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Study of background gas collisions in atomic traps Van Dongen, Janelle 2014

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Study of background gas collisions inatomic trapsbyJanelle Van DongenA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Physics)The University Of British ColumbiaJune, 2014c© Janelle Van Dongen 2014AbstractThis thesis describes an investigation and application of the loss of laser-cooled atoms from a trap induced by background collisions. The loss rateconstant depends on the density of background gas and the velocity aver-aged collisional loss cross section due to collisions. The velocity averagedcollisional loss cross section can be calculated and its dependence on trapdepth was verified using a magneto-optical trap. This verification involvedmeasurements of the loss rate constant for a quadrupole magnetic and amagneto-optical trap and measurement of the density of Ar background gasusing a residual gas analyzer. The second part of the thesis focuses on anapplication of these measurements of the loss rate constant to measure thepressure of the background gas. The experimental progress to date on theatom pressure sensor is provided.iiPrefaceThe work described in chapter 3 led to the publication titled ‘Trap-depthdetermination from residual gas collisions’ published in Phys. Rev. A 84,022708 (2011). My contribution to the publication was experimental setup,taking data and correcting the manuscript after the reviewer’s initial com-ments. Fig. 1.6 was produced by Dr. Daniel Steck. Fig. 2.4, 3.2 - 3.5, and3.8 - 3.13 are reproduced from that publication and were originally made byDr. James Booth. Tables 3.1 and 3.2 are also from this publication and dataanalysis performed for these tables was performed by Dr. James Booth.Chapters 4-7 describe work leading to a pressure sensor using trappedatoms. I was directly involved in the experimental design of the apparatus,ordering and assembly of the apparatus, and taking and analyzing precur-sory data. Fig. 4.4 is attributed to MKS Instruments and is used withpermission. Fig. 6.2 was produced by an image processing code written byDr. James Booth.Appendix A led to a publication titled ‘Magneto-optical trap loading ratedependence on trap depth and vapor density’ published in JOSA B, Vol. 29,Issue 3, pp. 475-483 (2012). I was listed as an author for background supporton the experimental apparatus. The figures and tables in the appendix arereproduced from that work and were originally made by Dr. James Booth.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Atomic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 3D magneto-optical trap . . . . . . . . . . . . . . . . 31.2.2 Magnetic traps . . . . . . . . . . . . . . . . . . . . . . 71.3 Trap dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Magneto-optical trap dynamics . . . . . . . . . . . . 121.3.2 Magnetic trap dynamics . . . . . . . . . . . . . . . . 162 Background gas collision induced loss . . . . . . . . . . . . . 182.1 A brief review of necessary scattering theory . . . . . . . . . 182.2 The velocity averaged collisional loss cross section . . . . . . 232.2.1 Calculation of 〈σlossvi〉X,i . . . . . . . . . . . . . . . . 232.2.2 Convergence criteria . . . . . . . . . . . . . . . . . . . 272.2.3 Interaction potential . . . . . . . . . . . . . . . . . . 282.2.4 The dependence of 〈σlossvi〉X,i on trap depth, U . . . 30ivTable of Contents3 Experimental verification of the dependence of the loss crosssection on trap depth . . . . . . . . . . . . . . . . . . . . . . . 343.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 343.2 Measurement of 〈σvAr〉87Rb,40Ar . . . . . . . . . . . . . . . . . 363.2.1 Measurement of Γ for a MOT . . . . . . . . . . . . . 373.2.2 Measurement of Γ for a magnetic trap . . . . . . . . . 373.2.3 Results of Γ vs nAr measurement . . . . . . . . . . . 393.3 MOT trap depth determination by the ‘catalysis method’ . . 433.4 Comparison of measurement with theory . . . . . . . . . . . 493.4.1 Magnetic trap data . . . . . . . . . . . . . . . . . . . 493.4.2 Magneto-optical trap data . . . . . . . . . . . . . . . 513.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Proposal for a trap depth measurement technique . . . . . . 534 Proposal for a cold atom based pressure sensor . . . . . . . 574.1 Pressure gauges . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.1 Capacitance diaphragm gauge . . . . . . . . . . . . . 584.1.2 Spinning rotor gauge . . . . . . . . . . . . . . . . . . 604.1.3 Bayard-Alpert ionization gauge . . . . . . . . . . . . 624.1.4 Residual gas analyzer . . . . . . . . . . . . . . . . . . 644.1.5 Pirani gauge . . . . . . . . . . . . . . . . . . . . . . . 654.2 Existing pressure standards . . . . . . . . . . . . . . . . . . . 654.2.1 Static expansion method . . . . . . . . . . . . . . . . 654.2.2 Orifice flow method . . . . . . . . . . . . . . . . . . . 664.3 A proposed density/pressure standard using trapped atoms . 684.3.1 Existing proposals for total vacuum pressure measure-ment using trapped atoms . . . . . . . . . . . . . . . 724.3.2 Magneto-optical trap versus magnetic-trap for a pres-sure sensor . . . . . . . . . . . . . . . . . . . . . . . . 734.3.3 Estimated uncertainty . . . . . . . . . . . . . . . . . . 745 Experimental apparatus for the pressure sensor experiment 765.1 High vacuum pumps . . . . . . . . . . . . . . . . . . . . . . . 805.2 2D MOT section . . . . . . . . . . . . . . . . . . . . . . . . . 815.2.1 2D MOT chamber region . . . . . . . . . . . . . . . . 815.2.2 2D MOT coils . . . . . . . . . . . . . . . . . . . . . . 825.3 3D MOT section . . . . . . . . . . . . . . . . . . . . . . . . . 875.4 UHV pumps and diagnostics . . . . . . . . . . . . . . . . . . 875.5 Vacuum apparatus assembly . . . . . . . . . . . . . . . . . . 895.6 Bakeout Procedure . . . . . . . . . . . . . . . . . . . . . . . 92vTable of Contents5.6.1 Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.6.2 Pre-bake preparation . . . . . . . . . . . . . . . . . . 935.6.3 Baking . . . . . . . . . . . . . . . . . . . . . . . . . . 945.6.4 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 955.7 Rb release from the ampoule . . . . . . . . . . . . . . . . . . 955.8 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.8.1 2D MOT optics . . . . . . . . . . . . . . . . . . . . . 995.8.2 3D MOT optics . . . . . . . . . . . . . . . . . . . . . 1006 2D MOT characterization . . . . . . . . . . . . . . . . . . . . 1046.1 Rubidium atomic beam divergence characterization . . . . . 1046.2 Atomic speed distribution characterization . . . . . . . . . . 1057 Loss rate measurements in a 3D MOT . . . . . . . . . . . . 1127.1 Loss rate variation with the total pressure of residual back-ground gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2 Loss rate variation with the pressure of Ar background gas . 1148 Future outlook and conclusions . . . . . . . . . . . . . . . . . 1188.1 Future outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.1.1 Magnetic trapping coils . . . . . . . . . . . . . . . . . 1188.1.2 NIST gauges . . . . . . . . . . . . . . . . . . . . . . . 1208.1.3 Technical challenges . . . . . . . . . . . . . . . . . . . 1218.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123AppendicesA Loading rate investigation . . . . . . . . . . . . . . . . . . . . 132A.1 Reif model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2 Experimental observables . . . . . . . . . . . . . . . . . . . . 133A.3 The dependence of loading rate on MOT trap depth . . . . . 134A.4 Trap depth determination using loading rates . . . . . . . . . 135A.5 The dependence of loading rate on rubidium density . . . . . 136A.6 Determination of b . . . . . . . . . . . . . . . . . . . . . . . . 138viList of Tables3.1 Measurements of the velocity averaged collisional loss crosssection, 〈σvAr〉87Rb,40Ar, for various MOT pump intensitiesand detunings. The velocity averaged collisional loss crosssection is the slope of the linear fit to measurements of theloss rate constant Γ of a 87Rb MOT versus the density of 40Arin the background gas. The values of the velocity averagedcross section and errors quoted are from the linear fit results. 553.2 3D MOT trap depths for various pump detunings and in-tensities. The trap depths, Ucat, were determined using thecatalysis method described in section 3.3. As a comparisonthe trap depths were also obtained by fitting the numericallycalculated 〈σvAr〉87Rb,40Ar vs U and then determining trapdepth from measured values of 〈σvAr〉87Rb,40Ar. Trap depthsdetermined numerically are denoted U〈σv〉. . . . . . . . . . . . 554.1 Percentage changes for the value of 〈σlossvAr〉87Rb,40Ar due to10, 50 and 100% increases in the values of C6, well depth ,temperature (T ), and trap depth (U). While one parameterchanged the other parameters were held fixed at T = 294 K,C6 = 336.4 a.u.,  = 50cm−1, and U = 5 mK . . . . . . . . . 745.1 Optical Components used in Fig. 5.9 . . . . . . . . . . . . . . 99A.1 MOT trap depths measured using the ‘catalysis method’ forvarious MOT settings. . . . . . . . . . . . . . . . . . . . . . . 136A.2 Parameters used in the calculation of b, the proportionalityconstant between the capture velocity vc and the escape ve-locity ve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139viiList of Figures1.1 A magneto-optical trap. . . . . . . . . . . . . . . . . . . . . . 41.2 Magnetic field lines from coils in anti-Helmholtz configuration. 51.3 Principle of how a magneto-optical trap traps atoms. . . . . . 61.4 The energy levels of the D2 transition for 87Rb. . . . . . . . . 71.5 The energy levels of the D2 transition for 85Rb. . . . . . . . . 81.6 Hyperfine energies as a function of magnetic field for 87Rb inits 52S1/2 ground state. . . . . . . . . . . . . . . . . . . . . . 101.7 Explanation of an ‘RF knife’. . . . . . . . . . . . . . . . . . . 121.8 Intra-trap loss mechanisms in a magneto-optical trap. . . . . 131.9 Shape of atom number loading versus time in a MOT. . . . . 152.1 Collision of two particles in the center of mass frame . . . . . 202.2 Classical collision picture. . . . . . . . . . . . . . . . . . . . . 282.3 Cartoon drawing of protons and electrons for two hydrogenatoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Velocity averaged collisional loss cross section versus trapdepth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Schematic of the optical setup for the the pump and repumplight of the MOT used for loss cross section measurements. . 353.2 Flourescence signal of a MOT loading. . . . . . . . . . . . . . 383.3 Example of experimental data taken to determine the lossrate in a magnetic trap. . . . . . . . . . . . . . . . . . . . . . 403.4 A decay curve for a magnetic trap. . . . . . . . . . . . . . . . 413.5 A plot of velocity averaged collisional loss cross section as afunction of argon density for a MOT and a magnetic trap. . . 423.6 Excitation of ground state atoms to a repulsive excited molec-ular potential with a ‘catalysis’ laser can cause loss of trappedatoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 The experimental setup used to measure the trap depth of aMOT using a catalysis laser. . . . . . . . . . . . . . . . . . . . 453.8 J = N0ssNss − 1 vs the catalysis laser duty factor d. . . . . . . . . 46viiiList of Figures3.9 J vs d shown in the linear region (i.e. small values of the dutyfactor, d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.10 A measure of the loss rate constant due to catalysis plottedversus detuning of the catalysis laser. . . . . . . . . . . . . . . 483.11 Measurement of three different MOT trap depths. . . . . . . 503.12 Velocity averaged collisional loss cross section versus trapdepth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.13 Velocity averaged collisional loss cross section versus trapdepth for the data taken with a MOT. . . . . . . . . . . . . 524.1 A mercury manometer. . . . . . . . . . . . . . . . . . . . . . . 584.2 A capacitance diaphragm gauge. . . . . . . . . . . . . . . . . 594.3 An AC bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4 A schematic of a spinning rotary gauge . . . . . . . . . . . . . 614.5 An ionization gauge. . . . . . . . . . . . . . . . . . . . . . . . 634.6 Schematic of an orifice-flow pressure standard. . . . . . . . . 674.7 A constant pressure flow meter. . . . . . . . . . . . . . . . . . 695.1 Vacuum apparatus for the pressure sensor experiment. . . . . 785.2 A different perspective of the vacuum apparatus for the pres-sure sensor experiment. . . . . . . . . . . . . . . . . . . . . . 795.3 The 2D MOT chamber. . . . . . . . . . . . . . . . . . . . . . 835.4 A cutaway of the differential pumping section. . . . . . . . . . 845.5 The 2D MOT coils installed on the 2D MOT chamber. . . . . 855.6 A drawing of the frame used to wind the 2D MOT coils. . . . 865.7 A picture of the titanium sublimation pump. . . . . . . . . . 885.8 A picture of the bakeout oven. . . . . . . . . . . . . . . . . . 925.9 The schematic of the optical setup used for the pressure sensorexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.10 A picture of the 2D MOT optics along one arm. . . . . . . . . 1015.11 A picture of the 2D MOT cloud. . . . . . . . . . . . . . . . . 1025.12 A picture of the 3D MOT. . . . . . . . . . . . . . . . . . . . . 1036.1 Experimental setup of atomic beam divergence measurement. 1056.2 A fluorescence image of the Rb beam. . . . . . . . . . . . . . 1066.3 Initial fluorescence signal in the 3D MOT from initial turn-onof 2D MOT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.4 The capturable speed distribution of the atomic beam comingfrom the 2D MOT. . . . . . . . . . . . . . . . . . . . . . . . . 110ixList of Figures7.1 An example loading curve taken with a 3D magneto-opticaltrap at a residual background pressure of 1.79 × 10−5 Pa(1.34 × 10−7 Torr). . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 The loss rate constant due to background collisions versustotal background pressure. . . . . . . . . . . . . . . . . . . . . 1147.3 An example decay curve of a 3D MOT in the presence of Arbackground gas. . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4 The MOT loss rate constant due to background collisionsversus background pressure of Ar. . . . . . . . . . . . . . . . 1168.1 A magnetic coil for the 3D MOT and magnetic trap. . . . . 1198.2 Drawing of apparatus with the NIST gauges. . . . . . . . . . 120A.1 Evidence that loading rate of a MOT is proportional to thesquare of trap depth. . . . . . . . . . . . . . . . . . . . . . . 135A.2 Predicted MOT trap depth using the ratio of loading ratesversus measured MOT trap depth. . . . . . . . . . . . . . . . 137A.3 Evidence that the loading rate of a vapour loaded Rb MOTis proportional to the background density of Rb. . . . . . . . 138A.4 Measurement of Mi versus the density of background gas,nRb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.5 A plot of MinRb versus U2i . . . . . . . . . . . . . . . . . . . . . . 141xList of Abbreviations3D three dimensional2D two dimensionalMOT magneto-optical trapRF radio-frequencyUHV ultra-high vacuumCDG capacitance diaphragm gaugeSRG spinning rotary gaugeRGA residual gas analyzerTSP titanium sublimation pumpNEG non-evaporable getterCF ConFlatTA tapered amplifierAOM acousto-optical modulatorNIST National Institute of Standards and TechnologyxiAcknowledgementsThank you to Dr. Kirk Madison for allowing me to be a part of the lab, forpatience in teaching, and for much help given to me. I thank you greatlyfor being my supervisor. Thank you to Dr. James Booth for participationin every aspect of this thesis and the work described in it. Thank you verymuch for the time you spend helping me and others in the lab. Thank youto Dr. Bruce Klappauf for getting the lab off to a good start and helping memany times. Thank you to Dr. Arthur Mills for help many times. Thankyou to Dr. Benjamin Deh for the important work he performed.Thank you to my commitee members Dr. Mona Berciu, Dr. Taka Mo-mose and Dr. Valery Milner for their time, help, and useful commentary.Thank you to the external examiner Dr. Paul Haljan for extremely usefulsuggestions and revisions, and for the time spent in reading and correctingthis thesis. Thank you to my university examiners, Dr Fei Zhou and Dr.Ed Grant, for the time spent reading my thesis, and useful commentaryduring the final oral exam. Thank you to the chair, Dr. Ezra Kwok, for histime spent. Thank you again to Dr. Mona Berciu and to Dr. Ovi Toaderfor help many times. Thank you to Dr. James Fedchak for the time spentcollaborating with our lab and for sending us calibrated gauges from NIST.Thank you to my physics professors and teachers: Mr. Bergen, Dr.Williams, Dr. Hayden, Dr. Ozier, Dr. Axen, Dr. McCutcheon, Dr.Halpern, Dr. Van Raamsdonk, Dr. Michal, Dr. Kiefl, Dr. Berciu, Dr.Zhou, Dr. Milner, Dr. Jones, Dr. Madison, Dr. Franz, Dr. Mattison, Dr.McKenna, Dr. Gay, Dr. Affleck, Dr. Hardy, Dr. Krems, Dr. Schleich,Dr. Rozali, Dr. Hickson, Dr. Rieger, Dr. Waltham, Dr. Kotlicki, and Dr.Carolan. Thank you to Dr. Blades. Thank you to Dr. Bates, Dr. Koster,Dr. Turrell, and Dr. Ives. A special thank you to Dr. Lam and Dr. Gerryfor their support and encouragement.Thank you to the physics electronics shop: Pavel, Stan, Richard, Gar,Marcel, Dave, Victor and Tom. Thank you to the physics machine shop:Martin, Dan, Nick, Matt, and Steve. Thank you to Domenic. Thank you toDoug, Joseph, and Tongkai. Thank you to the physics stores staff: Brian,Geoffrey, and Mark. Thank you to the physics department staff, especiallyxiiAcknowledgementsOliva, Bridget, Theresa, and Anilu. Thank you to the EngPhys project lab:Dr. Marziali, Dr. Nakane, and Bernhard. Thank you to the IT physicsdepartment: Ron, Mary Ann, Gerry, Hongyun, and Tom. Thank you tothe chemistry electronics shop staff: Brian, Benny, and Tony. Thank you tothe chemistry stores staff: John, Karen, Pat, Nate, Xin-Hui, and Sabrina.Thank you to the machinists at BCIT: Ernie and Glenn. Thank you tothe custodial staff, Harry, UBC security, UBC plant-ops, UBC buildingoperations, and building management for their help and encouragement.Thank you to all members of the lab, as well as thesis or project students,past and present. In particular, thank you to Tao, Jordan, Amy, Swati, Syl-van, Keith, Paul, Ray, Davey, Pavle, Bastian, Nina, Peter, Aviv, David,Alan, Bo, Magnus, Dallas, Chenchong, Gabriel, Tino, Theo, Ian, Weiqi,Haotian, Kousha, Nathan, Nam, Kais, Troy, Greg, Will, Will, Jon-Paul,Damien, Joshua, Kyle, Alec, Alysson, Brendan, Daniel, Mariusz, Gene,Tom, Zhiying, Felipe, Victor, Adrian, Glenn, Kahan, Kai, Koko, Steve,Calvin, Paul, Julien, Fumiei, Forrest, Blake, and Jeffrey. A special thanksto Mariusz, Will, Will, Gene, Tom, and Kais, who are the current core grad-uate group and have helped greatly. Another special thanks to Ian, Weiqi,Haotian, Kousha, Victor, Adrian, Dr. James Booth, Dr. Kirk Madison,Kais, Jeffrey, Blake and Forrest who have greatly contributed to the pres-sure measurement experiment. Thank you to the members past and presentof Dr. Momose’s group, Dr. Jones’s group, Dr. Milner’s group, Dr. Krem’sgroup, Dr. Shapiro’s group, and Dr. Grant’s group. Thank you to Dr.David Jones, T.J., Rob, Sybil, Matt, Yifei, Egor, Joshua, Emily, Carolyn,Marc, and Graeme. Thank you to Pavle for help many times. Thank youto those that attended my doctoral exams for their support. Thank youto the undergraduate and graduate students I have met, especially Lionel,Raveen, Anand, Dan, Sergei, Ryan, Jeff, Mandy, Eva, Adrian from chem-istry, Lara, Jasmine, Cecile, and Karla. Thank you to the Graduate Studiesstaff. Thank you to Murray McEwan at Agilent. Thank you to Randy,Jomy, Jinshan, Maria, Ian, Claire, and Patricia. Thank you to my Dad.Thank you to Jesus and Our Father.xiiiChapter 1Introduction1.1 Thesis overviewLoss of ultracold trapped atoms due to collisions with background (nontrapped) particles can be an unwanted feature that reduces the sample sizeof the trapped atoms to be studied. This thesis, however, shows that the lossrate of trapped atoms due to background gas collisions can be an importantand useful experimental observable. The first part of this thesis (chapters 1-3) describes how the velocity averaged collisional loss cross section, 〈σvi〉X,i,due to collisions between trapped atoms of type X and background gas ofspecies i, is calculated and measured. This quantity is related to the lossrate due to background collisions and is shown to have significant trap depthdependence when comparing trap depths of several mK to several K. Thistrap depth dependence can be important to take into account for collisioncross section measurements which use loss of trapped atoms due to collisionsas their measurement observable. A wide variety of such experiments existinvolving trapped ions [1–3], electron beams [4, 5], and atoms and molecules[6–10]. Photoionization cross sections have also been measured [11–13] usingtrapped atoms. As shown in Ref. [9] and Ref. [14], the measured collisioncross section based on loss from a trap can be lower than the total crosssection. This is because not all collisions impart sufficient kinetic energy tothe trapped atoms to cause loss from the trap.In our experiments 87Rb was used as the trapped species, X, and 40Ar asthe background species i. The velocity averaged collisional loss cross section〈σvAr〉87Rb,40Ar was calculated for a range of trap depths and measured pre-viously by members of our lab using a magnetic trap [14, 15]. The magnetictrap used could obtain trap depths up to 10 mK. The work performed forthis thesis was an extension of the measurement of 〈σvAr〉87Rb,40Ar to largertrap depths (0.5 to 2.2 K) using a magneto-optical trap (MOT). A techniquewas adapted from Hoffmann et al. [16] to measure the trap depth of a MOT.In this way a verification of the shape of the calculated 〈σvAr〉87Rb,40Ar ver-sus trap depth curve was performed for the trap depths attainable with ourmagnetic trap and magneto-optical trap. The results of this work are also1Chapter 1. Introductionreported in [17].Chapter 1 provides necessary background information on the atomictraps used in this work. Chapter 2 gives a basic understanding of the cal-culation of the velocity averaged collisional loss cross section. Chapter 3.1explains and presents results for the measurement of the velocity averagedcollisional loss cross section using a MOT. The measurement of the MOTtrap depth is also described in this chapter.The second, and related, topic covered in this thesis (chapters 4 - 8) isthe progress made towards using the background collision induced loss rateto determine the density of the background gas. Using trapped atoms as adensity (pressure) sensor would be a novel approach for a pressure standardin the range of 10−5 to 10−8 Pa (10−7 to 10−10 Torr). The potential advan-tages of such a standard are reproducibility from lab to lab, the potential ofminiaturization and portability, and the possibility of externally calibratedgauges not being needed, or needed only seldomly.Chapter 4 gives background information on the commercial pressuregauges installed in our apparatus and existing pressure standards. Chapter 5gives a description of our experimental apparatus for pressure measurementand details about the assembly of this apparatus to date are provided. Chap-ter 6 describes characterization performed of the two dimensional magneto-optical trap that is part of the apparatus. Chapter 7 describes loss ratemeasurements using a 3D magneto-optical trap as pressure in the systemvaried. Chapter 8 contains conclusions as well as the future outlook for thepressure experiment.1.2 Atomic trapsThe “trapping of atoms” means to spatially confine them with light, electricfields, or magnetic fields so that the atoms are held in vacuum and therebyisolated from the walls of the vacuum chamber. The average temperature,T , of the trapped atoms is defined by the average kinetic energy of theatoms as kBT . Typical temperatures are in the µK to mK range. Everytrap has an associated trap depth, Utrap = 12mv2e , where ve is the escapespeed needed for an atom to leave the trap. Trap depth will depend on theparameters of the confining fields such as light intensity, light frequency, andmagnetic field gradient. Depending on what type of trap is being used, trapdepths can go up to several K. This section will discuss the principle of a3D magneto-optical trap and a quadrupole magnetic trap. Atom numberdynamics in these traps is discussed in the next section.2Chapter 1. Introduction1.2.1 3D magneto-optical trapA magneto-optical trap (MOT) provides a means of obtaining a sampleof ultracold atoms starting from a vapour or with a beam of atoms [18,19]. A magneto-optical trap both slows (cools) and traps (confines) atoms.In a 3D MOT (see Fig. 1.1) three orthogonal counter-propagating pairs oflaser beams with frequency, ω, slightly tuned below an atomic resonance,perform laser cooling [19]. Laser cooling requires the light to be belowatomic resonance because of the Doppler shift. In the Doppler effect an atomtravelling with velocity ~v will ‘see’ light with wavevector ~k and frequency ωas having frequency ω′ = ω − ~k · ~v. When an atom with velocity ~v absorbsa photon with wavevector ~k, the momentum of the atom is changed by ~~k.The atoms entering the intersection of the orthogonal laser beam pairs willbe slowed down by photons with ~k · ~v < 0. Photons with ~k · ~v > 0 wouldspeed up the atoms. In the second case these photons have their frequencyshifted farther away from resonance so that they have a smaller probabilityof being absorbed. In this manner the atoms preferentially absorb photonsthat slow them down. Atoms spontaneously emit the photons in randomdirections so that, averaging over many absorption and emission events, themomentum gain from emission is zero and the atoms are slowed down.Laser cooling does not slow the atoms down completely to zero velocity.Momentum kicks imparted to the atoms from absorption and emission ofphotons result in residual motion of the atoms. Laser cooling also doesnot form a trap. The slowing force depends on their velocity and not theirposition so the atoms will diffuse out of the intersection of the laser beams.To provide a position dependent force for trapping, a magnetic field is addedproduced by two concentric coils in an anti-Helmholtz configuration. Theanti-Helmholtz configuration consists of coils that are parallel and spaced bytheir own radii, and have equal current through the coils running in oppositedirections. The direction concentric with the coils’ centers is called the axialdirection. For the direction of current shown in Fig. 1.2 the magnetic fieldalong a line parallel to, and half way in between, the two coils starts at zeromagnitude at the origin and increases linearly outwards pointing away fromthe center along the radial direction. Along the axial direction in Fig. 1.2 themagnetic field points toward the center of the MOT and the axial magneticfield gradient, d| ~B|dz is double the gradient along the radial directions [20].In a MOT the magnetic field in combination with the appropriate po-larization choice of the laser beams provides confinement of the atoms. Ina MOT atoms preferentially absorb light that pushes them to the magneticfield zero. The trapping region is at the intersection of the laser beams cen-3Chapter 1. IntroductionFigure 1.1: A magneto-optical trap (MOT). Three counterpropagating pairsof laser beams along three perpendicular axes are used along with two mag-netic coils in anti-Helmholtz configuration. The laser polarization is rightcircularly polarized (RCP) for beams travelling along the radial directionwhere the magnetic field is pointing radially outwards. The laser polariza-tion is left circularly polarized along the axial direction (concentric with thecoils) where the B field is pointing towards the center of the MOT. Thetrapping region is formed at the intersection of the six beams centered onthe zero of the magnetic field and a cloud of atoms will be collected there.tered on the zero of the magnetic field. Figure 1.3 explains the principle of aMOT. The laser light with energy hfL is detuned below the F = 0 to F ′ = 1transition, taken as an example. In the presence of the weak magnetic fieldof magnitude B generated in a MOT, the hyperfine mF sublevels changeaccording to ∆E = mF gFµBB, where gF is the Lande´ g-factor and µB isthe Bohr magneton [21, 22]. For gF ′ > 0 the mF ′ = −1 transition decreasesin energy away from the magnetic field zero position.Along the radial axes, right circularly polarized (RCP) light propagatingin the same direction as the B-field will drive ∆mF = mF ′−mF = +1, calledσ+ transitions. Right circularly polarized (RCP) light propagating in theopposite direction to the B-field will drive ∆mF = mF ′ −mF = −1 , calledσ− transitions. In Fig. 1.3 the mF = 0 to mF ′ = −1 transition is closest tothe laser frequency so the atoms would preferentially absorb the σ− light. Ifσ− light is travelling toward the center on either side of the magnetic fieldzero then an atom which goes away from the center will be pushed backtowards the center. Along the z axis, concentric with the coils, the B field ispointing inwards (as shown in Fig. 1.2). For this case left circularly polarized4Chapter 1. IntroductionFigure 1.2: The magnetic field lines generated by passing current I inopposite directions through two coils spaced by their radii (known as anti-Helmholtz configuration). The magnetic field is zero at the center betweenthe coils and increases linearly in magnitude pointing away from the centerin the radial direction and towards the center in the axial direction. Thez-axis is the axial direction. The x-axis (coming out of the page) and they-axis are pointing radially.light must be used along the z axis. Left circularly polarized light travellingin the same direction as the magnetic field will drive ∆mF = mF ′−mF = −1transitions and ∆mF = mF ′ −mF = +1 transitions when anti-aligned withthe magnetic field.For the work presented in the body of this thesis, 87Rb was primarilyused as the trapped atom species. The system used also has the abilityto trap 85Rb, which was used as the trapped atom of choice for the workpresented in the appendix. The choice to use one species or the other is oftena practical matter of availability of the lasers used for each species. For workinvolving magnetic trapping, as will be discussed in the next section, it ismore convenient to use 87Rb in order to trap atoms that are all in the samehyperfine sublevel.The laser light used for the trapping in the MOT is called the pumplight. A secondary laser light, called the repump light, is also needed toprevent atoms from pooling in a hyperfine state that cannot be excited bythe pump laser. The 52S1/2 to 52P3/2 transition is used for the pump and5Chapter 1. IntroductionFigure 1.3: Hyperfine sublevels undergoing a continuous energy shift withchanging magnetic field in a magneto-optical trap. The vertical axis, notlabelled, is energy. Atoms that move to the right or left of the magneticfield zero have a greater probability of absorbing light on a σ− transitionthat pushes the atoms back towards the magnetic field zero.repump processes (see Figs. 1.4 and 1.5). The pump is chosen to be resonantwith the F = 2 to F ′ = 3 transition for 87Rb and F = 3 to F ′ = 4 transitionfor 85Rb. The repump is chosen to be resonant with the F = 1 to F ′ = 2transition for 87Rb and F = 2 to F ′ = 3 transition for 85Rb. Taking 87Rbas an example, F = 2 to F ′ = 3 light can also drive off-resonant transitionsto the F ′ = 2 or F ′ = 1 hyperfine level. These levels can decay to the F = 1hyperfine state of the 52S1/2 ground state. The atoms will not stronglyabsorb the pump light once in that state and laser cooling and trappingwill not be accomplished. In order to get the atoms out of this ‘dark’ state,repump light driving the F = 1 to F ′ = 2 transition is needed so that thereis some probability of atoms decaying back to the F = 2 ground state.6Chapter 1. Introduction2/12S52/32P5 780.24 nm(384.23 THz)266.65 MHz156.95 MHz72.22 MHz6.83 GHzF'= 3F'= 2F'= 1F'= 0F= 2F= 1Figure 1.4: The energy levels for the D2 transition for 87Rb. The F = 2to F ′ = 3 transition is used for the pump light of the magneto-optical trap.The F = 1 to F ′ = 2 transition is used for the repump light. Numericalvalues and concept of figure from [21].1.2.2 Magnetic trapsPrinciple of a magnetic trapAfter cooling and collecting atoms in a MOT they can be transferred to amagnetic trap for further study or cooling. The initial cooling stage in aMOT is necessary because magnetic traps provide much weaker confinementforces than a MOT and do not provide cooling. A quadrupole magnetic trapcan be formed using the same coils as used for the MOT. The laser lightis shut off and a higher current than used for the MOT is run through thecoils to produce a higher magnetic field gradient. The magnetic field zero iswhere the middle of the trap is. The magnetic field increases in magnitudeaway from where the magnetic field zero is located.The contribution to the hamiltionian due to the electron spin, ~S, elec-7Chapter 1. Introduction2/12S52/32P5 780.02 nm121.0 MHz63.4 MHz29.3 MHz3.04 GHzF'= 4F'= 3F'= 2F'= 1F= 3F= 2Figure 1.5: The energy levels for the D2 transition for 85Rb. The F = 3to F ′ = 4 transition is used for the pump light of the magneto-optical trap.The F = 2 to F ′ = 3 transition is used for the repump light. Numericalvalues and concept of figure from [23].tron orbital angular momentum, ~L, and nuclear spin, ~I, interacting with anexternal magnetic field, ~B, is expressed asHB =µB~ (gS~S + gL~L+ gI~I) · ~B. (1.1)Here µB = 9.274× 10−24 J/T is the Bohr magneton, and gS , gL, and gI arethe spin, orbital and nuclear g-factors, respectively.For weak magnetic fields, HB is a perturbation to the hyperfine en-ergy levels, and the hyperfine levels change according to ∆E = gFmFµBB.Atoms in states that have Zeeman shifts resulting in higher energy as mag-netic field increases (states with gFmF > 0) will tend to stay near the zero8Chapter 1. Introductionof the field and are called ‘low field seekers’. Atoms that are ‘high fieldseekers’ will leave the trap.Referring to the energy level diagram in Fig. 1.4, for example, gF =−1/2 for the F = 1 hyperfine level of the 52S1/2 ground state for 87Rb andgF = +1/2 for the F = 2 hyperfine level. This means the trappable state forthe F = 1 level is mF = −1 and for the F = 2 state the sublevels mF = 1, 2can be trapped. The atoms can be prepared in the lower F = 1 or upperF = 2 state by the order of turn-off of the pump and repump when loadingthe magnetic trap. If the repump is turned off a few ms before the pumpthen the atoms will be in the lower hyperfine level. Conversely, if the pumpis turned off first the atoms will be in the upper hyperfine level of the 52S1/2ground state.87Rb is preferable for magnetic trapping since its lower hyperfine groundstate (F = 1) only has one trappable state (mF = −1). For 85Rb, theF = 2 lower level of the 52S1/2 ground state has gF = −1/3 and so boththe mF = −2 and mF = −1 state are trappable. Trap depth depends onthe mF state and applications where trap depth is important requires asingle mF state to be trapped. Gravitational filtering can be used to isolatethe F = 2,mF = −2 state for 85Rb. This is performed by lowering themagnetic field gradient (and hence trap depth) so that the mF = −1 stateis not supported against gravity while the mF = −2 state is.The expression ∆E = gFmFµBB for the energy change of the hyperfinelevels with magnetic field is only valid for weak magnetic fields ( . 0.001 Tor . 10 G [22]). For larger magnetic fields where the HB term of the hamil-tonian is comparable to the hyperfine term, the two contributions togetherare a perturbation on the fine structure states. The energy shifts on thefine structure in this case are given by diagonalizing the hyperfine and mag-netic field contribution. For J = 1/2 the energy shift is conveniently givenby the Breit-Rabi formula [22]. For strong fields the HB term is treatedas a perturbation on the fine structure which cause the energy of the finestructure states to change according to ∆E = gJmJµBB. The hyperfineterm is a further perturbation on those fine structure states which causesfurther splitting and energy level change for different mI values. Figure 1.6shows the change in the hyperfine levels of 87Rb in the 52S1/2 ground state[21]. For weak magnetic fields the energy shift is linear with magnetic fieldmagnitude. For intermediate fields the energy levels can change non-linearlywith magnetic field and states that were low field seeking at weak fields (forexample the |F = 1,mF = −1〉 state) can become high field seeking. Simi-larly a state that was high field seeking at weak fields can become low fieldseeking at higher fields (for example the |F = 2,mF = −1〉 state). For9Chapter 1. Introductionsufficiently large fields this must be taken into account when determiningthe trap depth and which states are trappable.0 15000magnetic field (G)10000500025-25E/h (GHz)F = 1F = 2mJ = -1/2mJ = +1/20Figure 1.6: Energy level changes for the 52S1/2 ground state of 87Rb as afunction of increasing magnetic field. Reproduced with permission from Dr.Steck [21].For this work we typically work at up to a maximum of 400 G in theweak and intermediate magnetic field regimes. Taking the weak field case,the trap depth of the magnetic trap, along a given direction, for an atomin state mF will be ∆E = gFmFµB(Bmax − Bmin) where Bmax(Bmin) isthe maximum (minimum) magnetic field a trapped atom can access alongthat direction. For quadrupole magnetic fields created by the coils in anti-Helmholtz configuration, Bmin = 0 and Bmax is limited either by the sizeof the vacuum cell or the roll-off of the magnetic field. In our case the sizeof the cell walls limits the maximum magnetic field the trapped atoms canexperience. The cell walls are at room temperature and so when a trappedatom hits the cell wall it gains a large amount of kinetic energy and will notbe trapped anymore.The magnetic field gradient is different in different directions away from10Chapter 1. Introductionthe magnetic field zero which means the trap depth is not the same in alldirections. For the work described in chapter 3, the trap depth, U , assignedto the magnetic trap used was taken as an average of the maximum energiesof a trapped atom along the downard axial direction and the radial direction.kBU =[E(B(zw))−mgzw] + E(B(yw))2 (1.2)Here zw and yw are the distances from the magnetic field zero to the cell wallalong the axial and radial directions, respectively. For the work in chapter 3these were both 0.5 cm. B(zw) and B(yw) are the magnetic field magnitudesat the cell wall along the axial and radial directions, respectively. E(B(zw))and E(B(yw)) are the energies of the atoms at the cell wall along the axialand radial directions, respectively. Themgzw term accounts for the decreasein trap depth along the downward axial direction where m is the mass ofa trapped atom and g is the gravitational acceleration. For a distance ofzw = 0.5 cm this gives mgzw/kB = 0.5 mK, which lowers the trap depthalong the downward axial direction for 87Rb. Magnetic traps have typicaltrap depths on the order of a few mK.The magnetic field from a pair of coils in anti-Helmholtz configurationcan be approximated asBz u 3µ0DR2(D2 +R2)5/2 Iz (1.3)andBy u32µ0DR2(D2 +R2)5/2 Iy. (1.4)Here µ0 is the permeability of free space, 2D is the spacing between the topand bottom coils, R is the coil radius, and I is the current. The coordinatesy and z are the distances from the origin along the y and z directions,respectively, as shown in Fig 1.2. The predicted gradient along the axialdirection, dBdz , is double that along the radial direction, dBdy . This meansthe trap depth along the axial direction is approximately double that of theradial direction.For the expressions for magnetic field given in Eq. 1.3 and Eq. 1.4 themagnetic field increases linearly in magnitude away from the magnetic fieldzero. A more accurate non-linear expression for the magnetic field for acircular loop is given in Ref. [24]. A discussion of the error this introducesis given in section 1. IntroductionRF knifeA useful way of controlling the trap depth of a magnetic trap is with a‘Radio Frequency (RF) knife’ [25]. The concept is explained in Fig. 1.7.An RF frequency is introduced while the magnetic trap is on. When thexBBm F  = 0m F  = -1m F  = +1m F  = -1m F  = 0m F  = +10F=1Figure 1.7: Hyperfine sublevels undergoing a continuous energy shift withchanging magnetic field in a magnetic trap. The vertical axis, not labelled,is energy. The trappable state shown here is the mF = 1 state (assuminggF > 0 here). An RF frequency can cause transitions to the untrappablemF = 0 state and atoms will be lost from the trap.splitting between levels of a trappable state (e.g. mF = 1 in Fig. 1.7) and anuntrappable state (e.g. mF = 0 in Fig. 1.7) corresponds to the RF frequency,atoms that are in the trappable state can transition to an untrappable state.This limits the maximum energy an atom can have in the trap and thereforethe trap depth.1.3 Trap dynamics1.3.1 Magneto-optical trap dynamicsIn a magneto-optical trap the number of atoms in the trap as a function oftime, N(t), can be given as [26]dNdt = R− ΓN − β∫n2(~r, t) d3~r (1.5)with t = 0 being when both the magnetic field and light have been turnedon [27]. R is the rate of capture, which is the number of atoms per second12Chapter 1. Introductionentering the intersection of the six beams and being slowed down and con-fined. ΓN is the rate of loss due to collisions of background gas atoms withthe trapped atoms. Γ has units of s−1 and is called the loss rate constantdue to background collisions. τ = 1/Γ is called the lifetime of the trap. Γhas typical values of 0.1 to 2 s−1 for the experimental setup described inchapter 3 and the loading rate, R, is around 106 s−1 to 107 s−1.The last term of Eq. 1.5 describes losses due to the collision of twotrapped atoms, where n(~r, t) is the density of the trapped atoms at position~r and time t. The origin is placed at the center of the trap where theatoms collect in a roughly spherical ball. β is the loss rate constant due totwo-body ‘intra-trap’ collisions. It is on the order of 10−11 to 10−13cm3s−1and is mediated by radiative escape, fine-structure changing collisions andhyperfine changing collisions [27, 28]. Fig. 1.8 shows the mechanisms ofFigure 1.8: Radiative escape occurs when two ground state atoms are ex-cited by a photon of energy hf and gain kinetic energy ∆RE before emittinga photon of energy hf ′ < hf to return to the ground state. Fine structurechanging collisions occur when atoms in the 52S1/2 + 52P3/2 excited statetransfer over to the 52S1/2 + 52P1/2 state. They then emit a photon of en-ergy hf ′′ to return to the ground state and gain kinetic energy ∆F . Figurepatterned after Fig. 14 in [27].radiative escape and fine-structure changing collisions. In the presence oflight, ground state atoms (for example, both atoms in the 52S1/2 state )can be excited by a photon of energy hf at a certain internuclear separationto an excited molecular potential (for example, the 52S1/2 + 52P3/2 state).13Chapter 1. IntroductionThe atoms can accelerate to an internuclear separation where the potentialenergy is lowered and emit a photon of lower energy hf ′ back to the groundstate (52S1/2+52S1/2). The kinetic energy picked up by the atoms is ∆RE =hf − hf ′ and if the kinetic energy imparted to each atom is greater thanthe depth of the trap, loss will result. This mechanism of loss is calledradiative escape [27, 28]. Fine-structure changing collisions occur when theatoms excited to the 52S1/2+52P3/2 state switch over to the 52S1/2+52P1/2state at the cross-over between the two states. The atoms return to largeinternuclear separations along the 52S1/2 + 52P1/2 state and emit a photon(hf ′′) back to the ground state. In this process ∆F of energy is gained,which is the fine structure energy splitting between the 52P1/2 state and the52P3/2 state. For homonuclear collisions each atom gains ∆F/2 in kineticenergy. For 87Rb the energy gained by each atom in a fine-structure changingcollision, in terms of temperature, is ∆F2kB = 171K. This kinetic energy gainfar exceeds typical MOT trap depths of several K.A loss mechanism that can also occur is hyperfine changing collisionswhere atoms colliding can change their hyperfine state, i.e. their F level.For example (referring to Fig. 1.4) two 87Rb atoms colliding in the F = 2state of the 52S1/2 ground state can during a collision change to the F = 1level. The F = 1 level is lower in energy than the F = 2 level and theenergy difference goes to the kinetic energy of the atoms. One or both ofthe colliding atoms can change their F level [27] while the sum of the mFvalues remains the same.If β∫n2(~r, t) d3~r is negligible compared to the background loss rate termthen the solution to Eq. 1.5 isN(t) = RΓ(1− e−Γt). (1.6)The steady state solution in this case is N(t = ∞) = RΓ . Typically for aMOT operated in this regime the steady state number in a MOT is around107 to 1010 atoms. An example plot of N(t), given by Eq. 1.6, is shown inFig. 1.9.Another case to consider is when β∫n2(~r, t) d3~r is not negligible, butthe atom number density is less than approximately 1010 atoms per cm3[29]. In this case the atom cloud in the MOT has a gaussian density pro-file of n(~r, t) = n0(t)e−(|~r|w)2, where n0(t) is the peak density at |~r| = 0,and w is the width of the gaussian distribution which is taken to be time-independent. For this density profile the integral in Eq. 1.5 is∫n2(~r, t) d3~r =[n0(t)]2(w√pi2)3. The total number in the trap taken by integrating the den-14Chapter 1. Introduction0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0time (s) 1.9: A plot of atom number, N(t), versus time, t, from initial MOTloading. The plot uses the model given in Eq. 1.6 with R = 2.2×107 atoms/sand Γ = 2.1 s−1. The steady-state atom number is N(∞) = RΓ = 1.05×107.sity is N(t) = n0(t) (w√pi)3. Using these results Eq. 1.5 simplifies todNdt = R− ΓN − aN2 (1.7)where a = β(w√2pi)3 . The solution to Eq. 1.7 isN(t) = Nss( 1− e−γt1 + ξe−γt)(1.8)where γ = Γ + 2βnss and ξ = βnssΓ+βnss [17]. The steady state number Nss isNss =RΓ + βnss(1.9)where the average steady state density of the MOT, nss, isnss =(∫n2 d3r∫n d3r)ss. (1.10)When the density of trapped atoms in a MOT is large the probabilityincreases for the trapped atoms to absorb photons emitted by other trapped15Chapter 1. Introductionatoms (called multiple scattering [29]). This results in an outward forceon the trapped atoms which sets a limit on the maximum density of thetrapped atoms at around 1010− 1011 cm−3 [30]. One approximation used inthis regime is that the trapped atom density is a constant, n, and Eq. 1.5becomesdNdt = R− ΓN − βnN. (1.11)The solution to this equation can be solved asN(t) = RΓeff(1− e−Γeff t)(1.12)whereΓeff = Γ + βn. (1.13)To determine the background pressure range at which the intra-trap termcan be neglected we use approximate values of β = 10−11 cm3s−1, a totalatom number N = 108, and a trapped atom density of n = 1010 cm−3. Thisgives βnN ≈ 107s−1. In order for ΓN to be an order of magnitude greaterthis means Γ > 1 which corresponds to pressures in the mid 10−6 Pa (10−8Torr) range and above.1.3.2 Magnetic trap dynamicsMagnetic traps are loaded once with an initial number of atoms that thendecays due to losses such as background collision induced loss. The numberof atoms, N , in a magnetic trap after a time, t, from initial loading followsdNdt = −ΓN − β∫n2(~r, t) d3~r −MN. (1.14)As in Eq. 1.5 for a MOT, Γ is the loss rate constant due to background colli-sions. The β term is the loss rate constant due to intra trap collisions wheren is the density of the trapped atoms. Here this intra-trap term includesonly hyperfine changing collisions. Fine structure changing collisions andradiative escape are not loss mechanisms in a magnetic trap because thosemechanisms involve the presence of light. M is the loss rate constant due toMajorana spin flip losses. This loss mechanism can occur when a trappedatom travels past the zero in the magnetic field, present in a quadrupoletrap, where the magnetic field changes directions. In this case the mF valuecan change to the opposite sign and the atom will no longer be trappable[28].16Chapter 1. IntroductionThe hyperfine collision loss rate for 87Rb for the F = 2, mF = 2 groundstate is estimated to be on the order of 10−15cm3/s [31]. To compare thesize of the intra-trap loss term to the background loss term, we estimatethe initial number in the magnetic trap to be on the order of 108 atoms. Γis approximately 0.01 s−1 for background gas pressures in the mid 10−8 Pa(10−10 Torr) range. This gives ΓN ≈ 106 s−1. Assuming a 1 cm3 magnetictrap size, an order of magnitude estimate for the intra-trap loss rate isβnN ≈ 10−15 cm3s × 108cm−3 × 108 ≈ 10s−1. The intra-trap loss rate is wellbelow the background loss rate expected in the 10−8 Pa (10−10 Torr) range.The decay time due to Majorana spin flips is estimated as τ = ml23~ wherem is the mass of a trapped atom, and l is the radius of the trapped atoms[28]. Taking 87Rb as the trapped atom and the trap size to be l = 1 cm, thisgives a decay time of around 12 hours, which is far greater than the typicaldecay time 1/Γ ≈ 100 s at the mid 10−8 Pa (10−10 Torr) range.If we assume then that the intra-trap and Majorana loss terms are neg-ligible, the solution to Eq. 1.14 becomesN(t) = N(0)e−Γt (1.15)where N(0) is the initial number in the magnetic trap.This thesis will focus on the loss rate constant due to background colli-sions, Γ. Its dependence on trap depth and density of background gas arethe two main topics presented. A pressure standard based on measurementsof Γ is proposed and the current experimental progress to that end is de-scribed. To start, the next chapter explains the loss rate constant Γ in termsof its dependence on density of background gas and its dependence on thecollisional loss cross section.17Chapter 2Background gas collisioninduced lossThe situation of interest is that of an elastic collision involving a backgroundgas atom and a trapped atom. The background gas particle could be resid-ual background gas from the outgassing of vacuum parts, atoms that areof the trapped species that are not trapped, or a purposefully introducedbackground species such as Ar, N2 or He.The loss rate constant due to background gas collisions, Γ, introducedin the last chapter, can be written asΓ =∑ini〈σlossvi〉X,i (2.1)where ni denotes the density of a particular background species i. Thesymbol σloss is the collisional loss cross section between a trapped atom oftype X and a background species particle of type i [32, 33]. As we willsee later in this chapter σloss is dependent on the relative kinetic energy ofthe colliding particles. The background gas speed, vi, should actually bethe relative gas speed, vr. For our case our trapped atoms are assumed tobe stationary, in the lab frame, with respect to the background gas atoms.In this case, the relative speed, vr, of a trapped atom and a backgroundgas particle is the speed of the background gas particle before collision, vi.The brackets indicate a Maxwell-Boltzmann average over all possible speeds,vi = 0 to vi = ∞, of background species i. Because we are describing lossfrom a trap, σloss also depends on the depth of the trap. If the trap depthis anisotropic then an average trap depth is taken. The calculation of thevelocity averaged collisional loss cross section, 〈σlossvi〉X,i, is described indetail in [14, 15]. This chapter serves as an overview of those works.2.1 A brief review of necessary scattering theoryTo properly describe the meaning and calculation of 〈σlossvi〉X,i, a basicreview of quantum scattering theory is needed. The following is based pri-18Chapter 2. Background gas collision induced lossmarily on the honours thesis of David Fagnan [15] who performed the calcu-lation of 〈σlossvi〉X,i for our group. The calculation was for 87Rb in its groundstate as the trapped species and 40Ar in its ground state as the backgroundspecies. The following quantum textbooks and scattering theory notes arealso helpful: Refs. [34–38].The hamiltonian describing two interacting particles of mass m1 and m2isH = |~p1|22m1+ |~p2|22m2+ V (|~r1 − ~r2|) (2.2)where ~r1, ~p1 = m1~v1, ~r2, ~p2 = m2~v2 are the positions and momenta ofparticle 1 and 2 respectively.The coordinate of the center of mass is~R = m1~r1M +m2~r2M (2.3)and the velocity of the center of mass is~vR =m1~v1M +m2~v2M (2.4)where M = m1 + m2 so that M~vR = ~p1 + ~p2 = ~P where ~P is the totalmomentum. By conservation of total momentum ~vR is a constant. In thecenter of mass frame ~vR = 0 so that the total momentum is zero and theparticles have equal and opposite momenta. It is also useful to use therelative coordinate ~r = ~r1 − ~r2 and the relative velocity ~vr = ~v1 − ~v2. Therelative velocity vector is the same in the lab and in the center of mass frame.Under conservation of momentum and total kinetic energy the magnitude ofthe relative velocity, |~vr| = |~v1 −~v2|, is a constant before and after collision.This is most easily seen in the center of mass frame where the speeds ofparticles 1 and 2 are unchanged by an elastic collision. The direction of ~vrdoes change before and after collision. Using the center of mass coordinateand the relative coordinate the hamiltonian given in Eq. 2.2 can be connectedto quantum mechanical operator form:H = − ~22M∇2R −~22µ∇2r + V (r) (2.5)where µ = m1m2m1+m2 is the reduced mass and ∇2R and ∇2r are the laplacianswith respect to the center of mass and relative coordinates, respectively. Inthis form the hamiltonian is composed of a sum of a center of mass part anda relative part.19Chapter 2. Background gas collision induced lossA time independent approach will be used because a time dependentwavepacket treatment given in [36] gives the same result for the differentialcollision cross section that is derived later in this chapter. A solution to thetime independent Schro¨dinger equation, Hψ = Eψ, is ψ = ψRψr where ψRsatisfies the equation− ~22M∇2RψR = ERψR (2.6)and ψr satisfies[− ~22µ∇2r + V (r)]ψr = Erψr (2.7)where E = ER + Er. The equation involving the center of mass coordinateis just that of a free particle of mass M where ER = P2R2M and PR is themagnitude of the momentum of the center of mass. In the center of massframe with PR = 0 we have ER = 0 and E = Er. It also suffices to havePR to be a constant so that ER is a constant. The second equation has theform of a single particle of reduced mass µ that is subject to a sphericallysymmetric potential V (r). A collision of two particles is shown in the centerFigure 2.1: In the center of mass frame particles 1 and 2 travel with equaland opposite momenta. After collision they travel along a line that makesan angle θ with respect to the original line of incidence.of mass frame in Fig. 2.1 where the two particles approach each other withequal and opposite momenta, interact and then recede from each other withequal and opposite momenta. The path of the receding particles makesan angle θ in the center of mass frame with respect to the original line of20Chapter 2. Background gas collision induced lossincidence. Note the angle θ is the angle between the relative velocity vectorsbefore and after collision and is the same in both the lab and center of massframes.The assumption is now made that the change in speed (in the lab frame)of the trapped atom due to collision is much greater than the initial speed ofthe trapped atom before collision. With this assumption the kinetic energyimparted to the trapped atom with mass m1 in the lab frame is [15]∆E ≈ µ2m1|~vr|2(1− cos(θ)). (2.8)If ∆E is greater than the trap depth, U0, then the trapped atom involved inthe collision will be lost from the trap. According to Eq. 2.8 this correspondsto collisional angles, θ, greater thanθmin = cos−1(1− U0m1µ2|~vr|2)(2.9)As discussed above, in the center of mass frame the collision between thetwo particles can be equivalently thought of as a single particle of reducedmass µ approaching a radially symmetric potential which goes to zero asr →∞. The solution ψr is expected to have the formψr ∼ ei~k·~r + f(k, θ, φ)eikrr (2.10)to describe the particle when it is far from the potential region after collision[38]. The first term describes that the particle is initially free, having planewave behaviour. The second term describes scattering as a spherical wavewhere the probability of scattering into angles θ and φ is dependent on thescattering amplitude, f(k, θ, φ). The wavevector ~k is related to the energyin the relative motion (i.e. the energy of the incident reduced mass particle)given by Er = ~2k22µ where k = µ|~vr|/~. Note this is the energy for theincident particle approximated as a free particle when it is far away fromthe potential region. We are interested in the situation after an elasticcollision when the particle is again far from the potential region so that theenergy Er is the same as the incident energy. The second term in Eq. 2.10describes scattering as a spherical wave, where f(k, θ, φ) indicates that thereis a different probability of scattering in different directions, (θ, φ).Classically one considers an incident beam of finite width consisting ofmany incoming particles centered on the target region. A detector collectsany scattered particles that travel into some solid angle, defined by the area21Chapter 2. Background gas collision induced lossof the detector, in the direction [θ,φ]. The differential cross-section, dσdΩ , isdefined asdσdΩ =flux scattered into the direction θ, φ per unit solid angleflux in the incident beam per unit area (2.11)where the differential solid angle element is dΩ = sin θdθdφ. Classically, fluxhere has units of number of atoms per second. The number of atoms persecond passing through a differential area d~S is ~J · d~S where ~J is the masscurrent density, which describes the number of particles passing through aunit area per unit time. In the absence of sources or sinks (no particles arecreated or destroyed) ~J obeys the continuity equationdρdt +~∇ · ~J = 0 (2.12)where ρ is the density of particles (the number of particles per unit volume).To determine a quantum mechanical expression for the differential crosssection, consider again our case of a single particle subject to some localizedpotential. |ψr|2 is a probability density ( the probability of finding theparticle at a particular place per unit volume) and can replace the classicaldensity ρ in Eq. 2.12. In this case since |ψr|2 has units of probability perunit volume, ~J will have units of probability per unit time per unit area.With this interpretation it can be shown [38] that ~J for a particle of massµ has the form~J = ~2µi(ψ∗∇ψ − ψ∇ψ∗) (2.13)= ~µ Im(ψ∗∇ψ)The denominator of Eq. 2.11, using a plane wave ψr = eikz in Eq. 2.13 andd~S = dxdyzˆ, is~J · d~SdS =~kµ . (2.14)The numerator of Eq. 2.11 can be found using ψr = f(k, θ, φ)eikrr in Eq. 2.13and d~S = r2 sin θdθdφrˆ = r2dΩrˆ as~J · d~SdΩ =~k|f(k, θ, φ)|2µ . (2.15)Taking the ratio of Eq. 2.15 and Eq. 2.14 the differential cross section isdσdΩ = |f(k, θ, φ)|2. (2.16)22Chapter 2. Background gas collision induced lossHereafter cylindrical symmetry is assumed so f is not a function of φ.The total cross section is given by integrating the differential cross sectionover all solid anglesσ = 2pi∫ pi0|f(k, θ)|2 sin θ dθ. (2.17)The collisional loss cross section given in Eq. 2.1 for the loss rate constant,Γ, is similar to the total cross section but only accounts for collisions thatinduce loss from the trap. This means that instead of starting at zero for thescattering angle θ we start from the minimum scattering angle that resultsin loss. This was expressed as θmin in Eq. 2.9 and gives a cross section forloss asσloss = 2pi∫ piθmin|f(k, θ)|2 sin θ dθ. (2.18)The cross section for heating collisions that do not result in trap loss is givenasσheat = 2pi∫ θmin0|f(k, θ)|2 sin θ dθ. (2.19)Note that the expressions given in Eq. 2.18 and Eq. 2.18 depend on therelative speed of collision through k = µ|~vr|/~. For our consideration ofa room temperature background gas interacting with an ultra-cold atomiccloud, there are many possible k values for collisions between a backgroundgas particle and a trapped atom.2.2 The velocity averaged collisional loss crosssectionThis section explains: the calculation of the velocity averaged collisional losscross section, 〈σlossvi〉X,i; the convergence criteria used in the calculation ofthe scattering amplitude, f(k, θ); the form of the interaction potential; andthe dependence of 〈σlossvi〉X,i on trap depth.2.2.1 Calculation of 〈σlossvi〉X,iThe section explains how 〈σlossvi〉X,i is calculated. The beginning treat-ment follows Refs. [14, 15, 38]. According to Eq. 2.18, the calculation ofσloss involves the scattering amplitude f(k, θ). The following consists of23Chapter 2. Background gas collision induced lossa derivation for the expression given in Eq. 2.31 for the scattering ampli-tude. Eq. 2.7 for ψr(r, θ, φ) written in terms of spherical coordinates andthe angular momentum operator squared, Lˆ2, is [34]~2µ[− 1r2∂∂r(r2∂ψr∂r)+ Lˆ2r2 ψr]+ V (r)ψr = Eψr. (2.20)The solution ψr(r, θ, φ) = Rl(r)Yl,m(θ, φ) is comprised of a radial and an-gular part. The angular part, called a spherical harmonic, Yl,m(θ, φ), is aneigenfunction of Lˆ2 with Lˆ2Yl,m = l(l + 1)Yl,m. Substituting this solutionfor ψr(r, θ, φ) into Eq. 2.20 gives [34]1r2ddr(r2dRldr)− l(l + 1)r2 Rl +2µ~2 [E − V (r)]Rl = 0. (2.21)In our case we assume cylindrical symmetry so that ψr is a function of rand θ only. For this case the spherical harmonics are proportional to theLegendre polynomials Yl,0 ∝ Pl(cos θ). Taking linear combinations for themost general solution of ψr(r, θ) givesψr(r, θ) =∞∑l=0(2l + 1)ilAlRl(r)Pl(cos θ) (2.22)where Al = 1 for a free particle and otherwise is a constant to be determined.Eq. 2.21 for the radial part Rl can be expressed in terms of ψl(r) =krRl(r) as[ d2dr2 +W (r)]ψl(r) = 0 (2.23)withW (r) = k2 − 2µ~2 V (r)−l(l + 1)r2 . (2.24)Note ψl(r) is not to be confused with ψr(r, θ).When V (r) = 0 the solution to Eq. 2.21 is [38]Rl(r) = cos(δl)jl(kr)− sin(δl)nl(kr) (2.25)where jl(kr) and nl(kr) are the spherical Bessel and Neumann functions, re-spectively, and δl is a real number called the partial-wave dependent phaseshift. This phase shift is key for interpreting elastic scattering. For a freeparticle Rl(r) = jl(kr) because the potential is zero at r = 0 and the Neu-mann part of the function is discounted because it blows up as r approaches24Chapter 2. Background gas collision induced losszero. For our case we are interested in the solution for V (r) → 0 for r →∞.At r = 0 we have V (r) 6= 0 and so the Neumann term is retained. Theasymptotic form of jl(kr) as r →∞ isjl(kr) →sin(kr − 12pil)kr . (2.26)The asymptotic form of nl(kr) as r →∞ isnl(kr) →− cos(kr − 12pil)kr (2.27)so that the asymptotic form of Rl(r) isRl(r) →sin(kr − 12pil + δl)kr . (2.28)The asymptotic solution for ψr given in Eq. 2.10 can be matched withEq. 2.22 with the form of Rl(r) given in Eq. 2.28. The free particle partof the solution ei~k·~r can be replaced by Eq. 2.22, with Al = 1 and with theform of Rl(r) given in Eq. 2.26. This gives∞∑l=0(2l + 1)ilPl(cos θ)Alsin(kr − 12pil + δl)kr (2.29)=∞∑l=0(2l + 1)ilPl(cos θ)sin(kr − 12pil)kr +f(k, θ, φ)eikrrThe sin and cos functions can be expressed as complex exponentials andcoefficients of e−ikr and eikr can be matched on either side of the expressiongiving [38]Al = eiδl (2.30)andf(k, θ) = 1k∞∑l=0(2l + 1) sin(δl)eiδlPl(cos θ). (2.31)To determine f(k, θ), complex T-matrix elements need to be determinedTl(k) ≡ eiδl sin δl. (2.32)The Tl(k) values can be found from the complex S-matrix elementsSl(k) ≡ e2iδl = 1 + 2iTl(k). (2.33)25Chapter 2. Background gas collision induced lossFinally Sl(k) can be found from the real K-matrix elementsKl(k) ≡ tan δl(k). (2.34)The S-matrix and K-matrix elements are related bySl(k) =1 + iKl(k)1− iKl(k). (2.35)To find Kl(k) Eq. 2.23 is expressed in terms of the logarithmic derivativeyl(r) =ψ′l(r)ψl(r)(2.36)to givey′(r) + y2(r) +W (r) = 0. (2.37)Here the prime in Eq. 2.36 and Eq. 2.37 means differentiation with respectto r. The values yl(r) are solved for in the limit of large r using numericalmethods described in [15, 39]. The asymptotic form of the solution for ψl(r)given in Eq. 2.25 can be expressed in terms of Kl(k) asψl(r) = Bl[jˆk(kr)−Kl(k)nˆl(kr)](2.38)where Bl = cos(δl) andjˆl(kr) = krjl(kr) (2.39)nˆl(kr) = krnl(kr). (2.40)(2.41)Substituting Eq. 2.38 in the definition of yl(r) given in Eq. 2.36 and solvingfor Kl(k) givesKl(k) =yl(r)jˆl(kr)− ddr (jˆl(kr))yl(r)nˆl(kr)− ddr (nˆl(kr)). (2.42)Once Kl(k) is determined then Sl(k) can be found from Eq. 2.35. Tl(k)can then be found from Eq. 2.33. The scattering amplitude, f(k, θ), for aparticular k and θ can be found by substituting Tl(k) into Eq. 2.31 and sum-ming over l until the convergence criteria mentioned below is met. f(k, θ)is then substituted into Eq. 2.18 to obtain the loss cross section, σloss, andthe integral is performed numerically over θ . This gives σloss for one par-ticular k = µvr/~ ≈ µvi/~ value where vi is the background particle speed26Chapter 2. Background gas collision induced lossin the lab frame and we used the assumption that the trapped atom is ini-tially stationary in the lab frame. To determine 〈σlossvi〉X,i for the trappedspecies X and background species i an average over all speeds vi using aMaxwell-Boltzmann distribution is performed〈σlossvi〉X,i = 4pi(mi2pikBT)3/2∫ ∞0v3i e− miv2i2kBT σloss(µ|vi|~ )dvi (2.43)which requires another numerical integration over vi. Here σloss given inEq. 2.18 depends on the background gas speed vi through θmin and thedependence of the scattering amplitude, f(k, θ) on k.In summary, the physical parameters to input into the calculation for〈σv〉 are the masses of the background gas species and the trapped atomspecies, the interaction potential V (r), the temperature of the backgroundgas, and the trap depth. There are no free parameters in the calculation,however there are choices regarding convergence criteria for the summationof Eq. 2.31 and the integrals in Eq. 2.18 and Eq. 2.43. The starting andending interatomic distance, typically 0.5 and 50 Bohr radii, respectively,and the number of steps, typically 10000, to use in the algorithm to solvefor yl(r) must also be specified in the code.2.2.2 Convergence criteriaThe total cross section can be expressed as [37]σtot =4pik2∞∑l=0(2l + 1)|Tl(k)|2. (2.44)The total cross section is used to determine the maximum l, denoted lmax,that is needed for calculation of Eq. 2.31. Successive summations for thetotal cross section were compared as a convergence criterion using(lmax∑l=0(2l + 1)|Tl(k)|2 −lmax−1∑l=0(2l + 1)|Tl(k)|2)< 10−2lmax∑l=0(2l + 1)|Tl(k)|2.(2.45)Another convergence criterion used together with Eq. 2.45 to determine lmaxfor a given k is based on the argument found in [37]. Consider a classicalparticle coming towards the potential region with some impact parameter, b,as shown in Fig. 2.2. The angular momentum ~L = ~r× ~p has magnitude L =pb. Quantum mechanically L = ~√l(l + 1) ≈ ~l and the momentum p = ~k.Equating the classical and quantum expressions for angular momentum and27Chapter 2. Background gas collision induced lossFigure 2.2: An incident particle with impact parameter b comes towards apotential region of size r0 and scatters. Classically if b > r0 then the particlewill not scatter.momentum gives ~l ≈ pb ≈ ~kb. Let r0 be the radius, centered at thescattering potential, beyond which the potential is negligible. A classicalpicture is that the impact parameter, b ≈ ~lp needs to be smaller than r0 forscattering to occur. For low energy, b is large except for l = 0, so there isonly s-wave (l = 0) scattering and only l = 0 has a non-zero phase shift. Forhigh energy, the maximum l needed in Eq. 2.31 to satisfy b > r0 is larger.In other words, more l values contribute to the scattering and have non-zerophase shifts. An impact parameter, b > r0, corresponds to l > kr0. Thiscondition can be used to determine the maximum quantum number l neededfor convergence in the summation of Eq. 2.31. Here ~k = µvr, where vr isthe relative speed (taken to be the incoming background gas speed). Thisgives l > µvrr0~ . The criterion l > [0.5 s · m−1]vr was used, meaning thatµr0~ = 0.5 s/m which corresponds roughly to r0 of 25 bohr radii, or 1 nm,using a reduced mass between 87Rb and 40Ar. A criteria of l > 2vr was alsotried but the velocity averaged collisional loss cross section given differed byless than 0.1%, calculated for a 2.2 K trap depth and also for the total crosssection. A typical maximum velocity for these calculations is 900 m/s andthe maximum l needed to satisfy Eq. 2.45 for that velocity is around 700.2.2.3 Interaction potentialThe form of the potential used in the calculation of 〈σlossvi〉X,i with bothcollisional partners in their ground state is V (r) = ((rm)12r12 − 2(rm)6r6)=28Chapter 2. Background gas collision induced lossC12r12 −C6r6 . Here r is the internuclear separation,  is the depth of the po-tential well, and rm is the internuclear separation at which the minimum ofthe potential occurs [40]. The C6 and C12 coefficients are more commonlyreferred to. This form is called a Lennard-Jones potential. The r6 term de-scribes an attractive force and arises due to long range interaction betweenthe atoms. The r12 term describes a short range repulsive term and is cho-sen to have a r12 dependence for computational ease since it is the squareof r6 [41].The following is a simple argument from [42] for why the attractive partof the interaction potential energy term, V (r), has a 1r6 dependence for twoatoms in their ground state. Fig. 2.3 shows two hydrogen atoms where theprotons are situated at A and B and their respective electrons are at ~rA and~rB . The distance between the protons is given by ~R. The potential energyFigure 2.3: A cartoon drawing of protons and electrons for two hydrogenatoms. This figure was patterned after that found in [42]. The protons,A and B, are separated by a distance, |~R|. The electrons associated withproton A or B are a distance |~rA|, or |~rB | away, respectively.arising from interaction between the hydrogen atoms is given asV = e2|~R|+ e2|~R+ ~rB − ~rA|− e2|~R+ ~rB |− e2|~R− ~rA|. (2.46)Taking |~R| to be much larger than | ~rA| and | ~rB | allows Eq. 2.46 to be ex-panded using | ~rA||~R| and| ~rB||~R| being small. This expansion gives [42]V = e2~rA · ~rBR3 −3(~rA · ~R)(~rB · ~R)R5 . (2.47)29Chapter 2. Background gas collision induced lossThe interaction potential V when ~R is taken to be along the z-axis isV = e2R3 (xAxB + yAyB − 2zAzB) . (2.48)For simplicity we will denote the ground S state of atom A and atom B tobe |SA〉 and |SB〉, respectively. The next excited P state we will label simi-larly |PA〉 and |PB〉. The first order correction to the energy with both atomsbeing in the S (ground) state is 〈SA|〈SB |V |SB〉|SA〉. For the sake of argu-ment we will just take the xAxB term of V giving 〈SA|xA|SA〉〈SB |xB |SB〉.This means the first order energy correction is zero since the S state issymmetric. To get a non zero correction to the energy second order per-turbation theory is needed which will involve, for example, a term like| e2R3 〈PA|xA|SA〉〈PB |xB |SB〉|2. This term is non zero and gives a1R6 de-pendence for the energy correction due to interaction between two atomsin their ground states. A physical interpretation of the 1/r6 dependence ofthe potential between ground state atoms is that it is the interaction energybetween two induced dipoles.The C6 constant can be found from the dynamic electric polarizabilities,αA(iω) and αB(iω) of atom A and B, respectively, asC6 =3pi∫ ∞0αA(iω)αB(iω)dω. (2.49)The atomic polarizabilities can in turn be found from semi-empirical oscil-lator strength distributions [43, 44].2.2.4 The dependence of 〈σlossvi〉X,i on trap depth, UIt is important to note that 〈σlossvi〉X,i is a function of trap depth throughthe limits of integration in Eq. 2.18. Intuitively one would predict that〈σlossvi〉X,i will decrease as trap depth increases. This is because the largerthe kinetic energy needed for a trapped atom to escape the trap the lessprobable it is that a collision with a background gas particle will cause loss.In the limit of zero trap depth the loss cross section becomes the total crosssection. Figure 2.4 shows the variation of 〈σlossvAr〉87Rb,40Ar with trap depth.For this calculation both species are taken to be in the ground state andthe potential used is V (r) = C12r12 −C6r6 . The coefficients were taken to beC6 = 280EHa6B and C12 = 8.6 × 107EHa12B , where EH = 4.35974 × 10−18 Jand aB is the Bohr radius [14, 45]. The C6 value given here is calculatedin Ref. [45] from optical data. The C12 coefficient is chosen to correspondto a 50 cm−1 well depth as per rough estimate from Dr. Roman Krems30Chapter 2. Background gas collision induced lossin the UBC chemistry department. As shown in Refs. [14, 15], the valueof 〈σlossvAr〉87Rb,40Ar is fairly insensitive to large variations in the C12 valuechosen. The points superimposed on the curve are experimental data whichwill be discussed in the next chapter.The following few paragraphs discuss the need for a quantum mechanicalscattering theory calculation at low trap depths. For large trap depthsclassical calculation predicts a U−1/6 dependence of the loss cross sectionon trap depth for a long range potential interaction of V (r) = −C6r6 [32, 46,47]. For small trap depths, where small angle scattering can induce loss,a quantum calculation is needed for the loss cross section. This is due tothe fundamental uncertainty in the scattering angle so that scattering atangles close to this uncertainty needs to be treated quantum mechanically[48]. The kinetic energy, d, imparted to a trapped atom for angles withinthe quantum regime is approximated in [49] asd =4pi~2mtσ(2.50)where mt is the mass of the trapped atom species and σ is the total crosssection. For collisions between 87Rb and 40Ar, both in their ground state,d is estimated to be 8.9 mK.For hard sphere scattering the scattering potential is infinite inside someradius, d, and zero outside this radius. The classical total collision crosssection is σ = pid2 [48]. The quantum mechanical total cross section is twoor four times the classical value, depending on the collisional energy [48].The fact that the quantum collisional cross section is larger than the classicalcross section is in analogy to light scattering off of a ball bearing. Classicallya shadow the size of the ball would be observed, but quantum mechanicallyfringes are observed past the size of the ball bearing. Our case is not thatof hard sphere scattering but this idea is useful to understand why below dthe slope of the 〈σlossvAr〉87Rb,40Ar versus trap depth curve is greater thanfor the classical regime of the curve. The loss cross section is limited to thetotal cross section so the slope must then decrease as trap depth tends tozero.An interesting consideration for the shape of the 〈σlossvAr〉87Rb,40Ar versustrap depth curve is to consider the trap loss process as a chemical reaction.The reaction could be expressed as A + B → C + B where A is a trappedatom, B is a background gas particle, and C is the trapped atom beinglost from the trap. The rate constant, k, for a chemical reaction dependson the activation energy, Ea, according to k ∝ e−Ea/RT where T is thetemperature of the reactants and R is the gas constant. If 〈σlossvAr〉87Rb,40Ar31Chapter 2. Background gas collision induced loss10-1100101102103104U (mK)<v>(109cm3s1)Figure 2.4: The velocity averaged collisional loss cross section plotted ver-sus trap depth showing a decrease in the loss cross section with increasingtrap depth. As trap depth increases the kinetic energy required to leave thetrap grows and the probability of a collision with a background gas particleimparting sufficient energy to leave the trap decreases. The curve is gen-erated by numerically calculating 〈σlossvAr〉87Rb,40Ar at discrete trap depthsfor 87Rb as the trapped species and 40Ar as the background species both intheir ground state. The potential used in the calculation is V (r) = C12r12 −C6r6 .The coefficients were taken to be C6 = 280EHa6B and C12 = 8.6× 107EHa12Bwhere EH = 4.35974 × 10−18 J and aB is the Bohr radius [14, 45]. Thepoints superimposed on the curve are experimental data.32Chapter 2. Background gas collision induced lossis likened to the rate constant and the trap depth to the activation energythen the expected dependence of 〈σlossvAr〉87Rb,40Ar on trap depth would be〈σlossvAr〉87Rb,40Ar ∝ e−U . In the case of collisions of background gas particleswith trapped atoms the temperature of the background gas ( ≈ 294 K ) isvery different from the trapped atom temperature (150 µ K). This meansthat the reactants in our proposed chemical reaction are not in thermalequilibrium with each other and we do not see an exponential dependenceof 〈σlossvAr〉87Rb,40Ar on trap depth.The calculation of 〈σlossvAr〉87Rb,40Ar depends on what state the back-ground atoms and trapped atoms are in. The long-range form of the poten-tial when 87Rb is excited and 40Ar is in its ground state is the same formas when both atoms are in their ground state, V (r) = C12r12 −C6r6 , howeverthe C12 and C6 values are different. If a is the fraction of trapped atoms inthe excited state, in for example a MOT, then the expected collision crosssection is〈σlossvAr〉87Rb,40Ar = (1− a)〈σloss,gvAr〉87Rb,40Ar + a〈σloss,evAr〉87Rb,40Ar.(2.51)The subscript e and g refer to when 87Rb is in the excited state or theground state, respectively.The next chapter describes the experimental verification of the depen-dence of the velocity averaged collisional loss cross section on trap depth.87Rb was the trapped atom species and 40Ar was the background species.Further reference to σ implicitly means the loss cross section σloss unlessotherwise noted.33Chapter 3Experimental verification ofthe dependence of the losscross section on trap depthThe first section of this chapter describes how the velocity averaged colli-sional loss cross section, 〈σvi〉X,i, due to collisions between trapped atoms oftype X and background gas of species i is measured. In our experiments weused 87Rb as the trapped species, X, and 40Ar as the background speciesi. The velocity averaged collisional loss cross section 〈σvAr〉87Rb,40Ar wascalculated for a range of trap depths and measured previously by membersof our lab using a magnetic trap [14, 15]. The magnetic trap used couldobtain trap depths up to 10 mK. The work performed for this thesis wasthe measurement of 〈σvAr〉87Rb,40Ar at larger trap depths using a MOT. Atechnique adapted from Hoffmann et al. [16] to measure the trap depth ofa MOT is described. It is also proposed that the dependence of 〈σvi〉X,i ontrap depth can be used as a measurement technique for trap depth. Thework described in this chapter is also reported in [17].3.1 Experimental apparatusOur apparatus consisted of optics and a vacuum apparatus to produce a 3DMOT as well as to introduce a background gas to the 3D MOT. The 52S1/2to 52P3/2 transition was used for trapping of either 85Rb or 87Rb. The pumpwas chosen to be resonant with the F = 2 to F ′ = 3 transition for 87Rbor the F = 3 to F ′ = 4 transition for 85Rb. The repump was chosen tobe resonant with the F = 1 to F ′ = 2 transition for 87Rb or the F = 2 toF ′ = 3 transition for 85Rb. A schematic of the optical setup is shown inFig. 3.1. On a separate table, shared among several experiments, externalcavity diode lasers generate laser light locked 180 MHz below the pump andrepump transitions using saturated absorption signals [50, 51]. The pumplight was fibered over to the experimental table where it injected a diode34Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthlaser amplifier (fiber FC1) to provide more power for the experiment thanwas sent in the fiber. The repump light in fiber FC2 was not amplified.Double pass acousto-optical modulator (AOM) setups were used to bringthe pump and repump frequencies from the fibers to the values used inthe experiment [52]. The frequency shifted pump and repump light werecombined together and sent to the MOT optics.Figure 3.1: A schematic of the optical setup used to produce the pumpand repump light for the MOT used in the loss cross section measurementexperiment. Light from the master table was used to inject a diode am-plifier via fiber FC1. An acousto-optical modulator (AOM) in double passconfiguraion was used to bring the output of the amplifier from 180 MHzto 12 MHz below the pump resonance. Repump light from fiber FC2 fromthe master table was also brought up to resonance using an AOM in doublepass configuration. The pump and repump beams were combined and senttowards the MOT optics. M: mirror, L: lens (PCX 300 mm), OI: opticalisolator, PM: parabolic mirror 300mm focal length, Q: quarter wave plate,H: half wave plate, PBS: polarizing beam splitter.A retroreflected 3D MOT configuration was used with a maximum sixbeam total power of 18.3 mW for the pump and 0.3 mW for the repump.The 1/e2 horizontal (vertical) diameter of the MOT beams was 7.0 (9.5) mmand was measured with a Coherent LaserCam-HR beam diagnostic camera.This corresponds to a maximum available pump intensity of 34.5 mW/cm2.Further increases in pump and repump intensity would increase the MOTatom number. As saturation occurs and light assisted collisional losses grow35Chapter 3. Experimental verification of the dependence of the loss cross section on trap depththe atom number would eventually decrease with increasing intensity.The axial magnetic field gradient used for the MOT was 27.9(0.3)G/cm(2.79(0.03) × 10−3 T/cm). To achieve different MOT trap depths differentpump intensities and detunings were selected with the AOM used for thepump light. A rectangular glass cell of dimensions 1 cm by 1cm by 3.5cm under vacuum was used. Rb vapour was introduced into the system byrunning current through a Rb dispenser (Alvatec Rb-20). The system alsocontains an ion pump (PS-100 Thermionics) and a non-evaporable getterpump (SAES getters).To introduce Ar into the system, a portable station was attached toa valve on the MOT vacuum chamber by a flexible bellows. This station(called the ‘bakeout station’) has a turbo pump (TV-70 Varian), a scrollpump (SH 100, Varian), a residual gas analyzer (RGA) (RGA200, StandfordResearch Systems), an ion gauge (843 Varian), and a leak valve (951-5106Varian). Ar could be introduced in the MOT region through the leak valvewhile the pressure of Ar was measured with the RGA. The residual gasanalyzer was about 1 m away from the trapped atoms. This could result inthe pressure of Ar at the trapped atoms not being the same as read by theRGA. This effect, however, is expected to be minimal since both the RGAand the trapped atoms were located at the same distance, of approximately 1m, from the leak valve where argon was introduced. Also, the RGA and thetrapped atoms were roughly the same distance from the turbo and scrollpumps. The ion pump on the apparatus close to the trapped atoms wasturned off while argon was being added to minimize pressure differentialsbetween the trapped atoms and the RGA.3.2 Measurement of 〈σvAr〉87Rb,40ArThe velocity averaged collisional loss cross section 〈σvAr〉87Rb,40Ar can bemeasured for a trap of a certain trap depth by measuring the backgroundcollision loss rate constant, Γ, at various measured densities, nAr, of 40Ar.Eq. 2.1 predicts that plotting Γ vs nAr will give a linear relationship with aslope of 〈σvAr〉87Rb,40Ar. The velocity averaged collisional loss cross section,〈σvAr〉87Rb,40Ar, was measured in this manner for different trap depths ofthe MOT. As already mentioned, the different trap depths were attainedby changing the intensity and detuning of the pump light. The dependenceof trap depth on detuning and intensity is discussed further in section 3.5.The technique for measuring the loss rate constant, Γ, in our magnetic andmagneto-optical trap will be discussed next.36Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth3.2.1 Measurement of Γ for a MOTIn a MOT when the magnetic field is first turned on, atoms will start to ac-cumulate in the intersection of the six laser beams comprising the MOT. Thefluorescence from the trapped atoms can partially be captured by a lens andfocused onto a photodiode. The photodiode voltage, V (t), is proportionalto the number of the atoms in the MOT, N(t), where we assumeV (t) = αγscN(t). (3.1)Here α is a proportionality constant relating the efficiency of collecting thephotons being fluoresced by the atoms onto the photodetector and the con-version efficiency of photons to voltage by the photodetector. The scatteringrate, γsc, is the rate at which an atom in the MOT scatters photons and isdependent on the frequency and intensity of the light.For the purposes of obtaining the loss rate constant, Γ, we recorded thefluorescence voltage on the photodetector as a function of time as the atomsaccumulate in a MOT until a steady state voltage (atom number) is reached.Time t = 0 was set to be when the magnetic field was turned on with therepump and pump light already on. Fig. 3.2 shows an example of a MOTloading curve from which Γ can be determined [17]. This loading curve canbe fit to an equation proportional to Eq. 1.6 and Γ can be determined. Fordetermination of Γ we do not need to know α and γsc.3.2.2 Measurement of Γ for a magnetic trapA magnetic trap is initially loaded by turning off the MOT light and turningup the current to the magnetic coils to increase the magnetic field gradient.A magnetic trap starts off with its maximal atom number which then decaysover time. Eq. 1.15 models this decay withN(t)N(0) = e−Γt (3.2)where N(t) is the number of atoms in the magnetic trap after a hold time, t,from initial loading of N(0) atoms. Consider instead the ratio N(t)NMOT whereNMOT is the steady state number of atoms in the MOT before loading intothe magnetic trap. The initial number in the magnetic trap is proportionalto the steady state value in the MOT, N(0) ∝ NMOT, so thatN(t)NMOT∝ N(t)N(0) = e−Γt. (3.3)37Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth012 34 5678 9time (s) 3.2: The fluorescence signal of the atoms accumulating in a MOTas a function of time from the turn on of the magnetic field.Fig. 3.3 shows an example set of data taken to determine N(t)NMOT for a par-ticular hold time, t. The experimental sequence is as follows:(a) First the fluorescence voltage due to the steady state atom number inour MOT was recorded.(b) The light was turned off and the magnetic field was increased to loadatoms into a magnetic trap. The atoms were held in the magnetic trapfor some hold time, t.(c) The MOT light was turned back on and the magnetic field was put backto the settings used for the MOT. The MOT was then allowed to loadfor a short time.(d) The magnetic field was turned off to let the trapped atoms escape. TheMOT light was left on to record a background level.(e) The magnetic field was turned back on allowing the MOT to reload38Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthcompletely before starting at step (a) again. Each cycle a different holdtime was used.Step c is needed because a magnetic trap has no light and the atoms do notfluoresce so that only background light is detected on the photodetector. Inorder to see how many atoms are left in the magnetic trap we turned theMOT conditions back on to detect the fluorescence. Because there are someatoms there to start with from the magnetic trap the fluorescence voltagewill be higher than when loading the MOT initially.We are interested in the ratio of the voltage corresponding to the numberof atoms in the magnetic trap, VMT(t), to the voltage in the steady stateMOT, VMOT. Both of these observables are labelled in Fig. 3.3. From Eq. 3.1and Eq. 3.3 we haveVMT(t)VMOT= N(t)NMOT∝ N(t)N(0) = e−Γt. (3.4)By measuring VMT(t)VMOT for different hold times and then fitting to an equationproportional to e−Γt we could determine Γ. An example result is shown inFig. 3.4. Each data point in Fig. 3.4 came from a sequence as describedabove, and as shown in Fig. 3.3. This sequence requires loading of theMOT, transfer to the magnetic trap, waiting a certain hold time, and thenmeasuring the fraction VMT(t)VMOT . The process is then repeated for the nexthold time. The data shown in Fig. 3.4 was not averaged. This method ofdetermining the trapped atom number decay rate in a magnetic trap assumesthat the same fraction of atoms from the MOT are loaded into the magnetictrap for each hold time sequence. Typically approximately 50% of the atomscan be transferred from the MOT to the magnetic trap. The fraction loadedinto the magnetic trap depends on various experimental factors such as trapdepth, the overlap of the MOT trapped atom cloud with the exact locationof the magnetic field zero, experimental timing, and the energy states ofthe atoms. These factors should be consistent through out the experiment.Averaging could, however, reduce the error in the loss rate constant, Γ, dueto this assumption, as well as any random noise.3.2.3 Results of Γ vs nAr measurementFig. 3.5 shows the loss rate constant versus argon density, Γ vs nAr, fortwo different trap depths obtained using a magnetic trap and a MOT. Therelationship is linear and the slope, 〈σvAr〉87Rb,40Ar, is larger for the smallertrap depth of the magnetic trap as predicted. Each loss rate constant, Γ,39Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth V  MOTVMT  (t)Figure 3.3: An example of data taken to determine N(t)NMOT for a certain holdtime, t, in the magnetic trap. A) First the fluorescence voltage is recordedfor a MOT in steady state. B) Atoms are loaded into the magnetic trap byturning off the MOT light and ramping up the magnetic field. The atomsare held in the magnetic trap for a hold time, t. C) The MOT light is turnedback on and the magnetic field set to the MOT setting. The MOT is allowedto load for a short amount of time in order to get a line that one can use toextrapolate the fluorescence voltage when the MOT light was first turnedback on. D) The magnetic field is turned off while the light is left on whichdumps the atoms from the trap. E) The magnetic field is turned back onto its MOT setting and the MOT is reloaded back to its steady state atomnumber. The difference between voltage levels D and the start of C, labelledVMT(t), is proportional to the number of atoms in the magnetic trap, N(t),after hold time, t. The difference between voltage levels A and D, labelledVMOT, is proportional to the number of atoms in the MOT, NMOT. Asshown here the hold time was long and not many atoms remained in themagnetic trap.40Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth0.0 0.5 2.5 3.0 3.5time (s) 3.4: The number of atoms in the magnetic trap divided by theoriginal steady state number in the MOT, N(t)NMOT , measured as a functionof hold time, t. Each point is generated as described in Fig. 3.3. The datais fit to an equation proportional to e−Γt to determine the background lossrate constant, Γ.measurement shown for a magnetic trap is determined from a fit of the atomnumber to a single decay curve as described in subsection 3.2.2. The errorbars for Γ are based on the fit results for Γ. The error bars increase withincreasing argon density because the atom number in the MOT decreasesas the density of argon increases. This means that fewer atoms are loadedinto the magnetic trap as argon density increases and the signal to noiseratio decreases. No averaging was performed in the data shown in Fig. 3.5.Improvements in the error of Γ for the magnetic trap could be made byaveraging many decay curves.The loss rate constant for the MOT data shown in Fig. 3.5 is determinedby fitting a single loading curve, as described in subsection 3.2.1, to a formproportional to (1 − e−Γt), following Eq. 1.6. Though not visible on thegraph, the error bars for Γ for the MOT data shown increase with increasingargon density. Again this is because the MOT atom number decreases andthe signal to noise ratio of the loading curve decreases. No error bars were41Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthassigned to the density of argon because the calibration error of the residualgas analyzer is not known and drifts with time. The error in argon densitymeasurement causes error in the absolute measurement of 〈σvAr〉87Rb,40Ar.Measuring Γ vs nAr provides a method of measuring 〈σvAr〉87Rb,40Ar for a0.6 0.8 1.0 1.8 2.0nAr(109cm3) 3.5: The background loss rate constant, Γ, is measured for a MOT(squares) and a magnetic trap (circles) as a function of the density of ar-gon nAr. The density of argon was measured using a residual gas analyzeras described in section 3.1. The slope of Γ vs nAr is equal to the velocityaveraged cross section for loss between trapped 87Rb atoms and 40Ar back-ground atoms 〈σvAr〉87Rb,40Ar. The slope for the magnetic trap is greaterthan for a MOT because the magnetic trap has a smaller trap depth andthe probability of background collisions causing loss is greater.given trap depth. 〈σvAr〉87Rb,40Ar was measured for a range of trap depthsaccessible by our magnetic trap (0 - 10 mK) and our MOT (500 mK to 2K). The trap depth of a magnetic trap can be calculated as described insection 1.2.2. By varying the magnetic trap gradient different trap depthscould be chosen. The trap depths for various MOT settings was determinedusing the ‘catalysis method’ described in the next section.42Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth3.3 MOT trap depth determination by the‘catalysis method’Fig. 3.6 shows a mechanism for imparting kinetic energy to trapped atoms[16]. Two cold ground state atoms in the trap separated by a certain inter-nuclear separation are photo-associated by a ‘catalysis’ laser to a repulsivemolecular potential. The atoms quickly move apart picking up kinetic energyand then spontaneously emit back to the ground state. The kinetic energypicked up by each atom in the case of homonuclear collisions is h∆/2. ∆is the detuning of the catalysis laser above the atomic resonance betweenthe ground and the excited state of an isolated atom (in our case 52S1/2 to52P3/2 for 87Rb). If h∆/2 > U , where U is the trap depth, then the catalysislaser will cause loss of the atoms from the trap.Figure 3.6: A ‘catalysis laser’ excites the ground state atoms to a repulsiveexcited molecular potential. The atoms quickly repel each picking up h∆/2in kinetic energy for the homonuclear case. ∆ is the detuning of the catalysislaser above the 52S1/2 to 52P3/2 atomic resonance for 87Rb. If the kineticenergy imparted to the atoms is greater than the trap depth loss will result.This figure is patterned after that found in [16].Fig. 3.7 shows the experimental setup used to measure the trap depth43Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthof a MOT. The MOT setup is described earlier in section 3.1. A CoherentTitanium:Sapphire Ring laser (899-01) pumped by a Verdi V10 Coherentlaser was used as the catalysis laser. The linewidth and frequency jitter ofthe catalysis laser was approximately 1 MHz. The frequency of the catalysislaser was set by a dc input voltage, from a frequency generator, fed to thecontrol box. This voltage was stepped through discretely to increment thecatalysis laser frequency. A small portion of the catalysis laser light wassent to a wavemeter (Bristol 621). The rest of the light was sent throughan AOM (IntraAction Corp. ADM-602AF3) and the first order used. TheAOM was used to turn the catalysis laser on and off at a duty cycle givenby a frequency generator (Standford Research Systems DS345) sent to theAOM driver (IntraAction Corp. DE-603H6). The first order of the AOMwas coupled into a single mode polarization preserving fiber and transferredto the MOT setup. The catalysis light was focused to ≈ 1 mm, with anintensity of ≈ 2 W/cm2, and overlapped with the trapped atoms in theMOT. A photodiode was used to monitor the fluorescence of the MOTin steady state without the catalysis light and also in the presence of thecatalysis light with a certain duty cycle. A LabVIEW program was used tocontrol the frequency generators responsible for stepping through differentcatalysis laser frequencies and for turning on and off the catalysis light beingsent to the MOT. The LabVIEW program was used to capture fluorescencedata from an oscilloscope (Tektronix TDS3014) as described later in thissection, as well as to record the catalysis light frequency from the wavemeter.Part of the catalysis light (not shown in Fig. 3.7) was sent to a Rbcell (Triad Technology Inc. TT-Rb-75-V-P). This was so that the catalysislaser could be scanned over the Rb absorption lines to see how the laserwas behaving and to position the frequency at an appropriate starting placefor the catalysis laser experiments. As mentioned, the catalysis laser lightwas focused onto the trapped atoms to be roughly the same size as thecloud of atoms. For initial alignment of the catalysis laser onto the trappedatoms in the MOT the catalysis laser was tuned to the MOT pump atomicresonance. During this alignment process the intensity was attenuated byneutral density filters so that the atoms could still be trapped with thecatalysis laser on. In the presence of resonant catalysis light the fluorescencewas lowered. The mirrors sending the catalysis laser light to the MOT wereadjusted to minimize the fluorescence of the MOT so that the catalysis beamhad maximal overlap with the trapped atoms.The presence of the catalysis laser affects the loading dynamics in a MOT44Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthCATLASERWavemeterAOMAOMDriverFreqGen 2ControlBoxFreqGen 1MOTPDOscilloscopeFigure 3.7: The experimental setup used to measure the trap depth of aMOT using a catalysis laser (CAT laser). The catalysis laser frequency ismeasured with a wavemeter. The frequency of the CAT laser is controlledby external control of the laser control box with a frequency generator. Thecatalysis light was sent through an acousto-optical modulator (AOM) andthe first order used for the experiment. The AOM driver had a TTL input sothat a function generator could be used to turn the catalysis light on and offat a given duty cycle. The catalysis light was fibered over to the MOT andaligned onto the MOT atom cloud. The fluorescence of the MOT in steadystate with no catalysis light and with catalysis light on at a certain dutycycle was recorded with a photodiode and an oscilloscope. A Labview codewas used to control the function generator and read from the wavemeter andoscilloscope by GPIB.so that Eq. 1.5 becomes [16]dNdt = R− ΓN − (β + d · βcl)∫n2(~r, t) d3~r (3.5)where βcl is the contribution to the two body intra-trap loss constant due tothe presence of the catalysis laser and β is the contribution from all othertwo body intra-trap losses not mediated by the catalysis laser. We considerthe case where the catalysis laser is modulated on and off with duty factor,d, where duty factor is the percentage of on time.A measure of βcl was determined, which involved the steady-state num-ber of the trapped atoms with and without the catalysis laser present. Thesteady state number for the trap when the catalysis light is present withduty factor, d, using Eq. 1.9 isNss =RΓ + (β + dβcl)nss(3.6)45Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthwhere nss is the average steady state density of trapped atoms. Taking theratio of Nss with the steady state number, N0ss, when the catalysis laser isnot present (i.e. d = 0) givesN0ssNss= 1 + βclnssdΓ + βnss. (3.7)Experimentally the ratio of N0ssNss was determined fromV 0ssVss as per Eq. 3.1. HereV 0ss is the steady state fluorescence voltage from the trapped atoms when thecatalysis laser is off. Vss is the steady state voltage when the catalysis laseris on with duty factor, d. Rearranging Eq. 3.7 we define the parameter J asJ = N0ssNss− 1 = ( βclnssΓ + βnss)d. (3.8)As long as d is not too large so as to change the steady state density nss, therelationship between J and d will be linear. Fig. 3.8 shows J as a functionof d for different catalysis laser detunings. Fig. 3.9 shows the portion over0.0 0.1 0.2 0.3 0.4 0.5Duty Factor, d0123456JFigure 3.8: J = N0ssNss − 1 vs the catalysis laser duty factor d. For small dutyfactors the steady state density of the MOT, nss is unaffected and J vs d islinear. As d increases, nss becomes dependent on d and the curve becomesnon linear. Each of the curves is for a different catalysis laser detuning.46Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14Duty Factor, d0. 3.9: J vs d taken from Fig. 3.8 in the linear region. The slope asgiven by Eq. 3.8 is proportional to the loss rate constant βcl for the repulsiveloss mechanism induced by the catalysis laser. Each of the curves is for adifferent catalysis laser detuning.which J versus d is linear. The slope of the linear portion of J vs d for agiven catalysis laser detuning is proportional to the βcl value for that detun-ing. Plotting this slope as a function of catalysis laser detuning provides ameasure of βcl as a function of detuning.Fig. 3.10 shows a plot of our measure of βcl versus detuning, ∆ for aMOT whose average trap depth was determined to be U = 0.64(0.12)K. Tounderstand the qualitative behaviour of βcl with changing ∆, we follow theargument made in Ref. [16]. It is argued in that work that βcl ∝ σP (h∆/2),where σ is the photoassociative cross section and P (h∆/2) is the probabilityof escape of a trapped atom with kinetic energy h∆/2. The cross section canbe written as σ = pir2f where r corresponds to the internuclear separationat which the catalysis laser transition occurs for a given detuning ∆. Theexcitation probability, f , is inversely proportional to dVdr . This is becausethe interaction time of the atom pair with resonant light at a given internu-clear separation decreases as dVdr increases. For a potential of V (r) = −C3r3and for V (r) = h∆ this gives σ ∝ ∆−2 and βcl ∝ ∆−2P (h∆/2). The MOT47Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth0 10 20 30 40 50Detuning (GHz)05101520clnss/(+nss)Figure 3.10: The slope of the J vs d curve (as shown in Fig. 3.9) is plottedas a function of catalysis laser detuning. The J vs d curve is proportionalto the loss rate constant βcl associated with the repulsive loss mechanisminduced by the catalysis laser. The detuning, ∆, at which βcl is maximalcorresponds to h∆/2 = U where U is the depth of the trap. In this mannerthe trap depth can be measured for a MOT. The data shown is for a 87RbMOT with a pump detuning of -5 MHz and a pump intensity of 2.7 mWcm−2.trap depth is anisotropic and was found to be approximately double alongthe axial direction [53]. This means that P (h∆/2) will not be a sharp stepfunction but will gradually increase as h∆/2 increases from the minimumtrap depth to the maximum trap depth. A gradual increase in P (h∆/2) pre-dicts a gradual increase in βcl. Because of the additional 1/∆2 dependenceof βcl, a peak in βcl occurs around h∆/2 = Uavg. A decrease in βcl is seenfor further increases in detunings. The decrease of βcl with detuning, ∆,shown in Fig. 3.10, seems to be slower than a 1/∆2 dependence. A possiblereason is because past the peak in βcl, P (h∆/2) continues to increase untilh∆/2 is the maximal trap depth. In summary, the peak of our measure ofβcl versus detuning is used to determine the average MOT trap depth. Theuncertainty in the detuning at which βcl is maximized was used to assign anuncertainty for the trap depth measurement. We note that that Hoffmann48Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthet al. [16] plot the probability of a trapped atom escaping as a functionof ∆ to determine trap depth . They interpret the detuning, ∆, at whichthis probability starts to increase past approximately 30 % to correspond tothe trap depth. For the trap depth measurement example given in [16], ourinterpretation of choosing the detuning at which βcl is maximized predictsa trap depth about 35 % higher than their prediction for their data. Ourmeasurement technique of MOT trap depth seems to be more correct basedon the data taken in the next chapter (shown in table 3.2).Fig. 3.11 shows the measure of βcl plotted versus detuning, ∆, for threedifferent MOT trap depth conditions. As trap depth increases, the widthof the curve increases, although the width relative to the trap depth staysapproximately the same. This is due to the trap depth anisotropy discussedin the above paragraph. This increase in width causes the absolute errorin the trap depth measurement using the catalysis method to increase withincreasing trap depth. The measure of βcl also decreases with increasingtrap depth because of the decreasing probability of excitation for increasingcatalysis detunings, ∆.3.4 Comparison of measurement with theoryMeasuring Γ versus nAr provides a method of determining 〈σvAr〉87Rb,40Ar.The theoretical predictions of 〈σvAr〉87Rb,40Ar versus trap depth U can nowbe compared with experimentally measured values. Fig. 3.12 shows that theexperimental results follow the predicted dependence (this is the same figureshown in Fig. 2.4). The error bars for the velocity averaged collisional losscross section, 〈σvAr〉87Rb,40Ar, come from linear fits of the loss rate constant,Γ, versus the density of argon. The error bars on a Γ measurement for amagnetic trap or a magneto-optical trap come from fitting a decay or loadingcurve, respectively, as described in section Magnetic trap dataThe data in Fig. 3.12 below 10 mK was taken previously by members ofour lab with a magnetic trap and is reported in [14]. For these data, eitherthe |F = 1,mF = −1〉 or the |F = 2,mF = 2〉 states of the 52S1/2 groundstate for 87Rb was used for magnetic trapping. Only the second lowest trapdepth recorded used the |F = 2,mF = 2〉 state. The |F = 1,mF = −1〉 wasused preferentially because it is easier to isolate experimentally. 40Ar has anuclear spin I = 0 so hyperfine changing collisions will not occur. The only49Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth0 20 40 60 80 100 120Detuning (GHz)05101520clnss/(	+nss)Figure 3.11: A plot of our measure of βcl versus detuning, ∆, for three differ-ent MOT trap depth conditions. The relative height of the curves decreasesas trap depth increases because the probability of excitation by the catalysislaser decreases as detuning increases. The curves widen with increasing trapdepth due to the trap depth anisotropy being approximately double alongthe axial direction compared to the radial direction. The width of each curverelative to the detuning at which the peak occurs is approximately the same.expected difference between the two states is that for a given magnetic fieldgradient the trap depth will be double for the |F = 2,mF = 2〉 state.As explained in section 1.2.2, the trap depth assigned for the magnetictrap data was an average of the trap depth along the axial and radial di-rections. The error bars assigned to the magnetic trap depth span the twovalues used in the average. The magnetic trap depth along a given directionwas calculated, as discussed in section 1.2.2, by diagonalizing the hyper-fine and magnetic field contribution for the intermediate B-field case. Themagnetic field used was from Eq. 1.3 and Eq. 1.4, which assumes the fieldmagnitude increases linearly with distance away from the center of the trap.A more precise expression for the B-field for a circular coil can be found inRef. [24]. Using the linear approximation predicts a magnetic field magni-tude that is 4% higher than the more accurate expression, at the edge of the50Chapter 3. Experimental verification of the dependence of the loss cross section on trap depth10-1100101102103104U (mK)<v>(109cm3s1)Figure 3.12: The velocity averaged collisional loss cross section for 87Rb asthe trapped species and 40Ar as the background species plotted versus trapdepth. The plot shows a decrease in the loss cross section with increasingtrap depth. As trap depth increases, the kinetic energy required to leavethe trap grows and the probability of a collision with a background gasparticle imparting sufficient energy to leave the trap decreases. The curve isgenerated by numerically calculating 〈σvAr〉87Rb,40Ar at discrete trap depths.The points superimposed on the curve are experimental data.cell at 0.5 cm from the B-field zero. This was not accounted for in the trapdepth error bars.3.4.2 Magneto-optical trap dataThe catalysis laser provides a means of determining the trap depth of aMOT. A zoomed in portion of the curve for the data taken with a magneto-optical trap is given in Fig. 3.13. The error bars on the trap depth aredetermined according to the width of the curve, as discussed in section 3.3.The error bars for the measurements of 〈σvAr〉87Rb,40Ar are larger for smallertrap depths. This is because for a lower trap depth MOT there are less atoms51Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthtrapped and the signal to noise ratio for the fluorescence voltage decreases,which increases the error when determining Γ from loading curves. The data0.5 2.5Trap Depth (K)<v>(109cm3/s)Figure 3.13: The velocity averaged collisional loss cross section plottedversus trap depth for the data taken with a magneto-optical trap. The trapdepth of the trap was changed by varying the pump detuning and intensity.The data presented here is similarly presented in tables 3.1 and 3.2.in Figs. 3.12 and 3.13 are also presented in tables 3.1 and DiscussionIt should be noted that measurements of 〈σvAr〉87Rb,40Ar and U did not nec-essarily fall on the predicted curve before a corrective factor was appliedto the 〈σvAr〉87Rb,40Ar measurements. This corrective factor corrects inaccu-racy in determining nAr. This inaccuracy is due to calibration uncertaintyand drift of the residual gas analyzer. In addition, as discussed in section3.1, there are possible differences in argon density at the RGA comparedto the location of the trapped atoms. To account for nAr measurement in-accuracy, 〈σvAr〉87Rb,40Ar for a magnetic trap was measured along with theMOT measurements. The loss rate constant, ΓMT, for a magnetic trap, withtrap depth 3.14(0.84) mK, was measured at each nAr where ΓMOT for a par-ticular MOT setting was also measured. The trap depth of the magnetic52Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthtrap was calculated and the theoretical value of 〈σvAr〉87Rb,40Ar for that trapdepth was calculated. The ratio of the measured and calculated value of〈σvAr〉87Rb,40Ar for the magnetic trap provided a correction factor for all the〈σvAr〉87Rb,40Ar values taken for the MOT settings. This correction factorwas less than 10 %, which is within the expected inaccuracy of the RGA.For the magnetic trap data taken by previous memebers of the labs, shownin Fig. 3.12, the trap depths were calculated and the data was scaled downto the theoretically predicted curve. In summary, the experimental datashown in Fig. 3.12 taken with both the magnetic trap and MOT were bothscaled to lie on the theoretical curve. The magnetic trap data and the MOTdata shown in that figure were taken at different times. For the MOT data,data was also taken for one trap depth with a magnetic trap to provide thescaling factor to apply to the MOT data.3.5 Proposal for a trap depth measurementtechniqueThis chapter has focused on confirming that the shape of the calculated〈σvAr〉87Rb,40Ar vs U curve agrees with measurement. It is proposed thatthis dependence can be used to determine the depth of a trap by measuringthe velocity averaged collisional loss cross section and determining based oncalculation what trap depth this corresponds to. The loss rate constant, Γ,can be measured for a trap as a function of ni where ni is the density ofbackground species of choice i. The slope of Γ versus ni gives 〈σvi〉X,i. Thevalue of 〈σvi〉X,i can be calculated for a sufficient number of trap depthsin the region where the depth of the trap is estimated to be and can benumerically fit to give a dependence of 〈σvi〉X,i on U . This dependence of〈σvi〉X,i on U can then be inverted to give U as a function of 〈σvi〉X,i. Themeasured 〈σvi〉X,i from the slope of Γ versus ni can be used to determinetrap depth. The accuracy of this trap depth determination would be limitedby the accuracy to which ni is known. As described above, inaccuracies indensity measurement can be corrected for when measuring 〈σvi〉X,i for atrap whose depth is unknown if 〈σvi〉X,i is also measured simultaneously fora known trap depth.Table 3.2 shows trap depths measured using the catalysis technique forvarious pump detunings and intensities. The trap depths determined fromthe numerically calculated 〈σvAr〉87Rb,40Ar vs U curve are also included. Ta-ble 3.1 and table 3.2 together contain the same information presented inFig. 3.13. The trap depth measurement data in table 3.2 confirms that the53Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthpeak of βcl as a function of catalysis laser detuning corresponds to the aver-age trap depth, as discussed in section 3.3. One may wonder how the MOTtrap depth depends on intensity and detuning. The values of trap depthfor various detunings and intensities, given in Table 3.2, follow the qualita-tive behaviour that was numerically simulated in [47]. In those simulationstrap depth increases approximately linearly with detuning until trap depthstarts to decrease. The trap depth initially grows rapidly with intensity, butas intensity increases, the increase in trap depth slows down as saturationoccurs.A complication of the trap depth measurement technique using back-ground collisions is that the 〈σvi〉X,i value changes depending on the stateof the trapped atom X. For both Rb and Ar in their ground state, Mitroyand Zhang [44] give C6 = 336.4 in a.u. For Rb in its np excited state andAr in its ground state the long range potential is given as V (r) = −C6r6 . Inthis case C6 = 924.1 for the Σ molecular state, and for the Π molecularstate it is C6 = 545.1, both in atomic units. This gives a total velocity aver-aged collision cross section, 〈σvAr〉87Rb,40Ar, in the three cases of 2.8× 10−9,4.2 × 10−9, 3.4 × 10−9 cm3 s−1, respectively. For 40Ar as the backgroundchoice an estimate of excited state fraction in our MOT at a maximumof 15% did not cause significant error in the agreement of the measured〈σvi〉X,i with the 〈σvi〉X,i calculated for rubidium in its ground state. Forexample, for a MOT setting with −12 MHz pump detuning and 34.5 mWcm−2 pump intensity, the measured value of 〈σvAr〉87Rb,40Ar given in table3.1 is 0.598 ×10−9cm3s−1 and the calculated value shown in Fig. 3.13 is0.594 ×10−9cm3s−1.As mentioned in section 3.4.3, it was necessary to correct the measure-ments of 〈σvAr〉87Rb,40Ar to account for pressure measurement inaccuraciesin the density of 40Ar. Present pressure gauges which can measure in the≈ 10−6 Pa (10−8 Torr) range, as will be discussed in the next chapter, aresubject to calibration drift. Calibration is difficult and expensive and de-pends on gas type. It is proposed by our lab (Dr. James Booth and Dr.Kirk Madison), that the atoms themselves could serve to provide accurateand stable measurements of the density of argon, nAr, as well as possibly ofother desired background gases. The idea is based on the relationship be-tween the loss rate constant, Γ, due to background collisions, the density ofbackground gas, and knowledge of the velocity averaged collisional loss crosssection. The remaining chapters of this thesis focus on the proposal of usingtrapped atoms as a pressure sensor. The next chapter introduces existinggauges and standards, and explains more fully the concept of the ‘trappedatom pressure sensor’. The chapters afterwards describe the experimental54Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthTable 3.1: Measurements of the velocity averaged collisional loss cross sec-tion, 〈σvAr〉87Rb,40Ar, for various MOT pump intensities and detunings. Thevelocity averaged collisional loss cross section is the slope of the linear fit tomeasurements of the loss rate constant Γ of a 87Rb MOT versus the densityof 40Ar in the background gas. The values of the velocity averaged crosssection and errors quoted are from the linear fit results.MOT detuning (MHz) Intensity (mW cm−2) 〈σvAr〉87Rb,40Ar (×10−9cm3s−1)-5 2.7 0.780 (0.043)-8 2.7 0.737 (0.033)-10 2.7 0.696 (0.031)-12 6.9 0.637 (0.008)-12 9.6 0.615 (0.006)-12 34.5 0.598 (0.003)Table 3.2: 3D MOT trap depths for various pump detunings and intensities.The trap depths, Ucat, were determined using the catalysis method describedin section 3.3. As a comparison the trap depths were also obtained by fittingthe numerically calculated 〈σvAr〉87Rb,40Ar vs U and then determining trapdepth from measured values of 〈σvAr〉87Rb,40Ar. Trap depths determinednumerically are denoted U〈σv〉.Detuning (MHz) Intensity (mW cm−2) U〈σv〉 (K) Ucat (K)-5 2.7 0.55 (0.15) 0.64 (0.12)-8 2.7 0.77 (0.17) 0.88 (0.12)-10 2.7 1.05 (0.22) 1.03 (0.12)-12 6.9 1.64 (0.10) 1.80 (0.18)-12 9.6 1.93 (0.07) 1.99 (0.18)-12 34.5 2.20 (0.05) 2.23 (0.24)55Chapter 3. Experimental verification of the dependence of the loss cross section on trap depthprogress made to date on the trapped atom pressure sensor.56Chapter 4Proposal for a cold atombased pressure sensorThe cold atom based sensor that is proposed here would measure the localdensity of a background gas where the trapped atoms are situated. Forsufficiently low pressures and high temperatures, the pressure, P , of a gasrelates to its density by the ideal gas law. The ideal gas law is PV =NkBT = nRT , where N is the total number of atoms in volume V , kB isBoltzmann’s constant, T is the temperature, n is the number of moles, andR is the Rydberg constant. The official SI unit of pressure is the Pascal(Pa). In North America (particularly in the US) a commonly used unit isTorr where 1 Torr = 133.3 Pa. Ranges of vacuum are defined as low/roughvacuum ( 105 to 102 Pa), medium/fine vacuum (102 Pa to 10−1 Pa), highvacuum (HV)(10−1 to 10−6 Pa), ultra-high vacuum (UHV)(10−6 to 10−10Pa) and extremely high vacuum (XHV)( 10−10 Pa and below) [54].There are a variety of pressure measurement devices (pressure gauges)on the market, most of which have to be calibrated by a primary or a trans-fer standard. A primary standard is a ‘measurement standard establishedusing a primary reference measurement procedure, or created as an artifact,chosen by convention’. A primary reference measurement procedure gives‘...a measurement result without relation to a measurement standard for aquantity of the same kind’. Quantities of the same ‘kind’ require the sameunits (for example diameter and circumference are of the kind length) [55].An example of a primary pressure standard is a mercury manometer(Fig. 4.1). The pressure difference, P1−P2, between two connected columns,partially filled with mercury, is determined by the height difference h of themercury in each column. The relation is P1 − P2 = ρgh where ρ is thedensity of mercury. By pumping down on one side so that P2  P1 we haveP1 ≈ ρgh.Primary standards are normally quite involved both in apparatus andtechnique so that they are not employed as gauges in a commercial way.Instead they are maintained by national laboratories who use their primarystandards to calibrate commercial gauges that can then be used as transfer57Chapter 4. Proposal for a cold atom based pressure sensorPumpPumpGas InputhP1 P0P1 P2Figure 4.1: A mercury manometer is used as a primary pressure standardrelating the height difference, h, of mercury in a connected tube to thepressure difference by P1 − P2 = ρgh. Both the density of mercury ρ andthe height difference h are traceable to primary standards in length andmass.standards. This chapter begins with a discussion of some existing pressuregauges for the HV and UHV range and some existing primary pressurestandards for the UHV vacuum range. At the end of the chapter a proposedpressure standard based on cold trapped atoms is discussed.4.1 Pressure gaugesIn this section we describe several pressure gauges used in the HV and UHVrange. The focus is on pressure gauges that are used in our experimentalapparatus for our proposed pressure standard.4.1.1 Capacitance diaphragm gaugeFig. 4.2 shows a schematic representation of a capacitance diaphragm gauge.The capacitance diaphragm gauge consists of a sealed chamber divided intotwo sections by a thin inconel tensioned sheet of metal (the diaphragm).For absolute pressure measurement one side is evacuated to low pressure≈ Pref = 0 and sealed. A chemical getter is included to absorb any particlesthat outgas from the gauge material in order to maintain low pressure afterthis section is sealed. On the other side of the diaphragm there is an inletto accept the gas whose pressure Px is to be measured. The diaphragm58Chapter 4. Proposal for a cold atom based pressure sensorC 1C 2P xP reftensionedinconel diaphragmgasinputABCFigure 4.2: A capacitance diaphragm gauge (CDG) consists of an enclo-sure divided into two sections by a thin metal sheet called the diaphragm.The diaphragm is typically inconel, an alloy of predominantly nickel andchromium. For absolute pressure measurement one side of the enclosure isevacuated to a very small pressure so that Pref ≈ 0. The other side of thediaphragm will be deflected according to the pressure Px to be measured.At different positions of the diaphragm the capacitance C1 and C2 formedbetween the diaphragm and electrodes will change. These capacitances (C1and C2) are fed into an AC bridge (see Fig. 4.3) and the voltage V acrossthe bridge related to the pressure Px.will deflect by an amount related to the pressure of the gas introduced.On the evacuated side several electrodes are placed so that the diaphragmforms a capacitor with each electrode. These two capacitance signals, C1and C2 in Fig. 4.2, are fed into an AC bridge via connections A, B, andC. Fig. 4.3 shows an AC bridge where the voltage V across the bridge iszero if the impedance Z1, Z2, Z3 and Z4 satisfies Z1Z3 =Z2Z4 . For capacitorsZ = 1iωC where ω is the frequency of the AC source. As the diapraghmdeflects with pressure the capacitances C1 and C2 change which changesthe voltage across the bridge. The voltage across the bridge versus pressurecan be characterized and the device made to output a voltage that varieslinearly with pressure. The 615A from MKS instruments for the 1 Torr (133Pa) range has a claimed accuracy of ±0.25% [56]. The pressure range forthe gauge is 1×10−5 to 1 Torr (1.33×10−3 to 133 Pa). The lowest pressure59Chapter 4. Proposal for a cold atom based pressure sensorV ~  C 1 C 2C 3 C 4Figure 4.3: An AC bridge. The voltage, V, across the bridge is zero if theimpedances satisfy Z1/Z3 = Z2/Z4.reading of the gauge is derived from the second last digit in the pressuredisplay on the controller front panel. The pressure reading of a capacitancediaphragm gauge can change with temperature changes so that often theyare temperature controlled [57].4.1.2 Spinning rotor gaugeThe spinning rotor gauge consists of a set of magnets and magnetic coils inthe midst of which a magnetized steel ball is levitated (shown in Fig. 4.4).The ball (R) is levitated by two permanent magnets (M). Using drive coils(D) the ball is set to spin at a certain number of revolutions per secondas detected by two pick-up coils (P). Two stabilization coils in the verticaldirection (S) and four in the horizontal direction (L) are used to minimizedeviations in the position of the rotor. The drive coils are then turned offand the rate of angular deceleration is measured. The relative rate of an-gular deceleration ( 1ω dωdt ) is proportional to the pressure of the environmentsurrounding the ball. The ball is within the vacuum surrounded by a steeltube. The magnets and sense coils slip over the tube and are external tothe vacuum environment. When the ball is spinning eddy currents induced60Chapter 4. Proposal for a cold atom based pressure sensorFigure 4.4: A schematic of a spinning rotary gauge. Two permanent mag-nets (M) are used to levitate a stainless steel ball (R). Four drive coils (D)are used to spin the ball (R) at a certain angular frequency. Vertical devia-tions of the ball are suppressed by two stabilization coils (S) and horizontaldeviations are suppressed by four coils (L). Pickup coils (P) are used tosense the angular frequency of the ball and the rate of angular decelerationwhen the ball is allowed to spin without being driven in the presence of gas.From the relative rate of angular deceleration the pressure of the gas canbe determined. The ball resides in a steel tube connected to the vacuumapparatus while the magnets and coil assembly slide over the tube externalto the vacuum. This figure is attributed to MKS Instruments and is usedwith permission. This figure source is Ref. [58] .61Chapter 4. Proposal for a cold atom based pressure sensorin the ball and surrounding structures will cause a ‘residual drag’ whichresults in a pressure reading even for zero pressure of gas and needs to becorrected for. Fluctuations of this residual drag limit the lowest pressurethat can be detected by the SRG. The SRG3 from MKS instruments whichis currently in our setup can read a pressure range of 5×10−5 Pa to 100 Pa.The quoted stability is ≤ 1% per year [58]. To read accurately, the devicemust be calibrated since exposure to air, travel, baking and remounting ofthe magnetic sensing head all effect the original calibration [59, 60].4.1.3 Bayard-Alpert ionization gaugeThere are two main categories of ionization gauges, hot cathode and coldcathode. These categories come from whether the source of electrons toionize gas whose pressure is to be measured comes from heating a filament(hot cathode) or from a high (kilovolt range) voltage cathode (cold cathode)[61–63]. Examples of hot cathode gauges are the Bayard-Alpert ionizationgauge and an extractor gauge. Examples of cold cathode gauges are the Pen-ning gauge, inverted magnetron gauge (Redhead) or the magnetron gauge.In our system a Bayard-Alpert ionization gauge is used which is part of thehot cathode category. In a Bayard-Alpert ionization gauge (see Fig. 4.5)current is sent through a thin tungsten filament heating the filament. Thefilament emits electrons which are accelerated towards an anode grid at +180V and follow trajectories around and through the grid until they hit thegrid. As the electrons travel they can ionize gas they collide with formingpositive ions. The grid surrounds a thin wire at 0V towards which any ionsformed inside the grid will go to. This wire is called the ion collector. Ionsformed outside the grid are accelerated towards the walls of the gauge (alsoheld at 0V) and do not contribute to the ion collector current. The ratioof ion collector current to electron emission current is proportional to thepressure of the gas in which the gauge is situated and the gauge can becalibrated to give pressure readings. The calibration constant will changefor different gas types because the ionization probability is different.This gauge though popular has several unwanted features [61, 65]:1. The calibration of the ion gauge can drift with time due to the saggingof elements in the ion gauge [66]. Any relative movement of the fila-ment to the grid affects the electron trajectories either by changing theelectron path length or by changing the amount of time the electronspends within the grid where detectable ions can be formed. Changesin electron trajectories will therefore change the ion collector current62Chapter 4. Proposal for a cold atom based pressure sensorAA(+ 180 V)grid(+30 V)filamentioncollector(0V) + 30 V+ 180 Vi+ i -Figure 4.5: A schematic of an ion gauge. A filament (typically tungsten oryttria-coated iridium) is heated and electrons released. The electrons traveltrajectories towards the positively charged grid. Ions that are formed fromcollisions of the electrons with the gas present will travel towards the ioncollector and a current will be detected on the ion collector ( i+ ). Thepressure, P of the gas being measured is proportional to the ratio of theion collector current and the electron emission current (i−) so that P ∝ i+i− .Figure patterned after that found in [64].63Chapter 4. Proposal for a cold atom based pressure sensorfor a given pressure. The calibration can also change as the filamentwears with time. ‘Poisoning’ of the filament surface by chemical re-actions and by adsorption of gases can change the work function ofthe filament and can change the emission current [67]. The NationalInstitute of Science and Technology (NIST) estimates the calibrationfactor over the period of several years to be stable within a standarduncertainty of 1.9 percent (k=1) [68]. This result came from calibra-tion analysis of 9 gauges, though an individual gauge calibration mayvary more than this (on the order of 5 or more percent).2. X-rays are produced when the electrons hit the grid. These x-rayscan strike the ion collector and cause electrons to be liberated fromthe collector. This is indistinguishable from an ion current and isnot related to the presence of gas being measured. This places alimit (called the ‘X-ray limit’) on the lowest pressure that can be readreliably. The ion collector is made to be a very thin wire so as tominimize this effect by intercepting as little x-rays as possible.3. The gauge changes the local pressure in which it is in by ionizingthe gas being measured and by heat-induced outgassing of the gaugecomponents, especially the filament and the surrounding structures.Chemical reaction of the gauge components with the gas being mea-sured can occur [67]. The electron emission current is kept low in aneffort to minimize these effects.The ion gauge installed on our system is the Granville-Phillips Series 370Stabil-Ion Gauge with Yttria-coated Iridium filaments. The measurementrange is 2× 10−11 to 5× 10−4 Torr (2.7× 10−9 to 6.7× 10−2 Pa). The x-raylimit is 2×10−11 Torr so measurements near this limit will not be repeatable[69]. The quoted accuracy for N2 is ±4% of the reading from 1 × 10−8 to1 × 10−4 Torr (1.3 × 10−6 to 1.3 × 10−2 Pa). The quoted repeatability is±3% of the reading from 1×10−8 to 1×10−4 Torr (1.3×10−6 to 1.3×10−2Pa)[64].4.1.4 Residual gas analyzerSimilar to an ion gauge, a residual gas analyzer (RGA) uses an electron beamto ionize gases present. A quadrupole RGA in addition uses an arrangementof electric fields to select which charge to mass ratio of ions will be detected.In this way by sweeping over different charge to mass ratios to be detectedthe RGA provides information on the composition of the gas being measured64Chapter 4. Proposal for a cold atom based pressure sensorand the relative amount of different gas species. Residual gas analyzers aremost easily used for qualitative detection of the presence of various speciesin the vacuum system. For quantitative work they must be calibrated forthe species that is to be measured. The RGA on our system is the QMG220M2 PrismaPlus from Pfeiffer Vacuum (part number PTM06241213) whichhas a minimum detection limit less than 2 × 10−12 mbar and a mass rangeof 1-200 amu [70].4.1.5 Pirani gaugeIn a Pirani gauge, changes in resistance of a heated wire with pressure aredetected. One of these gauges ( a ‘convectron’ gauge accompanying the 370Stabil-Ion gauge, part number 275256) is installed on our system to detectfailures in the turbo and scroll pumps. The pressure range they read is inthe 1× 10−4 to 1000 Torr range (10−2 to 105 Pa range).4.2 Existing pressure standardsThis section outlines some of the existing pressure standards that are usedto calibrate gauges in the high and ultra-high vacuum pressure range.4.2.1 Static expansion methodIn the static expansion method [71] the pressure in a chamber of knownvolume, V1, is measured at a pressure P1. The pressure P1 is high enoughthat it can be accurately measured with a gauge calibrated on a high pressurestandard (such as a mercury manometer). The gas is then expanded intovolume V1 and V2 so that P1V1 = P2(V1 +V2) where P2 is the pressure afterexpansion. Provided that the volumes V1 and V2 are accurately known andP1 is accurately measured then P2 can be determined. A gauge connected tothe volume V1 or V2 can be calibrated at lower pressures than the gauge thatwas used to measure P1. This could also be used to extend the calibrationrange of the gauge that measured P1. The static expansion method typicallyis used as a standard down to 10−5 Pa, though it can be extended for inertgases down to 10−7 Pa [72]. The lower pressure limit is mainly determinedby the outgassing from the vacuum chamber.65Chapter 4. Proposal for a cold atom based pressure sensor4.2.2 Orifice flow methodFor calibration of gauges at a pressure range of (10−1 to 10−10 Pa), orifice-flow standards (also known as dynamic expansion methods) are used [54,71, 73–75]. Ion gauges are typically calibrated by this method. The typicalsetup of the orifice flow method is shown in Fig. 4.6. Two chambers areseparated by an orifice with gas being fed into chamber 1 and pumped outat chamber 2. The flow of gas through the orifice can be characterized bythe ‘throughput’. Throughput, Q, also sometimes called flow or flow rate,of gas into/out of a volume V with pressure P is defined as Q = d(PV )dt . Forthe ideal gas law PV = nRT andQ = d(PV )dt = RTdndt (4.1)so that throughput is proportional to the rate of molar increase/decrease inthe volume V . Throughput has units of Pa·m3s . The molar rate of change dndtis also commonly called flow or flow rate and has units of mol/s. Throughthe orifice there will be a throughput ofQ = C(P1 − P2) (4.2)where P1 and P2 are the pressures in chamber 1 and 2 respectively. Theconductance C depends on the dimensions of the orifice, and the gas typeand temperature [61, 65]. The conductance is independent of pressure forsufficiently low pressures (i.e < 100 Pa) [76]. Chamber 1 in Fig. 4.6 has anet throughput of zero and its pressure, P1, is in steady state.In the schematic of an orifice flow standard shown in Fig. 4.6 two gaugesare installed at the upper chamber. One of the gauges (Ga) is calibrated onanother standard in a higher pressure range, and another gauge (Gb) is tobe calibrated at a lower pressure range than Ga. Dividing Eq. 4.2 above byP1 we haveQP1= C(1− r) (4.3)where r = P2/P1. In the molecular flow regime (where the gas atoms ormolecules hit the walls before hitting each other) the ratio r is a constantat different flow rates, Q. The ratio r can be determined by generating alarge flow so that the pressure, P1, in chamber 1 is high enough that it canbe measured with gauge Ga. Gauge Ga is typically a spinning rotary gauge.In addition, throughput Q is measured with a flowmeter so that the ratio rcan be solved from Eq. 4.3 with a known conductance.66Chapter 4. Proposal for a cold atom based pressure sensorFlowMeterP1 QG aG bCP2Pump QFigure 4.6: Two chambers are separated by an orifice of known conductanceC. The throughput through the orifice is Q = C(P1−P2). This throughputwill be the same as the throughput input to chamber 1 providing the pressureP1 in the upper chamber is in steady state and outgassing is negligible. Twogauges are connected to the upper chamber. Gauge A (GA) is calibrated onanother higher pressure range standard and is typically a spinning rotarygauge. Gauge B (GB) is the gauge that is to be calibrated by the orifice flowstandard. Q is measured with a flowmeter and C is either independentlymeasured or calculated. The ratio r = P1/P2 is determined by sending alarge enough throughput through the system so that gauge A can measureP1. The flow is then cut back so that the pressure P1 in the upper chamberis lower than the calibration range of gauge A. The ratio r is constant withinthe molecular flow regime so that the pressure P1 in the upper chamber canbe determined from Q, C and r and can be used to calibrate gauge B.67Chapter 4. Proposal for a cold atom based pressure sensorTo calibrate the gauge Gb in Fig. 4.6, the flow can then be cut back sothat the pressure P1 is below the measuring range of the calibrated gaugeGa. With measured Q, r, and known C, the pressure P1 can be determinedfrom Eq. 4.3.For a constant pressure flowmeter, as shown in Fig. 4.7, the volume ofa gas-filled region is changed at a constant rate, dVdt , driving gas out of thevolume through a leak valve [77, 78] . The pressure, P , is kept constantin the volume and is measured by a calibrated pressure gauge. With thepressure, the rate of volume change, and the temperature T , the molar flowrate dndt is given asQ = P dVdt =dndt RT. (4.4)The conductance of an orifice can be measured using a constant pressureflow meter. The orifice is used as the outlet for the flowmeter instead of aleak valve [78]. The basic idea is that for a known flow Q produced by theflowmeter, a known pressure P inside the flowmeter, and a small pressureon the output side of the orifice, the conductance C of the orifice is givenas C = Q/P . The conductance can also be calculated using Monte-Carlosimulations [78].A separated flow method can also be used to extend the range of aflowmeter [73, 76]. A known throughput Q from a flowmeter is put into avolume with two orifices with conductances C1 and C2 . The flow throughthe orifice of conductance C1 will be Q1 = Q C1C1+C2 . The flow through theorifice of conductance C2 will similarly be Q2 = Q C2C1+C2 . Smaller knownflows enables calibration of smaller pressures P1 using Eq. 4.3.Ar and N2 gas are typically used in orifice flow standards. When askedwhy, NIST [79] responded that reactive gases are not favourable, nor are‘sticky’ gases that are hard to pump, such as H2O. Also, if the gas moleculeis too complex (for example a polyatomic molecule) then calculations of theconductance of the orifice become difficult.4.3 A proposed density/pressure standard usingtrapped atomsIt is proposed that measurement of background gas collisional loss rate canbe used to measure the density of a given background gas surrounding acloud of trapped atoms. Assume for the moment that there is only onebackground species, for example 40Ar, and that the trapped species is 87Rb.68Chapter 4. Proposal for a cold atom based pressure sensorFigure 4.7: A constant pressure flowmeter used to generate a knownflowrate. The volume of a chamber is changed steadily while the pres-sure is kept constant with respect to a reference chamber allowing a precisedetermination of flowrate out of the variable volume chamber.69Chapter 4. Proposal for a cold atom based pressure sensorFrom Eq. 2.1 the loss rate constant can be expressed asΓ = nAr〈σvAr〉87Rb,40Ar. (4.5)The velocity averaged collisional loss cross section, 〈σvAr〉87Rb,40Ar, was de-scribed in Chapter 2. The proposed density measurement involves measur-ing Γ for trapped 87Rb atoms. With known Γ and 〈σvAr〉87Rb,40Ar for aknown trap depth of the trap, nAr can be determined. This proposal is notrestricted to our particular choice of trapped atom or background gas. Adiscussion of the advantages of using a MOT or a magnetic trap is discussedin section 4.3.2.Trap loss due to background gas collisions and its dependence on back-ground gas pressure is well established [27, 80–82]. The novelty of the studyof trap loss due to background collisions expressed here is the proposed appli-cation of the loss rate constant due to background collisions, Γ, to precisionmeasurements of background gas density. Also, as mentioned in chapter3, the loss rate constant, Γ, and the velocity averaged collisional loss crosssection can be applied to the measurement of trap depth of a trap.A decay time constant of 0.01 to 100 s, corresponds to a loss rate con-stant, Γ, of 100 to 0.01 s−1, which corresponds to a pressure measurementrange of 10−4 to 10−8 Pa (10−6 to 10−10 Torr). It is within vacuum tech-nology to achieve ultimate background pressures orders of magnitudes lowerthan this pressure range so that the assumption that 40Ar, for example,could be taken as the dominant species is reasonable. The expected pres-sure range of the atom gauge on the lower pressure side depends on theresidual background pressure in our system. Our apparatus is describedin the next chapter, and the background pressure is on the order of thelow 10−8 Pa (10−10 Torr) to 10−7 Pa (10−9 Torr) depending on the pumpsthat are used. In section 1.3.1 it is estimated that the intra-trap loss ratefor a MOT becomes comparable to the background collisional induced lossrate below 10−6 Pa (10−8 Torr). As also mentioned in section 1.3.2, theintra-trap loss rate for a magnetic trap should be negligible compared to thebackground loss rate even below 10−6 Pa (10−8 Torr). The higher pressurerange measurable is around the low (< 2× 10−4 Pa ( 2× 10−6 Torr)) rangebecause our magneto-optical trap does not form above that.The proposed density measurement method potentially also allows den-sity measurement of multiple species in the background (i.e. differentialpressure measurement), provided that what those species are is known. Sup-pose the background composition is known to contain N different background70Chapter 4. Proposal for a cold atom based pressure sensorspecies. The loss rate Γ will have a contribution from each of these speciesΓ =N∑i=1ni〈σvi〉X,i. (4.6)As discussed in previous chapters, the loss cross section depends on trapdepth. Measurements of Γ could be taken at N different trap depths andthe values of 〈σvi〉X,i at each trap depth determined. Eq. 4.6 then wouldgive N equations and N unknowns and the densities n1...nN could be solvedfor.It is hoped that the method of density measurement using trapped atomscould serve as a pressure standard. Currently the pressure standards for theUHV pressure range rely on measurements using gauges calibrated at higherpressures. The proposed method has the possibility of not having to use cal-ibrated gauges. For example, for Ar or noble gases the dispersion coefficientsfor the interaction potential between trappable alkali metal atoms have beencalculated [43–45] which allows the velocity averaged collisional loss crosssection to be calculated for a known trap depth. With a known velocityaveraged collisional loss cross section, an externally calibrated gauge shouldnot be necessary. There is also the possible advantage of improved inter-laboratory agreement for pressure calibration because atomic interactionsare fundamentally the same in any lab. The velocity averaged collisionalloss cross section should be the same in all laboratories using the same trapdepth and room temperature.There are efforts to make trapped atom technology more portable, whichwould help with providing on-site calibration. Semi-conductor manufactur-ing processes involving thin film deposition [83], ion implantation [84] andatomic layer deposition [85, 86] use residual gas analyzers and benefit fromaccurate calibration of these devices. In-situ calibration techniques havebeen developed which rely on externally calibrated ion gauges and flow rates[87]. Having an on-site calibration of RGAs or ion gauges using trappedatoms may be a possibility. The technique proposed of using trapped atomsto perform differential pressure measurement would help with RGA calibra-tion where more than one species is desired to be calibrated simultaneously.Trapped atoms could also potentially be applied to measurement of pressurefor gases that are currently not used in existing standards.There is still merit to the proposed method even if it is not desired orit is difficult to calculate the velocity averaged collisional loss cross sectionbetween the trapped atom species and the background gas species chosen.A gauge that is calibrated on another standard could be used to measure71Chapter 4. Proposal for a cold atom based pressure sensorthe density of the desired background gas. The loss rate constant, Γ, forthe trapped atoms could be measured for various background gas densities.As seen in subsection 3.2.3, the slope of the plot of Γ versus backgroundgas density yields the velocity averaged collisional loss cross section. Oncethe velocity averaged collisional loss cross section is known for a given trapdepth it does not change as long as the trap conditions stay the same. Thetrapped atoms once calibrated could serve as a transfer standard which if putin a miniaturized and commercially viable package could enable customersto calibrate their gauges on site. The trapped atoms could also be used toextend gauge calibration to lower pressures and serve as a standard at lowerpressures, much in the same way that orifice flow standards rely on a gaugethat is calibrated on a higher pressure standard.4.3.1 Existing proposals for total vacuum pressuremeasurement using trapped atomsIf the gas species whose pressure is to be measured is unknown, unfortunatelythe technique of using trapped atoms cannot give high accuracy pressuremeasurements. However, Arpornthip et al. [46] proposed that a MOTcould provide rough measurements of background pressure (within a factorof two). As seen in Eq. 2.1 the loss rate due to background collisions, Γ, canbe expressed asΓ =∑ini〈σvi〉X,i (4.7)where ni is the background density of species, i, and 〈σvi〉X,i is the velocityaveraged collisional loss cross section for collisions between the backgroundspecies i and trapped atoms (labelled type X). Writing Eq. 4.7 in terms ofpartial pressure Pi = nikBT givesΓ =∑iPikBT〈σvi〉X,i. (4.8)Arpornthip et al. calculated that 〈σvi〉X,i varies roughly within a factorof two independent of background species choice i and can be pulled out ofthe summation. For Rb as the trapped species, they estimate for backgroundgases such a H2, He, H2O, N2, Ar, and CO2, that the quantity a = 〈σvi〉X,ikBThas an an approximate value of a = 1.7 × 105 to 3.7 × 105 Pa−1 s−1 (a =2.3×107 to 4.9×107 Torr−1 s−1). This is for Rb as the trapped atom species.Assuming a is roughly the same for the different background species then72Chapter 4. Proposal for a cold atom based pressure sensora can come out of the summation. This gives a sum over partial pressureswhich adds to the total pressure, P , givingΓ = aP. (4.9)By taking measurements of Γ and using the rough values given for a one candetermine the background pressure in their system using a MOT. This ideais very similar to our proposed pressure measurement using trapped atoms.Their technique is different in that it is applied to lower precision pressuremeasurements of the total background pressure, whereas we are proposingusing trapped atoms as a pressure standard for a certain known species.Similar ideas to that of Arpornthip et al. of using a MOT to measure thesurrounding total vacuum pressure is also shown by Refs. [88, 89]. Theseideas are motivated by being able to have a more compact and portableMOT setup without the need for an ion gauge to measure the backgroundpressure.4.3.2 Magneto-optical trap versus magnetic-trap for apressure sensorEach of the two atomic traps have advantages and disadvantages for use asa high accuracy (< 5 % uncertainty) sensor. The magnetic trap has theadvantages of being able to trap atoms all in one state, being able to setthe trap depth with an RF knife, and not having light mediated losses. Themagneto-optical trap has some fraction of the atoms in the exicted state,and it has light mediated losses. The magneto-optical trap has the advan-tage of cooling down trapped atoms that were collided with but were notlost from the trap, whereas for a magnetic trap collisions that did not leadto loss lead to heating. A magneto-optical trap also has a simpler and fasterexperimental sequence to determine the loss rate constant, Γ. The signal tonoise for a single Γ measurement is also better for a MOT. Our aim is toexplore both options for pressure measurement, possibly using a combina-tion of magnetic trap at low pressures where light mediated losses becomecomparable to Γ for a MOT. A MOT could be used at high pressures wherethe signal would be small because the steady state atom number decreasesas pressure goes up. For differential pressure measurements, as describedabove, a magnetic trap may be better because the velocity averaged crosssection changes more with the trap depths accessible with a magnetic trap.However, it could also be useful for differential measurement to have accessto a wide range of trap depths provided by both a magnetic trap and aMOT. In summary, both the magnetic trap and MOT have advantages for73Chapter 4. Proposal for a cold atom based pressure sensoruse with pressure measurement, with the MOT being perhaps more suitedfor high pressure measurement and the magnetic trap for low pressure mea-surement. In the range of pressure overlap between the two types of traps,the pressure measurements should be shown to agree.4.3.3 Estimated uncertaintyThe uncertainty in the proposed pressure measurement using trapped atomsdepends on the uncertainty in the measurement of the loss rate constant,Γ, and the calculation of the velocity averaged collisional loss cross section〈σv〉. Γ has an uncertainty of roughly 3% on a single loss rate measurementwhich can be reduced by multiple measurements, hopefully to below 1%.As discussed in chapter 2, the physical parameters to input into thecalculation for 〈σv〉 are the masses of the background gas species and thetrapped atom species, the interaction potential V (r), the temperature of thebackground gas, and the trap depth. For 87Rb as the trapped species, and40Ar as the background gas species, the percentage change of the value of〈σlossvAr〉87Rb,40Ar was calculated for a 10 percent, 50 percent and 100 per-cent increase in the C6 coefficient, well depth, background temperature, andtrap depth. The original input parameter values used were a backgroundgas temperature of 294 K, a potential with V (r) = C12r12 −C6r6 with C6 = 336.4a.u., a 50 cm−1 well depth used for C12, and a 5 mK trap depth. The resultsof the calculation are shown in Table 4.1, where the values in the tables areexpressed in percentages and under each parameter is the percentage changeof 〈σlossvAr〉87Rb,40Ar due to 10, 50 and 100 percent increases in the parame-ters while the other parameters were held at the original values. Given thatTable 4.1: Percentage changes for the value of 〈σlossvAr〉87Rb,40Ar due to 10,50 and 100% increases in the values of C6, well depth , temperature (T ),and trap depth (U). While one parameter changed the other parameterswere held fixed at T = 294 K, C6 = 336.4 a.u.,  = 50cm−1, and U = 5 mKPercentage change C6  T U10 3 -0.02 4 -2.350 13 -0.06 15 -10100 23 0.03 26 -17the temperature variations expected in the room should generally exceed nomore than 2 K the room temperature variation should contribute negligiblyto the uncertainty in 〈σlossvAr〉87Rb,40Ar, i.e < 0.1%. Another source of error74Chapter 4. Proposal for a cold atom based pressure sensoris temperature variations at different locations in the vacuum apparatus.The temperature variation, except right near an ion gauge, is less than 5K.At an ion gauge flange the exterior of the gauge is ≈ 50 to 60 ◦C . Therest of the apparatus is at room temperature. As described in section 4.2.2,the ion gauges are typically calibrated by a gauge such as a spinning ro-tary gauge that is nearby but not at the same local temperature as the iongauge. This means although locally at the ion gauge the temperature ishigher than room temperature the gauge is calibration is performed withinan environment that is at room temperature.Changes in the well depth,  contribute negligibly. The most significantcontribution to uncertainty in 〈σlossvAr〉87Rb,40Ar are errors in the C6 coeffi-cient and trap depth. The C6 given is accurate to within 1% [43, 44], whichcontributes an error to 〈σlossvAr〉87Rb,40Ar of roughly 0.3%. It should benoted that there are dipole-quadrupole (C8/r8) and quadrupole-quadrupole(C10/r10) contributions to the interaction potential that should also be in-cluded to attain the best accuracy possible. If the trap depth can be setto within approximately 5%, then the resulting uncertainty contribution oftrap depth to 〈σlossvAr〉87Rb,40Ar is approximately 1.5%. To get a sense ofrealistic trap depth errors, we refer to the measurements described in chap-ter 3 in Fig. 3.12 and Fig. 3.13. There the estimated trap depth error forthe magnetic trap data was around 30% and the MOT trap depth error wasaround 15%. For the magnetic trap the trap depth error may be reducedby using an RF-knife (see section 1.2.2). An RF-knife was not used for themeasurements in chapter 3. Note also that the trap depth of 5 mK chosenfor these uncertainty estimates is approximately at the steepest point on the〈σlossvAr〉87Rb,40Ar versus trap depth curve in Fig. 3.12. This means the con-tribution of trap depth uncertainty to the uncertainty of 〈σlossvAr〉87Rb,40Arshould be an upper bound. In summary, an estimate of < 5% is hoped forfor the uncertainty in nAr based on uncertainty in 〈σlossvAr〉87Rb,40Ar and Γ.Another possible source of error is pressure differences at the trapped atomsversus the gauge locations. This is further discussed in section 8.1.The National Institute of Standards and Technology (NIST) performscalibration of ionization gauges at pressures of 10−3 to 10−5 Pa (10−5 to10−7 Torr) with an uncertainty of 0.7% or less with the uncertainty risingto 2% at 10−7 Pa ( 10−9 Torr) [90].75Chapter 5Experimental apparatus forthe pressure sensorexperimentThis chapter provides details on the design and construction of the exper-imental apparatus for taking pressure measurements using trapped atoms.A solidworks drawing of the vacuum apparatus is given in Fig. 5.1. Fig. 5.2shows the same apparatus from a different point of view. Vacuum created inthe system starts with the high-vacuum (HV) pump region of the apparatusconsisting of scroll and turbo pumps. The 2D MOT section consists of achamber with viewports for the 2D MOT light to enter. A Rb source is alsoattached to the 2D MOT chamber to load the 2D MOT from Rb vapour.The 2D MOT section is connected to the 3D MOT section through a seriesof tubes used for differential pumping. The 2D MOT provides an atomicbeam of 85Rb or 87Rb atoms to the 3D MOT section. The 3D MOT sectionconsists of a glass cell for input of laser beams and surrounding magneticcoils to create a 3D MOT and a magnetic trap. The 3D MOT cell hasultra-high vacuum (UHV) pumps attached to it to maintain low pressures(i.e. in the 10−9 Pa (10−11 Torr) range). Diagnostics (pressure gauges) arealso attached to the 3D MOT cell to compare pressure measurements ofcommercial gauges with measurements taken using trapped atoms. Belowthe 2D MOT section is a leak valve to input gases whose pressures are tobe measured. This chapter will provide more detail on all of these differentsections of the experimental apparatus. Not shown in Fig. 5.1 and Fig. 5.2are the optics used to generate the 2D and 3D MOT light. These optics willalso be discussed in this chapter. Throughout this chapter any abbreviatedlabelling refers to Fig. 5.1 unless otherwise noted.Several features of our apparatus that are not common in experimentsinvolving ultra-cold atoms are the presence of a leak valve to introducebackground gas, the design of our differential pumping system, and thenumerous pressure gauges installed on our apparatus. A new apparatus76Chapter 5. Experimental apparatus for the pressure sensor experimentwas built, different from the one used for chapter 3, to implement thesefeatures. A motivation for a brand new apparatus is that the 3D MOT usedin chapter 3 is vapour loaded and the implementation of a 2D MOT allowsthe pressure in the 3D MOT region of Rb to be low. Also, in a 2D MOTsetup, the Rb source is protected from reactive gases that may be studiedlater. The optical setup used in chapter 3 does not generate enough powerfor a 2D MOT and a 3D MOT, and a new optical setup with a taperedamplifier was needed. Another reason for a new setup is physically the newapparatus takes more room than is available on the optical table occupiedby the setup used in chapter 3.77Chapter5.ExperimentalapparatusforthepressuresensorexperimentFigure 5.1: The vacuum apparatus for the pressure sensor experiment. SP: scroll pump, TP1, TP2: turbopumps, MIG: mini ion gauge, CG1 and CG2: convectron gauges, SV: solenoid valve, V1-V9 (V8 not shown): allmetal valves, IP1-IP3: ion pumps, LV: leak valve, LD: linear drive mechanism, RbA: rubidium ampoule, 2D: 2DMOT chamber, GV: gate valve, 3D: 3D cell, RGA: residual gas analyzer, CDG: capacitance diaphragm gauge,SRG: spinning rotary gauge, IG: ion gauge, RbD: rubidium dispensers, TSP: titanium-sublimation pump, NEG,non-evaporable getter.78Chapter5.ExperimentalapparatusforthepressuresensorexperimentFigure 5.2: Another perspective of the vacuum apparatus for the pressure sensor experiment. The apparatusconsists of different main sections. At the 2D MOT section a 2D MOT is created to form an atomic beam to loada 3D MOT. A Rb source is connected to the 2D MOT region to load the 2D MOT from Rb vapour. The 3D MOTsection consists of a glass cell. UHV pumps are used to maintain ultra-high vacuum pressure in the 3D MOT cell.Pressure gauges are attached to the 3D MOT cell to provide measurements of pressure to compare to the pressuremeasurements taken with trapped atoms.79Chapter 5. Experimental apparatus for the pressure sensor experiment5.1 High vacuum pumpsOur experiment uses a variety of vacuum pumps to establish ultra-highvacuum (UHV) in our apparatus (pressures < 10−6 Pa or < 10−8 Torr).Starting from atmospheric pressure we pump with a scroll pump (SH-110,Agilent) labelled SP. The scroll pump has two circular mechanisms thatmove with respect to each other and compress gas that enters the mech-anisms towards an exhaust output. The scroll pump has a specified basepressure of 7 Pa (5× 10−2 Torr) [91]. The measured base pressure with aneyesys mini-BA ion gauge from Varian connected directly to the scroll pumpwas 4.8 Pa ( 3.6 × 10−2 Torr ). To get lower in pressure, turbomolecularpumps (TV 70 , Agilent) labelled TP1 and TP2 were attached in series andbacked by the scroll pump. Turbomolecular pumps have a series of bladeswhich are angled. When spinning, the blades impart momentum to the gasthat hit the blades towards another set of blades etc. The gas makes its wayto the exhaust which is pumped away by the scroll pump. The specifiedbase pressure of the turbo pump is 5.1 × 10−7 Pa (3.8 × 10−9 Torr).During initial evacuation and while baking we attached to the vacuumsystem the ‘bakeout station’ described in sec. 3.1. This station has anotherscroll pump (SH-100, Agilent) and a turbo pump (TV 70, Agilent). Duringinitial pump down and baking, the turbo and scroll pumps were the onlypumps operating since the other pumps in the apparatus operate in theUHV regime and need ‘cleaning’ by baking.An electromagnetic solenoid valve labelled SV (SA0150EVCF, Kurt Lesker)was placed after the turbo pump in the high-vacuum pump region in case ofa power failure to prevent air leaking into the vacuum chamber through thescroll and turbo pumps. The bakeout station also has a pneumatic valve forthis purpose. Two manual all metal valves, V1 and V2 (9515027, Agilent)can be used to valve off the HV pumps for further protection and for removalor repair of the pumps.The high vacuum pumps are connected to the main apparatus by abellows. This bellows connects to ion pump IP1 (9191145, Agilent) throughall-metal valve V3 (9515027, Agilent). This bellows is also attached by atee to all-metal valve V4 and V5. Valve V4 (9515017, Agilent) connectsthe high vacuum pumps to the 2D MOT chamber portion up above. ValveV5 (9515027, Agilent) connects the 3D MOT and UHV pump region ofthe apparatus to the high vacuum pumps. Valve V5 also separates the 3DMOT region from the leak valve LV (9515106, Agilent). A convectron gauge(labelled CG2) is present after valves V1 and V2 but can only read down to10−2 Pa (10−4 Torr).80Chapter 5. Experimental apparatus for the pressure sensor experiment5.2 2D MOT sectionA 2D MOT is formed from two orthogonal pairs of counterpropagating laserbeams travelling along, for example, the ±y and ±z directions. In addi-tion, magnetic field gradients along those directions are needed, and alsoappropriate right or left circular polarization choices for the light. Thereare no trapping laser beams along the x direction and the field gradientalong the x direction is small compared with the y and z direction. Thismeans Rb atoms entering the intersection of the trapping laser beams froma background vapour of Rb will have their y and z velocity componentsdecreased in magnitude but not their x velocity component. As a result anatomic beam will form exiting the trapping region (the intersection of thelaser beams) along the ±x axis. This atomic beam travels through smalltubes connected to the section of the vacuum apparatus where the coils andoptics for the 3D MOT reside. A ‘push’ laser beam is sent through the 2DMOT to push atoms in the positive x direction towards the tubes. The pushbeam is a collimated laser beam that is approximately the diameter of thetubes, points along the atomic beam axis, and is blue detuned to acceleratethe atoms towards the 3D MOT region. The small tubes separating the2D and 3D MOT regions provide differential pumping where the pressureon the 2D MOT side of the tubes can be high (e.g. 10−6Pa or 10−8 Torr)and on the 3D MOT side it can be low (e.g. 10−8 Pa or 10−10 Torr). Thetubes also act as a speed filter so that the speed distribution of the atomicbeam travelling to the 3D MOT is significantly below the speed distribu-tion expected from a thermal gas. The 2D MOT is loaded from Rb vapour.An advantage of loading a 3D MOT from a 2D MOT is that the 3D MOTvacuum region can be orders of magnitude lower in pressure. For our casethis is important since we want the residual background pressure that thepressure measurements with trapped atoms are performed in to be as low aspossible, while still maintaining a sizeable 3D MOT. Another advantage ofseparating the Rb source from the 3D MOT region is to protect the sourcewhile adding in background gas that is reactive with Rb.5.2.1 2D MOT chamber regionThe 2D MOT chamber (labelled 2D in Fig. 5.1 and also shown in Fig. 5.3)was custom built (Johnsen Ultravac) with seven ports. Four of these portsform a four-way cross with 4.5 inch CF viewports for light to enter for the 2DMOT. One port along the atomic beam axis is connected to a six way cross(shown in Fig. 5.2). This six way cross connects to the Rb source through81Chapter 5. Experimental apparatus for the pressure sensor experimentvalve V9 (9515014, Agilent). The Rb source consists of a glass ampoule with1g of Rb (K4584x from ESPI metals). This ampoule has a custom madeholder (Johnsen Ultravac). A linear drive mechanism LD (KLPDAA, KurtLesker) was installed for the purpose of breaking open the ampoule afterbaking. The six way cross also connects to the HV pumps when valve V4 isopen. Three viewports are installed on the six way cross for looking for Rbfluorescence when filling the 2D MOT chamber with Rb. The viewports inline with the atomic beams axis is also used for viewing the 2D MOT downthe atomic beam axis, and for sending in the push beam mentioned above.The port of the 2D MOT chamber in line with the beam axis, and closestto the 3D MOT section, contains a differential pumping section vacuum-welded into the port. Fig. 5.4 shows a cut away of the differential pumpingsection of the chamber. The first tube in the differential pumping designseparates the 2D MOT chamber, where the Rb vapour resides, from anion pump (IP2 9191145, Agilent). A second tube then connects the regionwith the ion pump to the 3D MOT side of the apparatus. The second tubehas a graduated opening to allow for divergence in the atomic beam as itpropagates towards the 3D MOT. Having the ion pump separated from theRb vapour helps to preserve the lifetime of the ion pump. The two tubesare lined up along the x direction (the atomic beam axis). Atoms from the2D MOT propagate with high directionality along the x direction and willmake it through both tubes. Atoms that randomly make their way into thetubes from the Rb background vapour within the 2D MOT chamber willtend not to be travelling with such high x directionality and will bouncearound in the region between the two tubes until they are removed by theion pump. The distance from the entrance of the differential pumping tubein the 2D MOT chamber to the center of the cell at the 3D MOT region isapproximately 55 cm.The design of the 2D MOT chamber as well as preliminary design, andordering, of the vacuum apparatus is credited to Ian Moult, Weiqi Wang,and Haotian Pang. The mechanical drawings for the 2D MOT chamber andthe Rb ampoule holder, as well as differential pumping analysis are foundin their report [92]. The Rb ampoule holder design is credited to Dr. BruceKlappauf.5.2.2 2D MOT coilsFigure 5.5 shows a picture of the 2D MOT coils installed on the 2D MOTchamber. Four rectangular coils are used to produce a magnetic field gra-dient along the two perpendicular arms of the large four-way cross of the82Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.3: The custom made 2D MOT chamber. The differential pumpingtube (see Fig. 5.4) can be seen to protrude slightly into the chamber. Thistube then leads to an opening where an ion pump is attached on the portthat is coming out at a 45 degree angle. A second series of tubes connectsthis opening to the 3D MOT chamber. The four large ports are for theviewports through which the 2D MOT laser beams are sent. The port onthe left in the figure is attached via a six way cross (not shown) to the Rbsource and to viewports. A viewport in line with the atomic beam axis isused both to view the 2D MOT and to send a push beam that sends theatoms towards the 3D MOT region.83Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.4: A cut away of the differential pumping section. Atoms travelfrom left to right through one tube and then through another series of tubesbefore exiting and going to the 3D MOT cell. Atoms in the atomic beamhave a high directionality and will make their way through the tubes. Atomsthat randomnly enter the tubes from the vapour on either side of the 2D or3D MOT regions will tend to bounce around in the section between the twodifferent tube sections and be pumped away by an ion pump.2D MOT chamber. Each coil is parallel to the viewport it fits around. Theposition of the 2D MOT beam axis with respect to the differential pumpingtube is optimized by controlling the current to each of the four coils. Theinside dimensions of each coil is 8 by 26 cm. The coil pair surrounding theviewports in the y direction has an inside separation of approximately 8.5cm and the pair surrounding the viewports in the z direction is separatedby approximately 12.5 cm. There are approximately 10 layers with 12 turnsper layer totalling 120 turns. The wire used was 16 AWG magnet wire fromSuperior Essex (H GP/MR-200). The gradient with 5A going through allthe coils was measured using a gaussmeter (model Bell 620) to be 16.6 G/cm(1.66×10−3 T/cm) along the transverse direction and 0.29 G/cm (2.9×10−5T/cm) along the axial direction.The inner width of the coils was less than the width of the viewportsthey slip over. For this reason the 2D MOT rectangular coils were wound ona separable frame (Fig. 5.6) that could be used to wind the coils and thencould come away from the coils leaving just the coil. The coil shape onceseparated from the frame was maintained by wrapping around the coil crosssection in a few positions either with high temperature kapton tape or with84Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.5: Four rectangular coils are slipped around each of the four view-ports of the 2D MOT chamber and provide the magnetic field gradient forthe 2D MOT.wire. This allowed the coils to be wound separately first and then installedlater which is more convenient than winding the coils in place. This waspossible because the inner coil width was large enough, and the coils flexibleenough, that the coils, with slight deformation, could be slipped over theviewports of the four-way cross of the 2D MOT chamber. The design andmanufacture of this frame as well as initial simulations of the 2D MOT coilsis credited to Kousha Talebian. Electrical tape was placed on the framesurface to prevent the wire from scratching on the frame when winding.Scratches could lead to the removal of the insulating layer on the wires andshorting. The coils were wound by hand with feet placed on a rod goingthrough the wire spool providing tension. The start of the first winding wassecured so that the wire did not move around when the coil winding was85Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.6: A drawing of the frame used to wind the 2D MOT coils. Theframe is made to separate away from the coils once the coils are wound. Thecoils are held together by wrapping high temperature kapton tape aroundthe coils in a few places. The coils could then be slipped over the viewportsof the 2D MOT chamber.getting started. Plastic wrap was placed over the viewports and over thebolts securing the viewports to the 2D MOT chamber when putting on thecoils to prevent either the coils or the viewports from being scratched.One of the coils needed to be mounted before the ion pump used for thedifferential pumping in the 2D MOT chamber could be installed. This coilcan not be removed and was present during baking. The three others coilswere not present during baking and are removable. The magnet wire usedfor the coils has an insulating layer of polyester/polyamideimide (which isnot kapton). This coating has a limited lifetime above 215◦C and Kaptoncoated wire should be used if a higher temperature limit is needed. Afterbaking for a long period of time (a month) near the upper temperature limitand then cooling down to room temperature, small cracks formed at stresspoints so that extreme care should be taken in handling these coils so thatthe coating is not rubbed off causing shorts.Credit goes to Vandna Gokhroo and Wolfgang Mu¨ssel for helpful ad-vice concerning the alignment and detection of the 2D and 3D MOT. AlsoRefs. [93–95] the following papers were useful in the design of the 2D MOTmagnetic coils and the optics planning.86Chapter 5. Experimental apparatus for the pressure sensor experiment5.3 3D MOT sectionThe 2D MOT chamber is separated from the 3D MOT section by an all-metal gate valve (48132-CE01-0002, VAT valve). After the gate valve is a six-way cross with viewports for diagnosing the 2D MOT beam characteristicsexiting the differential pumping tubes. The six way cross connects to an ionpump, leak valve and turbo and scroll pumps through valve V5 in Fig. 5.1.This six way cross is also attached to a glass cell into which laser beams forthe 3D MOT are sent. This glass cell is rectangular in shape where the laserbeams enter. The cross section of the cell is square with inner dimensions of40 mm by 40 mm and the cell is 90 mm long. Magnetic coils are also neededfor the 3D MOT and are representatively shown in Fig. 5.1 and Fig. UHV pumps and diagnosticsSeveral gauges are attached to the UHV region (the part numbers and spec-ifications are given in chapter 4). A capacitance diaphragm gauge (labelledCDG in Fig. 5.1) was installed to calibrate a spinning rotary gauge, whichin turn was installed to calibrate an ion gauge. Pressure measurements withthe ion gauge (IG) will be compared with the measurements provided bytrapped atoms. A residual gas analyzer (RGA) is also installed for analy-sis of the different species present in the background, for background gasdetection, and for comparative differential pressure measurements.The UHV section has a titanium sublimation pump attached to it (la-belled TSP, 916-0050, Agilent). A non-evaporable getter pump (NEG, C400-2-DSK ,SAES getters) and an ion pump (IP3, VacIon Plus 20, 9191145,Agilent) are also attached to achieve a low background pressure. Thesepumps are behind a valve (V6, VZCR60R, Kurt Lesker) so that they canbe isolated from the 3D MOT region during any pressure measurements toreduce pressure gradients and to protect the pumps. An elbow is installedbetween the valve V6 and the TSP, NEG, and IP3, to prevent Titaniumfrom the Ti-sub pump coating the valve which is exposed to the 3D MOTcell when the valve is closed.The titanium sublimation pump and the non-evaporable getter in oursystem work by chemisorption [61]. Chemisorption is when a material (ti-tanium or Zr V Fe in the case of our NEG) has a high binding energy foractive gases such as O2, N2, CO2 and H2 so that these gases adhere to thematerial. The titanium sublimation pump shown in Fig. 5.7 consists of ti-tanium filaments that are heated. At a sufficiently high temperature the87Chapter 5. Experimental apparatus for the pressure sensor experimenttitanium evaporates forming a film on the surrounding walls of the chamberwhich can then pump gases from the system. The titanium is evaporated atintervals forming a fresh layer over trapped material adsorbed onto previouslayers.Figure 5.7: A picture of the titanium sublimation pump. The wavy metalis three titanium filaments. They are mounted such that high currents canbe put through them sublimating the titanium and coating surroundingstructures in the vacuum apparatus. This thin titanium layer coating actslike a pump because gases bind to it.Our non-evaporable getter is ZrVFe arranged in disks to have a largesurface area. The getter is activated occasionally by heating causing theexternal monolayer of gases adhering to the surface to move within thegetter material allowing a fresh surface for continued pumping. The getteris then cooled because the lower the temperature of the getter, the lower thebase pressure of H2 realized in the system. About 90-100 activations can beperformed, with the time between activations depending on the gas inputto the system, how often the system is baked, and how often it is ventedto atmosphere. The getter is reaching its end of use when the hydrogenpressure slowly rises and re-activation leads to a baseline hydrogen pressurethat is higher than for previous re-activations. The getter pumps hydrogen88Chapter 5. Experimental apparatus for the pressure sensor experimentand active gases well: however, the getter pumps do not pump inert gases.These pumps are typically used to achieve and maintain UHV but can onlybe activated once a sufficiently low pressure has already been achieved byother pumps such as the scroll, ion, and turbo pumps.Ion pumps (VacIon Plus 20, 9191145, Agilent) are placed in several lo-cations of our apparatus. Ion pumps have an ultimate pressure of less than10−9 Pa (10−11 Torr) and should not be turned on above 10−1 Pa (10−3Torr). In ion pumps electrons are discharged from cathodes towards an an-ode. Magnetic fields are used to maximize the electron trajectory length byproducing spiral trajectories. During its trajectory an electron can ionizegases found in the system creating positive ions which are then acceleratedtowards a negatively charged cathode. This cathode is typically made oftitanium so that the ions react with the titanium and are removed from thesystem. Also neutral gases can react with the titanium as in a titanium sub-limation pump. When the ions hit the cathode they have sufficient energy toknock out or ‘sputter’ titanium from the cathode which coats surfaces withfresh titanium [61, 65]. Ionized noble gases can be pumped by being accel-erated towards the cathode and being buried within the cathode material.They are further buried by incoming sputtered cathode material.Part of the 3D MOT section is also a connection to Rb dispensers (RbD, AS-3-Rb-50-V , Alvatec) which can be valved off with valve V7 (9515027,Agilent) if not needed. These were installed for trouble shooting purposesin case a 3D MOT could not be achieved easily at first using the 2D MOT.5.5 Vacuum apparatus assemblyThis section describes the assembly steps of the vacuum apparatus. Thevacuum apparatus was assembled from the 2D MOT side towards the 3DMOT side. All the standard stainless steel vacuum components were as-sembled first, except for the gauges, pumps, and Rb sources. Mounts for2.5 inch ConFlat (CF) flanges and 4.5 inch CF flanges were machined andmounted on to 1.5 inch diameter posts to hold up the vacuum apparatus.The CF flanges are secured to the mounts using hose clamps that are cutand screwed to the mounts. The height from the optical table top to thecenter of the cell where the atoms are trapped in the 3D MOT is 17.5 inches.The reason the trapped atom location quite high relative to the table top isthat the apparatus was designed to be as narrow as possible, leaving roomfor surrounding optics. This pushed part of the apparatus upwards.Each piece used for ultra-high vacuum has one or multiple CF flanges89Chapter 5. Experimental apparatus for the pressure sensor experimentto connect to other pieces. A CF flange consists of a sharp knife edge. Acopper gasket is placed between two CF flanges and a seal is formed whenthe flanges are bolted and tightened together. If the connection is ever takenapart (vacuum is broken) then a new gasket must be used. Scratches in thegaskets can prevent proper sealing. It is sometimes hard to hold the gasketin place while holding pieces together and putting in screws and bolts. Ahelpful suggestion from a labmate was to put kapton tape on the edges of thecopper gasket to secure it to the conflat flange while connecting flanges. Thisprevents the gasket from falling out while tightening the bolts. The tapeis removed after the flanges and bolts have been finger-tightened together.The flange bolts are tightened in opposing pairs to maintain uniform cuttingof the flanges into the gasket. The bolts are tightened until a sliver of coppergasket can still be seen so that, in the case of a leak, the bolts can be furthertightened and to allow for expansion when baking. Wherever possible silverbolts ( eg. TBS25028125P from Kurt Lesker) are used to prevent seizingafter baking. Otherwise an anti-seize compound (VZTL from Kurt Lesker)can be used but care should be taken not to introduce the compound to thevacuum side of the flanges.All CF connections on the tee and valve V9 that are attached to the Rbampoule RbA have Ni annealed gaskets (GA-0275NIA and GA-0133NIA,Kurt Lesker) since this region comes in the highest contact with Rb. Theviewports on the apparatus have annealed copper gaskets (VZCUA38 andVZCUA64, Kurt Lesker) rather than regular copper gaskets to put less stresson the viewport which reduces the chance of the viewports developing a leak.No precleaning of vacuum parts was used (they are precleaned from thefactory). Gloves were used at all times and changed frequently. Care wastaken not to talk or breathe into the vacuum apparatus. The bellows on theRb cell was scrubbed with a bottle washer with alconox, sonicated in alconoxfor 1 hr, then in distilled water, then in acetone and then in methanol. Thebellows side of the cell was attached to the 2D MOT side of the apparatusto strain relieve the connection of the cell to the 2D MOT region whileall the components on the other side of the cell were being added. Thecompensation coil pair, shown in Fig. 5.12, that encircle the cell were put inat this stage. There are three pairs of compensation coils in total to allow amagnetic field to be added to the magnetic field from the 3D MOT magneticcoils. These coils were installed for planned optical-dipole trapping in thefar future.Another part that had to be cleaned was the electrical feedthrough whichwas machined (cut down and holes inserted) so that the Rb dispenser tabscould be put through. The feedthrough was cleaned with methanol sonica-90Chapter 5. Experimental apparatus for the pressure sensor experimenttion after machining. The NEG holder was also sonicated in methanol. Boththe NEG holder and the feedthrough have ceramic parts deeper inside thepieces and it is recommended not to get these wet with methonal or othersolvents since the ceramic will absorb it and outgas later under vacuum.The apparatus was leak-tested periodically as it was being built by blank-ing off all openings, pumping down using our bakeout station, and thenspraying He around the CF flange joints. The residual gas analyzer on thebakeout station described in section 3.1 was used to detect any He enter-ing the system. Once the main vacuum apparatus frame was in place, theTi-sub pump was installed.The ion pumps were installed next with the magnets left on. As men-tioned, the 2D MOT coil which wraps around the ion pump port on the2D MOT chamber was installed prior to connecting the ion pump. The ionpumps are shipped under vacuum, sealed off with a CF flange. This flangehas a bolt hole in it into which one can insert a bolt and use the bolt to re-move the flange, which is held tightly onto the pump due to the low pressureinside and the high pressure outside. The bakeable cables were attached tothe ion pumps since the ion pumps needed to be turned on towards the endof the baking phase (described in the next section). The optical table sur-face is grounded using a metal braid attached to grounded electrical pipingin the lab. This provides grounding to the vacuum apparatus through themounts holding the vacuum apparatus up.Pressure gauges were installed as per their manuals consisting mostly ofroutine connections of CF flanges. The spinning rotary gauge was installedusing a level since the axis of the measurement head needs to be verticalwithin ±1◦. After the gauges were installed two Rb dispensers were installed.Finally the NEG was installed. It was clear that the standard 2.75 inch CFnipple was too small in diameter as the NEG scrapes the walls. If vacuum isever broken, it is recommended to buy and install the manufacturer designedpump body for the NEG or to install the NEG in a larger diameter flange sothat the pumping capacity is not limited as it is now. Also the capacitancediaphragm gauge should be oriented so that the housing can come off andthe cables removed. Consideration of putting on a valve on any of thegauges should be made in case one wants to disconnect them and send themaway for calibration or repair. Additional recommended changes include theinstallation of a cooling sleeve on the Ti-Sub (9190180, Agilent). Credit isgiven to Ian Moult, Weiqi Wang, and Haotian Pang, Dr. James Booth, Dr.Kirk Madison, and Kousha Talebian for work on planning of the apparatus,purchasing, assembly and baking out of the apparatus.91Chapter 5. Experimental apparatus for the pressure sensor experiment5.6 Bakeout ProcedureBaking out a vacuum apparatus involves heating the apparatus to high tem-peratures (above 100◦C) to drive the water and other gases like H2, CO2,and O2 off of the vacuum apparatus walls and to pump them away.5.6.1 OvenTo bakeout the chamber, an oven was formed around the vacuum apparatuson the optical table (see Fig. 5.8). Firebricks (K23 Firebrick and 3 feet by 1Figure 5.8: A picture of the oven built around the setup to bakeout thevacuum apparatus.feet by 1 inch Fibre Block Insulation from Greenbarn Potters Supply) wereused to form the oven. They were wrapped in aluminum foil to prevent thedust from the porous firebrick material from getting everywhere. Two layersof firebricks were put down on a layer of aluminum foil on the optical tablearound the base of the vacuum setup. A few bricks were cut to try to fillin gaps in this bottom layer and aluminum foil was scrunched up and putin to fill remaining gaps. Five infrared heaters (900 W Infrared Salamanderheaters from Mor Electric Heating Assoc., Inc) powered individually by avariac were placed on this base layer of bricks at roughly equally spaced92Chapter 5. Experimental apparatus for the pressure sensor experimentlocations. Thermocouple gauges were also placed at various positions of thechamber to monitor the temperature while baking out. The thermocou-ple gauges were assembled from Newark part numbers (93F9305, 93F9313,50B5932, 93B0462). The firebricks have dimensions of roughly 9" by 4.5"by 2" and the overall dimensions of the oven was approximately 7 feet by 3feet by 3 feet and required around 420 firebricks. The lid of the oven wasmade from 3 feet by 1 feet by 1 inch Fibre block insulation with doublelayers put on top.5.6.2 Pre-bake preparationBefore any of the oven was formed we removed the wooden platform abovethe optical table and the HEPA filter to reduce the amount of heat trapped,to protect the filter and to allow the oven to be built tall enough to sur-round the vacuum apparatus. In preparation for the bakeout aluminum foilwas placed in thick layers around the viewports and cell to protect againsttemperature gradients and anything melting onto them. The plastic handlesfor the gate valve (labelled GV in Fig. 5.1), 4.5" CF valve (labelled V6) andthe linear drive mechanism (labelled LD) were removed. The linear drivemechanism was locked so as not to break the Rb ampoule. Any electricalconnections to the ion gauge, residual gas analyzer, capacitance diaphragmgauge as well as the spinning rotary gauge measurement head were not in-stalled prior to baking and must be removed for any future baking. Anystickers or plastic were also removed. The all-metal valves as well as thegate valve were open fully. The leak valve (labelled LV) was kept closed.Heater tapes powered by variacs were also used for heating the parts ofthe vacuum apparatus not contained in the oven such as the bellows con-necting the apparatus to the bakeout station. The lab’s ‘bakeout station’consists of a scroll and turbo pump for pumping, an ion gauge to indicatethe pressure, a residual gas analyzer, and a bank of inputs for thermocou-ple gauges. A labview program collected the temperature readings of thethermocouple gauges and the pressure from the bakeout station ion gauge.Before starting to bake the system we pumped on the vacuum apparatususing the bakeout station to a pressure of 9.3 × 10−6 Pa (7 × 10−8 Torr)as read by the ion gauge on the bake-out station. Initially we ran 30 Athrough each Ti-Sub filament just to test the Ti-Sub filament controller; thepressure rose to the high 10−2 Pa (10−4 Torr) range and then decreased.The Ti-Sub filaments were then turned off. After pumping down to the10−6 Pa (10−8 Torr) range again, the NEG was activated. To activate theNEG, DC voltage was applied in steps of 1V/min up to 16V which is then93Chapter 5. Experimental apparatus for the pressure sensor experimentleft for 60 min. The heating curves provided by SAES Getters give a gettertemperature of 475◦C at 16V. Again pressure in the range of 10−2 Pa (10−4Torr) is reached when the NEG is being heated due to outgassing. It wasnoted that when first applying voltage, the current predicted through theNEG by SAES Getters was about a factor of two off of their provided curves.This is because the resistance of the NEG increases as it is being heated sothat the current will match the measurements provided by SAES Gettersafter some wait time. While the NEG was at 16 V the TSP filaments wereoutgassed by running each in turn at 37 A for 1.5 min. Once the NEGwas activated the voltage was brought down to 7V which corresponds to agetter temperature of 250◦C and was left at that voltage for the durationof the bakeout so that the NEG would be hotter than the rest of vacuumapparatus. The coldest spot in the apparatus is where outgassed materialpreferentially sticks. The TSP controller can only send current through oneof the three filaments at a time, so for the remainder of the bake-out wecycled from filament to filament running 30A continuously through them.When changing from one filament to another the pressure increased by about10 times and then decreased.5.6.3 BakingThe actual baking started by increasing the voltage to the infrared heaters in5V intervals waiting for the temperature inside given by the thermocouplesto stabilize in between. The time to reach a steady temperature after eachincrease of variac voltage to the heaters was around 12 hours. A temperaturegoal of 180◦C in the oven was set since the RGA, linear drive mechanism, andthe capacitance diaphragm gauge cables all have a maximum temperatureof 200◦C. At the bottom of the oven the temperature attained to was 157◦Cwith temperature rising up to 185◦C at the top of the 3D MOT cell andthen decreasing to 179◦C at the top of the RGA since this is close to thetop cover of the oven where heat escapes. The hottest place in the ovenwas on top of the 4 way cross containing the NEG and Ti-Sub filamentsat 200◦C. This is because the NEG and Ti-Sub filaments were additionallybeing heated by running current through them.The oven was brought up first in temperature with the outside bellowslagging behind in temperature so as not to outgas material from the hot bel-lows into a colder main apparatus. Since the bakeout was low temperaturecompared to typical bakeout temperatures, we baked out for a longer timeperiod of about 1 month. During bakeout the maximum pressure attainedat maximum temperature was 2.9×10−4 Pa (2.2×10−6 Torr). While pump-94Chapter 5. Experimental apparatus for the pressure sensor experimenting, it decreased to 3.3 × 10−5 Pa (2.5 × 10−7 Torr) with both the bakeoutstation (one turbo, one scroll pump) and the main apparatus pumps (twoturbo’s and one scroll pump). The Ti-sub filaments were run at 40 A for 1.5min. After that 30A was used continuously, cycling through the filamentsoccasionally.At this stage the ion pumps were turned on and baked for another week.Though others [96] valve off their turbo pumps at this point we found thatvalving them off caused the pressure to rise 2 to 5 times and so we kept theturbo pumps and scroll pumps pumping on the system while the ion pumpswere pumping.5.6.4 CoolingCool down occurred slowly over the course of two days where the bellowswere kept colder than the main chamber. When the experimental chamberwas below 100◦C, the NEG was reactivated at 16V for 60 min and thenturned to zero. The current to the Ti-sub was also shut off. The systemthen was allowed to completely cool down. When cooled the ion gauge onthe bake-out station ‘flat lined’ at the lowest reading of 1.35 × 10−6 Pa(1.01 × 10−8 Torr). The ion current readout from the ion pumps, which isproportional to the pressure in the system, went down when the turbos werevalved off. This means that at that stage the ion pumps were doing a moreeffective job of pumping than the turbos and the scroll pumps. At this stagewe closed valve V1 to the bakeout system and the solenoid valve SV to thehigh vacuum experimental pumps. 48 A was put through one of the Ti-sub filaments several times with a maximum duration of two minutes and aminimum of 30 seconds to sublimate the titanium and coat the surroundingchamber walls. The ion gauge pressure reading on our experimental appa-ratus dropped to 2.3× 10−8 Pa (1.7× 10−10 Torr), over the course of a day.The next step was to break the Rb ampoule and see if the system neededmore baking due to the gases released in the ampoule.5.7 Rb release from the ampouleWhile baking, the Rb for the 2D MOT was inside of a sealed glass ampoule.The UHV section was valved off from the pumps before the ampoule wasbroken. The gate valve GV was closed connecting the 2D and 3D MOTsections and the valve V3 to the ion pump IP3 closest to the high vacuumpumps was also closed. The reason for this was to protect the 3D MOTregion and UHV pumps from contamination while breaking the ampoule.95Chapter 5. Experimental apparatus for the pressure sensor experimentValves V1, V4, and V9 were left open so that the 2D MOT section could bepumped on while the Rb ampoule was being broken and heated. Breakingthe ampoule involved pushing the linear drive mechanism over the top of theRb ampoule to break the glass which was prescored so that the top snappedoff easily. When the ampoule broke the Ar in the ampoule was released andthe pressure went above 10−2 Pa (10−4 Torr) dropping rapidly as the Arwas pumped away.Once the ampoule was broken, the custom holder and the tee holdingthe Rb ampoule was wrapped in heater tape and heated up over severalhours to 75 ◦C. This was to release Rb vapour into the 2D MOT chamberso that there is Rb to trap in the 2D MOT. The 2D MOT chamber waspumped on using the bakeout station and several (5-10) Ar bursts fromAr trapped inside the Rb were seen on a RGA. A laser beam was shonethrough a viewport close to the ampoule region to detect Rb release fromthe ampoule. The laser frequency was scanned over the 52S1/2 to 52P3/2transition of 85Rb. A photodetector was also placed close to the beam todetect fluorescence and the photodetector readout on an oscilloscope wasaveraged to reduce noise. Once the ampoule region was at 75 ◦C, it tookseveral hours to see fluorescence close to the ampoule region. Upon coolingof the ampoule, the valve connecting the 2D MOT chamber to the turboand scroll pump (V4), and the valve for the Rb ampoule (V9), were closedby hand.A few days later the ampoule was heated again up to 86 ◦C while pump-ing on the 2D MOT chamber using the turbo and scroll pump on the bake-out station. Again valve V4 and V9 were open for heating and closed whilecooling.Several weeks later we opened up the Rb ampoule valve V9 (no heating).The gate valve (GV) was open also and the pressure in the UHV region shotup to 10−3 Pa ( 10−5 Torr). The UHV section was then opened to thebakeout station pumps and the UHV region recovered to the 10−7 Pa (10−9Torr) range from the 10−8 (10−10 Torr) range previous to the leak. The leakwas traced to a seal at the top of the Rb ampoule valve V9. Tightening thisseal stopped the leak as determined by a He test detected with the RGA onthe bakeout station. The ampoule valve V9 had its top uncovered to air toallow access to tighten it while the bottom portion was heated so it seemsthat the temperature differential on the valve caused the leak. The heatertape was moved away from the valve and the ampoule was heated to 95 ◦Cfor about an hour. After this, valve V4 connecting the 2D MOT chamberwas closed and the rubidium ampoule valve V9 was left open.After the leak was fixed, running the Ti-Sub pump several times for 1-96Chapter 5. Experimental apparatus for the pressure sensor experiment2 min at 48 A allowed the pressure to decrease to around 2.7 × 10−8 Pa(2 × 10−10 Torr). The NEG was also reactivated but it is not clear thathelped as the baseline pressure was initially higher than before with veryslow decrease. Again the Ti-Sub was run around 1.5 min several timesand then the UHV section pressure slowly made its way to 1.25 × 10−8 Pa(9.4 × 10−11 Torr) over the course of several months with valve V5 closed.This is from an original low of 1.2× 10−8 Pa (8.9× 10−11 Torr) prior to theleak. Currently the base pressure with only the UHV pumps is 8.9 × 10−9Pa (6.7 × 10−11 Torr).After breaking and heating the ampoule, the next step was to try toproduce a 2D MOT. The next section explains the optical setup used forcreating a 2D MOT and for the first version of the 3D MOT.5.8 OpticsThe 52S1/2 to 52P3/2 D2 transition is used for trapping of either 85Rb or87Rb. The laser pump is chosen to be the F = 2 to F ′ = 3 transition for87Rb or F = 3 to F ′ = 4 transition for 85Rb. The repump is chosen to bethe F = 1 to F ′ = 2 transition for 87Rb or F = 2 to F ′ = 3 transition for85Rb. Light that is 180 MHz below the repump and pump transitions, foreither 87Rb or 85Rb, is brought over by optical fibers to the experimentaltable from a central optical table. This central table provides the initialfrequency stabilized light for all of our experimental tables [50, 51]. Thefibers used between the central and experimental tables are from OZ optics.The main optical setup to provide light for the 2D and 3D MOT is shownin Fig. 5.9. A list of optical components used in the setup with part numbersis given in Table. 5.1For the repump, light arriving by fiber from the central table (1.5 mW)injects a slave laser diode (MeshTel MLD780-100S5P at 18 ◦C). The slavelaser diode output is 44 mW. This laser-diode light is shifted up to anexperimental frequency of 4.5 MHz below resonance using an acousto-opticalmodulator (AOM) in a double pass configuration. Part of the repump lightis used for the 2D MOT (5.7 mW in each arm to total 11.4 mW). Theremainder of the repump is coupled into a fiber that leads to a 2 by 6 fibersplitter (Evanescent Optics Inc) used for the 3D MOT. There is 0.25 mWrepump light in each of the six fiber outputs of the splitter.The pump light, arriving by fiber from the central table (1.1 mW), in-jects a slave (again MeshTel MLD780-100S5P at 18 ◦C). This slave light iscoupled into a fiber and about 10.3 mW is sent into a tapered amplifier (TA).97Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.9: A schematic of the optical setup. QWP: quarter wave plate.HWP: half wave plate. AOM: acousto-optical modulator. L: plano-convexlens (f = 300 mm unless otherwise noted). PBS: polarizing beam splittingcube. F: fiber output/input using a fiber collimator. M: mirror. The re-pump slave light frequency is set by an AOM in double pass configuration(REPUMP DP). The repump light is sent to a fiber (3D MOT repump)for use in the 3D MOT. The rest is sent to the 2D MOT to form the twoarms of the 2D MOT. The pump slave light is sent to a fiber (TA IN) whichis sent to the tapered amplifier (TA). The output of the tapered amplifier(TA OUT) is sent to three AOM double passes (2D PUMP DP, PUSH DP,and 3D PUMP DP). The 2D pump light produced by 2D PUMP DP is sentfree space to the 2D MOT chamber and is combined with the repump light.Push light for the 2D MOT is produced by PUSH DP and is coupled intofiber PUSH. The 3D pump light is produced by 3D PUMP DP and is sentinto a fiber labelled 3D MOT PUMP. The 3D MOT PUMP and 3D MOTREPUMP fibers are coupled to a 2 by 6 fiber splitter (Evanescent Optics)used for the 3D MOT. Pump diag and Repump diag: fibers to send slavelight to a fabry-perot cavity and absorption signal from a Rb vapour cell toensure the slaves are injected. OI: optical isolator. Irises (not shown) areused to block unwanted orders from the AOMs.98Chapter 5. Experimental apparatus for the pressure sensor experimentTable 5.1: Optical Components used in Fig. 5.9Component Part Number VendorQuarter wave plate WPL1212-L/4-780 CasixHalf wave plate WPL1212-L/2-780 CasixAcousto-optical modulator ATD-801A2 IntraAction Corp.Polarizing Beam Splitter BPS0202 CasixFiber collimator F230FC-B ThorlabsMirror 45606 Edmund OpticsOptical Isolator I-80-T4-H IsowaveTapered Amplifier TEC-400-0780-2500 Sacher LasertechnikLaser diode MLD780-100S5P MeshTelLenses various e.g. LA1484-B ThorlabsIrises 53914 Edmund OpticsFiber collimator F230FC-B ThorlabsThe TA generates approximately 1W out of a high-power output fiber. Atapered amplifier is a semiconductor gain medium with a tapered shape. Asmall sized input beam to the tapered amplifier with reasonable powers canachieve significant amplification. The tapered amplifier (TA) is from SacherLasertechnik (TEC-400-0780-2500) with a fiber-coupled input and ouput.The TA light is sent to three double pass AOMs. One double pass AOMis for the 3D MOT pump light, which is fiber-coupled into the 2 by 6 splitter.There is 15-18 mW 3D pump light in each of the six fibers. The 3D pumpis operated at 11.2 MHz below resonance. A second double pass AOM isfor the push beam for the 2D MOT, which is also fiber-coupled. The pushbeam has 0.4-0.6 mW power and is approximately 2 mm diameter. A thirddouble pass is for the 2D MOT pump light. The 2D pump light is combinedwith the repump and is sent free space to the 2D MOT chamber. The nextsection describes the optics used to expand the 2D MOT pump and repumpto the size needed for the 2D MOT.5.8.1 2D MOT opticsThere is 54 mW pump light in each 2D MOT arm (108 mW total). The2D MOT pump operates 12 MHz below resonance and the push beam is10.6 MHz above resonance. The 2D MOT beam for each arm of the 2DMOT is increased in size with several lenses to a size of approximately 25mm diameter. Fig. 5.10 shows the lenses and mirrors used to expand the99Chapter 5. Experimental apparatus for the pressure sensor experiment2D MOT pump and repump beams and direct these beams into the 2DMOT chamber. Only one arm of the 2D MOT is shown, but the other armhas identical optics. The pump and repump beams for the 2D MOT areexpanded using a -50 mm focal length plano concave lens ( 1 inch diameterlens), labelled L1 in Fig. 5.10, then a -75 mm focal length plano concave lens(1 inch diameter), labelled L2, and finally a 3 inch diameter plano convex+200 mm focal length lens, labelled L3. The mirrors used to direct theexpanded laser beams to the 2D MOT chamber are 75 mm by 75 mm andare from Edmund optics (part number 45341). The direction of the pushbeam from a fiber collimator is shown in Fig. 5.10 as well.The 2D MOT was aligned so that the outer edge of the beams clip onthe entrance of the differential pumping tube. This ensures that the 2DMOT cloud is close to the tube. Looking down the axis of the atomic beamwith a camera focused on the differential pumping tube entrance, we wereable to see the flourescence of the laser beams and atoms collected in the2D MOT (see Fig. 5.11). With independent control of each of the four 2DMOT coils, we aligned the 2D MOT to visually overlap with the differentialpumping tube entrance. With the camera removed, the push beam was sentdown the atomic axis. The push beam was aligned to pass through thedifferential pumping tubes by sending a flashlight down from the other endof the apparatus and aligning the push beam to that.5.8.2 3D MOT opticsThis section describes the optics used for the 3D MOT, as shown in Fig. 5.12.Setting up the 3D MOT turned out to be the best way to initially detect fluxcoming from the 2D MOT and to optimize the 2D MOT. The initial signalsfrom methods to detect the flux, that are described in the next chapter,were too weak to detect the presence of the beam.For the 3D MOT, three of the six fibers from the fiber-splitter were usedto make a retroreflection MOT. The other three fibers were used for generalpurposes such as the beam divergence measurement described in the nextchapter. Each arm of the 3D MOT consists of a bare fiber output followedby a lens (100 mm PCX) and a 1-inch quarter waveplate (see Fig. 5.12).The optics are mounted using cage mount components from Thorlabs. Toprovide a retroreflection, a quarter waveplate and mirror are placed on theopposite side of the cell. The 3D MOT optics are mounted on an 80 20 framethat is used to support the compensation coils. These coils are not currentlyin use, but would be for loading into an optical dipole trap, which may beimplemented in the future. There are six compensation coils in total, two100Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.10: A picture of the lenses and mirrors used for expanding the 2DMOT pump and repump along one arm and sending them to the 2D MOTchamber. L1 is a f = -50 mm plano concave lens, L2 is a f = -75 mm planoconcave lens and L3 is a f = + 200 plano convex lens. On the opposite sideis a quarter waveplate and a mirror for retroreflection. Only one arm of the2D MOT beams is shown. The other arm is identical coming into the 2DMOT chamber perpendicularly to the laser beams depicted in this figure.The push beam orientation and path of travel is also shown.101Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.11: A picture of the 2D MOT cloud. The differential pumpingtube can be seen in the background. The fluorescence from the laser beamscan also be seen.of which encircle the cell and had to be installed prior to the cell duringthe vacuum assembly. The 3D MOT loading rate was optimized using aparameter search over the 2D MOT coil currents, 3D MOT coil current,and the detuning and power of the 2D pump, repump, 3D pump and pushlight. Credit goes to Victor Barua for writing the optimization code.102Chapter 5. Experimental apparatus for the pressure sensor experimentFigure 5.12: A picture of a 3D MOT loaded from the 2D MOT.103Chapter 62D MOT characterizationThis chapter provides some characterization of the Rb beam coming fromthe 2D MOT, such as the beam divergence, the flux in the atomic beam,and the capturable speed distribution.6.1 Rubidium atomic beam divergencecharacterizationFig. 6.1 shows a schematic of the experimental setup used to measure thebeam divergence of the atomic beam. An approximately 5 mm diameterdiagnostic probe laser beam with 18 mW pump and 0.7 mW repump wasintroduced perpendicular to the direction of the atomic beam and retrore-flected. The purpose of the retroreflection is to decrease the deflection ofthe atomic beam by the probe laser. A camera (PixeLink PL-B741EF)was placed perpendicular to both the diagnostic laser beam and the atomicbeam. This camera recorded the atomic beam fluorescence and an imageof the background scattered light with no atomic beam present. The twoimages were subtracted using python code and an example result is shownin Fig. 6.2. A lens in a lens tube was attached to the camera to focus atroughly the intersection of the laser beam and the atomic beam, and toreduce stray light. A slice of the subtracted images taken going through theatomic cloud was fit to a gaussian, ae−((x−u)/w)22 , giving w = 133 pixels. Thediameter of the atomic beam is 2w = 266 pixels. Based on taking a camerapicture of an object of a known size, a 10 mm width corresponds to 372pixels. This gives an atomic beam size of ≈ 7 mm in diameter at the placewhere the fluorescence picture was taken. The distance from the entrance ofthe differential pumping section to where the fluorescence picture was takenis 258 mm. This gives a full angle divergence of 28 mrad. The distancefrom the tube entrance to the 3D MOT center is 55 cm so that the atomicbeam size at the 3D MOT is approximately 15 mm in diameter. The size ofthe probe beam is not critical, as long as it is not so small that it does notintersect the atomic beam fully. Ideally the probe beam would have been a104Chapter 6. 2D MOT characterizationatomicbeamlaserlightlaserlightatomicbeamcameraFigure 6.1: The setup used to measure the atomic beam divergence. Res-onant laser light (pump and repump) was incident on the atomic beamperpendicular to the atomic beam motion. The laser beam was retrore-flected. A picture with a camera was taken with and without the atomicbeam present.thin sheet that intersects the atomic beam in a narrow slice. It is noted thatthe probe beam intensity used here is larger than the saturation intensity.That is not expected to have an effect on the beam width measurementsbecause even if all the atoms are saturated there is still more fluorescencein a region of space where the atomic density is higher.6.2 Atomic speed distribution characterizationTo measure the speed distribution coming from the 2D MOT we used the3D MOT as a diagnostic tool. The fluorescence of the trapped atoms inthe 3D MOT was recorded as a function of the time after the 2D MOT was105Chapter 6. 2D MOT characterizationFigure 6.2: A fluorescence image of the Rb beam. The result of subtractingpictures taken with a camera as shown in Fig. 6.1, with and without theatomic beam present. A portion of the atomic beam is shown. The directionof travel of the atomic beam and the probe laser beam are labelled. Thoughnot visible in the image shown, the center of the beam seemed to be pushedto the side slightly with reference to the viewport that the camera waslooking through. This could be due to the push beam being aligned offcenter of the viewport and also possibly due to the camera perspective beingat an angle with respect to the viewport.turned on and the atomic beam established. The 2D MOT was turned onby energizing the 2D magnetic field coils with the 2D MOT light already on.The 3D MOT light and magnetic field were already on when the 2D MOTwas turned on. Fig. 6.3 shows the resulting 3D MOT fluorescence data. Thevoltage on the photodiode from the fluorescence of the atoms trapped in the3D MOT, as already given in Eq. 3.1, isV (t) = αγscN. (6.1)As also given in Eq. 1.5 we have that the number of atoms in a 3D MOT,N(t), from initial loading followsdNdt = R− ΓN − β∫n2(~r, t) d3~r (6.2)106Chapter 6. 2D MOT characterization0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045time (s) 6.3: A plot of the 3D MOT fluorescence captured on a photodiodeas a function of time from initial turn on of the 2D MOT. The 2D MOTwas turned on with the 3D MOT light and magnetic field already on. Thecurvature gives information about the speed distribution of the atomic beam.where Γ and β are loss rate constants due to background collisions and intra-trap collisions respectively. The density of the trapped atoms at position~r from the center of the trap at time t is n(~r, t). For short times followinginitial loading of the 3D MOT the loss terms can be neglected since thereare not many atoms accumulated yet, meaning both N and n(~r, t) are small.This means we can approximatedNdt ≈ R. (6.3)Combining Eq. 6.3 with Eq. 6.1 we havedVdt = αγscdNdt = αγscR. (6.4)This links a measurable quantity, the slope of the fluorescence voltage curvefor small times from initial 3D MOT loading, to the loading rate, R.The 3D MOT has some maximum capture speed, vc, so that atomstravelling to the 3D MOT from the 2D MOT that are exceeding this speedwill not be trapped. There is a distribution of speeds, f(v), coming from107Chapter 6. 2D MOT characterizationthe 2D MOT. The loading rate, R, of the atoms into the trap will initiallychange as a function of time as atoms of different speeds arrive at the 3DMOT region. The curve in the fluorescence data shown in Fig. 6.3 relates tothe speed distribution of the atomic beam arriving at the 3D MOT. If thespeed probability distribution coming from the 2D MOT is f(v) then theloading rate as a function of time, t, from initial turn on of the 2D MOTwill followR(t) = φ∫ vcd/tf(v) dv. (6.5)In Eq. 6.5, d is the distance from the exit tube of the 2D MOT to the 3DMOT capture region, and t ≥ (tc = d/vc). The total number per secondfrom the 2D MOT is φ. When the curve in Fig. 6.3 becomes linear thenRmax ≈ φ∫ vc0f(v) dv (6.6)meaning that the majority of all speed classes have reached the 3D MOTregion from initial turn on of the 2D MOT.To determine the shape of f(v) a few mathematical rearrangements areneeded. First the derivative of R(t) from Eq. 6.5 is taken givingdR(t)dt = φf(d/t)(dt2 ). (6.7)Next dR(t)dt is obtained from Eq. 6.4 and inserted into Eq. 6.7 giving1αγscd2Vdt2 = φf(d/t)dt2 . (6.8)Now we divide both sides by Rmax given in Eq. 6.61Rmaxαγscd2Vdt2 =f(d/t) dt2∫ vc0 f(v) dv(6.9)Finally we apply Eq. 6.4 again to write Rmax in terms of voltage giving1(dVdt )maxd2Vdt2 =f(d/t) dt2∫ vc0 f(v) dv(6.10)Solving for f(d/t) givesf(d/t) = V′′V ′maxt2d∫ vc0f(v) dv (6.11)108Chapter 6. 2D MOT characterizationwhere the prime indicates differentiation with respect to time and the inte-gral involving f(v) is a constant.The goal now is to extract the speed probability distribution, f(v), fromthe data shown in Fig. 6.3. A python script was used to numerically extractthe second derivative, V ′′, and the maximum slope, V ′max, from the 3D MOTfluorescence data as a function of time from the 2D MOT turn on. Themethod used in the python script was to fit the 3D MOT fluorescence datashown in Fig. 6.3 to a line over a time interval of 0.007 s starting at t=0. Thetime interval was then shifted by 300 µs and then the 3D MOT fluorescencedata was fit again. The slope of the linear fits was recorded and assigneda time at the center of each interval. This provided the first derivative, V ′,of the 3D MOT fluorescence data. The second derivative, V ′′, was obtainedby repeating this procedure on the first derivative results. A plot of V ′′V ′maxt2dversus speed v = d/t is given in Fig. 6.4. The cutoff in the plot at lowvelocities is due to the fact that for the data shown in Fig. 6.3 times onlyup to t = 0.06 s were used (not shown in that figure). After this time theatom number is sufficiently large that the approximation used in Eq. 6.3 isno longer valid.Looking at Fig. 6.3 the 3D MOT fluorescence starts to rise approximatelyat 0.021 sec from initial turn on of the 2D MOT. The distance from theentrance of the differential pumping tube on the 2D MOT side to the 3DMOT center is 55 cm. This gives an approximate capture speed, vc, of the 3DMOT as 26 m/s. From Fig. 6.4 one can see the bulk of the speed probabilitydistribution of the atomic beam lies below 26 m/s so that we approximate∫ vc0 f(v) dv ≈ 1. With this approximation Eq. 6.6 gives Rmax ≈ φ. UsingRmax = 1αγscat(dVdt)max ≈ φ we can find an approximate value for φ. Themaximum slope(dVdt)max was taken as the slope in the linear portion of the3D MOT fluorescence data. The coefficient α can be given asα = (rlens)24(dMOT)2ηhcλ . (6.12)The first part of the expresssion for α gives the fraction of the total solidangle collected by a plano-convex lens of focal length 60 mm with radius,rlens, placed a distance dMOT from the 3D MOT. The power to voltageconversion factor of the photodiode is η and hcλ is the energy of a fluorescedphoton.  describes the transmitted fraction of photons as the fluorescentlight travels through the cell once. In our case rlens = 11.5 mm, dMOT =14.5 cm, η = 4.68V/µW, and  =√11.2/11.5. The  was determined bymeasuring one of the 3D MOT incoming beams as having 11.5 mW power109Chapter 6. 2D MOT characterization101520 25 30 35 40speed (m/s) 6.4: The speed distribution of atoms in the atomic beam from the2D MOT that is capturable by the 3D MOT. This distribution is extractedfrom the fluorescence of the 3D MOT recorded from initial turn on of the2D MOT.before it entered the cell and having 11.2 mW power after it exited the cellafter passing through two sides of the cell. This gives 11.52 = 11.2.The scattering rate (number of photons emitted per atom per second)can be expressed asγsc =Γ2(s1 + s+ (2δΓ )2)(6.13)where δ is the detuning of the MOT pump light from resonance which for ourexperiment was 12 MHz. Γ is the natural line width of the pump transition[23] which is 2pi × 6.07 MHz. The parameter s = IIsat where I is the totalintensity of the MOT pump beams and Isat is the saturation intensity ofthe pump transition which is 3.9 mW/cm2 [23]. In the pressure sensorexperiment, with each of the three arms having 18 mW initially travellingto the cell and getting retroreflected, the total pump power in the MOTis Ptot = 3(18 × ) + 3(18 × 3). The intensity is taken as I = PtotA withA = pirMOT2, where rMOT is the radius of the MOT pump beams. For ourexperiment the 3D MOT beams were roughly 23 mm in diameter.110Chapter 6. 2D MOT characterizationCombining our values for α and γsc gives αγsc = 2.9×10−8 V. The valueof (dVdt )max = 6.13 so that Rmax ≈ φ = 2×108 atoms/s total comes from the2D MOT beam. This number could be increased, for example, by makingthe 2D MOT beams larger in diameter [93, 95]. The speed distribution datain this section was taken with 85Rb but similar results were obtained with87Rb.The speed distribution measured here is only for speeds capturable byour 3D MOT. Schoser et al. [93] measured the total speed distribution fromtheir 2D MOT with and without a push beam. Without a push beam theirspeed distribution was quite broad, peaking around 50 m/s and extending toaround 200m/s. The peak of the distribution and the total flux was shownto depend on factors such as the volume of the 2D MOT cooling region,the power and detuning of the 2D pump light, and the pressure of the Rbgas. The speed distribution from the 2D MOT is not a Maxwell-Boltzmanndistribution. They showed that the presence of a push beam added a narrowand tall peak to the broad velocity distribution, centered at around 20 m/s.The peak position due to the push beam depended on the push power anddetuning. For our speed distribution measurement, we are likely only seeingthe narrow distribution due to the push beam because we are only detectingspeeds lower than the capture velocity of the 3D MOT.111Chapter 7Loss rate measurements in a3D MOTThis chapter describes two experiments where the loss rate Γ due to back-ground collisions was measured in a MOT as pressure in the 3D MOT cham-ber varied. The first experiment was when the residual background pressurewas varied and the second was when Ar was introduced into the system.7.1 Loss rate variation with the total pressure ofresidual background gasPressure measurements were taken with the Stabil-Ion gauge installed onour apparatus as the pressure varied due to outgassing of a residual gasanalyzer (RGA 200, Stanford Research Systems) installed on the bakeoutsystem which was connected to our apparatus. When the SRS RGA wasturned on it caused the pressure to rise significantly. With the UHV pumpsclosed off the base pressure was around 1× 10−6 Pa (1× 10−8 Torr). Whenthe RGA was turned on from the bakeout station the pressure rose to around2.7 × 10−5 Pa (2 × 10−7 Torr) and over the period of 5 hours dropped to6.0 × 10−6 Pa (4.5 × 10−8 Torr). The SRS RGA was then shut off and thesystem pressure decreased down to the 1 × 10−6 Pa (1 × 10−8 Torr) rangeafter another 3 hours.As the pressure in the system was varying loading curves of the 3D MOTwere recorded. The loading curves were recorded by turning off the 3D MOTmagnetic field and then turning it back on. With the 3D MOT light on atall times, the 3D MOT would start to fill again when the 3D MOT coilswere turned back on. The fluorescence was captured onto a photodetectorin the same way as described in section 3.2.1. The loading curve voltagewas fit to V (t) = A(1− e−Γt)+B. Fig. 7.1 shows an example loading curvetaken with the fit superimposed. The coefficient A = αγscatRΓ converts thesteady state atom number RΓ to a steady state voltage. The coefficient Baccounts for any offsets which commonly occur in experimental data due to,112Chapter 7. Loss rate measurements in a 3D MOTfor example, scattered laser light and background light in the room. Fig. 7.20.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8time (s) 7.1: An example loading curve taken with a 3D magneto-optical trapat a residual background pressure of 1.79×10−5 Pa (1.34×10−7 Torr). They axis is the fluorescence signal captured on a photodiode as a function oftime from initial 3D MOT turn on. The curve is fit to V (t) = A(1−e−Γt)+Bto extract Γ, which in this case gave Γ = 4.17 ± 0.01s−1 as the fit result.shows a plot of Γ versus pressure from fitting the loading curves taken whilethe pressure varied.Arpornthip et al. [46] performed semi-classical calculations for the slopeof Γ versus P for a 1K trap of Rb atoms and various background gases at atemperature of 300K. The results were 2.0×105 Pa−1 s−1 or 2.6×107 Torr−1s−1 for N2 and for CO2 while H2 was 3.7× 105 Pa−1 s−1 or 4.9× 107 Torr−1s−1. RGA scans taken with the Pfeiffer RGA installed on our apparatusshowed that when the SRS RGA was first turned on the dominant specieswere N2 or CO, CO2 and H2. As the pressure decreased H2 became thedominant species. This means as the background pressure decreased in oursystem and hydrogen became the dominant species the slope of Γ versuspressure P became steeper. Fitting the low pressure data to a line givesa slope of 2.5 × 105 Pa−1 s−1 (3.4 × 107 Torr−1 s−1) and for the higherpressure data the slope was 1.7×105 Pa−1 s−1 (2.3×107 Torr−1 s−1) which is113Chapter 7. Loss rate measurements in a 3D MOTreasonable compared to the slopes calculated by Arpornthip et al. mentionedabove. Comparing their calculation of the velocity averaged collisional losscross section to ours for a temperature of 295K background gas and a C6value of 336.4 a.u. [44] gives a difference of at most 5 percent in the rangeof trap depths from 1 to 2 K.0.0 0.5 (Torr)1e70123456(s1)Figure 7.2: The loss rate constant, Γ, due to background collisions ver-sus total background pressure. The pressure increased after a residual gasanalyzer was turned on and then decreased. As the pressure decreased load-ing curves of the 3D MOT were taken. The loading curves (photodiodefluorescence) were fit to V (t) = A(1− e−Γt) +B to extract Γ.7.2 Loss rate variation with the pressure of Arbackground gasThe loss rate constant, Γ, due to background collisions was measured forvarious Ar pressures. While the 3D MOT conditions were on, the 2D pumplight and push light were turned off and the push shutter was closed. Theatom number in the 3D MOT then decayed exponentially and a fluorescencesignal of the 3D MOT was recorded using a photodiode. The fluorescencevoltage signal was fit to extract Γ. Fig. 7.3 shows an example of the data114Chapter 7. Loss rate measurements in a 3D MOTtaken at a total pressure of 2.1× 10−5 Pa (1.61× 10−7 Torr). The data wasfit from a time of 30 ms after initial turn off of the 2D MOT to account forthe time needed for the atomic beam to stop travelling to the 3D MOT. A2.0 2.5 3.0 3.5 (s) 7.3: An example decay curve taken with the 3D MOT after the 2DMOT had been turned off. The total pressure of the system with Ar presentin this case was 2.1 × 10−5 Pa (1.61 × 10−7 Torr). The curve was fit to adecaying exponential and the loss rate constant Γ determined for this caseto be 2.60 s−1.decay curve was used to determine the loss rate constant, Γ, rather than aloading curve. This is to avoid the loss mechanism of the 2D MOT beamitself colliding with the trapped atoms and causing loss.The total pressure in the system as Ar was added was measured usingthe Stabil-Ion gauge installed on the apparatus. The background pressurebefore Ar was added was 3.95 × 10−7 Pa (2.97 × 10−9 Torr). Ar was addedthrough a leak valve installed on the bakeout station which was connectedto the apparatus. The original leak valve on the apparatus failed and needsreplacing. The ion gauge and SRS RGA on the bakeout station were turnedoff during measurement. The Pfeiffer RGA on our apparatus generally in-creases the residual background pressure by 1.5 to 2 times so it was alsoturned off to keep the residual background pressure as low as possible. Themeasurements of Γ in the 3D MOT as a function of Ar pressure are shown115Chapter 7. Loss rate measurements in a 3D MOTin Fig. 7.4 on a log-log plot. At each Ar pressure, five decay traces weretaken at each pressure and averaged and then fit. The loss rate constant Γversus the pressure of Ar was fit to a line and the slope extracted giving avelocity averaged collisional loss cross section of 0.634× 10−9 cm3/s. VictorBarua is credited with developing the software system that made taking themeasurements displayed in Fig. 7.4 possible. Adrian Cavailles wrote theoriginal code to take this data. Both helped make the gas manifold to addgases into the system, and also optimized and improved the performance ofthe magneto-optical traps.8.5 6.5 6.05.5log10(Pressure) (Torr) 7.4: Measurements of loss rate constant, Γ, in a MOT at various Arpressures. The Ar pressure was measured with an ion gauge.The velocity averaged collisional loss cross section, from the measure-ment of Γ vs PAr described here, corresponds to a trap depth of approx-imately 2.95 K for the 3D MOT in its present setup. This is based oncalculation of the velocity averaged collisional loss cross section as a func-tion of trap depth. The capture speed, vc, of 26 m/s given in section 6.2can also be used for a rough estimate of trap depth measurement. The trapdepth is given by 12mv2e , where ve is the escape speed. The capture speedwill be larger than the escape speed because the effective length over whichan atom can be slowed down is the whole diameter of the laser beams. To116Chapter 7. Loss rate measurements in a 3D MOTescape, the atoms start off at the middle of the laser beams and only haveto travel the radius of the laser beams. For an estimate of the relationshipbetween vc and ve, the kinetic energy for escape (or trap depth) can beexpressed as 12mv2e = Fr where F is the average force on an atom over theradius r of the beam. The maximum kinetic energy that an atom enteringinto the 3D MOT can have and be trapped can similarly be expressed as12mv2c = F (2r). Taking the ratio of these two expressions gives vc =√2ve.For vc = 26 m/s, a trap depth of 1.8 K is predicted along the radial directionof the 3D MOT (the atomic beam axis direction). The MOT trap depthis anisotropic and was found to be approximately double along the axialdirection [53]. This gives an estimate of the average trap depth of our 3DMOT to be (1.8 + 1.8 + 2× 1.8)/3 = 2.4 K. This is in reasonable agreementwith the trap depth estimate based on calculation given above as a roughestimate of trap depth.117Chapter 8Future outlook andconclusions8.1 Future outlookSo far the work on the atom pressure sensor has shown that the 2D MOTis operational and that a 3D MOT can be loaded from it. Preliminary dataof the dependence of the loss rate constant, Γ, with argon pressure has beenmeasured with a 3D MOT. An ion gauge was used to measure the argonpressure. Further experimental goals are to trap atoms in a magnetic trap,introduce gases into the system, and take measurements of pressure withthe trapped atoms. These pressure measurements would be compared withpressure readings from commercial gauges and calibrated gauges sent fromthe National Institute of Standards and Technology.8.1.1 Magnetic trapping coilsThe same magnetic coils used for the 3D MOT of the pressure sensor appa-ratus will be used for magnetic trapping. The 3D MOT coils currently inplace do not provide a gradient deep enough for magnetic trapping of atomsfrom the 3D MOT. To perform pressure measurements using the loss ratein a magnetic trap, another set of coils have been made from PVC coatedhollow core copper tubing. The hollow core allows water to be run throughthe tubing for cooling.The replacement coils have 8 axial windings and 8 radial windings. Theyare constructed from quarter inch outer diameter copper tubing with aquoted 0.03 inch wall thickness and a 0.032 inch thick PVC coating. Themeasured outer diameter of the copper tubing including the PVC is approx-imately 8.3 mm and the inner hollow core is 4.5 mm. The coil has a 188mm outer diameter and 38 mm inner diameter. The height of each coil is 73mm. The coil windings are secured by wrapping fiber glass tape around thecross section (see Fig. 8.1). The PVC coating is so that the coil windings donot short on each other. The tubing is from Alaskan Copper (part number118Chapter 8. Future outlook and conclusions142797). The coils will be driven using a 60 V, 250 A power supply. Theresistance of each coil is 38.5 mΩ.Figure 8.1: The new magnetic coils and mount for the 3D MOT and mag-netic trap of the atom pressure sensor experiment.Running water at 70-80 psi from the tap in parallel through the coilsgave a flow rate of 36 seconds per litre for one coil and 29 seconds per litrefor the other. At 230 A, the temperature of the water exiting the coils was28◦C . The incoming water temperature was 11◦C . Ref. [97] was invaluablefor designing magnetic coils using water cooled hollow core tubing. The coilswill be mounted by sandwiching them in between delrin plates with the topcoil sitting on spacers in between the two coils (see Fig. 8.1).The predicted magnetic field gradient is approximately 0.57 G/cm (5.7×10−5 T/cm) per A radially which would lead to a maximum radial magneticfield gradient of 125 G/cm (1.25 × 10−2 T/cm) at 250 A. From the centerof the cell, where the magnetic field zero is located, to the outer edge of thecell is 2 cm. The maximum magnetic field occurring radially at the edge ofthe cell would then be 250 G . The trap depth, as given in section 1.2.2, canbe estimated as ∆E = gFmFµBB for the mF = −1, F = 1 , level of the52S1/2 ground state for 87Rb. Using gF = −1/2 and µB = 9.274×10−24 J/Tfrom [21] and B = 250 G this gives a maximum radial trap depth of around119Chapter 8. Future outlook and conclusions8 mK. The measured coil gradient in anti-helmholtz configuration is 0.58G/cm (5.8×10−5 T/cm) per A along the radial direction and approximatelydouble along the axial direction.To trap atoms in a magnetic trap, coils for magnetic trapping and thesupporting infrastructure such as breadboards, mounts, electrical and waterconnections, must be installed.8.1.2 NIST gaugesTowards the completion of this writing the NIST gauges arrived and havebeen installed on the apparatus. This required removing Rb D in Fig. 5.1and installing the gauges, as in Fig. 8.2. NIST sent two spinning rotarygauges (MKS SRG-2), labelled SRG2 and SRG3 in Fig. 5.1, and one iongauge (370 Stabil Ion), labelled IG2 in Fig. 5.1.Figure 8.2: The experimental apparatus with the NIST gauges installed.Two spinning rotary gauges (labelled SRG3 and SRG2) and one ion gauge(labelled IG2) were added to the system. They will be calibrated usingour apparatus with trapped Rb atoms and then shipped back to NIST andrecalibrated on their orifice flow standard. Our spinning rotary gauge andion gauge are labelled SRG1 and IG1, respectively.120Chapter 8. Future outlook and conclusions8.1.3 Technical challengesA leak valve is used to introduce the gas into the apparatus. For the leakvalve used in the flowmeter at NIST a wait time of approximately 5 hoursis given once setting the leak valve to allow creep to settle [77]. If the pres-sure is changing during measurement that would add error to the pressuremeasurement using trapped atoms. This becomes more of an issue at lowpressures (< 10−7Pa or 10−9 Torr) where the decay time τ = 1/Γ is greaterthan 10 s. The time to take multiple sets of pressure measurements withthe magnetic trap would take around 20 min.Another difficulty may be the presence of pressure gradients in the ap-paratus so that the pressure at the gauges is different than at the atoms. Anattempt has been made to minimize this by being able to valve off any pumpswhich may cause pressure gradients and by situating the gauges as close tothe region where the atoms are trapped as possible. The gauge readingswith and without pumps present will give an indication of the contributionof a pressure differential due to pumping. The distance between the NISTion gauge and our ion gauge installed on our apparatus is approximatelythe same distance between our ion gauge and the trapped atom location.This should be incorporated into the calibration of the ion gauges with thetrapped atoms. We could also explore taking measurements with all thepumps closed off to decrease pressure gradients. This would be possible aslong as the residual background gas is significantly smaller than the pressureof Ar to be measured.8.2 ConclusionsThis thesis started with a brief description of magneto-optical trapping andmagnetic traps. These traps are the tools we used to prepare and studysamples of ultra cold atoms. A key parameter of interest for these traps is theloss rate constant due to background collisions, Γ. The loss rate constant wasrelated to the density of the background gas species. This relation involvesthe velocity averaged collisional loss cross section between the trapped atomsand the different background species. Quantum scattering calculations forthe loss cross section were described and performed earlier in our group. Theloss cross section depends on the trap depth and experimental verificationof this dependence was shown previously for trap depths attainable withthe magnetic trap used (up to 10 mK). This work provided verification fortrap depths for a MOT (≈ 1K). For this verification, a measurement of trapdepth adapted from Hoffmann et al. [16] using photoassociative loss was121Chapter 8. Future outlook and conclusionsused. The dependence of the velocity averaged collisional loss cross sectionon trap depth can be applied to trap depth measurements. This is desirablebecause trap depth is a difficult parameter to determine, especially for aMOT. Laser beam imperfections in shape, polarization, and alignment, aswell as the presence of interference fringes prevent accurate calculation ofMOT trap depth. The ‘catalysis’ method described in this work to measuretrap depth requires specialized equipment such as tunable lasers and is quiteinvolved. Another application proposed in this work is density measurementof a background gas species based on measurements of Γ for an atomic trapand calculation of the velocity averaged collisional loss cross section.Part of the apparatus for the pressure sensor experiment was designed,assembled, baked out, and a 2D MOT was shown to be operational. 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Magneto-optical trap load-ing rate dependence on trap depth and vapor density. J. Opt. Soc. Am.B, 29(3):475–483, Mar 2012.[99] F. Reif. Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York, 1965.131Appendix ALoading rate investigationA.1 Reif modelThis appendix describes an investigation of the loading rate of a vapourloaded MOT performed in our lab by Magnus Haw, Nathan Evetts andDr. James Booth described in [98]. It is included because many of themeasurement techniques used in this investigation are described in detail inthis thesis. The trap loading rate, R, for a magneto-optical trap loaded froma vapour can be modeled by what we call the ‘Reif model’ [82, 99]. The Reifmodel states that any atom entering the trap volume with speed less thanthe ‘capture velocity’, vc, will be cooled and trapped. This model leads toa prediction of loading rate asR = 2Av4cnpi2v3th(A.1)where vth =√8kBTpim is the mean thermal velocity of the background parti-cles, A is the surface area of the trap region, and n is the background densityof the species being trapped. The trapping volume is taken as the region ofintersection of the laser beams forming the MOT.The depth of a trap can be written as U = 12mv2e , where ve is the escapevelocity. We make the assumption that the capture velocity is proportionalto the escape velocity, vc = bve. This allows us to write the loading rate, R,asR =( 8b4Api2m2v3th)U2n. (A.2)This model had never been tested experimentally and has several attractivefeatures such as relating the loading rate to difficult to find parameterssuch as n, and U . There were four main goals of this work. The first andsecond were to test that the loading rate, R, is proportional to n and toU2 as predicted in Eq. A.2. The third goal involves a determination of trapdepth for different settings of MOT pump detuning and intensity based onknowledge of the trap depth for one particular pump detuning and intensity132Appendix A. Loading rate investigationsetting. The fourth goal of this work was to estimate the proportionalityconstant, b, between the escape and capture velocity.A.2 Experimental observablesAs seen previously in section 3.2.1, a portion of the photons being emittedby the atoms in a MOT can be collected onto a photodetector. The voltageoutput by the photodetector, V (t), will be proportional to the number ofatoms in the trap, N(t). Specifically,V (t) = αγscN(t). (A.3)γsc is the rate at which an atom scatters photons and depends on the detun-ing and the intensity of the light. α is the proportionality constant betweenthe number of photons emitted per second by the trapped atoms and thephotodiode voltage produced, and can be expressed as α = hcλ ( r24d2 )η. Thefactor η is the optical power to voltage conversion factor of the photodiode. describes the transmission of the glass and lens which the photons travelthrough to arrive at the photodetector. r is the radius of the lens that fo-cuses the fluorescent light onto the photodetector. d is the distance fromthe trapped atoms to the lens. The ratio ( r24d2 ) involving r and d accountsfor the solid angle of photons that are collected onto the detector.The loading rate equation of a MOT is usually modelled as (see Eq. 1.5in section 1.3)dNdt = R− ΓN − β∫n2(~r, t) d3~r. (A.4)This work focuses on the loading rate of the MOT, R. For short times frominitial loading the number of atoms trapped is very small so that one canapproximatedNdt |0 = R. (A.5)In terms of photodetector voltage Eq. A.5 becomesdVdt |0 = V˙0 = αγscR (A.6)where V˙0 is the rate of change of V for small times after the initial turnon of the MOT. V˙0 is proportional to R which changes with different MOTsettings, such as different pump light detunings and intensities, differentgeometries (e.g. beam sizes), and background gas density.133Appendix A. Loading rate investigationTo eliminate uncertainties in the quantities α and γsc, we define a newexperimental parameterMi =(VstdVi)V˙ i0 (A.7)where Vi is the steady state voltage when the MOT is fully loaded withsettings i. Vstd is taken by switching quickly from MOT setting i to somepre-selected ‘standard’ setting. The voltage immediately after the switch,Vstd, is recorded before the atom number has had time to change. V˙ i0 is thevoltage rise after initial turn on for MOT setting i. Using the fact that theatom number is the same for Vi and Vstd, Eq. A.3 givesVstdVi= γstdscγisc. (A.8)Substitution of Eq. A.8 and Eq. A.6 into Eq. A.7 givesMi =(VstdVi)V˙ i0 =γstdscγisc(αγiscRi)= αγstdsc Ri (A.9)so that Mi ∝ Ri with the same proportionality constant for different MOTsettings i.The MOT apparatus used was the same one described in section 3.1.85Rb was used as the trapped species. The total standard six beam pumppower was 18.3 mW and 0.56 mW for the repump. The beams had a 1/e2horizontal (vertical) diameter of 7.4 (8.4) mm. This corresponds to a pumpintensity of 37.5 mW cm−2. The MOT was operated with an axial gradientof 27.9(0.3) G/cm (2.79(0.03)×10−3) T/cm. This maximum pump intensityand a 12 MHz pump detuning was used as the standard MOT setting.A.3 The dependence of loading rate on MOTtrap depthTo determine if the loading rate, R, is proportional to U2 measurements ofMi were taken for various MOT settings, i. Ui were measured independentlyfor the various MOT settings via the catalysis method described in section3.3. The results, shown in Fig. A.1, support a linear relationship betweenMi ∝ Ri and U2i . Table A.1 gives the measured values of trap depth forvarious MOT pump detunings and intensities.134Appendix A. Loading rate investigation0.0 0.5 2.5 3.0 3.5 4.0U2trap(K2) A.1: Evidence that the loading rate of a MOT is proportional tothe square of trap depth. The quantity Mi for different MOT settings i isproportional to the loading rate Ri. Plotting Mi versus the trap depth ofeach MOT setting U2i indicates a linear relationship.A.4 Trap depth determination using loadingratesIt is proposed that the trap depth for different settings of MOT pump detun-ing and intensity can be determined based on knowledge of the trap depthfor one particular pump detuning and intensity setting. The approach forthis goal makes several assumptions. The surface area of the trap region isassumed to stay the same for different MOT pump detuning and intensitysettings. It is also assumed that the proportionality constant, b, between vcand ve stays the same for different settings. Finally it is assumed that thebackground density, n, also is a constant. With these assumptions the ratioof Eq. A.2 for two different MOT settings givesU2 = U1√R2R1. (A.10)To determine the trap depth of MOT setting 2, a measurement of the ratioof loading rates for setting 2 and another setting 1 is needed. The additional135Appendix A. Loading rate investigationTable A.1: MOT trap depths measured using the ‘catalysis method’ forvarious MOT settings.Pump Detuning (MHz) Pump Intensity (mW cm−2 ) U (K)-5 2.9 0.52 (0.12)-8 2.9 0.74 (0.12)-10 2.9 0.86 (0.12)-12 7.5 1.34 (0.12)-12 10.4 1.44 (0.12)-12 37.5 1.77 (0.19)knowledge of the trap depth for setting 1 provides the trap depth, U2.Using measurements ofM1 andM2 for two MOT settings and a catalysismeasurement of trap depth for setting 1 we can find the trap depth for MOTsetting 2 usingU2 = U1√R2R1= U1√M2M1. (A.11)Fig A.2 shows agreement between the predicted trap depth, Upred, deter-mined from Eq. A.11 and the measured trap depth, Umeas, from the catalysismethod, also given in table A.1.Note that one should not extrapolate this trap depth determination fora MOT too far from the known trap depth.A.5 The dependence of loading rate on rubidiumdensityTo show that Ri is proportional to nRb we show Mi ∝ nRb. To do this wemeasure the loss rate constant, ΓMT, due to background collisions betweenthe background gas and the trapped Rb atoms in a magnetic trap. The|F = 2,mF = −2〉 state was used for the 52S1/2 ground state of 85Rb. Asdescribed previously in chapter 2, ΓMT can be expressed asΓMT =∑jnj〈σvj〉Rb,j (A.12)where nj is the density of background species j. The term 〈σvj〉Rb,j is thevelocity averaged collisional loss cross section between the trapped Rb andbackground species j. Isolating the dependence of ΓMT on the background136Appendix A. Loading rate investigation0.0 0.5 A.2: MOT trap depth is predicted based on the ratio of loadingrates and the knowledge of a comparison trap depth. The predicted trapdepth for various MOT settings is plotted versus the measured trap depthsshowing good agreement between the two.density of Rb, nRb, we haveΓMT = nRb〈σvRb〉Rb,Rb + Γa (A.13)where Γa is the contribution to ΓMT due to background species other thanRb.From Eq. A.2 and Eq. A.9 our model states Mi = kinRb where ki isa proportionality constant. Rearranged slightly we have nRb = Miki andinserting this into Eq. A.13 givesΓMT =Miki〈σvRb〉Rb,Rb + Γa. (A.14)Measurements of ΓMT in a magnetic trap andMi for one particular MOTsetting, i, at various Rb densities should give a linear relationship for ΓMTversus Mi. The MOT setting, i, chosen was the standard setting. The Rbdensity was varied by filling the MOT region by running current througha Rb dispenser and then letting the density decay over time. ΓMT was137Appendix A. Loading rate investigationmeasured as described in section 3.2.2. Mstd for the standard MOT settingwas also measured at the same time. Fig. A.3 shows the results and verifiesthat Mstd ∝ nRb so that Rstd ∝ nRb. 2.5 3.0 3.5 (V s1) A.3: Evidence that the loading rate of a vapour loaded Rb MOT isproportional to the background density of Rb, nRb. The loss rate constantof a magnetic trap, ΓMT, varies linearly with Rb background density. Ifthe measure M , which is proportional to the loading rate of a MOT, is alsoproportional to the Rb background density then ΓMT will vary linearly withM . For this measurement the ‘standard’ setting of the MOT was used forM.A.6 Determination of bThe last part of this investigation is to determine the proportionality con-stant b. Using Eq. A.2 and Eq. A.9 and dividing by the background rubidiumdensity, nRb, we haveMinRb= αγstdscRinRb= αγstdsc( 8b4Api2m2v3th)U2i . (A.15)138Appendix A. Loading rate investigationThis equation predicts that MinRb should be linearly related to U2i with a slopefrom which b can be extracted.To obtain MinRb for different settings i, Mi was measured as the density ofrubidium changed. The MOT region was filled with Rb vapour. As the den-sity of Rb was slowly decreasing from initial filling, ΓMT for a magnetic trapwas measured. Mstd and Mi was also measured for various MOT settingsas the density of Rb changed.To determine the Rb densities, a plot of ΓMT versus Mstd was used. Theslope is 〈σvRb〉Rb,Rbkstd where Mstd = kstdnRb. If 〈σvRb〉Rb,Rb is calculated, asdescribed in chapter 2, then kstd can be determined and measurement ofMstd provides the density of rubidium. Mstd as a function of time as the Rbdensity decreased was fit so that nRb could be determined when each Mimeasurement was made.Fig. A.4 shows a plot ofMi vs nRb. The slopes of these plots are Ei = MinRbwhich can be plotted versus U2i as shown in Fig. A.5. The slope of the plotin Fig. A.5 allows b to be estimated from Eq. A.15.To determine b the values of α and γstdsc are determined as describedin section 6.2. The value of A was estimated as the surface area of theintersection of three perpendicular cylinders, A = 3(16− 8√2)r2, where r isthe laser beam radius averaged across the horizontal and vertical directions.The factor 8pi2m2v3th was computed for85Rb vapour of temperature T = 300K. Table A.2 gives the calculated values for all these quantities. From theTable A.2: Parameters used in the calculation of b, the proportionality con-stant between the capture velocity vc and the escape velocity ve.Parameter Value UncertaintyA 2.08 cm2 10 %α = (rlens)24(dMOT)2 ηhcλ  7.84 ×10−15 V s 20 %γstdsc = Γ2(s1+s+( 2δΓ )2)6.8× 106s−1 20 %8pi2m2v3th3.85 × 10−2 cmK2s 5 %quantities given in Table A.2 the proportionality constant b was determined.The relationship between the capture velocity, vc, and the escape velocity,ve, was found to bevc = 1.29(0.12)ve . (A.16)139Appendix A. Loading rate investigation0.4 0.5 0.6 0.7 0.8 0.9 1.01.1nRb(108cm3)01234M(Vsff1)Figure A.4: Measurement of Mi for various MOT settings, i, versus thedensity of background gas, nRb. The slopes should be linearly related to thetrap depth squared for the different MOT settings. The density of Rb wasdetermined by measurement of the loss rate constant of a magnetic trap,measurement of Mstd, and calculation of the velocity averaged collisionalloss cross section, 〈σvRb〉Rb,Rb.140Appendix A. Loading rate investigation0.0 0.5 2.5 3.0 3.5U2trap(K2) A.5: A plot of MinRb versus U2i . The proportionality constant, b,between the capture and escape velocity can be extracted from the slopewith the estimation of various coefficients.141


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