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Progress towards a primary pressure standard with cold atoms Jooya, Kais 2017

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Progress Towards a Primary PressureStandard with Cold AtomsbyKais JooyaB.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate Studies and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2017c© Kais Jooya 2016AbstractThis thesis describes a method of using an ultra-cold ensemble of atoms con-fined in a trap as an atomic primary pressure standard. The developmentof the standard and its current status are described in detail. This standarduses a 3D MOT to trap 87Rb and then transfer them to a quadrupole mag-netic trap where the atoms undergoes collisions with a background gas. Bymeasuring the number of atoms left in the magnetic trap as a function oftime one extract a loss rate and from this rate determine the backgroundgas density. This loss rate is a product of the density of the background,multiplied by the loss cross section averaged over the velocity distributionof the background gas. By computing the average loss cross section in themagnetic trap and measuring the loss rate, the density of the backgroundgas can be determined. This gives a calibration free measurement of densityof a background gas in the UHV range (10−6− 10−9) Torr or (10−4− 10−7)Pa which allows for it to be used as a standard. In conjunction with this,preparation of the atoms prior to the loss rate measurement is investigatedto ensure accuracy and reproducibility of the standard. Finally a comparisonbetween UBC’s atomic standard and NIST’s (National Institute for Stan-dards and Technology) orifice flow standard is conducted via an ionizationgauge which employed as a transfer standard. All measurement are carriedout using Argon gas as the background gas of study.iiPrefaceThis works in done in collaboration with James Fedchak of the NationalInstitute of Standards and Technology in the USA. He supplied an ionizationgauge and spinning rotor gauge. Chapter 3 and chapter 4 describes the allthe data and results for the experiment. I was responsible for adding themagnetic trapping coil to the setup which was done with the help of JanelleVan Dongen. I added the RF coil system to the setup. I collected andanalyzed all the data for this thesis. I also optimized and created the recipefor the experimental procedure. Fig. 3.1 and Fiq. 3.2 were adapted from theoriginal produced by Janelle Van Dongen.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . 62 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Magneto-Optical Traps . . . . . . . . . . . . . . . . . . . . . 82.1.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . 92.1.2 Loading Dynamics . . . . . . . . . . . . . . . . . . . . 112.1.3 Atom Number Calibration . . . . . . . . . . . . . . . 122.2 Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Gravitational Sag . . . . . . . . . . . . . . . . . . . . 152.2.2 Gravitational Filtering . . . . . . . . . . . . . . . . . 162.2.3 Majorana Losses . . . . . . . . . . . . . . . . . . . . . 182.2.4 Magnetic Trap Depth . . . . . . . . . . . . . . . . . . 182.3 Elastic Scattering theory . . . . . . . . . . . . . . . . . . . . 212.3.1 Two Body Problem . . . . . . . . . . . . . . . . . . . 222.3.2 Scattering Cross Section . . . . . . . . . . . . . . . . 222.4 Loss Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26ivTable of Contents3 Experimental Apparatus and Procedure . . . . . . . . . . . 293.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . 293.1.1 Optical Setup . . . . . . . . . . . . . . . . . . . . . . 303.1.2 RF Knife . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . 353.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 The 2D MOT . . . . . . . . . . . . . . . . . . . . . . 383.2.3 Sensor Ensemble Preparation in the 3D MOT andTransfer to the MT . . . . . . . . . . . . . . . . . . . 393.2.4 Atom Number Measurement . . . . . . . . . . . . . . 403.2.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . 433.2.6 Ensemble Preparation . . . . . . . . . . . . . . . . . . 454 Pressure Measurement Results . . . . . . . . . . . . . . . . . 484.1 Loss Rate Measurement . . . . . . . . . . . . . . . . . . . . . 484.2 Argon Measurement . . . . . . . . . . . . . . . . . . . . . . . 535 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62vList of Tables4.1 Table of gas calibration factors. . . . . . . . . . . . . . . . . . 564.2 Table of gas calibration factors with N2 contamination. . . . 58viList of Figures1.1 A Bayard-Alpert ionization gauge diagram . . . . . . . . . . . 31.2 Spinning Rotor Gauge . . . . . . . . . . . . . . . . . . . . . . 41.3 Orifice Flow Standard Diagram . . . . . . . . . . . . . . . . . 52.1 MOT Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 MT Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Zeeman Energy vs. Position . . . . . . . . . . . . . . . . . . . 152.4 Zeeman Energy vs. Position with Gravitational Potential . . 162.5 Recaputre voltage in a MT for different magnetic sub-levels . 172.6 Setting trap depth using RF knife . . . . . . . . . . . . . . . 212.7 Elastic Scattering of two particles COM . . . . . . . . . . . . 252.8 Elastic Scattering of two particles LAB . . . . . . . . . . . . . 262.9 〈σv〉 vs. Trap Depth . . . . . . . . . . . . . . . . . . . . . . . 283.1 Overview of Experimental Apparatus . . . . . . . . . . . . . . 303.2 Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Block diagram of PID control . . . . . . . . . . . . . . . . . . 323.4 Block diagram of Rf knife . . . . . . . . . . . . . . . . . . . . 333.5 RF knife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 RF Reflection vs. Frequency . . . . . . . . . . . . . . . . . . . 353.7 Timing Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 373.8 Loading curve of MOT . . . . . . . . . . . . . . . . . . . . . . 383.9 Loading to a constant density regime . . . . . . . . . . . . . . 403.10 Absoprtion meausrements . . . . . . . . . . . . . . . . . . . . 413.11 Difference of absorption measurement . . . . . . . . . . . . . 423.12 Fluorescence vs. Atom Number . . . . . . . . . . . . . . . . . 433.13 Cooling time and detuning Optimization . . . . . . . . . . . . 443.14 Hyperfine Pumping Optimization . . . . . . . . . . . . . . . . 453.15 Fraction in MT vs. RF frequency . . . . . . . . . . . . . . . . 463.16 Atom Number vs. RF time . . . . . . . . . . . . . . . . . . . 474.1 Experimental Run . . . . . . . . . . . . . . . . . . . . . . . . 49viiList of Figures4.2 Zoomed in veiw of experimental run. . . . . . . . . . . . . . . 504.3 Atom Number vs. MT time . . . . . . . . . . . . . . . . . . . 514.4 MOT Decay vs. Time . . . . . . . . . . . . . . . . . . . . . . 534.5 〈σv〉 vs. Trap Depth . . . . . . . . . . . . . . . . . . . . . . . 544.6 Loss Rate vs. Argon Pressure without RF . . . . . . . . . . . 554.7 Cold Atom Pressure vs. Argon Pressure without RF . . . . . 564.8 Loss Rate vs. Argon Pressure with N2 Contamination . . . . 574.9 Cold Atom Pressure vs. Argon Pressure with N2 Contamination 584.10 RGA trace of setup . . . . . . . . . . . . . . . . . . . . . . . . 594.11 RGA trace after faulty leak valve was replaced . . . . . . . . 60viiiAcknowledgementsI would like to thank Professor Kirk Madison and Dr. James Booth, fortheir guidance and supervision in the time that I have been in the lab. Iwould also like to thank Marisuz, Kahan, Koko, Gene, Will, William as Ihave been lucky enough to learn from them and a special thanks to JanelleVan Dongen for her guidance and patience in training me.ixDedicationTo my parentsxChapter 1Introduction1.1 MotivationThe ability to measure and communicate physical properties is the essence ofexperimental science. The ability to quantitatively express these properties,requires the use of units. This makes defining a unit an essential part ofexperimental science. To define a unit one needs a standard. A standardis an artifact, system or process that defines units of measure and is thereference for other measurements. This thesis describes work to establish anew primary pressure standard using ultra-cold atoms. This new standardties pressure or flux to the standard unit of time, the second.Experiments with ultra-cold atoms have become a pillar in Atomic Molec-ular and Optical (AMO) physics, offering a variety of different physical phe-nomenon to explore. The field began with Ashkin’s idea of using a laser toinfluence atoms and molecules using optical transitions[1]. Two years laterin 1972 the first deflection of an atomic beam was produced[13]. Hansch andSchawlow proposed the use of light to manipulate atoms further by exploit-ing the Doppler effect to slow atoms down. By creating a velocity dependentradiation pressure on the atoms, one can slow atoms down to hundreds ofmicro-kelvins[12]. This reduction of the velocity of atoms using light can beseen as the beginning of laser cooling. From here the use of laser light with amagnetic field produced the first magneto-optical trap (MOT), allowing forthe study of cold and localized atomic samples[22]. The ability to cool sam-ple led scientists to investigate how cold they could make these atom clouds.The first lower limit was broken when sub-doppler cooling was achieved byusing an optical molasses, scientists where were able to produce atomic en-sembles at tens of micro-kelvin[18]. Evaporative cooling took temperaturesdown to hundreds of nano-kelvin pushing the lower limit of ultra-cold atomsfurther[21]. Cooling further and further led to the achievement of a Bose-Einstein Condensate (BEC) in atomic gases. The possibility of a BEC wasfirst proposed by Bose and Einstein in 1924 and realized independently in1995 by Ketterle and by Cornell and Wieman[7][2]. This achievement gavethe field of ultra-cold atoms a lot of promise, by showing its capabilities11.1. Motivationin simulating condensed matter systems and culminating in the 2001 Nobelprize[15][6]. The field went on to explore other areas of interest includingplacing atoms in an artificial lattice produced by laser light, phase transitionsfrom different states of matter such as super-fluid to Mott insulators, as wellas looking at strongly correlated systems[11]. Furthermore ultra-cold atomshave been proposed as quantum simulators and quantum computers[3][23].Ultra-cold atoms have not only led to contributions in fundamentalphysics but have also enabled many advances in applied physics. Cold atomsoffer a unique tool to physics - a physical system in which all of the quantummechanical degrees of freedom can be controlled. This control makes coldatoms an excellent candidate for a measurement device. Cold atoms haveand currently are being used to measure many different physical quantitiesranging from electric and magnetic fields to gravitational fields and time.They allow for creation of devices with extraordinary sensitivity[16]. Oneof these devices is a cold atom based atomic clocks, in which frequencies ofatomic transition can be measured with such high precision that it is thetime standard across the world[5].One area that ultra-cold atoms are poised to make a significant impactis in the area of pressure measurement. Pressure has been studied sinceantiquity, but it wasn’t until recently that it has crossed paths with the fieldof AMO physics. The common method for measuring pressure in the ultrahigh vacuum range is with the use of ionization gauges. Ionization gaugesuse electrons boiled off a hot filament to ionize gas particles.21.1. MotivationFigure 1.1: Schematic of a Bayard-Alpert ionization gauge. The gauge op-erates by boiling off electrons from the filament and having them acceleratetowards the anode grid. As the electrons move through grid they ionize theatoms/molecules as they encounter them. If they do not encounter any gas,they continue along their path then exit the grid and are brought back due tothe coulomb force. This results in multiple passes for the electrons throughthe gas in the detection volume.As ions are produced they are collected bythe ion collector wire which results in a current that proportional to thepressure. This proportionality constant depends on the gauge temperature,the electron emission current, and the electric field within the gauge.31.1. MotivationBy collecting and counting these ions, and comparing this to the electroncurrent, an indirect measure of the background pressure can be made. Thereare significant limitations of ionization gauges. First the ionization poten-tial differs from species to species requiring a calibration for every type ofgas measured. Second is that if these x-rays hit the ion-collector wire, theycan produce electron emission resulting in a current that is indistinguish-able from the current of arriving gas ions and the associated the pressuredetermination is confounded. For higher pressures a spinning rotor gauge isused to measure the pressure. It consists of a magnetized stainless steel ballthat is spun up to a high rotation speed, and its deceleration is measuredto determine the pressure.Figure 1.2: Cross section diagram of a spinning rotor gauge. It consists of;a pair of permanent magnets to suspend a stainless steel ball in the vacuum,pairs of stability coils to maintain orientation of the ball and drive coils tospin up the ball.41.1. MotivationFinally both the ionization gauge and spinning rotor gauge need to becalibrated to a pressure standard to ensure the accuracy of the reading. Forthis purpose, pressure standards are maintained by standard laboratories allover the world.Figure 1.3: The orifice flow standard consists of two vacuum chambersseparated by an orifice, which allows gas to flow from one chamber to theother. Gas enters with a constant volume flow rate through a flowmeterand exits through a turbo-molecular pump. Two gauges are attached tothe side of the standard, one to measure the differential pressure in the twochambers, the other to be calibrated. There is a plate in the lower chamberto baffle the gas flow before it enters the turbo pump.One type of vacuum pressure standard is an orifice flow standard such asthe one at the US Nationals Institute of Standards and Technology (NIST).An orifice flow standard consists of two chamber attached by a small pre-cisely machined orifice. Gas is flowed from the upper high pressure chamber51.2. Thesis Overviewthrough the orifice to the lower low pressure chamber. By knowing the con-ductance of the orifice and the pressure ratio one can determine the pressurein the lower chamber. There a few issues with this type of system: Firstthe out gassing of the chambers sets the base pressure that is attainable.Second, with lower pressure the assumption of a Maxwell-Boltzmann dis-tribution of the gas, one of the crucial assumptions for proper operationbecomes invalid. For these reason a new standard is needed, especially forlower pressure operation.Ultra-cold atoms offer an alternative to the current pressure standardsince pressure can be measured by the collisions between sensor atoms andthe atoms or molecules in the background gas. Thus, by studying the inter-action physics between trapped atoms and background gases one can devisea new pressure standard. This new standard is thus based on fundamen-tal and immutable laws of nature and relies only on the knowledge of thelong-range interaction potential between the trapped atom and the collidingparticle.1.2 Thesis OverviewThis thesis describes progress towards the realization of this new standardand is organized in the following manner: chapter 2 begins by describingthe physical basis of the two types of atom traps being used, a magneto-optical trap (MOT) and a magnetic trap (MT), and includes a discussionof the scattering theory for collision of Ar onto Rb. The chapter ends witha discussion of the trapping potential of the MT. In Chapter 3, the experi-mental setup is discussed including our use of a 2D MOT for loading a 3DMOT to allow for low pressure experiments. We also discuss the trappingcoils for the MT, the RF-knife used to set the trap depth, and details of theoptics used. Chapter 3 also describes the various methods for preparing theatoms before the loss rate measurement is performed. These preparationsteps include cooling and optically pumping the atoms and then preparingthe energy distribution in the MT using an RF-knife to remove the mostenergetic atoms. Chapter 4 goes on to discuss the results of the experiment;atom number measurements as a function of time, loss rate measurementsas a function of argon pressure and finally the measure of the gas calibra-tion factor the ionization gauge. The future work is proposed in chapter5 were measurements of other noble gases and nitrogen are recommended.Measurements of gas mixtures are discussed as they offer an opportunity toexplore the residual gas analysis capabilities that the cold atom gauge could61.2. Thesis Overviewoffer. Finally cross section measurements of different Rydberg states couldbe measured.7Chapter 2TheoryIn this chapter, we discuss various aspects of the theoretical basis of thiswork. The topics include laser cooling, magnetic trapping, and the physicsof collisions. We focus on the details of the MOT and magnetic trap that arerelevant to pressure measurements as well as how the loss rate is determinedfrom a quantum scattering calculation.2.1 Magneto-Optical TrapsMagneto-Optical traps (MOTs) are the starting point of the experiment.They capture atoms from the background vapour and produce the coldsensor ensemble. Here we discuss three aspects of MOTs relevant for theiruse for pressure measurements. These topics are the mechanism for cooling(Doppler cooling) and the typical ensemble temperature in the MOT, theloading dynamics of a MOT, and methods for measuring the atom numberin the MOT. An understanding of these processes is required for precise andaccurate pressure sensor measurements.82.1. Magneto-Optical TrapsFigure 2.1: This is a pictorial representation of a MOT. The MOT consistsof six counter propagating laser beams, a pair along each orthogonal spatialaxis. Each pair has a certain polarization to ensure that the correct atomictransition is driven. On the x and y axes, we have right circularly polarized(RCP) light and on the z axis we have left circularly polarized (LCP) light.A pair of anti-Helmholtz coils are also used to create the magnetic fieldrequired for the trap. Due to the fact that the magnetic field directions arein the opposite directions for x and y as opposed to z leads to the need forthe opposite polarization along z.2.1.1 Doppler CoolingOne mechanism of slowing atoms in a MOT is Doppler cooling. It involvesusing off resonance light to slow down a moving atom, reducing its kinetic92.1. Magneto-Optical Trapsenergy. The idea can be illustrated in 1D by placing an atom in between twocounter propagating laser beams. The force that each beam exerts on theatom in the low intensity limit, in which the intensity of the laser I is smallcompared to the saturation intensity Isat can be described by the followingwhere (I < Isat)[20];F± =h¯~kγ2s1 + s+ [2(δ∓kv)/γ]2 (2.1)Here s is I/Isat, γ is the linewidth of the transition and δ is the detuningfrom resonance. Now considering the two laser beams, the total force fromthem up the first order in v is[20]F = 4h¯~k2sδvγ(1 + s+ 4δ2/γ2)2(2.2)If the light is tuned below resonance then this force can appear as a velocitydependent dissipative force. This can be expressed in terms of a dampingcoefficient, β.F = −βv (2.3)Looking at the energy we find that the light acts as a dissipative mechanism,removing kinetic energy from the atom according to Eq. 2.4.E˙ = −βv2 (2.4)By using six beams, a pair along each orthogonal coordinate axis, one canreduce the velocity of the atom very near to zero, producing an opticalmolasses. In experiments a zero velocity is never achieved, this is due toresidual heating from photon re-emission occurring in all directions. A lowerlimit is achieved when |δ| = γ/2kBT =h¯γ2(2.5)This limit is called the Doppler-cooling limit[20]. While there are othercooling mechanisms in the MOT that can lead to temperatures below theDoppler cooling limit, this limit is nevertheless a good estimate of the tem-perature of the atoms in the MOT[20]. This residual thermal motion of theatoms is important to keep in mind since it limits the temperature of theatoms when transferred into a magnetic trap and leads to a reduction ofthe momentum gain required to escape the trap due to the non-zero initialkinetic energy.102.1. Magneto-Optical Traps2.1.2 Loading DynamicsThe loading dynamics of a MOT play an important role in the applicationof a MOT for pressure measurements. The loading dynamics of a MOT canbe modeled byN˙ = R− ΓN − β∫n2(~r, t)d3~r (2.6)The loading rate R is the number of atoms entering the trap per second,ΓN is the loss rate due to collisions with atoms in the background andβ∫n2(~r, t)d3~r encompasses losses due to radiative escape, fine-structure col-lisions, hyperfine collisions, intra-trap collisions and is dependent on thedensity of the atoms in the MOT , n(r,t). The solution to this differentialequation is examined in two limiting regimes that a MOT can reach. Thefirst regime is the constant volume regime. The constant volume regime isone in which the MOT cloud is dilute enough that atom-atom repulsive in-teractions from light rescattering are negligible and atoms are confined in aspatial volume of fixed size. In this regime, the MOT grows in number witha constant volume. However, once the density of the MOT is high enough,repulsive forces between the atoms cause the MOT to grow in volume withincreasing number such that the peak density does not change. The con-stant volume regime can be modeled by assuming the atomic density has aGaussian profile n(~r) = n0e− r22w2 where n0 is the peak density at the centerof the trap ~r = 0 and w is the width (assumed here to be a constant). Thisallows Eq. 2.6 to be written as;N˙ = R− ΓN − βw(2pi) 32N2 (2.7)Solving this equation results in[8]N(t) =RΓ + βn′(1− e−(Γ−2βn′)t1 + χe−(Γ−2βn′)t)(2.8)Where χ equals βn′Γ+βn′ andn′ =(∫n2d3r∫nd3r)(2.9)is the average steady state density.With a sufficiently large population the MOT will enter the constantdensity regime. Applying the constant density condition to Eq. 2.6 resultsin the follow,N˙ = R− ΓN − βnN (2.10)112.1. Magneto-Optical TrapsThe solution to the above differential equation isN(t) =RΓ + βn(1− e−(Γ+βn)t) (2.11)Aside from understanding the changing behavior of the 2-body losses in theMOT, understanding these regimes is useful for loading atoms into the MT.As the MOT is filled and enters the constant density regime, the number ofatom per volume in the center of the trap begins to reach a constant. Thisconstant density allows the transfer of an exact number of atoms to the MTfrom the MOT if a certain central volume is selected.2.1.3 Atom Number CalibrationCounting the number of atoms left in the MOT or the MT trap is the mainmeasurement of the experiment. Therefore, the ability to accurately andprecisely measure the atom number is of utmost importance. There are twomain methods of measuring the atom number in a trap. The first methodinvolves the fluorescence measured on a photodiode which is directly relatedto the number atoms. This relationship is seen through Eq. 2.12 where themeasured photodiode voltage (V) is related to the atom number (N) , thescattering rate (Γ) and a conversion parameter (α). The parameter α takesinto account the transfer efficiency of a photon emitted by a single atom tothe voltage measured after the amplified photodiode.V = αΓN (2.12)Γ =γ2I/Isat1 + I/Isat +(2∆γ)2 (2.13)The fluorescence method requires measuring both the scattering rate of theatoms and α, the conversion parameter. This measurement can be made,however it requires a more elaborate procedure[? ]. These issues can beavoided by using a different technique called, optical pumping. Opticalpumping involves using a separate beam to optically pump atoms from onestate to another and measuring the photon loss[4]. For 87Rb, atoms startin the F = 2 state where they are pumped to the excited state F′= 2.Following this excitation, the atoms relax into either the F = 2 or F = 1ground state by the emission of a photon. By measuring the number ofphotons required to transfer all the atoms into the F = 1 state, and knowing122.2. Magnetic Trapsthe average number of photons needed to pump a single atom into the F = 1state, one can determine the atom number by the following formulaN =AφhνP (2.14)where A the integrated attenuation of the voltage signal, φ the average num-ber of photon scatter per atom , hν the photon energy, P the optical powerto the photodiode. By taking multiple measurements of various atom num-bers using optical pumping and their corresponding fluorescence voltages,the relation between voltage and atom number can be determined. The mostimportant outcome of such a comparison of the atom number as determinedby optical pumping and by fluorescence is to verify that the fluorescencesignal is linear in the atom number. Linearity of the signal is absolutely keyto a correct determination of the loss rate and corresponding pressure.2.2 Magnetic TrapsAlongside the MOT, the MT is the main trap used to study the collisionsof the trapped particles and colliding background gas. The magnetic trapis created by a pair of coils in an anti-Helmholtz configuration, seen inFig. 2.2. This type of trap is the simplest kind that one can use to trapatoms. This quadrupole magnetic trap has a zero field at the center and alinearly increasing field away from the center. The magnetic field to firstorder in the axial and radial direction is,Bz = 3µ0DR2(D2 +R2)5/2Iz (2.15)Bρ =32µ0DR2(D2 +R2)5/2Iρ (2.16)Thus, the resulting field has twice the gradient in the axial direction com-pared to the field in the radial direction. The field gradient creates a spatiallyconfining force that will trap the atoms in a particular hyperfine state.132.2. Magnetic TrapsFigure 2.2: Magnetic field lines generated by pair of anti-Helmholtz coilsused to produce the field for the magnetic trap.As the atoms move in the magnetic trap, their magnetic potential energyvaries with spatially varying magnetic field. This potential energy is differentfor the different atomic hyperfine energy levels: some levels are trapped(diamagnetic states) while others are not (paramagnetic) states. Atoms willstay trapped as long as the local magnetic field doesn’t change too rapidlyas the atoms move. The potential energy that the atoms experience isU(r) = mFµBgF |B(r)| (2.17)where mF is the hyperfine state magnetic sub-level, µB is the Bohrmagneton and gF is the Lande´ g-factor which is -1/2 for the F=1 groundstate of 87Rb.142.2. Magnetic TrapsFigure 2.3: The energy splitting of the different sub-levels for the groundstate F = 1 as a function of position. In the absence of a magnetic fieldthese levels are degenerate.2.2.1 Gravitational SagThe magnetic potential is not the only potential that influences atoms ina magnetic trap. Gravity also plays an important role. Fig. 2.4 shows thepotential energy that the atoms experience including gravity. This shift isa result of the gravitational field present along the vertical symmetry axisof the B-field.Upotenital = UZeeman + UGravity (2.18)Upotenital = −~µ · ~B(~r, I) +mgz (2.19)The potential energy is now modified and this leads to two main effects:First, the asymmetrical trapping potential can lead to an anisotropic trapdepth, second, a minimum B-field gradient is required to overcome the grav-itational force, which would otherwise cause the atoms to drop out of thetrap. The trapping asymmetry can be seen where the atom below the zero152.2. Magnetic Trapsposition (~r = 0 which coincides with zero of the B-field) have a lower barrierof escape, while atoms above the zero position see a barrier higher than thatproduced by the magnetic field alone.Figure 2.4: Energy level structure of the F = 1 states in the presence ofgravity. Gravity causes an asymmetry in the trap potential that leads to ananisotropic trap depth.2.2.2 Gravitational FilteringThe presence of gravity appears as an issue for magnetic traps, however, onebenefit it offers is gravitational filtering. Gravitational filtering uses the factthat different magnetic sub-levels have different potential energy surfaces inthe same B-field. Eq. 2.17 shows that the mF = 2 requires half the fieldstrength than the mF = 1 to achieve the same magnetic potential energy.Thus, there is a magnetic field gradient at which mF = 2 state is trappedbut mF = 1 is not.162.2. Magnetic TrapsFigure 2.5: Plot of recapture voltage in the MT vs. the current of the MT.The plot shows that for current below 11.2 A there is no recapture in theMT. This is because the the magnetic force of the MT is not enough toovercome gravity for any atomic states. As the current is increased past11.2 A there appears to be some recapture in the MT and then a levelingoff, this signal is of the mF = 2 state. As the current is increased past22.4 A the recapture signal increases again and levels off, this signal is ofthe mF = 1 state which requires twice as much current as the gradient isnow able to support atoms in either the mF = 2 or the mF = 1 hyperfinesublevels. As expected, the threshold currents scale as the mF value..The difference in potential energy allows for the magnetic state purifica-tion through the use of gravity as a filtering process. One limitation of thismethod is a lengthy waiting period for filtration. In our setup, we must wait70 ms for the atoms in the un-trapped spin state to fall out of the MOTrecapture region.172.2. Magnetic Traps2.2.3 Majorana LossesQuadrupole Magnetic traps are a convenient tool for trapping atoms. How-ever, one issue with this type of trap is a zero field in the center. This zerofield point can lead to atoms not having an axis of quantization to follow,resulting in loss from the trap. This loss is called spin flip loss or Majo-rana loss. These spin flips, as the name suggests, occur when atoms passthrough the center of the magnetic trap and experience an abrupt changein the magnetic field direction. This abrupt change leads to the possibilityof flipping from a trappable state to an untrappable one. The condition fora spin flip involves the local change in the magnetic field to be larger thanthe atom’s Larmor frequency. Mathematically the loss rate due to spin flipscan be estimated by the follow equation[21]ΓMajorana ≈ h¯ml2(2.20)where l is the radius of the cloud and m is the mass of the atom. We canexpress this condition in terms of the magnetic field gradient B’ and thetemperature as [19]ΓMajorana ≈ h¯m(µB′kBT)2 (2.21)where B′is the local gradient of the magnetic field. These equationsallow for the estimate of the losses due to spin flips, if we take T to be150µK and B′to be 99 Gcm then ΓMajorana is 40 seconds. The long lifetimeensures that spin flips will will not dominate the pressure measurements forpressures above 10−11 Torr .2.2.4 Magnetic Trap DepthThe magnetic trap depth is defined by the energy the atoms must have toescape the trap and be lost. One method for setting the trap depth in amagnetic trap involves using a RF coil. The RF coil produces an oscillatingB-field which can couple adjacent magnetic sub-levels in the trap and allowfor magnetic dipole transitions (∆mF = ±1). By driving transitions fromtrappable to untrappable states, the RF radiation can be used to set themaximum value of the energy distribution of the atoms alongside fixing thetrap depth. The potential energy of atoms inside the magnetic trap is givenbyUZeeman = −~µ · ~B = µBgFmF bI√x24+y24+ z2 (2.22)182.2. Magnetic Trapswhere bI is the magnetic field gradient(∂B∂z)along the z-axis. Adding thegravitational potential energy results inUpotenital = µBgFmF bI√x24+y24+ z2 +mgz (2.23)The application of an RF signal to the atoms in a gravity free situationproduces a transition between trapped and untrapped states on the surfaceof an ellipsoid specified by the locus of points xd, yd, and zd satisfyingEq. 2.24. As atoms travel in the magnetic trap and pass a certain point inspace, they interact with the RF field and may spin flip from a trappablestate to an untrappable one. This interaction surface traces out an ellipsoid,therefore it is named the ”ellipsoid of death”.∆UZeeman = ±µBgF bI√x2d4+y2d4+ z2d = hν (2.24)Atoms that do not possess enough energy to reach the ”ellipsoid of death”remain in the trap. By tuning the RF frequency, the volume and the po-tential energy the atoms occupy can be controlled. Atoms that reach thesurface are lost from the trap, setting the upper limit on the potential energyto be the following:Ulimit ≤ µBgFmF bI√x2d4+y2d4+ z2d +mgz (2.25)In order to find the points where the transitions occurs, it is useful to derivethe minimum current required to create the field gradient that can supportthe atoms against gravity. This current is when the gravitational force equalsthe Zeeman force.|~Fg| = |~FZeemam| (2.26)mgbI0= µBgF |mF | (2.27)where I0 is the minimum current and equal to 22.4 A in our setup. Thisprovides the useful relationshipµBgF b =mgmfI0(2.28)Combing Eq. 2.25 and Eq. 2.27 we have192.2. Magnetic TrapsUlimit ≤ mg II0√x2d4+y2d4+ z2d +mgz (2.29)wherezd =hν|mF |I0mgI(2.30)for the z-axis and for the x and yxd =2hν|mF |I0mgI(2.31)yd =2hν|mF |I0mgI(2.32)Gravity modifies the trapping potential and the minimum potential theatoms encounter along the z direction.On the z-axis the potential becomesUz,limit ≤ mg II0zd +mgzd (2.33)By re-writing the expression in terms of the RF energy we haveUz,limit(min) ≤ hν|mF |(1−I0I) (2.34)Uz,max ≤ hν|mF |(1 + I0I) (2.35)Ux,y,limit ≤ hν|mF | (2.36)The anisotropy introduced by the gravitational field is clearly illustrated byEq. 2.34 and Eq. 2.35. The consequence of which is if atoms have enoughtime to explore the entire trap they will encounter the minimum in the z -direction.Uz,limit ≤ hν(1− I0I) (2.37)By using the RF knife to eject atoms at a specific energy, tailoring of theenergy distribution can be achieved. Furthermore by setting the RF knifeto interrogate a range of frequencies from the trap depth and above, atomswith higher energies will efficiently be removed.202.3. Elastic Scattering theoryFigure 2.6: This figure shows the coupling of the different magnetic subs-levels to each through a radio-frequency signal. This signal is swept fromhigh to low frequencies to eject atoms out of the trap within a range ofpotential energies. This RF signal can thus be used to prepare the sensorensemble by removing atoms above a certain energy.In the presence of grav-ity the potential energy of the atoms are asymetric in space pushing theatoms towards a non-zero point as a potential energy minimum. This shiftcauses a spatial and energy asymmetry to the atomic distribution in themagnetic trap.2.3 Elastic Scattering theoryThe topic of particle-particle scattering is ubiquitous in ultra-cold physics.It is the mechanism behind the thermalization of quantum gases and a lossmechanism in traps. Here scattering is examined in the context of elasticscattering of particles at large relative velocities.212.3. Elastic Scattering theory2.3.1 Two Body ProblemThe simplest scattering model is a two-body collision between two particlesthat interact through a potential. The Hamiltonian of the system isH =~p212m1+~p222m2+ V (|~r1 − ~r2|) (2.38)and includes three terms; the first two terms are the kinetic energy termsof each particle and the third is the potential energy of interaction. Thisproblem can be solved in the center of mass frame. To change to the centerof mass frame a few new variables are needed. These include; total massM, total momentum P, and relative position r, relative momentum p andreduced mass µM = m1 +m2 (2.39)~P = ~p1 + ~p2 (2.40)~r = ~r1 − ~r2 (2.41)~p = µ(~v1 − ~v2) (2.42)By making the substitution the Hamiltonian becomes;H =~P 22M+~p22µ+ V (~r) (2.43)The Hamiltonian now describes a particle moving with mass M and onewith the reduced mass µ describing the relative motion of the particles. Byworking in the center of mass frame, the momentum of the center of massin the Hamiltonian becomes a constant, reducing Eq. 2.43 toH′=~p22µ+ V (~r) = H −HCM (2.44)This picture gives a single particle of mass µ moving in a potential V (~r)2.3.2 Scattering Cross SectionWe now wish to describe the calculation of the scattering cross section. Webegin with the Schrodinger equation(~p22µ+ V (~r))ψk(r) = Ekψk(r) (2.45)222.3. Elastic Scattering theoryand the ansatzψk(r) = ψinc(r) + ψsc(r) (2.46)which includes an incident wave ψinc(r) ∝ ei~k·~r and scattered wave. Thescattered wave can be found by investigating certain symmetries. For po-tentials falling off faster that r−2 and as r →∞, the solution for the scatterwave is of the formψsc(r) = f(k, θ, φ)eikrr(2.47)where f(k, θ, φ) is the scattering amplitude, which depends on k, the wavevec-tor of the incoming particle and θ, φ the direction of scattered wave. If thepotential is isotropic the Schrodinger equation is re-written in terms of thespherical coordinates and the angular momentum operator as [17]h¯2µ(− 1r2∂∂r(r2∂ψr∂r)+Lˆ2r2ψr)+ V (r)ψr = Eψr (2.48)where V(r) has no angular dependence. The wavefunction can now also beseparated in terms of a radial and an angular part.ψr(r, θ, ψ) = Rl(r)Yl,m(θ, φ) (2.49)Substituting this wavefunction into the Schrodinger equation leaves us with1r2ddr(r2dRldr)− l(l + 1)r2Rl +2µh¯2(E − V (r))Rl = 0 (2.50)The scattering problem is cylindrically symmetric along φ, allowing us tochange ψ from spherical harmonics to Legendre polynomials, leaving it inthis new formψ(r, θ) =∞∑l=0(2l + 1)ileiδlRl(kr)Pl(cosθ) (2.51)By re-expressing the radial part of the solution Rl in terms of ψl = krRl(r)we have[17] (d2dr2+W (r))ψl(r) = 0 (2.52)whereW (r) = k2 − 2 µh¯2V (r)− l(l + 1)r2(2.53)232.3. Elastic Scattering theoryBy setting up a new differential equationy′(r) + y2(r) +W (r) = 0 (2.54)whereyl(r) =ψ′lψl(2.55)andψl(r) = cos(δl) (krjl(kr)−Kl(k)krnl(kr)) (2.56)solving for Kl is achievable[14].The scattering amplitude can also be expressed in terms of LegendrePolynomials asf(k, θ) =1k∞∑l=0(2l + 1)eiδl sin(δl)Pl(cos(θ)) (2.57)where δl is the phase shift for the lth partial wave and is related to theT-matrixTl(k) = eiδl sin(δl) (2.58)The T-matrix can solved from the S-matrixSl(k) = e2iδl = 1 + 2iTl(k) (2.59)which in turn needs the K-matrixSl(k) =1 + iKl(k)1− iKl(k) (2.60)Kl(k) = tan δl(k) (2.61)In this work, a Lennard-Jones potential is used to model the interactionpotential to solve for the K-matrix. It is a potential which describes aninduced dipole-dipole interaction between colliding particles. Its long rangeinteraction is modeled by a r−6 and we model the short range repulsive partas a r−12.V (r) =C12r12− C6r6(2.62)By using numerical methods the solution for yl(r) can be obtained. What isrequired for this process is the potential and the reduced mass. The solutionallows for the K-matrix to be found which gives the T-matrix, from whichthe total elastic cross section can found.σ =4pik2∞∑l=0(2l + 1)|Tl(k)|2 (2.63)242.3. Elastic Scattering theoryFigure 2.7: This figure shows the an elastic collision in the center of massframe between Ar and Rb showing their initial and final momentum states.252.4. Loss RateFigure 2.8: In the lab frame a collision can be seen with the Rubidiumappearing stationary and the Argon atom moving towards it. As the Argonpasses the Rubidium atom it does so traveling off at an angle, this anglecan be used to parametrize the momentum transfer from the Argon atomto the Rubidium atom. Angles that are above a certain threshold resultin a momentum transfer large enough to induce loss from the trap, whileangles that are below result in heating. This threshold (minimum angle) isdirectly related to the trap depth. This threshold is set by the minimumenergy needed to escape the trap.2.4 Loss RateBackground collisions are an undesirable phenomenon in ultracold systems,limiting both the signal integrity (e.g. in atomic clocks) and lifetime ofa cold ensemble. In this work this undesirable effect is utilized to betterunderstand the composition of the background gas in the vacuum system.The loss rate of atoms in a MT can be expressed asΓ =N∑ini〈σlossvi〉(Rb,i) (2.64)262.4. Loss Ratewhere the loss rate, Γ, is the sum over the background species of densityni, multiplied by the loss cross section σloss and relative velocity v, of thebackground gas. The brackets represent an average taken over the velocitiesof a Maxwell-Boltzmann distribution. This 〈σv〉 term quantifies the interac-tion of the trap atoms and colliding background particles. The interactionis not only species dependent but also depends on the depth of the trappingpotential demanding that the dependencies be explored. The transfer ofenergy from the background particle to the trapped particles (assuming thetrapped particle initially has no kinetic energy) is[10]∆E =µ2mt|~vr2|(1− cos(θ)) (2.65)Here the energy transfer depends on the reduced mass µ, the relative velocity~vr, mass of the trapped particle mt and the center of mass frame collisionangle θ. By rearranging the equation and setting the amount of energy equalto the trap depth U0, a minimum angle that the scattered particle requiresto make for an atom to be lost from the trap, can be found.θmin = cos−1(1− U0mtµ2| ~vr|2)(2.66)Recalling that the cross section depends on the scattering amplitude bydσdΩ= |f(k, θ, φ)|2 (2.67)the total elastic collision cross section can be re-written as;σ =pi∫02pi|f(k, θ)|2 sin θdθ (2.68)Using the minimum angle as an indicator on whether or not a particle es-capes the trap, allows for the definition of the loss cross section asσloss =pi∫θmin2pi|f(k, θ)|2 sin θdθ (2.69)and the heating cross sectionσheating =θmin∫02pi|f(k, θ)|2 sin θdθ (2.70)272.4. Loss RateUsing Eq. 2.69 and modeling the background gas as a Maxwell-Boltzmanndistribution we get the 〈σlossv〉 term to be〈σlossv〉 =pi∫θmin2pi|f(k, θ)|2 sin θdθ∞∫0√m2pikT34piv2emv22kT dv (2.71)which which can solved using the scattering amplitude |f(k, θ)|2 describedin the previous section.Figure 2.9: Cross section averaged over the velocity of the backgroundgas for Ar and Rb collisions vs. the trap depth. This illustrates how theloss cross section varies with trap depth. The potential used is a Lennard-Jones potential. The C6 used in this calculation is 280EHa6B and C12 is8.6× 107EHa12B , where EH = 4.35974× 10−18 J and aB is the Bohr radius.28Chapter 3Experimental Apparatus andProcedureThis chapter is comprised of two sections: the experimental apparatus,where the setup and its components are discussed, and the experimentalprocedure, where the method of creating and preparing the cold atoms aswell as implementing the measurement scheme is examined. The experimen-tal apparatus consists of many different sections ranging from the optics thatcontrol the light used in creating the sample, to the vacuum required to trapand maintain the sample, and finally to the devices used to measure fluo-rescence, vacuum pressure, contamination, etc. Moreover the experimentalprocedure focuses on the preparation of the atoms.3.1 Experimental ApparatusThe experimental setup is comprised of a vacuum chamber which has threemain regions: a 2D MOT, a 3D MOT and the gauge region. The 2D MOTgenerates a cold beam of atoms from a Rb vapour and is created by twopairs of countering propagating laser beams and a quadrupole magnetic fieldcreated by 4 separate race-track coils. The atoms are accelerated out of the2D MOT region by a ”push beam” detuned to the blue of the resonance.The atoms are then trapped in the 3D chamber by a 3D MOT and it ishere where the pressure measurement is made. The gauge region includesa Residual Gas Analyzer (RGA), which measures the composition of thebackground in the setup, as well as two spinning rotor gauges (SRG) andtwo ionization gauges (IG). One of the IGs and SRGs were provided andcalibrated by NIST. Two types of gauges are employed because of theirdifferent operational ranges. The SRG (1× 10−2 to 5× 10−7 Torr ) is usedin the high vacuum range and the IG (1× 10−3 to 2× 10−11 Torr)is used inthe ultra-high vacuum range. The Fig. 3.1 shows the three different regionsof the vacuum chamber in detail as well as the port used to introduce gasesfor the pressure measurement.293.1. Experimental ApparatusFigure 3.1: The experimental apparatus consists of three main sections; a2D MOT, a 3D MOT and a Gauge region. A cold flux of atoms is created bya 2D MOT. This cloud is set in motion by a ”push” beam to the 3D MOTsection. In the 3D MOT section atoms are held in both a MOT and MT.The gauge region next to the 3D MOT is where all the gauges are located[9].3.1.1 Optical SetupThe optical setup of the experiment is illustrated in the Fig. 3.2. The light isgenerated on a separate ”master” table where it is then sent through fibersto the optical table for the experiment. The light is then amplified by otherlaser diodes and a tapered amplifier (TA). The polarization and frequencyare controlled by additional optics following amplification. After this stagethe light is split and sent to the different regions of the apparatus, to supplythe light needed to trap the atoms.303.1. Experimental ApparatusFigure 3.2: Diagram of the optical setup used to generate the laser lightfor the experiment. We take light from a frequency stabilized master table,amplify it, shift its frequency, and then send the light to the 2D and 3DMOTs[9].The intensity of the 3D MOT pump light is stabilized using a feedbackloop consisting of a PID controller and a photodiode (PD).The light thatis sent to the 3D MOT region is sampled by a beam splitter and sent to aphotodiode. The reading is compared to a preset voltage to generate theerror signal for the PID controller. The controller maintains a constantintensity by feeding back to an r.f. attenuator that regulates the r.f. powersent to the AOM that sets the 3D MOT power. A schematic of the intensitystabilization system is shown in Fig. 3.3.313.1. Experimental ApparatusFigure 3.3: Block diagram of the intensity stabilization system of the 3DMOT laser. The setup consists of the following: an Acousto-Optical Modu-lator (AOM), Photodiode (PD), Analog Output (AO), Differential Amplifier(Diff. Amp.), Pre-amplifier (Pre Amp.),RF Attenuator (RF Atten.), Directdigital synthesizer (DDS). The DDS generates the r.f. signal that is used todrive the AOM.3.1.2 RF KnifeThe RF knife is a key component for setting the trap depth in the magnetictrap. The parts necessary to create the oscillating B-field are shown inFig. 3.4. The r.f. system begins with a direct digital synthesizer (DDS) thatgenerates the RF signal that is sent to the coil. This signal is amplified witha pre-amplifier and post-amplifier where it is sent through the coil and thenattenuated and terminated with a 50Ω load. The attenuator and terminatingload are used to minimize reflections back into the post amplifier.323.1. Experimental ApparatusFigure 3.4: Block diagram of of the RF knife used to set the trap depth ofthe atoms in the magnetic trap.The RF coil is created by taking a RG-174 coax cable and stripping itsouter sheath to expose the inner conductor. The sheath is reconnected at theloop beginning and end. A single loop is used to maximize the bandwidth ofthe antenna, given that a large frequency range (500kHz-140MHz) is desired.333.1. Experimental ApparatusFigure 3.5: Picture of the RF coil used in the experiment. It is a singlelooped stripped BNC whose outer sheath is removed and the ends of thesheath are soldered together to continue the cable intact to the terminationresistance.The transmission characteristic of the RF coil can be seen in the fig-ure below. Here one end of the coil was connected to a network analyzer(HP8753E) and other end terminated with 50Ω. The reflectance as a func-tion of the frequency was measured. From the Fig. 3.6 there is a clear rolloff of reflectance for lower frequencies, where the coil acts as a short.343.2. Experimental ProcedureFigure 3.6: This is a plot of the reflection characteristic of the RF coil. Themeasurement was made by connecting the the RF coil to a network analyzer(HP8753E) and recording the reflection as a function of frequency. One cansee a drop off of the reflected (hence increase in transmitted) power in thelower frequencies, where the coil acts as a short. On the high end, the loopreflects a majority of the power acting as a high impedance.3.2 Experimental ProcedureThe experimental procedure involves the method for producing, controllingthe cold atoms and executing the experiment. The procedure has a fewkey steps that are necessary to prepare the atoms and to ensure the repro-ducibility of each measurement. Also each step is optimized to maximizethe number of atoms in the trap and the signal to noise ratio.353.2. Experimental Procedure3.2.1 OverviewThe atoms are first collected from a vapour and loaded into a 2D trap wherethe are sent to the 3D trap. From here they are further cooled, opticallypumped into the correct quantum state and captured in the magnetic trap.Then, the most energetic atoms are ejected from the MT and the remainingatoms are left to interact with the background gas. As measurements aremade of the remaining atoms in the trap for various holdtimes, a loss ratecan be extracted. This loss rate is then used to determine the backgrounddensity of gas.The timing diagram is shown in Fig. 3.7 and indicates the optimal tim-ings found for each stage that minimize heating and atom loss in the prepa-ration steps.363.2.ExperimentalProcedureFigure 3.7: Timing diagram of the experiment showing the optimal timings found for each stage that minimizeheating and atom loss in the preparation steps.373.2. Experimental Procedure3.2.2 The 2D MOTThe atoms are loaded into the 3D MOT from the 2D trap. The 2D MOTconsists of a pair of crossed beam. Each beam passes through a pair ofcoils which generates the magnetic field for the 2D MOT. The atoms arecooled and trapped from a room temperature gas. The atoms that enterthe center of the 2D MOT region are accelerated in the direction of the 3DMOT by a separate push beam. The loading rate from the 2D limits howhigh in pressure a loss rate measurement can be performed, because thesteady state atom number is dependent on the ratio of the loading rate andloss rate. During this loading the fluorescence of the 3D MOT is monitoredand a typical loading signal is shown in Fig. 3.8.Figure 3.8: Fluorescence vs. time of a 3D MOT. The curve can be modelby the loading Eq. 2.8 or Eq. 2.11 depending on which regime the MOT fillsto. Here the MOT turns on after an initial 2 second wait period.383.2. Experimental Procedure3.2.3 Sensor Ensemble Preparation in the 3D MOT andTransfer to the MTWhen atoms are loaded into the 3D MOT, measures are taken to ensure theconstant density regime is reached in the MOT since this produces the lowestshot-to-shot variation of atom number. The atoms are cooled and opticallypump to the lower ground state (F = 1), the optimization of these stages arediscussed in section 3.2.5. Atoms are loaded for different amounts of timeand transfered into the MT and then recaptured again in the 3D MOT todetermine the number transfered. The number in the MT saturates abovea certain number in the 3D MOT and therefore the MOT is always loadedto an atom number beyond this point to minimize number fluctuations.Fig. 3.9 shows that, initially, as the MOT atom number increases so doesthe recaptured amount. However once 6×108 atoms in the MOT is reached,the recaptured number levels off. Fig. 3.9 also shows that the maximumnumber loaded into the MT and the saturation point depends on the depthof the MT as set by the RF knife. For a fixed magnetic field gradient, moreatoms are captured by the MT for a larger trap depth because the spatialvolume is larger for a deeper trap.393.2. Experimental ProcedureFigure 3.9: Atom number loaded into and recaptured from the MT as afunction of the initial number in the 3D MOT for different MT depths setby the RF knife. This figure illustrates when the constant density regime hasbeen reach. The GREEN and RED plots correspond to a 5MHz trap depthand the BLUE corresponds to 10MHz. It can be seen that after 6×108atoms the MOT enters a constant density regime because the number ofatoms recaptured is becoming insensitive to the total atom number. Alsothe steady state value for the 5MHz and 10MHz trap depths differ becausethe 10MHz trap is spatially larger.3.2.4 Atom Number MeasurementThe measurements of sensor particle loss rate depends on measuring theatom number left in the MT over time, and it is critical that our measure-ment signal is strictly proportional to the atom number as any non linearitywill result in a systematic error in the loss rate. Fluorescence measurementsof the recaptured atoms in the 3D MOT are the most convenient; however,we need to calibrate the fluorescence signal to the atom number and verify403.2. Experimental Procedurethat the signal is linear in the atom number. Therefore an optical pump-ing technique is employed as an independent method to count atom number.The technique involves taking an absorption trace with and without an atomcloud in the path of a probe beam[4]. Fig. 3.10 shows the traces of the probebeam with (RED) and without (BLACK), the difference of the two is shownin Fig. 3.11. Fig. 3.11 represents the amount of photons scattered out ofthe path of the probe beam. Since the probe beam is set to the 52S1/2to 52P3/2 transition and each atom scatters a finite number of photons Nγbefore falling into the dark F = 1 ground state, the atom number is simplyNatoms =SNγwhere S is the total number of photons scatter out of the probebeam[4][? ].Figure 3.10: Plot of the transmitted intensity of the probe beam used tomeasure the atom number. The red trace corresponds to a signal with atomspresent in the beam path and the black corresponds to one without. Theround edge of the red signal is a result of atoms scattering the probe beam,thus reducing the transmitted light. The slight discrepancies in the steadystate signal is a result of the background level changing from shot to shot.413.2. Experimental ProcedureFigure 3.11: Plot of the difference of the probe beam transmitted intensitywith and without atoms in its path. We can see that over time atoms arebeing pumped out of the bright F = 2 state by the probe light allowing formore light to transmit resulting in a decrease in the absorption signal. Thearea under the curve is related to the total number of atoms in the path.By measuring the atom number and fluorescence for different amountsof atoms a calibration curve between atom number and fluorescence canbe obtained. This curve shown in Fig. 3.12 allows for the determinationof atom number solely from fluorescence. The advantage of a fluorescencemeasurement versus an optical pumping measurement is that the signal tonoise for a fluorescence measurement is much higher than optical pumping.It is for this reason that all measurements are made using fluorescence.423.2. Experimental ProcedureFigure 3.12: Atom number as measured by optical pumping induced by aprobe beam versus fluorescence of the same atoms in a MOT. We can see alinear relationship between them for atom numbers in the range from 1×109to 5×107 atoms. This provides a calibration for fluorescence signal to atomnumber. The calibration factor is approximately 5×108 per volt.3.2.5 OptimizationThere are many parameters to optimize in the experiment. In particularhyperfine pumping and cooling stages are looked at to maximize the numberand minimize temperature of the atoms loaded into the MT and to ensurestate purity. Cooling of the atoms before transfer in MT is accomplished bydetuning the light away from resonance and holding the atoms in that lightfor a period of time. The optimization was carried out at several differentfrequencies for various holdtimes. The results are summarized in Fig. 3.13.It appears that holding the atoms for 75 ms at a 60 MHz detuning maximizesthe number of atom recaptured in a MT trap depth of 180µK. These valuesare used in the experiment.433.2. Experimental ProcedureFigure 3.13: Atom number loaded into and recaptured from the MT as afunction of cooling duration for various detunings: 60 MHz(blue circles) 50MHz(red squares)and 40 MHz (green triangles). One can see a maximumoccurring at a duration of 75 ms for a detuning of 60 MHz.To ensure that atoms that are transfered into the MT are in the lowerhyperfine state, the repump is turned off for a period of time before the pumplight is extinguished allowing the atoms to pump to the F = 1 ground state.The amount of atoms captured in MT is measured and the results are shownin Fig. 3.14. From the results 2 ms was chosen as the optimal value for theexperiment.443.2. Experimental ProcedureFigure 3.14: Atom number in the mF = 2 state loaded into and recapturedfrom the MT as a function of the hyperfine pumping time. Spin filtering(discussed in section 2.2.2) was used to remove any atoms in the mF = 1state and detect only those atoms in the mF = 2 state. Since atoms in themF = 2 state can only be in the upper hyperfine state (F = 2), this signalprovides a measure of the hyperfine pumping efficiency.3.2.6 Ensemble PreparationThe atom number in the MT is not the only parameter which needs to be op-timized. In particular, the energy distribution plays a key role in determiningif the cloud is suitable for the measurement. In short the atom temperatureshould be small compared to the depth of the MT for a loss measurement.The atoms in the MT can be modeled by a Maxwell-Boltzmann distribu-tion with an offset in the energy. This offset is the minimum energy for theensemble set by Majorana losses and any mis-match between the MOT andMT centers. Here Fig. 3.15 shows a typical potential energy distribution forthe experiment. Here the average temperature is 161µK and the offset is453.2. Experimental Procedure1.45MHz or (72.5µK). This temperature is cold enough to have a suitableamount in a shallow magnetic trap.Figure 3.15: Trapped fraction in the MT versus the lower frequency in aradio frequency sweep used to remove atoms with energy above the cutoff Ec= hflower. From the profile one can extract the temperature (energy distri-bution) and the energy offset of the atoms in the magnetic trap. This profilecorresponds to a temperature of 161µK and an offset of 1.45 MHz(72.5µK).To verify the efficiency of the RF knife in clearing out atoms above acertain energy level, the amount of time it takes to empty the trap wasexamined. The time it takes for the trap to be emptied at various powersis shown in Fig. 3.16. It shows that at 30 W it takes about 0.7s seconds toempty out the trap, this sets the minimum time that the measurement cantake place.463.2. Experimental ProcedureFigure 3.16: Recaptured atom signal from the MT versus the RF exposuretime for various RF powers. The RF consists of a ”triangle frequency ramp”from 140 MHz to 1 MHz at rate of 100 GHz/s used to empty atoms fromtrap. The various powers correspond to RF powers of 0.74 W(Black), 4.65W(Blue), 8.26 W(Red), 12.5 W(Green) and 30 W(Yellow). Here one cansee that the recapture voltage decreases until it levels off after roughly 0.6seconds providing an empirical measure of the minimum RF time requiredto eject atoms from the trap and thus to set the maximum ensemble energylevel in the trap.47Chapter 4Pressure MeasurementResultsThis chapter reports on measurements of the decay of trap population overtime (here after called a decay trace) with and without argon added to thevacuum chamber. The measurements without Ar are needed to determinethe baseline and to assess the performance of the system. Following this,argon was added to the system and loss rates were measured. The measure-ment technique is as follows:for each pressure being studied a decay trace atthree different trap depths in the MT is recorded as well as a MOT decaytrace. From each trace a loss rate is extracted and a pressure is inferred fromthe 〈σv〉 calculation. This pressure is compared to the pressure measuredusing an ionization gauge (IG) allowing us to extract the IG calibrationfactor for that gauge for the gas being introduced (argon).4.1 Loss Rate MeasurementFigure 4.1 illustrates how the measurement is executed. Fig. 4.1 shows thecomplete experimental run, while Fig. 4.2 shows a zoomed in view whichcan be examined more carefully. As seen in the figure, atoms are loadedinto the 3D MOT and held with the 2D MOT off so there is no longer anyloading. The trapped atoms are then cooled and optically pumped in thelower hyperfine state F = 1. From here they are transferred to the magnetictrap where they are held for another period of time and exposed to an RFsignal setting the trap depth. Next the atoms are re-captured in the 3DMOT and re-imaged to see how many are left in the trap, after which thetrap is emptied by turning the magnetic field off. Finally a background signalproduced by scattered light is measured and the experiment is repeated.484.1. Loss Rate MeasurementFigure 4.1: MOT fluorescence versus time for a single magnetic trap lifetimemeasurement. Atoms are loaded in to the 3D MOT then transfered to theMT and held for some time during which losses occur, recaptured in theMOT, re-imaged, then emptied out.494.1. Loss Rate MeasurementFigure 4.2: Shows a zoomed in view of the experimental run. In stage(a), the 2D MOT is turned off and a drop in 3D fluorescence is seen. Instage (b) the atoms are cooled and optically pumped for about 100 ms andtransfered to the MT. In the MT, the light is off and the atoms are held forvarious amounts of time. In stage (c), the atoms remaining in the MT arerecaptured in the 3D MOT re-imaged by the trapping light. In stage (d),the MOT is emptied. In stage (e) the MOT light is turned on but with theMOT magnetic field off to obtain a scattered light background reading. Thefraction recaptured is determined by the ratio of the voltage in stage c tostage a.Fig. 4.3 shows the recaptured atoms from the MT as a function of theholdtime in the MT. The fraction is given byf =Vc − VeVa − Ve (4.1)The trap population is given by Eq. 2.6 In the dilute limit (i.e. when intra-504.1. Loss Rate Measurementtrap loss is negligible) Eq. 2.6 reduces to a simple differential equationN˙ = −ΓN (4.2)whose solution isN = N0e−Γt (4.3)The data is fitted to this exponential decay function and Γ is extracted.Figure 4.3: Semilog plot of the number of atoms remaining in the magnetictrap as a function of time. An exponential decrease in the atom numberas a function of time is observed. The lifetime is 2.5s and the backgroundpressure at which the trace was taken is 5.0×10−9 Torr. The background ismostly H2 in this case.A loss rate can also be determined by measuring the decay of the 3DMOT population when the loading from the 2D MOT is turned off. Fig. 4.4shows the decay of fluorescence from the atoms in the 3D MOT. For largeMOTs, the dilute approximation can usually not be applied. Therefore, a514.1. Loss Rate Measurementmore general approach is taken to model the decay, described in Eq. 2.6.N˙ = −ΓN − β∫n2(~r, t)d3~r (4.4)If, as the MOT grows in number, it grows in volume at a constant densitythe the above expression can be rewritten in the constant density regime asN˙ = −ΓN − βnN (4.5)wheren =1N∫n2(~r, t)d3~r (4.6)and the solution asN = N0e−Γeff t (4.7)Here, the effective loss rate is Γeff = Γ + βn. For sufficiently small MOTnumbers, the density is low enough and the loss is dominated by backgroundcollisions. Here the intra-trap two body loss contribution to the decay isnegligible reducing Eq. 4.7 back to the dilute limit. However the loss rate inthe MOT is not the same as the loss rate in the MT even in the dilute limitsince the cross section for loss is different. In the MOT, atoms are bothin the ground state and excited state, and each state has its own uniqueloss cross section. This difference in cross section occurs because the intra-molecular potential for the background gas particle with the Rb atom in theexcited state and the ground state differ. In the case of an excited state Rbatom colliding with an Ar atom, the potential energy surface is anisotropicand the collision cross section depends on the orientation of the Rb atom.524.2. Argon MeasurementFigure 4.4: Decay of the 3D MOT fluorescence as a function of time. Fromthis one can extract the loss rate due to intra-MOT collisions and due col-lisions with the background vapour, by looking at the different regions ofthe curve. The first region (before 50s) is dominated by both loss due tointra-MOT collisions and background gas collisions the second region(after50s) is predominately due to background gas collisions.To determine if the MOT is or is not in the dilute limit, a trace of thedecay is taken as in Fig. 4.4. The figure is examined carefully to establishif the decay can be modelled by a single exponential decay. The tail end ofthe data (50s and longer) is fit to extract the loss rate mainly due to thebackground collisions.4.2 Argon MeasurementThe argon pressure measurement requires a knowledge of the loss cross sec-tion of an argon and rubidium collision which is summarized in Fig. 4.5,showing the loss rate coefficient, 〈σv〉 for multiple trap depths. For deep534.2. Argon Measurementtraps (> 1K) the loss rate coefficient decreases with increasing trap depthbut as the trap depth becomes shallower the loss rate coefficient eventuallylevels off approaching the total loss rate.Figure 4.5: Loss cross section averaged over the velocity of the backgroundgas vs. the trap depth. This illustrates how the loss cross section varies withtrap depth. The three trap depths used are indicated by the filled circles:180µK (RED) 500µK (BLUE) 1mK (GREEN).To extract a pressure from the loss rate measurement the formula beloware used.Γ = Γ0 + ΓAr (4.8)Γ = Γ0 + nAr〈σv〉U (4.9)P = nkBT (4.10)P =Γ− Γ0〈σv〉 kBT (4.11)544.2. Argon MeasurementHere Γ0 is the background loss rate with no argon to the test chamber, kBis Boltzmann’s constant and T is the temperature of the background gas.Fig. 4.6 shows the loss rate as function of the added argon gas for threedifferent trap depths. The three different slopes correspond to the different〈σv〉 values. The trap depth mentioned below is the average trap depth thatthe atoms experience in the magnetic trap, due to the anisotropic nature ofthe MT.Figure 4.6: Plot of the loss rate as function of the NIST ion gauge readingas Ar was leaked into the system for three different trap depths resulting indifferent slopes. Each color corresponds to a different trap depth, 0.52 mK(RED) 1.29 mK(BLUE) 2.88 mK(GREEN). Here the trap depth is set bythe magnetic field gradient and the laser recapture volume. No RF knifewas used.By applying Eq. 4.11 to the data in Fig. 4.6, Fig. 4.7 is generated. Thisplot shows a comparison of the pressures measured by the cold atom lossrate and by a NIST calibrated ionization gauge.From the plot the gaugefactor can be extracted. The gauge factor is the correction factor needed to554.2. Argon Measurementconvert the gauge reading to the actual pressure when gases species otherthan N2 are measured (i.e. the correction factor for N2 is 1) The results aresummarized in the Table 4.1 below :Figure 4.7: Plot of the pressure measured by the cold atoms vs an ionizationgauge. The different lines correspond to the data of different trap depthsshown in Fig. 4.6, and all collapse into a single curve because the 〈σv〉term has been divided out . This line now represents the gas calibrationfactor of the ionization gauge for Ar, and shows that it is independent ofthe measurement trap depth.Trap depth (mK) Gauge Factor0.52 1.22(5)1.29 1.33(3)2.88 1.30(3)Table 4.1: Table of gas calibration factors.564.2. Argon MeasurementFrom the data in Table 4.1 the average gauge correction factor over alltrap depths is 1.29(1), which is in agreement with literature values of 1.3[24].One simple way of improving the measurement is by using an RF knife toset the trap depth. By using the RF knife, the precision of setting the trapdepth is improved and allows for the setting of trap depth to low values wherethe 〈σv〉 term becomes less sensitive to depth. Here we remind the readerthat the trap depth is not isotropic due to gravitational sag and significantlyfewer atoms are transferred to the MT, limiting the SNR. Fig. 4.8 shows thedata taken with the depth set by an RF knife.Figure 4.8: Trap loss rate versus NIST ion gauge reading. Here the trapdepth for the different traces is set using the RF knife. The trap depths hereare 0.18 mK(RED) 0.5 mK(BLUE) 1.0 mK(GREEN).In using the RF signal to set the trap depth, a slightly different set oftrap depths are used. This is because shallower traps are possible and allowfor loss measurements in the regime where 〈σv〉 is closer to the loss rate fora free particle subjected to the argon flux.574.2. Argon MeasurementFigure 4.9: Plot of the pressure measured by the cold atoms vs an ionizationgauge. The different lines correspond to the data of different trap depthsshown in Fig. 4.8, and they all collapse into a single curve because the 〈σv〉term has been divided out. This line now represents the gas calibrationfactor of the ionization gas for Ar, and shows that it is independent of themeasurement trap depth.The table below shows the gas calibration factors determined from thedata in Fig. 4.9 for each trap depth when the RF knife is used.Trap depth (mK) Gauge Factor0.18 1.047(34)0.5 1.073(28)1.00 1.078(26)Table 4.2: Table of gas calibration factors with N2 contamination.Clearly there is a large difference in the gauge factor reported here com-584.2. Argon Measurementpared to those found before in Table 4.1 for the two trials. We discoveredthat this difference can be attributed to a gas contamination introduced bya faulty leak valve. This was discovered when an RGA was used to checkall contaminants. The results are below;Figure 4.10: RGA trace of the contaminants in the setup when leaking inpure argon gas through a faulty leak valve. We observe many contaminantsbeing added to the system. The main contaminants are at 2,20,28,40 amu.These peaks likely correspond to H2, Ne, Ar+, N2, Ar. The black tracewas taken when the leak valve was closed. Therefore all the peaks herecorrespond to gas coming in through the leak valve.Here we see many contaminants effecting our setup specifically contami-nation from hydrogen, neon and nitrogen. The source of this contaminationcan be two-fold. First, the the argon line is made of a tygon tubing whichis porous enough to allow these elements into the vacuum system. Second,the leak-valve used in the experiment had shown weakness in maintainingvacuum previously. To address this issue both the leak valve was replaced594.2. Argon Measurementand the tygon tubing was replaced with a copper one. These changes greatlyremedied the situation as seen in the RGA trace below.Figure 4.11: An RGA trace of the contaminants in the setup when leakingin pure argon gas at different pressures after changing the leak valve andusing copper tubing. One can see the contaminants are negligibly small forAr pressures above 10−8 Torr.In Fig. 4.11 there appears to be a great reduction in the contaminants,giving more confidence that subsequent measurements will be of argon only.As a standard procedure, the RGA traces will be run for all future Ar/gasstudies.60Chapter 5Conclusion5.1 Future WorkThis experimental method can be used to explore other aspects of pressuremeasurements. Pressure measurements of other gases including N2 andthe other noble gases is high on the list of priorities and could provide anindependent verification of their gas calibration factors. In addition, morereactive gases (such as O2) can also be studied. Comparative studies oftrap loss from the MT and MOT could be used to differentiate ground andexcited state collisions and thus shed light on the excited state collisioncross sections. Also, sensor atoms other than Rb could be used. Finally,by using a more dense sensor ensemble (e.g. a BEC) the sensor sensitivitycould potentially be enhanced through an avalanche atom loss process. Inaddition, collisions with a macroscopic quantum mechanical object mightoffer additional opportunities to enhance the sensor sensitivity through amany-body quantum decoherence process.5.2 SummaryUltracold atoms provide a new way of measuring pressure in the ultra-highvacuum range. This thesis explores this idea and the possibility of usingultracold atoms as a new type of pressure standard. The operation of thetwo types of traps used (the MOT and MT) is examined and instructions areprovided for making background collision induced loss rate measurementsin each. We compare the pressure measured by cold atoms to the valuereported by a NIST calibrated ionization gauge and find the gas calibrationfactor for Ar in agreement with the accepted value. Because this methodoffers a calibration free way of determining the pressure of a gas based on afundamental atom-atom or atom-molecule collision process (assumed to bean immutable laws of physics), we believe that it is a good candidate for aprimary standard in the UHV.61Bibliography[1] A.Ashkin. Acceleration and trapping of particles by radiation pressure.Physical Review Letters, 24(3):156–159, 1970.[2] M.H. Anderson, J.R. Ensher, M.R. Mattews, C.E. Wieman, and E.A.Cornell. Observation of bose-einstein condensation in a dilute atomicvapor. Reviews of Modern Physics, 269:198–201, 1995.[3] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimbne. Quantumsimulations with ultracold quantum gases. Nature Physics, 8:39–44,2002.[4] Ying-Chen Chen, Yean-An Liao, Long Hsu, and Ite A. Yu. Simpletechnique for directly and accurately measuring the number of atomsin a magneto-optical trap. Physical Review A, 64:031401–1–031401–4,2001.[5] A. Clairon, P. Laurent, G. Santarelli, S. Ghezali, S. N. Lea, and M. Ba-houra. A cesium fountain frequency standard: Preliminary results.IEEE Transactions on Instrumentation and Measurement, 44:128–131,1995.[6] E.A. Cornell and C.E. Wieman. Noble lecture: Bose-einstein conden-sation in a dilute gas, the first 70 years and some recent experiments.Reviews of Modern Physics, 74:875–893, 2002.[7] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Drunten, D.S. Dur-fee, D.M. Kurn, and W. Ketterle. Bose-einstein condensation in a gasof sodium atoms. Physical Reveiw Letters, 75:3969–3973, 1995.[8] J. Van Dongen, C. Zhu, D. Clement, G. Dufour, J.L. Booth, and K. W.Madison. Trap-depth determination from residual gas collisions. Phys-ical Review A, 84:022708–1–022708–11, 2011.[9] Janelle Van Dongen. Study of background gas collisions in atomic traps.PhD thesis, University of British Columbia, 2014.62Bibliography[10] D. Fagnan. Study of collision cross section of ultra-cold rubidium usinga magneto-optic and pure magnetic trap. PhD thesis, University ofBritish Columbia, 2009.[11] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hnsch,and Immanuel Bloch. Quantum phase transition from a superfluid toa mott insulator in a gas of ultracold atoms. Nature, 415:39–44, 2002.[12] T.W. Hnsch and A.L. Schawlow. Cooling of gases by laser radiation.Optics Communications, 13(1):68–69, 1975.[13] J.L.Picqu and J.L. Vialle. Atomic-beam deflection and broadening byrecoils due to photon absorption or emission. Optics Communications,5(5):402–406, 1972.[14] B.R. Johnson. The multichannel log-derivative method for scatteringcalculations. Journal of computational physics, 13:445–449, 1973.[15] Wolfgang Ketterle. 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