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.
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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 ...... 108
6.4 Spectrometer performance for Gaussian noise ...... 109
6.5 Exponentially decaying noise in frequency space ...... 111
6.6 Performance of spectrometer on CSL with exp. decaying noise ...... 112
6.7 Performance of spectrometer with different temperatures ...... 113
6.8 Parameter space {λcsl, rc, ωm} for a fixed ϑ ...... 113
6.9 Parameter space {λcsl, rc, ϑ} with optimal ωm ...... 114
7.1 Circuit diagram for resistive cooling ...... 118
7.2 Simulations of resistive cooling ...... 124
7.3 Circuit diagram for feedback cooling ...... 125
7.4 Simplified circuit model ...... 126
7.5 Simulations of feedback cooling ...... 127
xii Chapter 1
Introduction
Quantum mechanics, as a mathematical structure, stands as an extremely effective tool for making predictions regarding quantum physics. The distinction between the two is important, but often overlooked. In physics, we generally follow a deceptively simple procedure:
1. We make abstractions about things in the world such that we can represent them sym- bolically.
2. We then manipulate these symbols using the array of mathematical techniques at our disposal.
3. We then project these abstractions, in their new form, back into the world. We interpret them so as to either make predictions about the way physical events will unfold, or to gain a deeper insight into physical phenomena around us as they are.
Step 2 here is entirely mathematical. It is steps 1 and 3 which distinguish physics from mathe- matics – the real task of the physicist is to make abstractions from the world around her (step 1), and to interpret the new abstractions presented to her by her calculations (step 3).
The product of this procedure is twofold. On the one hand, we have the predictive power of the theory - its ability to tell us precise things about what will happen, or what is happening, in certain physical systems. The other thing which it gives us is a story, a narrative about the
1 2 Chapter 1. Introduction processes which things undergo, and how the world works. A theory finds itself in trouble when either of these fails. Where the output abstractions fail to match up in a meaningful way with observations, there is a clear problem. Similarly, when a theory paints a picture of the world which is internally inconsistent, it becomes clear that it cannot be, in some fundamental sense, true – although it may still be accurate.
When we turn to the measurement problem, the paradox at the heart of the problems with quantum theory, we can see that different approaches to its resolution have generally taken issue with the story it tells over the predictions it makes. They have tried to identify the error in the theory as having occurred at different stages of the above procedure. For Bohmians, it occurs at step 1; the wrong ontological object has been identified as the protagonist of the theory. For Everettians, it has occurred at step 3; the theory is entirely correct, and if it does not seem to match the world around us, that is simply a result of us having limited access to the world; which is, in some sense at least, actually many worlds. For the Copenhagen school of thought, something strange occurs somewhere between steps 2 and 3, and the process should not be examined too closely. For the quantum Bayesians and instrumentalists of various other stripes, the goal of an internally consistent narrative is simply not worth pursuing - so long as certain strictures can be employed such that the predictions of a theory match the observations of experiment – understanding is itself understood to simply mean the ability to make accurate predictions, and not to build a narrative.
In this thesis we take up the work of dynamical reduction models. These theories identify step 2 as the weak link; the ontology on which they function is similar to that of orthodox quantum theory1, but the mathematical structure at its heart is fundamentally altered. Here it is quantum mechanics itself which is under attack, not just its interpretation.
As we will explore throughout this thesis, such an approach offers great rewards. In chapter 2 we will explore the logic which motivates the development of these models, with particular attention being paid to the measurement problem, and the possible relationships between the physical world and our mathematical representations of it. As we shall see, the development of
1The nature of this ontology will be the subject of section 3.5. 3 collapse models comes as a response to the crisis which emerges when we try to reconcile the image of the world which we glean from quantum mechanics with the world in which we seem to live. Collapse theories aim to deliver a new, unified theory which aims to undo this crisis. They posit a fundamental stochastic modification to the Schr¨odingerequation, engineered in such a way as to bring about quantum and classical mechanics at the appropriate scales, and thereby resolve the measurement problem.
This resolution comes, however, at a price. Dynamical reduction models raise questions of their own, and on multiple fronts. They come with questions of ontology, questions regarding their form, questions of falsifiability. They do not easily extend to relativity, and it’s not entirely clear that they even present a picture of the world with a clear narrative. In a certain light they appear to reach beyond physics itself, and posit elements which are necessarily non-physical. In other aspects, it appears that they conjure a ghost long since banished from physics – the aether – via the inclusion of a universal noise field. They threaten certain results from quantum information theory by virtue of their intrinsic non-unitarity, allowing for the deletion of information. The most pressing problem with these theories is the nature of the noise field upon which they rely, whose origins must necessarily lie outside the remit of description by quantum mechanics, posing serious problems for the status of the field within physics. In another light, it can be said that collapse theories simply displace the unknown; moving it from the measurement process to a new noise field. Far from resolving the crisis, it can be argued that these theories merely complicate it.
It is a confrontation with these problems which motivates chapter 3. We will take a close look at the structure to which collapse theories must adhere, and consider what such a structure might be saying about the world.
In chapter 4, we will examine what bearing dynamical collapse models might have upon time. We will argue that the introduction of indeterministic physics at the most basic, fundamental level allows for the construction of an arrow of time which is qualitatively different than that which we get from thermodynamics, giving an objective and ontological direction as opposed to one which is emergent and perspectival. We will also argue that such indeterminism may 4 Chapter 1. Introduction also be able to make sense of causation in a way which orthodox quantum theory forbids.
In chapter 5, we will argue that the nature of the noise field proposed by continuous collapse models might become known through a combination of two modes of enquiry. On the first, theoretical models can be developed which would be consistent with the demands of a collapse theory, and which would have some physical motivation behind them; for example, the Diosi Penrose model. On the second, the fullest range of possible models which might meet these demands can be identified, and then probed for experimentally. As will be shown, any valid collapse theory must produce certain experimentally detectable effects. By leaving the models as general as possible, we allow ourselves to affirm, or to falsify, collapse model en totalis. We demonstrate how a levitated nanosphere would serve as an ideal candidate for such an experiment, capable of probing the available parameter regime for such models to unprecedented limits.
In chapter 6 we extend our proposal. By creating experimental protocols which are both versatile and sensitive, we intend to create a testing ground capable of more than simply testing collapse theories per se. We develop a mathematical formalism by which a quantum harmonic oscillator may be used as a spectrometer for noise with arbitrary spectra – allowing for any temporal correlation functions to be reconstructed via the heating rate of the oscillator at different resonant frequencies. The assumptions in this derivation are standard and quite minimal. We go on to show how such a protocol, when applied to the experimental scenario of a levitated nanosphere, would enable the testing of collapse models with non-white noise fields. In this way, through the detection and characterization of the spectrum of the noise field, we propose that the development of specific collapse theories containing explanations of the noise field would be greatly bolstered.
In chapter 7 we report on work regarding all-electrical cooling techniques for levitated charged nanospheres. We show through numerical simulations that cooling to quantum states using electric feedback is in principle possible – a crucial prerequisite for the experimental schemes outlined in this the thesis.
Throughout this text we will often need to define terms, and will strive to be precise in our 5 deployment of certain phrases. Definitions will appear in boxes; by way of example, here is how we will use two very common, and very important, terms.
Quantum Mechanics is used here to refer strictly to the mathematical formalism which is generally agreed upon. It is axiomatised by the first four Von Neumann postulates, and is explained in various textbooks such as [10].
Quanum Physics is used here to refer to a range of physical phenomena which are not adequately explained by classical mechanics or relativity. It is the range of phenomena which a quantum theory aims to capture. Chapter 2
Why Collapse?
2.1 Our Philosophical Outlook
Though a philosophy chapter is perhaps unusual for a physics thesis, we feel that a proper discussion of the philosophy at stake here is completely necessary to both motivate and to properly understand collapse theories. We wish to begin by saying something about the philo- sophical outlook from which we will be approaching the questions of the coming chapter. For this work we will adopt the perspective of Scientific Realism2. Realism is sometimes taken to simply mean the metaphysical commitment to an objective world which exists independently of our experience – that there is a world out there whether we are in it and experiencing it or not. But the crucial claim of scientific realism is the one which goes beyond this, the claim that this world is knowable. Knowledge of the world may be changeable, contingent, and difficult to gain; nonetheless it remains possible. The kind of access to the world with which we are concerned lends itself best to an intersubjective agreement. In general we will believe things to exist, objects to have certain properties and so forth so long as others agree with us, and that we will hold the results of experiments to be true so long as they are reproducible. Entities which are not directly observable are permitted into our ontology on a case by case basis. We will take it that the world follows patterns or laws which are, at least on some level, consistent across time and space3 – and that these patterns or laws may be, at least in principle, and at
6 2.2. The Measurement Problem 7 least approximately, known, or in the very least described. And that the work of knowing or describing these patterns is the work of physics. The discovered or described laws and patterns ought not to contradict one another, being (at least ideally) drawn from one, internally con- sistent, world. This outlook is given here without further justification, and serves as a starting point from which to assess theories in physics.
2.2 The Measurement Problem
The measurement problem is a conceptual crisis implicit within quantum theory which appears to make us choose between two options; we can have a coherent narrative about the physical world, or we can have accurate predictions about what will occur in it. But we can’t have both. The problem involves, at root, a decision about where we draw the dividing line between epistemology and ontology; between what we know and what is. It is impossible to follow the procedure laid out on page 1 whilst maintaining a coherent narrative about the aspects of the world which the theory is describing.
The problem can be formulated in a number of ways:
• “What constitutes a ‘measurement’ in quantum theory?’
• “Where does the boundary lie between the quantum and classical worlds?”
• “How does the linear and deterministic evolution described by the Schr¨odinger equation give way to the probability distribution over outcomes described by the Born rule?”
Though the problem is well known, it will serve here to briefly summarise it.
2Scientific realism is a fairly standard approach which is used, mostly implicitly, throughout physics. A good pedagogical guide can be found at [11]. Not that this perspective is not to be confused with scientific materialism – the claim that all aspects of all things in the world are fully reducible to the fundamental elements of one’s chosen ontology in physics. We will have more to say on reduction in section 3.5. 3That the laws or constants of physics might change across time and space is no trouble to this view – so long as those changes themselves are subject to patterns which are consistent; or at least, at some nth level of regression. 8 Chapter 2. Why Collapse?
Suppose we begin with some system S in some state SA, which we represent via a statevector
|Ψi = c1|ψi1 + c2|ψi2 living in a Hilbert space HS. The statevector will evolve according to the Schr¨odingerequation ∂ i |Ψi = H|Ψi (2.1) ~∂t in which H is the Hamiltonian for the system. We wish to measure an observable O, where
|ψi1 and |ψi2 are eigenvectors of O; hψ1|O|ψ1i = λ1, hψ2|O|ψ2i = λ2. We would calculate that the measurement result for O through the Born rule; yielding outcome λ1 with a probability
2 2 |c1| and outcome λ2 with a probability |c2| . We say that at the time of measurement the system was projected into state |ψ1i or |ψ2i depending upon the result which we’ve received from the measurement.
We have made an abrupt transition from the unitary and deterministic evolution given by the Schr¨odingerequation above to a probabilistically determined outcome, and we have labelled this transition ‘measurement’. This mathematical transition is identified with a physical process - the word measurement is used to label both, and thereby conflates them. We might do better to refer to the mathematical process as a ‘stochastic projection’. The physical process is much less easily identified, as we shall see.
The above description gives measurement predictions with probabilities which will correspond to those found in experiments. This is, of course, no coincidence. But, as we argued in the introduction, we want more from our theories than that they be accurate. We want that they also be, in some sense, true. We want them to be able to help us to understand the world, to be able to paint a picture of it which is at once consistent and insightful. And what is the picture painted, implicitly, here? A rather strange one, in which the very nature of reality appears to make an instantaneous jump from one form of evolution to another, completely different one, at the moment of ‘measurement’. The scenario in which this is taken to occur is typically something fairly literal - the interaction of the quantum system under study with some measurement apparatus in the lab. Implicit in this leap from unitary and deterministic evolution to probabilistic outcomes is a transition in physics. The act of measurement has radically altered the flow of events, or at least the mathematical language in which we are 2.2. The Measurement Problem 9 describing them.
Something, evidently, is occurring (or in the very least becoming visible) through the process of measurement. This measurement, at its root, is an interaction between the measurement apparatus and the quantum system under study. But what is the measurement apparatus, if not an assemblage of quantum systems? Or rather, is it not simply a large quantum system itself? Whether it be a Geiger counter, a photographic plate, a photo-multiplier, or any other object capable of recording information with the necessary resolution; it must ultimately be made up of the same particles as the those which quantum mechanics was developed to describe. It must be, simply, a large quantum system.
And yet even this is too narrow. Quantum theory is a theory of reality, not of laboratories – it aims to describe the behaviour of all matter4. If the world is to be describable in terms of quantum theory, then we must say that the measurement process must be occurring all the time, between all things, in order to deliver the classical world with which we are so familiar.
Let’s revisit the above measurement process with this in mind. We now include the mea- surement apparatus as a quantum object which begins in its ready state |ΦiR. Including a dynamical description of the measurement process, we move from the system in its original superposition |Ψi|ΦiR → |Ψi|ΦiM, in which |ΦiM indicates that the apparatus has recorded a measurement. We can write this again as
(c1|ψ1i + c2|ψ2i)|ΦiR → c1|ψ1i|φ1i + c2|ψ2i|φ2i (2.2)
in which |φii indicates that the apparatus has recorded result i. The state on the right-hand side above is of course not a clear, final state, but rather an entangled superposition. The measurement apparatus cannot be said to have measured either result. But nor can it be said to have measured both, or neither – it is in a quantum state which is entirely alien to such classical statements. The situation, of course, doesn’t end here. Including an observer in the
4The failure to properly reconcile with relativity notwithstanding. We will return to this problem 10 Chapter 2. Why Collapse?
description, who begins in a ready state |ΘiR, we have
(c1|ψ1i + c2|ψ2i) |ΦiR|ΘiR → (c1|ψ1i|φ1i + c2|ψ2i|φ2i) |ΘiR → c1|ψ1i|φ1i|θ1i + c2|ψ2i|φ2i|θ1i (2.3) where |θii indicates the observer having read result i. It is clear from this point how the problem proliferates to the very edges of the visible universe – the environment surrounding the apparatus is drawn into the superposition, and then the larger environment surrounding that, and so on. The Everett interpretation leads on from this picture; a grand universal wavefunction which is forever dividing. Though elegant and poetic, we do not consider this viewpoint tenable. It does, however, under a certain light, produce a viable narrative about the world which also agrees with the predictions of quantum mechanics. It is also second only to the Copenhagen interpretation in terms of popularity amongst physicists [12, 13]. A detailed counter-argument to the Everett interpretation is outside the scope of this work. Nonetheless, we include a brief overview of our objections in appendix A.
The Bohmian approach, on the other hand, avoids these problems by issuing an ontology in which the superposition never represented a situation alien to classical metaphysics – |Ψi always represents one particle in one place, but one whose evolution is governed by a quantum potential which replicates the wavefunction. Like the Everettian approach, the Bohmian one offers its own resolution and comes at its own price. It bears little relation to collapse theories, and will not occupy us in this work. We simply note it as a contending interpretation worthy of mention.
2.3 Decoherence
One proposed resolution to the measurement problem is the mechanism of decoherence. This is a viable approach granted an Everettian quantum theory – indeed, decoherence is an essential component of any coherent Everettian interpretation. However, decoherence is also sometimes touted as a direct resolution to the measurement problem, something which addresses the issue without requiring any additional statements about the ontology of the quantum state, the 2.3. Decoherence 11 nature of probabilities, the issue of determining which basis (bases) the wavefunction branches ‘in’, and so forth. Such a claim is simply false, as was emphasized by Joos – one of the founders of decoherence theory – in his widely quoted statement [14]:
Does decoherence solve the measurement problem? Clearly not. What decoherence tells us, is that certain objects appear classical when they are observed. But what is an observation? At some stage, we still have to apply the usual probability rules of quantum theory.
Decoherence gives us a very clear mechanism by which some of the effects unique to quantum mechanics would become so small as to be undetectable upon certain scales. In order to understand why this is not sufficient to resolve the measurement problem, a brief account of decoherence will be useful.
A given environment will decohere a system coupled to it at a rate proportional to the number of degrees of freedom it possesses by the environment1 [15, 16, 17]. As a quantum system interacts with any other quantum system it will generally become correlated with it, which is to say that the two constituent systems will share mutual information through entanglement.
The claim of decoherence is that the two configurations of the environment are genuinely distinguishable, and as such that the inner product hφ1|φ2i will tend quickly to zero. Given that the system here is, by construction small, and the environment, by the same logic large, it is not intuitively obvious why this must be the case, why the action of the small thing should have such bearing on the dynamics of the large. In order to see why it is so, a simple illustration will suffice. For the sake of simplicity of calculation we will employ an especially straightforward model of decoherence. We will again label the system of interest |Ψi, but now the ‘measurement apparatus’ will refer to an element of the environment – for example the state |φi of a single photon. These are known as pointer states.
Say our system is a single mote of dust. If we begin with our system in some state |ψni and
1Though not all degrees of freedom are created equal, as has been stressed by Zurek [18]. Quantum Darwinism however doesn’t fundamentally alter the nature of decoherence, it just advances the description of the rate at which it occurs given certain couplings to certain baths. 12 Chapter 2. Why Collapse? allow it to interact with an element of the environment, such as a single photon – the interaction will leave the dust mote almost completely unchanged, whilst the same could not be said of the photon. In this sense, the measurement can be approximated as ideal, and the interaction characterised by X Hint = |ψnihψn| ⊗ An n where An is an operator on the photon state which conveys information about n. Then the interaction will proceed as
t −iHintt |ψni|φ0i −→ e |ψni|φ0i (2.4)
−iAnt = |ψnie |φ0i (2.5)
= |ψni|φn(t)i. (2.6)
Of course, if the object of interest itself begins in a superposition then we will have a corre- sponding evolution X t X cn|ψni|φ0i −→ cn|ψni|φ(n)i. (2.7) n n If we wish to now examine the state of the system alone after this interaction, we must trace out the environment to yield the reduced density matrix
X ∗ ρS(t) = cmcnhφm|φni|ψmihψn| (2.8) m,n
Now, on the assumption that the pointer states are roughly orthogonal, which is to say than hφm|φni ≈ δmn, then we have
X 2 ρs(t) ≈ |cn| |ψnihψn|. (2.9) n The situation is of course compounded by the fact that there will be many pointer states, not simply one. The overall effect on reducing the off-diagonal elements of the reduced density matrix can be easily calculated by taking a simplified model for the scattering of the pointer- state particles, such as assuming a constant flux and homogeneous wavenumber. Depending upon the nature of the pointer-state particles and environmental conditions, one can produce 2.3. Decoherence 13 an appropriate scatterring matrix in a fairly straightforward way [19, 20]. If the superposition is in the position basis, the off-diagonal elements of the reduced density matrix will evolve as
2 ρ(m, n, t) = ρ(m, n, 0)e−Λt(xm−xn) , (2.10)
2 where xm and xn are the positions corresponding to |ψmi and |ψni, and Λ = −k ϕ gives the decoherence rate; k being the wavenumber of the pointer-state photons and ϕ being the flux rate. For a dust particle of radius R = 10−7 m, (on a par with the size of particle which we will be considering in chapters 5 and 6), we would expect a decoherence rate of Λ = 1017 Hz from sunlight on Earth, and 1032 Hz from collisions with gas particles at room temperature and pressure.
So, is the superposition lost? Not quite. The density matrix of (2.9) still describes a super- position, only one who’s coherence has been displaced to the environment. The description of (2.2) still applies – one can always return to a larger Hilbert space in which the coherence of the superposition is fully apparent.
The fundamental mistake made in the argument for the emergence of objective classicality from decoherence is that it confuses its concepts. It switches an epistemic state for an ontological one, and this switching is made possible only because the formalism of the density matrix elides the difference between the two. Specifically, when looking only at the density matrix, an improper quantum mixture can be conflated with a proper quantum mixture, though they are ontologically very different1. This confusion is made possible by the formalism of the density matrix, which masks the ontology implicit in the theory even as it simplifies the calculations. We explain ourselves by way of example.
Returning to our example of a dust mote held in a spatial superposition, after some fraction of a second we will have 1 1 0 ρS(t) ≈ . (2.11) 2 0 1
1Or at least, are different on a scientific realist account of the world. It is possible that they may be reconciled under an anti-realist ontology, but in the very least some colour and creativity would be required. They may also be reconciled under an instrumentalist approach to quantum mechanics. 14 Chapter 2. Why Collapse?
This is often taken to signify a classical probability distribution. In a certain sense, it absolutely does – the diagonal elements in this matrix correspond to the probabilities with which we will find the system to be in a certain state, and show that we would expect no quantum interference effects to be visible. However, the above density matrix, whilst corresponding to only one probability distribution over measurements, is in fact infinitely degenerate with respect to to the wavefunctions of which it can be made. For example, it can be recovered from either of the following quantum ensembles:
half in |ψai = |0i
half in |ψbi = |1i which gives us a density matrix
1 1 ρ = ρ + ρ 2 a 2 b 1 1 0 1 0 0 = + 2 0 0 2 0 1 1 1 0 = ; (2.12) 2 0 1 or
1 half in |ψai = √ (|0i + |1i) 2 1 half in |ψbi = √ (|0i − |1i) 2 2.3. Decoherence 15 which gives us
1 1 ρ = ρ + ρ 2 a 2 b 1 1 1 1 1 −1 = + 4 1 1 4 −1 1 1 1 0 = . (2.13) 2 0 1
These two situations are wildly different, yet lead to the same final density matrix – and both are profoundly different to the situation which produced (2.11), yet again share the same form of density matrix! In the case of 2.12 we have a scenario which may as well be classical; an ensemble of states which, measured in this basis, are not in superpositions. The scenario of 2.13 is fundamentally quantum - we have an ensemble of two superpositions; states which in themselves have no classical analogue, but when taken together produce a ‘classical’ probability distribution – which is to say, the same diagonalised density matrix as the first scenario5. We spell this out for the purpose of making clear the fact that the density matrix is capable of disguising a great many things about the quantum state, or the ensemble of quantum states – first and foremost the difference between these two. By taking the density matrix to stand in for the wavefunction we are confusing our epistemology with our ontology – we are confusing what we know with what is. This may seem a bold statement – we devote the next section to qualifying it.
First though, we might ask why a fully diagonalised density matrix would be expected to return to us a classical world. We could simply answer that if the density matrix is taken to represent a probability distribution, then the condition that it be diagonal is exactly that by which it will mirror a classical probability distribution. But achieving such a density matrix through decoherence requires that we make a cut between a system and its environment. A density matrix of a superposition whose entanglement with the environment can be reduced to a diagonal density matrix represents a superposition, a state with no classical analogue –
5Both of these are proper quantum mixtures – ensembles of pure quantum states. The density matrix of (2.11) is an improper mixture – one whose mixedness is attained through tracing out an environment. 16 Chapter 2. Why Collapse? only, one in which information about that superposition has been relegated to an environment which we are choosing to excise from our narrative. Even in the story told by decoherence, there is a leap whereby the density matrix goes from representing the state of the world to the probabilistic outcomes of measurement.
If the density matrix is proper and pure then it must face the same problems, directly, as interpreting the wavefunction. If it is pure and improper, i.e. mixed, then it either represents an ensemble, or an epistemic restriction which corresponds to the ‘tracing out’ procedure. As such, it is telling us about what we can know, not what is. A density matrix of a higher dimension is always available, one which includes the environment, which will restore off-diagonal elements and quantum effects to the scenario. If the density matrix represents an ensemble, then it can always in principle be separated into a series of separate density matrices which are each pure and proper. The move by which the density matrix might allow us to avoid a confrontation with the ontology of the wavefunction is a flourish – decoherence conceals, rather than addresses, the problem.
So there we have it. It would appear that taking quantum mechanics to be a theory of reality leads us directly to paradox, to a world of increasingly many superpositions which propagate without any sign of letting up. Given that we, for the most part, seem to live in a classical world, such superpositions cannot be allowed on any theory of reality which aims to match the world. This is the moral of the story of Schr¨odinger’s cat.
Before progressing, we feel it necessary to address what we see as a particularly popular mis- conception about the measurement problem in the current discourse. A certain narrative holds that it is only with the advent of increasingly large superposition states being confirmed in the lab [21, 22] that the measurement problem has become a pressing issue [18]. Implicit here is the idea that Schr¨odinger’s cat initially had no real bearing on quantum theory, and that we are only now having to take it seriously since the superposition principle shows, so far, no sign of letting up. This misunderstands the problem. The conflict between quantum and classical mechanics is not fundamentally a dispute over the territories of scale, in which quantum is slowly winning ground by accurately predicting the behaviours of larger and larger objects. 2.4. The Ontology of the Quantum State 17
Rather it is a dispute over the fundamental nature of reality, which appears to have a stark dichotomy to its nature, curiously aligned with scale. It’s worth remembering that scale plays no particular explicit role in classical mechanics, and a limited one in quantum mechanics6.
2.4 The Ontology of the Quantum State
One may object that the wavefunction is, in fact, not the correct ontic object to be considering. That after all, every detectable element of reality will make itself known through some element of the density matrix or other, since the probability of every outcome of every measurement in any basis may be represented in a density matrix. We may say that the wavefunction is a useful calculational tool, but its ontological status as something fundamentally real is a relic from the early days of quantum theory. Such a viewpoint is often associated with an observables based ontology, in which the Heisenberg picture is esteemed over the Schr¨odinger– we consider such a perspective incoherent, and address it in appendix B. A more meaningful objection to wavefunction realism which nonetheless holds to a realism regarding other quantum elements can be found in relational quantum mechanics [23, 24].
We will argue instead that something close to wavefunction realism is essential for forming a metaphysics of quantum mechanics without contradiction. We say ‘something close to’ because, in a fairly literal sense it is meaningless to be a realist about the wavefunction itself. Perhaps the most common misstep in the interpretation of physics is the consistent failure to distinguish between things and their representations7. This distinction is clearer on classical terrain – if we take a realist approach to the trajectory of a cannonball for example, we would not describe that approach as ‘quadratic formula realism’. The quadratic formula would, of course, be a simple and useful mathematical tool in our representation of the cannonball, but no one would confuse it for the thing itself. However, it is exactly this sort of confusion which is implicit in the term wavefunction realism. As has been emphasised by David Albert [25, 26], what we
6It might be claimed that, on the contrary, h is in fact of vital importance to quantum theory, since it will determine the visibility of interference fringes which reveal superpositions in an experiment. In answer to this, we simply note that the problem of Scr¨odingerscat was never that we couldn’t construct a cat-interferometer, rather the problem is that quantum theory would appear to be describing a world which is completely different to the one in which we live. 18 Chapter 2. Why Collapse? ought to be realists about is, of course, the thing represented by the wavefunction. And that thing is a field – the wavefunction describes a field in a space which is characterised at every point by two quantities; an amplitude and a phase.
The Wavefunction is the familiar mathematical object. It describes a field, char- acterised at each point by an amplitude and a phase.
The Quantum State is the putative thing which quantum theory is seeking to cap- ture, whether through the wavefunction or another mathematical object. It names the element(s) of reality towards which we are reaching with our descrip- tions.
It might be tempting to simply turn away from the term ‘wavefunction realism’, and replace it with something more direct, such as ‘quantum state realism’. The problem with this move is that it would erase a crucial issue – that of trying to determine exactly what the relationship of representation is between the quantum state and the wavefunction, of trying to understand how much of the quantum state is represented in the wavefunction, and how well.
In order to examine this question, let us introduce a taxonomy8 for basic positions which can be taken with respect to the ontology of the wavefunction:
1. Anti Realist The wavefunction is simply a useful tool for making calculations. There is no underlying reality and no quantum state.
2. Ontological Hidden State The wavefunction captures some element of reality, but also describes an epistemic restriction upon our access to some deeper, underlying reality (the quantum state).
3. Ontologically irreducible Wavefunction The wavefunction captures every aspect of the quantum state, and can be thought of as a direct representation of an ontological element of reality.
7For a clear exposition on this problem by way of example, see Maudlin’s treatment of the hole argument in [27]. 8This echoes a taxonomy used by Leifer [28]. 2.4. The Ontology of the Quantum State 19
The first account above is characteristic of, for example, quantum Bayesianism (QBism)[29, 30], and is perhaps best summarised by Asher Peres when he states that [31]:
... quantum phenomena do not occur in a Hilbert space, they occur in a laboratory.
Or by Bohr, when he said that [32]
There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.
There are several accounts on which we disagree with this perspective. For brevity, however, we will simply state that approaches of this kind fail to meet one of our principle criteria from the previous chapter – they do not seek to form theories of reality, only methods by which we might master it. Since instrumentalist approaches need not be internally consistent, arguing against them draws more on ideas about the purpose of science itself than the specific content of the theories. As such, we will leave them aside.
The second and third approaches are not so simple. On approach 2 from above, the quantum state is described by the wavefunction in a manner analogous to the relationship between the microstate and the macrostate in classical thermodynamics. The analogy is instructive. Any classical thermodynamical system is presumed to be in a unique and distinct microstate at any given moment; it occupies an infinitesimal point in phase space. Our epistemic restrictions, however, prevent us from properly accessing this – instead we name a macrostate, a region of phase space which must contain the microstate. Thermodynamics then gives us the toolset with which to predict the behaviour of the macrostate, and the behaviour of any and all microstates within it must be coarsely described by the same dynamics. On the ontic hidden state picture, the wavefunction is something like the macrostate, and quantum mechanics is something like thermodynamics [33]. The wavefunction is here pictured as a coarse description of something unique beneath it, and the quantum state it tries to capture is analogous to the microstate. The probabilistic outcomes of the Born rule can here be interpreted in two ways: 20 Chapter 2. Why Collapse? they can simply reflect our ignorance, in the same way that we can only predict the weather (a thermodynamic system) probabilistically; or else the hidden quantum state can still contain something fundamentally stochastic within it.
There are some immediate problems with this approach, and some more subtle ones. First, we can quickly see that a naive theory of hidden variables, in which the state of the system is fully determined prior to any measurement, runs in direct contradiction to experimental evidence and quantum theory as it stands. By way of example, consider the Stern Gerlach experiment. Taking an input beam of particles which have not been prepared in any particular state, we can rotate the measurement apparatus about its axis continuously, and yet we still measure two discrete measurement outcomes. There is no way in which a continuously variable measurement criterion can return a discrete outcome without either influencing the outcome or somehow influencing the input state. Given this, we might amend the position to state that the quantum state is influenced by the measurement process, but that the wavefunction nonetheless represents an epistemically restricted view of an underlying ontological state, which is fully determined. Such a view is not tenable, as shown by the PBR theorem found in [34], a pedagogical overview of which can be found in [35]. Beyond this, it has been known for decades that a hidden variable model in which the information describing the state is contained locally is not viable [36]. All of this would seem to leave us with no option but to accept the wavefunction as an ontological object.
However, the third approach – that of an ontologically irreducible wavefunction – cannot be quite right either. Or at least, it would require a peculiar ontology. As described in [37], it is completely possible to make changes to a wavefunction which have no representation of any sort in the physical world. Specifically, we can change the phase of any wavefunction without altering the predictions it makes regarding physical observations. If the quantum state is confined to elements of reality which are at least in principle physically detectable, we must say that a given quantum state may be mapped to infinitely many wavefunctions, each differing from one another by some ‘global’ phase.
The above might seem like a pedantic point, and perhaps it is. Modulo phase, we might be 2.4. The Ontology of the Quantum State 21 able to say that each quantum state maps to a single wavefunction. The proper way of saying this would be that the quantum state would correspond to a ray in Hilbert space, not a vector.
But there are other problems which arise when we try to take the wavefunction as a literal representation of an ontological object. Some of these problems we reserve for section 3.5, but the most striking problem arising from wavefunction ontology is of course the measurement problem. The linear and deterministic nature of quantum mechanics leads inexorably to ever propagating superpositions9, a situation which would appear to be wildly at odds with the reality we live in. We summarise the possible paths one can take when considering these problems in figure 2.1.
Where now?
The contradictions, then, are fairly clear. If we start from a perspective of scientific realism, we have a choice regarding how we view the quantum state. To steer clear of the shores of Stern Gerlach, PBR and Bell, whilst staying within the realms of realism and rejecting instrumentalism, and further giving an explanation of the quantum to classical transition and answering the measurement problem, we appear to have only two choices: collapse theories or an Everettian approach. In this work we opt for the former – models which alter the fundamental dynamics of quantum mechanics by introducing stochasticity and non-linearity to the evolution of the wavefunction. Such theories have radical implications for physics, and the stories it would tell us about the world.
9We again refer the reader to our objections to the Everett interpretation in Appendix A. 22 Chapter 2. Why Collapse?
Your outlook is incompatible with Is the state of the system under study fully yes quantum physics, as shown by the determined prior to the measurement? Stern Gerlach experiment
no
Properties of quantum systems are formed relationally or contextually.
Your outlook is incompatible with Is there an underlying ontic state, of which ψ yes quantum theory, as shown by merely represents an incomplete knowledge? PBR.
no
The quantum state represents an element of reality in some sense.
You are an Everettian. Your Accepting that the quantum state represents yes position can be clearly an element of reality, is the process of formulated, but you have serious measurement fully described by decoherence? questions to answer. no You have chosen the Copenhagen A physical process occurs during the act Interpretation. Your position of measurement, breaking the linearity of no cannot be formulated in the form QM and replacing it with the Born rule. of a scientific theory, and does not Do you modify QM to account for this? offer insight. yes
You have chosen collapse theories.
Figure 2.1: A ‘choose your own adventure’ style summary of the approaches we might take to interpreting quantum theory, premised upon a stance of scientific realism. Chapter 3
Collapse Models
The Structure of Collapse Theories
We turn now to the kind of model which might be able to produce a resolution to the mea- surement problem via an intervention into the dynamics of quantum theory. In this chapter, we will build up an understanding of the dynamics of collapse theories and then return to questions of their metaphysics. To desire a departure from determinism and the introduction of stochasticity at the fundamental level is all well and good; but in order to make such a move and yet retain the bulk of physics, retain the vast and effective bodies of theory which we cherish so dearly, it must be made carefully. The first proposal which fits into the general form this chapter will address was made in 1952 by L. Janossy [38], and soon forgotten. Since then a wide range of collapse models has sprung up, and with a corresponding range of reasoning behind them. We’ll begin here with the model of Ghirardi Rimini and Weber (GRW) model [39, 40]. Though this fits into a chronology, in that the GRW model laid the foundations upon which following models were built; our motivation is rather that this model offers the clearest and direct example of what collapse models try to achieve, and how they do so.
23 24 Chapter 3. Collapse Models
3.1 QMSL - A simple model
The GRW is an incarnation of a type of collapse theories known as Quantum Mechanics with Spontaneous Localisation (QMSL) – this being the name which was originally given to the GRW model. The title of the original paper introducing the model was ‘Unified dynamics for microscopic and macroscopic systems’ – a mission statement for collapse theories which still stands. On this theory, a wavefunction |ψi describing N distinguishable particles is struck at random moments by an operator
1 2 2 −(xi−xf ) /(2rc ) Li(xf ) = 2 3/4 e (πrc )
th where Li is a localisation operator acting on the i particle, forming a Gaussian in three dimensions centred at xf . The wavefunction is otherwise entirely governed by the ordinary Schr¨odingerequation ∂ i |ψi = H|ψi ~∂t in which H gives the Hamiltonian. The frequency of occurrence of the localisation operator is given by an intensity λ, and such occurrences are uniformly distributed over time as a Poisson distribution. The occurrence of a localisation operator at location xf on particle i described by a wavefunction |ψi sends it to
i |ψx i |ψi → f |||ψi i|| xf i Lx |ψi = f . ||Li |ψi|| xf
The probability density for the position at which the localisation operator acting on particle
2 i is centred is given by P (x) = ||Li(x)|ψi|| , and the constraint that the probabilities sum to unity Z Z 3 3 i 2 d xP (x) = d x||Lx|ψi|| = 1 ensures that the collective action of the collapse operators reproduces the Born rule. In figure 3.1 we see a depiction of how the action of the GRW model suppresses superpositions. Beginning 3.1. QMSL - A simple model 25
a a
b b
x (a) Wavefunction in original superposition.
a a
rc b b
x (b) The wavefunction is struck by a collapse operator centred at +a.
a
rc b
x (c) The wavefunction has been localised about a, and the superposition sup- pressed.
Figure 3.1: The process by which the GRW localisation operator suppresses superpositions. 26 Chapter 3. Collapse Models with a wavefunction in a clear superposition, composed of two Gaussian wavelets of width b peaked at x = a and x = −a respectively
1 − 1 (x+a)2 − 1 (x−a)2 ψ(x) = (e 2b2 + e 2b2 ) N we then introduce a localisation operator centred at a,
1 2 2 −(a−x) /2rc La(x) = e 2 3 (πrc ) 4 which acts on the wavefunction to give
1 2 1 2 1 1 1 2 1 − 2 (x−a) − (x+a) − 2 ( 2 + 2 )(x−a) ψ (x) = (e 2rc e 2b2 + e b rc ) a N 0 1 − 1 (x−a)2 ≈ e 2b2 N 00 resulting in a wavefunction which is effectively localised10.
There are a few important things to note about this modification to quantum mechanics at this stage, and we will go through them now.
Predictive Agreement
The first is that (so long as rc and λ remain within certain bounds) it fully reproduces the predictive content of orthodox quantum mechanics. This is most clearly seen in the density matrix formalism. Whilst in the above example we had knowledge of the location of the localisation operator, and could thus map from a pure state to a pure state; in general, the locations at which the operators act are unknown, and as such their action is better described by the probability density function. The density matrix of an N particle system undergoing a
10Though clearly the wavefunction is still non-zero everywhere. We will return to this point in section 3.5. 3.1. QMSL - A simple model 27 localisation then is given by
N Z X |ψxihψx| |ψihψ| → d3xP (x) (3.1) i |||ψ i||2 i x N Z X 3 = d xLi(x)|ψihψ|Li(x) (3.2) i = ρ0. (3.3)
So that ρ −−−−−−→localisation ρ0, where ρ0 denotes the density matrix of a system which has undergone a localisation with a probabilistic distribution over the location of that localisation. Looking at the effect this has on the predictions such a density matrix would yield, we have that
2 2 0 −(x1−x2) /4rc hψxl |ρ |ψxj i = e hψxl |ρ|ψxj i (3.4)
Clearly, if x1 = x2 then
0 hψxl |ρ |ψxj i = hψxl |ρ|ψxj i, (3.5) and our diagonal elements are left unharmed. Now, since the occurance of the localisation events is probabilistic, the density matrix between times t and t + dt has the statistical description of either undergoing a localisation event (with a likelihood λ(t + dt)); or not (with a likelihood (1 − λ)(t + dt)). This gives us a master equation of
N ! ∂ i X Z ρ = − + λ dx L (x )ρL (x ) − ρ . (3.6) ∂t f i f i f ~ i
By equation (3.5), we can see that (3.6) will preserve trace. However, looking at the evolution of the off-diagonal elements, we have
∂ i 2 2 −(xi−xj ) /4rc hxi|ρ|xji = − hxi|[H, ρ]|xji − Nλ(1 − e )hxi|ρ|xji (3.7) ∂t ~ where N is the number of particles in the wavefunction, which is to say that
N Y |ψi = |ψii. i 28 Chapter 3. Collapse Models
So in the limit that |x1 − x2| rc, the off-diagonal elements of the density matrix will be suppressed at a rate of Nλ.
This, of course, is reminiscent of decoherence. However, the differences between collapse and decoherence are crucial, as we have stressed. Here the density matrix is being driven to properly represent a statistical mixture; a probability distribution over the likelihood of the occurrence of mutually exclusive scenarios which exist. As we demonstrated in figure 3.1, the kind of reduction driven by these localisation events is fundamentally different to that of decoherence. It requires no ‘tracing out’, no subjective distinction between system and environment, and no measurement. It transforms pure states into statistical mixtures of pure states (proper mix- tures). Decoherence, by contrast, transforms pure states to pure states, which, when analysed at the level of the density matrix and with the added move of tracing out the environment, appear as improper mixtures. Again we emphasise that these two kinds of mixture produce density matrices of identical form, but which signify profoundly different worlds11.
Let’s revisit the measurement problem from section 2.2. Instead of moving straightforwardly from (2.2)
(c1|ψ1i + c2|ψ2i) |ΦiR (3.8) to
c1|ψ1i|φ1i + c2|φ1i|φ2i, (3.9) and then from there towards an ever-growing superposition, we would have, at any moment, the probability for a collapse event to occur. From equation 3.7, we have that the system would be driven into a single, final state at a rate Nλ. And the presence of N here is crucial – it serves as the amplification mechanism which will deliver to us a classical world in the appropriate limit, and a quantum one too, in its own limit. To demonstrate, we take the |Ψi of equation (3.8) to represent a single particle, and the |Φi to represent some large system containing, say, 1023 particles. Then in equation (3.8) we have two localisation rates - one for the particle |Ψi, at
11Contingent, of course, upon some level of commitment to wavefunction realism of the third type described on page 19. 3.1. QMSL - A simple model 29
λ, and one for the measurement apparatus |Φi, at λ × 1023. Now, so long as the two systems remain factorisable, the collapse rates of each remain un-related. Taking the suggested value
−16 7 of λ of λGRW = 10 Hz, the system Φ will localise 10 times per second, and the system Ψ only once per 1016 seconds. Φ will behave essentially as a classical object, whilst Ψ will, most likely, maintain its superposition. As the systems interact, however, and we move to equation (3.9), a collapse for any of the particles in Φ will come to induce a collapse for Ψ, and vice versa. The entanglement in the position bases of the constituent particles makes the composite system Ψ + Φ equivalent to a single object, comprised of 1023 + 1 particles, with an effective localisation rate of λ(1023 + 1) Hz. So as soon as the interaction has occurred, the macroscopic superposition of a system having ‘measured’ two results is destroyed within ∼ 10−7 seconds,
2 2 delivering an outcome of either |ψ1i|φ1i or |ψ2i|φ2i with probabilities |c1| or |c2| respectively. There is no need for terms like ‘measurement’ or ‘observation’ – there is only the unfolding stochastic dynamics of two systems of different scales, and the dynamics of a new, composite system as they interact. The quantum superposition of Ψ may persist up until the moment of this interaction – the superposition then being suppressed at a rate proportional to both the number of particles with whose positions it is becoming entangled, and the rate at which that entanglement grows as per the interaction Hamiltonian.
This serves to demonstrate the basic functionality of collapse theories. However, looking at the construction of the QMSL model a few questions are immediately brought to mind: Why do the collapse operators function on the position basis? Could they function on another basis? Why are the particles taken to be distinguishable, isn’t this counter to quantum field theory? What are the other implications of introducing these collapse operators?
Some of these are interpretational, and will be examined in section 3.5. Some of these can be answered by reviewing the models which have gone beyond QMSL. Here we will introduce a more contemporary model which addresses some of these issues; beginning with the structure of the model itself, before returning to the above questions. 30 Chapter 3. Collapse Models
3.2 Continuous Spontaneous Localisation (CSL)
The CSL model [41] can be derived either to be linear or non-linear in |ψi. Both formalisms will bring about the same master equation, but if the equation is linear in |ψi then the noise field itself must carry the non-linearity. Here we will introduce both forms. We begin with a very general12 form for the evolution of the wavefunction
" # d −i X X |ψ(t)i = H + A w (t) − γ2 A2 |ψ(t)i (3.10) dt i i i ~ i i
where wi(t) is a stochastic field which will drive the collapse into a basis determined by Ai; a set of commuting self-adjoint operators. The collapse will occur with a strength given by γ. This is related to the collapse rate for a single nucleon held in a widely separated superposition, λ, in a
γcsl way which is linearly proportional but model dependant. For CSL, it is given by λcsl = 3/2 2 . 8π rc The norm preserving version of this is non-linear in |ψ(t)i, and given by
" # d i X X X |ψ(t)i = − H + (A − R )w (t) − γ (A − R )2 + γ (Q2 − R2) |ψ(t)i (3.11) dt i i i i i i i ~ i i i in which R = hψ|A|ψi and Q2 = hψ|A2|ψi.
Both give the corresponding master equation
d i X γ ρ(t) = − [H, ρ(t)] + γ A ρ(t)A − A2, ρ . (3.12) dt i i 2 i ~ i
The eigenmanifolds of the operators {Ai} determine the statespace into which the statevector will be stochastically driven; which is to say that Ai selects the basis for the ongoing collapse of the wavefunction.
Let’s notice a couple of things about the model at this stage. First of all, equation (3.10) is linear in |ψ(t)i, and (3.12) is linear in ρ(t). Both of these are important.
As shown by Gisin [42], when introducing non-linearities into the evolution of the wavefunction
12 Though not quite as general as can be, since we have assumed the operators {Ai} to be self-adjoint. 3.2. Continuous Spontaneous Localisation (CSL) 31 one must be very careful. Arbitrarily allowing for non-linear evolutions of the wavefunction allows for superluminal communication, and this can only be offset by tying the non-linearities to stochasticity. We can check for this very easily: if linearity is preserved at the level of the master equation, then the modifications will not allow for signalling, so equation 3.12 is fine on these grounds.
Given the general form of (3.10), we are faced with a choice about what operators to pick for
Ai, and further questions about what the stochastic processes wi(t) are meant to represent. The specific form allocated to the collapse operators will, of course, depend upon exactly what sort of narrative we are trying to get the theory to tell. Since this is typically one in which we are trying to recover a familiar classical world from modified quantum dynamics, the argument for the choice of something like a position basis for Ai follows from the simple observation that we do not find classical scale objects to be in superpositions in space, and that this would serve as a direct basis through which to attain a physics which accords with this. The typical formulation of CSL in fact replaces the summation in (3.10) with an integral over a continuous variable, and the stochastic processes wi(t) with a noise field which is continuous over time and space. Specifically, as our basis for collapse we have the smeared mass density operator
Z X 3 † N(x) = d yg(y − x)ak(y, s)ak(y, s) (3.13) k,s
† (in which ak(x, s) and ak(x, s) give the creation and annihilation operators for a particle of type k, of spin s at location x respectively, and g(x) gives an envelope function which we define below). For the mass density operator we require that
Z d3N(x) = N (3.14) which is satisfied by taking 3 2 1 2 1 − 2 (x) 2rc g(x) = 2 e . 4πrc
It comes as small surprise that the form of this function echoes that of the localisation operator L from section 3, giving a Gaussian in space. It’s worth noting, however, that although this 32 Chapter 3. Collapse Models form for g(x) is conventional it is not the only possible choice – we need only satisfy (3.14). For our stochasticity, we have a classical white noise field ξ(x, t) characterised by
hξ(x, t)ξ(y, s)i = γδ(t − s)δ3(x − y).
Applying these changes to (3.10) lands us with a modified Schr¨odingerequation of
d −i Z Z |ψ(t)i = H + d3x N(x)ξ(x, t) − γ d3x N 2(x) |ψ(t)i, (3.15) dt ~ and the corresponding master equation
d −i Z γ Z ρ(t) = [H, ρ(t)] + γ d3x N(x)ρ(t)N(x) − d3x N 2(x), ρ(t) (3.16) dt ~ 2 Z i λcsl = − [H, ρ(t)] − 3/2 3 dx[N(x), [N(x), ρ(t)]]. (3.17) ~ 2π rc
γ in which λcsl = 3/2 2 Now, equation (3.15) is clearly linear in |ψi, but it does not preserve the 8π rc norm. An alternate form, which is norm preserving but comes at the price of being non-linear in |ψi is given by the Ito equation
Z 2 −i 3 γ 2 d|ψti = Hdt + γ d (N(x) − hN(x)it) dξt(x) − (N(x) − hN(x)it) dt |ψti. (3.18) ~ 2
In which h·it represents the expectation value of · at time t, and in which dξ(x) is the increment of the noise field, with
hdξ(x)i = 0 and hdξ(x)dξ(y)i = δ(3)(x − y)dt.
These forms – linear and non-linear in |ψi – both lead to the same equation for the density matrix; that given in (3.17). However, as we have argued earlier, when trying to construct a quantum theory with a clear and meaningful ontology it is not enough to settle at the level of the density matrix – the meaning of the evolution of the wavefunction itself must be examined.
If we are to choose between equations (3.15) and (3.18) for representing the dynamics of this 3.2. Continuous Spontaneous Localisation (CSL) 33 ontology, we have to make a choice between what we are to sacrifice: the linearity of the dynamics in the statevector |ψi, or the norm of |ψi. In making this choice, we are compelled to pay close attention to the ontology which we would be attaching to the noise field – something which we have not yet considered – and at the same time return to considerations about the ontology of the quantum state and its relationship with the wavefunction (since a failure to preserve the norm of the statevector will correspond to a failure to preserve the norm of the wavefunction). Upon sacrificing the norm of the wavefunction, it would appear at first that we might be sacrificing any strong relationship between the wavefunction and the quantum state itself; if the world is the quantum state, and the quantum state is strongly represented by the wavefunction, then wouldn’t a reduction in the wavefunction be, somehow, a deletion or diminishing of the world itself? Alternatively, sacrificing the linearity of the theory in terms of the wavefunction would seem to imply something which is otherwise notably absent from quantum physics – self-interaction.
As we shall see, the arguments which compel us towards either choice here will depend strongly upon exactly what sort of space we think we live, and how many dimensions it has. It will also depend upon the physical nature of the putative noise field. We will engage with this choice directly in section 3.5. But first, we re-visit some of the strange features of QMSL, and see how they have been adjusted in CSL:
The distinguishability of the constituent particles has been addressed in CSL by replac-
ing the set of localisation operators Li, each tied to a specific particle, with a mass density operator N(x). This mass density operator clearly operates in the second quantization picture, and as such does away with distinguishability completely.
The localisation operators Li were introduced without any explanation beyond that – at least at the level of phenomenology – they would allow for a resolution of the measurement problem. Exactly why they ought to occur physically, or indeed what sort of physical thing they might represent, was not articulated. In CSL these operators have been replaced by the mass density operator, and the Poisson distribution governing their occurrence has been replaced with a classical noise field ξ(x, t). Such a noise field arrives with a host of 34 Chapter 3. Collapse Models
its own questions. However, in the very least, we might say that such a field is intuitively closer to being a physical process which is at least in principle describable.
The basis for collapse was taken, without enquiry, to be position. In CSL it is taken to be the mass density in space – a close analogue, but one which might in principle more easily extend to relativity. The choice of basis here is however not above criticism.
These are the principle reasons why CSL is considered to have superseded QMSL. Primarily the continuous model appears to be closer to something physical – less heuristic, more real. It remains, of course, purely phenomenological, with no justification provided for the modifications to quantum mechanics other than ‘were it to be like this, instead, then it would resolve the measurement problem’. These models are, at this stage, conjecture. As we have stated, they set themselves apart from other resolutions to the measurement problem in that they are not an interpretation of quantum mechanics, they are a modification to it attached to an implicit interpretation. This is at once their strength and their weakness. Strength, in the sense that this renders them testable, as will be the subject of chapters 5 and 6. Weakness, in that they introduce elements which demand an ontology and explanation; namely, ξ and the choice of basis for collapse.
3.3 What is Real?
We return here to the demands made in the introduction – that a physical theory must be able to accurately predict the physical behaviour of the things which it models whilst also providing a clear narrative of the parts of the world which it is describing. On the face of things, collapse models might appear to have done just this. By selecting a form for the evolution of the wavefunction which produces a linear master equation, the theory remains non-signalling. By the careful selection of the action of the collapsing operators and the coupling to the stochastic element, the theory has ensured that the Born rule is reproduced – that the diagonal elements of the density matrix are not distorted by the action of the noise field. Through a judicious choice of the values for the parameters introduced in the model, it can be made to agree with 3.4. The Dimensionality of Reality 35 all hitherto results of standard quantum mechanics, whilst still introducing a localising effect which is strong enough to prevent macroscopic superpositions from persisting for perceptible amounts of time.
But, does this really give us a picture of the world in which we live? We have succeeded in creating a framework within which wavefunctions will tend towards single peaks in a certain basis, and do so at a rate proportional to their mass. This, however, is not enough. It still leaves us a long way from a coherent description of the relationship between the representations in our theories and the world itself. The question incumbent upon collapse theories is an old one – how can the world which we experience arise from the kind of objects described by wavefunctions, or by a universal wavefunction?
Recalling the arguments summarised in figure 2.1, the reasoning by which we embark upon a collapse type theory in the first place necessarily takes the quantum state to be an ontological element of reality, and further takes the wavefunction to capture some irreducible facts about it. If the wavefunction were merely a calculational tool then none of this would be necessary, since standard quantum theory already produces correct predictions of physical phenomena. And what we want to argue is that, taking the wavefunction to be – at least in some sense – representational of something fundamentally real, and taking it to undergo a continuous and spontaneous process of localisation, does not, immediately, give us a picture of the world in which we seem to live13.
3.4 The Dimensionality of Reality
In order to get an understanding of the ontology of collapse theories – or at least, of the kind of ontologies which might be possible – we are going to need to return to the questions which we introduced in chapter 2. Specifically, we will need to examine the relationship between a thing and its representation14, and what this relationship looks like when we are talking about
13Which is not to say that such theories are wrong, by any means. As we shall see, it is not so hard to argue that the world in which we appear to live might well be some restricted, or perspectival, component of a larger, or different, fundamental reality. 36 Chapter 3. Collapse Models spaces. For the sake of clarity, we will freshly define some familiar concepts.
Configuration Space - A 3N dimensional space. The locations of N objects in three dimensional space can be represented by a single point in this space.
Phase Space - A 6N dimensional space. Similarly, we could say that both the positions and the velocities of N objects could be represented by a single point in this space.
Hilbert Space - A complex vector space upon which an orthogonal coordinate struc- ture can readily be placed with an infinite degeneracy. It is equipped with the inner product structure |o|2 = ho|oi, such that all inner products are positive
definite; and ho1|o2i = ho2|o1i, and that this =0 iff o1 or o2=0. Often Hilbert spaces will be of infinite dimensionality. It is the form of space used for describ- ing statevectors.
A Field as we will use it means an ontological entity which cannot be decomposed into anything more fundamental – a field cannot be said to be an emergent property of the behaviour of smaller things. By way of example, the wind would not qualify as a ‘field’ here, though it might be useful mathematically to describe it as such. The electromagnetic field (as it appears in classical electromagnetism), on the other hand, is the type of thing to which we refer – something which is considered fundamental. A field is defined over a space, and assigns a value (or values) to each point in that space.
A quick point about the spaces used in quantum mechanics. Discussions on quantum mechanics typically move between two representations of the quantum state – between the wavefunction ψ(x) and the statevector |ψi. Although a perfectly clear mapping exists between the two representations: ψ(x) = hx|ψi, |ψi = R d3x ψ(x)|xi, it is important to remain clear on which we are discussing. This is especially true when we are constructing an ontology for our theories,
14Outside of physics, the problematic nature of this relationship is often summarised in the famous phrase ‘the map is not the territory’ [43]. Though such language gives a common anchor from which to hang discussions about representation, we will avoid using it here since, of course, the word ‘map’ has a rather specific meaning in physics already. 3.4. The Dimensionality of Reality 37 since the protagonist objects of the theory will need to live in a space, and we will ultimately need to need to relate that space to the physical one in which we live. For example, the space required to represent a single particle on a single dimension in physical space can be one dimensional when working with the wavefunction, but would typically be of a (countably) infinite dimensionality if we work with the statevector formalism, with some attendant choice about how we discretise the first space. Because of this, although the statevector formalism is far and away the most common representation used in physics, it is the wavefunction and the space in which it lives which are most discussed in the philosophical literature [44]. Now, the space in which the wavefunction lives is usually named to be a configuration space. However, this strongly implies that the dynamics of the wavefunction are ultimately about the dynamics of particles – an ontological commitment which is not a necessary component of a wavefunction realism of the kind that we are considering (the third kind from page 19). This point made, we will nonetheless keep with convention and call the space of the wavefunction a configuration space for the sake of simplicity.
We now progress to examining the relationship between the world, our representations thereof, and the spaces in which those representations live, beginning with an example15. Let’s say that we have a system of N classical particles arranged in physical space, which we’ll call S. We can pick a coordinate system16 for that space, and yield a description of N vectors, each with three elements: {{x1, y1, z1}, {x2, y2, z3},..., {xN , yN , zN }}. We could just as easily represent the same situation with a single vector in a 3N dimensional configuration space C such as
{r1, r2, r3, . . . , r3N }. Clearly, these two descriptions are equivalent. If we wish to describe the motions of the particles we could similarly either give N 6 element vectors, or a single vector on a 6N dimensional phase space. Similarly, we could assign a Hamiltonian to either of these descriptions which would generate the dynamics – either of our N particles in a low dimensional space, or of our single ‘particle’ in a high dimensional space. Translating between these two pictures is a fairly trivial operation.
15This argument (including the form of the figures) is borrowed from Peter Lewis, [45, 46]. 16Of course, if our choice is a valid Euclidean set of coordinates then there must be an infinite number of possible valid choices, related to one another by rotations and translations. Whenever we say ‘coordinate system’, we will mean a set of mutually orthogonal dimensions unless explicitly stated otherwise. 38 Chapter 3. Collapse Models
So what? The point here is that when we deal with particles, we are free to make transitions between descriptions without loss of information. Naively, in the above situation C appears simply as a calculational tool. Any claims that C is the real, actual, ontological space of the world (instead of the familiar S) would need some serious justification. However, if we’re describing fields, the situation is very different.
Say we again have a three dimensional space, but now it contains two fields, F1 and F2; each of which can be given as a function over its spatial coordinates f1(x, y, z), f2(x, y, z) respectively.
Now say that F1 is peaked in regions R1 and R2, and negligible elsewhere, whilst F2 is peaked in regions R3 and R4. An illustration of this situation can be seen in figure 3.2a. However, when we want to represent this scenario as a single field on a higher dimensional space – the equivalent of moving from S to C above – we are confronted with a degeneracy. In figure 3.2b, each of the possible constructions shows a representation in which the fields F1 and F2 are each assigned to three orthogonal dimensions, and a single field exists in this new, six dimensional space. In each of these, the field is non-zero on the F1 dimensions in regions R1 and R2, and non-zero on the F2 dimensions in regions R3 and R4.
Since the mapping from the six dimensional representation to the three dimensional repre- sentation is many-to-one, it is safe to say that there is information contained in the higher dimensional representation which is not present in the lower dimensional one.
Let’s look back at these three representations in 3.2b. In the first one, the field’s amplitude on the F1 dimensions in the regions R1 and R2 is independent of it’s amplitude on the F2 dimensions, and visa versa for F2 in R3 and R4. However, in the second construction, the field is only non-zero in F1 around R1 when F2 is non-zero in R4 – in the third, when F2 is non-zero in R3. Thinking in terms of the original two fields, we could say that these various higher dimensional representations give us different versions of entanglement between F1 and F2 – all information about which is lost when we go to a three dimensional representation.
And this is exactly the case with quantum mechanics. We cannot simply reduce the dimen- sionality of our representations – we would lose information about the relationships between whatever ‘subfields’ we had decided would form our actors in the lower dimensional space, and 3.4. The Dimensionality of Reality 39
F1 F2
R1 R2 R3 R4
(a) The two fields F1 and F2 seen as peaks on their respective regions in three dimensional space – which is represented on the x axis, while field amplitude is given on the y.
f2 f2 f2
R4 R4 R4
R3 R3 R3
f 1 f f R1 R2 R1 R2 1 R1 R2 1 (b) The same situation can be represented in a number of ways on a higher dimensional space. Here the dark spots represent non-negligible amplitude, the x axis represents the coordinates of field F1 and the y axis the coordinates of field F2. Figure 3.2: A scenario of two fields which has a single, straightforward representation in a three dimensional space (as seen in 3.2a) has a degenerate description when viewed in a six dimensional space (as seen in 3.2b.) our theory would no longer be capable of making accurate predictions. Even if we can dismiss the high or infinite dimensionality of Hilbert space as an artefact of the calculational simplicity of vector spaces, we would nonetheless seem to have some ontological duties towards the 3N dimensional configuration space of the wavefunction. Any attempt to simply reduce this to a three dimensional physical space populated by N particles would erase information about entanglement, and so contradict known results of quantum physics. It would seem that there is an intrinsic conflict between the dimensionality of the world in which we seem to live, and the dimensionality of the space in which the wavefunction lives. 40 Chapter 3. Collapse Models
Now, as we have argued previously, and as we summarised on page 2.1, the philosophical outlook which motivates the very development of collapse theories is one in which the wavefunction is taken to be, at least at some level, a representation of an ontological entity. And it would be a very strange theory indeed which took the wavefunction to be ontological, whilst taking the space upon which it lives to be something else, something less than real. In the words of John Bell [47]:
There is nothing in this theory [GRW] but the wavefunction. It is in the wave- function that we must find an image of the physical world, and in particular of the arrangement of things in ordinary three-dimensional space. But the wavefunction as a whole lives in a much bigger space, of 3N dimensions.
And so, if we are to take collapse theories as actual theories of reality – which is exactly what they aim towards being – then we are compelled to take reality itself to be something of a staggeringly high dimensionality. And perhaps that’s OK – physics has delivered us very surprising news about the world in which we live before – but for it to be an acceptable claim, there is a very obvious question which the theory must be able to answer. If reality is really an incredibly high dimensional space, upon which lives a single wavefunction, endlessly contorting through time; why does it so consistently seem to be a three dimensional space, populated by objects?
In other words, how does the universal wavefunction ground, or give rise to, or fundamentally explain the appearance of three dimensional world?17
3.5 Collapse Ontologies
Let’s construct a first-order ontology for collapse theories, and see how far we can go in re- covering a picture of the world which might ground our experience. The totality of existence
17Note that this is not just a problem for collapse theories – it is a problem for any quantum theory which takes the wavefunction to directly represent something ontological; including, of course, the Everett interpretation. 3.5. Collapse Ontologies 41 comes down to four things: A high dimensional space H, a universal wavefunction which lives in that space Ψ, a universal noise field ξ, and a dynamical law which we will label Q which (stochastically) governs the evolution of Ψ and its interaction with ξ over time (where we may choose between equations (3.15) and (3.18) as options for Q; more on this choice soon). Let’s examine these things in turn.18
The dimensionality of the space H here is usually called 3N again, with N being the number of particles in the universe. We object that this reflects the very same particle-based thinking by which H is also so often called a configuration space. The dimensionality here is better thought of as the number of distinct degrees of freedom available to the universe – though this is perhaps a little troubling, since the number of available degrees of freedom may change over time. For example, an entangled system of two ‘particles’ may be accurately described on a space of lower dimensionality than that required for a proper description of the same two particles unentangled. We might do better to say that the dimensionality of the space is the lowest number of degrees of freedom available to the universe in a ‘worst case scenario’ – when the mutual information between any and all elements is zero. By saying this, however, we are inviting the snake to eat it’s own tail; since the premise under which we are working is that there are no discreet ‘elements’ at all – rather there is simply a single, grand wavefunction evolving over some high-dimensional space. We will leave the exact number of dimensions of said space, then, as an open and interesting problem for another day.
We return to the problem of how to ground the world in the wavefunction. The usual way to phrase such a question is in terms of reduction – how can a world of tables, chairs, stars and literature be reduced to a single wavefunction living in a configuration space? The word reduction can be used for a variety of purposes, so we will clarify a distinction19, and a further term, here:
18Gravity is, as ever in quantum theories, conspicuous by its absence. There are theories which try to recover gravity from collapse itself, which we will visit in section 3.6, but for now we restrict ourselves to considering more typical collapse theories. 19This distinction is common in the philosophical literature. For an example see [48]. 42 Chapter 3. Collapse Models
Inter-Theoretic Reduction refers here to the type of reduction in which one theory can be reduced to another. An example here would be how aerodynamics can be reduced to classical mechanics.
Ontological Reduction refers here to how an entity can be reduced to another, more fundamental entity. For example, we can reduce wind to the collective action of gas molecules.
Physical Property We say that a thing, with a wavefunction ψ, has a well defined physical property P if ψ is in an eigenstate of an observable O – of which P is an eigenvalue. This is known as the eigenstate-eigenvalue rule20.
Note that in both kinds of reduction described above, the things which are being reduced (aero- dynamics, wind) both remain useful at the appropriate level21. Nonetheless, we are interested here in the second kind of reduction – in explaining how the world in which we live might be understood to be fundamentally reducible to the four elements outlined above. Although, that said, it strikes us that the language of reduction might be misleading here – since we might just as well ask how a world as vast and complex as that comprised by H, Ψ, ξ, and Q might be reduced to one which appears to only have three dimensions, and in which objects mostly tend to be well localised in those three dimensions. There are two principle problems here:
1. How does a field (such as Ψ) correspond to configurations of objects?
2. How, and why, do three dimensions specify themselves within H as somehow special?
Grounding the World in the Wavefunction
In answer to the first, we must begin by saying that our basic intuitions regarding the makeup of reality are simply wrong – which is hardly a bold statement given that we are discussing a 20This rule is sometimes contested – for example by David Wallace [49]. It is not necessary under the Everett interpretation, however some version of it is essential to the ontologies of collapse theories [50, 51]. An overview of the discussion can be found in [52]. 21Although this is not by any means necessary. To give an example for each kind of reduction creating redundancy: the ontological reduction of the caloric fluid to the dynamics of microscopic particles rendered the notion of caloric fluid useless; the theoretic reduction of the Ptolemaic geocentric model of the solar system to the theory of Newtonian mechanics rendered the former model worthless. 3.5. Collapse Ontologies 43 quantum ontology. We start with observables. The locations about which the wavefunction peaks yield high probabilities for recording certain observable outcomes; which is to say, of things having certain properties22. Now, it is clear that each observed property of the world (the distance between this table and that chair, the colour of this apple, etc) is massively degenerate with respect to the exact configuration of its components. This is on some level an echo of the discussion of macrostates from the previous chapter. An important distinction obtains, however. That in the case of thermodynamics the choice of observables is on some level subjective. In quantum theory two observables {O1, O2} may not commute, meaning
a a that even upon the return of two measurement outcomes (physical properties) {P1 , P2 } at infinitesimally separated times, we still have to conclude that an immediate repeat of both
b b measurements might return different values, different properties {P1, P2}; and as such that the wavefunction always has some spread about its eigenvectors. This is profoundly different to thermodynamics. If even after the identification of some property P a system is free to take on another, mutually exclusive property P0 immediately (granted a non-commutative measurement is made between the two), then how do we ground the persistence of some properties of the world in the wavefunction?
Moreover, even though the wavefunction might be compelled into peaks, it is clearly never driven to points. Taking the example of a superposition of two Gaussian peaks being driven to a single Gaussian, as in figure 3.1, the end result is of course a single Gaussian. Which is to say it is non-zero everywhere. This is known as the ‘tails problem’[51], – that while things might appear to be here, in a place, they are apparently also, actually, everywhere, all the time. Not just that they have some probability of ‘jumping’ somewhere, (although they do), but that since the wavefunction is the thing, and the wavefunction is non-zero everywhere, the thing itself is in a sense always everywhere. Everything is. This problem is often identified as a weakness for collapse theories, though in truth it is a challenge for any wavefunction realism. We can easily exacerbate it in the case of collapse theories; post localisation, the resulting Gaussian could be decomposed into a sum of smaller functions, each over different regions. How are we to make sense of the apparent location of things in places, if they are in some sense everywhere? And we cannot emphasise this strongly enough: on a wavefunction realist account, the spread of 44 Chapter 3. Collapse Models the wavefunction does not simply give a non-zero probability of measuring something to be somewhere we wouldn’t expect; rather it literally means that the thing is, at least in some sense, always everywhere.
This is a problem for our grounding. The interpretation of a theory involves, but is not limited to, the mapping between statements about the world and the abstract mathematical represen- tations of the theory. Whilst collapse theories have delivered an evolution for the wavefunction which appears to avoid some of the gravest paradoxes intrinsic to orthodox quantum theory, it does not appear to have delivered an easy ontology. But this is not to say that it has delivered none at all. What we require for our grounding is some way of mapping the representations from the theory into statements about the world. If these statements are at odds with classical ones, as it appears they must be, then so be it. So, what can we say?
Something like this: that each of the points within the regions of high-amplitude of the wave- function correspond to, or bring about, or are equivalent to, or simply are; configurations of the world (corresponding to configurations of classical particles) which are commensurate with appropriate observations; that is, with the outcomes of certain eigenvalues from the application of an ‘observable’ matrix to the appropriate statevector – which is to say, commensurate with things having certain properties. And that this sort of description is going to have to be able to contend with both the state of everything, via a universal wavefunction Ψ on H, and with the states of smaller things, via component wavefunctions ψi ∈ Ψ on subspaces Hi ∈ H.
On this very rough account of grounding, what we might say is that the world which is supported by a wavefunction gathered in a single peak – as opposed to being gathered in several, as a superposition – is obviously more in a single physical state than the alternative. And that by embracing wavefunction realism in the first place, we are compelled to relinquish too close an adherence to our classical understandings of how the world ought to be altogether. That our understanding of things having properties which are definite might be wrong – that things
22The reader might note that the class of ‘observables’ in quantum mechanics (self-adjoint operators with orthonormal eigenbases) is far larger than the class of things which might actually be observed. This is an interesting point, and again invites us to consider the possibility that reality may be at root richer than the world to which we have access (that things might possess more physical properties than we can witness, even in principle). But for now we will put this question aside, and pretend that an ‘observable’ in quantum mechanics corresponds to an ‘observable’ in ordinary parlance. 3.5. Collapse Ontologies 45 might have properties which are, to some degree, somewhat obscure, or fuzzy23, or ambiguous. That the world might be, at root, somehow vaguer than it appears to be. And that this seems to be broadly in line with the spirit of quantum theories of any flavour (except Bohmian) – that some things simply aren’t fully definite.
Recovering Physical Space
And what of the second problem? If H is the fundamental ontic space of reality, why do we seem to live in a space with three dimensions? In [26] and [25], Albert argues that the answer ought to lie in the structure of the Hamiltonian. That the Hamiltonian ought to have a specific structure which groups dimensions in threes – that the generator of the dynamics of the world needs to be of a form which will specify interactions between the different dimensions of the configuration space. By way of analogy, he gives the example of a classical Hamiltonian on an N dimensional configuration space (where N is factorizable by 3)
N 2 2 2 X d x3i−2 d x3i−1 d x3i H = + + dt2 dt2 dt2 i N X 2 2 21/2 + Vij (x3i−2 − x3j−2) + (x3i−1 − x3j−1) + (x3i − x3j) (3.19) i6=j
in which Vij represents some interaction which scales quadratically in physical space such as the Coulomb interaction. It’s immediately clear on this picture how the presence of three physical dimensions could be said to emerge from the dynamics – the behaviour of a single particle moving on this high dimensional configuration space would be equally well described by N/3 particles moving on three dimensions, and with energy freely exchanging between kinetic and potential forms in the usual way. The claim here is that even if the familiar three dimensional
23David Albert has embraced this by suggesting that we replace the eigenstate-eigenvalue link with a ‘fuzzy link’ [51]:“ ‘Particle x is in region R’ iff the proportion of the total squared amplitude of x’s wave function which is associated with points in R is greater than or equal to 1-p.” Here p is a parameter selected for the ontology, a probability threshold which allows for translation between the mathematics of the wavefunction and statements about the world. This is proposed as a resolution to the ‘counting anomaly’ – a problem which emerges when trying to ground the world in the wavefunction. An interesting resolution which doesn’t require a ‘fuzzy link’ and is specific to collapse models using a mass density operator has been put forward [53], though this resolution is not without its own critics [54]. 46 Chapter 3. Collapse Models space in which we seem to live is not the fundamental space of reality, it ought to be perfectly possible for it to emerge as a robust and meaningful structure given a certain dynamics on the fundamental space.
The next questions are obvious: What is the quantum version of this argument? And can it recover a relativistic spacetime, or only a Galilean space? In the next section we will look at some approaches to this problem which are specific to collapse theories, and which attempt to marry its resolution to the solution of the principal problem in the ontology of collapse theories; namely, what is the noise field?
3.6 The Nature of the Noise
The ontological nature of the noise field ξ is a veritable can of worms. The central problem here is that the very positing of this field in some sense reaches oustside of physics itself. Let’s be clear about this.
The inclusion of noise in modified Schr¨odingerequations is nothing new (and, for that matter, neither is the inclusion of apparent non-linearities), and by no means unique to dynamical reduction models. The field of open quantum systems24 [55, 56, 57] is precisely the method of modelling the interactions between an environment and a quantum system when the specific state of the environment is unknown. This approach picks up the threads of decoherence and goes much further with them. By including the environment as a bath which interacts with the system and taking account of the specific form of the coupling in the interaction Hamiltonian, we can recover models of the interaction which describe heating, cooling, dissipation and diffusion, amongst other effects. By including a backreaction of the system upon the environment, we can yield an apparently non-linear evolution for the system. Upon tracing out the environment from the picture, we invariably represent its interactions via a noise, one which can look the same as that of collapse theories. However, as we noted in section 2.3, what this ultimately yields is decoherence, not true collapse.
24We will develop and utilise this formalism in chapter 6. 3.6. The Nature of the Noise 47
And this is what’s so different about the noise posited by collapse theories – by its very nature it cannot be attributed to the averaged action of some larger quantum system. There is no ‘church of the larger Hilbert space’ here. Because if there were, then the alleged collapse about which the theories are built would reduce to nothing more than ordinary decoherence, and they would be completely undone. The result of this conundrum is somewhat strange, and as such it’s worth dwelling on for a moment.
With the very notable exception of gravity, quantum theory has so far served to be an accurate description of every physical phenomenon to which it has been applied25. It seeks to be a physical theory capable, at least in principle, of describing everything physical 26. And so if we’re positing the existence of something which fundamentally cannot be described by standard quantum mechanics, it would appear that we are positing the existence of something non- physical. Again, we emphasise that if the origin of ξ were physical – if it was some physical process which was in any sense ordinary and familiar and relatable to concepts with which we were already acquainted, then it would always be possible to include a description of that process, and its influence upon our system, in a single wavefunction on some larger space. And then we would again have no collapse, no stochasticity, no macro-realism; only, once again, the unitary and deterministic evolution of standard quantum mechanics describing superposition states which persist and propagate from everywhere and everything to everywhere else and everything else. We would be back in an Everettian world, and we would have a host of different questions to answer.
So what are we to make of ξ? Is it acceptable for a theory of physics to postulate something outside of its own remit, and is this clause of indescribability absolutely necessary? Upon the face of it, it might appear that all collapse theories have achieved is a ‘bait and switch’. All of the confusion and paradox of the measurement problem has simply been bundled up and displaced to the noise field ξ. We are reminded of the Homunculus fallacy, in which that which is not yet understood is explained in terms of the phenomena one is seeking to understand27.
25Obviously notwithstanding the measurement problem, and other philosophical or interpretational conun- drums mentioned throughout this text. 26Whether or not we include gravity in here is open to discussion. 27Such as trying to understand the mind and consciousness through positing a smaller, equally conscious mind inside your skull witnessing all that you witness, and thinking all that you think. 48 Chapter 3. Collapse Models
If the existence of ξ is to be taken seriously then it demands an explanation. Let’s categorise the possible explanations into three kinds:
1. Perhaps the simplest option would be to simply call ξ something like a fact of nature. Something which is simply fundamental to the dynamics of the universe, and which has no root in anything outside itself. The values of fundamental constants, the initial conditions for the big bang, the number of quark varieties, and the distance scaling of the four forces are all things which spring to mind as being of this type (which is not to say, of course, that efforts cannot be made to explain or reduce those things).
2. Collapse theories could themselves be approximate and emergent out of some deeper theory. Trace dynamics [58, 59, 60] is the main example of this type, although Flash Ontology theories are arguably of the same ilk.
3. We could take the objection raised above – that anything describable by quantum me- chanics cannot be the source of ξ, and use it to identify a possible source by elimination. This leaves, of course, only gravity.28
There is little to say about the first of these. Such a choice would not invalidate collapse theories – they would still resolve the measurement problem in a consistent way, and deliver a unifying dynamics. But there can be little doubt that such a choice would not be philosophically satisfying.
The second is a fascinating direction of study, however, even an overview of Trace Dynamics is beyond the scope of this work. We leave it un-assessed.
The third option is, in our view, very promising. Historically, gravity has been touted as the possible origin of a collapse noise, and for obvious reasons; it scales with mass, it dynamically affects position, and so far it remains outside the remit of a fully quantum description. The question here is whether or not this effect would be decoherence in the ordinary sense, or whether
28In recent years there has been a growing interest in gravitational decoherence [61, 62, 63]. Though interesting in their own right, such arguments are not to be confused with gravitational collapse models, since they do not threaten the unitarity of ordinary quantum evolution. 3.6. The Nature of the Noise 49 it might be capable of causing genuine collapse, in the sense which we have discussed. To begin with, it might seem impossible for gravity to do this, since general relativity is decidedly deterministic. At the same time, however, we lack any proper understanding of how quantum mechanics relates to general relativity. Or rather, we only have a one-way description; we have a theory of quantum mechanics in curved spacetime, but no quantum theory of spacetime. Speculatively: if it were to be the case that the project to properly quantise gravity fails – that gravity is somehow emergent, yet distinct, from quantum mechanics – then any fluctuations in the local gravitational field might seem to a quantum system to be truly external in origin. There might be no proper quantum description which included them, and as such, we might be able to say that such fluctuations were truly stochastic from the perspective of the quantum system, even if they were deterministic from the perspective of GR. This is, of course, pure speculation. But we note it here as a possible source for ξ because it seems in many ways natural: it couples to mass, which gives a basic grounding for the amplification mechanism; it gives a clear motivation for identifying the position basis as that which the noise would couple to; and, since gravitation is the one phenomenon not yet adequately described by quantum theory, it is also the one plausible physical source for the noise field ξ by a process of elimination.
The best known of the gravity based models is that put forward by Diosi [64] (though it’s often referred to as the Diosi-Penrose model – in our view a misnomer). This model gives a modified form of QMSL which Diosi called Quantum Mechanics with Universal Density Localisation (QMUDL) which tries to avoid the introduction of new physical parameters, relying instead on G/~ to provide the effective strength of the localisation process. This stems from an earlier argument by the same author that any attempt to measure the Newtonian gravitational field locally would suffer from an intrinsic uncertainty (∆g)2 ≥ ~G/V t, in which V gives the volume over which it is being measured and t the time. It was subsequently shown by Ghirardi et. al [65] that the model requires a length scale parameter in order to avoid unacceptable predictions, such as the spontaneous breaking apart of nuclei. Something interesting to note about this program is that the noise here stems from the inability of the quantum state to characterise the gravitational potential (and hence its own Hamiltonian) with infinite precision, and as such an effective noise arises at the level of the quantum dynamics; the noise is not present in gravity 50 Chapter 3. Collapse Models itself29.
A different approach has recently been taken in [67], in which Gasbarri et. al instead endeavour to source the noise from fluctuations in the spacetime metric itself, and find the requisite form which such fluctuations would have to take in order to satisfy the constraints of the CSL model.
Beyond these, there are models which seek to go the other way – seeking to source gravity [68, 69] or even spacetime itself [70], from the action of the collapse. The last of these seems to be building in the direction suggested by Albert which we introduced on page 45, whereas the first two take the existence of a substantival spacetime background as given. Models in which spacetime itself emerges from interactions in Hilbert space would appear to move in a strongly positive direction towards building a dynamics with a satisfying ontology; one which can unite the quantum and classical worlds not only in terms of introducing a mechanism which disallows for superpositions at macroscopic scales and thereby unifies the dynamics, but which also yields an explanation of the ontic spaces for quantum and classical mechanics, and why one would emerge from the other. At the level of premise then, this seems an excellent candidate towards the kind of model which we are seeking. It comes, however, at a price. By generating spacetime (and presumably in the future a curved spacetime, and thereby gravity) from the action of the localisations themselves we have sealed off the last possible source for a physical explanation of the collapse itself, and would have to return to option 1 or 2 from above. Further, if we don’t presume a spacetime in the first place then we need some other way to produce the ‘t0 with which the dynamics of the (modified) Schr¨odingerequation are with respect to. We might naively simply replace the parameter t with some other parameter τ, and recover t via an operator t(τ) = hoˆiτ , and do something equivalent with distances. However, in order to interpret τ here without falling back on the idea of an underlying spacetime, it must be taken to indicate some other dynamical quantity within the Hilbert space. And to understand the evolution of that quantity, of course, we would need to refer to yet another clock. This is the beginnings of the argument towards a relational spacetime, a position which was famously put forward by Leibniz and contested by Newton [71], and later taken up by Mach. Its chief contemporary proponent is Julian Barbour [72, 73].
29This is also the thinking behind the model put forward by Karolyhazy much earlier, in [66]. 3.7. Remarks on Collapse 51
We note a couple of points about this. First of all, an emergent and relational spacetime is not simply a matter of perspective, it requires an attendant model of mechanics which can accommodate all of the phenomena which we can explain by assuming an absolute spacetime. Primarily, it must account for rotation, which is famously difficult to do on a relational approach. But there’s also something slightly strange about arriving at an argument towards a relational spacetime via the route that we have taken. Usually, the proponents of relationality build their arguments upon a perspective of hard empiricism. To summarise the typical argument: if the only things we can measure are duration and distance, not time and space themselves, then why suppose time and space exist at the level of basic ontology? But we have come from a completely different direction. Our realist approach to the quantum state is some fair distance from the positivism of Mach. The argument towards an emergent, relational spacetime here would come from taking a strong realist stance about the nature of the space required for a thorough quantum theory, and then trying to recover the familiar spacetime from that space. This discrepancy in the philosophical stance underlying the argument towards relationality is not a problem per se, more a point of interest.
3.7 Remarks on Collapse
As we have argued, there are two fundamental problems standing in the way of the interpreta- tion of quantum mechanics, standing as obstacles before the construction of a quantum theory which delivers a meaningful image of the world which is free of the contradictions we have described. The first is how we recover classical mechanics from quantum mechanics – how we solve the measurement problem. The second is how we reconcile the physical world of three dimensions in which we seem to live with the dimensionality of the space in which our represen- tation of the quantum state lives; which, as we have seen, cannot be easily dismissed as a mere mathematical tool, but appears to hold some aspects of the nature of the quantum state in its very structure. And of course these issues are not completely separable – if we take the wave- function to have a representational relationship to the quantum state which is at least partially irreducible, and take the quantum state itself to be ontic, then we are compelled to accept the 52 Chapter 3. Collapse Models existence of a higher dimensional space as something more than a mathematical convenience. Collapse models solve the first problem by breaking the unitarity of the Schr¨odinger equation, and doing so in such a way as to yield increasingly classical dynamics at correspondingly large scales. This comes at the price of having to announce the existence of an entirely new, and completely universal noise field. The question remains as to whether or not such a field might be brought to bear on the second of these obstacles, whether it might be capable of simultane- ously explaining the emergence of a classical mechanics and a classical, physical spacetime of 3+1 dimensions. The most pressing question for collapse theory ontologies, however, is that of the noise field itself.
If collapse models are to be more than sleight of hand, if they are to truly resolve the mea- surement problem and deliver at the same time a comprehensive metaphysics for quantum mechanics, and indeed the world – then they must provide an ontology for the stochasticity that they introduce. Even if all this amounts to its assertion by fiat (option 1 from above), then the assertion must be made. Else, a physical argument must be made for its existence, and in particular for how it might exist outside of quantum mechanics whilst still reaching into it and affecting the dynamics. Two options present themselves: the construction of a deeper, fundamentally stochastic theory, or else sourcing of the noise from gravity. The first of these would seem a daunting prospect, the second attacks the very project of quantum gravity. Nonetheless, these are the projects which must be undertaken.
We will circle back to these considerations in our concluding chapter. First, however, we proceed to an analysis of some of the other philosophical problems which might be resolved by the introduction of fundamentally non-deterministic physics. Chapter 4
Indeterminism
In the preceding sections, we have attempted to lay out a clear foundation for the motivations behind the development of collapse theories, to explain their form, and assess the philosophical problems which they introduce. The resolution of the measurement problem is without a doubt the driving force behind the development of these theories, and the reason they exist at all. In having such a clearly defined purpose for these ideas, however, we run the risk of myopia – of failing to see the other interesting things they might tell us. In this chapter we bring collapse theories to bear on two other problems in the philosophy of physics – the nature of arrow of time, and the meaning of causation. We will report much of what follows in [1], in which we examine some of the opportunities which arise from the most fundamental break collapse theories make with orthodox quantum mechanics, and indeed with classical mechanics as well – that of replacing determinism with indeterminism at the roots of nature. Although this move is, as we have discussed, not without its own intrinsic problems; it nonetheless has a significant bearing on other fundamental questions which we might ask about the world in which we live.
We want to begin by demonstrating that there are contradictions between the standard for- mulations of theories within physics – or rather, that there are contradictions between what they are generally understood to mean. We aim to shed light on two specific contradictions; between determinism and a particular definition of causation, and between determinism and an ontological arrow of time. We will further show how surrendering determinism rescues mean-
53 54 Chapter 4. Indeterminism ingful notions of both time and causation. To be clear; the kind of interventions we will be making into the narratives of time’s arrow and the nature of causation are not strictly limited to the effects of collapse theories per se, rather they are the results of replacing ontologically deterministic fundamental physics with ontologically stochastic fundamental physics – a move which is exemplified in, but not limited to, collapse theories.
The most common argument for the arrow of time in physics derives from two things: statistical mechanics applied to classical mechanics to yield thermodynamics, and the past hypothesis; a boundary condition applied to the early universe stipulating that it was of low entropy. Our first contention is that this only explains an apparent passage of time – not an ontological one. This definition of time is built upon the notion of macrostates, which describe a range of possible ontological states for a system with which some macroscopic observations are commensurate. The first cleavage is here, between what we consider our fundamental theories in physics to be about and what the standard account of time in physics is about. The former: the actual state of things independent of us; the latter: an apparent effect which is perspectival, not intrinsic to the dynamics which govern reality. The second cleavage is between the determinism of our fundamental laws and the sense in which we talk about causation in physics – as a ‘real’, non-illusory concept. Determinism is at odds with this picture of causation simply because on a fully determinist account of reality, all things are immanently caused by the state of the universe at any given moment and the structure of the dynamical laws governing the evolution of said state.
4.1 Time - the Standard Account
A thorough account of how an oriented time is most commonly understood to emerge from fundamental physics can be found in [74, 75, 76], and a recent popular account in [77]. Insightful discussions can be found in [78, 79], and an excellent contemporary discussion of the philosophy at stake can be found in [80], whilst the canonical philosophical assessment was given by Reichenbach in [81]. We summarise the proposal, first given by Boltzmann, here. 4.1. Time - the Standard Account 55
We start with some closed system G. This could be anything – let’s say it’s the entire universe.
30 At some time t0, G has an observer who can identify some set of macroscopic properties
M1 = {p1, p2, ...}, in which pi is a property (such as temperature). If we define a state space H (such as phase space) with a basis {d} for our description of G to live in, there will be a region G1 bounded by M1 ∈ H such that any point within G1 would describe G to be in a state commensurate with properties M1. We call this region G1 the macrostate of G at time t0. p d 1 j p2
G1 p3
Microstate
Macrostate (grey region) p 5 p4
di
Figure 4.1: The observable macroscopic properties {pi} bound the area of phase space which we call the macrostate. The unique ontological state which the system is actually in is the microstate – a single point which is contained in the macrostate.Note that although here we are depicting two dimensions of the state space, the illustration should be understood to represent a scenario of arbitrarily high dimension.
If we take any point – any microstate – from G1, we can also calculate its state at all other times, owing to the determinism of our fundamental laws. For a classical system we simply solve the equations of motion for all the constituent components; for a quantum system, we solve the Schr¨odingerequation for a wavefunction describing the whole system (and neglect any
30It may be objected that it is meaningless to consider an observer for the entire universe, since the observer will be, of course, part of the universe. This point is very valid elsewhere, but not here. Since we are considering an apparent and perspectival effect, the argument remains the same if we change the system from ‘universe’ to ‘universe minus the observer’. 56 Chapter 4. Indeterminism
dj dj dj dj
d d d d i t i t i t i t0 1 2 3 Figure 4.2: How entropy grows over time according to thermodynamics. The black dot gives the microstate at ti and the red line its historical path. The grey region gives the macrostate, and the dotted line the volume which it would be approximated to occupy when coarse-graining is employed. The actual volume of the macrostate is constrained to be constant as described by the Liouville theorem. measurement process). However, if we wish to see the evolution of a macrostate over time, we have to use statistical mechanics based upon the underlying laws of motion. Doing this, and defining the entropy of this macrostate as a measure of the total volume |G1| in phase space which this macrostate occupies, S = k log |G1|, we find that the volume occupied in phase space
31 increases for an increasing t (on the condition that G1 is not at thermal equilibrium), as per the famous second law of thermodynamics. The increasing volume occupied in phase space then entails an increasing entropy over time as depicted in figure 4.2.
There are some significant problems with this picture. Of these, the most widely known is the Lochshmidt paradox [74]; namely, that the same effect arises if we insert a negative t – the entropy appears to grow both forward and backwards in time. If all we know is the set of properties M1 characterising the state of G at t0, we find that it is overwhelmingly likely that it was in a state of higher entropy in for some t < t0. In other words, if we are constrained only by knowledge of the state the system at one moment, entropy appears to grow in both the future and the past, as shown in figure 4.3. If we are to orient our arrow of time towards a growing entropy, it would seem that we would need to associate this with both an increasing and decreasing t, which evidently ruins the attempt to match a fully asymmetric, directed time with the parameter t.
On the standard account, this problem is remedied by the introduction of the past hypothesis;
31Although not by all measures – a point of contention we will pick up in the next section. 4.1. Time - the Standard Account 57
d dj j dj dj
d d d d -t i -t i -t i i 3 2 1 t0 Figure 4.3: Entropy growing over a decreasing time parameter from the same initial macrostate. we conditionalise the path of the macrostate from the past to the present upon a lower entropy state for the past, and thereby ensure a non-negative gradient for the entropy at all times [82]. This boundary condition is typically taken to be a restriction over the possible states of the early universe. Such a move, however, is not without its problems [83]. For one, the primary evidence which we have regarding the state of the early universe is that of the cosmic microwave background radiation – a record which tells us that the universe was extremely close to a thermal state. Labelling this as a low entropy boundary condition seems, on the face of it, somewhat absurd, since of course the thermal state is that of maximum entropy [79]. The argument can be recovered by substituting Boltzmann entropy for other measures – yet this very freedom of choice in entropy measure is itself a problem, and the topic of the next section.
This concludes the standard account of time asymmetry within physics. The schema begins with a macrostate description of physical systems, and then applies a boundary condition to its past, and statistical mechanics to its future. Macrostates form the fundamental basis here, and it is worth examining exactly what they mean.
A Lack of Objectivity in the Fundamentals
The notion of macrostates might seem, at first, very appealing. They allow us to do simple and effective physics with such chaotic and complex systems as gases, simply by identifying some readily accessible macro-properties. However, if we want to utilise these same notions to define time for all things (not just gases) when they must be readily generalisable and applicable to 58 Chapter 4. Indeterminism any kind of system at appropriate scale.
Pressure, volume and temperature are reasonable macroscopic properties by which to bound the macrostate of a gas. But for more general systems, we need to discern which macroscopic properties define the state of a system at a given time – the measure of entropy for some system will still be drawn from the volume of statespace constrained by these properties. This allows for some curious situations to arise, as argued in [84]. We summarise the argument here:
Say two observers OA and OB look upon a single system F which is in an ontological mi- crostate Fx at time tx. The observers each have their own measure of macro-properties, and as
such describe the set of macroscopic properties which define F at t as {MA, tx } and {MB, tx } respectively. These give two different macrostates ΓA, tx and ΓB, tx , each of which have their own entropies S(F )A, tx and S(F )B, tx . Now, let the same, single system F be in an ontologi- cal microstate Fy at ty. Again, our two observers will ascribe a different series of macroscopic
properties as above, and yield different entropies to one another again – S(F )A, ty and S(F )B, ty . Which state occurred first? What is the time ordering here? On the account of time asymme-
try given above, all we need do is check whether which is larger between S(F )tx and S(F )ty .
However, it is quite possible that S(F )A, tx > S(F )A, ty whilst S(F )B, tx < S(F )B, ty . This would mean that time flows in opposite directions for two observers watching the same system!
To be absolutely clear; given a system, a set of macroscopic variables, and a choice of entropy measure, there are objective facts at any given time about the macrostate of the system and the entropy of that macrostate. If two observers agree on their choices of variables and their entropy measure, there is no way they can disagree about the entropic ordering of the macrostates of the system, and as such no way that they can disagree about the direction of time. Our contention is simply that there is no clear reason as to why, in general, we ought to select any particular set of macroscopic properties and any corresponding measure of entropy, nor any selection which would be general enough to capture all phenomena – whereas time itself does seem to be universal.
On the standard account of temporal asymmetry, it is the monotonic gradient of the entropy of systems (and this alone) which gives the passage of time. That this gradient depends upon 4.2. Causation – Some Minimal Criteria 59 a non-objective choice of ascribing macro-properties to the system does not mean that the resultant temporal arrow is illusory, but it does mean that it is perspectival and non-objective.
The arrow of time given above is, as we have stressed, a property of the macrostates which an observer would draw around the microstate which the universe is actually in. As shown above, there are different possible ways of doing this – the identification of macro-properties and the following designation of macrostates with attendant entropies is at the very least perspectival; at worst, subjective. This is at odds with the microstate which is supposedly underlying this coarse-grained picture of reality – there is presumed to be a single, definite state the system (universe) is actually in. The physical state of all things is fundamentally objective on the standard view within physics, yet the directionality of the time governing the dynamics of those things is not.
We might ask whether or not we need a direction for time which is fundamental. Why not perspectival? Perhaps such a theory is the best we can do. If so, we would nonetheless desire that it be acknowledged for what it is: a theory of the way things appear, not the way they are.
In section 4.3 we will see how the above story might be changed under a physics with funda- mentally non-deterministic dynamics. First, however, we will raise a parallel objection to the form of causation possible within standard physics.
4.2 Causation – Some Minimal Criteria
Consider two events, A and B with (discrete) outcomes {ai} and {bi} respectively. Under what circumstances can we meaningfully say that A was a cause of, or at least had some causal influence over, B? Drawing on Reichenbach [81], we propose the following as candidates for insufficient but necessary criteria C:
C(1) : There must multiple possible outcomes for each event, i.e |{ai}| ≥ 2, |{bi}| ≥ 2
C(2) : The two can be unambiguously time-ordered, such that A occurs before B. 60 Chapter 4. Indeterminism
C(3) : The probability distributions cannot be marginalised, i.e. P (B|A = ai) 6= P (B).
C(4) : There exists no ‘mutual cause’, C, such that P (C|A, B) = P (C|A) · P (C|B).
In order to explicate the bearing of determinism on this notion of causation, at this point we draw a distinction between apparent and ontic causation. This reflects a distinction which we draw between epistemic and ontic probabilities, and echoes the discussion of the density matrix from chapter 2.
Consider two coins: one which has been flipped and has landed behind an opaque screen, and one which is about to be flipped. Presuming the coins to be fair, an observer would ascribe
1 the two coins the same probability distributions P (H) = P (T ) = 2 . However, the meaning of the probability distribution in each situation is clearly different. In the first, the probability distribution reflects the ignorance of the observer, a simple look behind the screen would reveal the result. We call this form of probability distribution epistemic. In the second, (again, presuming a fair coin), we would naively say that the only way to determine the outcome of the next flip is to wait for it to be determined. We call this ontological probability – it is a true probability in that it reflects that the result is not just unknown, but unknowable at any time before the toss.
On a determinist account of reality all probabilities are epistemic. The state of the universe at all times is determined by its state at any other time, plus the nature of the microdynamical laws governing its change. All possibilities which seem to be immanent and real turn out to be nothing more than a statement of the ignorance of the observer. All coins are already flipped – they just lie behind a screen.
Again, we are not claiming that this is wrong. We are simply highlighting the fact that ontic probabilities, and hence ontic causation, are incompatible with fully deterministic fundamental physics. 4.3. An Ontological Arrow for Time 61
4.3 An Ontological Arrow for Time
At the heart of the argument we make here is the status of the fundamental laws of physics as deterministic, yet it is often claimed that quantum mechanics is anything but. Some clarification is in order.
As we stated in chapter 2, we are taking realism as a premise from which to explore the implica- tions of the theories with which we work. As such, an operational view of quantum mechanics under which it is only a system which is useful for calculating the probability distributions over possible outcomes of specific scenarios is insufficient. Rather, we would look to the interpre- tations of quantum mechanics in which the formalism captures something essential about an external reality. This is broader than the strong realism regarding the wavefunction described on page 19, since it would include, for example, Bohmian approaches. And in all such interpre- tations of quantum mechanics, the formalism itself remains untouched (or the chosen formalism is equivalent). And as such, all evolution through time is unitary, and therefore fully reversible. Excepting collapse models, only on the Copenhagen, or other instrumentalist approaches is irreversibility introduced, and this comes at the price of any meaningful ontology.
What would it mean to have a physical theory of time whose arrow was ontologically intrinsic31? To answer this question, we can first identify two asymmetries in time which are widely regarded as the most fundamental, and which such a theory would need to recover physically:
• Causal asymmetry; in that we can affect the future but not the past.
• Epistemic asymmetry; in that we have access to recorded information regarding the state of things in the past, but only predictive access to the state of things in the future. This predictive access is correct up to a probability, whereas the recorded information about the past is correct up to the fidelity of the recording.
31In [85] Tim Maudlin has argued for an entirely new approach to attaining an arrow of time, by replacing the basic elements from which we construct a geometric spacetime with ones which are intrinsically directed. This is orthogonal to the approach which we take here, but we note it as an interesting path towards the same goal. 62 Chapter 4. Indeterminism
Introducing stochasticity at the level of the fundamental dynamics – as is done in collapse models – would grant both of these properties. Returning to the criteria C from section 4.2 for allowing ontic causation32;
•C (1) is granted by the very nature of stochastic dynamics.
•C (2) is unaffected - time ordering was never at risk under determinism, only a direction- ality to that ordering.
•C (3) is now possible – the stochastic dynamics of event B will now no longer be limited to {0, 1}, and can be correlated to earlier states of the same system, or any other system with which it has been connected.
•C (4) can now also be satisfied. Under determinism, the state of the system at any previous time would have formed a mutual cause C; now, whilst mutual causes are still possible, they are by no means endemic.
By meeting these criteria, such a theory of physics makes ontic causation possible. This is a yield in itself, but of course it also meets the first criterion above which we require for an ontic arrow of time. Yet from this asymmetry in causation, the second criterion – that of epistemic asymmetry – naturally follows. On this picture, the state of things in the past is knowable through record because it is fully determined; the universe contains a single history. The state of things in the future is knowable only through prediction, and hence through probabilities, because it is only defined up to those probabilities. Time’s directionality is recovered as an ontological aspect of the universe. Moreover, the present, which is very hard to make sense of under determinism, becomes precisely the dividing line between those events which are determined and those which are not. 32We are saying nothing here about agency, only causal structure (though this might be regarded as a precondition for agency). 4.4. Summary 63
4.4 Summary
The ontology of the arrow of time in physics, as it stands, is at odds with the ontology of our fundamental theories; it is epistemic, whereas the theories are ontological. Given how fundamental the arrow of time is, this is a cause for concern. Faced with this discrepancy, we have a choice: We can take our theories as operational, be satisfied so long as they work, and not trouble ourselves over the picture of reality painted by such theories – let alone whether or not different theories within physics are even painting the same picture. Else, we can take seriously the ontological implications of physics. If we take this road, we face yet another choice; either we can satisfy ourselves with a time whose directionality is at root perspectival (and after all, we only have access to the universe through some perspective), or we can seek a theory of time in which no two observers could ever disagree upon its directionality, and in which we need make no further choice whatsoever as to what variables and so forth we consider important. As it happens, the stochastic dynamics which grant this view are the very same which are proposed as a resolution to the measurement problem. Such dynamics would give us a clear arrow of time, a meaningful notion of causation, and a clear explanation for the transition between quantum and classical mechanics. Chapter 5
Testing Collapse
As we have emphasised throughout this text, one of the defining features of collapse theories is that they make testable predictions about the world. Since through these theories we are seeking to reproduce both quantum and classical mechanics at the appropriate scales, and these theories are remarkably accurate within their territories, any deviation from their versions of events which we are seeking to glean must be exceptionally subtle. When we say that collapse theories are testable, it does not mean that they are easily testable. As we shall show, some rather specific criteria need to be met in order for the signatures of collapse effects to become visible, let alone differentiable from other, more mundane processes. In particular, since quantum mechanics is so established in the microscopic regime, and classical mechanics in the macroscopic; we will find that collapse theories make their mark most clearly in the mesoscopic regime, in between these two.
In this chapter we will explain our proposal for testing collapse theories using levitated nanospheres, as we reported in [2]. In section 5.8 we will go on to introduce some ideas on testing a variant of collapse theories which avoid the divergent energy predictions of the standard theory.
64 5.1. Finding the Effects 65
5.1 Finding the Effects
We now want to look at how the dynamics of large rigid bodies are affected by the introduction of the noise of CSL, as described in [86]. Let’s say that we have an object which is constructed of N particles in the first quantization picture, or that R d3x hψ|N(x)|ψi = N in the second, Q given by the wavefunction |ψ(q, s)i = |ψi(qi)i. We can considerably simplify the description i d by considering the centre of mass (COM) position of the object by assuming dt N = 0, and working in the first quantisation picture. Here, we can then express the position operator for
th each particle as qn = qn,0 + ˜q + Q, where qn,0 gives the equilibrium position of the n particle relative to the centre of mass position operator Q, and the operator q˜ gives the displacement from said equilibrium position. Such language is possible only because we are modelling stable, solid objects. We take the statevector to be an eigenfunction of this operator, and neglect the internal dynamics such that we can neglect q˜. We next make the assumption that ∆|hQi| rc, which allows us to make the approximation
Z −(x−y)2 1 2 2 2rc N(x) ≈ N0(x) + 2 dy µ(y)e (x − y) · Q (5.1) rc
(3) in which µ(y) = m0δ (y − qn,0) gives the system’s mass distribution.
Taking these approximations and moving to one dimension allows us to re-write the master equation (3.17) as d i 1 X ρ(t) = − [H, ρ(t)] − η[Q, [Q, ρ]], (5.2) dt 2 ~ i,j in which η is giving the effective diffusion caused by CSL:
3 λ r Z 2 2 csl c 3 −rc k 2 η = 3/2 2 d ke |%(k)| (5.3) π m0 where %(k) is the Fourier transform of the mass distribution for the object µ(y), given above.
This form for the master equation exactly matches that which we would find if the object were coupled to a Markovian noise source of infinite temperature32. Because of this, we can treat it as a simple heating source, and search for it accordingly. This marks something of a conceptual 66 Chapter 5. Testing Collapse leap – this noise, which was introduced specifically to suppress macroscopic superpositions, becomes detectable through dynamics which, on the face of them, have no obvious impact on superpositions.
Since the evolution operator no longer commutes with the free-particle Hamiltonian, it can eas- ily be seen that energy will not be conserved. Taking the expectation value for the Hamiltonian over time with the above solid-body approximations, the energy increase over time is given by [87, 88, 89] dE 3 ~2 M = λcsl 2 2 . (5.4) dt 4 rc m0
It is worth noting some of the rather specific features of this effective diffusion η from equation 5.3. In particular, its inclusion of %(y) gives us a dependence on material and form which is peculiar, as pointed out in [90]. If we focus on spherical objects of mass m and radius R, whose Fourier transformed mass density is given by
sin(kR) − kR cos(kR) %(k) = 3m , (kR)2 we find
4 2 R2 2 R2 6λcslrc m − 2 R − 2 rc rc η = 6 2 e − 1 + 2 e + 1 . (5.5) R m0 2rc
We can now look closely at how this quantity scales with both density and the radius of the object – evidently non-trivially for the radius. This is shown in figure 5.1, in which we can see that the effect scales sub-linearly with size in both the high and low limits for radius, specifying an effective resonance in size around R = rc. 32We will derive this in chapter 6. 5.1. Finding the Effects 67
10-16
10-18 η(R) (Hz) 10-20 R3 10-22
10-24
10-26 10-8 10-7 10-6 10-5 R (m)
Figure 5.1: The effective noise on a sphere of radius R, as compared to a line ∝ R3. The dashed red line gives rc. As we can see, η has a sub-linear scaling in mass beyond 2R = rc.
This change in the scaling factor was first noted in [90], and it tells us something rather interesting. Naively, we might think that collapse effects ought to become ever more visible for larger and larger objects. After all, this is an effect which is meant to affect objects at a rate proportional to their size. However, although the effect does continue to grow past R = rc, it does so at a lower rate than the volume scaling typical of most environmental noise – this tells us that it will be most visible for objects on the scale of rc. 68 Chapter 5. Testing Collapse
Figure 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 experiments 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].
When we consider the parameter range available for CSL, shown in figure 5.2, we can see
−8 −3 that this preference for objects of a scale with rc selects objects in the range 10 − 10 m, depending upon the collapse strength λcsl. This scale selection, which springs from both the theory of CSL and from experimental results, itself specifies a range of experiments which are particularly suited to testing the model. Specifically, those dealing with objects of the nano-to-micro scale make promising candidates. 5.2. Levitated Nanospheres 69
5.2 Levitated Nanospheres
The logic behind working with levitated nanospheres, and in particular those levitated using the hybrid-type technique which will be outlined in this chapter, is summarised as follows. The na- ture of collapse models which are introduced with the intention of addressing the measurement problem is that they are necessarily constrained in their form by the existing body of evidence from both quantum and classical experiments; they must be both non-linear and stochastic, and the parameters which introduce these two properties are constrained to lie within a certain range. The introduction of these two properties to the Schr¨odingerequation brings, however, additional baggage. The evolution operator for a free particle no longer commutes with the basic Hamiltonian, and as such the dynamics no longer conserve energy as per equation 5.4. This corollary effect of dynamical wavefunction localisation, profoundly unattractive at first glance, can in fact be viewed as a bonus feature of such models, since it renders them testable in a distinctive way. This opportunity for testing can be compared with the more direct form of testing offered by matterwave interferometry experiments.
The aim of matter-wave interferometry experiments such as [91, 92, 93] is to create spatial superpositions of objects large enough and for long enough times that they ought to begin experiencing the spontaneous localisation of a given model. However, although this does provide a very clear and explicit testing ground, the process of creating, maintaining and measuring superpositions of large particles is a notoriously difficult task [21, 22, 94, 95, 96, 97]. Other, more subtle methods for testing these theories are also possible, such as those proposed in [98, 99, 8, 100]. The notion of measuring the presence of a collapse field via its heating effects was first proposed by Adler [101], though a similar idea was mooted by Collett and Pearle in 2003 [102]. Other, similar schemes are found in [103, 104, 7, 105, 106] – and, of course, in our work [2]. Now, although all objects will experience an ineradicable heating on account of the localisation operator, an additional component of CSL is that the heating effect itself carries a further feature: it scales sub-linearly with mass in both limiting scales – large and small – but finds something akin to a resonance when the object of interest is of a scale comparable to its length parameter rc. This parameter is fixed to lie in a range which specifies objects of 70 Chapter 5. Testing Collapse
∼ 100nm as ideal test objects.
A range of experimental settings could be used to examine objects around this scale. The criteria for the selection between these candidates is, thankfully, fairly clear and simple. We require that:
1. The decohering noise which affects the experiments must be as minimal as possible.
2. The accuracy with which information can be read from the experiment must be as high as possible.
3. The object must be cool-able as far as possible (so as to make the heating effects more visible).
4. (optional but preferable) The size of the object be manipulable, so that the scaling shown in figure 5.1 can be made apparent.
All of these are desiderata are met to a high degree with the choice of levitated nanospheres.
As we can see in figure 5.1, the effects of the noise find a peak around 2R = rc, making something akin to a resonance between the size of the object and the length parameter of the model. It is this size resonance which selects nanospheres as such ideal testing grounds for collapse theories – large enough for the collapse effects to be visible, but small enough that those effects are not drowned out by conventional decoherence – nanospheres are a Goldilocks object here. They also boast some of the highest quality factors (∼ 1012) of any resonator experiments [107]. We propose a specific experimental protocol which is in principle capable of testing the CSL model
−14 −7 as far down as λ = 10 Hz at rc = 10 m, utilising only lab techniques which are already available. 5.2. Levitated Nanospheres 71
Optical Paul Trap
Field Potential ωm
Laser
Trapped nano-particle Detector
Figure 5.3: Schematic for the experiment. A charged nanoparticle is levitated by the electric field of a Paul trap whilst being cooled and measured by the field of an optical cavity. The use of two potentials creates a ‘hybrid’ type of trap, in which the benefits of each method (optical trapping and electrical trapping) can be utilised, whilst their respective shortcomings can be minimised.
Our proposal rests upon a hybrid type trap [108] as depicted in figure 5.3, in which an electric Paul trap and an optical cavity are used in conjunction to trap and cool the particle respectively. Each of these trapping techniques rely upon specific features of the particle: the optical trap requires that it be transparent with respect to the frequency of the light to be used in the cavity, and that it have a high dielectric permitivitty; the electrical trap relies upon the fact that the particle has been prepared with a non-zero charge. By using the two techniques in conjunction, the particle can be readily levitated by an electric field and cooled by a resonant optical cavity. The two fields work in concert to control and manipulate the dynamics of the particle, with the optical field offering the additional capability of accurate sensing. Optical cooling typically requires two fields [109]. Here, with one of those fields replaced by the electrical trap, the dynamics become simplified and the noise reduced. As was shown in [110], the coupling between the cavity field and the motion of an optomechanical oscillator carries enough information that the output cavity field can be used to accurately reconstruct the motion of the oscillator33. We outline the dynamics of said levitation in the next section, and give some numerical simulations 72 Chapter 5. Testing Collapse of the situation in chapter 7.
Conceptually, the experiment is extremely simple. The presence of a collapse noise affecting the dynamics of the sphere will heat it, and this heating is a detectable effect. If the particle can be prepared in a particularly cold state at some initial time t0, and then its occupancy
† n1−n0 n = a a measured at some later time t1, we can infer the heating rate Γheat ≈ . Now, t1−t0 if this heating rate matches that which we would predict in accordance with the conventional sources which will be coupled to the sphere (such as electric field noise and collisions with the background gas) then it would seem no collapse effects are present. If, however, Γheat exceeds what we would conventionally predict Γheat > Γconv, then we can infer the presence of some new, novel heating source. The lower we can make Γconv, the weaker a heating effect from this novel source we can detect. In other words, the level to which we can minimise conventional decoherence effects will determine the degree to which we can test the parameter ranges for λcsl and rc.
We can divide the procedure then into two phases; a cooling phase to prepare the particle in hni = n0, and a period of free evolution in which all cooling is switched off, in order to see how much it heats. The switching off of the cooling will prove to be important since the scattering of laser photons from the optical cavity is the dominant conventional noise source in the hybrid trap scheme [111]. In order to calculate the precise ranges for λcsl and rc which might be tested with this scheme, we need to pay close attention to both the capabilities of the cooling scheme and the effects of conventional noise upon the system.
5.3 Dynamics of the Sphere
The gains of the hybrid scheme over more traditional all-optical schemes are multiple. We avoid the complexities of having two optical fields (both in terms of the mathematical non-linearity,
33Note that in [110], the authors are considering the case of an oscillating mirror which forms one end of the optical cavity, not a nanosphere levitated between two fixed mirrors. The Hamiltonian, however, is of an identical form, and their analysis generalises to our case without complication. The oscillator mass, mechanical frequency, optomechanical coupling, and nature of the noise are all that change. 5.3. Dynamics of the Sphere 73 and the experimental challenges of managing multiple co-propagating beams). The deep po- tential of the Paul trap also provides more stability than can be generated by cavity traps; this is especially useful, because at lower pressures the particles are all too easily excited out of shallow traps. The hybrid trap discussed here allows for indefinite trapping of nanoparticles.
Figure 5.4: Illustration of the potential felt inside the hybrid trap, taken from [112]. The wide quadratic corresponds to the low frequency potential of the Paul trap, whilst the high frequency oscillations are the result of the periodic potential of the cavity field.
The electric field of the Paul trap creates a potential of
1 U (x, y, z, t) = mω2 (x2 + y2 − 2z2) sin(ω t), (5.6) E 2 m d with the mechanical frequency given by
2 2QV0 ωm = 2 . (5.7) mr0
Here ωd and V0 are the frequency and amplitude of the AC voltage applied to the trap electrodes, √ Q the charge of the nanosphere, m the mass, and 2r0 is the distance between two electrodes in the trap. Depending upon how these parameters are selected, the mechanical frequency range available to the particle spans from 2π × 102 Hz through to 2π × 106 Hz. The optical field of the cavity, aligned along the x axis, provides a potential of
2 2 2 2 −2(y +z )/Wc 2 Uc(x, y, z) = ~An¯c e cos klx (5.8) 74 Chapter 5. Testing Collapse
2 withn ¯c the mean number of resonant photons in the cavity, whose beam waist is of width Wc at the focus (which will be very near the mimimum of the electric trap potential). The remaining parameter is given by A = 3V (( − 1)/( + 2)) ν , in which V is the volume of the sphere, 2Vc r r l 2 Vc = πwc Lc is the cavity mode volume given a cavity length of Lc, νl is the angular frequency of the laser and kl its wavenumber, and r is the relative permittivity of the material of the levitated object. This potential creates the second, higher mechanical frequency for the sphere at 2 k2A ω2 = ~ l n¯2 cos 2k x (5.9) c m c l 0 with x0 the mean displacement from the equilibrium position in the trap.
For small oscillations, ignoring input noise and cavity leakage, the quantum Hamiltonian for the system is given by
ω H = − ∆ c†c + ~ m (x2 + p2) + g (c + c†)x, (5.10) ~ c 2 ~ in which c and c† give the creation and annhilation operators of cavity photons of frequency
νc, which are detuned from the laser by an amount ∆c = νl − νc. The effective length of the cavity is affected by the motion of the dielectric sphere, and this coupling is characterised by the parameter
2 2 2 2 ~k A n¯c 2 g (t) = sin (2klx0(t)) 2mωm which can be time averaged to 2 ωm 1 − J0(4k 2 x0) 2 ωc 2 2 2 g = ~k n¯c A (5.11) 2mωc
with J0 being the zeroth Bessel function.
Cooling the Motion of the Sphere
The cavity cooling technique makes use of the interaction between the position of the oscillator and the intensity of the intracavity field. Since the resonant frequency of the cavity depends 5.3. Dynamics of the Sphere 75 upon the position of the oscillator (as can be seen in the final term of equation(5.10)), the oscillations of the sphere can be used to affect the intensity of the intracavity field. By red- detuning the input laser (with a positive ∆c), the motion of the sphere can periodically bring the cavity closer to resonance. In particular, the further it is from an antinode of the periodic optical potential, the larger the effect. As the motion of the sphere varies the effective cavity length and periodically brings the cavity closer to resonance with the input laser, the amount of light coupling into the cavity increases. Because of the high finnesse of the cavity, the decay of this increased intensity is delayed with respect to the motion of the sphere, which causes cooling. This effect would, on its own, cause not insignificant cooling – but it would be limited, since the strength of the effect is proportional to the distance of the particle from the antinode, and the cooling would be reducing the amplitude of excursions from there. The potential of the Paul trap, however, can fix x0 to be non-zero, meaning that there is always some cooling effect.
In [108] the authors achieve a centre of mass temperature for a nanosphere levitated in a hybrid type trap of 10 K, which is improved to the level of milliKelvin in [113]. However, the same techniques ought in principle to allow for ground-state cooling, and are limited only by technical factors such as the finesse of the cavity mirrors.
In our scheme we will be cooling the motion in ωm, though the following methods could equally be applied to cool ωc. The steady-state phonon number achieved through cooling is
2 κ + κsc Γsc + Γothers nss = + (5.12) 4ωc Γ−
in which Γsc gives the heating rate due to the scattering of cavity photons, Γothers gives the heating rate due to all other processes, κ is the rate at which photons leak from the cavity, κsc the rate at which they leave the cavity via interactions with the sphere, and Γ− is the cooling rate given by " # 2 1 1 Γ− = g k − . (5.13) 2 κ2 2 κ2 (∆ − ωc) + 4 (∆ + ωc) + 4 For a finnesse of 105, ground state cooling within the optical potential should be achievable 76 Chapter 5. Testing Collapse
−4 −Γt Γt using an input laser power of 10 W. Using hnit = hnitherme +nss 1 − e where hnitherm is the mean phonon number the oscillator would have if thermalised with its environment, we can see that hnit ≈ hniss within about a microsecond, so there is no time impediment to achieving the cold states we require.
Cooling to a given phonon number in the optical potential does not translate directly into that phonon number being occupied in the Paul trap once the cavity field is turned off, due to the different trap frequencies. Supposing the antinodes of the optical well and Paul trap are perfectly aligned, when we turn the optical potential off the sphere should have n0 = Nssωc/ωs, where n0 is the phonon number in the Paul trap immediately after the cavity is switched off and Nss is the phonon number in the optical trap immediately before. However, if we take into account some displacement between the two trap centres δx, then we would expect that
2 n0 ≥ Nssωc/ωs + mωsδx/(2~), where the inequality accounts for the fact that it is also possible for the optical field to impart momentum to the sphere as it is turned off. The ability for this displacement to impart phonons requires that we ensure δx . 0.5nm.
When we consider these factors, it becomes clear that our predictive knowledge of the phonon number at the beginning of the free evolution n0 is far from perfect. This is important of course, because it is by comparing n0 with nf that we hope to learn anything. However, it is possible for us to gain experimental, not just predictive knowledge of n0. A ‘dry run’ is possible, in which rather than turning the cavity field completely off we reduce it to such a low power that it no longer traps the particle, but does remain coupled to the particle’s position.
As such we can use it to measure n0 via the techniques laid out in [110]. This information can be used heuristically to better align trap centres, and also to build up a statistical picture of n0, providing a benchmark for the actual experiment in which the cavity field is turned off completely.
We now turn to a precise treatment of the various conventional noise sources which we would expect to heat the particle. 5.4. Noise Sources 77
5.4 Noise Sources
Gas Collisions
The major appeal of the levitation scheme is, of course, the loss of the mechanical connection to the environment. The background gas nonetheless provides an environment in its own right. If the particle’s centre of mass motion is hot enough, the background gas can provide significant damping. In the regimes we are considering, however, the effect is negligible, since the gas will be modelled at temperatures of over 4 K, whilst the particle will be cooled to nano-Kelvin temperatures. This leaves us with a heating rate, which is given by
Dg = 2γgmkBT, (5.14) where γ = 4πP R2 wherev ¯ is the RMS velocity of gas particles in the chamber, and P is the g mv¯g g pressure [114]. The heating rate in terms of phonons per second is given by
0 Dg Dg = . (5.15) 2mωm
Blackbody Radiation
The contribution from blackbody radiation is comprised of two components; emissive and absorptive noise. The emissive rate from a spherical object in the Rayleigh regime is calculated
4 6 2π (kB T ) −1 by Chang et. al in [111] to be Dbb = 5 5 Im ; where is the permittivity of the object 63 c ~ ρω +2 and ρ the density. This value is approximately 10−10ω−1 for our parameters, which is negligible compared to noise from gas collisions or E field fluctuations.
When considering absorptive noise, however, our case is quite different to that of Chang et. al in [111], since we must consider the blackbody radiation from the nearby surface of the trapping electrode, not just free space. An analysis of environmental blackbody noise above trap electrodes in [115], analysing both surface emissions and those from free space, yields a 78 Chapter 5. Testing Collapse heating rate of 2 0 Q T 1 −27 Dbba = 2 × 10 , 4mdz~ωm 3 which gives an effect which ranges from eight to twelve orders smaller than the other noise sources considered over the range of ω which we examine, rendering it also negligible.
Electric Field Noise
The models built up around electric field noise in quadrupole traps are a mixture of heuristic description and theory. In general, such models are tested by measuring the heating rate of an ion held in the trap operating in the MHz range. One of the difficulties in attempting to build a coherent model of such noise is that when measuring the noise via the heating rate of an ion, it is impossible to distinguish between the different origins of the noise. As outlined in [115], there are many possible sources of noise, including patch potentials on the surface of the electrodes, Johnson-Nyquist noise, and interference with the equipment from other fields in the lab.
When modelling our electric field noise we face a dual difficulty. First, the noise sources - as far as they are understood - have primarily been studied in the MHz range, whereas we are interested in a wider range. Secondly, the variation in experimentally detected values for heating effects from electric field noise and the lack of a total and coherent theoretical framework for treating such noise makes a complete model impossible; either a heuristic one based on data from other experiments or a predictive one based on theory.
Our approach then will not rely on such. Instead, we will propose a generic model for electric field noise (not distinguishing between the possible origins of the noise) and describe a means by which we might fit the parameters of such a generic model to experimental data.
Following the convention in the literature, we model the electric field noise as Ohmic, an approximation which is legitimate when the correlation time of the field is considerably shorter than the heating time for the oscillator (as will be the case in our considerations.)
A generic model for the electric field noise density spectrum at the centre of the trap takes the 5.4. Noise Sources 79 form [115]
−α −β χ SE(ω) = gEω d T (5.16) in which V gives the voltage applied to the electrodes, α, β and χ are parameters to be fit to the specific trap, and gE is a scaling factor. The coupling for an object with charge Q held the centre of the trap and oscillating with mechanical frequency ωm then is
2 Q SE(ω) γE = . (5.17) 4mωm~
Drawing on [115], we take generic values of α = 1, β = 3, χ = 0.57,34 Again, the heating rate takes the form
0 DE DE = (5.18) 2πmωm~ where
DE = 2mkBT γE. (5.19)
Drawing from the data aggregated in [115] and assuming a cryogenically cooled trap we can
−17 estimate our parameters to be α = 1, β = 3, χ = 0.57 and gE = 1.55×10 , which corresponds to a heating rate of 1 quanta per second for a Ca40 ion trapped at ω = 2π × 5.5 kHz.
Micromotion
The trap is driven by an AC field at Ωrf . This field is null at the centre of the trap, but excursions by the trapped particle can lead to heating from the noise on this field. Specifically, the heating rate is given by
2 2 00 2 Q qx(Φrf ) SV (Ωrf ± ω) 2 Γrf,± = 2 (∆z) (5.20) 16m~ω Vrf 34Note that α can go as low as 0.5, and as high as 2, and β can range from 3.5 to 5. χ changes around t = 70 K to ∼ 2. 80 Chapter 5. Testing Collapse where Q|E | ∆z = static mω2 is the mean displacement of the particle from the null point of the rf field [115]. In our case, the static field along the z axis will be zero, entailing no effect from micromotion. However, even if we applied a static E field along this dimension with a voltage equal to that of the rf
−18 2 −18 field, we would have ∆z ≈ 10 /ω , and Drf ≈ 10 DE, meaning that we can safely ignore it.
Rotational Dynamics
Another effect of the rf field is worth taking note of however. The heating discussed above is due to noise on the rf field acting on the centre of mass motion of the particle, but there is also another mechanism capable of transferring heat to the mechanical frequency of the particle. In the case of a trapped nanosphere the charge can be distributed anisotropically over the surface of the sphere, unlike a single ion. An anisotropic charge distribution can lead to a torque on the particle as it passes through the rf field gradient, which can induce rotation. The energy from this mode can then couple into the mechanical frequency, causing heating. We have explored this effect thoroughly through numerical simulations. However, since we concluded that the effect is negligible, we relegate our treatment of the problem to Appendix E.
5.5 Testing Collapse
So, having built up a model incorporating all the significant sources of decoherence which our nanosphere might encounter, we can return to the question of exactly how far it can test collapse theories, specifically CSL. We recall the experimental overview given on page 72. The particle will be trapped by the two fields working in conjunction with one another, and cooled similarly by the methods described above. Once it has been cooled to a mean occupation of nss, the optical field will be turned off entirely. Since the noise from optical scattering is the dominant by several orders of magnitude [111], it is worth removing it entirely. The particle then remains levitated by the Paul trap potential, and will be heated. It will be heated by the noise sources 5.5. Testing Collapse 81 named above and, if it exists, by the effects of a localising field such as that of CSL. The optical field will then be turned back on, and the fluctuations in its intensity will be used to measure hni as per the methods described in [110]. If the measured quantity exceeds that which we would predict according to ordinary environmental noise, we have possible evidence in support of collapse theories.
An added bonus of removing the cavity field entirely from the period of free evolution is that the dynamics are made considerably more concise. The master equation34 (including noise) for our system is given by [114]
d i ρ(t) = − [H0, ρ(t)] − Dp[p, [p, ρ(t)]] − Dq[q, [q, ρ(t)]] − Γ[q, {p, ρ(t)}] (5.21) dt ~
in which Dp and Dq give momentum and position diffusion respectively. Given our parameters, the Dq term is negligible compared to the Dp, as is usual, and we can safely ignore it. The momentum diffusion here is given both by our conventional and unconventional sources: Dp =
η + Dconv, where Dconv is the sum of the heating rates due to the sources described in the previous section. From this master equation we can extract a heating rate
d hn(t)i = −Γhni + D0 (5.22) dt p where D0 = Dp . Equation (5.22) has the solution p 2~ωm
D0 D0 hn(t)i = e−Γt n − p + p , (5.23) 0 Γ Γ
n0 being the initial occupation of the oscillator, which will be equal to nss in our case.
In the regime in which Dcsl & Dconv, the effects of collapse will make themselves known through a heating rate for the particle which cannot be explained in ordinary terms – which cannot be attributed to the ordinary environmental sources. And of course the ratio between Dcsl and
Dconv depends on a number of things, namely the parameters λcsl, rc, and the environmental
34This form of master equation is well known. We will derive a more general form in the next chapter, where it can be seen that this form would emerge as a special case in which the bath is taken to be fully Markovian. See also appendix D. 82 Chapter 5. Testing Collapse
2500 - 8 λcsl=10
2000 λcsl=0
1500 〉 n 〈 1000
500
0 0.0 0.2 0.4 0.6 0.8 1.0 Time (s) Figure 5.5: The expectation of the number operator after a second of free evolution, given a preparation in the state n0 = 50. The mechanical frequency here is ωm = 5 kHz, the particle 3 −8 density is that of silica ρd = 2300 kg/m . We have taken rc = R = 100 nm and λcsl = 10 Hz. As we can see, the scenario including collapse effects heats the particle drastically more. factors which contribute to the conventional heating sources.
5.6 Differentiating Collapse from Decoherence
There is an obvious flaw to this scheme. If we were to conduct the experiment and measure a heating rate which could not be entirely attributed to the conventional noise sources, we might be tempted to claim the detection of some collapsing field commensurate with CSL, or some similar model. The objection, of course, would be simply that we had mischaracterized one of our noise sources, or indeed that we had neglected to model one altogether. The problem of distinguishing the effects of collapse from those of ordinary environmental noise is a vital one, and generic to any test of collapse theories. We can address this problem through parametric variations. 5.6. Differentiating Collapse from Decoherence 83
- 8 λcsl=10 105 λcsl=0 s 1
〉 4
n 10 〈
103
10- 11 10- 10 10- 9 10- 8 10- 7 Pressure (mbar)
(a) Heating rate as we vary the pressure.
106 λ - 8 - 8 csl=10 106 λcsl=10 5 λ 10 csl=0 105 λcsl=0
s 4 s 1 1 4 〉 10 〉 10 n n 〈 3 〈 10 103 102 102
102 103 104 105 106 107 10- 9 10- 8 10- 7 10- 6 ωm (Hz) Radius (m)
(b) Varying the mechanical frequency. (c) Varying the radius of the trapped sphere.
Figure 5.6: Expectation of the number operator evaluated at t = 1s, under a range of parametric variations which enable us to distinguish collapse from environmental noise. For each plot we see how the heating rate for the sphere would vary in response to a particular parameter, both with and without the effects of collapse.
In figure 5.6, we see how the heating rate for the object will respond to three significant parametric variations. As we can see, for each parametric variation there is a clear difference between the scenario in which collapse is present or not. Taking the example of varying the pressure – if there is no collapse, then the heating of the sphere is dominated by collisions with the background gas. As such, as soon as we increase the pressure we immediately increase the heating rate of the particle too. However, if the heating rate is indeed dominated by the presence of collapse, then there is a range in which the effects of the gas collisions remains irrelevant, since it is dwarfed by the effect of the novel noise. This region of immunity would have no explanation under ordinary circumstances. All of the three parameters investigated 84 Chapter 5. Testing Collapse
10-4 -12 10-6 P=10 Pa ( Hz ) 10-8 P=10-10 Pa csl
λ -10 10 P=10-8 Pa 10-12 P=10-6 Pa 10-14 10-10 10-9 10-8 10-7 10-6 10-5 rc (m) Figure 5.7: The limits of how far into the parameter range of CSL the experiment would be 5 able to probe. Here we have taken R = 100 nm, ωm = 10 Hz, a charge for the sphere of 5 eV, an evolution time of 1s and an initial phonon count of n0 = 0, idealising a perfect ground state. The results here promise a marked improvement upon what has been achieved already, as seen by comparison to figure 5.2. in figure 5.6 demonstrate a clear deviation between the mundane and the collapsing scenario. Plot 5.6c however, shows an entirely different shape in the presence of the proposed noise field.
We see something like a resonance around the size here, peaked at R = rc/2, such that the object itself is of a scale with the correlation length for the collapsing noise, which echoes what we saw in figure 5.1.
This then offers us a vital power – to be able to really attribute a detected effect to collapse. Or, at least, something with a scaling peculiar to collapse models. If we do indeed detect an unconventional heating rate, we can go through the parametric variations in figure 5.6 to isolate the effect which cannot be attributed to environmental noise. We can then repeat the experiment with varying sizes of particles to produce a plot like that in 5.6 (d), and check if the scaling of the noise matches the predictions of collapse theories. If it did, this would be extremely strong evidence for such a theory. In this sense this protocol goes beyond testing the parameter ranges for the two free variables of the theory, and into the domain of truly affirming it. 5.7. Testable Parameter range 85
5.7 Testable Parameter range
In the high-frequency regime the dominant source of heating is that of collisions with the background gas. As such, the parameter range which can be effectively probed by the lev- itated nanosphere will be determined in large part by the pressures which can be achieved experimentally. However, pressures lower than the most optimistic presented here have been reported in particle traps years ago [116], making the exploration of collapse frequencies as low
−14 as λcsl = 10 Hz possible.
This compares very favourably with other proposals for constraining these models, and would probe ∼ 2 orders deeper into λcsl than the best results available so far from cantilever experi- ments.
5.8 Constraining The Dissipative Collapse Model
As we have shown, the dynamics of collapse models will cause a spontaneous heating through momentum diffusion. This, of course, violates the conservation of energy – divergently so, in fact. In the above section, we showed a clear way in which such a violation could be tested and measured. In [117], however, Smirne et. al develop an alternative incarnation of CSL (which we will call CSLD, for continuous spontaneous localisation with dissipation) which would avoid the divergence of this violation. They postulate a dissipative effect, constituted in proportion to particle’s motion against a cosmic backdrop. This backdrop would give a rest frame to the universe in a way reminiscent of Newton’s ideas regarding absolute space, and in a sense echoes the thinking behind the aether. Particles moving too fast relative to this frame would be slowed by a dissipation, whereas those moving too slow would be heated by ξ in the ways that we have explored.
At first glance this might sound somewhat outlandish. But in a certain sense the model is a very well motivated attempt to give some physicality to collapse models. The fluctuation-dissipation theorem [118], as is well known, allows for no fluctuation without dissipation. Which is to say, 86 Chapter 5. Testing Collapse if we feed noise into our system without also incorporating a dissipative mechanism, we are positing something rather un-physical. The only scenario under which this would come about ordinarily would be if our noise source was at an infinite temperature. Otherwise, physically, it is very difficult to imagine a coupling which would allow energy to flow one way but not the other, which would correspond to fluctuation without dissipation. Small wonder, then, that the standard energy production under collapse theories is divergent.
By introducing a universal frame, is this model really committing a sacrilege further than that of ordinary CSL? Further than the positing of a universal noise field which couples to all things, but only allows energy to flow one way? And on the question of a universal rest frame, we recall that the CMB already gives some form of this as it is. Here we propose a pair of simple methods for examining the model.
Dissipation and Cosmic Rays
The first is through the examination of observational data regarding cosmic rays. Cosmic rays avail themselves to this purpose for two reasons; they have high enough energies that they would feel the dissipative effect, and they might travel for long enough before reaching us that the dissipative mechanism would appreciably affect their velocities.
We have that the Hamiltonian for a free particle under the dissipative model is given by [117]
−χt H(t) = e (H(0) − Has) + Has (5.24) where 4kλm2 χ = 5 2 , (5.25) ((1 + k) m0)
~ m is the mass of the particle, m0 is the mass of a proton, k = is a new constant introduced 2mvdrc by the model, characterising the damping rate in proportion with a speed vd, which sets the benchmark for the model. λ is the frequency of collapse for a single particle as per usual for CSL. 5.8. Constraining The Dissipative Collapse Model 87
The essential property here is that the damping rate is proportional to the mass of the particle. So, if two particles are emitted from some star at the same time and with the same velocities but different masses, they will undergo different levels of damping over their lifetimes and arrive at Earth as cosmic rays with corresponding energies.
Energy
2.5× 10 -9
2.× 10 -9 m=1m 0
-9 1.5× 10 m=2m 0
1.× 10 -9 m=3m 0
5.× 10 -10
Time(years) 104 105 106 107
Figure 5.8: Energies of particles with different masses but the same initial kinetic energy over time as their energy is damped
5 In figure 5.8 we look at a basic demonstration of this where the damping frequency νd = 10 Hz, corresponding to a bath temperature of 1.9 K. Clearly, the particles of different masses damp at different rates and hit their asymptotic energies at different times.
If we detect a cosmic particle with mass m and energy E0 on its own, of course there is no way of determining that it once had energy E and had been damped at a rate χ over a period t. However, the thinking behind this proposal is similar to how we detect redshift in light from distant stars; there would be no way to infer a given shift from a single photon. Rather, to understand redshift we look at a detected spectrum and compare it to processes which might have generated a transformed version of that same spectrum, and infer both the original spectrum and the transformation it has undergone (i.e redshift) from this. Likewise, if we were to look at observed cosmic rays the process would be as follows:
1. - Identify some cosmic event which ought to eject a known range of masses with known initial energies.
2. Identify the energies of incoming cosmic rays over time. 88 Chapter 5. Testing Collapse
3. See if the spectrum of arrival energies as a function of mass matches the initial ejection spectrum, but transformed according to the dissipative model.
4. Further check if there is a delay in arrival time determined by the mass of incoming particles as corresponding to the model.
This is a rough sketch of one method for putting this model to the test, as it were. In order to fulfil it, we would need to identify a class (or some classes) of cosmic event(s) which would produce ejecta with a known spectrum of mass and velocity. We would also need to square a certain circle regarding relativity. The CSLD model is thoroughly non-relativistic, just like CSL. The energies which we’ve input for figure 5.8 however, are relativistic.
It’s possible though that we might not need trouble ourselves over these contradictions. In Appendix C, we pick up the work done by Lochan et. al in [119], which examines the effect that including CSL dynamics would have on the CMB. By following a similar route, but for CSLD, we find that we can constrain the CSLD model through the condition that its inclusion in early universe cosmology would not predict a CMB other than we measure today. Doing so, we find that the effective temperature of the field would need to be ∼ 1036 K. On the face of it, this seems too high to be plausible. Either the CSLD model would compel us to accept a noise field with a temperature that is exceptionally high, or else it might be rescued by further modification – either by allowing the temperature of the field to change over time, or by relaxing the stipulation that its noise is characterised by a white spectrum. It is the characterisation of such spectra which will be the topic of the next chapter. Chapter 6
A Quantum Spectrometer for Arbitrary Noise
The development of a thorough treatment of the scenario of a levitated nanosphere described in the last chapter has led us into a broader analysis of noise and how it interacts with quantum systems. In [3] we explain our proposal for a quantum spectrometer for arbitrary noise. In this chapter we will relay these ideas, explaining how a spectrometer might be constructed from a quantum harmonic oscillator – one which is capable of characterising the spectrum of a noise source with an arbitrary temporal correlation function; given certain (fairly forgiving) assumptions about the nature of the noise source, the oscillator, and the coupling between the two. This would constitute a powerful tool for the study of noise in and of itself. Further, we expect that the ability to accurately characterise the environmental noise spectrum would find broad application. In the context of quantum computing, such knowledge would enable the development of optimised dynamical decoupling protocols tailored to the specific environment of the qubit(s)[120]. It may be useful in short-range force sensing [121, 122], where the spectrum characterising the interaction between the force being studied and the behaviour of the oscillator can be subjected to a similar treatment. Beyond these directly practical applications, it holds strong promise for foundational physics. In section 6.5 we will circle back to collapse theories, and demonstrate how this spectrometer might be used to advance the project central to this
89 90 Chapter 6. A Quantum Spectrometer for Arbitrary Noise thesis – that of promoting collapse theories from the phenomenological to the physical.
6.1 Non-White Noise
The presence of noise in experimental scenarios is an unavoidable fact. Though noise may be minimised and diminished, it remains an ever-present component of all physical dynamics. No system is every truly de-coupled from the world; and if it were, noise would inevitably make itself known at the moment of ‘measurement’. Even experiments designed to occur in deep space suffer from the unwanted, unpredictable, influences of the universe at large [123]. The impact of noise on measurements is of central importance to quantum metrology [124], and indeed quantum sensing more generally. However, in most theoretical models which include noise, it is typically either characterised as simply ‘Markovian’ or ‘non-Markovian’; the first being an idealisation, and the second being a general statement that said idealisation does not hold. What is generally lacking is any methodology for understanding how the unique and specific nature of the noise in any particular scenario will imprint upon the dynamics of the system under study. Such an understanding is by no means impossible – one simply needs an accurate model of the noise in order to get started.
As such, we expect that the techniques we report here may have wide application. Since so many experiments and sensing protocols rely upon the use of a quantum harmonic oscillator, it may be entirely possible, and even simple, for those experiments and protocols to incorporate our spectrometer technique as a precursor phase, in which the very system which is to be studied can first be used to characterise its environment, and thereby make possible a more advanced model of its dynamics, enhancing the analysis of the data it will then create. 6.2. Formalism 91
6.2 Formalism
We begin with a system S coupled to a bath B spectrally decomposed into an infinite set of bosonic harmonic oscillators, each with frequency ωα. The total Hamiltonian is given by
H = HS + HB + HI (6.1) in which
1 H = ω (a†a + ) (6.2) S m 2 X 1 H = ω (b†b + ) (6.3) B α 2 α
† X X ∗ † HI = a gαbα + a gαbα (6.4) α α where the above terms give the system, bath and interaction Hamiltonians35 respectively. a and
† † a are the creation and annihilation operators for the system, bα and bα being the equivalents for the bath mode α, and gα being the coupling between the system and the α mode of the bath.
We introduce the interaction picture
˜ i(Hs+HB )t −i(HS +HB )t HI (t) = e HI e , (6.5)
ρ˜(t) = ei(HS +HB )tρ(t)e−i(HS +HB )t (6.6)
with the master equation d ρ˜(t) = −i[H˜ (t), ρ˜(t)]. (6.7) dt
We make the standard assumption that at t = 0 the system is seperable, which is to say that it
35The form of the interaction Hamiltonian in (6.4) corresponds to having made the rotating wave approxima- tion – we have a direct energy exchange between the system and the bath, instead of the usualy position-position coupling. Strictly speaking, this approximation is not necessary for the technique which we lay out here. We make it because it allows for a very clear and intuitive derivation of the formalism. For completeness, we include an alternative derivation sans this approximation in Appendix D. 92 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
can be factorised as ρ(0) =ρ ˜S(0)⊗ρE(0), in which ρS and ρE are the reduced density matrices for the system and environment respectively. We make the Born approximation, which extends this for all times, and assumes that the environment is left relatively unchanged by its interactions with the system:ρ ˜(t) ≈ ρ˜S(t)ρE, rendering the reduced density matrix for the environment R t ˜ stationary. This gives a solution to equation (6.7) ofρ ˜(t) = ρ(0) − i 0 ds [HI (s), ρ˜S(s)ρE]
Next, we want to re-insert this solution back into equation (6.7), and make the Markov as-
36 sumption as we do so. This is to say that the future behaviour ofρ ˜S(t) will depend only on its present state, and not upon its past. This assumption is justified if the dynamics of the environment are evolving on much faster timescales than those of the system – something already implicit in us making the Born approximation. Using this assumption we get
Z t dρ˜S(t) ˜ ˜ = − ds TrE[HI (t), [HI (s), ρ˜S(t)ρE]]. (6.8) dt 0
where the Markov approximation has enabled us to make the replacementρ ˜S(s) → ρ˜S(t) in the second commutator.
We now introduce the grand operators for the bath
X † X ∗ † B = gαbα,B = gαbα, α α possessing two-time correlation functions
1 † Ct,s = TrE[B (t)B(s)ρE] (6.9)
2 † Ct,s = TrE[B(t)B (s)ρE] (6.10)
Ft,s = TrE[B(t)B(s)ρE] (6.11)
∗ † † Ft,s = TrE[B (t)B (s)ρE]. (6.12) 36Note that this does not equate to Markovian dynamics – it is a necessary but insufficient condition to attain them. 6.2. Formalism 93
We also notice that
i † † † Cs,t = TrE[B (s)B(t)ρE] = TrE[ B (t)B(s) ρE]
† ∗ = TrE[ B (t)B(s) ρE]
i∗ = Ct,s.
1 2 At this point, a Markovian bath would correspond to taking Ct,s = Ct,s = Jδ(t − s) and
∗ Ft,s = Ft,s = 0. However, our intention is to recover information regarding these functions through the dynamics of the system, and as such we leave them general for now.
Putting the above correlation functions into Eq (6.8), expanding and re-arranging we arrive at
Z t h d 1 iωm(s−t) † 1∗ −iωm(s−t) † ρ(t) = −i[H, ρ(t)] − ds Ct,se [a, a ρ] + Ct,se [ρa, a ] dt 0
2 −iωm(s−t) † 2∗ iωm(s−t) † + Ct,se [a , aρ] − Ct,se [a, ρa ]
iωm(t+s) † † ∗ −iωm(s+t) + Ft,se [a , a ρ] + Ft,se [ρa, a] i iωm(t+s) † † ∗ −iωm(s+t) + Fs,te [ρa , a ] + Fs,te [a, aρ] (6.13)
where for simplicity we have substituted ρS → ρ. Now, if we name the above to be a Liouvillian
d super-operator dt ρ(t) = Lρ(t), then we can examine the time evolution of the average of any operator of any operator O acting on the system via
d hO(t)i = Tr[OLρ(t)]. (6.14) dt
Using this, we can find an equation of motion for the number operator n = a†a:
d Z t h † † 1 iωm(s−t) 1∗ −iωm(s−t) ha ai = −i[H, ρ(t)] + ds ha ai + 1 Ct,se + Ct,se dt 0 † 2 −iωm(s−t) 2∗ iωm(s−t) − ha ai Ct,se + Ct,se
iωm(s+t) † † ∗ −iωm(s+t) + Ft,se ha a i + Ft,se haai i iωm(s+t) † † ∗ −iωm(s+t) − Fs,te ha a i − Fs,te haai (6.15) 94 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
This gives us our basic mechanism through which we might learn the nature of C1,C2 and F . By monitoring ha†ai and how it changes over time we can infer exactly what these correlation functions are doing. As it stands however, equation (6.15) contains too many variables for us to make proper sense of. We can simplify it without placing too much restriction on our model by making the assumption that the bath correlations are symmetric with respect time
i i i reversal; i.e. that Cτ1,τ2 = C (|τ1−τ2|) = Cτ2,τ1 and Fτ1,τ2 = Fτ2,τ1 , and time translation; i.e. that Ci(t, s) = Ci(t+τ, s+τ) and F (t, s) = F (t+τ, s+τ). Let’s break these assumptions down a bit. The time translation symmetry should be fairly uncontroversial – it’s physically equivalent to saying that our environment will be characterised by the same correlation function at any given time. This would not account for, for example, any drift in the parameters determining the behaviour of B. But in the scenario of a stable environment it is an entirely justified assumption. The time reversal symmetry is not saying very much at all. The correlation function asks us: if we sample a value, a point, at time t and we are then going to take a second sample at time s, what do we expect the product of these two sample to be? If what we are sampling is completely random, then there will be no correlation whatsoever, and C(t, s) ∝ δ(t − s). But if we think there is some rhyme, reason or regularity to what we are sampling, then we expect C(t, s) to be a function of the distance between the two times |t − s|, which, of course, = |s − t|. Not having this symmetry would be equivalent to saying that by having taken the first sample at t we ought further to know exactly where on the function this point corresponds. For example, if we were stochastically sampling a sine wave of amplitude 1 and we measure a value of .8; then removing the time symmetry from our expectation value for the correlation function would be equivalent to saying that we knew whether this value of .8 had been measured on the rising or falling edge of the sine wave – asserting the symmetry is equivalent to saying that all we know is the value of .8, and not where on the function it was sampled from.
i i∗ So, if we allow for these assumptions, then it will also be the case that Cτ1,τ2 = Cτ1,τ2 , since
i † † † Cτ1,τ2 = TrE[B (τ1)B(τ2)ρE] = TrE[ B (τ2)B(τ2) ρE]
† ∗ i∗ = TrE[ B (τ2)B(τ1) ρE] = Cτ2,τ1 . 6.2. Formalism 95
This allows us to simplify eq. (6.15) to
d Z t n o hn(t)i = ds cos ωm(s − t) 2(hni + 1)C1 − 2hniC2 . (6.16) dt 0
We can further adjust this. Using y = s − t:
Z t Z t ds Ci(s − t) cos[ωm(s − t)] = dy C(y) cos ωmy 0 0 1 Z t Z 0 = dy C(y)eiωmy + dy C(y)e−iωmy 2 0 −t 1 Z t Z 0 = dy C(y)eiωmy + dy C(−y)eiωmy 2 0 −t Z t 1 iωmy dy Ci(y)e . 2 −t
To this form, we now introduce the Fourier transform of C(y);
Z +∞ C˜(ν) = dy C(y)eiνy (6.17) −∞ 1 Z +∞ C(y) = dν C˜(ν)e−iνy, (6.18) 2π −∞ with which we can re-write Eq. (6.16) as
d Z t Z ∞ ˜ ˜ iy(ωm−ν) hn(t)i = dy dν (hni + 1)C1(ν) − hniC2(ν) e . (6.19) dt −t −∞
˜ ˜ ˜ Introducing C3(ν) = C2(ν) − C1(ν) and integrating over y we get
Z ∞ d sin t(ωm − ν) ˜ ˜ hni = 2 dν × C1(ν) − hniC3(ν) (6.20) dt −∞ (ωm − ν) which fits the very familiar form
d hni = A(t) − γhni. (6.21) dt
Here, γ provides a damping effect, and A(t) the heating. At this point we make a further 96 Chapter 6. A Quantum Spectrometer for Arbitrary Noise assumption, this one a little stronger than those that precede it. We neglect the damping effect ˜ ˜ by setting γ = 0, entailing that C1(ν) = C2(ν), such that we can drop the subscript and just use C˜(ν). If the bath were strictly thermal, this would be equivalent to taking it to be of infinite temperature. However here we have still not assumed a form for the spectrum of the bath, and it remains general (barring the assumptions already made). Instead, the assumption here should be read as meaning that the equilibrium energy to which the harmonic oscillator is tending is orders larger than its starting point, and that as such the damping rate is negligible compared to the heating rate.
This then allows us to take equation (6.20) to
Z ∞ d 1 ˜ sin[(ωm − ν)t] hnit = dν C(ν) (6.22) dt 4πmωm −∞ (ωm − ν) which has solution
Z ∞ 2 1 ˜ sin [(ωm − ν)t/2] hnit = hni0 + dν C(ν) 2 . (6.23) 2πmωm −∞ (ωm − ν)
Equation (6.23) gives the central result of the spectrometer, and as such it bears some exami- nation.
On the face of it, it may look as though we’ve resolved nothing, since (6.23) contains an integral which is unsolveable given, of course, that we don’t know what C˜(ν) is. However the sin2 term,
2 sin [(ωm − ν)t/2] 2 , (ωm − ν)
provides us with an envelope about ωm which behaves somewhat akin to a delta function. The ˜ term depicted in figure 6.1 will effectively select the effects of C(ν) around ωm, in the region of
2π 2π ν = [ωm − t , ωm + t ]. Since ωm is a changeable parameter for the harmonic oscillator, this will allow us to scan through the available range of frequencies, sampling the Fourier transform of the correlation function at each point. This gives us the protocol for the spectrometer then – to sample C˜(ν) at various mechanical frequencies, and use the results to reconstruct C˜(ν), and thereby C(t, s). 6.2. Formalism 97
2 1 sin t(ω m-ν) 2 2 (ωm-ν)
ωm/2 ωm 3ωm/2 2ωm ν ˜ 4π Figure 6.1: The function tied to C(ν) forms a peak of width t about ν.
In our setup, we have included a single bath B. In practice however, the system may be coupled to a number of baths, and all of these will contribute to its heating. To work this into our formalism is a straightforward task, simply taking
X i HB → HB
† X i i X i,∗ i,† Hi → a gαbα + a gα bα . α,i α,i
Taking the other baths to be Markovian, we would end up with a modified version of equation (6.23) for the expectation value for the number operator given by
Z ∞ 2 0 1 ˜ sin [(ωm − ν)t/2] hnit = hni0 + Dpt + dν C(ν) 2 (6.24) 2πmωm −∞ (ωm − ν)
0 in which DP gives the heating rate due to all baths other than the one of interest. If the harmonic oscillator used for the spectrometer were a levitated nanosphere, Dp would be given by the equations for the noise sources described in the previous chapter. The protocol is, of course, much broader than this specific experimental instantiation. In order to assess how effective a particular platform will be for enacting this technique, we must consider the following factors:
• The range over which the mechanical frequency ωm can be adjusted.
• The initial occupation number n0 in which the oscillator can be prepared. This has a 98 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
dual significance. Firstly, the colder the oscillator, the more accurate the approximation
that we can neglect the damping effects of the bath. Secondly, the lower hn0i, the lower
its spread, and the more accurately ∆n = n(t) − n0 can be estimated.
• The accuracy with which any other baths can be modelled. If the total effect of the environment is the object of study then, of course, this point is moot. But if there is a particular field which is to be characterised whilst in the presence of other fields or other noisy effects, then the careful characterization of those noises must be given due attention. Typically we would consider them to be Markovian and delta-correlated in time, making them much easier to model. If they are non-Markovian, and their spectra are unknown; it would in principle be possible to distinguish between the effects of more than one bath with an un-known correlation function. Something akin to the methods outlined in section 5.6 would need to be employed, and the spectra of the different baths ascertained at each frequency in ratio to one another. This sort of disambiguation would require, of course, parametric control over the couplings to each bath.
• The accuracy with which hni can be measured.
• The duration of the free evolution t in each measurement.
This last point bears further consideration. The question of the optimum magnitude of t lands us in a double bind. On the one hand, by increasing it we increase the accuracy with which we are estimating the value of C˜(ν) at a given frequency by narrowing the envelope which selects it. However, if t grows too large, we risk escaping the regime in which the damping can be ignored, and in which we can simplify equation (6.20) by taking C1 = C2. Ideally, we would compute the limit in which this regime is crossed. The problem, of course, is that in order to compute the point at which the damping effect becomes non-negligible we would need to know the value of nsteady, the steady state for the number operator on the harmonic oscillator. And to calculate this, we would need to know C1 and C2, which is an ouroboros.
In resolution to this, we propose that nsteady can be measured in situ, and that the optimal t can be calculated based on the difference between this and n0. 6.3. An Analytic Solution for Gaussian Noise 99
6.3 An Analytic Solution for Gaussian Noise
Equation (6.24) is the central result of the spectrometer protocol. It contains an integral which is, of course, unsolvable, since by construction C˜(ν) will not be known. Once data is gathered however, one might wish to fit it to a model. In this case different candidates for C˜(ν) will need to be plugged into (6.24) and compared to experimental results. This is not necessarily a straightforward task owing to the non-triviality of the function. Numerical solutions might not always be avoidable. Nonetheless, some analytic solutions are possible. Here we give an analytic solution for a common form of noise density function; a Gaussian.
We start with a Fourier transformed correlation function of
2 − (ν0−ν) C˜(ν) = ηe 2γ2 , (6.25)
in which η gives the strength of the coupling, ν0 gives the resonant frequency for the noise and γ gives the width of its spectrum.
Putting this correlation function into equation (6.24) gives us
0 hnit = hn0i + Dpt + Yt (6.26) in which
Z ∞ (ν −ν)2 2 η − 0 sin [(ωm − ν)t/2] Y 2γ2 t = dν e 2 2πmωm −∞ (ωm − ν) Z ∞ [y−(ω −ν )]2 2 η − m o sin (yt/2) 2γ2 = dy e 2 . (6.27) 2πmωm −∞ y
We can simplify this considerably. First we note that
sin(yt/2) 1 Z t = ds eiys/2 (6.28) y 4 −t meaning that we can re-write the second term of the integral as 2 t t sin (yt/2) 1 Z Z y 0 0 i 2 (s+s ) 2 = ds ds e (6.29) y 16 −t −t 100 Chapter 6. A Quantum Spectrometer for Arbitrary Noise which we can sub into equation (6.27) to get
2 Z ∞ Z t Z t (y−(ωm−ν )) y η 0 − 0 +i (s+s0) Y = dy ds ds e 2γ2 2 (6.30) 32πmωm −∞ −t −t which, if we perform the integration over y gives
Z t Z t ηγ 0 − 1 γ2(s+s0)2+ i (ω −ν )(s+s0) Y = √ ds ds e 8 2 m 0 . (6.31) 16 2πmωm −t −t
0 0 In order to proceed with this integration we will use a change of variables x2 = s+s , x1 = s−s . Using the Jacobian for this change of variables and correspondingly transforming the area over which we’re integrating: which is a square whose sides have length 2t in the original variables, and a rotated square whose sides have lengths 4t in the new variables. Collecting the integrand into a function f(s, s0), we have four distinct integrals:
Z t Z t Z 2t Z 0 Z 2t Z 2t−x2 0 1h ds ds f(s, s ) = dx2 dx1f(x1, x2) + dx2 dx1f(x1, x2) −t −t 2 0 x2−2t 0 0 Z 0 Z x2+2t Z 0 Z 0 i + dx2 dx1f(x1, x2) + dx2 dx1f(x1, x2) −2t 0 −2t −x2−2t (6.32)
which, if we change x2 → −x2 in the last two of these integrals and rearrange a little, gives
Z t Z t Z 2t Z 2t−x2 0 ds ds f(s, s ) = dx2 dx1 (f(x1, x2) + f(x1, −x2)) (6.33) −t −t 0 x2−2t
So we have
2t 2t−x+2 ηγ Z Z 1 2 2 1 2 2 − γ x +i(ωm−ν0)x2 − γ x −i(ωm−ν0)x2 Y = √ dx2 dx1 e 8 2 + e 8 2 32 2πmωm 0 x2−2t Z 2t ηγ − 1 γ2x2 1 = √ dx2(4t − 2x2)e 8 2 · 2 cos( (ωm − ν0)x2). (6.34) 32 2πmωm 0 2
x2 Introducing z = γ , we get
r Z tγ 2 η 1 − z z(ν0 − ωm) Y = dz (γt − z)e 2 cos . (6.35) 2γmωm 2π 0 γ 6.4. A Practical Application – Electric Field Noise In Paul Traps 101
This integral can be solved explicitly and left general, but the outcome is rather bulky. It is more instructive to consider two specific limiting cases. Recalling that we are examining a peak of width γ, and how the spectrometer would be impacted by this width if measuring it for time t, we can look at both the limits in which γ 1/t and γ 1/t. For the first of these, we can
− 1 z2 approximate the Gaussian in 6.35 as e 2 ' 1, which gives
ηγ r 1 1 − cos(ν − ω )t Y 0 m = 2 (6.36) 2mωm 2π (ν0 − ωm)
The above formula gives the largest contributions when |ν0 − ωm| ≤ 1/t (this is reasonable: if the distance between ν0 and ωm is larger than the probe precision 1/t, the probe cannot ‘see’ q ν ) in which case we can further approximate N ' 1 ηγ t2. However, all this also holds 0 t 2π 4mωm whether |ν0 − ωm| γ or |ν0 − ωm| γ, whereas a good probe requires that for |ν0 − ωm| γ the above should be zero. As expected, if γ 1/t then the probe is not effective for the bath considered here.
However, if γ 1/t i.e. the error in probing the noise spectrum is smaller than its width, then
r Z tγ 2 η 1 − s (ν0 − ωm) Nt ' γt ds e 2 cos s (6.37) 2γmωm 2π 0 γ
r Z ∞ 2 η 1 − s (ν0 − ωm) ' γt ds e 2 cos s 2γmωm 2π 0 γ 2 ηt − (ν0−ωm) = e 2γ2 , 4mωm which is exactly what we would expect from a good probe: the contributions are relevant only in the region ωm ' [ν0 − γ, ν0 + γ] where the bath correlation is non zero.
6.4 A Practical Application – Electric Field Noise In
Paul Traps
The analysis we have presented so far in this chapter is truly general – it would apply to any harmonic oscillator with a changeable mechanical frequency which is weakly coupled to a 102 Chapter 6. A Quantum Spectrometer for Arbitrary Noise stationary bosonic bath and prepared in a low occupancy in which damping effects are negligible. As it happens, the experimental scenario of a levitated nanosphere, presented in the previous chapter, can qualify as just such a system. Here we will give a demonstration of a real-world problem to which this system, coupled with the spectrometer protocol, could be turned.
The specific structure of the electric field noise which affects the levitated particles in Paul traps remains unknown [115]. It is believed to comprise of at least three distinct sources: Johnson noise, coming from thermal fluctuations in the circuitry; technical noise, coming from a wide range of processes in the controlling electronics; and adatom noise, coming from fluctuations in the configurations of atoms on the surface of the electrodes. The accurate reconstruction of the spectrum of the noise present here would serve at least two functions. Firstly it would be a scientific achievement in its own right, and bring an advancing clarity to a fairly well-developed field of study. It would constrain models of the three above-mentioned noise sources, and hence advance the study of those processes. Beyond this, it would have direct implications for the experiments being carried out in the settings of Paul traps. These constitute a veritable plethora of investigations, from quantum chemistry[125] to quantum computing [126, 127, 128, 129], to biological studies [130], to foundational experiments such as [131] and that described in the previous chapter. Here we create a non-trivial fictional structure for this electric field noise and demonstrate how our protocol could be deployed to reconstruct it.
In the previous chapter we modelled the electric field noise using equation 5.16, which makes an Ohmic approximation for the noise. Here, we have removed the straightforward ∝ 1/ω scaling of the noise density spectrum and replaced it with a sum over several Gaussians, collected in a ˜ function CE(ν). This constructs a fictional noise density spectrum in frequency space with non- trivial structure. The amplitudes and widths of the Gaussians are chosen in concert with the coupling gE such that the mean of the spectrum is in agreement with typical behaviours from ion traps reported in [115]. In other words, if the electric field noise were to have the chosen fictional structure and scaling, it would not contradict known results. Taking this new structure for the electric field noise, whilst treating all other noise sources as properly Markovian and 6.4. A Practical Application – Electric Field Noise In Paul Traps 103
collected as usual in the term Dp, gives us an expectation value for the number operator
2 Z ∞ 2 0 Q −β γ ˜ sin [(ωm − ν)t/2] hnit = hni0 + Dpt + d T gE dν CE(ν) 2 . (6.38) 2πm~ωm −∞ (ωm − ν)
One of the principle strengths of the levitated nanosphere experiment is that its mechanical frequency may be varied over a wide range – roughly from 102 − 106 Hz, enabling us to probe a correspondingly wide range of the noise spectrum. Taking achievable parameters for other experimental factors (such as background gas pressure, electric field noise, and environment temperature), and using an example of a R = 50 nm sphere with the density of silica ρ = 2300 kg/m3 and a charge of Q = 104 eV we can expect a heating rate from conventional sources of
Dp < 1 phonon/s at higher frequencies.
Figure 6.2 demonstrates the capabilities of the technique for recovering a fictional complicated structure assigned to the electric field noise. Further, it gives a clear demonstration of how the resolution changes with the measurement time. For each of the solid lines the measurement time at each frequency is made inversely proportional to that frequency, but by different factors.
The solid red line, which gives a measurement time of t = 1000/ωm achieves a fairly accurate reconstruction of the noise spectrum; whereas the shorter adaptive measurement time of t =
10/ωm depicted on the solid purple line attains only the basics of the shape. The minima of the function are missed by both attempted reconstructions owing to the obscuring role of the other noise sources present – in particular, the collisions with the background gas overshadow the effects of the electric field noise in these regions.
Again, we wish to emphasise that the simulation seen in figure 6.2 was constructed using realistic parameters, and that gE has here been scaled such that if the actual spectrum of electric field ˜ noise were given by CE(ν), it would be consistent with averaged results drawn from [115]. 104 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
106
4
t ˜ 10 C(ν) 〈 n 〉 〈n(t)〉|t=10/ωm 2 10 〈n(t)〉|t=1000/ωm
100 103 104 105 ωm (Hz) Figure 6.2: Performance of the spectrometer as enacted on a levitated nanosphere system with realistic parameters, deployed to measure a fictional and non-trivial structure to the electric field ˜ −1 10 noise. The dashed blue line gives the noise spectrum to be measured, scaled as C(ν)ωm × 10 for clarity of comparison. The two solid lines show what the spectrometer would reconstruct. The vertical axis gives hnit for the solid lines, and simply the value of the function for the dashed line. The solid purple line shows the performance when the measurement time at each frequency is set to t = 10/ωm, whilst the more accurate reconstruction seen on the red line is achieved with t = 1000/ωm. The minima of the function are missed by both reconstructions owing to the obscuring effects of other noises, in particular, collisions with the background gas. The parameters utilised for the model are achievable with present-day technology, and the scaling of the E field noise has been chosen such that even with the noise density structure shown it would yield heating rates commensurate with those typically measured in the literature – which is to say that Paul trap electric field noise could have this structure, and if it did our technique would recover it as seen in the solid line plots. 6.5. Using the Spectrometer to Test Non-White Models of Wavefunction Collapse 105
6.5 Using the Spectrometer to Test Non-White Models
of Wavefunction Collapse
In this section we will circle back to the central topic of this thesis. As we have explored in chapter 3, the collapsing field ξ presents a host of problems. Amongst these is the assumption of its Markovian nature, which is always an approximation when considering real-world noise sources.
The journey towards a physically plausible collapse model can be pursued in two principal directions. We can think of noise sources (such as the classical gravity mentioned on page 48), develop models based on these, and then design experiments capable of probing the specific noise spectra which such models would predict. Alternatively, we can develop general experimental tools which would be capable of discerning the spectrum of a detected collapse noise, and then use this data to direct the exploration and development of models which would correspond to it, identifying potential noise sources along the way. If we were to not only discover evidence in support of collapse models, but that the noise field causing such collapse was of a specific spectrum – then this spectrum would massively constrain, and thereby enhance, any attempts to build up more physical collapse models.
Either way, the specific nature of the proposed noise field is an object worthy of study. If collapse models are ever to be pushed beyond the phenomenological and into the physical, we will need to develop models of the noise field which paint a picture from start to finish; identifying an origin for the noise source which makes testable predictions about the specific characteristics of said noise, and provides a satisfying ontology explaining why this noise source can affect quantum objects without itself being reducible to quantum mechanics.
Towards this end, it behoves us to develop collapse models which go beyond the white noise approximation towards a more physical noise. No noise field is genuinely white in its spectrum (as this would correspond to a process occurring infinitely fast), and by limiting our collapse models to white-noise approximations we preclude ourselves from the being able to develop a model which could correspond to a physical source. We can develop general experimental 106 Chapter 6. A Quantum Spectrometer for Arbitrary Noise tools which would be capable of discerning the spectrum of a detected collapse noise, and then use this data to direct the exploration and development of models which would correspond to it, identifying potential noise sources along the way. One example of this is the recent work by Curceanu et. al [8], which more or less rule Ornstein-Uhlenbeck type noise processes with cut-off frequencies above 1018 Hz. More recent work looks at probing non-white noise models possessing exponential cut-offs; both with interferometry [132], and with non-interferometric techniques [133] similar to those of the previous chapter.
The goal of this section is to demonstrate that the spectrometer we have described is capable of adding to these, not just by increasing the power of probes into noise models with exponential cut-offs, but also by extending the experimental capabilities into the realm of noises whose correlation functions are arbitrarily complex. By way of example we look at two generic corre- lation functions for ξ; a Gaussian correlation function in frequency space as given by equation (6.25), echoing the mathematics section 6.3; and an exponential cut-off in frequency space of the Ornstein-Uhlenbeck type. Each of these extends the parameter space for the model. In the first instance, by introducing a resonant frequency ν0 and a width γ; and in the second by introducing a cut-off frequency ϑ.
The theoretical basis for this exploration can be found in the two papers [134, 135] which explored collapse models with non-white noises. The authors found that relaxing the condition of making the Markov approximation from the noise field ξ to a more general form
hξ(t)i = 0 (6.39)
hξ(t)ξ(s)i = C(t, s) (6.40) does not negatively impact upon the function of ξ; which is to say that the wavefunction is still driven to eigenstates in the chosen basis, and with the probabilities which would accord with the Born rule.
The master equation for a CSL type model with non-white noises when we consider a solid 6.5. Using the Spectrometer to Test Non-White Models of Wavefunction Collapse 107 object moving in one dimension is given by [134, 135]
d i Z Z t ρ(t) = − [H, ρ(t)] − η d3x ds C(t, s)[Q, [Q(s − t), ρ(t)]]. (6.41) dt ~ 0 where we have taken the first order of coupling to ξ. If we then include that our object is a harmonic oscillator with frequency ωm, we have that
sin(ωmτ) Q(τ) = cos(ωmτ)Q + P, (6.42) mωm with P being the centre of mass momentum operator for the object. Using this, (6.41) becomes
d i ρ(t) = − [H, ρ(t)] − ηA(t)[Q, [Q, ρ(t)]] − ηB(t)[Q, [P, ρ(t)]] (6.43) dt ~ in which
Z t A(t) = dsC(t, s) cos[ωm(s − t)] (6.44) 0 1 Z t B(t) = ds C(t, s) sin[ωm(s − t)]. (6.45) mωm 0
This is fully equivalent to the alternative treatment for the master equation for the spectrometer given in Appendix D, and if we again neglect damping it leads directly to the formula a hnit which is equivalent to equation (6.24):
Z ∞ 2 η ˜ sin [(ωm − ν)t/2] hnit = hni0 + dν C(ν) 2 . (6.46) 2πmωm −∞ (ωm − ν)
As such, the spectrometer which we have described gives a direct method for testing collapse models with non-white spectra. As we have stated, this could be used in two directions. Firstly it can be used to put specific models to the test – models which have a form for C˜(ν) based on physical reasoning. And secondly, it can be used to discover C˜(ν), and thereby point the way towards the development of models which might be able to explain any structures discovered in the spectrum. 108 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
One of the complexities of exploring non-Markovian noise here is that the environmental noise is not itself flat. The model of electric field noise which we employ (and which we suspended in section 6.4) is given by equation 5.16, which scales ∝ 1/ωm. This renders it Ohmic, which is not the same as non-Markovian [136]. A constant influx of energy at a given rate will yield a heating rate in terms of hni which is ∝ 1/ωm, owing to the fact that the energy per phonon scales with the mechanical frequency. Taking this into account, the heating in terms of hni from
2 background gas collisions scales ∝ 1/ωm, whereas the electric field noise scales ∝ 1/ωm owing to the additional factor from its Ohmic density noise spectrum. This can be seen in figure 6.3, in which the dominant source of heating transitions over the range of frequencies.
102 Gas E field Dp'
100 phonons / sec
10- 2 102 103 104 105 106
ω m Figure 6.3: Heating rates from the most significant mundane sources over the range of me- chanical frequencies available to the spectrometer, taking a silica particle of radius R = 100 nm, a charge of Q = 100 eV at a pressure of 10−10 Pa and in an environment at 4 K.
This, of course, impacts on how visible the effects of any collapsing noise are. Figure 6.3 shows how the effects of a collapse model with a white noise spectrum become more visible at higher frequencies. This will inform the following discussion, in which we examine the visibility of collapse models with non-white spectra of two archetypal forms. 6.5. Using the Spectrometer to Test Non-White Models of Wavefunction Collapse 109
106
105 t=0.001 s t=0.01 s 104 t=0.1 s
〈 n 〉 ˜ 103 C(ν)× 105
102
101 102 103 104 105 106 ωm Figure 6.4: Performance of the spectrometer for reconstructing the spectrum of a collapsing noise field with a peaked Gaussian structure. The noise spectrum itself has been scaled for clarity. Increasing the experiment time here increases the fidelity of the reconstruction, as in the previous section. The impact of the noise has been calculated using a typical value of −7 −5 4 rc = 10 m, and an atypical value of λcsl = 10 Hz. The noise is peaked about ν0 = 10 Hz and has a width of γ = 103 Hz.
Gaussian Structure
We progress to a version of CSL characterised by Gaussian spectrum version of ξ, as given in (6.25). We want to consider how well this correlation function could be characterised by the levitated nanosphere scenario. This functions similarly to the electric field noise spectrometer, except here the noise of interest is sourced from a collapsing field, and the electric field noise is reintroduced as an ordinary Markovian source as in chapter 5 and the resulting mundane heating rates are shown in 6.3.
Utilising the analytic solution for Gaussian type noises given in (6.35), we have an exact form for the expected phonon number after some time t given by
r Z tγ 2 η 1 −z z(ν0 − ωm) 0 2 hn(t)i = hni0 + Dpt + dz (γt − z)e cos . (6.47) 2γmωm 2π 0 γ 110 Chapter 6. A Quantum Spectrometer for Arbitrary Noise
The effects of the collapse noise are diminished by the confining effects of the Gaussian structure.
As such, they only become appreciable (compared to environmental noises) with a higher λcsl
−5 than would be typical for white noise models – in figure 6.4 we have used λcsl = 10 Hz. We should be hesitant, however, about comparing this too directly with the performance of the apparatus on the white noise model, where it would appear to be able to probe a full ∼ 9 orders of magnitude further into λcsl. Here we are exploring a four dimensional parameter space instead of the usual two, having added γ and ν0 to rc and λcsl. This makes too direct a comparison meaningless.
High-Frequency Cut-Off
The second generic form of correlation function we want to consider is that of an exponential cut-off, ϑ C (t, s) = e−ϑ|t−s|, (6.48) E 2 which corresponds to an Ornstein Uhlenbeck process. This is often taken as a ‘first order’ example of what the spectrum of ξ could look like, since noise processes which appear to have white spectra in some region of frequency space are often found to have exponential cut-offs when a larger range is taken into account; this is why it is the form of non-white noise considered in [132] and [133]. The fact that the spectrum can be modelled with the introduction of only a single new parameter – ϑ, the cut-off frequency – is attractive. In order to find the effects of the spectrometer, we first need to find the C˜(ν) corresponding to (6.48). Letting τ = t − s,
Z ∞ ˜ −iντ CE(ν) = dτCE(τ)e −∞ ϑ Z ∞ = dτe−ϑ|τ|−iντ 2 −∞ ϑ Z ∞ Z 0 = dτe−ϑτ−iντ + dτeϑτ−iντ 2 0 −∞ ϑ 1 1 = + 2 ϑ + iν ϑ − iν ϑ2 = . (6.49) ϑ2 + ν2 6.5. Using the Spectrometer to Test Non-White Models of Wavefunction Collapse 111
1
0.1
ϑ/10 ϑ 10ϑ ν
˜ ϑ2 Figure 6.5: A simple illustration of CE(ν) = ϑ2+ν2 with ν shown as multiples of ϑ.
This mimics the exponential decay itself, as seen in 6.5.
In order to properly model how a collapse model whose noise possesses this particular structure would affect our chosen experiment, we would, of course, need to substitute this expression into equation (6.46), include our environmental noise, and solve for
Z ∞ 2 2 0 ϑ sin ((ω − ν)t/2)) hnit = hnit + Dpt + η dν 2 2 2 . (6.50) −∞ ϑ + ν (ω − ν)
The integral here does have an analytic solution (barring the limits), but it is so large and unwieldy that it would serve little purpose to reproduce it here. Including the limits, we find the solution to be convergent numerically.
How far might a model with a given value of ϑ be probed in terms of λcsl? This is the question answered in figure 6.6. Here we have ignored the initial phonon number and solved numerically for values of λcsl such that the heating due to CSL is equal to the heating from environmental noise at different frequencies37. As expected, collapse noises with higher cut-off frequencies can be probed to lower values of λcsl. We also see again that the Ohmic scaling of the electric field noise makes higher frequencies preferable, up to the point at which the respective cut-off of the collapse noise ϑ determines an effective limit. Because of this, the lowest probeable value of
λcsl for each noise function can be found near the cut-off, around ωm ≈ ϑ. The behaviour of the particle evidently changes over the range of available frequencies, reflecting the diminishing
37Because we have set the heating rates to be equal, the plot should be interpreted as optimistic. It more indicates the level at which collapse effects would be relevant than where they would be strictly discernable. 112 Chapter 6. A Quantum Spectrometer for Arbitrary Noise effect of the noise at higher frequencies.
10-9 10-10 3 10-11 ϑ=10
csl 4 λ 10-12 ϑ=10 5 10-13 ϑ=10 6 10-14 ϑ=10
101 102 103 104 105 106 ωm
Figure 6.6: The minimal values of λcsl which could be detected as a function of mechanical frequency, given a noise structure of the form (6.49) and differing cut-off frequencies ϑ. Here −12 −20 we have used the parameters P = 10 Pa, R = 100 nm, Q = 10 ev, gE = 1.15 × 10 and T = 4 K.
Figure 6.6 shows the probeable range for λcsl with an optimal environmental temperature of T = 4 K. This, of course, might not be straightforward to attain in the lab. In figure 6.7 we see what the corresponding range would be for a given cut-off of ϑ = 104 Hz, but with different environmental temperatures.
All of the above analysis has taken place with a fixed spatial correlation length for the collapsing
−7 noise field, taking rc = 10 m, as is conventional. However rc is, of course, a free parameter itself. Typically studies which compare the results of experiment to the predictions of collapse theories do so in a two-parameter space comprised of rc and λcsl, such as we presented in figure 5.7. Here, however, we have introduced a third parameter, and as such a proper analysis of the capabilities of the experiment will take place in a three dimensional parameter space. In anticipation of this, we first look at the values λcsl which could be probed for a given value of
ϑ, but with a free rc and over the available range of ωm 6.5. Using the Spectrometer to Test Non-White Models of Wavefunction Collapse 113
10-8
10-9
-10 10 T=4 K
csl -11 T=100 K λ 10 T=300 K 10-12 T=1000 K 10-13
101 102 103 104 105 106 ωm
Figure 6.7: The range of λcsl which could be probed as a function of frequency for differing environmental temperatures. The T χ dependance of the electric field noise makes this a non- trivial question, hence the inclusion of this plot. The parameters used here are the same as in figure 6.6, with the choice of ϑ = 104 Hz and differing environmental temperatures.
Figure 6.8: The limits of λcsl and rc which can be probed for a fixed cut-off frequency of ϑ = 104 Hz, with variable mechanical frequency.
Figure 6.8 tells the same story as 6.6 – that the optimal place to probe a model with a cut-off frequency ϑ is at ωm = ϑ. Taking this as a constraint, we are now able to free ϑ. Figure 6.9 shows the performance of the experiment in three parameter space – given a pair of values for rc and ϑ, what’s the lowest value of λcsl which would be detectable over the environmental noise 114 Chapter 6. A Quantum Spectrometer for Arbitrary Noise in the system? Here we have fixed the radius at R = 100 nm, and the other parameters are the same as those in figure 6.6.
Figure 6.9: Selecting the optimal mechanical frequency of ωm = ϑ, we see how far into λcsl the system could probe for a variable ϑ and rc
Looking at figure 6.9 we can again see something that we saw in chapter 5 – there is a resonance between the size of the sphere and rc. As such, our choice of R = 100 nm selects a minimum for that dimension. As we’ve seen throughout this section, higher cut-off frequencies allow for lower values of λcsl to be probed – essentially because they remain visible at higher frequencies, where the environmental noises are diminished due to the Ohmic scaling of the electric field noise.
6.6 Conclusions on the Spectrometer
We have demonstrated that information regarding the spectrum of bosonic baths or noises cou- pled to harmonic oscillators will be represented in the motion of those oscillators. Further; that given some fairly benign assumptions regarding the environment and the noise field under study, 6.6. Conclusions on the Spectrometer 115 such correlation functions may be reconstructed via repeated experiment on low-temperature oscillators, granted that the mechanical frequencies of these oscillators may be varied.
We have gone on to show how the system of a levitated nanosphere is particularly well suited to this task. Possessing a wide range of operational frequencies and an exceptionally low conventional noise floor, it makes the ideal candidate setting for this protocol. We have then demonstrated a specific application of the protocol, demonstrating how a highly structured electric field noise spectrum could be reconstructed to a fidelity proportional to the time allowed per measurement. Finally, we have shown that non-white models of wavefunction collapse may be probed with such a system, and that the spectrum of the acting noise field in such a scenario can become known.
The true strength of the protocol lies in its potential breadth of applicability. Whether for the noise fields invoked by collapse theories or for more generic non-Markovian quantum noises, this method may be used to detect and characterise the fields of interest. This may be of use in a range of technologies, including quantum sensing and improving quantum computing architectures. Further, such a detection and characterisation would shine a clear light for theorists trying to determine what the origins of such a field could be – since the specifics of the field will tell tales of its genesis. Chapter 7
Feedback Cooling
So far we have broadly followed a trajectory from the conceptual to the practical; from phi- losophy, to theory, to experimental proposals and calculations. In this chapter38 we will delve further into the specificities of levitated nanoparticles, and report on our adaptation of the two kinds of electrical cooling schemes from the world of levitated ions to that of levitated nano and micro spheres. We will use numerical simulations of the stochastic dynamics to demonstrate the efficacy of the procedures, and to show that cooling to the quantum regime is possible in principle.
The control of nano- and micro-scale objects under vacuum conditions is of great importance for a wide range of applications [138, 139], from the study of individual proteins [108], viruses [96] and bacteria, to the detection of tiny forces [140, 122]. Existing cooling techniques, as well as those which we present here, make it plausible to claim that the motion of levitated nano and micro particles will likely controllable at the quantum level [115, 141] in the near future, enabling studies of macroscopic quantum physics. This also paves the way for exploiting the especially high quality factors of levitated objects for other purposes in quantum technologies, such as using NV centres for coherent storage or as signal transduction devices.
In terms of our project here, all-electrical cooling schemes may prove vital to the experiments
38Much of what follows has been reported in [137]. Some of the figures shown here, and elements of the wording, will be in common between the two.
116 117 described in the previous two chapters. Although the optical cooling of levitated nanospheres is a well demonstrated fact [108], as a technique it faces limitations which are not yet entirely clear. Particles held in intense optical fields are wont to ‘disappear’ from time to time – possibly breaking down or melting as their internal temperature gets too high. The steady state of an actively cooled particle will involve three distinct temperatures [140] – that of the environment, the internal temperature of the material itself, and the effective temperature of the COM motion. The non-equilibrium steady states which are achieved through cooling schemes are characterised by this range of competing temperatures; with energy flowing from mode to mode, bath to bath, drawn along by entropy and resisted, redistributed by our interventions. It is a virtue of the all-electrical schemes that we can simplify to two temperatures – that of the environment, and that of the COM motion – since the internal temperature will match that of the environment, being absent any reason (such as the absorption of an intense laser field) to depart from it. The avoidance of the laser field is in fact doubly useful, since it not only escapes the internal heating which can potentially melt particles, but also allows for COM cooling un-plagued by the optical scattering which, if present, is the dominant form of heating [111]. The use of all-electrical schemas further opens up the range of materials which might be cooled. Optical methods require, of course, dielectric materials – and further, the relationship between the wavelength of the field and the diameter of the object may require that the entire set-up be exchanged if a new item is to be levitated and cooled; here, a simple change in the amplitude and driving frequency of the field will suffice. 118 Chapter 7. Feedback Cooling
7.1 Set up and Detection
VI
z Re
Figure 7.1: Circuit diagram for the electronic detection of charged particle motion. The trapped particle induces a current i in the endcap electrodes, generating a voltage V across the effective resistance R of the circuit. Resonant detection is possible by forming a parallel RLC circuit, where the capacitance C is due to the trap electrodes plus any parasitic capacitance in the circuit, and L tunes-out this capacitance.
We consider a scenario which closely echoes that of chapter 5. We have a particle of charge
Q levitated in a Paul trap by a voltage V (t) = VDC + V0 cos(ωdt), with VDC being the DC voltage (which we will keep at 0), V0 being the amplitude of the AC voltage, which oscillates √ at a frequency ωd. The particle is held at an average distance of r0/ 2 from either endcap electrode, and its motion induces a current
Q z˙ i = √κ (7.1) 2r0 in the circuit pictured in figure 7.1, wherez ˙ is the velocity of the particle relative to the trap along the axis aligned with the trap electrodes, and in which κ is a geometric factor equal to 0.8 in our setting.
The voltage given on a simpler circuit would of course just be V = iRcirc. We can increase this however by tuning the circuit to the motion of the particle, using a capacitor and an inductor 7.1. Set up and Detection 119
−2 to create an LRC circuit or resonant frequency ωres = (LC) , which can of course be tuned to the motion of the particle by selecting L and C such that ωm = ωres. This gives an effective resistance on resonance which is boosted to
2 Rres = Qf Rcirc (7.2)
where Qf = ωresL/Rcirc is the quality factor of the circuit. The signal we get from the motion of the particle will scale quadratically in Qf , meaning that the attained quality of the circuit will strongly determine the effectiveness of the protocols we propose. Factors of Qf = 25000
+ have been attained [142] and used to cool N2 ions, and higher values ought to be attainable [143].
Given a completely thermal state for the particle, it’s maximum velocity will be given by
r k T z˙ = B , max m yielding a maximum induced current of
r Qκ kBT imax = r0 2m
√ which, we note, has the scaling imax ∝ Q/ m. So the schemes which we are about to discuss, which of course revolve around this induced current, depends strongly upon the charge of the particle.
The charge of the Levitated Particle
The theoretical maximum charge which a sphere39 can hold is given by the Pauthenier equation [144, 145]
2 Qmax = 4π0R pE (7.3) 120 Chapter 7. Feedback Cooling
in which p = 3 for a conductor, or p = 3r/(r + 2) for a dialectric; and E is the magnitude of the electric field. How high can this go, and how is the high E to be achieved?
The charging of micro and nano scale particles is a field of study in its own right. The maximum attainable charge will depend upon the material of the particle, its surface area, any anisotropies or defects in its shape, and of course upon the method used to affix the charge to it. Methods in use today include electron bombardment [146, 147], corona discharge [145] and adhesion of charged droplets.
The surface potential for a sphere is given by
Q φ = (7.4) 4π0R which for dialetric materials can exceed 5 × 108 V/m. As this potential gets high, it is possible for a sphere to lose charge through a spontaneous emission. As it stands, we lack any proper, more detailed model of what charges can be attained on what materials and at what sizes, and have instead to rely somewhat on what has been attained in the laboratory [148, 149]. For our discussions, we will keep Q to be within bounds which have already been attained, noting that for a R = 1 µm sphere, charges of over 106eV have been attained [150].
7.2 Simulating the dynamics
The equation of motion for the levitated sphere in dimension xi is given by
mω2 mx¨ − d (a − 2q cos(ω t))x = 0 (7.5) i 4 x x d i 39Both of the electrical cooling techniques which we describe here would apply, of course, to non-spherical trapped objects too. Nonetheless, for simplicity as well as relevance to the previous chapters, we will restrict our discussions here to spheres. 7.2. Simulating the dynamics 121 where
4VDCκQ ax = 2 2 (7.6) mωdV0 −2V0κQ qx = 2 2 (7.7) mωdV0
where for us az = 0 since we have VDC = 0. When we incorporate the effects of noises on our system we need to account both for their damping and heating effects, represented by γ and F (t) respectively, and related as per the fluctuation dissipation theorem by
hF (t)F (τ)i = 2mγkBT δ(t − τ). (7.8)
We incorporate these effects into the overly idealised equation (7.5) to get
mω2 mx¨ + mγx˙ + d cos(ω t) = F (t). (7.9) 2 d
Now, in our case, we will want to model both the effects of resistive and feedback cooling. Resistive cooling is an entirely passive process, and the aforementioned model is adequate to properly describe it. Feedback cooling, on the other hand, is somewhat more complicated. We want to vary the voltage being applied to the trap in response to the current being induced by the motion of the particle, so as to slow it as much as possible. In section 7.4 we will introduce a simplified model, but for now we will treat the feedback in full.
The voltage on the electrodes is modified such that
V = V0 cos(ωdt) + Vfb (7.10)
where Vfb = Vind(1−G), in which G is the gain applied to the feedback signal, and Vind = iindRres is the voltage induced across the circuit by the motion of the particle. Taking into account that this voltage is a function of the velocity of the particle, to modify equation (7.5) would clearly take us to a highly non-linear situation. Further, if we take into account that not only will F (t) be composed of the familiar noise sources (gas collisions, Johnson noise on the electronics, and 122 Chapter 7. Feedback Cooling technical noise); but will, in fact, include noise on the feedback itself, then we are left with a system with no clear analytic solution.
The bulk of the results in this chapter come from numerical simulations which we built up to model the non-linear dynamics of the particle. Taking all three dimensions into account, we have a system of six coupled, stochastic, delayed differential equations (SDDE’s). For such systems there is no guarantee of a stable solution, let alone a general method for finding one [151, 152].
Our solution was to build a Matlab code based on the Euler Maruyama numerical integration technique. This consists of vectorising each element of integration into N discrete elements, such that t → {t1, t2, . . . , tN },z ˙(t) → {z˙1, z˙2,..., z˙N } and so forth in which ti+1 − ti = δt is the timestep for the simulation. The system is then modeled through iterative solutions to the differential equations, where
dXt = a(t, Xt)dt + b(t, Xt)dW t (7.11)
is the basic Ito equation for the system, in which dWt is the increment of a Weiner process (in our case the noises affecting the particle), b(t, Xt) the coupling thereto, and a(t, Xt) describes the deterministic components of the equation. The equations are then iteratively solved by
Xi+1 = Xi + a(ti,Xi)δt + b(ti,Xi)(Wi+1 − Wi). (7.12)
This method is not guaranteed to converge upon correct solutions. However, in our case it will, since it meets the conditions that [153]
• a and b are differentiable up to fourth order, and their first derivatives are bounded (in our case by the stability criteria of the trap).
• They have low-order dependance (the highest being quadratic) upon their determining parameters.
The code by which we enacted this method is, of course, too lengthy to reproduce here. We turn instead to the results of our simulations, and the light which they shed upon the efficacy 7.3. Resistive Cooling 123 of the cooling schemes which we set out to test.
7.3 Resistive Cooling
As the particle moves and induces current in the circuit, the current will dissipate as heat
2 through the resistance of the circuit at a rate i Rres, which gives a momentum damping rate of
2 Qκ Rres γres = . (7.13) r0 2m
This is easily incorporated into equation (7.9) by subbing γ → γres. The associated noise, characterised by
hFres(t)Fres(τ)i = 2γresTresmkBδ(t − τ), (7.14) effectively represents the Johnson noise which the circuit imparts, modelled here as white noise (as is standard). Tres is here the physical temperature of the circuit. Lacking any other sources, the interplay between this heating and the cooling effect of the resistive dissipation will take the particle to an equilibrium temperature matching Tres at a rate determined by
Rres. Clearly, based on (7.13), this depends linearly upon the effective resistance of the circuit and quadratically upon the charge of the particle. We can see this process of equilibration in figure 7.2, in which the particle thermalises with the circuit at a rate dependant upon the circuit resistance (in (a)), and the charge (in (b)). We further see from (d) that the stochastic simulations ratify γres as a damping rate, with a very close agreement between theory and simulation. In all of these, we have calculated the temperature of the particle as T =
2 2 hz imωz /kB as per the equipartition theorem. These results show that resistive cooling is useful, but only insofar as the physical temperature of the circuit can be lowered. This, however, need not be such a prohibitive limit. Using a dilution refrigerator the physical temperature of the circuit could be brought as low as 5mK. Combining this with a microparticle of R = 1µm with
6 a charge of Q = 10 eV at a mechanical frequency of ωm = 2π × 1 MHz, we would yield an equilibrium occupancy of k t hni = B < 100 ~ωm 124 Chapter 7. Feedback Cooling
a) b) 4 1 M 200 M Q = 1 10 e
5 (nm) Position 1000 Q = 4 10 e 50 50 0 0 0
-50 -50 Position (nm) Position
Position (nm) Position -1000 0 20 40 60 0 50 100 c) Time (ms) d) Time (ms) T = 5 mK 10 3 circ 100 2 0 10
-100 10 1 (Hz) T = 300 K 0 100 circ res 10 -1
Position (nm) Position 0 10 -100 -2 10 0 50 100 150 200 10 2 10 3 10 4 10 4 10 5 10 6 Time (ms) R ( ) Q (e) circ Figure 7.2: (Figure taken from [137].) Motional damping simulated for two different values of a) circuit resistance Rcirc and b) particle charge Q. c) Simulated dynamics for two different circuit temperatures Tcirc, illustrating that the particle thermalises with the circuitry. d) Varia- tion in damping rate, extracted by fitting the simulated dynamics, with Rcirc (left hand panel) and Q (open circles: simulation, solid line: theory). Simulations produced with initial particle temperature Tin = 1000 K, rS = 1 µm, Qf = 100, r0 = 0.5 mm. 7.4. Feedback Cooling 125
placing us well within reach of the necessary n0 for the protocols of chapters 5 and 6.
7.4 Feedback Cooling
V I G
Re z
GVI
Figure 7.3: Circuit diagram for the feedback cooling protocol. A voltage v is induced across the effective resistance of the circuit R, which is then amplified by a gain G and fed back to one of the electrodes.
Using the active method of feedback cooling we ought to be able to reach temperatures lower than that of the circuitry.
As we have stated, a full model of the effects of feedback requires the solving of a system of coupled SDDE’s. A simpler model is, however, easily attained[138]. From the perspective of the trapped particle, the circuit in figure 7.3 is equivalent to simply seeing an enhanced resistance, as seen in figure 7.4. In which Re = (1 − G)Rres, which gives the damping rate γ = (1 − G)γres.
Accordingly, the temperature to which the particle will thermalise goes as T = (t − G)Tcirc. So, in the limit of limG→1, the particle will cool to absolute zero, but over an infinite time.
A more realistic model would include the noise present in the amplifier[138]. The signal which is fed to the amplifier will, like all signals, carry its own noise, and this noise will set a limiting temperature on the steady state of the particle which is above absolute 0. The amount of noise which the circuit which the amplifier picks up, however, will depend upon the bandwidth of
2 the circuit as hvn,fbi = 4kBTRampζ, in which ζ is the bandwidth and Ramp is the resistance of 126 Chapter 7. Feedback Cooling
Re z
VI
Figure 7.4: On a minimal model, the particle simply sees an increased resistance Re from the feedback. the feedback amplfier. This will add to the noise from the amplifier itself as
q 2 2 2 2 vn,total = (1 − G) vn + G vn,fb (7.15) which gives a final temperature for the particle of
G2 T = (1 − G) + T . (7.16) 1 − G n,fb
This is minimised when
rT G = 1 − n,fb (7.17) Tcirc giving a minimum temperature of p Tmin = 2 Tn,fbTcirc. (7.18)
The simulated dynamics of the particle can be seen in figure 7.5, in which the numerical simulations broadly ratify the simplified models we have laid out here. Using cyrogenic SQUID amplifiers which can have noise temperatures as low as 20µ K, it would be possible to reduce the temperature of the particle to the microKelvin level, even if the physical temperature of the circuitry is at 5 mK. For this temperature to acheive the quantum ground state of hni ≤ 1, we would require that ωm > 130 kHz, which is a readily achievable frequency. 7.4. Feedback Cooling 127
a) b) c) 150 400 100 G = 0 100
0 200 T (K) T T (K) T 50 -100 0 0 100 G = 0.7 0 2 0 0.5 1 Phase Gain 0 d) e)
-100 4 Position (nm) Position v = 10 nV T = 300K 0.3 n,fb 10 circ 100 G = 0.99 m) 0 T = 4K 2 circ -0.3 10 0 v = 10 V 0 2 n,fb 10 -100 0 -2 Position ( Position 10 -2 (K) Temperature 0 500 0 500 10 -12 10 -8 10 -4 Time (ms) Time (ms) v (V) n,fb Figure 7.5: The results of simulations of the feedback cooling scheme. In a) we see the fluctations in position over time being tamed as the gain of the feedback is increased. In b) we see that the feedback is most effective when the phase is set to zero (or almost the same at any multible of 2π. c) shows the equilibrium temperatures attained for differing values of G, in which the red line describes equation (7.16) and the circles report simulation results. d) shows the impact of of amplifier noise voltage upon the dynamics (with G=.95). e) quantifies this effect, showing the equilibrium temperature attained by the particle given over a range of vn,fb, with G = .95 and two circuit temperatures. 128 Chapter 7. Feedback Cooling
7.5 Conclusions
We have demonstrated that levitated charged particles can be efficiently cooled using both resistive and feedback based techniques. In the resistive case, to the temperature of the elec- tronics; in the feedback case, we predict ground-state cooling to be possible using cryogenic circuitry. These techniques can be used in concert with optical ones, as in the hybrid type trap described in chapter 5, or else could be used to do away with optics altogether. Chapter 8
Conclusions
The development of a theory which can describe the world in which we seem to live, whilst at the same time maintaining the capability to produce predictions which match the results of quantum physics, is no small task. Considering how wildly different the quantum world seems to be from the classical one, it is perhaps unsurprising that the work of interpreting quantum mechanics is still ongoing, or that it has yielded so many different theories. Somewhere along the way such a theory must reconcile the local, physical world with the fact of non-local correlations; it must allow for superposition states and interference whilst also giving something approaching a definitive state of things; it must make sense of a three dimensional physical world, whilst also giving a meaning to the space in which the fundamental elements of reality live – a space which may not be the same as that physical space. Ultimately it must unite the quantum and the classical, and explain how the everyday objects with which we are familiar can be decomposed into constituent parts which behave in a way that is so completely and profoundly alien to them that such a unity would seem, on the face of it, to be impossible. And yet it must be possible, since the world is here.
Collapse theories, as we have shown, offer a strong possibility. They allow for the objectivity of the classical world without destroying the relationality of the quantum one. They give a scaling to fundamental dynamics which could almost be called natural, one which delivers our well loved theories of quantum and classical mechanics as simple limits of a single, deeper
129 130 Chapter 8. Conclusions theory. They hold that, at root, the world might not be as clean as we had hitherto supposed, that it might be ultimately governed by an irreducible, ontological randomness. And if we take up that randomness we can see that other, age-old questions about the way of the world – in particular, the nature of the time’s directedness, and the meaning of causation – might simply dissolve.
There is a price to pay, of course. The introduction of a universal noise field is a convenient piece of mathematics, but if it is to solve problems of metaphysics then it must come with an ontology of its own. And yet in ascribing such an ontology we are faced, from the outset, with serious problems. In particular, the admission that by its very nature, such a noise must be outside the remit of a fully quantum description; the recognition that the requisite approach to the relationship between the wavefunction and the quantum state entails that we take a space of incredibly high dimensionality to be our fundamental, ontic space; and the necessary surrendering of any ontology for objects in which their properties are truly, finally, and absolutely definite. Even so, the situation is not hopeless. By relinquishing some of our classical intuitions, we can recover an ontology which relates wavefunctions to configurations of classical objects with more than enough specificity to satisfy our classical intuitions. The other two problems – the high dimensionality of the ontic space, and the ontology of the noise field – may find a kind of harmony with one another. Speculatively, we might be able to deliver an emergent world of 3+1 dimensions by the actions of the correct operator(s) on a larger space, recovering spacetime from collapse. Else, we might be able to source the noise field itself from fluctuations in the metric, or at least the bearing the metric has upon the quantum state. These directions of study are bold. But if they bear fruit they may be able to settle questions of physics and metaphysics all at once, delivering an ontology for the quantum state and the space in which it lives, a unified dynamics for matter at all scales, and an understanding of the relationship between quantum mechanics and the fundamental nature of spacetime.
Such promises have been sounded before, of course. What truly sets collapse theories apart is their capacity for falsification. We have given a thorough treatment of a particular experimental scenario which, owing to a resonant effect between the size of the oscillator and the length parameter of CSL, would be particularly well suited to testing collapse models. More, we have 131 shown that the same scenario makes for an ideal implementation of a generalised quantum spectrometer, one which would not only be capable of detecting collapsing noise fields with non-white spectra, but of characterising those spectra too. In this way, the line joining theory to experiment might become a loop, with lab results constraining and enhancing the development of increasingly physical collapse models. Appendix A
Objections to the Everettian School of Thought
A thorough examination of the Everettian approaches to interpreting quantum mechanics would be content enough for a thesis in its own right. Here we will do little more than name the main problems. In terms of reading, a good overview of the thinking and history behind Everettian interpretations can be found in [154]. For a thorough discussion of the issues at stake we recommend the essays collected in [155]. For further objections, see [156].
Interpreting the Probabilities
The principle problem when trying to make sense of the world on an Everettian interpretation is that of assigning meaning to the wavefunction amplitudes. One can simply state that they signify the same thing as they do in orthodox quantum mechanics, and that by treating them in the usual way they lead us to probability distributions over states of the world as per the Born rule. This, however, would seem to be at odds with the basic ontology of the interpretation.
Consider a superposition state which is initially seperable from the environment
! r1 r1 |Ψ, Φi = |ψ i + |ψ i ⊗ |φ i, (A.1) 2 1 2 2 r
132 133
in which |φri indicates some measurement system (here including an observer) in a ready state. Now, by the same sort of interaction which we introduced in chapter 2, the two systems will become entangled, producing
r1 r1 |Ψ, Φi = |ψ , φ i + |ψ , φ i. (A.2) 2 1 1 2 2 2
We could of course include a larger environment, propagate the superposition into it, and then trace it out to yield the decohered and diagonalised reduced density matrix of this system. The strict unitarity of the dynamics here tells us that the superposition never breaks down, it simply grows and grows. Very well – here, there is not one final observer-environment |φii measuring a single outcome; there are two, yet they do not interfere with one another on account of displacing their coherence to a still larger environment. Both worlds exist simultaneously.
The problem here is that this only really makes sense when we give an equal weighting to the branch amplitudes. If we revisit the same scenario, but with different coefficients, for example √ √ .9 and .1, then we would end up with
√ √ |Ψ, Φi = .9|ψ1, φ1i + .1|ψ2, φ2i. (A.3)
Now, what exactly do the amplitudes mean here? As we have stated, on this interpretation of quantum mechanics both branches definitely exist. Both occur, with probability 1. And yet, somehow we need to assign each branch a different probability; .9 and .1 respectively. This would seem like a contradiction.
There have been some creative solutions put forward to address this problem. For example, in [157], Carroll posits that the above scenario would in fact lead to ten branches of the world
– nine of which would be in state |ψ1, φ1i, and one of which would be in state |ψ2, φ2i. It would seem that one must go to considerable lengths in order to rescue the probabilities on this account. 134 Appendix A. Objections to the Everettian School of Thought
The Branching and Basis Problems
In the above scenario we considered a superposition state. Implicit in writing down this state was the assumption that we were working in a Hilbert space whose basis was chosen such that the superposed states were eigenstates. This choice of basis however, would require some justification. This choice echoes that of the choice for the collapsing operators which we had to consider in chapter 3. Analagously, position is usually settled upon as a choice of preferred basis for the Everett interpretation, and by similar reasoning.
Yet the selection of the position basis here is not without its own attendant problems. Position is, of course, a continuous variable. In the above example we considered discrete and identifiable eigenstates of some operator. If, instead, we considered a wavefunction which was not in a superposition per se, but instead had a Gaussian form in the position basis – how many branches would the world take when this wavefunction became entangled with its environment? One? Infinite? Some finite number N, determined by a choice of coarse graining?
In a way, this problem echoes one which we discussed in chapter 3 – the problem of grounding the world in the wavefunction. The task here for Everettians is to produce an interpretation which can marry an ontology of distinct branches to the mathematics of the continuous variables upon which those branches are based. Appendix B
Heisenberg Picture Ontology
There is a loose interpretation of quantum mechanics which ascribes ontological status to the observables of quantum mechanics, but not to the wavefunction. The initial argument for this makes a very clear kind of sense: the Heisenberg formalism works just as well as the Schr¨odinger,and can can be used with comparable ease – in many situations it is a much more concise and economic presentation. Since in this picture it is the observables which appear as the protagonists of the theory, they are sometimes accorded ontic status over and above the quantum state. The reasoning in support of this move is that the observables are observable, and as such there is an empirical basis for according them ontological status. There are some immediate problems with this approach.
The Heisenberg picture of course actually deals with the operators which represent observables - which will take on values and qualify as properties under one’s chosen criteria for measurement. Since the mathematical class of operators which fit the criteria of the picture (i.e. self-adjoint and Hermitian) is much larger than the set of operators which we might in practice consider to actually be observable (such as spin, position, momentum etc), we are led directly to a choice: either we accord an ontic status to every operator in the class, or else we restrict the ontology to those which we know can be enacted in the world.
On the first, we clearly land once again with an ontology which contains elements which are not empirically verifiable. We have given an ontic status to things represented by operators, where
135 136 Appendix B. Heisenberg Picture Ontology we have no expectation of actually seeing these things in the world – this of course undermines the empiricist basis by which the Heisenberg picture was favored in the first place.
On the second, we find ourselves in the rather strange position of having a set of ontic ob- servables represented by a class of operators where the former might be expanded by new experimental techniques, as ever more complicated measurement procedures make new opera- tors enact-able in the lab. On this picture we might be compelled to update our fundamental view of reality every time the some new lab equipment is deisgned; not because we have found out some new aspect of nature, but simply because we have expanded our capabilities and so grown the set of practically enactable observable operators, which would be somewhat absurd.
It is also worth noting that |ψihψ| is amongst the class of observable operators, so in some sense the wavefunction is included in this ontology anyway. Appendix C
The Cosmic Temperature of CSLD
In [119], Lochan et. al investigate the implications of the CSL model upon the CMB. The presence of a universal heating effect ought to raise the temperature of the matter in the early universe, and distort the CMB. Here we perform similar calculations to those in [119], and examine what would happen if CSLD were true – how that would affect the measured CMB, and how this might be used to constrain the model.
We begin with the the relaxation rate from equation (5.25). We want to add this into eq(26) from [119]. Doing this gives us
2 2 dρM (z) X 3λ α 4kλnM (z) = −3H(z)ρ (z) + ~ n (z) − , (C.1) dt M 4m M m2(1 + k)5 s=e,p s s
in which H(z) is the Hubble parameter as a function of the redshift z and nM is the number density for protons and electrons, presumed to be the same – and which gives the mass density
2 ρM = nM (me + mp)c .
Following equivalent steps to the original derivation, we arrive at
dρ 3 A ρ B k n ρ = ρ − 3 + 3 5 , (C.2) dt 1 + z mcR(1 + z) R(1 + z) (1 + k) | {z } =Θ
√ 2 2 2 P 3λ~ α P 4λ where H = R(1 + z) ,R = H0 Ω0, mc = (me + mp)c ,A = ,B = 2 , and in s 4ms s msmc
137 138 Appendix C. The Cosmic Temperature of CSLD
k which we’ve expanded to first order in k. We then take the first order expansion (1+k)5 → k, and sub this into the third term from the above equation to give
B n ρ k Θ = R(1 + z)3
We then put in k = r where r = ~ , and n = ρ , which gives us n 2vrc mc
B ρ r Θ = . R(1 + z)3 so that we get dρ ρ A f ρ B r ρ = 3 − 3 + 3 (C.3) dt 1 + z mcR(1 + z) R (1 + z) where we have introduced the ratio f, as in the original argument. This solves for
3 A f − B mc r 1 1 (1 + zf ) ρf = exp 2 − 2 3 ρi. (C.4) 2 mc R (1 + zf ) (1 + zi) (1 + zi)
and we recover the standard equation if λ = 0, and eq (35) from [119] if vd = ∞.
Now we can bound v. Since the argument of the exponent is no longer guaranteed positive, we
0 2 B 1 1 1 0 ~Bmc go with the condition − Af)C 1 where C = 2 − 2 and B = . vd 2mcR (1+zf ) 1+zi) 2rc This gives us B0C B0C v ≈ (C.5) d AfC ± 1 ±1
Taking the positive root, this gives us
v B0C (C.6) λ λ 3.6 × 1055. (C.7)
Via eq (8) in [117] and taking λ = 10−10,this gives us a lower bound for the temperature of the dissipative field of 1040 K.
v 50 36 Similarly, for the matter dominated era, we get λ 10 and a temperature of T = 10 K. 139
This seems very very high. It would seem though CSLD is more or less disallowed by the CMB. This could be remedied either by accepting an extremely high temperature for this field, or by positing that the value of vd could change over time – that it could have been very high in the early universe and then cooled down with everything else. Appendix D
Alternative Formalism for Spectrometer
Here40 we re-derive the central result of chapter 6: the equation of motion for the number operator of the spectrometer. However, here we make a more thorough model by dropping the rotating wave approximation. This gives us a position-position operator coupling between the oscillator and the bath modes, rather than the exchange coupling which we took in the main text.
We begin with the same overall structure for Hamiltonian which describes an harmonic oscillator S coupled to a bosonic bath B;
H = HS + HB + HI (D.1) in which the three terms above again represent the system, bath and interaction Hamiltonians respectively. Setting ~ = 1, they are given by 1 H = ω (a†aˆ + ) (D.2) S m 2 X 1 H = ω (b† b + ) (D.3) B α α α 2 α X HI = −x gαxα (D.4) α
40This appendix closely echoes the structure of the ‘formalism’ section from out upcoming paper [3].
140 141 where a†, a, x give its creation, annihilation and position operators respectively for the oscilla- tor. Similarly, we again decompose B into α modes, whose creation, annihilation and position
† operators are bα, bα and xα respectively. The interaction between the system and the α mode of the bath is now described through a position-position coupling whose strength is given by
2 gα = mαωα whose value can be settled by an appropriate choice of the bath oscillator masses.
The first description of the dynamics of this type of system was given by Caldeira and Leggett in their seminal paper [158], in which they derived an appropriate master equation using the Born-Markov approximation and in the limit of high temperatures of the bath. This result was improved by Hu, Paz and Zhang [159], who derived a master equation which is exact, and valid for any temperature. The subject has more been recently studied in [160], where the Hu, Paz and Zhang master equation was derived in the form41
d ρ = − i[H − Ξ(t)q2, ρ ] + Γ(t)[q, [q, ρ ]] dt t S t t
+ Θ(t)[q, [p, ρt]] + iΥ(t)[q, {p, ρt}], (D.5) and it is this form which we shall use as a jumping off point for developing this formalism. It is completely general – for example, the bath need not be thermal for the equation to be valid.
The exact definitions of the time-dependent coefficients Ξ(t), Γ(t), Θ(t) and Υ(t) are given through recursive series expansions and can be found in [160]. However, in the limit of a weak coupling between the system and bath, one can safely make a first order approximation which greatly simplifies the expressions for these coefficients. Further to this, we will consider only the regime where the damping effects of the bath upon the system will be negligible compared to its heating effects, just like in the main text. Here, the assumption is mathematically equivalent to assuming that the bath correlation function is real [160]. Taking these simplifications, one gets Ξ(t) = Υ(t) = 0 and
Z t Γ(t) = − ds C(t, s) cos[ωm(t − s)] (D.6) 0
41 2 Note that in [160], gα = mαωα. 142 Appendix D. Alternative Formalism for Spectrometer
Z t sin[ω (t − s)] Θ(t) = ds C(t, s) m (D.7) 0 mωm
in which C(t, s) = Tr[B(t)B(s)ρB] is the two-time correlation function for the bath operator P B = gαxα. The Fourier transform of this function – the spectrum of the noise function in α frequency space – is the object which our spectrometer will ultimately uncover through a study of its impact upon the system. Using these simplifications, eq. (D.5) becomes
d ρ = −i[H , ρ ] + Γ(t)[q, [q, ρ ]] + Θ(t)[q, [p, ρ ]]. (D.8) dt t S t t t once again, we select n = a†a as our observable to monitor, and use
d hni = Tr[nLρ ] (D.9) dt t t in which Lρt is a super-operator on ρt which summarizes the right-hand side of Eq. (D.8). Using this, the cyclicity of the trace, and a little algebra we find that
d Γ(t) 1 Z t hnit = − = C(t, s) cos[ωm(t − s)]. (D.10) dt 2mωm 2mωm 0
Now, if we again assume that the correlation function is invariant with respect to both time reversal and time translation, i.e. that C(t, s) = C(|t − s|), we can rewrite the right hand side of equation (D.10) in a more convenient form by using the relation
Z t Z t 1 iωmy ds C(|s − t|) cos[ωm(s − t)] = dy C(y)e . (D.11) 0 2 −t
From this point, we can simply use the same techniques as in the main text. Fourier trans- forming the correlation function and doing some algebra, we end up once again with
Z ∞ 2 1 ˜ sin [(ωm − ν)t/2] hnit = hni0 + dν C(ν) 2 . (D.12) 2πmωm −∞ (ωm − ν) Appendix E
Rotational Dynamics
In the case of a trapped nanosphere the charge can be distributed anisotropically over the surface of the sphere, unlike a single ion. An anisotropic charge distribution can lead to a torque on the particle as it passes through the rf field gradient, which can induce rotation. The energy from this mode can then couple into the mechanical frequency, causing heating.
The anisotropy of the charge distribution on the sphere will cause a dipole moment, and this dipole moment will experience a torque due to the electric fields, and further it will affect the centre of mass motion. The equations of motion for the centre of mass are given by
m¨r − q E(r, t) − ∇[µ · E(r, t)] = 0 (E.1) where r = (x, y, z) is the position of the nanosphere in three dimensions, µ is its dipole moment, and E(r, t) is the electric field at point r at time t. Clearly we can see a coupling between the orientation of the dipole moment and the centre of mass motion. In order to understand whether this coupling leads to heating of the centre of mass, and what this possible heating rate might amount to, we need to model the rotational dynamics of the dipole together with the translational motion of the sphere and see what the effect of the dipole moment is upon the centre of mass motion.
This situation is very similar to that of diatomic molecules held in equivalent traps, and as such
143 144 Appendix E. Rotational Dynamics we can draw upon the relevant literature. In modelling the rotational dynamics for systems of this type, it is a well known problem that the equations of motion cannot be solved using spherical polar coordinates [161]. Working with the quaternion formalism [162], we built a numerical simulation in XMDS [163] which modelled the coupled dynamics of the rotary and centre of mass motion, and experimented with the effects of different dipole moment magni- tudes. We expect that packing the maximum possible charge onto the surface of the sphere would naturally lead to the most isotropic distribution. We find that the effect of the rotary motion does cause an appreciable heating effect for a large enough dipole moment, which oc- curs when µ0 = R × 5 eV. This limit on the dipole magnitude µ0 is perhaps more clearly expressed as a ratio qedge/qcore ≤ 1/300, where we have simplified the charge distribution into q = qcore + qedge, qcore being the charge which is completely isotropic and effectively bunched at the centre of mass, and qedge being the charge which is bunched at one position of the surface, and hence causes the dipole moment.
Spectra of different dipole moments 1.× 10 -7
8.× 10 -8
Dipole Moment and plot area 6.× 10 -8 q=0, Area=7.1⨯10-12 q=1b, Area=4.1⨯10-12 q=10b, Area=7.5⨯10-11 -8 4.× 10 q=100b, Area=1.9⨯10-9
2.× 10 -8
0 0 1000 2000 3000 4000 5000 Figure E.1: Sample plot exhibiting the results of some of the numerical simulations which we ran. In the legend, q refers to the value of qedge, where b = 1e + V , serving as a measure of the dipole moment. The value of qcore for each plotted spectrum is 100. As can clearly be seen, the introduction of a dipole moment leads to a noisier spectrum. We can also see the area enclosed under each plot, giving us an indication of the total energy of the oscillator.
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