reaching this quantum regime requires both relatively high density and low temperatures1.While many experimental techniques can produce cold atoms and molecules, laser cooling1In this context, high density truly is a relative term. In ultra-cold experiments, “high density” meansa density on the order of 1013 /cm3. If this density is compared to that of air (on the order of 1019 /cm3),the relativity of the term “high” is evident!1Chapter 1. Introductionis unique in its ability to achieve phase space densities on the order of, or larger than, one.For example, buffer gas cooling and stark deceleration can reach typical temperature of tensto hundreds of mK while achieving densities on the order of 106 /cm3 to 108 /cm3, whichcorresponds to phase space densities on the order of 10−15 to 10−13 [6, 7, 8]. In contrast,laser cooling can reach temperatures on the order of tens to hundreds of nK with densitieson the order of 1013 /cm3, corresponding to phase space densities greater than one.In ultra-cold experiments, control is achieved over both the internal structure (i.e., thequantum state of the atom) and the external degrees of freedom (i.e., the translationalmotion). In addition, because they are almost stationary in the lab frame, the atoms canbe trapped and investigated for long times. Somewhat unique to these systems is that thetrapped sample can be considered a “dillute” gas, in the sense that the distance betweenparticles (on the order of 100 nm) is much larger than the characteristic size of the particle(typically on the order of one a0 ≈ 0.05 nm). Because of this, collisions between particlesare dominated by binary encounters and thus two-body interactions. Moreover, in thelimit of low energy, the interactions can be characterized by a single parameter (the s-wavescattering length) which describes the effective range of the interactions2. In the absense ofany resonance effects, this background scattering length is on the order of 10-100 a0, or afew nm. The extended range of the interactions (relative to the size of the atoms) is due toa dipole-induced dipole interaction, which results in a Van der Waals potential that scaleslike −C6/r6 at long range [9].Ultra-cold gases can be controlled by tuning the strength of the interactions (i.e., mod-ifying the s-wave scattering length) with external fields (for example, with Feshbach reso-nances (FRs) in the case of magnetic fields [10]) or by changing the dimensionality of thesystem through the use of optical lattices[11]. This control allows the ultra-cold atomic sys-tems to enter a regime were many-body physics such as Bardeen-Cooper-Schrieffer (BCS)pairing and superfluidity can be studied [12, 13].More recently, experiments have started to extend the control achieved in atomic gasesto molecules, in both homonuclear systems (for example, Cs2 [14, 15], Rb2 [16, 17] andSr2 [18]) and heteronuclear systems (for example,40K87Rb [19, 20], 87Rb133Cs [21, 22],2In ultra-cold gases, p-wave (and higher) collisions are strongly suppressed if the collision energy is lessthan the potential barrier associated with the rotational energy required by angular momentum conservation.The p-wave barrier height for bi-alkali collisions is typically on the order of a few mK, which leads to s-wavecollisions dominated at the typical collision energies on the order of a few µK or less.2Chapter 1. Introduction7Li133Cs [23], and 23Na40K [24]). Broadly speaking, research into ultra-cold molecules issignificant because they have much richer and more interesting interactions compared toatoms, which arise from their magnetic and electric dipole (and higher order) moments. Inaddition, molecules possess a richer internal structure that makes them even more usefulthan atoms as tools to, for example, study many-body quantum dynamics [25, 26, 27],realize quantum simulators [28, 29], and test fundamental laws of nature [30, 31, 32].The interest in dipole-dipole interactions lies in their longer range character, as theinteraction energy scales like 1/r3. This is in contrast to the 1/r6 scaling of the Van derWaals interaction between ground state atoms, which are short range in comparison tothe interparticle spacing that can be realized in ultra-cold systems (in light of this, theinteraction between atoms is typically described by the “contact interaction”). Moreover,the dipole-dipole interaction is anisotropic, and the interaction can be attractive or repulsivedepending on the relative orientation of the electric dipoles during a collision. These long-range dipole-dipole interactions are required for many theoretical proposals for quantumcomputation [28, 29], or proposals to study phenomena in other realms of physics. Forexample, it is possible to simulate more complicated many-body Hamiltonians importantto condensed matter physics [25, 26, 27].While the dipole-dipole interaction can be either magnetic or electric in nature, thestrength of the achievable interaction energies varies considerably. For example, the ratioof interaction energies between two electric (µel) and two magnetic (µmg) dipoles scales likeEdd,elEdd,mg∝ c2(µelµmg)2, (1.2)where c is the speed of light. Some of largest magnetic moments in ultra-cold systems havebeen realized using Dysprosium [33, 34] where µmg ≈ 10µB (and µB is the Bohr magneton),or Chromium [35, 36] where µmg ≈ 6µB. On the other hand, typical bi-alkali polar moleculeshave electric moments on the order of 1 D [37, 38]. Comparing these two cases using Eq. 1.2,it can be seen that the strength of the electric dipole interaction is more than two orders ofmagnitude stronger than the magnetic dipole-dipole interaction at a similar distance.Although polar molecules are said to have a permanent electric dipole moment, thispermanent moment exists only in the molecule frame. In the laboratory frame (and in theabsence of an external electric field) there is no fixed orientation of the molecular axis, andthe average electric dipole moment is zero. Therefore, an external electric field is required to3Chapter 1. Introductionpolarize the molecule along the field direction. The advantage of polarizing an ensemble ofpolar molecules over, for example, atoms, lies in the magnitude of the required electric field.As the effect of a dc electric field is to couple rotational states with different parity [39, 40],the required field strength to polarize an ensemble can be estimated by comparing the sizeof coupling term due to the electric field to the energy difference between adjacent rotationalstates. In this case, the coupling term is [39]H = ~d · ~E = ez , (1.3)where ~d = e~z is the electric dipole moment, ~E = zˆ is the dc electic field (applied along thez-axis), e is the electron charge, and  is the magnitude of the electric field. In this case, theBohr radius is used as an estimate of z. In the case of atoms, the coupled rotational statesare states of different electronic orbital angular momentum, where the splitting is on theorder of hundred of THz, and the required electric fields are on the order of 100 MV/cm.However, in heteronuclear molecules, the electric field can couple states of different nuclearorbital angular momentum where the splitting can be on the order of tens of GHz. Thisleads to required electric field strengths on the order of a few to tens of kV/cm, which is inthe range of electric fields that are experimentally realizable.Aside from the control over both the external and internal degrees of freedom provided byultra-cold systems, the need for ultra-cold systems to study much of the physics associatedwith long-range dipole-dipole interactions can be seen by considering the typical energyand time scales associated with these interactions. For example, the interaction energyassociated with electric dipole of 1 D separated by 266 nm (this length scale is chosenas it is the lattice spacing associated with a 532 nm optical lattice) is E ≈ ~ × 10 kHz≈ kB × 500 nK. Ultra-cold experiments based on laser cooling therefore represent an idealsystem, as this range of temperatures is currently experimentally achievable3.Ultra-cold molecules are also a useful tool in the study of ultra-cold chemistry [41].With polar molecules at ultra-cold temperatures, the outcome of a reaction can depend onthe orientation and movement of the molecule in three dimensions. For example, it hasbeen shown that it is possible to suppress chemical reactions using the repulsive nature ofthe “side-by-side” long range dipole-dipole interaction [42]. For molecules with a non-zero3For example, see ground state KRb molecules which have been formed with a peak density of 1012 /cm3and at a temperature of 350 nK [19, 20].41.1. Motivationspin, it has also been predicted that the elastic and inelastic collision rates can be modifiedby an external magnetic field [43, 44]. The density of these molecule-molecule resonancesis expected to be substantially higher than for atoms due to the more complex internalstructure of molecules, related to the large number of rovibrational states.The creation to ultra-cold molecules can generally follow two paths. The first (andperhaps most obvious) method is to directly laser cool molecules [45]. However, due totheir complex internal structure, it is often hard to find a closed transition on which to lasercool. Although it has been demonstrated that laser cooling is possible for SrF [46, 47], thesemolecules are far from the ultra-cold regime, and it is still not a widely used technique for thecreation of ultra-cold molecules. The second approach is to form ultra-cold molecules fromultra-atoms using a coherent population transfer via stimulated raman adiabatic passage(STIRAP) [48]. Here, ultra-cold atoms are associated into loosly bound Feshbach molecules,either through three body recombination or by adiabatically sweeping across a FR [10].Then, the loosely bound Feshbach molecules are coherently transferred (i.e., transferredwithout heating) into a deeply bound molecular state using two coherent lasers. In thiscase, the binding energy of the molecule is carried away by the laser fields. This methodhas proved enormously successful for both the formation of homonuclear and heteronuclearmolecules4.1.1 MotivationThe long term goal of our experiment is the creation and study of LiRb molecules. Thesemolecules are of particular interest because they are expected to have a large electric dipolemoment in the ground singlet potential (4.2 D [37]) and a moderately large moment in thelowest lying triplet potential (0.37 D [38]). This is the largest permanent electric dipolemoment of all the bi-alkali molecules, save for LiCs [37]. Moreover, the size of the electricdipole moment in the triplet state is on the order of the size of the dipole moment of theground singlet state of KRb, while the triplet LiRb molecule still has a non zero spin. Polarmolecules with a non zero spin degree of freedom are crucial, for example, in proposals tostudy lattice-spin models [26].In addition to LiRb molecules, we also have a strong interest in studying heteronuclear4In fact, all of the homonuclear and heteronuclear molecules referenced in this section have been createdusing STIRAP.51.1. Motivation6Li2 molecules. Fermionic atomic systems such as6Li are unique in the respect that looselybound Feshbach molecules can be produced with a higher efficiency than in bosonic gases(as the Pauli exclusion principle reduces collisional relaxation of these Feshbach moleculesto deeply lying state). In addition, 6Li2 is chemically stable with respect to collisions withother Feshbach pairs. These properties make it relatively simple to achieve a molecular Bose-Einstein condensate (BEC) of 6Li2 Feshbach molecules and, by extension, possible to achievea BEC of ground state molecule using STIRAP. In addition, the 6Li2 system is a goodcandidate to, for example, study the stability of alkali-metal dimers in the lowest lying tripletstate with respect to spin relaxation to the ground singlet state or with respect to trimerformation [38], and to search for magnetically tunable molecular scattering resonances [43,44].Experimentally, the recipe for forming ultra-cold molecules is similar for both homonu-clear and heteronuclear dimers. The first step is the formation of loosely bound Feshbachmolecules, which requires the knowledge of the location of the Feshbach resonances for theparticular mixture. These Feshbach molecules are then coherently transferred to a groundbound molecular state using a two-photon Raman transition. This process uses an inter-mediate state which, in this case, is an excited electronic state of the molecule. Therefore,spectroscopic knowledge of the energy levels in the accessible excited and ground molecularstates are essential and must be determined (if not already known) using one-color and two-color photoassociation spectroscopy. In addition, the two photon process must be coherent,and therefore the two lasers must be phase coherent.However, the technical challenges of forming LiRb or 6Li2 molecules vary considerably.Specifically, Feshbach molecules of LiRb are inherently unstable [49], and the ground statemolecules themselves are unstable with respect to chemical reactions that form alkali-metaldimers and trimers [38]. Therefore, the creation of (long lived) LiRb molecules requiresthat the molecules be separated in a three-dimensional optical lattice. For this reason, wechoose to first focus on working towards 6Li2 molecules and experiments with mixtures of(as opposed to molecules) of 6Li and Rb.As it turns out, there are many interesting opportunities that arise working towards theformation of ground state molecules. In the context of this thesis, the dual species workinvolves experiments with mixtures of 6Li and Rb (rather than a direct focus on moleculeformation), and we have observed FRs in the LiRb mixtures that are some of the broadest61.1. Motivationknown in heteronuclear mixtures [50, 51, 52]. These resonances are also an effective toolthat can be used to efficiently sympathetically cool 85Rb using 6Li. Additionally, the LiRbsystem has a large mass imbalance between the bosonic (Rb) and fermionic (Li) species. Oneimplication of this mass imbalance is that it may be more feasible to experimentally observethe appearance of Efimov states, which manifest themselves as a series of minima or maximain the three body loss coefficient that are equally spaced on a ln a scale [53, 54]. In addition,there are many proposals to study novel quantum phases, pairing phenomena and quantumimpurity problems using Bose-Fermi mixtures with tunable interactions [55, 56, 57].With respect to the formation of 6Li2 molecules, FRs in6Li have been well studied,and there exists a broad Feshbach resonance near 832 G which is ideal for the creationof loosely bound Feshbach molecules [58, 59, 60]. These Feshbach molecules are an idealstarting point for the formation of ground state molecules. Therefore, the next step is todetermine the location of the bound excited state molecular levels that can be used as asuitable intermediate state. To this end, we performed spectroscopy on the excited tripletc(13Σ+g ) and excited singlet A(11Σ+u ) potentials and the lowest lying triplet a(13Σ+u ) andground singlet X(11Σ+g ) potential to which the excited states can (respectively) couple.The focus of the 6Li work in this thesis is on the spectroscopy of these potentials, and thecreation and understanding of the atom-molecule dark states. These dark states arise in thepresence of fields which couple the excited state to both the initial state (either a two-atomscattering state or a loosely bound Feshbach molecule) and to a bound molecular state. Ofnote is that the a(13Σ+u ) potential is very shallow, supporting only ten bound vibrationallevels5. Spectroscopy of all ten vibrational levels of this potential offers a fully characterizedmolecule potential in the 6Li2 system, which is interesting to the ab initio community asfew-electron atoms represent a tractable system with which experimental measurementsand theory can be rigorously compared6.Atom-molecule dark states are based on the principle of EIT [61] and are a vital firststep to coherent population transfer via STIRAP [48]. They are also an incredibly usefultool in the high resolution spectroscopy and lifetime measurements of bound states in theground singlet X(11Σ+g ) and lowest lying triplet a(13Σ+u ) potentials [62, 63]. In addition,5The binding energy of the ground vibrational level is on the order of 9 THz, and therefore all 10vibrational levels are accessible with our Ti:Sapphire based laser system.6Note, however, that even knowledge of the energies of every bound state in the potential (a discrete set)does not allow one to uniquely determine the (continuous) potential energy curve.71.2. Outline of Thesisdark-states and photoassociation near the broad Feshbach resonance at 832 G can be usedas a probe of the many-body state of paired 6Li atoms that exist in the BEC-BCS crossoverregime [64, 65].There is also wide interest in the ability to control the scattering length on a fast timescale and with a high degree of spatial resolution in order to study, for example, non-equilibrium dynamics in a strongly interacting Fermi gas [66]. However, these proposalsrequire control of interactions on a timescale τF ≈ ~/EF ≈ 10 µs for EF ≈ kB × 1 µK [67].An obvious tool that can achieve both high spatial resolution and fast temporal control isoptical fields. Optical fields have been used to control the s-wave scattering length near amagnetic Feshbach resonance [68, 69] and to induce optical Feshbach resonances [67, 70,71, 72]. However, many of the practical applications of these methods are limited by lightinduced inelastic losses. The use of a dark state can allow for control of the scatteringlength while suppressing the spontaneous scattering through quantum interference [67, 70],and has been recently demonstrated using 6Li [73].1.2 Outline of ThesisThe goal of this thesis is, first and foremost, to present (and provide context for) the resultsof our experiment and describe the methods used in obtaining these results. However, asecondary goal was to write a document that I would have found helpful to read in the earlystages of my graduate career. That is, a document that outlines what we do, why we do it,and most importantly how we do it. While some information may appear to be obvious or“known”, many are ideas, thoughts or explanations that I would have found helpful (andoften still do find helpful) to have in a single place.Chapter 2 describes the important “hardware” required for the production of ultra-cold 6Li and Rb. This chapter briefly introduces the atomic structure of 6Li and Rb, anddiscusses the impact of the atomic structure on the requirements for the laser cooling light.Section 2.1 details how we generate the laser cooling light for the 6Li and Rb magneto-optical trap (MOT), as well as the light used in the Zeeman slower, for optical pumping,and for absorption imaging. In late 2014 (and into 2015), we made substantial changes toour apparatus to include a Zeeman slower in order to make the system more robust fordual species experiments. Section 2.2 introduces and discusses the design and construction81.2. Outline of Thesisof our vacuum system, the effusive sources and the magnetic coils used to create externalmagnetic fields.One of the key requirement for experiments with dark-states and STIRAP is that thetwo fields are phase coherent during the time in which the light is exposed to the atoms [74].When the binding energy of the ground state is small (on the order of 100s of MHz) it ispossible to use an acousto-optic modulator (AOM) to shift the frequency of a single laserto create the desired frequency difference between the two beams [75]. However, when thetarget ground state is deeply bound (or even moderately bound), two separate lasers mustbe used. To achieve phase coherence between the two lasers. In our experiment, the lasersare individually phase locked to a femtosecond frequency comb [76]. The comb which we usewas originally built by the lab of David Jones at The University of British Columbia [77], butwas rebuilt and modified to improve the self-referencing branch, and to add a measurementbranch used to stabilize the Ti:Sapphire lasers7. Section 2.3 describes the details of the thisfrequency comb. In addition, this section discusses the two Ti:Sapphire lasers, their lockto the frequency comb, and a novel “ratchet” technique which we use to keep one laser ata fixed frequency while the other is scanned when both lasers are locked to the frequencycomb [78].Chapter 3 explains the preparation and detection of cold atoms (that is, the sequence ofevents that occur in a single experimental cycle). This chapter begins where nearly all cold-atom experiments begin - with a MOT. As part of the upgrade to our apparatus, we nowuse a Zeeman slower as the atom source for the MOT. Section 3.1 outlines the operatingprinciples of Zeeman slowers, and discusses the features that are unique or important toour implementation. This section also includes a characterization of the MOT and Zeemanslower8. Section 3.2 describes the two optical dipole traps (ODTs) that are used in oursetup, and discusses strategies for the transfer of 6Li and Rb from the MOT to the ODT, andthe subsequent evaporation. One of the advantages of ultra-cold systems, and specificallytrapping in an ODT, is the ability to prepare an atomic ensemble in a particular hyperfine orZeeman state. This allows for experimental control over the scattering state which is usefulin, for example, our Feshbach resonance experiments. Section 3.3 describes the procedurefor optically pumping 85Rb, and for the state selection of 6Li in large magnetic fields.7In this rebuild, the oscillator itself remained the same.8While the details of laser cooling are not discussed, this information is readily available in textbooks,for example [79].91.2. Outline of ThesisFinally, Section 3.4 describes our imaging techniques - namely, fluorescence and absorptionimaging. This section also discusses imaging 6Li at high magnetic fields, as well as Zeemanstate selective imaging of 85Rb.Before our experiment began to focus on 6Li2 molecules, we were motivated to studythe effects of electric fields on atomic collisional resonances [80, 81], guided by a proposalregarding the effects of electric fields on FRs in ultra-cold LiRb mixtures [82]. Theseeffects required electric fields in excess of 20 kV/cm, and we set out to build transparentelectrodes that could be added to our system outside of the vacuum. As the electric fields ofinterest are greater than the dielectric breakdown of air, we had to find a way to physicallyprevent the flow of charge through air. The solution was to imbed an indium tin oxide(ITO) coated dielectric substrate inside of stack of two more transparent substrates, wherethe outer layers have a much higher dielectric strength than air. Chapter 4 is focusedon a discussion of the design, and the implementation of these field plates in our setup.Specifically, Section 4.1 details the design of the electric plates and the associated highvoltage components, and Section 4.2 discusses the testing and characterization of the platesusing spectroscopic measurements of the dc Stark effect in 85Rb. While we were able toproduce fields of up to approximately 18 kV/cm, we were severely limited by a large lossof atoms that occurred when we applied large voltages to the plates. In addition, weobserved a decay of the electric field strength over time (when the plates were held at ahigh voltage), as well as a residual field that remained after the plates were grounded.Section 4.3 describes these effects, with we attribute to free charges (primarily electrons)produced by field emission of grounded metal parts within the vacuum chamber. Due tothese limitations, we were unsuccessful in observing the proposed effects on LiRb FRs.However, these electric field plates will play an important role in experiments with polarmolecules, where electric fields on the order of 10 kV/cm are required to achieve partial labframe alignment of the electric dipole moment of the molecules [83].Chapter 5 shifts the focus of this thesis to the 6Li and 6Li2 system. This chapter servesas an overview of the theory and background necessary for the discussion of the resultspresented in Chapter 6. Section 5.1 introduces the two-atom scattering state which servesas the initial state for many of our photoassociation and dark state experiments, as well asthe bound states that exist in the 6Li2 molecule. This section also discusses s-wave Feshbachresonances in the mixtures of the |1〉 and |2〉 states, and the experimental realization of a101.2. Outline of Thesismolecular BEC and BCS-like pairs in our setup. Section 5.2 outlines the different types ofspectroscopy experiments that we perform, and discusses dark-states and STIRAP. Thissection also introduces and discusses the usefulness and limitations of the standard threelevel model for dark-states that is used in the analysis of much of our results. Additionally,this section discusses a mechanism that we believe suppresses the effects of laser decoherenceon the atom-molecule dark state in a regime where the initial state is only weakly coupledto the excited state.At long last, Chapter 6 presents the new and novel observations that have resulted fromour work with 6Li and 6Li2 molecules. The single color spectroscopy of the c(13Σ+g ) andA(11Σ+u ) potential are discussed in Section 6.1.1 and 6.1.2 respectively. Our measurementsof the c(13Σ+g ) potential are the first to observe and quantify the spin-spin and spin-rotationcoupling constants. In the A(11Σ+u ) potential, we also check the viability of performingSTIRAP by measuring the coupling strength of the |a〉 → |e〉 transition as a function ofvibrational level, and as a function of magnetic fields above and below the broad Feshbachresonance near 832 G (i.e., in the BEC-BCS crossover regime). In both the c(13Σ+g ) andA(11Σ+u ) potential, we observe ro-vibrational levels that have not yet been observed incold-atom experiments, and have resulted in significant improvements to the theoreticalpotentials for these states9. Section 6.1.3 presents dark state spectroscopy of the two leastbound levels (v′′ = 37 and v′′ = 38) of the X(11Σ+g ) potential. In particular, the v′′ = 37level is measured at magnetic fields above and below the Feshbach resonance. As thev′′ = 38 vibrational level is not bound at these magnetic fields, the v′′ = 37 vibrational levelis necessary for dark-state studies of the BEC-BCS crossover regime and for dark-statecontrol of FRs [67, 70, 73]With the relevant levels in the 6Li2 system, it turns out that is not possible to exper-imentally realize a system that contains only three levels (i.e., the standard system usedto describe STIRAP and dark-states). Section 6.2 discusses how the presence of these de-generate levels leads to unexpected profiles to appear in the Autler-Townes and dark-statespectrum. Specifically, we observe an anomalous third feature that appears in the Autler-Townes spectrum (which is typically associated with only two features), which we attributeto an interference between possible excitation pathways. However, this interference is dis-tinct from that which leads to dark-states as it does not require phase coherence between the9The theory work based off our experimental data was performed by Nike S. Dattani, and is not discussedin this thesis. Details of this work, and of the updated model potentials can be found in [84, 85].111.2. Outline of Thesisprobe and control fields. In addition, we observe a poor dark-state revival when particularstates are used in the a(13Σ+u ) and c(13Σ+g ) potentials. We attribute this to a decoherencemechanism which results from additional couplings of the dark state to decaying excitedstates that provide a mechanism for loss. We show that both of these effects are dependenton the polarization of the pump and control field. In the context of STIRAP, an under-standing of these decoherence mechanism is vital, as the maximum transfer efficiency isdirectly related to the fidelity of the dark state.Section 6.3 presents a measurement of the N ′′ = 0 and N ′′ = 2 rotational levels of all10 vibrational levels in the lowest lying triplet a(13Σ+u ) potential using two-color photoas-sociation spectroscopy. For the least bound v′′ = 9 vibrational level, we use dark-statespectroscopy to measure the binding energy of all three hyperfine levels in the N ′′ = 0 ro-tational level. The absolute uncertainty of these measurements represents an improvementby a factor of over 40 compared to previous measurements [86, 87]. We find that the mea-sured energy difference between hyperfine manifolds deviates from the standard treatmentof hyperfine structure in high lying vibrational levels [88, 89]. The dark-state spectroscopyof the F ′′ = 1 hyperfine manifold displays an experimental demonstration of the decoher-ence mechanism discussed in Section 6.2. The measurements of the binding energy, andunderstanding of the dark-state behavior is crucial in the work towards STIRAP and thecreation of ground triplet state molecules. In the N ′′ = 2 ro-vibrational levels, we observe aset of three features, separated by hundreds of MHz that, for the deeply bound vibrationallevels, correspond to the three J ′′ states. For the high lying vibrational levels, the size ofthe hyperfine coupling and spin-spin coupling are of similar magnitude, and the eigenstatesof the system appear to be a linear superposition of all three J ′′ states.Chapter 7 shifts the focus back to our work on mixtures of 6Li and Rb. Specifically,Section 7.1 discusses our strategies for the production of ultra-cold Bose-Fermi mixtures,and Section 7.2 reports the position, width and open channel atomic states associated withsix FRs in the LiRb mixture for magnetic fields between 300 and 1000 G. Knowledge ofthe location of these Feshbach resonances will guide future work with these mixtures, andare required to form loosely bound Feshbach molecules that are a vital first step towardsmolecule production.Finally, Chapter 8 summarizes the results of this thesis and outlines the future directionsof the experiment. The results of this thesis provide a clear path towards the production of121.2. Outline of Thesis6Li2 molecules in the near term. In addition, the upgrades and development of the appratusmotivate working in earnest with LiRb mixtures.On the Topic of NotationIn order to be consistent throughout, this thesis uses lowercase letters to represent quantumnumbers in atoms and uppercase letters to represent quantum numbers in molecules. Inaddition, quantum numbers with a single prime (i.e., f ′ or F ′) represent quantum numbersof an excited state of an atom or molecule. Likewise, quantum numbers with a double prime(i.e., F ′′) represent quantum numbers of a molecule in either the ground singlet or lowestlying triplet potential. Unless otherwise noted, quantum numbers without a prime refer toground state atoms or molecules that are associated with the two-atom scattering state orFeshbach molecules (i.e., the initial state used in photoassociation experiments).13Chapter 2Experimental Setup2.1 Laser Cooling LightThe focus of our experiment is laser cooling and trapping of two alkali metals: 6Li andRb10. As an introduction to these atoms, this section briefly discusses the contributionsto the atomic structure. The obvious starting point of this discussion is the electronicconfiguration. For 6Li, which consists of 3 electrons and 3 protons, the ground state electronconfiguration is (including the empty first excited electronic orbital)1s22s12p0 , (2.1)where the number represents the principal quantum number n, the letter represents theelectron orbital angular momentum l, and the superscript represents the number of electronsin each orbital. The empty p orbital represents the first excited state, and is shown inreference to the following discussion. For Rb, which consists of 37 electrons and 37 protonsthe ground state electon configuration is1s22s22p63s23p63d104s25p65s15p0 . (2.2)For both 6Li and Rb (and all the alkali metals), the ground state electron configuration hasa single valence electron in an s orbital. The first available excited state of the electron isa p orbital, as indicated by the first empty electronic orbital in Eqs. 2.1 and 2.2.Coupling between the spin of the electron s and the orbital angular momentum of theelectron l gives rise to fine structure, where the additional term in the Hamiltonian has theform ~l · ~s. In this case, it is useful to work in the basis where the total electronic angularmomentum ~j = ~l+~s is a good quantum number, and can take on values ranging from l+ s10An excellent resource for more detailed information on these atomic species has been written by DanSteck for 85Rb [90] and 87Rb [91], and by Michael Gehm for 6Li [92].142.1. Laser Cooling Lightdown to |l − s| in integer steps. Each fine structure state is labeled by the term symbol2s+1lj . (2.3)For the alkalis, the ground electronic state has s = 1/2 and l = 0, which leads to asingle fine structure state labeled by j = 1/2. The first electronically excited state is ap orbital, where l = 1. Therefore, j can take on two values, either j = 1/2 or j = 3/2.The 2S1/2 → 2P1/2 transition is referred to as the D1 transition, while the 2S1/2 → 2P3/2transition is referred to as the D2 transition. In6Li , these two transition frequenciesare seperated by approximately 10 GHz [92], while in Rb the splitting is on the order of7 THz [90, 91].Additionally, coupling between the spin of the nucleus i and the total angular momentumj leads to hyperfine structure, where this additional perturbation has the form ~j ·~i. Similiarto the fine structure case, it is useful to work in the basis where the total angular momentumincluding nuclear spin ~f = ~j+~i is a good quantum number, and f can take on values rangingfrom j + i down to |j − i| in integer steps.More specifically, the hyperfine structure is due to an interaction of the magnetic fieldfrom the proton (which constitutes a magnetic dipole in the case of a non-zero nuclear spin)with the electron spin. Given the proton dipole ~µ, the associated magnetic field is~B =µ04pir3[3(~µ · rˆ)rˆ − ~µ] + 2µ03~µδ3(~r) . (2.4)The first term leads to an energy correction that is proportional to the expectation valueof 1/r3 and vanishes for any wave function that is spatially symmetric (i.e., a state withl = 0). The second term leads to the Fermi-contact interaction and is non-zero only if theelectron wavefunction is non-zero at the position of the nucleus (for example, the l = 0state). When considering the hyperfine splitting of the ground (l = 0, a s orbital) and firstexcited (l = 1, a p orbital) states, the hyperfine interaction in the ground state is muchlarger than in the first excited state.The details of the hyperfine structure of 6Li and Rb directly impact the requirementson the laser cooling light. These details, along with the generation of the laser cooling light,are discussed in the following two sections. Of note is that the majority of our laser coolinglight is generated on a single “master table” and fiber coupled to all of the experiments152.1. Laser Cooling Lightin the lab for amplification. This details in this thesis focuses on the amplification andfrequency shifting schemes for both species. Full details on our home built masters lasers,and the saturated absorption spectroscopy which we use to reference and stabilize the laserscan be found the Masters thesis of Keith Ladoucer [93], Janelle van Dongen [94] and SwatiSingh [95].2.1.1 Lithium LightA 6Li atom has a nuclear spin i = 1. Therefore, in the 2S1/2 fine structure manifold, thisgives rise to two hyperfine levels, namely f = 1/2 and f = 3/2. These two hyperfine manifoldsare split by 3/2 · a2S ≈ 228 MHz, where a2S is the magnetic dipole hyperfine constant whichdetermines the strength of the hyperfine interaction (see Table 2.1). Likewise, in the firstelectronically excited state, the 2P1/2 fine structure manifold also has two possible hyperfinemanifolds, corresponding to f = 1/2 and f = 3/2. The 2P3/2 fine structure manifold hasthree possible hyperfine manifolds, corresponding to f = 1/2, 3/2, 5/2. The magnitude ofthe splitting of the hyperfine levels in the excited state is also determined by the magneticdipole hyperfine constant and, in the case of the 2P3/2 manifold, the electric quadrupolehyperfine constant. The value of these constants is summarized in Table 2.1.Property Symbol Value (MHz)22S1/2 magnetic dipole constant a2S 152.136840722P1/2 magnetic dipole constant a2P1/2 17.38622P3/2 magnetic dipole constant a2P3/2 -1.11522P3/2 electric quadrupole constant b2P3/2 -0.10Table 2.1: Magnetic dipole and electric quadrupole hyperfine constants of the ground stateand first excited state of 6Li. Note that only the 22P3/2 fine structure manifold has a nonzero electric quadrupole hyperfine constant. Values from [92, 96, 97].Our MOT operates on the D2 transition, with the “pump” light near the f = 3/2 →f ′ = 5/2 transition frequency and the “repump” light near the f = 1/2→ f ′ = 3/2 transitionfrequency. However, the splitting of the hyperfine levels in the 2P3/2 is on the order ofa few MHz, which is less than the natural linewidth of the transition. In this case, thepump light excites all three hyperfine manifolds with similar rates. Since the f ′ = 3/2 andf ′ = 1/2 excited manifolds can decay to the lower f = 1/2 ground hyperfine manifold, thepopulation in the upper f = 3/2 ground hyperfine manifold is rapidly depleted. Therefore,162.1. Laser Cooling Lightthe repump light must have a similar scattering rate (i.e., intensity) as the pump light, andconsequently also contributes to cooling and exerts a radiation pressure force on the MOTwhich is comparable to the pump light.As the splitting of the ground hyperfine levels is approximately 228 MHz, both thepump and repump light can be derived from the same master laser. In our case, our masterlaser is Toptica DL Pro, which we lock 50 MHz blue of the MOT pump transition. Thelocking error signal is generated with saturated absorption spectroscopy using a frequencymodulation scheme [93, 94, 95]. Details of specific beam paths and layouts can be foundin the PhD thesis of Mariusz Semczuk [98]. After optical isolators, the output power ofthe laser is about 15 mW, which is split three ways between the error signal generation,the offset lock for the high field imaging master and amplification for the experiment. Thelatter two pathways are discussed below.MOT and Zeeman SlowerThe portion of the master light destined for the experiment is used to seed a “slave” ampli-fier, which is based off of a Mitsubishi ML 101J27 laser diode. These diodes are specifiedto emit at a wavelength of 660 nm at 25◦C with a driving current of 120 mA. However,around 72◦C the free running wavelength is between 670 and 671 nm (near the requiredwavelength for 6Li). Although running the laser diode at higher temperatures significantlyreduces the output power, we have found that we can achieve reasonable output powers (onthe order of 60 mW) by increasing the current to a range between 350 and 450 mA. Whilethis current is well above the damage threshold at room temperature, it does not appearto damage the diodes at the high temperatures at which we operate. The output of thisslave is split into two arms using a 50/50 fiber splitter from Evanescent Optics, where eachoutput of the fiber splitter has a power of 15 mW.Light from one of the output ports is used to generate the light for the Zeeman slower.The light is used to seed a home-built tapered amplifier (TA) based off of an EagleyardEYP-TPA-0670-00500-2003-CMT02-0000 chip. The output of the TA is fiber coupled intoa 50 m long single-mode, polarization-maintaining fiber and sent to the experiment table.On this table, the light is frequency shifted to match the desired detuning for the slowingbeam using an AOM in a double pass configuration. We operate this AOM with an RFdriving frequency of 63 MHz, and use the negative first order diffraction such that the172.1. Laser Cooling Lightslowing beam is 76 MHz red of the pump transition. The output mode of the TA is quitepoor and, as such, we typically only couple approximately 110 mW out of 350 mW throughthe long fiber. After the double pass AOM we have approximately 30-35 mW available forthe slowing beam11.Light from the second output port is used to generate the MOT pump and repumpbeams. The slave light is used to seed a second slave, and the output of this slave is coupledinto a short fiber. The output of the fiber is used to seed two TAs from Sacher Lasertechnik(TPA-0670-0500). We fiber couple seeding light so that the source of seed beam can bechanged without impacting the alignment of the seed beam for both TAs. When new,these TAs output about 450 mW. However, we have found that the power degrades ratherquickly over time and we typically achieve 300 to 350 mW output power. We have alsofound that the output spectrum of the TAs varies with the seeding power, as shown inFig. 2.1. Typically, we use 18-20 mW to seed both amplifiers.To generate the MOT pump light, the seed beam is up shifted by 108 MHz using asingle pass AOM, and the output of the TA is downshifted by a double pass AOM to givea frequency that is centered 45 MHz red of the pump transition. To generate the MOTrepump light, we only up shift the output of a second TA with a double pass AOM to give afrequency that is centered 40 MHz red of the repump transition. Both beams are coupled inseparate 50 m long single-mode, polarization-maintaining fibers and sent to the experimenttable where they are combined and then split three ways for each of the orthogonal axesof the MOT. When the TAs were new, we were able to achieve about 100 mW of powerin each beam on the experiment table. After multiple years of operation, we typically haveabout 40 mW of each beam. However, we recently replaced the chip in the pump TA andcurrently achieve 115 mW of pump and 35 mW of repump power12. Although we have amechanical shutter on the experimental table, we use the two double pass AOMs for precisetiming of the MOT beams. A summary of this layout is shown in Fig. 2.2.In late 2015, we made a modification to our system in order to increase the poweravailable for the slowing beam, and to ensure that we had adequate power to seed both ofthe commercial TAs with 20 mW. The output of the first slave is still split with a 50/5011Note that the double pass efficiency is low because we are using an 80 MHz AOM, and operating at theedge of the region where high efficiency is possible.12Although more power is nice, we found that the additional power in the pump beam does not increasethe loading rate of steady state number in our MOT. This is discussed in Section Laser Cooling Light660 665 670 675 680Wavelength (nm)−60−50−40−30−20−10Power(dB)Figure 2.1: Spectrum of light from the Sacher TAs when the seed power is 20 mW (solidblack line) and 12 mW (dashed red line). We have found that the MOT and ODT performbetter when the TAs are seeded with 20 mW, which may be due to the improvement in theoutput spectrum at high seed powers.fiber splitter, and one output continues to seed the home-built TA. However, the output ofthis TA is now used to seed both of the commercial TAs. The second output of the fibersplitter is sent directly to our experiment table, and through the double pass AOM used toset the frequency of the slowing beam. The light after the double pass seeds a slave and theoutput of this slave is used for the slowing beam. The slave is seeded with approximately2 mW of power, and we are able to achieve approximately 60 mW of power in the slowingbeam. These changes are summarized in Fig. 2.3.ImagingFor our standard absorption imaging at zero magnetic field, we image on the MOT pumptransition. For this light, we pick off a portion of the MOT pump beam and couple it intoone of two import ports of a 50/50 fiber splitter. When an image is taken, we use thedouble pass AOM that controls the frequency of the light to set it on resonance with thef = 3/2→ f ′ = 5/2 transition. Similarly, the repump light used in imaging is picked off from192.1. Laser Cooling Lightsingle-pass AOMdouble-pass AOMlockingHF offset lockTopticaS SHome-Built TAto MOTRepump TAPump TAto Zeeman SlowerFigure 2.2: Layout of amplifier setup for the generation of 6Li light. The Toptica masterseeds a slave laser, and the output of this slave is used to seed a home-build TA and asecond slave. The output of the home-built TA is used for the Zeeman slower light, andthe second slave is used to seed two commercial TAs which amplify the MOT pump andrepump light. See text for details. The light is generated on our “master table” and is sentto the experiment table through 50 m long single-mode, polarization-maintaining fibers.single-pass AOMdouble-pass AOMlockingHF offset lockTopticaSHome-Built TAto MOTRepump TAPump TAto Zeeman SlowerSFigure 2.3: Updated layout of the 6Li amplifier setup to reflect changes made as of late 2015.These changes were made to increase the power of the slowing beam and to ensure thatthere was adequate power to seed both commercial TAs. In this new layout, the home-builtTA is now used to seed the commercial TAs. Additionally, light from the first slave is sentdirectly to our experiment table. Here, it is frequency shifted before used to seed a secondslave which generates the amplified slowing beam light.202.1. Laser Cooling Lightthe MOT repump beam and coupled into a second fiber. The output of this fiber is sentcounter-propagating to the imaging beam (see Section 3.4 for details).We also take absorption images of 6Li atoms in the presence of a large magnetic field(on the order of 500-800 G), where the imaging transition is on the order of 1 GHz red ofthe pump transition at 0 G13. To generate this light, we use a separate home-built extendedcavity diode laser (ECDL) based on the Roithner RLT6720MG diode laser [98]. This laseris offset locked to the Toptica master, where the offset frequency is set with a VCO (theADF4350 chip from Analog Devices). The output of this master laser is amplified by anotherslave laser and the slave output is frequency shifted with a double pass AOM. This AOMserves two purposes. First, it allows fast timing control of the light and power adjustment.Secondly, the AOM allows the frequency of the laser to be shifted by upwards of 80 MHz,which enables us to reach the required transition frequency to image two separate spinstates in 6Li without the need to change of the offset lock frequency.Given that the Toptica master (to which the high field imaging master is locked) is50 MHz blue of the pump transition, the offset beat between the two lasers is given byfbeat = |δ − 2faom − 50| , (2.5)where δ is the detuning of the high field imaging master after the double pass AOM,measured relative to the pump transition frequency. The driving frequency of the AOM isfaom, and all frequencies are measured in MHz.The light after the double pass AOM is coupled into a 50 m long single-mode, polarization-maintaining fiber which sends the light to the experiment table. On this table, the fiberis butt-coupled to the second input port of the 50/50 fiber splitter used for the absorptionimaging light. One of the outputs of this fiber splitter is used to send light down the imagingaxis. The use of the fiber splitter allows us to switch between imaging at high field and zerofield without any physical modifications to the apparatus. Details of the offset lock circuitcan be found in the PhD thesis of Mariusz Semczuk [98].13The details of the high field imaging setup are discussed in detail in Sec. 3.4. The specific transition areshown in Fig. 3.19 and the associated transition frequencies are given in Eq. Laser Cooling Light2.1.2 Rubidium LightIn our experiment, we can work with both 85Rb or 87Rb. The two isotopes differ in thenuclear spin, where the i = 5/2 for 85Rb and i = 3/2 for 87Rb, and this leads to a differinghyperfine structure.In 85Rb, the 2S1/2 manifold is split into two hyperfine manifolds corresponding to f =2, 3 where the splitting between these manifolds is ≈ 3 GHz. In the excited state, the 2P1/2manifold is also split into two hyperfine levels where f = 2, 3. The 2P3/2 manifold hasfour possible hyperfine manifolds corresponding to f = 1, 2, 3, 4. Details of the hyperfinestructure and energy splitting for 85Rb can be found in [90].In 87Rb, the 2S1/2 manifold is also split into two hyperfine manifolds corresponding tof = 1, 2. In this case, the splitting between these manifolds is ≈ 6 GHz. In the excitedstate, the 2P1/2 manifold is also split into two hyperfine levels where f = 1, 2 and the2P3/2manifold has four possible hyperfine manifolds corresponding to f = 0, 1, 2, 3. Details ofthe hyperfine structure and energy splitting for 85Rb can be found in [91].For both the 85Rb and 87Rb MOT, we operate on the D2 transition, where the “pump”light for 85Rb (87Rb) is near the f = 3 → f ′ = 4 (f = 2 → f ′ = 3) transition frequencyand the “repump” light is near the f = 2→ f ′ = 3 (f = 1→ f ′ = 2) transition frequency.Unlike 6Li, the energy difference between adjacent hyperfine manifolds in the excited state ismuch larger than the natural linewidth of the transition. Therefore, the “pump” transitionis (nearly) closed. However, off resonance transitions to the f ′ = 3 (f ′ = 2) manifold candecay to the f = 2 (f = 1) ground hyperfine manifold. Since the rate of these off resonantexcitations is suppressed due to the large detuning (on the order of hundreds of MHz), onlya weak repump beam is necessary to repump atoms back into the cooling transition.Because the spitting of the ground state hyperfine manifolds is on the order of GHz,it is not practical to use a single laser for both the pump and repump light. Therefore,we have four separate masters lasers (two for each of the pump and repump for both Rbisotopes). We use an identical locking scheme for both 85Rb or 87Rb and as such, thefollowing discussion is applicable to both isotopes.MOT and Zeeman SlowerEach of the masters lasers is a home-built ECDL based off of the GH0781JA2C chip fromSharp, and is locked to the atomic transition using saturated absorption spectroscopy com-222.1. Laser Cooling Lightbined with a frequency modulation scheme [93, 94, 95]. The output from each master laseris used to injection lock a slave laser, which is based on the Intelite MLD790100S5P chip.The output of each slave laser is fiber coupled into a distribution box, which uses fibersplitters from Evanescent Optics to provide four output ports for each master laser. Thisdistribution scheme allows multiple experiments in our lab to work with either 85Rb and87Rb at the same time, and easily switch between the two species by simply changing theport to which a fiber is connected14. The master lasers for both the pump and repump lightare locked 180 MHz red of resonant transition frequency.For our experiment, the pump light from the distribution box is used to seed a secondslave, and the output of that slave is coupled into a 50 m long single-mode, polarization-maintaining fiber which sends the light to the experiment table. On the experiment table,this light is used to inject a fiber coupled TA from Sacher Lasertechnik (TEC-400-0780-2500). We use approximately 14 mW of power to seed the TA and the output power istypically 1 W when the TA is run at 2800 mA15. The output of the TA is split using apolarizing beam cube between a path for light for the MOT pump beam and the slowingbeam. In each path, a double pass AOM is used to up shift the frequency such that it hasthe desired detuning from the pump transition. This layout is summarized in Fig. 2.4.The repump light from the distribution box is immediately fiber coupled to the exper-iment table using a 50 m long single-mode, polarization-maintaining fiber. On the experi-ment table the output of this fiber is used to seed an additional slave. The output of thisslave is up shifted in frequency with a double pass AOM such that it is on the repump tran-sition resonance. After the double pass AOM, the light is split on a polarizing beam cubefor distribution to the MOT and Zeeman slower axes. For each of the two axes, the repumplight is then combined (using another beam cube) with the pump light destined for theMOT or Zeeman slower. Note that the repump light that is sent down the Zeeman sloweraxis also acts as a repump for the MOT as the Zeeman slower passes directly through theMOT16. After the double pass AOM we typically have 30 mW of repump power available.This layout is also summarized in Fig. 2.4.14That said, all of the experiments discussed in this thesis use 85Rb.15We normally run the TA at lower currents the power requirements (about 100 mW between the slowinglight and the MOT liight) after frequency shifting for the experiment are much lower than the maximumoutput of the TA.16In fact, we found no decrease in loading rate or steady state number is all of the repump light is sentdown the Zeeman slower axis. This is discussed in greater detail in Section Laser Cooling LightAOMDistribution BoxPump SRb TASto MOTto ZSSReump SFigure 2.4: Layout of the amplifier setup for Rb light generation. We use a separate laserfor the pump and repump light, which is amplified and sent to a fiber based distributionbox. For our experiment, the pump light is amplified using a second slave, and is sent to theexperiment table where it is amplified again by a commercial TA. The output of this TA issplit into two paths, one for the Zeeman slower slowing beam and one for the MOT pumpbeam. Each path is frequency up shifted by a double pass AOM. The repump light is sentto our experiment table, where it is used to seed a slave laser. The output of the slave isfrequency shifted by a double pass AOM and then split to send light to both the MOT andZeeman slower. The pump and repump light destined for the MOT and Zeeman slower arecombined on a final beam cube. For clarity in this schematic, after the final amplificationstep the repump is shown as a dashed line while the pump is shown as a solid line. Thecombined pump and repump beams are as a thick solid line.Imaging and Optical PumpingFor the Rb absorption imaging light, we pick off a portion of the MOT pump beam afterit has been frequency shifted by the double pass AOM and couple the light into the samefiber as the 6Li absorption imaging light. This fiber is one of two input ports for a 50/50fiber splitter that is designed for 670 nm light. At 780 nm, over 90% of the power goes intoone of the two output ports from the splitter17. For hyperfine pumping of the atoms toupper hyperfine manifold before the image is taken, we simply use the MOT repump lightdirected along the MOT axis.The light that we use for optical pumping is also picked off from the MOT pumpand repump beams and coupled into a single fiber18 . At the input of the fiber, use we17Therefore, obviously, we choose to use this output port for the imaging beam.18The MOT pump and repump beams are are used as the optical depump and optical pumping beamsrespectively. The details of our optical pumping scheme are discussed in detail in Section 3.3242.2. Vacuum System and Apparatususe a half-wave plate and a Glan-Thompson polarizer to ensure that both beams havethe same polarization. The output of this fiber is sent down the same axis used for thephotoassociation laser light, and is collinear with one of the arms of the ODT (see Fig. 3.9).2.2 Vacuum System and ApparatusOur experimental apparatus is constructed of two sections: a “science” section which con-tains the MOT region and the ODT, and a “source” section which contains the effusivesources for both 6Li and Rb. The two sections are separated by a differential pumpingtube, which helps maintain a pressure differential between the sections. On overview of theappartus can be seen in Fig. 2.5, which shows the Soldiworks drawing of the experiment.This section presents an overview of the construction of the apparatus, the effusive sources,and details of some of the hardware used in the experiment. Additional information can befound in the Master’s thesis of William Bowden [99].Source SideOn the source side, the two effusive ovens are attached to two ports of a standard 2.75-inchfour-way cross. In addition, the top port of the cross is used to attach an ion pump (VarianVacIon Plus 20 Starcell), a non-evaporable getter (NEG) (SAES CapaciTorr D 400-2) anda bakeable all metal valve via another four way cross. The all metal valve is used to connectthe source side to our pumping station during the bake-out. A gate valve (VAT VacuumValves Series 010) separates this source chamber from a standard 6-way 2.75-inch cross.The 6-way cross is used to add a mechanical shutter which can block the atomic beamwhen the apparatus is not in use, and a copper feed through which we use as a cold finger toreduce the background pressure of Rb. On the two unused horizontal ports perpendicularto Zeeman slower axis we attach viewports such that we can see the atomic shutter andhave optical access to the atomic beam before it enters the slowing region. The layout ofthis 6-way cross section is shown in Fig. 2.6. Optical access to this section also helps inthe alignment of the slowing beam, which can be reflected off of the mechanical shutter.Additionally, we have used this optical access to send in repump light for both 6Li and Rbin an attempt to hyperfine pump the atoms into the slowing transitions before they enter252.2. Vacuum System and ApparatusGate ValveZeeman SlowerMOT CoilsIon PumpNEGLi OvenRb OvenSlowingBeamFigure 2.5: Solidworks drawing of the experiment apparatus. The apparatus is split intotwo sections by a gate valve. On the left hand side is the “science” section, which includesZeeman slower and the optically polished quartz cell containing the MOT. On the right handside is the “source” section, which contains the effusive ovens for 6Li and Rb (see Fig. 2.7).These two sections are separated by a differential pumping tube (see Fig. 2.6), which helpsmaintain a pressure differential between the sections. Not shown are the compensation coilslocated around the quartz cell (see Fig. 2.10).262.2. Vacuum System and ApparatusFigure 2.6: Cross-sectional view of the 6-way cross which contains the atomic beam shutter,cold finger and differential pumping tube. This cross is separated from the effusive sources(to the right, not shown) by a gate valve. The differential pumping tube is constructedfrom a stainless steel rod which is bolted to the zero-length CF reducer.the slower19. The 6-way cross is attached to the Zeeman slower via a 2.75-inch to 1.33-inch CF reducer. Bolted to this reducer is a differential pumping tube constructed from a12 cm stainless steel rod with a diameter of 6 mm (see Fig. 2.6). The conductance of thedifferential pumping tube and other components in the system are given in Table 2.2.The Zeeman slower itself is constructed from a 17 mm stainless steel tube, onto whichwe wound eight segmented coil sections. Each coil section is separated by a large metal19Unfortunately, this did not show any improvement in the loading rate. Details of the effect of repumplight on the loading rate of 6Li and Rb is discussed in Section Vacuum System and ApparatusPart Length (cm) Diameter (cm) C (L/s)Differential Pumping Tube 12 0.6 0.85Zeeman Slower 30 1.8 92.75-inch Cross 15 3.6 143Quartz Cell 35 3 42Table 2.2: Calculated hydrogen conductances for various vacuum components in our exper-iment.plate which helps to cool the coils. The diameter of the Zeeman slower tube is chosen suchthat the coil windings are as close to the slowing axis as possible in order to decrease thecurrent requirements. In addition, the small diameter increases the differential pumpingbetween the source and science sections, and is also compatible with 1.33-inch CF parts.We found that the metal cooling plates were sufficient to cool the Zeeman slower even incontinuous operation at highest currents required for 6Li. Each coil segment consisted of20 axial windings and 25 radial windings with 16 AWG Kapton coated wire. The currentin each coil segment can be controlled “on-line” through a computer controlled currentdriver [99].Effusive SourcesOur effusive sources were designed with one key requirement in mind: maximize the centerline intensity of the atomic beam (i.e., maximize the flux of atoms that are emitted directlydown the Zeeman slower axis) while at the same time limiting the off-axis intensity in orderto preserve the lifetime of the sources20. One possible way to achieve this requirement is touse a recirculating source, where thermal gradients wick back and recollect atoms which areemitted off-axis [100]. We choose an alternative technique, where microtubes are press fitinto the opening in the effusive source21. These microtubes effectively collect the majorityof atoms exiting the source off-axis without compromising the intensity of the center linebeam.In a dual species Zeeman slower, the design is complicated by the fact that both effusive20We intentionally isolated the sources from the rest of the vacuum system with a gate valve such that itis possible to replace the sources while minimizing the effect on the rest of the vacuum system. However,this replacement is a non trivial task and should be avoided if possible. Hence, the emphasis on maximizingthe source lifetime.21Other designs that use microtubes require a retaining bar or plug [101]. However, found that pressfitting the tubes held them sufficiently secure.282.2. Vacuum System and Apparatussources must emit down the same axis. Additionally, the temperature requirement of thetwo sources are often very different and care must be taken to prevent backflow betweenreservoirs [102]22. In our design, we use two effusive ovens which are thermally and physi-cally isolated, while the output of each sources is vertically offset from one another other,as shown in Fig. 2.7. This design is based on that by Wille et al. [103, 104], except thatour sources are made from standard commercially available vacuum parts.The 6Li oven consists of a 1.33-inch nipple which we lined with nickel mesh to ensurethat 6Li does not leak out of the oven23. It was also crucial to use nickel gaskets in anypart of the source that may come in contact with melted 6Li, as copper gaskets are veryquickly corroded by 6Li. A cleaned chunk of enriched 95% 6Li is placed inside the nipple,and the oven is heated with a band heater on the front and back flange of the nipple. Moreinformation on our cleaning technique is given in the following section on the bake-out.The Rb effusive oven consists of a Rb ampoule from ESPI Metals that is held in a flexiblebellows. After the bake-out, the bellows is used to break the ampoule. Care must be takento ensure that the ampoule is not broken before the bake-out, as doing so will result in anear total loss of Rb during the bake-out. Similar to the 6Li source, we heat the Rb sourcewith two band heaters on the bellows and the flange located near the exit of the source.Science SideOn the science side of the experiment, the MOT region is centered on an optically polishedquartz cell from Helma, which is not AR coated. The dimensions of the cell are 30× 30×100 mm with a wall thickness of 5 mm. The cell is connected at both ends to 2.75-inch CFflanges where, on one end, this connection is made through a stainless steel bellows.The fixed end of the cell is connected to the output of the Zeeman slower through aconical 2.75-inch to 1.33-inch reducer. We had hoped to use a zero-length reducer in order tominimize the distance between the end of the Zeeman slower and the MOT region. However,because the zero length reducer uses a threaded hole to attach to the 1.33-inch side, we didnot have enough clearance between the final metal fin and the 1.33-inch flange to fit therequired screw. However, we were able to minimize the impact of the additional length byusing the MOT coils as the final slower segment (see Section 3.1 for more details).22For example, in our case, the Rb source is held near 100◦C and the 6Li source is held around 400◦C23In tests without the nickel mesh, we found that the 6Li was wicked out of the oven within a few hours.292.2. Vacuum System and ApparatusFigure 2.7: Solidworks drawing of the 6Li and Rb effusive source. The 6Li source is attachedto the right port of the 4-way cross, and consists of a nipple lined with nickel mesh. Themicrotubes are inserted into a small hole drilled in the CF blank sandwiched between thetwo nipple flanges. The Rb source is attached to the bottom port of the 4-way cross. Aglass Rb ampoule is placed inside the flexible bellows and is broken after the bake-out. Themicrotubes are press fit into a small hole milled from the oven “cap” that extends into thecross. The inset shows a view of the oven outputs looking down the Zeeman slower axis,and demonstrates that the output of each source is vertically offset from one another.The bellows end of the cell is connected to a 2.75-inch four-way cross. We use a viewporton the opposite port of the cross for the Zeeman slower beam. On the top port, we attacheda second four way cross to which an ion pump and a NEG are mounted. Attached to thebottom port of the cross is an all-metal-valve which is used to connect our pumping stationduring the bake-out. The conductance of the cell and the 4-way cross are given in Table 2.2.302.2. Vacuum System and ApparatusOven Loading and Bake-outAs 6Li is highly reactive with both organic and inorganic reactants (including nitrogenand hydrogen), the 6Li that we purchase comes packed in oil in order to protect it fromreaction with air. To load 6Li into the effusive source, we first clean the 6Li chunk withpetroleum ether to remove any residual oil, and then use a clean razor blade to remove anexterior layer from all sides of the chunk. This is performed in a pure argon (or helium)environment to minimize the exposure of the 6Li chunk to air24. After cleaning, the 6Li isplaced in the oven nipple and immediately attached to a separate chamber on our pumpout station. At this point, we heat the 6Li oven to 500◦C for about six hours to removeany remaining containments. The oven is then backfilled with argon and transferred to themain apparatus. We estimate that we load about 3 g of 6Li , which is expected to yield alifetime of four years of continuous operation at 450◦C.The maximum temperature of our bake-out is limited to just under 200◦C by the rotaryfeed through for the atom beam shutter. We attached our pump-out station (which containsa turbo pump, a backing scroll pump and a residual gas analyzer (RGA)) to the all-metalvalves on both the science and source side. We use two connections to the apparatus sothat we are not pumping one of the sections through the differential pumping tube. Weperformed an initial bake-out for about 5 days during which we heat the NEGs to increasethe partial pressure of H2 in the system. This is done in order to increase the pumping ofH2 from the system, and to regenerate the NEGs. This first stage takes about a week, andis performed without the 6Li effusive source attached such that the 6Li is not contaminatedby the high hydrogen background pressure. At this point, the Rb ampoule has not brokenand therefore the Rb is unaffected.After this initial stage, we backfill the system with argon and attach the loaded 6Li effu-sive source. This second bake-out stage also took place at a temperature just under 200◦Cand lasted about two weeks. After this bake-out, we initially had issues with the pressure inour source chamber. In our attempts to diagnose the issue, we had the pumping station (andtherefore the turbo pump and RGA) attached only to the science side of the experiment.However, over the course of about a month, we found that the pressure issues went away.We attribute this to additional pumping by the turbo pump, ion pump and NEG during24We originally used a glove box, but we found that it was easier to work in a large clear plastic bag thatis filled with a positive pressure of either argon of helium.312.2. Vacuum System and Apparatusthis time25. We are now able to obtain MOT lifetimes on the order of 100 s26 and lifetimesin the low power ODT on the order to tens of seconds, which is more than sufficient for ourexperiment.MOT and Compensation CoilsWe generate the quadrupole magnetic field for the MOT and the homogenous magnetic fieldused during the experiments with a set a water cooled coils run in an anti-Helmholtz orHelmholtz configuration, respectively. We use a 100 V / 50 A power supply from Sorensen(SGI 100/50), and the resistance of our coils is designed to match the limits of the powersupply in order to maximize the magnetic fields that we can generate. As the coils arerun in series, this limits the resistance of each coil to be slightly less than 1 Ω27. Physicalconstraints in the experiment limited the inner radius of the coils to 4 cm, the outer radiusto 8 cm and the coil separation to 4.5 cm. Through a simulation of the magnetic field, wefound that best resistance match used 14 AWG wire with 21 radial windings and 10 axialwindings, gaving a coil resistance of 0.88 Ω. A summary of the achievable magnetic fieldand field gradient is given in Table 2.3.Coil Homogenous Field (G/A) Axial Gradient (G/(A·cm))MOT 25.8 6.15X-Compensation 2.67 –Y-Compensation 1.45 –Z-Compensation 13.9 3.6Table 2.3: Magnetic field produced by the MOT and compensation coils. The x and y compare only used to apply a homogenous offset field. The x and y axes are defined to lie in thehorizontal plane, with the y-axis collinear with the slowing beam axis, see Fig. 2.10.To provide cooling for the coils, the coils are immersed in water by winding them insidea watertight housing. The coil housing is constructed from Delrin, and consists of a lid,base and a retaining ring. The retaining ring is necessary to prevent the inside walls of25And just a little bit of luck...26This lifetime is the decay rate of a very small MOT with the atom shutter closed. In a small MOT, lossesthat are density dependent (for example, light assisted collisions) are small, and the dominant source of lossis through collisions with background gases. Nevertheless, light assisted losses do still contribute to the lossrate, and therefore this measured decay time provides an upper lime on the loss rate due to backgroundcollisions.27While the resistance at room temperature can be 1 Ω, running the coils at a large current can increasethe temperature of the wire, which in turn increases the resistance.322.3. Photoassociation Laser Systemthe housing from bulging at high water pressures. A water tight seal is achieved throughthe use of two o-rings between the lid and the outer wall and the base and the inner wall.The inside surface of the lid and base of the coil housing is milled to have dams in orderto force the water to flow through all parts of the coil, as shown in Fig. 2.8. In addition,we placed a spacer between every two radial windings to ensure that each winding was indirect contact with water. A cross sectional view of the coil design is shown in Fig. 2.9. Wetested that the coil housing was able to handle a pressure of 80 PSI, which corresponded toa flow rate of 8.6 L/min. At this flow rate, the temperature of the coils rose to 41◦C whenrun at 50 A. Full details of the coil design and testing can be found the Master’s thesis ofWilliam Bowden [99].The apparatus also has a set of three orthogonal compensation coils, which we use toapply a small (¡10 G) offset homogenous magnetic field. This field allows us to move thezero of the MOT quadrupole field to help balance the radiation pressure from the Zeemanslower beam, and to move the positive of the MOT to overlap it with the focus of the ODTduring the transfer from the MOT to the ODT. These coils are also used to cancel forany residual magnetic field at the location of the atoms. The compensation coils along thehorizontal x and y axes are constructed by winding Kapton wire on a rectangular plastichousing28. The vertical compensation coils are concentric with the MOT coils, and locatedjust above (below) the top (bottom) MOT coil. They have an inner diameter of 9.5 cmand an outer diameter of 13 cm and there are approximatly 15 radial windings and 5 axialwindings. The z-compensation coils are also used to provide a magnetic field gradient tocounter any inhomogeneities in bias field applied during experiments, or to compensate forgravity at low ODT depths29. The location of the compensation coils is shown in Fig. 2.10.A summary of the achievable magnetic field and field gradient (as a function of current) isgiven in Table Photoassociation Laser SystemThe light that we use for photoassociation must meet two key requirements: first, the lightneeds to be widely tunable and second, our uncertainty on the frequency determination28In a previous iteration, we used a metal housing. However, we found that the metal would knick theKapton wire and cause the coils to short.29Without this gradient field, both field inhomogeneities and gravity lead a tipping of the trapping potentialand a loss of atoms from the ODT. This is discussed in more detail in Section 3.2.332.3. Photoassociation Laser SystemFigure 2.8: View of the coil housing without the lid. The inside surface of the base (andlid, not shown) of the coil housing is milled to have dams in order to force the water toflow through all parts of the coil. The water flows into the ‘V’ shaped regions, and then isforced up through the coils. The lid has a similar ‘V’ shape pattern, but offset such thatthe water is forced down through the coils and into the next ‘V’ shaped section on the base.Figure 2.9: Cross sectional view of the MOT coil design. The coil housing in constructedfrom Delrin, and consists of a lid, base and a retaining ring. A water tight seal is achievedthrough the use of two o-ring between the lid and the outer wall and the base and the innerwall. A spacer is placed between very two radial windings to ensure that each winding wasin direct contact with water.342.3. Photoassociation Laser Systemy -Coilsx-CoilsFigure 2.10: Location of the compensation coils relative to the MOT coils and the locationof the MOT. The x-compensation coil produces a field that runs perpendicular to thecell axis in the horizontal plane, while the y-compensation coil produces a field that runsparallel to the cell axis. The z-compensation coil is the black object mounted concentric tothe MOT coils, and produces a field that in the vertical direction. Not shown is the secondz-compensation coil which is mounted in a similar fashion below the bottom MOT coil.needs to be at a level that is below the natural linewidth of the photoassociation transitions(on the order of 10 MHz). For experiments that require coherence (for example, darkstate spectroscopy or STIRAP, discussed in section 5.2.2) we require that the lasers havea narrow relative linewidth such that they are phase coherent during the illumination timeof the atomic ensemble30. We meet these requirements by employing two cw Ti:Sapphirelasers which are phase locked and referenced to the same femtosecond optical frequencycomb.30In the context of our experiments, “narrow” means a relative linewidth of less than 100 kHz. However,a relative linewidth that is lower than this relaxes other requirements, and can make such experiments lesstechnically challenging. This is discussed in greater detail in Section Photoassociation Laser System2.3.1 Femtosecond Optical Frequency CombThe femtosecond optical frequency comb (OFC) that we use was originally built by the lab ofDavid Jones at UBC [77] several years before we started our photoassociation experimentsin earnest. Later, we rebuilt and modified some parts of the comb and added a secondmeasurement branch which is used to stabilize the Ti:Sapphire lasers. However the oscillatoritself remained the same. A detailed description of OFCs and ultrafast physics can be foundin [105, 106].Unlike cw lasers, which are forced to lase on only one longitudinal cavity mode, pulsedlasers lase on many cavity modes at the same time, and rely on interference between cav-ity modes to create a pulsed output. This interference requires a fixed phase relationshipbetween cavity modes, and is achieved through mode locking. In our setup, the oscillatoris based on an erbium-doped fiber laser (similar to [107, 108, 109]) where mode locking isachieved through nonlinear polarization rotation [110]. This technique relies on nonlinearintensity-dependent rotation of the polarization in the optical fiber, combined with polar-ization optics within the oscillator, to create an artificial saturable absorber. This saturableabsorber has an intensity dependent loss which is low for high intensities and high for lowintensities such that the maximum gain occurs for short pulses. Mode locking is achievedempirically by tuning two sets of wave plates (a half-wave plate and a quarter-wave plate)placed on either side of an optical isolator31, situated in the small free space section of theoscillator (see Fig. 2.11). Our oscillator produces sub 100 fs pulses at a repetition rate offrep = c/L = 125 MHz, where L is the total roundtrip length of the cavity.In the time domain, the output pulse (of any pulsed laser) can be split into a carrierand envelope component (see Fig. 2.12), where the envelope frequency is 2pifrep, the carrierfrequency is ωc. There also exists a phase offset between the peak of the envelope and thepeak of the nearest carrier wave, which is called the carrier envelope offset phase ψceo. Thisoffset phase evolves as a function of time due to differences in the group velocity vg andphase velocities vp in the fiber, owing to its dispersive nature32. This results in a change of31An optical isolator is used instead of a polarizer to ensure that light only travels in one direction andthere is no standing wave in the cavity.32The group velocity and the phase velocity determine the velocity of the envelope and carrier waverespectively. They are related by vg = vp(1− k/ndn/dk). The two velocities are equal only in the case of zerodispersion, where dn/dk = 0.362.3. Photoassociation Laser SystemOscillatorPumpDiodeλ4λ2λ4λ2EDFAPZT90/1090 101090WDMRFSourceRFSourcePZTDriver ServoBPFPCMB50/50AOM0-1 EDFA WDMPumpDiodeHNLF to TS lockPCSREDFA WDMPumpDiodeHNLFPPLNInt. FilterBPFPFDServoSlow∫Figure 2.11: Schematic of the frequency comb and associated locking electronics. Shownare fiber segments (maroon), free space segments (red) and electronic connections (blue).Mode locking of the oscillator is achieved using nonlinear polarization rotation, and themode locking position is tuned using a pair of half and quarter wave plates on either sideof an optical isolator. These components are located in the short free space section of thecavity. The repetition rate of the oscillator is controlled by a fiber collimator mounted ona PZT, while a fiber AOM external to the oscillator is used for fast control of fceo. We alsouse slow feedback on the oscillator pump diode to ensure that the correction does not exceedthe AOM tuning range, where the diffraction efficiency of the AOM is small. The negativefirst order of the AOM is split into the measurement branch (MB) and the self-referencingbranch (SR), where the comb light is amplified and broadened (see text for details). Inboth branches, we use a fiber polarization controller (PC) to optimize the broadening inthe highly nonlinear fiber (HNLF). The Erbium doped fiber amplifiers (EDFA) are pumpedwith 980 nm pump diode, and coupled into the erbium doped fiber using a wavelengthdivision multiplexer (WDM). In the measurement branch, a f −2f interferometer based ona PPLN crystal is used to measure and stabilize the carrier envelope offset phase. For thelock of frep and fceo the RF signal from the photodiodes is sent through a bandpass filter(BPF), and the error signal is generated using a phase frequency discrimator (PFD) or amixer.372.3. Photoassociation Laser Systemψceo from pulse to pulse of [105]∆ψceo =(1vg− 1vp)Lωc . (2.6)In the frequency domain, the output of the oscillator produces a comb of frequencyelements which are spaced by frep and centered on ωc over a frequency range given bythe inverse of the pulse width in the time domain, as shown in Fig. 2.12. The frequencyelements, if extrapolated to zero frequency, are offset from zero by the carrier envelope offsetfrequencyTimeE(t)ψ = 0 ψ = ∆ψ ψ = 2∆ψT = 1/frepτFrequencyI(f) fceo = ∆ψ2pi · frep frep1/τfn = nfrep + fceoFigure 2.12: Output of the frequency comb in the time domain (top panel) and frequencydomain (bottom panel). In the time domain the carrier envelope offset phase ψ evolves at aconstant rate due to the difference in group and phase velocity in the dispersive elements ofthe oscillator. From pulse to pulse, the phase difference between the peak of the envelope(dashed line) and the nearest peak of the carrier (solid line) changes by an amount givenby ∆ψ. The output pulses have a width of τ , and the period of the pulses is related theinverse of frep. In the frequency domain, the output of the oscillator is a comb like structure,where the comb elements (solid lines) are separated by frep with a bandwidth related tothe inverse of the pulse width τ . If the comb elements are extrapolated back to near zerofrequency (dashed lines), the offset of the first comb element from zero frequency is thecarrier envelope offset frequency fceo. Therefore, the frequency of the nth comb element isfn = nfrep + fceo.382.3. Photoassociation Laser Systemfceo =(12pi)∆ψceoT=(∆ψceo2pi)frep . (2.7)Therefore, the frequency of nth comb elements is given byfn = nfrep + fceo . (2.8)In order for the OFC to be used to reference and stabilize our Ti:Sapphire lasers, bothfceo and frep must be known and actively stabilized. Both frep and fceo can be stabilizedvia feedback to the power of the pump laser [111, 112]. Alternatively, frep can be stabilizedby adjusting the round trip cavity length either by using. This can be achieved by using afiber stretcher [108] or, if the oscillator has a free space section, mounting one of the fibercollimators on a PZT [107]. However, these methods suffer from the disadvantage that thefeedback for frep and fceo are coupled. We tried to reduce the coupling by using an singlepass fiber AOM (Brimrose AMM-100-8-70-1550-3FP) placed external to the oscillator toprovide fast corrections for fceo. Stabilization of frep was achieved by adjusting the oscillatorlength with a fiber collimator mounted on a PZT. However, we found that long term driftsin fceo required corrections larger than the bandwidth of the AOM (on the order of a fewMHz). This required adding a slow feedback onto the pump current, where the correctionsignal is generated by integrating the output of the first servo loop. In addition, due toimperfections in the mounting of the fiber collimator on the PZT, the alignment of theoscillator is slightly changed when the cavity length is varied, which influences fceo. Whilewe were unable to completely decouple the stabilization of frep and fceo, this setup hasproven to be quite stable and the comb will typically stay locked over the course of a day.Light is coupled out of the oscillator using a 90/10 fiber splitter. This out coupledlight is split again by a second 90/10 splitter such that we have two output ports from theoscillator. We use the low power output to stabilize frep, and the high power output is sentthrough the fiber AOM and then split with a 50/50 fiber splitter. The two outputs of thissplitter are used for the self-referencing branch (to determine and stabilize fceo) and themeasurement branch (to reference and stabilize the Ti:Sapphire lasers). Note that the effectof the AOM is to shift every comb element by the same frequency. Both the referencing andmeasurement branch occur after the AOM, therefore this frequency shift can be absorbedinto fceo. In our case, we use the negative first order from the AOM, which is driven withapproximately 300 mW of RF at 100 MHz, generated by a homebuilt tunable VCO.392.3. Photoassociation Laser SystemRep Rate LockWe stabilize frep by sending the oscillator light from the low power output to a photodiodewhich produces RF signals at integer multiples of frep. The photodiode output is sentthrough a band pass filter centered at 3frep = 375 MHz and then mixed with the output ofstable RF source33 which determines the set point of frep. The RF source itself is referencedto a GPS disciplined quartz oscillator (Menlo Systems GPS 6-12). The mixed error signalis sent to a servo loop which produces the correction signal sent to the PZT inside theoscillator. This lock is summarized in Fig. 2.11. While the stability of frep is determinedby the stability of the RF source, the stability of a comb tooth in the optical frequencyis on the order of 106 larger, due to the multiplicative n-factor in Eq. 2.8. However, thismultiplicative factor can be reduced by a factor of m by locking to the mth harmonic of therep rate. It is for this reason that we choose to lock the third harmonic34.Referencing BranchStabilizing the carrier envelope offset phase, and thus fceo (see Eq. 2.7), is slightly moreinvolved than stabilizing frep. One possible approach is to use another stable optical ref-erence to stabilize the comb element nearest to the optical frequency of the stable source.However, the major difficulty is that this method itself requires a very stable optical ref-erence. Instead, stabilization of fceo is typically achieved via self-referencing in an f − 2finterferometer [113, 114]. In this method, the comb light is broadened to cover an octave(i.e., a factor of two in frequency), and a heterodyne measurement between the direct andfrequency doubled light on the blue end of the octave is used to extract fceo.In our case, the octave spans from 1 µm to 2 µm as the comb light is centered at1550 nm. Considering the octave-spanning light from the comb, let the frequency of thenth comb element at 2 µm be fn = nfrep + fceo. The corresponding comb element at 1 µmwill be given by the 2nth element - that is, f2n = 2nfrep + fceo. By frequency doubling thedirect comb light at 2 µm, one can create a copy of the comb at 1 µm, where the frequencyof the comb elements are given by 2fn = 2nfrep + 2fceo. The difference between the direct33For all the data in this paper, we used an RF synthesizer from HP (model number 8663A). However, itrecently stopped working and has been replaced with another HP synthesizer (model number 8660B) whilewe attempt to repair it.34In addition to reducing the multiplicative factor, locking to a higher harmonic also reduces the 1/f noisepresent in the signal. Ideally, one would lock to the highest harmonic possible. Our choice of the thirdharmonic was limited by the availability of band pass filters in the lab at the time the comb was constructed.402.3. Photoassociation Laser Systemand doubled comb light at 1 µm is then2fn − f2n = (2nfrep + 2fceo)− (2nfrep + fceo) = fceo , (2.9)which gives the carrier envelope offset frequency. With fceo determined, it can then bestabilized to a stable external RF source.To generate the octave spanning comb light (often called a supercontinuum), we usean erbium-doped fiber amplifier (EDFA) to amplify the light, which is then broadenedin a highly non-linear fiber (HNLF)35. After the 50/50 splitter after the AOM at theoutput of the comb we typically have a few mW of average power, which is amplified toapproximately 100 mW when measured after the HNLF. A single mode (SM) “pre-chirp”fiber is spliced before the EDFA, where the length of the fiber is set to make the outputspectrum from the amplifier as broad as possible (typically on the order of 50-100 nm). Thispre-chirp fibers compensates for the pulse chirp out of the oscillator, and compresses thepulse before it enters the EDFA. This high peak power in the EDFA causes a broadening ofthe pulse spectrum due to (in addition to standard chromatic dispersion effects) self-phasemodulation36. Therefore, the broadest spectrum out of the EDFA is achieved when thepre-chirp fiber is set such that the pulse into the EDFA has minimum pulse width. TheEDFA is pumped with light at 980 nm, which is coupled into the erbium doped fiber usinga wavelength division multiplexer (WDM).To generate the supercontinuum, the amplified light from the EDFA is broadened inthe HNLF. Because the broadening in the HNLF is strongly dependent on the intensity ofthe light into the fiber, another segment of SM fiber is spliced between the EDFA and theHNLF in order to further compress and maximize the intensity of the pulse into the HNLF.This method works because the single mode fiber (SMF-28) has anomalous dispersion at1550 nm, while the EDFA has normal dispersion. We also use a fiber polarization controller,placed before the pre-chirp fiber, to optimize and tune the supercontinuum generated bythe HNLF. While ideal SM fibers have cylindrical symmetry, typically the fibers exhibitsome birefringence (for example, due to bending of the fiber). The result is that the orthog-35For a review of supercontinuum generation and the associated non-linear effects, see [115]36Self-phase modulation is the phenomenon where a pulse travelling through a medium interacts withthe medium and imposes a phase modulation on itself (hence the name), which results from an intensitydependent refractive index. The amount of broadening is dependent on the magnitude of time derivative ofthe phase change and the peak intensity, and therefore is the amount of broadening is larger for short pulses.412.3. Photoassociation Laser Systemonal polarizations in the fiber exhibit a different group delay. The polarization controllerhelps compensate for the polarization mode dispersion and is simply tuned empirically tomaximize the spectral broadening out of the HNLF. This layout is summarized in Fig. 2.11.The length of the initial pre-chirp fiber and second compression fiber are on the order of1 m.Frequency doubling the comb light is done using a periodically poled lithium niobatecrystal (PPLN) from Covesion (model number SHG7-0.5-10). The crystal is 0.5 mm thickand 10 mm in length. It contains poling periods of 29.5 µm to 32.5 µm in 0.5µm steps,which allows frequency doubling of light in the range of about 2000 nm to 2250 nm. Coarsephase matching is achieved by cycling through the poling periods by translating the crys-tal perpendicular to the input beam. Fine tuning of the phase matching is achieved viatemperature control. For this, we used a general purpose temperature controller (ThorLabsTC200) which gives a temperature stability of 0.1◦C. The direct and doubled light is sentthrough a narrow interference filter in order to remove frequencies that do not contributeto the heterodyne signal, and then through a focusing lens (mounted with xy translation)and onto an avalanche photodiode for detection. We did not find that we had to make anyspecial effort in order to ensure a spatial overlap between the direct and doubled light, asidefrom making small adjustments to the position of the focusing lens on occasion.In our modifications to the existing comb, we added the AOM and the 50/50 fibersplitter, and spliced the existing self-referencing branch onto one of the splitter outputs.Although this branch already contained the pre-chirp and compression fiber, it was notoptimized for the current pulse shape out of the oscillator, nor for the change in poweras a result of adding the AOM and 50/50 splitter. We choose to optimize the broadenedoutput from the HNLF by making modifications to only the pre-chirp fiber. To do this, wespliced an additional long length (a few meters) of SM fiber into the existing fiber path,and slowly reduced the length of the fiber by cutting out segments a few cms in lengthand re-splicing. The final length of this fiber was set such that we achieved the highestamount of power in both the direct and doubled comb light near 1 µm. Unfortunately,our optical spectrum analyzer was only sensitive up to 1750 nm, so it was not possible todirectly measure the power on both the red and blue end of the supercontinuum at the sametime. Instead, we used a sensitive spectrometer to monitor the direct and doubled light at1 µm. We differentiated between the direct and doubled light by inserting and removing422.3. Photoassociation Laser System1000 1100 1200 1300 1400 1500 1600 1700 1800Wavelength (nm)−60−55−50−45−40−35−30−25Power(dB)Figure 2.13: Comb spectrum our of the oscillator (red dashed line) and after supercontinuumgeneration in the HNLF (black solid line). The supercontinuum extends up to 2 µm, butour optical spectrum analyzer is not sensitive to wavelengths above about 1700 nm. Wegenerate our self referencing signal using direct and doubled comb light at 1017 nm.the PPLN crystal used for frequency doubling. Empirically, we found that we achievedthe largest heterodyne signal using an interference filter centered at 1017 nm. Figure 2.13shows the spectrum of the output of the comb at the output of the oscillator and after thesupercontinuum generation in the HNLF.In reality, the heterodyne measurement between the direct and doubled comb lightdoesn’t just produce one beat note, as suggested by Eq. 2.9. Instead, we observe a forestof beat notes owing to interference between comb elements separated by n other elements,where n is an integer. If one considers the lowest frequency beatnote (fbeat < frep/2)), itsfrequency isfbeat =fceo if fceo < frep/2frep − fceo if fceo > frep/2 . (2.10)432.3. Photoassociation Laser SystemLikewise, the third beat frequency (which is the one we lock and readout) isfbeat =frep + fceo if fceo < frep/22frep − fceo if fceo > frep/2 . (2.11)The ambiguity in the beat note can be solved by increasing the carrier-envelope offsetfrequency and noting the change in the beat note frequency. In our setup, the AOMprovides an ideal means by which to do this: the AOM diffracts into the minus first order,so increasing the VCO frequency acts to increase the RF frequency driving the AOM butdecrease fceo. Therefore, if the AOM driving voltage is increased and the third offset beatmoves right (left) then fceo > frep/2 (fceo < frep/2). With this ambiguity solved, the uniquevalue of fceo isfceo =fbeat + frep if fceo < frep/22frep − fbeat if fceo > frep/2 . (2.12)We choose to stabilize the carrier-envelope offset frequency to 172.1 MHz, such that thebeat note is far from the rep rate signal (which occur at integer multiples of 125 MHz), andall other harmonics of the beat frequency. The output of the avalanche photodiode is sentthrough a band pass filter centered at this frequency, with a 3 dB bandwidth of 20 MHz.This output is compared with a reference RF frequency provided by an Agilent (modelnumber 8648A) synthesizer37 using a phase frequency discriminator (PFD). A correctionsignal is generated by a loop servo which feeds back onto the VCO modifying the RF drivingfrequency sent to the fiber AOM. We also use this correction signal as an error signal for aslow integrator, the output of which feeds back onto the pump current of the oscillator. Thisensures that there is no dc correction sent to the AOM, in order that the driving frequencyof the AOM remains within a window (about 1 MHz centered on 100 MHz) where thediffraction efficiency is high. The locking electronics are summarized in Fig. 2.11.Measurement BranchThe other output from the 50/50 fiber splitter is used for the measurement branch. In thisbranch, we use an identical fiber setup to that used in the referencing branch (described37Note that we use the same synthesizer to reference both Ti:Sapphire lasers, see Section 2.3.2 for moredetails.442.3. Photoassociation Laser Systemabove). We generate a broadened comb spectrum such that it can be frequency doubledto match the scanning range of our Ti:Sapphire lasers. Since our Ti:Sapphire lasers havea tuning range of about 760 nm to 820 nm, we only want to slightly broaden the combspectrum to span from approximately 1520 nm to 1640 nm. To start, we tried to adjust thelength of the pre-chirp fiber to achieve the broadest spectrum out of the EDFA, withoutusing any HNLF afterwards. However, we were only able to achieve reasonable doubledpower (i.e., power in the scanning range of the Ti:Sapphire lasers) in the range 770 nm to800 nm, corresponding to 1540 nm to 1600 nm in the comb spectrum38. We made someattempts to compress the pulse width with additional SM fiber before the HNLF. However,since we didn’t need to achieve a large broadening, carefully minimizing the pulse widthwasn’t crucial to achieving the required spectrum39.The output of the measurement branch is split into two independent locking arms foreach Ti:Sapphire with a polarizing beam cube. The comb light in each arm is frequencydoubled by a PPLN crystal with an input wavelength range of 1530 to 1620 nm (CovesionMSHG1550-0.5-20) or 1600 to 1720 nm (Covesion MSHG1650-0.5-20) depending on thedesired frequency of the Ti:Sapphire laser. The crystals have a length of 20 mm. Notethat the doubling efficiency in the PPLN crystal scales with the square of the intensity andlinearly with length. As such, to achieve a large intensity it is crucial to focus the beaminto the crystal. However, focusing to too small of a waist results in a large divergence suchthat the beam is only intense for a small portion of the crystal length. Maximum doublingefficiency can be achieved when the Rayleigh length of the focused beam is on the orderof half of the crystal length. With this setup, we have been able to achieve greater than0.5 mW of doubled power across the wavelength range of the Ti:Sapphire lasers. The detailsof the Ti:Sapphires and of their lock to the comb are discussed in the next section.2.3.2 Ti:Sapphire LasersAll of our photoassociation measurements use light from two cw Ti:Sapphires ring lasers(899-21) made by Coherent40. These lasers are an ideal light source of photoassociation38So close!39To quote from my lab book: “Got a little frustrated, just spliced the HNLF into the setup to see whatwould happen” followed some time later by “Spectrum actually looks pretty good!”.40In truth, one of the lasers was originally an 899-01, which comes without a tweeter mirror for activestabilization of the output frequency. However, we purchased a tweeter mirror and effectively upgraded thislaser to an 899-21.452.3. Photoassociation Laser Systemexperiments because the output is a high power, well polarized, single longitudinal modethat is widely tunable over a wide range of wavelength. The tunability of the laser isimportant so that it is possible to access multiple vibrational levels in a molecular potential,and also because the exact location of the molecular levels is not always well known andcan require searches spanning a few hundred MHz to a GHz.The wide tunability of the laser comes from the gain medium41 which is pumped witha narrow linewidth 532 nm laser (Coherent V18) and emits in a wavelength range of about670 nm to 1100 nm. The largest limitation on the effective output wavelength range is themirror set in the cavity. Coherent supplies three typical mirror sets which allow a broadbandtuning range of 700-825 nm (“short”), 790-930 nm (“medium”) and 925-1000 nm (“long”).The optics in our lasers are a mix of the short and medium sets, and provide a broadbandtuning range of about 760 nm to 820 nm.Because the ring cavity can support many longitudinal modes at the same time, thelaser contains elements that suppress the gain at all but one longitudinal mode42. Tohave the laser lase on a single longitudinal mode requires that the transmission of all thefrequency selective elements be maximum (or close to maximum) at the same frequency,as demonstrated in Fig. 2.14. These elements include a birefringent filter, and a thin andthick etalon.The birefringent filter is a birefringent plate with thickness l where the index of refractionof the ordinary and extraordinary input polarization components is (in general) different43.In this case, the filter is a uniaxial crystal, where the index of refraction is different alongone axis of the crystal, referred to as the optics axis. The ordinary polarization refers tolight polarized perpendicular to the plane that contains the propagation vector ~k and theoptic axis and extraordinary polarization refers to light polarized in the plane containing ~kand the optics axis. Due to the difference in the index of refraction between the ordinaryand extraordinary axis ∆n, the filter effectively acts like a phase retarder which rotates thepolarization of the input light. This polarization rotation is dependent on the wavelength ofthe light and, in particular, certain wavelengths (where l∆n/λ = m, where m is an integer)41The gain medium is a is a sapphire (Al2O3) crystal doped with Titanium ions (Ti3+).42An excellent resource on the details of these elements, and the experimental realization of single modelasers, see [116].43The type of birefringent filter in the Ti:Sapphires is a “Lyot filter”, made from a stack of three birefringentplates. This has the effect of producing a more narrow transmission peak than just a single plate. However,general operation principle is the same as in the case of single plate.462.3. Photoassociation Laser Systemwill experience no polarization rotation.The birefringent filter is inserted in the cavity at Brewster’s angle, such that the trans-mission through the output facet of the filter is polarization dependent. In this way, particu-lar wavelengths of light will experience the least loss, and the Ti:Sapphire will preferentiallylase at these particular frequencies. Wavelength tuning is accomplished by rotating thefilter about an axis perpendicular to its surface, such that the index of the refraction ofthe extraordinary polarization component is changed. In turn, this influences ∆n and thus,the wavelengths for which the transmission loss is minimum. In the Ti:Sapphires, thetransmission function of the filter is periodic, with a spacing of approximately 225 GHz.Housed within the intercavity assembly (ICA) are the thick and thin etalon. The thinetalon is made from a glass plate where the angle of the plate with respect to the cavitybeam (and thus, the effective thickness of the glass) is controlled with a galvo. Tuningthe thin etalon angle allows the preferred lasing mode (i.e., frequency of the laser) to bechanged in 10 GHz steps. The thick etalon is constructed from two prisms mounted on aPZT which can be used to adjust their separation. Tuning the thick etalon plate separationenables one to select the specific cavity mode on which the laser lases, where the spacing ofthe cavity modes in the Ti:Sapphire lasers is 180 MHz. The transmission of the frequencyselective elements is summarized in Fig. 2.14.Each of the etalons acts like a small optical cavity which has a transmission maximumat frequencies which satisfy [116]νn =nc2nt cos θ, (2.13)which depends on the thickness t of the element (or the separation of the elements in thecase of the thick etalon), the angle θ between the normal to the surface and the cavity beamand the index of refraction n of the element. It is from the transmission function of theetalon elements that the naming of the etalons becomes clear: increasing the thickness (orseparation) of the elements means a closer spacing of the frequency transmission maxima.Therefore, the “thin” etalon has a small thickness, which results in a larger spacing betweenthin etalon modes, while the “thick” etalon has a large thickness and results in a smallerspacing between thick etalon modes.The output mode of the laser (i.e., the laser frequency) can be selected using (in order)the birefringent filter, thin etalon, and thick etalon to move the laser frequency within472.3. Photoassociation Laser SystemFrequencyTransmissionFigure 2.14: Transmission of the frequency selective elements in the Ti:Sapphire lasers.Shown (not to scale) in order of the separation of the transmission maximum are the mirrorset (black line), birefringent filter (blue line), thin etalon (red line) and thick etalon (greenline). The black dashed line represents the net gain of all elements combined, and illustratesthat a single longitudinal mode (thick vertical lines) can be selected out of all the possiblelongitudinal modes (other short thin vertical lines) by proper alignment of all frequencyselective elements.180 MHz of the desired value. Final adjustments can be made by changing the cavitylength (and sometimes making small corrections to the thick etalon if the laser hops to adifferent cavity mode). This final frequency tuning is accomplished by translating one ofthe cavity mirrors mounted on a PZT (called the “tweeter mirror”). Most 899-21 lasers alsohave an additional glass slab mounted on a galvo inside the cavity which can be used tomake larger adjustments to the cavity length. We noticed an unexpected decrease in powerwhen we operated the laser with this element in place, so we removed it from both lasers44.While this element is important to scan over a tuning range greater than the throw of thePZT will allow, almost all of our scans required a range of less than 1 GHz (or could easilybe split up in multiple scans each with a range on the order of 1 GHz or less), which canbe done using only the PZT.Both Ti:Sapphire lasers are coupled to the experiment through an optical fiber. In orderto reduce the effect of polarization drifts in the fiber, we use a Glan-Thompson polarizer44If only all problems were this easy to solve...482.3. Photoassociation Laser Systembefore the fiber to ensure both lasers have the same linear polarization, and are launchedonto one the two orthogonal fiber axes 45. Using the same fiber also assures that the twofields have the same spatial mode at the experiment. The beams are overlapped with theODT beams using a dichroic beam splitter (see Fig. 3.9) and are focused to a waist ofapproximately 50 µm, which is slightly larger than the waist of the ODT beams. Thisensures that all the atoms held in the cross are illuminated by the PA light.Frequency StabilizationThe 899-21 has a control box which can be used to stabilize the laser frequency using anexternal reference cavity (which is also available from Coherent). However, we chose to useour own locking circuitry, in part so that we had greater control over the lock and becauseone of our lasers was an upgraded 899-01 laser, which did not have its own control box 46.The PZTs that control the thick etalon and tweeter mirror are driven by a home built highvoltage PZT driver, which outputs up to 400 V. The galvo on which the thin etalon ismounted is driven by a commercial dc power supply, and we limit the output current to amaximum of ±0.5 A (the galvo is bidirectional and can be driven with current running ineither direction).The error signal used to stabilize and reference the Ti:Sapphire laser frequency to theFFC is generated through a heterodyne beat between the frequency doubled light from themeasurements arm of the comb (see Section 2.3.1) and the cw laser. This heterodyne beatis compared to the output of a stable RF synthesizer using a phase/frequency discriminator(PFD) based on the AD9901 chip, and we feedback a correction signal onto the tweetermirror PZT. The correction signal is derived from a commercial laser servo from VescentPhotonics (D2-125) and amplified by our home built high voltage PZT drivers, as shown inFig. 2.15.Due to the comb like frequency structure of the OFC, the optical heterodyne of thecw laser with the comb will produce a series of RF beat notes at fbeat = ±f (0)beat + mfrep,where m = 0, 1, 2 and f(0)beat is the frequency difference between the laser frequency and thenearest comb element. In addition to the heterodyne beats, the photodetector output will45We use polarization-maintaining fibers, where the fibers are made to have a strong birefringence. There-fore, linear polarization will only be maintained during propagation through the fiber if the light is launchedinto the fiber along one of the two orthogonal fiber axes.46We also starting experiencing some issues with the Coherent lock box which we could not easily debug,and it became easier to work with our own locking system.492.3. Photoassociation Laser SystemCW Ti:SapphireThick Et.PZTfeed forward AOMto exp.BPFPFDServoBPFPFDServoto WMLock InAmp.Slow∫+ 2 kHzratchet AOMfast lock AOMRFSourcePPLNfromcombFigure 2.15: Layout of the beam paths and locking electronics for stabilizing and referencingthe Ti:Sapphire lasers to the FFC. A heterodyne measurement is made between light fromthe Ti:Sapphire and the frequency comb, which is frequency doubled in a PPLN crystal. Theheterodyne beat is referenced to a stable RF source, and a correction signal is fed back ontoa PZT mounted cavity mirror. In order to scan the laser without changing the repetitionrate of the comb, we employ ratchet lock scheme (see text), where the Ti:Sapphire light isfrequency shifted by a double pass AOM (the ratchet AOM) before comparison with thecomb. The fast lock AOM and the feed forward AOM allow us to implement a second fastlock on Ti:Sapphire light, where fast corrections are made to the pre-stabilized light from theTi:Sapphire. We also actively stabilize the thick etalon in the Ti:Sapphire cavity in order toincrease the long term stability and the mode hope free scanning range of the laser (see textfor details). A small portion of the light out of the laser is sent to a commercial wavemeter(WM) to aid with frequency determination. Free space beam paths are represented in red,while fiber segments are magenta and electronic connections are blue.also contain RF frequencies at integer multiples of frep. In order to minimize noise in thelocking signal, it is advantageous to choose fbeat such that it is well separated from theharmonics of frep and all other beatnote signals. The ideal spacing is achieved by settingf(0)beat = frep/447. In our case, we use the third beatnote in the series at 281.25 MHz. Theoutput of the photodetector is immediately sent through a band pass filter centered at thesame frequency, with a 3 dB bandwidth of 34 MHz (KR Electronics 3032). This effectivelyfilters out all other RF signals, and the remaining RF signal is amplified before it is send47This choice of fbeat is also critical in our ratchet lock scheme, which is discussed in the next section.502.3. Photoassociation Laser Systemto the PFD.It is also crucial to apply feedback to the thick etalon in order that the laser remainslocked and mode hop free for long times, or to be able to scan the laser frequency more thana few tens of MHz. Temperature fluctuations, or a change in the laser frequency (i.e., whenthe laser frequency is scanned) can cause the transmission maximum of the thick etalonmode to drift away from being centered on the desired laser frequency (see Fig. 2.14). Toensure that the thick etalon mode follows the frequency changes of the laser, we built aseparate lock closely based on the design used by the commercial Coherent control box.We apply a 2 kHz dither signal to the thick etalon control, and monitor the output powerof the laser on a photodiode. This signal is sent to a lock in amplifier which demodulatesthe signal. The output from the lock in amplifier is related to the slope of the thick etalontransmission function. We feedback on the thick etalon driving voltage with a correctionsignal obtained by integrating the output of the lock-in amplifier. This locking scheme issummarized in Fig. 2.15.With the locking scheme described above, we are able to achieve a relative linewidth of∆ν = 160 kHz between the two lasers. This relative linewidth is measured using a hetero-dyne beatnote between the two lasers where the frequency difference of the Ti:Sapphires isset to be less than 24 GHz, such that is observable on a fast photodiode.We have recently implemented a “fast” lock (similar to the design presented in [117]),where we use an external single pass AOM to make short time scale corrections on the outputof the laser. The error signal for this lock is derived in a similar fashion to that of our PZTlock, but the “fast” AOM is placed in the beam path after the heterodyne beat for the PZTlock (see Fig 2.15). In this way, the “fast” lock is responsible for narrowing the linewidthof a pre-stabilized laser. This is advantageous because the achievable frequency excursionof the single pass AOM (on the order of 1 MHz) limits the size of the corrections that canbe made with this lock. Another approach is to integrate the output of the AOM lock andfeedback on the PZT in order that the PZT handles the dc (or close to dc) correction signal.However, our preliminary testing indicated that this serial method much less successful andless stable than employing two separate locks. The details of this additional lock will bediscussed in the PhD thesis of Gene Polovy. With this additional lock, our preliminaryresults show that we can achieve a relative linewidth on the order of tens of Hz.512.3. Photoassociation Laser SystemFrequency DeterminationThe frequency of the laser can be determined with an accuracy of 60 MHz using a commercialwavemeter (Bristol 621A-NIR). However, our photoassociation experiments require that weare able to determine the frequency of the laser with an uncertainty that is less (ideallymuch less) than the natural linewidth of the transitions (on the order of 10 MHz). For theserequirements, the OFC is an ideal metrology tool [118].Using the FFC as a reference, the frequency of the laser is given by,fCW = fn,780 ± fhet= mfrep + 2fceo ± fhet ,(2.14)where fhet is the frequency of the heterodyne beat between the laser and the comb lightand m is an integer which represents the comb line multiple to which the laser is locked(see Eq. 2.8). The ± sign refers to whether the laser is locked to a comb line that is higher(minus sign) or lower (plus sign) in the frequency of the laser. It is possible to determinethe correct sign by monitoring the beat frequency and slightly increasing or decreasing frep.If the beat signal increases (decreases) when frep is increased (decreased), then the combline is located at a higher (lower) frequency than the Ti:Sapphire light.The value of the integer m can be determined from the wavemeter frequency readout offCW48 and the known information about frep, fceo and the heterodyne beat frequency viam =[fCW ∓ fhet − 2 · fceofrep]. (2.15)Again, the ± symbol refers to whether the frequency of the comb line is higher (plus sign)or lower (minus sign) than the frequency of the cw light. Note that the sign here is oppositeto that in Eq. 2.14.In order to determine the uncertainty on the frequency determination using the comb, wemeasured the 5S1/2 → 5P3/2 atomic transition frequencies of 85Rb using a vapor cell. In amethod similar to that described in [77], we compared the known values of these transitionswith the frequency determined using the method described above. To reduce systematiceffects due to residual magnetic fields, we enclosed the vapor cell and associated optics in48With the requirement that the uncertainty of the frequency readout from the wavemeter be less thanhalf of the repetition rate of the comb.522.3. Photoassociation Laser Systema µ-metal tube. From these measurements, we determined that the absolute frequency ofthe readout was ±600 kHz.Frequency Scanning and Ratchet LockThe most straightforward way to change the frequency of a laser referenced to an OFC isto change the value of frep, which modifies the frequency of the mth comb element to whichthe laser is referenced. This allows for a scanning range of many GHz, typically limitedby the mod hop free tuning range of the laser or the extent to which frep can be variedin the OFC. In our experiment, we can use an external control of the RF synthesizer towhich frep is referenced to change the reference frequency. At the optical frequencies atwhich we typically operate, n ≈ 3× 106 and, since we lock frep using its third harmonic, achange to the reference frequency of 1 Hz corresponds to a change of approximately 1 MHzin the optical frequency of the Ti:Sapphire. However, this method does not allow for theindependent referencing and tuning of two (or more) lasers as all of their frequencies will beaffected by the change of frep. In our case, many of our photoassociation experiments thatinvolve both Ti:Sapphires rely on keeping one laser at a fixed frequency while the other isscanned (for more details on these experiments, see Section 5.2).There are many examples of more universal schemes that overcome this issue. Forexample, instead of scanning frep it is possible to scan fhet. However, this approach iscomplicated by the degeneracy of the beatnote when the laser frequency approaches amidpoint between comb teeth, and when the laser frequency approach one of the combteeth. It is possible to jump over these “dead zones”, but more complicated techniques arerequired (see, for example [119, 120, 121]). An alternative approach is to use an EOM toshift fceo by changing the optical phase of the light during the time between subsequentcomb pulses, which has been demonstrated in [122, 123].The approach that we employ uses an AOM the shift the frequency of the laser beforeit is compared to the OFC. The distinct advantage here is that the frequency of the lasercan be changed without changing fbeat, and thus RF electronics only need to work well overa narrow frequency band. However, the scanning range of an AOM is typically limited to,at most, a few hundred MHz. This limitation is overcome through a ratchet scheme whichclimbs the lock point up or down a ladder of comb teeth, as shown in Fig. 2.16. In general,532.3. Photoassociation Laser SystemFigure 2.16: Illustration of the ratchet scanning method. In (a) the laser frequency fL(vertical thick green line) is shifted down in frequency by 2faom before comparison with thecomb element at fn (vertical thin black line). In (b) the AOM tuning bandwidth spansfrep and near the high end of the range the scan is stopped and faom is returned to itinitial value and the laser re-establishes the lock to the line fn+1. In (c) the AOM tuningbandwidth is only frep/2 and both the AOM frequency and the polarity of the error signalmust be changed in order for the lock to be re-established. In this case, the lock alternatesbetween being referenced to a comb tooth that is high in frequency, and one that is lower infrequency than the Ti:Sapphire. In our case, we employ the method in (c) as the scanningrange of our double pass AOM is only frep/2. See text for details. Figure and caption(modified) from [78].with this scheme, the locked lasers frequency isfL = nfrep + fceo ± fhet + 2faom , (2.16)where we set up the AOM in a double pass configuration such that the laser frequency isshifted down before the heterodyne comparison (i.e., we use the negative first order). Assuch, a change in faom results in a change of fL because the lock electronics adjust fL inorder to keep fbeat constant. In our setup, we lock using the third beat note in the seriessuch that fbeat = 2frep +frep/4, and the comb tooth to which we lock is higher in frequencythan the laser frequency. We use negative servo polarity such that if the laser frequencyincreases, the beat frequency will decrease and the servo responds by decreasing the laserfrequency. At this starting point, faom = 75 MHz and the laser frequency isfL = fn+2 + 2faom − fbeat= (n+ 2)frep + fceo + 2faom − fbeat= nfrep + fceo + 2faom − frep/4 ,(2.17)542.3. Photoassociation Laser Systemwhere fn is the frequency of the comb tooth nearest to the laser frequency and, correspond-ingly, fn+2 is the frequency of the comb tooth to which we lock. At this point, we can scanthe laser frequency by changing the driving frequency of the AOM until we have reached∆faom = frep/2 = 62.5 MHz which corresponds to faom = 106.5 MHz. At this point, weswitch the polarity of the error signal49 and, at the same time, the AOM driving frequencyis returned to faom = 75 MHz. Now, the frequency of the laser light isfL = fn−2 + 2faom + fbeat= (n− 2)frep + fceo + 2faom + fbeat= nfrep + fceo + 2faom + frep/4 ,(2.18)where the laser is now locked the a comb tooth that is at a lower frequency than the laserfrequency. Although the servo polarity is still negative, because the error signal has beeninverted, if the laser frequency increases and thus the beat frequency increases, the servo willstill respond by decreasing the laser frequency. The ratchet can be continued in a similarfashion such that when the error signal polarity is inverted again (such that it returns to theinitial setting), the laser will now be locked to the fn+3 comb tooth, and the laser frequencywill be higher than the starting value by frep.By repeating this process, fL can be moved an arbitrary amount while maintaining thelock to the comb. This scheme also allows a second laser to also be referenced to the comband held at a constant frequency. The key to this method is exploiting the finite responsetime of the PID controller. This allows us to change the comb element to which the laseris locked without disengaging the phase-locked loop and without affecting fL , due to thefact that faom is also changed at the same time.An illustration of this technique is shown in Fig. 2.17. Here, we show the measuredheterodyne beat note between the fixed frequency laser and the laser tuned using thisratchet method, both phase locked to the OFC comb, as a function of the driving frequencyof the double pass AOM. Although the frequency difference for this measurement was10 GHz in order that the heterodyne beat was visible on a fast photodiode, we have alsoused this method to bridge a 58 GHz gap (our spectroscopy of the X(11Σ+g ) state) and49We switch the polarity of the error signal by inverting the output of the PFD. We do this instead ofinverting the correction signal so that in the event there is any integrated correction signal the drivingvoltage of the tweeter mirror PZT does not undergo a discrete jump in position.552.3. Photoassociation Laser System75 80 85 90 95 100 105AOM Frequency (MHz)9.909.9510.0010.05HeterodyneBeatnote(GHz)Figure 2.17: Measured heterodyne beat note (black dots) between the fixed frequency laserand the variable-frequency laser, both phase locked to the OFC. The beat note is shown as afunction of the driving frequency of the double-ass AOM. The solid (dashed) lines representthe expected heterodyne beat note frequency with a negative (positive) servo polarity. Thehorizontal dashed-dot lines represent the jump of the AOM driving frequency, where thedifference in the doubled-passed AOM shift is frep/2 and the resulting fbeat is unchangedexcept for the comb element responsible for the beat not changes. See text for details.Figure and caption (modified) from [78].an 8 THz gap (our spectroscopy of the a(13Σ+u ) state). In our setup, discrete jumps arelimited to be at most 2∆faom = 20 MHz by the bandwidth of the band pass filters. Forthese jumps, we find the lock is re-established within 6 ms. For continuous sweeps whichinclude the discontinuous lock point jump we can achieve rates up to 200 MHz/s and forsmall continuous sweeps using only the double pass AOM (i.e., without the need to switchthe polarity of the error signal) we can achieve sweep rates of > 3 GHz/s.The major limitation of this method is that the phase coherence between the two lasersis not maintained during the frequency scan at the point when the hand-over between modesof the comb is performed. In theory, this hand over can be done is a fully phase coherentway by executing the frequency change in a time that is short compared to the 1/frep andtimed with the bright interval of the pulse train from the OFC, for which all the comb modeshave the same optical phase. However, if this phase coherence is required, it is likely thatthe EOM based methods [123] would be preferable because the update and settling time of562.3. Photoassociation Laser Systeman EOM is typically much shorter than that of an AOM. However, for all of our work, weonly require phase coherence when the lasers are kept at a fixed frequency difference andtherefore this ratchet scheme is applicable and was much more easily implemented.57Chapter 3Preparation and Detection of ColdAtomsA day in the life of a cold atom experiment may appear to be rather repetitive. As mostof the common techniques to extract information from the system are destructive, almostany measurement (whether the goal be optimization, calibration, or scientific in nature)requires many repeated runs. In each repeated run, an identical (or nearly identical, givenshot to shot fluctuations that can exist in the system) sample is created, and the effect ofa particular setting of one parameter is observed. For each subsequent run, the value ofthe parameter is changed such that its effect on the ensemble over a given range can beobserved.In our experiment, each individual run follows the same basic recipe: hot atoms leave aneffusive oven and are slowed into a cold atomic beam in the Zeeman slower. This cold atombeam acts as the atom source for the MOT, which is the standard starting point for almostall ultra-cold atom experiments. After the MOT is loaded, the atoms are transferred to theODT. In the ODT we perform an additional forced evaporation step in order to further coolthe atoms. At this point, we can also prepare the atoms in particular spin states, dependingon the requirements of the experiments. At the conclusion of this preparation stage, weperform the planned experiment on the atoms. For example, in a search for Feshbachresonances we may apply a homogenous magnetic field. Or, in the case of photoassociationor dark state measurements, we illuminate the atoms with the photoassociation light. Afterthis “science” step, we take an image of the atoms, which gives us a method of quantifyingthe number of atoms that remain and, in some cases, the spatial profile of the ensemble.This entire procedure produces a single data point at a single parameter, and is repeated asmany times as necessary to map out the behavior with respect to some changing parameter.In general, the entire procedure takes between 5 and 10 s to produce a single data point.583.1. MOT and Zeeman SlowerThis section covers the details of these steps for both 6Li and Rb. Section 3.1 discussesthe Zeeman slower, which is used as the cold atom source for our MOT. Section 3.2introduces ODTs, and provides the details surrounding the transfer of 6Li and Rb intothe ODT, and the subsequent evaporation. Section 3.3 provides the details of our opticalpumping scheme for Rb, and discusses how we prepare our 6Li ensemble in a particular spinstate (or states). Finally, Sec. 3.4 describes how we image the atoms through absorptionor fluorescence imaging, and the techniques we use to perform state selective imaging witheither species.3.1 MOT and Zeeman SlowerAt its inception, our ultra-cold 6Li and Rb machine was built to be extremely simple andcompact. The atomic source for the 6Li MOT was an effusive oven placed 10 cm from thecenter of the MOT, and the MOT captured the low velocity tail of the atomic velocitydistribution from the oven. The Rb MOT was loaded from an atomic vapor provided bycommercial 85Rb vapor dispensers. The details of this setup can be found in [93, 98, 124].Although this apparatus proved extremely successful and robust for single species opera-tions, the background 85Rb pressure required to load a sufficiently large 85Rb MOT was highenough that it severely limited the size of the 6Li MOT. A more in depth discussion of thisissue can be found in Chapter 7, which details our experiments with the 6Li +85Rb mixture.In an effort to improve the pressure in the science chamber (i.e., our glass vacuum cell)for dual species experiments, we rebuilt our apparatus to include a Zeeman slower thatprovided a cold atomic beam of 6Li or Rb to act as the atom source for our MOT. Thissection briefly outlines the operating principle of Zeeman slowers, and discusses the specificsthat are unique or important to our implementation. Full details on the construction ofthe Zeeman slower and the effusive sources can be found in the Master thesis of WilliamBowden [99] and in [125].Zeeman Slower TheoryA Zeeman slower relies on the photon scattering force to decelerate atoms along a longitu-dinal axis to create a cold (i.e., slow) atomic beam with a high flux. The photon scattering593.1. MOT and Zeeman SlowerforceFsc =~ωΓ2c· s1 + s+(2δΓ)2 , (3.1)is a function of the frequency of the slowing light (which depends on the atomic speciesbeing slowed), the saturation parameter s = I/Isat, and the detuning of the slowing beamfrom the atomic resonanceδ = δL +ωvc− µB~. (3.2)This detuning has contributions from three different sources. The first term in Eq. 3.2accounts for the detuning of the slowing beam from the atomic resonance. The second termaccounts for the Doppler shift of the slowing light due to the velocity of the atoms. In thiscase, it is assumed that v is positive, and the Doppler shift blue-shifts the slowing lightbecause the atom’s velocity is opposite to the propagation direction of the slowing beam.Finally, the third term represents the shift in the atomic transition frequency due to theZeeman shift in a field of strength B, where µ is the relative magnetic moment betweenthe ground and excited state used as the slowing transition. If µ is positive (negative),the transition energy increases (decreases) with increasing field, which results in a red shift(blue shift) of the slowing light.In order for an atom to be effectively slowed over the entire length of the slower, the netdetuning of the slowing beam must be close to zero along the length of the slower. If thedetuning of the slowing beam is fixed (i.e., δL is constant), then the magnetic field profilemust change in order to compensate for reduced blue shift from the Doppler effect as theatoms slow.The required field profile can be found by considering that the velocity of an atom withan initial velocity vi, which undergoes a constant deceleration a, at a distance z along theslower (where z = 0 is defined to be the start of the slower) isvf =√v2i − 2az . (3.3)If this velocity is used in Eq. 3.2, along with the assumption that the net detuning of theslowing beam is zero, the required magnetic field profile is found to beB(z) =~ωvicµ√1− 2azv2i− ~δLµ. (3.4)603.1. MOT and Zeeman SlowerFrom the term under the square root in Eq. 3.4, it can be seen that the length of the sloweris limited to z0 ≤ v2i /(2a). Physically, this is the length where the atoms final velocity iszero, and the magnetic field compensates only for the detuning the slowing light. In a realslower, if this length is reached before the atoms are captured in a MOT, the slowing lightwill accelerate the atoms in the opposite direction (i.e., the atoms will turn around).The deceleration of the atoms is limited by the finite scattering rate (Eq. 3.1) and takeson a maximum value of amax in the limit where s 1 and δ = 0. The acceleration that theatoms undergo is typically some fraction of amax, such that a = ηamax and 0 < η < 1. Thisacceleration is often chosen to ensure that the field gradient satisfies the adiabatic slowingcondition [99] ∣∣∣∣dB(z)dz∣∣∣∣ ~ωamaxcµv(z) , (3.5)throughout the entire length of the slower50.This poses a problem for simultaneously slowing two (or more) species in a fixed lengthslower. That is, although the scattering force may be similar for different species, themaximum acceleration may not be (because the acceleration is inversely proportional tothe mass of the atom). This means the required field gradient may vary substantially forthe different species. In fact, it can be shown that for efficient simultaneous slowing of twospecies, the ratio [99]η1η2=m1µ1ω2Γ2m2µ2ω1Γ1(3.6)must be on the order of one. For 6Li and 85Rb , this ratio is equal to 0.04, owing largely tothe large mass difference between the two species. This limitation has been overcome by thegroup of Stamper-Kurn [126] using a three stage design which creates a different effectivelength of the slower for 6Li and Rb. However, even if efficient slowing of both 6Li and Rbcan be achieved, the optimal parameter space for the 6Li and Rb MOTs do not overlap,and running a dual species 6Li +Rb MOT can diminish the MOT size for both species. Forthese reasons, we use a sequential loading technique, which requires the Zeeman slower toonly slow a single species at a time51.50Note that this statement isn’t entirely true. In his thesis, William Bowden writes: “The maximumgradient is inversely proportional to velocity, which makes the adiabatic condition more stringent for fastermoving atoms. Therefore, the strategy of uniformly stretching the field by changing η is not ideal, rather itis better to vary η depending on velocity at that particular location in the slower” [99].51More details about the full sequential loading technique, including the transfer to the ODT can be foundin Chapter 7.613.1. MOT and Zeeman SlowerIt is important to note that a Zeeman slower generally only provides slowing along theaxis of the slowing beam. When the transverse velocity spread of the atomic beam becomessignificant compared to the longitudinal velocity of the beam, beam blooming can occur. Ifblooming begins to occur far from the MOT region, the atomic beam can diverge enoughthat a large fraction of the atoms in the beam do not reach the MOT, which limits thecapturable flux. Therefore, it is important to minimize the distance between the end of theslower and the MOT52.Details of Our DesignOur design is similar to that of the group of Hackermu¨ller [128], where the magnetic fieldfor the Zeeman slower is generated with a segmented coil design. The current in each ofthe separate segments of coil is computer controlled, which allows us to easily optimize therequired current in each coil and to switch between the optimal coil gradient for each species.Additionally, we partially mitigate the problem of beam blooming by disengaging the atomsfrom the Zeeman slower with a large field gradient created by the final coil segment, whichwe refer to as the “disengagement coil”. This forces the atoms to exit the Zeeman slowermoving at a velocity where blooming is not a major issue. The final slowing occurs inthe magnetic field produced by the MOT coils, such that the atoms are captured in theMOT before the effects of blooming have a significant impact on the flux. The approximatemagnetic field produced by the Zeeman slower coils and the MOT coils used when slowing6Li and 85Rb are shown in Fig 3.1.As the magnetic field decreases along the length of the slower, we require that therelative magnetic moment of the slowing transition be positive such that the Zeeman shiftof the slowing transition decreases as the atoms propagate through the slower. This energyshift matches the decreasing blue shift due to the Doppler shift of the atoms as they areslowed. The consequence of this choice is twofold: first, we must cool on a σ+ transition,such that the excited state energy decreases faster than the ground state energy in thedecreasing magnetic field. Second, since both the Zeeman shift of the slowing transitionand the Doppler shift are small at the end of the slower, the detuning of the slowing beammust also be small, and is typically on the same order of magnitude as the detunings of the52Alternatively, it has also been shown that an additional beam can be used to provide transverse coolinginside the slower [127]. However, this is not used in our design.623.1. MOT and Zeeman Slower0.0−0.2−0.4−0.6Distance (m)−50050100150BField(G) Rb−2500250500750BField(G) LiFigure 3.1: The magnetic field profile produced by the Zeeman slower coils and the MOTcoils used for slowing 6Li (top panel) and 85Rb (bottom panel). The distance is measuredrelative to the position of the MOT. The region between the two vertical dashed linesindicates the area between the end of the Zeeman slower and the region where the finalslowing stage occurs. This final slowing stage uses the magnetic field produced by theMOT magnetic field coils.MOT beams53.In our slower, we slow on the D2 transition between stretched states in the ground and ex-cited hyperfine manifolds. At zero magnetic field, these states connect to the |f = 3,mf = 3〉and |f ′ = 4,m′f = 4〉 states for 85Rb and |f = 3/2,mf = 3/2〉 and |m′f = 5/2,m′f = 5/2〉 statesfor 6Li. In 6Li, the magnetic field in the majority of the slower is large enough that the atomsare in the hyperfine Pachen-Back regime, and the slowing transition is correctly labeled bythe projection of the nuclear spin mi and the total orbital angular momentum mj on themagnetic field axis. In this regime, the states used for slowing are the |mj = 1/2,mi = 1〉and |m′j = 3/2,m′i = 1〉 states. The energies of the relevant states and the slowing transitionare shown in Fig. 3.2 for 6Li and Fig. 3.3 for 85Rb.We performed a full optimization of the loading rate for both 6Li and 85Rb. We startedby setting the slowing beam detuning and magnetic field gradients based on a “virtual53The choice of the actual value of the detuning depends on a combination of the slower length, the capturevelocity of the MOT, and the highest initial atom velocity that you want to slow. This is discussed in moredetail in the Master’s thesis of William Bowden [99].633.1. MOT and Zeeman Slower0 1 2 3 4−10010Energy(MHz) 2P3/20 50 100 150 200−400−2000200400Energy(MHz) 2S1/20.0 0.2 0.4 0.6 0.8 1.Magnetic Field (G) 3.2: 6Li hyperfine structure in an external magnetic field. Shown with thick redlines are the states that we use as the slowing transition in our Zeeman slower. At zeromagnetic field, these states correlate to the |f = 3/2,mf = 3/2〉 and |m′f = 5/2,m′f = 5/2〉states. At high field (greater than 100 G) they correlate to the |mj = 1/2,mi = 1〉 and|m′j = 3/2,m′i = 1〉 states. Note the magnetic field range for the ground and excited state ischosen such that the energy splitting between states that differ in mi is visible. The givenenergy is relative to the hyperfine center of gravity.slower” simulation [99]. We then scanned the current of each coil about the calculatedvalue to find the current through each coil that gave the maximum loading rate. Werepeated this procedure for different choices of η and MOT parameters until we foundan overall maximum in the loading rate. Because many of the parameters are stronglycoupled, the optimal settings we found may only correspond to a local maximum. However,the loading rates were comparable to that reported in prior work on multi-species Zeemanslowers operating at similar oven temperatures and are more than sufficient to meet ourrequirements. The final loading parameters for the 6Li and 85Rb MOTs are given in Tab. 3.1,and the currents that we used in each coil segment are given in Tab. 3.2.Once we had determined the optimal settings, we measured the loading rate of both643.1. MOT and Zeeman Slower0 10 20 30 40 50 60 70 80−2000200400Energy(MHz) 2P3/2f = 1f = 40 10 20 30 40 50 60 70 8011501200125013001350Energy(MHz)2S1/2f = 30.0 0.2 .4 0.6 0.8 1.Magnetic Field (G) 3.3: 85Rb hyperfine structure in an external magnetic field. Shown with thick redlines are the states that we use as the slowing transition on our Zeeman slower. At zeromagnetic field, these states correlate to the |f = 3,mf = 3〉 and |f ′ = 4,m′f = 4〉 states.Not shown is the f = 2 manifold in the 2S1/2 level, which is approximately 3 GHz lower inenergy than the f = 3 manifold at zero field. The given energy is relative to the hyperfinecenter of gravity.species as a function of the slowing beam power. The result of this measurement is shownin Fig. 3.4. Although on the surface it may seem that increasing the slowing beam powerwould be beneficial, there are two important considerations.First, in the case of a σ+ slower, the slowing beam is not far detuned from the atomicresonance, and therefore imparts a non-trivial radiation pressure force on the MOT. Atlarge beam intensities, this force can push the MOT outside of the region where the MOTbeams overlap. In our setup, we attempt to compensate for this by focusing the slowingbeam through the slower, so that the beam intensity at the MOT is much less than thebeam intensity inside the slower. Additionally, we use the compensation coils to move thezero of the quadrupole field in order to compensate for the radiation pressure force, and653.1. MOT and Zeeman SlowerRb LiSlowing Beam Pump Detuning (MHz) -85 -76Slowing Beam Pump Power (mW) 15 60Slowing Beam Repump Detuning (MHz) 0 –Slowing Beam Repump Power (mW) 12 –MOT Pump Beam Detuning (MHz) -15 -45MOT Pump Beam Power (mW) 40 30MOT Repump Beam Detuning (MHz) 0 -40MOT Repump Beam Power (mW) 10 40MOT Axial Gradient (G/cm) 15.4 67.4Table 3.1: Optimal loading parameters for the 6Li and 85Rb MOT. The pump detuning for85Rb is with respect to the f = 3 → f ′ = 4 D2 transition while for 6Li it is with respectto the f = 3/2→ f ′ = 5/2 transition at zero magnetic field. The repump detuning for 85Rbis with respect to the f = 2 → f ′ = 3 D2 transition while for 6Li it is with respect to thef = 1/2 → f ′ = 5/2 transition at zero magnetic field. The MOT beams have a radius of9 mm, and the power is split between three retroreflected arms of the MOT.shift the MOT so that it is well centered on the three orthogonal MOT beams. However,as Fig. 3.4 shows, the loading rate of the Rb MOT is severely diminished at large slowingpowers. We believe this is, in part, due to this effect.Second, too large of an intensity can actually result in a decrease in the loading rate ifthe atoms are stopped (or turned around) prior to reaching the MOT. We believe this effectis also partially responsible for the rollover in the loading rate of 85Rb shown in Fig. 3.4. Webelieve that we see this rollover for 85Rb and not for 6Li because 85Rb exits the slower ata lower velocity and encounters a smaller magnetic field. This means that at high slowingbeam intensities, the slowing transitions can be power broadened such that an atom whichshould be moving slowly enough to disengage from the slowing field will actually continueto be slowed, and eventually stopped or turned around.Empirically, we also found that the 85Rb loading rate was greatly improved when weadded a repump beam that was collinear with the slowing beam. The loading rate as afunction of the repump power along the slowing beam axis is shown in Fig. 3.5. We alsotried to send the repump light down an axis transverse to the slowing beam near the outputof the effusive source. With this configuration, we did not notice any improvement to theloading rate. This suggests that the increase in the loading rate was due to the repumpbeam pumping atoms back into the slowing transition within the slower, and not simplyoptically pumping the atoms into the slowing transition before they entered the slowing663.1. MOT and Zeeman SlowerCoil Num. ILi (A) IRb (A)1 6.7 0.812 4.2 0.613 3.8 0.504 3.2 0.435 3.0 0.366 2.3 0.287 2.0 0.298 3.2 0.00Table 3.2: The current used in each coil segment for the 6Li and 85Rb Zeeman slower. Coilone is the coil that is closest to the oven (i.e., at the start of the slower). The eighth coil isthe “disengagement coil” (see text). The field profile produced by these currents is shownin Fig. 3.1.0 10 20 30 40 50 60Slowing Beam Pump Power (mW) 3.4: The effect of slowing beam power on the MOT loading rate for 6Li (red squares)and 85Rb (black dots). We attribute the rollover of the Rb loading rate at high slowingbeam powers to a combination of two effects: first, the atoms being stopped or turnedaround prior to reaching the MOT and second, the radiation pressure of the slowing beampushing the MOT outside of an optimal region.field.We suspect this is the case because the polarization of our slowing light is not perfect, andthe slowing beam likely also drives pi and σ− transitions. Therefore, off resonance transitionslikely occur from the stretched ground state to the |f ′ = 4,m′f = 3〉 or |f ′ = 4,m′f = 2〉673.1. MOT and Zeeman Slower0 5 10 15 20Slowing Beam Repump Power (mW) 3.5: The effect of adding hyperfine repumping light to the slowing beam on theloading rate of the 85Rb MOT. These data were taken using the parameters listed inTable 3.1, and the rate is normalized to the peak loading rate. Figure from [125].states or to the |f ′ = 3,m′f = 3〉 or |f ′ = 3,m′f = 2〉 states. Although the transition rateto the |f ′ = 3〉 states is suppressed due to the large detuning (on the order of hundreds ofMHz), decay to the |f = 2〉 manifold is possible. In this case, repump light is required torepump the atoms back into the |f = 3〉 ground state manifold.However, in the 6Li slower, we did not see any improvement in the loading rate withthe inclusion of a repump beam. We believe that this is the case for two reasons. First,at the large magnetic fields in the 6Li slower, the system is well within the Paschen-Backregime. In this regime, the projection of the nuclear spin must be conserved during anelectric dipole transition. Therefore, transitions to any of the other excited states withm′j = 3/2 are disallowed because each other state differs in the projection of the nuclearspin. Second, even if the polarization of the light is not perfectly circularly polarized, theonly other possible transition that the slowing light can excite is to the |m′j = 1/2,m′i = 1〉state. However, this transition is detuned from the slowing beam by over a GHz for themajority of the slower, which greatly suppresses the transition rate to this state. In addition,while the detuning of this transition is much smaller in the final slowing stage which takesplace in the magnetic field provided by the MOT, repump light is already present due to683.1. MOT and Zeeman Slowerthe MOT beams.Characterization of MOT and Zeeman SlowerTo characterize the Zeeman slower, MOT, and the pressure in the science chamber, wemeasured the loading rate, MOT lifetime, and steady state atom number for both speciesas a function of the temperature of the effusive source54. In addition, we measured thelifetime of 6Li atoms in our low power IPG ODT (see Section 3.2 for more information). Inorder to estimate the quality of the vacuum, it is better to rely on the ODT lifetime, as itprovides a more true measure of the loss rate due to collisions with background gases. Inthe MOT, the lifetime is also impacted by light assisted collisions and these effects becomemore pronounced for large MOTs where the density is high. We used the 6Li ODT forboth the 85Rb and 6Li measurement in order that that the measured lifetimes were directlycomparable.Figure 3.6 shows these measurements for the 6Li MOT as a function of the temperature ofthe 6Li effusive oven. In this case, the 85Rb MOT was “off” (i.e., held at room temperature).For low source temperatures, the MOT lifetime is much larger than the ODT lifetime dueto fact that the MOT trap depth is more than three orders of magnitude larger than theODT depth [129]. As the oven temperature is increased, the MOT lifetime decreases due toincreasing MOT density, and the ODT lifetime begins to decrease due to increased collisionswith background gases emitted by the oven. We choose a standard operational temperatureof about 400 ◦C in order strike a balance between a reasonable loading rate and the ODTlifetime55.Figure 3.7 shows these measurement for the 85Rb MOT as a function of the temperatureof the 85Rb effusive source. In this case, the 6Li effusive source was held at 400 ◦C. In theRb MOT, high densities are achieved at low oven temperatures, and we find that MOTlifetime is always less than the ODT lifetime. We typically operate the 85Rb effusive sourceat a temperature of about 80-100 ◦C, again chosen to balance the 85Rb loading rate andthe ODT lifetime.The Zeeman slower and the MOT represent the first step in an ultra-cold atom exper-54Note that the MOT lifetime presented here is the characteristic loading time of the MOT, found froma fit of the atom number as a function of loading to N(t) = N0(1− exp(−t/τ)), and not the decay time ofthe MOT.55The limit on the lifetime of the ODT is that the lifetime needs to be longer than the timescale for ourforced evaporation ramps, which are on the order of 2-4 s.693.1. MOT and Zeeman Slower360 380 400 420 440 460Li Oven Temperature (oC)024681012LiLoadingRate(107atoms/s)LiSteadyStateNumber(108atoms)05101520253035LiODTLifetime(s)LiMOTLifetime(s)Figure 3.6: The effect of the 6Li source temperature on the 6Li loading rate (black dots),steady state atom number (blue triangles), MOT lifetime (red diamonds), and the Li ODTlifetime (green squares). For this measurement, the Rb source was “off’ (i.e., held at roomtemperature). Figure from [125].40 60 80 100 120 140Li Oven Temperature (oC)02468101214RbLoadingRate(108atoms/s)RbSteadyStateNumber(108atoms)0510152025303540LiODTLifetime(s)RbMOTLifetime(s)Figure 3.7: The effect of the 85Rb source temperature on the 85Rb loading rate (black dots),steady state atom number (blue triangles), MOT lifetime (red diamonds), and the Li ODTlifetime (green squares). For this measurement, the 6Li source was held at 400 ◦C. Figurefrom [125].703.2. Optical Dipole Trapsiment. Typically, the MOT temperature is on the order of a few hundred µK, which isapproximately 3 orders of magnitude higher than the temperatures required for degeneracy,and at which we perform the majority of experiments. In order to further cool the atoms,we transfer the atoms from the MOT to an ODT. The details of these traps and theirimplementation in our apparatus are discussed in the following section.3.2 Optical Dipole TrapsOptical-dipole traps (ODT) are an ideal tool for ultracold experiments, as they have theability to trap any polarizable particle, atom, or molecule. Straightforward experimentalcontrol of the trapping potential is provided via the intensity of the laser, which easilyallows for forced evaporative cooling. Additionally, because the trapping potential relieson off-resonant light, atomic samples trapped in an ODT can be prepared in a particularspin configuration and external electric and/or magnetic fields can be easily applied. Anexcellent review of dipole traps can be found in [130], while the Quantum Optics notes byD. Steck [131] provide a detailed discussion of the complex polarizability, dipole force andphoton scattering rate.The dipole potential in an ODT results from the interaction of the induced dipolemoment (induced by the trapping field) interacting with the trapping field to produce adipole potential given byUdip =−~d · ~E2, (3.7)where the factor of 1/2 comes from the fact that the dipole is induced. The induced dipolemoment is defined as~d(+) = −e~x(+) = α(ω) ~E(+) (3.8)where α(ω) is the frequency dependent polarizability56, and gives a measure of how easilythe incident light induces a dipole moment.In addition to the dipole force (which is responsible for the trapping potential) theinteraction of the trapping light and the atoms results in a radiation pressure force. Thisforce is due to photon scattering, which can lead to a heating of atoms held within the trapand can result in trap loss. The dipole potential and the photon scattering rate are related56In general, this polarizability is complex.713.2. Optical Dipole Trapsto the real and complex part of the polarizability, respectivelyUdip = − 120cRe[α(ω)]I(~r) (3.9)Γsc =1~0cIm[α(ω)]I(~r) (3.10)where the intensity is related to the electric field amplitude in the usual wayI(~r) = 20c| ~E(+)|2 . (3.11)Note that the force resulting from the dipole potential is proportional to the gradient of theintensity, and acts to trap atoms in a potential well.During each photon scattering event in the dipole trap, the atom will gain, at most,4Erecoil = 4(~2ω2)/(2mc2) of energy57. Therefore, one can define the heating rate due tophoton scattering asE˙heat = Γsc × 4Erecoil = Γsc~2ω2mc2. (3.12)In most cases, the dipole potential and scattering rate can be well approximated usinga two level model, where the energy levels are separated by ∆E = ~ω0, and the detuning ofthe dipole trap light (with frequency ω) from the atomic resonance is ∆ = ω − ω0. In thecase of large detuning, where ∆ is much larger than the natural linewidth of the two leveltransition γ, the dipole potential from Eq. 3.9 and photon scattering rate from Eq. 3.10 canbe expressed asUdip = −3pic2γ2ω30(1ω0 − ω +1ω0 + ω)I(~r) (3.13)Γsc =3pic2γ22~ω30(ωω0)3( 1ω0 − ω +1ω0 + ω)2I(~r)=(ωω0)3( γω0 − ω +γω0 + ω)Udip~.(3.14)In the general case, where the dipole trap light couples more than two levels (for example,couplings to bound molecular levels in the excited states or higher lying excited atomic57The actual gain in energy will depend on the direction of the emitted photon. However, to estimate theheating rate, we will assume that the atom gains the maximum kinetic energy possible.723.2. Optical Dipole Trapsstates), then Eqs. 3.13 and 3.14 must be summed over all applicable levels.The scaling of the dipole potential and the photon scattering rate with intensity anddetuning become more apparent in the limiting case where the rotating wave approximationcan be applied. This approximation holds when the frequency of the dipole trap light is farenough from resonance such that |∆|  γ, but is close enough to ω0 such thatωω0≈ 1 and ω0 − ω  ω0 + ω . (3.15)In this regime, the expression for the dipole potential and the scattering rate becomesUdip =3pic2γ2ω30· I(~r)∆(3.16)Γsc =3pic2γ22~ω30· I(~r)∆2=( γ∆)(Udip~), (3.17)and it can be seen that the dipole potential scales like I/∆, while the scattering rate (whichis responsible for heating within the trap) scales like I/∆2. Therefore, in order to create thelargest trapping potential while maintaining a long lifetime (i.e., a small scattering rate),one should operate with large detunings and high intensity. In most cases, this can beachieved using infrared fiber lasers where the wavelength in the range of 1100 nm, or withCO2 lasers where the wavelength is 10.6 µm.It should be noted that the rotating wave approximation made in Eqs. 3.16 and 3.17 arenot well justified in our case, given that the wavelengths of the ODTs used in this apparatusare 1090 nm and 1064 nm, while the transitions wavelengths of 6Li and Rb are 671 nm and780 nm, respectively. Specifically, Eq. 3.16 underestimates the trap depth by approximately20%, while Eq. 3.17 overestimates the scattering rate by about a factor of two. Therefore,the full form given in Eqs. 3.13 and 3.14 in the absence of the rotating wave approximationmust be used for any quantitative estimations of trap depth and scattering rate.Optical dipole traps are formed (typically, and in our case) by focusing a Gaussian beamwith a total power P to a minimum beam radius (i.e., beam waist) of w0. The intensity ofthis focused beam, as a function of the radial and axial coordinate, isI(r, z) =2Ppiw2(z)exp(− 2r2w2(z)), (3.18)733.2. Optical Dipole Trapswhere the beam radius as a function of longitudinal position isw(z) = w0√1 +(zz0)2(3.19)and z0 is the Rayleigh range, given byz0 =w20piλ. (3.20)With this, the trap depth U0 is defined as the dipole potential (see Eq. 3.13) wherer = z = 0 and the intensity of the dipole trap light is I = 2P/piw20. Likewise, the photonscattering rate can be defined in a similar fashion using Eq. 3.14.When the atomic ensemble is at a temperature kBT  U0, it can be assumed thatthe atoms are concentrated near the focus of the beam, and the ODT potential can beapproximated as harmonic. Taking zz0  1 and rw  1, the trapping potential becomesU(r, z) ≈ U0[1− 2 2rw0− zz0]= −U0 + 12mω2rr2 +12mz2ω2z , (3.21)where the right hand side is in the form of a standard harmonic potential. Comparing theleft hand and right hand side of Eq. 3.21 gives the trapping frequencies for the radial andaxial directions to beωr =√4U0mw20and ωa =√2U0mz20. (3.22)One downside to a standard optical dipole trap produced from a single focused Gaussianbeam is that the trapping potential is much weaker along the longitudinal axis compared tothe radial axis. A small beam waist (on the order of tens of µm) is required to produce suf-ficiently deep traps to capture atoms from a MOT. However, despite the tight focusing, theRayleigh length is still long, and results in a much lower axial trapping frequency relativeto the radial trapping frequency. Given that the collision rate scales with the trapping fre-quency58, weak confinement along the axial direction can negatively impact thermalizationtimes and evaporation efficiency in addition to decreasing the critical temperature required58This can be seen by noting that the collision rate is given by γ = nσv¯, where n is the density, σ is thecollisional cross section and v¯ is the average thermal velocity. For a Gaussian density distribution, the peakdensity (and therefore the peak collision rate) n0 ∝ N/(σxσyσz) where σi =√kBTmω2iand ωi is the trappingfrequency along the i = x, y, z axis. Therefore, γ ∝ ωxωyωz, given the collisional cross section and averagethermal velocity are constant.743.2. Optical Dipole Trapsfor BEC formation (see Section 5.1.3 for more detail).To increase the axial confinement, one can employ a crossed optical dipole trap (CDT)where two focused beams cross at some small angle α (see inset to Fig. 3.8). The secondfocused beam is often the first beam recycled after its first pass through the trapping region,which has the advantage that it increases the total available power and creates a trap thatis tightly confining in all three dimensions. Given that the radial symmetry of the trap isbroken, one must now define trapping frequencies ωx, ωy, and ωz along three orthogonaltrap axes. These trap frequencies, normalized to the frequency when the crossing angle iszero, are shown in Fig. 3.8. From this, it is clear that even a small crossing angle providesa large increase in confinement. Note that the trapping frequency ωz does not change as afunction of angle, as this axis is perpendicular to the propagation direction of the beam. Ata crossing angle of approximately 20◦, we expect a factor of 30 improvement of confinement(see Fig. 3.8).0 10 20 30 40 50 60 70 80 90Crossing Angle (deg)ωyωxωzFigure 3.8: CDT trap frequencies as a function of the crossing angle. If the crossing angleis zero, ωy corresponds to the axial trap frequency and ωx,z corresponds to the radial trapfrequency (see Eq. 3.22), as shown in the inset. As the crossing angle is increased, ωxand ωy are modified as these axes lie in the same plane as the two crossing beams, whileωz remains at the same value. When the beams are collinear, the normalized ωy = 0.006.In our setup, we use a crossing angle of roughly 20◦ which corresponds to a normalizedωy = 0.177 and an increase in confinement of a factor of 30.753.2. Optical Dipole TrapsThe trapping frequencies can be measured through parametric excitation [132], in whicha modulating force (provided by varying the power of the trapping laser at a set frequency)acts on the potential. When the modulation frequency is equal to twice the trapping fre-quency (or any sub harmonic), the trapped atoms are efficiently heated. This manifestsitself as an observable decrease in the atom number as atoms are heated enough to escapethe trap. Therefore, by varying the modulation frequency one can map out the trappingfrequencies of the trap. This type of measurement also allows for an experimental determi-nation of the beam waist using Eq. 3.13 and 3.22, given the power in the dipole trap beamis well known. Knowledge of the trap frequencies is also crucial to determine the criticaltemperature for BEC formation or the creation of BCS pairs.An added level of complexity when working with two (or more) species in the same ODTresults from the scaling of trap parameters with the frequency of the atomic transition andthe detuning of the dipole trap beam from the atomic transition. In the case of 6Li and Rbwith a dipole trap wavelength of 1090 nm, the ratio of trap depths between the two speciesis URb/URb ≈ 2.5, due to the fact that the trapping laser is further detuned from the atomicresonance for 6Li (671 nm) than Rb (780 nm). Moreover, because the photon scatteringrate scales like the detuning squared, ΓRb/ΓLi ≈ 5.859. However, because Rb is much moremassive than Li, the ratio of the heating rates (see Eq. 3.12) is only E˙Rb/E˙Li ≈ 0.4.Experimental Trap SetupIn our experiment, we employ two CDTs that lie in a plane horizontal to the optical table.The beam paths of both lasers are shown in Fig. 3.9. The arms cross at an angle ofapproximately 20◦, and each laser propagates collinear with the other (i.e., they are spatiallyoverlapped). The first (high power) trap is formed from a 100 W multi-longitudinal modelaser (SPI Lasers, SP-100C-0013) with a center wavelength of 1090 nm and a linewidthexceeding 1 nm. Each arm is focused to a waist of approximately 45 µm. The second (lowpower) trap is formed from a 20 W single longitudinal mode laser (IPG Photonics, YLR-20-1064-LP-SF) with a center wavelength of 1064 nm and a linewidth of less than 10 kHz. Eacharm of this laser is focused to a waist of approximately 35 µm. In the following discussion,the high power laser will be referred to as the “SPI” laser, and the low power laser will be59The actual ratio depends on the wavelength of the laser, and if one is using the rotating wave approxi-mation.763.2. Optical Dipole Trapsreferred to as the “IPG” laser. Table 3.3 gives the calculated trap depth, scattering rate,heating rate, and estimated trapping frequencies for 6Li and Rb in the high power SPI andlow power IPG trap.While Rb atoms in the MOT are easily capture is easily trapped in a low power trap,the higher temperature of the 6Li MOT, combined with a lower ODT depth at a fixed laserpower (compared to Rb), means that efficient transfer of 6Li from the MOT to the ODTrequires a deep trap, which we generate using high power. Therefore, we initially transfer6Li from the MOT to the high power SPI trap, and later transfer into the low power IPGtrap after an evaporative cooling stage. We choose to work in the low power trap becauseit is well polarized, has a narrow linewidth, and the trap depth can be precisely tuned evenat very low powers via an AOM.SPI: 1090 nm (45 µm) IPG: 1064 nm (35 µm)6Li Rb 6Li RbU (µK/W) 18.5 44.4 31.5 77.4Γ (s−1/W) 0.0038 0.0220 0.0072 0.0434E˙ (µK s−1/W) 0.4016 0.164 0.7964 0.3417ωx,y,z (Hz/√W ) (1110, 200, 1120) (450, 80, 460) (1860, 330, 1890) (770, 135, 780)Table 3.3: Properties of the high power (SPI) and low power (IPG) dipole traps. The trapdepth (U), scattering rate (Γ) and heating rate (E˙) are calculated using Eqs. 3.13, 3.14 and3.12 respectively. The trap frequencies (ωx,y,z) are calculated using the curvature of thepotential at the trap center. These values assume a crossing of 20◦ and a waist of 45 µmfor the SPI and 35 µm for the IPG. The power is the total power used in the trap. That is,the sum of the power in both arms, or equivalently, twice the power delivered by the laserto the optics.The SPI has a built-in power and modulation control, where the output of the lasercan be enabled with a digital TTL signal and the output power of the laser can be setvia a scaled input analog voltage. When the SPI laser is initially turned on, the outputpower can experience relaxation oscillations of up to 20 times the set point power. In orderto suppress these oscillations, the turning on the laser to some set power is accomplishedby first turning on the laser to 2 W (regardless of the final power), waiting approximately300 µs, and then increasing the analog input voltage to the final value corresponding to theset power [133]. In all, the laser reaches the set point power in approximately 400 µs afterthe turn-on command is issued. To turn off the laser the TTL input is set to low and the773.2. Optical Dipole TrapsBottom LevelSPIaIPG75-25ShutterAOM1st Order(100 MHz)0th Order-50250bTop Level250baDichroc BST:1090 - R:1064 250Dichroic BST:780 - R:1090250cc200PA500HWPIMGFibre CouplerCameraFigure 3.9: Schematic of the beam paths for the high power (SPI) and low power (IPG)dipole traps and photoassociation (PA) laser. Also shown is the axis used for absorptionimaging (IMG, shown with a dotted line). The optics on the top level are built on a raisedbreadboard (large solid rectangles) for increased stability. On the bottom level, the SPIoptics are mounted on the optical table. The IPG optics are mounted upside down to thebottom of breadboard (large dashed rectangle) with the exception of the two lenses and theshutter, which are mounted on the optical table, but at the same beam height as the otherIPG optics. The focal length of each lens (in mm) is indicated by the number near eachlens. The three periscope pairs are labeled by a,b, and c. The dashed beam path betweenperiscope (c) is raised above the height of the experiment.laser power is extinguished after a short delay of 15 µs.The IPG laser power is controlled with a high efficiency AOM from Gooch and Housego(part number: 97-01672-11) operated in a single pass configuration. With this AOM, weachieve a maximum efficiency of about 90%. We drive the AOM with an RF frequency of110 MHz and a maximum RF power of 2.5 W. As the output of the IPG has a well definedlinear polarization, we orient the output fiber of the IPG such that the field is linearlypolarized in the vertical direction at the AOM, which is important to achieve the highestdiffraction efficiency. Although the zero order from the AOM is sent into a beam dump, wealso have a water cooled mechanical shutter (based off of a solenoid) which we use to shutter783.2. Optical Dipole Trapsthe IPG beam when the trap is not in use. We use a right angle periscope60 to match thebeam height to that of the SPI trap. This periscope rotates the polarization of the IPGto lie in the horizontal plane61. The IPG trap is then overlapped with the SPI beam ona dichroic beam splitter (Semrock LPD01-1064RS-25) designed to transmit 1090 nm andreflect 1064 nm, and afterwards follows the same beam path as the SPI laser. This ensuresa good spatial overlap between the two dipole traps and improves the transfer efficiency of6Li from the SPI to the IPG. The details of the transfer from the MOT to the ODT andbetween the two dipole traps are discussed below.Mitigating Thermal LensingIn a previous iteration of this apparatus, in which we used a Pyrex cell, we noticed significantthermal lensing and beam distortion. and permanent damage spots on the cell when theSPI laser was operated at powers greater than 20 W. Initial testing on a quartz slab showedno lensing effect, which motivated switching to a quartz cell in the current iteration of thesetup. In order quantify the lensing effect between the two cells, we measured the beamradius after a single pass through each cell. The horizontal and vertical radii as a functionof the laser power are shown in Fig. 3.10, which indicates that the lensing effects aresignificantly reduced with the quartz cell. However, a more precise measurement can bemade by tracking the location of atoms as a function of time while held in the trap. Evenwith the quartz cell, we found that the focus of the beams still move by approximately1.3 mm with a time constant of 1.4 s when the laser power is set to 100 W. Therefore,we make an effort to operate the SPI laser at high powers for only short times in order tominimize this effect.Transfer of 6Li to the ODTTo transfer 6Li from the MOT to the ODT, we use a two stage process where we compressand cool the MOT. We start this process by turning off the Zeeman slower light andincreasing the coil gradient from 67 to 92 G/cm (11 A to 15 A), while at the same timeshifting the frequency of the pump and repump to 11 MHz below resonance and decreasing60That is, a periscope in which the direction of the output beam differs by only 90◦ to the output.61Historically, the IPG laser was sent to the cell at Brewster’s angle in order to minimize the reflectionsfrom the cell. This required that the polarization of the beam lie in the horizontal plane. At the smallangle of incidence that we use currently, the difference in the reflection coefficient between vertically andhorizontally polarized light is minimal.793.2. Optical Dipole Traps200250300350400450Horizontal0 20 40 60 80 100200300400500600Vertical0.0 0.2 0.4 0.6 0.8 .Power (W)µm)Figure 3.10: Effect of thermal lensing on the ODT beam shape. Shown is the SPI beamradius through a Pyrex cell (red squares) and Quartz cell (blue diamonds) as a function ofpower along the horizontal (top) and vertical (bottom) axes. Also shown is the beam radiuswith no cell (black circles). This indicates a strong lensing effect with the Pyrex cell whichmodifies the beam radius. This effect does not exist with the Quartz cell.the power in each beam to approximately 20 mW. During this initial stage, the SPI laseris turned on and we apply a small compensation field to adjust the position of the MOTto overlap with the crossing of the dipole trap beams. These parameters are held for10 ms. In the second stage, we decrease the detuning of the pump and repump to 3 MHzbelow resonance and further reduce the power in each beam to approximately 5 mW. Wealso make another slight adjustment to the compensation field to ensure the MOT centerremains overlapped with the crossing of the dipole trap beams. After another 2.5 ms, weturn off the magnetic field (both the quadrupole and compensation field) and we opticallypump to the lower hyperfine state (f = 1/2) by turning off the repump light 400 µs beforethe pump light. The produces an incoherent mixture of roughly equal populations of atomin the mf = 1/2 (|1〉) and mf = −1/2 (|2〉) states. Without this optical pumping stage, weobserve extremely rapid trap losses (on the order of a few ms) due to hyperfine relaxation.803.2. Optical Dipole TrapsWe find the optimal compensation field empirically, by varying the size and direction of thefield along three axes (which moves the MOT position along the three axes) and optimizingthe number of atoms transferred into the trap.The number of atoms that we transfer to the SPI at 100 W trap saturates at about1.5× 106 atoms, and becomes independent of the size of the MOT when the MOT numberexceeds 30×106 atoms. We believe this saturation is due to density dependent losses62 thatoccur during the transfer of atoms from the MOT to ODT, which are more severe in theODT because the largest achievable ODT depth is orders of magnitude (typically in themK range) lower than the MOT (typically in the K range). For this reason, we typicallyload a small MOT in a short time in order to reduce the cycle time of the experiment, andtransfer about 1.3× 106 atoms to the cross SPI trap at 100 W.Although we have made an effort to optimize the transfer parameters to produce themost efficient transfer, many of the parameters are coupled (for example, the power anddetuning of the pump and repump beams) and the number of free parameters is large enoughthat doing a full N dimension search is not possible. However, in general, we found thatthe transfer efficiency is much more sensitive to the settings during the short second stageof the transfer. In particular, the transfer was found to be most sensitive to the detuningsof the pump and repump and the compensation coil settings.A final consideration during the transfer from the MOT to the ODT arises due to thenature of the Zeeman slower beam, which is only a few tens of MHz further detuned fromthe atomic transition than the MOT beams. Since the Zeeman slower beam imparts a largeradiation pressure force on the MOT, it is possible that when the slowing beam is shutteredthe MOT moves to an unstable position. Therefore, it is important to visually ensure thatthe MOT is stable in both the standard loading location, and the location where the transferoccurs.Evaporation of 6Li in the ODTDirectly after the transfer of atoms to the SPI trap, the trap depth is approximately 3.7 mKand the ensemble temperature is on the order of 500 µK. In order to produce an ultra-coldensemble, we perform free and forced evaporative cooling in order to decrease the temper-62These losses include spin exchange collisions, hyperfine changing collisions, and light-assisted losses thatinclude photoassociation and fine-structure changing collisions. A discussion of cold collision processes canbe found in a review by Julienne et. al. [134].813.2. Optical Dipole Trapsature by about three orders of magnitude. An excellent description of forced evaporativecooling can be found in the thesis of K. O’Hara [135], in which it is shown that forcedevaporative cooling in an ODT follows a general scaling lawNfNi=(UfUi) 12(3η−3), (3.23)where N is the atom number in the dipole trap and η = U/kBT is the ratio of the ODTdepth to the ensemble temperature. For 6Li, efficient cooling occurs when η ≈ 10. Thisscaling law can be used as a benchmark for the efficiency of our forced evaporative coolingramps.As efficient evaporation requires fast thermalization, we therefore perform our forcedevaporation ramps in the presence of a homogenous magnetic field in order to increase thecollisional cross section between the atoms63. Although it is possible to achieve very largescattering lengths near the broad s-wave Feshbach resonance at 832 G (see Section 5.1.2),we perform our evaporation at 300 G, and see no change in the evaporation efficiency athigher fields. We believe this may be the case because, in the unitary limit where k2a2  1,the scattering cross section is independent of the magnetic field (on which the scatteringlength depends), but is limited by the de Broglie wavelength. That is,limk2a214pia21 + k2a2=4pik2=λ2dBpi=2~2mkBT. (3.24)For high temperatures, the unitary limit clamps the collisional cross section to that whichis achievable near 300 G. At lower temperatures, the gain in cross section by evaporating atfields close to the Feshbach resonance only impacts the evaporation efficiency if the ramprate is too fast compared to the thermalization rate. It is also possible that in our setupthere are additional losses that turn on at high fields due to, for example, three-body lossesor losses induced by magnetic field noise. We do, however, see a minimum in the evaporationefficiency near the zero crossing of the scattering length near 530 G and when the magneticfield is less than 150 G.In spite of our optimization attempts, we found that the evaporation in the SPI laser at63It is possible to compensate for low thermalization times by performing a slower evaporation ramp.However, the timescale of the evaporation is limited by the finite lifetime of the trap. This finite lifetime isimposed by the photon scattering rate and the background collisional losses.823.2. Optical Dipole Trapshigh power was always less efficient than evaporation in the IPG64. Due to the inefficientevaporation and to minimize the time that the laser was on at high power, we perform afast evaporation from 100 W to 50 W of power in 100 ms. This typically results in a loss ofabout 25% of the trapped atoms (from 1.3 × 106 atoms to approximately 1 × 106 atoms),corresponding to η = 7. At this point, we begin the transfer of atoms to the IPG trapby turning on the IPG trap to its set power (typically 15 W) in 10 ms, and then turningoff the SPI in 500 ms in a linear fashion. We found that the transferred number was veryinsensitive to the parameters of the transfer, including the IPG turn on time, the power ofthe SPI when the transfer begins, and the time over which the SPI turns off. Our transferprocess typically results in about 250 × 103 atoms in the IPG trap at a temperature ofapproximately 55 µK. A timing diagram of this evaporation can be seen in Fig. 3.11.After the transfer to the IPG trap, we perform a series of linear ramps of differentdurations to reach the desired final temperature (see Fig. 3.11). Each linear ramp wasoptimized by choosing a final power and modifying the duration of the ramp in order tomaximize the remaining number. At low trap depths, we find that a large loss of atomsoccurs due to a “tipping” of the potential due to the residual magnetic field gradient andthe force of gravity. To compensate for this, we turn on an additional “gradient” coil. Forthis, we use the compensation coils which is concentric with the upper Feshbach coil.If the goal of the experiment is to produce ultra cold Feshbach molecules, we switch themagnetic field to a set value near the Feshbach resonance at a trap depth which correspondsto a temperature greater than the binding energy of the Feshbach molecules. In this way,when the final evaporation is performed, the atoms will form Feshbach molecules via threebody recombination as the ensemble energy becomes lower than the molecular binding en-ergy (see Section 5.1.3. With this method, we are able to achieve efficient evaporation rampswhere η ≈ 10. For example, evaporation to degeneracy is shown in Fig. 5.6 and discussedin detail in Section 5.1.3. Figure 3.11 also shows typical atom number and temperatures(measured via a time of flight measurement) at different points on the evaporation ramp.Note that the atom number listed in Fig. 3.11 is the number measured using our high fieldimaging setup. This technique only measures the population of one of the two spin states,and undercounts this state by approximately a factor of two due to polarization considera-tions (see Sec 3.4 for a detailed discussion). Therefore, the total atom number in the trap64One possible reason for this may be intensity noise the laser, as we do not perform any intensitystabilization at this time.833.2. Optical Dipole Traps0 500 1000 1500 2000 2500Time (ms)0.1110100Power(W)55µK / 250k2µK / 110k.311µK75 kFigure 3.11: Timing diagram of the transfer from the SPI trap (dashed line) to the IPG trap(solid line), and the following evaporation to degeneracy in the IPG. The vertical dashed dotline represents the point in the evaporation when we switch the magnetic field from 300 G to754 G such that the following evaporation cools the atoms in the Feshbach molecule state.At the end of the evaporation ramp, the magnetic field can be changed to the value at whichwe want to perform our experiment. The typical temperature and atom number at a fewcharacteristic locations along the evaporation are given. Note that the atom number listedis the number measured using our high field imaging setup. This technique only measuresthe population of one of the two spin states, and undercounts this state by approximately afactor of two due to polarization considerations (see Sec 3.4 for a detailed discussion). Theatom number listed in the figure is the inferred total atom number, based off of the highfield imaging measurements.is approximately four times larger than the number calculated from the high field imagingmeasurement. The atom number given in Fig. 3.11 is the inferred atom number based offof these measurements.For our spectroscopy work at 0 G, we turn off the homogenous field at the end of theevaporation ramp. To compensate for the residual magnetic field that persists due to theearth’s magnetic field and any other magnetic sources near the cell we apply a compensationfield using the compensation coils. It is not possible to use the gradient coil to help opposethe tipping of the trapping potential and, at the same time, minimize the residual field. Thisleads to a large loss of atoms at very low trap depths where the “tipping” is significant.843.2. Optical Dipole TrapsHowever, we were still able to achieve an atom number of a few 104 at a temperature of800 nK, while ensuring that the residual magnetic field was less than 20 mG65. In manycases, we stop the evaporation ramp at higher trap depths in order to increase the atomnumber. This has little effect on the spectroscopy results as the Doppler broadening is onlyapproximately 200 kHz at temperatures as high as 10 µK. This broadening is still muchsmaller than the natural linewidth of the transitions, and is a few times smaller than thetypical width of our observed dark state features.Transfer of Rb to the ODTThe transfer of Rb to the ODT proceeds in a much simpler (although similar) way thanthe 6Li transfer. The transfer takes place in one stage, where we slightly increase themagnetic field gradient from 15 to 25 G/cm (2.5 to 4 A) while simultaneously applyingsmall homogenous compensation field to move the center to the MOT to the location ofthe dipole trap. We typically leave the detuning of the pump (≈ 3Γ) and repump (onresonance) beams at the values used for the MOT loading, but we decrease the power inan attempt to reduce light assisted losses during the transfer. We hold these parametersfor 50 ms, and then turn off the magnetic field. In order to optically pump the Rb atomsinto the lower hyperfine state, we turn off the repump light 400 µs before the pump light.Because we can achieve a colder final temperature of the Rb MOT than the 6Li MOT andthe ODT depth for Rb is more than twice as deep as for 6Li, we can typically transfera large number of atoms without much optimization. For example, when we work withsmall MOTs, where density dependent losses in the transfer process are small, we transferapproximately 350× 103 atoms into a 15 W ODT from a MOT of 720× 103 atoms, whichgives a transfer efficiency of nearly 50%. Much of our work with 85Rb was in the contextof mixtures with 6Li, and a more detailed discussion on loading both 85Rb and 6Li into theODT can be found in Chapter 7.We typically transfer the Rb MOT directly into the low power IPG trap (rather thaninto the high power SPI trap) for three reasons. First, we can easily achieve the requiredtrap depths (and atom number) in the IPG trap without the complications of having totransfer the atoms from the SPI to the IPG for further evaporation. Second, we found65More details on this measurement, and the procedure to minimize the residual field can be found in thePhD thesis of Mariusz Semczuk [98].853.3. Optical Pumping and State Selectionthat loading Rb into a deep trap resulted in a higher starting temperature relative to thetrap depth, which resulted in a lower evaporation efficiency. In fact, we found that wecould achieve larger atom numbers at the same final temperature by loading directly intoa weaker trap than through evaporation from a deeper trap. Finally (and perhaps mostimportantly) we found that the SPI laser drove hyperfine changing transitions between thetwo ground hyperfine states.To investigate these transitions, we measured the number of the atoms in the f = 3state of 85Rb using absorption imaging without the standard step of optically pumping thetrapped atoms into this state (see Section 3.4 for a discussion of the imaging procedure).This way, any atoms that populate the f = 3 state when the image is taken are drivento this state by the SPI66. The population in this state as a function of the hold time inthe SPI is shown in Fig. 3.12. The transition follows an exponential curve, with a timeconstant of 120 ms when the power of the SPI is set to 20 W. After long times in the SPI,the population of atoms in the f = 3 state tends to approximately 35% of the total atomnumber. We found that we could effectively stop this population transfer by continuously“depumping” the ensemble, using light tuned near the f = 3→ f ′ = 3 resonance (see Fig.3.12). With this depumping light, we also see an increase in the overall lifetime in the SPI(from 1.3 s to 2.9 s), and no observable change in the overall atom number. We believe thatthe SPI laser has longitudinal modes which have a frequency separation close to the energydifference between the ground hyperfine levels in 85Rb (≈ 3 GHz), which drives two-photonRaman transitions between the levels. Note that we do not see a similar effect in 6Li, wherethe separation of the ground hyperfine levels is only ≈ 228 MHz.3.3 Optical Pumping and State SelectionAfter the transfer of atoms from the MOT to the ODT, we perform a short hyperfinepumping procedure (discussed in Section 3.2) to ensure that the atoms are in their groundhyperfine state (f = 1/2 in the case of 6Li and f = 2 in the case of 85Rb). After thispumping stage, the atoms typically populate all possible Zeeman states with roughly equalpopulation. However, one of the major advantages of trapping in an ODT is that the lack ofnear-resonant light and external dc fields allows for the ability to prepare an atomic ensemble66We have confirmed that in the IPG trap we see no atoms in an absorption image without the opticallypumping step.863.3. Optical Pumping and State Selection0 100 200 300 400 500Time (ms)0102030405060AtomNumber(×103)Figure 3.12: Evidence of two-photon (stimulated) hyperfine changing transitions for85Rb driven by the SPI ODT. The atom number in the f = 3 ground state is shownas a function of the hold time in the SPI trap with (red squares) and without (black circles)a “depump” beam tuned near the f = 3→ f ′ = 3 atomic resonance. As atoms are initiallypumped into the f = 2 ground state during the loading of the ODT, we believe that theappearance of atoms in the f = 3 state is due to two-photon Raman transitions betweenthe f = 2 and f = 3 hyperfine levels, which are driven by the SPI laser. The time constantof the fit to the f = 3 population as a function of time (black line) is 120 ms. We cansuppress these transitions using a “depump” beam. Note that this beam does not causeany change in the overall atom number (i.e., it does not cause any additional light assistedloss in the ODT).in a particular Zeeman state. This is advantageous because it allows experimental controlover the atomic scattering states. For example, this is useful in our Feshbach resonanceexperiments because the resonances occur for particular pairings of the 6Li and Rb Zeemanstates (see Chapter 7 and in particular Table 7.1). The ability to prepare an atomic ensemblein just one particular state (or explicitly not in a particular state) gives control over whichFeshbach resonances are possible. Additionally, preparing a system in the stretched statesZeeman states, where |mf | = f , ensures that the ensemble will be stable with respect totwo-body spin relaxation due to conservation of the total angular momentum projectionduring such a collision.This section describes our optical pumping scheme for 6Li and 85Rb. While the data873.3. Optical Pumping and State Selectionpresented is for 85Rb, the method discussed is easily applied to 87Rb, accounting for thedifferent atomic transition frequencies. It should be noted that for 6Li, because the groundhyperfine state has only two levels, we typically do not optically pump as much as “opticallyreduce”, where we selectively remove the population of one of the two projection states.Optical Pumping of 85RbIn the presence of a small external magnetic field67, each of the hyperfine levels is split into2f+1 degenerate levels, where the energy of each mf projection state relative to the energyof the state in the absence of the magnetic field isEZ = gfµBBmf (3.25)wheregf = gjf(f + 1) + j(j + 1)− i(i+ 1)2f(f + 1), (3.26)andgj = 1 +j(j + 1) + s(s+ 1)− l(l + 1)2j(j + 1). (3.27)In Eq. 3.25, µB is the Bohr magneton, which is approximately equal to the electron’sintrinsic spin magnetic moment. It’s useful to note that a magnetic moment of µB cor-responds to an frequency shift of approximately 1.4 MHz/G. For Rb, the g-factor for thef = 2 (f = 3) manifold is gf = −1/3 (gf = 1/3), which corresponds to an energy shift of−0.47 · mf MHz/G (0.47 · mf MHz/G) [90]. When we optically pump 85Rb, we apply asmall dc magnetic field in order to set the quantization axis, and to break the degeneracy inthe hyperfine manifolds. Typically this field is on the order of 1 G, such that the maximumenergy shift of any of the stretched states is only a few MHz, and is less than the naturallinewidth of the atomic transition.For 85Rb, we optically pump using light driving the D2 transition between the f = 2hyperfine level in the 2S1/2 manifold and one of the hyperfine levels in the2P3/2 manifold.The possible hyperfine levels in the 2P3/2 to which the light can couple is limited to f′ =1, 2, 3 by electric dipole selection rules. If the pumping light is polarized such that it drivesσ+ (σ−) transitions, each photon absorption will increase (decrease) mf by one unit. On67For Rb, the magnetic field can be considered small when it is less than a few hundred Gauss. When weoptically pump Rb, we typically use fields on the order of 1 G, which is safely within the low field regime.883.3. Optical Pumping and State Selectionthe other hand, each photon emission will, on average, not change mf . The net result ofmultiple scattering events is that the atomic population will tend to populate one of thestretched states, either mf = 2 or mf = −2. For the following discussion, we will assumethat we are optically pumping to the mf = 2 state using σ+ polarized light.Ideally, the final stretched state to which we optically pump into would be a darkstate (i.e., a state for which photon scattering is disallowed due to electric dipole selectionrules). This can be achieved through the correct choice of the excited hyperfine state towhich the optical pumping light couples. For σ+ polarized light, the f ′ = 2 manifold issufficient, because some pi polarized light would be required to drive a transition out ofthe |f = 2,mf = 2〉 state. The ideal case, however, is to use the f ′ = 1 manifold, wherethe stretched state is also dark with respect to pi transitions. However, note that if thef ′ = 1 hyperfine manifold is used, the optical pumping light must include some pi light suchthat the |f = 2,mf = 1〉 state is not also dark. Another significant advantage of using thef ′ = 1 manifold is that decay to the f = 3 ground hyperfine level is disallowed and this,aside from off-resonant transitions to the f ′ = 2 hyperfine manifold, ensures the opticalpumping transition is closed.In our experiment, we use the f = 2→ f ′ = 2 transition, in part because the light thatwe use for optical pumping is picked off from the same light that we use for the MOT repumpbeam. This light is set up to have a frequency that is resonant on the f = 2 → f ′ = 3transition after up shifting by 180 MHz in a double pass AOM, but the bandwidth of theAOM is not large enough to allow us to bridge the 93 MHz gap required to reach the f ′ = 1state. As a consequence of using the f = 2 → f ′ = 2 transition, decay to the f = 3state is possible which causes the atoms to eventually populate the upper (f = 3) groundhyperfine state. This requires the use of a “depump” beam68, for which we use light pickedoff from the MOT pump beam, resonant on the f = 3 → f ′ = 4 transition in the absenceof any ac Stark shifts. We are unable to tune the depump beam to the f = 3 → f ′ = 3transition for similar reasons as the optical pump beam: the “depump” is set to be nearthe MOT pump transition (f = 3 → f ′ = 4) using a double pass AOM, which does nothave the bandwidth to bridge the 120 MHz gap to the f ′ = 3 state. Therefore, we rely onoff-resonance transitions to the f ′ = 3 state for depumping, from which decay to the f = 2state is permissible. We couple both the optical pumping light and the “depump” light into68Named as such because this beam does the opposite of the repump beam in the MOT. That is, the“depump” light depumps atoms from the f = 3 to the f = 2 ground hyperfine level.893.3. Optical Pumping and State Selectionthe same optical fiber. Out of the fiber, the light propagates along the same path as thephotoassociation light (see Section 2.3), and is focused to a waist of approximately 50 µm.We use a half-wave plate and a quarter-wave plate at the output of the fiber to ensure thelight is circularly polarized.When we perform optical pumping in the ODT, we have to compensate for the ac Starkshift of the trap unless we momentarily turn off the ODT. With Rb, it is possible to turn thetrapping light off and on fast enough such that the majority of the atoms are recapturedand therefore there is minimal loss of atoms. For Rb, we find that an off period of upto 20 µs results in no loss of the Rb atoms from the ODT. Due to timing limitations,the shortest modulation period we could achieve was about 3 µs. However, even for thisshortest modulation time we experience a large loss of 6Li atoms from the trap. This islikely because 6Li is much lighter than Rb, and therefore the atomic cloud expands muchfaster during the time in which the ODT is off69. For our experiments with 6Li and Rbmixtures, the loss of 6Li is an issue because we load Rb into an ODT that already contains6Li (see Chapter 7). Therefore, we perform all of our optical pumping in the presence ofthe ODT and compensate for the associated ac Stark shift by frequency shifting our light.In order to determine the correct frequency for the light, we used the optical pumpinglight to pump atoms from the f = 2 to the f = 3 ground hyperfine state via the accessiblehyperfine states in the 2P3/2 manifold. We then took an absorption image of the atomsthat populated the f = 3 state, and measured this population as a function of the opticalpumping beam detuning. The result of this measurement for two trap powers, and the casewhere the laser was modulated off and on (such that there should be no ac Stark shift)is shown Fig. 3.13. Here, we can see the f = 2 → f ′ = 3 and the f = 2 → f ′ = 2transitions, split by approximately 61 MHz, roughly consistent with the expected splittingof 63.4 MHz [90]. The effect of the ac Stark shift due to the ODT is to blueshift thetransition, requiring a correspondingly larger frequency of the optical pumping beam. Inthe case where the laser is modulated off and on, it appears that there is still some shiftof the transition. This may be a result of an offset in the locking frequency of our masterlasers, or an indication that the ODT light is not completely extinguished during the entire69Of course, the rate at which the atomic ensemble expands is dependent on the temperature of theensemble. For Rb, these test were done at a temperature of approximately 50 µK. For Li, these tests wererun at a temperature of 10 µK. It is possible that this modulation technique would be more effective aftera further forced evaporation. However, overcoming the ac Stark shift of the ODT was not a problem, so wedid not investigate this issue further.903.3. Optical Pumping and State Selectiontime the optical pumping light is on.Critical to the efficiency of the optical pumping scheme is that the magnetic field isaligned along the propagation axis of the optical pumping beam. This ensures that thecircularly polarized light drives σ transitions. Figure 3.14 shows the population in thestretched states as a function of the magnetic field angle, where the field is defined to liein the horizontal plane. We also optimize the pumping efficiency by empirically tuning thepower in the optical pumping and depumping beam. Although these powers are stronglycoupled to the time in which the optical pumping light is on, we found that a time of 10 msand powers on the order of 50-100 µW gives us the highest efficiency. More power or longertimes resulted in a loss of atoms from the trap, presumably due to heating.To measure the efficiency of our optical pumping scheme, we used the state selectiveimaging technique described in Section 3.4. This technique relies on RF transitions toselectively transfer atoms from the f = 2 to f = 3 hyperfine level before an absorptionimage is taken. From the 11 possible RF transitions, we can determine the populationin the each of the five mf projection states in the f = 2 manifold (see Fig. 3.20 and theaccompanying discussion in Section 3.4). Without optical pumping, we find that 85Rb atomsare roughly equally distributed between the five f = 2 projection states, as expected. Afteroptical pumping, we find that approximately 60% of the population is in the target stretchedstate, while almost no population remains in the opposite stretched state. The total atomnumber in both cases (with and without optical pumping) is the same. Figure 3.15 showsthe raw population measurement using the state selective imaging technique, as well as thecalculated population in each of the projection states. We believe one reason that we are notable to achieve higher efficiencies is the fact that the stretched state is not a dark state, andwe are unable to fully eliminate the polarization components that can drive pi transitionsfrom the pumping light. Nevertheless, if a pure state population is required, it is possibleto selectively clean out the remaining population in the unwanted states by applying an RFfield to transfer atoms in a particular projection state in the f = 2 manifold to the f = 3manifold, while at the same time applying a strong depump beam. The strong depumpbeam acts to heat the atoms such that they leave the ODT.913.3. Optical Pumping and State Selection01Trap Off01PIPG = 7 W−40 −20 0 20 4001PIPG = 15 W0.0 0.2 .4 0.6 0.8 1.0Detuning (MHz) 3.13: 85Rb atom number in the f = 3 ground hyperfine manifold, measured with anabsorption image immediately after using the optical pumping beam to pump atoms fromthe f = 2 to the f = 3 manifold. The resonant frequencies shift as a function of the trappower due to the ac Stark shift in the ODT. The detuning is relative to the f = 2→ f ′ = 3atomic transition. Therefore, the single large peak in the top panel and the right peak inthe bottom two panels represents the f = 2→ f ′ = 3 transition. Likewise, the left peak inthe bottom two panels represents the f = 2 → f ′ = 2 transition. The measured splittingbetween these transitions is 61.6 MHz and 61.2 MHz when the trap power is 7 W and 15 Wrespectively, which is roughly consistent with the expected difference of 63.4 MHz [90]. Inthe case where the laser is modulated off and on, the peak atom number appears at adetuning of -6.7 MHz, which suggests that that there is still some shift of the transition.This may be the result of an offset in the locking frequency of our master lasers, or anindication that the ODT light is not completely extinguished during the entire time theoptical pumping light is on.923.3. Optical Pumping and State Selection0 50 100 150 200 250 300B-Field Angle (degrees)0102030405060AtomNumberinState(%)|2, 2〉|2,−2〉Figure 3.14: Percentage of 85Rb atom population in one of the |f = 2,mf = 2〉 and|f = 2,mf = −2〉 stretched states after optical pumping as a function of the magnetic fieldangle. Efficient optical pumping to the |2, 2〉 (|2,−2〉) state occurs when the optical pumpinglight drives σ+ (σ−) transitions. The polarization of the optical pumping beam is set to becircularly polarized using a half-wave plate and a quarter-wave plate. Therefore, this lightwill drive σ transitions when the magnetic field is directed along the propagation axis ofthe light (either co- and counter-propagating). This is consistent with the measured differ-ence in magnetic field angle (183◦) between the maximum populations in the two stretchedstates. The angle is measured in the horizontal plane relative to an axis that is collinearwith the long axis of the vacuum cell.State Selection of 6LiAs previously noted, we have not done any true optical pumping of 6Li, aside from hyperfinepumping to the f = 1/2 hyperfine level at the conclusion of the transfer from the MOT tothe ODT. This results in an incoherent mixture of roughly equal population in the twoprojection states. At magnetic fields greater than approximately 100 G, the Zeeman energydominates over the hyperfine energy and a three state manifold exists that consists of the|1〉, |2〉, and |3〉 states (see Fig. 5.1 and the more detailed discussion of the 6Li system inSection 5.1). The energy splitting between these three states is approximately 80 MHz. The|1〉 and |2〉 states asymptotically connects to the |f = 1/2,mf = ±1/2〉 hyperfine states andare roughly equally populated after the ODT transfer is complete. The |3〉 state asymptot-933.3. Optical Pumping and State Selection−3 −2 −1 0 1 2 3Detuning (MHz)AtomNumber(arb.units)−3 −2 −1 0 1 2 3Detuning (MHz)−2 −1 0 1 2State (mf )0204060Population(%) Without Pumping−2 −1 0 1 2State (mf )0204060With PumpingFigure 3.15: 85Rb state population with (right side) and without (left side) optical pumping.Without optical pumping, we have a roughly equal population in each Zeeman state. Withpumping, we find that approximately 60% of the population is in the target stretched state,while almost no population remains in the other stretched state. The total atom numberin both cases is the same. The top two panels show the result of the state selective imagingtechnique discussed in Section 3.4. In this technique, an RF pulse is used to transfer atomsfrom a particular projection state in the f = 2 manifold to the f = 3 hyperfine level. Then,an absorption image is used to measure the population in the f = 3 level. From the 11different RF transitions, the relative population of each mf state in the f = 2 manifold canbe extracted (see Section 3.4). The result of this calculation is shown in the bottom panel.For this measurement, we used a magnetic field of approximately 1 G which corresponds toan energy difference of 470 kHz between adjacent projection states (see Eq. 3.25).943.4. Imaging Techniquesically connects to the |f = 3/2,mf = −3/2〉 hyperfine state, which is initially unpopulatedafter the transfer to the ODT.State selection in 6Li is achieved using a “spin cleanup” technique, where we use lightto blow out the atomic population from any of the |1〉, |2〉, and |3〉 states. For this, weuse the same light that is used for high field imaging of 6Li (see Section 3.4 for details).This effectively removes all of the population from the target state, with no effect on thepopulation in either of the other two states.Using RF pulses, population can be transferred between the |1〉 and |2〉 states or the |2〉and |3〉 states. Additionally, it is also possible to create a coherent ensemble between two ofthese spin states. The energy difference between the |1〉 and |2〉 states and the |2〉 and |3〉states as a function of magnetic field is shown in Fig. 3.16. These RF transitions are quitenarrow, on the order of a few kHz. However, it is relatively easy to find the correct transitionfrequency by sweeping the RF frequency (rather than irradiating the atoms with a singlefrequency pulse) and looking for atom loss out of the state which is initially populated70.This allows one to quickly cover hundreds of kHz in only a few experimental cycles. Once losshas been observed from one sweep window, the precise transition frequency can be found byscanning over the frequency range of that window with single frequency RF pulses. Detailsof the RF antenna can be found in [98].3.4 Imaging TechniquesWith as many moving parts as there often are in ultra cold experiments, extracting quan-tifiable information from such a system is surprisingly simple. There are only two standardobservables in ultra cold atom experiments from which almost every other quantity of in-terest can be determined: atom number and temperature71. These quantities are typicallydetermined by taking a picture of the atoms, where the two most common imaging tech-niques are fluorescence imaging and absorption imaging.70The maximum range of the sweep window that will result in observable loss is dependent on the totalavailable RF power. Typically, we have found that a sweep range of ±10 kHz strikes a good compromisebetween loss and the ability to cover a large range in a short time given our (not so great) antenna and 2 WRF amplifier.71Most other quantities require knowledge of the atom number or temperature as a function of someother parameter. For example, the location of a photoassociation resonance or a Feshbach resonance can bedetermined by monitoring the atom number as a function of the frequency of the photoassociation light ormagnetic field, respectively.953.4. Imaging Techniques500 600 700 800 900 1000−4.8−4.6−4.4−4.2−4.075.45275.51575.720 75.793|1〉 → |2〉|2〉 → |3〉0. 0.2 0.4 0.6 0.8 1.Magnetic Field (G)−80MHz(MHz)Figure 3.16: RF frequencies of the |1〉 → |2〉 (bottom panel) and the |2〉 → |3〉 (top panel)transition in 6Li as a function of magnetic field. The RF transition frequencies at selectmagnetic fields are given in MHz. More information on the |1〉, |2〉, and |3〉 states be foundin Section 5.1 and specifically Fig. 5.1, which shows the magnetic field dependence of thesix 6Li Zeeman levels.Fluorescence imaging captures photons scattered by the atomic ensemble, and providesa highly sensitive measurement of the atom number. In our case, we perform fluorescenceimaging by recapturing atoms in the MOT and therefore this technique is unable to pro-vide information about the spatial distribution of the atoms, which is necessary for simplemethods of temperature determination72. On the other hand, absorption imaging providesan in situ measurement of the optical density of the cloud, from which it is possible toextract both atom number and temperature. With absorption imaging, it is also possibleto make state selective atom number measurements, or to image the atoms in the presenceof an external magnetic or electric field. This section briefly discusses each imaging sys-72That is not to say that fluorescence imaging in general cannot provide information about the spatialdistribution. For example, in quantum gas microscope experiments use fluorescence imaging, but the atomsare held fixed in an optical lattice [136].963.4. Imaging Techniquestem, and the implementation of each in our apparatus. For all of the imaging discussed inthis section, we use a Pixelink camera (model number PL-B771G) that images along thehorizontal plane (see Fig. 3.9).3.4.1 Fluorescence ImagingFluorescence imaging allows a determination of the atom number by counting the numberof photons scattered by the atoms on a camera or photodiode. Given known quantities ofthe system, a calibration factor can be used to convert the total pixel count (PC) in theimage to a true atom number. This calibration factor, which has units of atoms/PC, isgiven byCatoms =1L · η ·K · Patom , (3.28)where L is the loss of light due to transmission through the cell, K is the camera calibrationwhich provides a conversion from power to pixel counts, Patom is the total power scatteredby a single atom, andη =AcameraAtotal=pir2lens4pil2= 2.1× 10−3 (3.29)is the solid angle captured by the camera. In our case, rlens = 22/2 mm is the effectiveradius of the imaging lens73, which is placed l = 120 mm from the cell.We determine the camera calibration constant K by shining a laser beam of a knownpower into the imaging system and measuring the total pixel count in the resulting imageof the beam. We include the dependence on the image exposure time in the calculationthrough the camera calibration constant. That is, we use a different camera calibrationconstant for each exposure time. The camera calibration constant for both 6Li and Rb forrepresentative exposure times is given in Table 3.4. We typically use exposure times onthe order of a few to tens of ms. The fact that the exposure time appears to be non-linearin time may indicate that there is an offset between the set exposure time and the trueexposure time. This is not an issue for our atom number calibration because we determinethe calibration factor based off of the set exposure time.The total power scattered by a single atom is given by the photon energy times the73The lens is a standard one inch diameter lens, but the effective area is reduced due to a threaded ringused to mount the lens.973.4. Imaging TechniquesExposure Time (ms) KRb (PC/nW) KLi (PC/nW)1 7.4× 103 8.3× 1032.5 16.8× 103 15.4× 1035 26.8× 103 27.3× 10310 – 51.2× 103Table 3.4: Pixelink camera calibration factors for 6Li and Rb. The calibration factors weredetermined by shining a laser beam of known power into the imaging system and measuringthe total counts in the resulting image of the beam. The units of the calibration factor arepixel counts (PC) per nW. Although we did not measure a calibration constant for Rb withan exposure time of 10 ms, it (and any other time) can be extrapolated (or interpolated)from the measured values. The fact that the exposure time appears to be non-linear in timemay indicate that there is an offset between the set exposure time and the true exposuretime. This is not an issue for our atom number calibration because we determine thecalibration factor based off of the set exposure time.photon scattering ratePatoms =hcλΓsc =hcλΓ2· s1 + s+ (2δΓ )2, (3.30)where Γ is the natural linewidth, s = I/Isat is the saturation parameter, and δ is thedetuning of the light from the atomic resonance. The sensitivity (and thus the calibration)of the measurement at a fixed exposure time can be adjusted by changing the total powerand detuning the light. For measurements with 6Li, we set the power and detunings of theMOT pump and repump beams to be equal. The power that is used in the calculation isthe power of one of the pump or repump beams, as the atoms will only scatter photonsfrom either the pump or repump beams at a single time. In the case of Rb, we consideronly the power to the MOT pump beam.To determine the intensity of the light illuminating the atoms in the MOT, we measurethe power using a power meter that has an aperture diameter of .95 cm and assume thatthe intensity is constant over that area. Given that the MOT beams are approximately oneinch in diameter, and the atoms are located near the center of each beam, we believe thisapproximation is reasonable. We account for the loss of light due to the quartz vacuum cellby measuring the power before and after the cell, and calculate the loss coefficient for eachinterface. We find that the transmission per interface is approximately 93% along the twohorizontal axes, and approximately 95% along the vertical axis.We primarily use MOT recapture fluorescence imaging as a means to measure the atom983.4. Imaging Techniquesnumber in the MOT, as it is technically challenging to take an absorption image of a MOT(see the next section on absorption imaging for details). To take an image of a 6Li MOT, wetend to decrease the detuning to a few natural linewidths and decrease the power in orderto compress the size of the MOT, such that it fits on the camera chip. For Rb, we oftenkeep the detunings similar to that of the MOT and simply adjust the power or the exposuretime to ensure the image is not saturated. After this image, we turn the MOT light offand on to ensure that no atoms are held in the MOT and take a second background image.This background image is subtracted from the atom image, and we sum over the subtractedimage to determine the total pixel count and, using the calibration constant in Eq. 3.28,the atom number. An example of these images for a 6Li MOT is shown in Fig. 3.17.Atom Background SubtractedFigure 3.17: Example of a fluorescence image of a 6Li MOT. The background image (center)is subtracted from the atom image (left) to account for any scattered light that enters theimaging system during the exposure. The background image uses the same parameters asthe atom image, except that we ensure there are no atoms by turning the MOT beams offand on before the image is taken. The atom number is found by summing the pixel countin subtracted image (right), and using the correct calibration factor from Eq. 3.28. Notethat in this example, the gain of the atom and background image was increased after theimage was taken in order to illustrate the background light that is common to both images.In the true image, the saturation of the atom cloud is not present.To take a fluorescence image of the atoms trapped in the ODT, we recapture the trappedatoms in a MOT by turning off the ODT laser while simultaneously turning on the MOTbeams and the quadrupole field. For this step, the MOT beams and quadrupole field areset to the settings that we use for the fluorescence image (discussed above) rather thanthe settings that we use in the standard operation of the MOT. Immediately after, we993.4. Imaging Techniquestake an image of the recaptured atoms followed by a background image. In this case, it ispossible that a small number of atoms may load into the MOT during the image exposuretime. However, the same number of atoms will load into the MOT during the backgroundimage exposure time, such that the subtracted image will only show those atoms that wereoriginally trapped in the ODT. We can also close the atom shutter such that there is nocold atom beam source, which can greatly minimize this issue.3.4.2 Absorption ImagingAbsorption imaging detects the presence of atoms by measuring the decrease in light inten-sity in a near (or on) resonance beam that passes through the atomic cloud. The decreasein intensity is described by the Beer-Lambert lawdIdz= −n(z)σ(z)I(z) , (3.31)where z is the propagation axis of the imaging beam. The decrease in intensity is relatedto the atomic density n, absorption cross section σ, and the intensity of the light I. Ingeneral, the density, cross section and intensity can vary as a function of z. The intensityof the imaging beam at a particular position z can be found by integrating Eq. 3.31, whichgivesI(z) = I0 exp(−∫ z−∞n(z′)σ(z′)dz′). (3.32)By definition, the scattering rate is the absorption cross section times the incomingphoton flux per unit time (i.e., I/~ω). Therefore, the absorption cross section can bewritten asσ =Γ~ω2I· s1 + s+ (2δΓ )2=σ01 + s+ (2δΓ )2, (3.33)whereσ0 ≡ ~ωΓ2Isat=3λ22pi. (3.34)If one sets the absorption imaging beam to be on resonance (i.e., δ = 0) and the intensitylow enough such that s  1, then the absorption cross section can taken to be σ0 (andis independent of z). In addition, we ensure that the diameter of the absorption beam ismuch larger than the size of the cloud, such that we can assume that the beam intensityis constant over the size of the cloud. In this limiting case, the intensity of the absorption1003.4. Imaging Techniquesbeam at the detector74 (from Eq. 3.32) simplifies toId = I0 exp (−σ0n˜)→ ln(IdI0)= −σ0n˜ , (3.35)where n˜ is the atomic column density. The quantity ln(Id/I0) is the column optical densityof the cloud, and is a function of the two axes transverse to the propagation direction.Experimentally, the column optical density can be measured by taking an image withatoms (where the camera will observe a decrease in intensity at locations where the beampasses through the cloud), and a background image without atoms. In order to account forany dark counts or stray light in the system, we subtract from both images a “dark” imagetaken in the absence of the imaging light, but with all other parameters the same as theatom and background image. The measured column optical density is thenσ0n˜ = − ln(Iatoms − IdarkIbkg − Idark). (3.36)The column optical density is calculated for each camera pixel. Therefore, the atomnumber per pixel can be found by multiplying the column density by the effective area ofthe camera pixel, and taking into account the magnification of the imaging system. Theatom number per pixel is thenNpixel = − ln(Iatoms − IdarkIbkg − Idark)· ApixelM2σ0, (3.37)where M is the magnification of the imaging system. In our current setup with the Pixelinkcamera, A = 4.952 µm2, and we use a one-to-one imaging setup where M = 175. To find thetotal atom number, we sum the atom number per pixel over a small region of interest thatcontains the atomic cloud, and subtract from it the summed atom number in an identicallysized adjacent region. We also normalize the light level in the atom image to that of thebackground image to avoid over or undercounting due to intensity fluctuations. Figure 3.18gives an example of the atom, background, and false colour image showing the calculatedatom number per pixel.It is crucial that during the exposure time of the image only the light from the absorption74Assuming that the detector is placed somewhere after the beam passes through the atomic cloud, whichseems pretty reasonable.75We also verified the pixel size by fitting the displacement of the cloud as it falls due to gravitationalacceleration.1013.4. Imaging TechniquesAtom Background DividedFigure 3.18: Example of an absorption image of a 6Li ODT. In the atom image (left), theatoms scatter light from the imaging beam, which decreases the intensity of the beam at thelocation of atoms. The background image (center) is taken with the same parameters, butwithout any atoms present. The column optical density can be found by dividing the atomimage by the background image (right). The atom number per pixel can be calculated fromthe optical column density using Eq. 3.37, and the total atom number is found by summingover a small region that contains the atom cloud.imaging beam is incident on the atoms. However, for both 6Li and Rb the imaging beam ispicked off from the MOT beam after the double pass AOM used to set the frequency (seeSec. 2.1). Therefore, the MOT beams must be blocked with a physical shutter to ensurethat no light is incident on the atoms through the MOT axis during imaging. In the caseof the dipole trap, this is easily achieved by closing a mechanical shutter after the trap hasbeen loaded, and well before an absorption image is taken. However, an absorption imageof the MOT requires the ability to block the MOT in a time scale that is faster than theexpansion time of the cloud76. In our current system, the fastest shuttering times we havebeen able to achieve are on the order of a few hundred µs, limited by the swing speed of themechanical shutter arm and the beam size. The maximum suitable expansion times for theMOT are less than 100 µs, a limit which is placed by the high temperature and (relatively)low density of the atoms in the MOT. For this reason, we predominantly use absorptionimaging for the dipole traps, and recapture fluorescence imaging for the MOT. In general,we do not need to carefully measure the temperature or spatial distribution of the MOT,76In addition, the beam should be shuttered symmetrically so that the radiation pressure from the MOTbeams does not push the MOT to an unstable region. This can either be achieved with an AOM or with ashutter that is placed at the focus of a beam.1023.4. Imaging Techniquesso this is not an issue in our experiment.Another important consideration is that the absorption beam stay on resonance for theduration of the image. In general, the atoms will be accelerated along the propagationdirection of the imaging beam, which will act to red shift the frequency of the light. Thisissue is more pronounced with 6Li than with Rb, owing to the large mass difference betweenthe species. To counteract this effect, the exposure time of the image is kept short, such thatthe acceleration of the atoms is small and the absorption beam is not noticeably detuned.Although the general method for absorption imaging of 6Li and Rb (discussed above) issimilar, there are a few crucial differences.In Rb, we image on the f = 3→ f ′ = 4 (the MOT pump) transition. However, becausewe hyperfine pump to the f = 2 state after loading the ODT, we must repump the atomsinto the imaging state prior to taking the image. To do this, we turn on the MOT repumplight, which is tuned on resonance to the f = 2 → f ′ = 3 transition, for approximately50 µs before the image is taken. After the pumping stage, the MOT repump light is turnedoff and we take an absorption image of the ensemble. For Rb, we use an exposure time of80 µs.In 6Li, we also image on the MOT pump transition, which drives the f = 1/2→ 2P3/2transition. However, as the hyperfine splitting in the 2P3/2 manifold is less than the naturallinewidth of the transition, the pump beam excites transitions to all three (f = 1/2, 3/2, 5/2)hyperfine manifolds in the 2P3/2, some of which can decay back to the f = 1/2 groundhyperfine manifold. Therefore, a repump beam tuned on the f = 1/2 →2 P3/2 transition isnecessary. This beam is aligned such that it is roughly counter-propagating to the absorp-tion imaging beam to help counter the acceleration of the atoms due scattering from theabsorption imaging beam (see Fig. 3.9). For 6Li, we use an exposure time of 60 µs.It is also possible to image 6Li at large magnetic fields, where a closed imaging transitiondoes exist, and repump light is not necessary. This is discussed in more detail in the highfield imaging section at the end of this chapter.Temperature MeasurementAbsorption imaging can also be used to measure the temperature of the atoms along aparticular axis by monitoring the spatial width of the atomic cloud as a function of free1033.4. Imaging Techniquesexpansion time77. An in situ (that is, in the presence of the ODT, or immediately afterthe ODT has been turn off such that the atoms have had no time to expand) absorptionimage can be fit to a Gaussian profile which represents the initial position distribution ofthe atomsPi(x) =1√2piσ0exp(−x22σ20). (3.38)The velocity distribution of the atoms is given by the Maxwell-Boltzmann distributionf(vx) =(m2pikBTx)exp(−mv2x2kBTx), (3.39)where Tx is the temperature of the atomic ensemble along the axis of interest (in this case,whatever the x-axis represents, which in our case is either the horizontal or vertical axis)and m is the mass of a single atom (or molecule, in the case of a temperature measurementof, for example, Feshbach molecules). The position distribution after some expansion timeis given byPf (x, t) =∫ ∞−∞Pi(x)f(vx)dx=∫ ∞−∞Pi(x)f(xf − xt)dx=1√2piσf (t)exp(−x22σ2f (t)), (3.40)whereσf (t) =√σ20 +kBTxt2m. (3.41)Here, the integral is noted to be a convolution of the initial position distribution andthe velocity distribution. As such, the width of the resulting Gaussian is simply the widthsof the initial position distribution and velocity distribution added in quadrature. Thetemperature can be found by fitting the measured spatial width of the cloud as a functionof the expansion time, using the temperature as a free parameter. As the absorption imageis a 2D representation of the cloud, we find a temperature along the horizontal and vertical77This method is often called a “time of flight” temperature measurement because it is based on mea-suring how far an atom moves in a given time period. From this information, one can extract the velocitydistribution of the atomic ensemble and temperature of the atom.1043.4. Imaging Techniquesaxes of the cloud, which are defined with respect to the camera axis.It is important to note that this method, as described above, is only valid under threeassumptions: the velocity distribution is well described by a Maxwell-Boltzmann distribu-tion, the position distribution is well described by a Gaussian, and the atoms or moleculesare non-interacting during the free expansion. For 6Li, we can ensure the third assump-tion by performing this measurement at the s-wave scattering length zero crossing near528 G [137]. However, we have not seen a significant deviation in temperature for mea-surements (at typical densities between 5 × 1011 cm−3 and 1012 cm−3) performed at thezero crossing compared to the magnetic fields (near 750 G) that we use for 6Li high fieldimaging.6Li High Field ImagingAt large magnetic fields78the eigenstates of the 6Li system are well described by the quantumnumbers mj and mi, which represent the projection of the orbital and electron spin angularmomentum on the magnetic field axis respectively, as shown in Fig. 3.19. The eigenstatesform natural triplets, where the three states in each manifold have the same projection oforbital angular momentum mj , and differ only in the projection of the nuclear spin mi.The three lowest hyperfine levels that we trap in our ODT (the |1〉, |2〉, and |3〉 states)are separated by approximately 80 MHz. Since this separation is much greater than thenatural linewidth of the transition to the 2P3/2 state, this allows for state selective imagingof these states at high magnetic fields. At zero field these three states correspond to the|f = 1/2,mf = 1/2〉, |f = 1/2,mf = −1/2〉 and |f = 3/2,mf = −3/2〉 states respectively, whileat large field they correspond to the |mj = −1/2,mi = 0,±1〉 states.In the hyperfine Paschen-Back regime, the selection rules for an electric dipole transitionare simply that∆mj = 0,±1 and ∆mi = 0 , (3.42)which ensures that there is only a single allowable transition to the 2P3/2 state. Namely,a transition to the m′j = −3/2 manifold if we use σ− polarized light. Due to the small hy-perfine constant in the 2P3/2 manifold, the three mi projection states are only separated by78Here, large refers to a field larger than approximately 100 G, for which the 6Li system is within thehyperfine Paschen-Back regime, where the hyperfine interaction is disrupted and s and i are decoupled andindividually coupled to the axis of the external magnetic field. More details specific to the 6Li system canbe found in Chapter 5.1053.4. Imaging Techniques−2000−1000010002P3/20 100 200 300 400 500−750−500−25002S1/2|1〉 |2〉 |3〉0.0 0.2 0.4 0.6 0.8 1.Magnetic Field (G) mi3212-12-3210-1-101-12Figure 3.19: Magnetic field dependence of the Zeeman levels in the 2S1/2 manifold (bottompanel) and 2P3/2 manifold (top panel). In the2S1/2 manifold, the three states which makeup the mj = 1/2 manifold at high field are shown with dashed lines, while the mj = −1/2manifold is shown with a solid line. In the 2P3/2 state each of mj manifolds contains threelevels which differ in the projection of the nuclear spin. The splitting between these statesis approximately 1.7 MHz. The thick arrows indicate the three transitions used to imagethe |1〉, |2〉 and |3〉 states. The detuning of these transitions from the zero field transitionenergy are given in Eq 3.43.approximately 1.7 MHz (much less than the natural linewidth of the transition). However,due to the selection rule that ∆mi = 0 (i.e., the photon cannot couple to the spin degreeof freedom), each of the |1〉, |2〉, and |3〉 states has only one state to which it can couple inthe m′j = −3/2 manifold. This state is the state with an identical nuclear spin projection.Therefore, at high field the imaging transition for each of the three states is closed, and norepump light is required. The thick black arrows in Fig. 3.19 indicate these closed imagingtransitions for the |1〉, |2〉 and |3〉 states. The detuning of these three transitions (in MHz)relative to the zero field f = 1/2 → 2P3/2 (i.e., the MOT pump) transition frequency f0 as1063.4. Imaging Techniquesfunction of magnetic field aref|1〉 − f0 = −1.4 ·B + 158f|2〉 − f0 = −1.4 ·B + 82f|3〉 − f0 = −1.4 ·B ,(3.43)where the subscript indicates the state being imaged and B is the magnetic field measuredin Gauss.Note that these imaging transitions require the imaging field be circularly polarizedsuch that it drives a σ− transition. However, we currently image along an axis which isperpendicular to the vertical bias magnetic field. At best, the polarization of the imagingfield can be made up of equal parts LCP and RCP light if the field is linearly polarizedperpendicular to the magnetic field axis. This means that only half of the light has thecorrect polarization, and results in an undercounting of the atom number in a particularstate by roughly a factor of two.To understand this undercounting, consider the atom number calculated per pixel usingEq. 3.37. The intensity of the imaging light for each pixel is directly related to the numberof photons incident on each pixel. For an imaging beam where all the photons can bescattered by the atoms (i.e., all the photons have the correct polarization and the correctfrequency) thenNpixel ∝ ln(natomsnback), (3.44)where natoms and nback are the number of photons incident in each pixel in the atom andbackground image respectively. If additional light is added to the imaging beam which doesnot interact with the atoms (i.e., has the wrong polarization or the wrong frequency) thenthe number of photons incident on each pixel is increased by an amount naddn and thecalculated number becomesN ′pixel ∝ ln(natoms + naddnnback + naddn). (3.45)As natoms ≤ nback, the ln term in Eq. 3.45 will always be smaller than in Eq. 3.44 (in-dependent of naddn) and therefore the atom number will be undercounted. However, themagnitude of this underestimation depends on natoms/nback and naddn. In the specific caseof high field imaging, half the light is of the correct polarization, and half is of the wrong1073.4. Imaging Techniquespolarization. Therefore, nback = naddn and Eq. 3.45 simplifies toN ′pixel ∝ ln[0.5(1 +natomsnback)]. (3.46)In the limiting case where natoms/nback ≈ 1 (i.e., the limit of low optical density, where veryfew photons are scattered out of the beam) then N/N ′ ≈ 2, and we therefore undercount bya factor of 2. When natoms/nback < 1 then N/N′ > 2 and the atom number is undercountedby more than a factor of 2. Empirically, we find that that the atom number determinedfrom high-field imaging is approximately 4.5 times smaller than in the standard absorptionimaging case at 0 G. Given that a factor of two can be accounted for by the fact that weare only imaging half of the atoms (either the |1〉 or |2〉 state), this suggests that the under-counting due to the incorrect polarization component in the imaging light is approximatelya factor of 2.2579.This high field imaging technique can also be used to image Feshbach molecules formednear the broad Feshbach resonance at 832 G. This is because the Feshbach molecules areso weakly bound that the energy needed to break the bond shifts the absorption transitionfrequency by an amount that is less than the natural linewidth of the transitions [138].In fact, this high field imaging technique is crucial to image Feshbach molecules. If themagnetic field is ramped to 0 G for imaging, the Feshbach molecules follow the boundmolecular state responsible for the Feshbach resonance and become deeply bound, andtherefore “invisible” to the absorption imaging beam.Rb State Selective ImagingIn our standard absorption imaging, we use the MOT repump beam to pump atoms fromthe f = 2 to the f = 3 hyperfine manifold before the image is taken. However, it isalso possible to use an RF field to transfer atoms between these two hyperfine manifolds.Unlike 6Li, magnetic fields greater than 1000 G are required in order to be in the hyperfinePaschen-Back regime. Therefore, for the fields of interest in our experiment (and specificallythose used in this state selective imaging procedure), the Zeeman effect breaks the 2f + 1degeneracy of each of the hyperfine levels, and results in a linear energy shift proportionalto mf , where the energy shift between adjacent mf levels in the f = 2 and f = 3 hyperfine79In general, the undercounting of the atom number is not just limited to incorrect polarization. Anyoff-resonant frequency components in the imaging beam will have a similar effect.1083.4. Imaging Techniques-3-2-10123-2-1012abbccddeeff = 2gf = −1/3f = 3gf = 1/3Figure 3.20: Possible RF transitions between the f = 2 and f = 3 states in 85Rb. The thindashed lines show the energy of the two hyperfine manifolds in the absence of a magneticfield, while the thick solid lines show the energy of each state in a small magnetic field. Theenergy splitting between adjacent mf states is .47 MHz/G. The possible RF transitions arelabeled with dashed arrows (representing pi transitions) and solid arrows (representing σtransitions). These RF transitions are used instead of the MOT repump beam to pumpatoms out of the f = 2 manifold and into the f = 3 manifold when we perform stateselective imaging.levels is approximately .47 MHz/G80.In the presence of a small magnetic field, the energy splitting between adjacent hyperfinemf states is much larger than the linewidth of the RF transitions (which we observe to beon the order of a few kHz81), such that there are 11 distinct transitions frequencies thatwill couple specific projection states in the f = 2 manifold to the f = 3 manifold, as shownin Fig. 3.20. Therefore, using one of these RF transitions instead of a repump beam allowsfor a measurement of the number of atoms in one (or two) projection states in the f = 2manifold.One way to measure the population in each of the five mf states is to drive only the80This is described in detail in Sec 3.3 in the context of optical pumping.81Note that the natural linewidth of these transitions is exceedingly small, as the lifetime of the groundhyperfine states is nearly infinite. Therefore, the observed linewidth is set by the magnetic field fluctuationsin our setup.1093.4. Imaging Techniquespi transitions (the dashed lines in Fig. 3.20) where ∆mf = 0 due to selection rules. Acomplementary method is to determine the population from the remaining six transitionswhere ∆mf = ±1. If the measured atom number from a specific RF transition is Ni,where i = a, b, c, d, e, f labels the RF transitions as in Fig. 3.20, then the population in eachprojection state isN−2 = NaN−1 = Nb −NaN0 = [(Nc −Nb +Na) + (Nd −Ne +Nf )]/2N1 = Ne −NfN2 = Nf ,(3.47)where the subscript on the left hand side represents the mf projection state in the f = 2manifold.An example of such a measurement is given in Fig. 3.15, where this technique is usedto measure the efficiency of our optical pumping technique. While the actual atom numbercalculated using the pi transitions and the σ transitions may not agree (if, for example,the polarization of the RF field is such that it does not drive pi and σ transitions withequal probability), the relative population of each state (i.e., the percentage of the totalpopulation in each state) should agree. Figure 3.15 also shows the relative atom numbercalculated with this technique.110Chapter 4Electric Field PlatesIn many AMO experiments that work with (or plan on working with) polar molecules, it isessential to be able to apply dc electric fields greater than 5 kV/cm. The interest in polarmolecules arises from the existence of a permanent electric dipole moment, which lead tointeractions that are long range and spatially anisotropic. Electric dipole moments in thelab frame are induced by an external electric field82 because, in the absence of electric fields,there is no fixed orientation of the molecular axis and the average electric dipole momentin the lab frame is zero.The largest effect of a dc electric field with strength E is a mixing of closely adjacentrotational states where ∆N = ±1 [40]. For small fields, this mixing leads to an inducedelectric dipole which is parallel to the electric field and has a magnitude〈D〉 ∝ DEB, (4.1)where D is the magnitude of the permanent electric dipole moment and B is the rotationalconstant of the molecule. For large fields, where the field strength is larger than some criticalvalue (Ec ≡ B/D)83, the result in Eq. 4.1 is not valid, and 〈D〉 approaches its maximumvalue D. For example, for KRb in the singlet rovibrational ground state level E ≈ 4 kV/cm,and a calculation done by K. Ni [83] shows that at E = 100Ec = 400 kV/cm the inducedelectric dipole moment is about 92% of the permanent electric dipole moment. However,all hope is not lost: A field of 10 kV/cm can achieve partial lab frame alignment, andabout 50% of the maximum electric dipole moment [83]. For the LiRb system, we expect torequire approximately the same magnitude of electric field in order to realize a substantial(relative to the maximum value) induced electric dipole moment. In the rovibrationalsinglet ground state of LiRb (the X1Σ+ state) the rotational constant is about 6.5 GHz82In the frame of the molecule, there always exists a permanent electric dipole moment owing to thepreference of the elections to be nearer to one molecule than the other.83If B is in GHz and D in Debye, then Ec = (B/D) · 1.986 in kV/cm.111Chapter 4. Electric Field Plates[139] while the permanent electric dipole moment is expected to be 4.2 Debye [140], whichgives Ec ≈ 3 kV/cm.In addition to future work with polar molecules, the primary motivation for the cre-ation of these electric field plates was a proposed experiment which involved measurementsof the effect of electric fields on atomic and molecular collision resonances [80, 81, 82]. Theproposal suggests that an electric field can shift the position of existing magnetic Fesh-bach resonances or induce a Feshbach resonance for a particular partial wave collision thatpreviously only existed for a different partial wave (i.e., an s-wave induced resonance thatappears at the same location as an intrinsic p-wave resonance). The shift and width of theinduced resonance is a function of the applied electric field, and the size of the permanentelectric dipole moment. In particular, studying the effect of electric fields on heteronuclearFeshbach resonances in 6Li +Rb mixtures requires electric fields in excess of 20 kV/cm.Specifically, an electric field of 20 kV/cm will produce a shift of approximately 15% of theexperimental width of the s-wave Feshbach resonance at 394 G or 938 G (see Chapter 7)between 6Li and 85Rb. Even higher fields are needed for less polar mixtures like KRb [19],RbCs [21, 22], or NaK [24].Introduction and Design ChallengesMany AMO experiments involving the application of electric fields share the same require-ments - namely, electrodes that do not limit optical access and that have extremely lowresidual magnetism. Thin transparent conducting films (TCFs), such as ITO, applied toa transparent dielectric substrate satisfy these requirements. The ability to place the elec-trodes outside of a vacuum chamber allows for the addition of electric field capability afterthe design or construction of the apparatus. In addition, placing the electrodes in air sim-plifies the vacuum design, and can increase the flexibility of an experiment (most obviouslybecause they can be added and removed when necessary).The primary limitation to creating large electric fields is dielectric breakdown. Thisphenomenon occurs when free electrons are accelerated to sufficiently high energies by anelectric field such that they impact ionize atoms or molecules in the surrounding mediumor in the electrode itself. This leads to an avalanche production of many more electronsand ions (by electron impact ablation) generating an electric arc that rapidly reduces thefield. This process, known as Townsend discharge, limits the maximum electric field that112Chapter 4. Electric Field Platescan be generated when the flow of charge through the medium exceeds the current suppliedby the high voltage source. Breakdown also leads to material damage of the electrodes andother surfaces due to ablation and X-ray generation through bremsstrahlung which can alsodamage materials and sensitive electronics. One obvious challenge to creating large electricfields with a TCF is that electric discharge and the ion ablation that accompanies it caneasily destroy a thin film (see Fig. 4.1c). Thus, suppressing discharge of any kind is of keyimportance.Even when complete dielectric breakdown does not happen (and an electric arc is notpresent), discharge can still occur and is often evident from a corona discharge occurringnear sharp edges (see Fig. 4.1). The large curvature associated with an edge of a con-ductor causes a large potential gradient (i.e. a large local electric field) which can lead tofield electron emission followed by a local dielectric breakdown. Field emission and coronaformation can be suppressed by designing electrodes without sharp edges (i.e. by usinglarge-diameter, round-shaped conductors) and by polishing the surface to remove micro-scopic surface roughness. However, without physically stopping an ion avalanche resultingfrom Townsend discharge, the maximum achievable field is still limited by the dielectricbreakdown of the medium.Previous WorkTransparent electrodes capable of creating moderately high fields (on the order of 5 kV/cm)for experiments with ultra-cold molecules have been realized with ITO coated glass (#CH-50IN-S209 from Delta Technologies) by the group of Ye and Jin [19, 27, 141]. The ITO glasswas left exposed and the dielectric breakdown and discharge from the plates was limited bycovering all nearby conductors with many layers (4-5) of Kapton tape. With this systemthey claim to be able to apply up to ±5 kV on each plate without breakdown, generating a9 kV/cm field with a plate separation of 1.36 cm. They observed that for applied voltagesof ±3 kV on each plate (corresponding to 5.2 kV/cm field) a residual electric field wouldremain after the plates were discharged [83]. They conjectured that their pyrex glass cell(made from Borofloat by Starna Cells) developed a residual electric polarization.Other groups working with polar molecules have produced moderate electric fields ofup to 2 kV/cm using either ITO slides [24] or using four parallel rods placed at the cornersof the vacuum cell [21]. In both cases, the maximum achievable field (on the order of a113Chapter 4. Electric Field Platesfew kV/cm) is smaller than the fields needed to achieve large lab frame alignment of polarmolecules (> 10 kV/cm) or for experiments involving electric fields and Feshbach resonances(> 20 kV/cm). More importantly, the maximum achievable field in all three cases is limitedby the maximum voltage that can be applied to the plates (or rods) without causing damage(via dielectric breakdown or corona formation) to nearby experimental components or theplates or rods themselves.Figure 4.1: Left: Illustration of the effects of Townsend discharge and ion avalanche. In(a) an electric arc is established due to the breakdown of air across a 2 cm gap between agrounded metal rod and the tip of an exposed wire near 45 kV. In (b) the output currentof the high voltage supply is limited and the electric arc is not sustained. Nevertheless, thefaint blue glow of a corona discharge near the tip of the wire is visible indicating field electronemission. In (c), an ITO coated slide is shown, and the deleterious effects of discharge areapparent. At (1) the ITO coating appears darker and was damaged by corona dischargeat the corners where the local electric field is highest. At (2), the glass cracked and brokedue to electrical arcing. Conductive epoxy (3) was used to glue and electrically connect theconductor to the ITO coating and was insulated using RTV silicone adhesive. Right: Effectof discharge through glass insulator near connection of high voltage cable to a prototypeelectric field plate. In this case, the thickness of the glass (2 mm) was not enough to preventelectrical breakdown to a ground point placed against the glass.1144.1. Design of Electric Plates and HV Components4.1 Design of Electric Plates and HV ComponentsIn our design for the electric field plates, we also used a TCF of ITO. However, sincewe wanted to achieve electric fields larger than the dielectric breakdown of air (roughly20 kV/cm), we had to find a way to physically prevent the flow of charge through air. Todo this, we embedded the ITO coated dielectric substrate inside of a stack of two moretransparent substrates (making a glass sandwich84) where the outer layers have a muchhigher dielectric strength than air.The largest field that can be achieved with two parallel plates with a potential differenceV between them while separated by a distance d is Emax = V/d85. This holds in thelimit where the side length of the plates is much greater than the separation of the plates.However, in our setup the plates will need to be separated by about 4 cm (given thedimensions of the vacuum cell and the thickness of the glass used to shield the ITO), whichis on the order of a reasonable side length for the plates. In order to estimate the reductionin field strength due to the finite size of the plates we calculated the expected electric fieldat a location centered between the plates, corresponding to field that the trapped atomswould experience. Figure 4.2 shows the expected field as a function of side length for asquare plate and for a rectangular plate where the length of one side is fixed at 7.5 cm.This fixed side length was chosen as it was largest standard commercially available length ofITO. We decided to design the final version of the plates to have a conductive surface witha dimension of 7.5 cm × 10 cm, as this size maximized the electric field strength (≈ 85%of the theoretical maximum strength) and homogeneity inside the cell given the limitationsimposed by other elements in the apparatus. We also built a “trial” set of plates (beforecommitting to the larger design) which had a conductive surface that measured just under3 cm × 3 cm. With this size plate, the maximum achievable field was limited to about 60%of the theoretical maximum value.To build the plates, an ITO coated glass slide from SPI Supplies (5 cm × 7.5 cm ×0.7 mm) was laid onto a borosilicate glass flat (5 mm thick) with the coating side downand partially overlapping two strips of aluminum foil, which extend the conductive layerarea to a 7.5 cm × 10 cm area. We used aluminum foil to extend the conductive surface84While electrifying delicious, we did not get FDA approval for its public consumption.85The maximum field also depends on the dielectric constant of the material(s) between the plates. Forthis discussion, assume k = 1. The effect that the dielectric constant of the plate themselves and the vacuumcell have on the largest achievable field is discussed in greater deal in Section 4.21154.1. Design of Electric Plates and HV Components0 2 4 6 8 10 12Side Length (cm) 4.2: Expected electric field strength normalized to the largest achievable field as afunction of plate size. Black (solid) line is a square plate with the given dimensions and theblue (dashed) line is a rectangular plate with one fixed side length of 7.5 cm. The horizontalblack (blue) dot-dashed lines and corresponding circle (diamond) show the expected fieldstrength for the prototype (final) plate design.because optical access was not required far away from the cell and plate center. The ITOslide was glued to the aluminum foil using a silver conductive epoxy. Next, we applied5-minute epoxy made by Devcon to the area just outside of the ITO slab up to the edge ofthe bottom glass plate and another glass plate was pressed to the top to complete the stack.At least a 1 cm border filled with epoxy existed between the conductor and the edge of theglass sandwich, which ensured that a dielectric breakdown would not occur through theepoxy. A long flat conductor embedded in the sandwich connects the foil strip to the centerconductor of a high voltage coaxial cable (polyethylene RG-8U). Fig. 4.3 shows a pictureand schematic of the entire assembly. We found that the connection to the plate was aweak spot for breakdown. To make this connection more stable and well insulated, the endof the high voltage coaxial cable was stripped of its ground shield and inserted into a 10 cmlong cylindrical tube made of either Teflon or acrylic86 with a wall thickness of 5 mm thatextended down to the lower glass plate. We used 5-minute epoxy to affix the tube to the86Both Teflon and acrylic were found to be suitable although it was important to roughen the Teflonsurface before gluing to improve the bond.1164.1. Design of Electric Plates and HV Componentsglass, and to fill the inside of the tube to act as electrical insulation for the exposed wire.To further secure the assembly, rubber mastic tape (.065 thickness from 3M/Scotch) waswrapped around the tube and the areas where the tube and glass assembly contacted (notshown in Fig. 4.3).(6)(1)(4)(2)(5)(3)(7)(5)(6)(1)(4)(3)(2)(7)(a)(b)Figure 4.3: (a) Picture and (b) schematic (not to scale) of the electric field plate assembly. Ahigh voltage coaxial cable (polyethylene RG-8U) with a 20 cm length stripped of its groundshield (1) is inserted into a 10 cm cylindrical tube of either Teflon or, here, acrylic (2) thatextends down to the lower glass plate, and the center conductor connects to a flat metalconductor (3) that extends to aluminum foil strips (4) that contact the ITO coated glassslab (5). Insulating epoxy (6) glues the heterostructure together, maintains the positionsof all elements, and fills the gap between the external glass plates (7).We also built a high voltage switching network (similar to an H bridge) to control thepolarity of the voltages applied to the plates. The network is constructed using four singlepole double throw Gigavac high voltage relays (G71C771) and two Glassman 60 kV supplies1174.2. Testing and Characterization(EH60R01.5), as shown in Fig. 4.4. Four 1 MΩ Ohmite ceramic resistors (OY105KE) wereplaced in series between the high voltage supplies and the electric field plates in order to limitthe current spikes that occur during charge and discharging. In addition, each of the fourpossibles pathways from the voltage supplies to either of the electric field plates is connectedto ground through a 75 kV, 16 W, 1 GΩ Ohmite high voltage resistor (MOX96021007FTE).These resistors ensured that all non-operational arms of the network are grounded and allowa path for a (potential) rapid discharge of the charges stored on the electric field plates.The effect of these resistors on the charging and discharging time of the plates are discussedmore in Section 4.2.In addition to the electric plates, care had to be taken to insulate all electrical connec-tions, including those to or between resistors or those between lengths of cable. This wasachieved by embedding all connections inside of an extruded 1 inch diameter acrylic rodor block. To make the connections, a through hole was drilled along the axis of the rodthrough which the two cables were inserted. A perpendicular access hole was drilled, andused to solder the connection. Care was taken to ensure that solder connections had nosharp edges, and Super Corona Dope (from MG Chemicals) was applied to the connectionto help insulate and resist corona formation. Finally, all the holes were filled with 5-minuteepoxy or melted resin wax87. An example of a high voltage connection is given in Fig. 4.5.Due to the lack of availability of commercial high voltage connectors, almost all of the highvoltage connections used in this apparatus were permanent and hand-made.4.2 Testing and CharacterizationTo test the plates for weak spots where dielectric breakdown might occur, we placed aground cable at various locations around the plate, and slowly energized the plates up toa maximum of 60 kV (limited by the high voltage supplies). Using the design presentedabove, we observed that no arcing would occur when a grounded conductor was placedtouching any part of the assembly with the plate energized to either ±60 kV. Given theminimum distance from the inner conductor surfaces to the outside of the assembly was5 mm, this corresponded to a maximum electric field strength sustained across the outer87We found that filling volumes through narrow channels was more easily done with wax than the viscousepoxy. However, all the connections were made using 5-minute epoxy. Melted resin wax was only used toinsulate the 1 GΩ resistors in the high voltage switching network.1184.2. Testing and CharacterizationA A B B B B R2 R1 Figure 4.4: Bottom: Schematic for the high voltage switching network for the field plates.The R1 = 4 MΩ resistor in series with the electric field plates helps limit the instantaneouscurrent spikes during the charging and discharging of the plates, while the R2 = 1 GΩresistor in each connection arm ensures that all non-operational arms are well grounded,and allows for a fast discharge of the plates when the supply is disconnected. The highvoltage relays are controlled via an optically isolated digital switch. Top Left: Assemblyfor the first two HV switches (labeled B) directly after the power supplies. The four blacksquares (labeled R2) are acrylic blocks that contain the connections to the 1 GΩ resistors,which extend vertically downwards and are attached to a ground plane. Top Right: Housingfor the second set of high voltage relays (labeled A), along with the 4 GΩ current limitingresistor (labeled R1) are enclosed inside a grounded metal box. The resistors are embeddedinside an acrylic block, which is filled in with 5-minute epoxy. A second identical housingholds the second high voltage relay and current limiting resistor used for the other electricfield plate.1194.2. Testing and CharacterizationFigure 4.5: Example of a high voltage connection between two polyethylene RG-8U highvoltage cables. The stripped cable is inserted into a through hole drilled along the axis ofan acyclic rod. The conductor is soldered through a perpendicular access hole. This hole isthen filled in with 5-minute epoxy to insulate the connection.glass substrate of 120 kV/cm.During testing of the switching network, we noted that charging and discharging theplates occasionally had detrimental effects on electronic equipment in close proximity to theswitching network. We attributed this to transient spikes and/or radiation produced duringthe charging period. To avoid these effects on our experimental electronics, the switchingnetwork was placed 10 m from the experimental setup in an adjacent room, and the highvoltage relays were controlled via optically isolated digital switches in order to isolate themfrom our experimental control systems. Near the experiment (that is, in the same room asthe experiment), we ran as much of the high voltage cable as possible through groundedmetal pipes.Figure 4.6 shows how we incorporated two of these field plates into our system. Althoughwe experienced a loss of power due to reflections from the transparent substrates (whichwere not AR coated) and the ITO substrate (for which the transmission around 670 nmand 780 nm is about 80%), we did not experience any significant decrease in the MOTperformance. When the electric field plates were operated in the laser cooling setup, weplaced a grounded cable next to the connection between the high voltage cable and theplate to ensure that, in the event of a failure of this component, the arc would not be tothe optical table or to other equipment in the apparatus.Note that the measurements taken with the electric field plates were done in the previous1204.2. Testing and CharacterizationFigure 4.6: The placement of the field plates on our quartz vacuum cell. A groundedsupport rod holding the atomic beam shields is near to the top plate and is, unfortunately,a source of electrons by field electron emission when the top plate is positively biased.iteration of the setup, without the Zeeman slower. While Rb atoms were introduced intothe vacuum with an atomic dispenser placed 30 cm from the trapping region, 6Li atomswere supplied to the trapping region by an atomic beam formed directly from an effusiveoven. In order to prevent the buildup of 6Li emitted by the oven within the main sectionof the chamber, a series of beam shields were used. These shields were mounted on a metalsupport rod that ran along the top of the cell, as pictured in Fig. 4.6. More detail on thissetup can be found in the PhD thesis of M. Semchzuk [98].4.2.1 Timing of Charging and DischargingIn this high voltage assembly, we used two 60 kV / 1.5 mA power supplies from GlassmanHigh Voltage (model number EH60R01.5) with one set to negative polarity, and the otherto positive polarity. These power supplies have a total internal HV assembly capacitanceof 900 pF (not including the output cable), a 1.5 kΩ series limiting resistor88 and a 1.5 GΩbleed resistor89. The power supply has a specified rise time constant of 60 ms and, witha 75% resistive load, the fall time constant will be the same as the rise time constant.However, with no external load resistance the fall time constant is on the order of seconds(roughly τ = RC = 1.35 s in the absence of any load).Additionally, the rise time of 60 ms only holds if the total capacitance (including both88This resistor is internal to the power supply, and is placed in series with the load in order to limit theoutput current of the power supply. We added an additional 4 MΩ resistor in order to further limit themaximum current during charging and discharging.89This is the resistance that the power supply (and any associated load) will discharge through.1214.2. Testing and Characterizationthe load and internal capacitance) is low enough that the maximum current the supply candeliver will charge the net effective capacitance faster than the time constant. This timescale is given byt =CVImax. (4.2)For our supplies, the quoted rise time holds if the total capacitance is less than 1.25 nF,which limits the load capacitance to about 350 pF. The components which make up theload capacitance are the electric fields plates, the high voltage cable, and any parasiticcapacitances (for example, from the high voltage connections). The capacitance of the highvoltage cable is about 30 pf/ft, which is much larger than the capacitance of the electricfield plates (on the order of 1-5 pF from geometric considerations). However, because weplaced the power supplies and the switching electronics far away from the experiment, thisrequired a long length of high voltage cable. We measured the total capacitance of thesystem by limiting the output current of the power supply and measuring the time for thesystem to reach the set voltage90. With a current limit of 75 µA, the system took 720 ms toreach 20 kV which gives a total capacitance of 2.7 nF. This corresponds to approximately60 ft or 18 m of high voltage cable, ignoring parasitic capacitance. We estimated there is50 ft of high voltage cable between the power supply and the plates, so this is a reasonablemeasured capacitance. However, this means that the time to charge the plates (to 60 kV)is limited to a minimum of a few hundred ms. Note that we have also neglected the risetime for the load capacitor and load resistance, since this is on the order of 7 ms, given theload resistance is 4 MΩ and the load capacitance is 1.8 nF.A slightly faster approach to charging the plates is to first charge up the power supplywith the high voltage switch open (see Fig. 4.4), and then close the switch. This way, whenthe switch is closed the instantaneous charging current is provided by the energy storedin the capacitance of the HV supply. This will cause an instantaneous voltage drop thatdepends on the ratio of the two capacitances. After the voltage drop, the power supply willthen recharge the voltage at the time scales discussed above. Given our estimation of thetotal cable capacitance, this drop will be on the order of 60%. This large drop is due to thelarge capacitance from the high voltage cable that is place after the switch. This drop couldbe made much smaller either by placing the switch closer to the experiment (and moving90Note that we did not monitor the voltage of the plates directly. Any measurement of the voltage wasdone via an analog output on the high voltage power supply itself. When the electric plates were connectedto the power supply, we assume the plates are at the same potential as the supply.1224.2. Testing and Characterizationthe cable to the power supply side of the switch) or by stripping the cable of the groundedshielding to reduce to eliminate the cable capacitance.If the plates (and all the HV cable) are discharged through the power supply, the timeconstant for discharge is about 4 s. An alternative is to flip the high voltage switch todisconnect the plate and power supply, and connect each separately to ground. Now, theplate will discharge through the 1 GΩ resistor in the switching network. Due to the largecable capacitance in our setup, the time constant for this discharge is still about 2.7 s. Thisdischarge can be made faster (and, theoretically, much faster than the limit of 1.3 s if theplates discharge through the power supply, which is limited by the internal capacitanceand bleed resistor) in a similar way to improving the charge time: by decreasing the loadcapacitance on the plate side of the switch.In reality, we were not too concerned with the discharge time of the plates as we canimage the atoms in the presence of an electric field by detuning frequency of the imagingbeam (see Section 4.2.2). Therefore, we could turn off the field and wait for a completedischarge after a run was completed. In addition, we found that using a turn on time fasterthan a few hundred ms (typically when the field was switched in instantaneously using thehigh voltage relays) caused a larger atom loss than a slow turn on. We believe this effectis likely due to a rapid acceleration of electrons inside the vacuum chamber, which collidewith atoms in the ODT resulting in loss91. Given this, we typically turn the field on inabout 250 ms which was easily achievable even with the additional load capacitance of thesystem.4.2.2 Electric Field MeasurementIn our setup, the minimum field plate conductor separation is limited to 4 cm by the vacuumcell (3 cm) and the outer glass plate on the electrode assembly (5 mm). However, becauseof the high dielectric constant of the glass on the electrode assembly and the quartz cell, thefield is larger than simply the voltage difference ∆V divided by the physical distance d. Inaddition to the modification of the field strength due to the dielectrics, the field strength isalso affected by the finite size of the plates. In the case of a single dielectric92 the strength91Details on the loss that we observe from both the MOT and ODT as a function of the plate voltage canbe found in Section, in the case where the dielectric constant is the same for both the glass on the electrode assemblyand the cell.1234.2. Testing and Characterizationof the electric field at the location of the atoms (in vacuum) is|E| = K · ∆Vd· dd− t(1− 1/κ) = K ·∆Vd· η , (4.3)where K is a dimensionless factor to account for finite size of the plates (see Fig. 4.2, t is thetotal thickness of the dielectric layers, d is the physical separation between the conductinglayers and κ is the dielectric constant of the dielectric between the plates. When there aretwo dielectric layers with thickness t1 and t2 and dielectric constant κ1 and κ2 the strengthof the electric field is|E| = K · ∆Vd· κ1κ2dκ1κ2d+ t1(κ2 − κ1κ2) + t2(κ1 − κ1κ2) = K ·∆Vd· η , (4.4)which simplifies to Eq. (4.3) when κ2 = 1, equivalently t2 = 0 or κ1 = κ2.In our setup, the quartz cell (κ ≈ 4.5) walls have a total thickness of 1 cm, and theborosilicate glass plates (κ ≈ 4.6) the dimensionless quantity η is approximately 1.6, andK = .87 for the large plates. Thus, by applying ± 60 kV to each plate, we expected toobtain a field of approximately 42 kV/cm.In order to measure the electric fields produced by the plates, we performed in situspectroscopy on a sample of laser cooled 85Rb atoms inside the vacuum cell with the platesenergized, and measured the dc Stark shift of the 85Rb pump transition. In the limit of weakcoupling (where the Stark shift of each hyperfine level is small compared to the hyperfinesplitting) the dc Stark shift is [90]∆E|j,i,f,mf 〉 = −12α0E2 − 12α2E2[3m2f − f(f + 1)]k , (4.5)wherek =[3x(x− 1)− 4f(f + 1)j(j + 1)](2f + 3)(2f + 2)f(2f − 1)j(2j − 1) , (4.6)andx = f(f + 1) + j(j + 1)− i(i+ 2) , (4.7)and α0 and α2 are the scalar and tensor polarizability respectively. Since the ground statehyperfine splitting in 85Rb is 3 GHz, the limit on the weak field limit comes from thehyperfine splitting of the excited state. The pump transition couples to the f = 4 state1244.2. Testing and Characterizationof the 52S3/2 manifold, so Eq. 4.5 is valid when ∆E is less than about 100 MHz, whichcorresponds to fields of less than approximately 30 kV/cm93.The tensor polarizability is only non zero for the excited 52S3/2 state, so the effect ofan electric field is to shift the ground f = 3 hyperfine level down by a fixed amount andto break the mf degeneracy in the excited state. In the presence of an electric field, theenergy difference between the ground and excited states is changed by∆E =12(α(P )0 − α(S)0)E2 − 12α(P )2 E2[3m2f − f(f + 1)]k , (4.8)where the quantum numbers refer now to the excited state.From Eq. 4.8, it can be seen that the “pump” transition is broadened and shifted tolower frequencies (see Fig. 4.7). The relevant polarizabilities in this case are α(P )0 − α(S)0 =h·0.01340 Hz/(V/cm)2 and α(P )2 = h·−0.0406 Hz/(V/cm)2 [90]. An example of the expecteddc Stark shift as a function of applied voltage for the small prototype plate is shown in Fig.4.8. In order to determine the value of the electric field strength given a measured dc Starkshift, we assumed that the state population is evenly distributed among the mf states, andwe use the weighted mean Stark shift of the pump transition. Any associated uncertaintyon the determination of the field strength is due to the unknown state populations. As areference, if the spin population were all in the |mf | = 3 states, we would underestimate thefield by about 20% and if the population were all in the mf = 0 we would overestimate thefield by about 10%.We performed two types of spectroscopic measurements, an example of which are shownin Fig. 4.9 for a potential difference across the plates of ∆V = 40 kV. In each case, we usedthe shift in the resonance frequency of the pump light from the “on resonance” case, definedthe be the detuning in the absence of a Stark shift (i.e., with the plates off)94. In the firstcase, we measured the 85Rb MOT fluorescence as a function of the detuning of νpump fromresonance with νrepump set on resonance. In the presence of an electric field, the largestdetuning from the resonant case for which the MOT will operate shifts to lower frequencies,as shown in Fig. 4.9. However, an accurate determination of the electric field from the93Although we had hoped to measure fields larger than this, technical issues prevented us from measuringfields larger than 18 kV/cm. At the very least, this made the analysis of the Stark spectroscopy morestraightforward.94In some cases, this was not always at a detuning of zero which is likely due to a small offset in thelocation of the lock.1254.2. Testing and CharacterizationFigure 4.7: The dc Stark shift of the ground (f = 3) and excited state (f ′ = 4) levelsis illustrated. The scalar polarizability is larger in the excited state and it therefore shiftsdown more than the ground state. The tensor polarizability (zero for the ground state) shiftsthe excited state levels up by an amount proportional to m2f . Thus the pump transitionmoves to lower frequencies and also broadens somewhat indicated by the “min” and “max”transition energy differences.overall shift is confounded by the broadening of the operating point of the MOT and duethe unknown residual Zeeman shift resulting from light pressure imbalances which push theMOT center into a region where the magnetic field is non-zero.In the second case, we performed a magnetic-field-free measurement of the absorption ofthe atoms. For this, we imaged the shadow cast by the 85Rb atomic cloud in a laser beamtuned to the “pump” transition with the MOT light and magnetic field extinguished. Theabsorption profile was integrated, and the total integrated signal is plotted as a function ofthe imaging light detuning. In the presence of an electric field, the maximum signal is seento shift to lower frequencies (see Fig. 4.9). In addition, the transition is expected to broadendue to the broken degeneracy of the excited hyperfine manifold (see Fig. 4.7). However, thestandard deviation of the spread in transition frequencies at 14 kC/cm is only 2 MHz (seethe grey shaded region in Fig. 4.8 and therefore, the transition is only expected to broadenby approximately 10%. This is consistent with the observed widths in Fig. 4.9 - that is, awidth of 4.2 MHz in the absence of an electric field and a width of 4.5 MHz at 14 kV/cm.1264.2. Testing and Characterization0 10 20 30 40 50 60Applied Voltage (kV)−20−15−10−50DCStarkShift(MHz)Figure 4.8: Expected dc Stark shift as a function of applied voltage. The solid line representsthe mean shift assuming that the state population is evenly distributed among themF states.This is calculated by computing finding the weighted average of the five distinct transitionfrequencies (see Fig. 4.7). The grey shaded region is the standard deviation of this weightedaverage, and represents the expected broadening of the transition. The dashed lines arethe maximum minimum shift assuming the atomic population is entirely in the mf = 0 and|mf | = 3 state respectively. The electric field at a given voltage is determined using Eq. 4.4where K = .58 and η = 1.6. The black dots are the measured dc Stark shift at the givenplate voltage. These measurements were taken using the smaller prototype plates. Theerror on the measured dc Stark shift is smaller than the size of the dots. Figure from [142].One disadvantage of this method is that it requires multiple experimental cycles in orderto determine the field strength (here, each integrated signal for a fixed detuning is a separateexperimental run). This is in contrast to the first method, where the frequency of the pumpbeam can be swept from large to small detuning, while the MOT fluorescence is capturedon a photodiode (or a camera with a very fast exposure time). For the majority of thedata presented in the following sections, we were concerned with how the magnitude of themeasured electric field varied with time. Therefore, the first method (i.e., quickly sweepingthe MOT pump field) was used to measure the strength of the electric field.When we first installed the prototype (small) plates into the system for testing wemeasured the dc Stark shift as a function of the applied potential difference. The result is1274.2. Testing and Characterizationno atoms(a)−20 −15 −10 −5 0no atoms(b)(c)−35 −20 −5 10 25(d)0.0 0.2 0.4 .6 0.8 1.0MOT Detuning (MHz) 0.2 0.4 0.6 .8 1.0Imaging Detuning (MHz) 4.9: The influence of the dc Stark effect on the operation of the 85Rb MOT (aand b) and on the integrated absorption of a laser beam by the cold 85Rb atomic cloudin the absence of a magnetic field versus detuning (c and d). In (a) and (b), the MOTfluorescence is plotted versus the pump detuning. With no electric field present, the MOToperates for detunings as small as -2 MHz, whereas this transition point shifts down by morethan 11 MHz with a field of 14 kV/cm applied by the plates. The dashed line indicatesthe fluorescence level consistent with no atoms trapped. In (c) and (d) the integratedabsorption of a imaging beam (with the MOT light extinguished) is plotted as a function ofthe detuning of the imaging light from resonance. A repumping beam was simultaneouslyapplied during the absorption measurement. The frequency at which the absorption ismaximum shifts down by 13.9 MHz corresponding to the expected shift from the appliedelectric field of 14 kV/cm. The solid line is a fit to the data where the mean value for thefits are −0.6 MHz in (c) and −14.5 MHz in (d). Figure from [142].1284.3. Field Shielding and Trap Lossshown in Fig. 4.8. Overlaid is the theoretical value of the mean, maximum and minimumexpected dc Stark shifts given the expected electric strength at each potential differenceusing Eq. 4.4 where K = .58 and η = 1.6. In this case, it is interesting to note that theelectric field (and thus, the dc Stark shift) scaled as expected with the applied potentialdifference. This indicates that there is no shielding mechanism that turned on at large fieldstrengths. This observation is extremely different then what we observed when we installedthe larger field plates. The next section discusses these observations.4.3 Field Shielding and Trap LossWhen we tried to apply large voltages, corresponding to fields in excess of 15 kV/cm, wenoticed a large loss of atoms from the MOT and ODT. This loss was the primary reasonwhy we could not verify fields larger than 18 kV/cm using in situ dc Stark measurements.In addition, we observed the decay of the electric field strength over time while the plateswere held at a high voltage, as well as a residual field that remained after the plates weregrounded. We believe the bulk of these effects were due to field emission of electrons withinthe vacuum from the support rod on the beam shield. Our observations suggest that itis critical to ensure that the vacuum chamber near the field region is free of any chargesources.4.3.1 Electric Field ShieldingInitially, we studied the reduction of the electric field due to shielding as a function of thetime the plates were connected to the high voltage supplies. We first grounded one plateand applied either a positive or negative bias voltage of 40 kV to the other plate. Thebehavior shown in Fig. 4.10 suggests that the applied voltage is shielded by the movementof free charges and the number of these charges, and thus the extent of the shielding, isinfluenced by the polarity of the bias voltage as well as the choice of which plate is energizedand which is grounded. Based on the data, we believe that the free charges responsible forthis shielding behavior are primarily electrons produced by field emission of grounded metalparts within the chamber. For sufficiently high voltages, we believe that these free electronscan be accelerated to energies high enough to produce, upon collision with the cell walls,secondary positive ions that can also migrate and lead to shielding.1294.3. Field Shielding and Trap LossFigure 4.10: The measured electric field strength at the center of the cell versus the timethe plates are kept energized. In all four cases, one plate is biased at either ±40 kV, whilethe other plate is grounded. The low initial value and rapid reduction in the voltage occurswhen the top plate is biased with +40 kV (red squares). We note that the top plate is closeto the metal support of the beam shield, and this behavior suggests that field emission ofelectrons is responsible for the rapid build-up of shielding charges. In the other cases, wherethe field emission rate of electrons should be lower, we observe a slower and less completeshielding of field at the center of the cell. Figure from [142].Fig. 4.10 shows the electric field strength measured some time after the plates wereconnected to the high voltage supply. Each data point is an average of four measurements.In order to minimize the effects of accumulated charges from previous measurements, theplates were run with the opposite polarity between each measurement, and we waited ap-proximately 20 minutes before starting a data run for the next time in the series.When the upper plate is positively biased, the shielding is the most rapid and dramatic.This large and rapid shielding is in contrast to the moderate and slower shielding in allother configurations even when the lower plate is positively biased (with the upper plategrounded). We believe that this asymmetry is due to the atomic beam shield (see Fig. 4.6)which has a metal support rod that runs along the top of the cell. This support rod ismade from stainless steel and is grounded by virtue of its contact with a CF flange that alsosupports the lithium oven. Moreover, this rod was within a few cm from the conductive1304.3. Field Shielding and Trap Losslayer inside the top electrode. A large positive voltage applied near the grounded supportrod is expected to lead to a much larger electron field emission than an equally large negativevoltage.We also observed shielding in the other configurations including those with a negativebias. Any electrons on the surface or shallowly embedded inside the quartz cell wall willbe accelerated away towards the grounded support rod or towards the opposite groundedplate when a negative bias is applied to a plate. These electrons will acquire a considerablekinetic energy and could produce secondary positive ions when they impact the oppositecell wall or grounded support rod. These positive ions will be accelerated back towards thenegative plate and will also create secondary electrons when they impact the cell wall. Webelieve that this process is what leads to more electric field shielding for longer “on times”.Based on the shielding behavior, the rates of secondary ion and electron production appearto be similar for the other three configurations. In summary, the shielding behavior appearsto be distinct in the case when the upper plate is positively biased and field emission ofelectrons from the grounded support rod is most probable.The electric field plates were also operated with a positive bias on one plate and anegative bias on the other. Figure 4.11 shows the field strength as a function of time whenthe total potential difference is 40 kV and 60 kV for either field polarities. Even when thetotal potential difference is the same as in Figure 4.10 where only one plate is biased, thefield decay as a function of time is much weaker. Only when the total potential differenceis increased to 60 kV (such that each plate is now biased with ±30 kV) does the shieldingbecome significant. This behavior continues to suggest that the shielding behavior is mostdependent on the bias of the top plate (strongest when large and positive) and more weaklydependent on the total potential difference between the two plates.To simulate using the plates as part of an experiment that requires multiple repeatedruns, we measured the field strength as a function of the run number with total potentialdifference of 40 kV. In this case, the expected electric field at the center of the cell was14.0 kV/cm. During each experimental cycle, the plates were energized for 500 ms, andthe time between each run was approximately 20 s. The top panel of Fig. 4.12 shows themeasured field strength after each run. It is evident that some of the shielding charges thatare produced while the plates are on linger in the cell region during the time the plates aregrounded leading to more and more free charges available for shielding and lower measured1314.3. Field Shielding and Trap Loss60 kV 40 kV Figure 4.11: The measured electric field strength at the center of the cell versus the timethe plates are kept energized, in the case that one plate is positively biased and the othernegatively biased. The top (bottom) panel is the case where the total potential differencebetween the plates is 60 kV (40 kV), such that each plate is biased at ±20 kV (±30 kV).The solid lines indicate the expected electric field for each case, and the dashed (black) anddot-dashed (red) lines indicate the measured residual field when the plates where turned offfor an upwards (downwards) field.fields for subsequent runs. The configuration with the upper plate positively biased (redsquares) leads to a much larger number of shielding charges such that after the first run,with the plates on for just 500 ms, the field is already shielded to a value below 10 kV/cm.This is consistent with our observations in the case of a single positively biased top plate,shown in Fig. 4.10. By the 20th run, the measured field in this configuration was only7.5 kV/cm - approximately half the expected field. When the upper plate is negativelybiased and the lower plate positively biased (black circles), we observe, for the first fewruns, a field magnitude very close to the expected field for the potential difference betweenthe plates. However, with each subsequent run, the measured field is less as a result of the1324.3. Field Shielding and Trap LossFigure 4.12: The effects of shielding on the electric field strength over sequential experimen-tal runs. The field is either kept in the same polarity for each run (top panel) or the polarityis switched between each run (bottom panel). In each run, the plates were energized for500 ms before the field strength was measured. There is approximately 20 s between eachrun. The shielding observed in the top panel can be partially mitigated by alternating thepolarity of the field between sequential runs. Figure from [142].lingering of the shielding charges during the off time of the plates.It is neither helpful to have an electric field that changes in strength during the courseof an experiment, nor to have a field that is weaker than theoretical achievable value dueto a large amount of shielding. We attempted to fix both problems by flipping the fielddirection between each experimental run. As shown in the bottom panel of Fig. 4.12, wefound that after a few polarity changes, the field strength reached a steady state value of12 kV/cm. This is larger than the maximally shielded value, though not as large as theexpected field of approximately 14 kV/cm.1334.3. Field Shielding and Trap Loss4.3.2 Residual FieldsNot only did free charges within the chamber act to shield the electric field during thetime in which the plates were energized, they also resulted in a residual electric field thatpersisted within the vacuum cell after the electrodes were grounded.As in the case of field shielding, the residual field was greatest when a large positive biaswas applied to the top plate. For example, 250 ms after a 40 kV positive bias voltage wasapplied to the top plate (while keeping the bottom plate grounded), the measured electricfield strength in the center of the cell was 9 kV/cm, and 250 ms later after the top plate wasgrounded, a 5 kV/cm residual electric field was observed in the center of the cell. Althoughwe did not verify this, we assume that the residual electric field was oriented antiparallelto the expected field direction and was equal to the difference between the expected fieldmagnitude and the measured field magnitude for each configuration. Figure 4.11 shows theresidual field after the plates had been energized for 10 s in the case where a total potentialdifference of 40 kV or 60 kV was applied, using both a positive and negative bias. Noticethat the residual field is larger for the case where the positively biased plate is at the topof the cell.These observations suggest that the shielding charges that are created and accumulatewhile the plates are energized actually become embedded in the quartz cell walls such thatthey do not immediately leave when the plates are grounded. Nevertheless, we observedthat it was possible to remove the residual field by running the electric field plates in theopposite polarity. This was the primary motivation for the construction of the high voltageswitching network and appears to be critical when free charges (or, more importantly, asource of free charges) exist near the electric field region.4.3.3 Trap LossIn addition to the shielding effects, we also observed a loss of 85Rb atoms held in theMOT when large voltages were applied to the plates. This observed atom loss was theprimary technical limitation preventing us from measuring fields with a strength greaterthan 18 kV/cm. In addition, we also observed a loss of 6Li atoms held in the MOT andODT. Since our interest was in applying large electric fields to ultra-cold atoms held in anODT, we studied the loss of 6Li from the ODT.As Fig. 4.13 shows, we found that the 6Li atom loss was larger, and became observable1344.3. Field Shielding and Trap Lossat a smaller plate voltages, when using a positive bias voltage versus a negative bias voltage.Moreover, the loss due to the positive bias was larger when the upper plate was chargedand the lower plate was grounded. This asymmetry is consistent with our field shielding ob-servations where the accumulation of shielding charges is largest when the upper plate waspositively biased. In the case of a negatively biased plate (with the other plate grounded),there is almost no difference in the atom loss associated with the two configurations. Alongwith the observations of field shielding, this suggests that collisions with accelerated elec-trons and secondary ions within the chamber are responsible for the atom loss from theODT.Figure 4.13: Loss of 6Li atoms from the ODT as a function of the negative (top panel)and positive (bottom panel) plate voltage with the second plate grounded. The atom lossis much stronger when a positive bias is used (especially when the top plate is positivelybiased) due to strong field emission of electrons of the support arm of the beam shield. Theweaker loss when a negative bias is used is attributed to the acceleration of free electronsthat already exist within the vacuum or that are partially embedded and subsequentlyejected from the quartz cell during the on time of the plate bias. Figure from [142].1354.3. Field Shielding and Trap LossAlthough we did not see observable loss from the 6Li MOT when only one plate wascharged even up to 40 kV (corresponding to an expected electric field of 15 kV/cm), we didobserve 6Li atom loss from the MOT when both plates where charged (one with a negativebias, and the other with a positive bias) such that the total potential difference betweenthe plates was larger than 50 kV. The loss in this configuration for both the MOT and theODT is shown in Fig. 4.14. It is interesting to note that loss from the ODT for either fieldpolarity is approximately the same in the case where only the bottom plate is positivelybiased while the top plate is held at ground (see Fig. 4.13) for the same plate voltage,even though the total potential difference is twice as large. Potentially, this suggests amodification of the field lines in the cell (because the beam shield is always held at ground)such that less of the field emitted electrons predominantly travel through a region of spacethat is not occupied by the ODT.It is interesting to note that the loss from the ODT for the same potential bias of a singleplate is approximately the same in the case where only the bottom plate is positively biased(see Fig. 4.13), even though the total potential difference is much greater. In addition,the loss is weaker in the case where the plates have an opposite bias when the top plateis positively biased. Potentially, this suggests a modification of the field lines in the cell(because the beam shield is always held at ground) such that less of the field emittedelectrons do not travel through the ODT. In general, these loss observations indicate thatthe loss rate (similar to that for elastic collisions with neutral atoms [129]) is smaller froma deeper trap (for the same electric field plate potential difference), and suggests that notall collisions ionize atoms (which would result in trap loss regardless of the trap depth).4.3.4 Tests with Small Prototype PlatesIn our prototyping stage, some initial tests were conducted with smaller electric field plates.In particular, the conductive layer was constructed out of a much smaller ITO slide, mea-suring just under 3 cm × 3 cm. Using these smaller plates, even at similar applied voltagesand observed fields up to 15 kV/cm, we did not observe field shielding or the trap losseffects described above. This is clear in Fig. 4.8 where the measured electric field scales asexpected with the applied voltage. In this case, we concluded that no shielding effects turnon as the voltage is increased. As both plates were centered on the quartz cell, we attributethis difference in behavior to the larger distance between the plates and the beam shield1364.3. Field Shielding and Trap LossODT MOT Figure 4.14: Loss of 6Li atoms from the ODT (top panel) and MOT (bottom panel) as afunction of the absolute value of the plate bias. In this case, one plate has a positive bias,and the other has a negative bias. Figure from [142].support rod. We believe that this observation re-enforces the hypothesis that field emissionof electrons from the top of the beam shield is largely responsible for the fast shielding of theelectric field and the atoms loss. Therefore, in order to achieve large electric field strengths,commensurate with the potential difference applied across the plates, it is crucial to ensurethat the region inside of the vacuum chamber near to the plates is free of any charge source(for example, grounded metal parts).137Chapter 5Overview of 6Li2 System andPhotoassociationThe homonuclear Li2 system provides a relatively easy (as compared to LiRb) system inwhich to build our knowledge base and techniques for making ultra-cold molecules. Atthe same time, 6Li pairs play a role in many-body physics of the BEC-BCS crossoverregime, and can be used to improve our understanding of coherent processes with respectto atom-molecule dark states. The broad Feshbach resonance in 6Li allows for very efficientevaporation in ODTs from which we can access many different regimes of interest in ultra-cold physics, such as weakly or strongly interacting degenerate Fermi gases, to molecularBECs and BCS pairs.5.1 Introduction to 6Li and 6Li2 SystemIn the absence of a magnetic field, the 22S1/2 ground state of6Li has two hyperfine levels(f = 1/2 and f = 3/2) separated by approximately 228 MHz. In the presence of a magneticfield, the spin dependent part of the single atom Hamiltonian isHint = ahf(~s ·~i)+ ~B · 2µe~s− µN~i~, (5.1)where the first term represents the atomic hyperfine interaction whose strength is governedby the atomic hyperfine constant ahf . In the case of the 22S1/2 ground state of6Li, ahf =a2S ≈ 152.2 MHz (see Section 2.1.1 and Table 2.1). The second term represents the Zeemaninteraction, which can be treated as a perturbation to the system when it is much smallerthan the hyperfine interaction. This occurs for B  a2S/µe ≈ 10 G, the so called “low fieldlimit”95.95For 6Li and most of the alkali atoms, µe  µN .1385.1. Introduction to 6Li and 6Li2 SystemThe six Zeeman sublevels from the two ground hyperfine states in 6Li (two from thef = 1/2 manifold and four from the f = 3/2 manifold) are often labeled as states |1〉 to|6〉 in order of increasing energy, as shown in Fig. 5.1. At low fields the Zeeman statescan be written in the hyperfine basis, where the states are labeled by |f,mf 〉. However,at large magnetic fields when the Zeeman splitting is comparable to (or larger than) thehyperfine splitting, the hyperfine coupling is broken by the external magnetic field, and fis no longer a good quantum number. In this case, the atoms are treated as being in a pureproduct state basis labeled by |i,mi〉 |s,ms〉. In general (that is, at any magnetic field) theeigenstates of Eq. 5.1 can be written as a linear superposition of the |i,mi〉 |s,ms〉 states inthe form [143]c− 12∣∣∣∣ms = −12 ,mi = mf + 12〉+ c 12∣∣∣∣ms = 12 ,mi = mf − 12〉, (5.2)where s = 1/2 and, for 6Li , i = 1. Each of the six eigenstates can be written in the formof Eq. 5.2 where mf takes on the six allowed values from the projection of f = 1/2 andf = 3/2 onto the magnetic field axis [143]. This implies that, in general, only the total spinprojection mf is a good quantum number. Table 5.1 summarizes the quantum numbersassociated with each of the six states in the low field and high field regimes.State Label High Field |mi,ms〉 Low Field |f,mf 〉|1〉 |1,−1/2〉 |1/2, 1/2〉|2〉 |0,−1/2〉 |1/2,−1/2〉|3〉 |−1,−1/2〉 |3/2,−3/2〉|4〉 |−1, 1/2〉 |3/2,−1/2〉|5〉 |0, 1/2〉 |3/2, 1/2〉|6〉 |1, 1/2〉 |3/2, 3/2〉Table 5.1: State labels for the six Zeeman sublevels in the ground 2S1/2 manifold of6Li.Note that i = 1, s = 1/2, and for the ground state l = 0. The high field limit is valid forB  10 G.In our experiment, the three states of most interest are the states which, at high magneticfield, make up the ms = −1/2 manifold. The states |1〉 and |2〉, which correspond to thetwo degenerate spin projections of the f = 1/2 hyperfine level at 0 G, are the states initiallytrapped in the ODT. The third state, |3〉 is often used in RF spectroscopy at high magnetic1395.1. Introduction to 6Li and 6Li2 System0 20 40 60 80 100 120 140 160Magnetic Field (G)−300−200−1000100200300Energy(MHz)F=3/2F=1/2|6〉|5〉|4〉|3〉|2〉|1〉Figure 5.1: The six Zeeman states corresponding to the two ground hyperfine states in 6Li.At large magnetic fields, f is no longer a good quantum, and the states are labeled by|i,mi〉 |s,ms〉, see Table 5.1. As shown, the six Zeeman sublevels are often labeled as states|1〉 to |6〉 in order of increasing energy. In our experiment, 6Li atoms are prepared in thetwo lowest hyperfine states |1〉 and |2〉.fields, or experiments involving three-component Fermi gases [144]. The general form ofthese three states, following from Eq. 5.2 are [143]|1〉 = sin θ+∣∣∣∣12; 0〉− cos θ+∣∣∣∣−12; 1〉(5.3)|2〉 = sin θ−∣∣∣∣12;−1〉− cos θ−∣∣∣∣−12; 0〉(5.4)|3〉 =∣∣∣∣−12;−1〉, (5.5)1405.1. Introduction to 6Li and 6Li2 Systemwheresin θ± =1√1 + (Q± +R±)/2(5.6)Q± =(µN + 2µe)Bahf± 12(5.7)R± =√(Q±)2 + 2 . (5.8)When B = 0 G, it can be seen that sin θ+ = cos θ− =√1/3 and cos θ+ = sin θ− =√2/3,while at high magnetic fields (satisfying B  a2S/µe) sin θ± = 0 and cos θ± = 1. Notethat, using the appropriate Clebsh-Gordan coefficients, the eigenstates in Eq. 5.3–5.5 canbe written in the |f,mf 〉 basis, which are good quantum numbers at small magnetic fields.5.1.1 Two-Atom Scattering StateMost relevant to much of the discussion in this chapter is not the uncoupled free atom states,but rather the two-atom scattering state. At low magnetic fields (typically B ≈ 0 G wherewe work with a weakly interacting Fermi gas) this scattering state acts as the initial state inour photoassociation (PA) experiments. At large magnetic fields, this two-atom scatteringstate is coupled to a bound state in the X(11Σ+g ) potential, resulting in a broad FR nearB = 832 G. At these high fields, the dressed molecule state (which is a superposition ofthe v′′ = 38 bound state in the X(11Σ+g ) potential and the two-atom scattering state) isthe initial state for photoassociation. The details for the Feshbach resonance and scatteringstate at high fields is discussed in greater detail in Section 5.1.2.In the ODT, we trap atoms in an incoherent mixture of the |1〉 and |2〉 states. At lowmagnetic fields, this is the only stable hyperfine mixture because it cannot decay throughinelastic two-body collisions due to the conservation of mf . At high magnetic fields, it ispossible to create stable mixtures of the |1〉, |2〉 and |3〉 states. However, our photoasso-ciation experiments to date have focused solely on |1〉 and |2〉 state mixtures. Therefore,the two-atom scattering that we will consider is a mixture of these two hyperfine states.Because the atoms are fermions, the two-atom scattering state must be antisymmetric, andtherefore the incoming channel for two-atom scattering must be|12〉 = 1√2[|1〉1 |2〉2 − |2〉1 |1〉2] , (5.9)1415.1. Introduction to 6Li and 6Li2 Systemwhere the subscripts label the first and second atom.The scattering state in Eq. 5.9 is more illuminating when written in the |S,mS ; I,mI〉basis using Eqs. 5.3–5.5 (and the applicable Clebsh-Gordan coefficients). Here, ~S = ~s1 + ~s2and ~I = ~i1 + ~i2 are the total electronic and nuclear spin respectively. Note that the totalelectronic spin can take on two possible values (S = 0, 1) which represent a collisional statethat has singlet (S = 0) and triplet (S = 1) character. Written in this basis, the scatteringstate is [143]|12〉B = sin θ+ sin θ− |1, 1; 1,−1〉+ sin θ+ cos θ−(√13|0, 0; 0, 0〉 −√23|0, 0; 2, 0〉)(5.10)+ cos θ+ sin θ−(√13|0, 0; 0, 0〉+√16|0, 0; 2, 0〉 −√12|1, 0; 1, 0〉)(5.11)+ cos θ+ cos θ− |1,−1; 1, 1〉 . (5.12)At B = 0 (where sin θ+ = cos θ− =√1/3 and cos θ+ = sin θ− =√2/3), the two-atomscattering state reduces (quite nicely) to|12〉B=0 =√29(|1, 1; 1,−1〉+ |1,−1; 1, 1〉 − |1, 0; 1, 0〉) +√13|0, 0; 0, 0〉 , (5.13)which has both triplet (first term) and singlet (second term) character.Foreshadowing the labeling of the 6Li2 molecular states (see Section 5.1.4), it is usefulto once again rewrite the scattering state in Eq. 5.13, this time in the molecular basis|NSIJF 〉 where N is the nuclear orbital angular momentum, ~J = ~N + ~S is the totalangular momentum apart from the nuclear spin (recall that L = 0 for the ground state of6Li), and ~F = ~J+~I. At the temperatures where photoassociation is performed in our system(typically < 2 µK), p-wave (and higher-order) collisions are greatly suppressed. Therefore,the initial two-atom scattering state has N = 0, which corresponds to an s-wave collision.Although s-wave collisions are disallowed in a spin polarized fermionic system due to thePauli-exclusion principle, they are permissible using the |12〉 hyperfine mixture. In thismolecular basis, the scattering state at 0 G is|12〉B=0 =√23|0, 1, 1, 1, 0〉+√13|0, 0, 0, 0, 0〉 , (5.14)1425.1. Introduction to 6Li and 6Li2 Systemwhich follows directly from Eq. 5.13 and the appropriate Clebsh-Gordan coefficients. Thevalue of F in both admixture states is constrained to be F = 0 due to symmetry consid-erations. That is, states with a total spin (~G = ~f1 + ~f2) of G = f1 + f2, f1 + f2 − 2, . . .are symmetric while states with G = f1 + f2 − 1, f1 + f2 − 3, . . . are antisymmetric [89]. Inorder that the scattering state be antisymmetric, the total spin G must be antisymmetricbecause in an s-wave collision (for which N = 0) the spatial part of the wavefunction is, bydefinition, symmetric. For atoms in the |1〉 and |2〉 state, f1 = f2 = 1/2 and this thereforerestricts the total spin to G = 0 (the other possibility from angular momentum addition,G = 1, is symmetric). For this scattering state, since N = L = 0, the total spin G isthe same as F . The details of the symmetry considerations for the molecular states arediscussed in greater detail in Section 5.1.4.At high magnetic fields (satisfying B  a2S/µe) where sin θ± = 0 and cos θ± = 1, thetwo-atom scattering state in Eq. 5.10 written in the |S,MS ; I,MI〉 basis is|12〉B = |1,−1; 1, 1〉 . (5.15)The pure triplet character of this two-atom scattering state can be understood because theuncoupled free atom states (|1〉 and |2〉) have the same spin projection (ms1 = ms2 = −1/2),and only vary in the nuclear spin projection. Therefore, the total spin of this spin polarizedensemble must be S = 1. In the molecular basis |N,S, I, J, F 〉, the scattering state at largemagnetic fields is|12〉B =√16|0, 1, 1, 1, 2〉 −√12|0, 1, 1, 1, 1〉+√13|0, 1, 1, 1, 0〉 , (5.16)where mF = 0 for all three states because mS + mI = 0. However, this scattering state isnot the actual eigenstate at high magnetic field because of coupling to other states. In 6Li,two s-wave Feshbach resonances occur for the |1〉 and |2〉 state mixtures due to a couplingof the scattering state with two bound states in the X(11Σ+g ) potential. The couplingof the scattering state with the bound molecular state modifies the initial state used inphotoassociation. It’s important to note that the entrance channel state is not a singlestate, but rather a continuum of states that differ in the relative momentum of the collidingatomic pair. The origins and implications of these Feshbach resonances are discussed in thefollowing section.1435.1. Introduction to 6Li and 6Li2 SystemAlthough the two-atom scattering state exists in the continuum of the 22S1/2 + 22S1/2interatomic potential, the energy of the initial state differs from the asymptotic energy of thepotential (from which the binding energy of the bound states are defined). In the absenceof magnetic field, the energy of the f = 1/2 hyperfine level is lower than the asymptoticenergy by (see Eq. 5.1)∆E = ahf(~s ·~i) = a2S2[f(f + 1)− i(i+ 1)− s(s+ 1)] = −a2S . (5.17)Given this, the initial unbound two-atom scattering state is −2a2S below the hyperfinecenter of gravity of the 22S1/2 + 22S1/2 threshold. The energy of the hyperfine center ofgravity corresponds to the same energy as the 22S1/2 + 22S1/2 asymptotic energy. In thepresence of a magnetic field, the scattering state energy is further modified by the Zeemanshift of the |1〉 and |2〉 levels. This energy shift has been calculated, and is shown (as afunction of magnetic field) in Fig. 5.2. Note that this energy shift means that the bindingenergy of the molecular state cannot be inferred directly from the photoassociation laserfrequency. In the case of the excited states, the binding energy is computed by adding 2a2Sto the D1 transition energy [145] and subtracting the measured photon energy for the PAloss feature. For the ground states, the binding energy is computed by adding 2a2S to thedifference in photon energy between the two PA lasers at the frequencies which correspondto the two-photon resonance.5.1.2 s-wave Feshbach Resonances in the |12〉 Hyperfine MixtureThis section introduces the basic physics of FRs and some of the more important andrelevant properties of the resonances and the associated Feshbach molecules. Following this,the details of the s-wave FRs in the |12〉 hyperfine mixture and the resulting modificationsto the initial photoassociation state |a〉 are discussed. More detailed information regardingFeshbach resonances can be found in two excellent reviews by Chin et al. [10] and Julienneet al. [146]. In addition, there are many useful references which discuss the specifics ofFeshbach resonances in the Li system [67, 70, 143, 147, 148, 149, 150].Feshbach resonances in 6Li have been very well studied experimentally, starting withthe measurement of the scattering length zero crossing near 528 G in 2002 [137]. Thiswas closely followed by a demonstration of magnetic field control of elastic scattering [151]1445.1. Introduction to 6Li and 6Li2 System0 200 400 600 800 1000Magnetic Field (G)−3000−2500−2000−1500−1000−5000Energy(MHz) |2〉|1〉Figure 5.2: Magnetic Field dependence (solid lines) of the |1〉 and |2〉 states. The dashedline is a sum of the energies of the |1〉 and |2〉 states, and represents the energy differencebetween the initial unbound two-atom scattering state and the hyperfine center of gravityof the 2S1/2 + 2S1/2 threshold. The energy of the hyperfine center of gravity correspondsto the same energy as the 2S1/2 + 2S1/2 asymptotic energy. This energy must be accountedfor in the determination of the binding energies of the bound molecular states.and a measurement of the interaction energy [152] near the s-wave Feshbach resonance at832 G. Shortly after, three groups reported the conversion of a atomic Fermi gas to a stablemolecular Bose gas [58, 59, 60]. Finally, RF spectroscopy on weakly bound molecules wasused to precisely determine the Feshbach molecule binding energy in 2005 [153] and toprecisely characterize the Feshbach resonance in 2013 [154]. For mixtures of the |1〉 and|2〉 states, there also exists three p-wave resonances between 150 and 220 G and a narrows-wave resonance near 543 G, which are discussed in [155].Feshbach resonances can be understood by considering a scattering event with twochannels96: the open (or entrance channel) of the two colliding atoms, and a closed channelwhich represents a molecular potential that can support bound states at energies near the96The internal states of a particle in the initial (before) and final (after) scattering states are known asa channel. Since atoms have internal structure, the initial and final scattering channels can, in theory, bedifferent.1455.1. Introduction to 6Li and 6Li2 SystemFigure 5.3: Two channel model of a Feshbach resonance. Vbg(R) represent the potential ofthe two colliding particles, with collisional energy E. Vc(R) represents an molecular statepotential which supports a low energy bound state at an energy Ec, near the collisionalenergy. If the two channels have different magnetic moments, the energy difference E-Eccan be tuned with a magnetic field. In the case of the s-wave resonance in 6Li, the entrancechannel near the resonance is a two-atom scattering state of nearly pure triplet character,and the closed channel the ground singlet potential. At zero magnetic field, the entranceand closed channel asymptotes are degenerate. However, in the presence of an externalmagnetic field, the entrance channel shifts down in energy relative to the closed channel.Figure from [10].collision energy (see Fig. 5.3). In the presence of an external magnetic field, the energy ofthe two atom system is modified. The size of the energy shift is governed by the ZeemanHamiltonian, where the energy shift is proportional to the magnetic field strength and themagnetic dipole moment. If the two channels have different magnetic moments, then theenergies of the two states will move relative to each other. In this way, an external magneticfield can be used to vary the energy difference between the collision energy and the energyof the weakly bound closed channel state. When the energy difference between the twostates is small, coupling between the states can lead to strong mixing between the channelsand new eigenstates with energies different from the uncoupled channel energies.Physically, when the energy of the weakly bound state approaches the collisional energy1465.1. Introduction to 6Li and 6Li2 Systemfrom below (above), the coupling pushes the energy of the new dressed colliding stateupwards (downwards), which acts like an effective repulsive (attractive) interaction, andmodifies the scattering length to be more positive (negative). A FR occurs when themagnetic field is tuned such that the weakly bound state is degenerate with the energy ofthe uncoupled entrance channel state. Near the FR, the scattering length (as a function ofmagnetic field) can be modeled asa(B) = abg(1− ∆B −B0), (5.18)where abg is the scattering length associated with the entrance channel far from the Feshbachresonance, ∆ is the resonance width and is defined with respect to the zero crossing of thescattering length, which occurs at a magnetic field ∆ away from the location of the FR B0.Properties of Feshbach ResonancesIn the discussion of Feshbach resonances and scattering physics, a useful set of quantitiesare the van der Waals length and energy [10]RvdW =12(2µC6~2)1/4and EvdW =~22µ1R2vdW, (5.19)which are related to the C6 coefficient that characterizes the long range part of the inter-atomic potential. A closely related alternative length and energy scale is the mean scatteringlength and energy (introduced in [156]), wherea = [4pi/Γ(1/4)2]RvdW = 0.955978 . . . RvdW (5.20)E =~22µa2= 1.09422 . . . EvdW , (5.21)and Γ(x) is the gamma function. Additionally, a dimensionless background scattering lengthcan be defined by rbg = abg/a. Physically, RvdW approximates the outer turning point forlow l molecules, beyond which the wavefunction decays exponentially, and therefore charac-terizes extent of the vibrational motion (i.e., the “size” of the molecule). The exception isthat of a universal “halo” molecules, which are discussed below. For 6Li: C6 = 1393.39 a.u.,RvdW = 31.26a0 and EvdW = 29.47 mK or 614 MHz.Near a FR, the coupling between the closed and entrance channel mixes the entrance1475.1. Introduction to 6Li and 6Li2 Systemchannel into the closed channel, forming a “dressed” state. In the limit that a  a and|Eb|  EvdW (where |Eb| is the binding energy of the weakly bound state responsible for theresonance), the physics takes on universal behavior. Here, the details of the interaction (forexample, the C6 coefficient that defines the long range part of the potential) are irrelevant.In this regime, the interaction is characterized solely by the scattering length a, and thebinding energy of the (universal) dressed molecule state is simplyEb =~2µa2∝ (B −B0)2 , (5.22)where µ is the reduced mass of the two atom system. Note that in this regime, the bindingenergy scales like the square of the magnetic field, rather than the linear scaling that existsfar from resonance. These molecules form a quantum “halo” state, where the molecularwavefunction extends to distances well past RvdW.Due to the mixing of the entrance channel into the closed channel, the dressed moleculeeigenstate is a mixture of the closed and entrance channel|Ψb〉 =√Z |c〉+√1− Z |e〉 , (5.23)where Z defines the closed channel fraction.A two-channel square well model (for details, see [10, 157, 158]) can be used to modelthe physics of FRs, and specifically the near-threshold bound and scattering states. Thismodel is particularly useful because it is analytically solvable. With this, near the Feshbachresonance, where B → B0 and a→ +∞, the closed channel fraction is [10]Z =2sresrbg∣∣∣∣B −B0∆∣∣∣∣ , (5.24)wheresres =abgaδµ∆E(5.25)and δµ = µe − µc is the difference in the magnetic moment between the colliding atoms inthe open channel and the magnetic moment of the weakly bound state in the closed channel.The resonance strength sres provides a convenient way to classify FRs into two differenttypes: “narrow” resonances (where sres  1) which are closed channel dominated, and“broad” resonances (where sres  1) which are entrance channel dominated. It is important1485.1. Introduction to 6Li and 6Li2 Systemto note that the bound state is universal (that is, the binding energy is solely a functionof the s-wave scattering length) only for Z  1, when the closed channel fraction is small.From Eq. 5.24 it can be seen that Z  1 when |B−B0|  sresrbg|∆|. This implies that forentrance channel dominated resonances, the bound state is universal over a large fraction ofthe resonance width. Correspondingly, the bound state is universal over a small fraction ofthe resonance width for closed channel dominated resonances. Regarding the nomenclature“broad” and “narrow”: although “broad” resonances often have a large resonance widthand “narrow” resonances often have a small resonance width, this is not always the case97.Therefore, the classification of resonances is best done using the resonance strength, ratherthan just the resonance width. Table IV in [10] provides a summary of the properties forselect Feshbach resonances in both homonuclear and heteronuclear system. The propertiesFeshbach resonances in the |12〉 hyperfine mixture are also given in Table 5.2.channel B0 (G) l ∆ (G) abg/a0 δµ/µB sres rbg|12〉 834.1 s -300 -1405 2.0 59 -47|12〉 543.25.1 s 0.1 60 2.0 0.001 2.0|11〉 159.14 p – – 2.0 – –|12〉 185.09 p – – 2.0 – –|22〉 214.94 p – – 2.0 – –Table 5.2: Properties of s and p-wave Feshbach resonances in 6Li for a hyperfine mixture ofthe |1〉 and |2〉 states. For p-wave resonances, several properties are not defined (indicatedby a –). Numbers from [10].Origin of the s-wave Resonances in 6LiThe origin of the s-wave Feshbach resonance in 6Li lies in a coupling between the leastbound v′′ = 38 level of the X(11Σ+g ) potential, and the two-atom scattering state which,at high magnetic fields, has almost purely triplet character (see Eq. 5.14 and surroundingtext). The least bound state in the X(11Σ+g ) potential has a total nuclear spin of I = 0 or2, a binding energy of approximately 1.38 GHz, and a vanishing magnetic moment nearzero field (because S = 0). Thus, with increasing magnetic field, the energy of the entrancechannel state relative to the closed channel varies with a slope of dE/dB = −2µB solely97For example, the 7Li 737 G resonance where sres = 0.8, but ∆ = −192.3 G [10].1495.1. Introduction to 6Li and 6Li2 Systemdue to the Zeeman shift of the energy of the two-atom scattering state98.Following the description of the Feshbach resonances given by Simonucci et al. [147],in the absence of any coupling between the closed and entrance channels the energy ofthe bound states crosses the threshold of the entrance channel at about 550 G (see Fig.5.4). However, the hyperfine interaction (which is not diagonal in the |S,mS ; I,mI〉 basis)induces a coupling between states with ∆I = ∆S = 1. In this case, these two states arethe closed channel singlet states and the entrance channel scattering state. However, it ispossible to write a linear combination of the two v′′ = 38 singlet bound levels|ΦU〉 = 13(|0, 0; 0, 0〉+ 2√2 |0, 0; 2, 0〉), (5.26)that completely decouples from the triplet state. The narrow s-wave resonance at 543 Gis due a coupling of this state through a second order process to the entrance channel [67].This state maintains its singlet character as it crosses the entrance channel threshold.By virtue of orthogonality, the other linear combination is|ΦC〉 = 13(2√2 |0, 0; 0, 0〉 − |0, 0; 2, 0〉)(5.27)which couples to the triplet entrance channel and undergoes an avoided crossing. Thiscoupling strongly modifies the character of this weakly bound state beyond the avoidedcrossing, and the state takes on nearly full triplet character. Above approximately 650 G,this state is a weakly bound halo molecule state (defined by Eq. 5.23) where Z  1. Theenergy of this state moves almost parallel to the entrance channel, crossing the entrancechannel threshold at the location of the broad s-wave resonance at 832 G (see inset in Fig.5.4). Note that the triplet potential (due to the large background scattering length) isknown to have a virtual state very near the threshold energy99. It is to this virtual stateenergy that the Feshbach molecule state asymptotically tends to above the FR at 832 G(see Fig. 5.4).98Note that µB = 9.274× 10−24 J/T = 1.3996 MHz/G.99Marcelis et al [150] give an excellent description of a virtual state: “The energy associated with [a]virtual state is negative, but there is no proper physical bound state associated with this energy. A virtualstate can be regarded as a nearly bound state that behaves much like a real bound state in the inner regionof the interaction potential. Only in the asymptotic region (r → ∞) does the virtual state discover it doesnot quite fit to the size of the interaction potential, and the virtual state exponentially explodes”. One hasto feel some empathy for the poor state that, as far as I know, never did anything to deserve its fate.1505.1. Introduction to 6Li and 6Li2 SystemFigure 5.4: Origin of s-wave Feshbach resonance in 6Li. In the absence of coupling, theclosed channel v′′ = 38 level of the X(11Σ+g ) state (dashed blue line) crosses the thresholdenergy of the entrance channel two-atom scattering state (black dotted line) near 550 G.When coupling between the two channels is turned on via the hyperfine interaction, onelinear combination of the singlet states decouples from the entrance channel (i.e., still followsthe blue dashed line), and is responsible for the narrow s-wave resonance at 543 G. Theother orthogonal superposition state couples to the triplet entrance channel and undergoesan avoided crossing (solid red line). The inset shows that the energy of the Feshbachmolecule state (solid red line) tends towards the energy of the triplet virtual state (longdashed green line), crossing the entrance channel threshold energy (black dotted line) near832 G, resulting in the broad Feshbach resonance at this magnetic field. Figure from [147].In the context of photoassociation near the broad Feshbach resonance (at magnetic fieldsabove 650 G), there are two possible initial states to consider. The first is the two-atomscattering states, which is an entrance channel state with momentum k that has nearly puretriplet character (see Eq. 5.14). The second is the weakly bound dressed Feshbach molecule1515.1. Introduction to 6Li and 6Li2 Systemstate which, written in the |S,MS ; I,MI〉 basis, is|Ψb〉 =√Z |c〉+√1− Z |e〉=√Z |ΦC〉+√1− Z |e〉 (5.28)=√Z3(2√2 |0, 0; 0, 0〉 − |0, 0; 2, 0〉)+√1− Z |1,−1; 1, 1〉 .In the molecular basis |NSIJF 〉, the Feshbach molecule state can be written as|Ψb〉 =√Z3(2√2 |00000〉 − |00200〉)+√1− Z(√16|01112〉 −√12|01111〉+√13|01110〉)(5.29)where the entrance channel state in this basis is from Eq. 5.16. The closed channel fractionZ is much less than one for fields above 650 G. This fraction has been measured by Partridgeet al [64] and our group [98, 159], with both results closely matching theoretical predictionsdone by Romans and Stoof [160]. The measured and theoretical value for Z are shown inFig. 5.5.There is a weak hyperfine coupling (and therefore a weak mixing) between the continuumstates and the dressed bound state. Therefore, it is possible to photoassociate to boththe A(11Σ+u ) and c(13Σ+g ) potential from either the scattering state or from the Feshbachmolecule state. Experimentally, we can determine whether the initial state is the looselybound Feshbach molecule, or a thermal ensemble (i.e., the two-atom scattering state) by thevarying the final temperature of the ensemble, and the magnetic field at which we evaporate.The creation of Feshbach molecules (and the following evaporation to degeneracy), and thecreation of BCS pairs (on the high magnetic field side of the FR) are discussed in the nextsection.5.1.3 Degenerate GasesThe broad s-wave Feshbach resonance acts as an excellent tool to control the s-wave scat-tering length from large and positive on the low magnetic field size to large and negativeon the high magnetic side. In the fermionic 6Li system, a > 0 allows for the creation ofweakly bound Feshbach molecules which have bosonic character, and the ability to create1525.1. Introduction to 6Li and 6Li2 System600 650 700 750 800 850 900magnetic field (G)10−610−510−410−310−210−1ZFigure 5.5: Closed channel fraction (Z) of Feshbach molecule state. Red squares are datafrom Partridge et al. [64], and black circles are data from our experiment [98, 159]. The dash-dotted line shows theoretical predictions done by Romans and Stoof, digitized from [160].The grey region corresponds to the strongly interacting regime where kF|a| > 1, whichoccurs near the Feshbach resonance at 832 G. This magnetic field is indicated by the verticaldashed line.an molecular Bose-Einstein condensate (mBEC). When a < 0 on the high magnetic fieldside, cooling below the critical temperature allows for the formation of BCS-like pairs. Inboth cases, the system is in the strongly interacting regime. It is also possible to create aweakly interacting degenerate Fermi gas where a is small (or zero). This is often done ata zero crossing of the s-wave scattering length at B = 527.5 G. However, degeneracy canalso be achieved at (or near) 0 G, which is useful because it allows for the elimination ofmagnetic fields (and the associated systematic shifts on the photoassociation resonances).In our experiment, we have the ability to create any of the above systems. They are brieflydiscussed below, and more detail can be found in PhD thesis of M. Semczuk [98].Near the Feshbach resonance, the binding energy of the Feshbach molecule is a functionof the s-wave scattering length,Eb =~22ma2, (5.30)and is typically on the order of a few tens of kHz at the fields of interest. For example,at 770 G (where the scattering length is assumed to be 200 nm), the binding energy is1535.1. Introduction to 6Li and 6Li2 Systemapproximately 40 kHz or 2 µK [138].A convenient method of forming Feshbach molecules in 6Li is atom-molecule thermaliza-tion, which relies on three-body recombination. This method was employed by S. Jochimet al in 2003 to create a pure gas of optically trapped molecules [59] and soon after anmBEC of 6Li2 molecules in the group of Grimm [161] and Ketterle [138]. In this method,three free atoms collide to form a bound Feshbach molecule, and the remaining free atomcarries away the binding energy in the form of kinetic energy. A simple description of thisprocess assumes that the three body recombination rate scales like the square of the totalscattering cross section for a two body collision. In this caseγ3B ∝ σ2s =(4pia21 + k2a2)2, (5.31)where k is the wave number, given byk =√2mE~. (5.32)It can be seen that in the low temperature limit (k2a2 < 1) that γ3B ∝ a4, while for higherequilibrium ensemble temperatures (k2a2 > 1), γ3B ∝ 1/k4 ∝ 1/T 2. As might be expected,this simple model states that Feshbach molecule formation is enhanced close to the Feshbachresonance, and at lower temperatures100. In fact, the statement can be made more explicitby requiring k2a2 < 1 for efficient molecule formation which, using Eq. 5.30, can be writtenaskBTEb< 1 for efficient Feshbach molecule formation . (5.33)The result follows from a balance between the three-body recombination process thatforms molecules, and two-body dissociation processes that break the molecules. A moreformal description of this atom-molecule conversion process is presented by Cheng Chin andRudolf Grimm [164] which shows that for a non degenerate gas, the equilibrium between100It should be noted that the theoretical work of D.S. Petrov [162] and J.P. D’Incao and B.D. Esry [163]showed the relevant three body loss coefficient for a two component Fermi gas scaled like a6 in the lowtemperature limit, rather than the a4 scaling suggested by Eq. 5.31. Nevertheless, the conclusion thatmolecule formation is enhanced close to the Feshbach resonance and at low temperatures holds.1545.1. Introduction to 6Li and 6Li2 Systemthe phase space density of molecules φmol and atoms φat follows the relationφmol = φ2at exp(−EbkBT). (5.34)The Boltzmann factor is determined by the ratio of the binding energy of the Feshbachmolecule Eb (which is negative on the low field side of the FR, as the Feshbach moleculeis bound), and the thermal energy of the ensemble given by kBT . This factor can greatlyenhance the formation of molecules even for low atomic phase space densities, and is consis-tent with the statement in Eq. 5.33. However, this factor appears to suggest that moleculeformation would be greatly enhanced at large Eb and should be largest in the limit thatEb → ∞ (i.e., very far from the Feshbach resonance), where all atoms should accumulatein the molecule state. However, this argument neglects the effects of the energy releasedduring the molecule formation process. This energy (equal to the binding energy of themolecule) heats the sample and reduces the atomic phase space density, which in turnsreduces the molecular phase space density and thus the number of molecules that form.This motivates striking a balance between using a magnetic field which correspondsto a small Eb and the achievable final ensemble temperature and atom/molecule number.Therefore, our recipe for Feshbach molecule formation is to cool a thermal ensemble of6Li in an ODT at 754 G such that the condition in Eq. 5.33 can be satisfied given our finalequilibrium temperature, which is on the order of 300 nK at the lowest temperatures wecan achieve.Molecular Bose-Einstein-CondensatesSince the Li2 Feshbach molecule is bosonic, a mBEC is formed if the molecule density inthe trap exceeds the critical density,nc =2.612λ3c, (5.35)whereλ3c =√2pi~2mkBTc. (5.36)1555.1. Introduction to 6Li and 6Li2 SystemTc is the critical temperature,kBTc = 0.94 · ~ω¯N1/3 , (5.37)which is related to the trapping potential through the mean trap frequency, ω¯ = [ωxωyωz]1/3and the total molecule number N. Figure 5.6 shows an example of the trap temperaturerelative to the critical temperature during a standard evaporation ramp used to achievea mBEC in our system, as well as the formation of an mBEC in our system, highlightedby the appearance of a bi-modal distribution, where the spatial width (in the absence ofinteractions) of the condensed fraction is determined by the trap geometry (i.e., the groundstate wavefunction of the harmonic potential formed by the ODT).Figure 5.6: Left: Atom number and temperature as a function of the optical dipole trappower during a standard evaporation ramp to form a mBEC, which happens when the en-semble temperature falls below the critical temperature (given by Eq. 5.37). The efficiencyof the evaporation is characterized by a scaling law which depends on by η = U0/kBT (theratio of the trap depth to the ensemble temperature), see section 3.2. Right: Formation ofan mBEC in our system, highlighted by the appearance of a bi-modal distribution. Fromtop to bottom, the temperature of the cloud is T = 710 nK (T/Tc ≈ 1.2), T = 230 nK(T/Tc ≈ 0.9), T = 110 nK (T/Tc ≈ 0.8), T = 65 nK (T/Tc ≈ 0.6). Figure from [165].1565.1. Introduction to 6Li and 6Li2 SystemDegenerate Fermi Gases and BCS PairsOn the high magnetic field side of the FR, the Feshbach molecule state is unbound andtherefore the system remains fermionic, forming a degenerate Fermi gas if the equilibriumtemperature falls below the Fermi energy, which is given byEF = ~ω¯(6N)1/3 . (5.38)Unlike a bosonic system, no phase transition occurs unless an attractive interaction ispresent. It just happens that the high magnetic field side of the Feshbach resonance has anegative scattering length, corresponding to an attractive interaction. In this case, BCS-likepairs will form when the equilibrium temperature is below the critical temperature for pairformation,T ∗ = 0.61TF exp( −pi2kF |a|), (5.39)where the Fermi wave number is related to the Fermi energy in the usual way (see Eq.5.32). In the strongly interacting regime, where |kFa|  1, the critical temperature for pairformation is simply related to the Fermi temperature. Following a similar evaporation asin the mBEC case (see Fig. 5.6), except at a different magnetic field (typically 840 G), wecan form BCS pairs, which show the characteristic pairing gap in RF spectroscopy shownin Fig. 5.7).At 0 G, the singlet s-wave scattering length is aS ≈ 46a0 and the triplet s-wave scatteringlength is aT ≈ −2160a0 [86, 135, 166], where a0 = 0.529 A˚ is the Bohr radius. In thisregime, kF |a|  1 (termed the “weakly-interacting” regime) and, although the criticaltemperature for BCS-like pair formation is extremely low (see Eq. 5.39), it is possibleto cool to temperatures much lower than the Fermi temperature and achieve a degenerateFermi gas. More information on the details of BCS pairing, the BEC-BCS crossover regime,and the relationship to superconductivity can be found the review by Chen et al. [167].5.1.4 6Li2 Bound Molecular StatesThis section serves as an introduction to the state labeling and properties of the boundmolecule states of a homonuclear molecule, as well as the selection rules which govern thephotoassociation processes. The angular momentum coupling schemes vary from moleculeto molecule, depending on the strength (and type) of the involved perturbations. For linear1575.1. Introduction to 6Li and 6Li2 SystemFigure 5.7: Emergence of BCS Pairing signal in RF spectroscopy. Atoms in the |2〉 state areflipped to the |3〉 state by the application of an RF field at 81.34 MHz (which correspondsto a field of 839.2 G). Free |2〉 atoms undergo the transition at a zero offset, whereas anadditional energy corresponding to the pairing gap must be added to flip those bound to a|1〉 state atom as a BCS pair. Figure from [165].molecules these different cases are summarized by Hund’s coupling cases, characterized bythe choice of basis corresponding to the “good” (well-defined) quantum numbers. A moredetailed discussion of the Hund’s coupling cases can be found in many texts on molecularsystems (for example, [168, 169, 170]). For diatomic molecules the most common are Hund’scase (a) and (b), the latter being most applicable in the case of 6Li2, and are discussed below.1585.1. Introduction to 6Li and 6Li2 SystemHund’s Case (a)In Hund’s case (a), the total orbital angular momentum of the electrons in the unfilledorbital electrons ~L = ~l1 + ~l2, and the total spin ~S = ~s1 + ~s2 are strongly coupled to theinternuclear axis101. In this case, the projection of ~L (denoted by mL) and ~S (denotedby mS) on the internuclear axis are well defined. To classify the electronic states of themolecule, these projections are typically labeled using the quantum numbersΛ = |mL|, Λ = 0, 1, 2, · · · , L (5.40)Σ = mS , Σ = S, S − 1, · · · ,−S . (5.41)The electronic state is classified by the absolute value of mL because the energy of a partic-ular mL projection state does not depend on the direction of ~L with respect to the electricfield direction (that is, the internuclear axis). This is in contrast to the projection of mS ,where the energy of a particular mS projection does depend on the orientation of ~S withrespect to the axial magnetic field (which results from the orbital motion of the electrons).The total angular momentum (apart from nuclear spin) is~J = ~N + Ωzˆ , (5.42)where ~N is the rotational angular momentum of the nuclei perpendicular to the molecularaxis and Ω = Λ + Σ is the projection of the total electronic angular momentum ontothe internuclear axis. Therefore, in Hund’s case (a), the set of good quantum numbersis (n, J, S,Λ,Σ,Ω) where n represents all other quantum numbers of the electronic andvibrational state.Hund’s Case (b)In Hund’s case (b), the total orbital angular momentum of all electrons ~L is still stronglycoupled to the internuclear axis, but the total spin ~S is not. Formally, this is the case whenthe coupling of ~L with the molecular axis is stronger than the coupling of ~L with ~S. Thisholds when Λ = 0 (i.e., the projection of the orbital angular momentum of the electronsalong the internuclear axis is zero) and there is therefore no axial magnetic field to couple101By “coupled strongly”, one means that the individual coupling of ~L and ~S to the internuclear axis islarger than the direct coupling between ~L and ~S.1595.1. Introduction to 6Li and 6Li2 Systemthe total spin to the axis, or for molecules with small spin-orbit coupling (when Λ 6= 0). Inthis case, Σ and Ω are no longer good quantum numbers. Instead, Λ (the projection of ~Lon the internuclear axis) and the rotational angular momentum o