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Tools for trapping and detecting ultracold gases Dare, Kahan McAffer 2016

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Tools for Trapping and DetectingUltracold GasesbyKahan McAffer DareB.Sc., The University of British Columbia, 2014A 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)April 2016© Kahan McAffer Dare 2016AbstractWe construct a vertical imaging system designed to image along the quan-tization axis of the experiment. We demonstrate that it has a resolution onthe order of 1-2µm which is on par with previous characterizations of theconstituent components. We find that the inclusion of the vertical imag-ing system has a detrimental effect on the atom loading performance of theMOT. We show that this decrease is by approximately a factor of 2 downto 6.5 × 106 atoms per second and 8.1 × 107 atoms respectively. We subse-quently detail the design of a novel lattice apparatus capable of tuning thelattice spacing by many orders of magnitude on the timescale of a typicalexperimental cycle. A proof-of-principle for this so-called dilating lattice isrealized and the mechanism for variable lattice spacing is shown to work.Lastly, we cover our efforts towards measuring the effect of Feshbach reso-nances on collisional decoherence rates in 6Li. To this end, we show that theRabi frequency we can create given our current tools is approximately100Hz.A unknown strong mechanism for decoherence obstructs our experimentalsignature and a brief discussion of our attempts to discover its origin ispresented.iiPrefaceThis Master’s thesis contains some of the author’s research performed underthe supervision of Dr. Kirk Madison at the University of British Columbia.The author has chosen to leave out their work concerning the construction ofthe experimental apparatus as well as their involvement in the spectroscopyof 6Li, the measurement of Anomolous Autler-Townes splitting and the im-provement on the 2-photon linewidth of the photoassociation light. Detailsconcerning the construction can be found in William Bowden’s Master’sthesis [1]. A thorough discussion of the spectroscopic work, including theAnomolous Autler-Townes phenomena, is presented in William Gunton’sPhD thesis [2]. As for the work concerning the narrowing of the 2-photonlinewidth, the author has chosen to leave the reporting of those results toGene Polovy out of respect for his work. This thesis instead focuses onthe development of both a new imaging system and a novel lattice system.Michael Kinach was instrumental in the design of the mount for the imag-ing optics and its subsequent characterization discussed in Chapter 2. Thedilating lattice system described in Chapter 3 was built together with KaiOgasawara. The experimental apparatus used in Chapter 4 was built incollaboration with William Bowden, Will Gunton, Mariusz Semczuk, GenePolovy, and Koko Yu.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Why ultracold? . . . . . . . . . . . . . . . . . . . . . . . 21.2 Overview of the Experiment . . . . . . . . . . . . . . . . . . . . 31.3 Overview of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 62 The Vertical Imaging System . . . . . . . . . . . . . . . . . . . . 72.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Absorption Imaging . . . . . . . . . . . . . . . . . . . . 92.1.2 Fluorescence Imaging . . . . . . . . . . . . . . . . . . . 122.1.3 High-Field Imaging . . . . . . . . . . . . . . . . . . . . . 122.1.4 Imaging Resolution . . . . . . . . . . . . . . . . . . . . . 162.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24ivTable of Contents2.3.1 Effect of the Vertical Imaging Optics on the AtomLoading Performance of the MOT . . . . . . . . . . . . 293 The Dilating Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Plane Wave Interference . . . . . . . . . . . . . . . . . . 333.1.2 Gaussian Beam Interference . . . . . . . . . . . . . . . 343.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Beam Translation Options . . . . . . . . . . . . . . . . . . . . . 483.4.1 Linear Actuators . . . . . . . . . . . . . . . . . . . . . . 483.4.2 Rotation via an AOM . . . . . . . . . . . . . . . . . . . 513.4.3 Zaber’s T-NA Micro Linear Actuator . . . . . . . . . . 523.5 Design of the Monolithic Container . . . . . . . . . . . . . . . 524 Methods for Determining Collisional Decoherence Rates . 554.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Adiabatic Passage . . . . . . . . . . . . . . . . . . . . . 604.3 The Effect of Collisional Decoherence on Rabi Oscillations . 624.3.1 Experimental Results . . . . . . . . . . . . . . . . . . . 635 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68vList of Figures1.1 A labelled diagram of the experimental apparatus. . . . . . . . 42.1 A picture of the current imaging geometry . . . . . . . . . . . 72.2 A plot of the signal to noise ratio as a function of the intensityof the imaging light. . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The effect of magnetic field on the 22S1/2 and 22P3/2 levels of6Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 A diagram for illustrating the relationship between varioustransitions and their required polarizations of the couplinglight used in high-field imaging. . . . . . . . . . . . . . . . . . . 142.5 A diagram of the geometry used when calculating the impulseresponse function of a finite sized lens. . . . . . . . . . . . . . . 162.6 A cartoon comparing light rays focused through a perfect lenswith light rays focused through a lens with spherical aberra-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 A cross section of the science section of the experimental ap-paratus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 A drawing of the main components of the lens mount. . . . . 222.9 The final imaging mount design. . . . . . . . . . . . . . . . . . . 232.10 An exploded view of the final imaging mount design brokendown into the major 4 pieces. . . . . . . . . . . . . . . . . . . . 242.11 Images of a 1µm pinhole illuminated with 780nm light at ∼ 25times magnification. . . . . . . . . . . . . . . . . . . . . . . . . . 252.12 Images of the imaging system installed in the experiment. . . 262.13 A SolidWorks drawing of the quarter waveplate mount for theimaging system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27viList of Figures2.14 An image of the vertical MOT beam taken in the far-fieldafter having passed through the vertical imaging system. . . . 282.15 A comparison of atom loading performance of the MOT at380○C between the system with and without the imaging sys-tem installed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 A depiction of the interference mechanism. . . . . . . . . . . . 323.2 A depiction of the geometry associated with two interferingGaussian beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 The prediction for the intensity pattern created by interferingtwo Gaussian beams. . . . . . . . . . . . . . . . . . . . . . . . . 373.4 The original dilating lattice design. . . . . . . . . . . . . . . . . 383.5 The dilating lattice design from [28]. . . . . . . . . . . . . . . . 393.6 The final dilating lattice design. . . . . . . . . . . . . . . . . . . 403.7 A plot of a Gaussian beam’s spot size as a function of positionafter a lens and the initial beam waist’s position. . . . . . . . . 433.8 A picture of the prototype of the dilating lattice. . . . . . . . . 443.9 A diagram of the periscope shown in Figure 3.8. . . . . . . . . 453.10 A sample image showing the two dilating lattice beams onthe camera and the output of our peak finding algorithm. . . 463.11 A series of pictures demonstrating the variable lattice spacing. 473.12 A cartoon demonstrating the effect of imperfect alignment onthe interference axis. . . . . . . . . . . . . . . . . . . . . . . . . . 473.13 A picture of the method for rotating a mirror using a linearactuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.14 Comparison between the reflected beam’s alignment for twodifferent locations of a mirror’s axis of rotation. . . . . . . . . 513.15 A design for the monolithic container to hold the lattice optics. 544.1 A 2-level system coupled by a light field. . . . . . . . . . . . . . 564.2 A plot of the avoided crossing due to the coupling of two states. 594.3 A comparison of the effect the detuning has on Rabi flopping. 60viiList of Figures4.4 A depiction of the time evolution of an atom’s state’s compo-nents including collisions. . . . . . . . . . . . . . . . . . . . . . . 624.5 The efficiency of the transfer between the ∣1⟩ and ∣2⟩ statesfor various sweep rates. . . . . . . . . . . . . . . . . . . . . . . . 64viiiAcknowledgmentsGrowing up in this lab, I learned more than I could have ever anticipated.Looking back, I couldn’t have done it without all of the discussions andlaughs I had with the great people of our lab. The open and free atmospherethat my supervisor, Dr. Kirk Madison, has cultivated in our lab lead to acomfortable and relaxed environment. I will forever be grateful for theopportunity to pursue my own interests and voice my opinions as this hashelped me mature as a researcher and a person.Making the transition between the well-structured undergraduate lifeand the independent graduate life was made easy thanks to my peers thathelped me along the way. The long nights spent working with WilliamBowden showed me what real dedication and commitment are like. Hisrelentless yet friendly demeanour has stuck in my mind as a goal to strivetowards.Of course, I wouldn’t have been able to do much research without thementorship of Will Gunton. From my first day to my last, Will has con-sistently pushed me harder than anyone else to embody the spirit of a re-searcher. With his guidance, I have not only learned to build and run ourexperiment, but also to trust myself.Whenever I got tired or frustrated I could complain to my friends, andfellow Masters students, Koko and Kais. Whether it was classes, help fig-uring out problems with our experiments or just random discussions, Kokoand Kais were always there when I needed them.To Mariusz, I only wish you could’ve stayed in our lab for longer. Ialways enjoyed our talks about crazy experiments you were thinking up.One of the friendliest fellows I will ever meet, you could always make uslaugh no matter how tired or frustrated we were with the experiment. IixAcknowledgmentscan’t thank you enough for all that you’ve done for me and I wish you thebest in Poland.I couldn’t have asked for a better partner in crime than Gene Polovy.Forever playing the devil’s advocate, I couldn’t be more grateful for myfriendship with this magnificent fellow. Working together with him thesepast couple of years has been a pleasure and I can only hope that somedayhe realises that my hook idea is genius.Lastly, to my soon-to-be wife Zoya. I don’t know how you’ve put upwith me throughout my degree, let alone the past 7 years. The nights whereI had to stay all-night would’ve been terrible had it not been for your lovelycare packages. I’m glad that we could go through all of this together andlook forward to continuing along this journey with you.xChapter 1IntroductionWe will start the thesis with describing the author’s personal motivationfor pursuing research in the field of ultracold physics before discussing whatadvantages ultracold physics has when it comes to studying certain topics.This will lead to a more focused motivation for, and introduction of, thecurrent experiment and the work presented here.1.1 MotivationAs our understanding of physics progresses, so too do the problems we face.These questions we seek to answer become more numerous and what we callphysics grows and diversifies, slowly creating more subfields. In my opinion,one of the most interesting perspectives one can have on physics is comeswhen one straddles multiple subfields, transcending traditional knowledgeboundaries. After all, everything is physics and these distinctions betweennuclear, condensed matter, high energy, etc. are in no way hard boundaries.This perspective is why I am drawn to these ultracold experiments. Bytheir very nature, they act as clean testing grounds for researchers to pur-sue a variety of topics. Using the tools developed in the field of Atomic,Molecular and Optical (AMO) physics, one can study a large range of top-ics: the gravitational redshift [3], parity violating effects [4], EPR pairs [5]Kondo physics [6], precision spectroscopy [7, 8], and topologically protectedquantum qubits [9] to list a few. This is by no means an exhaustive list andbut the depth and breadth are impressive, especially considering it is onlya couple of decades old. It is this variety that motivates to me as it en-courages a broader perspective by allowing physicists to consider problemsoutside one’s realm of expertise.11.1. MotivationThe experiment we are currently pursuing is the formation of ultracoldLiRb molecules in their lowest lying triplet state. The motivation for thisexperiment lies in the recent realization that atoms in lattices can be usedto systematically engineer various types of Hamiltonians [10]. Even morecompelling is the proposal for using polar molecules in a lattice to simu-lated any permutation-symmetric two-spin-1/2 interaction[9]. The mecha-nism by which this works requires the molecules to have both an electricdipole moment (EDM) and a magnetic dipole moment to be able to createsuch anisotropic interactions.This flexibility and applicability of a single apparatus is a strong moti-vator, from an experimental standpoint. to construct such a system. Thepolar molecule we have chosen to pursue is LiRb which has an electric dipolemoment (EDM) of ∼ 4.15D[11–13]. Other heteronuclear molecules like KRband LiCs are also attractive options and have been formed by Dr. JunYe and Dr. Debbie Jin’s groups [14] and Dr. Matthias Weidemu¨ller’s [15]groups respectively. LiRb was chosen due to the strong knowledge of bothspecies within the group.1.1.1 Why ultracold?One might question the need for particles to be ultracold in order for re-searchers to study the topics mentioned above. After all, our lab is calledthe Quantum Degenerate Gases laboratory and yet degenerate Fermi gasesexist at temperatures over 105K in the form of white dwarves, dense starsnear the end of their lifetime. While it is true that one does not need to bein the ultracold regime to study such topics, it is extremely beneficial as itprovides a new approach to examining these systems. To study the physicsgoverning many aspects of a white dwarf, one need not create a star in thelab for probing but instead an ultracold plasma [16].For instance, it has been shown that that the Hubbard model, whichis believed to explain unconventional superconductivity, is impossible tosolve analytically or computationally. Various solutions have been positedunder various approximations but we still don’t have a complete picture as21.2. Overview of the Experimentto the mechanisms behind unconventional superconductivity. By trappingultracold atoms in optical lattices, physicists believe they can probe theHubbard model in a precisely controlled manner. In fact, Dr. RandallHulet’s group has managed to observe antiferromagnetic correlations in theHubbard model using 6Li at a temperature of ∼ 30nK [17]. It is believedthat by further decreasing the temperature by an order of magnitude, oneshould see evidence of D-wave pairing, giving condensed matter physicistskey insight into one of the outstanding problems in physics today.There are many more examples where lowering the temperature givesresearchers access to phenomena that are usually at too low of an energyscale to probe via other methods. This flexibility is supported by the manytools that have been developed over the past couple of decades to confineand control atoms.While various forms of traps have been developed, a more striking fea-ture of ultracold physics is the ability to tune interaction strengths betweenparticles. This tuneability most often comes in the form of Feshbach reso-nances whereby one can tune the scattering length across a resonance simplyby tuning the magnetic field in which the particles are held. The details ofthis phenomena are discussed in great detail in [2] but it suffices for ourmotivation to simply realize the experimental simplicity of this effect. Onecan suppress interactions, cause them to be attractive or repulsive, or somemixture in between if one considers the difference in the interactions betweenatoms of varying internal states, all by tuning the magnetic field.1.2 Overview of the ExperimentThe most common way to prepare ultracold atoms is to start with someform of laser cooling. Consider, for instance, an atom moving along the z-axis. If one were to shine a near resonant, red-detuned laser along the z-axis,counter to the motion of the atom, then the laser’s frequency will be Dopplershifted to a higher frequency. If the detuning is picked such that the Dopplershift shifts the laser into resonance then the atom can absorb a photon. Thenow excited atom will be moving slower to conserve momentum and after31.2. Overview of the Experimentsome time it will reemit a photon. However, this emission will be isotropicmeaning that on average the change in momentum due to the emission ofphotons will be zero. This means that the atom will have been slowed downby the laser. If this is now done in an ensemble of atoms, the temperaturewill have decreased. This is the principle behind laser cooling.Unfortunately, laser cooling molecules is not as easy because their com-plicated internal structure makes it difficult to make sure all of the atomsstay in the state which will be Doppler shifted correctly. Since this is one ofthe main tools in the AMO toolbox, for achieving ultracold temperatures,it makes an already involved experiment significantly more challenging. Analternative approach is to cool the constituent atoms to ultracold temper-Figure 1.1: A SolidWorks drawing of the major components of the experi-mental apparatus. For reference, the part of the apparatus to the left of thegate valve will be called the science section while the section to the rightwill be the source section. Differential pumping separates the two sections,allowing the science side to operate at a pressure of ∼ 5×10−9torr. An atomicshutter can be controlled electronically to block the direct path between thesource and science sections. Not pictured here are the compensation coils,the vertical imaging apparatus and optics. More detailed descriptions canbe found in [1, 2]41.2. Overview of the Experimentatures before pairing and transferring them to the desired molecular state.We have chosen to pursue the second method and the main steps for doingso are as follows: trap and cool Li and Rb atoms, transfer the atoms to anoptical lattice, pair the atoms together and transfer to a molecular state.To elaborate, we must first discuss the current experimental apparatusshown in Figure 1.1. The sources of our atoms are two chunks of metal (onefor each species) which are heated up to hundreds of degrees Celsius. Atthese temperatures, some portion of the metals are converted to a gaseousphase before they exit the source chamber and travel down the axis of theexperiment towards the science side. The atoms in these gases are movingat hundreds of meters per second which is far too fast to catch in any ofour traps. Hence, we employ a so-called Zeeman slower [18] to decreasethe velocity of the atoms down to captureable velocities for our Magneto-Optical Trap (MOT). These atoms are then cooled in the MOT via Dopplercooling down to temperatures on the order of ∼ 100µK. These atoms arethen transferred to a Crossed Optical Dipole Trap (CODT) before beingevaporated further down to anywhere from ∼ 100nK-10µK [19].In this current iteration of the experiment, we have achieved the firststep for creating LiRb molecules. The method for pairing Li and Rb atomsis tuning their interactions via the previously discussed Feshbach resonanceso that that it is effectively attractive. We have previously observed theseFeshbach resonances [19, 20] so all that is left is trapping the atoms in anoptical lattice and ultimately forming molecules. We are currently focusingon the latter, using only Li (forming Li2 molecules) as a testing ground forour experimental techniques. The method that we hope to utilize is calledStimulated Raman Adiabatic Passage (STIRAP) which uses two coherentlaser pulses to adiabatically transfer atoms between two states via somethird intermediate state.The reason we are taking this intermediate step is two-fold. First, whenthe experiment was first built, there was no high resolution spectroscopic forLiRb while Li2 was well studied, making it an ideal benchmark for a brandnew apparatus. Second, before this experiment, Li2 had only been welloutside of the ultracold realm, which means there is still interesting science51.3. Overview of this Thesisto be done. We therefore sought to create deeply bound Li2 molecules asquantum degeneracy of such molecules had never before been achieved.1.3 Overview of this ThesisChapter 2 discusses the theory, design and implementation of a verticalimaging system for increasing the detection efficiency of the current exper-imental apparatus. As we believe we will be producing, at least initially, asmall number of molecules, increasing the signal to noise ratio (SNR) of ourdetection scheme is of paramount importance for the development of ourmolecular formation techniques. A review of the initial work towards thisend [21] will be presented. The optomechanical constraints are discussed aswell as the consideration of a vertical lattice. A preliminary characteriza-tion of the vertical imaging system is subsequently presented as well as theproposed method for a more final verification of the resolution.Next, our efforts towards the realization of a novel lattice scheme willbe discussed in Chapter 3. We start with a discussion of interference andthe possible lattices spacings one can achieve for a given apparatus. Thisleads us to detail the various iterations of our design before we present ourfinal design for the dilating lattice system. The realization of this designwill then be examined, resulting in a thorough breakdown of the difficultiesassociated with building such a lattice. Finally, the merits and design for amonolithic container are presented as a way to improve the portability andapplicability of the system.Lastly, in Chapter 4, we will report on our preliminary work focused ona measuring the effect a Feshbach resonance has on collisional decoherencerates. We discuss the theory behind Rabi oscillations and adiabatic passagebefore discussing these concepts as experimental tools. We outline our ex-perimental procedure for using Rabi oscillations between hyperfine states in6Li to detect the rate at which decoherence occurs due to collisions betweenhyperfine states. We discuss some obstacles we encountered in trying to re-alize this proposal, and some open questions about the performance of ourRF spectroscopy equipment.6Chapter 2The Vertical Imaging SystemWhile the current iteration of the experimental apparatus utilizes a camerapositioned horizontally as shown in Figure 2.1, the development of a secondimaging system aimed along the vertical axis, parallel to the magnetic field,has become critical to our goal of creating ultracold polar LiRb molecules.The reason for this is twofold: the increase in resolution, and the ability toproperly image at high magnetic fields. Since we expect, at least initially,to be creating small numbers of molecules, we are aiming to increase oursignal to noise ratio (SNR) in order to amplify our weak signal.Figure 2.1: A depiction of the current imaging geometry. The green objectdepicts the current imaging systems orientation in the horizontal plane of theexperiment. The blue vertical arrow labels the orientation of the magneticfield due to the Feshbach coils. The red arrow shows the direction in whichthe absorption imaging beam propagates while the red sphere in the middleof the cell notes the location of the atoms.72.1. Theory2.1 TheoryOne of the most common methods for acquiring data in ultracold atomexperiments is via some form of imaging. Once calibrated, images provide astraightforward method for determining spatial densities of cloud of atoms.It is also an extremely adaptable method as the manner in which the cloudis prepared for the image can drastically change the outcome. For instance,the imaging light can be prepared in such a way that it only interacts witha certain atomic state, giving rise to state selective imaging.Another common technique, which is often times used for measuringthe temperature of a cloud of atoms, is time-of-flight imaging. Instead ofimaging the atoms while they are confined in some trapping potential, oneturns off the trap allowing the atoms to freely expand. To understand whathappens let us label the initial state of the atoms by ∣Ψ(t = 0)⟩ which hassome characteristic width L. Then if we time evolve the associated fieldoperator ψˆ„(x, t = 0) we find thatψˆ„(x, t) = Uˆ „(t)ψˆ„(x, t = 0)Uˆ(t),= ∫ L/2−L/2 dx′ψˆ„(x′, t = 0)∫ dk2piei[k(x′−x)−ω(t)t], (2.1)= ∫ L/2−L/2 dx′ψˆ„(x′, t = 0)I(x′ − x, t), (2.2)where I(x, t) = ∫ dk2piei[kx−ω(k)t] ≈ √ m2piith̵eimx22th̵ e−ipi/4. (2.3)Now if we imagined L were infinitesimally small then after some time eachparticle would classically be at a position given purely by its velocity and thetravel time. Therefore, if the momenta of the particles were quantized, onewould see shells of whose separation would be governed by the associatedvelocities and the expansion time. Now if the cloud originally had somefinite size then these shells would be blurred so one can naturally ask thequestion of how long does one need to let the cloud expand to resolve thesemomentum shells. To see this, we return to our example, taking now the82.1. Theorycommutator[φˆk, ψˆ„(x, t)] = ∫ L/2−L/2 dx′e−ikx′I(x′ −x, t) ≈ L exp(−i2th̵pi2k2m ) sinc(mxL2th̵ ) .(2.4)where we have assumed that x >> L4 . This is a valid assumption as we’reconcerned with the case where the cloud has expanded to be much largerthan its original size. From the sinc term, we can see that the characteristicwidth of this new distribution is W = 2pith̵mL . Given our assumption of theclouds expanded size, this requires an expansion time tTOF >> mL28pih̵ . Fora trapped Li cloud of size 20µm, we find that tTOF >> 1.5ms which is afeasible time expansion time for our experimental system. Hence, a simplemodification to normal imaging allows us to determine the momentum spacedensity which naturally leads to temperature information.Given these couple of examples we will proceed with discussing the twomain methods for taking images of atoms. The first, absorption imaging,revolves around the scattering of light away from the camera, casting ashadow on the detector that one can then use to infer the spatial density ofthe atomic cloud. This is the type of imaging we will be concerned with indiscussing the imaging system.2.1.1 Absorption ImagingAbsorption imaging is a popular imaging method as the atomic cloud neednot be held in place while the image is being taken. This means it can beused to take time-of-flight measurements for instance. While the versatilityis a strong benefit, it is heavily dependent on the atomic scattering ratewhich is defined byΓ = Γ02s1 + s + (2δ/Γ0)2 (2.5)where s = I/Is is a parameter which determines how close one is to saturatingthe transition, Γ0 is the bare scattering rate and δ is the detuning of thelight from a given transition. For most of our purposes, we can simplify thisequation by taking (2δ/Γ)2 ≈ 0, s ≈ 1 and Γ0 ≈ 6MHz for 6Li. This means92.1. TheoryΓ ≈ 1 − 2MHz.One then needs to compare this to the relevant noise levels in the imagingsystem. One inevitable source is shot noise. This simply comes about dueto the fact that light now acts like particles and their very discrete naturemeans they have an uncertainty governed by the Poisson distribution. Forlarge numbers of events, the standard deviation of shot noise approaches thesquare root of the number of events and therefore we can define a signal tonoise ratio (SNR) asSNR = Γ ⋅ t ⋅QE√A ⋅ I ⋅ t ⋅QE (2.6)where t is the exposure time, QE is the quantum efficiency of the imagingapparatus, I is the intensity of the light in photons per second, and A is thearea of a particle’s possible position. As can be seen in Figure 2.2, there isa maximum in the SNR as a function of s and one can show it occurs whens = 1 + (2δΓ0)2 . (2.7)The exposure time is effectively limited by the atomic drift, especiallyFigure 2.2: A plot of Equation 2.6 using δ = 10MHz, Γ0 = 6MHz which arestandard values for 6Li.102.1. Theorywhen one is concerned with light species. For instance, given a sample of6Li trapped at 10µK, the average velocity will be on the order of 0.1m/swhich means if one were to expose the atoms for even 0.5ms, the atom willhave moved 50µm which is larger than the size of our dipole trap (whichis on the order of 30µm). Hence, this would erase all spatial information.This is also not considering the recoil velocity imparted on the atoms by theimaging photons, which at this is also approximately 0.1m/s. In order forimages to retain their spatial information, one needs to at least decrease theexposure time such that the atoms only scatter photons within some smallspatial region on the order of the resolution of one’s imaging system.One should note that inverse relationship between the SNR and√A.While A is not usually a tuneable parameter, this relationship motivatesusing a pinning lattice to confine atoms to a well defined position. Toestimate the difference on the SNR for a lattice and a dipole trap one simplyneeds to consider the characteristic size of each trapping potential. For anormal 1064nm lattice and our dipole trap which is approximately 30µmin diameter, this ratio is approximately (30µm)2/(1064nm/2)2 ≈ 3000. Asthe quantum efficiency is usually anywhere between 5% − 90%, we see thatthis change between imaging atoms in a dipole trap as opposed to a latticemakes the most dramatic change in the SNR out of all of the parameters.For our current experiment, we are planning to use the Point Grey FL2G-13S2M-C which has a quantum efficiency of ∼ 40% at 671nm (Li light) and30% at 780nm (Rb light). For comparison, the Andor iXon Ultra 888 has80−90% at these wavelengths. This improvement would be great but the costof these products scales dramatically as one approaches a quantum efficiencyof 100% and since the SNR only scales like√QE, one should consider if thisimprovement is worth the expense.One other aspect to consider when looking at cameras is their dark cur-rent. This is another source of noise due to the small amount of currentthat flows through the device even when no photons are being registered.Fortunately, for the case of CCD cameras, this noise is negligible and wecan ignore it. It can be a significant problem for other types of detectorslike near-IR detectors.112.1. Theory2.1.2 Fluorescence ImagingThe main alternative to absorption imaging is fluorescence imaging wherebyatoms absorb light and re-emit it in a different direction. The emission lightcan be collected along some axis and the image will be bright if there is ahigh density. Since the captured solid angle is relatively small, the numberof photons that scatter towards the camera is low, requiring that the cloudbe exposed for a large duration to build up a detectable signal. These longdurations allow for atoms to drift and so one needs to confine them in orderto take an image. Some standard traps for this are MOTs or pinning latticeswhereby the new limiting factor is the timescale on which atoms would heatout of the trap.2.1.3 High-Field ImagingAtomic imaging relies on light being scattered and therefore that your imag-ing light is resonant, or at least close to resonant, with some atomic transi-tion. It must also be of the correct polarization to satisfy the dipole allowedtransition selection rules. As such, imaging light at low magnetic fields andhigh magnetic fields might have to be different and indeed that is the casewith 6Li.At zero magnetic field, we image on the ∣F = 1/2⟩→ ∣F ′ = 3/2⟩ transitionwhich counts all of the atoms since both the mF projections are degenerate.At small magnetic fields, hyperfine splitting causes us to select one of the twomF states to image meaning our effective SNR is decreased. Furthermore,as we increase the magnetic field, F and mF are no longer good quantumnumbers and replaced with mJ and mI . As is shown in Figure 2.3, the levelsseparate into mI = −1,0,1 triplets with different mJ numbers. We usuallyfocus on imaging the ∣1⟩, ∣2⟩, and ∣3⟩ states, which have mJ = −1/2 andmI = 1,0, and −1 respectively, using the m′J = −3/2 excited states. As wewill soon show, at high fields this transition is closed, allowing us to forgoa repump beam. However, there is still a differential Zeeman shift which isapproximately linear past 100G and can be written as122.1. Theory(a)(b)Figure 2.3: The effect of magnetic field on the 22P3/2 level of 6Li is shownin (a) and on the 22S1/2 level in (b). Note that in (a), the blue levels splitfrom the degenerate F ′ = 1/2 manifold while the red from the F ′ = 3/2 andthe green from the F ′ = 5/2.132.1. Theoryδ1 = −1.4MHz/G ⋅B + 158MHz, (2.8)δ2 = −1.4MHz/G ⋅B + 82MHz, (2.9)δ3 = −1.4MHz/G ⋅B. (2.10)where δi is the detuning of the transition ∣i⟩→ ∣m′J = −3/2⟩.This large frequency shift requires us to have a second laser system forgenerating the high-field imaging light and this is described Dr. Semczuk’sthesis [22]. Here we will focus more on the consequences of the hyperfinesplitting rather than discussing the laser system.Due to the splitting, one has to be careful of atoms decaying to a statewhich is transparent with respect to the imaging light as they will no longercontribute to the image. To understand how this might happen, we lookat a simplified 6 level system shown in Figure 2.4. The selection rules thatgovern which transitions are allowed are simply ∆mJ = 0,±1 and ∆mI = 0(the latter is what lets us only consider the 6 levels as each one is triplydegenerate with mI = 0,±1). In this case, we see that only the m′J = ±3/2m′J = −3/2 −1/2 +1/2 +3/2mJ = −1/2 +1/2Figure 2.4: A diagram for illustrating the relationship between various tran-sitions and their required polarizations of the coupling light. Note that thisis not meant to describe the energies of the states and one should consultFigure 2.3 if one is looking for that information. Furthermore, each levelhere actually represent a triplet with mI = 0,±1. The green, red and bluetransitions are those driven by pi, σ−, and σ+ polarized light respectively.Dashed transitions are those that a disallowed due to the lack of a level(shown as dashed levels). Note that any transition to either the m′J = ±1/2will not be closed due to there being two allowed decay channels. On theother hand, the only allowed transitions to the m′J = ±3/2 are closed as thereis only a single decay channel which returns the atom to its original state.142.1. Theorystates have closed transitions as they can only decay to the state which thelight is resonant with. As our atoms start in themJ = −1/2 states, this meansthe closed transition we are left with is mJ = −1/2 → m′J = −3/2 transition.Therefore, we require our light to be circularly polarized to conserve angularmomentum.In order for the particles to see purely right- or left-handed circularlypolarized light, the light must propagate along the quantization axis of theatoms. It is at this point that the orientation of our system is important.The magnetic field, generated by our Helmholtz coils, defines the axis ofquantization to be along the vertical axis, as depicted in Figure 2.1. Hence,an imaging system aligned along the vertical axis is required in order for theimaging light to efficiently interact with the gas cloud.At present, the system is aligned horizontally as shown in Figure 2.1.Light that is polarized in the plane perpendicular to the quantization axiswill have its polarization vector project onto the quantization axis, creatingand even superposition of right- and left-handed circularly polarized light.These different circular polarizations will then drive σ+ and σ− respectively.Alternatively, one can drive pi transitions by having light polarized along thequantization axis. Since we wish to drive σ− transitions though, we are forcedto linearly polarize our light in the plane perpendicular to the quantizationaxis. This results in the highest scattering rate for this orientation, howeverthis means that each atom only interacts with half of the photon due tothe even split in polarization. This causes a severe undercounting of thenumber of particles in the trap because much more light passes through thecloud than one might expect. One can correct for this undercounting inone’s analysis, however when one is working with small atom numbers othersources of noise can become comparable to the signal one is looking for.This can’t be corrected for in the analysis and therefore correcting for thisundercounting by imaging along the quantization axis becomes importantfor imaging small samples.152.1. Theory2.1.4 Imaging ResolutionAs we have decided to build a new imaging system we aimed to createa system that would be able to have single-site imaging capabilities for areasonably made lattice without spending an exorbitant amount of money.Our current plan is to build a lattice with a periodicity of 1 − 2µm andat these scales, the resolution of such a system is most likely diffractionlimited. When one is intending to image anything with great fidelity, onestarts thinking of how well your image will mirror your object. As is withall waves, diffraction will cause a blurring of edges in a realistic imagingsystem.To illustrate this, we will calculate the impulse response function of thesystem shown in Figure 2.5. The impulse can be described as a sphericalwave and thus, in the aperture plane, it can be approximated as a paraxialU(x, y)p(x, y)U1(x, y)d1 d2h(x, y)Figure 2.5: A standard diagram for a finite sized lens. We model the lensitself as an infinite lens but overlay it with an aperture, causing diffractionin the light.162.1. Theorywave. Therefore, in the aperture plane it has a complex amplitude given byU(x, y) ≈ U0exp(ikx2 + y22d1) . (2.11)After transmitting through the aperture and lens, it is now described byU1(x, y) ≈ U(x, y)exp(−ikx2 + y22f)p(x, y) (2.12)wherep(x, y) = ⎧⎪⎪⎨⎪⎪⎩ 1 ∶√x2 + y2 ≤ R0 ∶ otherwise (2.13)is the transmission function of the aperture and R is the radius of the aper-ture. If one then propagates this to a distance d2 which satisfies the imagingequation, then we get thath(x, y) = U0J1(2piRρ/λd2)piRρ/λd2 (2.14)where ρ = √x2 + y2 and J1(x) is the first order Bessel function. This in-tensity distribution is called an Airy pattern. The first zero-crossing of thisfunction occurs at ρmin = 1.22λd2/2R which gives a natural scale for the reso-lution of your system. The so-called Rayleigh criterion for distinguishabilityrequires that for two points to be resolved, one must have the separation oftheir respective Airy patterns satisfy ρ ≥ ρmin. Put another way, this statesthat the maxima of the Airy patterns created by two point sources must lieoutside of the others’ first zero-crossing. Assuming one were to image anobject at infinity, this criterion simplifies toρ ≥ 1.22λF# (2.15)where F# = f/2R is the F-number. Therefore, one wants as small an F-number as possible to increase the resolution of the system. Getting a largelens can help with this as well as decreasing the focal length.Unfortunately, in reality there are other effects that further limit the res-172.1. Theoryolution of one’s system. Two very common types of aberration are chromaticand spherical aberration.Chromatic aberration happens when different wavelengths of light focusat different positions. As a consequence of this, the imaging apparatus mustbe aligned separately for each 6Li and Rb since the depth of focus will be onthe order of the size of our atomic cloud. While this is unfortunate it alsomeans that chromatic aberrations won’t impact our resolution since we’llonly be using a single frequency of light at any given time.Spherical aberration, on the other hand, is a significant problem since itis inherent in almost all lenses. It is caused by the increased refraction oflight rays that strike the outer regions of the lens compared to the center.This results in the rays focusing at different points depending on where theyhit the lens as depicted in Figure 2.6.One can mitigate this effect by using a complex system of lenses, a spe-cially designed aspheric lens or an aperture. As our objective is to producean imaging system on a reasonable budget, we focus on utilizing the secondtwo options as they don’t require any costly components and are also rela-Figure 2.6: The top image is an example of what the rays would look likeif there was no spherical aberration. The bottom image depicts a lens withspherical aberrations where the rays focus at varying positions along thehorizontal axis.182.2. Designtively simple to implement. By placing the aperture at the Fourier plane,one can block the rays that aren’t focused at the focal plane. In principle,one can continue to aperture down until all but the perfectly focused raysare let through and hence get rid of all of the aberrations.In practice, there is some trade-off between reducing the spherical aber-rations and decreasing the resolution which will limit how much of the aber-rations can be eliminated. This limit can be found experimentally as it willdepend on each individual system. In certain systems, one can forego thisfiltering and attempt to correct for the spherical aberrations in the analysis,however it presents significant complications when looking at correlationsin an image. This is because the distortion in the image will cause devia-tions away from the true intensity of the pixels, which then can cause falsecorrelations to arise. These confounding artefacts are particularly harmfulwhen looking at small fluctuations in the image which is how one tests forentanglement as detailed in [6].2.2 DesignA thorough discussion of the basic design and characterization of the com-ponents of the imaging system was done in my undergraduate thesis [21]and I will only present the considerations we undertook when designing thissystem as well as the final design.2.2.1 ConsiderationsWhile designing an imaging system comes with its own inherent difficulties,our main constraints were due to the fact that the imaging system had tofit into an already built experiment. Furthermore, we had planned to adda lattice along the vertical direction of the existing apparatus, complicatingthe design further. Given the geometry of our experiment, shown in Figure2.7, we can see that the size of the lens must fit inside a 60mm cylindricalsection, limiting R. We then seek to place the lens as close to the cell aspossible and will henceforth discuss the restrictions on the lens placement.192.2. DesignThe major complication arises due to the fact that not only is the imaginglight and the lattice light travelling along the vertical axis, but the verticalMOT beam as well. This makes things more challenging as the MOT beamand imaging light are only detuned from one another by tens of megahertzand there are no available dichroic mirrors that have such a sharp cut-off at671nm.Hence, one needs to separate the imaging and MOT light from the lat-tice light and then separate them from one another. While this is normallynot difficult, the imaging lens will focus the MOT beam, causing it to sub-sequently diverge and therefore this diverging beam must be recollimatedafter the imaging beam is separated. This means that one has a path lengthof ∼ 2f , where f is the focal length of the imaging lens, to have the imagingand MOT beams completely separated. To increase our resolution, we wantFigure 2.7: A cross section of the science section of the experimental appa-ratus. Here the red dot signifies the location of a trapped cloud. Note thecylindrical regions directly above and below the clouds position are emptyand are there to allow for optical access along the vertical axis.202.2. Designto make f as small as possible so optimizing these parameters was the mainconsideration.The lab had previously purchased the ThorLabs AL5040-B 50mm lenswith imaging in mind. This lens happened to have a focal length that closelymatched what was possible for the imaging system to tolerate given its crite-rion. As such the lens, as well as other key components, were characterizedas part of my undergraduate thesis and basic design for the lens mount,shown in Figure 2.8, was made. This design would be attached to a 3-axistranslation stage for precise alignment. The dichroic mirror is meant to sep-arate the imaging and MOT beams (which are reflected) from the latticebeams (which are transmitted). A quarter waveplate and polarizing beamsplitter would be placed right after the circular aperture in the bracket toseparate the imaging and MOT beams.212.2. DesignFigure 2.8: A Solidworks drawing of the basic lens mount. The cell isdepicted as being stationed above the entire mount with the lens restingin a round hole. Inside the bracket part is a Thorlabs H45CN 45○ mountholding the dichroic mirror.222.2. Design2.2.2 ConstructionFrom the rather basic imaging mount design shown in Figure 2.8, we workedto build one monolithic mount to hold most of the imaging optics securely.This final design, shown in Figure 2.9 contains all of the separation optics.This mount is actually made up of 4 major pieces shown in Figure 2.10,upon which various optical elements are secured. For more details aboutthe imaging mount, one should refer to Michael Kinach’s report.Figure 2.9: The final imaging mount design. The green arrow labels thepath of the lattice beams, the red arrow is the path of the MOT beamand the orange is the path of the imaging light. This mount will fit on a3-axis translation stage with the cylindrical lens mount situated inside theHelmholtz coil mount.232.3. ImplementationFigure 2.10: The final imaging mount design broken down into the major 4pieces. Not pictured here are the triangle brackets used to add rigidity tothe top plate for the 3-axis translation stage.2.3 ImplementationAfter the design was completed we had the Physics Machine Shop build eachof the mount’s pieces out of aluminum. We then glued the cube and imagingoptic onto their respective parts before assembling the rest of the optics.Before introducing the imaging system into the experiment, we decided tocharacterize the assembly outside of the experiment as a final check. To doso, we repeated the procedure described in [21] that was used to characterizethe resolution before the mount had been made. We illuminated a 1µmpinhole, which was placed inside of a glass cell, with a collimated 780nmbeam and then using the imaging lens to create a magnified image whichwas captured on a CCD camera. This setup mimics that of our experiment242.3. Implementationwhere the 1µm pinhole is meant to act like a point source, meaningthe image we retrieve should characterize the impulse response function.After aligning the system so that it had a magnification of approximately25 times, we took the image shown in Figure 2.11b. Taking the FWHM ofthis airy pattern and converting this to an imaging resolution we find thatthe imaging resolution is ∼ 2µm which is slightly larger than what was foundpreviously but is still within the range of usable resolutions. For reference,we are planning on imaging a lattice with lattice spacing of 1−2µm and thissets the scale for our desired resolution. It is also important to note thatthe image has no noticeable astigmatisms.After these tests, we moved forward with adding the imaging system tothe experiment. In order to fit the mount into place, we had to removed theback plate and reattach it after the lens was inside the Helmholtz coil mountas the mount was too tall otherwise. Once the back plate was attachedwe bolted the mount onto a 3-axis translation stage. At this point it isimportant to note that our height measurements were off due to the lack(a) (b)Figure 2.11: Images of a 1µm pinhole illuminated with 780nm light at ∼ 25times magnification. Here each pixel is 6.8µm×6.8µm. Note the fringeswhich match the Airy pattern as expected. The alignment of the lightthrough the pinhole is responsible for the asymmetry of the Airy pattern’sintensity.252.3. Implementationof the bottom z-compensation coil. What this meant was that the entiresystem sits lower than originally planned for meaning we can only achieve amagnification of ∼ 4 times with the current lens. Since the system is alreadydesigned with the maximum amount of space between the lens and thedichroic, one cannot fix this problem without buying a new imaging optic.As this magnification is still better than that of our current imaging system,we elected to proceed without purchasing a new optic. Unfortunately, thefact that the imaging optic will not be placed approximately a focal lengthaway from the trap will present some complications for lattices that will bediscussed later.Since the holes for mounting to the 3-axis translation stage were made for(a) (b)Figure 2.12: Images of the imaging system installed in the experiment. Thelower side of the experiment is shown in (a) with the apparatus from Figure2.9. The green path will be that of a vertical lattice once installed, the redpath is for the MOT beam and the orange path is the imaging path. Notethat the second 2” mirror along the lattice path has not yet been installedas we currently don’t have a lattice. There is also no camera in place yetfor the imaging path as that will be installed once the ODT alignment iscompleted. The 50mm lens along the MOT path is used to counteract theeffect of the imaging optic on the MOT beam as together they form a 1-to-1telescope.262.3. Implementationa specific height we had to mill them out into slots to secure the mount. Afterwhich, we used the triangle brackets to attach the back plate onto the mountbefore securing it to the translation stage. Despite these modifications, theentire mount is very rigid. One can account for the lack of magnification witha second imaging optic, although this will further decrease the resolution ofthe system. A picture of the imaging mount and routing optics is shown inFigure 2.12a.Upon testing the components, we realized that the dichroic has a strongbirefringence which meant we had to modify the system further by placingthe quarter waveplate between the cell and the dichroic so that the lightthat was reflected from the dichroic was either “S” or “P” polarized.This can complicate the implementation of a vertical lattice as the wave-plate most likely won’t function properly at those wavelengths but this wasthe most obvious and immediate fix. Ideally, we would want the waveplateto not act as a waveplate at the lattice wavelength. To secure the waveplatewe made a simple device, shown in Figure 2.13, that would allow one tolock the waveplate in place while allowing a degree of tuneability. For MOTFigure 2.13: A SolidWorks drawing of the simple quarter waveplate mountfor the imaging system. The appendage on the left allows one to lock thequarter waveplate’s angle. The quarter waveplate sits inside the gear ona small circular ledge made by a thin ring attached below the gear. Thewaveplate is then glued to this ledge for stability. This entire mount is thinenough to slide into place between cell and Helmholtz coil mount.272.3. Implementationfunctionality, the quarter waveplate’s angle does not need to be tuned ex-tremely precisely and therefore the teeth on the gear were designed to givean angular precision of 10○.We also found that the vertical MOT beam no longer looked quite Gaus-sian after passing through the imaging system as can be seen in Figure 2.14.It seemed as though this was due to the large size of the MOT beam (∼ 1”in diameter) relative to the 1” mirrors. Since the periscope shown in Figure2.12a uses 1” mirrors, their vertical size is only 1/√2” ≈ 0.71” which meansthe vertical MOT beam is clipped. One should note that this is also an issuefor the other MOT beams as they use similar optics, however it seems asthough the imaging system amplifies this problem. As a consequence of thisaberration, it is no longer possible to fully collimate the beam. We tried tofind the source of the aberration but none of the optical components seemedto be individually responsible but rather the entire assembly. This could bedue to the large beam diameter relative to many of the optics, resulting ina non-negligible portion of the beam being lost. This effect is usually unno-ticeable when dealing with single optics but if at multiple points along thebeam path a portion of the beam is lost, this can result in a non-Gaussianbeam.After we added the imaging system into the experiment, we built thetop section pictured in Figure 2.12b to retro-reflect the MOT beam as weFigure 2.14: An image of the vertical MOT beam taken in the far-field afterhaving passed through the vertical imaging system. Note the ring patternwith the obvious intensity spike towards the edge. Various attempts toreshape the beam produced other odd patterns but none could transform itinto a Gaussian pattern.282.3. Implementationhad decided to introduce the vertical MOT beam from the bottom of theapparatus. The mirror for retro-reflection is placed inside of a threaded tubeto give fine control on its placement. This allows us to control the divergenceof the retro-reflected beam since it couldn’t be fully collimated.2.3.1 Effect of the Vertical Imaging Optics on the AtomLoading Performance of the MOTAfter reoptimizing the setup we achieved the MOT atom loading perfor-mance shown in Figure 2.15b. While this is worse than without the imagingsystem, it is important to note that the ODT saturates above ∼ 30 × 106atoms in the MOT. As such, we have deemed this change in MOT perfor-mance to be unfortunate but within reason and have not sought to furtherincrease our loading rate or steady state atom number.At this time, the dipole trap is being set up and so there are no imagesfrom the vertical imaging system with which we can analyze its final per-formance. However, we did perform some preliminary tests of the systemto see if its performance was in accordance with my data from my under-graduate thesis [21]. This characterization is described at length in MichaelKinach’s report and we found that the imaging system had a resolution ofapproximately 1.9µm at 671nm. While this is larger than what was initiallypredicted, a more thorough test should be done with atoms in a dipole trap.An image taken with the system using a 1µm pinhole is shown in Figure2.11a suggests that there are no major astigmatisms.292.3. Implementation(a)(b)Figure 2.15: A comparison of MOT loading curves at 380○C. The loadingcurve before the imaging system was added is shown in (a) while the curveafter it was added is shown in (b). One should note that for the data withthe imaging system, the Zeeman slowing beam had been telescoped up bya factor of approximately 2 which inevitably changes the intensity profilealong the slowing axis. This can in part be responsible for the differencebetween the two loading curves. 30Chapter 3The Dilating LatticeWithout the use of sophisticated optical arrangements, like those in [23–26],most imaging systems can achieve a resolution comparable to the 2µm reso-lution we achieved with the vertical imaging system. This can be a sufficientresolution for time-of-flight experiments, spectroscopy and even atomic in-terferometry. Certain experiments require the ability to detect single atoms.For a certain class of experiments one can use a micro-channel plate (MCP)based detection scheme to achieve such resolutions, although this methodis specific to metastable atoms and can’t be used to get information in situ[27].A recent development is the creation of the so-called quantum gas mi-croscope which is a high fidelity imaging system capable of resolving separa-tions between atoms less than a micron in size [23–26]. These high resolutionimaging systems require a pinning lattice to hold the atoms in place dur-ing the image. Imaging atoms in a lattice allows one to have well definedpositions for atoms which we showed in Section 2.1.1 can increase the SNRby 3 orders of magnitude. So far, these quantum gas microscopes are bothextremely expensive but also of limited use for imaging a quantum gas thatis not confined by a lattice. This specificity may narrow down the types ofexperiments one can perform with a given apparatus.An alternative to both of these schemes is to expand the spacing betweenthe particles so that their physical separation is greater than the resolutionof the imaging system. This second perspective has lead to the invention ofthe dilating lattice. Simply put, these lattices are created with a mechanismfor dynamically varying the lattice spacing. The is commonly achieved bychanging the separation of two parallel beams before they are focused bya lens to then create a lattice as depicted in Figure 3.1. While there are313.1. Theoryvarious schemes for doing so, we have modified the design used by MarkRaizen’s group [28]. This modification and proof-of-principle is detailed inthis chapter.Figure 3.1: A depiction of the mechanism responsible for the interferencecreating the lattice. If two beams intersect at some oblique angle, they cangenerate an interference pattern with a periodicity dependent on their angleof intersection. A lens will take two beams and intersect them at some pointon the focal plane, which means the angle of intersection is dependent onthe separation of the two beams d.3.1 TheoryAn optical lattice is generated by the interference of at least two light waves.Usually, this takes the form of two laser beams intersecting one another ata desired location. So, to start, we refer to the standard wave equation∇2u(x⃗, t) − 1c2∂2u(x⃗, t)∂t2= 0 (3.1)where u(x⃗, t) is the wavefunction and c = c0n is the reduced speed of thewave in the medium. Since this is a linear equation, we know that obeys thelaws of superposition, which is the key to understanding interference. Sincewe will be assuming our light is monochromatic, it makes sense to writeour wavefunction as the real part of some complex wavefunction U(t, x⃗). In323.1. Theoryparticular, we usually writeU(x⃗, t) = U0(x⃗)eiϕ(x⃗)ei2piνt (3.2)where ϕ(x⃗) is the phase of the wave, ν is the frequency and U0(x⃗) is somespatially varying amplitude. We often define the complex amplitude to beU(x⃗) = U0(x⃗)eiφ(x⃗) and therefore we can then define the optical intensityI ∣(x⃗)∣ = ∣U(x⃗)∣2.Now, these U(x⃗, t) also obey the wave equation and hence they also obeythe superposition principle. Therefore, the intensity of a wave composed oftwo waves with complex amplitudes U1(x⃗) and U2(x⃗) is given byI = ∣U1∣2 + ∣U2∣2 +U∗1U2 +U1U∗2 ,= I1 + I2 + 2√I1I2 cosϕ, (3.3)where ϕ = ϕ1 − ϕ2 is the phase difference between the two waves.3.1.1 Plane Wave InterferenceTo start with a more simple analysis of our system, we examine two planewaves intersecting at some angle 2θ. If we assume both plane waves arepolarized along the axis perpendicular to their plane of intersection, we candrop the polarization of the waves and simply writeU1 = √I1e−ik(cos θz+sin θx), U2 = √I2e−ik(cos θz−sin θx). (3.4)Using Equation 3.3, we find then that thatI = I1 + I2 +√I1I2ei2kx sin θ +√I1I2e−i2kx sin θ,= I1 + I2 + 2√I1I2 cos(2kx sin θ), (3.5)which in turn implies that ϕ = 2kx sin θ. Hence, the interference has aperiodicity of pi/k sin θ or, in terms of wavelength, λ/2 sin θ. If we refer again333.1. Theoryto Figure 3.1, if this is generated by a lens, we obtainsin θ = 1√1 + (2f/d)2 (3.6)where f is the focal length and d is the separation of the beams. We canthen see that the lattice periodicity is given bya = λ2¿ÁÁÀ1 + (2fd)2 (3.7)which means that if one were to generate a lattice using two 532nm beams,using a lens with a diameter of 50mm and a focal length of 40mm, we couldobtain a lattice spacing as small as 500nm. In the limit where d/2 << f wefind that a ≈ λf/d which means one could theoretically achieve an arbitrarilylarge lattice spacing. One needs to consider that, as a consequence of thedivergence of the lattice spacing as d → 0, a small change in d will resultin a large change in the lattice spacing. Therefore, the stability of thelattice spacing is becomes strongly correlated with the stability of the opticalcomponents. For instance, vibrations in mirror mounts can cause smallangular changes which turn into non-negligible changes in d and thereforelarge changes in a. Therefore, one will reach a point where the stability ofthe lattice spacing is predicated on the stability of the optical path.3.1.2 Gaussian Beam InterferenceWhen dealing with lasers, light is usually in the form of Gaussian beamsand so we will hereby take into account the higher order effects the comewith this more complicated situation. Recall that a Gaussian beam has acomplex amplitude given byU(r⃗) = U0 W0W (z)e− x2+y2W (z)2 e−i(kz+k x2+y22R(z) −ζ(z)). (3.8)If we use this expression in Equation 3.3, we can analytically determine theintensity field. This was done in [29] and assuming the beams are angled in343.1. Theorythe xz-plane, as shown in Figure 3.2, we can write that the phase differenceasϕ = −k(z1 − z2) + ζ(z1) − ζ(z2) − k2[x21 + y21R(z1) − x22 + y22R(z2) ] (3.9)Figure 3.2: A depiction of the scenario described in Section 3.1.2. Here wecan see the relationship between the lab coordinate system (x, z) and thebeam coordinate systems (x1, z1) and (x2, z2). Note the lab frame positionsof the beam waists shown as (xW1 , zW1) and (xW2 , zW2) in the figure. Lastly,the angle θ bisects the angle between the two beams defining a natural wayto orient the lab frame’s coordinate system.353.1. Theorywherex1 = x cos θ + z sin θ,x2 = x cos θ − z sin θ,z1 = −(x − xW1) sin θ + (z − zW1) cos θ,z2 = (x − xW2) sin θ + (z − zW2) cos θ, (3.10)and the subscript Wi denotes the location of beam i’s waist in space. Thisleads to a the lattice spacing being expressed asa = λ2 sin θ⎡⎢⎢⎢⎢⎢⎢⎣1 +( x1z1z21+z201 − x2z2z22+z202 )2 tan θ − ( x1z1z21+z201 − x2z2z22+z202 )⎤⎥⎥⎥⎥⎥⎥⎦ (3.11)where z0i = piW 20iλ . Note that the first term is the same as we found forplane waves, and it is the second term that is the deviation from this simplebehaviour. If both beams have the same waist size and position along thez−axis, Equation 3.11 simplifies toa = λ2 sin θ(1 + 1φ − 1) (3.12)whereφ = 2z1z21 + z20x1 − x2 tan θ. (3.13)This geometric factor φ now encodes all of the specifics of the Gaussianbeams. One should note that the case when θ → 0 means that the latticespacing scales twice as fast as in the plane wave case. Likewise if θ ∼ 45○ thenthe points where z1,z0,x1 and x2 all be comparable also have a lattice spacewhich is twice the plane wave case. As such, we can see quite a dramatic shiftaway from the simple plane wave case by including this geometric factor.An example of what the intensity pattern would look like for the caseof two equivalent beams is shown in Figure 3.3. These simulations will letus more accurately determine the lattice spacing as we change the angular363.2. Designseparation of the incoming lattice beams.Figure 3.3: The prediction for the intensity pattern created by interferingtwo identical Gaussian beams propagating relative to each other by 3○.The beams are at a wavelength of 532nm, with beam waists of 50µm at theinterference plane.3.2 DesignGiven our understanding of interference as a phenomena created by differ-ences in phase, the two main issues the dilating lattice design one mustaddress is how one will actually dilate the lattice as well as how the phase,from the path difference between the beams, will be kept constant. Thissecond aspect is in some ways secondary, however it is crucial to the actualimplementation of the system as a variation in the difference in path lengthwill cause the lattice to shift.We originally aimed to implement the design shown in Figure 3.4. It isa modification to a Michelson interferometer, whereby the lens L2 acts to373.2. Designflip the vertical position of the horizontal beam. As one can see, we shiftthe beam splitting cube labelled B1 along the vertical direction to changethe vertical beam’s horizontal placement. This is the mechanism by whichwe would change the separation of the beams before they are focused, andsubsequently interfered, by the lens L1.Unfortunately, the usage of L1 means that one arm of the interferome-ter has a changing path length with respect to the other as the angled pathbetween L2 and M2 varies as the beams are separated. One could con-ceivably account for this with some mechanism to shift M1 accordingly butthis solution seemed inelegant. We wished for a system that had a constantdifference in path length without the need for some active stabilization.To that end, we adopted the design in [28]. One can easily see that therelative path lengths of the two arms are always the same, within the uncer-MotionB1B2HWP1QWP1L2M2QWP2M1L1B: Polarizing Beam CubeL: LensM: MirrorQWP (HWP): Quarter (Half)WaveplateFigure 3.4: The original design for our dilating lattice. By translating theB1 polarizing beam splitting cube, we change the horizontal position of thevertical beam. We then use the L2 lens, along with the M2 mirror, toreflect one of the split horizontal beams about the lens’ axis. This createstwo parallel beams with a tuneable separation which we can then interfereusing the L1 lens.383.2. Designtainty due to components. One would need to implement a compensationpath to initially adjust the path lengths of the two arms so that the beamshave the same divergence and size at L1, in Figure 3.5, since we are assum-ing we will use Gaussian beams. Unfortunately for this design, to achieve alarge lattice periodicity, one must place the beams very close to one another.In this design, how close they can get is constrained by the quality of thebeam cubes at their edges.It is at this point we modified this design to account for this problem.The final design is depicted in Figure 3.6. The modification is done so thatthe two beams are recombined in one beam cube. This is beneficial becausewe can then move the beams arbitrarily close to one another, generating alattice with as large a periodicity as one would like. It limits how small alattice one can achieve, however increasing the size of the beam cube B3 canbe used to decrease this restriction. It also maintains the constant relativepath length property and as such, one can just stabilize the mirror M2 toFigure 3.5: An alternative design from [28]. This uses a horizontal (orvertical) translation of M1 to determine the location at which the beam issplit on B1. The symmetry of this design creates a second parallel beamand we implement a compensating path to adjust the relative path length.393.2. Designaccount for vibrations or other changes in the system.Using this design, we can reasonably achieve lattice spacings varyingfrom 0.8µm to 50µm using 532nm light and a lens with a numerical apertureof 0.55. To then determine the minimum sizes of the components we canrearrange Equation 3.7 to getd = ¿ÁÁÀ (2f)21 − (2aλ )2 (3.14)which means B3, P1 and L1 must be at least 28.2mm in diameter.To be able to take advantage of such a large lattice spacing we wouldrequire the beam diameters be large at the focal plane. We therefore analyzethe effect of the initial waist position on the beam size around the focal planeafter the lens L1. Recall that the Gaussian beam is characterized by its qparameter and for an optical system defined by its ABCD matrix, we haveFigure 3.6: A modified version of the design presented in [28]. This designrecombines the split beams in a third beam cube B3 to allow for arbitrarilysmall beam separation.403.2. Designthatq2 = Aq1 +BCq1 +D 1q2 = C +D1q1A +B 1q1 . (3.15)Now, if we first consider an incident beam on the lens that is then mod-ified by its transmission through the lens and then the free space after ituntil the focal plane, the ABCD matrix would be⎛⎝ A BC D ⎞⎠ = ⎛⎝ 1 f0 1 ⎞⎠⎛⎝ 1 0− 1f 1 ⎞⎠ = ⎛⎝ 0 f− 1f 1 ⎞⎠ . (3.16)Hence, we can see that1q2(z˜) = 1R2(z˜) − i λpiW2(z˜)2 (3.17)= 1f− q1f2= f − zf2− i z0f2(3.18)where ˜˜z is the distance between the focused beam’s waist and the focal planeof the lens. If we then take the imaginary part of Equation 3.17 and compareit to the imaginary part of Equation 3.18, we find thatz0f2= λpiW2(z˜)2 Ô⇒ W2(z˜) = λpi fW0 . (3.19)This tells us that the beam size at the focal plane is independent of theinitial beam’s waist location. Therefore, if we want W (z˜) ≥ 50µm, then thisrequires that W0 ≲ 135µm. As the lattice will exist in some spatial regionaround the focal plane of the imaging lens, we need to understand the trapshape for different input Gaussian beam parameters. Figure 3.7 summarizesthe complicated interplay between the initial beam waist, the initial waistlocation and the final beam size.In Figure 3.7d, we can see that the beam size at the image plane (whichis 4cm in this simulation) is independent of the initial beam’s waist location413.2. Designas we just derived. From Figure 3.7b, we can also see that the most col-limated beam occurs when the initial beam waist’s location is at the focallength of the lens (which is again 4cm in this simulation). Other beam waistlocations give rise to beam which have a larger variation in beam size alongthe propagation axis (here labelled as the imaging plane location). Thismakes sense as the lens then acts to collimate the beam. Needless to say,these parameters will be adjusted in the final setup to make sure that bothbeams have a beam size of at least 50µm at the focal plane of the lens.423.2. Design(a)(b) (c) (d)Figure 3.7: A depiction of the Gaussian beam size at different positionalong the propagation axis (labelled here as the imaging plane location) fordifferent initial beam waist sizes and locations. For these simulations, thefocal length of the lens is 40mm. The black line marks the focal plane ofthe lens. The entire surface relating the three parameters is shown in (a)while (b),(c), and (d) are projections along each axis to make reading theplot easier. A cross section of (a) taken parallel to the plane of (d) is theGaussian beam size as a function of z. As such, different cross sections relatewhat the different output Gaussian beams look like for different initial waistlocations. The projection of the surface in (b) gives you an idea of thecollimation of the output beams for different initial waist locations. Notethat the beam is tightly focused if the beam waist is located on the oppositeside of the lens compared to the beam was (i.e. at -10cm) while at an initialbeam waist location of 5cm the beam is significantly more collimated. Theprojection in (c) shows that the focus of the Gaussian beam changes as theinitial beam waist locations changes as we would anticipate. 433.3. Implementation3.3 ImplementationTo test the design depicted in Figure 3.6, we built a test system shown inFigure 3.8 using 780nm light due to its abundance in our lab. We useda simple translational stage for adjusting the beam separation which wecould then detect on the camera. To determine accurately the relative beamalignments, we set up a pair of flip mirrors to allow us to switch between aFigure 3.8: The setup pictured here is a realization of the design shown inFigure 3.6 with the main difference being the addition of the flip mirrors.This pair of mirrors allows us to alternate between the short green path andlong red path which travels out of the frame. Not pictured here are twomirrors that connect the long beam path at the bottom of the picture. Dueto the higher height of the camera, we needed to include a periscope P1to raise the beam heights to match that of the camera. A diagram of theperiscope is shown in Figure 3.9. The unlabelled optics before M1 are forshaping the beam and preparing its initial polarization.443.3. Implementationshort and long path before striking the camera.This allowed us to quickly determine if the beams were parallel by de-tecting their locations on the camera and looking for changes in their relativepositions. A sample image showing the two beams on the camera is shown inFigure 3.10. By iterating between the two paths, we could achieve a relativeshift of ∼ 0.5mm over an extra 1m of propagation (which equates to a 0.03○angular difference).Using the lens labelled L1 in Figure 3.8, we interfered the two beams andimaged the resulting intensity pattern. We then tuned the translation stageand took pictures for various translations. A selection of those pictures isshown in Figure 3.11.As a sanity check we unlocked the 780nm lasers and saw the interfer-ence disappear. This is due to the laser’s frequency drifting on a timescalemuch faster than the exposure time of the camera and thus any interferencepattern is blurred out.Once we locked the laser, we saw that we recovered the interferencepattern we expected to see. At this point, we translated M1 and saw thatthe interference pattern’s periodicity varied accordingly. This is shown inHWPCamera(a) (b) (c)(a)(b)(c)Figure 3.9: The diagram of the periscope P1 from Figure 3.8. We useda polarizing beam cube as the bottom mirror to project the two latticesbeams’ polarizations onto the same axis. At point (a) the polarizations ofthe two incoming lattice beams are orthogonal as is shown in the inset.We rotate the polarizations so that they’re 45○ to the conventional ”S” and”P” polarizations that way when the two beam reflect from the cube, theirpolarizations are parallel. If we did not include this cube, the lattice beamswould not interfere.453.3. ImplementationFigure 3.11. The rotation of the interference pattern is also expected becausethe system isn’t perfectly aligned. This is caused by the difference in beamheight’s as they hit the lens as shown in Figure 3.12. This can serve as atest of the positional alignment of the two beams relative to the lens. Onemight find this rotating behaviour undesirable but it should be noted thatit can be eliminated by using a cylindrical lens, although if two orthogonallattices were to be generated by a single lens this solution wouldn’t work.Figure 3.10: A sample image showing the two dilating lattice beams on thecamera and the output of our peak finding algorithm. The algorithm takesthe picture shown at the top and determines the center position of the twobeam. It does so by fitting a Gaussian intensity distribution to each regionof interest (as outlined in the top picture). This allows us to accuratelydetermine the beams’ positions.463.3. Implementation(a) Start: 0mm (b) 0.30mm (c) 0.60mm (d) 1.00mm(e) 1.30mm (f) 1.70mm (g) 2.00mm (h) 2.50mmFigure 3.11: A demonstration that the system performs as expected. With-out calibrating the entire system absolute positions are meaningless so wehave labelled each picture by the amount the translation stage was moved.We can see that the periodicity changes as we vary the M1 mirror’s place-ment. Note the orientation of the lattice rotating as the beam separationchanges due to the imperfect alignment of the system.Figure 3.12: A cartoon demonstrating the effect of imperfect alignment onthe interference axis. Here the pairs of coloured dots represent the positionsof the pairs of dilating lattice beams. The dashed lines that intersect thesepairs of points denote the axis along which the pairs of beams will interfereto form the interference pattern. Notice how, if the beams are misaligned,the axis along which interference occurs changes as the beams get closer.473.4. Beam Translation Options3.4 Beam Translation OptionsIn the above work we manually adjusted the beam’s position before it wassplit on the first beam cube B1. We wish to be able to vary the separationof the beams on the timescale of a typical experiment, which is ∼ 1s, in arepeatable fashion. To this end, we investigated a couple of different optionsfor translating the beam.3.4.1 Linear ActuatorsThe first linear actuator we considered attaching to a translation stage was aT-NA Micro Linear Actuator. As linear motion seemed like the most obviouschoice and we had this actuator in the lab we decided to test it first. Using anArduino UNO as a controller, we implemented a hall sensor on the actuatorto keep track of the position of the actuator. After calibrating the sensorand controller, we found that, although the rate of motion is fast enough forour purposes, there is a lot of jitter associated with the actuator. Since thestability of the mirror in motion will translate to the stability of the beams(and hence the dipole trapping mechanism itself), the presence of jitter inthe motion is thought to cause heating in the trap.We also tried the Newport CMA Linear Actuator. Instead of the Ar-duino, we could control the position and velocity via the Newport UniversalMotion Controller. This device allows us to carefully set the velocity of thestepper motor and displays on the screen the current location. Also, we maychoose to connect this to any computer and control it via a software, whichwill most likely be our choice for controlling the actuator. Furthermore, incomparison to the T-NA Micro Linear Actuator, the Newport actuator isnoticeably far more stable - there is no jittery motion associated with theactuation. While having all of the qualities stated above, we found thisactuator to be extremely slow. The CMA Linear Actuator has a maximumvelocity of 0.4mm/s, which means it will take 50s to travel the entire lengthof the beam splitter cubes, which have a side length of 2cm. 50s is longerthan the expected trapping time for the experiments the lab plans to con-duct, so this rate of motion will not work for the dilating lattice. After doing483.4. Beam Translation Optionssome research on these actuators, there seems to be no way of increasing thevelocity to above 0.4mm/s. If we were to utilize this linear actuator, thenwe would have to use it in some other, clever way, instead of simply linearlytranslating the mirror in the setup.We decided to investigate using a linear actuator to rotate a mirror aboutsome axis. To mimic the behaviour of the beam under the linear translationof a beam, we then place a lens after the rotating mirror to have the beamcome out parallel to our standard horizontal axis for each angle of rotation.The rotation was implemented roughly by removing the knob associatedwith the horizontal angling in a standard mirror mount, and replacing itwith the linear actuator as shown in Figure 3.13. The tip of the actuatorwill penetrate through the mount and press against the mirror side wherethe knob usually sits, allowing us to angle the mirror with the actuatorinstead of manually turning the knob.Next, we placed the rotating mirror between the two lenses of our tele-scope for collimating the beam, so that the second lens behaves as part ofFigure 3.13: The rotating setup with the Newport CMA Linear Actuator.As we can see, the knob associated with the horizontal motion is removedand the actuator tip is inserted to achieve the rotating motion.493.4. Beam Translation Optionsthe telescope while making the beams parallel as well. For our test, wefocused the beam with the initial lens such that the waist occurred at themirror, then placed the second lens with a focal length of 10cm after therotating mirror. We quickly realized that the careful positioning of this lenswas very important, since the focus needs to be right at the beam’s axisof rotation in order for this setup to behave properly. This sensitivity wasfurther amplified by the dilating lattice’s sensitivity to alignment. If thelens was either too close or too far from the mirror, the output beam fromthe lens is not going to be parallel for the entire length the mirror scans.To carefully adjust and achieve the focal length to coincide with the axis ofrotation, we placed this lens on a translation stage. After many iterations,the positioned lens was able to send the beams straight out of the plane ofthe lens as required over a distance of over one meter, thus verifying thatthe rotating mirror and lens combination is able to achieve what a singletranslation stage can do at directing the beam.The most significant benefit of this alternative design is the fact thata relatively slow actuator can be transformed into a fast, linear beam dis-placement, since we are adjusting a small angle over a rather long distance;however, along with some benefits, issues that cannot be ignored arise herewhich were not present in the original design. In order to get a reasonablylarge range of motion, one needs to either angle the mirror a substantialamount or the beam needs to propagate for a larger distance. The largerangle requires the actuator to act for longer, decreasing the benefit for slowactuators. The long distance can also have an impact on the stability of thesystem. Another major flaw is the sensitivity of the lens’ position as even asmall adjustment can destroy the parallel beam output.Lastly, we note that with the standard mirror mounts, the point at whichthe beam reflects from the mirror changes as the rotation takes place. This isdue to the mirror mount’s pivot point being offset from the mirror’s centeras shown in Figure 3.14. This is problematic as the distance to the lenswill shift as the rotation takes place, resulting in the beams which are notparallel. Although the beams were relatively parallel over about a meterdistance, we saw this as an issue. To solve this problem, we switched to a503.4. Beam Translation Optionsgimbal mount. Using this mount, we were able to achieve the mirror’s axisof rotation to be at the centre of the mirror - this is necessary in order tokeep the focus of the beam to be fixed such that the outgoing beam’s areparallel with respect to one another.(a) (b)Figure 3.14: (a) The ideal rotation of a mirror achieved using a gimbalmount. The blue dot is the axis of rotation. (b) Non-ideal rotation of amirror using a standard mirror mount. The blue dot is the axis of rotation.3.4.2 Rotation via an AOMInstead of physically rotating a mirror, the idea was to use the first orderbeam from an AOM to translate the beam in a similar setup to the rotatingmirror method. By changing the AOM’s drive frequency, we effectivelychange the angle at which the beam is reflected, hence allowing the AOMto behave the way our rotating mirror did. A benefit to the setup is thefact that we could eliminate the physical translation of an actuator in favourof changing an RF frequency input. This gives us a significant advantageas the AOM is significantly faster than any physical actuator and we cansmoothly sweep the frequency, eliminating any jitter in the motion.Unfortunately, we found that due to the small angular displacement, thechange in position was about 1cm over a path length of approximately 1m.Considering that we are aiming for the beam to translate a total of 2cm, thiswas a significant flaw. One could in principle use a telescope to magnify thisangular displacement but this added complexity was not ideal. However, if513.5. Design of the Monolithic Containerone were to consider a lattice which would have a lattice spacing varyingfrom 1µm to 5µm, this would be more feasible. In this case, if we solve ford in Equation 3.3 we find thatd = 2f√(2aλ )2 − 1 (3.20)If we then take δd to be the difference in space between the beams forthe two lattice spacings, then δd/2 is the distance a single beam must moveto be able to change from a separation of 1µm to 5µm. Recalling that wehave been considering our a 532nm wavelength beam with a lens with a focallength f=40mm, we can substitute in a1=1µm and a2=5µm into Equation3.20 to find that ∆d2 =8.9mm. Since we have already verified that the AOMcan displace the beam about 1cm if it is about a meter away, the AOMmethod may be viable for this type of small separation.3.4.3 Zaber’s T-NA Micro Linear ActuatorAfter we investigated all of these options, it seemed clear that the linearactuator was the best option if we could find a motor with enough rangeand speed. We found that Zaber’s T-NA Micro Linear Actuator satisfied allof our criteria as it is able to travel 25mm at a maximum velocity of 8mm/s.This linear actuator will be able to output up to a 50N thrust, so it will beable to handle any of the translation stages with which we decide to pairit. The quoted repeatability is less than 1µm which corresponds to a verysmall shot-to-shot variation of the beam separations and thus angles (andtherefore also lattice periodicities).3.5 Design of the Monolithic ContainerWhile the proof-of-principle was built on an optical table, one needs to makesure the entire apparatus is interferometrically stabilized to reduced jitterin the lattice spacing and phase. As such I hereby outline a design for amonolithic container for the lattice optics.523.5. Design of the Monolithic ContainerThe simplicity of the optics affords one to make a relatively portable andcompact system which outputs two beams with variable spacing which canthen be directed towards the location one wishes to generate a lattice. Inthis way, one can decouple the more complicated aspect of the design fromthe rest of an experiment’s optical system. Since alignment of this systemseems to be more crucial than most optical systems in ultracold systems,having the option of removing a box containing all of the elements so thatone can realign in a more suitable environment is a significant advantage.Over the course of several weeks we noticed no passive drift in the dilatinglattice system. We did so by periodically taking images of the two beams asshown in Figure 3.10 and analyzed their positions and found no discernabledrift within the uncertainty of our fitting routine which was on the order of300µm. This suggests that so long as one makes a rigid container, the opticswill remain aligned for an extended period of time.The design, shown in Figure 3.15, mimics our realization of Figure 3.6in a more compact manor with the optics designed for this application. Toreduce drift, we have made pedestals, upon which the cubes would sit in-stead of adjustable mounts. Some of the components must be on adjustablemounts so that we can achieve the very accurate parallel beam orientationthat we previously achieved on the optics table. To further increase theportability of the design, we have made it take a fiber for an input so thatone can easily add it to a system with minimal alignment. At the oppositeend, where the two parallel beams will be exiting the container, we will haveto slits for the beam to potentially be able to leave from - this is so that wecan use one of the exits for alignment purposes while the other one will befor the actual dilating lattice. As we lose half of the light by projecting thetwo beams’ polarizations onto the same axis, we have designed it to havetwo output slits - one for alignment purposes and the other for the actualdilating lattice.533.5. Design of the Monolithic ContainerFigure 3.15: A Solidworks drawing of the container. Note that the unlabelledmirrors are normal mirrors for the lattice wavelength and all of the cubes arepolarizing beam cubes. The reason we need two silver mirrors is because thehalf waveplate on the input will tune the polarization of the incoming lightto be an even split between ”S” and ”P” polarized. With normal dielectricmirrors, if the polarization isn’t either just ”S” or ”P” then the reflectedpolarization is not guaranteed to be linear. One should also note that theinput to the box is designed to be a fiber, fed through the hole in the bottomleft corner and mounted in the fiber mount. This makes the system muchmore portable. Also not drawn, there is a second rejection port which hasthe orthogonal polarization pair to the drawn output beams.54Chapter 4Methods for DeterminingCollisional DecoherenceRates4.1 MotivationAnother area of interest within our group is studying many-body physicsusing 6Li and Rb atoms. Within this broad area of interest, we became in-teresting in using Feshbach resonances to tune collisional decoherence ratesof superposition states after discussing our experiment with the authors of[30]. Decoherence is an important mechanism by which a system can tran-sition between quantum and classical realms, and it is therefore of funda-mental interest to physicists. Due to this effect, it is often times a limitingfactor for quantum experiments and collisional dechorence is often timesuncontrollable due to its stochastic nature. However, this proposal wouldgive researchers a simple method of suppressing decoherence using Feshbachresonances.To measure the decoherence rate, we had set about using microwavesfrom an RF coil to induce Rabi oscillations between hyperfine states of6Li. We would then look for decoherence in the oscillations which wouldmanifest itself as a decrease in the amplitude of the oscillations. We couldthen measure this decrease in amplitude as a function of interaction strength.This proposal will be discussed in more detail in Section 4.3.554.2. Theory4.2 Theory4.2.1 Rabi OscillationsRabi oscillations come from the coupling of at least two states via some timevarying electromagnetic field. To see how those oscillations come about, westart by considering the system shown in Figure 4.1. Here we have a lightfield that is detuned from the ∣e⟩ → ∣g⟩ transition by δ (i.e. ω = ω0 + δ).In the electric dipole approximation, we assume that the variation of theelectric field over the dipole is negligible and we can therefore writeE(t) ≈ E0 cos(ωt). (4.1)Therefore, the coupling term in the Hamiltonian can be written asHint = 2Ω cos(ωt) (∣e⟩ ⟨g∣ + ∣g⟩ ⟨e∣) (4.2)whereΩ = −E02⋅ ⟨g∣d ∣e⟩ (4.3)is called the Rabi frequency for reasons that will become apparent soon.The Hamiltonian takes this simple form because the diagonal terms are zerodue to the parity of the dipole operator and we can choose the phase of∣g⟩∣e⟩ωω0−δFigure 4.1: A standard 2-level system with a light field of frequency ω. Herewe note that the energy separation (in units of frequency) between the ∣1⟩and ∣2⟩ state is ω0 and the light field is detuned from this transition by δ.564.2. Theorythe dipole operator such that the off-diagonal matrix elements are real. Letus now consider some general unperturbed state ∣Ψ(t)⟩ = cg ∣g⟩ + cee−iω0t ∣e⟩where the time dependence comes from the unperturbed atomic HamiltonianHatom = h̵ω0 ∣e⟩ ⟨e∣.Note that although the time dependence of each state’s projection isnormally included in the coefficients cg and ce, I have elected to write itas a separate term for clarity in this short aside. If we then consider theexpectation value of the interaction Hamiltonian by this general state onefinds that this gives⟨Ψ(t)∣Hint ∣Ψ(t)⟩ = Ω (eiωt + e−iωt) (c∗ecgeiω0t + c∗gcee−iω0t)= Ω [c∗ecgei(ω+ω0)t + c∗gceei(ω−ω0)t + c∗ecge−i(ω−ω0)t + c∗gcee−i(ω+ω0)t](4.4)If we then focus on systems where ∣ω − ω0∣ = ∣δ∣ << ω + ω0 then we canneglect the rapidly oscillating terms of the form e±i(ω0+ω)t and focus onthe slower frequency terms. This approximation is called the rotating waveapproximation. As such, we can write our simplified Hamiltonian asHint ≈ h̵Ω2(e−iωt ∣e⟩ ⟨g∣ + eiωt ∣g⟩ ⟨e∣) (4.5)We can now write the total Hamiltonian of Figure 4.1 as H = Hatom +Hint. If one now considers some state ∣Ψ(t)⟩ = cg(t) ∣g⟩+ ce(t) ∣e⟩, taking theprojections against ⟨g∣ and ⟨e∣ of the time dependent Schro¨dinger equationgivesi∂tcg = Ω2ceeiωt (4.6)i∂tce = Ω2cge−iωt + ω0ce. (4.7)A standard method for solving these sorts of equations is to switch to arotating reference frame by defining c˜e = ceeiωt, In this new rotating frame,574.2. Theorythese equations can be written asi∂tcg = Ω2c˜e (4.8)i∂tc˜e = Ω2cg + (ω0 − ω)c˜e (4.9)or in matrix form asih̵⎛⎝ c˜ecg ⎞⎠ = H˜ ⎛⎝ c˜ecg ⎞⎠ (4.10)whereH˜ = h̵⎛⎝ −δ Ω/2Ω/2 0 ⎞⎠ . (4.11)To find the eigenstates of this system of equations one simply needs todiagonalize this rotating frame Hamiltonian, the results of which are∣E±⟩ = (δ ± Ω˜) ∣g⟩ +Ω ∣e⟩√2Ω˜(Ω˜ ± δ) (4.12)where the eigenenergies areE± = − h̵2(δ ∓ Ω˜) . (4.13)and Ω˜ = √δ2 +Ω2 is called the generalized Rabi frequency. The energy ofthe coupled states is shown in Figure 4.2 which demonstrates the avoidedcrossing characteristic of coupled states.To illustrate why Ω is called the Rabi frequency, let us consider the casewhere δ = 0. In this case, our eigenstates simplify to∣E±⟩ = ± ∣g⟩ + ∣e⟩√2(4.14)with E± = ± h̵2 Ω2. Then if we started with an atom in the lower state, one584.2. Theorycan write ∣Ψ(t = 0)⟩ = ∣e⟩ = ∣E+⟩ − ∣E−⟩√2(4.15)and therefore, the population in the excited state after some time t isPe(t) = ∣⟨e∣ Ψ(t)⟩∣2 = RRRRRRRRRRR⟨e∣ ⎛⎝e−iΩ2t ∣E+⟩ − eiΩ2 t ∣E−⟩√2⎞⎠RRRRRRRRRRR2= ∣−i sin(Ω2t)∣2= 1 − cos Ωt2. (4.16)Therefore, we see that Ω is the frequency with which the population os-cillates between the two states. This phenomenon is called Rabi flopping andFigure 4.2: A plot of the eigenenergies of the rotating frame Hamiltonianwritten in Equation 4.13. Note that as the magnitude of the detuning in-creases, the system increasingly acts like an uncoupled system as one wouldexpect.594.2. Theoryone can show that for δ /= 0, the population in the excited state (assuming∣Ψ(t = 0)⟩ = ∣g⟩) is given byPe(t) = (ΩΩ˜)2 1 − cos Ω˜t2. (4.17)We can see that if the light is off resonance, the amplitude of the oscilla-tions is not unity meaning the state of the atom is never fully in the excitedstate, as shown in Figure 4.3.Figure 4.3: A comparison of the effect the detuning has on Rabi flopping.Note that this scaling behaviour matches our intuition as a far-from-resonantlaser should cause almost zero coupling between levels in the system.4.2.2 Adiabatic PassageA useful consequence of the avoided cross shown in Figure 4.2 is the abilityto transfer atoms between the ∣g⟩ and ∣e⟩ by simply tuning the frequency.For instance, if the atom is in state ∣g⟩ and one were to sweep the frequencyof the light across the resonance slow enough so that it adiabatically followsthe coupled curves, then when the light is once again far detuned, the atomwill be in the ∣e⟩ state. This is called adiabatic passage and requires thatthe sweep rate is slow enough for the atom to stay in an eigenstate of the604.3. The Effect of Collisional Decoherence on Rabi Oscillationscoupled Hamiltonian.One then naturally asks how slow the sweep rate has to be for the popula-tion to be transferred with almost unit probability. This is what is called theLandau-Zener problem [31, 32] and it has been shown that the probabilityof the atom to not transfer to the desired state is given byPlost = e−pi2 Ω2∣∂tδ∣ (4.18)where ∣∂tδ∣ is the sweep rate. This assumes that you start and finish yoursweep far-detuned, otherwise you would project your initial or final stateonto the eigenstates of the system causing an extra decrease in efficiency.To have close to unit probability of transfer, one wants ∣∂tδ∣ << Ω2.One of the reasons this phenomena is so important is due to the robust-ness of this technique. For Rabi flopping, one is very sensitive to the energysplitting between the ∣g⟩ and ∣e⟩ states as this corresponds to a change inthe detuning. For instance, one can imagine shining an on-resonant laser(δ = 0) on some trapped atoms in the ∣g⟩ state. If one were to then leave thislaser on for a time t = pi/Ω then all of the atoms should be transferred tothe ∣e⟩ state. However, as is the case in many experiments, if there were toexist an inhomogeneous magnetic field in the region of one’s trapped atomsthen the energy splitting can vary as a function of location due to a differen-tial Zeeman shift. This would cause the generalized Rabi frequency to varywith position, quickly causing the different atoms in the trap to oscillateout of phase with one another, meaning that after the laser is turned off thepopulation will no longer be uniformly populating the ∣e⟩ state.Since one is just tuning the frequency across the resonance, shifts inthe resonance position make little difference to the efficiency of adiabatictransfer. This makes it much more robust to a large number of experimentaldisturbances.614.3. The Effect of Collisional Decoherence on Rabi Oscillations4.3 The Effect of Collisional Decoherence onRabi OscillationsAs previously mentioned, we had thought to use the sensitivity of Rabiflopping to our advantage when trying to measure collisional decoherence.The experiment we had in mind started with 6Li atoms in a dipole trapwith the population split evenly between the ∣1⟩ and ∣2⟩ states (as definedin Figure 2.3b). At magnetic field of at least 100G, these states are split byapproximately 76MHz. We can then use an RF coil to drive Rabi oscillationsin the atoms between the ∣2⟩ and ∣3⟩ states which have a splitting of 158MHz.We can then tune the collisional cross section of the ∣1⟩ and ∣3⟩ state atomsFigure 4.4: A depiction of the time evolution of an atom’s state’s compo-nents. Here the red (black) solid line is the population in the ∣2⟩ (∣3⟩) statewithout any collisions. As soon as a collision happens, the atom’s state isprojected onto purely the ∣2⟩ state as represented by the discontinuity be-tween the solid and dashed line. Now any of the atoms, either in the solidor dashed time evolving paths, can have a collision projecting them intothe ∣2⟩ again, this time drawn as the dotted line. One can then see thatquickly these different possible evolutions of state become out of phase withone another. This will then lead to the decrease in the amplitude of Rabioscillations averaged over the entire ensemble.624.3. The Effect of Collisional Decoherence on Rabi Oscillationsaround a Feshbach resonance and examine the effect this has on the rateat which dephasing occurs in the Rabi oscillations. This is because theinelastic scattering of ∣1⟩ and ∣3⟩ state atoms produce two ∣2⟩ atoms whichcarry away approximately 30µK of energy. As such, one can picture theatom superposition state changing because of this loss as shown in Figure4.4. Each time this atom experiences a collision with a ∣1⟩ state atom, the ∣2⟩and ∣3⟩ are set to 1 and 0 respectively due to the collisions products. Sincethe occurence of this collision will happen at different times for differentatoms, one will see that the resulting dephasing of the Rabi oscillations willscale with the collisional cross section. Thus we should be able to tune thedecoherence time of the Rabi flops with the Feshbach resonance.Before proceeding with our experimental results, we should note thata recent paper from Dr. Rudolf Grimm’s group [33] measures a similareffect except, instead of using three hyperfine states of a single species,they use a superposition of two hyperfine states in one atom interactingand decohering due to collisions with a second atom of a different species.Their experimental method uses a RF pulse sequence to interferometricallymeasure the decoherence rate.4.3.1 Experimental ResultsOur first objective was to see if we could see Rabi flopping between the ∣1⟩and ∣2⟩ states as their collisional rate should be small. To do so, we shonehigh-field imaging light (i.e. imaging light which has its frequency tuned tocorrect for the shift in transition frequency due to high magnetic fields) onthe 6Li atoms in the trap to empty one of the two hyperfine states beforeturning on our RF coil for some time. We would then image either of thestates after varying the duration that the RF coil had been on to reconstructthe population in each of the states as a function of time. Unfortunately,even with a 5W amplifier to increase the RF power, and therefore the Rabifrequency, we were unable to see any evidence for coherent Rabi flopping.We simply saw what looked like a random percentage of the total populationin each state. We also tried the transitions between the ∣1⟩ and ∣3⟩ states634.3. The Effect of Collisional Decoherence on Rabi Oscillationsand the ∣2⟩ and ∣3⟩ states but were met with the same noise-like data. Thisnoise would even persist on the shortest time between subsequent commandsfor our experiment (6µs) which seemed to suggest some other environmentalfactor was causing transitions.To test this, we repeated the exact same procedure but didn’t actuallyturn the RF coil on. In this case, we saw the populations in each stateremain fixed, suggesting that the noise was somehow due to the RF fieldwe were exposing the atoms to. We then considered the scenario whereour Rabi frequency was significantly larger than we had anticipated. Inthis case, the noise could be some sort of aliasing of rapid Rabi oscillations.To characterize our coil, and the Rabi frequency it produces, we swept theRF frequency across the resonance to adiabatically transfer atoms betweenFigure 4.5: The efficiency of the transfer between the ∣1⟩ and ∣2⟩ states forvarious sweep rates. We then fit this data using Equation 4.18 to determinethe Rabi frequency which is approximately 100Hz.644.3. The Effect of Collisional Decoherence on Rabi Oscillationsstates. We did this for various sweep rates and this data is shown in Figure4.5. From fitting this data, we determined that the Rabi frequency of ourcoil on the ∣1⟩→ ∣2⟩ to be on the order of 100Hz which was much too low tocause the transitions we were seeing.At this point the experiment started becoming more dedicated to spec-troscopic endeavours required for performing STIRAP, resulting in this re-search being postponed. We did however purchase the 30W Mini-CircuitsLZY-22+ amplifier and build a better RF coil to increase the achievableRabi frequencies for when we return to this research direction.65Chapter 5ConclusionThe tools we have developed over the course of this thesis have been builtto not only push forward towards our goal of STIRAP and the formationof ultracold molecules, but also to enhance the possibility of investigatingmany-body phenomena in a dual species experiment.The vertical imaging system can be used for both 6Li and Rb providedone has two cameras placed at different locations to account for the chro-matic aberration. This increase in the SNR by a factor of 2, due to thenew vertical orientation of our imaging system, will allow one to detect thesmall number of molecules created via STIRAP. The higher resolution andmagnification will also make it possible to image the cloud of atoms anddetect some spatial structure within the cloud. Currently, the entirety ofthe atomic cloud takes up approximately 20 pixels on our camera meaningone can only make gross measurements of our system without releasing andletting the atoms expand. In order to take advantage of these new opportu-nities though, one must finish the imaging system by introducing the camerainto the system and focusing it on our dipole trap. As soon as the dipoletrap is aligned and optimized, this will be one of the first things our labdoes.A longer goal is to realize the dilating lattice, prototyped here, along thevertical axis of the experiment. This will allow us to load all of the atomsinto one or two lattice sites, when the lattice is expanded, before decreasingthe lattice spacing and compressing our atoms. One could then investigate2D physics in these pancake potentials on top of the more standard latticephysics experiments to which the dilating lattice would provide access. Mostof the work presented on this lattice was discussed using 532nm light butwe are currently considering using 1064nm light from our dipole trap lasers66Chapter 5. Conclusionto form the lattice. In order to realize this design however, one needs towork out how to stabilize the lattice apparatus interferometrically in orderto minimize changes in the lattice’s position and spacing.Finally, the work towards observing Rabi flopping between hyperfinestates of 6Li remains unfinished. After some encouraging results, the focusof the experimental work shifted to making molecules. In preparation forthe resumption of this research, a new RF coil has been made and we havebought a 30W Mini-Circuits LZY-22+ amplifier. This coil has been designedto better impedance match our RF input, increasing the radiated powerand hence our Rabi frequencies. The amplifier was purchased with the samemotivation and together they should be able to generate Rabi frequencies onthe order of 100kHz which is 4 orders of magnitude larger than is currentlyaccessible. This then means one will be able to probe shorter timescales asthe oscillations will happen faster, giving one insight into the origin of ourobserved noise. Once this noise is identified and eliminated, one will haveall of the tools necessary to make investigate the effect Feshbach resonanceshave on collisional decoherence rates.Together, these three tools and techniques lay the foundation for ourresearch in many-body quantum effects. The lattice will act both as a trapand tuning parameter, the Rabi oscillations due to the applied RF radiationwill act as both a parameter and probe while the imaging apparatus isessential for the lattice and data acquisition. One of the motivating examplesof this many-body research is the ability to simulate a Kondo Hamiltonianusing the heavy Rb atoms pinned in a lattice immersed in a trapped gasof 6Li atoms. It has been shown that this system is analogous to electronsmoving in a metal [6] and therefore one can experimentally probe Kondophysics using cold atoms. This and other such experiments will now beavailable due to the work discussed in this thesis.67Bibliography[1] W. Bowden. An experimental apparatus for the laser cooling of lithiumand rubidium. Master’s thesis, UBC, 2014.[2] W. Gunton. Photoassociation and feshbach resonance studies in ultra-cold gases of 6li and rb atoms, 2016.[3] H. Mu¨ller, A. Peters, and S. Chu. A precision measurement of the grav-itational redshift by the interference of matter waves. Nature, 463:926–929, February 2010.[4] D. DeMille, S. B. Cahn, D. Murphree, D. A. Rahmlow, and M. G.Kozlov. Using molecules to measure nuclear spin-dependent parity vi-olation. Phys. Rev. Lett., 100:023003, Jan 2008.[5] Johannes Kofler, Mandip Singh, Maximilian Ebner, Michael Keller, Ma-teusz Kotyrba, and Anton Zeilinger. Einstein-podolsky-rosen correla-tions from colliding bose-einstein condensates. Phys. Rev. A, 86:032115,Sep 2012.[6] J. Bauer, C. Salomon, and E. Demler. Realizing a kondo-correlatedstate with ultracold atoms. Phys. Rev. Lett., 111:215304, Nov 2013.[7] Mariusz Semczuk, Xuan Li, Will Gunton, Magnus Haw, Nikesh S. Dat-tani, Julien Witz, Arthur K. Mills, David J. Jones, and Kirk W. Madi-son. High-resolution photoassociation spectroscopy of the 6li2 13Σ+gstate. Phys. Rev. A, 87:052505, May 2013.[8] Will Gunton, Mariusz Semczuk, Nikesh S. Dattani, and Kirk W. Madi-son. High-resolution photoassociation spectroscopy of the 6li2 a(11Σ+u)state. Phys. Rev. A, 88:062510, Dec 2013.68Bibliography[9] A. Micheli, G. K. Brennen, and P. Zoller. A toolbox for lattice-spinmodels with polar molecules. Nat Phys, 2(5):341–347, May 2006.[10] D. Jaksch and P. Zoller. The cold atom hubbard toolbox. Annals ofPhysics, 315(1):52 – 79, 2005. Special Issue.[11] G. Igel-Mann, U. Wedig, P. Fuentealba, and H. Stoll. Ground-stateproperties of alkali dimers xy (x, y=li to cs). The Journal of ChemicalPhysics, 84(9), 1986.[12] V. Tarnovsky, M. Bunimovicz, L. Vukovi, B. Stumpf, and B. Bederson.Measurements of the dc electric dipole polarizabilities of the alkali dimermolecules, homonuclear and heteronuclear. The Journal of ChemicalPhysics, 98(5), 1993.[13] M. Aymar and O. Dulieu. Comment on calculation of accurate perma-nent dipole moments of the lowest +1,3 states of heteronuclear alkalidimers using extended basis sets [j. chem. phys.122, 204302 (2005)].The Journal of Chemical Physics, 125(4), 2006.[14] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis,J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye. A highphase-space-density gas of polar molecules. Science, 322(5899):231–235,2008.[15] J. Deiglmayr, A. Grochola, M. Repp, K. Mo¨rtlbauer, C. Glu¨ck,J. Lange, O. Dulieu, R. Wester, and M. Weidemu¨ller. Formation ofultracold polar molecules in the rovibrational ground state. Phys. Rev.Lett., 101:133004, Sep 2008.[16] Y. C. Chen, C. E. Simien, S. Laha, P. Gupta, Y. N. Martinez, P. G.Mickelson, S. B. Nagel, and T. C. Killian. Electron screening andkinetic-energy oscillations in a strongly coupled plasma. Phys. Rev.Lett., 93:265003, Dec 2004.[17] Russell A. Hart, Pedro M. Duarte, Tsung-Lin Yang, Xinxing Liu,Thereza Paiva, Ehsan Khatami, Richard T. Scalettar, Nandini Trivedi,69BibliographyDavid A. Huse, and Randall G. Hulet. Observation of antiferromag-netic correlations in the hubbard model with ultracold atoms. Nature,519(7542):211–214, Mar 2015. Letter.[18] William Bowden, Will Gunton, Mariusz Semczuk, Kahan Dare, andKirk W. Madison. An adaptable dual species effusive source and zee-man slower design demonstrated with rb and li. Review of ScientificInstruments, 87(4), 2016.[19] B. Deh, W. Gunton, B. G. Klappauf, Z. Li, M. Semczuk, J. Van Dongen,and K. W. Madison. Giant feshbach resonances in 6Li-85Rb mixtures.Phys. Rev. A, 82:020701, Aug 2010.[20] Z. Li, S. Singh, T. V. Tscherbul, and K. W. Madison. Feshbach reso-nances in ultracold 85Rb-87Rb and 6Li-87Rb mixtures. Phys. Rev. A,78:022710, Aug 2008.[21] K. Dare. The design and characterization of a high-resolution imagingsystem, 2014.[22] M. Semczuk. 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