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Optical synthesis and ultracold reactions of triplet ⁶Li molecules Polovy, Gene 2018

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Optical Synthesis and UltracoldReactions of Triplet 6Li MoleculesbyGene PolovyB.Sc., The University of British Columbia, 2011A 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)August 2018c© Gene Polovy 2018The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Optical Synthesis and Ultracold Reactions of Triplet 6Li Moleculessubmitted by Gene Polovy in partial fulfillment of the requirements for the degree of Doctorof Philosophy in Physics.Examining Committee:Kirk W. MadisonSupervisorRoman KremsSupervisory Committee MemberValery MilnerUniversity ExaminerShahriar MirabbasiUniversity ExaminerAdditional Supervisory Committee Members:Sudip ShekharSupervisory Committee MemberLukas ChrostowskiSupervisory Committee MemberiiAbstractIn this thesis, we present the creation, via stimulated Raman adiabatic passage, and pre-liminary lifetime measurements of deeply bound triplet molecules of 6Li2 in several rovi-brabrational levels of the a(3Σ+u ) molecular potential, including the lowest lying state inthis potential. In addition to being the first experimental demonstration of the formation ofthese dimers, these results will serve as the basis for our ongoing efforts to reliably determinethe rate constants responsible for the finite lifetimes of these ultracold molecules and mayshed light on some of the mysteries surrounding the quantum state dependence of chemi-cal reactions in the ultracold regime. Moreover, all of the tools - including a robust lasercooling system for 6Li, an ultra-low phase noise Raman laser system and various softwarerecipes for the automation of data collection and analysis - and experimental techniquesdeveloped in this study can also be used for the creation of heteronuclear LiRb molecules.Unlike our 6Li2 homonuclear molecules which only have a magnetic dipole moment, theLiRb polar molecules are predicted to also have a large electric dipole in the lowest lyingtriplet state. These characteristics combined with a three dimensional optical lattice wouldgive us precise control over several degrees of freedom and enable us to perform quantumsimulations of exotic condensed matter systems.We also present the design and performance of a coherent source of Lyman-α radiationthat was used by the ALPHA collaboration at CERN for laser cooling anti-hydrogen forthe purpose of experimentally verifying the predictions of the standard model.iiiLay SummaryIn this thesis, we present the creation, using a specialized sequence of laser pulses, and pre-liminary lifetime measurements of Lithium-6 molecules that are 1,000,000 times colder thanouter space, in several quantum configurations. In addition to being the first experimentaldemonstration of the formation of these molecules, these results will serve as the basis for ourongoing efforts to reliably determine metrics that quantify the longevity of these moelculesand may shed light on some of the mysteries surrounding chemical reactions in the ultra-cold regime. Moreover, all of the instrumentation, software and experimental techniquesdeveloped in this study can also be used for the creation of ultracold Lithium-Rubidiummolecules, which have several quantum properties that could be used for performing quan-tum simulations of exotic condensed matter systems.We also present the design and performance of a laser system that was used by theALPHA collaboration at CERN for experimentally verifying the predictions of the funda-mental theories in particle physics that could help us to better understand the origin of theuniverse.ivPrefaceThe majority of the research and development presented in this dissertation relates to thesequence of steps we took to create and determine the loss rate constants of deeply bound6Li dimers in the triplet state. This work was carried out at the Quantum DegenerateGases laboratory at the University of British Columbia, Point Grey Campus. The Lyman-α generation project was conducted in the laboratory of Professor Takamasa Momose atthe same university.Optical Synthesis and Ultracold Reactions of Triplet 6LiMoleculesThe most impactful technical contributions I made to the creation of deeply bound 6Lidimers are listed below in chronological order:1. In collaboration with Kahan Dare, I diagnosed and solved a technical problem thatwas causing our injection locked diode lasers to lose lock frequently and operate athalf of their manufacturer specified optical power.2. With Kirk W. Madison’s guidance, I completely re-designed and re-built the lasersystem used for laser cooling and imaging 6Li.3. With Kirk W. Madison’s guidance and Julian Schmidt’s and Kahan Dare’s assistance,I designed and implemented of an ultra-low phase-noise Raman laser cooling systemfor the creation of deeply bound 6Li dimers.4. I wrote an image processing program that uses principal component analysis (PCA)to significantly improve the signal-to-noise ratio (SNR) we can achieve at low atomnumbers with absorption imaging.vVUV Coherent Source for Laser Cooling of Antihydrogen and Other Projects5. I wrote a substantial amount of software in Python to analyze data, program varioushardware devices and automate experimental sequences.The most significant scientific contributions I made to the creation of deeply bound 6Lidimers are listed below in chronological order:1. I implemented a time-efficient means of analyzing single-color, two-color, Autler-Townes and dark state spectroscopy data and collected a substantial portion of thedata presented in chapter 6 of this thesis.2. I identified and solved the problem that was preventing us from realizing stimulatedRaman adiabatic passage (STIRAP) to deeply bound molecular states in the a(3Σ+u )molecular potential of 6Li and was the first person to ever observe evidence of STIRAPto one of these deeply bound states1.VUV Coherent Source for Laser Cooling of Antihydrogenand Other ProjectsIn addition to creating ultracold 6Li dimers (discussed above), I was involved in the followingprojects:1. I played a crucial role in building a narrowband solid state VUV coherent source forlaser cooling of antihydrogen, analyzed all of the data shown in [104] and wrote themanuscript for this paper.2. I worked on a collaboration experiment whose intended purpose was to create RbHmolecules from a cold beam of atomic Hydrogen and 85Rb atoms in magneto-opticaltrap (MOT) by photo-association. My contribution was building and maintainingevery aspects of the MOT together with Thomas Prescott and later modifying theatomic beam portion of the apparatus in an effort to increase the flux.1After making several tweaks to the experimental sequence (see chapter 7 for details) to get unambiguousand accurate measurements of the decay rate constants for the deeply bound 6Li molecules mentioned aboveand those for the remaining four vibrational levels, we plan to publish this work in a peer review journal.viVUV Coherent Source for Laser Cooling of Antihydrogen and Other Projects3. To assist the pressure atom sensor (PAT) team at the QDG lab, I wrote a singlechannel scattering code capable of handling very large partial waves.4. I worked with Will Gunton to design optically transparent electrodes capable of pro-ducing electric fields as high as 120 kV/cm in air without arcing. These results arepublished in [63].viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Laser Cooling Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Fine and Hyperfine Structure of 6Li . . . . . . . . . . . . . . . . . . . . . . 52.2 Laser Cooling and Magneto-Optical Traps . . . . . . . . . . . . . . . . . . 62.2.1 Simple Description of Laser Cooling . . . . . . . . . . . . . . . . . . 62.2.2 Vacuum System and Zeeman Slower . . . . . . . . . . . . . . . . . . 82.2.3 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Lithium Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 6Li Laser System Design . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Gray Molasses Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Optical Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18viiiTable of Contents2.4.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Experimental Realization . . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 RF Spin Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Raman Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Introduction to Optical Phase Locked Loops . . . . . . . . . . . . . . . . . 303.2 Femto-Second Frequency Comb . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Ideal Optical Frequency Comb . . . . . . . . . . . . . . . . . . . . . 333.2.2 Erbium Doped Fiber FFC . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Performance Metrics Relevant to STIRAP . . . . . . . . . . . . . . 363.3 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Feshbach Resonances and the 6Li2 System . . . . . . . . . . . . . . . . . . 444.1 6Li Hyperfine Structure and Zeeman Splitting . . . . . . . . . . . . . . . . 444.2 Brief Introduction to Scattering Theory and Feshbach Resonances . . . . . 464.2.1 Single Channel Scattering . . . . . . . . . . . . . . . . . . . . . . . 474.2.2 Multi-Channel Scattering and Feshbach Resonances . . . . . . . . . 484.3 The Scattering State and Feshbach Molecule Formation . . . . . . . . . . . 504.4 Excited State and Deeply Bound State Labelling and Selection Rules . . . 525 Three Level Model: for Modeling Autler-Townes and Dark State Spec-troscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1 Autler-Townes Splitting and STIRAP . . . . . . . . . . . . . . . . . . . . . 565.1.1 Autler-Townes Splitting . . . . . . . . . . . . . . . . . . . . . . . . . 595.1.2 STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Derivation of an Analytical Fitting Function for Two-Color Photo-AssociationData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62ixTable of Contents5.2.1 Derivation of χ(1)(−ωp, ωp) in the Weak Probe Regime . . . . . . . 635.2.2 Relation to Number of Trapped Feshbach Molecules . . . . . . . . . 646 Molecular Spectroscopy at High Magnetic Fields . . . . . . . . . . . . . 666.1 Single Color Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.1.1 Finding Excited States . . . . . . . . . . . . . . . . . . . . . . . . . 686.1.2 Measuring the Rabi Frequency . . . . . . . . . . . . . . . . . . . . . 706.2 Two-Color Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.1 Finding Deeply Bound States . . . . . . . . . . . . . . . . . . . . . 726.2.2 Dark State Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 746.2.3 Autler-Townes Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 756.3 Tabulated Transition Frequencies and Coupling Strengths of All ConsideredLevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Creation and Lifetimes of Deeply Bound 6Li2 Molecules . . . . . . . . . 797.1 Experimental Implementation and Optimization of STIRAP . . . . . . . . 807.2 Derivation of the Molecule Lifetime Model . . . . . . . . . . . . . . . . . . 847.3 Deeply Bound Molecule Lifetime Measurements . . . . . . . . . . . . . . . 878 Narrow-Band Solid State VUV Source for Laser Cooling of Antihydrogen 938.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2.1 Ti:Sapphire Amplifier and SHG . . . . . . . . . . . . . . . . . . . . 958.2.2 Lyman-α Generation and Detection . . . . . . . . . . . . . . . . . . 958.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.3.1 Second Harmonic Generation (SHG) Stage . . . . . . . . . . . . . . 978.3.2 THG Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105xTable of ContentsA Beam Profile Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.1 SPI Beam Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.2 IPG Beam Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.3 Photo-Association Beam Profile . . . . . . . . . . . . . . . . . . . . . . . . 126B Trap Frequency Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 127B.1 SPI CODT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128B.2 IPG ODT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129C Passive Mode-Locking via Non-Linear Polarization Rotation . . . . . . 130C.1 Non-Linear Polarization Density in an Isotropic Medium . . . . . . . . . . 130C.2 Non-Linear Polarization Rotation in an Isotropic Medium . . . . . . . . . . 133C.3 Simple Ring Cavity Implementation . . . . . . . . . . . . . . . . . . . . . . 136C.3.1 Jones Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . 137C.3.2 Constructing an Artificial Saturable Absorber . . . . . . . . . . . . 139C.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142xiList of Tables4.1 Complete basis for the ground state in the coupled and uncoupled represen-tation for the atomic states of 6Li. . . . . . . . . . . . . . . . . . . . . . . . 466.1 Probe laser frequencies and Rabi frequencies for two excited states. . . . . . 776.2 Stokes laser frequencies and Rabi frequencies for five deeply bound states. . 787.1 Two body loss rate coefficients and related quantities for several v′′ and N ′′states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92xiiList of Figures2.1 Experiment Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Fine and hyperfine structure of 6Li. . . . . . . . . . . . . . . . . . . . . . . . 72.3 Laser cooling cartoon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 MOT cartoon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Master table layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 Amplification of cooling light and generation of high-field imaging light. . . 162.8 Energy eigenstates of 6Li as a function of magnetic field . . . . . . . . . . . 182.9 Crossed optical dipole trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.10 Cartoon representation of the change in the collision cross section of theatoms in the ensemble due to the Feshbach resonance near 832 G. . . . . . 242.11 Forced evaporation in the SPI and IPG. . . . . . . . . . . . . . . . . . . . . 253.1 Simplified block diagram of an OPLL in the phase-domain representation. . 313.2 A cartoon illustration of the phase noise of free running VCO or beat-noteand a phase-locked VCO or beat note. . . . . . . . . . . . . . . . . . . . . . 313.3 Ideal frequency comb with a repetition rate of frep, carrier offset frequencyof fceo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Phase noise spectrum of the third harmonic (376 MHz) of the FFC’s repetition-rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 System and component schematics for the AOM-based phase locked loop. . 383.6 Power spectrum and SSB phase noise of an out-of-loop heterodyne measure-ment between TS1 and the FFC. . . . . . . . . . . . . . . . . . . . . . . . . 41xiiiList of Figures3.7 Power spectrum and SSB phase noise of the heterodyne beat between theexperiment arms of TS1 and TS2 for three frequency differences. . . . . . . 424.1 Sketch of effective open and closed channel potentials. . . . . . . . . . . . . 495.1 Three level system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 STIRAP Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.1 Molecular potentials for 6Li2. . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Spectroscopy of the v′ = 20 manifold of the c(13Σ+g ) at 755 G. . . . . . . . . 696.3 Spectroscopy of the v′ = 20 manifold of the c(13Σ+g ) at 755 G. . . . . . . . . 706.4 Natural Linewidth of the Intermediate State at 755 Gauss. . . . . . . . . . 716.5 Single Color Lifetime at 755 Gauss. . . . . . . . . . . . . . . . . . . . . . . . 726.6 Spectroscopy of the v′′ = 0 manifold of the a(13Σ+u ) potential at 755 G (broadscan). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.7 Spectroscopy of the v′′ = 0 manifold of the a(13Σ+u ) potential at 755 G (finescan). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.8 Dark State Spectroscopy of the lowest lying v′′ = 0, N ′′ = 0 level in thea(13Σ+u ) potential we could access at 755 G. . . . . . . . . . . . . . . . . . . 756.9 Spectroscopy of the v′ = 20 manifold of the c(13Σ+g ) at 755 G. . . . . . . . . 767.1 Feshbach molecule number after a forward and reverse STIRAP sequence tothe lowest lying v′′ = 9 level as a function of the probe laser’s frequency . . 817.2 Oscilloscope trace of Gaussian pulses used for STIRAP. . . . . . . . . . . . 837.3 Lifetime comparison for a single arm ODT and a crossed ODT. . . . . . . . 897.4 Deeply bound molecule lifetime data for the v′′ = 0, v′′ = 5 and v′′ = 9 states. 918.1 Laser arrangement and non-linear optical stages. . . . . . . . . . . . . . . . 968.2 SHG pulse energy at 365 nm as a function of the pulse energy. . . . . . . . 988.3 SHG pulse energy with EP = 310 mJ as a function of time with the amplifiercavity locked. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.4 SHG temporal pulse shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99xivList of Figures8.5 Temporal and spectral profiles of the 364.7 nm and 121.46 nm pulses withthe amplifier cavity locked close to resonance with the seed laser. . . . . . . 1018.6 Temporal and spectral profiles of the 364.7 nm and 121.46 nm pulses withthe amplifier cavity locked far from resonance with the seed laser. . . . . . . 101A.1 Beam profile measurements for the first and second arms of the SPI CODT. 125A.2 Beam profile measurements for the first and second arms of the IPG CODT. 125A.3 Beam profile measurements for the photo-association beam at 798 nm. . . . 126B.1 Trap frequency measurements for the SPI laser. . . . . . . . . . . . . . . . . 128B.2 Trap frequency measurements for the IPG laser. . . . . . . . . . . . . . . . 129C.1 Simple polarization additive-pulse mode locking structure. . . . . . . . . . . 137xvList of AbbreviationsAOM acousto-optic modulatorBCS Bardeen-Cooper-SchriefferBEC Bose-Einstein condensateCODT crossed optical dipole trapDDS direct digital synthesizerDP double-passECDL extended cavity diode laserEDFA erbium-doped fiber amplifierEIT electromagnetically induced transparencyFC Franck-CondonFFC femto-second frequency combFR Feshbach resonanceHNLF highly non-linear fiberLO local oscillatorMB Maxwell-BoltzmannmBEC molecular Bose-Einstein condensateMOT magneto-optical trapODT optical dipole trapPA photo-associationPBS polarizing beam splitterPM polarization maintainingPZT piezo-electric transducerxviList of AbbreviationsREMPI resonantly enhanced multi-photon ionizationSA saturated absorptionSP single-passSTIRAP stimulated Raman adiabatic passageTA tapered amplifierTF Thomas-FermixviiAcknowledgementsFirst, I would like to thank Kirk W. Madison for the wealth of knowledge and skills he hasdirectly or indirectly bestowed upon me throughout my graduate career. Aside from thebasic theory I learned in my undergraduate physics classes and the (limited) experience Ihad with software development, I had virtually no practical skills when I started workingat the Quantum Degenerate Gases (QDG) Lab. Kirk’s tutelage and several engineeringclasses allowed to improve as a software developer and become a capable optical engineer,systems engineer and physicist - fully equipped for a career in either academia or industry.In addition to helping me develop technical expertise, Kirk facilitated my development asa leader and project manager by giving me the opportunity to train 13 students with awide range of technical backgrounds and supervise the projects they undertook. Becauseof my keen interest in the business aspects of the lab, I have also had the chance to play akey advisory role in discussions concerning equipment purchases, hiring, project planning,strategy and risk management.Second, I would like to acknowledge Mariusz Semczuk who started his PhD at the QDGlab just as I was finishing my undergraduate degree. Not only did he become a good friend,but he also introduced me to Kirk, encouraged me to join the lab and was one of the firstpeople to train me in the early days of my graduate career. I am thankful for this andfor the memories we shared outside of work, particularly those involving his prized leatherjacket.Third, I would like to thank everyone else at UBC who I have had the pleasure of workingwith and learning from throughout my years there: Julian Schmidt, William Bowden, WillGunton, Kahan Dare, Erik Frieling, Denis Uhland, Brendan Pousett, Tristan Calderbank,Roman Krems, David Jones, Takamasa Momose, Mario Michan, Sudip Shekhar, LukasChrostowski, Thomas Prescott, Glenn McGuinness, Isaiah Becker-Mayer, James Booth,xviiiAcknowledgementsJanelle Van Dongen, Kais Jooya and Komancy Yu. Each of the people in this list contributedin some way to forge the person I am today. While working with them, not only did I absorbsome of their technical expertise, but I also learned how to deal with failure, show humility,be patient, the value of hard work and what steps I need to take to achieve my life goals.For this, I am thankful.Finally, I would like to thank my parents, Sergiy Polovy and Larysa Shymko, for thelove, support and guidance they have given me throughout my life.xixChapter 1IntroductionRecent technological breakthroughs in laser cooling have led to the birth and rapid expan-sion of ultracold atomic and molecular physics. When the thermal or de Broglie wavelength,λdB =√2pi~2/mkBT , of particles at ultracold temperatures (T < 0.001 K) exceeds theinter-particle spacing, quantum mechanical effects can easily be observed [91]. Studies ofthe interactions between ultracold atoms have even led to the creation of completely newstates of matter: Bose-Einstein condensation of atoms and molecules [4, 19, 37, 57, 175] andBardeen-Cooper-Schrieffer (BCS) like pairs [13, 21, 26, 64, 138], which closely resemble theircondensed matter counterparts. The natural next level of complexity was the (laser induced)creation - from atoms cooled by relatively well established techniques - and study of ultracoldmolecules2. Stimulated Raman Adiabatic Passage or STIRAP is a technique first conceivedby Bergmann et al. [12] that relies on the creation and evolution of a coherent dark state.It has been the workhorse of high efficiency conversion of atoms to molecules in the ultra-cold community. Several groups have used this process to successfully create deeply boundhomonuclear and heteronuclear Alkali molecules [35, 36, 66, 87, 110, 114, 122, 148, 152, 165].In this dissertation, we focus primarily on the coherent population transfer via STIRAPof loosely bound ultracold Feshbach molecules of fermionic 6Li (created by evaporativecooling in an optical tweezer by methods similar to those described in [80, 81, 116, 117, 119])to deeply bound molecular states in the in the a(3Σ+u ) molecular potential and study thechemical properties of these states. In doing so, we aspire to gain insights into the quantumstate dependence of chemical reactions in the ultracold regime for these alkali dimers. As abonus, to create these dimers, we also had to build most of the hardware and develop the2Direct cooling of molecules is also an active research area. For more information about Zeeman andStark decelerators, direct laser cooling of molecules see, for example, [143, 156].1Chapter 1. Introductionhuman capital required for producing polar molecules of 6Li85Rb3 in bound states of thea(3Σ+u ) molecular potential - a longer term goal that is outside the scope of this dissertation.Unlike our 6Li2 triplet molecules which have a magnetic dipole moment due to their non-zerospin, but no permanent electric dipole moment, triplet 6Li85Rb molecules would have botha 0.37 Debye [155] electric dipole moment (which is comparable to that of KRb moleculesin their absolute ground state [114, 115]) and a magnetic dipole moment. Placing these(chemically unstable) polar molecules in a 3-dimensional optical lattice would enable usto study lattice-spin models and simulate exotic condensed matter phenomena by creatingsystems with equivalent Hamiltonians while maintaining a level of control over the systemthat is unavailable in more conventional condensed matter experiments [106].In the chapters that follow, we place emphasis on the following topics: the design andconstruction of a robust laser cooling system for 6Li atoms and a low phase-noise Ramanlaser system for quantum state manipulation, the search for and characterization of thedesired quantum states for coherent population transfer using single-color and two-colormolecular spectroscopy, our implementation and optimization of STIRAP for populationtransfer to deeply bound states in the a(3Σ+u ) molecular potential and measurements of thelifetimes of these molecules. Although the last topic is an ongoing project, the preliminarymeasurements demonstrated here will prove to be instrumental for the determination ofdecay rates and for understanding the underlying quantum chemical reaction mechanisms.In addition to using laser cooling techniques for the creation of ultra-cold 6Li dimers,we also discuss the application of the same technology to even more fundamental problemsin realm of particle physics. We focus on the design and construction of a narrowband solidstate vacuum ultra-violet (VUV) coherent source for laser cooling magnetically trappedantihydrogen at CERN [3, 104]. As we show in chapter 8, this pulsed VUV source was con-structed from an injection seeded Ti:Sapphire based unstable resonator and two non-linearstages - second harmonic generation in a BBO crystal followed by third harmonic generationin a mixture of Krypton and Argon gas. After we built this laser at UBC, it was transportedto CERN and used by the ALPHA collaboration to cool antihydrogen and establish new3Although we do not discuss the laser cooling and trapping 85Rb and 87Rb in this thesis because thistopic is irrelevant to its central theme, our laser cooling apparatus is fully equipped for working with thesespecies as well as 6Li.2Chapter 1. Introductionfrontiers in the search for violations of charge conjugation, parity transformation, and timereversal (CPT) symmetries as part of an ongoing validation effort for the standard model[3].3Chapter 2Laser Cooling ApparatusIn this chapter, our objective is to acquaint the reader with the operation of the laser coolingapparatus and several measurement techniques. We place strong emphasis on the detailsthat have not been published elsewhere and those that are most relevant to the second halfof this dissertation.To provide a roadmap for the reader before diving into the details, we show a simplifiedtiming sequence for collecting a single data point in typical experiment in Figure 2.1. (Everymeasurement is destructive, so we are forced to repeat this sequence for every data point.)First, we load atoms into a magneto-optical trap (MOT) in about 5 seconds. Second,we change several experimental parameters to compress the MOT and cool the atoms toabout 500 µK. This step takes about 5 ms. Third, we transfer the atoms to another kindof trap called an optical dipole trap (ODT), which confines the atoms via a conservativeforce created by a focused high power laser beam, which is detuned by about 400 nm fromthe nearest atomic resonance. Fourth, we use a technique called evaporative cooling tofurther cool the atoms. The length of this step ranges from about 10 ms to several secondsand we can reach ensemble temperatures below 100 nK. Fifth, we execute the“science”step, where we manipulate the atoms in some way. For example, this could be photo-association spectroscopy or the creation of deeply bound molecules via stimulated Ramanadiabatic passage (STIRAP). Finally, we use resonant light to take an absorption image ofthe atom cloud for the purpose of extracting the atom number or the spatial distributionof the cloud4. Depending on the experiment, the length of the entire sequence varies fromabout 10 s to 30 s. The steps described in the sequence above are programmed into python4We also have the ability to take fluorescence images of the atomic cloud. Although this technique wasused for optimizing the atom number in the MOT on numerous occasions, it was not used to collect any ofthe data presented in this dissertation, so we make no further mention of it. For more information, see werefer the reader to [17, 63, 139, 157].42.1. Fine and Hyperfine Structure of 6LiFigure 2.1: The timing sequence executed to collect a single data point. It takes about5 s to load the MOT, 5 ms to compress and cool it and transfer the atoms to the ODT,0-10 seconds (depending on the desired final temperature) to further cool the atoms usingevaporative cooling in the ODT, 0-30 seconds to manipulate the atoms, and 1-10 secondsto image and grab data from several measurement devices.classes as methods and can be enabled or disabled via booleans. The apparatus is controlledthrough this python front-end and a C back-end that interfaces with a National InstrumentsData Acquisition Card (NI-DAQ). The NI-DAQ then communicates with a parallel, uni-directional bus called the UTbus5, which controls the states and outputs of a variety ofdevices including home-built direct digital synthesizers (DDDs), digital outputs and analogoutputs. The outputs of these devices then control the states of or drive lasers, acousto-optic modulators (AOMs), current drivers, mechanical shutters and other hardware. Furtherdetails concerning the implementation of the control system can be found in [17, 63, 86, 139].In the sections that follow, we first briefly describe the hyperfine structure of 6Li atoms(a topic we will return to in chapter 4) and then explore the steps shown in Figure 2.1in greater depth. Thorough discussions of some topics (e.g. the Raman laser system andFeshbach resonances) will be deferred to later chapters. Finally, we will make no furthermention of the machine’s ability to cool 85Rb and 87Rb, which is covered in [16, 17], becausethis is irrelevant to the main theme of this dissertation: photo-association and quantum statemanipulation of 6Li.2.1 Fine and Hyperfine Structure of 6LiLithium-6 is an alkali metal with the following electron configuration: 1s22s12p0. Thus, ithas two electrons in its first s orbital, where the principal quantum number n = 1, andone electron in its second s orbital where n = 2. Like Hydrogen, it has a single valenceelectron and is therefore termed a hydrogenic. Due to coupling between the orbital l and5This is a reference to the University of Texas in Austin, where it was developed.52.2. Laser Cooling and Magneto-Optical Trapsspin s degrees of freedom, which gives rise to an l · s term in the Hamiltonian, we get thefine structure splitting of 10.056 GHz for the first electronically excited state. Because ofthis coupling, l and s are no longer good quantum numbers, so we define the total angularmomentum as j = l+ s and label the fine structure states using term symbols:2s+1lj (2.1)where s = 1/2 (for all of the alkalis), l is the orbital angular momentum and j varies from|l + s| to |l − s| in integer steps. For example, the ground state has l = 0 (s-orbital) andj = s = 1/2, so its term symbol is 2S1/2. The first excited state splits into the2P1/2and 2P3/2 states because l = 1 (p-orbital) and j = 1/2 or 3/2. These states and the finestructure splitting are shown in Figure 2.2. The associated spectroscopic lines are calledthe D1 and D2 lines. Similar to the l · s coupling, the total angular momentum j couplesto the nuclear spin i, which gives rise to a j · i term in the Hamiltonian and results in thehyperfine splitting of the j-levels. Again, we define a new quantum number f = j + i.Analogously to j, f can range from |j+ i| to |j− i| in integer steps. The hyperfine splittingsbetween the f -levels are also shown in Figure 2.2. For a discussion of the physical originof this term, see, for example, [53, 60, 131]. In addition, the hyperfine constants and otherproperties of 6Li can be found in [6, 55, 159].2.2 Laser Cooling and Magneto-Optical Traps2.2.1 Simple Description of Laser CoolingIn Figure 2.3, we show an alkali atom (e.g. 6Li) in its ground state with some velocity v.Suppose we launch a photon of frequency ν at the atom. In the atom’s frame of reference,this photon’s frequency will be Doppler shifted either to the red (lower frequency) or tothe blue (higher frequency) depending on the direction of travel of the atom. If the shiftedfrequency ν ′ of the photon is near resonant with the transition from the atom’s ground stateto its first excited state, the atom has some probability of absorbing it. After absorbinga photon, the atom gets a momentum kick in the direction of propagation of the incident62.2. Laser Cooling and Magneto-Optical Traps2P3/2f = 1/2f = 3/2f = 5/24.4 MHz2P1/2f = 3/2f = 1/2 26.1 MHz2S1/2f = 3/2f = 1/2 228.2 MHz10.056 GHzD2: 446.799 677 THzD1: 446.789 634 THzFigure 2.2: Fine and hyperfine structure of 6Li. The level splittings were retrieved from[55]. For the remainder of this dissertation, the D2 transitions will be most relevant. TheD1 transitions will be mentioned later in this chapter in the context of sub-Doppler coolingof 6Li - an upgrade that is still in progress.photon [124]. The atom then re-emits a photon in a random direction by spontaneousemission. While any individual re-emission event leads to a net change in the momentum,the momentum change due to spontaneous emission is zero on average because the re-72.2. Laser Cooling and Magneto-Optical Trapsemission direction is random. If we now add a second counter-propagating beam, the atomwill be subjected to a net radiation pressure force that opposes motion along one axis andis proportional to v. For a more detailed description of laser cooling and its limitations, see[29, 102, 124].Figure 2.3: Cartoon picture of laser cooling. An incident photon from a laser beam is ab-sorbed by an atom in its ground state and re-emitted in a random direction by spontaneousemission. The net change in the momentum of the atom is in the direction of the incidentphoton.2.2.2 Vacuum System and Zeeman SlowerIn addition to the hardware we need to implement laser cooling and trapping of 6Li in aMOT, we also need an ultra-high vacuum and source of 6Li atoms. The former is shown inFigure 2.4. The vacuum is maintained by means of two ion pumps and one non-evaporativegetter (NEG) pump and is on the order of 1 × 10−10 Torr on the left hand side of thechamber (the “science” side). By heating a solid chunk of Lithium, we generate a hot fluxof gaseous Lithium atoms that pass through a differential pumping stage made up of micro-tubes. These tubes collimate the atomic beam for the purpose of preventing the majorityof (untrappable) Lithium atoms from reaching the “science” side (left hand side) of thechamber and unnecessarily contaminating the vacuum. If we simply allowed this atomicbeam to propagate into the region with the glass cell where we trap the atoms, we would82.2. Laser Cooling and Magneto-Optical Trapsonly be able to trap a very small portion of the atoms emitted, those moving very slowlyat the tail end of the Maxwell-Boltzmann (MB) distribution. For this reason, we need tobegin cooling the atoms with a counter-propagating slowing beam before they enter thetrapping region. While it is straight-forward to slow a single velocity class with a singlefrequency laser source in the manner we described above, targeting many velocity classessimultaneously requires additional hardware. We do this by means of a Zeeman slower -a concept first devised by William Phillips et al. [102, 125]. The idea is to generate amagnetic field, which varies along the axis of propagation of the atomic beam, that shiftsthe energy levels of the atoms (via the Zeeman effect) in a way that makes a single frequencylaser source near resonant with a broad distribution of velocities throughout their journeythrough the chamber. At the entrance to the Zeeman slower, the magnetic field is such thatthe atoms at the highest velocity class are resonant and are slowed enough that they joinatoms in another velocity class just below the first. These atoms then enter the next spatialregion where the field is slightly lower and all the atoms in this slightly lower velocity classare slowed. This process continues in space and time as the atoms propagate down theslower until all of the atoms from the initial high velocity class down to some low velocityclass are swept down together to a velocity just below the lowest velocity class addressed,chosen to be below the capture velocity of the MOT.We have only provided a cursory overview of this system because the design and charac-terization of the vacuum chamber and Zeeman slower are well documented in [16, 17]. SinceZeeman slowers are the backbone of the majority of laser cooling experiments, the theoryand implementation of a wide variety of Zeeman slower designs are covered extensively inthe literature (e.g. [63, 102, 124, 125]).2.2.3 Magneto-Optical TrapThe elements we have described so far do not constitute a trap. Rather, they are a meansof slowing the atoms in one direction and do not spatially localize them. To create a MOT,we apply very similar concepts to those described above, but in a slightly different way.Two new ingredients are needed: cooling from all directions and a means of applying astronger force in the direction of the trap center when the atoms move away from this92.2. Laser Cooling and Magneto-Optical TrapsFigure 2.4: The design of the vacuum chamber showing the two ion pumps, NEG pump,Zeeman slower coils, MOT coils, the Lithium oven, glass cell and the slowing beam. Thered circle in the middle of the glass cell is meant to indicate the location where the 6LiMOT is formed. This figure was retrieved from [63].location. The former requirement is fulfilled through the use of three pairs red detunedlaser beams that intersect at the geometric center for the trap. These six beams and theatom cloud are shown in Figure 2.5. To satisfy the second requirement, we add spatiallyvarying magnetic field that is zero at the center of the trap and increases away from thecenter. (Experimentally, this field profile is obtained by running current through the MOTcoils in the anti-Helmholtz configuration.) If an atom moves away from the trap center,where the magnetic field is zero, its energy levels shift in a state dependent way due tothe Zeeman effect such that the laser beam opposing its displacement from the trap center(with a particular circular polarization) is closer to a transition resonance in the atom thanthe oppositely propagating beam (with the same circular polarization but traveling in theopposite direction). As a result, the radiation pressure due to photon scattering from thisbeam is larger than the opposing beam, and the atoms is pushed back to the center of thetrap. For a deeper discussion of the theoretical framework behind the operation of MOTs,we refer the reader to [30, 53, 101].So far, we have been implicitly making two assumptions: that the laser intensity is wellbelow the saturation intensity (so that stimulated processes can be ignored) and that thetransition is closed. The former is easily achieved by making the MOT beams large and102.3. Lithium Laser Systemsetting the laser powers appropriately. However, the second assumption is false when asingle frequency laser source is employed. For a 6Li MOT operating on the D2 line, a closedtransition does not exist because an excited state atom can decay to either the f = 1/2 stateor the f = 3/2 state (see Figure 2.2) at low magnetic fields. To get around this problem,we need to use two laser frequencies, which are often referred to as the pump and re-pump,separated by the hyperfine splitting of the ground state (228.2 MHz). The re-pump fieldre-excites atoms that fall into the f = 3/2 state, which would otherwise remain dark withrespect to the pump light, ensuring the continuation of the absorption and re-emission cyclewe need for efficient laser cooling. Due to the rapid depletion rate of the ground f = 3/2state in 6Li, which is a consequence of the very small splitting between the f -levels of the2P3/2 state, the re-pump intensity must be comparable to the pump intensity.2.3 Lithium Laser SystemWhile the addition of a Zeeman slower significantly improved the MOT loading rate andthe vacuum relative to what we had prior to the upgrade described in [17], the stability ofthe system degraded and the need for maintenance on a daily basis rose dramatically. Thishappened both due to elevated complexity of the system and because of aging equipment6.Hence, we designed and built a new system with emphasis placed on long-term stability andmodularity. These characteristics were achieved by limiting the lengths of free space regionsand de-coupling them through extensive use of fiber interconnects. In the subsections thatfollow, we describe the design and performance of the system in 2.3.1 and briefly mention asub-Doppler cooling mechanism that relies on the D1 transition and what we hope to gainby implementing it.2.3.1 6Li Laser System DesignIt is natural to divide the laser system into three parts based on their respective roles andphysical locations:1. Laser sources and initial amplification stage.6Due to budgetary constraints, we were unable to immediately replace our rapidly deteriorating taperedamplifiers and were forced to compensate for their poor performance with daily maintenance efforts.112.3. Lithium Laser SystemFigure 2.5: A sketch of the three pairs of counter-propagating MOT beams and the atomiccloud. In our setup, three of the six beams are retro-reflected versions of their counterparts.2. Frequency shifting and second amplification stage.3. Widely tunable optical circuit for imaging 6Li at high magnetic fieldsWe discuss each of them in detail below.Laser Sources and Initial Amplification StagesThe setup implemented on our master table is shown in Figure 2.6. It consists of twoTOPTICA DL Pro external cavity diode lasers (ECDLs) and the associated frequency122.3. Lithium Laser Systemstabilization schemes, two injection locked slave lasers (SLs) and two frequency shiftingstages.The D2 master is locked +50 MHz above the D2 pump transition (f = 3/2 → f ′ =5/2). The error signal is generated by demodulating - via lock-in detection - a sinusoidallydithered saturated absorption (SA) spectroscopy signal. A details of the optical layout,locking electronics and the construction of the Lithium heat-pipe used for the Doppler freespectroscopy can be found in [86, 139, 146, 157].Since the D1 transition is only ≈ 10 GHz away from the D2 transition (see Figure2.2), we generate and detect with the inexpensive HFD6180-421 package - which includesa GaAs PIN photodiode and a transimpedance amplifier (TIA) - a heterodyne beat notebetween the D1 and D2 master lasers. With a direct digital synthesizer (DDS), locked tothe Menlo Systems GPS-8 10 MHz reference, followed by a chain of non-linear conversionstages and amplifiers serving as a local oscillator (LO), this beat note is then mixed downto ≈ 400 MHz and directed to a home-built phase-frequency-discriminator (PFD), whichoutputs a signal proportional to the phase error. With this as the error signal, we phaselock the D1 master to the D2 master by feeding back on the laser diode current and thePZT attached to the grating in the D1 ECDL via the TOPTICA Digilock 110, an FPGAbased loop filter. We discuss the concept of phase locking in greater detail in chapter 3 inthe context of our spectroscopy system. The linewidths of the beat-note is approximately500 kHz, so the linewidth of each laser must be below this value. We do not seek to betterquantify the performance of each laser because the upper bound of 500 kHz would be morethan adequate performance for this application.The remainder of the light from the D2 master is used for injection locking slave lasersS1 and S3. (The latter is used to generate light for imaging at high magnetic fields and willbe discussed in greater detail in the paragraphs below.) Similarly, the D1 master injectsslave laser S2. The outputs of S1 and S2 pass through acousto-optic modulars (AOMs) thateach shift the frequency of the incident light by +108 MHz and can be switched on and offindependently. The former is required for generating the correct laser frequencies for theD2 magneto-optic-trap (MOT) and the latter is required for quickly (in a few microseconds)switching between the D1 and D2 light for the output of this stage and therefore also the132.3. Lithium Laser Systemsubsequent stage. The outputs of the AOMs are then combined on a polarizing beam split-ter (PBS). Next, we pass an equal proportion of the two beams through an optical isolator(resulting in a loss of 50% per beam) and couple the combined light into a polarizationmaintaining (PM) fiber which goes to the amplifier system, where, in addition to amplifi-cation, we shift the frequencies again with AOMs to generate the pump and re-pump lightfor the MOT and for absorption imaging at low magnetic fields.Figure 2.6: Master table layout. The abbreviations λ/2, PBS, Si, SP, PD and OI refer toa half-wave plate, polarizing beam splitter, injection locked slave laser i, single pass AOM,photo-detector and optical isolator, respectively. While we do not show this explicitly inthe figure, this system has one OI placed after every master laser and injection locked slavelaser to prevent unwanted reflections from disturbing or damaging the laser diodes. The OIdirectly before the fiber to TA0 serves two purposes: to combine the D1 and D2 light andto provide an additional 30 dB of protection for S1 and S2 against the back propagatinglight from TA0.Frequency Shifting and Second Amplification StageShown in Figure 2.7 (a), this stage receives seeding light (≈ 27 mW) for a home-builttapered amplifier (based on the Eagleyard EYP-TPA-0670-00500-2003-CMT02-0000 chip),142.3. Lithium Laser SystemTA0, from the appropriately labelled output of the previous stage. Next, the fiber-coupledoutput of TA0 is divided into the pump and re-pump paths and used to seed two TOP-TICA BoosTAs. (The astute reader may have noticed that the single-pass AOM shifts thefrequency of the re-pump path back to +50 MHz and initiatively conclude that the twosingle-pass AOMs in the previous stage are superfluous. While this would be true if our ob-jective was only to generate D2 light, these additional AOMs are necessary for the purposeof quickly switching between D1 and D2 light for the sub-Doppler cooling procedure weaim to implement.) Finally, the double-pass (DP) AOMs [98] (IntraAction ATD-1001A1and ATD-801A1) allow us to achieve detunings anywhere from -70 MHz to 0 MHz withrespect for the pump and re-pump D2 transitions. For the D1 light (which will be tunedblue of the pump and re-pump resonances), we have the same frequency scanning rangeand have the freedom to choose the optimal laser frequency for seeding TA0. (The latter isachieved simply by changing the frequency of the RF reference used for the PFD.) When weinitially built this part of the system, TA1 and TA2 were outputting 250 mW and 220 mW,respectively, and the power out of the fibers labelled ”MOT Pump” and ”MOT Re-Pump”were 135 mW and 90 mW7.Widely Tunable Optical Circuit for Imaging 6Li at High Magnetic FieldsThe energy eigenvalues of 6Li for the hyperfine manifolds of the 2S1/2 and2P3/2 levels areshown as a function of the magnetic field in Figure 2.8. These eigenvalues were calculatedusing the mathematica script included in [56]. For all of the photo-association experimentsdiscussed in this dissertation, we operate at high fields where the good quantum numbersare mj and mi. Specifically, we populate the two lowest energy levels: |1〉 = |1,−1/2〉 and|2〉 = |0,−1/2〉. To image states |1〉 and |2〉 using absorption imaging (more on this below),we must produce laser frequencies that couple them to the |1,−3/2〉 and |0,−3/2〉 levels(∆mi = 0), respectively, in the2P3/2 manifold. (In Figure 2.8, the splittings between thethree mj = −3/2 levels are not visible because are very small.) Unlike imaging close to 0 G,7Since then, the power has dropped signaling the imminent failure of the chips and the need for re-placements. Due to the moderate power requirements of our D2 MOT, this has yet to have an impact onthe number of atoms we can load into the optical dipole trap and therefore inconsequential for all of themeasurements discussed in this thesis. However, much higher intensities are required for D1 cooling, so thechips will need to be replaced before we can add this capability to our experimental sequence.152.3. Lithium Laser System(a) (b)Figure 2.7: Amplification of cooling light (a) and generation of high-field imaging light (b).The abbreviations λ/2, PBS, Si, TAj , SP, DP and OI refer to a half-wave plate, polarizingbeam splitter, injection locked slave laser i, tapered amplifier j, single pass AOM, doublepass AOM and optical isolator, respectively. While we do not show this explicitly in thefigure, this system has one OI placed after every injection locked slave laser and taperedamplifier to prevent unwanted reflections from disturbing or damaging the laser diodes andTA chips, respectively.a re-pump beam is not required at high magnetic fields because these transitions are closed[63]. To implement this system, we simply inject another SL, S3, with light directly out ofthe D2 master (+50 MHz), and shift its frequency three times with AOMs. The layout isshown in Figure 2.7 (b). The advantages of this design are listed below:1. An additional ECDL (or similar laser) and the associated locking electronics are notrequired.2. We can easily switch which state we would like to image every time we run theexperimental sequence.162.3. Lithium Laser System3. The wide scanning range of two cascaded DPs (IntraAction IntraAction ATD-2001A1)allows us to image the atoms over a ≈ 300 G range of magnetic fields or, equivalently,from -750 MHz to -1150 MHz with respect to the zero-field pump transition frequency.(The AOMs used for this setup are deflectors from IntraAction with a center frequencyof 200 MHz.)4. A mechanical shutter is not required because we drive both DPs with the same RFsource and have well under a nanowatt of leakage light when the RF is switched off.The only disadvantage is being unable (without adding another AOM) to extend the tuningrange further. So far, there has not been a reason for doing so. Finally, a small fractionof the unshifted light from S3 is coupled into the fiber that injects the last SL, S4, usedto generate light for the Zeeman slower. The frequency of the light out of S4 is shifted by-126 MHz via a DP. The remainder of optical layout required for producing a 6Li MOT isquite standard and has remained unchanged. We invite the reader to explore the detailspresented in [16, 17, 63].2.3.2 Gray Molasses CoolingMany Alkali metals can be cooled to sub-Doppler temperatures via the Sisyphus effect[24, 32, 125, 132] before they are transferred into an ODT and cooled further via evapo-rative cooling. This makes it much easier to produce degenerate quantum gases of thesespecies while retaining a large number of atoms. Due to the unresolved hyperfine structureof the 2P3/2 manifold, we cannot take advantage of this cooling mechanism for6Li. How-ever, several groups have implemented a gray molasses cooling stage that relies on the D1transition and leads to similar results [22, 50, 51, 59, 133, 144]. By virtue of gray molassescooling, these groups were able to load more than ten million atoms into their ODTs whilewe sometimes struggle to load two million atoms. It is for this reason that we purchaseda second ECDL and implemented the frequency switching scheme described above. Afterreplacing our dying TA chips, we hope to also take advantage of the five-fold increase inthe signal to noise ratio (SNR) we would gain by seamlessly adding gray molasses coolingto our toolbox.172.4. Optical Dipole Trap0 100 200 300 400 500 600 700 800 900 1000Magnetic Field (G)0−500−10005001000Energy(MHz)2S1/2mJ = −1/20−1000−200010002000Energy(MHz)2P3/2mJ = −3/2Figure 2.8: Energy eigenstates of the 2S1/2 (a) and2P3/2 (b) levels in6Li as a function ofmagnetic field. For imaging at high magnetic fields, we rely on transitions from the lowesttwo 2S1/2 levels with mj = −1/2 to the mj = −3/2 levels with the same mi values inthe 2P3/2 manifold. At high fields, these transitions are closed and no re-pumping laser isrequired.2.4 Optical Dipole TrapIn our discussion of laser cooling and MOTs, we were concerned with the dissipative forceassociated with atoms scattering near resonant photons. Since this is the dominant effectin the regime where the field intensity is low and the detuning is small, we ignored a secondconservative force that is dominant in the opposite extreme - where the field intensity ishigh and the detuning is very large. In this section, we will do the opposite by ignoring thedissipative force and concentrating on the conservative force. Specifically, we will demon-strate that high intensity far detuned lasers can be used used to create optical dipole trapsor optical tweezers that enable us trap atoms and cool them to sub-µK temperatures.182.4. Optical Dipole Trap2.4.1 Basic TheoryThe interaction Hamiltonian for a two-level system interacting with an electric field is givenbyHˆI = −µ ·E(r) (2.2)where µ is the dipole moment operator and E(r) is the electric field. For a very largenegative (or red) detuning, ∆, from the resonance frequency, ω0, it can be shown that theground state |g〉 shifts down while the excited state |e〉 shifts up in energy [61, 63, 157].These intensity dependent energy shifts or ac Stark shifts are given by∆Ee =3pic22ω30(Γ∆)I (2.3)∆Eg = −3pic22ω30(Γ∆)I (2.4)where Γ is the natural linewidth of the transition [61]:Γ =ω303pi0~c3|〈e|µ|g〉|2 (2.5)For a Gaussian beam incident on the atoms, the resulting ground state potential is givenbyUdip(r) =3pic22ω30(Γ∆)I(r) (2.6)whereI(r) = I(r, z) =2PpiW 2(z)exp(− 2r2W 2(z))(2.7)In equation 2.7, P is the optical power and W (z) is the beam waist given byW (z) = W0√1 +(zz0)2(2.8)where W0 is the minimum beam waist and z0 = W20 pi/λ is the Rayleigh length [147].The gradient of this potential, Fdip −∇Udip(r), is a dipole force that pushes atoms to the192.4. Optical Dipole Trapregion of highest intensity thereby creating an optical dipole trap. Since this trap is almostharmonic near its center and the atoms coalesce in the region where z  z0 and r  W0,we gain several experimentally relevant insights by Taylor expanding Equation 2.6 around(r, z) = (0, 0) and equating the resulting expression to that of an ideal harmonic trappingpotential:U(r, z) ≈ U0(1− 2r2W 20− z22z20)= U0 +12mω2rr2 +12mω2zz2 (2.9)where the trap depth U0 is the trapping potential at (r, z) = (0, 0) and ωr and ωz are thetrapping frequencies along the radial and axial directions, respectively. By equating powersof r and z, we get the following expressions for ωr and ωz:ωr =2W0√|U0|m(2.10)ωz =1z0√|U0|m(2.11)Finally, it is important to briefly mention that the dissipative force we have ignored so farin this section is responsible for heating in the trap. In the limit where ∆ Γ, the photonscattering rate is given byΓsc =1~(Γ∆)Udip (2.12)Above, we assumed that scattering losses are negligible for very large detunings becauseΓsc ∝ 1/∆2 while Udip ∝ 1/∆. To quantitatively confirm the validity of this assumption inthe context of our 6Li ODT, we can compute the scattering rate as a function of trappingpower using the 6Li D-line data from [55] and the results of our beam waist measurements(see appendix A). For the 6Li D lines, Γ/2pi = 5.9 MHz and ω0/2pi = 446.8 THz. Thedetunings and beam waists for our SP-100C-0013 SPI fiber laser (100 W, 1090 nm centerfrequency, 2 nm optical bandwidth and randomly polarized) and YLR-20- 1064-LP-SF IPGfiber laser (15 W, 1064 nm, single frequency and linearly polarized) are shown in the tablebelow:202.4. Optical Dipole TrapTrapping Laser W0 (µm) ∆/2pi (THz)SPI 40 -171.76IPG 58 -165.04Plugging these values into 2.12 yields ΓIPGsc = (0.041 s−1W−1)P and ΓSPIsc = (0.085 s−1W−1)P ,where P is the trapping power in watts. These calculations show that the scattering rate issmall on the time scale of a typical experiment (several seconds), even at a trapping powerof 100 W. However, since each scattering event heats the atoms by one photon recoil energyand we need to bring the atomic cloud to a much lower temperature in most experiments,we typically do the “science” step at low trapping powers.2.4.2 Experimental RealizationA careful inspection of equations 2.10 and 2.11 reveals that ωr  ωz because z0  W0.This implies a much weaker confinement in the axial direction than the radial direction. Toachieve strong confinement in all directions, we use a crossed optical dipole trap (CODT)with a crossing angle of α of 20 degrees. (For details concerning how this angle was chosen,see [63, 139].) A simplified schematic of this layout is shown in Figure 2.9 and applies toboth of our two trapping lasers8, which are overlapped spatially via a dichroic mirror (notshown) and propagate collinearly through this network of mirrors and lenses. Each trappinglaser is focused onto the compressed and cooled MOT (the vacuum chamber is omitted forsimplicity) with a 250 mm lens, re-collimated with another 250 mm lens and then focuseda second time on the MOT to increase the trap depth and to produce tighter confinementin the axial direction. We control the intensity incident on the atoms via an analog inputand a calibrated external AOM (Gooch and Housego 97-01672-11) for the 100 W SPI laser.For the second 15 W IPG laser, we rely exclusively on a second calibrated external AOM(same model). As a we show in appendix B, this arrangement results in ωy ≈ ωx andωz ≈ 0.15ωx. Relative to a single arm trap, the increase in confinement is about 20 timesfor the z-direction and the drop in confinement for in the x-direction is < 2% [63] relative to8For trapping 6Li, our 100 W SPI laser would be sufficient. The IPG laser is only required for trappingRb because trapping in the SPI results in loss of Rb due to multi-photon processes. We mention it herebecause the ability to trap using a laser with a different wavelength is advantageous for the verification oflifetimes of deeply 6Li2 molecules.212.4. Optical Dipole Trapwhat it would be for α = 0 degrees. Finally, we also show a photo-association (PA) beamthat originates from our spectroscopy system and is overlapped by means of a dichroic withthe second arm of the CODT. We will explore the origin of this beam in chapter 3 and itspurpose in the second half of this dissertation.Figure 2.9: Simplified schematic of our crossed optical dipole trap. The first arm is focusedonto the MOT with a 250 mm focal length lens. The second (re-cycled) arm is first re-collimated with another 250 mm focal length lens and focused a second time on the MOT.The angle between the two arms is 20◦. Also, we show how the photo-association (PA)beam is combined with the second arm of the CODT on a dichroic mirror and focused viathe same lens onto the atoms captured in the CODT.2.4.3 Evaporative CoolingAfter transferring approximately 2 million 6Li atoms from the MOT to the SPI CODT(U0/kB ≈ 3 mK at our maximum trapping power of 100 W), we initiate a process calledevaporative cooling. Through this process, we can cool the atoms from a temperature of≈ 500 µK to temperatures as low as ≈ 100 nK. In principle, this entails simply lowering thetrap depth such that the hot atoms escape the trap and the cold atoms remain in the trap.In practice, doing this while minimizing atom loss is quite challenging and has been thesubject of many experimental and theoretical studies including [1, 38, 39, 83, 92, 118]. Forthe purpose of achieving efficient evaporative cooling of 6Li in our setup, we have found thedoctoral thesis of Kenneth O’Hara [117] the most comprehensive reference. In particular,222.4. Optical Dipole TrapO’Hara shows that the evaporation follows a scaling law given byNfNi=(UfUi) 12(3η−3)(2.13)where Nf/Ni and Uf/Ui are the ratios of the atom number and trap depth, respectively, atthe end of the evaporation and the beginning of the evaporation and η := U/kbT ≈ 10. Froman experimental perspective, achieving efficient evaporative cooling requires two ingredients:a high thermalization rate and the freedom to adjust the speed and shape of evaporationramps. To achieve the former, we need to significantly increase the collision rate by adjustingthe effective collisional size or scattering cross-section σ of the atoms. We do this by meansof the broad Feshbach resonance in 6Li at 832 G [15, 27, 116, 136] - a quantum mechanicallever that allows us to tune the interactions between the atoms by changing the value of thehomogenous magnetic field produced by our Helmholtz coils. To avoid diverting the focusof this section, we postpone discussing this complex and rich phenomenon until chapter4 and illustrate the main point via the cartoon representation shown in Figure 2.10. Wefulfill the second requirement by calibrating the AOMs we use for the CODTs and using ourcontrol system to create piecewise intensity ramps (linear and exponential) into the DDSsdriving the AOMs via the python front end. Each ramp has a starting and stopping powerand a parameter (slope or time constant) that specifies the speed of the evaporation. Thevalues of these evaporation parameters and evaporation fields were optimized empiricallyin a sequential fashion. Despite our best efforts, we were unable to achieve the optimalefficiency (η ≈ 10) for the high power portion of the evaporation (possibly due to the heatingthat results from scattering and intensity noise at the highest powers). The evaporationefficiency improves significantly after most of the atoms coalesce in the cross (as opposedto the wings of the trap) at trapping powers below 12 W. This part of the evaporation isshown in Figure 2.11 for both the SPI trap and the IPG trap.232.4. Optical Dipole TrapFigure 2.10: Cartoon representation of the change in the collision cross section of the atomsin the ensemble due to the Feshbach resonance near 832 G. If we think of each atom as abilliard ball with some radius r at B = 0 G, then changing the magnetic field to 755 G(close to the Feshbach resonance near 832 G) is equivalent to increasing the radius of theballs to r′  r. As a result, the collision rate significantly increases.242.4. Optical Dipole Trap(a)(b)Figure 2.11: The Normalized atom number is plotted with respect to the normalized trapdepth for a forced evaporation in the SPI (a) and IPG (b). According to the fits to theO’Hara scaling law, η = 10.07 + / − 0.13 for evaporation in the SPI and η = 10.00 ± 0.11for evaporation in the IPG. In both cases, the maximum trapping power was 12 W and themagnetic field was held at 755 G for this portion of the evaporation ramp. The dashed redline seen in both plots is the “ideal” (η = 10) evaporation ramp.252.5. RF Spin Mixing2.5 RF Spin MixingFor the efficient production of Feshbach molecules - without which we would be unable toperform most of the photo-association experiments discussed later in this thesis - we needa 50:50 mixture of |1〉 and |2〉 state atoms (see chapter 4 for details about state namingconventions) at the end of the forced evaporation. To correct for any population imbalance,we used decoherence to our advantage. For an atom fixed in space and in the presenceof a uniform magnetic field B, RF photons with frequency fRF = (E|2〉(B) − E|1〉(B))/hwould couple the |1〉 and |2〉 states and induce coherent Rabi flopping between them. Sincewe want to produce an incoherent mixture of the |1〉 and |2〉 states with equal population,this coherent Rabi flopping is not adequate. However, both of these assumptions are false:the atoms do move around the trap and the magnetic field has a small gradient. Hence,the energy difference and the precession speed are spatially varying. This means thatan ensemble of atoms that were initially in phase and thus constituting a pure state willeventually be completely out of phase and we end up with an incoherent 50:50 mixture of|1〉 and |2〉 state atoms.Experimentally, we do this by driving a home-built antenna at ≈ 80 MHz (several loopsof copper wire) with a 30 W RF power amplifier9. Because the source and load impedanceswere not matched, only a small fraction of the power incident on the antenna is radiated.This sequence is turned on during our forced evaporation at 755 G, which takes place overseveral seconds - more than enough time for the atoms to circulate around the trap andde-phase.2.6 Absorption ImagingWhen laser light resonant with an atomic transition passes through a cloud of ultra-coldatoms, some of it is absorbed by the atoms and the remainder can be captured on a CCDcamera. Thus, we can determine the atom number and the shape of the atom cloud from9This amplifier has protection circuitry built it that prevents reflections from damaging it.262.6. Absorption Imagingthe reduction in the pixel count. The Beer-Lambert law tell us that∂I(x, y, z)∂z= −σabsn(x, y, z)I(x, y, z) (2.14)where I(x, y, z) is the intensity, n(x, y, z) is the density of the atoms and σabs is the ab-sorption cross section. For intensities much lower than the saturation intensity Isat and onresonance, σabs is given by [147]:σabs =3λ22pi(2.15)Solving equation 2.14 leads to the following expression for the column density:n˜(x, y) =∫ d0n(x, y, z′)dz′ =1σabslog(I0(x, y)I(x, y))(2.16)where I0(x, y) is the intensity incident on the CCD in the absence of atoms. Then, the atomnumber that corresponds to pixel ij is Nij = n˜ave(xi, yj)Apixel, where n˜ave(xi, yj) is the meanvalue of n˜(x, y) on a square of area Apixel. Finally, the total atom number is N = ΣNij . (Ifthe magnification of the imaging system differs from unity, AMpixel = Apixel/M2 should besubstituted for Apixel. In our system, M = 1.)The most trivial implementation of absorption imaging entails getting I(x, y) and I0(x, y)by capturing an image with and without atoms, respectively. The latter is often referredto as the background image. While this simple approach had been sufficient for the experi-ments described in [63, 139], the signal to noise ratio (SNR) we got when the atom numberdropped below 30,000 was inadequate for our STIRAP experiments. The problem was theimperfect correlation between the interference fringes that show up in atom images andbackground images. This is because the images are taken at slightly different times and theinterference fringes are moving in time. (For example, a fringe may appear at one locationin the atom image and at a different location in the background image.) However, by con-structing an orthonormal basis from an array of background images, it is possible to createa better background for the region where atom cloud is located for a given atom image. (Inpractice, we found that an array of 30 background images is sufficient.) Using this basis,it is possible to calculate the coefficients needed to expand the portion of the atom image272.6. Absorption Imagingwithout atoms as a linear combination of basis images. With these coeffcients, we constructthe background image for the atom image that includes the region obscured by the atoms.The combination of this method (see [48] for the mathematical formulation) and using asmall circular region of interest (ROI) in the image processor led to a significant reductionin the apparent fluctuations in our atom number, enabling us to collect reliable data withjust a few thousands of Feshbach molecules. For example, the standard deviation of a set of40 identical experimental sequences, where mean the atom number is approximately 10,000,is a factor of 3 lower when the circular ROI and fringe removal algorithm are used.So far, we have been making the assumption that a quantization axis is not defined i.e.that we are imaging at B = 0 G. However, when we image atoms or Feshbach molecules- which are essentially atoms as far as imaging is concerned since their binding energy isnegligible and they fall apart into atoms after the first scattering event - at high fields, only50% of the imaging beam has the correct polarization and the two states we image (whichextrapolate to |1/2, 1/2〉 and |1/2,−1/2〉 in the |f,mf 〉 basis) are no longer degenerate. Itwas shown empirically that this results in under-counting the total atom number by a factorof 4.5 [63].28Chapter 3Raman Laser System 10Inducing stimulated Raman transitions in physical systems requires mutual phase coherencebetween two optical sources. Certain applications are particularly demanding with regard tothe degree of phase coherence required including quantum state control and quantum gateoperations with trapped ions [164], quantum control of molecular states [35, 36, 87, 114, 165],electromagnetically induced transparency (EIT) [70, 97], and atom interferometry [89, 111].In some cases, the two optical fields for the Raman transition can be derived from the samelaser source using frequency shifting elements, and thus the detrimental effect of laser phasenoise is minimized. When the frequency difference of the two fields exceeds the range offrequency shifting actuators, an optical phase-locked-loop (PLL) can be used to phase-locktwo independent optical sources. Optical PLLs require a measurement of the optical phase(typically by heterodyne) and feedback to a frequency or phase correction actuator eitherplaced inside [89, 111] or outside the laser cavity [163, 172]. Unlike the intra-cavity options,the addition of external actuation does not require redesigning an existing laser cavity andresults in modest optical power penalties (10-20%).In this chapter, we explore the level of phase noise reduction that can be achievedusing a simple modification to an existing laser locking setup. This modification involvessampling the optical field of a laser already locked to a frequency reference and using anexternal acousto-optic modulator (AOM) to further reduce the phase noise of the opticalfield by fast electronic feedback to the VCO controlling the AOM. In addition to beingstraightforward to add to an existing system, this secondary locking system is simple toconfigure and inherently stable since it is independent of the primary lock of the laser tothe reference. Moreover, we show that the VCO output can be used to simultaneously10A substantial portion of the text and figures presented here originate from a manuscript titled “Phasenoise reduction of mutually tunable lasers with an external acousto-optic modulator” that we are in theprocess of submitting to a peer reviewed journal. They have been re-printed here with permission.293.1. Introduction to Optical Phase Locked Loopsdrive a second AOM allowing fast intensity control of the phase locked optical field. Thismethod is also compatible with previously demonstrated schemes for realizing continuoustuning of a laser locked to a frequency comb [65, 107]. Using two independently tunablelasers equipped with this locking system, we demonstrate quantum state manipulation ofultra-cold 6Li dimers using Stimulated Raman Adiabatic Passage (STIRAP) in Chapter 7.3.1 Introduction to Optical Phase Locked LoopsAn optical phase-locked loop (OPLL) is a method for transferring the coherence of a stableoptical reference laser to another laser. In an OPLL, the reference laser (a femto-secondfrequency comb in our case) is used to down-convert the optical frequency of the laser wewish to lock down to the RF domain via optical heterodyning. This is done by spatiallyoverlapping the two lasers (which must have the same polarization) on an AC coupled photo-detector that outputs an RF signal at the difference frequency between the two sources -a beat-note. (The photo-detector acts as the optical analog of an RF mixer.) For readersfamiliar with traditional phase-locked loops (PLLs), it may be helpful to think of this signalas the analog of a voltage controlled oscillator (VCO).A simplified block diagram of a PLL (optical or RF) in the phase-domain representationis shown in Figure 3.1, where PD is a phase detector (or phase frequency detector), C(s)is the transfer function (in the Laplace domain) of the loop filter and G(s) is the transferfunction of the system we wish to control - the plant. The purpose of this negative feedbackloop is to make the phase difference ∆φ = φout−φref constant and, consequently, dφout/dt =dφref/dt or fout = fref . In an OPLL, φout is the phase of the RF beat-note between thereference laser and the laser we aim to lock and φref is the phase of an RF source, typically asynthesizer with very low phase noise. If the beat note is locked, then the optical frequencyof the laser we are controlling is offset locked to the reference laser. Thus, the absolutephase stability of the laser we are controlling is determined by the phase-noise suppressionprovided by the OPLL and the phase stability of the reference laser. When an appropriatefunction with optimized corner frequencies and gains is chosen for C(s), the PLL correctsfor the low frequency phase-jitter and can be thought of as a high-pass filter applied to the303.1. Introduction to Optical Phase Locked Loopsphase-noise of the laser. This is illustrated in Figure 3.2.Figure 3.1: Simplified block diagram of an OPLL in the phase-domain representation, wherePD is a phase detector (or phase frequency detector), C(s) is the transfer function (in theLaplace domain) of the loop filter and G(s) is the transfer function of the system we wishto control - the plant.fPhaseNoise(dBc/Hz)Free-Running VCOPhase Locked VCOFigure 3.2: A cartoon illustration of the phase noise of free running VCO or beat-note anda phase-locked VCO or beat note. A well designed and tuned PLL heavily suppresses thelow frequency components of the single-side band phase noise spectrum.For a typical PLL, the transfer function of the plant is simply an ideal integrator,GVCO(s) = KVCO/s and feedback is provided via the VCO’s control voltage. However,for the OPLLs we use to lock our Ti:Sapphires, we have two actuators and two feedbackloops that can be tuned independently for reasons that will become clear in section 3.3. For313.1. Introduction to Optical Phase Locked Loopsthe low bandwidth and high dynamic range PZT-based lock, we use a type II PLL with ahome-built phase frequency detector. For the high bandwidth and low-dynamic AOM-basedPLL, we use a type I PLL with a simple passive mixer in saturation as the phase detector.(Since the PZT-based loop keeps the frequency locked, a type II PLL is not required in thelatter case. For a discussion of the limitations of a type I PLL and advantages of a type IIPLL, see, for example, [129].) For the PZT-based loop, the transfer function of the plant isa simple low pass response:GPZT(s) =KPZTs+ ωPZT(3.1)where ωPZT ≈ 2pi ·10 kHz is the corner frequency determined by the capacitance of the PZTand the output impedance of its high voltage driver and KPZT is a constant. For this loop,we use an empirically tuned PI2 controller as C(s). For the AOM-based loop, the transferfunction is approximately given byGAOM =KAOMe−sτs+ ωAOM(3.2)where τ ≈ 80 ns is the loop delay and the ωAOM ≈ 2pi·1.5 MHz and KAOM is a constant. Thephysical origins of τ and ωAOM are time it takes for the acoustic wave in the AOM to reachthe laser beam and the time it takes to propagate across the beam. For this loop, we usedan ultra-low noise and high slew rate analog loop filter from Vescent Photonics (model D2-125). Of the four available options11, we achieved the highest loop bandwidth with the PIDconfiguration. (Contrary to our intuition and the result of a MATLAB simulation, settinga large derivative gain was crucial for maximizing the closed loop bandwidth. Perhaps, thisresulted in pole-zero cancellation and allowed us to set a higher overall gain before the onsetof oscillation.) A more appropriate loop filter transfer function may exist for this plant, butthis would require a custom design. To avoid this complexity, we chose to settle for whatmay be sub-optimal performance.In this section, we provided a cursory overview of PLLs and alluded to several conceptsin control theory. For a more thorough discussion of these topics, we refer the reader to[7, 43, 129].11PI, PI2, PID and PI2D323.2. Femto-Second Frequency Comb3.2 Femto-Second Frequency CombWhile the primary focus of this chapter is to discuss the development and performance ofour Raman laser system, we must take a short detour to briefly introduce the reader to ourfemto-second frequency comb (FFC) - the optical reference. We aim to provide only a high-level overview of the key features of this laser and highlight the performance limitationsthat are relevant to our Raman laser system. For an in-depth discussion complete with theoptical and electronic schematics for this laser, we refer the reader to the PhD thesis ofWilliam Gunton [63]. We begin by describing the properties of an ideal FFC in the nextsubsection and discuss the specifics our Erbium doped fiber FFC in the subsections thatfollow.3.2.1 Ideal Optical Frequency CombA cartoon representation of the spectrum of an ideal FFC is shown in 3.3. Unlike a contin-uous wave (CW) laser, this laser simultaneously oscillates on many frequencies separatedby integer multiples of the repetition rate, frep = c/2L, where c is the speed of light inthe laser cavity (which is primarily made of silica for the case of our fiber laser) and L isthe length of the cavity. The center frequencies of the elements or teeth of this idealizedDirac comb are given by νn = fceo + nfrep, where n is an integer and fceo is the carrieroffset frequency. For an ideal FFC, fceo and frep are simply constants that determine thecenter frequencies of an infinite series of delta functions. (As we will discuss in the nextsection, both of these quantities must be stabilized via OPLLs for our FFC.) Since the in-verse Fourier transform of a Dirac comb in frequency with periodicity frep is another Diraccomb in time with periodicity T = 1/frep, this laser must be pulsed.333.2. Femto-Second Frequency CombFigure 3.3: Ideal frequency comb with a repetition rate of frep, carrier offset frequency offceo.3.2.2 Erbium Doped Fiber FFCIn a real laser, the finite frequency bandwidth of the gain medium sets a lower bound onthe pulse duration. For our Er-doped fiber oscillator, this pulse duration is approximately77 fs [107]. Hence, there exist a νnmin and a νnmax and the comb teeth do not extend all theway down to fceo (or up to infinity) in frequency.Achieving pulsed operation or mode-locking in practice requires inserting a passive oractive element into the laser cavity that leads to high cavity losses for low intensities (CWoutput) and low cavity losses at high intensities (pulsed output). As a result, this elementchanges the laser dynamics such that the cavity modes are compelled to maintain a fixedphase relationship and interfere constructive only when t = t0 +mT , where t0 is a constantand m is an integer. For our FFC oscillator, we use an artificial saturable absorber for pas-sive mode-locking. This method relies on the non-linear (intensity dependent) polarization343.2. Femto-Second Frequency Combrotation that takes place in the single mode (SM) fibers as well as several wave-plates and aFaraday isolator in the free space region of the cavity (see [65]). For additional informationabout Er-doped fiber oscillators and this method of mode locking, we invite the reader toreview Appendix C and [2, 77, 153, 154, 160].To create a FFC from a mode locked laser, frep and fceo must be stabilized via activefeedback. (The exception to this are difference frequency FFCs, which have have no offset[173] and therefore only require stabilization for frep.) To stabilize the former, we samplethe field directly out of the oscillator and send it to a home-built low noise photo-detector(Hamamatsu S5973 photo-diode followed by two monolithic low noise amplifiers). In thefrequency domain, we observe frep and its harmonics. To derive the error signal, we usethe 8663A RF synthesizer (which receives an ultra-stable 10 MHz reference input from theMenlo Systems GPS-8 frequency reference) as the local oscillator (LO) to mix the harmonicat 3frep ≈ 376 MHz down to DC. With this intermediate frequency (IF) signal acting as theerror input for the loop filter (Vescent Photonics D2-125 Laser Servo), we lock the cavitylength by feeding back on a PZT in the free space portion of the cavity. Locking the fceois a more involved process. The first step is octave broadening the output of the oscillator(after amplifying it with an Er-doped fiber amplifier or EDFA) in a highly non-linear fiber(HNLF) such that the spectrum extends from 1 µm to 2 µm. The heterodyne beat notebetween the doubled (from 2 µm to 1 µm) and the fundamental comb at 1 µm is thus2nfrep + 2fceo− (2nfrep +fceo) = fceo 12. We convert this beat note into a phase error usinga home built phase-frequency detector and provide feedback via the current pumping the980 nm pump diode for the oscillator. (This laser diode is pumped by a home-built currentdriver with an analog modulation input.) In the next subsection, we conclude this overviewby discussing the performance of the PLLs used for the stabilizing frep and fceo and theirrelevance to STIRAP. For a more complete introduction to FFCs, we invite the reader toconsult [8, 112, 135, 150, 169].12In reality, we see copies of fceo beat notes at nfrep ± fceo. We can extract fceo by reading any of themusing a frequency counter.353.2. Femto-Second Frequency Comb3.2.3 Performance Metrics Relevant to STIRAPAs we mentioned above and will discuss in greater detail in the chapters that follow, STIRAPis very sensitive to the relative phase coherence between the two optical sources used for thepopulation transfer, but relatively insensitive to their individual phase coherence. When anFFC is used as an optical reference for a Raman laser system, the relative phase coherencebetween the Raman lasers can be no better than the relative phase coherence between thecomb teeth the Raman lasers are locked to. Noise in the fceo lock does not effect therelative phase coherence because it is common to every laser locked to the FFC. Thus, our≈100 kHz linewidth is perfectly adequate for the fceo lock. Since we lock frep using its thirdharmonic, we expect the phase noise between comb teeth m and n to scale monotonicallyas a function of the frequency separation |∆ν| = |νm − νn| = |m − n|frep. We attemptto indirectly estimate the frequency scaling of the phase noise in section 3.4, but observeno clear increase in phase noise from 0.3 GHz to 12.2 GHz. It is difficult to predict basedon this data what the increase in phase noise is at, for example, ≈9 THz (the frequencyseparation required for accessing the v′′ = 0 levels in the a(3Σ+u ) potential). To measure it,we would need an even more stable optical reference. However, we can predict how quicklyit would need to scale in order to have a significant effect on the Raman laser system viaa back-of-the-envelope calculation. In Figure 3.4, we show the phase noise of the comb at376 MHz. At a 10 kHz offset (noise at lower frequencies is irrelevant for STIRAP unlessthe transfer time is very long), the phase noise is approximately -132 dBc/Hz. In Figure3.7, the phase noise at the same frequency offset is -88 dBc/Hz. Thus, the phase noisewould need to increase at a rate of 4.9 dBc/Hz per THz. Regardless of the accuracy of thisestimate, we do not see a drop in STIRAP efficiency when we switch from the least boundvibrational level to the lowest vibrational level. Thus, it is clear that the performance ofthe comb is currently not a bottleneck for us. (This may change for deeply bound levelsin the lowest lying single potential.) More demanding applications may require a morestable FFC. To significantly improve the performance of our comb, we would need to lockit to an optical (rather than RF) reference and potentially introduce an intra-cavity EOMactuator to increase the closed loop bandwidth of the PLL used for the frep lock (see[11, 72, 74, 75, 113, 151]). The alternative is purchasing a commercial solution from Menlo363.2. Femto-Second Frequency CombSystems or TOPTICA Photonics.Figure 3.4: Phase noise spectrum of the third harmonic (376 MHz) of the FFC’s repetition-rate.373.3. System Design3.3 System DesignFigure 3.5: System (a) and component schematics (b) and (c) for the AOM-based phaselocked loop. See text for details.A schematic for the system with an additional external AOM-based phase-locked-loop (PLL)is shown in Figure 3.5 where (a) shows the entire system consisting of the laser to bestabilized, optical reference, locking branch and experiment branch. Subfigures (b) and(c) show the phase/frequency correction actuator and the associated locking electronics,respectively.In our experiment, the frequency of a Ti:sapphire laser (TS1) is pre-stabilized by elec-tronic feedback to a piezoelectric transducer controlling the cavity length. The PZT lockuses light sampled directly out of the laser and is thus independent and de-coupled fromthe AOM-based lock that follows. A phase-frequency-discriminator (PFD) is used insteadof a mixer to extend the capture range of the PZT lock. Pre-stabilization eliminates DCfrequency errors and minimizes the dynamic range requirements of the AOM-based lock.The decoupling of the two locks allows them to be independently optimized for performanceand stability and avoids complications associated with designing customized loop filters forcoupled feedback loops.The phase noise of the pre-stabilized light is further reduced using feedback to an AOMactuator as shown in figure 3.5(b). To achieve a high closed loop bandwidth, we focus theinput beam (1/e2 intensity diameter of 1.5 mm) with a 300 mm lens and minimize the383.3. System Designacoustic wave propagation delay by translating the AOM such that the beam is incident onthe crystal as close as possible to the source. This results in a propagation delay of 135 nsand a 10-90% rise time of 80 ns. We maximize the actuator dynamic range and minimizeresidual amplitude variation from fiber coupling the deflected beam by placing a collimatinglens as close to the AOM as possible. Here, 200 mm was chosen because of the need to blockthe un-diffracted beam. The AOM deflected beam is re-collimated and is, to first order,parallel to the lens optic axis independent of the deflection angle (i.e. AOM frequency). Asa result, the power through the fiber is almost constant over a driving frequency range ofseveral MHz and drops to 80% at ±6 MHz away from our center frequency of 63 MHz. Asshown in (b), the shifted beam is combined with a second laser (entering the L2-in port),passed through a Glan-Thompson (GT) polarizer, and coupled into an SM/PM fiber toensure mode overlap. The heterodyne beat is detected on a photodiode in block (c). Inthe AOM locking branch, we measure the heterodyne beat of TS1 entering the L1-in portwith the Fiber-based Frequency Comb (FFC) entering the L2-in port. In block (b) of theexperiment branch we combine TS1 with light from a second TS laser (TS2) or the FFC(not shown) to diagnose the phase noise of the light generated for the STIRAP experiment.In (c), a mixer generates an error signal by combining a reference RF signal and thefiltered and amplified heterodyne beat-note between the frequency doubled (via a PPLNcrystal) FFC and TS1. The output of the VCO (Mini-Circuits ROS-70-119+), whose controlinput is driven by a Vescent Photonics D2-125 loop filter, is used to issue a correction signalfor the optical field. The VCO output is split and amplified by two power amplifiers (PA).The amplified signals (RF1 and RF2) are sent to the locking branch AOM and to theexperiment branch AOM generating a copy of the corrected light field. The amplitude ofthe RF sent to the experiment arm AOM can be independently controlled via a voltagecontrolled attenuator (VCA) without perturbing the lock. We achieve pulse shaping in anopen loop configuration using an SRS DS345 arbitrary waveform generator. For applicationsthat require further reduction of the amplitude noise, a stabilization control loop can beadded.To perform STIRAP, a second Ti:sapphire (TS2) laser is used. The arrangement isalmost identical, but includes a double pass AOM immediately after the laser block which393.4. Performancepermits scanning the laser frequency without changing the FFC’s repetition rate (see [65]for details). For the diagnostics measurement in figure 3.7, TS2 is combined with TS1 inblock (b) of the experiment section.3.4 PerformanceOut-of-loop measurements of the power spectrum and single-sideband (SSB) phase noiseof the heterodyne beat-note between a locked Ti:Sapphire laser and the FFC are shownin Figure 3.6 under different locking conditions. We achieve a sub-Hz -3 dB linewidth(not resolved with our RF spectrum analyser) and a level of noise suppression for thethe experiment branch (AOME) similar to the locking branch (AOML) above 3 kHz. Forexample, at a 10 kHz offset, the SSB phase noise is −92 dBc/Hz and −88 dBc/Hz forthe locking branch and experiment branch, respectively. The larger phase noise of theAOME measurement below 3 kHz results from vibrations of optical elements that are notcommon to both the AOME and AOML paths. Similar low frequency noise is evidentin the heterodyne beat measurement between TS1 and TS2 (see Figure 3.7) using fieldsfrom the experiment branches. While this has a negligible effect on the STIRAP efficiencydemonstrated here, other applications including atom interferometry are more sensitiveto noise at these frequencies [89, 111]. Rearranging the optical components, using higherquality optomechanical mounts and adding acoustic dampeners would reduce this phasenoise generated after the lock.403.4. Performance(a) (b)(c) (d)Figure 3.6: Power spectrum and SSB phase noise of an out-of-loop heterodyne measurementbetween TS1 and the FFC. For the power spectrum measurements, the resolution bandwidthis 300 Hz for the wide span and 1 Hz for the narrow span. The results are shown for twolocking conditions: with only the lock of the TS1 cavity length engaged (PZT) and withboth the cavity length and the external AOM-based lock engaged (AOM). The subscripts’L’ and ’E’ correspond to two different out-of-loop measurements: one is a measurement ofthe heterodyne generated by a copy of the fields incident on the locking photodiode (L) andthe other is a measurement of the heterodyne between the FCC and a copy of the correctedTS1 field generated by a second AOM .413.4. Performance(a) (b)(c)Figure 3.7: Power spectrum and SSB phase noise of the heterodyne beat between theexperiment arms of TS1 and TS2 for three frequency differences, |∆f | = |f2 − f1| ≈ |N −M | · frep, where TS1 (TS2) is locked to the N th (M th) comb element and frep = 125.6 MHzis the FFC repetition rate. The exact separation depends on the frequency of the doublepass AOM we use for scanning one of the lasers. The resolution bandwidth is 300 Hz forthe wide span and 1 Hz for the narrow span. Jitter in the FFC pulse repetition rate, whichdefines the comb element spacing, is expected to produce frequency/phase fluctuations inthe heterodyne between the two Ti:Sapphire lasers that increases with |N −M | and thus|∆f |. However, we do not resolve an increase in the phase noise for frequency separationsfrom 0.3 GHz to 12.2 GHz. The elevated noise floor of the black trace and the slightasymmetry of the blue trace result from the frequency dependence of the detector noisefloor and a slow polarization drift in our fiber combiners, respectively.423.5. Summary3.5 SummaryIn this chapter, we demonstrated phase noise reduction using an external AOM for mutu-ally tunable Ti:Sapphire lasers phase-locked to an FFC. We added feedforward arms (withsimilar performance) whose amplitudes can be arbitrarily shaped without disturbing thephase lock. As we will discuss in chapter 7, we then used this system to perform quan-tum state control in 6Li2 dimers using STIRAP. While we were unable to achieve phasenoise-reduction down to the shot noise level because of the inherent acoustic delay of theAOM actuators used and because of the limited configuration options of our loop filters, webelieve that increasing the closed loop bandwidth of the OPLL is possible. This could beachieved with AOMs that do not require angle tuning and consequently have a significantlyshorter acoustic delay [162, 168] or with custom loop filters whose transfer functions areoptimized for this plant.43Chapter 4Feshbach Resonances and the 6Li2SystemIn this chapter, we aim to establish a naming convention for the single atom, two atomand molecular states (where possible), discuss Feshbach resonances and the criterion forcreating Feshbach molecules (which are essential for STIRAP), and state the selection rulesfor dipole transitions for 6Li dimers at high magnetic fields. We begin with an overview ofthe hyperfine and Zeeman splitting in 6Li atoms.4.1 6Li Hyperfine Structure and Zeeman SplittingIn chapter 2, we introduced the hyperfine and Zeeman splitting in 6Li for the purposeof determining the optical frequencies required for laser cooling these atoms. Here, weelaborate on the origins of these effects and set the stage for magnetically induced Feshbachresonances.Since the atoms in the ODT rapidly decay to the electronic ground state after the MOTlight is extinguished, l = 0. In the absence of external fields, we only need to considerthe hyperfine splitting, which takes place due to interactions between the total angularmomentum of the electron j = l+ s = s (because l = 0) and the angular momentum ofthe nucleus i. The hyperfine hamiltonian Hhf is given byHˆhf = ahfj · i = ahfs · i (4.1)where ahf is the hyperfine constant. Due to this coupling, j is no longer a good quantumnumber, so we let f = j + i = s + i be the total angular momentum. Then, f2 = f · f =444.1. 6Li Hyperfine Structure and Zeeman Splittings2 + i2 + 2s · i and we can express Hˆhf asHˆhf =ahf2(f2 − i2 − s2) (4.2)If we now apply this to an eigenstate of f2, |f,mf 〉, we getHˆhf |f,mf 〉 = ahf2[f(f + 1)− i(i+ 1)− s(s+ 1)] |f,mf 〉 = Ehf |f,mf 〉 (4.3)For 6Li, i = 1, s = 1/2, f = 3/2, 1/2 for the 2S1/2 levels. Hence, Ehf = ahf/2, −ahf and thehyperfine splitting is 3ahf/2 = (3/2)152.1 MHz = 228.2 MHz - the value shown in Figure2.2. (We could repeat this calculation with l = 1 to calculate the hyperfine splittings forthe excited state.)Introducing a uniform magnetic field pointed in the z direction gives rise to Zeemansplitting, which breaks the degeneracy between the mf levels. The Hamiltonian responsiblefor this interaction is given byHˆZ = B ·(2µes− µni~)=Bz~(2µesˆz − µniˆz)(4.4)where µe and µi (µi  µe for 6Li) are the electronic and nuclear magnetic moments ofthe atom, respectively, and B is the applied magnetic field. Since |i,mi〉 is an eigenstateof iz and |s,ms〉 is an eigenstate of sz, this Hamiltonian is diagonal in the |i,mi〉|s,ms〉basis, not in the |f,mf 〉 basis. Hence, for small magnetic fields (< 10 G for 6Li), we treatthe Zeeman splitting as the perturbation and label states in the coupled |f,mf 〉 basis. Forlarge magnetic fields, the Zeeman effect dominates, so we label the states in the uncoupled|i,mi〉|s,ms〉 basis. (For a general form that is valid for magnetic fields in the intermediaterange, see [49].) In table 4.1, we provide the shorthand notation for the states at low andhigh magnetic fields. They are written in ascending order in energy. At the end of thesequence for loading the dipole trap, we have a mixture of the |1〉 and |2〉 states. BecausemF = mf1 + mf2 is a conserved quantity (for spherically symmetric interactions), thesestates cannot decay via inelastic collisions making this a stable mixture at low fields. Athigh fields, the states |1〉, |2〉 and |3〉 are stable [120]. For the remainder of this dissertation,454.2. Brief Introduction to Scattering Theory and Feshbach ResonancesLabel Low Field |f,mf〉 High Field |i,mi〉|s,ms〉|1〉 |1/2, 1/2〉 |1, 1〉|1/2,−1/2〉|2〉 |1/2,−1/2〉 |1, 0〉|1/2,−1/2〉|3〉 |3/2,−3/2〉 |1,−1〉|1/2,−1/2〉|4〉 |3/2,−1/2〉 |1,−1〉|1/2, 1/2〉|5〉 |3/2, 1/2〉 |1, 0〉|1/2, 1/2〉|6〉 |3/2, 3/2〉 |1, 1〉|1/2, 1/2〉Table 4.1: Complete basis for the ground state in the coupled and uncoupled representationfor the atomic states of 6Li. In general, the atomic states are superpositions of these basisvectors. At very small (large) fields, the |f,mf 〉 (|i,mi〉|s,ms〉) states are approximatelyequal to the eigenstates of the Hamiltonian.we will be concerned with just the |1〉 and |2〉 states at high fields.4.2 Brief Introduction to Scattering Theory and FeshbachResonancesMany-body collisions are rare in a dilute ultracold gas because the atomic density is lowand the mean inter-particle distance is much greater than the size of the collision complex(or equivalently the scattering length). Hence, it is sufficient to consider only two-bodycollisions. In the center of mass coordinate frame, we split the space dependent part of theHamiltonian into two parts: a center of mass part and a relative motion part. The former isirrelevant in scattering calculations because it drops out [91]. The problem then reduces toconsidering a virtual particle moving in a centrally symmetric potential V (r) about a fixedcollision center. The Hamiltonian for this system (in atomic units) is given byHˆ = Hˆrel + Hˆhf + HˆZ (4.5)where µ = m1m2/(m1 +m2) = m2/(2m) = m/2 is the reduced mass and Hˆrel is the relativemotion Hamiltonian given by [91]Hˆrel = − 12µr∂2∂r2r +lˆ(θ, φ)22µr2+ Vˆ (4.6)464.2. Brief Introduction to Scattering Theory and Feshbach ResonancesIn this equationlˆ2(θ, φ) = −(1sin θ∂∂θ(sin θ∂∂θ)+1sin2 θ∂2∂φ2)(4.7)is the rotational angular momentum operator and Vˆ is the potential energy operator whichcouples the orbital and electronic spin degrees of freedom. It asymptotically approacheszero for large r.4.2.1 Single Channel ScatteringIf we ignore Hˆhf and HˆZ (for now), the problem reduces to a single collision channel andthe solution to the Schro¨dinger equation for r →∞ is a superposition of an incoming planewave and an outgoing spherical wave and is given byψ(r, θ, φ) = A(eikz + f(k, θ, φ)eikrr)(4.8)where f(k, θ, φ) is the scattering amplitude and k =√2µE/~ is the wavenumber [91]. Sincewe only consider cases where the scattered wave is symmetric with respect to the z-axis, thescattering amplitude is actually independent of φ and we can expand the scattered wavefunction as a sum of Legendre polynomials Pl such thatψ(r, θ) =∞∑l=0Fl(k, r)Pl(cos θ) (4.9)where Fl(k, r) is the radial part of the wave function [91]. Each term in the sum is a partialwave with orbital momentum quantum number l. (Here, we are only concerned with s-wave scattering, where l = 0.) The scattering amplitude can be determined by matchingsolutions to the Schro¨dinger equation in several spatial regions. We do not show that hereand instead invite the reader to see the complete derivation in either [91] or [124].The differential cross section is related to the scattering amplitude bydσdΩ= |f(k, θ, φ)|2 (4.10)where dΩ is a small solid angle through which particles scatter. The experimentally relevant474.2. Brief Introduction to Scattering Theory and Feshbach Resonancesquantity for us is the integral cross section, which is given byσ(k) =∫dΩ|f(θ)|2 (4.11)where we integrate over all solid angles. This quantity can be thought of as the collisionalsize of the particles. Thus, if we can increase σ(k) (the size of the “billiard balls”), we alsoincrease the collision rate. To do this, we need to have multiple collision channels whoseinteraction potentials can be tuned by means of a lever we can control experimentally. Thispowerful tool is called a Feshbach resonance and the lever is the magnetic field.4.2.2 Multi-Channel Scattering and Feshbach ResonancesWhen we include the hyperfine and Zeeman interactions (equation 4.5), we get one collisionchannel for each state, so two atoms can either enter and exit via the same channel (elasticcollision) or via different channels (inelastic collision). The second option results in traploss. By virtue of the Zeeman effect, it is possible to change the asymptotic energy of thepotential energy surface associated with each collision channel. In particular, when twochannels have different magnetic moments, it is possible to tune the continuum of the openchannel channel such that it is degenerate with a bound state (typically the least boundstate) of a closed channel, as shown in Figure 4.1. The resulting coupling between a boundmolecular state and a free atom state gives rise to a Feshbach resonance (FR). Near aFeshbach resonance and at ultra-cold temperatures, the cross section is given byσ = 4pia2 (4.12)where a is the real (or elastic) part of the scattering length which can be determined fromcoupled channels calculations (multi-channel scattering theory) provided the singlet andtriplet potentials used are known accurately in the vicinity of their least bound states [27].The real part of the s-wave (l = 0) scattering length a is given bya(B) = abg(1− ∆BB −B0)(4.13)484.2. Brief Introduction to Scattering Theory and Feshbach Resonanceswhere abg is the background scattering length (far from the resonances positions), ∆B isthe width of the resonance (see [27, 124] for precise definition) and B0 is the position ofthe resonance, where the elastic scattering length grows to infinity. Since the cross sectionσ ∝ a2 according to equation 4.12, we would expect it to diverge, at a Feshbach resonance.In reality, it does not. We can see this if we include a second order correction [27]:σ(k) = 4pia21 + a2k2(4.14)When the scattering length diverges, this expression simplifies toσ(k) =4pik2=λ2dBpi(4.15)where λdB is the de Broglie wavelength, which depends on the wave vector or, equivalently,the temperature. Thus, the cross section levels off and would only diverge if we managedto reach absolute zero.Figure 4.1: A sketch of the effective potentials for an open channel and a closed channel.When the energy of the least bound state of the closed channel is close to degenerate withthe threshold energy, a Feshbach Resonances occurs.494.3. The Scattering State and Feshbach Molecule FormationFeshbach resonances in 6Li and other species have been studied extensively [9, 14, 27,49, 58, 71, 79, 82, 84, 96, 119, 137, 145, 167, 174]. For enhancing the efficiency of forcedevaporative cooling and Feshbach molecule creation, we rely on the extremely broad (δb =−300 G) s-wave resonance at 832 G. On the low side of this resonance, the scattering length ais positive because there is a weakly bound molecular state just below the continuum, and itis possible to form these molecules by 3-body recombination and to make a molecular Bose-Einstein Condensate of them at sufficiently low temperatures. We note that the effectiveinteraction on this side of the resonance is repulsive since the scattering length is positive[34, 78, 149]. For magnetic fields below the resonance at 832 G, where the scattering lengtha is negative, it possible to create Bardeen-Cooper-Schrieffer (BCS) like pairs [15, 161, 171].Both of these regimes have been explored with our apparatus [64]. However, our goal in thisstudy is to determine the decay rate constants for deeply bound 6Li2 triplet molecules. Forreasons that will become clear in chapter 7, we operate on the BEC side of the resonance,but actively avoid creating a mBEC. Instead, we aim to produce a thermal gas of Feshbachmolecules without forming a mBEC. In the next section, we discuss these loosely boundmolecules and return to the idea of (not) forming a mBEC in chapter 7.4.3 The Scattering State and Feshbach Molecule FormationAs we mentioned above and in chapter 2, we have a 50:50 mixture of the |1〉 and |2〉 stateat the end of the forced evaporative cooling stage (at 755 G). Since 6Li is a fermion, thetwo atom wave function |12〉 must be anti-symmetrized:|ψ〉12 = 1√2(|1〉1|2〉2 − |2〉1|1〉2) (4.16)where the subscripts refer to atoms 1 and 2. In the absence of p-wave collisions, whichare suppressed for temperatures below 2 µK, the scattering state in the molecular basis|N,S, I, J, F 〉 is given by [63, 140, 141]|ψ〉12 =√16|0, 1, 1, 1, 2〉 −√12|0, 1, 1, 1, 1〉√13|0, 1, 1, 1, 0〉 (4.17)504.3. The Scattering State and Feshbach Molecule FormationThese quantum numbers originate from the rotational angular momentum of the nuclei, N ,the total electronic spin S = s1+s1, the total nuclear spin I, the total angular momentumexcept for the nuclear part, J = N + S (when L = 0), and the total angular momentum,F = N +S + I. As our discussion in the previous section suggested, the coupling betweenthe open channel and closed channel near a FR resonance results in a dressed state, whichis a superposition of a dimer (v′′ = 38, singlet potential) and the incoming state [49]. Theformer is a singlet state (S = 0) and the latter is a triplet state (S = 1). This means that itis possible to photo-associate these molecules into excited singlet and triplet states. In the|N,S, I, J, F 〉 basis, this state is given by|ψB〉 =√Z3(2√2|0, 0, 0, 0, 0〉 − |0, 0, 2, 0, 0〉)+√1− Z(√16|0, 1, 1, 1, 2〉 −√12|0, 1, 1, 1, 1〉√13|0, 1, 1, 1, 0〉)(4.18)where Z is the closed channel fraction measured in [123, 139, 140] and predicted in [130].Over the range of magnetic fields over which we typically do photo-association (700-755 G),Z is on the order of 10−3, so this state is open channel dominated and therefore almostentirely a triplet state. It is for this reason that we obtain much higher Rabi frequencieswhen we photo-associate these Feshbach molecules to excited triplet states.These loosely bound molecules are created by a three-body recombination process thattakes place during atom-molecule thermalization [28, 63, 78, 121]. When three atoms collide,two of them form a Feshbach molecule and the third flies way with the binding energyconverted to kinetic energy. Paintner et al. show using statistical arguments that theFeshbach molecule fraction NFM/Ntotal is given by the following transcendental equation[121]:(1−NFM/Ntotal)2NFM/Ntotal= 6(TTF)3exp(EbkBT)(4.19)where TF = ~ω¯ 3√6Ntotal is the Fermi temperature and Eb = −~2/2ma2 is the bindingenergy of the Feshbach molecule13. If the Feshbach molecule fraction NFM/Ntotal  1, then13This expression is only valid near a Feshbach resonance.514.4. Excited State and Deeply Bound State Labelling and Selection Rulesthe expression in equation 4.19 simplifies toNFM/Ntotal =16(TFT)3exp(− EbkBT)=16(TFT)3exp(~22mkBTa2)(4.20)Clearly, in this limit, we can increase the Feshbach molecule fraction by decreasing thescattering length14, ceteris paribus. Thus, a large Feshbach molecule fraction can still beobtained at a relatively high temperature by setting the magnetic field further away fromresonance so as to decrease the scattering length - a point we will return to in the context ofSTIRAP in chapter 7. Although we only showed this in limit where the Feshbach moleculefraction is small, this statement holds for larger Feshbach fractions.4.4 Excited State and Deeply Bound State Labelling andSelection RulesAngular momentum coupling schemes for molecules are highly complex and different forevery molecule. Even for a molecule as simple as 6Li2, fully characterizing molecular statesand the associated transition probabilities requires extensive numerical studies that arebeyond the scope of this dissertation. However, we can gain insight from the theoretical andexperimental results discussed in [63, 139, 141], which are valid for 6Li2 molecules in singletstates (whose magnetic dipole moment is zero) for any magnetic field and for 6Li2 moleculesin triplet states at B = 0 G. In addition, Hund’s coupling cases (a)-(e) [20, 41, 68] are astarting point for angular momentum couplings of diatomic molecules. They state whichterms in the molecular Hamiltonian dominate over others and, consequently, which quantumnumbers are approximately “good”. Hund’s coupling case (b) is the best match for the statesof 6Li2 we work with because the total electronic angular momentum L = l1+ l2 = 0, so itsprojection mL = 0 and the coupling of L to the total spin S = s1+s2 is weak. As a result,the total angular momentum (aside from the nuclear spin) J = S+N+Λzˆ = S+N , whereN is the angular momentum associated with the nuclear motion and Λ = |mL| = 0 [63]. Ifwe now add the nuclear spin, we get the total angular momentum F = J + I = S+N + I.Since the value of s for each atom is 1/2, the total spin S = 1 or 0. It is tempting to assume14Again, this is only valid near a Feshbach resonance.524.4. Excited State and Deeply Bound State Labelling and Selection Rulesthat a similar argument would hold for the total nuclear spin I. However, the value of Ifor a molecular state of 6Li2 depends on the parity of the electronic wave function uponreflection about a plane containing the internuclear axis, the symmetry of the molecularwave function with respect to an inversion of all of nuclear coordinates, and the value of N .For all of the triplet states considered in this dissertation, I = 1 [63]. The two symmetryconsiderations are typically included in the term symbol for a molecular potential. Similarto the term symbols for atoms, molecular term symbols are specified as follows:2S+1Λ+/−g/uFor all of the potentials we will be considering, Λ = 0. This is denoted by Σ in the termsymbol. The +/− in the superscript refers to parity of the electronic wavefunction withrespect to reflections through a plane containing the internuclear axis and u/g refers tocoordinate reversal symmetry with respect the center of the molecule, r → −r. (Theletters stand for gerade and ungerade, which mean even and odd in German, respectively.)Since a term symbol is not always unique, a letter often precedes it. For example, thelowest lying and second excited triplet potentials in 6Li2 are denoted a(13Σ+u ) and c(13Σ+g ).(Capital letters are used for the singlet potentials.) These are the only two potentials wewill be concerned with.The selection rules for molecular transitions in 6Li2 when spin-spin coupling and spin-rotation coupling are negligible are [63]g ↔ u, ∆N = ±1, ∆S = 0, ∆I = 0, ∆J = 0,±1, ∆F = 0,±1, ∆mF = 0,±1. (4.21)The first rule states that the parity of the wave function with respect to the center of themolecules must change in an electric dipole transition. The second rule states that theorbital angular momentum of the nuclei must change by one unit when a photon (whichcarries one unit of angular momentum) is absorbed. The third rule states that neither Snor I can change because the photon does not couple to the spin degrees of freedom. Theremaining rules are consequences of the first three. There are two exceptions: ∆J 6= 0 ifJ = 0→ J ′ = 0 and ∆mF 6= 0 if ∆F = 0.534.4. Excited State and Deeply Bound State Labelling and Selection RulesIn the presence of a large magnetic field, J and F are no longer good quantum numbers,but the remaining selection rules are still approximately valid (because a photon does notcouple to the spin degrees of freedom):g ↔ u, ∆N = ±1, ∆S = 0, ∆I = 0, ∆mF = 0,±1. (4.22)For the triplet part of the Feshbach molecular state in equation 4.18, N = 0, so mN = 0.Also, we have already stated that mL = 0, mS = ms1 + ms2 = −1/2 − 1/2 = −1, mI =mi1 + mi2 = 0 + 1 = 1 and mF = mN + mL + mS + mI = 0. We can excite the Feshbachmolecules to any vibrational level v′ in the c(13Σ+g ) potential that is accessible by our lasers.(For STIRAP, we chose v′ = 20 vibrational level because this choice results in a larger Rabifrequency than what we achieve with higher vibrational levels. We are unable to tune ourlasers to the wavelengths required for vibrational levels below v′ = 20.) Regardless of thevibrational level, N ′ = 1, so m′N = 0,±1. Because photons do not interact with the spindegrees of freedom, mS and mI remain unchanged, so m′F = m′N = 0,±115. As we showin chapter 6, we can choose which of these levels we couple to by changing the polarizationof the photo-association light. With pi polarized light, we target only the (lowest lying)m′F = m′N = 0 state, which will serve as the intermediate state |e〉 for STIRAP. Similarly,we can target any vibrational level in the a(13Σ+g ) potential. This time, we can targetN ′′ = 0 and N ′′ = 2 states. The laser frequencies required to reach these rotational levelsand the associated binding energies (at B = 0 G) are listed in [63]. For the N ′′ = 0 states,mN ′′ = 0, so only one level is accessible regardless of which of the three excited statelevels we choose. (For the N ′′ = 2, we presume it is possible to access several states, butonly dealt with the mN ′′ = 0 level in this study.) We believe that the v′′ = 0 level withS = 1, I = 1, mS = −1, mI = −1, mN = 0 is the lowest lying level in the a(13Σ+g ). anddiscuss the creation of molecules in this state (and others) and their lifetime in chapter 7.15Actually, we can couple weakly to other mS levels due to the spin-spin and spin-rotation interactions,which we have neglected here. With Feshbach molecules created from a different spin mixture, we believe itis possible to couple strongly to these states, but have not demonstrated this experimentally. For additionalinformation about these interactions, see [20, 99, 108].54Chapter 5Three Level Model: for ModelingAutler-Townes and Dark StateSpectroscopyIn this chapter, we discuss the three level model, which is a good approximation for thephotoassociation process we study using real molecular systems described in the next twochapters. Using this abstraction, we explain the origin of the Autler-Townes splitting, howstimulated Raman adiabatic passage (STIRAP) works and derive the Hamiltonian for athree level system driven by two fields is given byHˆ(t) = Hˆ0 + HˆpI + HˆsI (5.1)where H0 is the Hamiltonian for the bare states |g〉, |a〉 and |e〉 shown in Figure 5.1, HˆpI isthe interaction Hamiltonian associated with the probe field (which couples |a〉 and |e〉) andHˆsI is the interaction Hamiltonian associated with the Stokes field (which couples |g〉 and|e〉). These three Hamiltonians are defined as follows:Hˆ0 = ~((ωge − ωae)|a〉〈a|+ ωge|e〉〈e|) (5.2)HˆpI (t) = −(µˆ · ˆp)Ep0 cos(ωpt) = ~Ωp cos(ωpt)(|a〉〈e|+ |e〉〈a|) (5.3)HˆsI (t) = −(µˆ · ˆs)Es0 cos(ωst) = ~Ωs cos(ωst)(|g〉〈e|+ |e〉〈g|) (5.4)555.1. Autler-Townes Splitting and STIRAPwhere ωge and ωae are the resonance frequencies for the bare states, µˆ is the dipole momentoperation, ˆp and ˆs are the polarization vectors for the fields, Ep0 and Es0 are the fieldamplitudes, ωp and ωs are the driving frequencies and Ωp = −〈a|(µˆ · ˆp)|e〉Ep0/~ and Ωs =−〈g|(µˆ · ˆs)|e〉Es0/~ are the probe and Stokes Rabi frequencies. After we drop the counter-rotating terms, the full Hamiltonian (in matrix form) becomesHˆ(t) =~20 0 Ωse−iωst0 2(ωge−ωae) Ωpe−iωptΩseiωst Ωpeiωpt 2ωge (5.5)where rows (columns) 1, 2 and 3 correspond to |g〉, |a〉 and |e〉, respectively16.5.1 Autler-Townes Splitting and STIRAPTo explain the Autler-Townes doublet and dark states, it is helpful to get rid of the timedependence by working in the dressed state picture and sufficient (for our purposes) toimpose the restriction that ∆p = ∆s = 0. This is because we will always be in this regimefor STIRAP and the level shifts due to the probe field will be negligible in our Autler-Townes scans because we will use a weak probe Rabi frequency. With these simplifications,the dressed picture Hamiltonian is given by [31, 52]HˆD =~20 0 Ωs0 0 ΩpΩs Ωp 0 (5.6)The eigenvalues of this Hamiltonian are simply E0 = 0 and E± = ±~√Ω2p + Ω2s/2 and itseigenvectors are [12, 40, 52, 63]|a0〉 = cos θ|a〉 − sin θ|g〉, |a±〉 = 1√2(sin θ|a〉 ± |e〉+ cos θ|g〉) (5.7)16For an alternative derivation of the Hamiltonian in the dressed state picture, see, for example, [12, 63].565.1. Autler-Townes Splitting and STIRAPwhere tan θ = Ωp/Ωs. Of these three eigenstates, only a0 has no component in the |e〉state. This coherent superposition is known as a dark state because it has zero probabilityof excitation to the |e〉 state. This is the reason STIRAP is such an effective method forpopulation transfer. Before we explore this notion further, we will first discuss the Autler-Townes doublet and its value as a preparatory step for STIRAP.575.1. Autler-Townes Splitting and STIRAP|g〉|e〉Γloss|a〉Ωs, ωsΩp, ωp∆s ∆pFigure 5.1: Three level system dressed by a probe field and Stokes field. The probefield which couples the initial (Feshbach molecule) state |a〉 to an excited state |e〉. It ischaracterized by its frequency ωp, Rabi frequency Ωp and detuning ∆p with respect toωae. Similarly, the Stokes field, which couples the final state |g〉 to the excited state |e〉,is characterized by analogous variables: ωs, Ωs and ∆s. Of the three states, |e〉 has by farthe shortest lifetime or, equivalently, the highest loss rate Γloss, which is on the order of10 MHz. Hence, we can regard any Feshbach molecules transferred to the |e〉 state as lostbecause they are either ejected from the trap or fall stochastically into various bound states.Either way, they become invisible to our imaging light.585.1. Autler-Townes Splitting and STIRAP5.1.1 Autler-Townes SplittingAs we saw above, the two states with non-zero probabilities of excitation to the |e〉 statehave energy eigenvalues E± = ±~√Ω2p + Ω2s/2. In the limit where Ωs  Ωp (the weakprobe regime), we can neglect Ωp. Then, the energy splitting between these states is simply~Ωs. Hence, as we scan the frequency of the probe laser, we observe two loss featuresseparated by Ωs/2pi in laser frequency - the Autler-Townes doublet. For our purposes, thesignificance of this is a simple way to measure Ωs at some Stokes laser power and be thenable to calculate the laser power required for any desired Ωs. (There is no third loss featurebecause |a0〉 cannot be excited.)5.1.2 STIRAPStimulated Raman adiabatic passage is a clever sequence used for high efficiency populationtransfer. It exploits the fact that the projection of the dark state |a0〉 onto the |a〉 and |g〉states can be controlled by adjusting the probe and Stokes Rabi frequencies (laser powers).Before describing the laser pulse sequence and the resulting evolution of the dark state, itis helpful to look at the two extremes. When |Ωs| > 0 and |Ωp| = 0, |a0〉 = |a〉. Conversely,when |Ωp| > 0 and |Ωs| = 0, |a0〉 = |g〉. Since the sample of Feshbach molecules we prepareis the |a〉 state, the projection of this state onto the dark state 〈a|a0(t = t0)〉 = 1 if onlythe Stokes laser is turned on initially. As shown in Figure 5.2, we initially make Ωs largeand then slowly increase Ωp while slowly decreasing Ωs. In doing so, we allow the darkstate to evolve adiabatically from the |a〉 state to a coherent superposition of the |a〉 and |g〉states without populating the lossy |e〉 state (in the absence of decoherence mechanisms).Eventually, |Ωs| goes to zero and |Ωp| reaches its maximum, so the dark state becomes |g〉,completing the population transfer. In Figure 5.2, we also show the reverse process that wemust perform for detection purposes. While it is possible to detect deeply bound moleculesdirectly via resonantly enhanced multi-photon ionization (REMPI), our apparatus is notequipped for REMPI, which requires a pulsed laser and electrodes inside the vacuum. So,we instead re-create Feshbach molecules via reverse STIRAP after we finish interrogatingthe deeply bound molecules. For a review of REMPI, we refer the reader to [90].In the idealized picture we painted above, we left out a number of crucial experimental595.1. Autler-Townes Splitting and STIRAPdetails, which we will now discuss. Specifically, we mentioned that we must allow thedark state to evolve slowly from |a〉 to |g〉 to achieve efficient population without explicitlydefining what this means and we neglected to address the issue of laser phase jumps andother decoherence mechanisms. We can simultaneously address these issues by means of asimple adiabaticity condition derived in [12, 54, 85]:Ωeff∆τ =√Ω2s + Ω2p ∆τ > 10 (5.8)where ∆τ is the time during which the probe and Stokes pulses overlap and the numericalvalue 10 was determined via numerical simulations and empirically. This formula has thefollowing three key implications on the experimental realizations of STIRAP:1. Over the course of the STIRAP sequence, the effective Rabi frequency Ωeff shouldremain constant. Since Ωp and Ωs are zero at the beginning and end of the (forward)STIRAP sequence, respectively, their maxima must be equal to Ωeff . As we willsee in the next chaper, the Franck-Condon overlap between the |a〉 and |e〉 states issignificantly lower than that of any accessible |e〉 and |g〉 combination. Hence, themaximum Ωeff we can achieve is approximately equal to the maximum probe Rabifrequency we can achieve with available laser powers17.2. The minimum required duration of the sequence is determined by the Rabi frequencieswe can achieve. In previous studies [35, 36, 66, 87, 110, 114, 122, 148, 152, 165], thehighest achieved probe Rabi frequency Ωp has been on the order of 2pi · 10 MHzand the lowest was on the order of 2pi · 10 kHz. The optimal pulse durations anddelays that correspond to these Rabi frequencies have typically varied between severalmicroseconds to several milliseconds. We operate with Ωp ≈ 2pi4 MHz and obtainedthe highest transfer efficiency for ∆τ ≈ 10 µs.3. It is implicitly implied that Raman lasers must remain phase-coherent with each other17Technically, it is slightly lower than this because we couple the pump and probe beams into the samefiber with the same polarization. Hence, we must take away some power from the probe field to get therequired Stokes power. However, in practice, this amount is only 10% for the bound-to-bound transitionswith the lowest FC overlap and even less for the rest. Since the Ωp ∝ Pp, a 10% reduction in power resultsin just a 5% reduction in Ωp.605.1. Autler-Townes Splitting and STIRAPthroughout the sequence. Relative phase jumps prevent the dark state from evolvingadiabatically, cause projections onto the other two eigenstates and ultimately lowerthe efficiency of the process.The three points above and the fact that we restricted ourselves to ∆p = ∆s in this sectionsuggest that we need four pieces of information to implement STIRAP for every three levelsystem we will consider: Ωp as a function of the probe laser power, Ωs as a function of theStokes laser power, ωae and ωge18. This requires high resolution spectroscopy (the subjectof the next chapter) and a robust fitting model for analyzing the data. We derive this modelin the next section.18It is possible to perform STIRAP with ∆p 6= 0 and ∆s 6= 0 as long as the relative detuning δ =∆p−∆s ≈ 0. For example, this has been demonstrated by K. K. Ni et al. [115]. However, knowledge of ωaeand ωge is still required for setting δ ≈ 0. To maximize the coupling strength, we consistently operate withδ ≈ ∆p ≈ ∆s ≈ 0615.2. Derivation of an Analytical Fitting Function for Two-Color Photo-Association DataFigure 5.2: Ideal forward and reverse STIRAP pulse sequence and the resulting evolutionof the |a〉 and |g〉 state populations. First, we trigger the Stokes pulse. After a delay ofapproximately the full width at half maximum (FWHM) of this Gaussian pulse, we triggerthe probe pulse and initiate the population transfer. Once the amplitude of the Stokes laserapproaches zero and that of the probe laser approaches its maximum, the population of theinitial (Feshbach molecule) state |a〉 approaches 0 and that of the deeply bound |g〉 stateapproaches 1. After some delay, we then run the same pulse sequence in the reverse orderto re-create Feshbach molecules for absorption imaging purposes.5.2 Derivation of an Analytical Fitting Function forTwo-Color Photo-Association DataOur goal in this section is to derive an expression for fitting experimental data of photo-association induced trap loss. Since a molecule that absorbs a photon is lost from thetrap, we can solve this problem deriving an expression for the absorption coefficient, whichis related to the imaginary part of the first order susceptibility χ(1)(−ωp, ωp) (subsection5.2.1). Then, we will use this to calculate the absorption coefficient and obtain an analyticalexpression for the number of Feshbach molecules remaining in the trap as a function of the625.2. Derivation of an Analytical Fitting Function for Two-Color Photo-Association Dataprobe and Stokes detunings and Rabi frequencies and the exposure time (5.2.2). Thisexpression will be valid as long as the probe Rabi frequency Ωp is sufficiently weak.5.2.1 Derivation of χ(1)(−ωp, ωp) in the Weak Probe RegimeThe expectation value of the polarization density in the atomic cloud for our dressed three-level system is given by the following expressions [52, 127]:P (t) = %(µaeρeae−iωaet + µgeρege−iωget + c.c.)(5.9)=120(Epχ(1)(−ωp, ωp)e−iωaet +Esχ(1)(−ωs, ωs)e−iωget + c.c.)(5.10)where % is the particle density, µae and µge are dipole matrix elements, and ρea and ρeg areelements of the density matrix. Hence,χ(1)(−ωp, ωp) = 2%µae0Ep0ρ˜ea = − 2%µ2ae0~Ωpρ˜ea (5.11)where ρ˜ea = ρea exp(iωpt). To obtain an expression for ρ˜ea, we need to solve the Liouvilleequation [31, 127]:˙ˆρ =i~[ρˆ, Hˆ]− γˆρˆ⇔ ρ˙ij = i~ [ρˆ, Hˆ]ij − (γˆρˆ)ij (5.12)where ρˆ = ΣPi|ψi〉〈ψ| is the density operator and γˆ represents phenomenological decayrates. The coupled differential equations for the off-diagonal elements or coherences of thedensity matrix are given below:˙˜ρga = −(γga − iδ)ρ˜ga + iΩp2ρ˜ge − iΩs2ρ˜ea (5.13)˙˜ρea = −(γea − i∆p)ρ˜ea + iΩp2(ρee − ρaa)− iΩs2ρ˜ga (5.14)˙˜ρeg = −(γeg − i∆s)ρ˜eg + iΩs2(ρee − ρgg)− iΩp2ρ˜ag (5.15)635.2. Derivation of an Analytical Fitting Function for Two-Color Photo-Association Datawhere ∆s = ωge − ωs is the Stokes, ∆p = ωae − ωp is the probe detuning, δ = ∆p −∆s isthe two-photon detuning and we introduced the following substitutions the fast exponentialterms): ρga = ρ˜ga exp(−i(ωp−ωs)t), ρea = ρ˜ea exp(−iωpt), ρeg = ρ˜eg exp(−iωst). Since the|a〉 state is the only long lived state and the dark state is predominantly the |a〉 state whenΩs  Ωp, the populations must be ρaa ≈ 1, ρee ≈ ρgg ≈ 0. Plugging these approximationsinto 5.13, 5.14, 5.15 yields0 = −(γga − iδ)ρ˜ga + iΩp2ρ˜ge − iΩs2ρ˜ea (5.16)0 = −(γea − i∆p)ρ˜ea − iΩp2− iΩs2ρ˜ga (5.17)0 = −(γeg + i∆s)ρ˜ge + iΩp2ρ˜ga (5.18)ρ˜ea = −2iΩp(γeg(γga − iδ) + iγga∆s + δ∆s + Ω2p4)4(γga − iδ)(γea − i∆p)(γeg + i∆s) + (γea − i∆p)Ω2p + (γeg + i∆s)Ω2s(5.19)In the weak probe regime, we can neglect the terms with Ωnp where n > 1 in the numeratorand denominator. In this approximation, the expression becomesρ˜ea =−2iΩp(γga − iδ)4(γga − iδ)(γea − i∆p) + Ω2s(5.20)Finally, plugging this result into equation 5.11 and taking the imaginary part yields19Im{χ(1)(−ωp, ωp)}=2%µ2ae0~(8δ2γea + 2γga(Ω2s + 4γgaγea)|4(γga + iδ)(γea + i∆p) + Ω2s |2)(5.21)5.2.2 Relation to Number of Trapped Feshbach MoleculesThe (probe) absorption coefficient is defined as follows [147]:a(−ωp, ωp) = 2k0Im {n(−ωp, ωp)} ≈ k0χ(1)(−ωp, ωp) (5.22)19Note that it is common to include a factor of 1/2 in front of γˆ. To make equation 5.20 conform to thisconvention, we would simply need to make the following substitutions: γij → Γij/2.645.2. Derivation of an Analytical Fitting Function for Two-Color Photo-Association Datawhere n(−ωp, ωp) is the complex refractive index given byn(−ωp, ωp) =√1 + χ(1)(−ωp, ωp) ≈ 1 + χ(1)(−ωp, ωp)2(5.23)The Beer-Lambert law states that∂I∂z=∂∂t(∂Nphoton∂V)hν = −a(−ωp, ωp)I (5.24)Since a Feshbach molecule (FM) that absorbs a photon is either ejected from the trapor rendered invisible to our imaging light, dNphoton = dNFM. Next, we substitute Ωp =−µaeEp0/~ and I = 0c(Ep0 )2/2 into 5.24 and integrate over volume to getdNFMdt= −Ω2pξNFM (5.25)where ξ is the expression in the parentheses in equation 5.21. The solution to this ODE issimplyNFM(t) = NFM(0) exp(−Ω2pt{8δ2γea + 2γga(Ω2s + 4γgaγea)|4(γga + iδ)(γea + i∆p) + Ω2s |2})(5.26)This equation will prove to be an indispensable tool for fitting Autler-Townes and dark statescans in chapter 6, where we analyze a collection of spectroscopic measurements to extractthe parameters we need as a prelude to STIRAP to |g〉 states in the a(13Σ+u ) potential.65Chapter 6Molecular Spectroscopy at HighMagnetic FieldsTo create deeply bound molecules in the a(13Σ+u ) potential via STIRAP from from looselybound Feshbach molecules, we first needed to use high resolution spectroscopy to findthe desired states and determine the probe Rabi frequency Ωp and Stokes Rabi frequencyΩs as functions of their respective laser powers. The states we are able to access with ourTi:Sapphire lasers are shown in Figure 6.1. In the next chapter, we will use this informationto set Ωp ≈ Ωs for the purpose of preserving adiabaticity during STIRAP. To do this, wefollowed this sequence of steps (not necessarily in this order):1. Find a suitable |e〉 (intermediate state) using single color spectroscopy. The require-ments are a sufficiently large Frack-Condon (FC) overlap with both |a〉 (the Feshbachmolecule state) and |g〉 (the final state) and accessibility with available laser sources(a Ti:Sapphire ring laser in our case).2. Determine the natural linewidth of this state and use a lifetime measurement to getan expression for Ωp as a function of laser power using equation 6.1.3. Find a deeply bound state in the a(13Σ+u ) using two color spectroscopy.4. Use equation 5.26 to deduce the magnitude of Ωs as a function of laser power bysplitting the excited state into an Autler-Townes doublet and scanning the probelaser near the single color resonance.66Chapter 6. Molecular Spectroscopy at High Magnetic Fields5.1. Introduction to 6Li and 6Li2 System4 6 8 10 12 14Internuclear Distance (a0)10000500005000100001500020000Energy(cm1)X(1⌃+g )A(1⌃+u )B(1u)C(1⌃+g )a(3⌃+u )b(3u)c(3⌃+g )d(3g)Figure 5.8: The first seven potentials of Li2. The grey band represents the energy rangeaccessible by our current Ti:Sapphire lasers, which includes five potentials: c,b,A,B,C.Of particular interest is the c and A potentials (bold lines), due to their usefulness asintermediate states for molecule formation. The potentials in this figure are based on thenumerical calculations published in [174].when F = 0108. In addition, the parity of the electronic wavefunction must change, whichrestricts allowed transitions to those that satisfy g ! u or vice versa. This results fromconservation between the total angular momentum of the initial state plus photon system108The particular value of mF which is allowed in an electric dipole transition is determined by thepolarization of the light.166Probe FieldStokes FieldFigure 6.1: The seven lowest lying potentials of Li2. The gray bar shows which excited stateswe can access in the c(13Σ+g ) potential with our probe laser. We are able to access everystate in the (shallow) a(13Σ+u ) by virtue of these excited states via our Stokes laser. Thepotentials shown in this figure originate from [76]. This figure was copied, with permission,from [63] and annotated.5. Verify that the two-photon detuning δ = 0 when both the probe detuning ∆p ≈ 0 andthe Stokes detuning ∆s ≈ 0 by doing the same scan in the dark-state regime, wherea narrow revival is observed when the two-photon resonance condition is satisfied.676.1. Single Color SpectroscopySince the procedure is the same for the other states we targeted, we summarize the resultsfor these states in a condensed format at the end of this chapter6.1 Single Color SpectroscopyIn this section, we discuss items 1 and 2 of the list provided in the introduction: findingand characterizing an excited state to determine if it meets the requirements of STIRAP.6.1.1 Finding Excited StatesAs a starting point for finding a suitable excited state, we used two sets of data:1. The locations of seven vibrational levels in the 13Σ+g potential at B = 0 Gauss pub-lished in [141]. We choose this triplet potential for the following reasons: we knewthe approximately locations of the states, our goal was to form molecules in the lowertriplet state and our Feshbach molecular state is predominantly a triplet state. Hence,we can obtain much higher probe Rabi frequencies by coupling to excited triplet statesthan singlet states.2. An unpublished calculation of FC factors by Xuan Li, which predicts that the v′ = 20level has a larger FC factor than any other level we can access with our laser system.(We later confirmed empirically that the maximum Rabi frequency we can achieve islower for the v′ = 21 level than the v′ = 20 level and assumed that the trend wouldcontinue for higher vibrational levels.)By running the broad probe laser frequency scan - in the vicinity of the state locations atB = 0 Gauss for v′ = 20 vibration level - shown in Figure 6.2, we found three manifolds ofstates at B = 755 Gauss. Additional details about this scan are provided in the caption.686.1. Single Color SpectroscopyFigure 6.2: Spectroscopy of the v′ = 20 manifold of the c(13Σ+g ) at 755 G. Using the locationof this vibrational level at 0 G [141] as a starting point, we scanned the frequency of theprobe laser over a 12 GHz span. The polarization was aligned approximately diagonallywith respect to the quantization axis defined by the magnetic field direction to ensure thatwe had a mixture of pi, σ+ and σ− light (for the purpose of accessing the maximum possiblenumber of levels) and set the laser power to 170 mW (the maximum we can achieve).Because mS = −1 for the initial state, we observe strong coupling to the three m′S = −1levels - the large dip near the middle of the plot (not resolved in this figure due to powerbroadening) - in this vibrational manifold. The coupling to the higher energy states is weakbut not zero because although ∆mS = 0, the total spin projection is not perfectly welldefined for these states due to spin-spin and spin-rotation coupling. The three levels weobserve in each m′S manifold differ in m′N and m′F. For the three (unresolved) states inthe broad feature, m′I = 1, m′S = −1, so m′F = m′I + m′S + m′N = m′N = {0,+1,−1}. Them′F = m′N = 0 state is the lowest lying state, as we show in Figure 6.3.The coupling to the states in the two higher frequency manifolds is weak, but not zerobecause although ∆mS = 0, the total spin projection is not perfectly well defined for thesestates due to spin-spin and spin-rotation coupling. Because we observed strong couplingto the lower energy manifold, we decided to closely examine it. As shown in Figure 6.3,the broad feature actually consists of three states. By rotating the laser polarization to beparallel to the vertically aligned magnetic field to produce exclusively pi light, we were able696.1. Single Color Spectroscopyto completely ”turn off” the two transitions near 230 MHz and obtained a strongly coupledand isolated intermediate state for STIRAP. Next, we discuss several measurements fromwhich we can calculate the Rabi frequency as a function of laser power.Figure 6.3: Spectroscopy of the v′ = 20, mS = −1 levels in the c(13Σ+g ) at 755 G. For theselevels, m′I = 1, m′S = −1, so m′F = m′I +m′S +m′N = m′N = {0,+1,−1} in order of increasingenergy. By rotating the laser polarization such that it is parallel to the quantization axisset by the magnetic field, we can turn off the transition to the two higher energy states andobtain an isolated excited state for STIRAP.6.1.2 Measuring the Rabi FrequencyIn the absence of a Stokes field, our model simplifies to that of a two-level system with acharacteristic decay rate Γ(∆):N(t) = N(0) exp(−|ΩP|2Γt)(6.1)In the weak-excitation limit, the Fourier transform of this equation is simply a Lorentzian706.1. Single Color Spectroscopy[147]. Hence,N(∆) =A∆2 + Γ/22(6.2)where A is a constant. The measurement of the natural linewidth of the |v,N,mS,mF〉 =|20, 1,−1, 0〉 state is shown in Figure 6.4. Knowing Γ, we can obtain Ωp at this power bysetting ∆ = 0, scanning the exposure time and fitting the data to the exponential in 6.1.This is shown in Figure 6.5. Since Ωp ∝√Pp, we can now calculate Ωp for any probepower. That is, Ωp/2pi = 34.7 kHz√Pp/(8× 10−3 mW). With our maximum probe powerof 130 mW, Ωp/2pi = 4.4 MHz - similar to the value used by Ni et al in [114]. As we showin the next chapter, this state is adequate for implementing STIRAP. Next, we find andcharacterize a set of |g〉 states.−20 −15 −10 −5 0 5 10 15 20νP − ν0P (MHz)17.520.022.525.027.530.032.535.0NumberofAtomsin|1〉State(×103)Figure 6.4: Fine scan of the single color feature fit to Lorentzian lineshape at B = 755Gauss. (We later confirmed that the magnetic moment of this state is approximately equalto that of the initial state by repeating this scan at both 700 Gauss and 839 Gauss. Inboth cases, the ν0p was unchanged.) To prevent power broadening, the laser power was setto 4 µW. The exposure time was 1 ms. From the fit, we obtain, νp0 = 366861.2520 GHz,Γ/2pi = 6.7 MHz.716.2. Two-Color Spectroscopy0 250 500 750 1000 1250 1500 1750t (µs)051015202530NumberofAtomsin|1〉State(×103)Figure 6.5: Number of atoms in the |1〉 state as a function of exposure time with ∆ = 0,Pp = 8.0 µW at B = 755 Gauss. At this laser power, Ωp/2pi = 34.7 kHz.6.2 Two-Color SpectroscopyIn this section, we discuss the remaining three steps in the sequence: finding a deeply boundstate in the a(13Σ+u ) potential at high magnetic fields, using an Autler-Townes spectrumto determine the Rabi frequency of the Stokes laser as a function of power and, finally,confirming that the two-photon resonance condition (δ = 0) is satisfied when both ∆p ≈= 0and ∆s ≈= 0.6.2.1 Finding Deeply Bound StatesWith the probe laser frequency set such that ∆p = 0 with respect to the excited statediscussed above (we will refer to it generically as the |e〉 state for brevity here), we inducerapid atom loss; however, when we scan the frequency of the Stokes laser across the two-color resonance (where δ = ∆p = ∆s = 0) we couple the |e〉 and |g〉 states. This AC stark726.2. Two-Color Spectroscopyshift shifts or splits the |e〉 state such that the probe laser is no longer resonant with itresulting in suppression of loss.To narrow down the search, we use the locations of the hyperfine manifolds at B = 0Gauss, which are tabulated in [63]. If we assume that the states at high fields are located atmost a few GHz away, we can quickly find the new states by scanning the frequency of theStokes laser, νs. (We make this assumption because the magnetic moment of these statesshould be on the order of double the magnetic moment of a 6Li atom or 2 × 1.4 MHz/G.With this magnetic moment, the shift at 755 G would be 2.1 GHz.) A broad scan of νsis shown in Figure 6.6 . A fine scan of the more deeply bound broad feature is shown inFigure 6.7.Figure 6.6: Two-color spectroscopy of the v′′ = 0 manifold of the a(13Σ+u ) potential at 755G (broad scan). The probe power is set to 50 µW (just high enough to achieve full lossin the absence of the Stokes laser) with an exposure time of 1000 µs. A high Stokes laserpower of 60 mW is used to broaden the two-color resonances. The polarization of both laserfields is rotated to 45 degrees with respect to the quantization axis set by the magnetic fieldto ensure that we do not miss any two-color resonances we have the ability to access. (Welater confirmed by repeating a portion of this scan that the small feature seen in this plotwas an experimental artifact, not a second two-color resonance.)736.2. Two-Color Spectroscopy−150 −100 −50 0 50 100νS − ν0S (MHz)0510152025NumberofAtomsin|1〉State(×103)Figure 6.7: Two-color spectroscopy of the v′′ = 0 manifold of the a(13Σ+u ) potential at755 G (fine scan). Here, Pp = 500 µW, Ps = 12 mW, the exposure time is 100 µs and thepolarization of both laser fields is rotated to 45 degrees with respect to the quantization axisset by the magnetic field. We only observe a single two-color resonance corresponding tothe state with m′′I = 0 since our initial Feshbach molecular state is mI = 0 and the photonabsorption does not couple to the nuclear spin. The other two nearby states have a differentnuclear spin projection mI = ±1. From the Gaussian fit, we obtain ν0s = 375836.0223 GHz.6.2.2 Dark State SpectroscopyHaving found ν0p and ν0s using single color and two-color spectroscopy, it is useful to confirmthat these frequencies are accurate by setting the Stokes laser to resonance and scanningthe probe laser to produce a signature dark state spectrum. (This technique is also usefulfor accurately measuring the binding energies of bound states as demonstrated in [109,166].) When ∆p = ∆s = δ = 0, we expect to see almost complete suppression of lossor electromagnetically induced transparency (EIT) provided that Ωs  Ωp. (If Ωs is toolarge, however, we will instead observe an Autler-Townes doublet. Empirically, we foundthat Ωs/Ωp ≈ 10 produced the desired spectrum. Before determining the laser powerdependence of Ωs, finding the correct power ratio must be done empirically.) A dark state746.2. Two-Color Spectroscopyscan for the excited and bound levels discussed above is shown in Figure 6.8. We fit thedata to equation 5.26 20 to extract ∆s/2pi = 0.11 MHz. This shows that the aggregate errorof our single color and two color measurements was about 0.1 MHz. As we will show in thenext chapter, the width of our STIRAP features is 0.2-0.3 MHz, so being accurate to even±1 MHz and scanning over a 2 MHz span would be sufficient.−40 −30 −20 −10 0 10 20 30 40νP − ν0P (MHz)0510152025NumberofAtomsin|1〉State(×103)Figure 6.8: Dark state spectrum at 755 G with Pp = 8 mW, Ps = 15 mW, an exposuretime of 10 µs and a trapping power of 140mW. The excited state |e〉 = |v,N,mS,mF〉 =|20, 1,−1, 0〉 and the deeply bound state |g〉 is the lowest lying v = 0, N = 0 level we canaccess via this excited state in the a(13Σ+u ) potential. From the fit, we extract ∆S/2pi = 0.11MHz.6.2.3 Autler-Townes SpectroscopyThe final pre-requisite for STIRAP is finding Ωs as a function of laser power. We needthis information to set Ωs ≈ Ωp - a requirement for efficient STIRAP. Having confirmed20Technically, equation 5.26 is not completely valid because the probe power used for this scan is notsufficiently weak. While some of the calculated fit parameters may be inaccurate because of this, theparameter we are interested in here ∆s depends only on the symmetry of the spectrum with respect toνp − ν0p = 0. Hence, the equation is still adequate for this purpose.756.2. Two-Color Spectroscopythat the measured ν0p and ν0s are sufficiently accurate for our purposes via the dark statescan above, we can repeat this scan with a lower probe power to see an the Autler-Townesdoublet. As we showed in the previous chapter, when the Stokes laser is on resonance,the excited state splits symmetrically and the Stokes Rabi frequency Ωs is equal to thisenergy gap (in angular frequency units). An Autler-Townes scan - for the same |e〉 and|g〉 states as the dark state scan in Figure 6.8 - is shown in Figure 6.9. From the fit toequation 5.26, we extract Ωs/2pi = 22.51 MHz when Ps = 10 mW. Since |Ωs(Ps)| ∝√Ps,ΩS(Ps)/2pi = 22.51 MHz√Ps/(10 mW).−30 −20 −10 0 10 20 30νP − ν0P (MHz)051015202530NumberofAtomsin|1〉State(×103)Figure 6.9: Autler-Townes spectrum at 755 G with Pp = 450 µW, Ps = 10 mW, an exposuretime of 20 µs and a trapping power of 140mW. The excited state |e〉 = |v,N,mS,mF〉 =|20, 1,−1, 0〉 and the deeply bound state |g〉 is the lowest lying v = 0, N = 0 level we canaccess via this excited state in the a(13Σ+u ) potential. From the fit, we extract Ωs/2pi = 22.51MHz.766.3. Tabulated Transition Frequencies and Coupling Strengths of All Considered Levels6.3 Tabulated Transition Frequencies and CouplingStrengths of All Considered LevelsAbove, we established a straight forward procedure for finding excited and deeply boundstates and for characterizing the associated transition strengths. We followed the samesequence of steps for a number of other levels and summarize the results that are relevantfor STIRAP in tables 6.1 and 6.2. Since the measured Ωp and Ωs values are dependenton optical alignment and the accuracy of the laser power measurement, the values shownbelow should be considered approximate. The error could be as high as a factor of 2. Thislevel of accuracy is sufficient for implementing STIRAP because it is possible to see a signalprovided that the Stokes laser power is in the correct order of magnitude. (The probepower is always maximized.) As long as we can obtain a weak signal as a starting point,it is trivial to scan the amplitude of a laser over one order of magnitude to improve thetransfer efficiency. In the tables, the Rabi frequencies we provide were extrapolated to alaser power of 1 W for consistency. The data in appendix A in conjunction with these valuescan be used to calculate the expected Rabi frequencies for an arbitrary intensity. Havingmade these prerequisite measurements, we now proceed to discussing our implementationof STIRAP and the lifetimes of deeply bound 6Li2 triplet molecules.ν′ N ′ B (G) νp (GHz) Ωp(Pp = 1 W)/2pi (MHz)20 1 700, 755, 839 366861.2522 12.321 1 755 371574.5777 3.9Table 6.1: Probe laser frequencies and Rabi frequencies for two excited states.776.3. Tabulated Transition Frequencies and Coupling Strengths of All Considered Levelsν′ N ′ ν′′ N ′′ B (G) νs (GHz) Ωs(Ps = 1 W)/2pi (MHz)20 1 0 0 755 375836.0223 135.920 1 0 2 700 375780.2835 135.920 1 2 0 700 372303.578 4226.121 1 2 0 700 372303.578 3415.320 1 5 0 700 368668.3772 342.120 1 6 0 700 367898.7643 1316.120 1 7 0 700 367353.2512 779.420 1 8 0 700 367025.5601 803.620 1 9 0 755 366885.6354 425.6Table 6.2: Stokes laser frequencies and Rabi frequencies for five deeply bound states.78Chapter 7Creation and Lifetimes of DeeplyBound 6Li2 MoleculesIn this chapter, we demonstrate coherent population transfer from a loosely bound Feshbachmolecular state of 6Li (recall that the |FM〉 is an s-wave molecule, N ′′ = 0, with both spinsinglet and triplet charater) to more deeply bound states in the triplet a(3Σ+u ) potential.As the intermediate state |e〉, we use the lowest lying v′ = 20, N ′ = 1 level in the triplet13Σ+g potential [141]. We also discuss the measured lifetimes of deeply bound state to gaininsight into the mechanisms behind the chemical reactions that induce trap loss.In the previous chapter, we tabulated the laser frequencies and Rabi frequencies (as afunction of laser power) for several optical transition of interest for STIRAP to the deeplybound levels in the a(13Σ+u ) potential. To implement STIRAP, we simply needed to switchfrom the square pulses we used for two-color spectroscopy to appropriately timed adiabaticpulses (see chapter 5 for details) whose amplitudes correspond to (approximately) equalRabi frequencies. Since we have no means of detecting deeply bound molecules directly, weuse the reverse pulse sequence to convert the deeply bound molecules back into Feshbachmolecules (see Figure 5.2). In addition, we use fields resonant with the atomic transitions,which originate from the same optical source as the light we use for absorption imaging athigh magnetic fields, to expel any |1〉 or |2〉 state atoms (paired or not) that remain afterthe forward STIRAP sequence.797.1. Experimental Implementation and Optimization of STIRAP7.1 Experimental Implementation and Optimization ofSTIRAPTo create deeply bound molecules and later re-create Feshbach molecules, we executed aspecific sequence of steps for every deeply bound |g〉 state. First, we used the data from theprevious chapter to calculate the laser powers for which Ωmaxs = Ωmaxp . We set the maximaof the Gaussian pulses (programmed via the Stanford Research DS345 arbitrary waveformgenerators) to these values through a combination of manual half-wave-plates rotationsand programmed pulse amplitude settings. Second, we choose and set a pulse width thatsatisfies the adiabaticity condition for STIRAP (equation 5.8). Influenced by the similaritiesbetween our system and that of Ni et al in [114], we made our initial attempts with theiroptimal pulse width and delay - 20 µs. We later found that the optimal values for boththe pulse width and delay for our system is 10 µs. Third, it is crucial to ensure that theFeshbach molecules we image at the end of a forward and reverse STIRAP sequence (seeFigure 5.2) had been deeply bound molecules prior to the reverse STIRAP sequence. Thealternatives are free atoms and Feshbach molecules that were neither associated into deeplymolecules nor lost from the trap as a result of excitation to the |e〉 state. This could happendue to imperfect beam pointing for Feshbach molecules and would be guaranteed to happenfor free atoms that photo-associate on a 100-1000 ms time scale. The solution is to use lightthat is far off-resonant for the deeply bound molecules and resonant for the free atoms andthe Feshbach molecules (for both the |1〉 state and the |2〉 state) to induce loss after theforward STIRAP sequence. Conveniently, our imaging light for high magnetic fields meetsthese requirements and two 5 µs pulses (one for each state) are sufficient for accomplishingour objective. Fourth, the STIRAP resonance is very narrow and prone to shifting slightlyfor a variety of experimental reasons (e.g. alignment), so we normally set the frequency ofthe Stokes laser to the resonance frequency for the |e〉 state to |g〉 state transition and scanthe probe laser in fine steps (e.g. 50 kHz) over a several MHz span around the measured|a〉 to |e〉 resonance frequency. An example of a STIRAP lineshape (average of 10 runs)is shown in Figure 7.1 for a v′′ = 9 N = 0 level. Fifth, we set the probe laser frequencyto the optimal value, where the two-photon resonance condition is satisfied, and scan the807.1. Experimental Implementation and Optimization of STIRAPamplitude of the Stokes laser to optimize the STIRAP efficiency. (Empirically, we foundthat the STIRAP efficiency is not very sensitive to this parameter as long as it is above athreshold and not exceedingly high.) This step allows us to correct for slight experimentaldrifts and measurement uncertainties (with regards to Rabi frequencies) in chapter 6. Thisconcludes our sequence for achieving and optimizing STIRAP.Figure 7.1: Feshbach molecule number after a forward and reverse STIRAP sequence tothe lowest lying v′′ = 9 level as a function of the probe laser’s frequency. To transferthe population from the Feshbach molecule state, |a〉, to the deeply bound state, |g〉, wefirst trigger the Stokes pulse such that the dark state, |DS〉 = |a〉 initially. Then, as weslowly turn on the probe laser and simultaneously lower the Stokes laser’s power, the darkstate adiabatically evolves into the final (deeply bound) state, |g〉, when the two-photondetuning δ is close to zero. At this point, we turn on a field that is resonant with theatomic transition for the |1〉 state for 5 µs and then a second field that is resonant withthe atomic transition for the |2〉 state for 5 µs to expel any residual atoms or Feshbachmolecules from the trap. Finally, we repeat this sequence in reverse to re-create and detectthe initial Feshbach molecule state.After going through the procedure described above, we achieve a round trip STIRAPefficiencies of ≈ 50% or (presumably) ≈ 70% each way. We are still investigating why it is817.1. Experimental Implementation and Optimization of STIRAPnot higher. The possibilities include the following:1. The efficiency is actually higher than the apparent efficiency because molecules thatform in other deeply bound states (as a result of spontaneous emission from the|e〉 state) collide with the Feshbach molecules after reverse STIRAP inducing loss.(The undesired molecules that may be forming would also collide with deeply boundmolecules. However, by triggering the reverse sequence shortly after the forward se-quence ends, we can minimize the interaction time and the resulting losses.) Theimpact of this effect would be diminished if we minimized the time these moleculeshave to collide with Feshbach molecules by imaging the Feshbach molecules immedi-ately after reverse STIRAP. Currently, there is a ≈ 100 ms delay between these events- more than enough time for collisions to take place. If we make the assumptions thatall of our losses are caused by this effect and that any collision between a Feshbachmolecule and a deeply bound molecule results in loss, then it is easy to show that ourone way STIRAP efficiency could be as high as 85%.2. There could be residual laser phase noise that leads to decoherence and results inpopulating the short lived |e〉 state. As we saw in chapter 3, we were unable to bringthe phase noise of our photo-association laser down to the shot noise level for largefrequency offsets. Mitigating this problem would require faster AOM actuators orextensive changes to the laser system (either purchasing lasers with no phase noiseoutside the bandwidth of our AOM-based OPLL or constructing an OPLL based onfaster actuators).3. Projective losses of the dark state can occur due to the small discontinuous steps atthe beginning and end of the Gaussian pulses shown in Figure 7.2. Based on thesize of the intensity jumps (6-7% of the amplitude), we estimate that the projectivelosses are approximately 20%. We have recently replaced our Gaussian pulses withBlackman pulses that should eliminate this problem.4. Intensity noise in the photo-association fields. In the derivation of equation 5.8, it isassumed that the pulses are smooth. If high frequency intensity noise modulates thesmooth pulse envelope, the pulses may not be adiabatic. The best way to mitigate827.1. Experimental Implementation and Optimization of STIRAPFigure 7.2: Gaussian pulses used for STIRAP. Because these pulses are truncated, we sufferfrom projection losses at four instances during the sequence. We estimate that our roundtrip efficiency drops by about 20% in total as a result of these losses. By using Blackmanpulses, we can eliminate the discontinuities while preserving adiabaticity during the restof the sequence. For a deeply bound molecule lifetime measurement, we typically use RFswitches to quickly turn off the RF applied to both the shutter AOM and the pulse shapingAOM to prevent leakage of photo-association light. Leakage from the Stokes laser wouldresult in single-color photo-association of deeply bound molecules to the excited state andcould significantly impact the apparent lifetime. Leakage from the probe laser should noteffect the deeply bound molecules, but we shutter it completely as well as a precaution.For every measurement, we also trigger two ≈ 5 µs long pulses (not shown) resonant withthe |1〉 and |2〉 states, respectively, but invisible to the deeply bound molecules. Thesepulses eject any residual atoms or Feshbach molecules immediately after forward STIRAPsequence. In the absence of the STIRAP pulses, zero Feshbach molecules show up in theabsorption image thus confirming deeply bound molecule formation.this would be to produce the pulses via negative feedback rather than the open loopmethod we have been using. We believe this is unlikely to be the dominant problembecause the pulses appear smooth on an oscilloscope trace.5. Noise and inhomogeneity in the magnetic field. Differences in the magnetic momentsof the states involved in STIRAP give rise to a two-photon resonance condition that837.2. Derivation of the Molecule Lifetime Modeldepends on both time and space. Luckily, we observed that the shift in the two photonresonance is < 1 MHz when we change the field from 700 G to 755 G. If we assume thatit is 1 MHz (an upper bound), then the shift is about 20 kHz/G. Since the STIRAPlineshape is about 200 kHz wide, the temporal noise or field inhomogeneity wouldneed have to be quite large to produce a noticeable effect because it would take a≈ 5 G shift to “break” the two-photon resonance. Since we assumed a generous upperbound for the differential magnetic moment, this number is probably much higherthan 5 G for our sample. It is therefore unlikely that this has a significant impact onour STIRAP efficiency. We mention this factor for completion because it is a seriousconcern for STIRAP that involves states with radically different magnetic moments.For example, it is discussed in the context of RbCs in [40].6. Since our CODT is created by Gaussian beams, the AC Stark shift on the Feshbachmolecules in the wings of the trap is different than that of the atoms near the cen-ter. This is also true for the deeply bound molecules we create. As a result, thetwo-photon resonance condition is not satisfied for the atoms in the wings and theSTIRAP efficiency may be reduced. (This would also imply losses due to the creationof additional molecules in other deeply bound states.) Briefly turning off the CODTduring STIRAP and reverse STIRAP would reveal the severity of this problem andmay noticeably improve our round-trip STIRAP efficiency.7.2 Derivation of the Molecule Lifetime ModelTo extract physically meaningful information from the deeply bound molecule lifetime mea-surements presented in the next section, we first need to derive a model to which we canfit the data. We begin with the time derivative of the cloud density distribution, which isgiven, up to third order, by [61]n˙(r, t) = −αn(r, t)− βn2(r, t)− γn3(r, t) (7.1)847.2. Derivation of the Molecule Lifetime Modelwhere n(r, t) is the density distribution of the cloud at time t and α ≥ 0, β ≥ 0 and γ ≥ 0are the rate constants for one-, two- and three-body losses, respectively. Experimentally,it would be challenging for us to accurately capture the spatial distribution of the deeplybound molecules, especially for very dilute samples. It would be advantageous to mold thisequation into a form that would enable us to use the total atom number as the primaryobservable. Hence, we integrate equation 7.1 over all space to getN˙(t) = −αN(t)− β∫ ∞−∞n2(r, t)d3r − γ∫ ∞−∞n3(r, t)d3r (7.2)To obtain α, β and γ from this form of the equation, we would still need the densitydistribution (unless β = γ = 0). We can simplify this equation if we assume that thedeeply bound molecules move around the trap volume (and redistribute) much faster thantheir half-life such that n(r, t) is simply a 3-dimensional Gaussian distribution with a timevarying amplitude (for a thermal cloud) orn(r, t) = n0(t) exp(− x22σ2x)exp(− y22σ2y)exp(− z22σ2z)(7.3)where σi = ω−1i√kBT/mLi2 [61]21 With this ansatz, we can easily evaluate the integrals inequation (7.2) to get∫ ∞−∞n(r, t)d3r = 2√2pi3/2σxσyσzn0(t) = N(t) (7.4)∫ ∞−∞n2(r, t)d3r = pi3/2σxσyσzn20(t) =18pi3/2σxσyσzN2(t) (7.5)∫ ∞−∞n3(r, t)d3r =23√23pi3/2σxσyσzn0(t) =124√3pi3σ2xσ2yσ2zN3(t) (7.6)21Of course, this simplifying assumption is the most accurate for short hold times because the distributionshape at time t is only partially determined by that of the initial Feshbach molecule cloud (the initialcondition to the differential equation). Since the molecule-molecule interactions are different for the |g〉state, the distribution could evolve in time away from the initial distribution. A more sophisticated analysiswould involve capturing the shape of the distribution for every experimentally measured hold time.857.2. Derivation of the Molecule Lifetime ModelPlugging these expressions into equation (7.2) yieldsN˙(t) = −αN(t)− β8pi3/2σxσyσzN2(t)− γ24√3pi3σ2xσ2yσ2zN3(t) (7.7)= −α′N(t)− β′N2(t)− γ′N3(t) (7.8)While it is possible to solve this differential equation analytically, the solution for an initialmolecule number N0 = N(0) is a transcendental equation. Rather than solve the rootfinding problem, we chose to solve the ODE numerically using the module “ode” in scipy.So far, we have assumed that the atomic cloud follows Maxwell-Boltzmann statistics.However, this solution would be invalid if the molecules (which are composite Bosons) form amolecular Bose-Einstein condensate (mBEC). In terms of the chemical potential µ, trappingpotential U(r) and coupling constant from the Gross-Pitaevskii equation g [62, 126], thedensity distribution of a BEC is given by [80]n(r) =µ− U(r)gΘ (µ− U(r)) (7.9)whereµ =~ω¯2(15aNaho)2/5(7.10)U(r) =12mω¯2r2 (7.11)g =4pi~2am(7.12)and Θ(x) is the Heaviside function. In equations 7.10, 7.11 and 7.12, U(r) is assumed tobe a harmonic potential with a mean trap frequency ω¯ = (ωxωyωz)1/3, a is the scatteringlength and aho =√~/mω¯. If we again assume that n(r, t) = n0(t)n(r), we get the followingresult:N˙(t) = −αN(t)− β 152/5(aN(t))7/514pia2( ~mω¯)6/5 − γ 54/5(aN(t))9/556 5√3pi2a3( ~mω¯)12/5 (7.13)Given that σi ∝ ω−1i in equation 7.7, the two body loss term in this equation is propor-867.3. Deeply Bound Molecule Lifetime Measurementstional to ω¯2N2(t). Meanwhile, the three body loss term in equation 7.13 is proportional toω¯2.4N1.8(t). Due to the functional similarity between these terms, if we have a mixture of aBEC and a thermal cloud, it would be very difficult to determine whether we are observingtwo body or three body losses and even more difficult to reliably extract the values of β andγ from the data. Because β′ ∝ ω¯6 and γ′ ∝ ω¯12 for a thermal sample (and the differenceis less obvious for a mBEC), it is advantageous to work with a thermal sample and avoidforming a mBEC. The condensed fraction is given by [80]NBECNtotal= 1−(TTc)3(7.14)where Tc is the critical temperature:Tc ≈ 4.5(ω¯/2pi100 Hz)N1/3 nK (7.15)Hence, to have a purely thermal sample, we must operate above Tc. Although we intendedto operate in the regime where T > Tc, our quest for increasing the Feshbach moleculefraction by evaporating to lower trap depths may have inadvertently led us to the mBECregime22. For this reason, we extract decay rates for both cases (thermal and mBEC) in theanalysis that follows. While neither method will yield accurate values of the rate constantswhen the mBEC fraction is neither 0 nor 1 (unless the dominant loss mechanism is one-bodyloss), we can still gain valuable insight from this analysis and use the results to guide futuremeasurements.7.3 Deeply Bound Molecule Lifetime MeasurementsAs we saw in chapter 4, the N ′′ = 0 deeply bound states we can access (when |a〉 = |FM〉is the Feshbach molecule state specified by equation 4.18) via STIRAP have m′′I = 1, m′′S =−1, m′′N = 0, and m′′F = 0. No other state should be strongly coupled to our intermediatestate |e〉. This is also true for the only N ′′ = 2 level we accessed because we are usingpi polarized fields (m′N = m′′N ). Since every state can be specified by this common set of22This happened because we collected the lifetime measurements discussed below prior to characterizingthe dipole trap. In retrospect, it would have been prudent to reverse the order of these steps.877.3. Deeply Bound Molecule Lifetime Measurementsquantum numbers and v′′ and N ′′, we will omit, for brevity, the common part of the statelabels for the remainder of this section.Our goal is to gain insight into the physics behind the decay of the molecules. Whilewe will be unable to definitively identify the reaction mechanism responsible for the loss ofthe deeply bound molecules we create without theoretical support, it is possible eliminatesome candidates by determining experimentally whether the loss rate is dominated by aone-, two- or three-body mechanism. For most of the ultracold alkali molecules that havebeen probed experimentally to date, the dominant loss mechanism has been two-body losses[128]. If this is also the case for 6Li2, then the reaction mechanisms for the ground tripletstate areLi2(a3Σ+) + Li2(a3Σ+)→ Li3 + Li (7.16)Li2(a3Σ+) + Li2(a3Σ+)→ Li2(X1Σ+) + Li2(T ) (7.17)where T could beX1Σ+ or a3Σ+ [155]. For higher vibrational levels, rovibrational relaxationis another possibility.We now attempt to deduce whether one-, two- or three-body losses dominate by takingadvantage of the trap frequency dependence of the coefficients in equations 7.7 and 7.13.One way to do this is to compare the loss rate in a single arm ODT to that in a CODT23.The axial trap frequency for a single arm ODT is approximately 20 times lower than thelowest trap frequency component for a CODT with our crossing angle [63] if the totaltrapping power is unchanged. As a result, we expect a factor of√2× 3√20 ≈ 4 reduction inthe effective trap frequency ω¯, where the√2 comes from drop in the trapping power. (Theoptical power in the second trapping arm remained unchanged.) Figure 7.3 shows a lifetimemeasurement for the v′′ = 0, N ′′ = 0 state in a single arm ODT and a CODT. Detailsare provided in the caption. Since the loss rate changed significantly, the dominant lossmechanism is clearly not a one-body process. For a thermal cloud, the decay could not bedue to a three-body process because the difference between the two data sets would be evenmore dramatic, according to equation 7.7. For a mBEC, we cannot reach this conclusion23Since the photo-association beam is overlapped with the second arm, we had to misalign the CODT suchthat the first arm was not overlapped with the MOT and the pointing of the second arm was unchanged.887.3. Deeply Bound Molecule Lifetime Measurements0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t (ms)0.00.20.40.60.81.0NumberofAtomsinthe|1〉State(a.u.) Single Arm ODT (MB)Single Arm ODT (TF)CODT (MB)CODT (TF)Figure 7.3: Lifetime measurements for the lowest hyperfine state of the v′′ = 0, N ′′ = 0ro-vibrational level in a single arm ODT and a CODT (SPI laser). For each measurement,we used two models for fitting the data: the solutions to equations 7.7 (solid lines) and 7.13(dotted lines) with α = γ = 0. Since we did not measure the trap frequency for the singlearm trap, we assumed that ωr ≈ ωx = ωy and that the axial frequency is a factor of 20± 5lower in the single arm trap, for a fixed trapping power. (These assumptions are justifiedin [63].) The β coefficients determined from these least squares fits and several additionalparameters are shown in table 7.1 and agree within the bounds of the uncertainties in bothcases. The agreement is significantly better when we make the assumption that the sampleis a mBEC (Thomas-Fermi distribution).simply by inspection. Since γ is a constant, we should be able to extract the same value fromboth data sets or, equivalently, we expect γSA/γcross = 1, where the subscripts SA and crossrefer to a single arm ODT and a CODT, respectively. Since we used the same magnetic fieldfor both measurements, γSA/γcross = (γ′SA/γ′cross) (ω¯cross/ω¯SA)12/5 = 3.8±1.8 > 1, where γ′SAand γ′cross were determined by fitting the data to a pure three-body decay law for a mBEC.Thus, the dominant loss mechanism is a two-body process and we can assume that one- andthree- body losses are negligible (α = γ = 0 in equations 7.7 and 7.13). This simplificationallows us to extract the two-body loss rate β by fitting the solutions to these differential897.3. Deeply Bound Molecule Lifetime Measurementsequations to the v′′ = 0, N ′′ = 0 data in Figure 7.3, the v′′ = 0, 5, 9, N ′′ = 0 data in Figure7.4 and several additional data sets for other rovibrational levels. To differentiate betweenthe two-body loss rate predicted assuming a pure Maxwell-Boltzmann (MB) distributionor a pure Thomas-Fermi (TF), we will refer to them as βMB and βTF, respectively. Thesecoefficients and other relevant parameters for all of the states we populated are shownin table 7.1. First, it is clear that βMB is approximately an order of magnitude largerthan βTF for every state. Moreover, if we assume that every close range collision leads tochemically induced loss, then the temperature independent loss rate for s-wave collisions isapproximatelyβth = 2pi(2~C6µ3)1/4= 3.7× 10−10 cm3/s (7.18)where CLi26 ≈ 4 × CLi6 = 4 × 1389 = 5556 a.u. [44, 170] and µ is the reduced mass of6Li2 [73, 128]. This estimate of the upper bound for the loss rate is also an order ofmagnitude smaller than the values of βTF we calculated and makes their credibility highlyquestionable. With the exception of the entry in the third row (which is inconsistent withrows one and two), the values of βTF are approximately equal within the bounds of theuncertainties suggesting that we may be in the regime where every collision leads to loss.It is important to emphasize that the v′′ = 0, N ′′ = 0 state - which is different fromall the other states because it is the lowest energy (triplet) state and cannot decay byrovibrational relaxation mechanisms - is not exempt from this trend. However, we cannotdefinitively conclude that the cloud was a mBEC. According to our calculations, our cloudcould not have been fully condensed for any of these measurements. (The large uncertaintyin the condensed fraction is due to the large uncertainty in our temperature and trappingpower measurements. Using a time-of-flight measurement to determine the temperatureand measuring the trap frequencies at the trapping power used for the measurement wouldalleviate this problem.) making these preliminary results inconclusive. For this reason, wepropose a sequence of steps for unambiguously determining the rate constants in chapter 9.907.3. Deeply Bound Molecule Lifetime Measurements0 2 4 6 8 10t (ms)0.00.20.40.60.81.0NumberofAtomsinthe|1〉State(a.u.) v′′ = 0 (MB)v′′ = 0 (TF)v′′ = 5 (MB)v′′ = 5 (TF)v′′ = 9 (MB)v′′ = 9 (TF)Figure 7.4: Lifetime measurements of the lowest hyperfine states of three ro-vibrationallevels in the a(3Σ+u ) potential. All three levels have N′′ = 0 and vary in v′′, which takeson the values 0, 5 and 9. For these measurements, we used a CODT created by the IPGlaser. Again, we used two models for fitting the data: the solutions to equations 7.7 (solidlines) and 7.13 (dotted lines) with α = γ = 0. The β coefficients determined from theseleast squares fits and several additional parameters are shown in table 7.1.917.3. Deeply Bound Molecule Lifetime Measurementsv′′ N ′′ B (G) Conf. T (nK) ω¯ (krad/s) CF (%) βMB (cm3/s) βTF (cm3/s)0 0 755 (S,O) 126± 32 0.46± 0.06 0-36 (8.0± 5.0)× 10−9 (4.4± 1.0)× 10−100 0 755 (S,C) 270± 70 1.83± 0.21 0-94 (2.9± 1.5)× 10−9 (5.4± 0.8)× 10−100 0 700 (I,C) 337± 80 1.21± 0.14 0-37 (2.2± 1.1)× 10−8 (1.0± 0.2)× 10−90 2 700 (I,C) 337± 80 1.21± 0.14 0-37 (1.0± 0.5)× 10−8 (4.9± 1.0)× 10−105 0 700 (I,C) 271± 65 1.09± 0.12 0-66 (7.0± 4.0)× 10−9 (3.2± 0.7)× 10−106 0 700 (I,C) 271± 65 1.09± 0.12 0-66 (5.6± 2.8)× 10−9 (3.6± 0.5)× 10−107 0 700 (I,C) 271± 65 1.09± 0.12 0-66 (4.1± 2.4)× 10−9 (2.7± 0.7)× 10−108 0 700 (I,C) 271± 65 1.09± 0.12 0-66 (4.0± 2.0)× 10−9 (3.1± 0.5)× 10−109 0 755 (I,C) 271± 65 1.09± 0.12 0-66 (2.8± 1.4)× 10−9 (3.2± 0.5)× 10−10Table 7.1: Two body loss rate coefficients and related quantities for several v′′ and N ′′ statesunder two assumptions: the cloud is either a pure thermal MB distribution or a pure Bose-condensed sample with a Thomas-Fermi distribution. We denote the loss rates βMB andβTF, where the subscripts refer to Maxwell-Boltzmann and Thomas-Fermi, respectively. At700 G and 755 G, the scattering lengths are (3.15± 0.10)a0× 103 and (1.49± 0.10)a0× 103,respectively [117, 119]. These values were used for computing βTF. The column title “Conf.”refers to the configuration of the optical dipole trap. The first letter refers to the trappinglaser and could be S for SPI and I for IPG. The second letter denotes whether the trap was aCODT (C) or a single arm trap (O). The trap frequencies and temperatures were calculatedbased on the data in appendix B and the forced evaporation scaling law (equation 2.13).Using these quantities and upper and lower bounds for the Feshbach molecule number, wecalculate a range for the condensed fraction. Finally, we use equations 7.7 and 7.13, wedetermined βMB and βTF.92Chapter 8Narrow-Band Solid State VUVSource for Laser Cooling ofAntihydrogen24We describe the design and performance of a solid-state pulsed source of narrowband(< 100 MHz) Lyman-α radiation designed for the purpose of laser cooling magneticallytrapped antihydrogen. Our source utilizes an injection seeded Ti:Sapphire amplifier cavityto generate intense radiation at 729.4 nm, which is then sent through a frequency doublingstage and a frequency tripling stage to generate 121.56 nm light. Although the pulse energywas limited to 12 nJ with a repetition rate of 10 Hz when [105] was written, we expectedto obtain greater than 0.1 µJ per pulse after replacing damaged optical components in theamplifier cavity and improving the efficiency of the frequency tripling stage.8.1 IntroductionAs the simplest and best understood atom in the periodic table, Hydrogen appeared to bethe natural choice for early laser cooling experiments. Despite this, the first and only studyof the optical cooling of Hydrogen was published in 1993 [142], many years after severalalkalis were successfully cooled and trapped. The reason for this is the inherent difficultyof producing coherent radiation at the laser cooling transition for Hydrogen - 121.56 nmor Lyman-α. Numerous broadband pulsed Lyman-α sources were developed in the late1970s and early 1980s [10, 33, 69, 88, 93, 94, 95, 100, 158]. These were suitable for manyspectroscopic applications, but not for laser cooling, which requires, for efficient cooling, the24 c© of Hyperfine Interactions. Reprinted, with permission from [105].938.1. Introductionbandwidth of the source to be less than or equal to the natural linewidth (Γ ≈ 100 MHz) ofthe transition [142]. The development of the first narrow-band (∆ν ≈ 40 MHz) Lyman-αsource in 1987 [23] made it possible to laser cool Hydrogen, trapped in a magnetic trapand pre-cooled cryogenically, to less than 3 mK [142]. Nevertheless, low repetition ratepulsed sources cannot be used for conventional laser cooling experiments, which rely onmagneto-optical traps (MOTs) for efficient cooling and trapping of large atomic popula-tions, and interest in coherent Lyman-α sources subsided until a new application emerged -laser cooling of trapped antihydrogen. In particular, the ALPHA collaboration successfullytrapped antihydrogen in a magnetic trap at a translational temperature of 500 mK [5, 42]and, more recently, used the trapped antimatter to test for charge, parity and time (CPT)symmetry violations. Using two-photon spectroscopy, they found that the 1S-2S transi-tion frequency of antihydrogen is the same as that of Hydrogen at a relative precision of2?× 10−10 [3]. In other words, they saw no evidence of CPT violations. To reach the levelof accuracy required for these measurements, the trapped antihydrogen must first be cooledto a translation temperature of 20 mK and, for this, a high power (∼ 0.05 µW) Lyman-αsource with a 100 MHz linewidth is required, according to simulations [42]. Two contin-uous wave (CW) Lyman-α sources, which rely on a four-wave mixing scheme in gaseousMercury, have been realized [47, 134], but they are not yet capable of delivering sufficientpower (and one of them relies on dye lasers). Here, we present an alternative approach -a narrow band (< 100 MHz) pulsed solid-state Lyman-α source. Our motivation was tocreate a solid-state source to avoid the maintenance and additional safety issues associatedwith a dye-laser-based source. Simulations have shown that a narrow-band pulsed sourcethat produces 0.05 µJ per shot at a repetition rate of 10 Hz is capable of cooling a trappedantihydrogen atom down to 20 mK [42]. While the maximum Lyman-α output we havedemonstrated here - by frequency tripling a 7.2 mJ pulse of 364.7 nm light - is only 12 nJ,we are confident that more than 0.1 µJ is possible with this system as we have observedmore than 19 mJ per pulse of 364.7 nm radiation25. However, due to optical damage inthe 729.4 nm amplifier crystal, we were unable to maintain an output power greater than7.2 mJ per pulse at 364.7 nm. If the required average power is achieved, this Lyman-α25The authors of [3] glance over most technical details and we have been unable to find the power usedfor the experiment elsewhere in the literature.948.2. Apparatussource should be capable of laser cooling trapped anti-hydrogen, from 500 mK to 20 mK.8.2 ApparatusThe laser system shown in Fig. 8.1 consists of a pulsed Ti:Sapphire amplifier seeded by anarrow linewidth (< 1 MHz) semi-conductor laser (Toptica Photonics TA Pro) operatingat 729.4 nm, an anti-reflection coated type I BBO crystal for second harmonic generation(SHG) at 729.4 nm (Castech Inc.) and a third harmonic generation (THG) and detectionchamber.8.2.1 Ti:Sapphire Amplifier and SHGTo obtain the intensities required for efficient SHG and THG while maintaining a narrowlinewidth, we used an injection seeded pulsed Ti:Sapphire amplifier based on [46]. Thisamplifier is an unstable resonator that consists of two brewster cut Ti:Sapphire crystals(10 mm in diameter, 36.8 mm in path length, GT Advanced Technologies), a convex gradedreflectivity mirror (f = −5 m, INO) coated for 729.4 nm, a high reflector (HR) and twoisosceles brewster prisms for coarse wavelength selection. As demonstrated previously in[46], in order to approach the Fourier Limit, each crystal is pumped by a pulsed frequencydoubled Nd:YAG laser (532 nm, 10 Hz repetition rate) from both sides and injection seededwith a CW diode laser (∆ν < 1 MHz) operating at 729.4nm. The output of the amplifieris then sent into a coated type I BBO crystal (dimensions: 7 mm× 7 mm× 8 mm), toproduce 364.7 nm radiation. After the second harmonic (364.7 nm) is spatially separatedfrom the fundamental beam (729.4 nm) by means of two Pellin Broca prisms, it is sent tothe THG chamber, where we produce and detect Lyman-α radiation.8.2.2 Lyman-α Generation and DetectionTo generate Lyman-α radiation, we used non-resonant THG in a mixture of Krypton andArgon - an approach that has been demonstrated a number of times in the literature [10, 23,33, 69, 88, 93, 95, 158]. A resonant four-wave-mixing (FWM) scheme was also successfullydemonstated with a broadband dye-laser source by Michan et al. [103]. However, THG958.2. ApparatusTHGChamberL1 L2HRSeed Beam(729.4nm)BS(T=99.5%)Ti:SapphireCrystalTi:SapphireCrystal OC(R=30%)BS(T=99.5%)BBOPDPMTKr/ArInjectionFigure 8.1: Laser arrangement and non-linear optical stages. The Ti:Sapphire amplifiercavity is seeded by a narrow linewidth semi-conductor laser (Toptica Photonics TA Pro) viaa an uncoated CaF2 beam splitter (BS) with a reflectivity of ∼ 0.5%. The two Ti:Sapphirecrystals are pumped from both sides by a pulsed Nd:YAG laser (Quanta-Ray Pro). Thepump power is distributed evenly between the four arms. The amplified light exits thecavity via the output coupler (OC). It is then sent through a coated BBO crystal. Thesecond harmonic is separated from the fundamental beam (not shown), with two PellinBroca prisms and routed to the THG chamber. The beam is then focused by an MgF2 lensL1 (f = 100 mm) onto a mixture of Krypton and Argon gas. Finally, Lyman-α radiationis produced by non-resonant THG, recollimated by an MgF2 lens L2 (f = 200 mm) anddetected by a solar-blind photo-multiplier tube (PMT). Two Lyman-α filters are placedbetween L2 and the PMT to prevent 364.7 nm light from reaching the PMT (not shown).968.3. Results(which requires only one beam) was chosen here because of its technical simplicity andthe availability of high energy pump sources. In Fig. 8.1, we show that radiation fromthe SHG stage is sent into the THG chamber, which consists of an MgF2 focusing lens L1(f = 100 mm), an input port for the Krypton-Argon mixture, an MgF2 recollimating lens L2(f = 200 mm), two Lyman-α filters and a solar-blind photo-multiplier tube (PMT). In ourimplementation, the first lens L1 focuses the slowly diverging 364.7nm beam (with a verticaldiameter of 6mm and a horizontal diameter of 3mm, based on a burn card measurement)onto the gas mixture to reach the intensities required for efficient THG. The generatedLyman-α is then recollimated with lens L2, sent through two Lyman-α filters (to preventthe 364.7 nm beam from passing) and detected by the PMT. Using the response function,gain curve and quantum efficiency of the PMT (Hamamatsu R972) we extract from theraw signal the temporal profile of the Lyman-α beam and its pulse energy. To confirmthe production of Lyman-α, we verified that the PMT signal vanished in the absense ofKrypton.8.3 ResultsIn this section, we describe the results we obtained at the SHG and THG stages. A detaileddescription and characterization of the injection seeded Ti:Sapphire amplifier can be foundin [46] and [45].8.3.1 Second Harmonic Generation (SHG) StagePulse EnergyIn Fig. 8.2, the average pulse energy of the SHG beam (364.7nm) ESHG is plotted as afunction of pump pulse energy EP. This data was taken with the cavity unlocked and in anenrironment with an abundance of acoustic noise (generated by turbo-pumps from a nearbyapparatus). As a result, large fluctuations in the pulse energy were observed. We fit thisdata to the following model:ESHG = α · (EP − ETH)2 ·Θ (EP − ETH) (8.1)978.3. ResultsEP [mJ]0 50 100 150 200 250 300ESHG[mJ]024681012Figure 8.2: SHG pulse energy at 365 nm as a function of the pulse energy of the 532 nmpump for the 729.4 nm seeded amplifier.Shot Number0 200 400 600 800 1000 1200 1400 1600 1800 2000ESHG[mJ]101214161820Figure 8.3: SHG pulse energy with EP = 310 mJ as a function of time with the amplifiercavity locked. The observed power fluctuations are correlated with fluctuations in theNd:YAG pump laser. The mean pulse energy is 16.0 mJ and the stand deviation is 1.4 mJ.where α = (5.7± 2.8)× 10−4mJ−1 is a proportionality constant, ETH = 150± 27 mJ is thepump power at the 729.4 nm lasing threshold and Θ is the Heaviside step function. Furtheroptimizations of the amplifier cavity enabled us to reach the SHG pulse energies shown inFig. 8.3 with EP = 310 mJ, where the mean pulse energy is 16.0 mJ and the standarddeviation is 1.4 mJ. However, due to the environmental conditions of the facility at CERNand imperfections in one of the Ti:Sapphire crystals, we were forced to lower the pumppower to avoid further damaging the crystal. For this reason, all of the data presented in8.3.1 and in section 8.3.2 were taken at lower pump powers. After replacing the crystal andalleviating problems due to the environment, we expect to consistently achieve the powersshown in 8.3.988.3. Resultst [ns]150 200 250 300 350 400 450I(t)[a.u.]00.51(a)f [MHz]-200 -100 0 100 200|I˜(f)|[a.u.]00.51(b)t [ns]150 200 250 300 350 400 450I(t)[a.u.]00.51(c)f [MHz]-200 -100 0 100 200|I˜(f)|[a.u.]00.51(d)31.6 ±1.7 ns 26.1 ±1.2 MHz 24.7 ±0.5 ns 32.8 ±1.0 MHz Figure 8.4: (a) SHG temporal pulse shape with the cavity not in resonance with the seedlaser. (b) Spectrum of the pulse in (a). (c) SHG temporal pulse shape with the cavity nearresonance with the seed laser. (d) Spectrum of the pulse in (c).Pulse ShapeIn Fig. 8.4 (a) and (b), respectively, we show the intensity as a function of time for a364.7 nm pulse and the spectrum of this pulse when the amplifier cavity is not in resonancewith the seed frequency. The spectrum consists of a dominant peak (which corresponds tothe envelope of the temporal profile) and two side lobes, which account for the modulationseen in the temporal profile. The full width at half maximum (FWHM) of the temporalprofile ∆tFWHM and the FWHM of the spectrum ∆νFWHM are 31.6 ± 1.7 ns and 26.1 ±1.2 MHz, respectively. Similarly, the temporal profile and spectrum of a pulse generatedwhen the amplifier was tuned to be near resonance with the seed laser are shown in Fig.8.4 (c) and (d). As expected, the side-bands (which correspond to other cavity modes) aresuppressed. For this pulse, ∆νFWHM = 32.8 ± 1.0 MHz and ∆tFWHM = 24.7 ± 0.5 ns. Inthe next section, we find that operating the Ti:Sapphire amplifier cavity off-resonance withrespect to the input seed frequency results in higher pulse energies and more efficient THGthan operating near resonance.998.3. Results8.3.2 THG StageThe Lyman-α intensity is proportional to χ2N2I3SHGF , where χ is the third order suscepti-bility, N is the number density of the Krypton, ISHG is the intensity of the 364.7 nm beamand F is the phase matching factor. In the limit of a tight focus, the phase-matching factoris maximized when b·∆k = b·(k121.56 nm−3·k364.7 nm) = −2, where b is the confocal param-eter of the beam (assuming a TEM00 mode) [69]. However, because the Lyman-α intensityis also proportional to N2, adjusting the Krypton density to satisfy the phase matching con-dition alone is insufficient to optimizing the Lyman-α intensity and the problem becomes athree parameter optimization for the focal length of the focusing lens L1 and the pressuresof Krypton and Argon in the mixing chamber. Based on previous results from preliminarydye-laser experiments and some empirical optimization, we chose an f = 100 mm MgF2 lensfor L1 and set the Krypton pressure to 84 mTorr and the Argon pressure to 200 mTorr.With the amplifier cavity locked via a simple side-lock near resonance, we observed thetemporal profiles shown in Fig. 8.5 (a) and (c) for the 364.7 nm and 121.6 nm pulses,respectively. The corresponding spectra are shown in Fig. 8.5 (b) and (d). The tempo-ral and spectral widths of the 364.7 nm pulse are given by ∆tFWHM = 20.9 ± 3.0 ns and∆νFWHM = 44.3± 2.1 MHz, respectively. As shown in Fig. 8.5, this 5.0 mJ pulse producesa Lyman-α pulse with a temporal width of ∆tFWHM = 11.5 ± 0.5 ns and a spectral widthof 76.8± 2.9 MHz. The energy of the Lyman-α pulse is approximately 2 nJ. Remarkably,locking the cavity not near resonance with the seed laser resulted in higher THG efficiency.In Fig 8.6, we again show the temporal and specral profiles of the 364.7 nm and 121.46 nmpulses (same naming scheme as Fig. 8.5). Here, the side bands in the spectrum of 364.7 nmpulse are more pronounced with respect to the central peak, but the overall pulse energyis also larger - 7.2 mJ. Because of the shorter pulse duration and higher peak intensity inthe 364.7 nm pulse, we observe a slightly broader spectral profile - ∆ν = 83.2 ± 0.8MHz -and a higher pulse energy - 12 nJ - for the 121.46 nm pulse. Its temporal width ∆tFWHMis 11.2± 0.3ns.1008.3. Resultst [ns]200 250 300 350I(t)[a.u.]00.51(a)f [MHz]-400 -200 0 200 400|I˜(f)|[a.u.]00.51(b)t [ns]200 250 300 350I(t)[a.u.]00.51(c)f [MHz]-400 -200 0 200 400|I˜(f)|[a.u.]00.51(d)20.9 ±3.0 ns 44.3 ±2.1 MHz 11.5 ±0.5 ns 76.8 ±2.9 MHz Figure 8.5: Temporal and spectral profiles of the 364.7 nm and 121.46 nm pulses with theamplifier cavity locked close to resonance with the seed laser. (a) Temporal profile of the364.7 nm pulse. (b) Spectral profile of the 364.7 nm pulse. (c) Temporal profile of the121.46 nm pulse. (b) Spectral profile of the 121.46 nm pulse.t [ns]200 250 300 350I(t)[a.u.]00.51(a)f [MHz]-400 -200 0 200 400|I˜(f)|[a.u.]00.51(b)t [ns]200 250 300 350I(t)[a.u.]00.51(c)f [MHz]-400 -200 0 200 400|I˜(f)|[a.u.]00.51(d)21.3 ±2.0 ns 42.7 ±1.4 MHz 11.2 ±0.3 ns 83.2 ±0.8 MHz Figure 8.6: Temporal and spectral profiles of the 364.7 nm and 121.46 nm pulses with theamplifier cavity locked far from resonance with the seed laser. (a) Temporal profile of the364.7 nm pulse. (b) Spectral profile of the 364.7 nm pulse. (c) Temporal profile of the121.46 nm pulse. (b) Spectral profile of the 121.46 nm pulse.1018.4. Summary8.4 SummaryWe made significant progress toward developing a narrow-band solid state Lyman-α sourcecapable of laser cooling magnetically trapped anti-hydrogen to 20mK. Further optimizationof both the pulsed amplifier stage and the THG were required to achieve this at the timeof writing the manuscript for [105]. To reach a pulse energy of 0.05µJ (the value used inthe simulation [42]), we needed to achieve stable operation at pulse energies close to thoseshown in Fig. 8.3 and determine the focal length of the focusing lens and the Kryptonand Argon pressures that optimizes the THG conversion efficiency. We note that if theconversion efficiency remained constant, simply increasing the SHG pulse energy to 16 mJshould have resulted in approximately 132 nJ of Lyman-α.102Chapter 9Conclusion and OutlookThis study documents the first successful attempt to create, using STIRAP, deeply boundtriplet molecules of 6Li2 - in the lowest hyperfine states of several rovibrational levels ofthe a(3Σ+u ) potential - and examine their chemical properties. On our path toward thisgoal, we designed and built a robust laser system for cooling 6Li, which will allow usto implement an enhanced laser cooling mechanism that relies on the D1 transition, anda Ti:Sapphire-based Raman laser system with the best (to our knowledge) phase noisesuppression of any solid state laser system with only an AOM for fast actuation. Wealso added various automation routines that will continue to accelerate the rate of datacollection and experimental parameter optimization for years to come, wrote a programto process absorption images with high fidelity that will be used for virtually every futureexperiment conducted with this apparatus, and created a library of scripts for processingphoto-association and molecular lifetime data. These ingredients will prove useful not onlyfor future studies involving 6Li, but also greatly simplify the path toward creating polar6Li85Rb molecules - a long term goal for the QDG lab and a means of exploring exoticphenomena in many body physics. Finally, we built a solid-state VUV laser system thatallowed researchers in the ALPHA collaboration to use laser cooling in a novel way tocool magnetically trapped antihydrogen and make significant advances in their quest forviolations of the charge, parity, and time reversal symmetries. They did this by confirmingthat the 1S-2S transition frequency of antihydrogen is equal to that of Hydrogen at a relativeprecision of 2× 10−10 [3].While we made significant progress in our study of the chemical reactions that take placebetween the deeply bound 6Li2 molecules we create, a few crucial unresolved issues remain.To unequivocally determine the two-body loss rate constant for each molecular state, weneed to know whether the sample is a thermal gas or a mBEC and the trap frequencies103Chapter 9. Conclusion and Outlookof the deeply bound molecules. For the reasons discussed in chapter 7, it is advantageousto work with a pure thermal gas, so we should operate above the critical temperaturewhile maintaining a sufficiently large number of Feshbach molecules. Experimentally, thisentails terminating our evaporative cooling ramp at a higher trap depth and changing themagnetic field to a value at which the three-body recombination rate is relatively high(e.g. 700 G) such that we get efficient Feshbach molecule formation without crossing thecritical temperature26. Doing this and validating our temperature estimates via time-of-flight expansion measurements would ensure that the sample is thermal. Next, we needto investigate how the trap frequencies change after we form deeply bound molecules. Sofar, we have been implicitly assuming that the trap frequencies remain the same after thepopulation transfer. However, it would be quite coincidental if that were the case becausethe sum of the AC stark shifts associated with all of the excited states of 6Li2 would haveto add up to the same value as AC stark shift for 6Li atoms. 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Precise characterization of li 6 feshbach resonances using trap-sideband-resolved rf spectroscopy of weakly bound molecules. Physical review letters,110(13):135301, 2013. page 50[175] Martin W Zwierlein, Claudiu A Stan, Christian H Schunck, Sebastian MF Raupach,Subhadeep Gupta, Zoran Hadzibabic, and Wolfgang Ketterle. Observation of bose-einstein condensation of molecules. Physical review letters, 91(25):250401, 2003. page 1123Appendix ABeam Profile MeasurementsTo calculate the intensities of our dipole trapping lasers and photo-association lasers, weneed the powers and beam parameters. Here, we extract the latter for each beam. Thebeam waist of an ideal Gaussian beam is given by equation A.1.W (z) = W0√1 +(zz0)2(A.1)where the z0 = piW20 /λ is the Rayleigh length. In practice, aberrations increase the mini-mum beam waist to approximately 4W0/3 [147]. Hence, equation A.1 becomesW (z) = W ′0√1 +(zz0)2≈ 43W0√1 +(zz0)2(A.2)In the sections that follow, we will use this model to extract the beam parameters from leastsquares fits of beam profile measurements for the SPI CODT beams, IPG CODT beamsand photo-association beam.A.1 SPI Beam ProfilesThe beam waist of the SPI laser (1090 nm) along the x and y directions is shown as afunction of distance along the optical axis in Figure A.1 for both trapping arms. The beamparameters extracted from the fit are included in the caption.124A.2. IPG Beam Profiles(a) (b)Figure A.1: Beam profile measurements for the first (a) and second (b) arms of the SPICODT. For the first arm, W ′0x = 41.45 ± 0.58µm, W ′0y = 36.88 ± 0.91µm, z0x = 3.32 ±0.03 mm, z0y = 2.75 ± 0.04 mm and the distance between the x and y foci is ∆foci =0.6 ± 2.9 mm. For the second arm, W ′0x = 40.52 ± 0.58µm, W ′0y = 37.91 ± 0.72µm, z0x =3.35 ± 0.04 mm, z0y = 3.13 ± 0.06 mm and the distance between the x and y foci is∆foci = 1.1± 7.4 mm.A.2 IPG Beam ProfilesThe beam waist of the IPG laser (1064 nm) along the x and y directions is shown as afunction of distance along the optical axis in Figure A.2 for both trapping arms. The beamparameters extracted from the fit are included in the caption.(a) (b)Figure A.2: Beam profile measurements for the first (a) and second (b) arms of the IPGCODT. For the first arm, W ′0x = 53.10 ± 0.31 µm, W ′0y = 58.99 ± 0.13 µm, z0x = 4.59 ±0.02 mm, z0y = 5.12 ± 0.01 mm and the distance between the x and y foci is ∆foci =4.6 ± 4.1 mm. For the second arm, W ′0x = 47.85 ± 0.92µm, W ′0y = 55.65 ± 0.07µm, z0x =4.44 ± 0.08 mm, z0y = 5.88 ± 0.01 mm and the distance between the x and y foci is∆foci = 4.2± 5.5 mm.125A.3. Photo-Association Beam ProfileA.3 Photo-Association Beam ProfileThe beam waist of the photo-association beam along the x and y directions is shown as afunction of distance along the optical axis in Figure A.2. The beam parameters extractedfrom the fit are included in the caption.Figure A.3: Beam profile measurements for the photo-association beam at 798 nm. W ′0x =36.82± 1.01 µm, W ′0y = 39.08± 0.51 µm, z0x = 4.20± 0.12 mm, z0y = 4.94± 0.09 mm andthe distance between the x and y foci is ∆foci = −1.4± 13.4 mm.126Appendix BTrap Frequency MeasurementsOne of the most unambiguous methods27 of measuring the trapping frequencies of an ODTis perturbing the trap by changing the pointing of the trapping laser (by stepping thefrequency of an AOM or by other means) for some length of time and then returningit to its normal position. As a result of this step displacement, the center of the atomcloud begins to oscillate at the trap frequencies around the equilibrium position. Since thetrapping frequencies are proportional to√I, where I is the intensity of the trapping laserincident on the atoms, making a measurement at a single trapping power is sufficient. (Asa precaution, we validated these measurements by also measuring the trap frequencies atfour times the laser power and confirmed that the trap frequencies doubled, as expected.)We were able to use this technique for the CODT formed by the SPI laser, but not forthe CODT formed by the IPG laser because of the orientation of a periscope used for thisaligning this trap. Changing the AOM frequency results in the trap splitting into two armsrather than simply a shift in the cross. Hence, we made the measurement for the IPG trapin a single arm and used this in conjunction with insights from the SPI data to calculatethe trap frequencies for the IPG CODT. To extract the oscillation frequencies from thewaveforms we obtained, we use the following fitting function:x0(t) = mt+ b+A sin(2pift+ φ) exp(− tτ)(B.1)Next, we checked for consistency between equations 2.6 and 2.10. The former relies onthe accuracy of the power measurement while the latter relies on the accuracy of the trapfrequency measurement. The accuracy of both methods is strongly dependent on the validity27Parametric excitation induced by amplitude modulation [25] of the trapping laser is another way of mea-suring the trapping frequencies. It is more difficult to interpret and validate because an array of harmonicsand subharmonics of the trapping frequency are observed.127B.1. SPI CODTand interpretation (when there is strong astigmatism) of the beam waist measurements inAppendix A. (The functional dependency on W0 is not the same.) It is important to notean implicit assumption we are making: while equation 2.10 was originally derived for asingle arm trap (which has cylindrical symmetry), we are assuming that ωx ≈ ωy ≈ ωr forour small crossing angle in a CODT based on the simulation results presented in [63]. Thedata and analysis are shown in the sections that follow.B.1 SPI CODTAccording to the fit in Figure B.1, ωx/2pi = 552.2±5.9 Hz and ωz/2pi = 81.6±1.9 Hz whenthe trapping power is 70± 5 mW and 60± 4 mW in the first and second arms, respectively.The simulation described in [63] shows that ωy ≈ ωx. It is important to note that this trapis, in part, created by a small gradient in the magnetic field.(a) (b)Figure B.1: Trap frequency measurements for the x direction (a) and z direction (b) with70 ± 5 mW and 60 ± 4 mW of laser power in the first and second arms, respectively (SPIlaser). Based on the fit of the data in (a), ωx/2pi = 552.2± 5.9 Hz. For this measurement,the trap was displaced for 100 ms by changing the AOM frequency by 1 MHz, the cloudexpansion time was 2 ms and the magnetic field was set to 755 G. Based on the fit of thedata in (b), ωz/2pi = 81.6 ± 1.9 Hz. Here, the trap was displaced for 500 ms by changingthe AOM frequency by 0.5 MHz, the cloud expansion time was 2.5 ms and the magneticfield was set to 755 G.The astigmatism in the SPI beam is very mild and W0 ≈ 38 µm. Using equations 2.6and 2.10, we get U0/kB = 2.73 µK or U0/kBP = 21 µK/W and U0/kB = 3.14 µK orU0/kBP = 24.2 µK/W, respectively. The discrepancy may be due to the additional trapping128B.2. IPG ODTpotential generated by the slight magnetic field gradient, which is not accounted when weuse equation 2.6. If so, the depth of this additional trap is approximately 0.4 µK.B.2 IPG ODTAccording to the fit of the data in Figure B.2, ωy/2pi = ωx/2pi = ωr/2pi = 1400.3± 8.4 Hzat 5 ± 0 W. According to the above data for the SPI trap, we should expect ωz ≈ 207 Hzfor an IPG cross trap. At this large trapping frequency, the effect of the field gradient isnegligible.Figure B.2: Trap frequency measurement for the y direction with 5± 0.5 W of laser powerin the first arm (IPG laser). Based on the fit of the data, ωy/2pi = 1400.3 ± 8.4 Hz. Forthis measurement, the trap was displaced for 0.25 ms by changing the AOM frequency by0.5 MHz, the cloud expansion time was 0.3 ms and the magnetic field was set to 755 G.Since the IPG beam is astigmatic, we have to guess what the effective W0 is based on thedata and fits in Appendix A. With a choice of W0 ≈ 58 µm, equations 2.6 and 2.10 agreethat the trap depth U0/kB = 47 µK. This implies that the U0/kBP = 4.7 µK/W.129Appendix CPassive Mode-Locking viaNon-Linear Polarization RotationC.1 Non-Linear Polarization Density in an IsotropicMediumIf we neglect its small birefringence, we can treat an optical fiber (that is not polarizationmaintaining) as an isotropic and centrosymmetric medium. Since there is no second orderresponse in a centrosymmetric medium, the dominant non-linear term is the third orderpolarization density P (3). To determine the nature and strength of this effect, we needto first explore the symmetry properties of the third order susceptibility tensor χijkl =χ(3)ijkl(ω4 = ω1+ω2+ω3). We begin by determining which tensor elements are non-zero. Thiscan be done without knowing the precise quantum mechanical nature of the interaction andinstead we rely on the classical anharmonic oscillator model - a modified Lorentz oscillatormodel. (We caution the reader that this is just an outline of the complete derivationpresented in [18].) In this case, the restoring force is given byFr = −mω20r +mb(r · r)r (C.1)where r is the displacement, m is the mass of an electron, ω0 is the natural frequencyof oscillation in the absence of non-linear effects and b is a parameter that characterizesthe strength of the non-linear effects. The first term in equation (C.1) corresponds to aharmonic potential and the second represents the non-linear correction to this potential. (Ifwe set b = 0, this would reduce to the Lorentz oscillator model.) The equation of motion is130C.1. Non-Linear Polarization Density in an Isotropic Mediumthenr¨+ 2γr˙+ω20r−mb(r ·r)r = −eE˜(t) = −e(E1e−iω1t +E2e−iω2t +E3e−iω3t + c.c.)(C.2)where γ is the damping constant, e is the charge of an electron and E˜(t) is the drivingelectric field. To determine the third order response of this system this, we expand r(t) ina power series in λ as follows:r(t) = λr(1)(t) + λ2r(2)(t) + λ3r(3)(t) (C.3)After inserting this form into equation (C.2), it is relatively straight-forward to show thatthe r(2) = 0 and derive an expression for r(3) at an oscillation frequency ωq. Each componentof the third order polarization density at this frequency is then given byP(3)i (ωq) = −Ner(3)i (ωq) = 0∑jkl∑mnpχ(3)ijkl(ω4 = ω1 + ω2 + ω3)Ej(ωm)Ek(ωn)El(ωp) (C.4)where N is the number density of dipoles. Finally, the third order susceptibility is given byχ(3)ijkl(ω4 = ω1 + ω2 + ω3) =Nbe4 (δijδkl + δikδjl + δilδjk)30m3D(ωq)D(ωm)D(ωn)D(ωp)(C.5)where D(ωξ) = ω20 −ω2ξ − 2iωξγ. Since we are interested in the propagation of light thougha non-linear medium as opposed to harmonic generation, we can immediately simplify theresult in equation (C.5) as follows:χ(3)ijkl(ω = ω + ω − ω) =Nbe4 (δijδkl + δikδjl + δilδjk)30m3D(ω)3D(−ω)) (C.6)Clearly, δijδkl + δikδjl + δilδjk = 0 if the value of one of i, j, k or l only appears oncein the subscript of χ(3)ijkl. This leaves us with 21 non-zero elements. For i = j = k = l,δijδkl + δikδjl + δilδjk = 3. For every other non-zero element, δijδkl + δikδjl + δilδjk = 1.131C.1. Non-Linear Polarization Density in an Isotropic MediumExplicitly, this implies the following:χ1111 = χ2222 = χ3333 (C.7)χ1122 = χ1133 = χ2211 = χ2233 = χ3311 = χ3322 (C.8)χ1212 = χ1313 = χ2323 = χ2121 = χ3131 = χ3232 (C.9)χ1221 = χ1331 = χ2112 = χ2332 = χ3113 = χ3223 (C.10)It is important to note that, by using the model presented above, we have restricted our-selves to (highly) non-resonant electronic non-linearities - the case we are considering for anErbium-doped fiber at 1550nm. However, equations (7)-(10) are valid for all centrosymmet-ric isotropic media. For our choice of frequencies, the following relations are also generallyvalid [18]:χ1111 = χ1122 + χ1212 + χ1221 (C.11)χ1122 = χ1212 (C.12)and equation (C.4) simplifies toP(3)i (ω) = 30∑jklχ(3)ijkl(ω = ω+ω−ω)Ej(ω)Ek(ω)El(ω) = 60χ1122 (E ·E∗)Ei+30χ1221 (E ·E)E∗i(C.13)For the case of highly non-resonant electronic non-linearities, we can simplify this equationeven further since it is clear from equation (C.7) thatχ1111 = 3χ1122 = 3χ1212 = 3χ1221 (C.14)However, we will use the more general form shown in equation (C.13) in the derivations thatfollow and simply remember that, in our case, χ1122 = χ1221. In vector form, the non-linear(NL) polarization density is given byPNL = 0A (E ·E∗)E + 120B (E ·E)E∗ (C.15)132C.2. Non-Linear Polarization Rotation in an Isotropic Mediumwhere A = 6χ1122 and B = 6χ1221. (In our case, A = B.) Equipped with equation (C.15),we can now show that an isotropic Kerr medium (such as a silica fiber) rotates an incidentelliptical polarization state and that the angle of rotation depends on the intensity of theincoming light.C.2 Non-Linear Polarization Rotation in an IsotropicMediumFor the purpose of this derivation, it is convenient to express the complex amplitudes ofthe driving electric field and the non-linear polarization density in the circular polarizationbasis rather than the usual Cartesian basis:E˜(z, t) = E(z)e−iωt + c.c. = (E+σˆ+ + E−σˆ−) e−iωt + c.c. (C.16)P˜NL(z, t) = PNL(z)e−iωt + c.c. = (P+σˆ+ + P−σˆ−) e−iωt + c.c. (C.17)whereσˆ± =xˆ± iyˆ√2(C.18)In the discussion that follows, we will need the identities that are proven below:σˆ∗± =xˆ∓ iyˆ√2= σˆ∗∓ (C.19a)σˆ± · σˆ± = 12 1±i · 1±i = 0 (C.19b)σˆ± · σˆ∓ = 12 1±i · 1∓i = 1 (C.19c)Our goal is to solve the non-linear wave equation to obtain a simple input-output (IO) rela-tionship between the incoming field and the outgoing field (assuming no reflections). This133C.2. Non-Linear Polarization Rotation in an Isotropic Mediumwill enable us to add this element (Kerr medium) to our existing arsenal of (linear) polar-ization manipulation tools. We begin by massaging equation (C.15) into a more convenientform. Using the above identities, it is trivial to show thatE·E∗ = (E+σˆ+ + E−σˆ−)·(E∗+σˆ∗+ + E∗−σˆ∗−)= (E+σˆ+ + E−σˆ−)·(E∗+σˆ− + E∗−σˆ+)= |E+|2+|E−|2(C.20a)E ·E = (E+σˆ+ + E−σˆ−) · (E+σˆ+ + E−σˆ−) = 2E+E− (C.20b)Substituting these results into equation (C.15) yieldsP± = 0(A(|E+|2 + |E−|2)E± +B (E+E−)E∗∓)= 0(A(|E+|2 + |E−|2)+B|E∓|2)E±= 0(A(|E+|2 + |E−|2)+A|E∓|2 −A|E∓|2 +B|E∓|2)E±= 0(A|E±|2 + (A+B) |E∓|2)E±For an electric field, whose z dependence appears only in its phase, propagating in the +zdirection, the term in the parentheses is independent of z. Making this assumption makesphysical sense if we ignore losses (and gain) within the Kerr medium. (If there are no lossesor reflections, the input and output power should be equal.) We will henceforth refer tothis quantity as χNL± .χNL± = A|E±|2 + (A+B) |E∓|2 (C.21)We can now express the components of PNL (in the circular polarization basis) as follows:P± = 0χNL± E± (C.22)The non-linear wave equation for an isotropic medium is given by∇2E˜(r, t) = (1)c2∂2E˜(r, t)∂t2+10c2∂2P˜NL(r, t)∂t2(C.23)134C.2. Non-Linear Polarization Rotation in an Isotropic Mediumwhere (1) =(n(1))2and n(1) is the linear refractive index of the material. In our simplifiedpicture, the electric field has no x and y dependence, so we can simplify equation (C.23):∂2E±∂z2= −(1)ω2c2E± − ω20c2P± = −ω2c2((1) + χNL±)E± (C.24)If we assume that the wave just before the fiber is E±(z = 0) = A± (implying that there isno reflected wave), the solution is simplyE± = A±eik±z (C.25)wherek± =ωc√(1) + χNL± =n(1)ωc√1 +1(1)χNL± ≈n(1)ωc(1 +12(1)χNL±)(C.26)We assumed above that χNL± /(1)  1 because χ(3) = 2.5−22 m2/V2 for silica [18]. As wewill show below, it is instructive to express k± in terms of a common and differential part.Hence, we define two new quantities k and ∆k and let k± = k ±∆k, wherek =k+ + k−2=n(1)ωc(1 +14(1)(χNL+ + χNL−))=n(1)ωc(1 +14(1)(2A+B)(|E+|2 + |E−|2))(C.27)∆k =ω2cn(1)(χNL+ − χNL−)=ωB4cn(1)(|E−|2 − |E+|2) (C.28)135C.3. Simple Ring Cavity ImplementationThus, the electric field at the the output of the Kerr medium E(z = d) can be expressed interms of the input field E(z = 0) as follows:E(z = d) = E+(z = d)σˆ+ + E−(z = d)σˆ−= A+eik+dσˆ+ +A−eik−dσˆ−= eikd(A+ei∆kdσˆ+ +A−e−i∆kdσˆ−)= eikd(E+(z = 0)ei∆kdσˆ+ + E+(z = 0)− e−i∆kdσˆ−)= eikdei∆kd 00 e−i∆kdE(z = 0)The exponential eikd is just an overall phase term. The interesting physics is containedin the diagonal matrix whose physical meaning will become clear shortly. For now, it isimportant to emphasize that ∆k is a function of the intensity of the electric field. We canwrite the above results in the following short-hand notation:E(z = d) = MCBKERRE(z = 0) (C.29)C.3 Simple Ring Cavity ImplementationIn this section, we will show that the simple structure shown in Figure C.1 can be usedto construct an artificial saturable absorber. This structure represents a single round tripin a ring cavity. It consists of a polarizer, quarter wave-plate (QWP), the Kerr medium(optical fiber) and a half wave-plate (HWP). It is the simplest (and by no means the best)structure capable of achieving additive pulse mode-locking (APM) that relies on non-linearpolarization rotation in an isotropic optical fiber. In section 4.2, we develop the Jonesmatrix formalism required to analyze the structure shown in Figure C.1. We then use thismachinery developed in section 4.2 to demonstrate in section 4.3 that this structure canoperate as an artificial saturable absorber.136C.3. Simple Ring Cavity Implementationpolarizer polarizerQWP (θ ) HWP (θ )KerrMedium1 2E EE E E0 1 2 3 4Figure C.1: Simple polarization additive-pulse mode locking structure.C.3.1 Jones Matrix FormalismTo arrive at the result we seek, it is convenient to work with the Jones matrix formalism,so we devote this section to defining the matrices we will need for the remainder of thederivation. We will be expressing the polarization state of the light in the linear (Ex, Ey)and circular (E+, E−) bases. To convert between them, we will use the basis transformationU and its inverse U−1:U =1√21 ii 1 , U−1 = 1√2 1 −i−i 1 (C.30)Using U and U−1, we can use the relations below to convert Jones vectors from the linearbasis to the circular basis and vice versa.E+E− = UExEy ,ExEy = U−1E+E− (C.31)137C.3. Simple Ring Cavity ImplementationSimilarly, we can transform Jones matrices in the linear basis to Jones matrices in thecircular basis and vice versa as followsMCB = UMLBU−1, MLB = U−1MCBU (C.32)To mathematically describe the rotation of the wave-plates with respect to the x-axis, wewill also need the rotation matrixR(θ) = cos(θ) sin(θ)−sin(θ) cos(θ) (C.33)If we rotate an element represented by a Jones matrix M by an angle θ, we obtain (in thelinear basis) another matrix M(θ). This is just another linear transformation given byM(θ) = R(−θ)MR(θ) (C.34)In the linear basis, the Jones matrices for a half-wave-plate (HWP), quarter-wave-plate(QWP) and an x-polarizer are defined as followsM(LB)HWP =1 00 −1 , M (LB)QWP =1 00 i , M (LB)POL =1 00 0 (C.35)If we rotate the fast and slow axes of these elements by an angle θ, we obtainM(LB)HWP(θ) =cos(2θ) sin(2θ)sin(2θ) − cos(2θ) , M (LB)QWP(θ) = 1− i2i+ cos(2θ) sin(2θ)sin(2θ) i− cos(2θ) (C.36)Next, using equation (C.32), we find the circular basis representations of these matricesM(CB)HWP(θ) = 0 −iei2θie−i2θ 0 , M (CB)QWP(θ) = 1 + i2 1 −ei2θe−i2θ 1 , M (LB)POL = 121 −ii 1(C.37)138C.3. Simple Ring Cavity ImplementationThe last element we will need in the derivation below is the Jones matrix for the Kerrmedium (fiber) itself. In the previous section, we derived this matrix in the circular basisMCBKERR = eikdei∆kd 00 e−i∆kd (C.38)In the linear basis, this matrix is given byMLBKERR = ekdcos(∆kd) − sin(∆kd)sin(∆kd) cos(∆kd) (C.39)When written in this representation, it is clear that this matrix is just another rotationmatrix, but with a twist. Since the angle of rotation depends on ∆k, it is intensity depen-dent. It also depends on the input polarization state of the light. For example, if the inputpolarization state is linear, the angle of rotation will be zero regardless of the intensity. Thiscan be seen by examining equation (C.28), which shows that, for an even mixture of E+ andE−, ∆k = 0. This equation also suggests that it is advantageous to make the input statepurely right-hand-circularly polarized or purely left-hand-circularly polarized because thiswould maximize the angle of rotation. However, this is also not very useful (for the schemewe are considering) because one cannot distinguish with a linear polarizer between differentpurely circular states. Hence, we will need to make the input field elliptically polarized.The output polarization will then be the same polarization ellipse (its shape will remainunchanged) rotated by an intensity dependent angle.C.3.2 Constructing an Artificial Saturable AbsorberAt last, we have all of the ingredients necessary to construct a simple artificial saturableabsorber. We now return to Figure C.1 and begin by qualitatively predicting the outcomewe will derive below analytically. As shown in Figure C.1, we always start the trip aroundthe cavity with x-polarized light directly out of the polarizer. The QWP oriented at someangle θ1 transforms this polarization state E0 into an elliptically polarized state E1. Thisstate enters the fiber (Kerr medium) and gets rotated by an intensity dependent angle ∆kd.This polarization state E3 exits the fiber with a new orientation, but with its shape and139C.3. Simple Ring Cavity Implementationhandedness unchanged. Finally, the HWP oriented at some angle θ2 with respect to thex-axis rotates the polarization ellipse by an intensity independent angle and it returns tothe x-polarizer. The projection parallel to the x-axis passes and the remainder of the lightgets rejected. Since the angle by which the Kerr medium rotates the polarization elllipseis intensity dependent, we have the liberty of choosing angles θ1 and θ2 such that thelosses are low for high intensities and high for low intensities. In doing so, we have addedan intensity dependent lossy element into the cavity. In other words, we have created anartificial saturable absorber.To make the above qualitative discussion more concrete, we now proceed to derivethe output intensity after a single round trip assuming, for simplicity, that every elementis loss-less and ignoring the amplification produced by the Erbium doped fiber. Let thepolarization state directly out of the polarizer beECB0 = UA00 = A0√21i (C.40)After the QWP, the state is given byECB1 = M(CB)QWP(θ1)E0 =A0√2 eiθ1 (cos(θ1) + sin(θ1))ie−iθ1 (cos(θ1)− sin(θ1)) (C.41)To compute the angle ∆kd, we need to first determine |E+|2 and |E−|2 for the light enteringthe fiber from ECB1 . The result is|E±|2 = A202(1± sin(2θ1)) (C.42)After the fiber, the polarization state is given byECB2 = MCBKERRE1 =A0√2eikd ei(θ1+∆kd) (cos(θ1) + sin(θ1))ie−i(θ1+∆kd) (cos(θ1)− sin(θ1)) (C.43)wherek =n(1)ωc(1 +3A4(1)A20), ∆k = − ωA4cn(1)sin(2θ1)A20 (C.44)140C.3. Simple Ring Cavity ImplementationThe state after the HWP in the linear basis (LB) isELB3 = U−1M (CB)HWP(θ2)ECB2 =A02eikd e−iα(cos(θ1) + sin(θ1)) + eiα(cos(θ1)− sin(θ1))−ie−iα(cos(θ1) + sin(θ1)) + ieiα(cos(θ1)− sin(θ1))(C.45)where α = θ1 + ∆kd− 2θ2. Finally, the x-polarizer selects the x-component of ELB3 and weend up with the following state after a complete round tripELB4 = M(LB)POLELB3 = xˆA02eikd(e−iα(cos(θ1) + sin(θ1)) + eiα(cos(θ1)− sin(θ1)))(C.46)Using equation (C.46), we can easily compute the ratio of the output intensity and inputintensityIoutIin=12(1 + cos(2θ1) cos(2α)) (C.47)=12(1 + cos(2θ1) cos(2(θ1 + ∆kd− 2θ2))) (C.48)=12(1 + cos(2θ1) cos(2(θ1 − 2θ2)− ωA2cn(1)sin(2θ1)dA20))(C.49)This result may not appear to be particularly illuminating, but we can make several worth-while observations. First, it is clear that Iout/Iin ≤ 1, as expected. For the equality to hold,we would need to send into the Kerr medium a linear polarization. However, as we pointedout previously, this would not be useful because the result would not be an artificial sat-urable absorber. Second, for a fixed fiber length, the intensity depends on three variables:θ1, θ2 and the intensity we would like to optimize for, which is represented in the aboveexpression by |A0|2. Finding the optimal configuration for constructing a mode-locked laseris not trivial (it is not just a matter of finding the maxima of the above expression) andrequires a more sophisticated treatment that is beyond the scope of this report. We brieflydiscuss this in the section below.141C.4. Concluding RemarksC.4 Concluding RemarksWe have shown that an isotropic Kerr medium (such as an optical fiber) rotates the polar-ization state of an incoming elliptically polarized beam by an intensity dependent angle. Byplacing this element, a QWP, HWP and a polarizer into an optical cavity, we can constructan intensity dependent loss element - an artificial saturable absorber. To determine the op-timal wave-plate angles, we would need to consider the dynamics of the laser, dispersion, selfamplitude modulation (SAM), self phase modulation (SPM), the gain profile of the Erbiumdoped fiber, other lossy elements in the cavity and a number of other factors. To account fordispersion, we would require solve the non-linear wave equation in the slow varying envelope(SVE) approximation and take into account that the linear refractive index is wavelengthdependent. It would also require several other metrics that can be obtained from the MasterEquation for mode-locking [67, 154]. In practice, this optimization is often done empiricallywith the help of an optical spectrum analyzer. Also, we have only discussed the simplestimplementation of this type of passive mode locking. By including additional elements (e.g.wave-plates and Faraday rotators), it is possible to achieve mode locked operation withlower round trip losses. For a more involved treatment of this problem, we invite the readerto consult [67, 154].142

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