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Determination of the atom’s excited-state fraction in a magneto-optical trap Shen, Yue 2018

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Determination of the Atom’s Excited-stateFraction in a Magneto-optical TrapbyYue ShenA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2018© Yue Shen 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Determination of the Atom’s Excited-state Fraction in a Magneto-optical Trapsubmitted by Yue Shen in partial fulfillment of the requirements forthe degree of Master of Sciencein PhysicsExamining Committee:Kirk Madison, PhysicsSupervisorJames Booth, BCITSupervisory Committee MemberSupervisory Committee MemberAdditional ExaminerAdditional Supervisory Committee Members:Supervisory Committee MemberSupervisory Committee MemberiiAbstractThis thesis introduces an empirical method for determining and controlling theexcited-state fraction of atoms in a magneto-optical trap (MOT), which is essen-tial for the use of cold atoms as a sensor when they are held in a MOT since theirinteractions with other particles and fields are quantum state dependent. A four-level theoretical atomic model was used to describe the transitions of the atoms ina MOT, and the fluorescence emitted from a fixed number of atoms under differentlaser conditions were measured to determine the saturation parameters empirically.Two saturation parameters 푃sat = 1.15 (0.06) mW and 푃r,sat = 2.05 (0.59) mWwere successfully extracted from the model, and the excited-state fraction in thefour-level model was accurately calculated as a function of the MOT trap parame-ters, which ranges from 0.045 to 0.415 for the experimental settings currently avail-able. We also observed minor deviations from the four-level model for the photonscattering rate, and a hypothesis of atom pinning under high powers was proposedto explain the problem. We plan to use this simple excited-state fraction determi-nation method to distinguish the ground and excited state collision cross section ofRubidium atoms with species in residual gas of the vacuum. This is the first step toestablishing atom loss rates from a MOT as an atomic primary pressure standard.iiiLay SummaryThis thesis introduces an empirical method for determining the fraction of atomsleft in an excited state by the cooling and trapping lasers. In most cases, researchershave used the formulas derived from the simple two-level atoms interacting with amonochromatic light, which is not accurate to some extent. Thus we proposed andverified a four-level atomic model for atoms in a magneto-optical traps, predictinga more reliable value of the excited-state fraction. Such type of measurement isessential for the use of cold atoms as a sensor when they are held in a MOT sincetheir interactions with other particles and fields are quantum state dependent.ivPrefaceThis work is based on the experimental apparatus of Erik Frieling’s undergraduateproject, which is described in Section 3.1. The FPGA QDG-Bus driver systemmentioned in this section was built by Wenjun Wu. I have added new magneticcoils to the MOT chamber and made the intensity stabilization system with thehelp of Erik Frieling. The python scripts listed in the appendix were designed byVedangi Pathak and modified by me. All the data were collected and analyzed bymyself. None of the text of the thesis is taken directly from previously published orcollaborative articles.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Magneto-Optical Traps . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Trap Dynamics . . . . . . . . . . . . . . . . . . . . . . . 92.2 Photon Scattering Rate . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Two-Level Model . . . . . . . . . . . . . . . . . . . . . 112.2.2 Four-Level Model . . . . . . . . . . . . . . . . . . . . . 122.3 Excited-State Fraction . . . . . . . . . . . . . . . . . . . . . . . 222.4 Atomic Model Contrast . . . . . . . . . . . . . . . . . . . . . . 233 Apparatus and Procedure of Measurements . . . . . . . . . . . . . 263.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . 263.1.1 Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 MOT Chamber . . . . . . . . . . . . . . . . . . . . . . . 313.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Repump Saturation Determination . . . . . . . . . . . . 33viTable of Contents3.2.2 Photodetector Calibration . . . . . . . . . . . . . . . . . 363.2.3 Fluorescence Measurement . . . . . . . . . . . . . . . . 363.2.4 Atom Number Loss . . . . . . . . . . . . . . . . . . . . 434 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1 G Parameter Measurement . . . . . . . . . . . . . . . . . . . . . 484.1.1 Determining the True Pump Detuning . . . . . . . . . . . 494.2 Measurement of Saturation Parameters . . . . . . . . . . . . . . 524.3 Calculation of the Excited-state Fraction . . . . . . . . . . . . . 584.4 Hypothesis of Atom Pinning for High Laser Power . . . . . . . . 615 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70AppendixA Data and code address . . . . . . . . . . . . . . . . . . . . . . . . . 73viiList of Tables2.1 Standard and test settings of the lasers . . . . . . . . . . . . . . . 173.1 Values of the test pump power . . . . . . . . . . . . . . . . . . . 403.2 Timing and control settings of the experiment . . . . . . . . . . . 424.1 Measurement result . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Slopes and Intercepts from the high-power fitting . . . . . . . . . 64viiiList of Figures1.1 Loss inducing collision cross sections . . . . . . . . . . . . . . . 32.1 MOT diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Quadrupole magnetic field for the MOT . . . . . . . . . . . . . . 72.3 Zeeman effect in a MOT . . . . . . . . . . . . . . . . . . . . . . 82.4 A four-level model diagram . . . . . . . . . . . . . . . . . . . . . 132.5 G vs. pump saturation parameter . . . . . . . . . . . . . . . . . . 192.6 Estimation of the excited-state fraction for different repump satura-tion parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Estimation of the excited-state fraction for different pump satura-tion parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Experimental control diagram . . . . . . . . . . . . . . . . . . . 273.2 Lasers amplification setups . . . . . . . . . . . . . . . . . . . . . 293.3 Intensity stabilization diagram . . . . . . . . . . . . . . . . . . . 303.4 MOT setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Steady-state MOT vs. repump power . . . . . . . . . . . . . . . . 343.6 Fluorescence voltage vs. pump power . . . . . . . . . . . . . . . 353.7 Fluorescence during a typical MOT loading run . . . . . . . . . . 373.8 Fluorescence of the background scattered light . . . . . . . . . . . 383.9 Fluorescence of the trapped atoms . . . . . . . . . . . . . . . . . 393.10 Determine the fluorescence voltage 푉ss,mot . . . . . . . . . . . . . 443.11 MOT fluorescence change under standard settings . . . . . . . . . 453.12 Comparison of 퐺init and 퐺f inal . . . . . . . . . . . . . . . . . . . 474.1 G vs. pump power . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Intercept of G vs. pump detuning . . . . . . . . . . . . . . . . . . 494.3 Detuning corrections . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Empirical and theoretical values of a common scaling factor . . . 514.5 Intercept of G vs. A . . . . . . . . . . . . . . . . . . . . . . . . . 524.6 Intercept of 푏G vs. A . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Normalized slope vs. A/B . . . . . . . . . . . . . . . . . . . . . 54ixList of Figures4.8 Empirical pump saturation power . . . . . . . . . . . . . . . . . . 554.9 푘r vs. the inverse of repump power . . . . . . . . . . . . . . . . . 564.10 Empirical repump saturation power . . . . . . . . . . . . . . . . . 574.11 Determined excited-state fraction vs. repump power . . . . . . . . 594.12 Determined excited-state fraction vs. pump power . . . . . . . . . 604.13 G vs. pump power in a wider range of the pump power . . . . . . 624.14 Residuals of G from the linear fitting . . . . . . . . . . . . . . . . 634.15 Empirical pump saturation power for high pump power . . . . . . 654.16 푘r vs. the inverse of repump power for high pump power . . . . . 665.1 Prediction of the MOT loss rate . . . . . . . . . . . . . . . . . . . 68xAcknowledgementsI am grateful to all those who helped me during my master’s. I especially thank mysupervisors, KirkMadison and James Booth, for their patience, encouragement, andinvaluable supervision to my lab work. I would also like to thank my lab colleaguesPinrui, Erik, Denis, Vedangi, Wenjun, Gene, and Will, they are so friendly andhelpful. Finally, thank you to my beloved parents and friends, who have alwayssupported me and helped me through all the difficulties.xiChapter 1IntroductionLaser cooling has opened several exciting new chapters in atomic, molecular, andoptical (AMO) physics, due to its significant advances in the research areas in-cluding improved spectroscopy, ultracold collisions [1] [2], ultracold molecule for-mation, quantum degenerate gases (Bose and Fermi) [3] [4], atom optics [5], andquantum computation [6]. The idea to use laser radiation to cool and trap atoms wasfirst proposed in 1975 by Wineland and Dehmelt [7] and independently by Hanschand Schawlow [8]. The Doppler effect experienced by the moving atom rendersthe radiation force velocity dependent. This velocity dependence of the absorp-tion process leads to a dissipative force which can cool the atoms down to a fewmicrokelvins [9]. To compensate for the changing Doppler shift as the atoms de-celerated, Zeeman slower was introduced and firstly succeeds in slowing atoms in1982 [10], using a spatially varying magnetic field to tune the atomic levels alongthe beam path. Based on these investigations, the first magneto-optical trap (MOT)employing both optical andmagnetic fields was demonstrated in 1987 [11]. AMOTis an essential technique in the applications of ultracold atoms, including advancesin frequency metrology [12] [13], and the development of commercial cold atominstruments, such as clocks [14]. Due to the high sensitivity of the ultracold atomsto the interactions with their surrounding particles, ultracold atoms are also used insensors, such as gravimeters [15], magnetometers [16], and inertial sensor [17].This project investigates a primary pressure standard using theMOT. It is achievedby measuring the excited state collisions between the trapped atoms and the hotbackground gas particles. The background particles with high velocities run intothe trapping region and collide with the trapped sensor atoms, knocking the sensoratoms out of the trapping region and thus contributing to the atom losses in the trap-ping field. We can measure the atom number in the MOT to devise a new pressurestandard, which only relies on the long-range interaction between the trapped atomand the colliding particle. Pressure can be measured from the loss rate of atomsfrom a trap, Γ:Γ = 푛 < 휎loss푣 > (1.1)where n is the number of the background gas particles, < 휎loss푣 > is the veloc-ity averaged loss collision cross-section between the trapped atoms and the back-ground particles in the vacuum system. In a magnetic trap this loss rate coeffi-1Chapter 1. Introductioncient, < 푠푖푔푚푎loss푣 >, describes collisions between ground state trapped atoms andground state background particles. Using a MOT instead of a magnetic trap hassome advantages. First, the MOT has a much larger trap depth than magnetic traps,meaning that the loss rate coefficient is much lower, allowing higher pressures tobe measured. Second, the MOT is a richer environment as it contains both groundstate and excited state atoms. Taking 87Rb atom vapor as the test object, here Γ canbe expressed asΓ = 푛Rb < 휎푣 >Rb+Rb (1 − 푓e) + 푛Rb < 휎푣 >Rb+Rb∗ 푓e + Γother , (1.2)where 푓e is the excited-state fraction of the atoms in the MOT, and Γother is a con-stant rate due to other losses. The first term and the second term describe the lossrate due to collisions with the trapped atoms in their ground electronic state andtheir excited state, respectively. Therefore, it is necessary to differentiate these twocollisions by measuring the excited-state fraction of the atoms in the MOT.Fig. 1.1 shows the velocity-averaged collision cross section as a function of thetrap depth for Argon hitting Rubidium atoms in their ground state (red circles andblack squares) in a magnetic trap (MT). The prediction of what we would like tomeasure is plotted in the bright pink region, showing the same quantity for back-ground Rb hitting trapped Rb atoms in their ground state (blue squares) and theirexcited state (pink squares). The trap depth of the Rb atoms in the MOT displayedin the bright pink region is a hundred times larger than that in the MT displayed inthe bright green region.One challenge that remains is to determine and control the excited state fraction,which is the main goal of this thesis.Chapter 2 describes the basic theory of the physical apparatus, models, andconcepts used in the experiment. The working principle of the magneto-opticaltrap (MOT) and its trap dynamics are introduced first. Then a two-level model anda revised four-level model are proposed to explain the atomic transitions for 87Rb.The excited-state fraction is deduced and compared for the two models.Chapter 3 contains the experimental apparatus and the experimental procedure.The apparatus includes the controlling system for the measurement, the optical se-tups which provide the MOT lasers, and the MOT. In the procedure section, themeasuring method of the MOT fluorescence is presented.Chapter 4 discusses the result of the experiment, determining the amount ofthe frequency shift for the pump laser and verifying the prediction of the four-levelatomic model. The calculation of the excited-state fraction is also involved in thischapter.Finally, Chapter 5 is a conclusion of the experimental result, which points outthe next steps for establishing an atomic primary pressure standard.2Chapter 1. IntroductionFigure 1.1: A plot of the loss inducing collision cross sections between Ar andtrapped ground-state Rb atoms. This figure is plotted by my supervisor ProfessorKirk Madison. The proposed measurements for background Rb hitting Rb atomsin their ground state and their excited state in the MOT are predicted in the figureand labelled by blue squares and pink squares, respectively.3Chapter 2TheoryThis chapter describes the fundamental theory in the determination of the excited-state fraction of atoms in a magneto-optical trap (MOT). It involves the principlesof the MOT and its trapping dynamics, the estimates of the photon scattering ratewhere a four-level model is constructed to describe the cycling transitions of atoms,and the determination of the atomic excited-state fraction in the MOT.2.1 Magneto-Optical TrapsAmagneto-optical trap (MOT) is an apparatus that uses 3 pairs of counter-propagatinglaser beams crossing at the zero of an applied magnetic quadrupole field to captureand cool atoms to temperatures less than one millikelvin (see Fig. 2.1). The atomsin the MOT are cooled by a velocity-dependent force generated from the mecha-nism of Doppler effect, and trapped by a position-dependent force exploited by theZeeman effect [9].2.1.1 Doppler CoolingDoppler cooling is a technique for laser cooling of small particles, the basic ideais that absorption and subsequent spontaneous emission of photons lead to lightcooling forces, which are velocity-dependent through the Doppler effect.To describe the motion of the atoms in a MOT, the radiative force in the lowintensity limit (퐼 < 퐼sat so that stimulated emission is not important) is considered,the expression of the force from one of the counter-propagating laser beams on theatoms is given by [18]:퐹⃗± = ±(ℏ푘⃗훾2)[푠1 + 푠 + (2훿±∕훾)2], (2.1)where 훾 is the decay rate of the excited state atoms in the MOT, s = 퐼∕퐼sat is asaturation parameter, and 훿± is an effective detuning of the moving atom in the lightfield with wave vector 푘⃗ and detuning 훿 from resonance. If an atom is travelling at a42.1. Magneto-Optical TrapsFigure 2.1: A diagram of the magneto-optical trap. Three orthogonal counter-propagating pairs of laser beams along three orthogonal spatial axes get crossedat the zero of a magnetic quadrupole field generated by a pair of anti-Helmholtzcoils. The laser beams in the radial direction (in the x and y axes) are right cir-cularly polarized (RCP), as the magnetic field opposites the direction of the laserbeams. Also, the vertical beam gets left circularly polarized (LCP), where the mag-netic field points towards the center of the MOT along the z axis. A MOT is gener-ated in the crossing region from this optical molasses configuration with an appliedmagnetic quadrupole field.52.1. Magneto-Optical Trapsvelocity 푣⃗, the detuning 훿± entering in the expression of the force is Doppler shifted:훿± = 훿 ∓ 푘⃗ ⋅ 푣⃗. (2.2)Then the total force of a pair of counter-propagating laser beams can be composedas [18]퐹⃗ = 퐹⃗+ + 퐹⃗−≅ 8ℏ푘2훿푠푣⃗훾[1 + 푠 + (2훿∕훾)2]≡ 훽푣⃗. (2.3)This is a dissipative force proportional to velocity with a damping coefficient 훽. Ifa laser is detuned below the atomic resonance frequency, that is 훿 < 0, the atomsexperience a damping force opposing their velocities. By using three intersectingorthogonal pairs of oppositely directed beams, the movement of atoms in the beamoverlap volume is damped, slowing the atoms and creating an "optical molasses".In principle, the atoms’ velocity should be reduced to zero very quickly, result-ing in a temperature of T = 0 K. However, one should also consider some heatingcaused by the light beams, due to the discrete size of the momentum steps the atomsundergo with each emission or absorption [18]. The competition between this heat-ing with the damping force results in a nonzero kinetic energy in the steady state.The temperature is found by equating the average energy imparted per scatteredphoton to the atom:푇D =ℏ훾2푘B, (2.4)where 푘B is Boltzmann constant and 푇D is called the Doppler temperature [18].For the 퐷2 transition of 87Rb, this temperature is equal to 145 휇K, which gives anestimates of the ensemble energy in our experiment.2.1.2 Magnetic FieldDoppler cooling rapidly slows down the atoms in the intersection volume of thelaser beams, but the cold atoms are not trapped and will eventually diffuse out ofthe cooling region.. In order to trap atoms, a position-dependent force must beintroduced. A quadrupolar magnetic field superimposed on the molasses achievesthis.The magnetic quadrupole field is produced by a pair of coils in anti-Helmholtzconfiguration, where two identical concentric coils are shifted vertically and carry-ing the same current in opposite directions. Fig. 2.2 describes that this magneticfield goes to zero at the center between the coils and increases linearly in magni-tude away from the center. The magnetic field points outwards away from the cen-ter along the radial direction, and points toward the center of the MOT along the62.1. Magneto-Optical TrapsFigure 2.2: A diagram of magnetic field lines generated by a pair of coils in anti-Helmholtz configuration, where the current in the coils is circulating in oppositedirections. The magnetic field is zero at the center between the coils and increaseslinearly in magnitude away from the center, pointing outwards from the center alongthe radial direction and towards the center in the axial direction.72.1. Magneto-Optical Trapsaxial direction. For a spherical quadrupole configuration, the axial field gradient,푑|퐵⃗z|∕푑푧, is twice the radial field gradient, 푑|퐵⃗휌|∕푑휌.Figure 2.3: Principle of a MOT in 1D. The energy of the hyperfine sublevels islinearly shifting with the changing magnetic field because of the Zeeman effect.The 푚F = 0 → 푚F′ = −1 transition (휎−) is closest to the laser frequency labelledby the horizontal dashed line. Therefore, atoms not at the center of the trap prefer toresonance with the 휎− laser beam and are pushed towards the zero of the magneticfield at x = 0.By the optical pumping of slowly moving atoms in this magnetic field, theatoms are confined close to the zero of the magnetic field. A simple 1D schemeof the atomic transitions with three Zeeman components can explain the mecha-nism (see Fig. 2.3), where a laser light with energy ℎ휔l is detuned below the zerofield atomic resonance. When the atoms move in a weak magnetic field 퐵⃗ gener-ated in a MOT, the potential energy they experience for different hyperfine state82.1. Magneto-Optical Trapsmagnetic sub-levels 푚F is푈 = 푔F푚F휇B|퐵⃗|, (2.5)where 푔F is the Landé g-factor, and 휇B is the Bohr magneton. Therefore the excitedstate 푚F′ = +1 is shifted up for 푔F′>0, whereas the state with 푚F′ = -1 is shifteddown, because of the Zeeman shift. In Fig. 2.3, the transition of 푚F = 0 → 푚F′ =−1 (휎−) is tuned closer to the laser frequency than the 푚F = 0 → 푚F′ = +1 (휎+)transition, so the atoms will absorb more light from the 휎− beam. To push the atomstowards the center of the trap where the magnetic field is zero, the polarization ofthe laser beam incident from the same direction as the magnetic field is chosen to be휎+, which is right circularly polarized, and correspondingly 휎− for the other beamcounter propagating with respect to the B-field direction. In the z axis, B-field ispointing towards the center of the trap, the polarization of the laser light should bereversed (that is left circularly polarized), then the laser in z axis incident inwardsthe center also drives the 휎− transitions. As a result, the atoms preferentially absorblight that drives them to the zero of a magnetic field, which is the center of theMOT.2.1.3 Trap DynamicsThe dynamics of MOT loading and loss can be modeled from the rate equation푁̇ = 푅 − Γ푁 − 훽 ∫ 푛2(푟⃗, 푡)푑3푟⃗. (2.6)Here N is the number of atoms in the MOT, t = 0 is the time when both the mag-netic field and the light are turned on and the MOT starts loading, and 푟⃗ = 0 meansthe center of the MOT. The first term in the equation is the loading rate, R, whichdescribes the loading of atoms from the background vapor. It is directly propor-tional to the background atoms’ density and the square of the trap depth [19]. Thetrap losses are described by Γ, the rate constant for losses due to collisions be-tween the trapped atoms and the hot background gases. The third term describestwo body intra-trap losses, 훽 ∫ 푛2(푟⃗, 푡) describes the losses due to radiative escape,fine-structure collisions, hyperfine collisions, and intra-trap collisions, where 훽 isthe rate constant for losses due to inelastic two-body collisions within the trap, and푛(푟⃗, 푡) is the density of the atoms in the trap.K. R. Overstreet et al [20] pointed out that the trapped atoms are confined ina space with constant volume for a MOT with atom number N less than of order105, since the repulsive interactions from light scattering between atoms are weakand negligible in this regime. For a MOT with constant volume, the density of thetrapped atoms is modeled as푛(푟⃗, 푡) = 푛0(푡)푒−( 푟푎 )2, (2.7)92.2. Photon Scattering Ratewhere 푛0(푡) is the peak density of the MOT when 푟⃗ = 0, and a is a constant. Thusthe solution to Eq. (2.6) is [21]푁(푡) = 푁ss(1 − 푒−훾푡1 + 휒푒−훾푡). (2.8)Here 훾 = Γ+2훽푛ss and 휒 = 훽푛ss∕(Γ+ 훽푛ss), 푛ss is the average steady-state densityof the MOT expressed by푛ss =(∫ 푛2푑3푟∫ 푛푑3푟)ss. (2.9)In addition, the steady-state MOT number푁ss should follow푁ss =푅Γ + 훽푛ss. (2.10)In the limit where Γ≫ 훽푛ss, Eq. (2.8) simplifies:푁(푡) = 푅Γ(1 − 푒−Γ푡). (2.11)For larger N, the relative significance of the two-body losses is reduced. Lightscattering leads to a constant mean density of the trapped atoms, 푛, with the MOTgrowing in volume with the increasing atom number. The mean density 푛 is definedas 푛 = (1∕푁) ∫ 푛2(푟⃗, 푡)푑3푟⃗. In the constant density limit, the solution to Eq. (2.6)is [22]푁(푡) = 푅Γ + 훽푛(1 − 푒−훾푡), (2.12)where 훾 = Γ + 훽푛. If Γ≫ 훽푛, this solution has the same form as Eq. (2.11).2.2 Photon Scattering RateThe estimate of the photon scattering rate is quite useful in the calculations of thenumber of atoms in the MOT and the excited-state fraction. In most cases, a stan-dard two-level model is utilized to determine this value, with the saturation intensitycorresponding to the 퐹 = 2 → 퐹 = 3′ pump cycling transition while ignoring thescattering from any light tuned to the 퐹 = 1 → 퐹 = 2′ repump transition. Toproduce a more accurate photon scattering rate, the hyperfine pumping effects areconsidered in a four-level atomic transition model, and an experimental parameterG is measured using the fluorescence emitted from a fixed number of atoms underdifferent conditions.102.2. Photon Scattering Rate2.2.1 Two-Level ModelIn our experiment, a pump beam transfers atoms in the |퐹 ,푚F⟩ = |2, 2⟩ to the|퐹 ′, 푚F′⟩ = |3, 3⟩ state of the D2 (52푆1∕2 → 52푃3∕2) manifold transitions for 87Rb.The steady-state photon scattering rate per atom for a two-level atomic model, 훾sc,can be derived using the density matrix approach as [23]:훾sc =훾2푠1 + 푠 + (2Δ∕훾)2, (2.13)where 훾 is the natural decay rate of the atoms in the excited states, andΔ is the pumplaser detuning, which is the difference between the laser’s optical frequency and theresonance frequency of the F = 2→ F’ = 3 pump transition. Here 푠 = 퐼∕퐼sat , where퐼 is the intensity of the pump laser light experienced by the trapped atoms, and 퐼satis the saturation intensity of the pump transition, which is the intensity needed fora beam to excite the pump transition at a rate equal to one half of its natural linewidth. To investigate the behaviour of the atoms in the experiment, we focus thefluorescent light signal, which is emitted from the cold atoms in the MOT, on aphotodetector. The converted electrical signal, 푉f luo, can be expressed as푉f luo = 훼훾sc푁, (2.14)where 훼 is the photon collection efficiency of the optical system times the photon-to-voltage conversion factor for the detector, and푁 is the number of atoms that emitsphotons. This equation is under the condition that each photon is only scatteredfrom a single atom before leaving the dilute MOT.If we assume that the laser power measured outside theMOT, 푃 , is proportionalto the intensity of the laser light on the MOT, 퐼 , the parameter 푠 in Eq. (2.13) be-comes 푠 = 퐼∕퐼sat = 푃∕푃sat . 푃sat is a strictly experimentally determined parameter,simplifying the measurement procedure. That is, it is impossible to measure 퐼satat the location of the MOT. Thus, we choose to measure P at a convenient locationoutside the vacuum cell and 푃sat the corresponding experimental parameter used toestimate s. Therefore, 푃 and Δ are the parameters that can be controlled preciselyin the experiment. By contrast 훼 and N are difficult to measure precisely, so the ex-perimental method described here removes them from the measurements. With thehypothesis that Eq. (2.14) is a good approximation for the experiment, the MOT isloaded to equilibrium under some pre-determined laser settings (power, detuning),then the settings are quickly switched to a set of "standard" settings. As computedin Eq. (2.14), the theoretical ratio of the steady-state MOT fluorescence at the testparameter settings, V, to the steady-state MOT fluorescence using the "standard"112.2. Photon Scattering Ratesettings, 푉std, is푉std푉=훾stdsc훾sc=푠std푠1 + 푠 + (2Δ∕훾)21 + 푠std + (2Δstd∕훾)2=푃std푃1휉(2)std(퐴 + 푃푃sat),(2.15)where the switching time (<300 휇s) is short compared to the time for N to changeso that N is constant for it to cancel out. The detuning-dependent quantity 퐴 =1+(2Δ∕훾)2 is defined here for convenience. The term 휉(2)std = 퐴std+푠std is a commonscaling factor determined by the standard laser beam settings 푠std and Δstd, whichare constant values as the laser standard settings are fixed.Eliminating the ratio of standard and test powers from the signal ratio in Eq.2.15, one finally obtains:퐺2 =(푃푃std)(푉std푉)= 1휉(2)std(퐴 + 푃푃sat). (2.16)G is an experimental parameter constructed from four easily measured quantities푃 , 푃std, 푉 , and 푉std, which provides a method to determine the pump saturationpower by fitting experimentally determined values of G to the model. For a fixedpump detuning Δ, the relation in Eq. (2.16) from the two-level atom model showsa linear relationship between the empirical parameter G and the pump power P. Thededuced slope 푚(2)G is푚(2)G =1휉(2)std1푃sat, (2.17)and the intercept 푏(2)G is푏(2)G =1휉(2)std퐴. (2.18)Combining theses two quantities, one can determine 푃sat from the two-level model:푃sat =푏(2)G푚(2)G 퐴(2.19)2.2.2 Four-Level ModelA limitation of the two-level model is that it does not describe the fact that thepump light also non-resonantly excites transitions from ground state F = 2 to the122.2. Photon Scattering RateFigure 2.4: A schematic diagram of the four-level model for the atoms of 87Rb. 푛1,푛2, 푛3, and 푛4 represent four levels with increasing energy. Δ is the pump laser de-tuning from the 퐹 = 2→ 퐹 = 3′ pump transition; and Δhf is the energy differencebetween the 퐹 = 2′ and 퐹 = 3′ hyperfine atomic levels.132.2. Photon Scattering Ratehyperfine state F’ = 2. The excited atoms in this state then decay to the F = 1 andF = 2 state. In particular, atoms ending in the F = 1 ground state can not form acycling transition, requiring the addition of a repump laser. Therefore, the effect ofthe repump light has to be added to the model. The repump laser resonantly excitesatoms from the F = 1 to the F’ = 2 state, and finally leads to a decay of the atomsback to the F = 2 state, due to spontaneous emission, at a rate 훾 . The D2 transition(52푆1∕2 → 52푃3∕2) for 87Rb has 훾 = (2휋)6.065(9) MHz, and the energy differencebetween the 퐹 = 2′ and 퐹 = 3′ hyperfine atomic levels, Δhf , is equal to 266.650(9)MHz [24].For this cycling transition, a four-level atom model is constructed to describethe atoms of 87Rb as if they were confined to 4 states. Fig. 2.4 shows the schematicdiagram, where 푛1, 푛2, 푛3, and 푛4 refer to the states 퐹 = 1, 퐹 = 2, 퐹 = 2′, and퐹 = 3′ respectively. Atoms in 푛1, 푛2, 푛3, and 푛4 states can be described by thefollowing couple rate equations:푛̇1 = −Γ13푛1 + (Γ13 + 훾∕2)푛3,푛̇2 = −Γ23푛2 − Γ24푛2 + (Γ23 + 훾∕2)푛3 + (Γ24 + 훾)푛4,푛̇3 = Γ13푛1 + Γ23푛2 − (Γ13 + Γ23 + 훾)푛3,푛̇4 = Γ24푛2 − (Γ24 + 훾)푛4,(2.20)with Γij representing the rate at which atoms are excited from levels 푛i to 푛j. Inequilibrium 푛̇i = 0, and the total atom number푁 = ∑i 푛i is conserved. Solving therate equations in steady state where the derivatives vanish one obtains the steadystate populations of the F’ = 2 (푛3) and F’ = 3 (푛4) atomic hyperfine levels,푛3 =( 2Γ23∕훾1 + 2Γ23∕훾)(푁퐷),푛4 =(Γ24∕훾1 + Γ24∕훾)(푁퐷),(2.21)where D is a quantity defined by퐷 = 1 +( 2Γ23∕훾1 + 2Γ23∕훾)(1 + 4Γ13∕훾2Γ13∕훾)+Γ24∕훾1 + Γ24∕훾. (2.22)The measured voltage of the fluorescence due to the photons emitted by theexcited-state atoms is calculated from Eq. (2.14) as푉 = 훼훾(푛3 + 푛4)= 훼훾( 2Γ23∕훾1 + 2Γ23∕훾+Γ24∕훾1 + Γ24∕훾)(푁퐷).(2.23)142.2. Photon Scattering RateAnd the atom’s transition rate Γij can be expressed from Budker’s book [25]:Γ13 =(훾2)( 퐼r퐼1−2′sat)= 훾2푠r ,Γ24 =(훾2)( 퐼퐼2−3′sat)⎡⎢⎢⎢⎣11 +(2Δ훾)2⎤⎥⎥⎥⎦ =훾2푠퐴,Γ23 =(훾2)( 퐼퐼2−2′sat)⎡⎢⎢⎢⎣11 +(2(Δ+Δhf )훾)2⎤⎥⎥⎥⎦ =훾2휖푠퐵.(2.24)Here 퐼2−3′sat = 퐼sat , 퐼1−2′sat = 퐼r,sat , and 퐼2−2′sat are the saturation intensities for thepump transition (퐹 = 2 → 퐹 = 3′), repump transition (퐹 = 1 → 퐹 = 2′), andthe non-resonant transition (퐹 = 2 → 퐹 = 2′), respectively. The two detuning-dependent quantities 퐴 = 1 + (2Δ∕훾)2 and 퐵 = 1 + [2(Δ + Δhf )∕훾]2 are definedhere for simpler notation. 휖 is assigned to be the ratio of the saturation intensityfor the 퐹 = 2 → 퐹 = 3′ pump transition, 퐼2−3′sat , to the saturation intensity for the퐹 = 2→ 퐹 = 2′ transition, 퐼2−2′sat . It has a value 휖 = 퐼2−3′sat ∕퐼2−2′sat = 3.577∕10.01 =0.3572 in this situation [24].Combining Eq. (2.22), Eq. (2.23), and Eq. (2.24), the fluorescence signaldetected from the photodetector is푉 = 훼훾푁[푠2(퐴 + 푠)] (퐻푊)(2.25)with퐻 = 퐵 + 2휖(퐴 + 푠)퐵 + 휖푠,푊 = 1 +(푘r휖푠퐵 + 휖푠)(퐴 + 푠∕2퐴 + 푠).(2.26)To characterizes the effect of the repump laser on the observed fluorescence, thefactor 푘r is introduced, which provides the method to determine the repump satu-ration power. Its definition is푘r = 2 +퐼r,sat퐼r= 2 +푃r,sat푃r, (2.27)Using the same ratio-metric method as in the two level model, one can construct152.2. Photon Scattering Ratethe four-level empirical parameter G4 as퐺4 =(푃푃std)(푉std푉)=(1퐴std + 푠std)(퐻std푊std)(퐴 + 푠)(푊퐻)= 1휉(2)std(퐻std푊std)(퐴 + 푠)(푊퐻) (2.28)For a two-level model, the factor B goes to infinity, which leads to퐻 → 1 and푊 → 1. Therefore Eq. (2.28) reduces to the two-level parameter퐺2 = (1∕휉(2)std)(퐴+푠).Estimates of Saturation ParametersFor the four-level atomic model, the expansion equation for G4 is complicated:퐺4 =(1퐴std+sstd⋅퐻std푊std)[휖푠2(1 + 푘r∕2) + 푠[퐵 + 휖(1 + 푘r)퐴] + 퐴퐵퐵 + 2휖(퐴 + 푠)]= 1휉(4)std[휖푠2(1 + 푘r∕2) + 푠[퐵 + 휖(1 + 푘r)퐴] + 퐴퐵퐵 + 2휖(퐴 + 푠)] (2.29)Eq. (2.29) simplifies considerably in the case studied in the thesis where thedetuning Δ is much smaller than the hyperfine splitting in the excited state and s isnot huge. It can be motivated by estimating the values of the parameters achievedin the experiment. First, we need to estimate the theoretical values of 푃sat and 푃r,sat .C. Gabbanini 푒푡 푎푙. [26] pointed that one should calculate an average over all thetransitions between the various Zeeman sublevels in the ground and excited statesto include the effects of partial optical pumping of the atoms in the MOT. Thus anaveraged squares of the Clebsch-Gordan coefficients 퐶2 should be used to weightthe pump saturation parameter s in the scattering rate calculations, which is equal to0.46 for 87Rb. It is also stated that there is an uncertainty 훿(퐶2)∕(퐶2) = 25% for acesium MOT [27]. Moreover, the laser beam loses intensity when passing throughthe MOT apparatus so a factor 휂 (90%) is multiplied for the laser intensity after thetransmission; passes through the vapor cell sides also decrease the intensity of theMOT beams, the efficiency at each window is 훽 (90%).Hence a better description of the saturation parameter s is푠 = 퐼퐼sat= 퐶2푎퐼퐼sat. (2.30)162.2. Photon Scattering RateHere 푎 = 휂훽(1 + 훽2)∕2, the terms 훽 and 훽3 are owing to the incoming and theretroreflected beams in the MOT configuration, respectively. The intensity of thelaser beam measured outside the vacuum cell in Eq. (2.30) can be expressed as[28]:퐼 = 2푃휋푤2, (2.31)where w is the 1∕푒2 radius of the beam. Combining Eq. (2.30) and Eq. (2.31), andrecall the assumption that 퐼∕퐼sat = 푃∕푃sat , one can thus deduce the estimation of푃sat from the known 퐼sat value:푃sat =휋푤2퐼sat2퐶2푎(2.32)To calculate this value, the diameter of the beam circle with 95% power trans-mitted through it, 퐷푖, was measured, and the 1∕푒2 beam radius can be computedfrom 푤i = 퐷i∕(2 × 1.224) [28]. For the two horizontal beams paralleled to theMOT table, the diameter of the laser beam is 1.03 cm, and for the beam verticallypassing the MOT cell, the diameter is 0.95 cm. Using 퐼sat = 3.577 mW cm−2[24], the value of 푃sat can be estimated from Eq. (2.32) to be 푃sat = 1.994 mW.Similarly, the estimated value of 푃r,sat is 3.351 mW, knowing that the theoreticalrepump saturation intensity 퐼r,sat = 6.01 mW cm−2 [24]. However, these estimateshave large uncertainties, owing to the uncertainties in the beam widths (5 %), thetransmission losses through the MOT apparatus as well as the glass cells, and theuncertainty of the coefficient 퐶2 (25 %). Consequently, the estimates of the lasers’saturation powers are 푃sat = 1.99 (0.74) mW and 푃r,sat = 3.35 (1.25) mW.Pump Laser Repump LaserPower (mW) Detuning (MHz) Power (mW) Detuning (MHz)Standard Settings 18 -10 0.483 0Test Settings 7 to 28 -6 to -14 0.011 to o.483 0Table 2.1: Standard and test settings applied in the measurement for the lasers.Secondly, we can study the settings used in the experiment. For standard set-tings, the pump laser gives a total power of 18 mW to the MOT (푃std = 18 mW)and is red detuned by 10 MHz from the 퐹 = 2 → 퐹 ′ = 3 pump transition (Δstd =-10 MHz); while the full repump power going into the cell is 0.483 mW (푃r,std =0.483 mW), and is resonant with the repump transition 퐹 = 1 → 퐹 = 2′ (Δr,std =0 MHz). For test settings, the pump power 푃 is varied from 7 mW to 28 mW withthe detuning Δ range from -6 to -14 MHz, and the repump detuning is kept zero.The values of the laser settings are reported in Table. 2.1.Therefore one expects:172.2. Photon Scattering Rate• 5 < 퐴 < 22, 퐴std ≈ 11.• 6900 < 퐵 < 7400, 퐵std ≈ 7200.• 푠 < 100, 푠std ≈ 9.• 푘r = 2 for 푃r →∞, 푘r = ∞ for 푃r → 0. 푘r,std ≈ 9G SimplificationUsing the estimated parameters in section 2.2.3, one obtains that in the denominatorof G4 in Eq. (2.29), 퐵 ≫ 2휖(퐴 + 푠), the denominator [퐵 + 2휖∕(퐴 + 푠)] can beexpanded as1퐵 + 2휖(퐴 + 푠)≈ 1퐵[1 − 2휖(퐴 + 푠)퐵+ 4휖2(퐴 + 푠)2퐵2− 8휖3(퐴 + 푠)3퐵3+…]≈ 1퐵[1 − 2휖퐴퐵− 2휖푠퐵+ 4휖2퐴2퐵2+ 8휖2푠퐴퐵2+ 4휖2푠2퐵2− 8휖3퐴3퐵3−24휖3푠퐴2퐵3− 24휖3푠2퐴퐵3− 8휖3푠3퐵3+…].(2.33)Due the the fact that A∕B ≈ 0.002, we only keep the terms up to order A/B in thefunction, thus the parameter G4 from the four-level model is estimated as follows:퐺4 =(푃푃std)(푉std푉)=(1휉(4)std)⋅[퐴(1 − 2휖퐴퐵)+ 푠(1 + 휖(푘r − 3)퐴퐵)+휖푠2퐵(12(푘r − 2) − 휖(3푘r − 4)퐴퐵)].(2.34)Because of the parameter 퐵 in the denominator of the quadratic coefficient inEq. (2.34), it can be concluded that the term quadratic in 푠2 is much smaller thanthe other two terms. It can also be proved by the linearity in Fig. 2.5, where Gis plotted as a function of s, using the standard values of Δ, 푃r and the estimatedvalues of 푃sat , 푃r,sat . Therefore, the third term in Eq. (2.34) can be neglected fromthe equation. Finally we have the result of the simplified equation of G in a four-level atom model:퐺 =(1휉(4)std)[퐴(1 − 2휖퐴퐵)+ 푃푃sat(1 + 휖(푘r − 3)퐴퐵)]. (2.35)182.2. Photon Scattering RateFigure 2.5: A plot of G as a function of the pump saturation parameter s predictedby Eq. (2.34). The range of s is estimated from the test pump powers and theestimated pump saturation power 푃sat = 1.994 mW. G is composed by applyingthe standard values of Δ, 푃r and the estimated values of 푃sat , 푃r,sat to Eq. (2.34).The solid straight line indicates that G can be discussed as a linear function of s inthis situation.192.2. Photon Scattering RateWhen the test pump laser detuning Δ and the test repump laser power 푃r arefixed, Eq. (2.35) indicates a linear relationship between G and P, with the slope푚(4)G =1휉(4)std1푃sat(1 + 휖(푘r − 3)퐴퐵), (2.36)and the intercept푏(4)G =(퐴휉(4)std)(1 − 2휖퐴퐵)≈ 퐴휉(4)std. (2.37)One can neglect the 2휖퐴∕퐵 term in Eq. (2.37), considering that 2휖퐴∕퐵 ≈ 0.002 <<1, which leads to the same form as 푏(2)G in the two-level model. The intercept valueextracted from plotting 푏(4)G as a function of 퐴 should be zero, and the slope shouldbe the empirical value of 1∕휉(4)std. So far, a dependence of the slope푚(4)G on 푘r shownin Eq. (2.36) is the distinction between the four-level model prediction and the two-level model. 1∕휉(4)std is a scaling factor related to the standard laser beam settings.This value may fluctuate during the measurements due to instability in the laser de-tunings. It is hard to compute its theoretical value because of the unknown 푃sat and푃r,sat . Thus we can eliminate 1∕휉(4)std from the slope of G versus P using its interceptvalue:푌 =푚(4)G 퐴푏(4)G= 1푃sat[1 + 휖(푘r − 3)퐴퐵]. (2.38)This expression shows that at a fixed repump effect ratio 푘r , the normalized slopefor different pump laser detunings should be linear with A/B. Extracting the slope,E, and the intercept, F, from the linear relation, the empirical value for the parameter푃sat is then computed as푃sat =1퐹, (2.39)and the empirical repump saturation power 푃r,sat is푃r,sat = 푃r(푘r − 2) = 푃r( 퐸휖퐹+ 1). (2.40)Dealing With Errors in the Laser DetuningIt is important to use the true pump laser frequency detuning values in the calcula-tion since it is related to the values of Y, E, and F. However, the pump frequencymay shift over the duration of the experiment, due to the drift in the electronics usedto stabilize the laser frequency. So a pump detuning correction term Δ′ is added toΔ to define the true detuning value: Δt = Δ+Δ′. To determine this correction, the202.2. Photon Scattering Rateintercept of G versus P for different pump laser detunings, 푏G, can be expressed asa function of Δ′:푏(4)G =퐴휉(4)std= 1휉(4)std[1 +(2(Δ + Δ′)훾)2]. (2.41)휉(4)std has a complicated expression, its definition is휉(4)std = (퐴std + 푠std)푊std퐻std, (2.42)where the terms 퐻std = [퐵std + 2휖(퐴std + 푠std)]∕(퐵std + 휖푠std), and 푊std = 1 +푘r,std휖푠std(퐴std + 푠std∕2)∕[(퐵std + 휖푠std)(퐴std + 푠std)]. Concerning the relative sizesof the parameters used in the equation as mentioned before, one can obtain the resultthat퐻std∕푊std ≈ 1. Thus 휉(4)std can be expressed by the simple form:휉(4)std ≈ 퐴std + 푠std=4(Δstd + Δ′)2훾2+ 푠std + 1.(2.43)Eq. (2.43) has the same function as the two-level common scaling factor 휉(2)std. Thisis not surprising, since in the case of a two-level model퐻 → 1 and푊 → 1.The intercept value of G versus P is redefined by combining Eq. (2.41) and Eq.(2.43),푏(4)G =4(Δ + Δ′)2 + 훾24(Δstd + Δ′)2 + 훾2(푠std + 1). (2.44)It is noted that the simplified equations Eq. (2.43) and Eq. (2.44) have no depen-dence on 푘r , and are only related to the pump laser. The intercept values 푏(4)G is fittedas a function of the original pump frequency detuningsΔ referring to this equation,while the detuning correction Δ′ and the standard pump saturation parameter 푠satare treated as constant values. The actual detuning values are then calculated andused in the equations (2.38), (2.39), (2.40).On the other hand, the repump laser frequency may also shift during the ex-periment. This leads to a new definition of 푘′r with a parameter 퐴r added in theequation:푘′r = 2 + 퐴r푃r,sat푃r. (2.45)Here퐴r = 1 +(2Δr훾)2, (2.46)212.3. Excited-State Fractionwhere Δr is the detuning of the repump laser frequency from the repump transition퐹 = 1 → 퐹 = 2′. If the repump light gives a resonant excitation, Δr goes to zero,푘′r has the same form as 푘r in Eq. (2.27). As a result, we can only compute the valueof the repump saturation power 푃r,sat multiplied by the repump detuning-dependentparameter 퐴r , since the value of 퐴r is hard to estimate from the measurement.2.3 Excited-State FractionIn many experiments involving magneto-optical traps, it is imperative to know thefraction of atoms left in an excited state by the cooling and trapping lasers. In ourwork, we specifically would like to know the excited state fraction to allow us tomeasure the loss rate of the background atoms due to collisions with the trapped87Rb atoms in ground state and in excited state, which can help using the MOT asan atomic primary pressure standard. In the four-level atom model, the excited-state fraction of the atoms in the MOT can be determined from the ratio of atomnumbers: 푓 (4)e = (푛3 + 푛4)∕푁 . Eq. (2.21) and Eq. (2.24) are combined to get thecomplete form of 푓 (4)e as푓 (4)e =푛3 + 푛4푁= 1퐷( 2Γ23∕훾1 + 2Γ23∕훾+Γ24∕훾1 + Γ24∕훾)=[푠2(퐴 + 푠)] (퐻푊),(2.47)where H and W are defined in Eq. (2.26). Recall that 퐺4 = (1∕휉(4)std)(퐴+ 푠)(푊 ∕퐻)in Eq. (2.28), one can thus express 푓 (4)e as a function of G4 and use the simplifiedform of G4 in Eq. (2.35) to interpret 푓 (4)e . The obtained function is푓 (4)e =푠2휉(4)std퐺4. (2.48)Using the simplified equation of G4 in Eq. (2.35), the excited-state fraction can beapproximated as푓 (4)e ≈푠2[퐴(1 − 2휖퐴퐵)+ 푠(1 + 휖(푘r − 3)퐴퐵)] , (2.49)recalling that 푠 = 퐼∕퐼sat = 푃∕푃sat , and A, B are detuning-dependent parameters.This equation provides a method to determine the excited-state fraction by measur-ing the the powers and detunings of the pump and repump lasers, and it hinges onbeing able to properly measure 푃sat and 푃r,sat .222.4. Atomic Model ContrastFor comparison, in the two-level model B goes to infinity, the function of 푓 (2)egets more simple as푓 (2)e =푠2(퐴 + 푠), (2.50)which only depends on the pump saturation parameter s, and the detuning of thepump laser.2.4 Atomic Model ContrastIn section 2.2, a two-level model and a four-level atomic model are introduced todescribe the atomic transitions of the 87Rb atoms. The derived equations of theexperimental parameter G in Eq. (2.16) and Eq. (2.35) both indicates a linearrelationship between G and the pump power P, with the extracted intercept 푏G =퐴∕휉std. In particular, the four-level model which takes the repump transition intoaccount has a dependence on the repump effect ratio 푘r in the slope of G as 푚(4)G =(1 + 휖(푘r − 3)퐴∕퐵)∕(휉(4)std푃sat), while in the two-level model 푚(2)G = 1∕(휉(2)std푃sat) isa constant. This results to a dependence on 푘r in the excited-state fraction of thetrapped atoms as well.Fig. 2.6 predicts the behavior of 푓e in the two-level model (Eq. 2.50) andthe four-level model (Eq. 2.49), as a function of the repump saturation parameter푠r = 퐼r∕퐼r,sat . The standard pump laser setting is used in the calculation. The two-level excited fraction is a constant in the figure since it is not related to the repumptransition. By contrast, the four-level excited-state fraction has more dependenceon the low 푠r , and goes to be equal to the two-level value as 푠r increases. It is impor-tant to note that this increase in 푓 (4)e corresponds to a repump saturation parametersmaller than 0.1, and may be negligible for large 푠r .For comparison, Fig. 2.7 shows that the excited-state fraction in the two dif-ferent models tends to have different limiting values when s goes to infinity, witha small value of 푠r = 0.003. For 푓 (2)e = 푠∕2(퐴 + 푠), the excited-state fraction ap-proaches to 0.5 as s is much greater than A, while the limit of 푓 (4)e is lower than 0.5due to the repump term 휖(푘r − 3)퐴∕퐵 in Eq. (2.49). The inset figure in Fig. 2.7illustrates the behavior of 푓e in the range of the s value I can obtain, showing that푓 (4)e is 15% lower than 푓 (2)e for s = 30.232.4. Atomic Model ContrastFigure 2.6: Predictions of the atomic excited-state fraction in the two-level model(n) and the four-level model (l) for different repump saturation parameter 푠r =퐼r∕퐼r,sat . The calculation is based on the standard pump laser settings, 푃std = 18mW and Δ = -10 MHz. A dependence of 푓 (4)e on 푠r is shown in the figure, while푓 (2)e is a constant.242.4. Atomic Model ContrastFigure 2.7: Predictions of the atomic excited-state fraction in the two-level model(n) and the four-levelmodel (l) for different pump saturation parameter 푠 = 퐼∕퐼sat .In this estimation 푠r = 0.003,Δ= -10MHz. 푓 (2)e is higher than 푓 (4)e in this situation,and these two quantities go to different limiting values when s goes to infinity.25Chapter 3Apparatus and Procedure ofMeasurementsThis chapter contains two sections: the experimental apparatus and the experimen-tal procedures. The apparatus section describes the FPGA controller, the opticalsetup which produces the pump and repump light to the MOT table, and the MOTchamber and its surrounding components arranged to trap atoms. In the experimen-tal procedure, the preparation of the optics and electronic devices and the methodof measuring and fitting the fluorescence from the MOT are introduced.3.1 Experimental ApparatusThe MOT apparatus is composed of three systems: the optical setup which pro-duces and delivers the laser light, the MOT vacuum arranged to generate a trap,and the FPGA controller connected to a computer used to control the laser fre-quencies, intensities, magnetic field, and event sequence timing. The first two partsof the experimental apparatus are described in details in sections 3.1.2 and 3.1.3,respectively.Fig. 3.1 shows an overview of the apparatus and the command and controlrelationship between the devices. The optical setup amplifies narrow linewidth lightfrom the master table, and controls the laser intensities as well as detunings byacousto-optic modulators (AOMs) from IntraAction (model ATD-801A2 for pumplaser, model ATM-901A2 for repump laser). The direct digital synthesizers (DDS)generate the RF signals that are used to drive the AOMs. A shutter for each lasercontrols the access of the light to theAOM. The pump and repump laser beams, eachwith a specific intensity and a frequency, then get combined in beam splitters andare sent to the MOT in three dimensions. Next, a magneto-optical trap of rubidiumatoms is generated with a magnetic field produced by a pair of anti-Helmholtz coils.The background rubidium vapour is released from the rubidium dispenser.The system is controlled using custom built python scripts which translate theinstructions into time-ordered sequences of events run by the FPGA (Terasic, modelDE1-SoC). the timing of the sequency for each experimental run is accurate to263.1. Experimental ApparatusFigure 3.1: Block diagram of an overview of the experimental apparatus. To gen-erate a magneto-optical trap, narrow linewidth pump and repump light from themaster table is injected into separate slave laser amplifiers. The slave laser outputbeams’ intensities and frequencies are controlled by AOMs. A computer talks to theFPGA controller which sets the input signal of the DDS, the current in the magneticfield coils, and the opening/closing of two mechanical shutters. The DDS gener-ates the RF signals that are used to drive the AOMs. Rubidium atoms released fromthe dispenser thus get trapped in the overlapping region of the laser beams and theemitted fluorescence is measured from a photodetector connected to the computervia an ADC.273.1. Experimental Apparatuswithin 1 휇s. The AOM settings, the driving current of the magnetic coils, and themechanical shutters are preset in the python scripts and managed by the FPGA con-troller as well. A photodetector collects the fluorescence from the MOT and sendsthe voltage signal to the computer through an analog to digital converter (ADC).3.1.1 Optical SetupThe optical setup of the experiment is illustrated in Fig. 3.2. The seeding pumpand repump laser light with narrow linewidth is generated on a separate mastertable where it is then sent through fibers to the optical table for the experiment.The light is then coupled to the slave laser through an isolator and get amplified toaround 60 mW. The laser diodes used in the slaves are from Thorlab, part numberL785P090. A mechanical shutter controls the access of the light to an optical beamsplitter, where a small reflection of the light is coupled into a fiber as a diagnosticsignal to check the injection of the master light on an oscilloscope. The rest of thelight is transmitted through the beam splitter into a double-pass AOM setup.In the double-pass AOM setup, the intensity and frequency of the light canbe changed to certain values, which is achieved by an acousto-optic modulator(AOM), and the polarization direction of the light is shifted by a quarter waveplate.Light from the beam splitter passes through the AOM, after which the first-orderdiffracted light deviates from the zeroth-order light with an angle. A plano-convexlens of focal length f = 150 mm placed after the AOM ensures that the passedfirst-order diffracted light is parallel to the zeroth order. The zeroth order beam isblocked while the first order gets reflected back from a planar mirror in the originalpath. From these two passes through the AOM, the output frequency is increased bytwice the AOM frequency. Since the light from the master table is detuned 180MHzbelow the pump or repump transition, the RF frequency driving the AOM, 휈AOM,should be set as휈AOM =180 + Δ2, (3.1)where Δ is the desired detuning of the laser. The intensity of the light depends onthe amplitude of the RF driving signal sent to the AOM, and the RF signals aregenerated by direct digital synthesizers (DDS) and amplified before going to theAOMs. The amplitude and frequency of the RF signal operating the AOM, as wellas the status of the mechanical shutter, are set via the FPGA controller. Moreover,a quarter waveplate placed between the convex lens and the planar mirror shifts thepolarization direction of the light as it double passes the waveplate, and blocks thezeroth-order diffracted light. Therefore, the diffracted beam returned through theAOM is now reflected from the polarizing beam splitter and gets magnified by twolenses. The total power of the laser is measured behind the concave lens, which283.1. Experimental ApparatusFigure 3.2: A schematic diagram of the pump and repump laser amplification setups[29]. Each slave laser is seeded by a narrow linewidth light from the master tablethrough an isolator. The laser beam then gets through the shutter and is split by abeam splitter, the reflection from which is coupled into a fiber as a diagnostic lightto check the injection. Next, the transmitted light is sent to a double-pass AOMsetup, where the intensity of the first-order diffracted light is controlled by settingthe AOM amplitude, and its frequency is increased by twice the AOM frequency.A quarter waveplate shifts the polarization direction of the light as it double passesthe waveplate, and blocks the zeroth-order diffracted light. Consequently, the beamreturned through the AOM is reflected from the beam splitter and gets magnifiedby two lenses. The combination of these two beams with specific frequencies andintensities is sent to the MOT to trap atoms. The total powers in the pump andrepump beams before entering the MOT devices were measured by a power meter,whose sensor is conveniently placed at positions ¬ (pump) and ­ (repump) in thefigure.293.1. Experimental ApparatusFigure 3.3: Block diagram of the intensity stabilization system of the pump laser.The feedback in the loop is from a small reflection of the pump laser detected by aphotodetector. The PID controller reads the error signal from the differential ampli-fier and accordingly outputs a control voltage to the RF attenuator, which controlsthe RF signal from the DDS and thus pushes the laser intensity towards stability.303.1. Experimental Apparatusis a convenient location to place the sensor of the power meter (Coherent, modelLabMax T0).The intensity of the pump laser light is stabilized using a feedback loop consist-ing of a photodetector, a differential amplifier, a proportionalâĂŞintegralâĂŞderiva-tive (PID) controller, and a RF attenuator (see Fig. 3.3). The pump beam afterthe double-pass AOM arrangement is reflected by a piece of glass and sent to aphotodetector. The output voltage of the photodetector, as well as the preset refer-ence voltage from the FPGA controller, are compared in the differential amplifier,providing an error signal to the PID controller. The PID controller then applies acorrection to its control function according to the input signal and controls the RFdriving signal by a RF attenuator. Therefore, the controller continuously reads thefeedback and corrects the control function, until the detected light voltage is equalto the reference voltage so the laser intensity is stabilized.3.1.2 MOT ChamberTo generate a magneto-optical trap, a MOT system was set up, which consists ofa rubidium vacuum cell, MOT laser beams, magnetic coils, and a photodetector(see Fig. 3.4). The output light of the pump and repump lasers from the opti-cal setup in Fig. 3.2 is combined and sent to the MOT table in three dimensions,where the pump beam is evenly distributed in all three perpendicular directionswith the powers of 푃x = 3.72, 푃y = 3.50, 푃z = 3.26 mW. The repump beam isonly sent along the two horizontal directions with the powers of 푃r,x = 2.60 and푃r,y = 2.90 mW. The combination and the power distribution of the light are ac-complished by using optical beam splitters and half waveplates. After that, the lin-early polarized beam passes through a quarter waveplate, which converts the lightinto circularly polarized light. The light in each arm is then transmitted throughthe glass cell, and passes through a second quarter waveplate located in front ofa retro-reflection mirror, completing each beam arm. From this configuration, theretroreflected light has the same circular polarization as the incoming beam, thuscompletes a pair of counter-propagating laser beam. Consequently, three orthogo-nal counter-propagating pairs of laser beams along three orthogonal spatial axes getcrossed at the 1 cm length vapor cell, trapping atoms with the help of a magneticfield produced by the coils, as shown in Fig. 3.4 (a).The background rubidium vapor is released by energizing the rubidium dis-penser. It is typically loaded for 3 min at a current of 5.3 A, around once a week.After energizing the dispenser, the rubidium vapor density decreases for one daybefore coming to equilibrium when the test data can be acquired. An optical col-lection system consisting of a convex lens of focal length f = 35 mm set towards thevacuum cell, an optical iris, and a photodetector, collects the fluorescence through313.1. Experimental ApparatusFigure 3.4: Diagrams of the MOT setups. (a) is a schematic of the MOT systemwhich contains a vapor cell, MOT laser beams, mirrors, quarter waveplates, mag-netic coils, and a photodetector. The laser light shown as red lines is sent from theoptical setups in Fig. 3.2 and crossed in the cell. The two laser beams in x andy directions are mixture of the pump and repump light, while the vertical beam iscoming from the pump light. Two quarter waveplates and a mirror for each opti-cal path enable the same circular polarization for the incoming and retroreflectedbeams. An optical collection system collects the fluorescence through the end win-dow of the vacuum cell. An overview of the MOT system can be seen in a picture(b).323.2. Experimental Procedurethe end window of the vacuum cell. This system collects over a solid angle of 0.071sr and passes through an aperture of 6 mm to limit the scattered light transmittedto the photodiode. A CCD camera is used to observe the status of the MOT in thecell. A picture of the MOT system is displayed in Fig. 3.4 (b).3.2 Experimental ProcedureIn the experimental procedure, the background scattered light for different laser set-tings are firstly recorded, then the atoms in the cell are loaded into a MOT, and thefluorescence due to the trapped atoms under different laser frequency and ampli-tude is measured. Care was taken to insure the reproducibility and accuracy of themeasurements. Section 3.2.1 tells the determination of the useful repump powerrange, and section 3.2.2 introduces the calibration of the photodetector. The detailsof the measuring steps, as well as the recording of the fluorescence, are shown insection Repump Saturation DeterminationThe goal of the experiments is to verify the predictions of the four-level atomicmodel. In particular, we want to extract the empirical parameters, 푃sat , 푃r,sat , anduse these to estimate the fraction of atoms in electronic states as a function of theMOT trap parameters. Pump transition plays a lead role in the cycling transition ofatoms, while repump light is used to pump the atoms in the F = 1 ground state backinto the cycling transition, which is an occasion with small possibility. Therefore,recognizing the useful range of the repump laser power is essential in the experi-ment. Before the measurement of G parameter, the fluorescence voltage due to thesteady-state MOT is plotted as a function of different repump powers. For thesemeasurements the pump laser parameters were fixed at P = 18 mW, Δ = -10 MHz,Δr = 0 MHz, and B-field current = 0.5 A (field gradient to 12 G/cm). The resultsare given in Fig. 3.5. The steady-state MOT voltage increases rapidly when therepump power increases roughly from 0.2 mW to 0.5 mW, then it slows down andgets saturated as the repump power is greater than 0.8 mW. This implies that only푃r ≈ 0.5 mW is required for the repump light to saturate the cycling of atoms inthe dark state in the MOT. Consequently, the repump laser power is set lower than0.5 mW in the next experiment to observe the atom’s dependence on the repumppower.333.2. Experimental ProcedureFigure 3.5: A plot of the voltage of steady-state MOT versus the repump laserpower. The fluorescence voltage of the steady-state MOT stops its rapid growthafter the repump power reaches 0.5 mW; therefore the repump power should belower than 0.5 mW in the next experiment.343.2. Experimental ProcedureFigure 3.6: A plot of the voltage of the scattered light signal due to the pump laserversus the corresponding pump power when Δ = -10 MHz. There is a linear rela-tionship between the scattered light signal voltage 푉pump and the power 푃 . A simpleequation 푉f luo = (22.80 (0.09) P) mV is fitted to show the association.353.2. Experimental Procedure3.2.2 Photodetector CalibrationThe voltage signal due to the trapped atoms, 푉 , is equal to the fluorescence of thesteady-state MOT, 푉ss,mot , minus the baseline signal of the scattered light from thelasers, 푉ss,zero:푉 = 푉ss,mot − 푉ss,zero (3.2)for each setting. Accordingly, the voltage of the pump laser power is computedby subtracting the background light 푉off from the pump laser scattering into thephotodiode 푉pump on:푉pump = 푉pump on − 푉off . (3.3)The fluorescence from the MOT, which is our primary measurement quantityin the MOT experiment, is collected by focusing the light on a photodetector witha 1 cm × 1 cm sensor. Thus it is crucial to make sure that the reading of the pho-todetector is correct and reasonable. To test and calibrate the photodetector, thetotal power in the pump beam before entering the vapor cell was measured by acalibrated power meter as a function of the AOM amplitude setting, and comparedto the photodetector readings. The power meter was placed at a location before thelaser beams are split and sent to the MOT table, labelled in Fig. 3.2.Meanwhile, the voltages generated by pump light scattering corresponding tothe measured pump powers were recorded in the computer. The background lightwas subtracted from the pump laser scattering into the photodiode according to Eq.(3.3). We expect that the collected fluorescence of the pump laser is linearly re-lated to the pump power measured by the power meter, and Fig. 3.6 verifies thisprediction with a negligible intercept. Hence it is adequate to use the photodiodevoltages 푉 testpump∕푉 stdpump instead of measuring the actual optical powers 푃∕푃std to de-termine the empirical parameter G (once the calibration of P to V is known). Themeasured scattered light signal voltage 푉pump can be converted to powers 푃 as wellwhen concerning the relationship between G and P in Eq. (2.35).3.2.3 Fluorescence MeasurementFrom Eq. (2.34), one can conclude that the saturation power parameter, 푃sat , isquantified by setting arbitrary laser parameters and loading the MOT to equilib-rium, then quickly switching the settings to a test set of values to read. In the prepa-ration of the measurement, the slave lasers were well injected, and the master laserswere locked at the correct frequencies. The double-pass AOMs were well alignedso that the laser beams on the MOT table would not move when changing the laserdetunings from -30 MHz to 30 MHz. Particular attention was paid when align-ing the laser beams to generate a MOT under various laser settings. The spatial363.2. Experimental ProcedureFigure 3.7: A plot of the voltage signal collected from the photodetector versus time.In this figure, Δ = -8 MHz, 푃r = 0.009 mW, and the pump test power P was variedfrom 7 mW to 28 mW. Section A and section B labelled in the figure represent thebackground scattered light and the fluorescence signal due to the trapped atoms,respectively. In section A, step 2 is the signal of the background light while thelasers and magnetic field were all off, step 5 shows the scattered light of the pumplaser at the standard settings. In step 6 pump and repump lasers were all turned onunder the standard settings, then the magnetic field was turned on at 0.5 A to trapatoms in step 7. Measurements made in a short amount of time, which are in thesections A and B in the figure, are plotted in Fig. 3.8 and Fig. 3.9 respectively formore unobstructed view.373.2. Experimental ProcedureFigure 3.8: A plot of the details of section A in Fig. 3.7. The three signal levels aredifferent voltages of the pump scattered light with different pump powers when thepump detuning was set to -8 MHz.383.2. Experimental ProcedureFigure 3.9: A plot of the details of section B in Fig. 3.7. The main figure dis-plays the fluorescence voltage of the MOT when the power and detuning of thepump laser, as well as the repump power, were switched back and forth between13 different "test" values and 1 set of "standard" values, the procedure of which isdescribed in step 8 in the measurement method. The inset figure above the mainplot shows a specific fluorescence change from the MOT when the laser settingswere turned from "standard" (푃std = 18 mW, Δstd = -10 MHz, 푃r,std = 0.483 mW,labelled by blue color) to "test" (P = 13 mW,Δ = -8MHz, 푃r = 0.009 mW, labelledby green color), then back to "standard" (blue). During this period, the fluorescentvoltage before and after the quick laser condition change generates two experimen-tal parameters: 퐺init (switched from "standard" to "test") and 퐺f inal (switched from"test" to "standard"), labelled by the red circles in the inset figure. The fitting de-tails are explained in Fig. 3.10. Moreover, one can see a decaying trend in the MOTvoltage under test settings (green), which indicates an atom number loss due to thechanged trapping conditions.393.2. Experimental Procedureposition shift of the MOT for different settings was minimized to avoid variationsin the atom number due to changes in the laser parameters. Also, a dense MOTwas avoided as it could induce multiple scattering of photons, which reduces thedetected fluorescence signal from the MOT.In our work, the procedure of a loading run to measure the empirical parameterG with a typical test repump power is listed as follows:1. Values of Δstd = -10 MHz, 푃std = 18.0 mW, and 푃r = 0.483 mW are selectedas "standard" for these experiments. Choose a new detuning and 13 differentvalues of power for the pump laser as "test" settings (see Table. 3.1), whilethe repump laser also has a new power.2. Measure the background level 푉off when the lasers and magnetic field are off.3. Turn on the pump laser with test frequency and different test powers, whilethe repump laser is still off, and get 푉 testpump on.4. Turn on the lasers under various test settings to record the test baselines,푉 testss,zero.5. Turn off the repump laser, switch the pump laser under standard settings, andrecord the scattered light, 푉 stdpump on.6. Turn on the two lasers at standard settings and measure the standard baseline푉 stdss,zero.7. Set the driving current of the magnetic coils to 0.5 A, fill the MOT to equi-librium in 40 s, and record the fluorescence 푉 stdss,mot .8. Quickly switch the lasers to one set of test settings for 20 ms, then switchthem back to the standard settings for another 20 ms. Record the fluorescentsignals in the period to determine 푉 testss,mot and 푉 stdss,mot .9. Repeat step 8 for different test powers of the pump laser.10. Repeat step 8 and step 9 for 5 times in total, each time with random test pumppowers permutations, in order to increase the accuracy of the data.Test Pump Power (mW)3.79 5.60 7.77 9.78 11.96 14.19 16.00 18.10 20.36 22.29 24.31 26.32 28.22Table 3.1: 13 different values of the pump laser power as "test" settings.403.2. Experimental ProcedureThe timing of each step in the experiment and the corresponding control settingsare listed in Table. 3.2. One significant advantage of this measuring method ofquickly switching the laser settings back and forth between different "test" settingsand one "standard" settings is that it reduces the duration of each experiment. Onecan notice from Table. 3.2 that it only takes 48 seconds for such a loading run, so allthe measurement is expected to take less than two hours, which limits the shift in thelaser power and frequency, the room temperature, and other optical and electronicdevices used in the experiment, and reduces any effects related to variations in thebackground Rb vapour pressure.Fig. 3.7 shows the recording of the fluorescence voltage during a typical loadingrun. Measurements made in a short amount of time, which are in the sections Aand B in the figure, are plotted in Fig. 3.8 and Fig. 3.9 respectively. The red circlesin Fig. 3.9 represents two sets of voltages 푉 stdss,mot and 푉 testss,mot , which can be usedto calculate the experimental parameter G. Therefore, one 퐺init and one 퐺f inal aregenerated for each test MOT interval. We can collect 5 sets of 퐺init and 5 sets of퐺f inal from Fig. 3.7 in one loading run, and the average of these 10 data sets givesa reliable final result of the parameters G.413.2.ExperimentalProcedureBackground Test Pump Power Test Baseline Standard Pump Power Standard Baseline Loading Standard MOT Measuring 푉 testss,mot and 푉 stdss,motTime (s) 0.5 0.26 0.26 2 2 40 2.6Pump Shutter off onPump Laser off on at "test" on at "standard" switch between "test" and "standard"Repump Shutter off on off onRepump Laser off on at "test" off on at "standard" switch between "test" and "standard"B-field off on at 0.5 ATable 3.2: A table of the timing and control settings in the experiment. Pump and repump shutters were closed when turningoff the AOMs to avoid the leakage of the zeroth-order diffracted light to the MOT.423.2. Experimental ProcedureFluorescence FittingTo precisely determine the value of G, the measured signal voltage was recordedin the computer and analyzed in python scripts. The voltage of the backgroundscattered light, 푉off , 푉pump on, and 푉ss,zero, were extracted by taking the averagesfrom their corresponding light signal voltages. Fitting the MOT fluorescent volt-age is more complicated since the voltage reading we need is strict with time. Fig.3.10 shows a specific fluorescence change when the laser settings were turned from"standard" to "test". To ensure that the atom number in the MOT was the same forthe "standard" and the "test" measurements, 푉 testss,mot should be taken just after theswitch. Therefore, the time for the experimental system to read the script instruc-tions, change the laser settings, and record the voltage signal was determined in thefirst subplot in Fig. 3.10, where the scattered light voltage due to the pump laserchanged from 푉 stdpump on to 푉 testpump on in 300 휇s (3 points). This delay labelled in thelight blue region was then used in the second subplot to calculate the start point ofthe fluorescence due to the trapped atoms under test settings, eliminating the effectfrom the system delay on the fluorescence voltage change.Based on the assumption that the fluorescence measured in the photodiode isproportional to the atom number in the trap: V∝N, and the atom number dynamicsdescribed in Eq. (2.11), the fluorescence due to the trapped atoms, V, is fitted byan exponential model푉 = 푉0 + 퐴(1 − 푒−Γ푡), (3.4)where 푉0 is the initial voltage in the fluorescence measurement interval. A is apositive constant for a loading MOT and turns negative when the MOT is decaying.The red curve in Fig. 3.10 fits the decaying fluorescence according to this equation.Assuming that the fluorescence change during the system delayed time (300 휇s) isnegligible, the start point and the end point of the fitting curve were then used toobtain the best estimate of 푉 testss,mot , and the standard fluorescence signal 푉 stdss,mot wasmeasured in the same way.3.2.4 Atom Number LossOur experiment requires a fixed atom number in the MOTwhen changing illumina-tion conditions from the pump and repump lasers. However, in the zoomed figurein Fig. 3.9 a decaying trend is illustrated from the MOT voltage under test settings(green points). The fluorescence from trapped atoms for a longer time is given inFig. 3.11, where the laser settings were switched back and forth between different"test" values and one set of "standard" values. It is clearly shown that the MOT flu-orescence under standard settings is decreasing in (a) from the continuous changesof the laser settings, and it roughly keeps constant in (b). The only difference in433.2. Experimental ProcedureFigure 3.10: A plot of the scattered light voltage due to the pump laser and thefluorescence voltage of the MOT when the laser settings were changed from "stan-dard" to "test". The time for the scattered light voltage to change between differentsettings (in the light blue region) was used in the fluorescence voltage change to de-termine the start point of the test fluorescence measurement interval. The intervalwas then fitted by a red curve according to Eq. (3.2.3) to obtain 푉ss,mot .443.2. Experimental ProcedureFigure 3.11: A plot of the fluorescence due to the trapped atoms under differentillumination conditions. Figure (a) is an extension of Fig. 3.9, where the MOTfluorescence under standard settings is decreasing over time. In figure (b) the testrepump power is changed from 0.009 mW to 0.483 mWwhile the standard repumppower is the same as in figure (a), and the fluorescence of the standard MOT keepsconstant. The comparison of these two figures points that our assumption of con-stant atom number is valid for the applied pump detunings, but not entirely practicalfor a small repump power.453.2. Experimental Procedurethe laser settings between figures (a) and (b) is the test repump power, as figure (a)has a test repump power of 0.009 mW, and in figure (b) it is equal to the standardrepump power 0.483 mW. Therefore, one can say that the atom number loss duringthe experiment is due to the changed atom loading rate and loss rate when turningto the settings with a small repump power, which leads to a lower equilibrium atomnumber in that measurement interval.To test the influence on the measurement results from this phenomenon, twodata sets 퐺init and 퐺f inal generated from Fig. 3.9 are compared in Fig. 3.12 bycalculating the ratio (퐺f inal −퐺init)∕퐺init . Recalling that 퐺init is measured from thefluorescence voltages of the trapped atoms when the laser settings were switchedfrom "standard" to "test", and퐺f inal is measured when switching from "test" back to"standard". If the atom number loss we observed in Fig. 3.12 is negligible in eachfluorescence measurement interval, 퐺init and 퐺f inal should have approximately thesame values. The ratio in Fig. 3.12, as we expected, fluctuates around the zero level,showing that the atom number loss during the MOT fluorescence measurement didnot influence the experimental results.463.2. Experimental ProcedureFigure 3.12: A plot of the ratio (퐺f inal − 퐺init)∕퐺init for different test setting pumppower P from Fig. 3.9. 퐺init and 퐺f inal were measured at the start point and the endpoint of the fluorescence intervals under test settings, respectively. The ratio of Gfluctuates around the zero level, which is labelled by the horizontal line, showingthat 퐺init and 퐺f inal are approximately equal and are not influenced by the atomnumber loss.47Chapter 4Results4.1 G Parameter MeasurementFigure 4.1: A plot of G versus the test setting pump power P. Each set of data isfitted to a line and corresponds to pump laser detunings of Δ∕2휋 = -6 (l), -7 (n),-8 (⬟), -9 (H), -10 (u), -11 (©), -12 (:), -13 (6), and -14 (t) MHz. The testrepump power is held constant at 0.009 mW with no detuning.In our work, measurements of the experimental parameter G were taken fordifferent repump powers: 0.009, 0.011, 0.013, 0.017, 0.023, 0.029, 0.037, 0.073,and 0.483 mW, the order of which is chronological. For the repump power 푃r =0.009 mW, G is plotted as a function of the pump laser power 푃 for each pump laser484.1. G Parameter Measurementdetuning Δ in Fig. 4.1. The prediction from Eq. (2.35) and Fig. 2.5 give that G is alinear function of the test pump power, and its intercept value in Eq. (2.37) shouldincrease with the increase of the pump detuning. The measured result shows thatG values follow the predictions.4.1.1 Determining the True Pump DetuningFigure 4.2: A plot of 푏G as a function of the pump laser detuningΔ. The solid curvefits the intercept values using Eq. (2.44). The derived detuning correction value Δ′is utilized to modify the detuning value, and the standard saturation parameter value푠sat can be evidence of the accuracy of the detuning correction method, as discussedlater in Fig. 4.4.The expression of the intercept 푏(4)G derived from Fig. 4.3 simplifies in Eq.(2.44) as 푏G = [4(Δ+Δ′)2 + 훾2]∕[4(Δstd +Δ′)2 + 훾2(푠2std +1)], which is a functionof Δ with unknown parameters Δ′ and 푠std. Therefore, the detuning error for thepump laser, Δ′, can be determined by plotting the intercept 푏(4)G as a function of theoriginal pump laser detuning Δ, as shown in Fig. 4.2. The fitting curve accordingto Eq. (2.44) fits well with the intercept values, a detuning correction Δ′, as well494.1. G Parameter MeasurementFigure 4.3: A plot of the detuning correction valuesΔ′ for different repump powersas a function of the measured time. The test repump powers corresponding to themeasured detuning corrections are labelled in the figure. The solid straight linelinearly fits Δ′ by the equation Δ′∕2휋 = -0.0467 (0.0023) t + 1.83 (0.10) MHz,which predicts the accurate detuning value for each G versus P data set. It showsobviously that the pump laser frequency shifts linearly with time, around 3.8 MHzin 80 minutes after the laser frequency was locked to the pump transition.as a standard pump saturation parameter 푠sat , are hereby determined. For each testrepump power, one can collect a set of 푏G like this one, and the extracted Δ′ isplotted in Fig. 4.3, in the order of the data taken time. Fig. 4.3 illustrates a linearrelationship between the detuning correction value Δ′ and time, which indicatesthat the pump laser frequency is shifting with time, with a rate of 0.0467(0.0023)MHz/min. Though the test repump powers for the acquired detuning correctionsare different, there is no evidence that the pump frequency shift is related to therepump power. Considering that the time for one loading run with different testpump powers and a constant test pump detuning is 48 s, the corresponded pumplaser frequency shift is 0.037 MHz, which makes small effects on the trapped atomnumbers. Moreover, the frequency shift during the time of measuring all the Gvalues in Fig. 4.1 with a fixed repump power is 0.34 MHz, it is a tiny change504.1. G Parameter Measurementthat we do not need to worry about except when calculating correct values of Aand B. To produce light of stable frequency to the MOT setups, the laser is lockedto a frequency corresponding to the sharp edge of a transition peak in the errorsignal resulting from the saturated absorption spectrum. The phenomenon of thefrequency shift tells that the pump frequency is gradually pushed away from thelocking point due to the drift in the electronics. Nevertheless, based on the fittingresult in Fig. 4.3, one can compose the actual detuning value Δt for each G versusP data set measured during this period using Δt = Δ + Δ′.Figure 4.4: The slope of 푏G versus A as a function of the simplified theoreticalvalue of the scaling factor 1∕휉(4)std. The solid straight line fits the data points linearlywith an equation: y = 0.990 (0.025)x + 0.001(0.000). The slope of this equationis approximately 1, and the intercept is very close to 0, which indicates that theexperimental data points accept the theoretical values very well.To verify the validity of this detuning correction method, the common scalingfactor 1∕휉(4)std was found experimentally for various standard settings by plotting theintercept 푏G as a function of 퐴 = 1 + (2Δt∕훾)2 and founding its slope, recallingthat 푏G = 퐴∕휉(4)std in Eq. (2.37). These measured values were plotted versus thesimplified theoretical values calculated from Eq. (2.43), using the detuning correc-514.2. Measurement of Saturation Parameterstion value Δ′ and the standard saturation parameter 푠sat derived from Fig. 4.2. Fig.4.4 displays the relationship between these two quantities, which is expected to bein direct proportion if the fitted parameters Δ′ and 푠sat are accurate. In accordancewith expectations, the straight fitting line in Fig. 4.4 has a slope of approximately1 and a negligible intercept, thus provides evidence for the precision of Δ′ and 푠stdfitted from Eq. (2.44).4.2 Measurement of Saturation ParametersFigure 4.5: The intercepts from Fig. 4.1 as a function of the modified 퐴 = 1 +(2Δt∕훾)2. A solid line fits the intercept values by a linear equation 푏G = 0.0530(0.0020) A + 0.0212 (0.0195), which shows a linear relationship between the twoquantities as we expected in Eq. (2.37): 푏G = 퐴∕휉(4)std, and the intercept from thefigure is very close to zero.After correcting the parameters A and B using the actual detuning valuesΔ푡, 푏Gfor one test repump power 푃r = 0.009 mW is plotted in Fig. 4.5, and the interceptof 푏G from Fig. 4.5 and for other repump powers are plotted as a function of the524.2. Measurement of Saturation ParametersFigure 4.6: The intercept of 푏G versus A as a function of the measured time. Eq.(2.37) predicts that this value is roughly equal to zero. The figure indicates that itis independent of time as we expected, but the values are slightly higher than zero.푃r (mW) Slope of Y (mW−1) Intercept of Y (mW−1) 푘r 푃sat (mW) 퐴r푃r,sat (mW)0.009 71.3 (55.4) 0.815 (0.083) 248(192) 1.23 (0.12) 2.21 (1.71)0.011 51.9 (44.7) 0.840 (0.070) 176 (148) 1.19 (0.10) 1.91 (1.61)0.013 45.4 (39.0) 0.813 (0.067) 159 (135) 1.23 (0.10) 2.05 (1.73)0.017 50.9 (30.3) 0.779 (0.056) 186 (110) 1.28 (0.09) 3.13 (1.85)0.023 30.6 (10.8) 0.810 (0.035) 109 (72) 1.23 (0.05) 2.46 (1.63)0.029 17.3 (16.7) 0.826 (0.033) 61.7 (56.5) 1.21 (0.05) 1.73 (1.59)0.037 14.3 (16.9) 0.829 (0.033) 51.2 (57.1) 1.21 (0.05) 1.82 (2.03)0.073 -6.00 (15.5) 0.818 (0.031) -17.5 (53.2) 1.22 (0.05) -1.42 (4.33)0.483 -23.4 (11.8) 0.814 (0.025) -77.6 (40.6) 1.23 (0.04) -38.5 (20.1)Table 4.1: Slopes and intercepts obtained from the plots of Y versus A/B for ninedifferent repump powers. The experimental values of 푘r , 푃sat , and 푃r,sat are com-posed from the measured data in the the first three columns.534.2. Measurement of Saturation ParametersFigure 4.7: The normalized slopes 푌 = 푚G퐴∕푏G from the plots of G versus P asa function of the modified 퐴∕퐵 = (훾2 + 4Δ2t )∕[훾2 + 4(Δhf + Δt)2]. Five sets ofdata points are fitted linearly to straight lines, which correspond to repump powerof 푃r = 0.009 (l), 0.017 (n), 0.037 (⬟), 0.073 (H), and 0.483 (u) mW. The slopeof these fitted lines is decreasing as the repump power increases. It agrees with ourexpectation in Eq. (2.38), that the increasing 푃r lowers the value of 푘r , and thusdecreases the slope in the figure.544.2. Measurement of Saturation Parametersmeasured time in Fig. 4.6. Eq. (2.37) predicts that the intercept value 푏G and Ashould be linearly related with zero intercept, which is exactly in agreement withthe result in Fig. 4.5. Consistently, the intercepts of 푏G for all the repump powers,shown in Fig. 4.6, present nonzero but small positive values.Fig. 4.7 plots the normalized slopes Y for different repump powers. The linearrelationship between Y and A/B also coincides with the predictions in Eq. (2.38).Furthermore, the slopes of the data points in Fig. 4.7 decrease from positive tonegative, as we increase the repump power, which obeys the relationship betweenthe slope value 휖(푘r−3)∕푃sat in Eq. (2.38) and the repump power 푘r = 2+푃r,sat∕푃r .The slopes and intercepts of Y are extracted for the saturation power calculationsand listed in Table. (4.1), where the empirical pump saturation power 푃sat and therepump effect parameter 푘r are collected from Eq. (2.39): 푃sat = 1∕퐹 , and Eq.(2.40): 푘r = 퐸∕(휖퐹 ) + 3, respectively.Figure 4.8: A plot of the experimental pump saturation power 푃sat measured withdifferent repump powers 푃r . These two quantities are not related to each other aswe expected. The mean value of 푃sat is 1.23 (0.03) mW.Fig. 4.8 displays the empirical pump saturation power for different repumppowers. One can notice that the 푃sat value is independent of 푃r , which is in agree-554.2. Measurement of Saturation Parametersment with our prediction and shows the precision of the detuning correction values.The mean value in Fig. 4.8 returns 푃sat = 1.23 (0.03) mW, which falls within a rea-sonable range compared to the estimate discussed before (1.99 (0.74) mW).Figure 4.9: A plot of the repump parameter 푘r as functions of 1∕푃r . As we expected,푘r linearly increases with 1∕푃r , but non-linearity is observed for the last 3 points.The straight line is fitting 푘r and 1∕푃r with a constrained intercept of 2, generatinga result: 푃r,sat = 1.73 (0.43) mW.Moreover, The repump parameter 푘r is plotted as functions of 1∕푃r in Fig. 4.9.Recall that 푘r = 2+푃r,sat∕푃r , the calculated repump effect parameter 푘r is expectedto be linear with the inverse of repump power 1∕푃r , with a minimum of 2. It alsopredicts that 푘r = 3 when 푃r = 푃r,sat . Fig. 4.9 shows that 푘r is linearly increasingwith 1∕푃r . A straight line with a constrained intercept of 2 fits 푘r as a function of1∕푃r in the figure. It is inside the error of all the data points, except the first one.The slope of the fitting line provides a composed repump saturation power 푃r,sat =1.73 (0.43) mW, which is lower than the previously estimated repump saturationpower (3.35 (1.25) mW). However, in Fig. 4.9 a value of 푘r = 3 corresponds toa repump power falling between 0.037 and 0.073 mW (between the second pointand the third point), this leads to a totally different result: 0.037 mW < Pr,sat <564.2. Measurement of Saturation ParametersFigure 4.10: A plot of the experimental repump saturation power 푃r,sat times adetuning-related factor 퐴r = 1 + (2Δr∕훾)2 for different repump powers. The valueof 퐴r푃r,sat has no significant overall changes for the first seven measured data setsand drops to a negative value after that. The average value of 퐴r푃r,sat excluding thenegative point is 퐴r푃r,sat = 1.74 (0.16) mW.574.3. Calculation of the Excited-state Fraction0.073 mW. Besides, 푘r should always be positive, according to Eq. (2.27), whereasthe empirical values of the first two points in the figure are negative. The firstproblem could be due to the shifted frequency of the repump laser. As noted in Eq.(2.45), 푘r is only proportional to the inverse value of the repump power when therepump detuning is kept zero in the experiment. If the repump frequency changes,in a similar fashion of the drift in the pump laser detuning, the parameter 퐴r =1 + (2Δr∕훾)2 in Eq. (2.46) is no longer a constant value, which makes it hard toacquire the repump saturation power. The value of 퐴r푃r,sat are calculated from Eq.(2.45) and plotted in Fig. 4.10. It is shown in the figure that the measured value of퐴r푃r,sat fluctuates around 2 mW for the repump powers below 40 휇W, and dropsto a negative value for higher repump powers due to the drift in the repump laserdetuning. The average value of 퐴r푃r,sat excluding the negative one is 퐴r푃r,sat =1.74 (0.16) mW, which agrees with the value of the slope in Fig. Calculation of the Excited-state FractionFrom the analysis of the repump frequency shift in section 4.2, the detuning-relatedparameter퐴r is combinedwith푃r,sat bymultiplicationwhen determining the excited-state fraction of the atoms in the MOT. Thus Eq. (2.49) becomes푓 (4)e =푠2[퐴(1 − 2휖퐴퐵)+ 푠(1 + 휖(퐴r푃r,sat푃r− 1)퐴퐵)] . (4.1)Our empirical values of the saturation powers are 푃sat=1.23 (0.03) mW and퐴r푃r,sat= 1.73 (0.43) mW. Therefore 푓e can be expressed as a parameter dependent on Δ,푃 , and 푃r :푓 (4)e =푃∕(1.23 mW)2[퐴(1−(2×0.357)퐴퐵)+(푃∕(1.23 mW))(1+0.357((1.73 mW)∕Pr−1)AB)] , (4.2)while in the two-level model 푓 (2)e is only related to Δ and 푃 :푓 (2)e =푃∕(1.23 mW)2(퐴 + 푃∕(1.23 mW)). (4.3)The values of the excited-state fractions in the two-level model and the four-level model are plotted as a function of the repump laser power in Fig. 4.11. 푓 (2)eremains constant in this figure, since the effect of the repump transition is not takeninto account in this model. For comparison, 푓 (4)e starts from zero when the repumppower is zero, which is more reasonable than the two-level model. It then quickly584.3. Calculation of the Excited-state FractionFigure 4.11: A plot of the determined excite-state fraction in the two-level model(n) and the four-level model (l) for different repump powers. Here Δ=-10 MHzand 푃=18 mW. As 푃r increases, 푓 (2)e remains constant while 푓 (4)e increases fromzero to the same level as 푓 (2)e .594.3. Calculation of the Excited-state FractionFigure 4.12: A plot of the determined excite-state fraction in the two-level model(n), the four-level model with 푃r=0.009 mW (l), and the four-level model with푃r=0.003 mW (t) for different pump powers. Here Δ=-10 MHz. These threecurves saturate at different levels, since the atoms are spending less time in thedark state with a higher repump power. The inset figure shows a 20% discrepancybetween 푓 (2)e (n) and 푓 (4)e with 푃r=0.003 mW (t) when 푃=28 mW.604.4. Hypothesis of Atom Pinning for High Laser Powerincreases and reaches the same level as the two-level model, since the atoms arequickly transferred from the dark state to the excited state with a high repump power.We can also plot the excited-fraction as a function of the pump laser power inFig. 4.12 with Δ = −10푀퐻푧. The three curves shown in the figure representthe excited-state fractions with different repump powers (푓 (2)e , 푓 (4)e with 푃r=0.003mW, 푓 (4)e with 푃r=0.009 mW). They are all increasing as the pump power increasessince more atoms are transferred to the excited-state, and they saturate at differentlevels as P goes to infinity, which is due to the accumulated atoms in the dark statecontrolled by the repump transition. In the range of the pump power that we canreach in the experiment, there is a 20% discrepancy between 푓 (2)e and 푓 (4)e with푃r=0.003 mW. It is not a big difference, but considering that the excited-state losscoefficient < 휎푣 >∗ is large compared to the the ground-state loss coefficient, anaccurate excited-state fraction value is required.For comparison, Shah 푒푡 푎푙. [30] model-independently measured the excitedfraction of 87Rb atoms trapped in a MOT using a charge transfer technique, andbuilt a model which accurately estimates 푓e with the knowledge of only the pumplaser intensity and detuning. It is not surprising that Shah proposed that 푃 andΔ are dominant factors for 푓e, since repump transition becomes more significantwhen it is very weak. Therefore, we can conclude that the four-level atomic modelis reliable for the excited-state fraction, especially when the repump power is small.4.4 Hypothesis of Atom Pinning for High Laser PowerIn Section 4.1 we have discussed the behaviour of G as a function of the test pumppower, which is quite linear as we expected. However, there is also a tendencythat G is going to saturate as the pump power decreases. It is more obvious in Fig.4.13 where G values with lower pump powers are plotted. The value of G with highpump power is fitted linearly, which is shown by the solid straight lines in Fig. 4.13,and the residuals of G are plotted in Fig. 4.14. When the pump detuning is smallerthan -10 MHz, the residuals of G are independent of P, with a constant value ofapproximately zero. But for a further detuned pump laser light, the residuals in-crease dramatically as the pump power decreases, especially when Δ=-14 MHz.One hypothesis that can explain this phenomenon is that the atoms in the cyclingtransition can be seen as in small lattices. The trap depth of the atom, which isthe energy required to remove the atom from the trap, is proportional to the laserpower and inversely proportional to the laser detuning squared: 푈trap ∝ 푃∕Δ2 [21].Therefore, for large pump power and small pump detuning, the depth of the latticeis high, which means the atoms may not be easily transferred to other states and canbe pinned on certain transitions, corresponding to a lower saturation intensity. For614.4. Hypothesis of Atom Pinning for High Laser PowerFigure 4.13: A plot of G versus the test setting pump power P. The last 8 pointsin each set of data are fitted to a line and corresponds to pump laser detunings ofΔ∕2휋 = -6 (l), -7 (n), -8 (⬟), -9 (H), -10 (u), -11 (©), -12 (:), -13 (6), and -14(t) MHz. Here 푃r=0.483 mW.624.4. Hypothesis of Atom Pinning for High Laser PowerFigure 4.14: Residuals of G from the linear fittings in Fig. 4.13. Each data setcorresponds to a pump laser detuning ofΔ∕2휋 = -6 (l), -7 (n), -8 (⬟), -9 (H), -10(u), -11 (©), -12 (:), -13 (6), and -14 (t) MHz.634.4. Hypothesis of Atom Pinning for High Laser Powerexample, the saturation intensity for the |퐹 = 2, 푚F = ±2⟩ → |퐹 = 3, 푚′F = ±3⟩transition is 퐼sat(mF=±2→m′F=±3) = 1.669 mW∕cm2 [24], which is lower than the av-eraged pump saturation intensity 퐼sat = 3.577 mW∕cm2, and the correspondingestimated saturation power is 푃sat(mF=±2→m′F=±3) = 1.39 (0.52) mW.Detuning (MHz) A Slope (mW−1) Intercept-6 4.91 0.0280 (0.0006) 0.240 (0.011)-7 6.33 0.0271 (0.0006) 0.312 (0.012)-8 7.96 0.0268 (0.0003) 0.384 (0.007)-9 9.81 0.0262 (0.0003) 0.469 (0.007)-10 11.9 0.0268 (0.0002) 0.528 (0.004)-11 14.2 0.0261 (0.0006) 0.622 (0.014)-12 16.7 0.0263 (0.0007) 0.706 (0.015)-13 19.4 0.0267 (0.0006) 0.791 (0.013)-14 22.3 0.0256 (0.0008) 0.935 (0.017)Table 4.2: Slopes and intercepts obtained from the fitting lines for high power inFig. 4.13 when 푃r = 0.483 mW.The slopes and intercepts from the high-power fits in Fig. 4.13 are listed inTable. 4.2, where the slopes are all about the same value, and the intercepts areproportional to A. Assuming that the atom pinning hypothesis is true, the saturationparameters 푃sat and 푘r can be extracted using the same method in Section 4.2, butonly from the measured G values for high pump power (13 mW to 28 mW). Fig.4.15 shows the calculated pump saturation powers for different repump powers.The result gives an average value of 푃sat = 1.15 (0.06) mW, which is in the range ofthe estimated saturation power for the 휎± transitions (1.39 (0.52) mW). Moreover,the relationship between the repump effect parameter 푘r and 1∕푃r is plotted in Fig.4.16, where a straight line with a constrained intercept of 2 is fitting 푘r as a functionof 1∕푃r . This figure is more reasonable than Fig. 4.9, since the fitting line fallsinside all the data points, and its slope provides 푃r,sat = 2.05 (0.59) mW. Hence,it is possible that the saturation parameters are not constants, but can change withthe laser conditions. That makes our model more complicated and requires morecalculations in the next steps.644.4. Hypothesis of Atom Pinning for High Laser PowerFigure 4.15: A plot of the experimental pump saturation power 푃sat for differentrepump powers. The data gives a mean value of 푃sat = 1.15 (0.06) mW.654.4. Hypothesis of Atom Pinning for High Laser PowerFigure 4.16: A plot of the repump parameter 푘r as functions of 1∕푃r . The straightline is fitting 푘r and 푘r with a constrained intercept of 2, which gives 푃r,sat = 2.05(0.59) mW.66Chapter 5Conclusion5.1 SummaryIn this work we have demonstrated a simple method to experimentally determineand control the excited-state fraction of the atoms in aMOT. A four-level theoreticalatomic model was used to describe the transitions of the atoms in a MOT, and anexperimental parameter G is constructed and measured to determine the saturationparameters for the pump (퐹 = 2→ 퐹 = 3′) and repump (퐹 = 1→ 퐹 = 2′) transi-tions in 퐷2 line (52푆1∕2 → 52푃3∕2) for 87Rb. By fitting the intercept of G versus P,푏G, we successfully measured the accurate pump laser frequency, therefore deduceda reliable pump saturation power with precision (푃sat=1.23 (0.03) mW). However,because of the unknown frequency shift of the repump laser, only a product of therepump saturation power times a detuning-related factor is measured: 퐴r푃r,sat =1.73 (0.43) mW.The excited-state fraction 푓e can be calculated from these two measured sat-uration parameters for different laser powers and detunings. The two-level modelprediction is independent of the repump laser power being used. However, it showsthat the four-level model better explains themeasured result, and gives awider rangeof the excited-state fractions. For the experimental settings currently available, arange of the excited state fractions from 0.045 to 0.415 can be achieved.5.2 Future WorkIn the next step, we can apply the determination of the excited-state fraction tomeasurements of the trap loss rates while keeping the trap depth constant, since theloss rate coefficient is dependent on the trap depth. Fig. 5.1 predicts the relationshipbetween the loss rate of the atoms in the MOT, Γ, and the excited-state fraction, 푓e.When 푓e = 1, which means that all the atoms are in the excited state, the measuredtrap loss is only due to the excited-state collisions 푛Rb < 휎푣 >Rb+Rb∗. Similarly,when 푓e = 0, Γ is related to the ground-state cross section 푛Rb < 휎푣 >Rb+Rb. Onechallenge in this measurement is to determine different excited-state fractions withthe same trap depth, which needs to be modelled and tested.675.2. Future WorkFigure 5.1: A plot of the trap loss rate from the MOT as a function of the excited-state fraction with a constant trap depth. The extrapolation of Γ from 푓e=0 and푓e=1 indicate the ground-state cross section and the excited-state cross section forone trap depth, respectively.685.2. Future WorkMoreover, we can control the trap depth by varying the size of the laser beamsandmake the samemeasurement. The laser intensity should be kept constant, whichalso indicates a constant excited-state fraction. Such experiments will lead to a di-rect measurement of the excited-state cross sections. The result of particular inter-est is the measurement of the 87Rb hitting 87Rb atoms in their excited state, whichcan be distinguished from the collisions of 85Rb hitting 87Rb atoms in their ex-cited state. This will provide a first measurement distinguishing between isotopiccollision partners.69Bibliography[1] RS Schappe, T Walker, LW Anderson, and Chun C Lin. 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Density-matrix approach to dynamics of multi-level atoms in laser fields. Physics Reports, 365(2):65–143, 2002.[24] Daniel A Steck. Rubidium 87 d line data, 2001.[25] Dmitry Budker, Derek F Kimball, Derek Kimball, and David P DeMille.Atomic physics: an exploration through problems and solutions. Oxford Uni-versity Press, USA, 2004.[26] C Gabbanini, A Evangelista, S Gozzini, A Lucchesini, A Fioretti, JH Müller,M Colla, and Ennio Arimondo. Scaling laws in magneto-optical traps. EPL(Europhysics Letters), 37(4):251, 1997.[27] CG Townsend, NH Edwards, CJ Cooper, KP Zetie, CJ Foot, AM Steane,P Szriftgiser, H Perrin, and J Dalibard. Phase-space density in the magneto-optical trap. Physical Review A, 52(2):1423, 1995.[28] Melles Griot. Gaussian beam optics. Optics Guide, section, 2, 2008.[29] Alexander Franzen. Componentlibrary: a free vector graphics library for op-tics. Licensed under a Creative Commons Attribution-NonCommercial, 3.[30] MH Shah, HA Camp, ML Trachy, G Veshapidze, MA Gearba, and BD De-Paola. Model-independent measurement of the excited fraction in a magneto-optical trap. Physical Review A, 75(5):053418, 2007.72Appendix AData and code addressThe experimental data is stored in the following address: C:∖Users∖QDG∖repo∖QDG-bus-v1∖BCodeGen for TD∖MOT_Data∖Yue_document∖G_values_with_different_repump_powers_24.04.18.The experimental driving and analysis codes written in Python 3 are located in:C:∖Users∖QDG∖repo∖QDG-bus-v1∖BCodeGen for TD. The names and functionsof the codes are:• main_code.py: run and control the measurement of the scattered light andthe MOT fluorescence for one repump power.• Analysis_main.py: extract the determined voltages by fitting the fluores-cence data.• Analysis_class.py: define some classes used in Analysis_main.py.• G_parameter_different_repump_power.py: Use the values of Gmeasuredin Analysis_main.py to calculate the saturation parameters.• DefaultSettings.py: a dictionary of the controllable settings used in the mea-surement, the settings can also be reedited in the above scripts.73


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