Three-Dimensional Manipulation of Ultracold Atoms using Optical Tweezers Craig S. Chisholm A thesis submitted in fulfilment of the requirements for the degree of Master of Science in Physics The University of Otago, Dunedin, New Zealand August, 2018 Supervisor: Associate Professor Niels Kjærgaard Submitted for examination May, 2018 Abstract This thesis describes the design and construction of a three-dimensional optical tweezer system. The three-dimensional aspects were constructed as a modular upgrade to a previously existing one-dimensional system and were characterised using 87Rb with avenues for future research using 40K in mind. The three-dimensional optical tweezer system was used to demonstrate multiple Bose-Einstein condensation in horizontal and vertical planes and gravitationally driven atomic collisions. In order to take advantage of the three-dimensional nature of the optical tweezer system and to build capabilities with Rydberg quantum optics, vortex lattice studies and atom interferometry, a vertical imaging path was added to the existing experiment. Compared with the existing horizontal imaging path, the vertical imaging path has higher spatial resolution. A magnetic levitation scheme was implemented to prevent depth of focus issues due to atoms falling under the influence of gravity during ballistic expansion in time-of-flight. The levitation scheme was also used to increase the maximum time-of-flight using the horizontal imaging path by more than a factor of two. The improved resolution enabled observation of matter wave interference fringes for the first time in the Kjærgaard lab, which is a first step towards the study of quantum vortices. i Acknowledgements I would like to begin by thanking my supervisor Assoc. Prof. Niels Kjær- gaard, not only for the guidance and support throughout this project and the previous projects I have worked on in his lab but also for inviting me to join his group in the first place. A second person who must be acknowledged is Dr. Amita Deb. Amita was my co-supervisor during my Honour's project and has remained in a very strong and useful mentoring capacity for me since then. Beyond that, Amita has been great company in the lab from his renditions of \Counting the Cattle" by D-A-D to general chats about life, politics and just about anything else. I should also mention the other members of the Kjærgaard lab. Dr. Ryan Thomas has moved from senior PhD student to post-doc during my time in the lab and he has offered help and advice on a number of things, in particular with the radio frequency state preparation mentioned in Chapter6. Bianca Sawyer has provided a good deal of advice and comradery as well as the occasional loan of a text book. Matthew Chilcott has provided advice on various pieces of electronics and is always happy to engage in heated debates. Milena Horvath also provided a number of amusing anecdotes before she left to start a PhD in Innsbr¨uck. The newest member, John Chung, has not been around for long but so far he has ensured that I am no longer the first one to show up and wait in Niels' office for group meetings to begin. Kris Roberts, who left the group before I joined, kindly supplied me with a number of SolidWorks models and useful advice. Ana Rakonjac, who also left the group before I joined, suggested investigating whether the thin dichroic mirrors had become curved as mentioned in Chapter4. A number of technical and administrative staff also need to be acknowledged. In no particular order, I would like to thank: Peter Stroud for improving my de- signs and manufacturing various custom components for the lab, Peter Simpson and Simon Harvey for providing assistance with setting up and debugging the FlexDDS, Anne Ryan for cutting down the objective lens for vertical imaging to fit between the IP trap coils, Shae MacMillan, Diana Evans, Anita Foster, and Nick Theobald for coordinating equipment ordering and shipping. There are also a number of people who provided various other forms sup- port: Wolfgang Wieser supplied custom firmware for the FlexDDS allowing us to operate it outside of the specifications, Assoc. Prof. Jevon Longdell provided Python code for ray tracing, Prof. Brian Anderson, Dr. Tyler Neely, Dr. Mark ii Baker, Thomas Bell, Guillaume Gauthier, Dr. John Helm, Dr. Danny Baillie, and Michael Cawte provided useful discussions; in particular, Mark pointed me in the right direction for magnetic levitation of atoms and Tyler helped me to understand the physics of forming vortices by merging BECs. Special thanks to Jelena Rakonjac for proof reading this thesis and teaching me about the evils of comma splices. Last but not least, my parents provided moral, financial and food based support throughout my time in Dunedin as well as helping me out in all aspects of life in ways which I will never really be able to pay them back for. iii Contents 1 Introduction1 1.1 History of Optical Trapping and Manipulation..........1 1.2 This Thesis.............................2 1.2.1 Layout............................2 1.3 Division of Labour.........................3 2 Background5 2.1 Gaussian Beam Optics.......................5 2.1.1 Wave Equation Solution..................5 2.1.2 Propagation.........................6 2.1.3 Non-ideal Beams......................7 2.2 Dipole Trapping of Neutral Atoms.................8 2.2.1 Two Level Atoms......................8 2.2.2 Cross Beam Dipole Traps................. 11 2.2.3 Harmonic Oscillator Approximation............ 11 2.2.4 Time Averaged Potentials................. 12 2.3 Acousto-optic Deflection...................... 13 2.4 Quantum Gases........................... 15 2.4.1 Bose-Einstein Condensation................ 15 2.4.2 Degenerate Fermi Gases.................. 18 3 The Otago Ultracold Atom Machine 19 3.1 Magneto-Optical Trap....................... 19 3.1.1 Rubidium-87 MOT..................... 19 3.1.2 Potassium-40 MOT..................... 20 3.2 Ioffe-Pritchard Trap......................... 21 3.3 Evaporative Cooling........................ 22 3.3.1 Sympathetic Cooling.................... 22 3.4 Imaging............................... 23 3.4.1 Atom Counting....................... 23 3.4.2 Temperature Measurement................. 24 3.5 Cooling to Degeneracy....................... 25 3.5.1 Measurements of Condensate Fractions.......... 26 3.6 Optical Collider........................... 28 iv 4 Design and Construction of the Three-Dimensional Optical Tweezer System 31 4.1 Constraints and Computer Drawing................ 31 4.2 Construction............................ 32 4.3 ABCD Analysis........................... 34 4.3.1 Three Lens Telescope.................... 35 4.3.2 Geometric Ray Tracing from Fibre to AOD....... 37 4.4 Experimental Control and Power Delivery............ 38 4.4.1 Laser Heating in an Acousto-Optic Modulator...... 39 4.5 Calibration............................. 43 4.6 The Effect of Toggling AOD Driving Frequencies......... 50 4.6.1 Two-Dimensional Toggling Delay............. 52 5 A Showcase of Three-Dimensional Atomic Manipulation 59 5.1 Multiple Bose-Einstein Condensation in One Dimension..... 59 5.2 Multiple Bose-Einstein Condensation Beyond One Dimension.. 61 5.3 A Gravitationally Driven Collider................. 65 5.4 Chapter Summary......................... 68 6 Vertical Imaging System 71 6.1 Optics Selection........................... 71 6.2 Lens Position Tolerancing..................... 74 6.3 Levitating Atoms During Ballistic Expansion........... 75 6.3.1 The Levitation Coil..................... 77 6.3.2 Testing Magnetic Levitation................ 79 6.4 Calibration of the Imaging System................ 83 6.5 Chapter Summary......................... 85 7 Matter Wave Interference 87 7.1 Observation of Matter Wave Interference............. 87 7.2 A Simple Model of Matter Wave Interference........... 90 7.3 Analysis of Interference Data................... 92 7.3.1 Randomness of Relative Phase............... 92 7.3.2 Scaling of Fringe Spacing.................. 92 7.3.3 Ramp Rates and Hold Times............... 97 7.4 Chapter Summary......................... 97 8 Future Directions 99 8.1 Towards the Study of Vortices................... 99 8.2 Optical Ring Traps......................... 103 8.3 Rydberg Quantum Optics..................... 104 Summary and Conclusions 107 References 109 v Appendices 118 A Physical Constants of 87Rb and 40K................ 118 A.1 Rubidium-87........................ 118 A.2 Potassium-40........................ 118 B Temperature of Atoms Trapped by a Given Potential...... 122 C The effect of Gravity on a Dipole Trap.............. 123 C.1 An Analytic Expression for the Trap Depth of a Dipole Trap in the Presence of Gravity.............. 124 C.2 Correction to the Harmonic Oscillator Approximation.. 125 D Measurement of Vertical Trapping Frequency........... 126 E Power Spectral Density of Equation 7.7.............. 128 F The Probability of Observing One Vortex from the Merging of Four Independent Bose-Einstein Condensates........... 128 vi vii Chapter 1 Introduction 1.1 History of Optical Trapping and Manipu- lation Radiation pressure was discussed theoretically by James Clerk-Maxwell as early as 1876 and was used to explain the solar repulsion which causes comet tails to always point away from the sun by Pyotr Lebedew in 1892 [1]. How- ever, radiation on macroscopic objects and absorbing gases was not detected in a laboratory setting until the year 1901 when Nichols and Hull were able to measure the effect of a lamp on a torsion balance holding mirrors at various gas pressures [1]. The