The development of techniques to prepare and probe at single atom resolution strongly interacting quantum systems of ultracold atoms Martin David Shotter A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Oxford Christ Church University of Oxford Trinity Term 2009 Abstract The development of techniques to prepare and probe at single atom resolution strongly interacting quantum systems of ultracold atoms Martin David Shotter, Christ Church, Oxford University DPhil Thesis, Trinity Term 2009 Strongly interacting many-body quantum dynamics are inherently extremely complex. This makes strongly interacting quantum systems very hard to simulate, pre- dict or understand. In certain circumstances, strongly interacting quantum dynamics may be observed in samples of ultracold atoms. Ultracold atomic systems have unique advantages for the investigation of such dynamics: further to the long coherence times and extremely low temperatures, there is the potential to have a great deal of control over the basic quantum processes. This suggests the possibility using systems of ul- tracold atoms as a quantum simulator, in which the quantum dynamics of one system is simulated by an experiment on a different system, the quantum simulator. Quantum simulation is a part of the emerging field of quantum information processing. Up to the present time, observation of strongly interacting quantum states of ul- tracold atoms has been almost entirely limited to time-of-flight imaging of many thou- sands of atoms. Although one can learn much about the overall nature of the quantum states by this method, detail at the level of individual atoms is lost. My research has focussed on developing novel experimental techniques to make it possible to measure these complex states at the single atom level. In particular, I have developed methods which are capable of making unprecedentedly direct and detailed observation of hun- dreds or thousands of strongly interacting atoms, at the resolution of single atoms, and with a spatial resolution capable of reliably resolving atoms in individual wells of an optical lattice. In this thesis, I describe work that I have carried out during the period of my DPhil, primarily concerned with the design and simulation of these techniques. I have also investigated new methods to prepare the initial quantum states needed for quantum simulation. In addition, I describe earlier work in which I constructed apparatus for, and reached, Bose-Einstein Condensation. Acknowledgements I would like to thank my supervisor, Chris Foot, for his discussions and his sup- port, particularly during the period of my illness. I’d also like to thank all the members of the Clarendon BEC group over the years, particularly those who I’ve worked closely with: Rachel, who showed a newbie the ropes (or at least the coaxes); Andrian and Ben, fellow basement troglodytes; and Dimitris, my most recent co-conspirator. My friends at Oxford kept me in at least passing contact with reality, for which I’m very grateful. In particular, I’d like to mention Myrto, who stuck with me while I was ill, and everyone at KEEN, with whom I’ve shared many eventful and happy hours. But above all, I’d like to thank my parents for the love and support they’ve always shown me. Contents 1 Introduction and motivation 1 1.1 Structure of the thesis . 3 2 Survey of literature related to strongly interacting systems of ultracold atoms and observation of individual ultracold atoms 5 2.1 Ultracold atoms in optical lattices: the Bose-Hubbard model and a quantum phase transition . 5 2.2 Quantum simulation and quantum computation . 8 2.3 Detecting signatures of strongly interacting quantum systems of ultracold atoms . 11 2.3.1 Number squeezing and the Mott insulator transition . 11 2.3.2 Measurements of Mott insulator structure . 13 2.3.3 Direct measurement of sub-Poissonian statistics . 15 2.3.4 A lattice of double wells . 15 2.3.5 Strong interactions at low density in one dimensional systems . 16 2.4 The observation of low numbers of ultracold atoms using fluorescent imaging . 16 2.4.1 Fluorescent imaging in a three-dimensional optical lattice of period 5 microns . 16 2.4.2 Fluorescent imaging in a two-dimensional optical lattice of period 640 nanometres . 17 2.4.3 Fluorescent imaging of a single dipole trap immersed in a MOT 20 2.5 Other techniques used to observe low numbers of ultracold atoms . 21 2.6 Summary . 22 iv Contents 3 Constructing apparatus for Bose-Einstein Condensation and reaching quantum degeneracy 23 3.1 Cooling requirements for quantum degeneracy . 23 3.2 Apparatus . 25 3.2.1 The magneto-optical trap . 25 3.2.2 The magnetic trap . 25 3.2.3 The laser beams . 26 3.2.4 The cooling sequence . 26 3.3 Reaching quantum degeneracy . 27 4 A novel technique to probe strongly interacting ultracold atomic systems at high spatial and single atom resolution 36 4.1 The criteria for observation of individual atoms within a periodic optical potential . 36 4.2 Comparison of techniques . 37 4.3 Fluorescent imaging and the criteria for the observation of individual atoms in a strongly interacting quantum system . 39 4.3.1 Resolution, distinguishability and spatial period . 39 4.3.2 Signal-to-noise ratio . 41 4.3.3 Light-assisted collisions . 42 4.4 The accordion lattice . 44 4.4.1 Collimation of lattice beams for the counter-propagating accordion lattice . 48 4.4.2 Maximum distance between accordion lattice sites . 51 4.5 The imaging lattice . 52 4.5.1 Practical considerations . 54 4.6 Cooling while detecting in the imaging lattice . 55 4.6.1 Outline of cooling methods . 56 4.6.2 The choice of cooling method . 57 4.7 Fluorescent imaging of atoms trapped in the imaging lattice . 59 Contents v 4.7.1 Cooling in three dimensions . 60 4.7.2 Shared or separate optical elements . 61 4.7.3 Translation of the imaging lattice and three-dimensional imaging 62 4.7.4 Fluorescence from out-of-plane atoms . 62 4.7.5 Coherence and reabsorption of scattered radiation . 63 5 Theoretical and computational methods used to model the imaging lattice technique 65 5.1 The mechanisms of polarisation gradient cooling in free space . 65 5.1.1 Orthogonal linear polarisations . 66 5.1.2 Orthogonal circular polarisations . 67 5.2 The mechanisms of polarisation gradient cooling of confined atoms . 68 5.2.1 Orthogonal linear polarisations . 69 5.2.2 Orthogonal circular polarisations . 71 5.3 Three-dimensional polarisation gradient cooling of confined atoms . 71 5.4 Simplification of the simulated system . 72 5.4.1 The imaging lattice . 72 5.4.2 Restricting the number of participating states using repumping . 74 5.5 Quantum description of polarisation gradient cooling . 74 5.5.1 Unitary quantum dynamics . 75 5.5.2 Relaxation mechanisms of the internal states . 76 5.5.3 Explicit retention of the excited states . 76 5.5.4 The Wigner transformation . 77 5.6 The semi-classical approximation . 78 5.7 Conversion to Langevin form . 79 5.8 Unravelling the Optical Bloch equations . 81 5.8.1 Continuous and discrete recoil momentum terms . 82 5.9 Computational methods . 82 5.10 Comment on nature of the simulation . 83 5.11 Extended Monte Carlo analysis . 83 vi Contents 5.11.1 The ‘measure’ and ‘events’ . 84 5.11.2 System dynamics when representative start vectors are known . 85 5.11.3 Finding representative start vectors . 86 5.11.4 Several daughters vectors per mother vector . 88 5.11.5 Focussing the analysis on less frequented regions . 89 5.11.6 The choice of points on the measure . 90 5.11.7 Extended Monte Carlo simulation: summary . 91 5.12 Markovian and non-Markovian analyses . 92 5.12.1 Markovian probability analysis . 92 5.12.2 The Markovian master equation . 94 5.12.3 Outline of non-Markovian treatment . 94 5.13 Implementation of the extended Monte Carlo algorithm for the imaging lattice technique . 95 6 The simulated performance of the imaging lattice technique 97 6.1 Consistency testing . 97 6.2 Frictional energy loss . 99 6.2.1 Free space . 100 6.2.2 Confined atoms . 100 6.3 Specific example of atomic hopping dynamics . 103 6.3.1 Rationale . 103 6.3.2 Parameters used . 105 6.3.3 Discussion of results . 105 6.3.4 Coherence properties of the scattered light . 110 6.3.5 Reabsorption . 111 6.3.6 Summary . 113 6.4 Atomic hopping dynamics over a range of parameters . 113 Contents vii 7 Summary and extensions to the accordion and imaging lattice techniques; potential for novel ultracold physics 116 7.1 High resolution three-dimensional measurements at the level of individual atoms for arbitrary ultracold atomic configurations . 116 7.2 Minimisation of fluorescence from out-of-plane atoms whilst imaging three-dimensional samples . 119 7.3 Summary of the basic technique . 121 7.3.1 Capabilities . 121 7.3.2 An exemplar experimental design . 122 7.3.3 Practicalities . 124 7.3.4 Performance . 127 7.4 Variations on the basic technique . 128 7.4.1 Measurement of the internal state of the atoms at each site of the lattice . 128 7.4.2 Lattices and superlattices . 128 7.4.3 Generalisation of accordion lattice geometry . 129 7.4.4 Using the accordion lattice to prepare a small initial sample of atoms . 129 7.4.5 Longer time-of-flight measurements and higher group velocity samples . 131 7.4.6 The use of shorter-wavelength light . 132 7.5 Comparison with the recent experiment at Harvard . 134 7.6 Outline of prospects for novel physics .