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UBC Theses and Dissertations

Photoassociation spectroscopy of a degenerate Fermi gas of ⁶Li Semczuk, Mariusz 2015

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PHOTOASSOCIATION SPECTROSCOPY OF ADEGENERATE FERMI GAS OF 6LIbyMariusz SemczukMSc, University of Warsaw, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2015c©Mariusz Semczuk, 2015AbstractThis thesis describes a suite of experimental tools and spectroscopic measurementsperformed on ultra-cold 6Li molecules. The aim is to create the necessary ingre-dients for the coherent transfer of the population of ultracold, weakly bound, Fes-hbach molecules of 6Li to deeply bound ro-vibrational levels, making the eventualcreation of a Bose-Einstein condensate (BEC) of ground state lithium dimers a re-ality at the University of British Columbia. Some of the technological milestonesinclude the development of a unique laser system consisting of two Ti:Sapphirelasers frequency stabilized to a femtosecond frequency comb as well as the demon-stration of the first in Canada BEC of Feshbach molecules.To determine a suitable path for the coherent transfer using stimulated Ramanadiabatic passage (STIRAP) we measure the binding energies of the vibrationallevels v′′ = 20−26 of the c(13Σ+g ) and v′′ = 29−35 of the A(11Σ+u ) excited statesof lithium dimers by the photoassociation of a degenerate Fermi gas of 6Li atoms,achieving accuracy of 600 kHz. For each vibrational level of the triplet potential,we resolve the rotational structure using a Feshbach resonance to enhance the pho-toassociation rates from p-wave collisions. We also, for the first time, determinethe spin-spin and spin-rotation interaction constants for this state.Finally, we are the first to demonstrate exotic dark states in quantum gases offermionic lithium where atom-molecule coherence is produced between a deeplybound singlet (or triplet) molecular level and atomic pairs in a weakly interact-ing Fermi gas at zero gauss or in the BEC-BCS crossover regime (i.e. Feshbachmolecules or BCS-like pairs). We observe an abrupt and unanticipated change ofthe classic EIT signature (Electromagnetically Induced Transparency) of the dark-state (i.e. the suppression of single photon absorption to the excited state) in thevicinity of the broad Feshbach resonance at 832.2 G potentially indicating newphysics not previously considered.iiPrefaceChapter 2 extends the material that has been published in W. Gunton et al., Phys.Rev. A, 88, 023624, Aug (2013) of which I am an author. I was involved in everystage of the design of the experiment, data taking and analysis. I wrote parts ofthe manuscript and participated in editing. Figures 2.17, 2.18, 2.19, 2.20, 2.21 aremade by W. Gunton and come from W. Gunton et al. (2013).The results of Chapter 3 have been published in M. Semczuk et al., Phys. Rev.A 87, 052505, May (2013) and W. Gunton et al., Phys. Rev. A 88, 062510, Dec(2013) of which I am an author. I was involved in every stage of the design ofthe experiment, data taking and analysis. I wrote parts of the manuscripts andparticipated in editing. The theoretical analysis of molecular potentials presentedin the above papers was performed by Dr. N. S. Dattani and is not included in thisdissertation. Figures 3.4 and 3.9 are made by Dr. N. S. Dattani and Dr. X. Li,respectively and come from M. Semczuk et al. (2013).Chapter 4 is based on M. Semczuk et al., Phys. Rev. Lett. 113, 055302, July(2014). I was involved in every stage of the design of the experiment, data takingand analysis. I wrote and edited the manuscript incorporating feedback from otherauthors. I was responsible for submission process of the manuscript for publicationand prepared replies to referees’ comments. Figure 4.14 was made by W. Guntonand comes from M. Semczuk et al. (2014).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Projects prior to lithium spectroscopy . . . . . . . . . . . . . . . 51.3 List of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 72 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Vacuum system and magnetic trapping . . . . . . . . . . . . . . . 112.2 Trapping, imaging and state preparation . . . . . . . . . . . . . . 162.2.1 Magneto-optical trapping laser system . . . . . . . . . . . 162.2.2 High field imaging laser system . . . . . . . . . . . . . . 192.2.3 Dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.4 RF antenna for spin manipulation . . . . . . . . . . . . . 322.2.5 Photoassociation laser system . . . . . . . . . . . . . . . 342.3 Preparation of ultracold quantum gases . . . . . . . . . . . . . . . 40iv2.3.1 Trapping performance . . . . . . . . . . . . . . . . . . . 402.3.2 Molecular BEC . . . . . . . . . . . . . . . . . . . . . . . 462.3.3 Strongly interacting degenerate Fermi gas . . . . . . . . . 472.3.4 Fermi gas at 0 gauss . . . . . . . . . . . . . . . . . . . . 503 Single color photoassociation . . . . . . . . . . . . . . . . . . . . . . 553.1 A note about Feshbach resonances . . . . . . . . . . . . . . . . . 583.2 Theory of diatomic molecules . . . . . . . . . . . . . . . . . . . 623.2.1 Initial state of colliding atoms . . . . . . . . . . . . . . . 633.2.2 Final excited molecular states . . . . . . . . . . . . . . . 653.3 Spectroscopy of the c-state . . . . . . . . . . . . . . . . . . . . . 663.3.1 p-wave Feshbach resonance enhanced photoassociation . . 703.3.2 Spin-spin and spin-rotation constants . . . . . . . . . . . 733.3.3 Systematic shifts . . . . . . . . . . . . . . . . . . . . . . 783.4 Spectroscopy of the A-state . . . . . . . . . . . . . . . . . . . . . 813.4.1 v’=30 anomaly . . . . . . . . . . . . . . . . . . . . . . . 844 Dark state spectroscopy of bound molecular states . . . . . . . . . . 864.1 Two-color photoassociation and dark states: theory . . . . . . . . 884.2 Two-color photoassociation . . . . . . . . . . . . . . . . . . . . . 924.3 Dark state spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 v”=9,N”=0,F”=1 revival - role of the dipole trap . . . . . 994.3.2 Dark state lifetime . . . . . . . . . . . . . . . . . . . . . 1004.4 Binding energies of the least bound states . . . . . . . . . . . . . 1024.5 Binding energy of v”=37, N”=0 . . . . . . . . . . . . . . . . . . 1044.5.1 Unexpected features in the v”=37, N”=0 spectrum . . . . 1084.6 Exotic dark states in the BEC-BCS crossover . . . . . . . . . . . 1084.6.1 Dark states in a BEC of Feshbach molecules . . . . . . . 1104.6.2 Dark states in a degenerate Fermi gas . . . . . . . . . . . 1104.6.3 Dark states in the BEC-BCS crossover . . . . . . . . . . . 1135 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.1 Future experiments . . . . . . . . . . . . . . . . . . . . . . . . . 116vReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119viList of TablesTable 2.1 MOT loading parameters . . . . . . . . . . . . . . . . . . . . 41Table 3.1 Accessible potentials . . . . . . . . . . . . . . . . . . . . . . 57Table 3.2 Allowed rotational levels and corresponding nuclear spin con-figurations for 6Li2 . . . . . . . . . . . . . . . . . . . . . . . . 66Table 3.3 Photoassociation to (N′ = 1,J′ = 1), (N′ = 1,J′ = 2), and (N′ =1,J′ = 0), c-state . . . . . . . . . . . . . . . . . . . . . . . . . 70Table 3.4 p-wave enhanced photoassociation to N′ = 0, c-state . . . . . . 73Table 3.5 p-wave enhanced photoassociation to N′ = 2, c-state . . . . . . 74Table 3.6 Experimentally determined λv and γv . . . . . . . . . . . . . . 76Table 3.7 Photoassociation to (N′ = 1, J′ = 1), A-state . . . . . . . . . . 83Table 3.8 Photoassociation to N′ = 1, J′ = 1, A-state, field free values . . 83Table 4.1 Binding energies of the least bound ground state levels . . . . . 103Table 4.2 v′′ = 37, N′′ = 0 at non-zero magnetic field . . . . . . . . . . . 107viiList of FiguresFigure 2.1 Lithium oven . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.3 Lithium extraction . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.4 Lithium oven T vs. I . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.5 Lithium master table . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.6 Detunings from the cooling transition . . . . . . . . . . . . . 20Figure 2.7 High field imaging offset lock . . . . . . . . . . . . . . . . . 21Figure 2.8 High field imaging error signal . . . . . . . . . . . . . . . . . 22Figure 2.9 Dipole traps - current . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.10 Dipole trap beam arrangement . . . . . . . . . . . . . . . . . 27Figure 2.11 Dipole traps - past approaches . . . . . . . . . . . . . . . . . 31Figure 2.12 RF transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.13 6Li 2S Zeeman splitting . . . . . . . . . . . . . . . . . . . . 33Figure 2.14 RF antenna design . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.15 Coherent 899-21 lasers and the frequency comb . . . . . . . . 37Figure 2.16 PA light distribution . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.17 MOT and dipole trap lifetimes . . . . . . . . . . . . . . . . . 42Figure 2.18 MOT to ODT transfer . . . . . . . . . . . . . . . . . . . . . 43Figure 2.19 Evaporation to mBEC and DFG . . . . . . . . . . . . . . . . 45Figure 2.20 Emergence of the mBEC . . . . . . . . . . . . . . . . . . . . 48Figure 2.21 Emergence of the BCS pairs . . . . . . . . . . . . . . . . . . 49Figure 2.22 Atom number at 0 G . . . . . . . . . . . . . . . . . . . . . . 51Figure 2.23 6Li 2S Zeeman splitting at low magnetic fields . . . . . . . . 52Figure 2.24 Residual magnetic field . . . . . . . . . . . . . . . . . . . . . 54Figure 3.1 Potentials of 6Li2 . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.2 s-wave Feshbach resonances in 6Li . . . . . . . . . . . . . . . 60viiiFigure 3.3 p-wave Feshbach resonances in 6Li . . . . . . . . . . . . . . 62Figure 3.4 c-state potential with measured levels . . . . . . . . . . . . . 67Figure 3.5 13Σ+g N′ = 1,G′ = 0 photoassociation, three features . . . . . 69Figure 3.6 13Σ+g N′ = 1,G′ = 0 photoassociation, second feature . . . . . 71Figure 3.7 p-wave enhanced PA, c-state . . . . . . . . . . . . . . . . . . 72Figure 3.8 Spin-splitting parameter-theoretical . . . . . . . . . . . . . . 75Figure 3.9 λv, experiment and ab initio . . . . . . . . . . . . . . . . . . 77Figure 3.10 v′ = 24,N′ = 1,J′ = 1 B field dependence . . . . . . . . . . . 78Figure 3.11 v′ = 24,N′ = 1,J = 2 B field dependence . . . . . . . . . . . 79Figure 3.12 v′ = 24,N′ = 1,J′ = 0 B field dependence . . . . . . . . . . . 80Figure 3.13 11Σ+u ,N′ = 1,G′ = 0 photoassociation . . . . . . . . . . . . . 81Figure 3.14 11Σ+u v′ = 30, N′ = 1, G′ = 0 photoassociation . . . . . . . . 85Figure 4.1 Two-color spectroscopy - energy diagram . . . . . . . . . . . 90Figure 4.2 Two-color PA, X–state, v′′ = 38, N′′ = 0 . . . . . . . . . . . . 93Figure 4.3 Two-color PA, a–state, v′′ = 9, N′′ = 0 . . . . . . . . . . . . . 94Figure 4.4 Li molecules - energy levels . . . . . . . . . . . . . . . . . . 95Figure 4.5 Dark state v′′ = 38, N′′ = 0 . . . . . . . . . . . . . . . . . . . 97Figure 4.6 Dark state v′′ = 9, N′′ = 0 . . . . . . . . . . . . . . . . . . . 98Figure 4.7 Hyperfine splittings, triplet state . . . . . . . . . . . . . . . . 103Figure 4.8 Two-color PA, X-state, v′′ = 37,N′′ = 0 . . . . . . . . . . . . 105Figure 4.9 Dark state v′′ = 37, N′′ = 0 . . . . . . . . . . . . . . . . . . . 106Figure 4.10 Dark states in a mBEC . . . . . . . . . . . . . . . . . . . . . 109Figure 4.11 Revival height definition . . . . . . . . . . . . . . . . . . . . 110Figure 4.12 Dark state in a DFG . . . . . . . . . . . . . . . . . . . . . . 111Figure 4.13 RF spectroscopy of a dark state in a DFG . . . . . . . . . . . 112Figure 4.14 Dark state revival in the BEC-BCS crossover . . . . . . . . . 114ixList of AbbreviationsAOM acousto-optic modulatorBCS Bardeen - Cooper - SchriefferBEC Bose-Einstein condensateCDT/CODT crossed dipole trapFR Feshbach resonancemBEC molecular Bose-Einstein condensateMOT magneto-optical trapODT optical dipole trapPA photoassociationREMPI resonance-enhanced multiphoton ionizationSTIRAP stimulated Raman adiabatic passagexAcknowledgmentsA 5.5-year long journey has come to an end...Both personally and professionallyit has been a rollercoaster, but I am privileged to have shared both ups and downswith many amazing people.First and foremost, my great thanks go to my supervisor, Kirk Madison. Work-ing with him was a very enriching experience that let me grow as a scientist andas a person. It was overwhelming at times to witness how much he knew, not onlyabout physics, and how quickly he grasped the essence of a problem whenever hefaced one. I admire his attitude towards students and everyone working for himimmediately realizes that he really cares about you, that your growth and successare important to him and he would do anything to help you achieve your goals. Igreatly benefited from numerous discussions we had and many advices I receivedhelped me better shape my future. I owe most of my success to him.Postdocs Bruce Klappauf and Ben Deh were essential in introducing me to thefield of ultra cold atoms in the early stages of my graduate career. I appreciate theireffort to teach me experimental skills and I apologize for constantly interruptingtheir work by asking often silly questions - but in the end the time they spent as myteachers turned out to be a pretty decent investment.This thesis would not be what it is if it was not for Will Gunton. Working in ateam with him was very rewarding and enjoyable. I feel like in many aspects wecompleted each other and our differences were, in my opinion, the strongest assetof our collaboration. It was rare that we immediately agreed on things, but thisonly forced both sides to think a little deeper about a problem and I believe it madeboth of us much better scientists. Now the experiment is in his capable hands andthere is no doubt in my mind that he will lead it to greatness.From the very beginning of my stay in Canada I had a great pleasure to haveGene Polovy as a friend. I could always count on him and I learned a lot abouteveryday life in Canada from him. This added a lot of value to my experience as aforeigner. He helped to make my life outside of the lab more fun, maybe at timeseven more than it was initially anticipated. In the end we will both remember thexishenanigans we went through over the years and decades from now we will havememories that will bring smiles to our wrinkled faces.Janelle’s cookies fuelled us during countless hours in the lab, often servingas a substitute for a proper meal. Her kind personality contributes to the friendlyatmosphere in the lab. You rarely find people that nice and helpful and I am surethat generations of students can also testify to that. Her father, a frequent visitor,offered a very unique perspective on life and I appreciate his constant support andhis faith in me.I enjoyed many laughs and very fun discussions with Kais and Jim, the MAT(and then PAT) guys. I really hope that their experiment will produce great resultssoon and they will help establish the first ultra cold atom based pressure standard.A brand new addition to the team, Kahan and Koko, very quickly became in-dispensable members of the group. I am glad I had a chance to work with them andI am looking forward to seeing their growth - which, I believe, will be spectacular.Joining Psi Upsilon immediately after arriving in Canada let me feel like a partof a family of a sort. It gave me an opportunity to meet many amazing people whobecame my friends. The countless hours spent ”on the pine” with them helped meunderstand Canada and Canadians much better. It was a great getaway from thelab work and it will be always a source of great stories and fond memories.Last but not least, I want to acknowledge my parents and the rest of my familyfor their constant support. I hope I made you proud.xiiChapter 1IntroductionSpectacular progress in cooling of atoms resulted in a series of ground breaking ex-periments leading to the 1995 breakthrough: the production of Bose-Einstein con-densate of alkali atoms (BEC) by groups from JILA/University of Colorado [1],MIT [2] and Rice University [3]. A wealth of new opportunities to study Na-ture opened to researchers and for last 20 years the field has been only gainingmomentum, to the extend that nowadays some atomic physicists are even called“condensed matter physicists” (for example Markus Greiner from Harvard Univer-sity). Ultracold atoms and molecules trapped in optical lattices became ”quantumsimulators”, shedding light on many aspects of condensed matter systems [4] buttheir applications reach beyond that [5, 6].Many questions about fundamental concepts in physics can be nowadays ad-dressed in experiments using ultracold atoms, or relying on the techniques devel-oped for their studies. For example, the manipulation of single atoms with pho-tons and single photons with atoms was recognized with the 2012 Nobel prize inphysics awarded to Serge Haroche and David J. Wineland ”for ground-breakingexperimental methods that enable measuring and manipulation of individual quan-tum systems”.The laser-cooled atoms, among others, have been used to measure the Newto-nian gravitational constant [7, 8], parity nonconservation in cesium [9], predictedby general relativity gravitational redshift [10] and are expected to enable the real-ization of the original Einstein-Podolsky-Rosen proposal [11–13] where the entan-glement of the external degrees of freedom of freely moving massive particles canbe used to test the fundamental aspects of quantum mechanics.Some topics that emerged from the studies of ultracold matter have alreadyentered the realm of ”applied research”. Many national metrology institutes (forexample Physikalisch-Technische Bundesanstalt, Germany; National Institute of1Standards and Technology, USA; National Physical Laboratory, UK) have researchprograms focusing on the development of frequency standards using optical clocksbased on ions [14, 15] and atoms in optical lattices [16, 17]. Atom interferome-ters [18], where optical fields are replaced by interfering matter waves of Bose-Einstein condensates, have recently became contenders as a technology of the fu-ture. This approach has been recently validated by the British Ministry of Defenceby funding research projects that, amongst others, are expected to develop table-top atom interferometers to provide ultra-precise, highly reliable positional datafor submarines at depth, where traditional GPS satellite navigation systems willnot work.Applications of ultracold atoms for both fundamental science and emergingtechnologies cannot be underestimated and the future for the field is bright. It is amatter of time when newly developed techniques will be extended to more compli-cated systems like molecules or even macroscopic objects, hopefully proving onceagain that only the sky is the limit if it comes to progress.1.1 MotivationFrom its inception, the Quantum Degenerate Gases laboratory at the University ofBritish Columbia has been focused on investigation of ways to create ground statemolecules of LiRb which are expected to have a sizable permanent electric dipolemoment of µ ∼ 4.15× 3.33564× 10−30 Cm = 4.15Debye1 in the X1Σ+ groundstate [19–21]. This is a value in a frame where the molecular framework is fixed inspace. In the laboratory frame, however, a molecule in a given rotational state hasno permanent electric dipole moment. It is necessary, therefore, to apply an exter-nal electric field that couples states of opposite parity (different rotational states)and thereby polarizes molecules along the field direction [22]. In modern experi-ments with alkali molecules the electric dipole moment reached in the laboratoryframe is typically a fraction of the permanent value (< 40% [23]), limited by theability to create high electric fields that would enable full polarization.Our early results include a demonstration of a dual species magneto-optical1Electric dipole moment is frequently expressed in the units of ea0 with 1Debye= 0.39ea0. eand a0 are the elementary charge and Bohr radius, respectively.2trap of 6Li and 85Rb [24] as well as simultaneous trapping of both species in adipole trap which enabled measurement of magnetic Feshbach resonances, someof the broadest among reported in other heteronuclear mixtures [25].The long-range 1/r3 character and the angular dependence of dipole-dipoleinteractions (as opposed to 1/r6 character of van der Waals interactions betweenground state atoms) makes polar molecules an interesting object of studies in thecontext of strongly interacting systems. Especially, if these molecules are confinedin an optical lattice they are expected to enable novel ways to simulate quantummany-body systems, moving it beyond what can be achieved with ultracold atoms(see e.g. [26–28]).The ratio of interaction energies between two electric (µ) and magnetic (m)dipoles trapped on neighboring sites of an optical lattice isEd-d electricEd-d magnetic∼(cµm)2, (1.1)where c is the speed of light. Contrasting LiRb with dysprosium (m = 10µB = 10×9.274× 10−24 J/T), the most magnetic atom that has been cooled to the quantumdegenerate regime [29, 30] the ratio of electric to magnetic dipolar interactions∼ 2000 strongly favors molecules. An alternative system providing long-rangeinteractions, with electric dipole moments on the order of 1000s Debye, can beeasily obtained using ultracold Rydberg atoms [31, 32] due to the scaling of thedipole moment with the square of the principal quantum number n. These highlyexcited Rydberg states, typically with n > 50 are, however, usually short lived (∼50µs2), therefore they cannot be considered as a universal substitute for molecules.Ultracold molecules, the character of interactions aside, are interesting in theirown rights. They are, for example, an entry point to the exciting field of ultracoldchemistry [33–35], where chemical reactions can be controlled with electric andmagnetic fields to an unprecedented level [36]. When the de Broglie wavelengthincreases (as the temperature decreases), many-body interactions become signifi-cant such that the spatial confinement of molecules can influence the outcome ofchemical reactions, enabling chemistry in restricted geometries. At very low tem-2In the literature devoted to Rydberg gases these highly excited levels are treated as “long lived”,because they are typically contrasted with first excited states with lifetimes on the order of 500 ns.3peratures the thermal motion of molecules becomes insignificant thus the controlof bi-molecular reactions could be possible. Such a degree of control and flexibilityis not achievable with molecules in supersonic beams or in hot vapors.For the eventual transfer of loosely bound Feshbach molecules of 6Li85Rb tothe ground state, a fiber based frequency comb has been built [37] with an intent ofusing it to stabilize the frequency of two Ti:Sapphire lasers that would be used forthe stimulated Raman adiabatic passage (STIRAP).High resolution spectroscopic studies of LiRb had not been available until2011 [38, 39] and later [40–42] therefore we decided to test the feasibility of ournewly developed experimental setup by focusing on Li2, as this molecule had beenalready quite extensively studied, but primarily outside of the ultracold regime.This approach was meant to develop a know-how required for successful STIRAP.It was quickly realized that Li2 molecules would be a very interesting objectfor investigations instead of being only a test bed as initially assumed. Due to theirlongevity (caused by Fermi suppression of collisional losses) it is possible to createa molecular Bose-Einstein condensate of loosely bound Feshbach molecules andthis brings a question whether this could be used as a starting point for STIRAP,resulting in a Bose-Einstein condensate of ground state molecules. Such expec-tation is unique to fermionic species (6Li and 40K) and so far no Bose-Einsteincondensation of Feshbach molecules consisting of bosons has been demonstrated.Even though many experimental groups made significant progress towards thisgoal, quantum degeneracy of real (here: deeply bound) molecules has not beenachieved yet.Lithium dimers in the ro-vibrational ground state do not have a permanent elec-tric dipole moment, therefore the molecules in the lowest lying triplet potentialseem like a more interesting object of investigation. This stems from the fact thatthey possess a magnetic dipole moment, possibly enabling e.g. observation ofmolecule-molecule Feshbach resonances [43, 44]. Such resonances are expectedto be more numerous than in the atomic case due to the contribution to the scat-tering process from rotational and vibrational excitations. Addressing this in theultracold regime with a relatively simple molecule like lithium dimer would be amajor step in understanding collisional physics.The demonstration of atom-molecule dark states in fermionic lithium in the4BEC-BCS crossover (as discussed in Chapter 4) opens up exciting possibilities tooptically tune interactions between colliding atoms while suppressing spontaneousscattering. Due to the capability to control optical fields on a sub-microsecondtimescale it is conceivable to use this approach to study the dynamics of stronglyinteracting fermionic systems on time scales not easily accessible with standardmethods of magnetically tunable Feshbach resonances [45]. The use of opticalfields would also enable high-resolution spatial tunability of interactions. In mix-tures it might be possible to control interaction strength in a species-dependent way,for example by optically changing interactions between fermions in a 6Li85Rb mix-ture while boson-fermion and boson-boson interactions are fixed by the magneticfield.Finally, the ability to trap ground state lithium molecules would provide a sys-tem with a well defined initial population of ro-vibrational levels which could beused for studies of ultracold superrotors (fast rotating molecules whose rotationalenergy is comparable with the molecular bond strength), by utilizing methods de-veloped for molecular beams [46–48]. In these experiments the singlet moleculesneed to be sufficiently deeply bound such that femtosecond pulses used to preparerotating molecules cannot drive any transitions between the initial state and statesin excited molecular potentials. For pulses centered at ∼790 nm (typical for tita-nium sapphire laser systems) it requires lithium molecules to be prepared in theX(11Σ+g ) potential with vibrational quantum numbers at most v′′ = 12. This re-strictions assures that the photon energy is not enough to reach the minimum of thefirst excited singlet potential A(11Σ+u ).1.2 Projects prior to lithium spectroscopyHere is a (non-exhaustive) list of projects that I was involved in but which do notdirectly contributed to the research presented in this thesis. However, they eitherled to the improvement of the infrastructure of the laboratory or were a source ofknow-how that was useful when focusing on 6Li.• Feshbach resonance measurements in the 6Li85Rb mixture confined in adipole trap. It was the first such a measurement in 6Li85Rb and resultedin the discovery of the broadest Feshbach resonances among all commonly5studied heteronuclear mixtures. The results were published in [25].• attempt at measurement of electric Feshbach resonances in 6Li85Rb. It hadbeen predicted that DC electric field of about 30 kV/cm would induce anobservable shift of a magnetic Feshbach resonance. We built a high voltagesetup that was designed to produce electric fields up to 40 kV/cm at the loca-tion of the atoms. The fields could be turned on and off in about 100 ms. Theelectrodes were made from the indium tin oxide (ITO) coated glass and werelocated outside of the glass cell, 3 cm apart. After a series of experimentsunanticipated shielding effects were observed and as a result electric fieldshigher than about 17 kV/cm were impossible to achieve. Moreover, switch-ing the electric field caused a significant atom loss. As a result, we wereunable to create conditions required to successfully perform the planned ex-periment. The high voltage setup is still an integral part of our experimentand if needed can be used e.g. to study Li2 or, eventually, 6Li85Rb in highDC electric fields.• a laser system for simultaneous trapping and cooling of 85Rb and 87Rb wasdesigned for experiments in collaboration with prof. Valery Milner. Parts ofa vacuum chamber for this project were designed, and a dual species (Li andRb) 2D MOT chamber was built. This experiment was cancelled due to lim-ited funding but the design of the laser system was used by another graduatestudent to construct a Rb MOT for an experiment aiming at production ofultracold RbH, in collaboration with prof. Taka Momose.• mechanical shutters based on 2.5” hard drives were designed and built. Thisproject resulted in shutters that can be assembled even by a relatively un-skilled person, on a mass scale. At the same time they are maximally simpli-fied by using only essential elements of a hard drive (magnets and a swingingarm that moves the HDD readout head) and as a result are cheap to build andresemble a commercial product.61.3 List of papersThe following is the list of papers published during my time as a doctoral student.Publications 2-5 form the basis of this thesis.1. B. Deh, W. Gunton, B. G. Klappauf, Z. Li, M. Semczuk, J. Van Dongen andK. W. Madison, “Giant Feshbach resonances in 6Li-85Rb mixtures.”, Phys.Rev. A Rapid Communications, 82, 020701, August (2010) [25].2. M. Semczuk, X. Li, Xuan, W. Gunton, M. Haw, N. S. Dattani, J. Witz, A.K. Mills, D. J. Jones and K. W. Madison, “High-resolution photoassocia-tion spectroscopy of the 6Li2 13Σ+g state.”, Phys. Rev. A, 87, 052505, May(2013) [49].3. W. Gunton, M. Semczuk and K. W. Madison, “Realization of BEC-BCS-crossover physics in a compact oven-loaded magneto-optic-trap apparatus.”,Phys. Rev. A, 88, 023624, August (2013) [50].4. W. Gunton, M. Semczuk, N. S. Dattani and K. W. Madison, ”High-resolu-tion photoassociation spectroscopy of the 6Li2 A(11Σ+u ) state.”, Phys. Rev.A 88, 062510, December (2013) [51].5. M. Semczuk, W. Gunton, W. Bowden and K. W. Madison, “Anomalousbehavior of dark states in quantum gases of 6Li.”, Phys. Rev. Lett. 113,055302, July (2014) [52].1.4 Outline of the thesisChapter 2 of this work gives a description of the technical aspects of the experi-mental setup. All the essential elements of the vacuum system and the laser setupsfor cooling, trapping and photoassociation are discussed. The field of ultracoldatoms is mature enough to justify treatment of certain topics as established knowl-edge, therefore the theory of magneto-optical trapping, dipole traps and, in general,atom-light interaction which can be easily found in textbooks (e.g. [53]) is kept tominimum. The focus is on solutions that are unique to the approach taken by ourgroup or are essential to the proper interpretation of the experimental results.7Chapter 3 presents the results of the single color photoassociation spectroscopyof vibrational levels v = 20−26 and v = 29−35 in c(13Σ+g ) and A(11Σ+u ) excitedpotentials of 6Li, respectively. The experiments performed in a sub-µK Fermi gasconfined in a dipole trap at 0 G magnetic field report ro-vibrational levels that havenever been observed before resulting in a significant improvement of the modelsdescribing the c(13Σ+g ) and A(11Σ+u ) potentials. Building on the idea of s-waveFeshbach resonance enhancement of photoassociation [54] we show a similar en-hancement for the c-state spectroscopy to excited levels with N′ = 0,2 in the vicin-ity of the p-wave Feshbach resonance at 185 G. Furthermore, we discuss systematicuncertainties introduced by the residual magnetic fields, photoassociation laser andthe dipole trap. Unexpected splitting of the v = 30 level in the A-state potential ispresented and is attributed to the dipole trap induced coupling to an (unidentified)excited level when the trapping light intensity is high enough.Chapter 4 reports on the first ever observation of dark states in degenerate gasesof fermionic 6Li. Such states have been reported in the past in experiments withbosonic species and Bose-Fermi mixture of KRb but there is no prior work in Fermigases. Exceedingly long lifetimes of the dark states (as defined by the revivalheight decay time for extended exposure to the Raman lasers) are observed on thetwo photon resonance and are orders of magnitude larger than what would naivelybe expected given the relative phase coherence of the photoassociation lasers andother possible sources of decoherence. In the BEC-BCS crossover regime, close tothe B = 832.18 G Feshbach resonance, the dark states are produced in a partiallyBose-condensed gas of Feshbach molecules (below the resonance) and in a paireddegenerate Fermi gas (above the resonance). For magnetic fields below 829 G therevival height on the two-photon resonance unexpectedly and abruptly changes,providing so far the only observable in the BEC-BCS regime that does not exhibita monotonic behavior.In order to demonstrate the feasibility of dark states for precision spectroscopythe binding energies of the least bound vibrational levels in X(11Σ+g ) and a(13Σ+u )potentials are measured with an accuracy as high as 21 kHz. The accuracy of thelevel v = 9 in the triplet potential is improved 500-fold and its molecular hyperfinestructure is resolved. The v = 38 level in the singlet potential has not been observedat 0 G prior to our experiments therefore together with v = 9 it provides an input8for further refinement of the lowest lying molecular potentials of 6Li2 and, as aconsequence, could be used to improve the predictions for the singlet and tripletbackground scattering lengths of two component 6Li mixture. For experimentswith dark states at non-zero magnetic fields the field free location of the vibrationallevel v = 37 in the singlet potential has been also measured, providing the laserfrequencies needed for dark-state studies of the BEC-BCS crossover regime.Chapter 5 summarizes the results of the thesis and briefly outlines the currentstatus of the experiment. The impact this thesis could have on the field of ultracoldatoms and molecules is discussed with the focus on the experiments that build onthe methods and results that are the subject of this work.9Chapter 2Experimental setupThe main results of this Chapter have been published in• W. Gunton, M. Semczuk, and K. W. Madison, “Realization of BEC-BCS-crossover physics in a compact oven-loaded magneto-optic-trap appara-tus.”, Phys. Rev. A, 88, 023624, August (2013) [50].Even though the field of ultracold quantum gases is very mature there is nooff the shelf device that would allow production of degenerate quantum gases ondemand, satisfying the requirements of different research groups. There are certaincommon characteristics of the experimental setups built around the world but in theend each one needs to be customized depending on the specific goals of the plannedresearch program and the atomic species used. This significantly increases both thetime it takes to produce scientific results and the start up costs for every new projectin the field. In order to simplify the vacuum part of these setups the magneto-optical traps (MOTs) of alkaline atoms can be loaded directly from the backgroundgas as has been demonstrated for Cs [55], Na [56], K [57] and Rb [58] or fromthe effusive oven as shown for Li in [24, 59, 60]. This path is, however, rarelychosen for experiments with degenerate gases, mainly because the background gascollisions reduce the lifetime of the trapped samples.This Chapter describes the details of our experimental setup, with the emphasison these that are unique to the approach taken by our research group. The fermioniclithium source (an oven) loads the magneto-optical trap without being separatedfrom the trapping region with a differential pumping stage. This approach, contraryto previous beliefs, allows efficient creation of molecular Bose-Einstein conden-sates and BCS pairs, and is comparable in performance (atom number and lifetimeof the degenerate sample) to systems where the lithium MOT is loaded from a slowatom sources like a Zeeman slower [61] or a 2D MOT [62]. It results in a compact,10relatively inexpensive vacuum setup that can be easily miniaturized.2.1 Vacuum system and magnetic trappingThe loss induced by the interactions of ultracold atoms with the background gasin vacuum needs to be negligible on the time scales relevant to the experiments.For lithium, this condition can be satisfied for vacuum pressures on the order of10−10 mbar, higher than 10−11 mbar required in early experiments with Bose-Einstein condensates [1]. The difference is caused by the rapid evaporation pos-sible in optical dipole traps together with the high elastic collision rate enhancedwith a help of a Feshbach resonance. These two factors decrease the time of an ex-perimental cycle thus putting less stringent requirements on the quality of vacuum.The MOT trapping region is centered within an optically polished quartz cellbonded on both ends to glass to metal transitions, as shown in Fig. 2.1. The choiceof the cell’s material is mainly dictated by the quartz’s low absorption of light at thewavelength used for cooling and trapping of lithium. This minimizes the thermallensing effects that are often present when dipole traps are created with tightlyfocused, high power lasers. The atomic source is introduced on one end and issupported inside the cell by a 2.75 inch conflat electrical feedthrough. The otherend of the cell is connected through a stainless steel bellows and 6 inch conflat crossto a second, stainless steel chamber which serves as a port for a 20 L s−1 VarianStarCell ion pump and a SAES CapaciTorr NEG pump. The outer dimensions ofthe quartz cell are 30 mm × 30 mm × 100 mm with a 5 mm wall thickness.The atoms are confined by a quadrupole magnetic field created by a pair ofwater cooled coils enclosed in a PVC housing (Fig. 2.2). The same coils, set tothe Helmholtz configuration by reversing the current in one of the coils, are alsoused to create homogeneous magnetic fields up to 950 G. Cooling is achieved byrunning tap water at about 10 ◦C. The details of the design can be found in [63].At low dipole trap powers gravity distorts the trapping potential leading toatoms spill out. This effect can be enhanced by small inhomogeneities of the mag-netic field produced by the MOT coils in the Helmholtz configuration. To minimizesuch losses an additional coil was placed inside the top MOT coil to create a mag-netic field gradient. This coil was essential for creation of quantum degenerate11Figure 2.1: Drawing of the vacuum system showing the effusive atomicsource, the metal shield used to prevent Li from coating the inside ofthe quartz cell, and the MOT position, (inset) 3D model of the oven.Figure taken from [50].samples.Lithium ovenThe effusive lithium oven is located 10 cm from the trapping region in a configu-ration similar to that reported in Ref. [59] with the exception that here the trappinglight for lithium is single frequency and is not broadened in any way to enhancethe loading of the MOT. A small beam shield (see Fig. 2.1) is situated betweenthe oven and the trapping region in order to shield the quartz cell walls from thedirect output of the effusive oven. The lithium MOT captures atoms from the lowvelocity tail of the effusive oven output without any additional slowing stages.The lithium oven consists of a cylindrical reservoir (15 mm long, 7 mm innerdiameter, 1 mm wall thickness) enclosed by a screw cap with a 1 mm hole (Fig. 2.1,inset). It is made of non-magnetic stainless steel and the supporting element that isattached to the cap is made of a non-magnetic alloy of nickel and chromium (80%nickel and 20% chromium). It provides a mechanical support and electrical contactbetween the feedthru electrodes. The thin profile of the leads (1 mm) providesohmic heating for the atomic reservoir when current is applied. To ensure that hotlithium leaves only through the hole in the cap both the cap and the reservoir have12Figure 2.2: View of the vacuum system. (a) NEG pump, (b) ion pump, (c)Helmholtz (compensation) coils, (d) dipole trap optics, (e) rubidiumdispenser connectors, (f) black box housing the impedance matchingcircuit for the rf antenna, (g) MOT coils, (h) oven connectors, (i) glasscell.13Figure 2.3: Different views of the stainless steel press designed to create alithium rod by pushing chunks of lithium through a hole. The diameterof the hole matches the required diameter of the extruded lithium rodthat is put into the oven. A slot for a 13/16 wrench was cut to preventthe press from rotating when the piston is screwed in.tapered threads commonly used in plumbing to connect pipes. This is a significantimprovement over our previous design where the connection between the reservoirand the cap was not leak-tight and lithium vapor was found to escape from the sidesof the oven [24], effectively reducing the useful lifetime of the source.Achieving a good vacuum with a hot atomic oven less than 10 cm from theMOT requires careful preparation of the enriched (95% 6Li) sample and systematicdegassing of the oven. A nickel wire mesh acts as a wick preventing the moltenlithium from draining out of the hot oven. The mesh was rolled onto a 4 mm rodto create a cylindrical roll 10 mm long and with an outer diameter of 7 mm andwas then put inside the reservoir. The oven was cleaned with acetone and bakedin air at about 300◦C to remove contaminants from the machining process. Invacuum equilibrium temperature of the fully assembeled oven (without lithium)was measured as a function of current (Fig. 2.4 and it exhibits a linear dependence:T [◦C] = 46.5◦C/A× I−93.8◦C. (2.1)The initial sample of lithium was cleaned with petroleum ether to remove residual14Figure 2.4: The temperature calibration of the oven performed in a test cham-ber at 10−6 Torr. The oven is fully assembled but there is no lithiumpresent.oil from the surface in an argon filled glove-box. A clean razor blade was usedto remove the black exterior from all sides of the lithium sample. The designof the glove-box makes it hard to handle small elements (in this case the lithiumsample) therefore to simplify the reservoir loading process and maximize the fillingof the oven, the lithium sample was put in a home-built press that produced 10 mmlong cylindrical extrusion with a 4 mm diameter (Fig. 2.3). The extrusion was putinto the center of the wire mesh roll and covered on the top with an additionalround piece of nickel mesh. It is crucial to perform all these steps in an inert gasatmosphere to limit exposure of lithium to air thus limiting the contamination ofthe sample. In particular, we found that exposure to oxygen, water, or nitrogen wasparticularly problematic producing an oxide or LiN coating on the exposed metal.The full lithium oven was degassed by heating it to 200 ◦C within an auxiliarypreparation chamber such that the total pressure from outgassing never exceeded10−6 Torr. The treatment removed the majority of the trapped nitrogen, oxygen,and hydrogen. These were the primary outgassed contaminants as identified by aresidual gas analyzer (RGA). The oven was then heated until the pressure due tooutgassing at a current of 10 A (corresponding to an oven temperature of 640 K)15fell below the base pressure of the preparation chamber (5× 10−8 Torr; 10 l s−1pumping speed). After degassing, the preparation chamber was flushed with argon(the use of dry nitrogen was avoided since it was found to contaminate the lithium)and the oven was quickly moved to the experimental vacuum chamber where a6-day bakeout at 200 ◦C was performed.2.2 Trapping, imaging and state preparation2.2.1 Magneto-optical trapping laser systemThe laser system used to cool and image 6Li atoms, shown in Fig. 2.5, is builton a dedicated optical table and the required light is sent to the experiment via50 m long optical fibers, thus decoupling the light production from the possibleunwanted feedbacks from the experiment itself. 6Li is not an ideal 2-level system,therefore cooling requires the presence of two laser fields: “pump” light (near the2s1/2, F = 3/2 → 2p3/2, F = 5/2 transition frequency) and the “repump” light(near the 2s1/2, F = 1/2→ 2p3/2, F = 3/2 transition frequency). In the 2p3/2excited state, the hyperfine level spacings are so small that the pump light, which istypically tuned to the red of the transition by several linewidths, excites with similarrates all three excited states causing rapid depletion of the upper, F = 3/2, groundstate due to optical pumping. To mitigate this effect, the repump light must havea similar scattering rate as the pump light and it therefore contributes to coolingand exerts a force on atoms in the MOT comparable to that exerted by the pumplight. Consequently, both the pump and repump light must be introduced into thetrap along all three directions with similar intensities for the proper operation ofthe lithium MOT.16Figure 2.5: Trapping light production and distribution system. The shaded region hosts elements of the high fieldimaging setup. PD1,PD2: photodiodes; STA: laser diode seeding tapered amplifiers TA20 and TA21; DL670:master laser Toptica DL670; SH: laser diode injection locked to a high field imaging master laser MH; FC1in(out):single mode, polarization maintaining (SM-PM) fiber delivering light for seeding TA20 and TA21; FC2(4)in:diagnostics of STA(SH), connected to a fiber splitter; FC2f−s: output of a fiber splitter, sends light from FC2(4)into the F-P cavity; FC3in: fiber for creating a beatnote between DL670 and MH; FC5in: SM-PM 50 m long highfield imaging fiber; FC6(7)in: SM-PM 50 m long repump(pump) light fiber.17The master laser (Toptica DL Pro) uses as a reference the signal derived froma Doppler-free saturated absorption spectroscopy of lithium vapor performed in aheat pipe and is locked 50 MHz above the pump transition. 2.6 mW of this stablelight is used to injection lock a Mitsubishi ML101J27 laser diode (STA). Thesediodes emit 120 mW 660 nm light at 200 mA and a temperature of 25 ◦C butheating them to 72 ◦C changes their free running wavelength to 670-671 nm andreduces the output power to 60 mW at 314 mA. The diode is always kept on and weobserve that over time the output power decreases therefore to keep it at the 60 mWlevel the current needs to be increased periodically, usually until it reaches about450 mA. This, however, still allows us to use one diode for over a year. Turningthe diode off every time it is not in use might prolong its lifetime even further.The cooling light is derived from two tapered amplifiers (TA) from SacherLasertechnik which are seeded with 14 mW of light from the laser diode STA.Typically, the output power of each TA is around 450 mW. The beam quality varieswith the alignment of the seeding beams and it has been observed that the optimizedpower does not necessarily translates into the most efficient coupling into fibersFC6in and FC7in (see Fig. 2.5). To reach required frequencies for the atom trappingthe light seeding the pump TA is up-shifted by 108 MHz and then the output ofthe TA is down-shifted with an acousto-optic modulator (AOM) in a double passconfiguration. The output of the repump TA is up-shifted with a double pass AOM.Finally, the pump and the repump light are each coupled into a separate, 50 mlong, single mode, polarization maintaining fiber and sent to the experimental table.80 mW of pump and 60 mW of repump light is available for experiments.On the experimental table the pump and repump light for trapping is combinedinto a single beam and, by way of several polarization beam splitter cubes, is splitinto four beams. Three beams are then expanded to a 1/e2 diameter of 2.5 cm,and introduced into the MOT cell along three mutually orthogonal axes in a retro-reflection configuration. It maximizes the optical power available for trapping andso far it has proven to perform better than a six independent beam configurationtested in the early versions of this experiment. The fourth (slowing) beam is sentcounter-propagating to the atomic beam, and enters the cell through a viewport atthe opposite end of the chamber from the oven. We do not observe any coatingof the window by lithium, an issue encountered in many experiments involving18Zeeman slowers. This is primarily because the oven output is not an intense, col-limated atomic beam and because the distance from the source to the viewport islarge (on the order of 1 m). Effectively, our approach creates a 7-beam MOT, wherethe slowing beam increases the capture velocity of the MOT along the atomic beamaxis, leading to about two-fold improvement in the final atom number.The main observable in ultracold atom experiments is the atom number mea-sured by absorption imaging. The absorption probe beam uses pump light whilea counter-propagating repump beam is included to provide hyperfine repumpingwith a radiation pressure opposing the probe beam to limit the acceleration of theatoms during imaging.2.2.2 High field imaging laser systemDue to the limitations of our existing current driver, turning off the magnetic field toimage the trapped sample is a relatively slow process often introducing additionalatom loss. It also erases any information about the spatial properties of the trappedcloud at the fields where the experiments are performed. In order to mitigate theseissues both the experiment and the imaging can take place at the same (or nearby)magnetic field but because the differential Zeeman shift of 22S1/2 and 22P3/2 (seeFig. 2.6) is larger than the typical tuning range (few tens of MHz) of the pump andrepump light discussed in section 2.2.1 a separate imaging laser system is needed.At magnetic fields above a few tens of Gauss, the atomic hyperfine coupling isgreatly suppressed and the optical transitions used for imaging atomic states be-come closed and no repump light is required. Only one frequency is necessary andthis light is produced by an in-house-built extended cavity diode laser (based onthe Roithner RLT6720MG diode laser) that is locked with a large frequency offsetto the master laser (Fig. 2.7 shows the working principle of the offset frequencylock). The output is then amplified by injection locking a Mitsubishi ML101J27diode laser, and the output beam’s amplitude and fine frequency tuning is providedby an AOM in a double pass configuration (see Fig. 2.5).Typically, the imaging light is detuned 600–1000 MHz below the cooling tran-sition with the frequency offset lock and then precisely tuned on resonance withthe AOM. The double pass AOM also acts as a shutter and allows us to change the19Figure 2.6: Magnetic field dependent detuning of the imaging light from thecooling transition enabling imaging of each of the three high-field seek-ing states (called |1〉, |2〉 and |3〉). Due to the Zeeman splitting it ispossible to independently address these three states by driving opticaltransitions to m j = −1/2 (upper branch) or m j = −3/2 (lower branch)in the 22P3/2 level. The inset shows the range used for experiments inthe BEC-BCS crossover.beam frequency to image either of the two lowest spin states which are separatedby approximately 76 MHz (for fields 600–900 G, see Fig. 2.13). It can also beused to apply short light pulses for spin-selective removal of atoms. The beam isfiber coupled and combined with MOT pump light for imaging at low magneticfields using a fiber-based combiner (Evanescent Optics Inc.). The imaging axis isperpendicular to the magnetic field, and in this arrangement it is impossible to po-larize the light such that it drives only either σ+ or σ− transitions. To optimize theabsorption signal, the polarization of the imaging beam is set to be linear and alongthe magnetic field axis. This leads to the underestimation of the atom number ina particular spin state by a factor of two as the imaging light is a mixture of bothleft and right circular polarizations and only one of these polarizations drives the20Figure 2.7: A diagram of the frequency offset lock. See the text for the ex-planation of the working principle of the circuit. A more detailed dis-cussion of a circuit of this type can be found in e.g. [64].transitions of interest.Down to 200 G below the wide (300 G) resonance at B= 832.18 G it is possibleto create Feshbach molecules of 6Li. They are sufficiently loosely bound that it ispossible to image them with standard absorption imaging [65]. The binding energyis so small that it introduces a negligible frequency shift from the atomic transition,allowing the molecule’s atomic constituents to be imaged separately. As with freeatoms the absorption image of molecules underestimates the molecule number bya factor of two.Setting the high field imaging frequencyLet’s call δ the detuning of the high field imaging light from the cooling transition(frequency fcool), as shown in Fig. 2.6. The frequency fH of the high field imagingmaster is fH = fL± fbeat, where fL = fcool + 50 MHz is the frequency of TopticaDL670 and fbeat is the beatnote between the two master lasers. The double passAOM up-shifts fH by 2 fAOM = 130−270 MHz. The frequency fAOM of the double21Figure 2.8: The oscilloscope trace showing the error signal produced by thefrequency offset circuit. The horizontal axis shows the frequency dif-ference between the high field imaging laser and Toptica master laser.(Top): full scanning range of the high field imaging master laser. Theerror signal is symmetric and the shaded area corresponds to the fre-quencies that are lower than the frequency of the Toptica DL670 masterlaser. (Bottom): decreasing the scanning range zooms into the shadedarea. The high field imaging laser is locked to the highlighted slope(∼ 5.1 mV/MHz).pass AOM can be changed over 65− 135 MHz range, more than is required toimage both |1〉 and |2〉 states without changing fH . As a result, the beatnote fbeatbetween two masters required to achieve detuning δ is:fbeat = |δ −2 fAOM−50MHz| (2.2)The beams derived from the Toptica master laser and the high field imagingmaster laser are overlapped and then coupled into a common fiber. The output ofabout 150 µW per beam is directed into a photodiode which records the beatnote δ .22After amplification the beatnote is mixed with the output of the voltage-controlledoscillator. The signal is split into two equal parts which are recombined with afrequency mixer after one part has been delayed by ≈ 2 m of coaxial fiber. Theoutput of the mixer at twice the frequency is blocked by a low pass filter with acutoff frequency of 1.9 MHz. The resulting output voltage (Fig. 2.8), all else keptconstant, is a function of the beatnote frequency δ . The feature spanning fromabout -200 MHz to +200 MHz is due to the imperfections of the frequency mixerthat result in the transmission of the rf signal at low frequencies δ . The signalis cut off at 200 MHz by the low pass filter, but within this range it creates anerror-like feature proportional to cosΦ = cos(2piδ/Ω0), analogous to the desiredcosΦ= cos(2pi(δ − f )/Ω0) shown in Fig. 2.7.To create a high quality error signal, the frequency of the local oscillator needsto be set to fVCO = fbeat +92MHz 1 and the slope (∼ 5.1 mV/MHz) highlighted inFig. 2.8(bottom) should be used as a reference for the locking circuit. The choiceof this specific slope assures that the desired imaging frequencies can be achievedwith the frequency shift 2 fAOM introduced by the AOM. With a different AOManother slope would perform equally well. The part of the signal to the right of itssymmetry axis (Fig. 2.8(top)) corresponds to fH < fL such that the excited leveladdressed by the high field imaging light is m j =−3/2 in 2p3/2 state. The choiceof this level (instead of m j =−1/2 in the 2p3/2 state, which is less sensitive to themagnetic field 2) allows us to use the high field imaging laser for a coarse calibra-tion of the magnetic field. Once the fbeat is decided on, the imaging frequency canbe changed from the control software level only by changing the frequency fAOM.2.2.3 Dipole trapsThe temperature of atoms confined in a MOT is limited by the scattering of photonsfrom the cooling beams. Further decrease of the sample’s temperature required toreach quantum degeneracy is achieved with evaporative cooling, which for 6Liis typically done in optical dipole traps, where the elastic collision rate can bewidely tuned thanks to the presence of a wide Feshbach resonance at B = 832.18 Gbetween states |1〉 and |2〉 [66].1τ = (92MHz)−1 is related to the delay introduced by the 2 m long delay line shown in Fig. 2.7.2The magnetic dipole moment of m j =−3/2 is −2µ0B/h vs. −2/3µ0B/h for m j =−1/2.23The off-resonance scattering of photonsΓsc =3pic22h¯ω30(Γ∆)2I (2.3)and associated heating of the sample by the dipole traps is not an issue here as thelasers used in this work are detuned by more than ∆∼ 2pi×165 THz from both theD1 and D2 transitions, with transition frequencies ωD10 = 2pi × 446.789634 THzand ωD20 = 2pi × 446.799677 THz, respectively [67]. Γ = 2pi × 5.872 MHz is thenatural linewidth of the transitions [67] and I is the intensity of the trapping lightexperienced by atoms.For typical experimental conditions during photoassociation the total power(intensity) of the 1064 nm dipole trap rarely exceeds 1 W (30 kW/cm2), corre-sponding to the upper bound on the photon scattering rate of 0.14s−1. Only duringthe transfer of the atoms from the MOT into the high power dipole trap (1090 nm,total 200 W power and 6 MW/cm2 intensity) and subsequent fast evaporation (over-all some tens of ms) the photon scattering rate reaches 13s−1.The light for the crossed optical dipole trap (referred to as CODT) is derivedfrom a multi-longitudinal mode, 100 W fiber laser (SPI Lasers, SP-100C-0013)with a central wavelength 1090 nm and a spectral width exceeding 1 nm. TheCODT is comprised of two nearly co-propagating beams crossing at an angle of14◦ (Fig. 2.9). Recycling the first beam results in the total available power of200 W. The beams are focused to a waist (1/e2 intensity radius) of 42 µm (first)and 49 µm (recycled) which produces a 6 mK deep trap. The output power of thisSPI laser is varied by an analog input controlling the pump diode current of thepost amplifier stage.We limit the time the CODT operates at high power to mitigate the thermallensing effects which produce beam aberrations. With a quartz cell the position ofthe beam waists of the CODT arms moves at approximately 780 µm/s when thehigh power CODT is initially turned on with 100 W per beam 3. These thermaleffects were worse with a previously used vacuum cell made from borosilicateglass. Quartz has a higher transmission in the near IR (e.g. at 1090 nm), and the3The beam focus is observed to move 395 µm in the first 500 ms and 280 µm in the next 500 msdue to thermal lensing24Figure 2.9: The arrangement of the dipole traps used in current experiments.The low and the high power dipole traps are overlapped using a dichroicmirror BS2 from Semrock that transmits 1090 nm and reflects 1064 nmand 780 nm. The photoassociation light and the low power dipole trapare overlapped with a dichroic mirror BS1. This arrangement optimizesthe transfer between the dipole traps and provides superior stability overapproaches used previously. The dashed line is the part of the path thatis moved up to go above the cell.thermal expansion coefficient of quartz is ten times smaller than that of borosilicateglass.After initial pre-evaporation in the high power CODT atoms are transferredinto a lower-power, 1 mK deep CODT. The light for this second CODT is gener-ated by a single frequency, narrow-linewidth (< 10 kHz), 20 W fiber laser operatingat 1064 nm (IPG Photonics YLR-20-1064-LP-SF). This transfer is done to avoidensemble heating observed in the SPI laser CODT and allows further forced evap-orative cooling to much lower powers and thus ensemble temperatures. The 26 nmwavelength difference between the trapping lasers allows overlap of the low-powertrap with the SPI on a dichroic mirror (Semrock) that reflects 780 nm and 1064 nmand transmits 1090 nm. With this approach we assure good spatial overlap be-25tween the traps that results in improved transfer efficiency. Most importantly, thisapproach offers superior stability and minimizes the misalignment of the traps.Both IPG beams have the same frequency and are focused to waists similar tothat of the SPI (1/e2 intensity radius, 42 µm and 49 µm). The polarization of thefirst beam is chosen to lie in the horizontal plane to minimize reflections at the cell(the cell uncoated). The polarization of the recycled beam deviates from horizontalby a small degree. This results from the beam path the beam travels from one sideof the cell to the other and will be addressed in the planned upgrade of the setup.This small relative angle between polarizations of both beams does not limit theperformance of the trap.The IPG CODT beam power is controlled by a water cooled AOM (Intraac-tion ATD-1153DA6M). The diffraction efficiency of the AOM is 60%. Due to theexperimental constraints the polarization of the IPG CODT is perpendicular to thepolarization that, according to the manufacturer of the AOM, would lead to thediffraction efficiency of 75%. It is worth noting that the power of neither the SPInor the IPG laser is actively stabilized.Trapping potential and trap frequenciesThe trapping potential is created by two Gaussian beams crossed in the X −Yplane at an angle α (here ∼ 14◦) [Fig. 2.10]. However, if the thermal energy kBTof an atomic ensemble is much smaller than the potential depth U0, the trappingpotential experienced by a particle with a mass m can be well approximated by thethree dimensional potential of a harmonic oscillator [68]U ≈−U0 +12mν2x x2 +12mν2y y2 +12mν2z z2, (2.4)26Figure 2.10: Two crossed Gaussian beams creating the optical dipole trap.The elliptical shape in the crossing of the beams corresponds to theatomic cloud. The trapping potential used in the experiments resem-bles an oblate spheroid (ellipse rotated about one of its axes).with trap frequenciesνx =√4U0mw20(2.5)νy =√4U0mw20(α2)2=α2νx (2.6)νz =√4U0mw20(2.7)Here we assumed that both beams have the same waists w0 and powers, and α/2 isa small angle expressed in radians. U0 is related to the total power P of the crossedbeams and their waists,U0 =3c2Pω3671w20Γ∆(2.8)where ω671 = 2pi × 446.789634 THz and Γ = 2pi × 5.9 MHz are the frequencyand the natural linewidth of 6Li D1 line. ∆ = 2pi×165 THz is the detuning of the1064 nm dipole trap from the transition ω671.The frequencies of the IPG crossed dipole trap were determined by measuringthe trapped atom number after parametric heating. The best quality of the measured27signal was achieved in a four step excitation process. (1) The sample was cooledto an IPG power of 0.7 W which was then (2) increased adiabatically (in 1 s) to thedesired power. It was verified that this step did not change the T/TF ratio. (3) TheRF power of the water cooled AOM was then modulated for a fixed time, mappingonto power modulation of the dipole trap. (4) Finally, the dipole trap power wasturned back down to 0.7 W in 50 ms in order to convert heated atoms into lostatoms. The atom number was measured as a function of modulation frequency,with a loss showing up at twice the trap resonance (for reference see e.g. [69]).Two strong features were present in the spectrum, a low and high frequency onecorresponding to νy and νx ≈ νz, respectively. Assuming the trap is approximatelyharmonic, the frequency ν scales as a square root of the laser power P. Fittingν = A√P yields trapping frequencies:νy = 0.13kHz×√P/Watt (2.9)νx = νz = 1.9kHz×√P/Watt (2.10)Transition to degeneracyAs the sample confined in the dipole trap is cooled, its quantum nature becomesmore and more relevant. There are two important cases to consider when workingwith ultracold 6Li: (a) creation of a Bose-Einstein condensate of loosely boundFeshbach molecules and (b) creation of a degenerate Fermi gas. Case (a) is relevantfor magnetic fields below the Feshbach resonance at B = 832.18 G. When thephase space density n0λ 3dB = n0[2pi h¯2/(mkBT )]3/2≈ 2.61 the sample undergoes aquantum phase transition that occurs at the temperature TCTC = 0.94h¯ω¯kBN1/3mol . (2.11)The condensed fraction for T < TC isN0Nmol= 1−(TTC)3. (2.12)28In the above equations n0 is the density at the trap center, N0 is the population inthe BEC phase, Nmol is the number of noncondensed molecules, λdB is de Brogliewavelength, and ω¯ = (ωxωyωz)1/3 (where ωi = 2piνi). The above relations aretrue for an ideal (noninteracting) gas but even for real systems they are very goodapproximations.For the negative scattering lengths (attractive interactions) present near B= 0 Gand at magnetic fields larger than B = 832.18 G the case (b) takes place. The figureof merit here is the Fermi temperature TF which for the spin polarized sample of Natoms isTF =h¯ω¯kB(6N)1/3 (2.13)When the temperature of the sample drops below TF, the Fermi gas starts to exhibitquantum properties. For example, the average energy per particle starts to deviatefrom 3kBT expected from a classical gas [70] and the blocking of the elastic colli-sions due to the Pauli exclusion principle leads to the decrease in the evaporationefficiency of a two-component Fermi gas [66, 70, 71].If a two-component Fermi gas (with N atoms per spin state) at a temperatureT ≤ 0.5TF is converted into a gas of N bosonic molecules by adiabatically changingthe interactions to repulsive in the vicinity of a Feshbach resonance [72], the result-ing bosonic molecules will be below the BEC transition temperature, TC (Eq. 2.11).Additionally, when T is on the order of 0.5TF, the experimental signatures of thequantum nature of a Fermi gas can be more reliably observed (as in [70]). There-fore, in the field of degenerate Fermi gases T/TF ∼ 0.5 is treated as a limit thatneeds to be reached in order to consider the sample to be quantum degenerate 4.Previous dipole trap arrangementsIt became clear very early that efficient photoassociation of lithium in a single armdipole trap would be inefficient because the density of sample was not high enough.A cross trap was therefore necessary and we tested two approaches before arrivingat the solution described in the previous section. Due to the timing of the dipoletrap changes it happened that the data for each of the single color photoassociation4This is, by no means, a strict definition but more of a ”rule of thumb”, useful from the experi-mental point of view.29papers we published [49, 50] were taken with a different dipole trap arrangement,both shown in Fig. 2.11. For the sake of completeness both, now obsolete, ap-proaches are discussed.• in order to avoid the creation of a lattice the cross beam was derived fromthe 0-th order of the main IPG AOM and then down-shifted by 100 MHz(Fig. 2.11a). As a result, the frequency difference between the main and thecross beam was 190 MHz. This solution allowed us to independently controlthe power of the cross beam and thanks to the use of a fiber the necessaryday-to-day alignment was reduced. The main drawback of this solution wasthe loss of power due to the low efficiency (about 50%) of the AOM. Thefiber was a standard 1064 nm fiber from Thorlabs (P3-1064PM-FC-2) and itworked well with incident powers as high as 10 W, maintaining the couplingefficiency of 70% - even though it was not specified for high power aplica-tions. In this arrangement the photoassociation light was overlapped onlywith the main dipole trap arm. The section that utilizes the 0-th order of themain AOM will be used in the future to create a 1-D lattice and the fiber willguarantee high quality of the lattice beams.• the independent control of the cross beam power turned out not to provideany additional advantage and the best evaporation efficiency was actuallyachieved when the powers of both the main and the cross beam were bal-anced. This led to an alternative dipole trap design shown in Fig. 2.11b)where the IPG beam was recycled after passing through the cell. The re-cycled beam had the same polarization and is focused to approximately thesame size as the main beam. The main advantage of this design is that thephotoassociation beam is also recycled, doubling the power available forexperiments – an important feature in a situation when there is no reliabletheoretical predictions or previous experimental data that would lead to esti-mations of the power necessary for STIRAP of 6Li. Unexpected heating andloss was observed when the polarization of the second beam was fixed per-pendicular to the first (i.e. perpendicular to the horizontal plane and parallelto the magnetic field axis).30Figure 2.11: Dipole trap arrangements used in previous iterations of the ex-periment. a) the cross beam of the low power dipole trap is indepen-dently controlled with an additional AOM and is derived from the 0-thorder of the main IPG AOM. It is sent to the setup via fiber and en-ters the cell with the same polarization as the main beam. The pho-toassociation light is overlapped only with the main IPG beam. Thisarrangement was used to obtain results published in [49], b) the mainbeam of the low power dipole trap is recycled and both arms have thesame polarization. The photoassociation light follows the entire IPGpath. This arrangement was used to obtain results published in [51] aswell as for the production in the first molecular BEC in Canada [50].The dashed line corresponds to the recycled beam that is raised abovethe glass cell.31Figure 2.12: a) Magnetic field lines of the one loop antenna used for rf spec-troscopy. The rf field shows some degree of inhomogeneity but farfrom the antenna, where atoms (large, red dot) are located. Bres isthe uncompensated residual magnetic field. b) Rf transitions betweenZeeman sublevels of the 22S1/2 level relevant to the magnetic field can-cellation described in section 2.3.4. |1〉 to |6〉 at ∼ 0 G correspond to|F,mF〉 states.2.2.4 RF antenna for spin manipulationRadio frequency used for driving transitions between atomic or molecular levelsproved to be an effective technique for spectroscopy and spin state manipulation(e.g. [73–75]) in ultracold samples. Notably, an “rf knife” [76] was used to evap-oratively cool magnetically trapped atoms, an essential step in creating first Bose-Einstein condensates of rubidium [1] and sodium [2].There are two relevant frequency ranges for the spin state manipulation in 6Li.One is centered around the atomic hyperfine splitting, 228.205 MHz [77], and isused to drive transitions between hyperfine levels F = 1/2 and F = 3/2 of the22S1/2 level. Depending on the orientation of the residual magnetic field Bres withrespect to the field generated by the antenna, σ−, σ+ or pi transitions can be driven32Figure 2.13: Frequency difference between magnetic sublevels within the22S1/2 level for magnetic fields relevant to the rf spectroscopy of6Li inthe BEC-BCS crossover regime. The numerical calculations are donebased on [67, 77].as shown in figure 2.12(b). It is important to note that no matter what is the relativeorientation of the rf radiation and Bres there is always a transition available that issensitive to the magnetic field 5. This property can be used to probe the residualmagnetic field enabling magnetic field cancellation (see section 2.3.4).Magnetic field splits F = 1/2 and F = 3/2 into Zeeman sublevels, mj, thus it isalso possible to drive rf transitions between various mj’s, especially the lowest threewhich are separated by 76-84 MHz in the BEC-BCS crossover regime (Fig. 2.13).These transitions can be used to calibrate the magnetic field but, more importantly,rf provides a diagnostic tool for spectroscopic studies of Feshbach molecules [78]and BCS pairs [74].The rf signal is derived from a direct digital synthesizer (DDS) driven by an5For example, if the rf field is pointing along the residual magnetic field it induces pi transitions.If, however, this transition is insensitive to the magnetic field then the rf spectroscopy of this levelwould not show any field dependent behavior.33Figure 2.14: The design of the impedance matching circuit of the antennaused for driving rf transitions in 6Li. The dc voltages correspondingto “frequency” and “impedance” are used to change the resonant fre-quency and the impedance of the circuit, respectively. Their values areset between 0 V and 10 V, depending on the rf frequency required bythe experiments.atomic clock frequency reference at 10 MHz, frequency doubled and then ampli-fied to about 1 W. The spin manipulation is performed with a single loop antenna(R≈ 2 cm) located 10 cm away from the trapped sample (see Fig. 2.2). An elec-tronic circuit 2.14 is used to tune the resonance frequency (50 MHz to 250 MHz)as well as the impedance of the antenna coupled to the rf source, such that the re-flections of the rf signal are minimized over the range of frequencies relevant tothe experiments with 6Li. Such reflections result in a lower power emitted by theantenna and in principle could damage the rf source. The design is inspired byan impedance matching circuit used in experiments where rf was used to associateEfimov trimers [79]2.2.5 Photoassociation laser systemThe high resolution photoassociation spectroscopy in the ultracold regime requireslasers systems that allow users to determine the frequency of measured transitionswith sub-linewidth accuracy and precision. In alkali systems it usually means thatthe laser frequency needs to be known to within few MHz. This kind of accuracy israrely achievable by direct measurements of the light frequency with wavemeters,34which usually are accurate only to within few tens of MHz. A relatively new [80],but already widespread method of measuring optical frequencies uses frequencycombs where a cw laser used for spectroscopy is phase locked to a comb line [81]allowing accurate determination of its frequency. This approach, in principle, al-lows referencing multiple cw laser to one frequency comb assuring good relativephase coherence between these lasers even if their frequencies differ by tens ofTHz.Frequency combThe frequency comb is a femtosecond erbium-doped fiber laser oscillator basedon nonlinear polarization rotation (spectrum centered at 1550 nm) that producessub-100 fs pulses at a repetition rate of 125 MHz [37]. The free-space sectionsof the femtosecond oscillator and self-referencing optics are limited to make thesystem more robust, and less sensitive to the influence of the environment. Thelight is split into two amplified branches. The referencing branch is used to stabi-lize the offset frequency of the comb, fo, and the repetition frequency, frep. It isspectrally broadened with a highly nonlinear fiber (HNF) for self-referencing thecarrier-envelope offset frequency via an f –2 f interferometer. The signals contain-ing fo and frep are stabilized to separate rf synthesizers (Agilent 8648A and HP8663A, respectively) using phase/frequency discriminators (PFDs) and loop filtersto control the oscillator pump current and cavity piezoelectric transducer (PZT),respectively. These rf synthesizers are in turn referenced to a global positioningsystem (GPS) disciplined quartz oscillator (GPS 6-12 Frequency Standard, MenloSystems GmbH). The second, ”measurement”, branch is also spectrally broadenedwith HNF and then split in two paths for referencing two Ti:Sapphire lasers. Hereonly 1.5–1.65 µm range is of importance for our applications, as opposed to theentire 1–2 µm region required for the referencing branch.Titanium-sapphire lasersHigh output power (100 mW and more) and narrow linewidth are typically requiredfrom lasers used for photoassociation spectroscopy. Moreover, these lasers need tobe widely tunable because the theoretical predictions for the levels locations (at35least for lithium) are not always known precisely and frequency scans spanninghundreds of MHz are often necessary. We use two Coherent 899-21 ring cavityTi:Sapphire lasers (called TS1 and TS2 throughout the thesis, see Fig. 2.15) withcavity optics suitable for lasing in the 760–830 nm range 6. A Coherent Verdi V-18pumps each Ti:Sapphire laser with 9 W of 532 nm light. For some experiments therelative phase coherence of the two lasers is important therefore both are locked tothe same frequency comb.As described in the previous section there are two “measurement” paths pro-vided by the frequency comb and each is frequency doubled (second harmonicgeneration, SHG) in a separate periodically poled lithium niobate (PPLN) crystaland then overlapped with a respective cw laser. A heterodyne beat, fbeat, betweenthe SHG comb, a cw laser and a stable rf reference signal is used to generate anerror signal that after processing is fed back as a control signal such that eachTi:Sapphire is phase locked to a tooth of the frequency comb. The resulting fre-quency of a laser isflaser = m frep± fo± fbeat , (2.14)where frep, fo and fbeat are known radio frequencies, and m is a natural numberon the order of 3× 106. A preliminary frequency measurement using a standardwavemeter (with accuracy better than frep/2) enables the mode number m to bedetermined unambiguously.Directly locking two lasers to the same frequency comb makes it impossibleto change the relative frequency of the lasers over wide range. To circumvent thisissue, for one of the lasers (TS1) the beam used for the beat note generation is fre-quency shifted by an AOM in a double pass configuration before overlapping withthe comb. The technical details of this method will be discussed in the doctoralthesis of William Gunton and here we provide only a brief description. The laseris initially locked to a tooth n with the AOM and TS1 frequencies f iAOM and f+n ,respectively. f+n fulfills the condition 0 < f+n − fn < frep/2, where fn is the fre-quency of the tooth number n and frep is the repetition rate of the frequency comb.Because the beat note is fixed, changing the AOM frequency must result in the6We would like to acknowledge Takamasa Momose and Mark G. Raizen for the long term loanof these lasers.36Figure 2.15: a) Fiber based frequency comb. The box to the left with the topoff hosts the femtosecond oscillator. The green light comes from theerbium-doped fibers. b) Two Ti:Sapphire Coherent 899-21 ring lasersused for photoassociation are enclosed in a box with walls coveredwith a damping foam to reduce the influence of the temperature fluc-tuations and vibrations on the stability of the lasers. The accessiblewavelength range is 760 nm-830 nm and is limited only by the cavityoptics set.37change of the TS1 frequency. When the AOM is set to f f 1AOM the laser frequencybecomes f−n+1 and 0 > f−n+1− fn+1 =−( f+n − fn). Then the sign of the error signaland, simultaneously, the AOM frequency are changed such that the laser becomeslocked to the comb tooth n + 1 with frequency f−n+1 and the AOM is again setf iAOM. The AOM is further changed to ff 2AOM such that the laser reaches f+n+1 (and0 < f+n+1− fn+1 = f+n − fn) and the sign of the error signal is changed. Simulta-neously the AOM is set to f iAOM. This completes the cycle and the laser becomeslocked to the comb element n+1 with frequency f+n+1 = f+n + frep. The steps repeatuntil the predetermined final frequency is reached. This method exploits the finiteresponse time of the PID lockbox and allows continuous scanning of the frequencyof TS1 over the entire mode-hop free range while keeping it locked to the comb atall times.The absolute frequency uncertainty of the comb-referenced Ti:sapphires wasverified by measuring the resonant frequencies of the D2 line at 780 nm (the5s1/2,→ 5p3/2 transition) of85Rb atoms in a vapor cell and comparing them withtheir known values [82]. The test cell was enclosed in a µ–metal shield to limitthe influence of stray magnetic fields on the transitions. The measured absolutefrequencies differed from known values by amounts that depended on the probedtransition - some where below, some were above the expected values. For this rea-son the difference, that never exceeded 600 kHz, was not attributed to systematicshifts due to the frequency comb but rather to the quality of the test cell setup. Asa result, ±600 kHz was taken as the uncertainty of the absolute frequency determi-nation.The beatnote measurement between the two Ti:Sapphires allowed us to deter-mine the relative linewidth of the lasers. The two frequencies were set to differ byup to 10 GHz such that the width of the beatnote signal could be measured with aphotodiode and an RF spectrum analyzer. The resulting width ∆ν = 160 kHz wasconstant over the entire operational range of the photodiode and it is justified toassume that it does not change when the frequency difference increases to 24 GHz(required for the binding energy measurements). The actual value of the beatnotedetermined directly with a spectrum analyzer agrees to ±1 kHz with the value ob-tained by subtracting the frequencies of both lasers (calculated from measured rffrequencies frep, fo and fbeat). This small difference is due to the resolution of the38Figure 2.16: The distribution of the light derived from the Ti:Sapphire lasersTS1 and TS2. The light from TS1(TS2) is coupled into fibers(FCXin(out)) for the wavelength measurement, FC4in(out)(FC6in(out)),beatnote with the comb, FC1in(out)(FC2in(out)), and photoassociation(PA), FC5in. The PA light is shuttered with single pass AOMs. Thedouble pass AOM is used for scanning as described in the text. PD1,PD2, PD3 are photodiodes.frequency counter used in our setup.Figure 2.16 shows how the Ti:Sapphire light is distributed in order to monitorthe frequency and create beat notes to lock lasers to the frequency comb. The lightfrom both lasers is overlapped on a polarizing beam splitter cube, sent through aGlan-Thompson polarizer to assure that both beams have the same polarization,coupled into a single mode, polarization maintaining fiber and sent to the experi-mental table. Before overlapping with the dipole trap (see Fig. 2.9) the output of thefiber is filtered once again with a Glan-Thompson polarizer to assure a well definedpolarization of the photoassociation light. The beams follow the exact same pathas the IPG dipole trap and are focused such that the 1/e2 radius is about 50 µm.392.3 Preparation of ultracold quantum gasesThis section discusses the performance of our experimental setup and steps thatlead to the production of ultracold samples required for high-precision photoas-sociation spectroscopy. Furthermore, we demonstrate that both molecular Bose-Einstein condensate of Feshbach molecules and the so-called Bardeen-Cooper-Schrieffer pairs in a strongly-interacting, degenerate Fermi gas can be created usinga relatively simple design of the vacuum chamber and the atomic source.The development of our experimental setup is always a “work in progress”,especially regarding the dipole traps (Section 2.2.3). The data presented in thissection should be treated as a snapshot of the capabilities of our setup at a pointin time and an indication of the minimum performance level that we are able toachieve.2.3.1 Trapping performanceAn exhaustive search in the pump (∆νp) and repump (∆νr) detunings for a rangeof axial magnetic field gradients (∂z|B|) between 10 to 70 G cm−1 7 was performedto optimize the MOT numbers at an oven current of 9.5 A (corresponding to anoven temperature of 625 K). Table 2.1 contains a summary of the optimum MOTparameters. The highest atom number we were able to trap (at Ioven = 9.5 A) was(8±2)×107 atoms at ∂z|B|= 35 G cm−1 and using a detuning from resonance of∆νp =−8.5Γ and ∆νr =−5.1Γ for the pump and repump light, respectively. Thesaturation intensities given in table 2.1 are calculated for the case of an isotropicpump field with equal components in all three possible polarizations [82]. In ad-dition, the lithium interaction was assumed to include all of the allowed 22P3/2excited state levels due to our large detuning relative to the excited state hyperfinesplittings.Fig. 2.17 shows the behavior of this oven loaded lithium MOT at an axial fieldgradient of 35 G cm−1. The loading rate (R), the lifetime or inverse loss rate (γ−1loss)of both the MOT and ODT, and the steady state atom number (N∞) in the MOT areshown as a function of the oven temperature. These parameters were determinedby fitting the loading curve of the MOT to the solution, N(t) = N∞(1− e−γlosst)7The axial magnetic field gradient for our coils is 5.8 G/cm/A40Table 2.1: Optimal parameter settings for the MOT at an oven current of9.5 A (oven temperature of 625 K): Wavelength and width of the D2line, saturation intensities (assuming isotropic light polarization), pumpand repump intensities along all three principle axes, relative detuningsof the pump and repump light ( ∆νp and ∆νr), axial magnetic field gra-dient, initial load rate, decay rate, and steady state atom number. Tabletaken from [50].6LiλD2,vac (nm) 670.977Γ/2pi (MHz) 5.87Isat (mW cm−2) 3.8 [82]Ip/Isat 21Ir/Isat 16∆νp (Γ) -8.5∆νr (Γ) -5.1∂z|B| (G cm−1) 35R (1.3±0.3)×106 s−1N∞ (8±2)×107of the differential equation N˙ = R− γlossN. The steady state number is then theproduct of the loading and inverse loss rates, N∞ = R/γloss.This model does not include a two-body loss term to model particle loss due tolight assisted collisions between cold atoms within the MOT, and therefore the lossterm we report is an overestimate of the MOT losses due only to collisions withthe residual background vapor or atoms from the hot atomic beam. In addition,because the Li–Li∗ collision cross section (with one atom in the excited state) ismuch larger than for ground state collisions, the loss rate for the excited state atomsin the MOT will be significantly larger due to collisions with fast moving Li atomsin the atomic beam. These two effects make the MOT lifetime only an estimate ofthe expected lifetime for the ensemble in the CODT.For comparison, the inverse loss rate (i.e. lifetime) of a 50 µK deep CODT isalso provided for a few of the oven current settings. Since the MOT is significantlydeeper than the CODT, the MOT trap loss rate due to background vapor collisionsis expected to be smaller [83, 84]. In addition, residual losses from the CODT willoccur due to spontaneous emission and evaporation losses. Thus the lifetime in a41Figure 2.17: The atom loading rate (diamonds), the MOT lifetime (cir-cles), the ODT lifetime (triangles), and the steady state atom number(squares) of the MOT are shown as a function of the oven temperaturewith parameter values as defined in Table II.We typically operate thesource in the shaded region (between 590 K and 625 K), where a goodcompromise is found between a sufficiently large MOT atom numberand a long ODT trap lifetime. The trend lines serve only as a guide forthe eye. Figure taken from [50].shallow CODT will always be an overestimate of the total background collisionrate.At low oven current settings where the loss rate is dominated by collisions withatoms or molecules other than Li atoms, the MOT lifetime is, as expected, muchlarger than the lifetime in the CODT. Whereas at high oven current settings wherethe loss rate is primarily determined by the collisions with fast moving atoms in theLi atomic beam, the MOT lifetime is similar to the CODT lifetime. The longestinverse loss rate measured for the CODT was on the order of 40 s (at an ovencurrent of 8 A and a temperature of 550 K) corresponding to a background vaporpressure of approximately 10−10 Torr [53, 83, 84]. As the temperature of the ovenincreases so does the captured flux and the loss rate. At a current of Ioven = 11 A(corresponding to an oven temperature of 680 K), the steady state atom numberis maximized. There exists a clear tradeoff between the inverse loss rate of the42Figure 2.18: Atom number transferred into the CODT versus atom number inthe MOT. Plotted are the initial atom number in the high power CODTat 200 W total power (circles), the atom number in the high powerCODT after evaporation to 100 W total with 50 W per beam (squares),and the atom number transferred into the low power CODT (diamonds)for various initial MOT atom numbers. Figure taken from [50].lithium MOT and the steady state atom number, and by running the oven at 625 K(with Ioven = 9.5 A) instead of 680 K, the inverse loss rate can be increased by morethan a factor of four with only a factor of two reduction in the trapped number. Wenote that the MOT performance is similar to that previously reported in Ref. [24].The primary difference is that here the oven aperture is 30% larger and a portion ofthe MOT light is sent counter propagating to the oven flux. This additional slowingbeam enhances the loading rate by a factor of two.To reach degeneracy, atoms need to be transferred from the MOT into a trapthat allows evaporative cooling, like an optical dipole trap (as discussed in Sec-tion 2.2.3). The atomic cloud trapped in the MOT is compressed and cooled byincreasing the axial magnetic field gradient from 35 to 64 G cm−1, lowering theintensity and shifting the frequency of both the pump and repump light to 10 MHzbelow resonance. During this compression and cooling phase, a CODT of 200 Wtotal power is turned on and, in less than 10 ms, up to 10% of the 6Li atoms are43transferred into the CODT. Fine adjustments of the MOT position to optimize itsoverlap with the CODT are essential to achieving efficient transfer and are madeby adding a small (5 to 10 G) homogeneous magnetic field that shifts the magneticquadrupole center during the compression phase. This field is generated by threepairs of large (30 cm diameter) “compensation” coils for independent control ofthe field along each of the three orthogonal spatial axes.During the compression phase, we observe extremely rapid trap losses (withtrap lifetimes on the order of a few ms) due to light assisted collisions and hyperfinerelaxation, and we therefore optically pump to the lower hyperfine state (F = 1/2)during the transfer by extinguishing the repump light 400 µs before the pump light.As a result an almost equal population of the two sub-levels of the lower hyperfinestate: |1〉 ≡ |F = 1/2,mF = 1/2〉 and |2〉 ≡ |F = 1/2,mF =−1/2〉 is produced.After the MOT light is extinguished, the CODT beam power is ramped down lin-early in time to 100 W total (50 W per beam) in 100 ms while applying a homo-geneous magnetic field of 300 G produced by the same coils used to generate thequadrupole magnetic field for the MOT. Rapid thermalization occurs because of thelarge collision rate between the |1〉 and |2〉 states at 300 G [85]. At the end of thisforced evaporation stage, there are approximately 106 atoms remaining at a tem-perature of 200 µK (verified by a time-of-flight expansion measurement). Finally,a second dipole trap is turned on and as a result about 25% of atoms are trapped inlow power CODT (see Section 2.2.3). This transfer is done to avoid ensemble heat-ing observed in the SPI laser CODT and allows further forced evaporative coolingto much lower powers and thus ensemble temperatures.In the experiments that originally led to the creation of degenerate gases (andthe data presented in this Section) the trap depth was lowered using a series oflinear ramps of different duration, for a total evaporation time of 4 s.8 We use anadditional “gradient” coil, concentric with the top main magnetic coil, to add amagnetic field gradient to both compensate for the residual magnetic field gradientof our main magnetic coil pair and to provide a magnetic force on the atoms equaland opposite to the gravitational force. This additional coil is essential for theproduction of quantum degenerate samples as in this case the evaporation must8We have recently successfully reduced the total evaporation time to about 1.3 s without anymeasurable decrease in the evaporation efficiency thus reducing the data collection time.44Figure 2.19: Ensemble temperature (diamonds) and particle number (circles)as a function of the CODT laser power (i.e.,trap depth) in the finalforced evaporation stage at (a) B = 800 G (below the FR) and (b) B =900 G (above the FR). The number of ”spin-up” atoms (atoms in the|1〉 state) is detected at these high magnetic fields by absorption imag-ing. For the evaporation below the FR, when the temperature is wellbelow the binding energy all of the atoms form FR molecules and thenumber detected in the |1〉 state corresponds to the total FR moleculenumber. The solid line indicates the expected particle number for anefficient evaporative cooling. The dashed line indicates the Fermi tem-perature for the |1〉 component at each power computed from the par-ticle number and the trap frequencies. In (a) the dotted line shows thecritical temperature for BEC for the molecules. Figure taken from [50].45be continued to the very lowest optical trap depths where gravitational saggingsignificantly compromises the confinement.Figure 2.19 shows the result of evaporation both below and above the Feshbachresonance center at B = 832.18 G. Both the ensemble temperature and particlenumber are shown as a function of the trap depth during the final forced evaporationstage. The solid line indicates the expected particle number N at various trap depthsU obtained from the scaling lawNNi=(UUi)3/[2(η ′−3)](2.15)under the assumption that the ratio η of the trap depth U to thermal energy kBTis ∼ 10 (typical for evaporative cooling in optical traps). Here i denotes initialconditions and η ′ = η +(η − 5)/(η − 4). The detailed discussion of the scalinglaws for evaporative cooling in time-dependent optical traps, with emphasis on 6Li,can be found in Ref. [86].2.3.2 Molecular BECFor evaporation at B = 800 G, we observe the formation of FR molecules whenthe temperature nears the molecule binding energy (250 nK), and at the end of theevaporation ramp, we observe 2× 104 molecules at a temperature of 70 nK. Toimage the molecular cloud, we reduce the magnetic field from 800 to 690 G andrelease the ensemble from the dipole trap. After some fixed time-of-flight (TOF)expansion, we then take an absorption image of the molecular cloud. The spin-up(or spin-down) component of this very weakly bound molecule can be imaged asif it were a free atom since the binding energy is far below the excited state width.At the end of the evaporation ramp at B = 800 G the molecular gas is stronglyinteracting and deep in the hydrodynamic regime. In particular, the molecule-molecule s-wave scattering length is predicted to be on the same order as the atom-atom scattering length (amol = 0.6a [87]), and the atom-atom scattering length atB=800 G is a > 5000aB [88]. In order to reduce the interaction strength duringthe TOF expansion and thus increase the visibility of the characteristic bimodaldistribution of the mBEC, the TOF is performed at 690 G [72].46Figure 2.20 shows a series of absorption images and resulting horizontal den-sity profiles of these Feshbach molecules after a 3 ms TOF expansion at a magneticfield of B = 690 G for different final ensemble temperatures. The wings of the pro-files are fit to a Gaussian function and the bimodal character of the density profileis evident for temperatures below the critical temperature for Bose Einstein con-densation. For the coldest molecular cloud, we show in Fig. 2.20 the absorptionimages after different free expansion times. The in-situ shape of the molecularcloud (image taken at 0 ms) has the anisotropic shape of the CODT, and the cloudanisotropy reverses during the time-of-flight expansion. This anisotropy reversal isexpected for a BEC of non-interacting particles due to the larger confinement fre-quency along the vertical direction in these images. However, even at the magneticfield of B = 690 G, the molecules are still strongly interacting, and the inversionof the aspect ratio of the molecular cloud can also be the result of hydrodynamics[71, 89].2.3.3 Strongly interacting degenerate Fermi gasOne of the first experimental signatures of fermionic pairing above the FR wherethe atom-atom interactions are attractive was the observation of the pairing gap in astrongly interacting Fermi gas of 6Li atoms by radio frequency spectroscopy [74].While the correct interpretation of these spectra requires a full accounting of boththe final-state and trap effects to understand the contributions from the pairing-gap,pseudogap, and no-gap phases, nevertheless, it is a simple measurement that can beintuitively understood and that provides a clear signature of pair formation (bothso-called “pre-formed” and condensed pairs) [90].Here we perform the final evaporation stage at B = 839.2 G, just above the FR,to different final trap depths producing different final ensemble temperatures. Atthe end of this evaporation, we apply radio frequency (rf) radiation for 1 secondthat transfers atoms in state |2〉 to the unoccupied |3〉 state. We monitor the loss ofatoms from state |2〉 by state-selective absorption imaging, and in Fig. 2.21 we plotthe loss as a function of the rf radiation frequency detuning δ from the free atomresonance at 81.34 MHz (corresponding to the energy splitting between the |2〉 and|3〉 states at this magnetic field including the difference in the mean field interaction47Figure 2.20: Absorption images and resulting horizontal density profiles of6Li2 Feshbach molecules after a 3 ms time of flight expansion ata magnetic field of B = 690 G for different ensemble temperatures,(a) T = 710 nK (T//TC ∼ 1.2), (b) T = 230 nK (T//TC ∼ 0.9), (c)T = 110 nK (T/TC ∼ 0.8), (d) T = 65 nK (T/TC ∼ 0.6). The wings ofthe profiles are fit to a Gaussian function and the bimodal character ofthe density profile is evident for temperatures well below the criticaltemperature. For the coldest molecular cloud (d) we show in (e) the ab-sorption images after different free expansion times. The in-situ shapeof the molecular cloud (image taken at 0 ms) has the anisotropic shapeof the CDT, and the cloud anisotropy reverses during the time-of-flightexpansion. Figure taken from [50].48Figure 2.21: RF spectroscopy of the degenerate Fermi gas at 840 G. Initiallyonly free atoms peak is present (a) but as the final temperature of thesample decreases the atoms form BCS-like pairs which presence showsup as an additional loss feature [(b) to (d)]. Figure taken from [50].energies). We note that when the loss of atoms from state |1〉 is monitored insteadof the loss from state |2〉 the spectrum is the same. This is likely due to the loss ofatoms in state |1〉 due to the collisional instability of these mixtures when |3〉 stateatoms are present [91, 92].In Fig. 2.21(a) and (b) where the temperature is above the pair dissociationtemperature, T ∗, the rf-spectrum shows a single resonance peak at a frequency off-set of zero. This corresponds to the energy required to flip the spin of an unpairedatom. In (c) and (d), the ensemble temperature is T < T ∗, and a second maximum49appears at higher energy. Unpaired |2〉 atoms undergo the transition at a zero offsetwhereas an additional energy due, in part to the pairing gap, must be added to flipthose bound to a |1〉 state atom as a pair. Here the temperature is slightly abovethe critical temperature for pair condensation, and these so-called pre-formed pairsexist in what is known as the ”pseudogap” region [90, 93]. In Fig. 2.21(d) wherethe temperature is lower than in (c), the ratio of pairs to free atoms is larger asexpected; however, the gap energy is less because the Fermi temperature in (d) issmaller than in (c). In addition, the unpaired spin flip energy is shifted to a negativeoffset in (c) and (d). This is because we produce a colder ensemble by evaporat-ing to a lower trap depth and this changes the atomic density producing a differentdifferential mean field interaction energy for the unpaired atom transition [73].2.3.4 Fermi gas at 0 gaussThe most spectacular experiments with fermionic 6Li have been performed neareither B = 543.3 G [88, 94] or B = 832.18 G [85] Feshbach resonance but therehas been very little work done on the weakly interacting gas at 0 G beyond initialexperiments reporting fermionic degeneracy [66, 95, 96]. This regime does notseem to be of any significant interest, possibly because its main advantage, van-ishing interactions between states |1〉 and |2〉, can be also reproduced at the zerocrossing of the s-wave scattering length (B = 527.5 G [88, 97]), while benefitingfrom the rapid control of interactions as the magnetic field is varied in the vicinityof the crossing.Our experiments are the first that perform systematic photoassociation spec-troscopy of 6Li in a dipole trap. This allows us to remove magnetic field as asource of systematic shifts, a step impossible if photoassociation spectroscopy isdone in a MOT. This is especially important for high precision studies of molecularstructure using dark states.The sample confined by the IPG CODT is evaporatively cooled at 300 G to afinal trap depth at which the spectroscopy measurements are done. The MOT coilsand the gradient coil are then turned off, while simultaneously a uniform magneticfield is applied with a set of three pairs of compensation coils (the same that areused to move the MOT for dipole trap loading). This additional uniform field is50Figure 2.22: The evaporation curve at 0 G. The atom number drops suddenlywhen the trap becomes too shallow to support atoms against gravity.This sets a limit for the evaporation ramp and thus the atom numberand temperature that can be achieved with the degenerate Fermi gasat 0 G. Two different evaporation trajectories are clearly visible: thesteeper one corresponds to an inefficient evaporation with η = 4.5.used to compensate for any residual magnetic fields that might be present in thesystem, including the magnetic field of Earth. Fitting the equation 2.15 to the dataobtained for the evaporation at 300 G and subsequent turn off of the magnetic field(shown in Fig. 2.22) indicates that the process is very efficient for a range of finaltrap depths, with η =U/kBT = 11.At low laser powers the trap depth decrease due to the gravitational potentialbecomes significant. This results in spilling of atoms out of the trap and severelylimits the atom number at the lowest ODT powers. Turning off the gradient coileffectively corresponds to a sudden decrease of the trap depth which is a sum ofoptical and gravitational potentials as well as a contribution from the magneticgradient. Past certain point switching to 0 G and turning off the gradient coil dra-matically limits the trap depth leading to atom loss by spilling, seen as a change inthe slope of the evaporation curve in Fig. 2.22, corresponding to U/kBT = 4.5. Weavoid this problem at non-zero magnetic fields by using a gradient coil throughout51Figure 2.23: The frequency of the σ+, σ− and pi RF transitions between Zee-man sublevels of F=1/2 and F=3/2 manifolds of the 22S1/2 groundstate. The states |i〉 correspond to |F,mF〉 as shown in figure 2.12, Sec-tion 2.2.4. The grid indicates the magnetic field region below 20 mG- relevant to the compensation of the day-to-day magnetic field drift inthe experimental setup.the entire experimental cycle but it is not feasible at 0 G because the field couldnot be canceled across the entire cloud, distorting precision spectroscopy measure-ments. Fig.2.19 and Fig. 2.22 show the striking difference between evaporationand imaging at high magnetic fields, and evaporation at 300 G with subsequentswitching to 0 G.Cancellation of the residual magnetic fieldAs a result of the regular operation of the experimental setup there is always someresidual magnetic field present even if all the sources are off. This is mainly due tothe magnetic field of Earth but it is likely that the magnetization of some elementsof the setup also plays a role. Even a small magnetic field B of some tens of mGcan change the energies of the hyperfine levels in the 22S1/2 state according to∆E =µBh¯gFmFB, (2.16)52where µB = 9.27400968(20)× 10−24 J/T is Bohr magneton, mF the projection ofthe total atomic angular momentum F on the quantization axis and gF for the 22S1/2state is given bygF = gJF(F +1)−5/42F(F +1)+gIF(F +1)+5/42F(F +1)(2.17)with gJ = 2.002301 and gI = −0.0004476540 being the electronic and nuclear g-factor of the 22S1/2 state, respectively. This property is exploited to characterizethe magnetic field by driving σ+, σ− and pi rf transitions between |1〉, |2〉 and|F = 3/2,mF〉 Zeeman sublevels as shown in figure 2.12, Section 2.2.4).After major changes to the setup like coil replacement or dipole trap rearrange-ment the residual magnetic field can reach few hundreds of mG. Rf spectroscopyof the sample reveals multiple peaks [Fig. 2.24(a)] corresponding to transitionsshown in Fig. 2.23. A sequence: ”change magnetic field using compensation coilsand then perform rf spectroscopy” if repeated many times lets us track the behaviorof the peaks by comparing the measured frequencies with theoretical calculations.In practice, the |2〉 → |3〉. transition is followed because it is the most sensitive tothe magnetic field. As the residual field approaches 0 G the levels become degen-erate and there is only one transition, corresponding to the 228.205 MHz 22S1/2hyperfine splitting. A signal as shown in Fig. 2.24(b) is treated as ”good enough”because further cancellation becomes very time consuming without providing anyobvious benefits. We believe that a residual field of 15 mG is acceptable for ourapplications.53Figure 2.24: Loss induced by driving σ+, σ− and pi RF transitions betweenZeeman sublevels of 22S1/2 level. a) The magnetic field sources areoff but the residual field is not canceled. Signals like this, here corre-sponding to about 130 mG, are typically observed after major changesto the setup, like the magnetic coils replacement or rearrangement ofthe dipole trap. Zeeman splitting between sublevels can be resolved. b)At very low magnetic fields all transitions become nearly degenerate.From the shape of the loss feature the residual magnetic field could beestimated to be below 15 mG.54Chapter 3Single color photoassociationThe main results of this Chapter have been published in• M. Semczuk, X. Li, Xuan, W. Gunton, M. Haw, N. S. Dattani, J. Witz, A.K. Mills, D. J. Jones and K. W. Madison, ”High–resolution photoassocia-tion spectroscopy of the 6Li2 13Σ+g state.”, Phys. Rev. A 87, 052505, May(2013) [49].• W. Gunton, M. Semczuk, N. S. Dattani and K. W. Madison, ”High–resolutionphotoassociation spectroscopy of the 6Li2 A(11Σ+u ) state.”, Phys. Rev. A 88,062510, December (2013) [51].Single color photoassociation spectroscopy has proven to be a very powerfultechnique for precise measurements of binding energies of cold molecules [98],pairing in the BEC-BCS crossover regime [99], optical control of Feshbach reso-nances in bosonic [100] and fermionic [101] species. Recently, coherent one-colorphotoassociation of a Bose-Einstein condensate of 88Sr was demonstrated by Yanet al. [102].Under certain conditions adding a photon of a proper energy to a colliding pairof atoms can result in association of atoms and formation of an excited molecule.The required energy is the difference between the energies of the excited leveland the initial state (molecules or colliding atoms). In the ultracold regime thephotoassociation relies mainly on the s-wave collisions between atoms because atsub-µK temperatures higher partial waves do not play a significant role. In thiswork a spin mixture of 6Li is used because the photoassociation rate of a spinpolarized sample of fermions is highly suppressed at ultra-low temperatures. Thisis due to the Pauli exclusion principle that prevents a single component Fermi gasfrom interacting via s-wave collisions and all higher partial wave collisions aresuppressed by an amount proportional to T 2 [103].55The experimental work on the single color photoassociation spectroscopy of6Li2 preceding the research presented in this thesis has been done with atomsin magneto-optical traps (MOT) [104, 105] and was limited to the c(13Σ+g ) andA(11Σ+u ) potentials. In MOTs atoms are trapped in the two hyperfine levels of the2S1/2 electronic ground state. Additionally, MOT temperatures are typically on theorder of 400 µK and are high enough for p-wave (and sometimes higher order)collisions to take place. This allows access to multiple ro-vibrational levels in theexcited potentials but at the same time these traps suffer from hard to control sys-tematic effects induced by the presence of high magnetic field gradients and theMOT cooling light.In the experiments presented in this work we confine a two-component de-generate Fermi gas in an optical dipole trap, as described in Chapter 2. With nomagnetic field present the only systematics come from the frequency uncertainty ofthe photoassociation (PA) laser and the ac Stark shift of the initial and final levelsintroduced by the dipole trap and the photoassociation laser. The frequency uncer-tainty of the PA laser is estimated as discussed in Section 2.2.5. The ac Stark shiftsare quantified by performing spectroscopy at different trap strengths and for differ-ent photoassociation laser powers. Finally, the measurements are extrapolated tothe field free values.Figure 3.1 shows seven lowest lying potentials of 6Li. The specific vibrationallevels in the five potentials that can be probed with our PA laser system are listedin Table 3.1. It is worth noting that except for the c- and A-states these potentialshave never been studied with photoassociation spectroscopy. In fact, the experi-mental data (including other techniques) is rather scarce and it is not clear to whatdegree one can trust the theoretical predictions in the context of photoassociationspectroscopy. The characterization of these levels is expected to be very time con-suming and for the time being is not a part of our research program. Long term,however, these levels might be important because it is very likely that they couldbe used for direct absorption imaging of ultracold ground state molecules as hasbeen demonstrated in the case of KRb [106]. In this Chapter only two potentials,c(13Σ+g ) and A(11Σ+u ), are studied, mainly because these are the most relevant toour long term goals. The theoretical predictions guiding the initial measurementswere expected to be accurate to ∼1 GHz for the range of frequencies covered in56Figure 3.1: Seven lowest lying singlet (black) and triplet (grey) potentials of6Li2. The shaded area corresponds to the tuning range of our PA lasersystem, 760 to 830 nm. The energy corresponding to the dipole traplaser (1064 nm) is highlighted in maroon (dashed line). The 2s+ 2Pasymptote is shown in red, corresponding to the D1/D2 atomic transi-tion. Figure based on numerical form of potentials published in [107]Table 3.1: Potentials and the corresponding vibrational levels that can beprobed by photoassociation spectroscopy in the range of 760-830 nm.Molecular potential c(13Σ+g ) b(13Πu) A(11Σ+u ) B(11Πu) C(21Σ+g )Vibrational levels 20 – 26 32 – 38 29 – 35 0 – 4 0 – 4this work, making the initial search easier than it would be with other potentialsshown in Table 3.1.The measurements presented in this chapter probe not only the previously un-explored range of the c(13Σ+g ) and A(11Σ+u ) potentials but also enter a new regimeof atomic densities and temperatures, not addressed in previous photoassociationexperiments with 6Li. For these reasons in the initial measurements we employed57all the available techniques that we believed would increase the photoassociationrate:• the atoms were trapped in a cross beam to increase the density of the cloud(reaching 5×1011 cm−3),• the temperature was kept relatively high (about 20 µK),• the maximum available photoassociation power was used and the atoms wereexposed to the light for as long as it was reasonable (at times up to 10 s),• for the c-state the process took place at 800 G to enhance the collision ratedue to the proximity of a broad Feshbach resonance (FR) at 834 G and thenthe features were traced back to 0 G. The method resulted in very broad(several hundred of MHz) loss features, significantly broader than expectedbased on the predicted natural lifetimes of these levels [108]. To accessrotational levels N′ = 0,2 in the excited c-state the collisions in the initialstate needed to have a rotational number N = 1, corresponding to the p-wave collisions, which are strongly suppressed at sub-µK temperatures. Forthis reason the photoassociation rate to N′ = 0,2 was negligibly small andhad to be enhanced by increasing the p-wave collision rate in the proximityof a p-wave Feshbach resonance at 185 G [109]. The p-wave enhancementshould also work for the singlet A(11Σ+u ) state.The initial search for the photoassociation features and p-wave enhanced pho-toassociation measurements was performed with a Ti:Sapphire laser locked to itscavity and laser frequency was determined by a commercial wavemeter (Bristol621A-NIR) with an absolute accuracy of 60 MHz and a shot-to-shot repeatability(i.e. precision) of 10 MHz in the frequency range of this work. All the measure-ments (except for the p-wave enhanced photoassociation) were refined with thephotoassociation laser locked to the frequency comb (see Section 2.2.5).3.1 A note about Feshbach resonancesThe properties of magnetic Feshbach resonances [110, 111] are extensively used inexperiments with ultracold atoms. Since the initial observation in a Bose-Einstein58condensate of sodium [112] they have become indispensable for control of in-teractions between alkali atoms. The associated phenomenon of formation ofloosely bound states near such resonances, so-called Feshbach molecules, resultedin the production of Bose-Einstein condensates of weakly molecules of 6Li [113]and 40K [114], and proved to be an essential step in obtaining near degenerateground-state molecules of KRb [23]. Recently, magnetic Feshbach resonanceshave been also observed in dipolar gases of chromium [115], erbium [116] anddysprosium [117], further extending the scope of systems with magnetically tun-able interactions.The literature on Feshbach resonances in ultracold gases is very extensive, in-cluding an excellent review by Chin et al. [118]. For the purpose of this thesis onlya brief discussion is presented.A Feshbach resonance occurs when the energy of the bound molecular state inthe closed channel (e.g., ground state molecular potential) approaches the energyof the scattering state in the open channel (colliding atoms or molecules). Whenthe corresponding magnetic moments are different, the energy difference betweenthe open and the closed channel can be controlled via magnetic field, leading toa magnetically tuned Feshbach resonance. In 6Li, s-wave Feshbach resonanceslocated at magnetic fields ∼ 543 G and 832.2 G arise from the v′′ = 38, N = 0 ro-vibrational level in X(11Σ+g ) potential with nuclear spin I = 0 or I = 2 (Fig. 3.2).In the vicinity of a resonance located at B0 with a width ∆ the s-wave scatteringlength a changes as a function of the magnetic field Ba(B) = abg(1−∆B−B0), (3.1)where abg is the background (off-resonant) scattering length associated with theinteraction potential between colliding atoms. In this formalism both abg and ∆can be both positive and negative. The scattering length associated with the Fes-hbach resonance vanishes at a magnetic field Bcross = B0 + ∆. The parameterscharacterizing the broad Feshbach resonance in 6Li occurring at B0 = 832.18(8)are abg =−1582(1)a0 and ∆=−262.3(3) G, with a0 = 0.529177 nm [85].The tunability of the scattering length with the magnetic field is often exploited59Figure 3.2: Coupled-channels calculation of 6Li2 bound states, which giverise to s-wave Feshbach resonances at threshold. The two-atom states(dashed line) are indicated by their quantum numbes (F1,m f1 ;F2,m f2)and are: ab = (1/2,1/2;1/2,−1/2), ad = (1/2,1/2;3/2,−1/2),be = (1/2,−1/2;3/2,1/2), c f = (3/2,−3/2;3/2,3/2), de =(3/2,1/2;3/2,−1/2) while the bound states (blue and red lines) arelabeled by the molecular quantum numbers S and ML, where ML is theprojection of total angular momentum on the quantization axis. The ar-rows indicate the locations of the 543 and 832.2 G Feshbach resonances,where the binding energy of a threshold bound state equals 0. While thelow B field I = 2 X(11Σ+g )(v”=38) level retains its spin character as itcrosses threshold near 543 G, the I = 0 level mixes with the entrancechannel and switches near ≈550 G to a bound level with ab spin char-acter, eventually disappearing as a bound state when it crosses thresholdat 832.2 G. The figure and caption adapted from Chin et al. [118].to increase the elastic scattering cross-sectionσe =4pia21+ k2a2, (3.2)which enhances the elastic collision rate. This leads to faster thermalization andas a consequence enables rapid and efficient evaporative cooling of atoms confinedin a dipole trap. Equation 3.2 holds true for s-wave scattering of distinguishableparticles, e.g different spin states of an atom as is the case in our experiments. As60is shown in this chapter, such an enhancement was crucial in the early stages ofthe photoassociation spectroscopy experiments where it was used to enhance thephotoassociation rate [54, 108].The scattering process (here for distinguishable particles) is characterized byits cross-sectionσ(k) = 4pik2 ∑(2l +1)sin2 δl(k) (3.3)and can be described by a scattering phase shifts δl of different partial waves l. δlincorporate the effect of the whole potential on the collision event. In the ultracoldregime the relative momentum h¯k of colliding atoms typically tends to zero and forthe van der Walls potential one finds that tanδl varies as k and k3 for s(l = 0) andp(l = 1) waves, respectively [119]. For the s-wave collisions near the resonance amore precise statement relating tanδ0 and k is often required:k cotδ0(E) =−1/a+12r0k2. (3.4)Equation 3.4 is called the effective range expansion and r0 is the effective range- a parameter that takes into account that the scattering phase shift and thus thecollision cross section depends on the collision energy and the k 6= 0 condition isnot strictly fulfilled in experiments.p-wave Feshbach resonancesFor the case of p-wave resonances in fermionic systems the scattering in spin po-larized samples is possible, as opposed to s-wave case. For 6Li with f = 1/2, oneexpects three p-wave Feshbach resonances corresponding to three possible combi-nations of | f = 1/2,m f 〉+| f = 1/2,m′f 〉, denoted as (m f ,m′f ). They appear at fieldsB = 159.1 G, 185.1 G and 215 G corresponding to (1/2,1/2), (1/2,−1/2) and(−1/2,−1/2), respectively [109, 120], and are relatively narrow (∼ 0.4 G). Theirpresence is a consequence of crossing of the free atom threshold by the v′′ = 38level in the X(11Σ+g ) (S = 0) molecular potential with a rotational quantum num-ber N′′ = 1 (see Fig. 3.3). As a result of the symmetrization requirements of thetwo-body wave function this state has a nuclear spin I = 1 [120].At ultracold temperatures p-wave collisions are strongly suppressed and the61Figure 3.3: Coupled channels calculation of p-wave binding energies of 6Li2,which give rise to Feshbach resonances at threshold. The two-atomstates (full line) are indicated by their quantum number (m f1 ,m f2), whilethe bound state (dashed line) is labeled by the molecular quantum num-bers S, I, and N. The figure and caption taken from [120].only method to enhance them is by tuning the field to one of the resonances. Weexploited this property to perform single color photoassociation to the rotationalmolecular levels N = 0 and 2 that are inaccessible if atoms collide in the s-wavechannel. To our knowledge, this is the first example of FOPA (Feshbach-OptimizedPhotoassociation) [54] for a p-wave Feshbach resonance.3.2 Theory of diatomic moleculesMolecules in the 13Σ+g and A1Σ+u excited states are characterized by the Hund’scase “b” coupling scheme in which the total electronic (nuclear) spin ~S =~s1 +~s2(~I =~i1 +~i2) is completely uncoupled from the internuclear axis. Here ~s j (~i j) isthe electronic (nuclear) spin of atom “ j”. This occurs when Λ = 0, the projectionof the orbital angular momentum of the electrons along the internuclear axis iszero, and there is therefore no axial magnetic field to couple the total spin to theaxis. For “Σ” states, the orbital angular momentum of the electrons is zero andtherefore Λ is always identically zero; however, even in some cases where Λ 6= 0,62especially for light molecules, the coupling is sufficiently weak that Hund’s case“b” is still the appropriate scheme [121]. The total angular momentum, apart fromthe spin, is ~K ≡ ~N +~Λ, the vector sum of ~Λ and the rotational angular momentumof the nuclei ~N. Therefore for “Σ” states ~K = ~N, and thus ~K is perpendicular tothe internuclear axis. The total spin of the molecule is ~G = ~S +~I and is a goodquantum number so long as the hyperfine interaction and spin-rotational couplingsare small. The total spin combines with the total angular momentum apart fromspin ~K to result in the total angular momentum including spin as ~J = ~K + ~G. Forelectric dipole radiation, the selection rule is that ∆J = 0,±1 with the restrictionthat J = 0 = J = 0. In addition, under the emission or absorption of a photonthe parity of the electronic orbital must change (+↔−) and for a homonuclearmolecule, the symmetry of the coordinate function under interchange of the twonuclei must change from symmetric to anti-symmetric or vice versa (g↔ u). In thepresent scenario of Hund’s case “b” coupling, the spin is so weakly coupled to theother angular momenta that both quantum numbers S and K are well defined and wehave in addition the selection rules ∆S = 0 (or equivalently ∆G = 0) and therefore∆K = 0,±1 with the restriction that ∆K = 0 is forbidden for Σ→ Σ transitions.Since this chapter is only concerned with transitions to the A1Σ+u and c(13Σ+g )excited states, we have that ∆N =±1 and ∆G = Initial state of colliding atomsIn this work, we only consider collisions between two 6Li atoms, which are com-posite fermions (consisting of 9 fermions: 3 protons, 3 neutrons, and 3 electrons),and we note that the 2-body eigenstates, composed of a spin part and an orbital part,must be antisymmetric upon exchange of the two atoms. The consequence is thatonly certain spin states are possible given a particular orbital state. An importantexample of this constraint imposed by exchange symmetry is that the two-body po-sition wave function (sometimes called the “coordinate function” or orbital state)must be antisymmetric for a collision between two fermions in the same spin state(for which the spin wave function is manifestly symmetric). Thus a spin polar-ized Fermi gas can only have odd partial wave collisions (p-, f -, h-wave, etc...)corresponding to odd values of the rotational angular momentum of the complex63(N = 1,3,5 . . .), which are antisymmetric with respect to atom exchange. For a gascomposed of two distinct spin states, even partial wave collisions can occur (s-,d-, g-wave, etc...) so long as the spin wave function is antisymmetric upon atomexchange. The ability to turn off s-wave collisions by spin polarizing the gas is auseful feature of our system that we use to validate our assignment of the PA lines.The total spin angular momentum of the initial unbound molecular state isgiven by the vector sum of the f quantum numbers for the isolated atoms: ~G =~f1 + ~f2. Here ~f1 = ~s1 +~i1. In our experiment, the atoms are optically pumpedto the lowest hyperfine state before being exposed to the photoassociation light.Therefore we have that f1 = f2 = 12 and there are two allowed values of the totalspin: G = 0,1. Certain values of G (specifically G = f1 + f2, f1 + f2− 2, . . .) areassociated with spin states symmetric with respect to interchange of the atomswhile the orbital states with even values of N are symmetric under the interchangeof the atoms. Therefore all even partial wave collisions (N = 0,2,4, . . .) have atotal spin of zero (G = 0) and all odd partial wave collisions (N = 1,3,5, . . .) havea total spin of one (G = 1).Following [122], the initial antisymmetric unbound molecular state can be writ-ten in the |S,MS; I,MI〉 basis states, where MS and MI are projections of the totalelectronic end nuclear spin on the quantization axis, respectively. A general ex-pression that holds also for non-zero magnetic fields B|ΨB〉=sinθ+ sinθ−|1,1;1,−1〉+ sinθ+ cosθ−(1√3|0,0;0,0〉−2√3|0,0;2,0〉)(3.5)+ cosθ+ sinθ−(1√3|0,0;0,0〉+1√6|0,0;2,0〉−1√2|1,0;1,0〉)+ cosθ+ cosθ−|1,−1;1,1〉at B = 0 simplifies to|Ψ〉=√2/3(√1/3|1,1;1,−1〉+√1/3|1,−1;1,1〉−√1/3|1,0;1,0〉)(3.6)+√1/3|0,0;0,0〉.64Here sinθ± = 1/√1+(Q±+R±)2/2, Q± = (µN + 2µe)B/ah f ± 1/2, R± =√(Q±)2 +2, µN and µe are, respectively, the nuclear and electronic magnetic mo-ments, and a2S is the hyperfine constant.The state |Ψ〉 at B = 0 is a linear superposition of a singlet and a triplet state,therefore allowing photoassociation spectroscopy of excited molecular states withboth singlet and triplet characters. For the purpose of this thesis it is useful torewrite Eq. 3.6 in the molecular basis |N,S, I,J,F〉|Ψ〉=√2/3|Triplet〉+√1/3|Singlet〉 (3.7)=√2/3|0,1,1,1,0〉+√1/3|0,0,0,0,0〉.3.2.2 Final excited molecular statesThe final state is a molecule in the c(13Σ+g ) or A1Σ+u potentials. For the tripletstate, the total electronic spin is well defined (S = 1) and the “gerade” symmetrysignified by a sub-script “g” denotes that all states with an even rotational quan-tum number (N = 0,2,4, . . .) are symmetric under the interchange of the two nu-clei [104]. Because the electronic spin is well defined and fixed for this excitedstate, we now consider interchanging just the nuclei while leaving the electronsuntouched. There are three possible values of the total nuclear spin (I = 0,1,2)since the nuclear spin of each atom is i = 1. Similar to the symmetry of G, stateswith I = i1+ i2, i1+ i2−2 . . . (corresponding here to I = 0 and I = 2) are symmetricwith respect to interchange of the nuclei whereas the I = 1 state is antisymmetric.Since the nuclei are bosons the total wave function must be symmetric under theinterchange of the nuclei. Putting this together, we have that the even (odd) valuesof I occur with even (odd) values of N.The final singlet state A1Σ+u has a well defined total electronic spin S = 0.The sub-script “u” refers to the “ungerade” symmetry, therefore all states withan even rotational quantum number (N = 0,2,4, . . .) are antisymmetric under theinterchange of the two nuclei [104]. Due to the bosonic character of the final 6Limolecules the reasoning as presented for the triplet state leads to the conclusion thatI = 0,2 correspond to states with odd N and I = 1 corresponds to states with evenN. The total spin angular momentum quantum number G can take on all values65Table 3.2: Allowed rotational levels and corresponding nuclear spin config-urations for 6Li2 molecules in the limit that spin-spin and spin-rotationcouplings are small enough that G is a good quantum number.State Electronic Nuclear Allowed Totalspin spin rotational states Spinground states- - - N = 0,2,4 . . . G = 0- - - N = 1,3,5 . . . G = 1excited statesc(13Σ+g ) : S = 1 I = 0 N = 0,2,4 . . . G = 1I = 1 N = 1,3,5 . . . G = 0,1,2I = 2 N = 0,2,4 . . . G = 1,2,3A(11Σ+u ) : S = 0 I = 0 N = 1,3,5 . . . G = 0I = 1 N = 0,2,4 . . . G = 1I = 2 N = 1,3,5 . . . G = 2between and including |I +S| and |I−S|.The possible quantum numbers for the ground and excited states are tabulatedin Table 3.2. For a ground state s-wave collision (N = 0) we find that there is onlyone allowed value for the total spin: G = 0. From an initial state with N = 0 andG = 0, we see that there is only one possible transition to an excited “Σ” state:(N = 0,G = 0)→ (N′ = 1,G′ = 0). For a ground state p-wave collision, the initialstate is (N = 1,G = 1) and there are two possible transitions to the excited tripletstate: (N = 1,G = 1)→ (N′ = 0,G′ = 1) and (N = 1,G = 1)→ (N′ = 2,G′ = 1).In both cases, there are two possible values of the total nuclear spin: I = 0 or 2.The initial energy of the colliding complex is lower than the hyperfine centerof gravity by a factor of 2a2S. This extra energy must be added to the D1 transitionfrequency when determining the binding energies of each vibrational level in thestudied states.3.3 Spectroscopy of the c(13Σ+g ) potentialThe measurements of the seven vibrational levels v′ = 20− 26 of the 13Σ+g ex-cited state of 6Li2 molecules cover a completely unexplored spectral range for this66Figure 3.4: The 13Σ+g potential studied in this work (solid line). The horizon-tal lines indicate experimentally measured bound levels for 6,6Li2. Thepresent work includes high resolution data from seven new vibrationalstates (v′ = 20 to 26) including the N′ = 0,1,2 rotational states in eachcase. The theoretical long-range potential according to [126] is shownby the dotted line. Figure taken from [49].molecule. As shown in Fig. 3.4 we bridge a gap between measurements of thedeeply lying v′ = 1− 7 levels by Fourier transform spectroscopy (of both 7,7Li2and 6,6Li2 molecules) [123, 124] and measurements of the binding energies of lev-els v′ = 62−90 of 7,7Li2 and v′ = 56−84 of 6,6Li2 by photoassociation of atomsin a magneto-optic trap [125].The experiments are performed in an ensemble prepared as described in Sec-tion 2.3.4 using the trap arrangement shown in Fig. 2.11a. At the end of the finalevaporation step the magnetic field is lowered to a very small value and PA lightfrom a single-frequency, tunable Ti:sapphire laser beam illuminates the atomiccloud for a certain exposure time. The PA light is a single beam that propagatesco-linearly with one of the arms of the lower-power CDT and is focused to a waist(1/e2 intensity radius) of 50 µm. The light is linearly polarized and aligned alongthe direction of the bias magnetic field used for the measurements of p-wave Fes-67hbach enhanced PA. When the bias field is off, there persists a residual magneticfield below 400 mG. For these experiments, the power of the PA light is up to 100mW corresponding to an intensity of 1270 W/cm2. When the photon energy hνPAequals the energy difference between the unbound state of a colliding atomic pairand a bound molecular excited state, molecules form at a rate proportional to theatom-atom collision rate and atoms are subsequently lost from the trap. This lossoccurs because the excited state molecule either decays into the unbound contin-uum of two free atoms with sufficient energy to be lost from the shallow CDT or itspontaneously decays into a bound state molecule which is not detected in our atomnumber measurement. The probability of this latter event (called fluorescence) canbe quite high when exciting particular vibrational levels in the 13Σ+g excited state[127]. The excited lithium molecules gain kinetic energy on the order of the recoilenergy Erec = p2/2M = h2ν2/(2Mc2) as a result of the decay to the ground state.Here, p is the momentum of the emitted photon with frequency ν , M is the massof a lithium molecule, h and c are the Planck’s constant and the speed of light,respectively. In units of temperature we typically have Erec ∼ 1−4µK, dependingon the final state of the ground state molecule. Such energy is usually below thetrap depths used in the experiments (1−10µK) therefore the fluorescence decaysof the excited molecules result in trapped ground state molecules. After the atomsare illuminated by the photoassociation light for some exposure time, the numberof atoms remaining is determined by an absorption image of the cloud immediatelyafter the extinction of the CDT.We observed that the PA spectrum of each vibrational level had associatedwith it three narrow (below 10 MHz FWHM) features distributed across a rangeof 0.7 GHz as shown in Fig. 3.5. Figure 3.6 shows a higher resolution scan of thesecond feature shown in Fig. 3.5. In order to reduce as much as possible the thermalbroadening and the inhomogeneous AC Stark shift produced by the optical dipoletrapping potential, these data were obtained in a very shallow trap (Utrap/kB ∼8µK) and an ensemble temperature of 800 nK, a temperature well below the Fermitemperature for this two component Fermi-gas (T/TF = 0.4). We then verifiedthat these PA resonances arise from collisions between atoms in states |1〉 and|2〉 by using a state-selective resonant pulse of light to remove all atoms in eitherof the two states. The spin purification was done at the end of the preparation68Figure 3.5: Normalized 6Li atom number as a function of photo-associationlaser energy hνPA after a 2 second hold time with a residual magneticfield below 400 mG and a PA laser intensity of IPA = 635 W/cm2. Thesethree resonances correspond to a transition from an initial unboundmolecular state with N = 0,G = 0 to the v′ = 21 vibrational level ofthe 13Σ+g excited state with N′ = 1, G′ = 0. The ensemble temperaturewas 15 µK.sequence, and we observed the absence of these atom loss features with either oneof the states removed. The spin purification was performed at a high magneticfield (typically 700 G), where the optical transitions from the |1〉 and |2〉 states tothe excited 2p3/2 manifold are well separated. In addition, the field is sufficientlylarge to disrupt the hyperfine coupling and these transitions become approximately“closed” such that the excited state atom returns to the original ground state witha large probability allowing each atom to scatter many photons during the pulseand subsequently leave the trap. To rule out the absence of these loss featuresdue to a simple reduction of the density, we observed a reappearance of the PAfeatures when using an incoherent mixture of the |1〉 and |2〉 states with the sametotal number of particles and temperature as the ensembles after spin purification.Given that p-wave collisions are dramatically suppressed at these temperatures andthat these PA loss features were visibly enhanced by the s-wave FR, we inferred69Table 3.3: Experimentally measured PA resonances for s-wave collisions inan incoherent mixture of the |1〉 and |2〉 states of 6Li. These three PAresonances correspond to a transition from an initial unbound molecu-lar state with N = 0, G = 0 to the vth vibrational level of the 13Σ+g ex-cited state with N′ = 1. The spin-spin and spin-rotation coupling split theexcited state into three sub-levels producing the three PA features cor-responding to quantum numbers (N′ = 1,J′ = 1), (N′ = 1,J′ = 2), and(N′ = 1,J′ = 0) respectively. The absolute uncertainty in each of thesemeasurements is ±0.00002 cm−1 (±600 kHz).v′ 1st 2nd 3rdcm−1 cm−1 cm−1GHz GHz GHz20 12237.17755 12237.18587 12237.20126366861.3537 366861.6031 366862.064521 12394.39726 12394.40535 12394.42039371574.6820 371574.9245 371575.375422 12546.06767 12546.07552 12546.09025376121.6465 376121.8818 376122.323423 12692.17316 12692.18080 12692.19509380501.7789 380502.0079 380502.436324 12832.70080 12832.70820 12832.72214384714.6916 384714.9134 384715.331325 12967.64150 12967.64862 12967.66219388760.1018 388760.3254 388760.732226 13096.99114 13096.99804 13097.01125392637.9166 392638.1235 392638.5195that they arise from s-wave collisions between atoms in states |1〉 and |2〉. Thus,they correspond to a transition from an initial unbound molecular state with N = 0,G = 0 to an excited state with N′ = 1,G′ = 0 (assuming G is a good quantumnumber). The locations of these three features for each of the seven vibrationallevels is provided in Table p-wave Feshbach resonance enhanced photoassociationFor each of the vibrational states we probed in the c-state potential, we locatedthe PA resonances to the N’=0 and N’=2 rotational levels associated with p-wave70Figure 3.6: High resolution scan of the normalized 6Li atom number as afunction of photo-association laser energy hνPA after a 750 ms holdtime with zero bias magnetic field and a PA laser intensity of IPA =635 W/cm2. This is the second of the three resonances shown in Fig. 3.5corresponding to a transition from an initial unbound molecular statewith N = 0, G = 0 to the v′ = 21 vibrational level of the 13Σ+g excitedstate with N′ = 1, G′ = 0. The ensemble temperature was 800 nK. TheFWHM of this loss peak is 0.00048 cm−1 (14.4 MHz).ground-state collisions. However, these features were only observable in our ex-periment when measures were taken to enhance the PA scattering rate. In order toobserve these PA resonances, we enhanced the p-wave scattering rate by stoppingthe evaporation at an ensemble temperature of 250 µK and by holding the mag-netic field at 185 G during the PA stage. This magnetic field is near the p-waveFeshbach resonance between the |1〉 and |2〉 states at 185.1 G [109]. Due to theFeshbach resonance enhancement of inelastic ground-state collisions, the ensem-ble particle loss in the absence of the PA light was approximately 50% during the2 second hold time. Additional loss was induced when the light was near a PAresonance. Figure 3.7 shows the loss spectrum for a transition from an initial un-bound molecular state with N = 1,G = 1 to the v′ = 20 vibrational level of the13Σ+g excited state with N′ = 2,G′ = 1. For each of the seven vibrational levels,71Figure 3.7: Normalized 6Li atom number as a function of photo-associationlaser energy hνPA after a 2 second hold time. The circles are for anensemble temperature of 250 µK at 185 G, and four distinct featuresare observed. The diamonds denote the atom loss for an ensemble tem-perature of 15 µK and at a magnetic field of 184.7 G. At this lowertemperature, these loss features are seen to result from multiple PA res-onances that are unresolvable at 250 µK. These PA features arise fromp-wave ground-state collisions and are enhanced by proximity to a p-wave Feshbach resonance between the |1〉 and |2〉 states at 185.1 G. Thero-vibrational level shown here is v′ = 20,N′ = 2we observed at least 4 (3) distinct loss features for transitions to the N′ = 2,G′ = 1(N′ = 0,G′ = 1) final state. By evaporating the ensemble to 15 µK and holding themagnetic field at 184.7 G, we observed that each of these loss features results frommultiple PA resonances that are unresolvable at 250 µK. The locations of the lossfeatures observed at 250 µK for each of the seven vibrational levels is providedin Tables 3.4 and 3.5. These measurements were performed in the absence of thecomb stabilization. Instead, the Ti:sapphire laser was referenced to the wavemeterwhose uncertainty is 60 MHz.To our knowledge, the enhancement of photoassociation with a p-wave Fesh-72Table 3.4: Experimentally measured PA resonances for p-wave collisions inan incoherent mixture of the |1〉 and |2〉 states of 6Li held at a magneticfield of B = 185 G. Each of these values was extracted by fitting a lossspectrum like that shown in Fig. 3.7. These PA resonances correspondto a transition from an initial unbound molecular state with N = 1,G = 1to the vth vibrational level of the 13Σ+g excited state with N′ = 0,G′ = 1.While the precision in these measurements is 0.001 cm−1, the uncer-tainty, limited by the wavemeter, is ±0.002 cm−1v′ 1st 2nd 3rdcm−1 cm−1 cm−1GHz GHz GHz20 12236.388 12236.407 12236.424366837.68 366838.25 366838.7621 12393.629 12393.648 12393.664371551.65 371552.22 371552.7022 12545.320 12545.338 12545.355376099.23 376099.77 376100.2823 12691.446 12691.465 12691.480380479.98 380480.55 380481.0024 12831.995 12832.012 12832.029384693.53 384694.04 384694.5525 12966.957 12966.975 12966.991388739.59 388740.13 388740.6126 13096.326 13096.346 13096.362392617.98 392618.58 392619.06bach resonance is the first example of FOPA (Feshbach-Optimized Photoassocia-tion) [54] for a p-wave FR.3.3.2 Spin-spin and spin-rotation constantsIn order to properly label the three PA resonances (associated with ground states-wave collisions) observed for each ro-vibrational state given spin-spin and spin-rotational coupling, we redefine ~J to be the total angular momentum apart fromnuclear spin, ~J ≡ ~N +~S. Here, a magnetic coupling between ~S and ~N (involving aninteraction term of the form Hˆspin−rot = γv~N ·~S) as well as a spin-spin coupling term(of the form Hˆspin−spin = 2λv[Sˆ2z − Sˆ2/3]) cause a splitting of the rotational levels,73Table 3.5: Experimentally measured PA resonances for p-wave collisions inan incoherent mixture of the |1〉 and |2〉 states of 6Li held at a magneticfield of B = 185 G. Each of these values was extracted by fitting a lossspectrum like that shown in Fig. 3.7. These PA resonances correspondto a transition from an initial unbound molecular state with N = 1,G = 1to the vth vibrational level of the 13Σ+g excited state with N′ = 2,G′ = 1.While the precision in these measurements is 0.001 cm−1, the uncer-tainty, limited by the wavemeter, is ±0.002 cm−1v′ 1st 2nd 3rd 4thcm−1 cm−1 cm−1 cm−1GHz GHz GHz GHz20 12238.757 12238.772 12238.780 12238.795366908.70 366909.15 366909.39 366909.8421 12395.936 12395.951 12395.958 12395.973371620.81 371621.26 371621.47 371621.9222 12547.567 12547.579 12547.587 12547.601376166.60 376166.96 376167.19 376167.6123 12693.628 12693.642 12693.648 12693.665380545.39 380545.81 380545.99 380546.5024 12834.113 12834.128 12834.134 12834.150384757.03 384757.48 384757.66 384758.1425 12969.011 12969.026 12969.032 12969.047388801.17 388801.62 388801.80 388802.2526 13098.315 13098.332 13098.339 13098.355392677.60 392678.11 392678.32 392678.80previously labeled by N, according to the J quantum number, given by J = (N+S),(N + S− 1), (N + S− 2),· · · , |N− S|. Therefore, each level with a given N(≥ S)consists of 2S + 1 sub-levels, and the number of sub-levels is equal to the spinmultiplicity. However, for N < S, the number of sub-levels is equal to 2N + 1(the rotational multiplicity). Hence, all N = 0 levels do not split. For a particularro-vibrational state, |ν ,N 〉, with a total spin S = 1, the rotational energy is given74Figure 3.8: Spin-splitting parameter D = 2λ for the lowest triplet a(13Σ+u )state (dashed line) and for the c(13Σ+g ) state (solid line) of Li2 molecule.The figure taken from [129].by [121, 128]FJ=N+1 = BvN(N +1)+(2N +3)Bv−λv−√(2N +3)2B2v +λ 2v −2λvBv + γv(N +1)FJ=N = BvN(N +1)FJ=N−1 = BvN(N +1)− (2N−1)Bv−λv+√(2N−1)2B2v +λ 2v −2λvBv− γvN, (3.8)where λv and γv are constants. Here, λv is related to the spin-spin interaction andit describes the coupling between the total spin, ~S, and the molecular axis; γv isrelated to the spin-rotation interaction and it is a measure of the coupling between~S and ~N. Under most circumstances, these two constants describe small effectswhich are not spectroscopically resolvable and are typically ignored in the Dunhamexpansion. However, at the level of resolution in the current experiment, one needsto take into account these second-order perturbations.When spin-spin and spin-rotation couplings are small (Bv |λv|, |γv|) we can75Table 3.6: The values for the spin-spin interaction constant, λv, and thespin-rotation interaction constant, γv, determined from Eq. 3.9 and thepeak spacings reported in Table 3.3. The uncertainty in these values is±400 kHz. The λv values are plotted in Fig. 3.9 along with their expectedvalues determined from ab initio calculations.v′ λv (MHz) γv (MHz)20 -348.2 -14.521 -339.4 -14.522 -331.1 -14.723 -321.7 -14.224 -312.2 -14.425 -303.6 -14.026 -294.3 -14.3simplify Eq. 3.8 toFJ=N+1 = BvN(N +1)−2N +22N +3λv + γv(N +1)FJ=N = BvN(N +1)FJ=N−1 = BvN(N +1)−2N2N−1λv− γvN. (3.9)In addition, when spin-spin coupling is much more important than spin-rotationcoupling (|λv|  |γv|), the energy ordering results from the λv terms, and we canlabel these three peaks in Table 3.3, energetically from low to high, as (N′ = 1,J′ =1), (N′ = 1,J′ = 2), and (N′ = 1,J′ = 0) because λv is negative.Using the peak spacings reported in Table 3.3 and Eq. 3.9, we extract the twoparameters, λv and γv. The determined λv constants as a function of v′ are plottedin Fig. 3.9. The dashed line is provided to show its trend. These results agreewell with the previous ab initio calculation for lithium diatoms [129]. By usingFig. 3.8 (taken from Ref. [129]) and averaging λ (R) over the internuclear distanceR using the wave functions corresponding to the eigenfunctions of the excited statepotential curve we refined with our data, we estimate these ab initio λv constantsfor all v′ states and plot those also in Fig. 3.9. Note, the uncertainty of the ab initioresults given in Fig. 3.9 is at least a few tens of MHz. This results from the es-7620 21 22 23 24 25 26Vibrational quantum number-380-360-340-320-300Spin-spin interaction parameter: λss (MHz)ab initio  resultsExperimental ResultsFigure 3.9: The experimentally determined (circles) and ab initio computed(squares) spin-spin interaction constants, λv, as a function of the vibra-tional quantum number for the 13Σ+g electronic state. These constantswere determined from the frequency splittings of the three features ob-served for the N = 0→ N′ = 1 transition. The uncertainty in these val-ues is ±400 kHz. The dashed lines are guides to the eye. Figure takenfrom [49].timated error of the original ab initio calculation (a few percent corresponding to≈ 10−30 MHz) and the error (≈ 10 MHz) associated with our digitization of thedata in Fig. 3.8, as well as the fact that the ab initio calculation was likely donefor 7,7Li2 rather than 6,6Li2. This comparison of λv obtained from experimentaldata and that obtained from ab initio calculations clearly demonstrates the validityof the current model to label separate peaks in Table 3.3. The values for λv andγv determined from our data are provided in Table 3.6. The uncertainty in theseparameters is ±400 kHz and results from the uncertainty in the PA resonance po-sitions. Using Eq. 3.8, we verified that the uncertainty in the exact value for Bvdoes not contribute significantly to the uncertainty in these parameters. We notethat this is the first direct measurement of the spin-spin and spin-rotation coupling77Figure 3.10: v′ = 24, N′ = 1, J′ = 1, magnetic field dependence for low mag-netic fields. Around 10 G the line shows signs of splitting into Zeemansublevels (inset).constants in a diatomic lithium system.3.3.3 Systematic shiftsWhile the absolute uncertainty of our PA measurements made using the frequencycomb is ±600 kHz, the data was taken in the presence of a small but non-zeromagnetic field and in an optical dipole trap with a known intensity. These residualfields as well as the PA laser itself can lead to a systematic shift of the resonancepositions from their zero-field values. Therefore, in an effort to quantify the role ofthe PA laser intensity, the CDT laser intensity, and the residual magnetic field onthe PA loss features, we varied each one and measured the PA resonance positionand width for various excited states. In each case, we assumed a linear dependenceand determined a shift rate of the resonance position with the corresponding fieldstrength. The uncertainty in this rate is a one-sigma statistical uncertainty on theslope of the linear fit.When varying the photoassociation laser intensity from IPA = 0.19 kW/cm2to IPA = 1.27 kW/cm2 we observed that the centroid of the first feature (J′ = 1)78Figure 3.11: v′ = 24, N′ = 1, J′ = 2, magnetic field dependence for low mag-netic fields. The loss feature splits into peaks corresponding to Zeemansublevels (insets). To estimate the influence of the magnetic field onthe position of the photoassociation feature the center of gravity of theline at each magnetic field has been determined (near horizontal line).associated with the v′ = 26 excited state shifted to higher frequencies at a rateof 471± 433 kHz per kW/cm2. When the CDT laser (1064 nm) intensity wasvaried from 5.4 kW/cm2 (145 mW total CDT power) to 140 kW/cm2 (3.1 W to-tal CDT power) the PA feature centroid associated with the v′ = 24, J′ = 1 stateshifted down in frequency at a rate of −(19± 1.2) kHz per kW/cm2. The reso-nance positions reported in Table 3.3 were determined using a PA laser intensity ofIPA = 635 W/cm2, and a CDT intensity of 7.5 kW/cm2. Assuming the differentialac Stark shift is the same for all excited states, the reported values are thereforeshifted lower by 142±9 kHz due to the CDT and higher by 300±274 kHz due tothe PA laser than their extrapolated position at zero differential ac Stark shift. Theoverall AC Stark shift of the resonance positions is thus higher by 157 kHz withan uncertainty of ±274 kHz. Both this shift and uncertainty are small comparedto the absolute uncertainty of the frequency comb. For the resonance positions re-ported in Tables 3.4 and 3.5, the trapping power was larger (40 W total) and thedifferential AC Stark shift due to the CDT is estimated to be −(15±1) MHz.79Figure 3.12: v′ = 24, N′ = 1, J′ = 0, magnetic field dependence for low mag-netic fields. No Zeeman splitting has been observed, as opposed to themeasurements of the J′ = 1 and J′ = 2 levels. To estimate the system-atic shift a linear dependence near 0 G has been assumed.When the magnetic field was varied from 0 G to 10 G the PA features associatedwith the v′ = 24, J′ = 1, J′ = 2, and J′ = 0 states were observed to shift and, in thecase of J′ = 1 and J′ = 2, to broaden and eventually split into multiple resolvablepeaks (see Figs. 3.10,3.11,3.12). In each case, we measured the PA feature centerof mass and found that when the magnetic field was varied from 0 to 1 G, thebarycenter of the PA features moved by −(91.2± 18.3) kHz for the J′ = 1 state,+(46±28) kHz for the J′ = 2 state, and +(74.5±30.1) kHz for the J′ = 0 state.Since the resonance positions reported in Table 3.3 were determined in the presenceof a residual magnetic field below 400 mG, the uncertainty in their positions dueto the magnetic field was below 50 kHz for all J states and thus small compared tothe absolute uncertainty of the frequency comb.80Figure 3.13: High resolution scan of the normalized 6Li atom number asa function of photoassociation laser energy hνPA after a 1 s holdtime with zero bias magnetic field and a PA laser intensity of IPA =755 W/cm2. The feature corresponds to a transition from an initialunbound molecular state with N = 0, G = 0 to the v′ = 31 vibrationallevel of the 11Σ+u excited state with N′ = 1, G′ = 0. The ensembletemperature was 600 nK. The FWHM of the loss signal is 13 MHz.3.4 Spectroscopy of the A(11Σ+u ) potentialIn its gaseous form 6Li2 exists predominantly in the ground state, X(11Σ+g ). Forthis very reason various spectroscopy methods can be easily applied to probe ex-cited state levels of singlet potentials. The A(11Σ+u ) potential (in conjunctionwith the ground X-state), especially, has been extensively studied and many ro-vibrational levels in the range v = 0− 88 have been measured [130] prior to thiswork. However, the only photoassociation spectroscopy of this potential that pre-cedes our work has been done in a MOT and v′ = 62− 88 levels have been re-ported [104, 125]. Here we present the binding energy measurements of sevenvibrational levels v′ = 29−35 of the A(11Σ+u ) excited state of6Li2 molecules withan absolute uncertainty of ±600 kHz by photoassociating a quantum degenerateFermi gas of lithium atoms held in a shallow optical dipole trap, with the atomic81ensemble prepared as described in Section 2.3.4. These measurements representan observation of the N′ = 1 rotational level 1 for each of the reported vibrationallevels and have absolute uncertainties that are 25 to 500 times smaller than otherdata available for the A-state.The experiments are performed in an ensemble of 4× 104 atoms with equalpopulations in the states |1〉 and |2〉. After the preparation stage, we apply a ho-mogeneous magnetic field that cancels any residual field remaining at the locationof the atoms. The background field after cancellation is verified to be less than20 mG via rf spectroscopy between the F = 1/2 and F = 3/2 ground hyperfinelevels of 6Li (for details see Section 2.3.4). The dipole trap arrangement used forthese measurements is shown in Fig. 2.11b. The photoassociation light propagatescolinearly with the recycled crossed optical dipole trap (Fig. 2.11b) and illuminatesthe cloud for 2 s. The atom number is then measured and from the loss of atomsthe location of a PA resonance can be extracted. Figure 3.13 shows an example ofa resulting feature.The photoassociation spectrum observed for seven vibrational levels (v′= 29−35) of the A(11Σ+u ) state arises from s-wave collisions between atoms in states |1〉and |2〉; see Table 3.7. To reduce thermal broadening and the inhomogeneous acStark shift produced by the CDT potential, these data were obtained in a low inten-sity trap (ICDT = 9.6 kW/cm2). At this low trap power, the ensemble temperatureis 800 nK (T/TF = 0.4). In order to fully characterize the systematic shifts of theresonance location due to the CDT and PA laser, we varied the CDT intensity from9.6 to 55 kW/cm2 and the PA laser intensity from 65 to 760 kW/cm2 for each ofthe seven vibrational levels.Assuming that the ac Stark shift from each laser is independent, and the shiftof the resonance position is proportional to the laser intensity, the shift in the res-onance locations can be fit to extract the shift rate due to the dipole trap and PAbeam (Table 3.7). In addition, the field free resonance location can be inferred(Table 3.8).The 1σ statistical error on the fit to the resonance location, and to the extrap-olated field free resonance location is typically 250 kHz, which is small compared1For levels v′ = 29− 35 the rotational levels N′ = 1 have not been observed in previous experi-ments (Refs. [130, 131]).82Table 3.7: Experimentally measured PA resonances for s-wave collisions inan incoherent mixture of the |1〉 and |2〉 states of 6Li. These PA reso-nances correspond to a transition from an initial unbound molecular statewith N = 0, G = 0 to the vth vibrational level of the A1Σ+u excited statewith N′ = 1,G′ = 0. For these measurements, the CDT intensity was9.6 kW/cm2 and the PA laser intensity was 65 W/cm2. The absolute un-certainty in each of these measurements is ±600kHz. The ac Stark shiftof each resonance induced by the PA laser and the CDT laser is alsolisted, where the number in brackets is an estimation of the 1σ error onthe last digit(s).v Feature ODT Shift Rate PA Shift Rate(GHz) kHz / (kW/cm2) kHz / (kW/cm2)29 363113.1067 199(6) -745(661)30 368015.0436 -546(11) -2120(783)31 372780.6714 44(4) -73(228)32 377406.2393 -79(6) 803(707)33 381887.7859 100(5) 80(253)34 386221.1190 73(5) 272(265)35 390401.8749 -9(10) -826(437)Table 3.8: Extrapolated field free resonance locations based on the table 3.7.v′ Field free featurecm−1 GHz29 12112.14946 363113.105830 12275.66069 368015.049231 12434.62472 372780.670932 12588.91710 377406.240133 12738.40534 381887.784834 12882.94978 386221.118135 13022.40482 390401.87583to the absolute uncertainty of the frequency comb. Since the magnetic field is con-firmed to be less than 20 mG, any systematic shifts due to the residual magneticfield are negligible. Note that at low dipole trap intensities, the CDT potential istilted due to gravity which leads to a spilling of atoms out of the trap. Typically,this tilt can be offset using a magnetic field gradient. However, it is not feasible tocompensate for the tilt due to gravity and keep the magnetic field in (and across)the region of the atoms at a small (less than 20 mG) value. Therefore, 9.6 kW/cm2is the lowest CDT intensity we can use for PA spectroscopy without incurring largeatom loss.3.4.1 v′ = 30 anomalyUnlike for the other vibrational levels probed, for the v′ = 30 vibrational levelwe observe a splitting of the resonance into many distinctive features shown inFigs. 3.14a to d. The splitting does not depend on the PA light intensity as con-firmed by the measurements for different PA light intensities that varied by a factorof 30. For the lowest PA and dipole trap intensities, 0.13 kW/cm2 and 9.7 kW/cm2,respectively, the splitting is not present (Fig. 3.14d). When the dipole trap intensityis increased by a factor of five the level experiences a strong ac Stark shift and twodistinct loss features separated by 18 MHz appear (Fig. 3.14e to f). We attributethis splitting to a CDT laser-induced coupling of v′ = 30 to another molecular levelin one of the potentials dissociating to the 2p+ 2p asymptote. Performing spec-troscopy in a dipole trap operating at various frequencies would allow us to modifythe strength of the coupling thus verifying our assertion. In our setup this couldbe achieved by changing the temperature of the laser used for optical trapping, re-sulting in a frequency change on the order of 10 GHz. Such a test has not beenperformed yet.84Figure 3.14: Normalized 6Li atom number as a function of photoassociationlaser energy hνPA after a 1 s hold time with zero bias magnetic field.a)-c) The photoassociation laser is locked to its cavity and the accuracyis on the order of the wavemeter read out accuracy, 60 MHz. The datais taken for three different PA laser intensities IPA a) 0.13 kW/cm2,b) 1.3 kW/cm2 and c) 3.3 kW/cm2 at the dipole trap intensity of410 kW/cm2 with the sample temperature of 25 µK. d)-f) The pho-toassociation laser is locked to the frequency comb. The data is takenwith IPA = 0.13 kW/cm2 for three different dipole trap intensities d)9.7 kW/cm2, e) 32.5 kW/cm2 and f) 54 kW/cm2. The feature cor-responds to a transition from an initial unbound molecular state withN = 0, G = 0 to the v′ = 30 vibrational level of the 11Σ+u excited statewith N′ = 1, G′ = 0. The splitting has not been observed in other mea-sured levels.85Chapter 4Dark state spectroscopy of boundmolecular statesThe main results of this Chapter are based on• M. Semczuk, W. Gunton, W. Bowden and K. W. Madison, “Anomalousbehavior of dark states in quantum gases of 6Li.”, Phys. Rev. Lett. 113,055302, July (2014) [52].Coherent dark states [132] in ultracold atomic gases lie at the heart of phe-nomena such as electromagnetically induced transparency [133], slow light [134]and coherent population transfer [135, 136]. They are useful for precision spec-troscopy of molecular levels [137] and studies of their properties [138]. In recentyears such superposition states between atoms and molecules have been demon-strated in ultracold metastable 4He [137] and 23Na [139] as well as in a Bose-Einstein condensate of 87Rb atoms [140]. Other important examples of systemswhere molecule-molecule dark states have been created include 87Rb [141, 142],84Sr [143] and 133Cs [144]. Due to the nature of dark states, coherence is achievedbetween the initial and the final state and as a result of destructive interference, nointermediate molecular levels are populated. The experiments prior to our workfocused on bosonic atomic species with a notable exception of a non-degeneratefermionic molecules of KRb [23, 145]. Most of these experiments have been re-stricted to magnetic fields close to Feshbach resonances, and due to the increasedthree-body recombination [146] the sample would rapidly decay (typically in lessthan 10 ms).The ability to create atom-molecule or molecule-molecule dark states is a pre-requisite to coherently transfer weakly bound Feshbach molecules to a deeplybound level in the electronic ground state using stimulated Raman adiabatic pas-86sage technique (STIRAP) [136]. When the initial state is a Feshbach molecule andthe final state a deeply bound level in the lowest singlet or triplet potential (includ-ing the rovibratonal ground state), this method has been shown to work remark-ably well with 87Rb [141, 142], 84Sr [143], 133Cs [144, 147], 40K87Rb [23, 145],87Rb133Cs [148, 149] and 23Na40K [150]. The initial Feshbach molecules are,however, short lived (on the order of 10 ms) [151] if they consist of bosonic atoms;therefore to improve the efficiency of the transfer they needed to be separated ina 3D optical lattice to avoid three-body recombination. One of the main goals ofthese experiments has been to create a Bose-Einstein condensate (BEC) or a de-generate Fermi gas (DFG) of molecules in the absolute rovibrational ground state.The required temperatures and phase space densities have not been achieved yetand so far it is not clear if simply melting the 3D lattice as proposed by Jaksch etal. [152] can lead to the final goal.Theoretical proposals suggest that dark states can also be created in degener-ate Fermi gases [153, 154] and they have great potential as a probe of many-bodyphysics (e.g. BCS pairing and superfluidity) [153, 155–157] avoiding the final stateeffects that complicate the interpretation of the rf spectra of the BCS pairs [74, 90].As proposed in the context of 6Li, molecular dark states can be used for the op-tical control of magnetic Feshbach resonances (FR) [158, 159]. This method notonly suppresses spontaneous scattering but also provides larger tuning of the in-teractions than a single frequency approach [100, 101] while enabling independentcontrol of the effective range. The proposal applies to all alkali species and hasimportant consequences to anyone interested in the dynamics of Bose and Fermigases at unitarity.As opposed to molecules consisting of bosons, a two-component Fermi gasof 6Li forms Feshbach molecules that are long lived, with lifetimes approaching10 s [160]. This is a direct consequence of the Fermi statistics for the 6Li atoms thatleads to the suppression of the collisional relaxation of the weakly bound dimers todeep bound states [87]. Additionally, depending on the value of the magnetic fieldB with respect to the Feshbach resonance at B = 832.18 G, a Bose-Einstein con-densate of Feshbach molecules [65, 113] (B < 832.18 G) or BCS-like pairs [161](B > 832.18 G) can be created. The demonstration of dark states in the BEC-BCScrossover regime (Section 4.6) opens up possibilities of using STIRAP to transfer87the initial Bose-condensed state to a sample of ground state molecules that is ex-pected to be also Bose-condensed. 1 The sample is also “bulk” as no separationusing a 3D lattice is required, a feature not presently available with non-fermionicsystems currently under consideration and allowing greater flexibility of studies ofeventual ground-state molecules.Building on the method demonstrated by Moal et al. [137] we have used darkstates created in the non-interacting Fermi gas at 0 G to determine the bindingenergy of the v′′ = 9, N′′ = 0 ro-vibrational level of the a(13Σ+u ) potential with ac-curacy improved by a factor of 500 over [162] and [163]. The molecular hyperfinestructure of this level has been resolved and the binding energy of the v′′ = 38,N′′ = 0 ro-vibrational level of the X(11Σ+g ) potential has been measured for thefirst time in the ultracold regime. Measurements of this type do not require darkstate spectroscopy method but they benefit significantly from the superior accu-racy that is hard to reach with commonly used two-color photoassociation spec-troscopy [162–164]. Since both the hyperfine structure of v′′ = 9 and the bindingenergy of v′′ = 38 have not been observed prior to this work our result might beof interest to spectroscopists because Li2 is the simplest neutral diatomic moleculeafter H2. Additionally, the a(13Σ+u ) state of H2 is not bound [165] therefore Li2seems like the best candidate to study the properties of this potential. Our resultsmight have impact not only on the models for molecular potentials but, more im-portantly, on the molecular theory itself because the observed hyperfine splittingsdo not match what is expected based on the theory outlined by A. J. Moerdijk andB. J. Verhaar [166] (see Section 4.4).4.1 Two-color photoassociation and dark states: theoryHere we present a brief overview of the scattering process in the presence of twolaser fields based on the theory of laser assisted collisions developed by Bohn andJulienne (for a detailed discussion see [167, 168]). The analysis presented here islimited to three-level systems in the Λ configuration as shown in Fig. 4.1, whichare the most relevant to the experiments discussed in this Chapter.1The result depends, among other factors, on the polarizbility of the ground state molecules asthis influences the trapping frequencies and, consequently, changes the critical temperature.88In a collision process a pair of cold atoms with relative energy E (typically< 10µK in our experiments) approaches one another guided by their interactionpotential U0. In the presence of two laser fields the atom pair can be promoted bythe first laser with frequency ω1/2pi to a bound molecular state |b〉 with energyEb. From the level |b〉 the second laser with frequency ω2/2pi can drive the atompair to another bound molecular level |g〉 with energy Eg, here chosen to belong toeither singlet or triplet ground molecular potential.There exists the possibility of a spontaneous emission event in which the ex-cited molecule decays into bound ground states or into dissociative continuum oftwo ground-state atom which may have enough kinetic energy to leave the trap. Inboth cases the observed trapped atom number decreases. Another trap loss mech-anism is caused by nonradiative decay to other molecular states that cannot bedetected in a typical experiment.The formalism developed by Bohn and Julienne relates the two-color photoas-sociative loss to the thermally averaged value of the loss of population from theexcited state |b〉 (|Sb|2) and decay of molecules in the probed ground singlet ortriplet potential |g〉 (|Sg|2)2:K(T,∆1,∆2, I1, I2,γb,γg) = 〈pivk2∞∑l=0(2l +1)(|Sb|2 + |Sg|2)〉. (4.1)The brackets denote the appropriate thermal average over a distribution of relativevelocities v, at temperature T , and k =√2µE/h¯2, where µ and E are the reducedmass and kinetic energy of the colliding par. The detunings ∆1 and ∆2 of laserswith intensities I1 and I2 are defined in Fig. 4.1. Assuming Maxwellian distribu-tion at temperature T and taking into account that for lithium only s-wave (l = 0)collisions take place at temperatures in the microkelvin range (and below), we ob-tainK(T,∆1,∆2, I1, I2,γb,γg) =1hQT∫ ∞0(|Sb|2 + |Sg|2)e−E/kBT dE, (4.2)where the partition function for the reduced mass µ is QT = (2pikBTµh2 )3/2.From the scattering amplitudes Sb and Sg derived by Bohn and Julienne in2Sb and Sg are diagonal elements of the scattering matrix (S-matrix).89Figure 4.1: ∆1 and ∆2 denote the detunings from the one-photon and two-photon resonance, respectively, of lasers with frequencies ω1/2pi andω2/2pi and intensities I1 and I2. Ω1 and Ω2 are Rabi frequencies. Theexcited molecular state of energy Eb |b〉 spontaneously decays with arate γb to levels outside this scheme. The molecular state of energy Eg|g〉 is attributed a decay rate γb which phenomenologically takes intoaccount losses through inelastic collisions and laser induced dissocia-tion, e.g., when laser 1 couples |g〉 to the unstable state |b〉 . E is therelative energy of a colliding pair |a〉 .Ref. [168] we obtain scattering probabilities|Sb|2 =γbγs[(E−∆2)2 + γ2g/4]n(E,γs,γb,γg,∆1,∆2), (4.3)|Sg|2 =h¯2Ω22γgγsn(E,γs,γb,γg,∆1,∆2), (4.4)n(E,γs,γb,γg,∆1,∆2) ={[E− (∆1)](E−∆2)− (h¯Ω2)2− γg(γb + γs)/4}2+{(γb + γs)(E−∆2)/2+ γg[E− (∆1)]/2}2, (4.5)where the symbols are consistent with the convention used throughout the thesis(explained in Fig. 4.1) rather than Ref. [168].In the above equations γs represents the stimulated absorption/emission rate90generated by the laser due to coupling of levels |a〉 and |b〉 :γs = 2piV 2ab|〈a|b〉|2 = 2pi(h¯Ω1)2, (4.6)where Vab is a radiative coupling of levels |a〉 and |b〉 . The light shift induced bycoupling |a〉 and |b〉 by the laser with intensity I1 is incorporated into ∆1.In order to relate |Sb|2 and |Sg|2 to observables typically used in ultracold atomexperiments (usually the atom number N) let’s consider the loss of population fromthe dipole trap in the presence of two laser fields. It can be described by the evolu-tion of the atomic densityn˙ =−2Kn2−Γn (4.7)with Γ being the one-body loss rate due to background collisions and off-resonantscattering from the photoassociation lasers. K determines the spectrum of the pho-toassociative loss and is obtained from combining Eqs. 4.2, 4.3, 4.4 and 4.5. Theintegration of Eq. 4.7 gives the atom number N(t) as a function of the photoasso-ciation time t:N(t) =Nt1+ 2NtKeffVtΓV 20(eΓt −1)(4.8)where Nt is the atom number after a hold time corresponding to the photoassocia-tion time t but with no photoassociation light present. The volumes V0 and Vt arethe initial trap volume and the volume after time t, respectively. Keff is definedfollowing [164] asKeff =1Vt∫Vd3re−2U(~r)/kBTV ×1hQT∫ Umax−U(r)0dE(|Sb|2 + |Sg|2)e−E/kBT . (4.9)where U(r) is the trap potential. The first integral in Eq. 4.9 is the effective trapvolume. The kinetic energy integral is truncated by the local trap depth Umax−U(r).The analysis presented above can be used to model multiple physical realiza-tions of two-photon-assisted collision processes like Autler-Townes doublet, theelectromagnetically induced transparency or coherent population trapping reso-nance [133]. Complementary treatments extending the theory of scattering in thepresence of two laser fields can be also found in e.g. [169] and [170]. The semiclas-91sical density matrix representation describing a set of closed three-level Λ atomicor molecular states (Ref. [170]) is analogous to the treatment of atom-moleculedark states in a BEC of 87Rb used by Winkler et al. [140]. The analysis of our datafollows the same approach (Section 4.4).4.2 Two-color photoassociationThe single color photoassociation spectroscopy, discussed in Chapter 3, is an effi-cient method to determine binding energies of molecules in excited states. How-ever, probing the ground state molecular potentials is not as straightforward. Inprinciple, by measuring the frequency of the photons emitted when the excitedmolecules decay into the ground state molecules one could determine the energydifference between the rovibrational levels involved in the process and extract fromthat information about the ground state potential. This approach, the detectionof fluorescence, is typically used when studied large ensembles of molecules inmolecular beams or in hot vapors. However, the number of molecules involved inultracold experiments is so low that fluorescence measurements would require nearsingle photon detection capability. Moreover, the assignment of the frequencies ofthe fluorescence photons to the final molecular levels requires the knowledge ofaccurate models of molecular potentials. Alternatively, instead of looking at thefluorescence one could directly probe the population created from the decay ofthe excited state molecules using state selective ionization. REMPI (resonance-enhanced multiphoton ionization) was already demonstrated to work with ultra-cold RbCs [171] and LiCs [172] molecules trapped in a magneto-optic trap. Thedrawback of the above methods is that they require additional, quite often expen-sive, hardware (like tunable pulsed lasers for photoionization and ion detectors)that significantly increase the complexity of the experiments and limit the opticalaccess to the atoms. As a result, these solutions put constraints on the range of theexperiments that can be done in a given setup.In experiments with ultracold gases the most common method to measure bind-ing energies of molecular ground state levels is the so called two-color photoasso-ciation which relies on a 3-level Λ system as shown in Fig. 4.1, where |e〉 is oneof the excited molecular levels (see Chapter 3) and |g〉 is a level in the a(13Σ+u )92Figure 4.2: The normalized 6Li atom number in two color photoassociationspectroscopy of the v′′ = 38, N′′ = 0 ro-vibrational level in the X(11Σ+g )potential. The laser L1 with frequency ν1 induces constant loss bydriving transition from an initial unbound molecular state with N = 0,G = 0 to the v′ = 31 vibrational level of the 11Σ+u excited state withN′ = 1,G′ = 0 while the frequency ν2 of the laser L2 is varied across thev′ = 31 to v′′ = 38 transition. The bias magnetic field is set to 0 G.or X(11Σ+g ) potential which binding energy we want to measure (Fig. 4.4). Thestarting point here is a pair of colliding atoms confined in a ∼10µK deep dipoletrap, that are resonantly coupled by a laser L1 (frequency ω1/2pi) to some molec-ular level |e〉. The intensity of this laser is set such that during time t the inducedatom loss (due to single-color photoassociation) can be clearly distinguished fromthe shot-to-shot atom noise. In practice, the loss is typically kept in the 30%-50% range. The frequency ω2/2pi of the second laser L2 is then varied to matchthe resonance condition ωe-g/2pi corresponding to the energy difference betweenbound levels |e〉 and |g〉. When ω2/2pi approaches ωe-g/2pi it reduces the rate ofloss of trapped atoms increasing the trapped atom number. In our experiments theintensity of L2 is kept low such that the primary mechanism responsible for thesuppression of loss is the reduction of the population of molecules in the excitedmolecular state, reducing the rate of spontaneous emission events that lead to theatom loss [173].93Figure 4.3: The normalized 6Li atom number in two color photoassociationspectroscopy of the v′′ = 9,N′′ = 0 ro-vibrational level in the a(13Σ+u )potential. The laser L1 with frequency ν1 induces constant loss by driv-ing transition from an initial unbound molecular state with N = 0, G = 0to the v′ = 20 vibrational level of the 13Σ+g excited state with N′ = 1,G′ = 0, J′ = 1 while the frequency ν2 of the laser L2 is varied across thev′ = 20 to v′′ = 9 transition. The bias magnetic field is set to 0 G. Themolecular hyperfine structure is partially resolved.Figures 4.2 and 4.3 are examples of a two-color photoassociation spectroscopyapplied to determine the binding energies of v′′= 38, N′′= 0 and v′′= 9,N′′= 0 ro-vibrational levels in the X(11Σ+g ) and a(13Σ+u ) potentials, respectively. Here, bothlasers L1 and L2 are locked to the same frequency comb to increase the accuracyof the measurements but the scheme does not rely on the phase coherence of thelasers. In fact, the initial detection of these levels was done with the frequencyof the laser L2 stabilized to an external cavity while laser L1 was stabilized to thefrequency comb.The method described above supported by the theoretical framework presentedin Section 4.1 has been used by many experimental groups to determine the bind-ing energies of bound molecular levels, with the emphasis on the ground statemolecular potentials [164, 173–181]. In our case, however, this is only the firststep necessary for further experiments where the binding energies are measured94Figure 4.4: Energy levels at B = 0 relevant to the dark state experimentspresented here. At B = 0, the initial unbound two-atom spin state is|i0〉 =√1/3|00000〉+√2/3|01110〉, a linear combination of the sin-glet and triplet states shown lying 2a2S below the 2S1/2 +2S1/2 asymp-tote. The levels are labeled with the quantum numbers |NSIJF〉, where~N is the molecular rotational angular momentum, ~S is total electronicspin, ~I is total nuclear spin, ~J is total angular momentum apart fromnuclear spin and ~F = ~J +~I. Here, Ω1 and Ω2 are Rabi frequen-cies of transitions driven by the lasers L1 and L2, respectively, anda2S = 152.137 MHz is the atomic magnetic dipole hyperfine constantof 22S1/2. The naming convention of the lasers refers to their role inthe dark state spectroscopy experiments. The hyperfine splittings of thev′′ = 9 level shown here differ from our measurements, as discussed inSection 4.4. It should be also noted that a similar figure in [162] con-tains an incorrect hyperfine shifts of |01112〉 and |01110〉 levels in thea(13Σ+u ) potential.exploiting the coherence properties of atom-molecule dark states.4.3 Dark state spectroscopyAfter atoms are transferred into the low power CDT a uniform magnetic field isset to 300 G and the sample is evaporatively cooled to the desired temperature.The magnetic field is then turned off and three orthogonal pairs of compensationcoils are used to cancel the residual magnetic field to a level below 20 mG, as con-95firmed by the rf spectroscopy (section 2.3.4). After 1 s hold time (the timescale forthe photoassociation) at the trap depths used in the experiment there are typically40×103−60×103 atoms left. The lowest temperature the sample is cooled to is∼ 600 nK and T ≈ 0.6TF, where TF is the Fermi temperature of a one componentFermi gas. This temperature, and a corresponding trap depth of about 4 µK, isfundamentally limited because the shallow dipole trap is not able to support atomsagainst gravity which causes atom loss and deterioration of the quality of the spec-troscopic signal. Magnetic levitation, commonly used at non-zero magnetic fieldsto circumvent this issue, is not possible at low magnetic fields because the trappedstates have opposite magnetic moments. It is possible to decrease the T/TF ratioat 0 G by using a different design of the dipole trap and by improving the initialatom number, which would increase the Fermi temperature for a given trap depth.For example, Granade et. al [66] quotes T ≈ 0.48TF with about 105 atoms per spinstate.At B = 0, F and mF are good quantum numbers, but to explicitly show the sin-glet and triplet character of the initial two-atom spin state |i0〉 we follow Ref. [122]and represent it in the molecular basis (|NSIJF〉) shown in Fig. 4.4, such that|i0〉=√1/3|00000〉+√2/3|01110〉.The illumination time and light intensity I1 of laser L1 with frequency ν1 (seeFig. 4.4) is chosen such that after 1 s L1 induces a single color loss of 40% to80% for intensities 100− 500 W/cm2 corresponding to Rabi frequencies Ω1 1 kHz. The intensity of L2 (20−200 W/cm2) is chosen so that the correspondingRabi frequency Ω2  Ω1 but also low enough (Ω2 < 1 MHz) to avoid inducinga large Autler-Townes splitting of the excited state. The frequency of L2, ν2, isset to match the v′′ = 9↔ v′ = 20 (v′′ = 38↔ v′ = 31) transition between thetriplet(singlet) levels (Fig. 4.4) using the frequencies initially determined with thetwo-color photoassociation spectroscopy. The choice of the excited levels is basedon empirical evidence that their ac Stark shift due to the field from laser L1 is smallin comparison with other levels measured in [49, 51]. Because there has been noprior high precision spectroscopy of the levels investigated in this thesis, our choicehas been, to some extent, also a matter of convenience: the frequencies required forphotoassociation to v′ = 20 and v′ = 31 are close enough such that it is easy to tunethe Ti:Sapphire lasers when switching between the triplet and singlet potentials.96Figure 4.5: High resolution dark state spectra in a Fermi gas at 0 G withT ≈ 0.6TF . The observed feature corresponds to the v′′ = 38, N′′ = 0,F ′′ = 0 ro-vibrational level in the X(11Σ+g ) potential. The frequency ν2is fixed on the v′′ = 38, N′′ = 0 to v′ = 31, N′ = 1 bound-to-bound tran-sition whereas ν1 is scanned. The maximum revival appears when thetwo-photon condition is satisfied and the frequency difference ν2− ν1equals the difference between the energy of the initial free atoms stateand ground state molecules. The intensity of I2 of laser L2 is kept lowenough to avoid inducing a large Autler-Townes splitting.The frequency ν1 is scanned over a range that induces atom loss due to thesingle color photoassociation of free atoms to v′ = 20 or v′ = 31 vibrational levels.With L2 on, dark state spectra are observed as shown in Fig. 4.5 and 4.6. It isworth noting that if the laser L2 is detuned from a bound-to-bound transition thedark state spectra become asymmetric, but it has no influence on the determinationof the binding energies. When the two-photon resonance condition is fulfilled anatom-molecule dark state is created and the loss induced by L1 is almost completelysuppressed for all levels but |g1〉= |v′′ = 9,N′′ = 0,F ′′ = 1〉 where the suppressionis only partial (Fig. 4.6b). We do not expect that the poor revival is caused by thefinite collisional lifetime of |g1〉 as it would have to be unreasonably short, ∼ 1 µs.This estimation comes from the fit of the 3-level model from the Eq. 4.18 to thedark state feature which returns the decay rate of the molecular level γg ∼ 1 MHz.97Figure 4.6: High resolution dark state spectra in a Fermi gas at 0 G with T ≈0.6TF. The molecular hyperfine levels F (a–c), belong to the v′′ = 9,N′′ = 0 ro-vibrational level in the a(13Σ+u ) potential. The frequency ν2is fixed on a v′′ = 9,N′′ = 0, F ′′ = 2,1,0 to v′ = 31, N′ = 1 bound-to-bound transition whereas ν1 is scanned. The maximum revival appearswhen the two-photon condition is satisfied and the frequency differenceν2−ν1 equals the difference between the energy of the initial free atomsstate and ground state molecules. The intensity of I2 of laser L2 is keptlow enough to avoid inducing a large Autler-Townes splitting.98However, both γg and the Rabi frequency Ω1 cannot be reliably determined forexposure time of the atoms to the photoassociation light longer than about 10µs.For more details see Section |v′′ = 9,N′′ = 0,F ′′ = 1〉 revival - role of the dipole trapOff-resonant transitions driven by the optical dipole trap could be, in principle,responsible for the loss of population from |g1〉 and subsequent decrease in the re-vival of the dark state signal. The dipole trap (1064 nm, 9398.5 cm−1) can couplethe |g1〉 state to an excited level in a triplet character molecular potential c(13Σ+g )or b(23Σ+g ), as shown in Fig. 3.1. To the best of our knowledge the levels ener-getically closest to the frequency of the dipole trap can be found in the c(13Σ+g )potential: v′ = 5 and v′ = 6 with binding energies E5 = −5674.9324 cm−1 andE6 = −5436.1865 cm−1, respectively3. The dipole trap laser is therefore detunedfrom v′ = 5 by 170.3 cm−1 and from v′ = 6 by 68.4 cm−1. These are predictionsfor rotational levels with N′ = 0 but because the rotational constant is negligiblein comparison to the value of detunings, the energies E5 and E6 can be treated asenergies of states with N′ = 1.In order to estimate the scattering rateΓsc =3pic22h¯ω30(Γ∆)2I (4.10)caused by the off-resonance excitation of molecules due to the dipole trap photons,the on-resonance scattering rate ΓΓ=ω303piε0h¯c3|〈e|µ|g1〉|2 (4.11)needs to be calculated. Here ω0 is the angular frequency of the transition and ∆ isthe detuning of the laser with intensity I from this transition. c, h¯ and ε0 are thespeed of light, Planck’s constant and vacuum permittivity, respectively. To obtainthe dipole matrix element between the ground |g1〉 and the excited molecular state|e〉, |〈e|µ|g1〉|2, the wavefunctions for both states need to be known. However, to3Private communication, Dr. N. S. Dattani99estimate an upper bound on the scattering rate Γ induced by the dipole trap, weassume that |〈e|µ|g1〉|2 is on the same order of magnitude as matrix elements forthe 2S to 2P atomic transition at 671 nm (with frequency ω ′0). We haveΓ=ω303piε0h¯c3|〈g1|µ|e〉|2 ≈(ω0ω ′0)3 ω ′303piε0h¯c3|〈2s|µ|2p〉|2 =(ω0ω ′0)3Γatom. (4.12)As a result, Γ = 0.25Γatom = 1.5 MHz, where Γatom = 5.9 MHz [67]. For thedipole trap intensities used in dark state experiments (less than 13 kW/cm2), weobtain scattering rates of up to 8 Hz and 50 Hz for detunings 170.3 cm−1 and68.4 cm−1, respectively. The complete revivals on the two-photon resonance ob-served for |v′′ = 9,N′′ = 0,F ′′ = 0〉 and |v′′ = 9,N′′ = 0,F ′′ = 2〉 levels confirmthat these scattering rates are negligible in our experiment. Moreover, when thefrequency of L2 is tuned to the F = 1 hyperfine level, the probed bound-to-boundtransition differs by up to 152 MHz from the above mentioned cases. Unless thecoupling of F = 1 to the excited state is orders of magnitude stronger than that ofthe other F levels (e.g. due to different selection rules that do not allow couplingof F = 0 and F = 2 to the excited level), it does not influence the scattering rateas the dipole trap is already far-detuned from any transition. Based on our analysisthe dipole trap does not seem to be the cause of the observed poor revival of the|v′′ = 9,N′′ = 0,F ′′ = 1〉 level and further investigation of this effect is required.4.3.2 Dark state lifetimeThe recorded spectra reveal a complete suppression of loss on two-photon reso-nance even for illumination times close to 2 s. To create the dark states we turn thelaser fields on and off in the following order: L2 on, L1 on, L1 off, L2 off; wherethe turn on or off times are ∼150ns 4. The turn on or off of L1 is adiabatic for mostof the cases; however, when the turn on or off of L1 is non-adiabatic, we expect aloss of atoms due to the projection of the initial state onto the bright state (whichdecays immediately) and due to the projection of the dark state back onto the free4The AOM turn on or off times are ∼1µs. However, coupling the photoassociation light into afiber results in the decreased effective turn on or off time.100atom statePloss = 1−|〈DS|a〉|2|〈DS|a〉|2 = 1−(Ω22Ω21 +Ω22)2, (4.13)where |DS〉 is the initial dark state and |〈DS|a〉|2 is the probability that the darkstate will survive the turn on (turn off) of laser L1. When Ω2Ω1, this projectionloss is 2(Ω1/Ω2)2.Another loss mechanism results from non-adiabatic phase jumps of the lasersthat lead to the projection of the atom-molecule state|AM〉=Ω2|a〉−Ω1eiφ |g〉√Ω21 +Ω22(4.14)onto the new bright state|BS〉=Ω1|a〉+Ω2|g〉√Ω21 +Ω22. (4.15)Here, φ is the new relative phase between two lasers after the jump. The |a〉 and|g〉 are the initial atomic state and a molecule in the ground state, respectively. Thismechanism amounts to the loss of population from the initial state |AM〉 that equals|〈AM|BS〉|2 =2Ω21Ω22(1− cos(φ))(Ω21 +Ω22)2. (4.16)We can model the phase jitter of the lasers as producing a non-adiabatic jump of therelative phase by an average φ = pi/2 every dephasing time τ ∼ 1/∆ν = 6.2 µs.5After the time T this causes loss of the initial atom number NiPjitter =NiN=(1−2Ω21Ω22(Ω21 +Ω22)2)T/τ. (4.17)For the experiments in the weakly interacting regime we chose Ω2/Ω1 > 1000.5∆ν = 160 kHz is the relative linewidth of the photoassociation lasers used to create the darkstate, see Section this condition, the atom loss is expected to be on the order of 10% after350 ms. 6 However, this simple model underestimates the observed dark statelifetime, because even after 2 s we can still observe near complete revivals on thetwo photon resonance. We note that to the best of our knowledge the longest timesreported in experiments with other species rarely exceed few tens of ms.4.4 Binding energies of the least bound statesTo determine the ac Stark shifts induced by the lasers L1, L2 and the dipole trap,the dark state spectra are measured for several dipole trap depths with the con-trol(probe) beam intensity 50(350) W/cm2. At the lowest trap depth, additionalspectra are measured when both the control and the probe beam intensities arevaried. A three-level model (as in e.g. [140])ia˙ =−Ω1a∗b,ib˙ = [(∆+δ )− iγb/2]b−12(Ω1aa+Ω2g),ig˙ = (δ − iγg/2)g−12Ω2b, (4.18)describing the normalized field amplitudes a, b, and g of the initial state, the excitedmolecular and ground state, respectively, is numerically fitted to each data set toextract the two-photon resonance position. The resulting values for each set ofcontrol and probe beam intensities are used to extract the field free binding energieswhich are summarized in Table 4.1.Due to the long time scale of the experiment (1 s) which exceeds, by ordersof magnitude, the timescales associated with the decay rates γb, γg and Rabi fre-quencies Ω1, Ω2, simulating the full, 1 s long evolution of the system is too timeconsuming to be practical with a use of a desktop computer. For that reason theevolution of the system is simulated for up to 10 µs and as a result the 3-levelmodel (Eq. 4.18) cannot be reliably used to extract Ω1 and γg. However, the two-photon resonance is time scale independent therefore its determination is reliable.6In our setup, 10% loss of atoms is usually indiscernible due to the shot-to-shot atom numbervariation.102Table 4.1: Experimentally measured binding energies of the least bound vi-brational levels of the a(13Σ+u ) and X(11Σ+g ) potentials of6Li2. The fre-quency difference ν2−ν1 is extracted from the dark state spectra and cor-responds to the energy difference between the initial and the final state.The initial free atomic state is 2a2S below the hyperfine center of grav-ity of the 2S1/2 + 2S1/2 asymptote therefore the actual binding energyis computed by adding 2a2S = 304.274 MHz to the measured frequencydifference. The quoted uncertainties represent the statistical uncertaintieson the fits. The systematic uncertainty is below 1 kHz.v′′ F ′′ ν2−ν1 [GHz] Binding energy [GHz]X(11Σ+g )38 0 1.321671(21) 1.625945(21)a(13Σ+u )2 24.010649(46) 24.314923(46)9 1 24.163035(105) 24.467309(105)0 24.238372(54) 24.542646(54)Figure 4.7: The difference between the predicted (based on Eq. 4.20) andmeasured molecular hyperfine splittings in the v′′ = 9, N′′ = 0′′ levelin the a(13Σ+u ) potential. The horizontal line corresponds to a perfectagreement with theory and the uncertainties result from the uncertain-ties of the determination of the binding energies of molecular hyperfinelevels.103The dark state spectroscopy is a differential measurement; therefore only theabsolute frequency difference of the lasers L1 and L2 needs to be known precisely.The resulting systematic uncertainty is therefore on the order of 1 kHz, as discussedin Section 2.2.5. We note that the standard spin-dependent Hamiltonian of twonon-interacting atoms [104, 166]H =a2S2(~s1 ·~i1 +~s2 ·~i2) =a2S2(12~S ·~I +12(~s1−~s2) · (~i1−~i2))(4.19)with associated energies of a state |NSIJF〉E|NSIJF> =a2S4[F(F +1)−S(S+1)− I(I +1)] (4.20)predicts molecular hyperfine splittings (Fig. 4.4) that differ from those inferredfrom the binding energies listed in Table 4.1 by more than can be explained bythe uncertainties of our measurements, thus providing a possible reference for re-finements of molecular hyperfine structure models. The disagreement cannot becaused by a systematic shift of unknown origin; see Figure 4.7. In order for allof our data to agree with theory the uncertainties would have to be increased by afactor of seven, which cannot be justified.4.5 Binding energy of v′′ = 37, N′′ = 0At magnetic fields above the Feshbach resonance at B = 543.3 G the energy ofthe v′′ = 38, N′′ = 0 molecular level becomes nearly degenerate with the energy ofthe two colliding atoms in states |1〉 and |2〉. This leads to the coupling betweenv′′ = 38, N′′ = 0 level and the atoms in the entrance channel, resulting in a broadFeshbach resonance at B = 832.18 G [85] (see Section 4.6). Above this field v′′ =38, N′′ = 0 is no longer bound and v′′ = 37 becomes the least bound molecularlevel in the singlet potential. 7Theoretical studies of dark states in degenerate Fermi gases in the BEC-BCScrossover regime (e.g. [158, 159]) rely on molecular levels in the X(11Σ+g ) poten-tial therefore the knowledge of the binding energy of v′′ = 37, N′′ = 0 at various7see e.g., a review paper devoted to Feshbach resonances by Chin et al. [118]104Figure 4.8: The normalized 6Li atom number in two color photoassociationspectroscopy of the v′′ = 37, N′′ = 0 ro-vibrational level in the X(11Σ+g )potential. The data shows an early attempt to measure the binding en-ergy of this level and high dipole trap and L2 intensities are used tofacilitate the detection. The lasers L1 (I1 = 0.1 kW/cm2) with frequencyν1 and L2 (I2 = 2.5 kW/cm2) with frequency ν2 are both locked to thefrequency comb. Laser L1 induces constant loss by driving transitionfrom an initial unbound molecular state with N = 0, G = 0 to the v′= 31vibrational level of the 11Σ+u excited state with N′ = 1, G′ = 0. The fre-quency ν2 is varied across the v′ = 31 to v′′ = 37 transition. The dipoletrap intensity is 54 kW/cm2 corresponding to the sample temperature ofabout 2 µK. The bias magnetic field is set to 0 G. The lines are guidesfor the eye.magnetic fields is essential. This level has been treated as “known” because thetheoretical potentials for 6Li2 are quite well developed and some low accuracymeasurements (±150 MHz at best) are available [131]. It is worth noting that asrecently as in 2012 Wu and Thomas used for their calculations 53.5 GHz [158]and 55.8 GHz [159] as the energy difference between v′′ = 37 and v′′ = 38. Ourmeasurements, however, indicate that this difference at 0 G is 56.938(1) GHz (Ta-ble 4.2). The discrepancy does not change the conclusion of [158] and [159] butfrom the point of view of an experimentalist setting up an experiment the improvedmeasurements of the required binding energies are essential to making their pro-105Figure 4.9: High resolution dark state spectrum associated with the v′′ =37,N′′ = 0 ro-vibrational level in the X(11Σ+g ) potential. The frequencyν2 of the laser L2 is fixed on the v′′ = 37, N′′ = 0 to v′ = 31, N′ = 1transition while the frequency ν1 of the laser L1 is scanned over the freeto bound transition (free atoms to v′ = 31, N′ = 1). The intensity of L2is kept low enough to avoid inducing a large Autler-Townes splitting.posal more feasible. The experimental determination of the binding energy ofv′′ = 37, N′′ = 0 and v′′ = 38, N′′ = 0 performed by our group has been guidedby theoretical calculations performed by Dr. Nikesh S. Dattani. As an illustrationof the power of modern theoretical techniques it is worth mentioning that the en-ergy difference between v′′ = 37, N′′ = 0 and v′′ = 38, N′′ = 0 predicted by Dr.Dattani’s calculations is 57.002 GHz (predictions were independent of our highprecision experiments).The initial, very coarse, determination of the binding energy of v′′= 37, N′′= 0at 0 G has been performed using two-color photoassociation technique. The lasersL1 (I1 = 0.1 kW/cm2) with frequency ν1 and L2 with frequency ν2 are locked tothe frequency comb. Laser L1 induces constant loss by driving transition from aninitial unbound molecular state with N = 0, G = 0 to the v′ = 31 vibrational levelof the 11Σ+u excited state with N′ = 1, G′ = 0. The frequency ν2 of the laser L2(I2 = 2.5 kW/cm2) is varied across the v′= 31 to v′′= 37 transition. The dipole trap106Table 4.2: The energy difference between the initial state |ic〉 (Feshbachmolecules or BCS-like pairs, Eq. 4.21) and the final singlet state |g〉 (cor-responding to v′′ = 37, |00011〉) measured at selected magnetic fields.The uncertainties are conservatively estimated to be 1 MHz. Theoreticalvalues are obtain by adding the Zeeman shifts of states |1〉 and |2〉 to theexperimental value obtained at B = 0 G.B [Gauss] 0 754 804 838.8ν2−ν1 [GHz] (exp.) 58.260 56.364 56.225 56.127ν2−ν1 [GHz] (th.) – 56.3637 56.2242 56.1271(th.) - (exp.) [MHz] – 0.3 0.8 0.1intensity is 54 kW/cm2 corresponding to the sample temperature of about 2 µK.When the frequency ν2 equals the energy difference between v′ = 31, N′ = 1 andv′′ = 37, N′′ = 0 the level v′ = 31, N′ = 1 becomes detuned from the frequencyν1 resulting in the suppression of the atom loss, shown in Fig. 4.8. As a result,ν2− ν1 can be interpreted as a frequency difference between the free atoms andv′′ = 37, N′′ = 0. This measurement has been verified and improved upon by per-forming a dark state spectroscopy with L2 fixed on v′′ = 37, N′′ = 0 to v′ = 31,N′ = 1 transition and with the frequency of L1 scanned across the v′ = 31, N′ = 1photoassociation line, as in Section 4.3. The resulting dark state spectrum taken atthe sample temperature of 800 nK can be seen in Fig. 4.9. The measurement of thebinding energy of v′′ = 37 has not been performed as rigorously as for v′′ = 9 andv′′ = 38 described in Section 4.3 - the ac Stark effect induced by the photoassocia-tion lasers and the dipole trap has not been systematically studied. However, basedon our experience with the least bound states and given low intensities I1, I2 and thedipole trap, the measured value is very conservatively estimated to have accuracyof 1 MHz. This accuracy is sufficient for all practical purposes because it is muchless than the width of the excited state level v′ = 31 and as a result only fine tuningof frequency is required if the conditions change (if e.g., different dipole trap in-tensity is used). To the best of our knowledge the binding energy of this level hasnever been reported before in the context of ultracold lithium.The dark state spectroscopy has been performed for a range of magnetic fields(described in details in Section 4.6) and the energy difference between the |g〉 =107|v′′ = 37,N′′ = 0〉 and the initial state |i〉 at selected magnetic fields is summarizedin Table 4.2). Our observations are consistent with the expectation that the energydifference E|g〉−E|i〉 is caused solely by the Zeeman shift of the initial state |i〉since the bound molecular state is expected to have no magnetic moment.4.5.1 Unexpected features in the v′′ = 37, N′′ = 0 spectrumThe multi-peak structure observed in the two-color photoassociation spectrum ofthe v′′ = 37, N′′ = 0 level (Fig. 4.8) at 0 G came as a surprise. This level is notexpected to have any degeneracies that could be lifted by the high intensities ofL2 (due to F = 0) therefore the structure of the signal might come from couplingof v′ = 31 to some other excited molecular potential by laser L2. The multi-peaksignal, however, goes away when I2 is reduced by a factor of ≈ 20, resulting ina single peak corresponding to ν2− ν1 ≈ 58.260 GHz. The disappearance of theanomalous spectrum when I2 is decreased warranted focusing on other, more im-portant, scientific goals, therefore a detailed study of the observed spectrum hasnot been performed.4.6 Exotic dark states in the BEC-BCS crossoverWhen the final evaporation is done at magnetic fields close to the broad FR atB = 832.18 G, pairs form, here referred to as Feshbach molecules below (where atwo-body bound state exists) and BCS-like pairs above resonance (where pairingis a many-body phenomenon). In this case, the initial Feshbach-dressed moleculeor pair state is|ic〉=√Z|gclosed〉+√1−Z|gopen〉, (4.21)where the open channel has almost a pure triplet character and is strongly cou-pled to the closed channel molecular state responsible for the wide FR, |gclosed〉 ≡(2√2|00000〉− |00200〉)/3, a linear combination of the I = 0 and 2 (v′′ = 38) sin-glet states [122, 159]. To form dark states, we use the singlet levels |e〉 (v′ = 31,|00011〉) and |g〉 (v′′ = 37, |00011〉) mainly because of their insensitivity to themagnetic field.108Figure 4.10: Dark state spectra observed at selected magnetic fields in a par-tially Bose condensed sample of Feshbach molecules of 6Li usingtransitions between singlet molecular potentials. The photoassocia-tion light is on for 100 ms for measurements at magnetic fields (a)B = 754 G and (b) B = 804 G. (c) Magnetic field is set to B = 754 Gand the photoassociation lasers are on for 5 µs, the timescale relevantto STIRAP. The change of the exposure time by over four orders ofmagnitude does not seem to change the dark state feature. No parame-ters have been found that would improve the revival on the two photonresonance.109Figure 4.11: A generic dark state spectrum showing quantities defining therevival height as the ratio of the amplitude of the suppression signalAsuppr to the amplitude of the loss signal Aloss4.6.1 Dark states in a BEC of Feshbach moleculesThe evaporation on the molecular side of the B = 832.18 G FR allows us to cre-ate a mixture of Feshbach molecules and a Bose-Einstein condensate of Feshbachmolecules. For intensities I1 = 0.045-10 kW/cm2 and I2 = 0.040-0.3 kW/cm2,and illumination times 5µs to 100 ms (short times correspond to large I1) we ob-serve dark state features that show only partial loss suppression on the two-photonresonance. Figure 4.10 shows the observed features at 754 G and 805 G for thephotoassociation light exposure times of 5 µs and 100 ms. Both Ω1 and Ω2 as wellas the exposure time have been varied over a range where a full revival is expected.Nevertheless, we always observe revival heights below 50%, even for temperaturesabove the mBEC critical temperature (T > TC). For the purpose of further discus-sion we define the revival height as the ratio of the amplitude of the suppressionsignal Asuppr to the amplitude of the loss signal Aloss (see Fig. 4.11).4.6.2 Dark states in a degenerate Fermi gasWhen a degenerate Fermi gas is prepared above the resonance (here at 839 G) suchthat there are BCS-like pairs present we observe dark state features correspondingto a coherent superposition of these pairs and molecules in the |g〉 level of the110Figure 4.12: A dark state spectrum at B = 839 G observed after 5 µs expo-sure time to the photoassociation light in a BCS-like paired 6Li usingtransitions between singlet molecular potentials. Here Ω1/2pi = 5 kHzand Ω2/Ω1 ∼ 1000. The full revival is in stark contrast with the resultsobtained below the Feshbach resonance at B = 832.18 G.singlet potential (Fig. 4.12). As this is the first ever observation of such an exoticquantum superposition it has not been obvious a priori if dark states can be createdwithout disrupting the many-body pairing physics by the photoassociation lightfrom two lasers. The destruction of pairing is a possibility and the observed darkstate would then be between “regular” strongly interacting two component Fermigas and ground state molecules.To verify the persistence of the BCS-like pairing the frequencies ν1 and ν2are set to match the two-photon condition where negligible dark-state tuning ofthe scattering length is expected and rf spectroscopy with 200 ms long rf pulses isperformed, revealing a spectrum consistent with the presence of pairing [74]. Here,Ω1/2pi = 5 kHz and Ω2/Ω1 ∼ 1000, thus the BCS-like pairs were only weaklydressed with the |g〉 level, and the many-body interactions were, thus, negligiblyperturbed and no change of the spectrum that could be associated with the presenceof the dark state has been observed, as revealed in Fig. 4.13.111Figure 4.13: a) RF spectroscopy of BCS pairs of 6Li at 839 G, the narrowpeak at -1 kHz corresponds to the |2〉 → |3〉. transition whereas thebroad peak is associated with breaking BCS pairs, b) the frequenciesof the lasers L1 and L2 are set to a two photon resonance creating a co-herent superposition of BCS pairs and singlet ground state moleculesat 839 G. RF spectroscopy confirms the persistence of pairing for thisexotic BCS pair- ground state molecule dark-state. The lines shown onboth plots are the same functions and are intended only as a guide tothe eye.1124.6.3 Dark states in the BEC-BCS crossoverThe revival height on the two-photon resonance has been studied in the BEC-BCScrossover regime using Ω1 = 200 kHz, Ω2 ' 2 MHz, and exposure time of 40 µs.Figure 4.14(top) shows that the revival height changes abruptly for fields below829 G. This change, coincidentally, occurs at magnetic fields where the two-bodybound state is present. However, we observe a near full revival also below the Fes-hbach resonance center at B = 832.18 G [85]. This abrupt change in the revivalheight was unexpected, and we, therefore, independently checked that Ω1 is con-tinuous in this regime by performing single-color photoassociation |ic〉 to |e〉. Onlythe I = 0 part of the closed channel contributes to photoassociation to |e〉; thereforeΩ1(B) = 〈ic|~d ·~E|e〉=√Z〈gclosed|~d ·~E|10011〉v′=31 (4.22)=√Z√89〈00000v′′=38|~d ·~E|10011〉v′=31 =√8Z9Ω0,We determine Ω0 experimentally from a fit of the measured dark-state spectraat B = 0 shown in Fig. 4.5. In that spectra, Ω0 plays the role of the bound-to-boundcoupling Ω2. We observe that Z ≡ (9/8)(Ω1/Ω0)2 [shown in Fig. 4.14(bottom)]is continuous in the region where the dark state revival changes abruptly. Our ob-servations are consistent with those reported by Partridge et al. [99] where a muchmore weakly bound excited state v′ = 68 was used. However, our determinationof Z does not rely on the calculation of Ω0 and thus is independent of a theoreticalmodel for the molecular potentials.As of now, the source of the sudden change of the revival height in the vicin-ity of the Feshbach resonance cannot be explained and this issue requires furtherinvestigation, possibly leading to a new understanding of the BEC-BCS crossovertheory.113Figure 4.14: (top) The revival height of the dark state features defined as aratio of the suppression amplitude to the loss amplitude, measuredon both sides of the B = 832.18 G Feshbach resonance (dashed ver-tical line). The states involved are Feshbach molecules or BCS-likepairs (|ic〉), v′ = 31 in the A(11Σ+u ) excited molecular potential (|e〉)and v′′ = 37 in the X(11Σ+g ) ground-state potential (|g〉). For mag-netic fields above 829 G, we observe near complete revival on thetwo-photon resonance. Insets: dark states feature at selected magneticfields: 754 G, 829 G and 839 G. (bottom) Full circles represent theprobability Z of the dressed molecule to be in the bare molecular statein the closed channel of the Feshbach resonance. Squares (red) show Zmeasured by Partridge et al. [99] where the excited state |e〉 is v′ = 68[A(11Σ+u ) potential]. Dash-dotted line shows theoretical prediction forZ taken from digitized Fig. 8 in [182]. The grey region correspondsto kF|a| > 1. For these data, T/TF = 0.4± 0.15, EF/h = 11 kHz,Ω1 = 200 kHz, and Ω2 ' 2 MHz. Figure taken from [52].114Chapter 5Summary and outlookThe results presented in this thesis have led to the development of experimentaltools and methods that eventually will enable our laboratory to perform cuttingedge experiments with ultracold molecules formed from ultracold atoms. Trig-gered by the goal of creating ground state polar molecules of 6Li85Rb we used 6Lito test certain technical aspects of our setup and develop know-how that up untilthis work had not been available at the University of British Columbia (and formost part in Canada). These technical aspects cover primarily:• The demonstration of a Bose-Einstein condensate (of weakly bound Fesh-bach molecules), as well as a strongly interacting degenerate Fermi gas inthe regime where BCS-like pairing is present. This has been achieved in adrastically simplified experimental setup, where the magneto-optical trap isloaded from an oven located in the same chamber as the region where theexperiments are performed. It defied the generally accepted rule that theatomic source and the experimental section need to be separated by a differ-ential pumping stage for experiments with degenerate quantum gases.• The setup of a laser system for the single and two-photon manipulation ofatoms. It consists of two tunable Coherent 899-21 Ti:Sapphire lasers phaselocked to a GPS disciplined femtosecond frequency comb. The system hasbeen tested on 6Li and is now ready for experiments leading to the productionof ground state molecules. Long term, it will enable experiments with otherspecies, notably a 6Li and 85Rb mixture.The high precision single color photoassociation spectroscopy presented in thisthesis resulted in a significant improvement of the models describing the c(13Σ+g )and A(11Σ+u )6Li potentials. The experimental demonstration of the enhancement115of the photoassociation rate in the vicinity of a p-wave Feshbach resonance extendsthe scope of Feshbach Optimized Photoassoaciation (FOPA) method beyond the s-wave regime.The demonstration of atom-molecule and molecule-molecule dark states in de-generate gases of fermionic lithium extends the range of systems where such su-perposition states have been created and enables realization of theoretical proposalexploiting dark states as tools for optical control of interactions and probing of thesuperfluidity and pairing. For the past decade, 6Li has been a work horse in thefield of strongly interacting Fermi gases, therefore new probing techniques basedon dark states could potentially have a significant impact on the field.The measurement of the binding energies of the least bound states of the low-est singlet and triplet potentials X(11Σ+g ) and a(13Σ+u ), respectively, provides themost precise spectroscopic measurements of these levels so far, with accuraciesreaching 20 kHz. The observed anomaly of the molecular hyperfine splittings inthe v′′ = 9 vibrational level of the a(13Σ+u ) state will require additional investiga-tion to reconcile the observed difference between our measurements and theoreticalcalculations.Finally, this work puts lithium in the spotlight as a system that could be used toproduce a Bose-Einstein condensate of ground state molecules in the ro-vibrationalground state. As opposed to other species (bosonic atoms or Bose-Fermi mixtures)studied by many groups, the starting point for the transfer of the initial state us-ing stimulated Raman adiabatic passage (STIRAP), long-lived sample of a Bose-Einstein condensate of Feshbach molecules, can be relatively easily achieved. Thisapproach may result in a transfer directly into a degenerate gas of ground statemolecules, without a need for additional cooling of molecules. Furthermore, Li2 isone of the simplest dimers and creation of Li2 in the ultracold regime could lead toexperiments that would enrich our understanding of molecular physics, thereforealso potentially impacting chemistry.5.1 Future experimentsAfter over six years of service, we decided to upgrade the experimental setup usedin this thesis and as a result the rebuild of the vacuum system and the lithium laser116system is under way. The long term goal of creating ground state molecules of6Li85Rb will benefit from the separation of atom sources from the science chamber(this would be unnecessary if we were to work only with lithium). By loadingthe magneto-optical trap from a slow atom source (Zeeman slower) not only willwe increase the trapped atom number but, more importantly, we will decrease theexperimental cycle duration from 10−15 s to < 5 s, significantly decreasing dataacquisition cycle. The optimization of the dipole traps, the improvement of theoptical access and an improved imaging system will lead to a much larger samplesof degenerate gases, imaged with better resolution and signal-to-noise ratio. Thedetails of this upgrades will be presented in a Master’s thesis of William Bowdenand a Doctoral thesis of Will Gunton.From the scientific point of view, the work presented here has been a necessarystep towards experiments involving lithium molecules and, in a broader sense, con-trol of atomic (especially fermionic) systems with laser fields. The research that isexpected to be a natural continuation of this thesis will focus on the following:• Transfer of atoms from a standard magneto-optical trap based on the D2 lineinto a D1 line based trap, thus enabling sub-Doppler cooling mechanisms dueto well resolved hyperfine structure of the excited 22P1/2 level. This shouldlead to a decreased temperature of the sample and increased phase spacedensity, and as a result a more efficient transfer into the dipole trap. Similarapproaches applied recently to e.g. 39K [183], 40K [184] and 7Li [185, 186]show a great promise of this method.• We have performed preliminary measurements (with resolution∼5 MHz) ofthe binding energies of N′′ = 0,2 rotational levels of all vibrational levels(v′′ = 0 to v′′ = 9) in the a(13Σ+u ) molecular potential. It makes it one ofthe most accurately measured molecular potentials. Using dark state spec-troscopy described in Chapter 4 would allow further improvement of the de-termination of binding energies (with accuracy∼100 kHz), providing exper-imental data useful for tests of ab initio calculations of molecular energies.Moreover, the discrepancy between the theory and experiment observed inthe molecular hyperfine structure of the v′′= 9,N′′= 0 level (see Section 4.4)indicates that high accuracy (∼100 kHz) measurements could be a reference117for the refinement of the molecular theory.• Our laser system allows us to measure binding energies of the N′′= 0,2 rota-tional levels of v′′ = 26 to v′′ = 38 vibrational levels in the X(11Σ+g ) molecu-lar potential. The expected accuracy on the order of 20 kHz (as demonstratedfor v′′ = 38,N′′ = 0, see Section 4.4) would be at least four orders of mag-nitude better than the data available so far. With the precision spectroscopycovering almost 1/3 of the potential depth, this would make X(11Σ+g ) one ofthe best known molecular potentials.• The demonstration of the dark states in 6Li and the measurement of the bind-ing energy of the v′′ = 37 level in the X(11Σ+g ) potential (see Section 4.5)are the first (and absolutely necessary) steps towards experimental realiza-tion of a proposal utilizing dark states for optical control of Feshbach reso-nances [158].• Finally, converting a Bose-Einstein condensate of Feshbach molecules intoa Bose-Einstein condensate of the triplet ground state molecules is a majorresearch goal of our group. Since all the hardware is already in place, as wellas a necessary know-how has been acquired, performing STIRAP is one ofthe most important steps that will be taken after the upgrade is finalized.118References[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.Cornell. 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