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An experimental apparatus for the laser cooling of lithium and rubidium Bowden, William James 2014

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An Experimental Apparatus for the Laser Cooling ofLithium and RubidiumbyWilliam James BowdenEngineering Physics, The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics and Astronomy)The University of British Columbia(Vancouver)October 2014c©William James Bowden, 2014AbstractWe demonstrate a two species effusive source and Zeeman slower for lithium 6and rubidium 85. The fluxes produced by this slower allow for magneto-opticaltrap loading rates in excess of 108 atoms per second for both species. A detailedmodel is developed to predict the emission properties of the effusive source alongwith the flux of cold atoms produced by the slower. Novel to this design is themating of Zeeman slower magnetic field to the field produced by trapping coilswhich increases the effective length over which atoms are slowed. This allowsfor a smaller, more compact slower, without a sacrifice in performance. Detailsrelating to the design and performance of the vacuum system and magnetic fieldproducing coils are also covered. The apparatus can be easily adapted to operatewith different atomic species making it well suited for ultracold atomic physicsexperiments studying mixtures or as starting point for the creation of hetero-nuclearmolecules.iiPrefaceThis Master’s thesis contains the result of research undertaken in the QuantumDegenerate Gases Laboratory under the supervision of Dr. Kirk Madison at theUniversity of British Columbia. The laboratory is part of the Centre for Researchin Ultra Cold Systems founded in 2008 and funded by a major grants from theCanada Foundation for Innovation and the British Columbia Knowledge Develop-ment Fund. This thesis focuses on the development of an experimental apparatusfor the laser cooling of lithium and rubidium for the eventual study of ultra-coldmixtures and the creation of hetero-nuclear molecules. The laser systems and vac-uum system used in Chapter 4 were built in collaboration with Janelle Van Dongen,Will Gunton, Kahan Dare, Steven Novakov, and Mariusz Semczuk. None of thetext of the dissertation is taken directly from previously published or collaborativearticles.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Why Study Ultracold Polar Molecules? . . . . . . . . . . . . . . 21.2 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 32 Theory of Multi-species Atomic Source . . . . . . . . . . . . . . . . 42.1 Theory of Zeeman Slowers . . . . . . . . . . . . . . . . . . . . . 52.1.1 The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . 72.1.2 Alternative Field Profiles . . . . . . . . . . . . . . . . . . 122.1.3 Further Design Considerations . . . . . . . . . . . . . . . 142.1.4 Optimal Zeeman Slower Length . . . . . . . . . . . . . . 182.2 Theory of Effusive Sources . . . . . . . . . . . . . . . . . . . . . 192.2.1 Transparent Regime . . . . . . . . . . . . . . . . . . . . 20iv2.2.2 Opaque Regime . . . . . . . . . . . . . . . . . . . . . . . 232.2.3 Velocity Distribution in the Opaque Regime . . . . . . . . 243 Design of Multi-species Atomic Source . . . . . . . . . . . . . . . . . 293.1 Previous Experimental Setup . . . . . . . . . . . . . . . . . . . . 303.2 Simulation of Zeeman Slower Flux . . . . . . . . . . . . . . . . . 303.3 Design of a Zeeman Slower . . . . . . . . . . . . . . . . . . . . . 333.3.1 Review of Existing Designs . . . . . . . . . . . . . . . . 333.3.2 Construction of Zeeman Slower . . . . . . . . . . . . . . 363.3.3 Computer Controlled Current Driver . . . . . . . . . . . . 403.4 Design of an Effusive Source . . . . . . . . . . . . . . . . . . . . 463.4.1 Rubidium Source . . . . . . . . . . . . . . . . . . . . . . 473.4.2 Lithium Source . . . . . . . . . . . . . . . . . . . . . . . 473.5 Design of Experimental Setup . . . . . . . . . . . . . . . . . . . 483.5.1 Quadrupole Magnetic Field Coils . . . . . . . . . . . . . 483.5.2 Verification of Magnetic Fields from Compensation Coils 553.5.3 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . 574 Characterization of Multi-species Atomic Source . . . . . . . . . . . 644.1 Characterization of the Lithium Oven . . . . . . . . . . . . . . . 654.1.1 Measuring Angular and Velocity Distribution of the AtomicBeam with Florescence . . . . . . . . . . . . . . . . . . . 674.2 Characterization of Zeeman Slower . . . . . . . . . . . . . . . . 704.2.1 Characterization of the Slower with an Atomic Beam . . . 704.2.2 Assembling Experimental Apparatus and Bake-out . . . . 714.2.3 Characterization with MOT . . . . . . . . . . . . . . . . . 754.2.4 Comparison of Observed and Simulated MOT Loading Rates 804.2.5 Improving Lithium MOT Lifetime . . . . . . . . . . . . . 805 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A Dampening Posts for Optical Bread Boards . . . . . . . . . . . . . . 91vB Measuring Dipole Trap Frequencies . . . . . . . . . . . . . . . . . . 95B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.3 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . 96B.4 Experimental Parameters . . . . . . . . . . . . . . . . . . . . . . 97B.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97C Modeling Atomic Flux and Distribution from the Lithium Oven . . 100viList of TablesTable 2.1 Hyperfine constants for 6Li, 85Rb, and 87Rb. . . . . . . . . . . 8Table 3.1 Simulated magnetic field properties of the MOT coils. . . . . . 51Table 3.2 Magnetic fields per amp produced by the various coils in the ex-perimental setup. Gradients for x and y coils are not measuredas they are run exclusively in HC. . . . . . . . . . . . . . . . . 56Table 3.3 Conductance for hydrogen of the various vacuum components. 62Table 4.1 Loading parameters for lithium MOT. . . . . . . . . . . . . . . 78Table 4.2 Lithium MOT loading curve fit values. . . . . . . . . . . . . . . 78Table 4.3 Rubidium MOT loading curve fit values. . . . . . . . . . . . . 79Table 4.4 Loading parameters for rubidium MOT. . . . . . . . . . . . . . 80Table 4.5 Effect of viewport temperature and atomic beam on MOT lifetime. 82Table B.1 Experimental parameters for trap frequency measurements. . . 97viiList of FiguresFigure 2.1 Ideal magnetic fields to slow 6Li and 85Rb. . . . . . . . . . . 7Figure 2.2 Hyperfine splitting of ground state of 6Li in a magnetic field.Colors correspond to different F states which are mixed by theexternal field. . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.3 Hyperfine splitting of D2 manifold of 6Li in a magnetic field.Colors correspond to different F states which are mixed by theexternal field. . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.4 Hyperfine splitting of ground state of 85Rb in a magnetic field.Colors correspond to different F states which are mixed by theexternal field. . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.5 Hyperfine splitting of D2 manifold of 85Rb in a magnetic field.Colors correspond to different F states which are mixed by theexternal field. . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.6 Comparison of a Zeeman slower for 85Rb using the old designmethod utilizing η as a design parameter and the new methodusing α . For fast moving atoms, the field gradients are thesame for both methods, but the blue curve becomes steeper asthe atoms slow down. The larger initial magnetic field leads toa larger capture velocity. . . . . . . . . . . . . . . . . . . . . 16Figure 2.7 Angular distribution for an ideal cosine emitter compared tochannels with an aspect ratio of aspect ratio of 10 and 50. . . . 27Figure 2.8 Velocity distribution from a transparent channel (Max. BoltzDist.) and an opaque channel. . . . . . . . . . . . . . . . . . 28viiiFigure 3.1 Simulation of capturable flux of lithium entering MOT located20 cm from a slower of variable length operating with an η =.6 and final velocity of 20m/s. The atoms are emitted from asource placed 60 cm from the slower with 90 microtubes oflength 1 cm and inner diameter 200 µm. The source tempera-ture is 400◦C. . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.2 Simulation of capturable flux of rubidium entering MOT lo-cated 20 cm from a slower of variable length operating withan η = .6 and final velocity of 20m/s. The atoms are emittedfrom a source placed 60 cm from the slower with 90 micro-tubes of length 1 cm and inner diameter 200 µm. The sourcetemperature is 80◦C. . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.3 SolidWorks rendering of the Zeeman slower along with actualslower. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.4 Comparison of the ideal slowing field and theoretical field pro-duced by the segmented slower . . . . . . . . . . . . . . . . . 39Figure 3.5 Comparison of the maximum allowable gradient as defined byEquation 2.19 and the theoretical gradients produced by thesegmented slower . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 3.6 Magnetic field for the ideal case and hybrid case which com-bines both the MOT and slower field. . . . . . . . . . . . . . 41Figure 3.7 Phase space plot showing Li atoms being slowed by the fieldin Figure 3.6 by a laser with an intensity s0 = 7 and a detuning−80 Mhz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.8 Magnetic field profile for a single solenoid. . . . . . . . . . . 43Figure 3.9 Magnetic field profile inside the Zeeman slower. . . . . . . . 44Figure 3.10 Single channel on the current distribution board, not shown isthe bypassing capacitors on the op-amp (typical values usedare 10µF tantalum cap in parallel with 100nF ceramic cap. . 45Figure 3.11 Pin out for UT control bus. . . . . . . . . . . . . . . . . . . . 45Figure 3.12 Digital logic circuitry used to select DAC. . . . . . . . . . . . 46Figure 3.13 SolidWorks rendering of the two sources. Atoms leave thesources and travel towards the right into the science chamber. 47ixFigure 3.14 Rubidium effusive source is shown sub-figure A along witha close-up of the microtubes. The backstop is shown in sub-figure B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.15 CF blank which holds the microtubes for the lithium oven. . . 49Figure 3.16 Copper covered microtube after the gasket was dissolved bythe heated lithium. . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.17 Magnetic field produced by the Helmholtz configuration. Thefield exhibits a saddle point in the middle of the two coils. . . 52Figure 3.18 Magnetic field produced by the anti-Helmholtz configuration.The gradient is twice as large in the z direction due to ∇B = 0. 52Figure 3.19 SolidWorks rendering of the MOT coils. . . . . . . . . . . . . 53Figure 3.20 Coil heating as a function of current for various flow rates. Theslope of the line gives the thermal resistance. . . . . . . . . . 54Figure 3.21 The thermal resistance of the coil as a function of flow rate.Each point is from the slope of a curve in Figure 3.20. . . . . 55Figure 3.22 Magnetic field produced by the MOT coils with current of I =10.8A for the HC (red dots) along with the simulation (blue line). 56Figure 3.23 Magnetic field produced by the MOT coils with a current ofI = 10.8A for the AHC (red dots) along with the simulation(blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.24 Magnetic field produced by the z-axis compensation coils witha current of 1 amp for the HC (red dots) along with a quadraticfit (blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.25 Magnetic field produced by the z-axis compensation coils witha current of 1 amp for the AHC (red dots) along with a linearfit (blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.26 SolidWorks rendering of the differential pumping tube. . . . . 60Figure 3.27 Cross section of the experiment. The ion pumps (IP) are la-beled and used to estimate the pressures P1 and P2. The twosections are separated by a differential pumping tube labeledas DPT (6mm dia. 12mm length). . . . . . . . . . . . . . . . 61xFigure 3.28 Pressure ratio between source and science section of experi-mental apparatus during NEG heating tests used to infer con-ductance of differential pumping tube. . . . . . . . . . . . . . 63Figure 4.1 Experimental setup used to conduct the initial diagnostics testsof the atomic beam. The lithium oven is in the bottom rightcorner. Not shown is vacuum pump system. . . . . . . . . . . 65Figure 4.2 Florencense signal observed from the effusive source withoutthe nichrome mesh lining. The lithium source is at the top andthe atomic beam travels downwards while the probe beam goesfrom left to right. . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 4.3 Florence signal observed from the oven due to the probe beamwith the nichrome mesh lining. . . . . . . . . . . . . . . . . 67Figure 4.4 SolidWorks rendering of the apertures used to measure thethermal distribution of the atomic beam. The atomic beamgoes from right to left. . . . . . . . . . . . . . . . . . . . . . 68Figure 4.5 Fluorescence signal from the two probe beams used to measurethe thermal distribution emitted from the atomic source. . . . 69Figure 4.6 Fluorescence signal from a single probe beam transverse to theatomic beam at the output of the effusive source. The sourcewas operated at 450◦C. . . . . . . . . . . . . . . . . . . . . . 71Figure 4.7 Velocity distribution of the lithium atomic beam when the sloweris operational. The peak of slow atoms is centered at 240m/s 72Figure 4.8 Experimental apparatus enclosed in the bake-out oven. . . . . 73Figure 4.9 RGA trace of the lithium oven before, during, and after bak-ing. The dominant contaminates are hydrogen (2amu), water(18amu), carbon monoxide (28 amu), and carbon dioxide (44amu). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 4.10 RGA trace of the vacuum system oven before, during, and afterfirst bake-out. . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.11 RGA trace of the vacuum system oven before and after the sec-ond bake-out. . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.12 Loading curves for lithium MOT with and without the slower. . 77xiFigure 4.13 Loading curves for rubidium MOT with and without the slower. 79Figure 4.14 Effect of viewport temperature on MOT lifetime. . . . . . . . 82Figure 4.15 Effect of atomic beam on MOT lifetime. . . . . . . . . . . . . 83Figure 5.1 A lithium MOT with approximately half a billion atoms. . . . 86Figure A.1 The customs 80/20 post consist of a lead filled extrusion, iden-tical top and bottom caps, and two nitrile spacers . . . . . . . 92Figure A.2 Impulse response of the breadboard to a disturbance on theoptical table. . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure A.3 Frequency response of the breadboard to a disturbance on theoptical table. . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure A.4 Impulse response of the breadboard to a disturbance on thebreadboard directly. . . . . . . . . . . . . . . . . . . . . . . . 94Figure A.5 Frequency response of the breadboard to a disturbance on thebreadboard directly. . . . . . . . . . . . . . . . . . . . . . . . 94Figure B.1 Flow chart of measurement setup . . . . . . . . . . . . . . . 96Figure B.2 Transverse frequency loss peaks. . . . . . . . . . . . . . . . . 98Figure B.3 Longitudinal frequency loss peaks. . . . . . . . . . . . . . . . 98Figure B.4 Transverse frequency as a function of power. . . . . . . . . . 99Figure B.5 Longitudinal frequency as a function of power. . . . . . . . . 99Figure C.1 Experimental setup used to characterize atomic beam. . . . . 101xiiGlossaryList of abbreviations used in this thesis.AHC Anti-Helmholtz ConfigurationAOM Acousto-optic modulatorCF ConFlat Vacuum FlangeDAC Digital to Analog ConverterHC Helmholtz ConfigurationLIRB Lithium RubidiumMOT Magneto-optical TrapNEG Non-evaporable getterRGA Residual Gas AnalyzerxiiiAcknowledgmentsI entered my Master’s thesis with the goal of focusing on a specific problem inphysics culminating in me contributing in some meaningful way to society’s gen-eral body of knowledge. During this time, I had the privilege of working with someremarkable people. In the end, I found these two parallel processes equally valu-able and often thought it rather unfair that the former gets outlined in great detailin a 100 page thesis while the latter gets only a brief one page mention. However,I now realize that this is completely justified for if you were to ask me in the fu-ture “what was power of the MOT beam?” or “what is the ground state splittingof 6Li?” I would no doubt have to look it up, but if asked about the friendships Ifounded during these past two years I would be able to recall them instantly withgreat fondness.First, I would like to acknowledge my supervisor, Dr. Kirk Madison, whose enthu-siasm and insatiable curiosity for physics is truly inspiring. Under his guidance,I have acquired many technical skills as well as a scientific approach to thinkingwhich will be invaluable throughout my professional career. Your dedication toyour students is greatly appreciated.I would like to thank the entire QDG group as the friendships I made with youall are one of the most valuable things gained from my work in the lab. In par-ticular, I would like to acknowledge the two senior PhD students working on theproject, Will Gunton and Mariusz Semczuk, for their guidance and teaching. Oftentheir approach to experimental physics was as different as night and day (literallyand figuratively), but through working with both I gained valuable experience andxivinsight into experimental physics research. I always enjoyed conversing with Mar-iusz about physics as if it was a sport and about baseball, with Will, as if it were ascience.I would like to thank Janelle Van Dongen for her patience in showing me the insand outs of the lab when I first joined. I often told her that I would stop bother-ing her with my questions if she didn’t always give me the answer, but she neverwould take my advice. I would also like thank Gene Polovy for our many fruitfulengineering conversations which did wonders for teaching me things I thought Ialready knew. Finally, Kahan Dare and Komancy Yu, I would like to extend myappreciation for your constant willingness to lend a hand with whatever was caus-ing me to pull my hair out on that given day. It always amazed me how quickly“my” problems were solved once they became “ours.” Along with Kais Jooya, Ilook forward to following your future endeavors in the QDG lab.I would like to extend my deepest thanks to my parents for their love and guid-ance. They instilled in me the curiosity and confidence to pursue my interest inphysics. I commend your persistence in always asking how my day was only to bereturned with an unenthusiastic fine.Lastly, I would like to say thank you to Maggie Chao. I do not think I will ever beable to convey how much your support and love meant to me during these past twoyears.xvChapter 1Introduction11.1 Why Study Ultracold Polar Molecules?The field of ultracold atomic physics focuses on the formation and control ofultracold atoms through which scientists can study a range of physical phenom-ena. These systems have been fruitful in improving our understanding of ultracoldchemistry, few and many body physics, and quantum information theory. Suchexperiments are made possible by the arsenal of tools which have been developedto trap and cool various atomic species along with the ability to precisely controltheir internal and external degrees of freedom.Building upon the success of ultracold atoms research, scientists are now look-ing for new systems to apply these same methods and further develop this toolkit.An excellent candidate for such work are ultracold molecules as their complexenergy level structure, which includes rotational and vibrational levels, and longrange interactions provide scientists with a handle through which they can manip-ulate these complex quantum systems. In particular, the inherent dipole moment ofsuch molecules give rise to relatively strong, long range, dipole-dipole interactionswhich are easily tunable with external electric fields [1]. These interactions are ofparticular interest in the study of ultracold collisions between polar molecules asthey give rise to novel types of resonances [2] and the potential to control chem-ical reactions [3]. These same tunable dipole-dipole interactions make ultracoldmolecules well suited for quantum information applications requiring the creationand manipulation of entangled states [4, 5]. Last but not least, the long range natureof these interactions, in contrast to the typical contact interactions found in otherultracold systems, could be utilized in quantum gases of polar molecules to studyexotic phases of matter [6, 7].Polar molecules made from lithium and rubidium are a promising candidate forsuch applications as they possess a predicted dipole moment of appropriately 4Din the X(1)1Σ+g ro-vibrational ground state [8]. However before such experimentscan be conducted using ultracold Lithium Rubidium (LIRB) molecules, they mustfirst be made.21.2 Overview of ThesisThis thesis outlines the theory, design, construction, and characterization of anexperimental apparatus used to produce large samples of cold 6Li, 85Rb, and 87Rb.The samples are held in a Magneto-optical Trap (MOT) and are a necessary startingpoint for the formation of ultracold LIRB molecules. The main focus of this thesiswill be on a dual species Zeeman slower used to load the MOT and an analysis ofits performance. This thesis is divided into three main parts covering the theory,design, and characterization of the experimental apparatus.The theory chapter of the thesis focuses on the Zeeman slower and the effu-sive atomic sources. For the Zeeman slower, the principal of operation is coveredwith particular focus on slowing lithium and rubidium. With respect to the effusivesources, a model for accurately predicting the atomic flux, along with angular andvelocity distribution is highlighted. The principal operation of a MOT is not cov-ered as there exist numerous sources which present this information with excellentclarity and depth [9].The design section of this thesis focuses on the design of the effusive sources,Zeeman slower, and experimental apparatus. The beginning of the chapter offers areview of existing Zeeman slower designs and insight into the various challenges ofconstructing a slower for multiple atomic species. The design of the effusive sourcefor both rubidium and lithium is covered. The chapter ends with a discussion ofthe electronics used to control the Zeeman slower remotely and the design of thevacuum system.The characterization section of the thesis starts with the investigation of the per-formance of the effusive sources along with a comparison of the theoretical modeldeveloped in Chapter 2. Next the performance of the Zeeman slower is character-ized, first by spectroscopically probing the atomic beam to measure the modifiedvelocity distribution and then by loading a MOT of both lithium and rubidium.3Chapter 2Theory of Multi-species AtomicSource42.1 Theory of Zeeman SlowersThe Zeeman slower was first developed by William Phillips in 1997 and uses theradiation pressure of light to slow atoms within an atomic beam. The ingenuity ofthe device is in its utilization of spatially varying magnetic fields that Zeeman shiftthe atomic levels to compensate for the changing Doppler shift as the atoms areslowed. Without compensation for the Doppler shift, after slowing by a few tensof meters per second, the atoms would no longer be in resonance with the slowingbeam. However, by correcting for this shift, the Zeeman slower can slow beamsfrom 1000 m/s to 10 m/s which makes them ideal for loading a MOT.The scattering force, Fs, exerted on an atom moving with velocity v towards acounter propagating laser beam with wavenumber k and intensity I = s0Is 1, withIs being the saturation intensity of the atomic transition of natural width γ , is givenby:Fs =h¯kγ2s01+ s0 +(2δ ′/γ)2(2.1)The effective detuning, δ ′, accounts for the following three sources of detuning:(1) laser detuning from the cycling transition frequency ω given by δ = ω − ck,(2) the Doppler detuning which is given by kv, and (3) the Zeeman shift due to amagnetic field B given by µB/h¯ where µ is the magnetic moment of the transition.Combining these three effects gives the expression for δ ′:δ ′ = δ + krv−µ ′B/h¯ (2.2)The maximum force is achieved when the effective detuning is zero and results inconstant deceleration, a, given by:a = amaxs01+ s0(2.3)where amax is the largest possible deceleration in the limit of infinite intensity. Itis convention to write the deceleration as a = ηamax where η is a design parame-ter between zero and one which controls how aggressively the atoms are slowed.1s0 is a dimensionless quantity often referred to as the saturation parameter.5Larger values of η lead to larger initial capture velocities, but require more intenseslowing beams and decrease the allowable deviation from the ideal slowing field.Given that the effective detuning must always be zero, the ideal slowing field iseasily calculated under the assumption of constant deceleration. Using kinematics,the positionally dependent velocity is given by:v(z) =√v2c−2az (2.4)where vc is the initial velocity of the atomic beam which can be captured by theslower. Inserting this expression into Equation 2.2 and setting the detuning to zerogives the desired field parameter.B(z) =h¯kvcµ ′√1−2av2cz−h¯δµ ′ (2.5)Therefore, the desired field is uniquely defined by an amplitude Ba, length scale z0,and finally an offset B0 given by:Ba =h¯kvcµ ′z0 =v2c2νamaxB0 =h¯δµ ′(2.6)and such, the desired field can be rewritten as:B(z) = Ba√1−zz0−B0 (2.7)As an aside, it is now clear what effect η has on the field profile; it effectivelystretches the field allowing atoms to spend more time at a particular magnetic fieldto ensure they slow to the desired velocity at that position. Figure 2.1 shows theideal field for a 50 cm slower with η = .75 for both 6Li and 85Rb.6Figure 2.1: Ideal magnetic fields to slow 6Li and 85Rb.2.1.1 The Zeeman EffectIn order to calculate the desired magnetic field profile, the dependence of energyeigenstates on an external magnetic field must be calculated. For typical magneticfield strength, the energy splitting is on the order or greater than the hyperfinestructure, but much less than the fine structure. Therefore the relevant Hamiltonianis the one which takes into account both the effect of an external magnetic field andthe coupling of the nucleus to the internal electric and magnetic fields which givesrise to the hyperfine structure. The hyperfine Hamiltonian including the magneticdipole and electric quadrupole terms is given by:Hh f s = Ah f sI ·J+Bh f s3(I ·J)2 + 32 I ·J− I(I +1)J(J +1)2I(2I−1)2J(2J−1)(2.8)where Ah f s and Bh f s are the magnetic dipole and electric quadrupole constants,respectively and J and I are the total electronic angular momentum operator andthe nuclear spin operator, respectively. Table 2.1 lists these values for the variousspecies of interest in this thesis [10, 11].7Table 2.1: Hyperfine constants for 6Li, 85Rb, and 87Rb.Constant Value [MHz] Ref6Li 22S1/2 Magnetic Dipole 152.1368407 [12]6Li 22P3/2 Magnetic Dipole -1.155 [12]6Li 22P3/2 Electric Quadrupole -0.10 [12]85Rb 22S1/2 Magnetic Dipole 1011.910813 [12]85Rb 52P3/2 Magnetic Dipole 25.0020 [12, 13]85Rb 52P3/2 Electric Quadrupole 25.790 [12, 13]This coupling results in neither the projection of J nor projection of I being con-served quantities resulting in them being no longer good quantum numbers. How-ever, this coupling still ensures that total angular momentum F , given by the sum ofI and J, and its projection m f remain conserved quantities. Hence, F and its projec-tion mF remain good quantum numbers and the above Hamiltonian is diagonalizedin the |I,J,F,mF〉 basis.When a static magnetic field is applied, rotational symmetry is broken whichlifts the degeneracy in m f values. The Hamiltonian which describes the interactionbetween an atom and the external field is given by:HB =µBh¯(gsS+gLL+gII) ·B (2.9)where gS and gL are the g-factors for the electron’s spin and orbit, while gI accountsfor nuclear spin. The value for gs has been measured to a high degree of precision.The orbital g-factor to first order in the electron to nucleus mass ratio is:gL = 1−memn(2.10)The nuclear g-factor is challenging to calculate theoretically as it takes into accountthe complex structure of the nucleus and it is therefore best to use experimentallymeasured values. By convention, the magnetic field is taken to be along the z-axis.Making use of the fact the energy associated with the magnetic field within theZeeman slower is weaker than the spin-orbit coupling, J remains a good quantumnumber and HB can be written as:8HB =µBh¯(gJJz +gIIz)Bz (2.11)where gJ is the Lande g-factor given by:gJ = gLJ(J +1)−S(S+1)+L(L+1)2J(J +1)+gsJ(J +1)+S(S+1)−L(L+1)2J(J +1)(2.12)For calculating the slower field profile, a high degree of precision is not requiredand it is sufficient to neglect relativistic effects and mass corrections in which casegs = 2 and gI = 1 simplifying the above expression to:gJ = 1+J(J +1)+S(S+1)−L(L+1)2J(J +1)(2.13)To calculate the energy spectrum the complete Hamiltonian, which includesboth hyperfine and magnetic field effects, must be diagonalized. However, pertur-bation theory can be used to simplify this calculation in the two extremes whenthe B Ah f s/µB and B Ah f s/µB. In the former case, F and mF remain “prettygood” quantum numbers and the interaction Hamiltonian can be approximated:HB =µBh¯gFFzBz (2.14)where the hyperfine Lande g-factor is given by:gF = gJF(F +1)− I(I +1)+ J(J +1)2F(F +1)+gIF(F +1)+ I(I +1)− J(J +1)2F(F +1)(2.15)To first order in perturbation theory, the energy shift is linear in a magnetic fieldand depends only on the projection of F onto the magnetic field axis and is givenby:∆E|I,J,F,mF 〉 = µBgFmFBz (2.16)In the other extreme, large fields cause J and I to process about the z-axis whichleads to both mJ and mI being conserved quantities and the effect of the hyperfine9Hamiltonian can be calculated via perturbation expansion in the |J, I,mJ,mI〉 basis.In such a regime, first order perturbation gives the following energy spectrum:∆E|I,J,mJ ,mI〉 = µB(gJmJ +gImI)B+Ah f smImJ+Bh f s9(mImJ)2−3J(J +1)m2I −3I(I +1)m2J + I(I +1)J(J +1)(4J(2J−1)I(2I−1))(2.17)In the intermediate regime, the entire Hamiltonian must be diagonalized. This cal-culation is most easily done by first computing the two interaction terms in theHamiltonian in their respective diagonal bases, i.e. |J, I,F,mF〉 for the hyperfineterm and |J, I,mJ,mI〉 for the magnetic contribution. Next, either the hyperfine ormagnetic component of the Hamiltonian is transformed to its non-diagonal basisusing the Clebsh-Gordan coefficient which relates the two bases. The energy spec-tra are shown in the following figures along with the labeled slowing transition.Figure 2.2: Hyperfine splitting of ground state of 6Li in a magnetic field. Col-ors correspond to different F states which are mixed by the externalfield.10Figure 2.3: Hyperfine splitting of D2 manifold of 6Li in a magnetic field.Colors correspond to different F states which are mixed by the externalfield.The stretched states experience a linear shift for all magnetic fields. This couldhave been deduced earlier as the Clebsh-Gordan which relates the maximal valueof mF to maximal values of mI and mJ is one. Therefore, the transition magneticmoment used in Equation 2.2 to calculate the desired field profile is:µ ′ = µB(gF,emF,e−gF,gmF,g) (2.18)where the subscript denotes the excited and ground state levels of the slowing tran-sition. For 85Rb the slowing transition, in the low field basis, is from the 52S1/2(|J, I,F,mF〉=|1/2,5/2,3,3〉) level to the 52P3/2 (|3/2,5/2,4,4〉) level which hasa transition moment from Equation 2.18 equal to simply µB. For 6Li, the slowingtransition is from the 22S1/2 (|1/2,1,3/2,3/2〉) level to the 22P3/2 (|3/2,1,5/2,5/2〉)level which has an transition of µB as well.11Figure 2.4: Hyperfine splitting of ground state of 85Rb in a magnetic field.Colors correspond to different F states which are mixed by the externalfield.2.1.2 Alternative Field ProfilesThe theoretical profile presented in Section 2.1 assumes that the energy splittingbetween the ground state and excited state increases with magnetic field. However,it is possible to design a Zeeman slower which operates on transitions for which thelevels move closer in energy with increasing field. For such a slower, the requiredfield must increase as the atom slows in order to compensate for the decreasingDoppler shift. The various types of Zeeman slower are often referred to by thepolarization of light which drives the transition, namely σ+, σ−, and finally spin-flip. The major advantages and disadvantages of each type of slower are outlinedis this section.σ + Zeeman SlowerIn such a slower, atoms enter in a high field region which decreases as theyslow. As a result of the atoms being in a low field when moving the slowest, only12Figure 2.5: Hyperfine splitting of D2 manifold of 85Rb in a magnetic field.Colors correspond to different F states which are mixed by the externalfield.a small detuning from the zero velocity resonance is needed to maintain the reso-nance condition. This is convenient as such detuning can be derived from existingMOT beams using an Acousto-optic modulator (AOM). However, having a nearresonance slowing beam can interfere with the MOT. The small detuning leads toanother disadvantage as the atoms can be stopped completely by the slowing beambefore even reaching the MOT.σ - Zeeman SlowerOpposite to the σ+ design, the atoms experience an increasing field as theymove down the slower and experience the largest Zeeman shift at the end. As aresult, such a design requires a large laser detuning, typically on the order of GHz,in order to keep the atom in resonance. Such a detuning is beneficial as it will havelittle influence on the behavior of the MOT, although practically it typically requiresa more complicated scheme than simply frequency shifting the MOT lasers with an13AOM. The other benefit of having the field maximum at the end of the slower isthat it ensures the atoms will quickly fall out of resonance with the slowing beamwhich prevents them from being further slowed and potentially stopped completelybefore reaching the MOT.Spin-Flips Zeeman SlowerThe final field profile commonly implemented has zero crossing at some pointalong the slower. The field can either be increasing or decreasing as is the case withthe σ+ and σ−slowers, respectively, but the the detuning is changed to shift thefield downwards. Practically, such a slower has a smaller absolute field maximumand therefore less current is needed to generate the desired profile which in turnleads to less heating. The atoms leave the slower at a larger field which requiresgreater detuning and therefore it shares many of the same benefits as the σ− design.However, one drawback of the spin-flip design is that the atoms spend more timeat smaller fields. In such a regime, the energy levels are closer spaced and theslower transitions are not closed. This leads to atoms getting optically pumped todark states which stop the slowing process and requires additional beams to transferatoms back into the slowing state. However, for some atomic species repump lasersare also needed for σ+ and σ− designs as well.2.1.3 Further Design ConsiderationsThere are many key design considerations when determining the various featuresof the Zeeman slower. This subsection highlights a few key points which shouldbe considered during the design process.Adiabatic Slowing ConditionAs mentioned in Section 2.1, there is a maximum deceleration imposed by the finitescattering rate. This limit constrains the maximum magnetic field gradient alongthe slower axes above which will no longer slow atoms. Given that the accelerationcan be written as a = v dvdz , differentiating Equation 2.2 leads to the expression forthe maximum field gradient:14|dB(z)dz| h¯kamaxµv(z) (2.19)This criteria is typically referred to as the adiabatic slowing condition. Zeemanslowers are designed to slow at some constant fraction of amax. However, this isnot an optimal design strategy in terms of capturing the largest fraction of atomsfor fixed slower length. The maximum gradient is inversely proportional to veloc-ity which makes the adiabatic condition more stringent for faster moving atoms.Therefore, the strategy of uniformly stretching the field by changing η is not ideal,rather it is better to vary η depending on velocity at that particular location in theslower.An alternative design strategy was proposed that takes into account the velocitydependence in the adiabatic slowing condition [14]. The design method defines afixed difference, α , between the gradient at a particular location and the maximumgradient defined in Equation 2.19 given the velocity at that position. Furthermore,both the Doppler shift and Zeeman shift are linear in velocity and magnetic field,respectively. In order for the two to cancel each other the v(z) must be proportionalto B(z) with the proportionality factor being h¯k/µ . Hence, v(z) can be writtenin terms of the magnetic field which transforms Equation 2.19 into the followingdifferential equation:α = h¯kamaxµ ′B ±µ ′h¯k∂B∂ z (2.20)The ± factor corresponds to whether the slower is σ+ or σ−. The factor α isreferred to as the noise parameter and takes into account laser intensity fluctuations,current noise which produces the field fluctuations, or deviations in the magneticfield from the ideal profile due to winding. The differential equation is solved byusing the Lambert W function and has the following solution:B(z) =h¯kµ ′amaxα (W [(vcαamax−1)exp(zα2 + vcαamax−1)]+1) (2.21)where vc is the initial capture velocity of the slower. The value for η when theatoms first enter the slower gives the corresponding value for alpha:15α = (1−η)amaxvi(2.22)In the limit of α goes to zero, the above expression reduces to the one previouslyderived in Equation 2.7 by setting η = 1. For comparison, the capture velocity fora 50 cm slower for 6Li with a η = 0.5 and final velocity of 50 m/s would increasefrom 950 m/s to 1100 m/s and for 85Rb it increases from 240 m/s 270 m/s. Theseare improvements of approximately 15% which may not appear significant, butit should improve the loading by more than 50% given the v4 scaling which willbe derived in Section 2.1.4. Figure 2.6 shows the resulting fields from the twodifferent design methods: the red curve is derived from Equation 2.7 which tunesthe field using η and the blue curve is Equation 2.21 which uses α as a designparameter.Figure 2.6: Comparison of a Zeeman slower for 85Rb using the old designmethod utilizing η as a design parameter and the new method usingα . For fast moving atoms, the field gradients are the same for bothmethods, but the blue curve becomes steeper as the atoms slow down.The larger initial magnetic field leads to a larger capture velocity.16Zeeman Slower for Multi-speciesThe design of a multi-species slower requires different approaches depending onwhich combination of species is being slowed. Two questions must be addressedwhen designing a multi-species slower: (1) can the slower simultaneously slow thedesired species when operating at the same magnetic field and (2) can the slowerslow the different species when operating at different magnetic fields. Dependingon the species being slowed, the answer to the first question is maybe, but fortu-nately the answer to the second question for all commonly slowed species is yes.As derived in Section 2.1, the slowing profile is specified by the amplitudeBa and a length scale z0. The amplitude can be controlled via the current, but z0depends on the construction of the slower and cannot be easily changed in mostdesigns. The value of z0 is critical to the slower’s operation as it controls η . It canbe shown that if the final velocity is small compared to the initial velocity, whichis always the case, then z0 is a constant for all species. Let z0,1 and z0,2 be theslower length scale factors for atomic species 1 and 2. The ratio of these lengths,which can be taken to be greater than one without loss of generality, is given byEquation 2.6 as:z0,1z0,2=1+v2f2a1z1+v2f2a2z(2.23)where z is the slower length, a is the deceleration, and it is assumed that bothspecies have comparable final velocities. The quantity 2az is approximately thesquare of the initial capture velocity of the slower, vc, and the above expression is:z0,1z0,2≈ 1+v2f2za1−a2a1a2< 1+v2f2za1≈ 1+v2fv2c,1(2.24)For any effective Zeeman slower, the ratio of the final velocity to the initial velocityis greater than 10 which makes the largest difference between z0 values less thanone percent. Hence, if the Zeeman slower is designed for one species it can beused to slow any another atomic species by simply changing the amplitude Ba,most likely by increasing the current through the windings. In order to determineif a slower can effectively slow two species simultaneously one can look at the ratio17of η which should be on the order of 1. Given that Ba and z0 are the same for bothspecies, the ratio of η1 to η2 is given by:η1η2=M1µ ′1k2γ2M2µ ′2k1γ1(2.25)For 6Li and 85Rb, this ratio is 0.04 due to large mass imbalance which eliminatesthe possibility for simultaneous slowing. A complete table showing this ratio formost trappable elements can be found in [15].2.1.4 Optimal Zeeman Slower LengthArguably the most critical design choice when constructing a Zeeman slower isdeciding upon the length which in turn determines the maximal velocity which canbe slowed. However, increasing the length may not actually increase the size of theMOT as the divergence of the beam leads to a flux which decreases in accordancewith the inverse square law. The capture velocity, vc of a Zeeman slower of lengthL is given by√2ηamaxL and the fraction of atoms captured is the integral of thevelocity distribution function up to this capture velocity. For small velocities, thedistribution function grows as v3 and the integral grows as v4c . Therefore the flux,Φ is independent of the Zeeman slower length as:Φ ∝v4cL2∝√L4L2(2.26)This reasoning greatly oversimplifies the problem as it neglects that atoms do nottravel in straight trajectories as they are slowed leading to a faster divergence andthat the velocity distribution does not grow as v3 indefinitely. Both of these fac-tors motivate going to shorter slowers and begs the questions why have a slower atall? In practice, it is not desirable to have an atomic source in close proximity tothe MOT as it limits achievable background pressure. When the atomic source isplaced near the MOT and initially heated, the flux of alkali atoms increases. At lowtemperatures, background pressure typically stays unchanged as it is dominated byother sources rather than the outgassing of the oven. The MOT loading rate (i.e. thecapturable flux) increases along with the steady state captured atom number withincreasing temperature. Once the oven temperature is high enough that the out-18gassing of the source starts to dominant the background pressure, the atoms decayrate from the MOT increases. The steady state atom number, given by the ratio ofthe loading rate to the loss rate, would then be expected to saturate with increasingtemperature as both rates are proportional to the atomic density inside the oven.However, this has not been observed with our previous experiential apparatus. In-stead, the outgassing of the oven is primarily dominated by hydrogen or other gasesthat have a lower binding energy to the material and thus the MOT loss rate due tobackground gas collisions increases faster than the loading rate due to the flux ofalkali atoms. This results in a maximum in the steady state MOT atom number withincreasing oven temperature. [16, 17]. The Zeeman slower provides differentialpumping which isolates the source section of the experimental apparatus from thescience section and allows for lower background pressure without greatly reducingthe flux.2.2 Theory of Effusive SourcesThe majority of cold atom experiments start with an effusive source which consistsof a container containing a solid sample of the atomic species being investigated.The container has a small orifice through which atoms can leave. The flux ofatoms leaving the source is controlled by heating the sample to increase or de-crease the vapor pressure. In the molecular flow regime, the flux from a reservoirthrough a orifice channel is characterized by its Knudsen number, defined as theratio between the mean free path length (λ ) of atoms in the reservoir to the relevanttransverse dimension of the orifice, e.g. the radius for a cylindrical hole. Effusiveflow or molecular flow occurs when the Knudsen number is much larger than one.In this regime, the flux is proportional to the vapor pressure inside the reservoir.If the mean free path decreases below the channel radius, the flow transitions toa viscous or super-sonic flow characterized by a rapid increase in flux, quicklyexhausting the source material. Furthermore, in the viscous regime there is a de-pletion of the low velocity tail of the Maxwell-Boltzmann Distribution [18]. Theother relevant dimension is the channel length. For channels shorter than the meanfree path, the channel is characterized as transparent. The angular distribution ofthe flux from the channel is comprised of a sharp peak confined to the region which19has direct line of sight into the reservoir and background flux due to diffusion fromthe channel walls. The other regime, when the mean free path is less than the chan-nel length, is referred to as opaque flow. In this regime, inter-channel collisionschange both the angular and velocity distribution of the atomic flux.2.2.1 Transparent RegimeFor an effusive source, the most relevant quantity of interest is the total number ofatoms leaving the source, N, and the angular distribution in which they leave. Theangular distribution is described by an intensity function, I(θ), which has units ofatom per steradian per second. The angle θ is the angle from the normal of the exitchannel opening. I assume the opening is circular and the angular distribution isrotationally symmetric. The total flux and the intensity are related by:N =∫ΩI(θ)dΩ (2.27)The simplest transparent effusive source to model is an infinitely thin walledaperture. Such a source is often referred to as a cosine emitter as the angulardistribution function is simply cos(θ). Such an emitter is analogous to Lambert’scosine law in optics which states that the radiance from an ideal diffusive surfaceis proportional to the cosine of the viewing angle. The number of particles, dN,emitted into a solid angle in the direction defined by the angle θ from a reservoir ofatoms with density n and average velocity v¯ through an aperture of area A is givenby [19].dN = (dΩ4pi )nv¯cosθA (2.28)Integrating the above expression from θ equal zero to pi/2, an expression for thetotal flux from the cosine emitter is found:N = (14)nv¯A (2.29)The velocity distribution inside the oven is the standard Maxwell-BoltzmannDistribution (F3DMB(v)) for an ideal gas which is proportional to the velocity squared.20F3DMB(v) =4piv2(αpi)3/2exp−(vα )2 (2.30)where α is the standard expression for the most probable speed√2kT/m [19].However, this is not the distribution for the beam as the probability of an atomleaving the oven is proportional to its velocity. Simply put, faster moving atomshave a higher probability of leaving the oven per unit time and the velocity distri-bution inside the beam is given by:FBeamMB (v) =2v3α4 exp−(vα )2 (2.31)The pre-factor can be derived given the normalization condition. If we considera more realistic case for a finite length channel, the total flow is reduced by theClausing factor, W , which accounts for the fact that wall collisions lead to a fractionof the atoms returning to the source. This effect alters the angular distributionresulting in it being described by a new distribution function, typically written asκ f (θ)[20]. The number of particles emitted into a solid angle in a direction θ isnow:dN =dΩ4pi Wnv¯κ f (θ)A (2.32)For a circular channel with radius a and length L, the Clausing Factor is givenby[21]:W =8a3L(1+8a3L)−1 (2.33)The variable κ is referred to as the peaking factor and describes the ability of aspecific channel geometry to focus the beam. It is defined as the ratio of the centerline directional intensity (I(θ = 0)) to the total flux N.κ = pi I(0)N(2.34)The factor of pi is simply from convention such that the peaking factor for co-sine emitter is one. Since the center line intensity is unchanged in the transparentregime, it follows that the peaking factor for a transparent channel is simply the in-21verse of the Clausing Factor. Unfortunately, the nomenclature for effusive sourcesvaries significantly from reference to reference. I will follow the convention of Bei-jerinck and Verster and define the angular distribution as the product of the peakingfactor and the angular profile, f (θ), which has the following normalization integralover the entire solid angle [20]:∫Ωκ f (θ)dΩ= pi (2.35)For long channels which provide a high degree of focusing (L/a > 10), the angulardistribution function can be reasonably approximated. For angles which have directline of sight into the oven (θ < arctan(2a/L)) the angular distribution is [20, 21]:κ f (θ) = 2cosθpiW [(1−W/2)R(p)+23(1−W )×1− (1− p2)3/2p+piW4] (2.36)The values of of p and R(p) are defined as:p =L tanθ2a(2.37)R(p) = arccos p− p(1− p2)1/2 (2.38)For angles which do not directly extend into the reservoir the atoms observed areemitted from the walls and the distribution function is:κ f (θ) = 8acos2 θpiW sinθ (1−W )+cosθ2(2.39)Figure 2.7 shows the angular distributions for various peaking geometries multi-plied by the Clausing factor. In all cases, the centerline intensity is the same. Thevelocity distribution leaving channel is identical to that of a cosine emitter as theatoms which fly directly from the interior and those which are emitted from thewalls are in thermal equilibrium.The key assumption in this derivation is atomic collision with the wall willresult in the atoms being re-emitted in a random direction whose distribution isgoverned by the cosine law. Secondly, collisions within the channel are neglected.22If surface diffusion occurs such as wicking though the channel, the angular distri-bution will differ from that predicted above.2.2.2 Opaque RegimeThe opaque regime applies to the situation when the mean free path is less than thelength of the channel, but still much larger than the diameter, and inter-molecularcollisions within the tube modify the angular profile and velocity distribution. De-termining these two profiles is challenging and an initial ansazt of the density pro-file along the length of the channel must be made [22]. If the channel is long, thedensity varies linearly from the source density, n0, to the exit and any perturbationinduced from the channel opening and exit can be neglected. It is convenient to as-sume the exit density is close to zero. Therefore, the density as a function positionfrom the exit is x n0L . Using this density profile, the point in the channel at which themean free path is equal to the remaining length of the channel can be found. Thisremaining length in the channel is denoted as L′ and can be calculated by settingthe mean free path at a position L′ equal to L′ as follows [20, 22]:λ [L′] = 1√(2)n(L′)piσ= L′=L√(2)n0L′piσ= L′⇒ L′ = L√Kn(2.40)where Kn is also a Knudsen number, but it is now the ratio of the mean free pathin the reservoir to the channel length, and σ is the collisional cross section. Thesource density, n′ at the point L′ is given by:n′ = n0√Kn (2.41)If we now make the simplifying assumption that the opaque channel behaves asa transparent channel with an effective length L′ and source density n′, we cantreat the problem exactly as we did before for the transparent case. From thisassumption, we can define the peaking factor for an opaque channel:23κopaque = κtrans√Kn (2.42)This model will lead to underestimates of the total flux which results from theassumption of an isotropic velocity distribution at the distance L′. However, thechannel length prior to L′ will help to focus the beam and provide a net velocity inthe forward direction. On the other hand, if the channel was treated as transparentthe resulting flux would be over estimated. Therefore, modeling the channel usingboth the actual length and the effective length will provide upper and lower boundson the total flux. The lithium effusive source operates well within the transparentregime as the mean free path is on the order of meters. However, rubidium has amean free path of about of 1 cm at 120◦C which is approaching the opaque regimeand should be considered when modeling the source.2.2.3 Velocity Distribution in the Opaque RegimeThe collisions within the channel lead to a modification in the velocity profile,in particular slow moving molecules experience a larger collisional rate and aredepleted from the population. This is particularly concerning as the low velocityatoms which are trapped by the MOT. The mean free path within the channel de-pends on the density at that location and the atom’s speed. An atom at position xaway from the exit of the tube, will escape without any collisions with a probabilitygiven by:P(x) = exp[−∫ 0xdx′λ (x′) ] (2.43)Given the assumption of the linear density profile, the mean free path is pro-portional to the reciprocal of distance down the channel and the integral can beevaluated directly and the result is [23]:P(x) = exp[−(x/L)22Kn] (2.44)It is now important to modify the mean free path to account for the dependence onvelocity. In actuality, we should have done this earlier when we derived the angularprofile of the opaque channel which would have resulted in a velocity dependent24effect length. However, for practical purposes this can be neglected as experimentalresults are in good agreement with the results derived previously. The velocitydependent mean free path length written in terms of the normalized speed z definedas v/α is:λ (z,x) = λ (x)ψ(z) (2.45)where ψ(z) is given by the following equation for which a derivation can be foundin [24]:ψ(z) = zexp[−z2]+√pi(2+ z2)erf(z)√2piz2(2.46)The rate at which molecules moving at a reduced speed z at a position x collidewith a background gas with density n(x) is the reciprocal of the mean free pathmultiplied by the particle’s velocity. The number of particles in this reduced speedrange is n(x)F(z)dz where F(z) is the velocity distribution. As a first approxima-tion, we can assume collisions do not perturb the distribution significantly awayfrom a Maxwell-Boltzmann distribution and will remain a valid estimator of F(z).In accordance with the principle of detailed balance, the fraction atoms between zand z+dz entering into collisions must equal the number of molecules in the samevelocity range produced by collisions. Hence, the number of molecules being pro-duced due to collisions with velocity in range z+dz in a channel volume pia2dx isgiven by αzλ−1(x)F(z)n(x)pia2dx. Of these molecules, a fraction 1/4pi is emittedinto the solid angle along the tube axis. Of those molecules which are scattered inthe forward direction along the tube axis a fraction will escape with the probability:P(x) = exp[−ψ(z)(x/L)22Kn] (2.47)We can then integrate over the entire channel length to get the molecules which areemitted along the channel axis due to collisions. We also have to add the fractionthat are emitted directly from the reservoir and make it to the exit without anycollisions. This calculation was carried out in reference [23] and the resultingvelocity distribution is given in terms of a perturbation function P[Kn,Z]:25F(z) ∝ FBeamMB (z)P[Kn,Z] (2.48)The perturbation function depends on speed and channel geometry and is given by[23]:P[Kn,z] =pi2erf[√ψ(z)/2Kn]√ψ(z)/2Kn(2.49)To normalize the result, the above expression must be divided by integral overall velocities. Unfortunately, to the best of my knowledge this integral cannot besolved exactly, but can be easily computed numerically. As the channel length goesto zero, the perturbation function goes to 1 as expected. For large channel lengths,the contribution from molecules emitted directly from the reservoir goes to zeroand the result no longer depends on the channel geometry. As a result, the functiongoes to a constant. Figure 2.8 shows the modified velocity profile. The effect ismarginal and will be challenging to observe experimentally.26Figure 2.7: Angular distribution for an ideal cosine emitter compared tochannels with an aspect ratio of aspect ratio of 10 and 50.27Figure 2.8: Velocity distribution from a transparent channel (Max. BoltzDist.) and an opaque channel.28Chapter 3Design of Multi-species AtomicSource293.1 Previous Experimental SetupOur existing experimental setup has been in operation, with various modifications,for the past six years and has performed well. A detailed performance of the sys-tem can be found in [16, 17]. The system excelled in its simplicity as it did notrequire a Zeeman slower or a 2D MOT to load the 3D MOT. The lithium MOT isloaded directly from an effusive source placed inside the quartz cell while the ru-bidium MOT was loaded by a dispenser. However, with future goals of dual speciesexperiments it was unclear if such a system would produce sufficiently large sam-ples of ultracold atoms from which to form molecules. When working primarilywith lithium, the long MOT lifetime allowed for relatively large atomic samples(approximately 108 atoms) despite modest loading rates. The downside of sucha system is the wait time for loading the MOT decreases the repetition rate of theexperiment which slows data acquisition. Unfortunately, when adding rubidium tothe system the background pressure increases which in turn decreases the lithiumlifetime leading to a smaller MOT. To load a rubidium MOT of approximately 108atoms results in lithium lifetimes on the order of seconds which would not suffice.This shorter lifetime could be mediated by decreasing the background pressure ofrubidium, but this in turn results in a much smaller rubidium MOT. For exam-ple, reducing the rubidium background pressure to a level which only decreasedthe lithium lifetime by factor of two resulted in rubidium MOTS which containedon the order of 106 atoms. This motivated the addition of the Zeeman slower asit isolates the sources from the science section while allowing for larger loadingrates.3.2 Simulation of Zeeman Slower FluxBefore constructing a Zeeman slower, a virtual slower was created to investigatethe effect of various design parameters on the flux of cold atoms. The virtual slowerwas implemented in MATLAB and uses basic kinematic theory to determine theflux of cold atoms which can captured by the MOT.The effusive oven and Zeeman slower produce a focused beam of cold atomswhich can be captured by the MOT. Assuming the angular and velocity distributionof atoms leaving the effusive atomic source is known, the flux of cold atoms which30pass the through the MOT with a velocity less than the MOT capture velocity canbe determined from geometry and simple kinematics. The Zeeman slower onlyaffects the axial component of an atom’s velocity and leaves the radial componentunaffected. As a result, the trajectory of the atom will be parabolic and divergeaway from the trap which leads to diminishing improvements as the slower is elon-gated. As a result of this divergence, only a fraction of the atoms which leave theoven at an angle less than some critical value reach the trap. This critical angle,(θmax), can be determined as follows. Assuming the oven orifice is focused at theMOT, an atom will miss the trap if the radial displacement during the time it takesthe atom to reach the trap is greater than the MOT radius. The time needed to reachthe trap can be divided into four parts: the time needed to reach the slower (tb), thetime in the slower before the atom reaches the necessary magnetic field to bring itinto resonance with the slowing beam (to f f ), the time during which the atoms areslowed to the exit velocity (ton) and finally the time needed to reach the MOT afterleaving the slower ta. An atom, moving with velocity vz,i, enters resonance withthe slower beam a distance z down the length of the slower which is given by:z =v2z,i− v2z, f2a(3.1)The time of flight for the various stages can be calculated as follows:tb =lbvz,ito f f =lzvz,i−v2z,i− v2z, f2avz,iton =vz,i− vz, fata =lavz, f(3.2)where vz is the axial velocity given by vcosθ and vr is the radial velocity givenby vsinθ . The value lb, lz and la are the length atoms travel before, in and afterthe Zeeman slower, respectively and vz, f is the final axial velocity upon leaving theslower. The above expressions assumes that vz,i is greater than vz, f . If that is not the31case, then the Zeeman slower has no effect and the time of flight is simply the totaldistance to the trap divided by the axial speed. Using these values, the maximumangle, θmax can be determined using the following equation given a MOT radius,rMOT .θmax = arcsin [rMOTv(tb + to f f + ton + ta)] (3.3)For typical designs, the distance from the oven to the slower is much larger thanthe MOT radius and solving the above equation is greatly simplified by evoking thesmall angle approximation. This maximum angle depends on the initial velocityof the atoms. Now that the critical angle is determined, the angular and velocitydistribution for an effusive source operating in the transparent regime derived inSection 2.2 can be used to calculate the total number of atoms entering the MOTwhich have been slowed below the MOT capture velocity, denoted as Nc, is:Nc =Wnv¯ANt4pi∫ vc0F(v)dv∫ θmax0κ f (θ)dθ∫ 2pi0dφW : Clausing Factor (see Equation 2.33)n : Density Inside Sourcev¯ : Average VelocityA : Orifice AreaNt : Number of Microtubesvc : Capture Velocity of the SlowerF(v) : Velocity Distribution (see Equation 2.31)κ f (θ) : Angular Distribution of Emitted Atoms (see Equation 2.36)(3.4)If the source is operating in the opaque regime, κ and the velocity distributionmust be modified in accordance with Equation 2.41 and Equation 2.48, respec-tively. For the design of the slower, the effect of the slower length on Nc wasinvestigated. Figure 3.2 shows the flux of capturable atoms as a function of slowerlength. Once the length is comparable to the distance the end is placed away fromthe MOT, the benefit of increasing the slower length is suppressed. For example32increasing the length of the slower from 20 cm to 30 cm only leads to increase influx by 50%, while lengthening it from 10 cm to 20 cm leads to increase in flux by200%.This model neglects transverse heating due to the scattering of the slowing laserwhich results in the blooming of the atomic beam. This process can be thought of asa random walk resulting from the atoms uniformly remitting the absorbed photonsfrom the slowing beam in all directions [25]. The root-mean-square displacementof the atoms at the end of a slower of length L due to this process is given by:< x2 >=√4vrec9viL (3.5)where vrec is the recoil velocity due to the emission of a single photon and is givenby h¯k/m [26]. For a one meter long slower, this effect amounts to an expecteddisplacement of less than a millimetre for both lithium and rubidium which justifiesthis effect being neglected.3.3 Design of a Zeeman Slower3.3.1 Review of Existing DesignsBefore building a Zeeman slower, a review of existing designs was conducted. Thetwo key traits which differentiate the various designs is the mechanism which gen-erates the field, in particular whether it is current driven or uses permanent magnets,and if the field is static or can be dynamically changed to optimize loading. It isdifficult to directly compare the performance of Zeeman slowers as the flux of coldatoms is equally dependent on the atomic source which is coupled into the slower.Continuous Wire WoundContinuous wire wound slowers are the most commonly used design [25–27].They consist of a single continuously wound solenoid where the number of wind-ings is varied spatially to produce the desired field profile. The design benefits fromthe fact that only a single power supply is needed to power the slower. The down-side of such a design is that it allows for no adjustment of the field once wound,33Figure 3.1: Simulation of capturable flux of lithium entering MOT located 20cm from a slower of variable length operating with an η = .6 and finalvelocity of 20m/s. The atoms are emitted from a source placed 60 cmfrom the slower with 90 microtubes of length 1 cm and inner diameter200 µm. The source temperature is 400◦C.besides from increasing or decreasing the current. Furthermore, if the wire coatingis damaged leading to a short circuit it could require extensive rewinding.Helical Wound Zeeman SlowerUnlike the continuous wire wound design, the helical wire wound Zeemanslower is made from single winding with varied pitch to produce the desired fieldpattern [28]. The design is easier to assemble as it does not require many hours ofwinding coils. However, to compensate for fewer windings much larger currentsare needed which can be challenging to work with. Similar to the continuouslywire wound slower, this design does not allow for modifying the slowing fieldaside from changing the current.34Figure 3.2: Simulation of capturable flux of rubidium entering MOT located20 cm from a slower of variable length operating with an η = .6 andfinal velocity of 20m/s. The atoms are emitted from a source placed60 cm from the slower with 90 microtubes of length 1 cm and innerdiameter 200 µm. The source temperature is 80◦C.Segmented SlowerThis design consists of separate solenoids with each current being indepen-dently controlled [29]. This allows for the field profile to be controlled remotelyand changed in real time to optimize MOT loading rate. Furthermore, alternate de-signs such as σ+, σ−, and spin-flip magnetic fields to be easily tested. If a coil isdamaged it does require that the entire slower be rewound. The downside of thisdesign is the field will have small modulations in the magnetic field profile. It alsorequires a more complex current source.Transverse Field SlowerAnother design approach is to run high current parallel to the slowing axis toproduce the magnetic field transverse to the slowing beam. Such a design was35reported for the slowing of 85Rb and used two metal bars which carried the cur-rent [30]. The bars were machined with a specific profile such that they producedthe desired magnetic field. Similar to the helical design, large current in excessof 100A was needed to produce the field. Another drawback is the slowing beamcannot have a well defined circular polarization as it does not propagate along thequantization excess set by the magnetic field. This design is appealing as it is sim-ple to construct and can be easily removed or modified.Permanent Magnet DesignSeveral designs have emerged which use permanent magnets. Such designshave the benefit of requiring no power supply or cooling. A permanent magnetbased on the Halbach configuration has successfully loaded a MOT of 87Rb andhas the added benefit of being assembled after baking. Unfortunately, such a designwould not work for two species slowers as the field cannot be changed.To address the ability to modify the field, an alternative approach places the per-manent magnets on stepper motors which vary the distance of the magnets from theslowing tube. This allows the fields to be optimized in real time to maximize slow-ing. However, to switch between species would take much longer than a currentbased approach.3.3.2 Construction of Zeeman SlowerAfter reviewing existing design we decided to use a wire wound design consistingof eight separately wrapped sections. The rationale for such a design is it allowsfor modification of the slowing field quickly which is needed to switch betweenatomic species. Furthermore, the ability to independently control each coil allowsfor real time optimizing of flux. The slower length of 30 cm was decided uponbased on the simulations in Section 2.1 which indicated that longer slowers did notyield significant improvements in flux. Secondly, a longer slower would requirelarger fields leading to increased heating which would require water cooling.The slower consists of a 17 mm diameter tube divided into eight sections bymetal plates. The plates are secured by aluminum rings which are wedged be-tween the tube and plate. The tube diameter was chosen for three reasons: (1)36(a) SolidWorks Rendering(b) Finished Zeeman SlowerFigure 3.3: SolidWorks rendering of the Zeeman slower along with actualslower.the small inner diameter improves differential pumping, (2) the smaller radius ofthe solenoid leads to larger magnetic fields for a fixed current, and (3) it is com-patible with CF-133 flanges. The coils consist of 20 axial windings and 25 radialwindings of 16AWG insulating amide-amide polymeric magnetic1. The numberof windings was selected such that the maximum voltage drop over the coil pro-ducing the largest field was 10V and the current required to produce fields did notexceed that specified for the wire gauge. For this slower, the optimal wire gaugewas AWG16. After winding the coils, CF-133 flanges were welded to the tube. Thecomplete design is shown in Figure 3.3.Magnetic Field from Segmented SlowerThe magnetic field from a loop of wire with radius R carrying current I lying in inthe x-y plane at a height h and distance ρ away from z-axis is given by[31]:Bz =µI2pi1(R+ρ)2 +h2)1/2× [K(k2)+R2−ρ2−h2(R−ρ)2 +h2 E(k2)] (3.6)Bρ =µI2piρh(R+ρ)2 +h2)1/2× [−K(k2)+R2 +ρ2 +h2(R−ρ)2 +h2 E(k2)] (3.7)1Manufactured by Superior Essex37where K and E are the complete elliptical integrals with argument k2 equal to:k2 =4Rρ(R2 +ρ2)2 +h2 (3.8)In order to calculate the field from a single solenoid, the contribution from ev-ery coil loop, each having a slightly different radius and position, must be consid-ered. Once the field for a single solenoid is known, the field for the entire slowercan be calculated by convolving the single solenoid field profile with a train ofeight delta functions separated by the coil spacing scaled by the appropriate cur-rent. Finding the currents which minimize the difference between the actual fieldand the ideal field is made easy by the fact the magnetic field is linear in current andtherefore can be optimized using a linear least squares method. By defining a cur-rent vector, ~I, whose elements are the initial guesses for currents in the solenoids,the current,~Imin, which minimizes the least squares difference between the fields isgiven by:~Imin =~I− ([JJT ]−1J~r)T (3.9)~I and ~Im are both vectors of dimension n equal to the number of solenoids. Theresiduals, given by~r, is a vector of dimension m given by the difference between theideal field and field produced by the current vector I at position along the solenoid.The matrix J is the Jacobian with dimensions n×m with element Ji, j given by:Ji j =∂ r j∂ Ii(3.10)The magnetic field produced by this design always exhibits slight modulationsdepending on the coil spacing and as a result will never perfectly match the desiredfield profile. However, as long as the gradient does not exceed the maximum valuedefined by Equation 2.19, the adiabatic slowing condition will be met. Figure 3.4shows both the ideal field and the one returned from the least squares regression.The field gradient was also computed to ensure the adiabatic slowing condition wassatisfied, this result is shown in Figure 3.5.38Figure 3.4: Comparison of the ideal slowing field and theoretical field pro-duced by the segmented slowerMatching Slower Field to the MOT FieldDuring the design of the Zeeman slower, the possibility of combining the Zeemanslower field and MOT field was investigated. This would have the benefit of raisingthe capture velocity of the slower by increasing its effective length and help miti-gate the blooming of the beam by slowing closer to the trap. To verify the feasibilityof this design two things had to be checked: (1) is it possible to smoothly connectthe two fields and (2) would such a field be able to slow the atoms. To address thefirst question the same least squares algorithm was used, but this time the effect ofthe trapping field was included. The result of this fit is shown in Figure 3.6.To verify the atoms could be slowed by such a design, a virtual slower wasprogrammed which numerically solves the one dimensional equation of motiongiven the scattering force exerted by the slowing beam at a particular location.The program was run for different initial velocities and a phase space plot wascalculated for the slower which is shown in Figure 3.7. The simulation parameterswere: beam intensity of s0 = 7, a detuning −80Mhz, and η = 0.5. The virtual39Figure 3.5: Comparison of the maximum allowable gradient as defined byEquation 2.19 and the theoretical gradients produced by the segmentedslower.slower was able to slow atoms from 760 m/s to 30 m/s. The undulations in themagnetic field require a larger laser intensity than predicted by Equation 2.3 toensure the adiabatic slowing condition is still met.Verification of Magnetic FieldThe magnetic field from the slower was measured using a hall sensor and wascompared to the theoretically computed profile. Figure 3.8 shows a comparison ofthe theoretical slowing field to the field measured experimentally. The fields werecompared for the entire slower and shown in Figure 3.9.3.3.3 Computer Controlled Current DriverOne drawback of the slower design is that it requires multiple current sources todrive each solenoid. It is not feasible to purchase eight separate power supplies, in-40Figure 3.6: Magnetic field for the ideal case and hybrid case which combinesboth the MOT and slower field.stead we elected to build a MOSFET controlled current source which can be pow-ered from a single high current supply. In order to easily switch between atomicspecies and optimize load roads, the current source should have the ability to becontrolled via computer. Therefore the current controller consists of two parts: thehigh power current board which distributes the current to slower and the computercontrol board which interfaces with our existing experimental control system.Current Distribution BoardThe current distribution board consists of eight independent channels which drivespecified currents based on a set point voltage. Each channel consists of a cur-rent sense resistor, a MOSFET, and gate-driver op-amp. Figure 3.10 is the circuitschematic for a single channel.The circuit works by having the op-amp drive the difference between the setpoint voltage and voltage drop across the resistor to zero by controlling the currentthrough the MOSFET. We chose to use the OP275 as it contains two devices per41Figure 3.7: Phase space plot showing Li atoms being slowed by the field inFigure 3.6 by a laser with an intensity s0 = 7 and a detuning −80 Mhz.package which allows for the second op-amp to be used as buffered monitor. Italso has good low noise performance and can source considerable current whichhelps to quickly change the gate voltage. The combination of R3 and C1 form alow pass filter with a cut-off frequency of 100 kHz which limits the high-frequencygain and serves to stabilize the circuit. The current sense resistor is different forevery channel as each solenoid is operated within a different range of currents. Itis important to add a pin-wheel diode in parallel with the solenoid which clampsthe back-emf when the current is switched off to protect the power supply andMOSFET. A small resister (50Ω) is placed in series with the MOSFET gate. Thishelps limit the effect of the large gate capacitance at the output of the op-amp whichcan lead to ringing and oscillations in the output. The current sense resistors shouldbe chosen to have low sensitivity to changes in temperature as changes in resistancelead directly to systematic errors in current. Another important point is the PCBshould be routed to ensure minimum return path resistance for the sense resistorto ensure the grounds for the reference voltage and sense resistor do not deviate42Figure 3.8: Magnetic field profile for a single solenoid.significantly. Lastly, the MOSFET should be chosen to operate in saturation whenbeing used as a current source as it will minimize the effect of power supply voltagefluctuations. However, this last point is challenging when designing a slower formultiple species which require different magnetic fields. The MOSFETs and senseresistors dissipate significant power, approximately 300 W in total, and the entirecircuit must be water cooled.Computer Control BoardA common 50 pin utilities control bus (UT-BUS) is the communication backboneof our experiment. The bus consists of a 16-bit data bus, a 8-bit address bus, anda single strobe bit as shown in Figure 3.11. The strobe is clocked at 20 MHz andon every rising edge transition a new data word and address are written to the databus and address bus. Devices which are controlled by the bus have a local address,typically set by a dip switch, which is compared to the bus address. If the twomatch, the data word is latched through.For the Zeeman slower’s computer control board, the 12 least significant bits43Figure 3.9: Magnetic field profile inside the Zeeman slower.of the 16-bit data bus set the output voltage of a 12-bit Digital to Analog Con-verter (DAC) (model: AD5725 manufacturer: Analog Devices). Each board hastwo DACS, each of which has four output channels which drive the eight solenoidsfor the slower. The address bus selects a channel by first selecting the board (ad-dress bits 39-47), then one of the two DACS (address bit 37), then finally one of thefour output channels is selected (address bits 33 and 35). The digital logic whichselects a chip is shown in Figure 3.12. The strobe signal enters the comparator andpasses through if the two addresses match. The AND and NOT gates following thecomparator send the strobe bit to select one of the two DACS labeled as CS1 andCS2 in the schematic. The strobe bit is also delayed and is used to load the datainto the memory register of the DAC and onto the analog output. The delay ensuresthe data has reached the DAC and is given time to stabilize.The analog portion of the board consists of a 3.3V reference (model: ADR4533manufacturer: Analog Devices) which provides a reference for the DACS. Thevoltage reference determines the highest voltage which can be sent to the currentdistribution board which in turn sets the maximum current. The board has a 6V44Figure 3.10: Single channel on the current distribution board, not shown isthe bypassing capacitors on the op-amp (typical values used are 10µFtantalum cap in parallel with 100nF ceramic cap.Figure 3.11: Pin out for UT control bus.regulator which powers both DACS and voltage references. The board has a digitalground which is a reference for the digital logic circuitry and an analog ground forthe DACS and 3.3V reference. The DACS also have a digital ground, but for noiseperformance it is best to connect that to the analog ground. The digital and analogground should be connected at a single point via a ferrite bead and care should betaken to avoid ground loops.45Figure 3.12: Digital logic circuitry used to select DAC.3.4 Design of an Effusive SourceThe effusive sources are the starting point for the experiment and care should betaken to ensure they operate as intended. They are required to produce well colli-mated atomic beams of approximately 1015 to 1016 atoms per second. In order toproduce these fluxes, they are heated to achieve suitable vapor pressure within thesource. As a result, the sources must be able to withstand high temperature; thisis particularly true for lithium. As the lithium and rubidium atomic beams travelthrough the same Zeeman Slower, they must be emitted collinear. Finally, thevacuum system must baked at approximately 200◦C in order to reach sufficientlylow background pressures. At such temperatures, the rubidium reservoir would bequickly depleted. To prevent this, the rubidium is contained in ampule which isbroken in situ after the bake-out. This section describes the design of the sourcesin order to meet these specifications.The sources were built primarily from standard ConFlat Vacuum Flange (CF)vacuum parts. Figure 3.13 shows the two sources with lithium on the left andrubidium located underneath. The atomic beams exit the reservoirs through twooffset openings. The openings are pressed fit with approximately 80 microtubeswith 200µm (300µm) inner (outer) diameter.46Figure 3.13: SolidWorks rendering of the two sources. Atoms leave thesources and travel towards the right into the science chamber.3.4.1 Rubidium SourceThe rubidium oven consists of an ampule holder, bellows, ampule breaker, andtower. The ampule’s base is held in a cylindrical tube which is welded to a CF-133blank and the top is held in a hole milled into the CF blank. The two blanks areattached via a flexible bellows which can be bent in order to break the ampule.To prevent the ampule from breaking prematurely, a pipe is slide over the twoCF blanks to prevent the bellows from bending. Once broken, rubidium fills thebellows and tower where it then leaves via the microtubes. A removable backstopplaced inside the oven, which can be seen in Figure 3.14, was machined to easethe insertion of the microtubes. The top of the tower is kept as thin as possible tominimize the space between the openings of the two sources. The source is heatedby two band heaters, one of which heats the base of the rubidium tower to 100◦Cand the other heats the pipe which keeps the bellows rigid to 80◦C.3.4.2 Lithium SourceThe lithium source is much simpler than its rubidium counterpart as it is not re-quired to break an ampule. The lithium is loaded directly into a CF-133 nipplewhich is sealed from the experiment with a CF blank. Similar to the rubidium47Figure 3.14: Rubidium effusive source is shown sub-figure A along with aclose-up of the microtubes. The backstop is shown in sub-figure B.source, a hole is milled into the blank in which microtubes are pressed fit. This canbe seen in Figure 3.15 .One important detail is that copper gaskets cannot be used for the lithiumsource. Lithium, especially when heated, is very corrosive and will quickly dis-solve copper within a few hours. Figure 3.16 shows the end result of such process,the copper gasket was completely dissolved by the lithium and then condensedon the microtubes. Instead, annealed nickle gaskets should be used as they arenot eroded by lithium. The oven is heated by two band heaters and wrapped inaluminum foil.3.5 Design of Experimental Setup3.5.1 Quadrupole Magnetic Field CoilsThe ability to produce large magnetic fields and gradients is a prerequisite foratomic trapping and cooling. A MOT requires a field zero with a linear gradientto confine atoms while evaporation cooling in a dipole trap requires the tuning ofthe scattering lengths by a applying homogenous magnetic fields. The former field48Figure 3.15: CF blank which holds the microtubes for the lithium oven.can be produced by two loops offset by their radius carrying equal currents in op-posite directions, typically refereed to as the Anti-Helmholtz Configuration (AHC).The latter homogenous field is created with the identical coil configuration butwith currents flowing in the same direction. This configuration is referred to asthe Helmholtz Configuration (HC). Such configurations have the desired propertiesthat the gradient is linear to 3rd order for AHC while the 2nd order contribution tocurvature cancels for the HC. Using equation Equation 3.6 and Equation 3.7, toleading order the axial ,Bz, and radial fields, Bρ for the HC are:Bz = µI85√5R+ ...Brho = 0+ ...(3.11)49Figure 3.16: Copper covered microtube after the gasket was dissolved by theheated lithium.while for the AHC the gradient is given by:∂Bz∂ z = µI4825√5R2∂Bρ∂ρ =−12∂Bz∂ z(3.12)For our applications we desire homogenous magnetic fields in excess of 1kG. Inorder to maximize the magnetic field, the resistance of the coils should match thepower supply’s current and voltage limits. Our supply is a 100 V and 50 A linearsupply and the coils are driven in series, hence each coil should have a resistanceof 1Ω during maximum current operation. The rise in wire temperature should not50exceed 70◦C which leads to a room temperature resistance of 0.85Ω. 2. In order tofit in the existing coil mounts, the maximum outer diameter has to be less than 9cm. The coils are enclosed in a housing which allows for water cooling and woundwith Teflon spacers between every two windings to ensure all wires have directcontact with the water. The quartz optical cell limits the minimum spacing betweenthe coils to 4.5 cm which defines the coil radius for the Helmholtz configuration.However, in this ideal Helmholtz configuration the existing power supply wouldnot allow us to reach the desired fields. Therefore, the inner radius needs to bedecreased at the expense of field homogeneity. The inner radius is also limited bythe 2′′ vertical imaging optics to a minimum value of 30 cm. A computer programwas written which iterated through possible wire gauges and filled the availablespace inside the coil house with windings. It calculated the magnetic field and thehomogeneity in the HC along with the gradient for the AHC. Finally, it returned theexpected resistance of the coil given the length of wire needed for winding. Thebest resistance match was 14AWG with 23 radial windings and 11 axial windings.The properties of such a coil is listed in Table 3.1:Table 3.1: Simulated magnetic field properties of the MOT coils.Parameter ValueMaximum Field per Amp in HC 32G/AMaximum Gradient per Amp in AHC 3.8G/A · cmHomogeneity in HC over 1mm×1mm region 60 ppmResistance 0.88ΩBelow are contour plots showing the magnetic fields for both the Helmholtz andanti-Helmholtz configurations.Design of CoilsThe final coil design consists of three parts: a lid, a base, and a retaining ring. Allthree parts are made from Delrin. Barriers are machined into the inside of the lidand base to force water through the windings. The retaining ring has two semi-circle sections which bolt together to secure the inner wall and prevent the coils2The resistance of copper is given by R = R0(1+α(T −T0)) where α is 3.9×10−3 ◦C−151(a) XY-plane (b) XZ-planeFigure 3.17: Magnetic field produced by the Helmholtz configuration. Thefield exhibits a saddle point in the middle of the two coils.(a) XY-plane (b) XZ-planeFigure 3.18: Magnetic field produced by the anti-Helmholtz configuration.The gradient is twice as large in the z direction due to ∇B = 0.from bulging when under water pressure. Two O-rings, one between the lid andouter wall and the other between the base and inner wall, maintain a water tightseal. The coil housing was stress tested to 80 psi and did not leak. Figure 3.19shows a SolidWorks rendering of the coil.Thermal Testing of MOT CoilsThe coils were tested with flow rates from 2.8 L/min to 8.6 L/min while monitoringthe average wire temperature for currents ranging from 5 to 50 A. At the maximumtested flow rate of 8.6 L/min and 50 A, the average wire temperature was 41 ◦C. Atthis flow rate, the thermal resistance is 5.5◦C/kW. Figure 3.20 is a series of plotsshowing the coil heating as a function of flow rate, Q˙.52(a) Coil (b) Coil without lid showing dams(c) Bottom showing retaining ring (d) Cross section showing O-ring grovesFigure 3.19: SolidWorks rendering of the MOT coils.From these heating curves, the thermal resistance of the coils as a function offlow rate can be extracted. Note, the thermal resistance is the slope of the linewhich relates the rise in average wire temperature and power dissipation as follows(∆T = RthP). To first order, the thermal resistance scales linearly with the inverseflow rate as:Rth = Rth,WireToWater +mQ˙(3.13)The first term describes the thermal resistance between the wire and water. Physi-cally, it is the thermal resistance as the flow rate goes to infinity and measures theability to transfer heat from the wire to the water through the insulating coating.This number is of great interest as it determines the rate limiting step in heat flowout of the coils. The slope of the line, m, accounts for how an increase in averagewater temperature leads to an increase in average wire temperature. By extractingslopes from data in Figure 3.20, the thermal resistance from each flow rate can bedetermined. This result is shown in Figure 3.21.53Figure 3.20: Coil heating as a function of current for various flow rates. Theslope of the line gives the thermal resistance.The thermal resistance of the wire to water interface is 2.3 ◦C/kW. This is in rea-sonably good agreement with the theoretical resistance expected for the insulatingcoating of 1.5 ◦C/kW, in addition to any thermal contact resistance at the water tocoating and coating to copper interfaces.Magnetic Field Testing of MOT CoilsOnce completed, the MOT coils were tested to ensure the fields matched the the-oretical profile. Due to spatial constraints the final coil housing was only ableto accommodate 21 radial windings and 10 axial windings. Figure 3.22 and Fig-ure 3.23 show the measured field profiles compared to the theoretical profile for theHelmholtz and anti-Helmholtz configurations. A summary of the measured fieldsfor both MOT and compensation coils can be found in Table 3.2.54Figure 3.21: The thermal resistance of the coil as a function of flow rate.Each point is from the slope of a curve in Figure 3.20.3.5.2 Verification of Magnetic Fields from Compensation CoilsThe MOT requires three sets of compensation coils, one for each axis, to provideoffset fields to move the location of the zero field position. They are also used tomove the MOT when compressing it and loading into dipole traps. We plan to alsouse the z-axis compensation coils to provide a gradient to compensate for gravityand/or residual gradients resulting from the atoms not being well centered betweenthe coils. The required gradient to compensate a force F is given by:dBdz=Fµh (3.14)where µ is the magnetic moment (in units of frequency) and h is Planck’s con-stant. For lithium, the required gradient to compensate for gravity is 1 G/cm. Thez-axis coils are integrated into the MOT coil housing which helps with cooling. Fig-ure 3.24 and Figure 3.25 show the measured field profile compared to theoreticalprofile for the Helmholtz and anti-Helmholtz configurations.55Figure 3.22: Magnetic field produced by the MOT coils with current of I =10.8A for the HC (red dots) along with the simulation (blue line).The y-axis coils produce fields perpendicular to the cell while the x-axis coilsproduce fields parallel to the cell. The properties of all the compensation coils andMOT coils are outlined in the Table 3.2.Table 3.2: Magnetic fields per amp produced by the various coils in the ex-perimental setup. Gradients for x and y coils are not measured as theyare run exclusively in HC.Coil Field at Center in HC (G/A) Axial Gradient in AHC (G/(A·cm)MOT Coils 25.8 6.15Z-Compensation 13.9 3.6Y-Compensation 2.67 –X-Compensation 1.45 –56Figure 3.23: Magnetic field produced by the MOT coils with a current of I =10.8A for the AHC (red dots) along with the simulation (blue line).Maximum Homogenous Magnetic FieldThe maximum homogenous magnetic field capable of being produced by the MOTcoils is 1300 G and is limited by the maximum current of the power supply of 50A. Neglecting the limitation of the power supply, the maximum achievable fieldwhen constrained by keeping the average wire temperature below 70◦C at a flowrate of 8.5 L/s is 2100 G. This corresponds to a current of 80 A.3.5.3 Vacuum SystemThe experimental apparatus consists of two sections: a source side which has theRb and Li effusive sources and a science side which contains the quartz cell inwhich we perform experiments. The two sections are both pumped with an ionpump (model: VacIon Plus 20 Starcell manufacture: Varian) and a Non-evaporablegetter (NEG) (model: CapaciTorr D 400-2 manufacturer: SAES). The two sectionsare connected by a differential pumping tube with inner diameter of 6 mm and57Figure 3.24: Magnetic field produced by the z-axis compensation coils witha current of 1 amp for the HC (red dots) along with a quadratic fit (blueline).length of 12 cm. The differential pumping machined from a single stainless steelrod which is bolted to the CF reducer which connects the Zeeman slower to thesix-way cross and can be seen is Figure 3.26.The two sections are furthered isolated by a gate valve3 which allows thesources to be replenished without exposing the science section to atmosphere. Thegate valve is not all-metal valve and the air tight sealed is maintained with a VitonO-ring. Finally, there is a general purpose six way cross which allows for opticalaccess to the atomic beams prior to entering the slower via two view ports. Thecross also has a manual shutter and a copper feedthrough which can be used as acold finger to reduce the background pressure of Rb. The mechanical shutter canalso be used to reflect the Zeeman slowing beam out of the chamber which helpsduring alignment. Initial Residual Gas Analyzer (RGA) tests of the Li oven showedthat when operational, the background pressure in the source side rose to low 10−83Manufacture: VAT Vacuum Valves Model: Series 01058Figure 3.25: Magnetic field produced by the z-axis compensation coils witha current of 1 amp for the AHC (red dots) along with a linear fit (blueline).torr range. The target pressure in the science section glass cell, PC, is below the109 torr range and must be maintained by the differential pumping. Figure 3.27shows a cross section of the apparatus.To estimate the conductance between the two sections, the NEG on the sourceside was heated in order to increase the background pressure of hydrogen to 10−5torr. The pressure at the source side ion pump, P1, and science side ion pump P2were estimated from the ion current. The pressure P1, P2, and Pc are related by theconductance and the mass flow, Q, emitted from the heated getter via the followingexpression:Q = (P1−Pc)C1 = (Pc−P2)C2 = (P1−P2)C (3.15)where C1 is the conductance between the heated NEG and the cell and C2 is theconductance between the cell and the ion pump on the science side. The totalconductance, C, between the heated NEG and the science side ion pump is:59Figure 3.26: SolidWorks rendering of the differential pumping tube.C−1 =C−11 +C−12 (3.16)In steady state, the hydrogen which passes through the differential pumpingtube is pumped out of the system by the NEG and ion pump at a rate given by:Q = P2(ΓNEG +Γionpump) (3.17)where Γ is the pumping speed of the pump. Combining Equation 3.15 and Equa-tion 3.17 leads to relation between P1 and P2, under the assumption of P1  P2,given by:P2P1=CΓNEG +Γionpump(3.18)60Figure 3.27: Cross section of the experiment. The ion pumps (IP) are labeledand used to estimate the pressures P1 and P2. The two sections areseparated by a differential pumping tube labeled as DPT (6mm dia.12mm length).The relation between the cell pressure and source pressure is now:PCP1=C1 +P2P1C2C1 +C2≈=C1C2(1+C2ΓNEG +Γionpump) (3.19)The above approximation is valid for C2  C1. The conductance of a circulartube of length L in centimeters and diameter D in centimeters for a gas with meanvelocity v¯ is:Ctube = 2.6×10−4V¯D3L(3.20)The dominant contributor to the background pressure is hydrogen for which theconductance at room temperature is:Ctube = 47D3L(3.21)Table 3.3 lists the conductance for the various parts of the vacuum system.From Table 3.3, it is clear that C1 can be reasonably approximated as simplythe differential pumping tube while C2, which consists of contributions from thetwo CF-275 crosses and half the quartz cell, is approximately 40 L/s. The pumpingspeeds for hydrogen for the NEG and ion pump are 100 L/s and 15L/s respectively.61Table 3.3: Conductance for hydrogen of the various vacuum components.Part Length[cm] Dia.[cm] C [L/s]Differential Pumping Tube 12 0.6 .85Zeeman Slower 30 1.8 9Cf-275 Cross 15 3.6 143Quartz Cell 35 3 42Hence, the pressure ratio between the outlet of NEG (P1) and inlet of the scienceside ion pump (P2) as defined in Equation 3.21 is 130. The pressure differentialbetween the cell and source defined in Equation 3.19 is 0.03 which is a reductionis pressure by a factor of 30. This pressure ratio is hard to measure experimentally,but the ratio P1 to P2 can be inferred from the ion currents. Assuming the pressureinside the science section scales as:P2 = P2,0 +CΓNEG +ΓionpumpP1 (3.22)where P2,0 is the background pressure prior to heating up the NEG. In order toverify the differential pumping, the NEG was heated while the ion current fromboth pumps was measured at various hydrogen pressures. The result is shownin Figure 3.28 along with a fit to Equation 3.22. The fit returned the hydrogenconductance of the differential pumping tube of 1.1 L/s assuming a total pumpingspeed of 110 L/s. This is in reasonable agreement with the theoretical value of 0.85L/s.62Figure 3.28: Pressure ratio between source and science section of experimen-tal apparatus during NEG heating tests used to infer conductance ofdifferential pumping tube.63Chapter 4Characterization of Multi-speciesAtomic Source644.1 Characterization of the Lithium OvenPrior to installing the lithium oven into the experimental apparatus, it was impor-tant to verify that its operates as expected. In particular, we are interested in theatomic beam’s flux, i.e. how many atoms are leaving the oven per second, angulardistribution, and finally velocity distribution. Spectroscopic probing of the atomicbeam is the most convenient method to measure these desired properties. From theresulting florescence signal one can infer the flux and transfer Doppler broadening.Unfortunately, it is challenging to simultaneously measure the angle and velocitydistribution from the Doppler broadened spectrum as the two are inherently cou-pled.Figure 4.1: Experimental setup used to conduct the initial diagnostics tests ofthe atomic beam. The lithium oven is in the bottom right corner. Notshown is vacuum pump system.Figure 4.1 shows the experimental setup for fluorescence imaging. The lasersource for the diagnostic tests is from a home built external cavity diode laser [32].The florescence signal was detected with a high sensitivity photodiode. The probebeam had a total of power of 0.4mW and a beam waist of 0.25mm. Figure 4.265shows a picture taken of florescence emitted from the atomic beam when the laseris tuned to the D2 transition and scanned over the hyperfine ground state levels.Figure 4.2: Florencense signal observed from the effusive source without thenichrome mesh lining. The lithium source is at the top and the atomicbeam travels downwards while the probe beam goes from left to right.It was immediately evident that the atomic beam was not well collimated and uponcloser inspection it appeared that lithium had leaked out of the oven. This hadbeen observed with previous oven designs and was mediated by lining the ovenwith a mesh made from nichrome. The hypothesis for why this helps to ensure thelithium remains in the oven is that the mesh provides a scaffolding to which thelithium preferential adheres to. This prevents lithium from leaking out or beingdrawn out of the oven through the microtubes. Figure 4.3 shows the florescenceafter installing nichrome mesh which shows the improved collimation.The majority of atomic sources which utilize microtubes insist on keepingtubes much warmer (50◦C hotter) than other parts of the source to prevent clogging[33]. However, we chose to operate the oven with microtubes at same temperatureor slightly warmer (10◦C hotter) than the back of the oven. The reasoning is that atthe operational temperatures of the oven, lithium is in the liquid phase and thermalgradients will lead to a wicking force drawing the liquid toward the microtubes if66Figure 4.3: Florence signal observed from the oven due to the probe beamwith the nichrome mesh lining.they are the warmest. As this was previously observed to be an issue, the entiresource is operated at the same temperature between 400− 450◦C. The oven wasoperated for a week at this temperature while monitoring the florescence and nochange in flux brightness or angular distribution was observed.4.1.1 Measuring Angular and Velocity Distribution of the AtomicBeam with FlorescenceIt is difficult to determine the angular and velocity distribution with florescencealone as the Doppler broadening couples the two distribution via k · v. Therefore,it is best to try to measure the two independently. In practice it is much easierto produce a beam with minimal angular distribution using apertures compared totrying to narrow the velocity spread. If the atomic beam is well collimated, thena laser beam angled to the atomic beam directly probes the longitudinal velocitydistribution. To collimate the atomic beam, it is passed through the Zeeman slowerwhich acts as an aperture. The probe laser is then split in two with one beamintersecting the beam at 90◦ and the other at 45◦. The first beam can be used to67calibrate the laser’s frequency based on the hyperfine splitting while second can beused to measure the Doppler shift. The described setup is depicted in Figure 4.4.The geometric constraints limit the maximum divergence angle of the atomic beamto approximately 1◦.Figure 4.4: SolidWorks rendering of the apertures used to measure the ther-mal distribution of the atomic beam. The atomic beam goes from rightto left.The florescence signal, Fbeam, emitted from an atomic beam with zero angulardivergence from a laser beam bisecting at an angle θ with intensity I much lessthan Isat and frequency f is given by:Fbeam( f , f0,T,γ,θ) =C∫ ∞0f (v)L( f − f0−k · v2pi ,γ)dv (4.1)where the integral is over all velocities, f (v,α) is velocity distribution in an atomicbeam given by Equation 2.31, and L is the standard Lorentzian line shape cen-tered at f0. The temperature dependence enters through α in the velocity dis-tribution.The parameter C is a constant which depends on the intensity of light,scattering cross section, and density of atoms in the atomic beam. For extractingthe velocity distribution this constant is irrelevant, but will be important for extract-ing the beam flux. The florescence signal observed contains four peaks resultingfrom the two probe beams exciting the 2 hyperfine levels in the ground state. Thefeatures produced by the perpendicular beam are easily resolvable, while thoseproduced by the angled beam are not due to Doppler Broadening. The effusivesource temperature was measured using a thermal couple to be 475◦C and Fig-ure 4.5 shows the observed florescence signal along fit using Equation 4.2. Themodel also includes a linear offset to account for unwanted background signal dueto scattering of the probe beam. The frequency is changed by sweeping the diodecurrent which also leads to a ramp in intensity. The model neglects this effect andassumes the intensity is constant as the change is on the order of a few percent.68Figure 4.5: Fluorescence signal from the two probe beams used to measurethe thermal distribution emitted from the atomic source.Signal( f ) =C1Voight( f , fF=1/2,γ,Ttransverse)+C2Voight( f , fF=3/2,γ,Ttransverse)+C3Fbeam( f , fF=1/2,T,γ,θ)+C4Fbeam( f , fF=3/2,T,γ,θ)+A× f +B(4.2)The parameter fF=1/2 and fF=3/2 are the D2 resonance frequencies for the F = 1/2and F = 3/2 ground state levels, respectively. The Voight profiles correspond tothe signal produced by the transverse laser beam and the values returned by the fitfor the two peak locations are used to confirm the frequency ramp of the laser. Thetemperature inside the atomic source can be extracted using the signal from thebeam angled 45◦. The resulting temperature from fit was 475◦C which is in goodagreement with the measured value.69Once the temperature of the beam was confirmed, the angular distribution andatom number can be inferred. The atomic beam is probed by a transverse beamimmediately at the output of the oven as shown in Figure 4.1. The photodiodevoltage signal is shown in Figure 4.6 along with the result of a numerical simula-tion of the expected florescence signal. The simulation varies the total flux fromthe oven and angular distribution to best fit the observed data. A more detaileddescription of the simulation can be found in Appendix C. The atomic source wasoperated at 450◦C while the probe beam had a beam waist of approximately 2mmand total power of 450 µW. The total flux from the oven based on the simulationwas 9.2× 1015 atoms/s with the FWHM of the angular distribution of 2.9◦. Theexpected flux at this operational temperature is 2× 1015 atoms/s which is in verygood agreement with the measured value. The FWHM is approximately 2.5 timesbroader than the expected angular distribution based on the microtube geometry.Possible explanation for the discrepancy are: (1) microtubes are not all collinear,(2) microtubes are not well packed which lead to large gaps between tubes whichemit atoms, and/or (3) gaps between the tubes and the edge of the hole in the CFblank are emitting atoms. Regardless, the broader distribution will not significantlyaffect the performance of the slower and any decrease in center line intensity canbe compensated for by increasing the oven temperature.4.2 Characterization of Zeeman Slower4.2.1 Characterization of the Slower with an Atomic BeamThe slower was initially tested using the experimental setup depicted in Figure 4.4with a single probe beam angled at 45◦ to the atomic beam. The magnetic field anddetuning were set to slow to a final velocity of 350m/s and the resulting velocitydistribution is shown in Figure 4.7. The dip in the top of the peak was a result of thetapered amplifier which was used to generate the slowing beam being improperlyseeded resulting in significant amplified spontaneous emission.Although the slower was tuned to ensure the atoms leave with a final velocity of350m/s, the resulting peak of slow atoms is centered at 240m/s. The discrepancyarises because atoms still undergo slowing as the field decays to zero outside of the70Figure 4.6: Fluorescence signal from a single probe beam transverse to theatomic beam at the output of the effusive source. The source was oper-ated at 450◦C.slower. As the final velocity is tuned closer to zero, the peak disappears entirelyindicating that the atoms have been turned around prior to reaching the probe beam.This problem has been reported in similar slower designs [29].4.2.2 Assembling Experimental Apparatus and Bake-outPrior to testing the slower by loading a MOT, the vacuum system must be assembledand baked out. In order to achieve the desirable base pressure, it is important that allcomponents are clean. New parts are thoroughly cleaned by the manufacturer andare installed as is. Parts which have been previously used are sonicated in acetonefor one hour, followed by methanol for one hour. Acetone is a stronger solventthan methanol, but tends to leave a residue. Parts which are machined are typicallycontaminated with oils and cooling fluids and must be cleaned more thoroughly.Such parts are hand washed with a mild detergent1 and rinsed with water. They1For our application, Simple Green All Purpose Cleaner is typically used.71Figure 4.7: Velocity distribution of the lithium atomic beam when the sloweris operational. The peak of slow atoms is centered at 240m/sare then sonicated with the following solutions for a duration of one hour: (1)detergent, (2) distilled water, (3) acetone, and finally (4) methanol. The assembledapparatus is supported by an adjustable 8020 scaffolding and a temporary oven iserected from insulating fire bricks which is used for the bake-out. This can be seenin Figure 4.8.Loading LithiumThe lithium is stored in mineral oil which can easily contaminate the vacuum sys-tem. Therefore, the lithium oven is baked out separately from entire apparatusbefore being connected. Lithium oxides quickly and when exposed to air the sur-face will tarnish. The oxide layer can be easily removed with a razor blade, butwill quickly form again in seconds when exposed to air. To prevent this, a cleargarbage bag is filled with argon. The lithium is cut inside the bag and then trans-ferred to the CF-nipple which is kept at positive argon pressure as well. All cuttingtools are sonicated as well to minimize contamination. During the transfer process72Figure 4.8: Experimental apparatus enclosed in the bake-out oven.the lithium is sprayed by helium to prevent oxidization. Initially, we sprayed thelithium with nitrogen, but this lead to sample tarnishing presumably as result offorming a layer of LiN. The lithium was cooked at 500◦C for 6 hours followedby 400◦C for 12 hours. After cooling, the oven was back filled with argon andattached to main experimental apparatus. Figure 4.9 shows a RGA spectrum of thelithium loaded oven prior, during, and after the separate bake-out. We estimate thatapproximately 3g of isotope 6 enriched lithium was loaded into the oven.Bake-Out of Experimental ApparatusAs mentioned, the apparatus and lithium oven are initially baked out separatelythen attached for further baking. The first bake-out of the system was for 5 daysat 185◦C. The limiting factor for the temperature was the rotary feed through. Oneconcern was that the coated wire for the Zeeman Slower would degrade and shortafter baking out. However, this was not an issue as the coating only darkenedslightly during the bake-out. Figure 4.10 shows a residual gas analysis of the sys-tem prior, during, and after the first bake-out. During the first bake-out, the NEGSwere activated which increased the hydrogen pressure of they system to > 10−673Figure 4.9: RGA trace of the lithium oven before, during, and after baking.The dominant contaminates are hydrogen (2amu), water (18amu), car-bon monoxide (28 amu), and carbon dioxide (44 amu).torr. Having the lithium separate at during this step help to prevent contaminationby the large amount of hydrogen released during the activation.After attaching the lithium, the system was baked following the same procedureas the first bake-out. Figure 4.11 shows a residual gas analysis of the system priorand after the second bake-out.From the RGA traces, it is clear the largest contaminate in the system when thelithium is either hot or cold is hydrogen. Baking the lithium oven prior to connect-ing it to the apparatus helped eliminate the heavier species such as water, carbonmonoxide, nitrogen and carbon dioxide, but did little to change the hydrogen basepressure. It is unclear what heating did to heavier species such as mineral oil as itis beyond the mass range of the RGA. Most likely, the large lithium sample loadedinto the oven was completely saturated with hydrogen and trying to bake it all offis not feasible. Instead appropriate differential pumping and vacuum pumps mustbe installed in the apparatus.74Figure 4.10: RGA trace of the vacuum system oven before, during, and afterfirst bake-out.4.2.3 Characterization with MOTUpon completion of the bake out, the optics were setup for the dual lithium andrubidium MOT which was used to verify the final performance of the slower. Thelithium and rubidium slowing beams are passed through a double pass AOM fre-quency shifter to allow for optimization of the detuning frequency before beingcombined with a long pass dichroic splitter. 2. The slowing beam is expanded us-ing a telescope such that is slowly comes to focus immediately before the slower.The benefit of this is two fold. First, the curvature of the slowing beam helps tofocus the atomic beam and, secondly, the intensity at the MOT is less which helpsto reduce the radiation pressure. The slowing beams are aligned down the opticalaxis by looking at the scattering off the shutter arm.2Manufacturer: Edmond Optics Part, Number: #69−89275Figure 4.11: RGA trace of the vacuum system oven before and after the sec-ond bake-out.Lithium MOTThe lithium MOT was tested first with the Zeeman slower. In order for the ZeemanSlower to operate properly, the final coil had to be run backwards. In order for theZeeman Slower to operate properly, the final coil had to be run backwards. Thisproduced a rapidly decreasing magnetic field at the end of the slower whose gra-dient far exceeded the adiabatic condition thus disengaging the slowed atomic fluxfrom the slowing beam thus preventing the atoms from being turned around by theslowing beam. The current in the last coil is optimized such that atoms leave theslower at a velocity which can be further slowed by the MOT field. If they are trav-eling too fast they will pass directly through the trap, while if they are moving toslowly, only a part of the full MOT field will be used for slowing. We observed thatthe slowing beam, without the Zeeman coils activated, also provided an increasein captured MOT flux. This was due to the MOT field extending outside of trappingregion acting as a Zeeman slowing field. Loading curves for the MOT, MOT withslowing beam, and finally MOT with Zeeman Slower is shown in Figure 4.12. Be-76cause of delays in the code, the MOT starts loading approximately half a secondbefore t = 0.Figure 4.12: Loading curves for lithium MOT with and without the slower.The a sequential optimization of all the Zeeman Slower currents was performedand all values agreed with that predicted by theory. The other MOT parameters arelisted in Table 4.1. The detuning of both the pump and probe was also increasedduring the optimization of the Zeeman Slower. This did not affect the loading rate,but improved the MOT lifetime leading to larger steady state size.The loading curves for the lithium MOT was fit to the following model.N(t) =RΓ(1− exp−tΓ) (4.3)where N is the atom number, R is the loading rate, and Γ is the decay rate which isthe inverse of the MOT lifetime 3. The steady state atom number is given by R/Γ.The result of the fits is presented in Table 4.2.3This equation is the solution to N˙ = R−ΓN77Table 4.1: Loading parameters for lithium MOT.Parameter ValueSlowing Beam Detuning -85 MHzSlowing Beam Power 45 mWPump Beam Detuning -40 MHzRepump Beam Detuning -40 MHzMOT Axial Gradient 49 G/cmCoil 1I 6.7 ACoil 2 I 4.2 ACoil 3 I 3.28 ACoil 4 I 2.85 ACoil 5 I 2.52 ACoil 6 I 2.0 ACoil 7 I 1.74 ACoil 8 I -1.5 ATable 4.2: Lithium MOT loading curve fit values.Test R (106 atoms/s) Lifetime (s) R/Γ (106)Only MOT 4 3.5 14Slowing Beam 32 2.2 72Zeeman Slower 140 5.0 690Rubidium MOTAfter the verifying the operation of Zeeman Slower for the lihtium slower, loadingthe rubiudm MOT was verified. Because of the mass difference, slowing rubdiumis not as effective and a smaller field is needed. Rubidium exhibits much strongersaturation due to two body losses which limits the absolute size of the MOT. There-fore, the model which governs the loading of the rubidium MOT is given by Equa-tion 4.4.dNdt= R−ΓN−βN2 (4.4)where β is the two body loss coefficient. The solution to Equation 4.4 whosesolution is given by Equation 4.5.78N(t) =√−4βR−Γ2 tan [√−βR− (Γ/2)2(c1− t)]−Γ2β (4.5)The parameter c1 is arbitrary constant which sets the initial atom number. Theloading curves for rubidium are shown in Figure 4.13 for cases when only theslowing beam on and when the Zeeman slower is on.Figure 4.13: Loading curves for rubidium MOT with and without the slower.The fit parameters for both curves are given in Table 4.3.Table 4.3: Rubidium MOT loading curve fit values.Test R (106 atoms/s) Lifetime (s) Nt=∞ (106) β (atom· s)−1Slowing Beam 20 1.9 40 –Zeeman Slower 232 1.5 350 2.1×10−4The MOT and Zeeman Slower setting used to load the MOT are given in Ta-ble 4.4.79Table 4.4: Loading parameters for rubidium MOT.Parameter ValueSlowing Beam Detuning -85 MHzSlowing Beam Power 15 mWSlowing Repump Beam Detuning 0 MHzSlowing Repump Beam Power 3 mWPump Beam Detuning -18 MHzRepump Beam Detuning 0MOT Axial Gradient 12.3 G/cmCoil 1I 0.85 ACoil 2 I 0.62 ACoil 3 I 0.52 ACoil 4 I 0.5 ACoil 5 I 0.4 ACoil 6 I 0.3 ACoil 7 I 0.3 ACoil 8 I -0.5 A4.2.4 Comparison of Observed and Simulated MOT Loading RatesThe model developed in Section 3.2 was compared the observed loading rates todetermine its validity. For lithium, the model predicts a loading rate of 2.7× 108atoms per second at an operational temperature of 400◦C, while the observed valuewas 1.4× 108 at this temperature. The model predicts a rubidium loading rate of8.0× 108 atoms per second at an operational temperature of 80◦C while the ob-served value is 2.4× 108. The predicted flux is a factor of two higher for lithiumand slightly more than three times larger for rubidium. Given the simplicity of themodel, the agreement between the observed and simulated loading rates is verygood. The model completely neglects optical pumping effects which would de-crease the flux of cold atoms.4.2.5 Improving Lithium MOT LifetimeThe atoms cooled in the MOT will be eventually transferred to a high power dipoletrap. The dipole trap is shallower trap than the MOT and will therefore have ashorter lifetime. The duration of the evaporation in the dipole trap is typically be-80tween two and three seconds followed by any experiments which typically last forat most 1 second. Therefore, a reasonable upper bound for the dipole trap holdtime is about is about 5s. In order to minimize atom loss during this process, thelifetime in the dipole trap should be at least 5s, but ideally closer to 10s. Becausethe background collision limited loss rate of the MOT (trap depth of 1K) is typically5 times that of a shallow dipole trap (trap depth of less than 100 micro-Kelvin), theMOT lifetime should be between 60 to 100s to prevent significant atom loss duringthis hold time [34]. From the initial MOT loading, the lifetime is well below thistarget and must be improved by decreasing the pressure. The major contributorto background pressure is hydrogen which is measured to be 1.6× 10−9 torr. inthe science chamber. In order to find the majority contributor to hydrogen the gatevalve was closed which isolated the source and science sections. After pumpingthe system overnight while keeping the lithium at the operational temperature, thegate valve was reopened. The hydrogen pressure, as read by the RGA, only in-creased by 10% indicating that the source and science sections are well isolated bythe differential pumping. Another possible culprit was the viewport for the slowingbeam which is heated to prevent build up of lithium. Cooling the viewport to roomtemperature from 200◦C decreased the hydrogen pressure down to 1×10−9. Thisis alarming and indicated that the viewport or CF nipple to which it is attachedcould be contaminated. To investigate this possibility, a different CF flange washeated and pressure of hydrogen rose to 1.6×10−9 torr. level indicating compara-ble out gassing. Loading curves were taken with and without the viewport heatedto clearly show the effect of the improved vacuum on lifetime. The result is shownin Figure 4.14.The result is fit to a model which includes two body losses as the loading curvesexhibits saturation effects.Finally, the effect of the atomic beam on MOT lifetime was investigated byblocking the atomic beam and looking at the decay in atom number. The resultis shown is Figure 4.15. The data is fit to solution of Equation 4.4, but with Rset to zero. The MOT lifetime was the same with the beam blocked as unblockedand is thus unaffected by beam collisions. A summary of the tests is shown in thefollowing Table 4.5.Although the lifetime was improved, it still does not meet the required target.81Figure 4.14: Effect of viewport temperature on MOT lifetime.Table 4.5: Effect of viewport temperature and atomic beam on MOT lifetime.Test Hydrogen Pressure (10−9 torr.) Lifetime (s)Viewport Hot, MOT Loading 1.6 15Viewport Cold, MOT Loading 1 40Viewport Hot, MOT Loading 1.6 15Viewport Cold, MOT Decay 1 34Improving differential pumping would not help as the source of hydrogen appearsto be coming from the science side. Most likely another NEG must be added to thesystem.82Figure 4.15: Effect of atomic beam on MOT lifetime.83Chapter 5Conclusion and Future Work84We have presented the design, construction, and characterization of an experi-mental apparatus capable of producing cold samples of lithium and rubidium in amagneto-optical trap. This serves as a starting point for the study of ultracold mix-tures and the formation of hetero-nuclear molecules. This thesis primarily focusedon the design of a dual species effusive source and Zeeman slower. A physicalmodel was developed to predict several of the operational parameters of the sourcesuch as flux, angular distribution, and velocity distribution. The model was com-pared to the experimental results which showed good agreement to the theory. Theperformance of the Zeeman slower was also tested by loading a MOT. The slowerwas capable of producing fluxes of cold atoms leading to loading rates in excessof 108 atoms per second for both species. This was in reasonably good agreementwith the model developed to predict the performance of the slower. Along withthe performance of the slower, a detailed description of the vacuum system andmagnetic field coils was also presented.Moving forward, the main point in the design of the experimental apparatuswhich must be addressed in the short lifetime of atoms in the MOT due to thelarge background pressure of hydrogen. Currently, modifications to the system areunderway to reduce the background pressure of hydrogen by adding additional dif-ferential pumping and installing another NEG between the slower and the trappingregion.85Figure 5.1: A lithium MOT with approximately half a billion atoms.86Bibliography[1] C. Ticknor and J. L. Bohn, “Long-range scattering resonances instrong-field-seeking states of polar molecules,” Phys. Rev. A, vol. 72,p. 032717, Sep 2005. → pages 2[2] A. V. Avdeenkov, D. C. E. Bortolotti, and J. L. Bohn, “Field-linked states ofultracold polar molecules,” Phys. Rev. A, vol. 69, p. 012710, Jan 2004. →pages 2[3] R. V. Krems, “Cold controlled chemistry,” Phys. Chem. Chem. Phys.,vol. 10, pp. 4079–4092, 2008. → pages 2[4] D. DeMille, “Quantum computation with trapped polar molecules,” Phys.Rev. Lett., vol. 88, p. 067901, Jan 2002. → pages 2[5] S. F. Yelin, K. Kirby, and R. Coˆte´, “Schemes for robust quantumcomputation with polar molecules,” Phys. Rev. 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An optional note. → pages 7[12] E. Arimondo, M. Inguscio, and P. Violino, “Experimental determinations ofthe hyperfine structure in the alkali atoms,” Rev. Mod. Phys., vol. 49,pp. 31–75, Jan 1977. → pages 8[13] A. Banerjee, A. Das, and V. Natarajan, “Absolute frequency measurementsof the d 1 lines in 39 k, 85 rb, and 87 rb with 0.1 ppb uncertainty,” EPL(Europhysics Letters), vol. 65, no. 2, p. 172, 2004. → pages 8[14] B. Ohayon and G. Ron, “New approaches in designing a zeeman slower,”Journal of Instrumentation, vol. 8, no. 02, p. P02016, 2013. → pages 15[15] E. Wille, Preparation of an Optically Trapped Fermi-Fermi Mixture of 6Liand 40K Atoms and Characterization of the Interspecies Interactions byFeshbach Spectroscopy. PhD thesis, Universitat Innsbruck, Innrain 52, 6020Innsbruck, Austria, 2009. → pages 18[16] W. Gunton, M. Semczuk, and K. W. Madison, “Realization ofbec-bcs-crossover physics in a compact oven-loaded magneto-optic-trapapparatus,” Phys. Rev. A, vol. 88, p. 023624, Aug 2013. → pages 19, 30[17] K. Ladouceur, B. G. Klappauf, J. V. Dongen, N. Rauhut, B. Schuster, A. K.Mills, D. J. Jones, and K. W. Madison, “Compact laser cooling apparatus forsimultaneous cooling of lithium and rubidium,” J. Opt. Soc. Am. B, vol. 26,pp. 210–217, Feb 2009. → pages 19, 30[18] C. A. Stan and W. Ketterle, “Multiple species atom source for laser-coolingexperiments,” Review of Scientific Instruments, vol. 76, no. 6, p. 063113,2005. → pages 19[19] N. Ramsey, Molecular beams. International series of monographs onphysics, Clarendon Press, 1963. → pages 20, 21[20] H. C. W. Beijerinck and N. F. Verster, “Velocity distribution and angulardistribution of molecular beams from multichannel arrays,” Journal ofApplied Physics, vol. 46, no. 5, pp. 2083–2091, 1975. → pages 21, 22, 2388[21] P. Clausing, “The flow of highly rarefied gases through tubes of arbitrarylength,” Journal of Vacuum Science and Technology, vol. 8, no. 5,pp. 636–646, 1971. → pages 21, 22[22] G. R. Hanes, “Multiple tube collimator for gas beams,” Journal of AppliedPhysics, vol. 31, no. 12, pp. 2171–2175, 1960. → pages 23[23] D. R. Olander, R. H. Jones, and W. J. Siekhaus, “Molecular beam sourcesfabricated from multichannel arrays. iv. speed distribution in the centerlinebeam,” Journal of Applied Physics, vol. 41, no. 11, pp. 4388–4391, 1970. →pages 24, 25, 26[24] L. Loeb, The Kinetic Theory of Gases. Dover Phoenix Editions, DoverPublications, 2004. → pages 25[25] M. A. Joffe, W. Ketterle, A. Martin, and D. E. Pritchard, “Transverse coolingand deflection of an atomic beam inside a zeeman slower,” J. Opt. Soc. Am.B, vol. 10, pp. 2257–2262, Dec 1993. → pages 33[26] K. Gunter, “Design and implementation of a zeeman slower for 87rb,”Master’s thesis, Ecole Normale Superieure,, Paris, France, 2004. → pages 33[27] U. Schnemann, H. Engler, M. Zielonkowski, M. Weidemller, and R. Grimm,“Magneto-optic trapping of lithium using semiconductor lasers,” OpticsCommunications, vol. 158, no. 16, pp. 263 – 272, 1998. → pages 33[28] S. C. Bell, M. Junker, M. Jasperse, L. D. Turner, Y.-J. Lin, I. B. Spielman,and R. E. Scholten, “A slow atom source using a collimated effusive ovenand a single-layer variable pitch coil zeeman slower,” Review of ScientificInstruments, vol. 81, no. 1, pp. –, 2010. → pages 34[29] I. Kaldre, “A compact, air-cooled zeeman slower as a cold atom source,”Master’s thesis, Duke University, Durham, United States, 2006. → pages 35,71[30] P. Melentiev, P. Borisov, and V. Balykin, “Zeeman laser cooling of 85rbatoms in transverse magnetic field,” Journal of Experimental andTheoretical Physics, vol. 98, no. 4, pp. 667–677, 2004. → pages 36[31] T. Bergeman, G. Erez, and H. J. Metcalf, “Magnetostatic trapping fields forneutral atoms,” Phys. Rev. A, vol. 35, pp. 1535–1546, Feb 1987. → pages 37[32] S. Singh, “Progress towards ultra-cold ensembles of rubidium and lithium,”Master’s thesis, The University of British Columbia, Vancouver, Canada,2007. → pages 6589[33] R. Senaratne, Z. Rajagopal, S.and Geiger, V. Fujiwara, K.; Lebedev, andD. Weld, “Effusive atomic oven nozzle design using a microcapillary array,”arXiv, July 2014. → pages 66[34] J. Van Dongen, C. Zhu, D. Clement, G. Dufour, J. L. Booth, and K. W.Madison, “Trap-depth determination from residual gas collisions,” Phys.Rev. A, vol. 84, p. 022708, Aug 2011. → pages 8190Appendix ADampening Posts for OpticalBread BoardsThe new experimental setup requires optical breadboards which sit adjacent to theexperiment. These breadboards support sensitive optical components which areused for dipole traps and optical lattices and care should be taken to ensure they arestable. Inspired by commercially available posts which are designed to minimizedampening, we built custom posts. These posts use 80/20 aluminum extrusionsfilled with lead shot to minimize vibrations. These posts are compared to standard1′′ stainless steel posts which have been used previously to support bread boards.Each post consists of a 80/20 T-Slotted extrusion with dimensions 2′′×2′′×6′′.The extrusion has four holes which can be tapped for 1/4− 20 screws. Thesescrews are used to secure the end caps made from 3/8′′ aluminum which are usedto secure the post to the optical table and breadboard. Between the breadboard andendcap is a 1/16′′ cutout of nitrile rubber which acts as a spacer to further dampenvibrations. Prior to assembling the post, the inner cavity is filled with lead shot.In total, the entire post costs approximately $30 which is a third the price of acommercial post and clamp. The various parts of the post are shown in Figure A.1prior to assembly.In order to characterize the post’s performance two breadboards were assem-bled with one being equipped with 1′′ Thorlabs posts and the other with our custom8020 posts. In particular we were interested in two points: 1) how well isolated was91Figure A.1: The customs 80/20 post consist of a lead filled extrusion, identi-cal top and bottom caps, and two nitrile spacersthe breadboard from the optical table and 2) how quickly were vibrations damp-ened once they coupled to the boardboard? In order to address these points we at-tached a peizoelectric accelerometer1 to the breadboard and measured the impulseresponse to a disturbance on either the table or on the breadboard. In this case, thedisturbance was a ball driver being dropped on the handle end from a fixed heightas it reliably produced the same response. Figure A.2 is a plot showing the impulseresponse for disturbance on the optical table.It is constructive to look at the Fourier transform of the two signals whichclearly shows the improved frequency response. Unfortunately, the proportionalcalibration factor which related the voltage to acceleration is not known, howeverit is not needed for relative comparison of the two posts.Next we dropped the ball driver directly on the breadboard. Once again, thecustom posts are superior.1model: 7703 manufacturer: ENDEVCO92(a) 8020 Posts (b) Thorlabs PostsFigure A.2: Impulse response of the breadboard to a disturbance on the opti-cal table.Figure A.3: Frequency response of the breadboard to a disturbance on theoptical table.93(a) 8020 Posts (b) Thorlabs PostsFigure A.4: Impulse response of the breadboard to a disturbance on thebreadboard directly.Figure A.5: Frequency response of the breadboard to a disturbance on thebreadboard directly.94Appendix BMeasuring Dipole TrapFrequenciesB.1 IntroductionBelow is a brief summary on how to measure the trap frequency by modulating thepower of the dipole trap laser (here called the ”IPG”). The power dependenttransverse trap frequency was measured to be 1.9kHz/√W and the powerdependent longitudinal trap frequency was measured to be 0.13kHz/√W .B.2 SetupThe RF power controlling the optical power diffracted towards the atoms with anAOM was modulated using a voltage controlled attenuator1. The control voltageinput was attached to an adder circuit which combined a constant DC value of 10Vwith a sinusoid from a function generator 2. The function generator was controlledusing a USB port which was attached to the device via a USB to RS-232 converter.The sinusoid output was turned off and on by attaching a digital out port to theAM-Mod input on the back of the DS-345. For small sinusoid amplitudes, it wasdifficult to turn the signal completely off. Therefore, the output was attenuated1manufacture: minicircuits model: ZX73-2500-S+2manufacture: Stanford Instruments model: DS-34595by a 20-dbm attenuator before being attached to the adder. This allowed a largeramplitude to be set which could easily be shut off, while maintaining the correctmodulation power.The output of the DDS (the RF source generating 80 MHz) followed by theDDS-preamplifier was attached to the input of the RF attenuator, while the outputwas attached to the high power amplifier 3 IMPORTANT: Do not exceed 10 dbminput power to the high power amplifier. The amplifier power was comparedwith and without the attenuator. To compensate for the reduction in power bythe attenuator, the output power from the first DDS was scaled by changing theattenuation factor in the code from 0.4 to 0.69.Figure B.1: Flow chart of measurement setupB.3 Experimental MethodIt was found that the loss signal was greatly enhanced by having a three step ex-citation process. First the sample is cooled to an IPG power of 0.7 W (this cor-responds to an IPG set power in the code of 0.5). Then the power increased adi-abatically to the desired power (1 second ramp time). IIt was verified that whentrapping lithium alone, the ratio of T to Tf did not change significantly during thisstage. The IPG was then modulated at a given frequency for a fixed amplitude3manufacture: minicircuits model: ZHL-03-SWF96and duration. It was observed that the attenuation level was frequency dependentfor modulation frequency above 1 KHz. This is corrected for by dividing the de-sired function generator amplitude by a frequency dependent correction given byC=-0.0083713 f 2+0.290178 f +.77834. Finally, after the modulation the IPG poweris ramped quickly (50ms) back to 0.7 W. The atom number is then measured asa function of modulation frequency to determine the resonance frequency of thetrap. The strongest loss signal is observed at twice the trap resonance. A classicalexplanation follows from the fact if the trap is modulated at twice the frequency,the atoms are at the turning points during each field maximum and therefore expe-rience the largest energy transfer.B.4 Experimental ParametersThe transverse frequency is much larger than the longitudinal frequency and showsa much clearer loss feature. The best signal was achieved by using relatively lowexcitation power and large modulation times. Below is a table summarizing theexperimental parameters:Table B.1: Experimental parameters for trap frequency measurements.Mode Function Generator Amplitude (V) Hold Time (s)Transverse 2 4Longitudinal 4 8B.5 ResultsBelow are the standard loss spectra for various trap powers. All powers are thetotal trap power (i.e. the sum of both arms).Fitting the peak location as a function of power gives the expected square rootrelation. These plots are shown below.97Figure B.2: Transverse frequency loss peaks.Figure B.3: Longitudinal frequency loss peaks.98Figure B.4: Transverse frequency as a function of power.Figure B.5: Longitudinal frequency as a function of power.99Appendix CModeling Atomic Flux andDistribution from the LithiumOvenEstimating the flux and angular distribution from an effusive source can be dividedinto two steps. The first step is to calculating the total florescence emitted from theatomic beam when excited by the probe beam. For this, the signal produced fromthe entire interaction length over which the atomic beam interacts with the probemust be integrated over. The florescence of each point along the probe is differentfor two reasons: (1) the local density of atoms is different due to the angular distri-bution from the source and (2) the Doppler shift is angle dependent. If the atomicbeam is significantly dense or the probe beam is weak, one must take into accountthe abortion of the probe beam. The calculation is further complicated by the factthe atomic beam consists of many velocity classes, each of which experience a dif-ferent Doppler shift. Therefore, the contribution of signal from each velocity classmust also be integrated over.The experimental apparatus is shown in Figure C.1. The coordinate z denotesthe distance along the probe beam with z = 0 being immediately in front of thesource. The distance d1 and d2 are distances from z = 0 to the source and detector,respectively.I will consider only a two-level system with natural linewidth γ , but the result100Figure C.1: Experimental setup used to characterize atomic beam.can be easily extended to include the two hyperfine ground states of Li6. Thenumber of photons scattered, ds, from a length element between z and z+ dz byatoms moving with speed between v and v+dv due to a probe beam with relativeintensity I/Isatequal to s0, cross sectional area A, and detuning δ is given by: 1ds(δ ,z,v) = (n(z,v))γ2s01+4( δ+k·vγ )2Adzdv (C.1)where n(z,v) is the local density of atoms at position z moving with velocityv. Given the velocity distribution, f (v) of atoms leaving the oven, given by Equa-tion 2.31, and the angular distribution g(θ), given by Equation 2.36 and Equa-tion 2.39, n(z,v) can be calculated to be:n(z,v) = Ng(θ)r2vf (v) (C.2)where N is the total number of atoms leaving the source and r is the distance fromthe source to the element dz on the probe beam given by:d =√z2 +d21 (C.3)1This is only valid for the low intensity limit such that s0  1, else the intensity distribution inthe beam must be taken into account due to power broadening.101The Doppler shift as a function of position is given by:k · v = kvsinθ (C.4)where θ is the angle from the normal of the oven. The second part of the calculationis to determine what fraction of photons reach the photodiode and the power-to-voltage conversion factor of the detector which I denote as (α). If the detectorarea, Ad is small compared to the distance between the detector and probe beamthen the fraction of photons captured (Fc) is simply given by:Fc =Ad4pi(d22 + z2)(C.5)Hence, the voltage, dV , produced by the photodiode due to the scattering ds isgiven by:dV(δ ) =αds(δ ,z,v)AdEph4pi(d22 + z2)(C.6)where Eph is the energy of the photon. Integrating over z and v gives the totalvoltage signal as function of detuning. The limits on the z integral are typicallydefined by geometric constraints which limit the line of sight of the photodiode.The result of numerically integrating Equation C.6 is shown in Figure 4.6102

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