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Philosophical, Theoretical and Experimental Propositions on Wavefunction Collapse

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Philosophical, Theoretical and Experimental Propositions on Wavefunction Collapse Imperial College of Science, Technology and Medicine Department of Physics Philosophical, Theoretical and Experimental Propositions On Wavefunction Collapse Daniel Goldwater Submitted in part fulfillment of the requirements for the degree of Doctor of Philosophy in Physics of the University of London and the Diploma of Imperial College, September 2018 Declaration I, Daniel Goldwater, confirm that the work presented in this thesis is my own. Where infor- mation has been drawn from other sources, it has been appropriately labelled and referenced in the text. The work described in chapter 4 will appear in a forthcoming paper [1]; the results of chapter 5 were reported in [2]; whilst those of chapter 6 appear in [3], and were achieved in collaboration with Dr. Sandro Donadi. Chapter 7 recounts a collaboration with Dr. James Millen, and is reported in [4]. Though I wrote the code for the simulations presented in chapter 7, it was James who produced the plots from these simulations. Copyright declaration: The copyright of this thesis rests with the author. Unless otherwise indicated, its contents are licensed under a Creative Commons Attribution-Non Commercial 4.0 International Licence (CC BY-NC). Under this licence, you may copy and redistribute the material in any medium or format. You may also create and distribute modified versions of the work. This is on the condition that: you credit the author and do not use it, or any derivative works, for a commercial purpose. When reusing or sharing this work, ensure you make the licence terms clear to others by naming the licence and linking to the licence text. Where a work has been adapted, you should indicate that the work has been changed and describe those changes. Please seek permission from the copyright holder for uses of this work that are not included in this licence or permitted under UK Copyright Law. Abstract Collapse models are posited as a resolution to the measurement problem. On one level, they offer a clear, simple and testable resolution to an age-old problem. Yet at the same time, they raise many new questions of their own - what is the origin of the putative noise field? What are its properties, and why ought it couple to wavefunctions in this particular way, inducing collapse in some analogue of the position basis? How might these models be extended to the realm of relativity without incurring catastrophe? What sort of image of the world do they deliver? In this thesis we begin with a philosophical exploration of collapse theories. We examine, in detail, the relationship between stochastic noise fields and the evolution of the wavefunction - shedding light both on the solutions offered by collapse models and the new issues which they raise. We discuss the possibilities for constructing an ontology based on these theories, and look at possible implications for the arrow of time and the meaning of causation. This in turn motivates the development of protocols for experiments which might be capable of probing these models; to new degrees in some senses, and in new forms in others. We develop a comprehensive theoretical model of a levitated nanosphere held in an electric quadrupole trap, and find the limits to which this can probe the characteristic collapse rate λ and correlation length r of collapse models. Further, we develop a novel treatment of this scenario in the style of open quantum systems, and show that such an apparatus can constitute a general quantum spectrometer, capable of characterising arbitrary correlation functions for a noise source coupled to the oscillator - whether that noise be invoked by collapse models or other, more mundane sources. Finally, we utilise numerical simulations of trap dynamics to demonstrate the capabilities of electronic feedback cooling - showing that quantum states ought to be achievable without the use of optics. This work is motivated by a desire to understand the world, and specifically to address some of the paradoxes which arise when we try to use quantum mechanics to do so. We have aimed to follow what we see as best practice in physics - from a motivation within philosophy, to the development of theory capable of meeting that philosophy, to the design of experiments which would be able to speak to the relationship between that theory and the world. Acknowledgements Throughout the years I have spent working on on the topics presented here I have been excep- tionally well supported. Although my name appears as a single author on this thesis, the work contained herein would never have been possible were it not for the care and encouragement of my friends, family and community, and I am indebted to them. To the people I lived with over the last four years { from Oval to Tulse Hill, for their patience and support. To my mother, for her guidance on writing; my father, for his enthusiasm about my research; and my brother, who always offered a haven of escape. And to more friends than I could name, for helping me keep balance, and for their generosity in supporting something so personal, something I generally couldn't explain to them. I have benefited hugely from the learning structure of the Imperial Centre for Doctoral Training in Controlled Quantum Dynamics, which emphasised the social aspects of science from the outset. The directors of the program { Myungshik Kim, Terry Rudolf and the late Danny Segal { have my thanks, as does Richard Thompson for his sage advise. It was through the CDT that I met the members of my cohort, who gave me a fantastic experience of collectivity in science. They, and other friends I've made along the way, have been a wonderful part of the experience. In particular Jon Richens, Max Lock, Lia Li, James Millen, Sandro Donadi, Mauro Paternostro, Bryan Roberts, Uther Shackerley-Bennett and Alessio Serafini have been especially important { for guidance on physics, or friendship, or both. Most of all, of course, I am grateful to my supervisor, Peter Barker. Throughout the entirety of the PhD he has always, always been a source of encouragement and enthusiasm. His support for my ideas has given me license to pursue the projects I've found most exciting, whilst his guidance away from some of my less grounded proposals has saved me countless months. His care, wisdom and creativity have often made the PhD a joy to work on { and have been invaluable when it was not. Contents Abstract iv Acknowledgements v 1 Introduction 1 2 Why Collapse? 6 2.1 Our Philosophical Outlook . .6 2.2 The Measurement Problem . .7 2.3 Decoherence . 10 2.4 The Ontology of the Quantum State . 17 3 Collapse Models 23 3.1 QMSL - A simple model . 24 3.2 Continuous Spontaneous Localisation (CSL) . 30 3.3 What is Real? . 34 3.4 The Dimensionality of Reality . 35 3.5 Collapse Ontologies . 40 3.6 The Nature of the Noise . 46 vii viii CONTENTS 3.7 Remarks on Collapse . 51 4 Indeterminism 53 4.1 Time - the Standard Account . 54 4.2 Causation { Some Minimal Criteria . 59 4.3 An Ontological Arrow for Time . 61 4.4 Summary . 63 5 Testing Collapse 64 5.1 Finding the Effects . 65 5.2 Levitated Nanospheres . 69 5.3 Dynamics of the Sphere . 72 5.4 Noise Sources . 77 5.5 Testing Collapse . 80 5.6 Differentiating Collapse from Decoherence . 82 5.7 Testable Parameter range . 85 5.8 Constraining The Dissipative Collapse Model . 85 6 A Quantum Spectrometer for Arbitrary Noise 89 6.1 Non-White Noise . 90 6.2 Formalism . 91 6.3 An Analytic Solution for Gaussian Noise . 99 6.4 A Practical Application { Electric Field Noise In Paul Traps . 101 6.5 Using the Spectrometer to Test Non-White Models of Wavefunction Collapse . 105 6.6 Conclusions on the Spectrometer . 114 7 Feedback Cooling 116 7.1 Set up and Detection . 118 7.2 Simulating the dynamics . 120 7.3 Resistive Cooling . 123 7.4 Feedback Cooling . 125 7.5 Conclusions . 128 8 Conclusions 129 A Objections to the Everettian School of Thought 132 B Heisenberg Picture Ontology 135 C The Cosmic Temperature of CSLD 137 D Alternative Formalism for Spectrometer 140 E Rotational Dynamics 143 Bibliography 145 ix x List of Figures 2.1 Approaches to quantum mechanics { a decision tree . 22 3.1 GRW localisation process . 25 3.2 Representations of fields in low and high dimensional spaces . 39 4.1 Macrostates and microstates . 55 4.2 Growing Entropy for an increasing t ......................... 56 4.3 Growing Entropy for a decreasing t ......................... 57 5.1 Scaling of collapse noise with object size . 67 5.2 Figure taken from [5], showing the available parameter space for CSL. The blue, green and red lines in the upper section show the space expluded by space ex- periments such as LISA [6]. The purple line comes from cantilever experiments [7], which are fairly similar to the proposal we make here; whilst the grey line comes from X-ray experiments [8], which function because the non-conservation of energy predicted by collapse models ought to lead to spontaneous emission. We can see that the GRW parameter selection is almost ruled out, as are the parameters suggested by Adler [9]. 68 5.3 Schematic for nanoparticle experiment . 71 5.4 Hybrid trap potential . 73 5.5 Heating due to CSL . 82 xi 5.6 Distinguishing CSL via parametric variations . 83 5.7 Probe-able limits of CSL using levitated nanospheres . 84 5.8 Cosmic damping in CSLD . 87 6.1 Resolution of spectrometer . 97 6.2 Spectrometer reconstruction of electric field noise . 104 6.3 Conventional heating rates for a nanoparticle .
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