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Quantum Metrology with Bose-Einstein Condensates

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Quantum Metrology with Bose-Einstein Condensates Submitted in accordance with the requirements for the degree of Doctor of Philosophy SCHOOL OF PHYSICS & ASTRONOMY Quantum metrology with Bose-Einstein condensates Jessica Jane Cooper March 2011 The candidate confirms that the work submitted is her own, except where work which has formed part of jointly-authored publications has been included. The contribution of the candidate and the other authors to this work has been explicitly indicated below. The candidate confirms that the appropriate credit has been given within the thesis where reference has been made to the work of others. Several chapters within this thesis are based on work from jointly-authored publi- cations as detailed below: Chapter 5 - Based on work published in Journal of Physics B 42 105301 titled ‘Scheme for implementing atomic multiport devices’. This work was completed with my supervisor Jacob Dunningham and a postdoctoral researcher David Hallwood. Chapter 6 - Based on work published in Physical Review A 81 043624 titled ‘Entan- glement enhanced atomic gyroscope’. Again, this work was completed with Jacob Dunningham and David Hallwood. Chapter 7 - The research presented in this chapter has recently been added to a pre-print server [arxiv:1101.3852] and will be soon submitted to a peer-reviewed journal. It is titled ‘Robust Heisenberg limited phase measurements using a multi- mode Hilbert space’ and was completed during a visit to Massey University where I collaborated with David Hallwood and his current supervisor Joachim Brand. Chapter 8 - Based on work completed with Jacob Dunningham which has recently been submitted to New Journal of Physics. It is titled ‘Towards improved interfer- ometric sensitivities in the presence of loss’. For all the above work I was the main contributor to the research but sought as- sistance and guidance from the other authors. The results of Chapter 7, however, contain some numerical results which were obtained using code written in MATLAB. There were two main sections of code (one to create the initial wavefunction and one to model particle losses from the system) one of which was written by David Hallwood (the initial wavefunction code) and the other by myself (the loss code). This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. c 2011 The University of Leeds and Jessica Jane Cooper Acknowledgements I would like to begin by thanking Jacob Dunningham for his supervision during my PhD. He has provided me with an interesting project and has always been around to help me work through the many problems I have encountered due to my lack of understanding. He has patiently answered my repeated questions and his knowledge and passion for physics have often inspired me. I would also like to thank David Hallwood for answering all my trivial questions. He too provided me with an interesting project as the work presented in Chapter 7 is the result of his idea. I must also take a moment to thank both David and his current supervisor, Joachim Brand, for their hospitality on a visit to Massey University. They both made me feel very welcome and some promising results were obtained from this visit. On non-project related matters I would like to thank Neil Lovett for some great times over the last few years. He has been, and I’m sure will continue to be, a great friend - it has been a lot of fun sharing an office. I am especially grateful for the welcome distractions from thesis writing and the many games of WordTwist we have enjoyed. Neil has also patiently helped me with my computing problems and taught me many valuable computing related skills. I must also thank all the other members of the QI group, in particular, Michal, Andreas and Jonathan for some fun times. Finally, I would like to thank my friends and family for their continued support and belief in me. A special thank you must go to Simon who has, as well as showing interest in my work, put up with my worries about my PhD. He has provided great encouragement throughout. i ii Abstract The ability to make ultra-precise measurements is fundamentally important to sci- ence as it allows theories to be tested and refined. Interferometers offer unrivalled measurement precision and therefore form the basis of many metrology schemes. Research has shown that by using quantum states as inputs to interferometers, pre- cisions better than anything possible classically can be achieved. Nevertheless, these states are difficult to produce and fragile to particle losses. Consequently, classical inputs, which are extremely robust, are used in experiments. Here, however, we pro- pose experimentally accessible schemes to make quantum-limited measurements, in particular rotation measurements using Bose-Einstein condensates, that are robust to losses. We begin by describing how, by loading a Bose-Einstein condensate into an optical ring lattice, multiport beam splitters are created through a simple raising and lowering of potential barriers between sites. We then use these ‘splitters’ to create an atomic gyroscope. We demonstrate how to create several quantum states in the gyroscope, all capable of making rotation measurements. Whilst NOON states afford best precision in idealised set-ups, we find they are outperformed by ‘bat’ states for modest loss rates. However, bat states are not ideal as they are outperformed by classical states for large losses. A second gyroscope scheme is therefore developed. Using multiple momentum modes, rather than just two, we show quantum-limited precisions can be reached using states that have similar robustness to classical states. The final section focuses on the precision of linear interferometers. Recent work iii [1, 2] has calculated the theoretical optimum initial states for two-mode lossy inter- ferometers. Here we present an experimental way to produce initial states that afford similar precisions to this optimum. We also consider lossy multimode interferometry and demonstrate a potential advantage over two-mode systems. It is thought with further investigation other advantages will be found. iv Thesis publications 1. ‘Towards improved interferometric sensitivities in the presence of loss’, J. J. Cooper and J. A. Dunningham, submitted to New Journal of Physics (2011). 2. ‘Quantum metrology with rotating matter waves in different geometries’, J. A. Dunningham, J. J. Cooper and D. W. Hallwood, arxiv:1102.0164 (2011). 3. ‘Robust Heisenberg limited phase measurements using a multimode Hilbert space’, J. J. Cooper, D. W. Hallwood and J. Brand, arxiv:1101.3852 (2011). 4. ‘Entanglement enhanced atomic gyroscope’, J. J. Cooper, D. W. Hallwood and J. A. Dunningham, Physical Review A 81, 043624 (2010). 5. ‘Measuring atomic NOON states and using them to make measurements’, D. W. Hallwood, A. Stokes, J. J. Cooper and J. A. Dunningham, New Journal of Physics 11, 103040 (2009). 6. ‘Scheme for implementing atomic multiport devices’, J. J. Cooper, D. W. Hall- wood and J. A. Dunningham, Journal of Physics B 42, 105301 (2009). 7. ‘Single particle nonlocality with completely independent reference states’, J. J. Cooper and J. A. Dunningham, New Journal of Physics 10, 113024 (2008). v vi Contents Acknowledgements i Abstract ii Thesis Publications v List of Figures xiii 1 Introduction 1 1.1 The classical and quantum worlds . 1 1.1.1 The uncertainty principle . 3 1.2 Quantum metrology . 4 1.3 Thesis aim . 7 1.4 Thesis overview . 7 2 Bose-Einstein condensates 11 2.1 History . 12 2.2 Bose-Einstein condensation in an ideal gas . 13 2.3 Creating a Bose-Einstein condensate . 16 2.3.1 Laser cooling . 16 2.3.2 Evaporative cooling . 17 2.4 The Bose-Hubbard model . 18 2.4.1 Optical potentials . 18 2.4.2 Optical lattices . 20 vii Contents 2.4.3 Single particle system . 21 2.4.4 Derivation of the Bose-Hubbard Model . 22 2.5 Rotating the system . 25 2.5.1 Flow states . 26 2.5.2 Rotations . 29 3 Interferometry and quantum metrology 33 3.1 Interferometry . 34 3.2 Fisher information . 37 3.2.1 Classical Fisher information . 39 3.2.2 Quantum Fisher information . 41 3.3 Improving precision using entangled initial states . 43 3.3.1 Unentangled initial states . 43 3.3.2 Squeezed states . 44 3.3.3 NOON states . 47 3.3.4 Bat states . 50 3.3.5 Multiple pass strategies . 52 3.3.6 Nonlinearities . 54 3.4 Particle loss model . 55 3.4.1 Imaginary beam splitter model . 55 3.4.2 Master equation approach . 60 3.5 Uses of precision measurements in quantum technologies . 62 3.5.1 Frequency standards . 62 3.5.2 Lithography . 63 3.5.3 Gyroscopes . 63 4 An introduction to gyroscopes 65 4.1 Different types of gyroscopes . 66 4.1.1 The mechanical gyroscope . 66 4.1.2 The Sagnac effect gyroscope . 66 viii Contents 4.2 Atoms versus light . 69 4.3 Why use a BEC? . 69 4.4 Units of measurement . 71 4.5 Progress to date . 71 5 Multiport atomic beam splitters 75 5.1 Introduction . 76 5.2 The scheme . 77 5.2.1 Example: The Tritter . 78 5.3 Larger devices . 81 5.3.1 Producing a balanced splitter . 82 5.3.2 The difference between S even and S odd . 85 5.4 Inverse transforms and unbalanced splitters . 86 5.5 Practical limitations to the number of ports . 88 5.5.1 Intensity fluctuations . 88 5.5.2 Condensate lifetime . 89 5.5.3 Interactions . 90 5.5.4 Timing errors . 93 5.5.5 Loss of particles . 95 5.6 Conclusions . 96 6 A scheme to implement a two-mode atomic gyroscope 97 6.1 The system .
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