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Flux and profile measurements of an atomic beam using laser cooled atoms Prescott, Thomas 2016

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Flux and Profile Measurements of an Atomic Beam UsingLaser Cooled AtomsbyThomas PrescottB.Sc. Physics, The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)January 2016c© Thomas Prescott, 2016AbstractThe population dynamics of a Magneto-Optical Trap (MOT) make it a potentialcandidate for flux measurements of an atomic beam. This is achieved by determin-ing the collisional cross section of the trapped atom and the beam particle whichwould result in ejection of a trapped atom. Due to the properties of a MOT it is pos-sible to make spatial and time-of-flight profiles of the beam using this technique.The work of this thesis explores the collisional cross sections and flux profiles ofseveral gaseous beams with a MOT of 85Rb or 87Rb. Each of the beams, gener-ated through supersonic expansion, produced a loss cross section on the order ofthe combined van-der-Waals radii of the two particles. The flux and time-of-flightinformation of the beam was verified with a Residual Gas Analyser (RGA) andhigh beam rep rate pressure measurements. The MOT was characterized througha combination of fluorescence detection for population and a catalysis process forthe trap’s depth. A custom built translation mechanism for the MOT’s optics andHelmholtz coils was constructed to perform the profiling measurements.iiPrefaceThe machine used for the work described in this thesis was originally constructedfor a rubidium hydride production experiment which was modified after that projectwas discontinued. The original beam line was designed and constructed by MarioMichan with modifications done by visiting professors Dr. Eckart Wrede and Dr.Frank Stienkemeier. The MOT and it’s associated equipment was constructed byGene Polovy, Mathias Striebel and myself.After inheriting the machine all designs and modifications were done by my-self. This included the implementation of the two skimmer setup, the nozzle align-ment rig, and the translating optical table. Device communication scripts werewritten by myself with the assistance of Cameron Herberts. All data analysisscripts were written by myself with the exception of the loading curve and losscurve fitting routines which were written by Gene Polovy.The UBC Chemistry Technical staff built the CRUCS valve along with thecurrent supply and the DDG control system.This work is original and unpublished.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Atomic and Molecular Beams . . . . . . . . . . . . . . . . . . . 21.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Atomic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Trap Loss Conditions . . . . . . . . . . . . . . . . . . . . 62.1.2 Scattering Cross Section . . . . . . . . . . . . . . . . . . 72.1.3 Independent Events . . . . . . . . . . . . . . . . . . . . . 82.2 Supersonic Expansion and Atomic Beams . . . . . . . . . . . . . 92.2.1 Supersonic Expansion of an Atomic Gas . . . . . . . . . . 102.2.2 Collimation of the Atomic Beam . . . . . . . . . . . . . . 12iv3 The Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . 143.1 Overview of the MOT Trapping Mechanism . . . . . . . . . . . . 143.1.1 Separation of the Hyperfine Energy Levels . . . . . . . . 153.1.2 Selection Rules for the Optical Transitions . . . . . . . . 163.1.3 Full Ensemble for the Magneto-Optical Trap . . . . . . . 193.1.4 Population Dynamics . . . . . . . . . . . . . . . . . . . . 223.2 The Specifics for a Rubidium MOT . . . . . . . . . . . . . . . . 243.3 Trap Population Estimates and Measurements . . . . . . . . . . . 253.4 MOT Trap Depth Measurement . . . . . . . . . . . . . . . . . . . 263.4.1 Catalysis Effects on Trap Population . . . . . . . . . . . . 274 The Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1 Atomic Beam Setup . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.1 Atomic Beam Line . . . . . . . . . . . . . . . . . . . . . 294.1.2 Residual Gas Analyser . . . . . . . . . . . . . . . . . . . 324.1.3 The Hotwire Detector . . . . . . . . . . . . . . . . . . . 324.2 Rubidium Magneto-Optical Trap . . . . . . . . . . . . . . . . . . 344.2.1 Trapping Lasers . . . . . . . . . . . . . . . . . . . . . . . 354.2.2 Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . 414.2.3 Rubidium Source . . . . . . . . . . . . . . . . . . . . . . 444.2.4 Trap Fluorescence Imaging . . . . . . . . . . . . . . . . . 454.2.5 MOT Translation Mechanics . . . . . . . . . . . . . . . . 504.2.6 Catalysis Laser for Trap Depth Measurement . . . . . . . 535 Atomic Beam-MOT Collisions . . . . . . . . . . . . . . . . . . . . . 565.1 Collisional Losses . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.1 Measurement Procedure . . . . . . . . . . . . . . . . . . 585.2 Profiling an Atomic Beam . . . . . . . . . . . . . . . . . . . . . 595.2.1 Initial Proof of Concept . . . . . . . . . . . . . . . . . . 605.2.2 Coil Translation Results . . . . . . . . . . . . . . . . . . 615.2.3 Table Translation Results . . . . . . . . . . . . . . . . . . 635.3 Trap Depth Measurements . . . . . . . . . . . . . . . . . . . . . 706 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76vBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.1 Photodiode Detectors . . . . . . . . . . . . . . . . . . . . . . . . 81A.1.1 Photodiode Detector Response Times . . . . . . . . . . . 81A.1.2 Photodiode Calibration Values . . . . . . . . . . . . . . . 82A.2 Residual Gas Analyser . . . . . . . . . . . . . . . . . . . . . . . 83A.2.1 Pumping Speed Method . . . . . . . . . . . . . . . . . . 84B MOT Position Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 87B.1 Standard Imaging Procedure . . . . . . . . . . . . . . . . . . . . 87B.2 Translating Coils Position Measurements . . . . . . . . . . . . . 88B.3 Translating Table Position Measurements . . . . . . . . . . . . . 88C January 2015 Results Quirks . . . . . . . . . . . . . . . . . . . . . . 91C.1 Anomalies in Loss Measurements . . . . . . . . . . . . . . . . . 91C.2 MOT Population and Loading Times . . . . . . . . . . . . . . . . 91D Alignment Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 95D.1 Atomic Beam Alignment . . . . . . . . . . . . . . . . . . . . . . 95D.1.1 Skimmer Alignment . . . . . . . . . . . . . . . . . . . . 95D.1.2 Detection Chamber Adjustments . . . . . . . . . . . . . . 97D.1.3 Nozzle Alignment . . . . . . . . . . . . . . . . . . . . . 98D.2 MOT Trapping Laser Alignment . . . . . . . . . . . . . . . . . . 100D.2.1 Splitting and Polarization . . . . . . . . . . . . . . . . . . 100D.2.2 Laser Alignment . . . . . . . . . . . . . . . . . . . . . . 100E Residual Gas Analyser Fitting Function . . . . . . . . . . . . . . . . 102E.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 102E.2 Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . . . 102E.3 Lorentz Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 103F Trap Population ODE Solving Code . . . . . . . . . . . . . . . . . . 106viList of TablesTable 2.1 Expected Beam Velocities . . . . . . . . . . . . . . . . . . . . 12Table 4.1 Experiment Modification Log . . . . . . . . . . . . . . . . . . 30Table 5.1 Nov 2015 MOT Parameters . . . . . . . . . . . . . . . . . . . 64Table 5.2 Collisional Cross Section Results . . . . . . . . . . . . . . . . 68Table 5.3 RGA Signal Fitting Parameters . . . . . . . . . . . . . . . . . 69Table A.1 Detector Signal Conversion . . . . . . . . . . . . . . . . . . . 83Table A.2 Gas Pressure and Pumping Values . . . . . . . . . . . . . . . . 85Table A.3 Atomic Beam Pulse Particle Count . . . . . . . . . . . . . . . 86viiList of FiguresFigure 2.1 Supersonic Expansion Graphic . . . . . . . . . . . . . . . . . 11Figure 3.1 MOT Trapping Mechanic . . . . . . . . . . . . . . . . . . . . 20Figure 3.2 MOT Diagram and Image . . . . . . . . . . . . . . . . . . . 21Figure 3.3 MOT Loading Curve . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.4 85Rb and 87Rb D2 Transitions . . . . . . . . . . . . . . . . . 25Figure 3.5 Catalysis Process Schematic . . . . . . . . . . . . . . . . . . 28Figure 4.1 Vacuum Chamber Photographs . . . . . . . . . . . . . . . . . 33Figure 4.2 Vacuum Chamber Illustration . . . . . . . . . . . . . . . . . 34Figure 4.3 Nozzle Mount Diagram . . . . . . . . . . . . . . . . . . . . . 35Figure 4.4 Nozzle Mount Interior . . . . . . . . . . . . . . . . . . . . . 36Figure 4.5 RGA Inside the Detection Chamber . . . . . . . . . . . . . . 37Figure 4.6 Amplifier Table . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.7 Pump Light Amplifier . . . . . . . . . . . . . . . . . . . . . 39Figure 4.8 Rempump Light Amplifier . . . . . . . . . . . . . . . . . . . 40Figure 4.9 MOT Optical Table . . . . . . . . . . . . . . . . . . . . . . . 41Figure 4.10 MOT Optical Arrangement . . . . . . . . . . . . . . . . . . . 42Figure 4.11 Anti-Helmholtz Field . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.12 New Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . 44Figure 4.13 Rubidium Dispensers . . . . . . . . . . . . . . . . . . . . . . 45Figure 4.14 MOT Camera Setup . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.15 Photodiode Detector 1 . . . . . . . . . . . . . . . . . . . . . 46Figure 4.16 Photodiode Detector 2 . . . . . . . . . . . . . . . . . . . . . 48viiiFigure 4.17 MOT Compensation Coils . . . . . . . . . . . . . . . . . . . 51Figure 4.18 Sliding Helmholtz Coils . . . . . . . . . . . . . . . . . . . . 52Figure 4.19 Sliding Optical Table . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.20 The Catalysis Laser . . . . . . . . . . . . . . . . . . . . . . . 54Figure 5.1 Loss Curve Example . . . . . . . . . . . . . . . . . . . . . . 59Figure 5.2 Beam Profile Proof of Concept . . . . . . . . . . . . . . . . . 61Figure 5.3 Loss Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 5.4 Hydrogen Profile Results . . . . . . . . . . . . . . . . . . . . 64Figure 5.5 Helium Profile Results . . . . . . . . . . . . . . . . . . . . . 65Figure 5.6 Nitrogen Profile Results . . . . . . . . . . . . . . . . . . . . 65Figure 5.7 Argon Profile Results . . . . . . . . . . . . . . . . . . . . . . 66Figure 5.8 Krypton Profile Results . . . . . . . . . . . . . . . . . . . . . 66Figure 5.9 Overlay of All Profile Results . . . . . . . . . . . . . . . . . 67Figure 5.10 Beam Time of Flight Measurements . . . . . . . . . . . . . . 70Figure 5.11 Beam Velocity Distributions . . . . . . . . . . . . . . . . . . 71Figure 5.12 Catalysis Measurement Showing Loss . . . . . . . . . . . . . 73Figure 5.13 Catalysis Result from the Center of the MOT Cell . . . . . . . 74Figure 5.14 Catalysis Result from the Side of the MOT Cell . . . . . . . . 75Figure A.1 Homebuilt Detector Response . . . . . . . . . . . . . . . . . 82Figure A.2 Thorlabs Detector Response . . . . . . . . . . . . . . . . . . 82Figure B.1 MOT Vertical Images 1 . . . . . . . . . . . . . . . . . . . . . 89Figure B.2 MOT Vertical Images 2 . . . . . . . . . . . . . . . . . . . . . 90Figure C.1 Anomalous Loss Measurement . . . . . . . . . . . . . . . . . 92Figure C.2 Coil Translation Measurements MOT Statistics . . . . . . . . 93Figure C.3 Table Traslation Measurements MOT Statistics . . . . . . . . 94Figure D.1 Skimmer Alignment . . . . . . . . . . . . . . . . . . . . . . 96Figure D.2 Skimmer Alignment Setup . . . . . . . . . . . . . . . . . . . 97Figure D.3 Detection Chamber Alignment Before and After . . . . . . . 98Figure D.4 Nozzle Alignment Contraption . . . . . . . . . . . . . . . . . 99Figure D.5 MOT Alignment Setup . . . . . . . . . . . . . . . . . . . . . 101ixFigure E.1 RGA Fitting Normal Distribution Trial . . . . . . . . . . . . 103Figure E.2 RGA Fitting Maxwell-Boltzmann Trial . . . . . . . . . . . . 104Figure E.3 RGA Fitting Lorentz Trial . . . . . . . . . . . . . . . . . . . 105Figure E.4 RGA Fitting Lorentz Arctan Trial . . . . . . . . . . . . . . . 105xGlossaryAOM Acousto-Optic ModulatorCCD Charge-Coupled DeviceCFI Canadian Foundation for InnovationCRUCS Center for Research on Ultra-Cold SystemsDC Direct CurrentIR Infra-RedMOT Magneto-Optical TrapND Neutral DensityODE Ordinary Differential EquationPTFE PolytetrafluoroethyleneRGA Residual Gas AnalyserTOF Time of FlightxiAcknowledgementsI would like to thank everyone who has helped me over the past couple of yearsin helping me complete my work here at UBC. Thank you to Dr. Takamasa Mo-mose and Dr. Kirk Madison for taking me on as a student and for supporting andproviding guidance throughout my work here. Thank you everyone who I workedwith and sought advice from in the lab, Mr. Gene Polovy, Dr. Janelle van Dongen,Mr. Kais Jooya, Dr. Jim Booth and Dr. Mario Michan. Thank you to visitingprofessors Dr. Eckart Wrede and Dr. Frank Stienkemeier for aiding in the develop-ment and construction of this experiment. Thank you to my students Mr. CameronHerberts and Ms. Janet Leung for helping me with the construction and measure-ments made for this experiment. Thank you to Mr. Pavle Djuricanin and Mr. TonyMittertreiner for their technical support in keeping everything running smoothly onthe experiment. And a thank you to everyone in the Momose Group with whom Ihave worked alongside for these past two and a half years.This research was supported by grants provided by the Canadian Foundationfor Innovation (CFI) to the Center for Research on Ultra-Cold Systems (CRUCS)group at University of British Columbia (UBC).xiiChapter 1IntroductionThe Magneto-Optical Trap (MOT) has become an integral part of many cold atomphysics and chemistry labs since it’s development in 1987 [1][2]. A typical MOTof alkali atoms such as rubidium can easily achieve temperatures as low as 1 mKwith populations on the scale of hundreds of millions, and even billions or tens ofbillions, of atoms. With such cold temperatures and large sample sizes it becomesa fantastic toolbox for further cooling and cold atom physics and chemistry [3].In many experiments the change of the atom number in the trap over timeis used as an observable for some physical process. As such, trap losses due tobackground collisions are kept at a minimum to increase the sensitivity to the phe-nomenon studied. In recent years however, it has been proposed that a MOT can beused as a standardized measurement of pressure through the measurement of theatom loss rate from the MOT and a careful calculation of the collisional cross sec-tions of the trapped atoms with the atoms in the surrounding near-vacuum [4][5][6].Here the effects of a background gas, with a particle velocity following a Maxwell-Boltzmann distribution, on a MOT are measured and the ambient pressure of thatgas determined. One of the properties of this mechanism is that, provided that thetrap is characterized and the cross sections known, any trap of any population is apotential candidate for this type of measurement.One adaptation that can be this pressure measurement is to take advantage ofthe fact that the spatial distribution of the trapped atoms is small and can be madeto move to provide a local measurement of pressure or particle flux within the1vacuum. For example, the atom loss from the MOT can provide a measurement ofthe local flux produced by a molecular beam. Alternatively, if the particle flux isknown by other means the atom loss can be used to determine the absolute crosssection of the beam particles with the sensor atoms [7].In the case where the flux ofa molecular beam impinges on the trapped atoms, we can write the rate of changeof the population in a MOT asdNdt= R−GN−bnN−〈sv〉MBnbeam(r; t)N (1.1)where the first three terms follow the standard equations for a MOT [8] with thefourth term representing the additional losses produced by collisions with the molec-ular beam. The population of the trap N, of density n, is increased by the loadingrate R and decreased by the background collision rate G, the intratrap collisionrate b and the introduced loss term 〈sv〉MBnbeam(r; t). The cross section term〈sv〉MB can be determined from quantum mechanics [4][9] but the gas densityterm nbeam(r; t), which has a time and position dependence, may be unknown. Thisleads to the main proposal or hypothesis for this thesis which is that an atomictrap may be a potential measurement tool for time and spatial dependent flux of anatomic or molecular beam, such as those used for fundamental science and thoseused for technological applications.1.1 Atomic and Molecular BeamsLike the MOT, an atomic beam is a tool for many cold atom scientific applications.Through the supersonic expansion of a high pressure gas into a vacuum, the atomsand molecules of the gas are ejected with a high, but relatively uniform, velocitywhich can then be manipulated and applied to a testing environment. Deceleratorstaking advantage of the Zeeman and Stark effects of an atom or molecule control,contain and even trap a gas which was originally produced by supersonic expan-sion. Alternatively, one can produce a supersonic beam from an effusive sourcewhere a high temperature oven chamber vaporizes the beam material. This tech-nique is particularly useful when working with metallic sources, but is also quiteuseful for nonmetallic clusters, ionic compounds, and intermetallic species[10].On a related note, thermal processes such as molecular beam epitaxy, which has2seen remarkable growth in recent years due to the semiconductor industry, relies onthe emission of atoms in the form of a gas to build crystalline layers on a substrate[10][11].As with any experiment or industrial process an atomic beam has to be mea-sured and characterized to determine it’s flux and uniformity. Physical detectiontechniques such as a fast ionization gauge provide a time dependent measurementof the local pressure in the vacuum [12]. Chemical methods allow for a moretargeted approach to beam detection and can at times measure more difficult ma-terials, such as atomic hydrogen through chemical absorption in a semiconductor[13]. Laser-induced fluorescence is a particularly useful technique in that both thebeam density and it’s velocity can be characterized due to the velocity dependentDoppler Shift [14].Many of the techniques shown above have two key features: they have a degreeof spatial resolution and they can provide a time dependent signal with which thebeam velocity can be inferred through time-of-flight analysis. With these parame-ters in mind, it is here that the use of a MOT as a beam detector will be presented.The space and time requirements of a beam detector are met by the inherent prop-erties of the trap; A typical MOT will have the vast majority of the trapped atomswithin a volume smaller than a cubic millimetre and the time dependence comesfrom the continuous loading and ejection of atoms seen in Equation 1.1. The detec-tor would rely on the interaction cross sections of the various beams which couldbe put it’s way. The prior knowledge of those cross sections is the only prerequisitefor performing an accurate measurement. Moreover, because the atom loss processrelies on an unchanging law of nature (the interaction potentials between the sensoratoms and the atoms or molecules in the beam), this method provides a method toobtain absolute measurements of flux. Aside from the trapping atoms, a MOT doesnot introduce any additional equipment into the measurement area allowing for theinert testing of the beam.1.2 Thesis OverviewThis thesis will present the theory, construction, and testing of a rubidium MOTatomic beam detector. The measurement of the flux and profile of five separate3gases is performed where the loss-producing cross section is extracted from mea-surements made at the peak intensity of the beam. The trap depth for the MOT isalso determined in order for the cross sections to be compared to the theoreticallycalculated cross sections at a later date. Several iterations of the MOT translationmechanism and the resulting measurements are also presented.4Chapter 2Theory2.1 Atomic ScatteringIn the most simplistic sense, a collision event between two atoms can be thoughtof in what can best be described as “pool-ball mechanics:” a collision between twospheres resulting in predictable post-collision velocities. Of course, we don’t havethe luxury of perfect spheres when calculating the interaction of two atoms, but theproduct of the interaction can be calculated nonetheless.The use of the term “collision” is not fully descriptive of what is taking placeduring an interaction event. The electrical properties of an atom dictate that an in-teraction of two atoms or molecules must be treated as an interaction of two poten-tials pushing against the center of mass of each atom where there isn’t a classicalcollision but a region in which two species have a greater degree of interaction.Considering a single cold collision between these atoms, the interaction now re-quires the application of quantum mechanics to fully understand the interaction ofthe atoms at a particular relative velocity and interaction distance.For the purposes of this thesis we will be interested in the interactions of a“cold” atomic or molecular beam, which has a high-velocity in the lab-frame butlow relative velocity between the atoms or molecules in the beam, interacting withalmost stationary rubidium atoms held in a magneto-optical trap. More specifically,we will detect those collisions which result in an exchange of momentum largeenough that we observe a loss of rubidium atoms from the MOT. While the full5atom-atom interaction potential is required to exactly compute the differential crosssection and thus the trap loss rate, the vast majority of collisions that occur do soat very large impact parameters and the particles do not approach each other closeenough to probe the short-range part of the interaction potential. Thus, a very goodestimate of the trap loss can be made only knowing with high accuracy the longrange part of the potential. A corollary of the above observation is that a vastmajority of the collisions that occur impart only a small amount of momentum tothe sensor atoms and detecting these collisions requires differentiating this smallmomentum change before and after the collision. This latter requirement motivatesthe need for laser cooled atoms where the initial momentum is very well known andsmall changes can be measured by the induced trap loss from a shallow trap. [15].2.1.1 Trap Loss ConditionsIf we consider two atoms in a collision event there will undoubtedly be a transferof energy between those entities. Thinking of this event classically, as “pool ballmechanics”, this energy transfer will depend not just on the atoms and their kineticenergies but the relative angle in which the collision occurs. The shallower theangle, the smaller the energy transfer until the two particles miss entirely. Thishas an interesting application when we consider one atom to be contained withina trap (potential well) and a minimum energy is necessary to remove the atomfrom the trap. For collisions with a large angle of incidence, the trapped atommay simply be recaptured whereas a more direct collision will result in ejection.This loss-producing collisional cross-section is a fundamental property of collisionexperiments where trap loss is the measured parameter.The theory behind the collision mechanics is very similar to what was describedby Van Dongen et al. [4][9] with some adaptations for the work in this thesis.Beginning with a classical view of collision mechanics we can work in thecenter of mass frame of a trapped atom ‘a’ and perturbing atom ‘b.’ What we arelooking for is a collision event which gives the trapped atom enough energy suchthat it is able to escape a trap of depth Utrap.We begin by identifying the exchange of energy as a result of a collision eventbetween atoms ‘a’ and ‘b.’ This will occur at some relative velocity ~vr and at a6collision angle of q , which can be defined as the angle between the initial ~vr andfinal ~v′r relative velocities of the particles. As usual, m is the reduced mass of thesystem. The equationDE ∼ m2Ma|~vr|2(1− cosq) (2.1)gives the energy exchange between the atoms. This assumes that the initial momen-tum of the trapped atom, of mass Ma, is zero. For the application we are interestedin, there is a potential well U0 that we are are attempting to escape. Keeping therelative velocity of the particles constant this places a condition on the scatteringangle such thatqmin = arccos(1− MaU0m2|~vr|2)(2.2)gives the minimum angle at which a collision may take place that can meet thepotential trap ejection conditions. As such, we now have a condition for trap losswith U0 =Utrap2.1.2 Scattering Cross SectionSo far we have looked at the conditions necessary for removing a single atomfrom an atomic trap, but this still assumes that such an event will take place. Thisrelies on the two atoms passing within what can be called the loss-producing crosssection, sloss, of the two atoms. In a system with a large number of trapped atomsand a large number of collision events, we can then apply this cross section into aloss rate from the trapgloss = nbvprobsloss = nbvprob∫ pqmin(vprob)(ds=dW)dW: (2.3)For this, nb|~vr|(ds=dW)dW represents all scattering events into a solid angle dWfrom a perturbing particle beam of density nb. For now, vprob is a place-holder andis the most probable collision velocity of |~vr|. If we assume the atoms in the trapare stationary, then |~vr| is the beam velocity.Looking further into the cross sectional term we can redefine the differential7cross section as the quantum-mechanical scattering amplitude ds=dW= | f (k;q)|2with k as the collision wave vector k = m|~vr|=h¯. This gives the scattering crosssection for a particle of wave number k colliding with the atoms contained in a trapof depth Utrap assloss(k) =∫ pqmin(h¯k=m)2p(sinq)| f (k;q)|2dq : (2.4)Up until this point this section has followed very closely to the theory presentedin the papers by Van Dongen et al. [4][9]. While both the experiments performedby Van Dongen et al. and the work conducted for this thesis measure the lossesfrom an atomic trap, the method in which those loss-producing collisions are pro-duced differ substantially. By using a Maxwell-Boltzmann probability distributionfor the background atom velocity, the velocity-averaged cross section is given byVan Dongen et al. as〈vs〉VanDongen =( MB2pkBT)3=2 ∫ inf04psloss(k)v3be−Mbv2b=2kBT dvb: (2.5)For our application we will be using a beam of atoms or molecules producedby supersonic expansion with the ultimate goal of measuring the flux density andprofile of that beam with the MOT through collisions. We then define the velocityaveraged loss rate from the MOT-beam collisions as 〈g〉= nb〈vs〉 where〈vs〉MB =∫ inf0sloss(k)vb fbeam(vb)dvb: (2.6)The velocity-dependent term fbeam(vb) is the normalized velocity distribution ofatoms passing by the atom trap. Experimentally we can use the signal generatedby the Residual Gas Analyser (RGA) to retrieve the time-of-flight information ofthe beam and therefore determine the velocity distribution of the beam.2.1.3 Independent EventsOne concern with measuring the loss of atoms from a trap due to collisions is ifone collision event results in multiple atoms to leave the trap. This avalanche effectwould distort the measurement by dampening atoms with a low kinetic energy and8keeping them in the trap, or by causing high energy atoms to ricochet around thetrap and potentially causing a greater loss to be measured. To ensure that we canneglect any of these collisions, the mean free path of an atom in the trap must befar larger than the size of the trap itself. This will keep the number of collisionsthat result in a secondary collision to a minimum compared to the sample size.A typical MOT will contain approximately 108 atoms (N) contained within avolume (V ) of approximately 1 mm3. Using an estimate for the mean free path`=kBT√2pd2sP(2.7)with the ideal gas law PV = NkBT we get`=V√2pd2sN: (2.8)If we use the van-der-Waals radius for rubidium (303 pm) for the approximateinteraction distance ds we get a mean free path estimate of 24.5 m for the MOT.Although this is not necessarily an ideal gas, the distance calculated is far larger, bya factor of 105, than the size of the trap making it safe to assume that ejected atomshave a collision-free path to exit the trap and that the collisions are completelyindependent events.2.2 Supersonic Expansion and Atomic BeamsThe mechanism by which we produce the high-velocity atoms required in ourcollision-based experiments is through the rapid expansion of gas from gated con-trol point and the refinement of that gas into a narrow, but expanding, beam. Giventhe proper conditions, this process produces a beam of particles travelling fasterthan the local speed of sound, hence the name “supersonic expansion.”This section will contain the theory behind the supersonic expansion processas well as several numerical estimates for the resulting beam. This section will alsoinclude information regarding the skimmers and their effects on the beam. Muchof the theory discussed here can be referenced back to Giancinto Scolles’ Atomicand Molecular Beam Methods [10].92.2.1 Supersonic Expansion of an Atomic GasReleasing a gas of higher pressure P0 into a chamber at a lower background pres-sure Pb takes on much different properties once the pressure difference is above aparticular value. In this case the higher pressure gas will expand supersonicallyinto the lower pressure region. The exact condition for this process to occur iswhereP0=Pb > G: (2.9)The value of G is dependent onG =(g+12)g=(g−1)(2.10)where g is the ratio of the specific heats for the gas. This is 5/3 for a monatomicand 7/5 for a diatomic gas each giving values 2.04 and 1.89 for G respectively. Inour system G is far exceeded by P0=Pb, which to the order of 1010.When we have a supersonically expanding system it has a particular set ofproperties which are illustrated in Figure 2.1 which assumes a point source for theorigin of the high pressure gas. The gas expands into the high vacuum, radiatingaway from the source. The outward expansion from the source would continu-ously increase in Mach number, the ratio of the particle speed relative to the speedof sound, if it was not constrained by shock waves forming a bubble around the su-personic region. At it’s maximum, the thickness of this shock wave is on the orderof the mean-free-path of the background gas Pb. The effective volume in which thebeam is propagating supersonically is known as the “zone of silence.”At the end of the supersonic region, where the zone of silence meets the sub-sonic region beyond, we have the Mach disk. The location of this interface is givenbyxM =2d3(P0Pb)1=2(2.11)with d being the diameter of the nozzle’s aperture. For our experiment, with abacking pressure P0 of 5200 torr and a background vacuum pressure Pb of 5.1x10-7torr, the Mach disk appears 16.8 m from the nozzle aperture. This is longer thanthe entire beam line of our experiment by an order of magnitude.10M>1M<1M>>1MachpDiskShockCRUCSpValveSkimmerZonepofpSilenceP0,T0M<<1PbCompressionpWavesReflectedpShockBarrelpShockJetpBoundaryFigure 2.1: A diagram of a supersonically expanding gas [16]. A nozzle ofopening diameter d releases high-pressure gas into a vacuum. The gaspropagates supersonically through the zone of silence until a combina-tion of compression waves and the Mach disk reduce the Mach numberto subsonic levels. Skimmers can be placed within the zone of silenceto reduce the gas to a slowly diverging beam.Velocity estimates for the supersonic beam are given by the equation for max-imum terminal velocityV¥ =√2RW( gg−1)T0 (2.12)which is dependent on the average molecular weight W and the ideal gas constantR. T0 is the initial temperature of the backing gas (room). For the purposes of theexperiments described in this thesis, the calculated terminal velocity of the relevantatoms are shown in Table 2.1.One additional approximation that can be made is the ratio of the parallel tothe perpendicular beam velocities. This can be used to better understand how theshape of the atomic beam will deform in time. This approximated ratio is given byS‖ ≈ 5:4(P0 ·d)0:32: (2.13)For our system, Equation 2.13 gives a ratio of 0.00368. This ratio will be refer-11Gas Mass [g/mol] g V¥ [m/s]He 4.00 5/3 1736Ar 39.95 5/3 549.3Kr 83.8 5/3 379.3H2 2.01 7/5 2897N2 28.01 7/5 776.2Table 2.1: The expected maximum velocity of the atomic beam. The valueswere determined from Equation 2.12. The reservoir for the beam gas isa long supply tube kept at room temperature. Therefore, these estimatesare made with T0 = 293 Kenced to in Section 2.2.2 in the estimation of the beam size.2.2.2 Collimation of the Atomic BeamTo refine the supersonically-expanding gas into an atomic beam skimmers wereplaced between the beam source and the MOT. These skimmers made it very un-likely for any atoms to pass from one chamber to another unless the atoms weredeliberately directed through the skimmer. The effects of this were twofold: theskimmer would refine the gas into a beam and reduce the transmission of back-ground and excess gas between the chambers, creating a differential pumping in-terface. For the purposes of our experiment, up to two skimmers were placed toensure that the beam would follow a well defined path and to reduce the excess gasinterfering with the MOT.A condition for skimmer placement is that it must be within the zone of silencein order to properly reduce the outward spray of atoms and molecules to a beam.Fortunately, as described in Section 2.2.1, the Mach disk does not even appear inour experiment as the zone of silence extends for over 16 meters.There were two iterations of the nozzle and skimmer setup for this experiment.The first iteration had only a single 3 mm diameter skimmer bordering the sourceand differential chambers with the nozzle fixed on a place approximately 3 cm fromthe skimmer. It was found that this set-up would flood the MOT chamber with gaswith a significant portion of the gas bouncing off the exterior surfaces of the MOT12cell before interacting with the MOT atoms. The second arrangement had the 2 mmdiameter skimmer placed at the differential-MOT chamber border. At this time, thenozzle was attached to the alignment apparatus as described in Section 4.1.1 andAppendix D. Figure 4.2 provides an illustration of the beam line setup with all thedistances and skimmer sizes included.The arrangement of the skimmers provides the following distances: The firstskimmer is 12 cm from the nozzle and is 3.0 mm in diameter, the second skim-mer is 47 cm from the nozzle and is 2.0 mm in diameter, and the MOT is 99 cmfrom the nozzle. Effectively the beam is defined by the second skimmer diameterand distance as well as the distance to the MOT. Due to the transverse motion ofthe beam described in Equation 2.13 any beam widths determined will be roughestimates. With a purely geometric argument and assuming a point source for theatoms, the peak intensity of the beam is expected to be 2:0 9947 mm, or 4.21 mm indiameter at the MOT. Even with this broadening after the 2 mm skimmer, the beamwill be diverging further through the transverse motion of the atoms by Equation2.13. This will broaden the beam creating a taper out to a maximum distance fromthe peak intensity. Using the velocity ratio of 0.00368 we assume that the atomswill deviate at most 0:00368×52 cm, or 1.91 mm, from the peak intensity regionof the beam. This will result in a beam which will have a maximum measurableprofile width of 8.0 mm.13Chapter 3The Magneto-Optical TrapA magneto-optical trap MOT is a form of cooling and trapping atoms using a mag-netic field to split the Zeeman sublevels of the relevant atoms and polarized andfrequency-locked lasers to push those atoms towards the center of the trap. Theelements which are trapped are usually alkali atoms such as rubidium or lithium,but alkali-earth elements such as strontium [17] and cold polar molecules such asTiO are also possible [18]. Typically, trapped atoms will have a temperature onthe order of 1 mK, although measurements of sub-Doppler temperatures have beenmade [19][20].Our interest in cold atoms comes from the study of interactions where ther-mal effects are greatly reduced. In this regime, quantum phenomena become dis-cernible from the thermal broadening of the specific energy transitions which al-lows us to probe or push precise properties.3.1 Overview of the MOT Trapping MechanismA MOT operates on the conservation of momentum from the absorption and emis-sion of photons by an atom. This process is enhanced by circularly polarizingand red detuning the light whilst applying a magnetic field gradient to split thehyperfine structure of the target atom thereby creating an environment in whichthe atoms are “pushed” towards the center of the trap while simultaneously beingcooled. This section will give a generalized view of the trapping mechanism of a14MOT with later sections of this chapter providing the details specific for trappingrubidium.3.1.1 Separation of the Hyperfine Energy LevelsIn the absence of a magnetic or electric field, the hyperfine energy states remaindegenerate. The application of a magnetic field splits this degeneracy in a processknown as the Zeeman effect. Similarly, the Stark Effect will also split the hyperfinelevels, but this will not be discussed in this thesis.The full derivation for finding the energy shift due to the Zeeman effect can befound in David Griffiths’ Introduction to Quantum Mechanics[21]. The spin-orbitmagnetic-moment interaction is given by the HamiltonianHz =−(~ml + ~2ms) ·~B = eme (~L+2~S) ·~B (3.1)which affects the energy of the spin of an electron along the orientation of themagnetic field ~B. If this were oriented along a z-axis, the spin and orbital terms~S and ~L would be proportional to the Pauli sz matrix. This Hamiltonian is alsodependent on the electron charge e and rest mass me.In the weak-field Zeeman effect we can use first-order perturbation theory togive the correction to a transition energy asE1Z = 〈nl jm j|H1z |nl jm j〉=e2m~Bext · 〈~L+2~S〉: (3.2)Solving for the expectation value we getDE = mBgFmFB (3.3)where mB is the Bohr Magneton, mF is the projection of the magnetic quantumnumber, and gF is the lande` g-factor. The values for mF have the possible values(−l;−l+1; :::; l−1; l) for a total of l(l+1) states.While the theory that has been presented so far has dealt with a constant mag-netic field, what about a spatially-dependent field? The answer is a simple extrap-olation given the system that we wish to work in for this report.If we consider a magnetic field which has a gradient in the z direction dB(x;y;z)dz15we getDE =−mBgFmF dB(x;y;z)dz z (3.4)for the energy shift due to the magnetic field gradient. It is possible to supply such afield from a Helmholtz coil in the anti-Helmholtz configuration. In the applicationto the magneto-optical trap, the Helmholtz coils provide a magnetic field gradientwhich is radially symmetric in the plane passing between the two Helmholtz coilsand a gradient of opposite sign and twice the magnitude passing through the coils.This magnetic field will function in tandem with the red-detuned photons describedin the later sections of this report to go provide the trap environment.3.1.2 Selection Rules for the Optical TransitionsThe individual polarization of the photons affects, by means of the transition selec-tion rules, which energy states are coupled by the trapping photon. In this section Iwill discuss the selection rules governing the polarization of the photons as well ashow the orientation of that polarization integrates into the MOT. For this section, itwill be useful to describe the selection rules for hydrogen-like atoms as is the casefor the alkali-metals.The valence electron of hydrogen-like atoms follows similar wave functionsto the standard spherical and radial wave functions described by the three quan-tum numbers n; l;m which correspond to the principal, angular momentum, andmagnetic quantum numbers respectively. How the selection rules between two ofthese states comes into play can be shown from the interacting Hamiltonian on thespherical harmonics Y ml .If we consider a two state system, then the rate of decay between those statesis given by Fermi’s Golden RuleRi→ f =2ph¯2d (w f i−w)|〈 f 0|H ′|i0〉|2 (3.5)with the Dirac delta function limiting this transition to photons of frequencyw . TheHamiltonian term is the first order perturbation to an election around a hydrogen-like atom. This perturbation is the effect of an electromagnetic field (photon) on16the electron and is given byH ′ =ieh¯2mei~k·~r~A0 ·Ñ (3.6)where ~A0 is the vector potential [15]. Expressing this in terms of raising and low-ering operators, we get the interaction HamiltonianH ′ =− ieh¯m√V å~k;l√h¯2e0w(k)ei~k·~r~el (~k) ·Ñ[Al (~k)+A+l (−~k)] (3.7)where A;A+ are the photon raising and lowering operators and~el (~k) is the photonpolarization vector. For the equation, m is the electron’s rest mass, V is a volumeterm required for normalizing the expansion of the raising and lowering operatorsfrom quantum field theory, and e0 is the vacuum permittivity constant.Applying this to two the initial and final atomic wave functions gives the matrixelement〈yn′;l′;m′ |H ′|yn;l;m〉 µ 〈yn′;l′;m′ |ei~k·~r~el (~k) ·Ñ|yn;l;m〉: (3.8)This can be reduced, by Taylor expanding the exponential to~el (~k) · 〈yn′;l′;m′ |Ñ|yn;l;m〉 (3.9)and reduced further to a more convenient form by[H;~r] =−ih¯m~p=− ih¯mÑ〈yn′;l′;m′ |Ñ|yn;l;m〉=−mh¯2〈yn′;l′;m′ |[H;~r]|yn;l;m〉=mwh¯〈yn′;l′;m′ |~r|yn;l;m〉(3.10)which we can now solve for a specific matrix element for a set of initial and final17states.From the fundamental properties of photons, we are limited to Dl =±1. Whatwe have yet to determine is what the selection rules are for the magnetic quantumnumber m, which will be shown below.The vector~r can be written in spherical coordinates as~r = r(sin(q)cos(f);sin(q)sin(f);cos(q)) (3.11)which interacts with the radial and spherical wave functions. The determination ofthis matrix element can be separated into the radial and spherical parts and solvedindividually through integration:∫∫dWY m′∗l′ ~rYml : (3.12)The spherical wave functions have a particular f dependence depending on thechoices for m and m′. As an example the two wave functionsY 01 µ cos(q) (3.13)Y±11 µ sin(q)e±if (3.14)provide a f dependence if there is a Dm = ±1. Due to the oscillating nature ofthe exponential containing this dependence, the only nonzero matrix elements willoccur when this f dependence in the exponential is removed. Thus, the threepossible vectors become~r′ = (x;±iy;0) (3.15)and~r′ = (0;0;1) (3.16)which correspond to photons of right-hand and left-hand circularly polarized lightto drive the Dm = ±1 transitions and linearly polarized light to drive the Dm = 0transition respectively.183.1.3 Full Ensemble for the Magneto-Optical TrapNow that the basic theory of both the separation of the hyperfine energy levels andthe transition selection rules have been introduced, it is time to put everything intoaction for the MOT.The primary mechanism by which the MOT functions is the conservation ofmomentum due to the absorption of a photon. By absorbing a photon, the atom isgiven a mild kick along the same wave vector as the photon. The subsequent spon-taneous emission of a photon, which is in a random direction, gives a secondarykick to the atom opposing the emitted photon [10]. This secondary kick can be ig-nored down to the Doppler cooling limit as the sum of many (millions of) randomvectors is zero at such a large scale. This has the net effect of pushing the atomsalong from the radiation pressure of the laser. Having an identical laser opposingthe first will cause the atoms to effectively sit in place and fluoresce.It is at this point that we can apply external effects to the atoms and built atrapping mechanism. Figure 3.1 shows a simplified view of a two-level systemwhere a positionally dependent magnetic field has split the hyperfine levels. Inthis case, a hypothetical F = 0→ F = 1′ transition is shown and, since we havel(l+ 1) states, the F = 1′ state has been split in three where the field is non-zero.The view that is shown in this diagram is more representative of a 1-dimensionalcross-section of the trap with circularly polarized photons displayed as s+ and s−.In the lab frame both of these photons will have the same polarization (either right-hand or left-hand circular) but it is convenient to write them as s+ and s− due tothe chosen reference frame. The energy of the photons is red-detuned from theDm = 0 transition.The result of this is that the circularly-polarized photons are now preferentiallyabsorbed by atoms which are on the side of the trap which is closer to the photonsource. This is because, from the point of view of a photon, the atoms before theMOT are on-resonant and the atoms after the MOT are off resonant. This is due tothe magnetic field increasing away from the center of the trap and thus the magneticfield has an observed sign change at the center of the trap. The opposing photon inthis picture observes an identical effect. The radiation pressure from this preferen-tial absorption process will move atoms towards the center of the trap which slows19Energym=+1m=0m=-1m=0F=0F=1'σ+ σ-ћω0 zB(z)    zFigure 3.1: A simplified view of the photon interaction with the split hyper-fine levels of a hydrogen-like atom. The red-detuned light is circularlypolarized to preferentially excite a Dm =+1 or the Dm =−1 when thephoton is absorbed and brings the electron to the excited state.and cools the atoms in the process. Similarly to how it was described above, thespontaneous emission processes can be ignored but only until the Doppler coolinglimit is approached [10].The picture given here is for the one-dimensional case, but this picture can beeasily scaled up to three-dimensions. In a fully-functioning trap, the magnetic fieldhas a radially-symmetric gradient in the x-y plane with the axial (z) componenthaving the opposite sign. The result of this orientation is that each photon in thex-y plane must have the same polarization in the lab frame as it enters the MOTenvironment. The photons along the z direction must have an opposite sign tothose in the x-y plane.20Figure 3.2: Left: A diagram showing the six trapping lasers and theHelmholtz coils in the anti-Helmholtz configuration for the MOT. Right:An image of the MOT taken from the same point of view as the left im-age. The MOT is seen with a cloud of fluorescing atoms around it.A Technical Comment on MOT OpticsA description of the interaction of mirrors and polarisers with photon polarizationprovides an additional point-of-view on the cooling dynamics of a MOT. Each ofthe 3 spatial dimensions has an independent cooling beam which is passed overthe MOT twice. The photons are initially oriented with a quarter-wave plate tothe optimal polarization to interact with the MOT. After passing through the MOT,they are reflected and sent back to interact again but this time from the oppositedirection.If a single mirror with no other optics was placed to do this, the polarizationwould remain, from the lab reference frame, identical to the incident beam butpropagating in the opposite direction. This would negate the effects of the originalbeam by interacting preferentially with the atoms in the far side of the trap fromthe direction of propagation. Introducing a second quarter-wave plate before thismirror linearly polarizes the light before it is reflected and returns it to a circular,but opposite, polarization before interacting with the MOT again. This completesthe radial symmetry of the x-y plane of a MOT. The axial direction functions onthe same principle, but with oppositely-circular polarizations for the beam and itsreflection.213.1.4 Population DynamicsThe population of a MOT is subject to several factors which affects the loading andremoval of trapped atoms. These effects can be modelled as an ordinary differentialequation relating the change in the trap’s population as the sum of the effects onthe populationdNdt= R−GN−b∫n2mot(r; t)dV: (3.17)The total population and it’s density are given as N and nMOT (r; t). Here, the changein trap population dNdt is affected by the loading rate R adding to the populationwhile collisions with a hot background gas produces a trap loss rate G and intratrapcollisions b∫n2mot(r; t)dV also remove atoms. b is the intratrap collision coeffi-cient. The intratrap collision term completes the equation, but can be removed ifthe density of atoms is small enough such that the loss rate is independent of thenumber of trapped atoms[22][23].For our application we will be looking at shortterm changes to the MOT number produced by a brief transient loss with a muchlarger magnitude than any of these terms. Thus we can safely neglect them in ourloss model.Without the presence of an external perturbation, Equation 3.17 can be mod-elled as an exponential function approaching the steady-state MOT population. Byassuming that the atom density is constant we can rewrite Equation 3.17 asdNdt= R−GN−b∫n2motdV= R−GN−bn2V= R− (G+bn)N:(3.18)To simplify further calculations we will use G′ = G+bn. This ordinary differ-ential equation can be solved to yield22∫ dNR− (G′+bn)N =∫dt− 1G′ln(| −G′N+R−G′Nt=0+R |)= t−G′N+RR= e−G′tG′NR= 1− e−G′tN = Nss(1− e−G′t)(3.19)in which we define R=G′ = Nss as the maximum trap population after enoughtime has elapsed (the “steady-state” population). The exponential reached in Equa-tion 3.19 models the trap population when a trap is “turned on” at t = 0. Experi-mentally, this can be the switching on of either the coils or the trapping lasers.A complimentary measurement made during any particular experiment is whatis known as the “loading curve.” Such a measurement provides the information onthe unperturbed trap that is described in Equation 3.17 by fitting the exponentialdescribed in Equation 3.19. An example of one such measurement can be seen inFigure 3.3.In Figure 3.3 the trapping coils were switched on at time t=0s and rubidium al-lowed to load into the trap. As shown in Equation 3.19 the trap population asymp-totically approaches a maximum population value. The quantities we can extractfrom this fit are the total trap signal V0, the loading time constant G, and the off-set applied to the signal due to light from the trapping lasers reaching the detectorVbase. These values also allow us to determine the loading rate constant R.The trap loading equation and subsequent loading curve provides all the infor-mation relevant to the unperturbed population dynamics of the trap. For part ofthe work outlined in this thesis the Ordinary Differential Equation (ODE) given inEquation 3.18 will require a term representing the collisional effects of the atomicbeam on the MOT. Given the nature of the atomic beam that we will be using,this term will depend on both the time the interaction takes place and the locationwithin the beam the MOT will sit. Adding this term to Equation 3.18 givesdNdt= R−GN−bnN−〈sv〉MBnbeam(r; t)N (3.20)230.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time [s]0.20.00.20.40.60.81.01.21.4Population [Atoms]1e8MOT Loading CurveSample: Ar no. 1Figure 3.3: An example of the loading curve of the MOT. The trapping coilsare initially off and are switched on to begin trap loading. Eventually anequilibrium population is reached.where 〈sv〉MB is the velocity-averaged loss-producing cross section of theatomic beam on the MOT and nbeam(r; t) is the time-dependent density of the atomicbeam. Further details regarding the cross section of this interaction can be found inSection 2.1 on scattering theory. During experimentation, the atomic beam is mea-sured with a RGA and details regarding the beam and it’s detection can be found inSection 2.23.2 The Specifics for a Rubidium MOTApplying the principles of the MOT to any particular atom requires an understand-ing of the electronic transitions being driven. A typical trapping set-up for rubid-ium will drive transitions within either the D1 (52S1=2→ 52P1=2) or D2 (52S1=2→52P3=2) transition group. In this experiment we trapped 85Rb and 87Rb with elec-tronic transitions in the D2 line.For the trapping of 85Rb with the D2 line the F = 3→ F = 4′ is driven asour main trapping laser, known as the “pump” laser. For 87Rb this is the F =24F=4F=3F=2F=3F=2F=1PRP52P3/252S1/2384.2304 THzP = 384.2292 THzRP = 384.2322 THz85Rb 87RbF=3F=2F=1F=2F=1F=0PRP52P3/252S1/2384.2305 THzP = 384.2281 THzRP = 384.2347 THzFigure 3.4: The energies of the 85Rb D2 and 87Rb D2 transitions [24] [25].The frequencies listed are for the pump (P) and repump (RP) lasers.2 → F = 3′ line. Although this pumping laser provides sufficient containmentof the atoms, there is a non-zero probability that the atoms will decay throughthe forbidden F = 2→ F = 4′ (F = 1→ F = 3′ for 87) transition and becometrapped in a “dark” state [24]. For this reason, we also require a “repump” laser toexcite the F = 2→ F = 3′ (F = 1→ F = 2′) transition to allow decay through theF = 3→ F = 3′ (F = 2→ F = 2′) transition and return to the pumping process.3.3 Trap Population Estimates and MeasurementsThere are two measurement techniques for estimating the number of trapped atoms,or “population” of a MOT: atom fluorescence and an absorption signal techniquedeveloped by [19]. In the technique described in [19], the atoms are preparedinto the 5S1=2F = 2 state by switching off the pump light shortly followed by therepump light. A probe laser then drives the 5S1=2 F = 2→ 5P3=2 F ′ = 2 transition25and is turned on for anywhere from 5 ms to 500 ms just after the repump shuts off.The probe transmission to the detector is measured both with and without atoms inthe trap and the integrated difference in those signals gives us the number of atomsin the trap. This result is independent of detuning, intensity, and polarization of theprobe light.In the fluorescence technique, the photon decay rate of the transition driven bythe pump light is retrieved. Assuming a nearly instantaneous absorption processfollowing each emission, the population of the trap can be estimated from thisdecay rate. This was the primary technique used for MOT population estimates inthis work. The decay rates for the relevant rubidium transitions can be found in[24] and [25]. For the work outlined in this thesis the fluorescence technique wasthe preferred method. Details on the calibration of the detectors used can be foundin Appendix A. It should be noted that the absolute atom number of the MOT isirrelevant to the outcome of our beam profiling and cross-section measurements;only the relative changes to the trap’s population are important. More details onthis can be found in Section 2.1.3.3.4 MOT Trap Depth MeasurementOne property of a magneto optical trap is what is often referred to as the “depth”of the trap and is the depth of the energy well that the atoms rest in. Typicallythis value is expressed in kelvins. For our experiment, knowledge of the trapdepth allowed us to determine the energy needed to be transferred to the trappedatoms during a collision such that the trapped atom would be ejected from thetrap. As described in Chapter 2.1 the trap depth leads us to the velocity-averagedloss-producing cross-section of the atomic beam.A technique developed by Hoffmann, Bali and Walker in 1996 [26], and fur-ther developed by Van Dongen et al.[4], gives a detailed method of measuring thedepth of an atom trap through the photoassociation and dissociation of the trappedatoms. This technique is referred to as the “catalysis laser technique” for the workrelated to this thesis. In this process, the rubidium atoms are photoassociated intoa repulsive molecular potential curve which dissociates at the resonant frequency.The catalysis technique requires a tunable rubidium laser capable of sweep-26ing from an electronic resonance, in our case the 85Rb D2 and 87Rb D2 transitionregimes, upwards in energy by up to 200 GHz. While on-resonance, the laser willact as a push beam normally reserved for optical trapping and will violently per-turb the atom trap. As energy is increased away from resonance, photoassociativeprocesses take over as the dominant perturbation to the trap. If photoassociationoccurs, the excited state of the two atoms in this this process form a repulsive stateand will naturally dissociate due to the repulsive molecular potential curve. If theenergy provided to the atoms is greater than resonance, the atoms become moretightly bound in their excited state. The dissociation process of these two atomsnow emits a resonant photon with the remaining energy being split between thetwo atoms, which are fired apart from one another. If the excess energy providedby the initial photon is greater than twice the trap depth, hD> 2UtrapKB, the atomswill have enough energy to leave the trap and the catalysis process will result ina reduced trap population [4][26]. An illustration of this process can be found inFigure 3.5.3.4.1 Catalysis Effects on Trap PopulationThe catalysis laser alters the population of the trap which we can model by includ-ing a new term in Equation 3.17 to makedNdt= R−GN−b∫n2MOT (r; t)dV −bcat(z;D)∫n2MOT (r; t)dV: (3.21)This introduces the term bcat(z;D) representing the effect of the catalysis laser, withthe coefficient b dependent on the laser intensity z and detuning D, on the MOT. Ina similar manner to Equation 3.18 we assume a constant atom density and solvethe integrals to yielddNdt= R−GN−bnN−bcat(z;D)nN: (3.22)and a steady-state atom number ofNss+cat =RG+bn+bcat(z;D)n: (3.23)27Internuclear/Separation/(R)Potential/Energy/(V(R)) excitationdissociation hΔD2catalysisemission52S1/2/,/52S1/252P3/2/,/52S1/2Figure 3.5: An schematic of the catalysis process. The photoassociation en-ergy provided to the pair of atoms is greater than the on-resonant tran-sition energy. For the work of this thesis we will be using the the RbD2 transition. The unstable excited state decays with an on-resonant D2photon with the excess energy given to the two atoms. In the catalysisprocess, if hD2 >Utrap then the atoms will be ejected from the trap.28Chapter 4The Experiment SetupIn this chapter the details and specifications of the experiment’s apparatuses willbe presented. Table 4.1 gives a list of the major changes which were made to theexperiment from August of 2013 to December of 2015.This chapter is split into two sections representing the two components to theexperiment table: the beam and the MOT.4.1 Atomic Beam SetupThis section will describe the atomic beam line and all of it’s components. Thisincludes the vacuum chambers, beam source, alignment mechanics, and the beamdetection methods.4.1.1 Atomic Beam LineThe atomic beam line is spread across four vacuum chambers used for generating,refining, transmitting, and detecting the atomic beam. Figure 4.2 gives an overviewof the geometry of the atomic beam line. The individual chambers are separatedby the two skimmers and the aperture which, in addition to controlling the atomicbeam, provide differential pumping stages for the vacuum setup.The first vacuum chamber in the beam line housed the beam source and thefirst of two skimmers along with all associated mounting apparatus. The nozzleis a home-built Center for Research on Ultra-Cold Systems (CRUCS) valve made29Date Action Summary2013 Aug Addition Construction of a separate beam detectionchamber. Second gate valve, fast ionizationgauge, turbo pump added.2013 Aug Addition Installation of a second skimmer between thefirst skimmer and the first gate valve2013 Nov Alteration Removal of nozzle mounting plate. Construc-tion of adjustable nozzle mount.2013 Dec Addition 1 cm aperture at beam detection chamber gate2013 Dec Addition Hotwire Detector2014 Feb Addition Helmholtz coil sliding mount installed2014 May Addition Rotary feedthrough added to the adjustablenozzle mount manipulator system.2014 Aug Removal Tapered amplifier removed from amplifier ta-ble. Pump light frequency shifted before am-plification2015 May Addition Sliding optical table constructed and in-stalled. Helmholtz sliding mount removed2015 May Replacement Removal of the old Helmholtz coils, installa-tion of new coils2015 Jul Addition Catalysis laser constructed and installed2015 Jul Addition Repump light coupled into a separate fiber2015 Nov Replacement Homebuilt detector removed, Thorlabs detec-tor installedTable 4.1: A table listing all the major changes which were done to the ex-periment. This includes changes to the MOT light production, distribu-tion, and alignment to the MOT and everything associated with the atomicbeam source and detection.30by the University of British Columbia (UBC) Chemistry technical staff and basedoff of an Even-Lavie design [27]. Our valve has an opening diameter of 0.25 mm.More details about the supersonic expansion of an atomic or molecular gas can befound in Section 2.2.1.The mounting assembly for this valve is a series of custom-built parts whichwere added progressively as the experiment was upgraded. Initially, the valve wasmounted on a plate which was directly attached to the same flange as the firstskimmer, but this restricted the nozzle to a fixed distance from the skimmer andprevented any alignments from being made to the nozzle’s position. An illustrationof the upgraded system can be seen in Figures 4.3 and 4.4. The nozzle is mountedon a counter-balanced plate suspended from a rotary feed-through and bellowsassembly. By adjusting the rotation of the feed-through and the expansion and tiltof the bellows the nozzle can be aligned to the skimmers. The tension on the gasfeed-through from the back of the chamber also affects the alignment of the nozzle.The skimmer leading out of the source chamber leads into the second chamberwhich contains the second collimation skimmer. This chamber is part of the vac-uum assembly due to a previous application of the experiment in which a Lymann-a laser required access to the atomic beam. This second, or “differential” chamber,remained a part of the setup for two reasons. First, it provided an additional dif-ferential pumping stage between the beam source and the MOT which providedsome protection for the MOT from a build-up of excess gas. Second, it provided alocation for any future upgrades to the beam line to be housed.Two gate valves isolate the third chamber from the rest of the beam line becausethis chamber houses the rubidium dispensers and the working volume for the MOTas well as providing a pathway for the atomic beam. On the outsides of the gatevalues (outside of this chamber) a set of bellows provided some isolation to thischamber from outside disturbances. The support assemblies for this chamber wereplaced directly on the optical table which necessitated noise isolation between thepumping systems and the optical table. This chamber was the only chamber whichwas not directly pumped on by a turbo pump and had it’s own ion pump to providea sufficient vacuum for the rubidium MOT to form.A 1 cm-wide aperture provides an escape for the atomic beam from the thirdchamber while preventing most of the gas from escaping back through the setup31once it collides with the back wall of the detector chamber. This chamber is thelast in the beam line and housed the RGA, Section 4.1.2) and the hotwire detector(Section 4.1.3. A view through a window in the back of the chamber can be seen inFigure 4.5. The atomic beam would interact with the RGA as it passed close to theRGA filament. The remainder of the beam would be evacuated from the chamberthrough the turbo pump located below the hotwire detector. The repetition rate ofthe experiment required that most of the gas introduced to this detection chamberbe pumped out before the next pulse was activated, therefore this turbo pump wasthe only one on the experiment which was backed by a smaller turbo pump andmechanical pump.The source, differential, and detection chambers had independent pressure gaugesattached with the MOT chamber using a readout from the ion pump for pressuremeasurements. Details on those pressure measurements can be found in SectionA.2.4.1.2 Residual Gas AnalyserIn order to make a measurement of the interaction between the atomic beam andthe MOT, an independent detector was needed to characterize the time-of-flightprofile of the beam. This would provide information about the velocity profile ofthe atoms released with a nozzle pulse.The primary detection method for the atomic beam was a RGA, a quadrupolemass spectrometer. For our experiment we used the SRS RGA200 from StanfordResearch Systems. A programmable feature of the RGA allowed us to operate it asa single gas monitor and operate continuously[28]. This provided us with the time-of-flight measurement necessary to detect the beam. More details on the operationof the RGA can be found in the RGA manual[28].4.1.3 The Hotwire DetectorThe hotwire detector is a partially built Langmuir-Taylor surface ionization detec-tor which is an alternative method of detecting rubidium in the detection chamber[29][30][31]. The components of the hotwire can be seen in the foreground of Fig-ure 4.5. It’s placement in the beam line is approximately 40 cm after the RGA along32Figure 4.1: Several pictures depicting the vacuum chambers for the beamline. Top: The source (left) and differential (right) pumping chambers.The rotary feedthrough can be seen on top of the source chamber. Bot-tom Left: The optical table with the glass MOT cell running through themiddle. Bottom Right: A gate valve leads to an ion gauge (left) and theRGA (right). The hotwire detector can be seen on the far right of theimage.333mm 2mm 10mm12cm 35cm 52cm 25cm 41cmNozzle Sup.IExp. Skimmer Skimmer Aperture RGAMOT HotwireTurboIPump IonIPumpGate GateFigure 4.2: Diagram of the atomic and molecular beam line. There are fourmain chambers separated by the skimmers and aperture. From left:beam source chamber, differential chamber, MOT cell, and detectionchamber. Gate valves on either end of the MOT chamber isolate sec-tions of the experiment to avoid breaking vacuum if work needs to bedone on the system. The source and differential chambers are backedby a common scroll pump. The detection chamber turbo is backed bya smaller turbo and roughing pump. There are no additional pumps onthe MOT chamber.the beam path.The hotwire detector was built with the hopes of measuring the ejected ru-bidium atoms after the beam collided with the MOT. Unfortunately, the distancesinvolved are too great and the number of ejected atoms too few to retrieve anymeasurable signal. The detector is capable of measuring the presence or rubidiumwhen the gate valve to the MOT is opened from the background vapour.This detector was constructed in December of 2013.4.2 Rubidium Magneto-Optical TrapThis section will provide details regarding the rubidium MOT and it’s associatedhardware and mechanics. This will include all details related to creating the trapas well as the imaging systems, motion mechanics, and characterization measure-ments.34Figure 4.3: A diagram of the nozzle mount assembly. Starting from the topdown: A rotary feedthrough provides a pivoting axis for the mount as-sembly. It is attached to a bellows to provide an additional translationand rotation axis. Extending down into the chamber is a vertical con-necting rod down to a horizontal mounting plate. Attached to this plateis the copper nozzle holder (right) and a counterweight (left). The noz-zle is inserted into holder and directed at the skimmer (far right). Thegas supply for the nozzle is supplied from the opposite side of the cham-ber from the skimmer (far left).4.2.1 Trapping LasersThe laser light that was used for the trapping lasers in this experiment was splitfrom a seed laser from what is known as the “master table.” This table is a part ofthe Madison Lab and seeds laser light to multiple experiments throughout severallabs at UBC.The generation of the necessary laser energies, as described in Section 3 wasdone by locking an external cavity diode laser to the necessary rubidium transitionusing the saturated absorption technique. A detailed procedure on how this tablewas constructed can be found at [32].The light generated by these master lasers was amplified twice before beingsent to the experiment table. The amplifying lasers are operated at a constant tem-35Figure 4.4: Three images depicting the nozzle mount setup photographedfrom below. Left: The nozzle is mounted in a copper mount whichis suspended from a mounting plate. It is directed at a skimmer alongthe chamber wall. Center: The mounting plate is connected to a shaftgoing through the roof of the vacuum chamber to the bellows and ro-tary feedthrough. Right: A steel gas line feeds through the back of thechamber supplying the nozzle with the beam gas.perature and with a stable current source which allows them to accept and replicatethe seed laser light. On either end of the second amplifier, known as the amplifiertable (Section 4.2.1), there were two 50 m polarization-maintaining optical fibersfor the pump and repump light. The length of the fibers is due to the experimentbeing on the opposite side of a building from the trapping light source.The sections below describe the experimental setups for the trapping laserswhich were constructed for this experiment.Amplifier TableThe light produced by the master table was both too weak and red-shifted by 180MHz to function as trapping lasers. This was by design as the laser light was splitfor use in multiple experiments. This meant that the light used for this work neededto be amplified and frequency tuned to the specific parameters necessary for ourapplication.Due to the distance between the master lasers and the experiment, a dedicatedtable for laser light amplification was constructed at a midway point along the36Figure 4.5: A view through the window at the end of the detection chamber.The RGA can be seen in front of the exit aperture. The glow is fromthe RGA filament. Seen in the foreground is the hotwire detector. Therhenium wire is stretched between the two circular pins on the top andbottom. To either side of the wire are the charged plates with the chan-neltron sitting in the left of the image.optical fiber. The table accepted both the pump and repump seed light into twoindependent amplifiers and recoupled the light into either one or two fibers afterthe amplification process. Several technical requirements were set for the lightreaching the experiment table. For pump light, an optical power range was setat 1-15 mW at a frequency between 5 and 15 MHz red detuned from resonance.Repump required less power with as little as 40 mW of on-resonant light.Both amplifiers used an MLD780-100 Infra-Red (IR) laser diode. Each laserdiode was contained inside a custom-built laser house which also contained a ther-moelectric pad and thermistor which regulated the temperature of the diode as wellas a protection circuit which regulated the voltage applied to the diode.The amplification setups for pump and repump differed in the order in which37Figure 4.6: The amplifier table as seen in February of 2014. The pump lightsystem is to the left and, at the time this photo was taken, was using atwo stage amplification process with a laser diode and tapered amplifier.The repump light system is on the right.the light had it’s frequency shifted and the optical power amplified. As seen in Fig-ures 4.7 and 4.8, the pump light is amplified after frequency shifting and repumpamplified before. This was the result of the power requirements set for the experi-ment table. By removing the power loss from the Acousto-Optic Modulator (AOM)double-pass from the amplified pump light, a greater total power could be sent tothe MOT at the price of losing control of the power being sent. This was determinedto be an acceptable arrangement as power control could be regained by placing anadditional AOM after the laser diode to dump power through the ±1st order.The amplifier went through two major iterations in the course of this work.The final iteration for the pump and repump amplifiers can be seen in Figures 4.7and 4.8. At the time of writing, both amplifier systems independently amplifyand frequency shift the pump and repump light to the desired optical power andfrequency. The two independent fibers carry light to the experiment table as seenin Figure 4.10.An earlier version of the amplifier table had a double amplification of the pumplight through a pre-amplifier constructed in a similar manner to the repump ampli-fier, but with the output fiber going to a Sacher S-780 tapered amplifier. This was38originally installed to provide an excess of power necessary for constructing a 2-dimensional MOT. A second AOM was also installed after the tapered amplifierwhich resulted in operational frequencies of -200 MHz for the pre-amplifier, at+283-290 MHz for the post-amplifier AOMs. The tapered amplifier was operatedat a low output power ( 230 mW) to be compatible with the experiment. The excesspower was lost due to a low transmission efficiency across the post-amplifier AOM.The use of independent fibers to send light to the experiment table was in-troduced in June of 2015. Previously, pump and repump light were coupled intoa common beam through a Glan-Thompson prism before the common beam wascoupled into a polarization maintaining fiber. Pump light was coupled on-axis andrepump-light was off-axis.Laser diode: MLD780-10082-87MHzλ/2To DiagnosticsTo MOTFrom MasterGlassλ/2Figure 4.7: The optical ensemble for the pump light amplifier. Light is car-ried to the table by a polarization-maintaining optical fiber before beingfrequency shifted (AOM, purple) and amplified. The frequency shift isperformed before the amplification to increase the optical power reach-ing the experiment table. After the optical isolator (yellow) a piece ofglass picks off a small fraction of the optical power for laser lockingdiagnostics.39From1MasterTo1MOTTo1Diagnostics90MHzλ/2O.I. Laser1diode:1MLD780-100λ/2Figure 4.8: The optical ensemble for the repump light amplifier. As with thepump light, the light is carried to and away from the amplifier by apolarization-maintaining optical fiber. Here, the light passes through anoptical isolator before being amplified and sent to an AOM to be shiftedto resonance. Reducing the power to the AOM lowers the transmissionefficiency of the AOM’s double pass, reducing the total optical powerreaching the MOT.MOT OpticsThe setup used for the trapping lasers follows the usual construction methods of aMOT[22][33][34]. An image of the optical table can be seen in Figure 4.9 and adiagram of the trapping laser setup can be seen in Figure 4.10. Pump and repumplight are brought to the table by two polarization-maintaining optical fibers. Therepump light does not depend on the direction in which it interacts with the trapand is introduced at the second polarizing beam-splitter cube (pol. cube). Thepump light first passes through a Glan-Thompson prism to remove any unwantedpolarization drifts that may have become part of the beam. The light then passesthrough a series of half-wave plates and pol. cubes split the light into three beams.Each beam passes through a quarter-wave plate and a telescope to circularly polar-ize and expand the light to an 8 mm diameter. After passing the MOT, each beamis retro-reflected back to the MOT using a lens, quarter-wave plate, and mirror tocomplete the optical portion of the trapping setup.40Figure 4.9: The MOT optical table as seen in September of 2015.4.2.2 Helmholtz CoilsA set of Helmholtz coils, in the anti-Helmholtz configuration, provided the mag-netic field gradient necessary to produce the Zeeman shift described in Section3.1.1. The field gradient provided by the coils follows the equationsBz = mI3DR2(D2+R2)5=2z+O2Br =−mI 32DR2(D2+R2)5=2r+O2(4.1)where D and R are the coil’s axial and radial distances from the MOT and z andr are the axial and radial field gradients [35]. In an ideal setup, the radial fieldgradient is half of the axial.For our setup, the positioning of the coils is limited by the atomic beam lineand the 30 cm tall MOT vacuum cell. The orange copper coils are visible in mostimages of the optical table in this thesis.41λ/2 λ/4λ/4λ/2λ/4Pol.5BSTo5verticalPumpRePumpλ/4 λ/4Cat5diag.ShutterGlassGTf=150mmFigure 4.10: The optical arrangement for the MOT (not to scale). The pumpand repump light is brought from the amplifier table and split into threeperpendicular trapping beams. The catalysis laser diode, and associ-ated optics and hardware, is located on, or near, the MOT table. Notpictured are the two imaging cameras for determining the location ofthe MOT.42SolenoidMagnetic Field LinesFigure 4.11: The field lines produced by a pair of Helmholtz coils in an anti-Helmholtz configuration. The field gradient along the axial directionis twice that of the field gradient in the radial direction. The MOT willreside in the exact center between the coils.First Coils: December 2012 to April 2015The first set of Helmholtz coils were a set acquired from the Madison Group atUBC when the experiment was first constructed. The exact turn count is unknownbut the estimates are at approximately 300 turns with a median radius of 37 mm.The size of the torus’s hole is 32 mm in diameter. The field gradient produced bythis pair of anti-Helmholtz coils is 15.2 GA·cm axially.Replacement Coils: May 2015 to January 2016The existing coils proved difficult to work with after the initial loss curve mea-surements were generated. The inability to observe both edges of the glass cellmade measuring the position of the MOT more difficult. In addition, the coils wereconstructed around a thick shell which added to their physical dimensions at theexperiment table.New Helmholtz coils were constructed to remedy both of these issues by hav-ing an inner diameter large enough to image the MOT more easily and a mounting43technique with a small physical footprint.These coils are identical in construction with an inner diameter of 62.8mmand an outer diameter of 106.6mm with a thickness of 26mm. The wire is wound17x14 times with an extra 6 turns due to packing. The 17 turns are at a constantradius and there is a thickness of 14 rows. The wire used is a 1.5mm gauge coilwire from Essex. The coils are held together by a thermally conductive resin (partnumber: Duralco 132-IP from the Cotronics Corporation). Simulations of this coilconfiguration give an axial field gradient of 13.27 GA·cm .Figure 4.12: Helmholtz coils which were constructed and installed to the ex-periment table in May 2015. The compact design and large inner di-ameter allow the cameras to image the MOT and both edges of the cellto use as reference edges.4.2.3 Rubidium SourceRubidium was released into the chamber by passing electrical current through arubidium dispenser installed on the inside of the glass vacuum cell as seen in Figure4.13. The rubidium dispenser that we used was an RB/NF/7/25 FT10+10 fromSAES Getters.During operation, a constant current was passed through the dispenser whichreleased 85Rb and 87Rb into the glass cell at a fairly consistent rate. The continu-ous operation of the rubidium source was because of the need to maintain a high44pumping speed on the experiment. Shutting off the dispensers would cause thebackground rubidium pressure to deteriorate below a level needed to form a MOTwithin a minute.Figure 4.13: The rubidium dispenser inside the glass vacuum cell approxi-mately 4 cm from where the MOT is formed. The rubidium is on theend of the ‘V’-shaped clip attached to the copper rods. The copperrods carry the electrical current which causes the dispenser to heat upand release atoms of rubidium into the cell.4.2.4 Trap Fluorescence ImagingThere are several cameras and detectors to both measure and assist in aligning theMOT. This section will detail the three optical devices for this data taking andalignment making.PhotodiodesThe first instrument for measuring the fluorescence from the MOT is a homebuiltphotodiode assembly with an internal amplifier and a removable telescope attach-ment (Figure 4.15). An end piece was attached to the end of the telescope to blockreflections of the trapping beams from reaching the detector. Set to the lowest gainsetting, this detection system could measure the fluorescence from a MOT with up45Figure 4.14: Left:MOT position imaging camera (Section 4.2.4) pointed downthrough the Hemlholtz coils. An ND filter is placed directly after thecamera lens. Upper Right: Alignment camera (Section 4.2.4) pointedat the MOT opposite the photodiode. Lower Right: The distance cali-bration card for MOT position measurements (Appendix D).Figure 4.15: Left: The photodiode assembly is directed between theHelmholtz coils at the MOT. Right: Two views of the photodiode dis-assembled. A Canadian $2 coin is included for scale.46to 1010 rubidium atoms. This detector was mounted on a modified optical mirrormount. This detector was in operation from the start of the experiment until theend of October of 2015.The second detector, which replaced the homebuilt model in November of2015, is a Thorlabs DET100A/M Si biased detector with a 1 cm diameter visi-ble detection surface and is featured in Figure 4.16. For this detector a StanfordResearch Systems Model SR570 low-noise current preamplifier is used to amplifythe detector’s output signal. For all measurements made with this detector, theamplifier was set to provide no offset, no filter, and a sensitivity of 2x100 nA/V.The calibration procedures and figures of these detectors can be found in Ap-pendix A.These detectors had several issues with signal pickup and offset that had tobe addressed. Three of the most common frequencies that were picked up by thedetectors were at approximately 600, 2100, and 4100 Hz. The 600 Hz signal wascaused by a floating ground related to the rack-mounted power supply. 4100 Hz isgenerated by the power supply which supplies the nozzle with current. The 2100Hz signal appears on the MOT and at the time of writing is of unknown origins. TheDirect Current (DC) offset was a real signal that was detected by the camera, butwas due to reflected beams reaching the photodiode and pushing the signal reachingthe oscilloscope above a 1 volt limit on recording higher-accuracy measurements.The nose cone on the end of the telescope reduces this offset. For the homebuiltdetector, an op-amp circuit was implemented to offset and amplify the signal fromthe built-in amplifier on the detector. The Thorlabs detector did not require anyamplification beyond the setup described above.Data Acquisition HardwareWith the exception of the MOT position images, all the data collected throughoutthe work of this thesis was done through a Tektronix MSO 4034 oscilloscope.This included all signals generated by the MOT detectors and amplifiers as well asthe signal from the RGA and any diagnostic signals needed for characterization orconstruction of the apparatus.Data was transferred off of the oscilloscope in two manners. Early measure-47Figure 4.16: The photodiode assembly which replaced the homebuilt detec-tor. The telescope is the same one which was used on the homebuiltdetector with no changes to the distances or the 1 inch lens inside.ments, such as those with the sliding coils (Section 5.2.2) or the detector charac-terization (Appendix A) were done with a built-in direct-to-USB save function. Inthese instances, averaging was done inside the oscilloscope. For the table transla-tion results (Section 5.2.3) and the catalysis measurements (Section 5.3) the oscil-loscope was connected to a computer through an Ethernet connection and a Pythonscript was written to communicate with and acquire data from the oscilloscope.Position Imaging CameraHaving a translatable MOT required the position to be measured. Estimates couldbe calculated based on the motion of the translation mechanism, whether it wasthe coil translation or the full table, but a direct measurement would be required toretrieve the exact position of the MOT. During the installation of the coil translationstage (Section 4.2.5) a vertically-mounted Sony ExwaveHAD B&W video camerawas installed which was pointed down through the center of the coils. This locationwas chosen because the curved glass prevented shallow-angle imaging and lookingdown the beam axis was not possible due to the RGA obstructing the view towards48the MOT from the detection chamber window.Measuring the MOT at this angle had several issues that were addressed. Asshown in Figure 4.14 the vertical camera was mounted with a slight angle in orderto see past the mirror for the axial trapping beam. In addition to this, the axialbeam scattered a significant amount of light off of the glass which saturated a largeportion of the camera. Our solution to this was to place a Neutral Density (ND)filter immediately after the camera lens to cut down on the incoming light. Thescattering of this light off of the glass also highlighted small imperfections in theglass which appeared in the captured image as “MOT-like” objects which werecorrected for during image processing as described in Appendix B.The translation stage-mounted coils provided an additional challenge whichwas eventually corrected for with the coil redesign (Section 4.2.2). The narrowinner diameter of the coil restricted visibility of the glass cell such that only a singlewall was visible to any single measurement. There was also no guarantee that theentire 5 mm thick wall was captured in the frame. To provide a reference a papercard was placed below the cell and imaged before each MOT position measurement.An image of the MOT could then be displayed over this image to give the exactposition of the MOT within the cell. The driving force behind the coil redesignwas to increase the inner diameter of the Helmholtz coil to be large enough suchthat the camera could image both outer edges of the 20 mm-wide vacuum cell andremove the need for the alignment card.Alignment CameraA camera set opposite the photodiode provided a useful alignment tool as well as amechanism to image the vertical position of the MOT. The camera we are using isa Watec WAT-120+. This camera is a basic Charge-Coupled Device (CCD) camerawhich picks up 780 nm particularly well. This camera was a vast improvementover the IR viewers used previously on the experiment. Since all measurementswere made at the same vertical position, any measurements made by this camerawere to confirm that the MOT had not moved vertically between measurements.494.2.5 MOT Translation MechanicsTo demonstrate the effectiveness of the rubidium MOT as a beam profiler, a mech-anism for moving the position of the MOT relative to the beam had to be imple-mented. In addition, the trap depth of the MOT had to be maintained as to provideconsistent conditions for the experiment. The decision to move the MOT instead ofthe atomic beam was due to the large mass of the vacuum chambers and the threeturbo pumps attached to those chambers.As a prototype for the experiment, compensation coils were installed to thesystem to push and pull the MOT into different positions and can be seen in Figure4.17. While effective in nudging the MOT into different positions, this techniquedid not provide a means of measuring the distance the center of the trap had moved,nor could we guarantee a consistent measurement of the trap depth. It should benoted that during the time these coils were a part of the system, the vertically-mounted camera had not been installed.For future upgrades to the system, the decision was made to translate the MOTin the plane of the table perpendicular to the direction of travel of the beam. Thecompeting option was a vertical translation, but that was ruled out due to the diffi-culty of moving either the MOT or the beam in that direction. The MOT was limitedin that direction because the optical table was providing structural support for theatomic beam. It would have been possible to move the beam, but turning mountsupport nuts and counting the threads was determined to not be an accurate methodof measuring distance.Coil TranslationThe first design for a MOT translation mechanism was for only the Helmholtz coilsto be translated. This would have the effect of moving the magnetic focus of thecoils upon which the trapping lasers would load atoms.The design of the translation stage was a sliding aluminium plate guided by aflat edge mounted on the breadboard. The position was set by a translation stagewhich was adjusted with a micrometer. The construction can be seen in Figure4.18. This design allowed the coils to slide along an aluminium-aluminium inter-face. The 1/4-inch coil mounting plate allowed for a tapped hole for the thread50Figure 4.17: An early version of the experiment used individual coils to per-turb the trapping magnetic field and adjust the location of the MOT.The coils used, shown to the top and to the back of the MOT in the im-age, were large rectangular loops of unknown turns. These coils werenever used outside of proof of concept scenarios as the uniformity ofthe MOT was disturbed during operation.support rods while being thin enough to fit between the breadboard and the glassMOT cell. At this time, the vertically-mounted camera was installed to image theposition of the MOT within the glass chamber.Translating Optical TableTo provide a consistent environment for the MOT during translation, a fully translat-able optical table was designed to translate the Helmholtz coils, the trapping lasers,the catalysis laser, and the MOT imaging systems. Due to the manner in which theexperiment was constructed, the existing optical table was providing structural sup-port for the MOT and detection chambers. Because of this it was determined thatthe table redesign would require the existing table to remain in place.A set of images of the sliding optical table can be seen in Figure 4.19. Thelargest piece of the redesigned breadboard consists of two 3/8 inch thick half-tables51Figure 4.18: The MOT with the Helmholtz coils attached to a translationstage. This system moved the magnetic center of the trap, but leftthe optical components stationary.joined at the middle by connector plates. The two halves have cut-outs for themounting structures of the beam as well as holes for the trapping laser and coilwires. The tapped holes are standard 1/4-20 tapped holes with 1-inch spacing in asquare pattern. One of the table halves is 4 inches shorter than the other to allowspace for the translation mechanism. The two table halves were cut to includeoverhang sections to mount a table which hangs below the main optical table. Thislower table houses the retroreflection optics for the axial (vertical) trapping laser.Physical limitations of the setup necessitated a suitably thin translation mech-anism. A threaded rod system acted as a push and pull force on the upper opticaltable using the lower (original) table as an anchor. To reduce the force of frictionbetween the two tables, Polytetrafluoroethylene (PTFE) pads were adhered to thetop of the existing breadboard and the bottom of the new breadboard. This low-ered the coefficient of friction between the two breadboards while remaining thinenough to fit under the atomic beam and the glass cell for the MOT. During theinstallation of the new table the atomic beam, all four chambers associated withthe atomic beam had to be raised by a 1/2 inch.The construction and installation of this table happened concurrently with the52Figure 4.19: Clockwise from upper left: (1) The first half of the optical ta-ble containing the pump and repump fibers, optical telescopes, and theimaging camera and photodiode. (2) The second half of the opticaltable containing the catalysis laser, alignment camera, and the retrore-flection optics for the two horizontal trapping beams. (3) The under-carriage mounted to the sliding portion of the main table. This containsthe retroreflection optics for the axial beam. (4) One of two manipu-lators which moves the sliding table. The upper and lower tables areseparated by a series of PTFE pads.construction of the second set of Helmholtz coils (Section 4.2.2) and the catalysislaser (Section 4.2.6)4.2.6 Catalysis Laser for Trap Depth MeasurementThe catalysis laser, used for measuring the trap depth of the MOT as described inSection 3.4 and [26], was set-up using an Eagleyard Photonics EYP-DFB-0780-00080-1500-TOC03-000x laser diode contained within a home-built laser houseand mounted directly to the sliding optical table. A diagram of the set-up can befound in Figure 4.10.Inside the laser house, the laser diode is mounted to an apparatus containinga collimating lens. The collimation set-up was then mounted to the base of the53house which allowed for manual adjustments to the positioning of the laser withinthe house. The laser diode we were using had an on-board thermoelectric andthermistor which provided sufficient thermal management of the laser diode. Thisthermal management technique was conditional on the environment surroundingthe laser diode remained calm. To achieve this, the lid on the laser house wassealed with a sponge-like trim to reduce airflow. A Brewster’s window allowedthe laser light to pass to the outside of the house with minimal losses. Imagesof the catalysis laser with the roof on and off can be seen in Figure 4.20. As anadditional precaution to protect the laser diode, a protection circuit identical to theone described in Section 4.2.1 was included in the design.Figure 4.20: Left: The insides of the catalysis laser. An Eagleyard Photon-ics EYP-DFB-0780-00080-1500-TOC03-000x laser diode is mountedwithin the teal ceramic mount in the center of the image. The powerleads to the laser-diode pass through a protection circuit, designed bythe Madison group, to regulate the voltage across the diode. Right: Thelaser housing is sealed with sponge-like double-sided tape to insulatebetween the roof and the walls and a Brewster’s window to efficientlytransmit laser light to the outside of the house.The laser diode and associated collimating optics produced an approximatelyelliptical beam with a major axis of 5 mm and a minor axis of 3 mm. This beamwas focused immediately before the MOT with a 150 mm focusing lens which, bythe equationw1 =lpFw0(4.2)54produces a beam which has a major axis of 12.4 mm and a minor axis of 7.44 mmat the MOT [36]. At a 40 mW laser power, this gives us an intensity at the MOTof 1.37x108 W·m-2. In this equation, l is the wavelength of the laser, F the focaldistance of the lens, and w0 the beam waist of the incident laser.55Chapter 5Atomic Beam-MOT CollisionsThe measurements which were made in this experiment can be split into threephases with the first two providing a proof of concept for the flux measurementand profiling technique. The third phase contains the most accurate and most sup-ported version of the measurement. This third phase is associated with the catalysismeasurements in Section 5.3.5.1 Collisional LossesThe measurements taking in this experiment were of the MOT fluorescence overtime as the MOT was bombarded by a time and density dependent atomic, or molec-ular, beam. For the purposes of this discussion, the term “loss event” or “collisionevent” will be in reference to the period of time in which the atomic beam is passingthrough the MOT.The model which describes the behaviour of the trap’s population is given inEquation 3.20. This can be separated into two components: the parameters deter-mined independent of the beam and those dependent on the beam. The independentparameters, the loading rate R and background loss constant G can be determinedoutside of the measurements presented in this chapter. This is done using trap load-ing measurements as described in Section 3.1.4, Figure 3.3, with fitting proceduresperformed according to Equation 3.19. What is left is the 〈sv〉nbeam(r; t) whichbecomes the focus of the experiments and results outlined in this chapter.56From the experimental setup there are two measurements which can be madeduring a loss event. The photodiode provides a real-time measurement of the MOTin the form of a voltage signal, VMOT (t), which is proportional to the trap popula-tion. The RGA supplies the second measurement which gives the local density ofthe beam as it arrives at the RGA. This is also recorded as a voltage, VRGA(t). Itbecomes useful then to rewrite Equation 3.20 in terms of these measured signals asdVdt= RV −GV −TP(r)VRGA(t)V: (5.1)Here, G remains as it was from Equation 3.20 as it’s a rate and independent of scale.RV is the loading rate into the trap with the units of [Vs−1]. This leaves the lossterm from interactions with the beam. VRGA(t) has been presented already and isproportional to the time dependence of the density of the beam, nbeam(r; t) . TP(r)is a new parameter which represents the elastic collision loss rate where the MOTintersects the beam. The MOT will respond to the beam by losing atoms, the rate ofwhich depends on the local density of the beam and the interaction cross section.This parameter TP(r) carries information about that rate, but also retains a spatialdependence since the beam is not spatially uniform and will vary depending on theposition of the MOT.The loss rates due to the beam in Equations 3.20 and 5.1 represent the exactsame quantity. As such, they are equivalent to one another givingTP(r)VRGA(t) = 〈sv〉nbeam(r; t): (5.2)This provides the ansatz of this measurement. TP(r)VRGA(t) is a measurable quan-tity which can be determined through loss measurements similar to the one shownin Figure 5.1. The signal from the MOT is measured as the beam passes over it whilethe time-dependent signal profile of the beam is recorded by the RGA. Through anumerical solution to the ODE in Equation 5.1 performed by the Python code shownin Appendix F, an appropriate value for TP is determined for the MOT at positionr. By knowing TP(r)VRGA(t) and one of 〈sv〉 or nbeam(r; t), the unknown quantitycan be determined.There is one final calculation which must be made to complete the measure-ments which were made. The unknown value in this process is the scattering cross57section 〈sv〉 but the values estimated for nbeam(r; t) do not contain a time depen-dence. As shown in Appendix A, the beam was measured by determining the totalnumber of atoms passing through the MOT cell every time the nozzle fired. Assuch, the final calculation to determine the scattering cross section isTP(r)∫VRGA(t)dt = 〈sv〉∫nbeam(r; t)dt = 〈sv〉MBnbeam(r) (5.3)where nbeam(r) is the average density of the beam as it passes over the RGA. Fromthe results in this chapter and the beam atom count in Appendix A, a collisionalcross section s v¯ using the average velocity will be estimated for the system.5.1.1 Measurement ProcedureThe procedure for measuring a loss event was designed around performing an ac-curate measurement with sufficient averaging as well as working within the ex-perimental constraints of the apparatus. At the start of each measurement the trapbegins fully loaded and the MOT cell clear of all gas with the exception of the back-ground rubidium vapour. The nozzle (Sections 2.2 and 4.1.1) is pulsed with a totalon time of 50 ms to initiate the atomic beam. The beam travels approximately 99cm through two skimmers before interacting with the trap and passing through theexit aperture and recorded by the RGA. After this event, the experiment needs tobe reset for the next measurement to be taken by removing the excess gas from theMOT cell and reloading the MOT. This reset takes approximately 2 seconds anddefines the duty cycle of the experiment. The trap reloading time was determinedfrom the loading curves described in Section 3.1.4. Figure 5.1 is an example of asingle collision event measurement.The fluorescence from the MOT was measured with a homebuilt silicon photo-diode and amplifier setup for the initial test and coil translation measurements anda Thorlabs DET100A/M and an SRS SR570 current preamplifier. Details aboutthe detectors can be found in Section 4.2.4. The calibration information for thedetector can be found in Appendix A. In Figure 5.1 the MOT signal is shown as ablack curve. The red trace also represents the population in the trap but is the resultof applying a Butterworth filter to the MOT signal.The atomic beam was measured with the RGA and appears as the blue trace58in Figure 5.1. A description of the RGA can be found in Section 4.1.2 and thecalibration procedure and values in Appendix A. The measured peak is a Time ofFlight (TOF) measurement of the atomic beam measured at a point approximately165 cm from it’s source at the nozzle. As a result of this, the position of the atomicbeam was shifted in time by a factor of 99/165 to align the beam position with theinteraction event at the MOT.0 1 2 3 4Time [ms]7.627.647.667.687.707.727.747.767.787.80Trap Pop.1e7Rb PopulationNumerical FitAtom Flux0.50.00.51.01.52.02.5Atom Flux [Atoms/sec]1e16MOT Loses from Collisions with Ar BeamFigure 5.1: A graph showing an individual loss curve. The MOT data (black)is first put through a filter (red) before being fit (green) to the loadingequation, Equation 3.20, with the perturbing particle intensity profilebeing supplied by the atom signal from the RGA (blue).5.2 Profiling an Atomic BeamBy measuring the losses from the trap at different points across the beam path thespatial density profile of the beam can be measured. In addition to this, the in-teraction cross section can be measured and estimated from the peak intensity ofthe beam. There were three iterations of this experiment which were performedwith improvements to the apparatus being made in between. The largest improve-ments were made in the translation mechanisms (Section 4.2.5), imaging systems59(Appendix B) and alignment practices (Appendix D).5.2.1 Initial Proof of ConceptThe earliest observations and predictions that a MOT could be used to profile anatomic beam were made before the beam profiling experiment being formallystarted. These measurements were also the only ones made using the compensa-tion coils to adjust the position of the MOT. In these measurements, the effect of thebeam on the MOT could be altered by increasing the current to the compensationcoils and moving the MOT around the trap. Based on several rough calculationsshown in Section 2.2.2, it was predicted that it was possible to completely pro-file the atomic beam within the existing vacuum cell housing the MOT providedthat the MOT could be translated across the beam. Unfortunately, the data fromthese early measurements was unable to be located. There were also no positionmeasurements made of the MOT beyond what current was supplied to the coils.A formal proof of concept would require a more direct method in which toalter the position of the MOT. This resulted in the earliest modification to the setupwhich saw the replacement of the compensation coils with a translating stage forthe Helmholtz coils. This would directly move the magnetic zero of the system andnot “tug” on the magnetic field as the compensation coils had done. The positionof the coils could also provide a rough measurement of the position of the MOT.This setup is shown in detail in Section 4.2.5 and in Figure 4.18. Only a few pointswere taken, but the results of this proof of concept measurement can be seen inFigure 5.2.From this result, it was somewhat clear that it was possible to measure the pro-file of the atomic beam. The losses from the trap were observed to be under 1% ofthe total atoms in the trap, but the outline of a beam can be seen forming. This mea-surement would also come with several caveats based on the methodology chosen.Without a direct measurement of the MOT’s position and instead recording the po-sition of the coils with the translation stage’s micrometer, the exact position of theMOT is still unknown. The low loss fraction observed from the beam also increasedthe uncertainty in the voltage difference measured due to signal noise. Addition-ally, the parameters governing the MOT had not been optimized for signal stability60Figure 5.2: An early measurement of an atomic beam profile. This measure-ment took a few points across the MOT and produced several measure-ments similar to Figure 5.1 and compared the fraction of atoms lost dueto collisions with the beam. The profile of the beam is partially visiblewith only the few points available.but instead for total fluorescence signal and maximum population. Improvementswould need to be made to move beyond these early measurements.5.2.2 Coil Translation ResultsFollowing the initial proof of concept several modifications were made to achievemore accurate results. The inclusion of a vertically mounted camera, as shown inSection 4.2.4) allowed for an exact measurement of the position of the MOT withinthe vacuum cell. Details on how this measurement was performed can be found inAppendix B. The parameters governing the MOT were also adjusted to stabilize thepopulation. Another major addition to the experiment was the introduction of thefitting code shown in Appendix F which could plot loss curves like the one shownin Figure 5.1 and extract TP(r). A profile was taken using this new setup and canbe seen in Figure 5.3.From the results shown in Figure 5.3 there is a profile outlined between -1mm and 7 mm from the center of the cell. The lopsided nature of this result maybe due to a misalignment of the atomic beam as the methods used for aligning the614 2 0 2 4 6 8MOT Position [mm] (from cell center)0100200300400500600700800900TP [V−1s−1]Beam interaction constant TPmeasured across the cellFigure 5.3: A profile of the loss parameter TP used in the fitting equation asit varies in space. The lopsided nature of this measurement was due inpart to a misalignment of the atomic beam.skimmers and the exit aperture are not exact. More details on the alignment processcan be found in Appendix D.One particular criticism of the data taken for this plot is it does not have ameasure of the trap depth of the MOT. This is of particular concern as the 1.6cm diameter trapping beams may have problems loading and holding the trap dueto the proximity of the trap to the glass wall of the vacuum cell. An additionalproblem arose during the measurement process from the alignment of the trappinglasers. Having the Helmholtz coils move, but the optics stationary misaligned theMOT which then required realignment between measurements. Although the trapwould hopefully be very similar between measurements, this assumption is notnecessarily true. Both these hypotheses cannot be confirmed as the experiment hadbeen modified by the time the catalysis laser was in full operation. These inconsis-tencies, along with the misalignment of the atomic beam, leads to the conclusionthat we can profile an atomic beam with a MOT, but improvements need to be madebefore accurate information about the beam can be extracted from the experiment.More details on the coil translation results can be found in Appendix C.625.2.3 Table Translation ResultsAfter the results discussed in Section 5.2.2 had been analysed several changes weremade to the experiment to improve upon the procedure. The goals of those changeswas to provide a more reliable method to translate the MOT relative to the beam,provide a more accurate measure of the MOT’s position, and to develop a system tomeasure the depth of the MOT in order to compare with theoretical calculations fortrap loss.The most extensive change came with the implementation of the sliding opticaltable in order to preserve the optical alignments of the MOT between measurements.This table is described in Section 4.2.5. In parallel with the table changes, theupgraded Helmholtz coils described in Section 4.2.2 were constructed and installedto provide a better field of view for the position measurement camera. Details aboutthe upgraded camera images can be found in Appendix B. The catalysis laser wasalso built and installed prior to the next set of measurements. The background onthe measurement can be found in Section 3.4, the construction and setup details inSection 4.2.6 and the results found in Section 5.3. The MOT cell along with thedetection chamber were also shifted to align the beam more to the center of theMOT cell.With the changes in place, several atomic and molecular beams were profiledby measuring the collisional losses from the MOT. The technical parameters for theMOT used for these measurements can be found in Table 5.1. In total there were 5gases which were passed as beams through the setup and one gas (SF6) withdrawnfrom the experiment due to technical difficulties with producing an atomic beamfrom this species.The results shown for argon in Figure 5.7 provides a much clearer image of theatomic beam than what was shown in Figure 5.3 from Section 5.2.2. The differ-ence in the magnitude TP(r) is due to several factors. The 10 Hz rep rate pressuremeasurements made for the beam are the same across both measurements due tothe skimmers remaining in alignment which keeps the atom count passing by theMOT relatively steady. The integrated RGA signals from the coil translation resultsare greater than the table translation results by a factor of 1.40 which accountsfor some of the discrepancy. This could be caused by changes to the beam as is63Parameter Value UnitTrap Laser Detuning -6.0 MHzPower of Horizontal Trap Lasers 410 mWPower of Vertical Trap Laser 800 mWRepump Laser Power 180 mWTrapping Laser Beam Diameter 8.0 mmHemlholtz Coil Current 1.36 ATable 5.1: The parameters used for the MOT for the duration of the measure-ments made in November of 2015. The results of the measurements madewith these parameters are found in Section 5.2.3passes through the exit aperture or changes to the ionization efficiency of the RGA.The relative uncertainties of the measurements are also in question with the data inFigure 5.3 showing much larger relative uncertainties compared to 5.7.6 4 2 0 2 4 6 8MOT position [mm] from center0200400600800Loss Rate TP [V−1s−1]H2  Loss MeasurementsFigure 5.4: The flux profile of an atomic beam of hydrogen made with theMOT on a translating optical table.What is clear from the five loss curves and shown again in Figure 5.9 is theconsistency of the beam’s profile across all measurements. Calculations made in646 4 2 0 2 4 6 8MOT position [mm] from center01000200030004000Loss Rate TP [V−1s−1]He Loss MeasurementsFigure 5.5: The flux profile of an atomic beam of helium made with the MOTon a translating optical table.6 4 2 0 2 4 6 8MOT position [mm] from center050010001500Loss Rate TP [V−1s−1]N2  Loss MeasurementsFigure 5.6: The flux profile of an atomic beam of nitrogen made with theMOT on a translating optical table.656 4 2 0 2 4 6 8MOT position [mm] from center05001000150020002500Loss Rate TP [V−1s−1]Ar Loss MeasurementsFigure 5.7: The flux profile of an atomic beam of argon made with the MOTon a translating optical table. These results show a far more definedbeam than the one measured and shown in Figure 5.3.6 4 2 0 2 4 6 8MOT position [mm] from center010002000300040005000Loss Rate TP [V−1s−1]Kr Loss MeasurementsFigure 5.8: The flux profile of an atomic beam of krypton made with the MOTon a translating optical table.66Section 2.2.2 expected a beam width at it’s maximum extent to be 8.0 mm. All fiveof the plots have an outer width of at least 8 mm with some profiles as wide as 10mm (argon). These are all very acceptable beam widths which are very consistentin width and peak position at 1.5 mm. Not taken into account in these profiles is thefinite size of the MOT which can be about 1 mm wide which effectively measuresa convolution of the MOT with the beam.Also visible in each of the profiles is a plateau forming at the center of thebeam. This is starting to get a bit rough due to the distances travelled before theprofiling was performed, but they still remain partially intact in the profiles.6 4 2 0 2 4 6 8Position in the Cell [mm]010002000300040005000Normalized Loss Rate TP∗ [s−1]Normalized Loss Rates For All GasesH2HeN2ArKrFigure 5.9: A combination plot of all the beam profiling measurements. Thisshows a consistent profile outlined through all measurements.From the data collected for the 5 profile figures a collisional cross section canbe estimated. To do this, there are two values which need to be included with TP(r).The integrated beam density term nbeam is the number of atoms passing through anarea in the measurement time. It is found by taking the total pulse atom countfrom Appendix A and assumes it is uniformly distributed about a circular beamwith width 6±0.2 mm determined from the profile curves. ∫ VRGAdt comes fromthe average integration of the RGA traces used in a profile. Since we are interested67Gas nbeam TP∫VRGAdt 〈sv〉MBUnits [m-3s] [V-1s-1] [Vs] [m3s-1]H2 1.64(34)x1017 593(86) 2.93(18)x10-5 1.06(27)x10-19He 2.17(12)x1017 3733(239) 8.65(64)x10-6 1.49(17)x10-19N2 2.95(18)x1016 1443(175) 1.11(19)x10-5 5.4(12)x10-19Ar 5.40(31)x1016 2232(148) 2.30(52)x10-5 9.5(23)x10-19Kr 3.07(18)x1016 3740(173) 1.21(5)x10-5 1.47(13)x10-18Table 5.2: The measured values for the collisional cross sections for the fivecollision and profile experiments made. The beam density was calculatedfrom the total atoms in a pulse (Appendix A contained in a beam width of6(0.2) mm. The collisional cross section was found from solving Equa-tion 5.2 with the necessary integrals on VRGA and nbeam(r; t) made.in the peak beam density, a sample of TP values at positions around 1.5 mm weretaken. The results of this calculation can be found in Table 5.2.From the values given, the larger and heavier the noble gas, the greater thevelocity averaged cross section, 〈sv〉 ,that was measured. The nitrogen and hy-drogen values also follow a similar pattern with greater collisional cross sectionscompared to the noble gases with similar mass. It is clear that patterns are emerg-ing in the velocity averaged collisional cross sections measured in this experiment.The values determined here will hopefully be put to the test with simulations of thecross sections being done in tandem with this experiment.Atomic Beam Velocity DistributionsThe measurements made by the RGA produced a TOF profile of the beam whichcontains information regarding the velocity distribution of the particles in the beam.In this section a fitting function is presented for the RGA signal along with theconversion necessary for producing a velocity profile.The function which best fits the RGA was determined through trial-and-errorwith several of the functions tested shown in Appendix E. The function which waschosen is given by68Gas g t0H2 0.000055 0.00071He 0.0001 0.001006N2 0.00016 0.00217Ar 0.00018 0.002865Kr 0.00043 0.0040Table 5.3: The fitting values for the RGA signal as per Equation 5.4.f (t) =1pg[1+( t−t0g)2](arctan((t− t0)=g)p +0:5): (5.4)This function is a combination of a Lorentz distribution with an arctangent includedto adjust the fit to the beam. It should be noted that this function is not normalized.The fitting parameters t0, representing the time offset of the distribution, and g , thewidth of the function, for the RGA signal from each gas is given in Table 5.3. Aplot of each of these functions normalized can be found in Figure 5.10.Every position in Figure 5.10 corresponds to an arrival of a group of atoms ormolecules at the RGA. Since the travelled distance is known, this information canbe converted to show the velocity distribution of the beam by redefining the x-axisasv =1:65tarrival: (5.5)where tarrival is the arrival time of a particular atom or molecule and v is that parti-cle’s velocity. 1:65 is the total distance from the nozzle to the RGA and is measuredin meters. Figure 5.11 shows the velocity distributions of the five beams. The in-formation was determined numerically by maintaining the arrays used to generatethe plots in Figure 5.10 and applying Equation 5.5 to the array representing thetime points.690.000 0.001 0.002 0.003 0.004 0.005 0.006Time [t]Normalized DistributionBeam Arrival Time at RGAH2HeN2ArKrFigure 5.10: The five functions which fit the time of flight profiles for eachof the five beams. Equation 5.4 with the values in Table 5.3 was usedto produce the plots.5.3 Trap Depth MeasurementsOne of the parameters characterizing an atomic trap is the energy required to re-move an atom from the trapped population. This is known as the “trap depth.” Thedeeper the trap, the greater the energy required to remove an atom.This quantity is of particular relevance to this experiment as the mechanismby which our atomic beam is measured is through the removal of atoms from ourtrap. The quantity for trap depth, Utrap presents itself in Equation 2.2 where it playsa role in determining the loss-production scattering cross-section. Therefore, thisquantity must be measured as part of this experiment to connect our results to thetheoretical calculations for the scattering cross-section.Measurements of the trap depth also play a role in determining the consistencyof the beam profile measurement points as a check to confirm that the trap depthdoes not change with the position of the MOT within the cell. This concern is ofparticular relevance to the results in Section 5.2.2 where the inch-diameter trapping700 500 1000 1500 2000 2500 3000Velocity [m/s]Normalized DistributionBeam Velocity DistributionH2  2276 m/sHe 1596 m/sN2  745 m/sAr 566 m/sKr 400 m/sFigure 5.11: The velocity distributions of the five atomic beams used in ex-periments for this thesis. Listed in the legend are the most commonvelocities for each gas.lasers may have resulted in an overlap of the trapping area and the cell wall. Forsubsequent beam profiles the trapping lasers were reduced to one centimetre in di-ameter, but as a check to confirm that the trap depth remains consistent, a catalysismeasurement can be taken at the center of the cell and near one of the walls toconfirm that the trap depth does not change.If we consider the unperturbed steady state MOT population to be defined byEquation 3.19 and the steady state population of the MOT after the application ofthe catalysis laser to be defined by Equation 3.23 we can define an observable asNsscatNss=RG+bn+bcatnG+bnR=G+bnG+bn+bcatn: (5.6)By measuring the steady state population of the trap in both instances and takingthis ratio, any effect the catalysis measurement has on the trap will be reflected bya noticeable peak in the data.71The process of making a single measurement is shown in Figure 5.12. The traphelmholtz coils are switched off and the catalysis light blocked at the start of themeasurement to reset the trap. After one second the Helmholtz coils are switchedon and the trap loaded. This gives us our value for Nss. At the 6 second mark inthis measurement the shutter blocking the catalysis laser is removed and the laserallowed to interact with the MOT. The MOT is allowed some time to reach Nsscatbefore the coils are switched off and the trap reset. The difference between thecoils on and the coils off gives us the measured value for Nsscat .The measurements performed with this technique and discussed in this thesissupport the results described in Section 5.2.3From the measurements taken there are two key results which were found.Figures 5.13 and 5.14 show the results of two scans of the catalysis laser wherethe effect of the catalysis laser on the MOT population was measured for variousdetunings. Both graphs show a peak at 2D = 57GHz which, by Ttra = Utrap=kB,gives us a trap depth of 1.82 K. Having the same trap depth at both the center ofthe trap and at a position 2.5 mm from a wall shows that the trap remains constantthroughout all of our beam profiling measurements as this is the closest any ofthose measurements gets to the walls of the vacuum cell.The data that we have collected shows what appears to be a very jagged result.This is an artefact of the measurement process in which there was a drift in thelasers or the MOT. This is clearly seen in Figure 5.13 with the upper and lower datagroupings.720 2 4 6 8 10Time [s]0.340.360.380.400.420.440.460.480.50Signal [V]Catalysis Single MeasurementFigure 5.12: A single measurement of the catalysis process in which the ef-fect of the catalysis laser very apparent. A fully loaded trap (0.5s to 6s)is struck by the catalysis laser (on at 6s) and is allowed to decay to anew equilibrium population given by Equation 3.23. At 9s the trappingcoils for the MOT are switched off leaving only the DC signals due tothe trapping lasers and the catalysis laser (10s). The quantity extractedfrom this measurement is the ratio of the population in the trap with-out the catalysis laser (the difference of the fluorescence signals 0.5sand 6s) to the population in the trap with the catalysis laser interactingwith the trap (the difference of fluorescence signals at 10s and 9s). Theringing that takes place between 0 and 0.5 seconds is the shutter for thecatalysis light bouncing as it closes.730 20 40 60 80 100 120 140Detuning from Resonance (GHz)0.900.951.001.051.101.15Voltage ratio of No Cat laser to Cat laserCatalysis Measurement  MOT Position: 0 mmFigure 5.13: A catalysis measurement performed when the MOT was in ap-proximately the center of the glass vacuum cell. The peak of this mea-surement is close to 57 GHz. The measurement was taken with a dou-ble pass technique which resulted in an offset being measured in thesecond pass.740 20 40 60 80 100 120 140Detuning from Resonance (GHz)0.91.01.11.21.3Voltage ratio of No Cat laser to Cat laserCatalysis MeasurementTrap Position: -7.5mmFigure 5.14: A catalysis measurement performed when the MOT was approx-imately 2.5mm from the side of the glass cell. There is very littlechange in the location of the peak from Figure 5.13 indicating that thetrap maintains a consistent depth up until at least this close to the wallof the trapping area.75Chapter 6ConclusionThe use of a magneto-optical trap as a time-dependent probe of the spatial profileof an atomic beam has been demonstrated. The use of such a probe to determinethe absolute density of the beam, or the cross section of the collisions occurring, isshowing promise with work continuing on the project to see this become a reality.In the process of building the apparatus to make these measurements, a mechanismfor translating a MOT across a distance of 2 cm while maintaining alignment andtrap depth was developed and successfully implemented.76Bibliography[1] E. L. Raab, M. Prentiss, Alex Cable, Steven Chu, and D. E. Pritchard.Trapping of neutral sodium atoms with radiation pressure. Phys. Rev. Lett.,59:2631–2634, Dec 1987. doi:10.1103/PhysRevLett.59.2631. → pages 1[2] Steven Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable. Experimentalobservation of optically trapped atoms. Phys. Rev. Lett., 57:314–317, Jul1986. doi:10.1103/PhysRevLett.57.314. → pages 1[3] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.Cornell. 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Operating Manual and Programming ReferenceModels RGA100, RGA200, and RGA300 Residual Gas Analyzer, 1.8 edition,May 2009. http://www.thinksrs.com/downloads/PDFs/Manuals/RGAm.pdf,Retrieved 2014-5-1. → pages 32[29] F. Stienkemeier, M. Wewer, F. Meier, and H. O. Lutz. Langmuirtaylorsurface ionization of alkali (li, na, k) and alkaline earth (ca, sr, ba) atomsattached to helium droplets. Review of Scientific Instruments, 71(9), 2000.→ pages 3279[30] T Langmuir. The electron emission from thoriated tungsten filaments.Physical Review Letters, 22(4):357–398, 1923. → pages 32[31] J. B. Taylor and I. Langmuir. The evaporation of atoms, ions and electronsfrom caesium films on tungsten. Physical Review, 44(6):423–458, 1933. →pages 32[32] Janelle Van Dongen. Simultaneous cooling and trapping of 6li and 85=87rb.Master’s thesis, University of British Columbia, 2006. → pages 35[33] Keith Ladouceur, Bruce G. Klappauf, Janelle Van Dongen, Nina Rauhut,Bastian Schuster, Arthur K. Mills, David J. Jones, and Kirk W. Madison.Compact laser cooling apparatus for simultaneous cooling of lithium andrubidium. J. Opt. Soc. Am. B, 26(2):210–217, Feb 2009.doi:10.1364/JOSAB.26.000210. → pages 40[34] Krzysztof Kowalski, V Cao Long, K Dinh Xuan, M Gło´dz´, B Nguyen Huy,and J Szonert. Magneto-optical trap: fundamentals and realization. Comput.Meth. Sci. Technol. Special, (115):129, 2010. → pages 40[35] Todd P. Meyrath. Electromagnet design basics for cold atom experiments,2003. http://atomoptics-nas.uoregon.edu/∼tbrown/files/relevant papers/meyrath%20electromagnets.pdf, Retrieved 2015-4-25. → pages 41[36] Daniel A. Steck. Classical and modern optics, August 2013.http://atomoptics-nas.uoregon.edu/∼dsteck/teaching/optics/, Retrieved2014-2-20. → pages 55[37] Agilent Technologies. VacIon Plus 20 Pumps, May 2011. http://www.idealvac.com/files/ManualsII/Agilent VacIon Plus 20 pumps Manual.pdf,Retrieved 2016-21-1. → pages 85[38] Agilent Technologies. Agilent Ion Pumps, 2016.https://www.agilent.com/cs/library/catalogs/public/06 Ion Pumps.pdf,Retrieved 2016-21-1. → pages 8580Appendix ADetector CalibrationThis appendix gives the procedures and calibration constants for calibrating thedetectors used in this experiment.A.1 Photodiode DetectorsTwo separate photodiode setups were used in this experiment to measure the flu-orescence of the MOT. Determining the calibration constants required to convertfrom a fluorescence signal to a population figure required measurements of the pho-ton detection efficiency and the geometry of the detector as well as the scatteringrate of the rubidium atoms in the trap.A.1.1 Photodiode Detector Response TimesThe fastest feature that was to be measured by these detectors was the drop in MOTfluorescence due to collisions from the atomic beam. It was decided that a detectorwhich had a response speed greater than 20 KHz would be sufficient. The responseof the detector was determined by directing the repumping light directly onto thephotodiode and rapidly switching the repump AOM off and on to shutter the light.Figure A.1 gives the response time for the homebuilt detector which took datafrom the start of the experiment until the end of October 2015. In a fit of V =Vo f f +V0e−Gt the value of G indicates a reliable response of 92.5 KHz.Figure A.2 gives the response time of the Thorlabs detector which took data810.00010 0.00005 0.00000 0.00005 0.00010 0.00015Time [s]0.40.60.81.01.2Detector Signal [V]Response time of the Homebuilt DetectorResponse Speed: 92507 HzFigure A.1: Response time of the homebuilt detector and amplifier.from the start of November 2015 until the time of publication for this thesis. Thereliable response time of this detector is 36 KHz.0.0004 0.0002 0.0000 0.0002 0.0004Time (s)0.10.00.10.20.30.40.5Detector Signal (V)Response time of the Thorlabs DetectorResponse Speed: 36003 HzFigure A.2: Response time of the Thorlabs detector and bench amplifier.A.1.2 Photodiode Calibration ValuesThe detectors used in measuring the fluorescence from the MOT delivered a signalin volts which required calibration to give a trapped atom count. This sectiondetails that conversion with the results listed in Table A.1.The homebuilt detector was connected to the oscilloscope with a 1 MW input82Detector Watts/Volt Trapped Atoms/VoltHomebuilt 5.74x10-5 7.22x108Thorlabs 3.29-4 4.14x109Table A.1: The conversion numbers to translate the detector signal in volts toa MOT atom count.impedance and the amplifier on the detector set to the lowest detection setting. TheThorlabs detector was paired with the amplifier listed in Section 4.2.4 and wasconnected to the oscilloscope with a 50 W input impedance.Four values are needed for converting the signal needed. They are: The de-tector’s efficiency in watts per volt, the photon energy, the scattering rate, and thephoton capture fraction on the experiment table.Determining the detection efficiency was achieved by directing a low-poweredlaser of a known, stable power into the detector and measuring the resulting values.This was done with the repumping light by redirecting and attenuating the beam.Reducing the RF power to the AOM controlled the laser power by changing the1st order diffraction efficiency of the AOM. With the equation E = hn we get aphoton energy of approximately 2.5459x10−19J for the pump and repumping lightfor the rubidium D2 transitions. From the rubidium 85[24] and 87[25] data sheetsby Daniel Steck we get a similar decay rate of 38.117x106 for the 52S1=2→ 52P3=2transition for the both isotopes. Finally, the 1 inch diameter focusing lens in thedetector telescope is located 70 mm from the MOT capturing approximately 0.82%of the available light radiated by the trapped atoms. This gives the photon capturefraction on the table.A.2 Residual Gas AnalyserTime of flight information about the beam is recorded by the RGA as a signal re-ceived in volts. As is, it provides information about the velocity distribution of theparticles in the beam, but it requires a calibration constant to convert that signalinto a density measurement. For our application we are interested in the total fluxof atoms passing through a 2-dimensional cross section of the chamber.83A.2.1 Pumping Speed MethodOne method in which the RGA can be calibrated is by calculating the throughput ofthe beam gas through the second skimmer. This method is advantageous as it doesnot rely on the RGA providing it’s own calibration values for the measurements.The effects of the exit aperture as well as the ionization and detection efficiency ofthe beam do not factor in to the calibration values determined by this technique.Instead, the RGA signal is normalized during analysis to the expected beam density.The primary assumption that we make during this calibration process is thatwe are able to create a region of a constant pressure with only the atomic beamas the source and the ion pump as the sink. This was achieved by closing off thegate value to the RGA and operating the nozzle at a rate of 10 Hz. This produceda relatively constant pressure inside the MOT cell with only the ion pump and thereverse side of the 2 mm skimmer as exits for the trapped gas, with the amountremoved by the skimmer being negligible compared to the pump. The ion pumpfor this chamber is a Varian VacIon Plus 20 Starcell. We also assume that the ionpump operates at a constant speed.During the measurements made for Section 5.2.3 the pressure inside the MOTcell was recorded with the values available in Table A.1. Since the pressure mea-surements were made at the site of the pump, and the desired value is the numberof atoms removed from the system per second, no conversions were made for con-ductance. The pumping speed values available from Agilent only cover those fornitrogen and argon. Using the ionization potential for the remainder of the gasescan provide an estimate, but the exact per-pulse atom count will not be as accurate.Due to it’s similarities to argon, krypton is the likely the easiest of the three toestimate the atom removal rate.With the pressure (P) known the density can be calculated with the ideal gaslawPV = NkBT (A.1)to determine the number of atoms (N) per litre. Since the chamber walls are atroom temperature it is assumed that the gas will be approximately that temperatureby the time it reaches the ion pump. This density is given in table A.2.84Gas Pressure [mbar] Density [Atoms L-1] Eff. Ps. [L/s] [37]N2 1.50(5)x10-8) 3.71(12)x1011 22.5(4)[37]Ar 3.49(7)x10-8 8.63(19)x1011 17.7(4)[37]Kr 2.17(5)x10-8 5.35(12)x1011 16.2(4)[38]H2 5.4(11)x10-8 13.4(27)x1011 34.5(6)[38]He 12.2(2)x10-8 30.2(7)x1011 20.2(5)[38]Table A.2: The pressures measured in the MOT cell with the exit gate closedand the nozzle operated at 10 Hz along with the atom density and thepumping speed of the system. In this table, “Eff. Ps.” stands for “Ef-fective Pumping Speed” and takes into account the conductance of thechamber. The backing pressure for all the gases is the standard 100 PSIused in the table translation measurements.The pumping speeds were found for the five gases from the ion pump manuals[37][38]. The effective pumping speed was then found through the conductanceequation1Se f f=1Spump+1Caperture(A.2)whereCaperture =14Av¯ =14A√8kBTpm(A.3)is the conductance of the aperture of area A for a particle of mass m at temperatureT .With the pumping speeds known the pumping speed in terms of atom count canbe found. Table A.3 give the beam calibration values for the gases with a known(or reasonably estimated) pumping speed.85Gas Nozzle Pulse Particle Count (Number/pulse)N2 8.34(31)x1011Ar 1.53(5)x1012Kr 8.68(30)x1011H2 4.64(93)x1012He 6.13(20)x1012Table A.3: The number of atoms or molecules passing through the MOT cellfrom a single pulse of the nozzle.86Appendix BMOT Position ImagingSection 4.2.4 detailed the MOT camera equipment used to align, image and acquirethe position of the MOT. The purpose of this appendix is to detail the use of thesecameras in acquiring the MOT position for the results detailed in Sections 5.2.2 and5.2.3.B.1 Standard Imaging ProcedureEach measurement requiring the position of the MOT was accompanied by eitherthree or four position images of the MOT with the labels a, b, c, and d. Image ‘a’was the only image taken with the horizontal camera (Figure 4.14, upper right)and was primarily used for centering the MOT in the cell and for assisting in align-ments. This ‘a’ image was considered optional for the table translation measure-ments as the table translation mechanism preserved the trap and no realignmentswere needed for the duration of the measurement process. Images ‘b’, ‘c’ and ‘d’were taken from the vertical camera (Figure 4.14 upper left) and were processedinto a single image to give the MOT’s position within the cell. Only the verticalcamera saw any motion in the MOT as the horizontal camera was directed along theaxis of motion for the MOT.Image ‘b’ provided the reference image for the MOT images in ‘c’ and ‘d.’These images were of the MOT cell with the MOT off and a light illuminating theglass walls and reference features to be used for measuring the MOT position. The87remaining images were of the MOT as it would appear during an experiment (‘d’)and an identical image but with the Helmholtz coils switched off (‘c’). The pro-cessing procedure was to take the difference between the ‘c’ and ‘d’ images andremove the green and blue channels to get a red difference image. The red valuesgenerating that images were then added to the ’b’ reference image to produce animage in which the MOT appeared as red next to the reference features.B.2 Translating Coils Position MeasurementsThis section compliments the results in Section 5.2.2.At the time of this experiment, the Helmholtz coils being used obstructed theview of the MOT cell such that only one of the glass interfaces of the cell was visibleto the camera. To compensate for this, a printed card was slid under the glass andabove the lower Helmholtz coil to provide a distance reference for the ‘b’ image.The small inner diameter of the coils’ torus also forced the camera to be more inline with the trapping lasers which had to share the same hole. This caused a lot ofscattered light from the trapping lasers to reach the camera and necessitated the useof the ‘c’ and ‘d’ images to identify the MOT from the illuminated imperfections onthe glass. The images in Figure B.1 gives a sample of a measurement made withthis setup.B.3 Translating Table Position MeasurementsThis section compliments the results in Section 5.2.3.With the introduction of the upgraded Helmholtz coils the now larger innerdiameter allowed for a more streamlined and overall better measurement of theMOT’s position. The increased field of view allowed for both sides of the 20 cm-wide cell to be captured in the reference image. This removed the need for thereference card as the cell dimensions became a reliable method of measuring dis-tance. The increased inner diameter of the coil torus also allowed the camera to bemoved farther from the vertical trapping laser which drastically reduced the amountof unwanted light reaching the camera. This reduced the unwanted light by such adegree that the neutral density filter was removed entirely for the translating tablemeasurements. This contrast can be seen in the difference between Figure B.2 and88Figure B.1: Images used in analysing the position of the MOT for the resultsin Section 5.2.2. Upper Left: The reference image to provide a scale.The distortion towards the bottom of the image marks the barrier be-tween vacuum and glass. The vertical distance markers on this card are2 mm apart. A lamp is used to illuminate the reference card. UpperRight: Looking down on the MOT while the trap is running. LowerLeft: Looking down on the MOT while the trap is off. Lower Right:The difference of the upper right and lower left images is added in redto the upper left image.Figure B.1.As with the previous section, Figure B.2 provides an example of the measure-ments made with this setup. The lines in image (4) were placed on the image toretrieve the position of the MOT between the cell walls.89Figure B.2: Images used in analysing the position of the MOT for the resultsin Section 5.2.3. Upper Left: The reference image to provide a scale.The width of the vacuum is 20 mm and the walls 5 mm each. A lampis used to illuminate the cell. Upper Right: Looking down on the MOTwhile the trap is running. Lower Left: Looking down on the MOT whilethe trap is off. Lower Right: The difference of the upper right and lowerleft images is added in red to the upper left image. Lines are drawn tomeasure key distances. The walls are marked as the upper (red) andlower (green) with the MOT (blue) in the middle. There is a significantamount of noise in this image, but the MOT shines brightly nonetheless.90Appendix CJanuary 2015 Results QuirksIn Section 5.2.2 the results showed a profile of the atomic beam being made, butthere were several aspects of the data which lead to the upgrades made for theresults in Section 5.2.3. This appendix will outline the results which lead to theconclusions made regarding the accuracy of the data.C.1 Anomalies in Loss MeasurementsSeveral inconsistencies arose in the measurements taken for the Helmholtz coiltranslation set which resulted in many of the data points being discarded. In totalthere were originally 24 points which were reduced to 13. Many of the graphs hadanomalies like the one shown in Figure C.1 where a strange signal jump appearedimmediately before the loss event took place. The tail of the data also continueddownwards after the beam had passed over the trap. This is a stark contrast tothe “good” measurements such as the one seen in Figure 5.1 The cause of thisfeature was unknown and it’s appearance was not tied to any particular change inthe experiment. This feature would not show up in the measurements made withthe sliding table.C.2 MOT Population and Loading TimesThe beam profile measurements contain a large amount of data collected and fittedwhich either supplement or are not directly used in producing the final plots of910 1 2 3 4 5Time [ms]4.304.354.404.454.504.55Trap Pop.1e7Rb PopulationNumerical FitAtom Flux0.50.00.51.01.52.02.5Atom Flux [Atoms/sec]1e16MOT Loses from Collisions with Ar BeamFigure C.1: The fluorescence signal from the MOT gave a very strange result.Immediately before the atomic beam arrives at the MOT the fluores-cence detection setup recorded a spike in the signal. In addition to this,the fluorescence signal gradually tapers off instead of reaching a newequilibrium value. Several measurements had this feature but the causewas not determined before it had vanished. This data, and others likeit, were not included in the final result.Sections 5.2.2 and 5.2.3. Plotting some of that data reveals several issues with thesetup used in Section 5.2.2.Figure C.2 gives the MOT population and loading time constant as a function ofthe MOT’s position within the cell. The data was retrieved from the loading curvesfor the data sets. Over the course of the measurements the population of the trapchanges by an order of magnitude with the more populous traps occurring dispro-portionately towards one side of the cell. The loading time constant G follows areverse trend with the larger values appearing with the smaller traps. It is clearfrom these plots that there is a feature of the environment either generating theMOT or containing the MOT that is affecting the trap. Unfortunately, the catalysislaser was not available at this time and the trap depth could not be measured so theeffects on the results of the experiment are unknown.924 2 0 2 4 6 8Position in the Cell [mm]0.00.51.01.52.02.53.03.54.04.5Trap Population1e8MOT PopulationLoading Time Constant12345678Loading Time Constant Γ [s−1]MOT Statistics as the Trap is  Moved About the Vacuum CellFigure C.2: Graph showing the number of atoms trapped and the loadingtime constant for each MOT of the 24 points collected for the argon coiltranslation measurements. There is a clear trend in the loading rate, butthis may depend on several factors including trapping laser alignmentand the proximity of the MOT to the cell walls (at ±10mm). The valuefor G (Equation 3.19) has an approximate inverse relation to the MOTsize.The plots in Figure C.2 show that improvements would be needed before moremeasurements of the beam are made. After the improvements described in Section5.2.3 were made, similar plots to Figure C.2 were produced and are shown in FigureC.3. Here the MOTs are far more consistent between measurements and also spana wider distance within the cell.936 4 2 0 2 4 6 8Position in the Cell [mm]0.0180.0200.0220.0240.0260.0280.0300.0320.034Trap PopulationMOT PopulationTime Constant Γ1.41.51.61.71.81.92.02.1Loading Time Constant Γ [s−1]MOT Statistics as the Trap is  Moved About the Vacuum CellFigure C.3: Graph showing the number of atoms trapped and the loadingtime constant for each MOT of the 25 points collected for the argontable translation measurements. These statistics are in start contrast tothose shown in Figure C.2. The reduction in beam size as well as trans-lating both the trapping coils and the trapping lasers keeps the MOTconsistent across all measurements. This plot was generated from theloading curve data for the argon table translation measurements.94Appendix DAlignment MethodsThis appendix will cover the techniques and methods used for aligning the atomicbeam and the MOT.D.1 Atomic Beam AlignmentThere were several stages to the beam alignment procedure. The first stage, and theonly stage which required opening the vacuum chamber, was the skimmer align-ment. This was performed first and was not adjusted after being completed. Theadjustments made to the detection chamber were performed after a full profile ofthe atomic beam was made, but represented a semi-permanent change to the posi-tion measurement of the beam in the MOT vacuum cell. Nozzle alignments weremade regularly to maximize the throughput of gas to the MOT and detection cham-bers.D.1.1 Skimmer AlignmentThis section is written as a guide for how the skimmer alignment process wascompleted starting from a newly assembled vacuum setup. For the results shownin Section 5.2.2 and 5.2.3 the skimmer alignment was maintained and adjustmentsmade to other components of the experiment to optimize the path of the beam.More details on these adjustments can be found in Section D.1.2.Setting the skimmer alignment is something which was performed only a few95times and forms the base of all other alignments. For all profiling measurementsof the atomic beam the beam was assumed to remain stationary and so maintainingthis fundamental alignment structure was paramount. This mandated the use oftwo skimmers to set the trajectory of the beam.The diagram in Figure D.1 shows the major points of interest for skimmeralignment. Initially the telescope is aligned to the gas inlet port and the secondskimmer mounting plate or the reference dot on the MOT gate valve. This alignmentwill form the trajectory that the beam will take when the experiment is being run.The skimmers are then mounted one at a time and aligned to the reference dot orthe other skimmer already in place. Figure D.2 shows the telescope and severalportions of the experiment relevant to the beam alignment. The lower right imagegives an example of the view from a misaligned telescope looking into the chamber.In this instance, the telescope would have to be lowered.The technique for aligning the skimmers is to line up the circles created bythe various objects along the beam path. This is an iterative process in which theskimmer mounting plate will have to be removed to make an adjustment. It isinadvisable to skip placing an old copper gasket between the vacuum parts as thiscan cause the mounting plate to misalign to the vacuum chamber. Since the 3 mmskimmer is in place (1) in Figure D.1, the 2 mm skimmer is visible and could bealigned to both the 3 mm skimmer and the reference dot.(Z)(A) (B) (C) (D)(1) (2)Figure D.1: A diagram showing the key features of the skimmer alignment.A telescope (Z) is positioned such that the gas inlet port (A) is in align-ment with the second skimmer mount (C) by aligning the circular ports.The skimmers (1) and (2) are mounted onto ports (B) and (C) andaligned using the telescope and a mark made on the gate valve to theMOT (D).96Figure D.2: (Upper Left: The telescope pointing towards the gas inlet port.For alignment the telescope was placed approximately 3 meters fromthe source chamber. Upper Right: the gas inlet of the source chamber.During alignment the Klein Flange (KF) is removed to allow access tothe vacuum chamber. Lower Left: The bellows between objects (C)and (D) on Figure D.1. The bellows became kinked after the processdescribed in Section D.1.2 was completed. Lower Right: An illustra-tion of what is seen through the telescope. The focus is adjusted to seea series of circles representing different ports or skimmers. The circlesneed to be aligned with one another for a successful alignment. Youwould see an image similar to this when setting up the telescope andthe telescope is too high.D.1.2 Detection Chamber AdjustmentsAfter a profile had been made of the beam any errors in the alignment of the skim-mers became apparent. The red points in Figure D.3 shows the results of the firstprofile made of the beam after the final skimmer alignment had been made. Thecenter of the beam appears to be off of the center of the MOT cell by as much as 6mm. This placed the outer edge of the beam beyond where the MOT could make ameasurement at the time.Instead of performing the skimmer alignment again, it was simpler to move the976 4 2 0 2 4 6 8MOT Position [mm] (from cell center)02004006008001000Fitted K Value (red)050010001500200025003000Fitted K Value (black)A Before and After Comparissonof the Beam Alignment in the CellFigure D.3: Before and After profiles of the atomic beam showing thechanges as a result of adjustments made to the detection chamber andMOT cell position. The red trace is the data shown in Figure 5.3 and isthe ”before” image. The black trace is the data from Figure 5.7 and isthe ”after” image.vacuum cell and detection chamber by taking advantage of the two sets of bellowson either end of the MOT cell. The results of the first profile could then be used todetermine how far to make this adjustment. In addition to this, minor adjustmentswere made to the exit aperture to minimize the amount of gas remaining in theMOT cell after the nozzle fired. This was done by operating the nozzle at 10 Hzand adjusting the position of exit aperture while watching the change in pressurein the MOT cell.The combination of these techniques improved the alignment of the beam rel-ative to the MOT vacuum cell. The black points in Figure D.3 show just how pro-nounced this improvement is. It should be noted that the measurement techniqueswere also improved between the two measurements.D.1.3 Nozzle AlignmentThe contraption suspending the nozzle in place had two main mechanisms to movethe nozzle about the source chamber. This allowed the nozzle to be aligned to theskimmers to generate the atomic beam used in our experiments.98Figure D.4: The mounting plate for the skimmer is connected to a rotaryfeedthrough attached to a bellows. This contraption provides the noz-zle with enough translational and rotational freedom such that it can bealigned to the two skimmers.Figure D.4 shows the manipulator setup on top of the source chamber. Themounting plate for the nozzle is suspended below a rotary feedthrough which isitself mounted above a bellows and plate setup. By adjusting the rotation of thefeedthrough and the separation of the plates holding the bellows the nozzle couldbe moved about the source chamber.By running a gas such as argon through the nozzle at 10 Hz a relatively con-stant pressure could be read in each chamber. The nozzle can then be adjusted tooptimize the gas making it through to each chamber. Typically the gate valve at theexit aperture (Section 4.1.1) is closed since the atomic beam would exit the MOTcell once aligned properly and the pressure measurement at the MOT cell wouldshow very little gas.99D.2 MOT Trapping Laser AlignmentThis section will describe several processes related to aligning the trapping lasersfor the MOT.D.2.1 Splitting and PolarizationSplitting the laser light and tuning the polarization should only be done if the MOToptics have been disassembled or a major change has happened which requires afull realignment of the MOT.The splitting mechanism is a series of two l /2 waveplates and two polarizingbeam splitting cubes. The power ratios of the three trapping axis can vary any-where from 13 :13 :13 to14 :14 :12 with the12 being the axial trapping laser. The secondconfiguration listed was used for the measurements discussed in Section 5.2.3.After splitting the polarization is rotated to either right-hand or left-hand cir-cular with a set of three l /4 waveplates. The two radial trapping lasers must beidentical in polarization and opposite that of the axial polarization. The polariza-tion can be checked with a spare l /4 waveplate and polarization cube. This is doneby returning the light at the MOT to a linear polarization and checking the resultingpolarization with the cube. The l /4 waveplates on the retroreflection optics do notneed to be aligned.D.2.2 Laser AlignmentThere are several tricks to aligning the trapping lasers of the MOT.The magnetic center of the trap should be estimated before beginning the laseralignment. One trick that was used in this experiment was using the tapped holeson the optical table to draw two trajectories through the center of the trap. Thistechnique works because the Helmholtz coils are mounted to the same optical table.Standard telescoping procedures should be applied to setting the size of thetrapping lasers.Three irises can be quite useful in setting the alignment of the beams. If theirises are set at the correct height and position then they can be closed when neededto shrink the beams. The trap is quite robust and can be maintained with beamsonly a few millimetres in diameter. This also assists in aligning the retroreflection100Figure D.5: Left: The beam is directed to the MOT through a cardboard align-ment card which has been placed directly over a trajectory line drawnon the table. Right: An open iris is clamped into place to provide aconsistent reference point for the alignment of the trapping lasers.optics by providing a small target to aim the returning beam at. In the absence ofirises, a hole-punched cardboard card is sufficient for approximating the alignmentand aligning the retroreflected beam.Having the dispensers on and the lasers locked also assists in viewing the beaminside the vacuum cell. The locked lasers will cause the atoms in the cell to fluo-resce and give the appearance of a beam passing through the cell. This is particu-larly useful when aligning the vertical trapping beam as it can be quite difficult tosee through the Helmholtz coils.101Appendix EResidual Gas Analyser FittingFunctionThe data from the RGA is fit using a customized function which is a combination ofa Cauchy Distribution and an arctangent. This appendix will present four functionsand the reasoning for choosing the function shown in Section 5.2.3.E.1 Normal DistributionOne function which was tested was a normal distribution with the equationf (t) =1s√2pe−(t−m)22s2 ; (E.1)the results of which are presented in Figure E.1. For this distribution, the widthis given by s and the time offset by m . The inability of the function to properlyprofile the leading and trailing edge of the beam resulted in this distribution to berejected.E.2 Maxwell-Boltzmann DistributionA Maxwell-Boltzmann distribution, with a Heaviside step function, was testedwhich followed the equation1020.002 0.000 0.002 0.004 0.006 0.008 0.010Time (s)0.020.010.000.010.020.030.040.050.060.07RGA Signal (V)RGA Signal Function TrialNormal DistributionFigure E.1: The trial fit of the RGA signal with a normal distribution follow-ing Equation E.1. This function was not chosen.f (t) =√2p(t− t0)2e−(t−t0)2=(2a2)a3H(t− t0) (E.2)with t0 as the time offset, a defining the width, and H(t− t0) being the Heavisidefunction. The results of this trial can be found in Figure E.2. Similar to the normaldistribution, the leading and trailing edge of the beam is fit poorly discounting thisfitting routine.E.3 Lorentz DistributionThe Lorentz (Cauchy) distribution was the first distribution to show promise withfitting the RGA signal. It is given by the equationf (t) =1pg[1+( t−t0g)2] (E.3)with t0 as the time offset and g the width. Figure E.3 shows an example fit with thisfunction. Here the function is forming close to the signal produced by the RGA but1030.002 0.000 0.002 0.004 0.006 0.008 0.010Time (s)0.020.010.000.010.020.030.040.050.060.07RGA Signal (V)RGA Signal Function TrialMaxwell-Boltzmann DistributionFigure E.2: The trial fit of the RGA signal with a Maxwell-Boltzmann distri-bution following Equation E.2. This function was not chosen.with opposing problems for the leading and trailing edge of the beam.A modification was made to Equation E.3 to account for the problems shownin Figure E.3 by including an arctangent function. This is given byf (t) =1pg[1+( t−t0g)2](arctan((t− t0)=g)p +0:5): (E.4)The result of this modification can be seen in Figure E.4. The arctangent functioncauses the leading edge of the distribution to drop and allows the trailing edgeto rise in order to better represent the RGA signal. This function was chosen torepresent the RGA in the time and velocity plots in Section 5.2.31040.002 0.000 0.002 0.004 0.006 0.008 0.010Time (s)0.020.010.000.010.020.030.040.050.060.07RGA Signal (V)RGA Signal Function TrialCauchy DistributionFigure E.3: The trial fit of the RGA signal with a Lorentz distribution follow-ing Equation E.3. This function would be modified to generate a betterfit.0.002 0.000 0.002 0.004 0.006 0.008 0.010Time (s)0.020.010.000.010.020.030.040.050.060.07RGA Signal (V)RGA Signal Function TrialCauchy Arctan DistributionFigure E.4: The trial fit of the RGA signal with a Lorentz arctan distributionfollowing Equation E.4. This function follows the RGA fairly close andwas chosen to generate the fits.105Appendix FTrap Population ODE SolvingCodeThis chapter contains the code used for plotting Equation 3.20 and producing allthe loss graphs necessary for the results shown in Chapter 5.106# ! / usr / b in / pythonimpor t numpy as npimpor t mathfrom sc ipy . i n t e g r a t e impor t odeimpor t m a t p l o t l i b . pyp lo t as p l tfrom sc ipy impor t i n t e r p o l a t e , s t a t simpor t commandsfrom sc ipy . op t im ize impor t l eas tsqfrom sc ipy . s i g n a l impor t bu t te r , l f i l t e rimpor t os# 3dB POINT FOR LOW−PASS FILTER (10kHz )lowcut = 1e4# DISTANCES FOR TIME RE−SCALING (cm)d mot = 99.d rga = 165.def f i t l o s s ( t , mot , rga , the data no , t ime 00 , t ime 10 , f i t pa ram , load param ,save d i r , work d i r , f i t d i r , gas , e x p o r t d a t a l o c a t i o n ) :p r i n t ’ s t a r t f i t loss ’# EXPAND SIGMA AND LOADING VALUESdata no = the data norange s = t ime 00range e = t ime 10# F i t t i n g constant tempk i n i t = f i t p a r a m [ 0 ]#delayd i n i t = f i t p a r a m [ 1 ]#MOT Conversion f a c t o rmot conver t = f i t p a r a m [ 2 ]#RGA Conversion f a c t o rrga conver t = f i t p a r a m [ 3 ]#Loading valuesload vb = load param [ 0 ]load gam = load param [ 2 ]l o a d r = load param [ 1 ] load param [ 2 ]# SET GLOBAL VARIABLESg loba l R, gamma, V baseR = l o a d rgamma = load gam107V base = load vb# SET OTHER VARIABLESs t a r t i n d e x = range s+150end index = range ep0 = [ k i n i t , d i n i t ]mot V to atoms = mot conver trga V to atoms = rga conver t#Data f i lenamesinput name = ’ data ’+ s t r ( data no ) + ’ . csv ’output = s t r ( i n t ( data no ) ) . z f i l l ( 3 ) + ’ . pdf ’# T i t l e on top of p l o tp l o t t i t l e = r ’MOT Loses from C o l l i s i o n s w i th ’+ gas + ’ Beam’#RGA s i g n a l a r r i v e s i nve r t edrga = −rga# FILTER OUT HIGH FREQUENCY NOISEm o t f i l t e r e d = f i l t e r d a t a ( t , mot , lowcut )#Determine MOT s t a r t i n g po in tV mot0 = np . mean( m o t f i l t e r e d [ 1 : 2 0 0 0 ] )#####V base = V mot0−load param [ 1 ]# PRIOR TO SETTING VALUES, V0 MUST ME EXTRACTED FROM MOT FILTERED# SET DATA TO START AT INDEX s t a r t i n d e xt i m e o f f s e t = t [ s t a r t i n d e x − 1] − t [ 0 ]t = t [ s t a r t i n d e x : end index ]t = t − t [ 0 ]mot = mot [ s t a r t i n d e x : end index ]m o t f i l t e r e d = m o t f i l t e r e d [ s t a r t i n d e x : end index ]rga = rga [ s t a r t i n d e x : end index ]# RE−DEFINE TIME SCALE FOR RGA MEASUREMENTS TO ACCOUNT# FOR TIME LAGS DUE TO VELOCITY SPREAD AND SPACIAL SEPARATION# B/W THE RGA AND MOTt p r ime = ( d mot / d rga )  td t p r ime = t p r ime [ 1 ] − t p r ime [ 0 ]# RE−DEFINE OFFSETSmot = mot − V base108m o t f i l t e r e d = m o t f i l t e r e d − V baser g a o f f s e t = np . mean( rga [200 :600 ] )rga = r g a o f f s e t − rga# I n t e g r a t e RGA s i g n a l to d i sp lay lossd a t l en = len ( rga )i n t s t a r t = math . t runc ( d a t l en / 6 )i n t en d = math . t runc (5 da t l en / 6 )the dx = ( t [ i n t e nd ]− t [ i n t s t a r t ] ) / ( 2  da t l en / 3 )# INTERPOLATE THE RGA SIGNAL TO MAKE IT A TIME DEP. FUNCTION# SUITABLE FOR PYTHON’ S ODE SOLVERSt max = max( t p r ime )g loba l r g a i n tr g a i n t = i n t e r p o l a t e . I n t e r p o l a t e d U n i v a r i a t e S p l i n e ( t p r ime , rga )# INTERPOLATE THE FILTERED MOT SIGNALm o t i n t = i n t e r p o l a t e . I n t e r p o l a t e d U n i v a r i a t e S p l i n e ( t , mot )# INTERPOLATE THE FILTERED MOT SIGNALg loba l m o t f i l t e r e d i n tm o t f i l t e r e d i n t = i n t e r p o l a t e . I n t e r p o l a t e d U n i v a r i a t e S p l i n e ( t ,m o t f i l t e r e d )# ODE INPUTSt0 = t [ 0 ]d t = t [ 1 ] − t [ 0 ]V mot0 = np . mean( m o t f i l t e r e d [ 0 : 2 0 0 0 ] )# LEAST SQUARES OPTIMIZATION : NOT WORKING#plsq = leas tsq ( res idua ls , p0 , args =( t0 , dt , t max , V mot0 , m o t f i l t e r e d ) )p lsq = [ p0 , 2 ]# OPTIMIZED FIT PARAMETERSK = plsq [ 0 ] [ 0 ]delay = plsq [ 0 ] [ 1 ]# SOLUTION TO ODE WITH OPTIMIZED FIT PARAMETERSode time , ode soln = solve ode (K, t0 , dt , t max , V mot0 )ode t ime = ode t ime + delay# RESIDUALSchi2 = s t a t s . chisquare ( m o t f i l t e r e d i n t ( ode t ime ) , ode soln )109mot V to atoms = 100/ V mot0# PLOT RESULTSf i g = p l t . f i g u r e ( f i g s i z e = (7 ,5 ) )ax2 = f i g . add subplot (111)p l o t t i m e = ( ode t ime − ode t ime [ 0 ] )  1e3 # + t i m e o f f s e t )  1e3lns4 = ax2 . p l o t ( p l o t t i m e , r g a i n t ( ode time−delay ) (2 .23E13/0.00008) d rga / d mot ,’ b ’ , l i n e w i d t h =3 , l a b e l =r ’ Atom Flux ’ )ax2 . s e t y l a b e l ( r ’ Atom Flux [ Atoms / sec ] ’ )ax1 = ax2 . tw inx ( )lns1 = ax1 . p l o t ( p l o t t i m e , m o t i n t ( ode t ime )  mot V to atoms , ’ k ’ ,l a b e l =r ’ Rb Populat ion ’ )lns1 = ax1 . p l o t ( t , motmot V to atoms , l a b e l =gas+r ’ Atom Number ’ )lns2 = ax1 . p l o t ( p l o t t i m e , m o t f i l t e r e d i n t ( ode t ime )  mot V to atoms ,’ r ’ , l i n e w i d t h =3 , l a b e l =r ’ Rb Populat ion ’ )lns3 = ax1 . p l o t ( p l o t t i m e , ode soln  mot V to atoms , ’ g ’ , ms=2. ,l i n e w i d t h =3 , l a b e l = r ’ Numerical F i t ’ )ax1 . s e t x l a b e l ( r ’ Time [ms ] ’ )ax1 . s e t y l a b e l ( r ’ Trap Pop . ’ )ax1 . t i c k l a b e l f o r m a t ( s t y l e = ’ sc i ’ , ax is = ’ y ’ , s c i l i m i t s = (0 ,0 ) )l ns = lns2 + lns3 + lns4labs = [ l . g e t l a b e l ( ) f o r l i n lns ]ax2 . s e t x l i m ( min ( p l o t t i m e ) , max( p l o t t i m e ) )ax2 . legend ( lns , labs )p l t . t i t l e ( p l o t t i t l e )p l t . t i g h t l a y o u t ( )os . chd i r ( save d i r )p l t . save f ig ( output )p l t . c lose ( )r e t u r n in teg ra ted , ch i2 [ 0 ]def r e s i d u a l s ( p , t0 , dt , t max , V mot0 , mot ) :ode time , ode soln = solve ode ( p [ 0 ] , t0 , dt , t max , V mot0 )s t a r t = p [ 1 ]stop = s t a r t + t max110t d e l a y = np . arange ( s t a r t , stop , d t )i f s t a r t < 0. or stop > dt  l en ( mot ) :res = 1e99else :res = m o t f i l t e r e d i n t ( t d e l a y ) − ode solnr e t u r n resdef solve ode (K, t0 , dt , t max , V mot0 ) :# FOR A GIVEN K, SOLVE THE ODE AND RETURN RESULTso lve r = ode ( dV mot dt ) . s e t i n t e g r a t o r ( ’ vode ’ , method= ’ bdf ’ ,w i t h j acob ian =False )so l ve r . s e t i n i t i a l v a l u e ( V mot0 , t0 ) . se t f params (K)ode t ime = [ ]ode soln = [ ]wh i le so l ve r . success fu l ( ) and so l ve r . t < t max :so l ve r . i n t e g r a t e ( so l ve r . t + d t )ode t ime . append ( so l ve r . t )ode soln . append ( so l ve r . y [ 0 ] )r e t u r n np . ar ray ( ode t ime ) , np . ar ray ( ode soln )def dV mot dt ( t , y , K ) :# MOT FLUORESCENCE ODEr e t u r n R − (gamma + K  r g a i n t ( t ) )  ydef bu t te r lowpass ( lowcut , fs , order =3 ) :nyq = 0.5  f slow = lowcut / nyqb , a = b u t t e r ( order , low , btype = ’ low ’ )r e t u r n b , adef b u t t e r l o w p a s s f i l t e r ( data , lowcut , fs , order =3 ) :b , a = bu t te r lowpass ( lowcut , fs , order=order )y = l f i l t e r ( b , a , data )r e t u r n ydef f i l t e r d a t a ( t , data , lowcut ) :o f f s e t = data [ 0 ] # s h i f t data to s t a r t a t 0 f o r b e t t e r f i l t e r e d s i g n a ldata = data − o f f s e td t = t [ 1 ] − t [ 0 ]f s = 1 . / d t # sampling frequency111f i l t e r e d d a t a = b u t t e r l o w p a s s f i l t e r ( data , lowcut , fs , order =3)f i l t e r e d d a t a = f i l t e r e d d a t a + o f f s e tr e t u r n f i l t e r e d d a t ai f name == ’ main ’ : main ( )112

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