Probing Foundations of Quantum Mechanics: a Study Into Nonlocality and Quantum Gravity

Probing Foundations of Quantum Mechanics: A Study into Nonlocality and Quantum Gravity

Parth Girdhar

A thesis submitted in fulﬁlment of the requirements for the degree of Doctor of Philosophy

School of Physics University of Sydney

2020 2

Abstract

This thesis is about probing aspects of the foundations of quantum mechanics. Firstly, two notions of quantum nonlocality are explored: EPR-steering, the ability to control a remote quantum state, and Bell nonlocality, the inconsistency of a theory with local causality. A necessary and sufficient witness of Einstein-Podolsky- Rosen (EPR) steering is derived for a two qubit system employing only correlations between two arbitrary dichotomic measurements on each party. It is demonstrated that all states that are EPR-steerable with such correlations are also Bell nonlocal, a surprising equivalence between these two fundamental concepts of quantum mechanics. Next, testing modifications of the quantum mechanical canonical commutation relations is addressed. These are properties of some quantum gravity theories that involve an effective minimal length. It is shown that optomechanical probes of position noise spectrum of macroscopic oscillators can produce constraints on these theories. A comparison with current and future realistic experiments reveals the potential to beat constraints from direct experiments on elementary particles. Finally, it is studied how such modifications of quantum mechanics manifest in the theory of general continuous quantum position measurements. Several behaviours are found that deviate strongly from that of standard commutation relations. Contents

Abstract ...... 2 Statement of Originality ...... 5 Statement of Contribution and Attribution ...... 5 Acknowledgements ...... 6

1 Introduction 9 1.1 The Quantum Formalism ...... 10 1.2 Bell’s Theorem ...... 13 1.3 Einstein-Podolsky-Rosen Steering ...... 16 1.4 Generalisations ...... 18 1.4.1 Complexity of Determining Bell Nonlocality ...... 21 1.5 Modiﬁed Canonical Commutation Relations ...... 24

2 EPR-Steering and Bell Nonlocality in the CHSH Scenario 29 2.1 Necessary and Suﬃcient Steering Inequality ...... 31 2.2 Equivalence Classes of Measurements ...... 34 2.3 States Steerable via CHSH-type measurements are Non-local . . . . . 35 2.4 Discussion ...... 36 2.5 Appendix A ...... 38 2.6 Appendix B ...... 40

3 Probing Modified Commutators with Quantum Noise 47 3.1 Introduction ...... 47 3.2 Background ...... 48 3.3 Modified Noise Spectrum ...... 52 3.3.1 Setup ...... 52 3.3.2 Noise spectrum ...... 54 3.4 Constraints on the Modified Commutator ...... 57 3.4.1 Estimating current constraints ...... 59 3.4.2 Future constraints ...... 62 3.4.3 General driven oscillator ...... 68 3.5 Discussion ...... 69 3.6 Appendix ...... 72 3.6.1 First term of (3.13) ...... 73 3.6.2 Second term of (3.13) ...... 76

3 4 CONTENTS

3.6.3 Perturbed spectrum ...... 79 3.6.4 aLIGO modelling ...... 81 3.6.5 Steady-state expectation values without the adiabatic approximation ...... 83 3.6.6 Standard spectrum ...... 84

4 Modified Commutators and Continuous Position Measurements 87 4.1 Quantum Trajectory Equation ...... 87 4.2 Differentials of general mean & covariance matrix ...... 92 4.3 Modified canonical commutation relations ...... 93 4.4 Discussion ...... 99

5 Conclusion 103 CONTENTS 5

Statement of Originality

I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. This thesis has not been submitted for any degree or other purposes.

Parth Girdhar

Statement of Contribution and Attribution

There are three main chapters in this thesis. Appendices are presented immediately after the main contents of each chapter.

• Chapter 2 of this thesis is published as: Girdhar, P. and Cavalcanti, E.G., 2016. All two-qubit states that are steerable via Clauser-Horne-Shimony-Holt-type correlations are Bell nonlocal. Physical Review A, 94(3), p.032317. In the journal the publication is in two-column format but it has been adjusted to single column for this thesis. I am the lead author of the publication and derived all the results, advised by Eric Cavalcanti on the research questions. The question connecting Bell nonlocality and EPR-steering was prompted by Curtis Broadbent, I independently discovered the proof of the main theorem and its implications. I wrote the main drafts and minor edits were provided by Eric Cavalcanti.

• Chapters 3 with minor revision is published as: Girdhar, P. and Doherty, A.C., 2020. Testing generalised uncertainty principles through quantum noise. New Journal of Physics, 22(9), p.093073. Chapter 4 is currently unpublished. I derived all the results, advised by Andrew Doherty who suggested research questions and some of the techniques. I wrote both of these chapters and edited them in response to comments by Andrew Doherty.

Parth Girdhar 6 CONTENTS

Acknowledgements

Firstly I wish to thank my principal advisor Andrew C. Doherty and auxiliary advisor Eric G. Cavalcanti. I am grateful to have had the opportunity to discuss and learn from you about many strands of quantum mechanics (and what might be lurking behind it) and receive feedback about my work. I am also thankful to Curtis Broadbent, Cyril Branciard, Michael Hall and Howard Wiseman for discussions and feedback. I appreciate the support from Nicolas Brunner, Peter Graham, Achim Kempf, Antony Milne, Terry Rudolph and for being available to host my research visits, talks and the subsequent exchange of ideas. In particular I acknowledge Tim Ralph for hosting the presentation of my research at RQI-10. To my friends in the USyd quantum information theory group I oﬀer my thanks for the conversations on walks through physics and life, the group meetings (“best time of the week”) and seminars. Thank you to Stephen Bartlett, Andrew Doherty, Steven Flammia, Arne Grimsmo and Isaac Kim for leading the activities of the group that have broadened my vision of quantum information theory. I also appreciate the lessons on practical innovation from colleagues downstairs working on quantum experiments. I acknowledge administrative support from Geraldine Arriesgado, Alexis George, Lorraine Di Masi, Anthony Monger, Jeremy Platt, Leanne Price, Satpal Sahota, Fran Vega, Wicky West, Nicole Yang and Amy Zhu. I cherish the companionship of my friends in collectives outside of physics, especially all the volunteers of the USyd Vegesoc. Theodore, a constant source of positivity on my days on campus, did not survive to see the completion of this thesis: I will always value our friendship and your dedication to the great causes of the world. This work could not have happened without the support of my immediate family. I am especially grateful to my mother whose help is immeasurable. Finally, my ﬁrst gurus of physics Isaac Asimov, Albert Einstein and Stephen Hawking have guided me all this way, their writings shall inspire students eternally. CONTENTS 7

Mr W: Quantum mechanics is a real Bohr. Mr X: How dare you say that! It just shows you are as thick as a Planck! Mr Y: Oh Shut Up! Stop Bragg-ing about your knowledge Mr X! Mr X: Hey, you step out of this. You’re just jealous of the abilities I was Born with. Mr Y: Why you! (All three fight) Mr Z: Guys, cut it out! I’ll call the Pauli-ce if you don’t stop fighting! Mr A: Just being Curie-ous, aren’t you a Pauli-ce officer? Mr Z: No no. It seems my uniform has made me a Feyn-man (Girl comes by, Mr A offers flower to girl) Girl: Fer-mi? Mr A: Yes, if you A-Laue me to give it to you. Mr B: Hey, I’m her fiance! You’re just a Neumann. Mr A: We’ll see who’s the Neumann. Bring it on! (Mr A and B start fighting) (Mr W, X and Y start fighting with Mr A and B for no reason) (Mr Z joins in for no reason either) Random: To complement I shall blast this Bohm. Muhahaha! (All die) THE END

A QM skit, composed at age 14 8 CONTENTS Chapter 1

Introduction

Some may say quantum mechanics is the cherry on top of a creamy dessert called physics. Its predictive power has amazed us all, beginning with Planck’s realisation that the quantisation of energy absorption and emission of a blackbody explains the blackbody radiation spectral density, to the prediction in quantum electrodynamics (QED) of the electron anomalous magnetic dipole moment to a precision of better than one part in a trillion. In more recent years the success of quantum mechanics has opened the field of quantum technology, where the distinct features of quantum mechanics are exploited to bring enormous advantage in technological tasks like computation and communication. But there have been two questions about quantum mechanics that have been lingering since the birth of the theory itself: whether it is a complete framework of physics, and what its relationship is to gravity. Most physicists have segregated the questions by relying on the argument that the foundations of quantum mechanics are independent of the features that make it operational for statistical predictions. In principle these features could be combined with an operational theory of gravity. But even at an operational level it is generally regarded that experimentally probing the quantum mechanics-gravity relationship, the problem of quantum gravity phenomenology, is not currently within reach. In this thesis I follow an alternate philosophy. That is, I examine how the operational features of quantum mechanics can be used to constrain the possibility of a more complete framework of nature as well as the relationship to gravity. In this way the two questions above are amalgamated together. If quantum mechanics is taken to be a complete framework then the phenomenon of Einstein-Podolsky-Rosen (EPR) steering (the ability to control a remote quantum state via measurements) implies that the theory contains an apparent nonlocality, in the sense that systems arbitrary distances apart can affect each other instantaneously. Even if it is hypothesised to be incomplete then Bell’s Theorem shows there is still an apparent nonlocality. Many witnesses of Bell nonlocality have been found, but only recently has progress been made to find a necessary and sufficient inequality to witness EPR-steering. Such a witness is constructed in this thesis and in fact it is shown that it can be used to witness Bell nonlocality. This closes a gap between these two aspects of quantum foundations.

9 10 1. INTRODUCTION

Slight modifications of quantum mechanics, specifically of the canonical commutation relations, have been proposed as features of some theories of quantum gravity. These modifications imply a sort of minimal length in nature, which is found for example in string theory. The canonical commutation relations have been probed in a variety of physical systems, from electrons to the planets. But it is especially interesting to find low energy table-top experiments for this purpose. Here I look at cavity optomechanics, a type of controlled interaction between high power optical fields and macroscopic oscillators, as a means to achieve this goal. The modified quantum and thermal noise on oscillators subject to optomechanical interaction are used to find bounds on the canonical commutation relations, and future feasible experiments are explored. I also study the trajectories of systems subject to general continuous quantum position measurements (in the sense of [1]), not necessarily through optomechanical interactions, that are modified directly as a function of the commutation relations. The covariance matrix is analysed to show that there is a stark contrast in behaviour when the commutator is modified. This behaviour includes part of that seen in the optomechanical case, but also there exist other signatures like the exponential growth of the momentum moments that have large scope for improving commutator bounds. Firstly in the curent chapter the background of the thesis is laid. Here the history and formalism of quantum mechanics is established followed by an overview of relevant aspects of the foundations of quantum mechanics. In particular we will explore Bell’s Theorem and its generalisations, Einstein-Podolsky-Rosen (EPR) steering, and the possibility of modifications to the canonical commutation relations. In the next chapter 2 I derive a necessary and sufficient witness of EPR-steering from the statistics of general quantum states in a simple but commonly studied scenario (two qubits with the Clauser-Horne-Shimony-Holt correlations) and that all ‘steerable states’ of this type are also Bell nonlocal. In chapter 3 I examine how modifications of the canonical commutation relation could be witnessed through optomechanical measurements of noise on macroscopic oscillators. Finally in chapter 4 I explore the behaviour of the moments of position and momentum under general continuous position measurements assuming such a modified theory of quantum mechanics and the consequences, relevant to a variety of future experiments.

1.1 The Quantum Formalism

We will review the development of the formalism of quantum mechanics. A great introduction to the subject is the start of [2], and [3] is a collection of early works on quantum foundations. As noted above, the history of quantum mechanics be- gins with Planck’s derivation of the blackbody spectrum through the assumption of quantised energy. Einstein extended this idea and proposed to interpret light as composed of particles of energy, called photons, in order to explain the photoelectric eﬀect. Further examples shaped this new paradigm of ‘wave-particle duality of light’ and culminated with the Bohr model of the atom in which the electrons orbiting the nucleus of the atom jump between orbits via emission/absorption of photons. 1.1. THE QUANTUM FORMALISM 11

The model elegantly produced the spectral lines of hydrogen but to make sense of its assumptions required a generalistion of wave-particle duality to electrons (and possibly for all matter), initiated by de Broglie and experimentally shown by Davis- son and Germer. Schrödingerthen discovered a partial differential equation for the wave amplitude associated to any matter particle, inspired by the fact that classical waves can be modelled by the Hamilton-Jacobi equation for particles in the low wavelength limit. Heisenberg, Born and Jordan developed an alternate expression of Schrödinger’swave mechanics using the language of matrices, now called matrix mechanics. Following on a quantum statistical description of light-matter interactions by Einstein, Born found that the square modulus of the wave amplitude expresses the probability density of detecting a particle at a certain place, known now as the Born rule. This formed the statistical basis of quantum theory and allowed the prediction of intensities, not only frequencies, of atomic spectra. Both Schrödinger’sand Heisenberg-Born-Jordan’s approaches predict the same values a variable of a system can take as well as statistics of values observed. Emergent from this framework is the famous (or infamous) ‘uncertainty principle’ which says that the variances of two variables associated with non-commuting operators obey a relationship such that if one approaches zero the other must approach infinity (there are multiple ways of expressing this principle mathematically). The quantum theory also incorporated a new concept of ‘spin’, an intrinsic quantised degree of freedom of elementary objects similar to rotation about an axis. This was due to the identification by Pauli that the introduction of this concept explained the number of electrons found in shells of stable atomic systems, and the insight of Uhlenbeck and Goudsmit that such a degree of freedom can also explain the splitting of electron energy levels when subject to magnetic fields (anomalous Zeeman effect). Earlier evidence for spin was also present in the Stern-Gerlach experiment, which recorded the the deflection of silver atoms by a spatially varying magnetic field. The success of these principles in explaining atomic and thermal processes could be called the quantum revolution. A rigorous and unified mathematical framework of the theory was accomplished by John von Neumann [4] and Dirac[5]. Since then the postulates of the theory have been expressed in various ways [6, 7]. One formulation of quantum mechanics involves these five axioms [8]: • Every system is described by a ‘state’ |ψi, a ray in Hilbert space H (a vector space that posesses an inner product and complete with respect to the norm).

• An observable on a system is a Hermitian operator A i.e. A = A†.

• Measurement of an observable gives an outcome that is an eigenvalue of the observable and the state of the system is updated to an eigenstate associated with the eigenvalue. For the outcome a the new (normalised) state is Ea|ψi/ kEa|ψik where Ea is the orthogonal projection onto the eigenspace associated with a. The Born rule states that probability of obtaining that outcome is hψ|Ea|ψi. • A unitary operator (U such that UU † = I) describes the dynamics of closed systems. In the ‘Schr¨odingerpicture’ this means a quantum state evolves under 12 1. INTRODUCTION

the ‘Schrödingerequation’ d|ψi/dt = −iH|ψi/~ whiere H is the Hamiltonian observable and ~ is a real number called Planck’s constant. In the ‘Heisenberg picture’ the state is fixed over time but the observables evolve under conjuga- tion by the appropriate unitary such that probabilities values of all observables are the same as in the Schrödingerpicture.

• The Hilbert spaces of two systems compose via the tensor product i.e. HRS = HR ⊗ HS for systems R and S. As written the second and third axioms are strictly applicable for systems whose observables can potentially take only a finite number of values i.e as operators they act on a finite dimensional Hilbert space. Even though the quantum mechanics of matter systems began with an analysis of position and momentum variables, which hypothetically can take an uncountable infinity of possible values i.e. they act on infinite dimensional Hilbert space, it is actually far more complicated to construct a rigorous mathematical theory of such quantities (this is discussed by von Neumann in considerable detail in [4]). Nevertheless the predictions of experiments using these observables in the formalism above has stood scrutiny till date. Crucial here is the relationship between operator-valued observables as defined above and the real-valued variables in classical (non-quantum) physics [5]. For a classical position x of a mechanical system in one dimension the corresponding operator xˆ is defined to be the quantum mechanical position observable; it has eigenvalues consisting of all the possible real-valued position outcomes. In classical physics the fundamental concept of momentum is the generator of infinitesimal spatial translations. The analogous momentum operatorp ˆ in quantum mechanics is defined as the generator of translations between eigenstates of the position operator associated with infinitisemally close position outcomes. The canonical commutation relation [ˆx, pˆ] = i~ holds and corresponds to the classical Poisson-bracket relation {x, p} = 1. As in classical physics other important quantum observables like the Hamiltonian may be constructed via position and momentum observables. However in quantum mechanics there are certain observables like spin that do not (as far as we know) have straightforward classical counterparts. There is furthermore the fact that in a realistic experiment information about a system is learnt by measuring another system, a meter, coupled to the original target system. The axioms above can be used to show that for each outcome a from the 0 measurement of the meter there is an operator Ea acting on the target system Hilbert 0 space such that the probability of the outcome is hψ|Ea|ψi where |ψi is the quantum 0 0† P 0 state of the target system prior to measurement. Also: Ea = Ea , a Ea = I. 0 The {Ea} form a set called a POVM (positive operator-valued measure). Note that the POVM is not necessarily an orthogonal set and the state of the target system immediately after measurement is arbitrary if only given the measurement outcome. Further details on the POVM-formalism and the theory of general quantum measurements and evolutions can be found in [8]. As an aside, it should be noted that the fusion of quantum mechanics with the special theory of relativity (also discovered by Einstein), and its application to field 1.2. BELL’S THEOREM 13 theory like Maxwell’s electromagnetism, is called quantum field theory. The consistent construction of quantum field theory through ‘renormalisation’ may be labelled the second quantum revolution[2], though not to be confused with the more common usage of this term from [9] to describe the advent of technology based on quantum mechanics (like quantum computation). Despite the success of quantum field theory there is no well-established theory that applies it to Einstein’s general theory of relativity, which revolutionised our understanding of gravity. We will see later that some attempts at such a quantum theory of gravity suggest that the canonical commutation relation between position and momentum [x, p] = i~ should be modified, at least in some low energy limit.

1.2 Bell’s Theorem

I now introduce one of the topics of this thesis, the famous Bell’s Theorem of [10]. The essence of Bell’s work is to translate between the operational/experimental aspects of quantum mechanics and the language of any theory that contains additional objects or ‘pre-existing values’ usually termed hidden variables. Generalisations of Bell’s theorem are classified according to the relevant experimental scenarios. Each scenario can be expressed in terms of the number of subsystems n, number of measured variables/settings m(i) associated to each subsystem i ∈ n, and number of possible outcomes of each variable k(l) where l ∈ m(i). For simplicity here we mainly consider scenarios where m is independent of i (so m(i) = m) and k(l) is independent of l (so k(l) = k), so such scenarios may be labelled by (n, m, k). A good review on this subject is [11]. Firstly I will introduce the background behind Bell’s works. Ever since Max Born established the probabilistic description of quantum mechanics applied to matter there was a debate between physicists on the foundations of quantum mechanics. If quantum mechanics is a complete description of reality then it would mean, by definition, that there is no better description of reality beyond probabilities. Several founders of quantum theory like Einstein, Schrödingerand de Broglie could not digest this view. However others including Born, Pauli, Heisenberg and Dirac believed that indeed quantum mechanics implies the inherent uncertainty of reality or at least that there does not exist a better description for predicting the results of experiments. It was thought that if quantum mechanics is incomplete then reality must involve ‘hidden variables’. The opening paragraph of Bell’s first paper [12] on foundations of quantum mechanics (written in 1964 but published in 1966) gives a description of what this means: “To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical ‘dispersion free’ states would be specified not only by the quantum mechanical state vector but also by additional ‘hidden variables’ - ‘hidden’ because if states with prescribed values of 14 1. INTRODUCTION these variables could actually be prepared, quantum mechanics would be observably inadequate.” In a paper written in 1935 [13] by Einstein, Podolsky and Rosen an argument (now called the ‘EPR-paradox’) was given that hidden variables are required in order for quantum mechanics to be consistent with a locality principle (that an action at one place cannot instantaneously affect a system at another place). Firstly, it is possible to prepare a quantum state involving two systems such that the position and momentum of particles of one system are correlated with the same variables of the other system. Such a correlated quantum state is aptly called an ‘entangled state’ (the specific state is called an EPR-state named after the authors). It is argued that if the quantum state is an element of reality then since the act of measurement of any variable on one system causes collapse of the quantum state of the other system, due to the correlated nature of the state, locality is violated. More than that, one is able to ‘steer’ the state on one system to a position or momentum eigenstate depending on if position or momentum is chosen to be measured on the other system (this phenomenon dubbed ‘EPR-steering’ is further explored in the next section). Since locality is assumed to be a sacred principle, the alternative being in Einstein’s words “spukhafte Fernwirkung”(“spooky action at a distance”), the authors of the paper decided in favour of incompleteness of quantum mechanics and thus the existence of hidden variables. Whilst this is a compelling argument, several proofs aimed to show that quantum mechanics is inconsistent with any hidden variable theory without appealing to locality. Bell’s paper [12] contains a rebuttal of such proofs. Von Neumann’s proof [4] assumes that the value of a hidden variable associated with a sum of two arbitrary observables (Hermitian operators) must be the sum of the values associated with each observable. Bell identifies that quantum mechanics implies that only one of two non-commuting observables can be measured at a particular time, so any relation between the values of these observables is unjustified, and therefore von Neumann’s assumption does not hold in generality. To explicitly refute von Neumann’s proof Bell shows a simple hidden variable theory for a spin-1/2 system. A later proof by Jauch and Piron makes von Neumann’s assumption only for commuting operators, but does assume a relation between values associated with non-commuting variables. Similarly the collary of a work by Gleason says that if the dimension of the Hilbert space is greater than two then any hidden variable theory is inconsistent with von Neumann’s assumption applied only to commuting operators. Bell gives his own proof of this result but emphasises that it implicitly assumes non-contextuality. In a non-contextual hidden-variable theory for quantum mechanics the hidden variable value associated to a projector doesn’t depend on the complete commuting set with which it is analysed. This is used to arrive at the intermediary conclusion (labelled ‘B’ in the paper), that if the expectation values of two projectors onto orthogonal vectors are zero then the expectation value of the projector on to any superposition of the vectors is also zero. Applied to the hypothetical ‘dispersion free’ states (those with definite hidden variable values for all variables) it implies a relation between values for non-commuting variables. Thus non-contextuality leads to the fallacious 1.2. BELL’S THEOREM 15 assumption found in the other proofs. Bell’s proof is revisited by Mermin in [14] but this analysis of the intermediary step is omitted. An important aspect of Bell’s proof is that it holds independent of the state. Thus the underlying non-contextuality assumption is often referred to as state-independent. Later, Kochen and Specker also showed a proof of the inconsistency of hidden variables with quantum mechanics [15] if state-independent non-contextuality is assumed. Non-contextuality for a hidden variable theory in relation to an arbitrary empirical statistical theory can be defined using standard probabilistic notation [16]: Let λ be the hidden-variable, a be an outcome of an allowed measurement A of the empirical theory and B, C, ..., B0,C0, ... be other allowed joint measurements (so the probabilities p(A, B, C, ..., λ) > 0 and p(A, B0,C0, ..., λ) > 0 hold). The hidden- variable theory is non-contextual if for all possible values of these variables:

p(λ|A, B, C, ...) = p(λ|A0,B0,C0, ...) p(a|A, B, C, ...) = p(a|A, λ) (1.1)

The empirical theory will consequently obey p(a|A, B, C, ...) = p(a|A0,B0,C0, ...) and is then also termed non-contextual. Assuming the hidden variable theory is deterministic and reproduces quantum mechanics the notion of non-contextuality reduces to the one in [12] and the Kochen-Specker theorem[15]. We can now look at Bell’s famous theorem found in [10]. The essential result is that quantum mechanics is not consistent with any local hidden variable theory. To show this an experimental scenario is considered which can be expressed in the language described at the start of this chapter as n = 2, m(1) = 1, m(2) = 2, k = 2. The two subsystems share a singlet quantum state on which the measurements are performed. Bohm proposed this state as a two-outcome per variable version of the EPR-state. Suppose there is a set of hidden variable parameters λ that can describe the experiment. For a spin measurement in the direction along unit vector a on one system, and along b on the other system, the result of the measurement is A(a, λ) for a and B(b, λ) for b. All results are either 1 or -1. We can decompose the correlator (expectation value) P (a, b) of these observables via a probability distribution ρ(λ) over these hidden variables: Z P (a, b) = dλρ(λ)A(a, λ)B(b, λ) (1.2)

Importantly locality is assumed in the sense that the outcome of a depends only on λ and a but not b (though the λ determines outcome of b), and vice versa. Introducing another spin measurement direction c an inequality is then produced from these, which is violated by statistics of the singlet state:

|P (a, b) − P (a, c)| − P (b, c) ≤ 1 (1.3)

However specific observables on each party are not given to produce this violation. Instead the contradiction is that the absolute value sign in the inequality implies P (b, c) is non-stationary (derivative is non-zero) at b = c, unlike in quantum theory 16 1. INTRODUCTION where P (b, c) = − cos θ where θ is the angle between b and c. In fact Bell shows the quantum statistics cannot even be arbitrarily close to satisfying the inequality. By demonstrating this conclusion on a version of the EPR state Bell showed that somehow nature shows no escape from nonlocality or, more strictly put, no definite adherence to locality. To clarify, the perfect (expectation value 1 or -1) correlations in the EPR-paradox can be explained via hidden variables or pre-assigned reality that preserve the principle of locality, but from Bell’s theorem which examines other measurement correlations we find that if these hidden variables exist they must nonlocal. A variant of this point was spoken by John Bell in one of his final recorded talks [17], but he confesses he does not know the definite way out.

1.3 Einstein-Podolsky-Rosen Steering

I will now look in more detail into the phenomenon of Einstein-Podolsky-Rosen steering, another topic covered in this thesis. In the EPR paradox it was realised that via an entangled state it is possible for one party to control the state associated with another arbitrarily remote system. Of this Schrödingersaid “It is rather discomfort- ing that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter’s mercy in spite of his having no access to it.”[18]. The concept of steering, or even entanglement, was not a major focus of physics research till the advent of quantum information theory. One advance in [19] was introducing the possibility of experimentally testing the EPR-paradox, in the sense of showing almost perfect correlations involving conjugate variables. There it is argued that for two entangled photons made through a nondegenerate parametric amplifier both of the conjugate quadrature phase amplitudes are sufficiently correlated between the photons to demonstrate a (noisy) version of EPR-paradox (for quadrature phase components instead of position and momentum). It can be witnessed by the ‘Reid criterion’, essentially a violation of the Heisenberg uncertainty principle for probability distributions of the conjugate variables on one system con- ditioned on outcomes of measurements of the variables on the other system. For spin systems analogous criteria were derived in [20]. Later the concept of EPR-steering was revived as a whole field of study by Wise- man, Jones and Doherty in [21]. The key insight is that just as Bell’s theorem demonstrates the absence of a local hidden variable description of quantum mechanics, EPR-steering can be viewed as the absence of a ‘local hidden state’ model (or in some publications called local hidden state-local hidden variable model). The meaning of this is as follows. Firstly for a given quantum state W shared between A A two parties, Alice and Bob, letρ ã = Tr[W (Πa ⊗I)] be the (unnormalised) state projected on to Bob’s system as a result of Alice’s measurement A giving an outcome A a (associated with projector Πa ) . Then a local hidden state model exists for all A that Alice can measure and all outcomes a if there is an ensemble of states {℘ξρξ} 1.3. EINSTEIN-PODOLSKY-ROSEN STEERING 17 and ∀A, a there exists a non-negative distribution ℘(a|A, ξ) such that:

A X ρã = ℘(a|A, ξ)ρξ℘ξ (1.4) ξ The picture behind this equation is that Alice could reproduce the statistics of the projected state by sending the states of the ensemble to Bob with a probability weighted by the distribution. Note that the reduced density matrix of W for Bob’s P P A system is ρ = ξ ℘ρξ = a ρã . If such a distribution and ensemble do not exist then a local hidden state model cannot be constructed so EPR-steering exists. Another way of expressing EPR-steering is that it is not possible to have for all possible outcomes a, b of observables A, B by Alice and Bob that: X P (a, b|A, B; W ) = ℘(a|A, ξ)P (b|B; ρξ)℘ξ (1.5) ξ A particular application of the steering concept is on the set of bipartite mixed states on Cd ⊗ Cd called ‘Werner states’[22]. These are parameterised by a real number η ≤ 1. The value η = ηBell is defined such that a Bell inequality violation can be produced by measuring the Werner state iff η > ηBell. Werner proved that ηBell > 1 − 1/d [22]. It is shown in [21] that in fact this value is the same as ηsteer (the analogous value for EPR-steering). Thus generally ηBell > ηsteer so if a state has Bell inequality violation it also has EPR-steering, but not necessarily vice versa. In [21] it is also shown that for the EPR-state the Reid criterion for the EPR-paradox is equivalent to a criterion for EPR-steering. The hierarchy as established in [21] and [23] for general mixed states is: entangled states ⊂ EPR-steerable states ⊂ Bell nonlocal states. However even for general two-qubit mixed Werner states the exact value of ηBell is unknown and so is the divide between steerable and Bell nonlocal states. EPR-steering also has an interpretation in the language of quantum information theory. Suppose two parties share a quantum state but one of the parties is untrusted, for example has a faulty measuring device, and can only exchange information by classical communication. An external referee could efficiently test the entanglement of the state by checking for EPR-steering [21]. Several unique aspects of EPR-steering have been uncovered since the seminal work of [21]. It is possible to have one-way EPR-steering [24], in the sense that any projected state on one system can be explained using a local hidden state model but this is not the case for all projected states on the other system. Like Bell’s inequality there are also inequalities that can be used to witness EPR-steering (called steering inequalities) if they are violated [25]. It is generally easier to experimentally witness EPR-steering than Bell nonlocality, demonstrated in [26] through the evasion of loop-holes that are typically present in tests of Bell inequalities. Many experiments on entangled photons have demonstrated EPR-steering inequality violation [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. This experimental direction is also linked to the application of EPR-steering in entanglement verification for quantum key distribution. In one-sided QKD on one system a device can be trusted for 18 1. INTRODUCTION making quantum measurements to check if a state has been steered by untrusted measurements on the other system [38]. For further on EPR-steering the review [39] is helpful.

1.4 Generalisations

From Bell’s theorem further important generalisations, directly utilising probabilities of coincident outcomes of experiments, were discovered in other scenarios. These generalisations form a framework to accomplish a similar task for EPR-steering. We will go through a time-line of important papers in this direction. Clauser-Horne-Shimony-Holt 1969 [40]: The ‘CHSH inequality’ (named after its authors) is derived for correlators in the scenario (2,2,2) that arise from any local hidden variable theory. Unlike in Bells theorem this inequality does not apply to only a speciﬁc class of states. The inequality is generally expressed as:

|P (a, b) − P (a, b0) + P (a0, b0) − P (a0, b)| ≤ |P (a, b) − P (a, b0)| + |P (a0, b0) − P (a0, b)| ≤ 2 (1.6) where the triangle inequality is used on the middle inequality, which is the actual result found in the paper. It can also be violated by certain measurements on the singlet state. Also the inequality is generalised in the paper to an experimental scenario with detection inefficiencies e.g. due to lost photons in a light-based experiment. Wigner 1970 [41]: Here a simplified argument of Bell’s result is presented. It is in the (2,3,2) scenario on the singlet state. An important insight is that the conflict between local hidden variable theory and quantum mechanics arises when trying to impose a positive probability distribution over all possible outcomes, here on a 6-dimensional vector space (three possible measurements on each side), and so the introduction of negative probabilities can allow a consistent description. In fact Wigner recognises a set of expressions the positivity of which is necessary and sufficient for a description in terms of hidden variables. Also he shows that the constraints on these expressions are equivalent to an upper bound on the circumference of triangles with 2 side lengths 1/2 sin (θik/2) for angles θik between measurements on each system, as well as a triangle inequality. This quadratic dependence on angle is crucial, for a linear dependence would allow satisfaction of the triangular conditions. Finally, Wigner says that his version of Bell’s theorem can be applied to all pure entangled states (not just the singlet state). This fact was proved rigorously in a later note by Gisin [42]. In an insightful footnote Wigner claims that von Neumann’s belief in the inadequacy of hidden variables was not the one found in his textbook (which is the one Bell addressed). Instead, based on conversations with von Neumann, he thinks it was founded on an argument involving many sequential measurements of a spin system in different directions. As this is a tangential matter I leave it to the reader to explore this in further detail. Bell 1971 [43] and 1976[44]: The main contribution relevant for our discussion is that the CHSH inequality is extended to the scenario in which stochastic, not 1.4. GENERALISATIONS 19 necessarily deterministic, outcomes are assumed in the original setup. This allows for the scenario that the measuring apparatus has hidden variables. In [44] this is brought together with a more general definition of locality (it essentially says that the events of two systems can only be correlated if there is a common cause of them in the past) and called ‘local causality’ to derive the CHSH inequality. Bell also suggests in [43] that to experimentally test the inequality it could be more suitable to use two-photons or kaons instead of spin 1/2 particles (as in his original proposal), since certain decay processes involving the former produce robust entangled states. Indeed it is noted in [44] that the CHSH inequality can be trivially satisfied with non-ideal experiments. It is also shown that ordinary quantum field theory implies faster than light signalling is not possible, independent of Bell’s theorem, and that a different background causal structure or restricting freedom of settings could modify the conclusion. Local causality can been expressed in more generality as follows adopting standard notation [16][11]: Firstly let a, b, c, ... be outcomes of an experiment associated with measurements A, B, C, ... and λ a hidden variable. The hidden-variable model satisfies locality if for all a, b, c, ..., A, B, C, ...λ, given p(A, B, C, ..., b, c..., λ) > 0 (that is these measurements are possible under the hidden variable theory):

p(a, b, c...|A, B, C, ..., λ) = p(a|A, λ)p(b|B, λ)p(c|C, λ)... (1.7)

This deﬁnition will be used in the next chapter. Clauser Horne 1974[45]: The CHSH inequality is expressed in terms of probabilities of outcomes and their pairwise coincidences, instead of correlators. The new inequality (the ‘CH inequality’), using the notation as before and labelling outcomes j = ±1, k = ±1:

0 0 0 0 0 −1 ≤ pjk(a, b) − pjk(a, b ) + pjk(a , b) + pjk(a , b ) − pj(a ) − pk(b) ≤ 0 (1.8)

Furthermore, with an extra assumption, this is adjusted for experimental tests. Froissart 1981 [46]: It is shown that local causality implies that a vector of probabilities of coincidences in the CH inequality is a convex sum of vectors containing fully deterministic coincidences. Thus there is the theory of Bell inequalities is a special case of the theory of convex polytopes. Some of the faces of the convex polytope are the CH inequalities under permutations, expressed as trace matrix inequalities. This is generalised to 3 settings for 2 parties. It is however claimed that Bell inequality violating experiments from quantum statistics that implicitly depend on the quantum formalism (e.g. a quantum interpretation of polarisation) are inconclusive on the question of local causality, as they depend on a circular argument. Fine 1982 [47]: Here Fine elaborates in the (2,2,2) scenario the connection between Bell’s theorem and the existence joint (positive) probability distribution over observables, following Froissart and Wigner. For this scenario these statements are shown to be equivalent:

• existence of a deterministic hidden variable model 20 1. INTRODUCTION

• existence of a joint probability distribution over all four observables (two on each system) that returns experimental probability distributions (either of one observable, or two observables on different systems) as marginals • existence of a joint probability distribution involving three of the observables (one on system A and two on system B) and a distribution with a different observable on system A, that both produce experimental probability distributions as marginals and yield the same probability distribution over the two observables (which may not commute) on system B. • satisfaction of Bell/CH inequalities • existence of a factorisable stochastic model (which could be motivated by the local causality axiom). Emphasis is placed on the fact that the violation of CH inequalities involves non- commuting observables on each system and that the joint probability distribution is required to also be over these non-commuting observables “whose very rejection is the essence of quantum mechanics”. It is however not the case that this is a sufficient criterion, since in the EPR-Bohm experiment the statistics of the same perpendicular observables on each system (which have perfect correlations) can be explained using a local causal theory, nor is it necessary that all observables are non-commutative e.g. in Bell’s theorem one of the observables, on one system, commutes with the other two on the other system. Fine conjectures that this can be generalised to more than two settings per party. But Garg and Mermin have a counterexample [48]: more than two observables on each subsystem can be generated by linear combination, and then joint probability distributions over four observables sets (like one proposed by Fine) do not necessarily produce the correct marginal probability distributions, even though CH inequalities are satisfied. Garg and Mermin 1984 [49]: In this paper a mathematical framework is introduced on the topic of general Bell inequalities. Firstly a property in statistics is observed: a given set of probability distributions each over two observables are not guaranteed to be marginals of a higher order joint distribution, unlike if only given a set of single observable distributions. Indeed quantum mechanics passes through this open door. The authors show that the existence of a joint probability distribution implies solution of a single matrix equation, where the matrix contains only 0’s and 1’s, but Farkas lemma implies that a (non-negative) joint distribution exists iff the marginals are inside a polyhedral cone i.e. polytope whose facets are associated with certain inequalities (as realised in the simplest scenario by Froissart and Fine). More precisely, Farkas’ lemma states that, given a vector pα and a matrix vµα consisting of 0’s and 1’s, the conditions that are necessary and sufficient for the existence of P non-negative real numbers Pµ so that pα = Pµvµα are that all sets of cα satisfy- P µP ing, for all µ, α vµαcα ≥ 0 should also satisfy α cαpα ≥ 0. This is a central result in the theory of linear programming and convex optimisation, see [50]. In this way behind Bell’s theorem is what they call “Bells problem: “Given pair distributions for all pairs of N random variables, what are the necessary and sufficient conditions 1.4. GENERALISATIONS 21 on these distributions for there to be compatible, consistent, non-negative third order distributions for all triples of variables”. Note there is no mention of different parties here. The reason is that distributions over two observables on a singlet state, in which each one is on a separate party, can be regarded as a distribution over two (not necessarily commuting) observables on the same system due to perfect correlation between the same measurements on each system. The more restricted “Clauser and Hornes Problem” (relevant to the CH inequality) is similarly defined except the marginals are over pairwise observables on separate parties, and each third order distribution involves at least one observable on each party. From Farkas lemma all the inequalities for Bells problem for higher spin quantum scenarios that obeying certain symmetries, and for Clauser and Hornes problem for the spin 1/2 scenario, are derived. In the former the total range of violation of the inequalities diminishes from spin 1/2 to 1 but increases from 3/2 to 5/2. We conclude that coincidence probabilities from experiments can be combined into a vector that is a convex combination of 0-1 vectors (fully deterministic marginal and/or coincidences) and a way looking at non-classicality, whether or not it be from quantum mechanics, is that the probability vector lies outside the polytope. In other words it violates an inequality associated with any of the facets of the polytope. The particular reason for this can also be viewed in this way: each probability vector is a set of subjective probabilities associated with fully determined outcomes/coincidences, therefore it is a convex sum of fully determinate vectors. Only if the determinate vectors have different elements that are associated with non- commuting observables (in the language of quantum mechanics) may it be possible for the probabilities to lie outside the polytope. The generalisation of Bell’s theorem to arbitrary experimental scenarios means the exploration of convex polytopes of increasing dimension and vertices. It is shown in the next chapter that a witness for EPR-steering can be constructed analogous to the CHSH inequality, but interpreted geometrically has an ellipsoid structure unlike the CHSH polytope. Furthermore it can even be used to witness Bell nonlocality.

1.4.1 Complexity of Determining Bell Nonlocality We have glimpsed that given an experimental scenario the computation of all the Bell inequalities appears to be an expensive task, as noted by Garg and Mermin. This makes it a rich subject at the nexus of physics, mathematics and computer science. The precise difficulty of related tasks, from the perspective of computational complexity, was studied first by Pitowsky. In the following we look at the works of Pitowksy and others on such questions. This is non-essential to the main contents of this thesis but I provide it for the sake of completeness on the topic of this chapter. The techniques described here could be partially applied to corresponding questions in relation to EPR-steering, which are still open questions. It may be of help to the reader to refer to a standard text on computer science, for example [51]. Firstly in [52] it is noticed that a vector of probabilities over some propositions (labelled by numbers 1 to n) and a subset of coincidences of the propositions, which 22 1. INTRODUCTION are also consistent with laws of propositional logic (this assumes an existence of a joint distribution over the propositions), will be a member of a polytope whose vertices are the vectors containing possible truth values (0 or 1) for the propositions and coincidences. The facets of this correlation polytope are inequalities and the task of finding all such inequalities was in fact studied by George Boole over a century earlier in the field of logic. Such polytopes and their inequalities are the same as the polyhedra examined by Garg and Mermin except there only pairwise coincident distributions form the probability space. In an experimental context each proposition here represents the event of a certain outcome of an observable. A proof is then given that the problem of determining if an arbitrary vector of the above type lies in the corresponding polytope, in a scenario containing all pairwise coincidences or another scenario of all such coincidences except of form {i, n} from i = 1 : k where n = k + m + 1 for integers k and m explained below, is NP-complete (in n as well as the size of the vector). This implies that if the problem can be solved in time bounded by a fixed polynomial in the size of the vector then P=NP, which is considered to be the most important open question in computer science though there is a large consensus that it is likely false. The crux here is that if an arbitrary vector from a certain set can be determined to be in the corresponding polytope in polynomial time then, in the scenario containing all pairwise coincidences mentioned above, the NP-complete problem ‘1 in 3 3-SAT’ can be solved in polynomial time, and in the other scenario omitting certain coincidences, the NP-complete problem ‘3-Colourability of a Graph’ can be solved in polynomial time. [Technical details: In ‘1 in 3 3-SAT’ the problem is to determine value of a Boolean proposition (one that takes value 0 or 1) which is the intersection of m propositions, each composed of the union of 3 propositions, each of which is taken from a set of propositions of size k. k 3 Here, for a given k, it is the case that k/2 ≤ m ≤ 3 (which is of order k ). The vector of probabilities that is being tested is mapped to the main proposition in 1 in 3 3-SAT in such a way that either the vector contains n = k + m + 1 individual n n events and all 2 pairwise coincidences, or just 2 − k coincidences if those of the form {i, n} from i = 1 : k are removed. Then it is clear that a polynomial time algorithm in n or the size of the vector corresponds to polynomial time algorithm in k of the proposition]. A related problem is finding the minimum energy of an Ising spin model (which is used to model a variety of physical systems from spin glass to neural networks). This is equivalent to a linear program, that can be solved via usual linear programming techniques by first finding the facets of the polytope as before including all pairwise coincidences. It is shown that this minimum energy problem is NP-complete by mapping it to 1 in 3 3-SAT. Furthermore this implies, from a theorem by Karp and Papadimitiriou, that if the problem of checking if a certain inequality is a facet of the polytope can be solved in polynomial time i.e. if the facet problem is in NP, then NP=co-NP. Pitowsky’s book [53] reproduces the proof in [52] that reduces the polytope membership problem to 1 in 3 3-SAT, except the given vector has a (smaller) subset of the pairwise coincidences. So in fact testing membership in a polytope of lower dimen- 1.4. GENERALISATIONS 23 sion is sufficient; the probabilities for the extra coincidences in [52] follow from the same assumptions as in the book. In the book there are less than 7m coincidences, which is a smaller polynomial in k than those coincidences considered in the paper. The question of finding all Bell inequalities was revisited by Peres in [54]. In this paper the Garg-Mermin connection between Farkas’ lemma and Bell inequalities is translated into a colouring problem and applied to multiple parties. In terms of parties, tests and outcomes it is conjectured that all Bell inequalities in experiments involving more outcomes or parties than in the CH scenario can be reduced to CH tests by coarse graining outcomes together and via ‘distillation’ i.e. two parties perform the CH tests conditional on that the measurements by all the other parties obtain specific results. It is proven that under partial transposition of the quantum state Bell inequality violation is preserved, and the ‘PPT/Peres-Horodecki criterion’ is presented: that if a state is separable then under partial transposition the states eigenvalues must be positive i.e. it must be PPT. But the converse is not true i.e. there exist PPT states that are entangled, and in fact all such states are undistillable (defined as being unable to be distilled to singlets). The famous ‘Peres conjecture’ (acknowledged in the paper as being suggested by Tal Mor), a corollary of the conjecture above, is that these states allow no violation of a Bell inequality. This would mean Bell inequality violation involving a state, after any local opera- tions with classical communication (LOCC), can occur iff the state is distillable, and only if the state has non-positive partial transpose. The conjecture was disproved by Vertésiand Brunner via a counterexample produced through semi-definite programming [55]. In the appendix of the paper an algorithm is outlined that generates all Bell/facet inequalities, but it comes down to the problem of deciding which subsets of vectors defining a polytope are facets, which according to Peres is an NP-complete problem (refering to [52]). It is suggested that symmetries of the cone can simplify the problem. Notably a more detailed complexity theoretic analysis of Bell inequalities in the context of 2 parties with many 2-valued observables was studied by Avis, Imai, Ito, Sasaki [56] by a mapping to the well studied ‘cut polytopes’. It is recognised that [53] shows that in the (2,2,2) scenario the polytope of probabilities of correlated outcomes from measurements on each party that have a local causal explanation (called the B(2, 2, 2) Bell polytope) is affinely isomorphic (bijective transformation that preserves points, straight lines and planes) to a polytope of a complete bipartite graph K2,2 in a space involving individual events and their correlations (called the COR (K2,2) correlation polytope). By extending this to B (2, m, 2) scenario and invoking a theorem that shows an affine isomorphism of correlation and cut polytopes (these polytopes lie in a space with dimension equal to the number of edges of the event graph) it is presented that B(2, m, 2) is affinely isomorphic to a cut polytope. It is then shown that membership testing of this cut polytope is NP-complete which implies that membership testing of B(2, m, 2) is NP-complete. This is done by showing that the ‘weighted maximum cut’ problem on the graph associated with the cut polytope, a NP-complete problem, reduces to the membership test of this cut polytope. Indeed this is similar to Avis and Deza [57] which shows that the 24 1. INTRODUCTION membership test to cut polytopes of complete graphs is NP-hard. A fact noted in [56], often not acknowledged, is that Pitowsky’s similar result on the NP-completeness of membership testing is proved via ‘Karp reducibility’ from 1-in-3 3 SAT (distinct from ‘Turing reducibility’ used in [56]) and importantly only applies to correlation polytopes, which are strictly different from Bell polytopes. Here I note that, on its own, this is a significant difference as the size of the set of practically measurable probabilities of all correlated outcomes in a general system may be smaller than the dimension of the correlation polytope space. Indeed certain correlations of outcomes in this space may refer to measurements that are incompatible’. For example, single party spin measurements in different directions are thought to be incompatible according to quantum mechanics, and incompatibility can also be defined in general probability theories. To apply Pitowsky’s proof to membership testing in the Bell polytope it is required to use the affine isomorphism between correlation and Bell polytopes, which involves adding extra probabilities to the original vector that have not necessarily been measured and may not be practically attainable.

1.5 Modiﬁed Canonical Commutation Relations

So far we have assumed the validity of quantum mechanics but allowed the possibility of a more complete description of its predictions. In this thesis I also look at the possibility that quantum mechanics itself is modified and the observational consequences. Specifically I consider the modification of the canonical commutation relations, motivated by theories of quantum gravity. Various proposals exist for the deformation of quantum mechanics. One of the first proposals, by Heisenberg, was to introduce a minimal length. Originally Heisen- berg was motivated by the fact that Fermi’s explanation of β decay implies that in high energy collisions large numbers of particles must be produced, which he called ‘explosions’[58]. He argued that if this production actually exist then “measurements of positions are not possible to a precision better than r0”, where r0 is the length scale associated with the explosions and can effectively be treated as a minimal length. Later it was understood that Fermi’s theory is only an approximation so these explosions don’t actually exist (in the language of quantum field theory Fermi’s theory was ‘non-renormalisable’). Heisenberg had also used the concept to try to explain the discrete mass spectrum of elementary particles[59] and earlier attempted to incorporate modified commutation relations such that [xi, xj] 6= 0 [60] for position observables in orthogonal directions. A concept of minimal length can also be expressed by instead modifying the canonical commutation relations. The main such relation we will focus on is [61]: ! p 2 [x, p] = i~ 1 + β0 (1.9) MP c

2 2 where A = β0/MP c , MP is the Planck mass and c is the speed of light in vacuum. 1.5. MODIFIED CANONICAL COMMUTATION RELATIONS 25

This is the simplest modiﬁed isotropic translational invariant canonical commuta- 1 tor. When hpi = 0 the Robertson uncertainty relation ∆x∆p ≥ 2 | h[x, p]i |, where the terms on the left-hand-side refer to standard deviations of position and momentum operators respectively, implies the generalised uncertainty principle (GUP):

~ β0 2 ∆x∆p ≥ 1 + 2 (∆p) . (1.10) 2 (MP c) Notice on the right-hand-side the variance of√ momentum appears. This implies, minimising over all ∆p values, that ∆x ≥ LP β0, so there is a minimal length. For β0 = 1 it is the Planck length. This GUP is actually present in the string theory phenomenon of T-duality, where there is an equivalence of high energy (small radius) and low energy (large radius) string physics, as explained qualitatively by Witten in [62]. Several other commutators have been proposed for other theories of quantum gravity[63, 64, 65]. Experimentally probing this modified commutation relation, or its associated GUP, has been of much interest due to its connection with quantum gravity. Gener- ally it is thought difficult to constrain theories of quantum gravity since their unique −35 features are usually pronounced near the Planck length LP = 1.62 × 10 m. The reasoning given is that the standard Heisenberg uncertainty principle implies that to directly probe such a scale via a photons requires ∼ 1020 GeV which is far greater than energies achievable with current particle accelerators. However a priori it is not necessary that the Planck length is the minimal length and the modified canonical commutator allows for this possibility through the parameter β0. Also the modification of the momentum operator (which can produce the modified commutator) has implications for many phenomena. Indeed various bounds on β0 have been placed by looking at deviations in the behaviour of macroscopic systems like harmonic oscillators to high energy particle collisions and the Lamb shift. It is not straightforward to relate the β0 bound associated with one system to another. Several arguments, for example in [66], show that it is possible to model the Hamiltonian of a system with N components that each satisfy (3.1) with modification parameter β0 so that the corresponding β0 for the centre of mass is approximately rescaled by 1/N 2. Terms that couple the total momentum to the relative momenta of the components must be neglected for this to hold, a good approximation for a rigid body at sufficiently low temperature and small internal motion. One proposal [67] suggests using cavity optomechanics to probe the modified canonical commutators. Here a sequence of pulses of light are sent into a cavity to interact with an oscillator and impart a phase change to the output light that depends on the mechanical centre of mass canonical commutator. It is claimed that in future it could be experimentally feasible to reach bounds like β0 = 1 on a variety of commutators. This paper inspired experiments [68, 69] that probed harmonic oscillators for the amplitude-frequency effect (dependence of the resonant frequency of a harmonic oscillator on its amplitude of motion), a signature 6 of the commutator above. Bounds of down to β0 = 10 for the centre of mass have been reported this way. 26 1. INTRODUCTION

In chapter 3 the position quantum noise spectrum on an oscillator driven by a laser, assuming the modiﬁed canonical commutator above, is studied to reveal current and future β bounds achievable through realistic experiments. This is a special case of a continuous position measurement. A more general treatment of such measurements following the model of Caves and Milburn [70] with this commutator, applicable to many other experimental setups, is in chapter 4. 1.5. MODIFIED CANONICAL COMMUTATION RELATIONS 27 28 1. INTRODUCTION Chapter 2

EPR-Steering and Bell Nonlocality in the CHSH Scenario

We derive an inequality that is necessary and suﬃcient to show Einstein-Podolsky- Rosen (EPR) steering in a scenario employing only correlations between two arbitrary dichotomic measurements on each party. Thus the inequality is a complete steering analogy of the Clauser-Horne-Shimony-Holt (CHSH) inequality, a generalisation of the result of Cavalcanti et al. [E. G. Cavalcanti, C. J. Foster, M. Fuwa, and H. M. Wiseman, JOSA B, 32, A74 (2015)]. We show that violation of the inequality only requires measuring over equivalence classes of mutually unbiased measurements on the trusted party and that in fact assuming a general two qubit system arbitrary pairs of distinct projective measurements at the trusted party are equally useful. Via this it is found that for a given state the maximum violation of our EPR-steering inequality is equal to that for the CHSH inequality, so all states that are EPR steerable with CHSH-type correlations are also Bell nonlocal.

29 30 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

Einstein-Podolsky-Rosen-steering (EPR) characterizes the apparent ability to nonlocally affect a quantum state, the central problem in the infamous EPR argument [13], that aimed to show that quantum mechanics is incomplete. That argument considered an entangled state shared between two distant parties, and proceeded to show that by measuring one or another of two non-commuting observables on the local system, the distant system is left in different possible sets of quantum states, an effect that Schrödingerlater termed “steering” [18]. These allow the experimenter to predict the result of measuring one or another of two non-commuting observables at the distant system. But since the systems no longer interact, EPR argued, the local choice of measurement cannot affect the “elements of reality” associated with the distant system. Thus both quantities should have simultaneous reality, which EPR believed would be described by a theory more complete than quantum mechanics. However, the possibility of such a local hidden-variable (LHV) description was ruled out by Bell in 1964 [10]. In 1989 Reid derived variance-inequalities that are violated with EPR correlations for continuous variable systems [19] and this was extended to discrete variables in [20]. Wiseman, Jones and Doherty (WJD) introduced a notion of steering as the inability to construct a local hidden state (LHS) model to explain the probabilities of measurement outcomes [21]. In quantum-information terms, EPR steering can be defined as the task for a referee to determine whether two parties share entanglement, when one of the parties is untrusted and using only classical communication [21]. Based on this, EPR-steering inequalities were defined in [25], with the property that violation of any such inequality implies steering. It was shown in [21] and [23] that the set of steerable states, that is states for which there exist local measurements that produce violation of a steering inequality, are strictly a subset of entangled states and a superset of states that violate a Bell inequality (Bell nonlocal states). In particular the set of Bell local Werner states that are unsteerable was found but no clear connection between the set of mixed steerable states and Bell nonlocal states has been determined. Experiments on entangled photon pairs [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] have produced violations of steering inequalities thus demonstrating the EPR paradox; in particular [26] reported loophole-free steering inequality violation, analogous to the much sought-after loophole-free Bell inequality violation, that was reported for the first time recently [71]. The WJD formalism has also had application in quantum information theoretic tasks such as one-sided device-independent quantum key distribution, quantum teleportation and subchannel discrimination [38, 72, 73]. Recently, the authors of [74] derived an EPR-steering analogue of the Clauser- Horne-Shimony-Holt (CHSH) inequality, that is, an inequality that is necessary and sufficient to demonstrate EPR steering in a scenario involving only correlations between two dichotomic measurements on each subsystem. However, this inequality requires that measurements by the trusted party (the “steered” party) be mutually unbiased. Here we produce a necessary and sufficient steering inequality in the same CHSH scenario as [74], that applies to any pair of projective measurements at the trusted party. This is presented in Sec. II with a full proof in Appendix A. 2.1. NECESSARY AND SUFFICIENT STEERING INEQUALITY 31

In Appendix B the set of unsteerable correlations for arbitrary dichotomic positive operator-valued measures (POVM’s) is found though a simple necessary and sufficient inequality cannot be constructed for this case. In Sec. III it is shown that the inequality is violated if and only if an inequality involving mutually unbiased measurements is also violated. This fact is used in Sec. IV to find the maximum violation of this EPR-steering inequality for a given bipartite state, which turns out to be equal to the maximum violation of the CHSH inequality for the given state as calculated in [75]. The inequalities have the same right hand side hence we find an equivalence between steering and nonlocality for this scenario. Thus the known distinction between the sets of Bell nonlocal and steerable states cannot be determined with CHSH-type correlations alone.

2.1 Necessary and Suﬃcient Steering Inequality

Here we develop the EPR-steering formalism, following the notation of [74], and develop the necessary and sufficient EPR-steering inequality for the CHSH scenario with a full proof in Appendix A. Through a similar process the boundary of the set of unsteerable correlations can be found for dichotomic POVM’s as we show in Appendix B. We have a pair of isolated systems, one at Alice and the other at Bob. We denote a measurement at Alice’s (Bob’s) system as A (B), chosen from a the set of observables Dα,(Dβ) in the Hilbert space of Alice’s (Bob’s) system, with outcomes labelled by a ∈ Lα(b ∈ Lβ). A state W shared between Alice and Bob is defined as Bell local or it has a LHV model if and only if it is the case that ∀a, b, A, B the joint probability distributions can be written in the form: X P (a, b|A, B; W ) = ℘(λ)℘(a|A, λ)℘(b|B, λ) (2.1) λ where ℘(λ) is a probability distribution over hidden variables λ ∈ Λ, ℘(a|A, λ) is the probability of outcome a for measurement A given λ, and likewise for ℘(b|B, λ). A state W is unsteerable or it has a local hidden variable – local hidden state (LHV-LHS) model if and only if all joint distributions have the form: X P (a, b|A, B; W ) = ℘(λ)℘(a|A, λ)P (b|B; ρλ) (2.2) λ where now it is further assumed that λ determines a local quantum state ρλ for Bob, and P (b|B; ρλ) is the quantum probability of outcome b if B is measured on ρλ. Since those probabilities are given by a quantum state, they must be constrained by uncertainty relations. This scenario has an operational meaning: Bob wishes to verify if W is entangled given joint distributions of outcomes between Alice’s and Bob’s measurements, but assuming only Bob’s outcomes are “trusted” as arising from quantum measurements. Here it is not possible to determine entanglement via state tomography as only Bob’s measurements are trusted, however showing that not all distributions can be 32 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO expressed as Eq. (2.2) is sufficient to show entanglement. The scenario in which both Alice’s and Bob’s measurements are untrusted would require testing the joint distributions for Bell nonlocality. The set of correlations in Eq. (2.2) forms a convex set [25] so we can express it in terms of its extreme points as: Z X B P (a, b|A, B) = dξ℘(χ, ξ)δa,f(A,χ)hψξ|Πb |ψξi (2.3) χ

B where Πb is a projector for outcome b of measurement B, χ is a parameter that determines all values of A via a function f(A, χ) and ξ determines a pure state ψξ for Bob. In constructing an EPR-steering inequality analogous to the CHSH inequality, we assume Alice and Bob can choose between two measurements {A, A0} and {B,B0} respectively, with possible outcomes a, b ∈ {1, −1}. We consider the ordered set of correlations (hABi, hA0Bi, hAB0i, hA0B0i) obtained in such an experiment, where hABi = P (a = b|A, B) − P (a = −b|A, B) and similarly for the other terms. These are the same correlations appearing in the CHSH inequality, and we want to ask what we can say about the steerability of a state using only this information. A LHV- LHS model can reproduce these correlations if and only if there exists a probability distribution ℘(χ, ξ) such that they can be expressed as: Z X A B hABi = dξ℘ (χ, ξ) 2p1 (χ) − 1 2p1 (ξ) − 1 , (2.4) χ

A B where p1 (χ) = ℘(1|A, χ) and p1 (ξ) = P (1|B, ψξ). For Alice there are four extreme values of χ, which we label as χ ∈ {1, 2, 3, 4} A A0 A A0 A A0 corresponding respectively to p1 = p1 = 1, p1 = 1 − p1 = 1, p1 = 1 − p1 = 0, A A0 0 p1 = p1 = 0. {B,B } are quantum projective measurements, which can be written B B as B = 2 Π1 −I, where Π1 is the projector onto the +1 eigenstate of B, and similarly 0 B B0 for B . Following [74], let µ =Tr{Π1 Π1 } and the possible pairs of probabilities B B0 B B p1 (ξ), p1 (ξ) with p1 (ξ) = hψξ|Π1 |ψξi form an ellipse, which can be parameterised as:

B 2p1 (ξ) − 1 = cos(ξ + β) (2.5) B0 2p1 (ξ) − 1 = cos(ξ − β). (2.6) √ √ where β = arctan 1 − µ/ µ with 0 ≤ β ≤ π/2. It turns out that if {B,B0} are B B0 dichotomic POVM’s then p1 (ξ), p1 (ξ) also form an ellipse, as we show in Appendix B, and so via a proof similar to that presented here the set of unsteerable correlations can be found for POVM’s but an analogous inequality does not exist. Varying ξ and χ, the possible values for the integrands in Eq. (2.4) for each correlation, are given 2.1. NECESSARY AND SUFFICIENT STEERING INEQUALITY 33 by: χ = 1 χ = 2 hABi cos(ξ + β) cos(ξ + β) hA0Bi cos(ξ + β) − cos(ξ + β) . (2.7) hAB0i cos(ξ − β) cos(ξ − β) hA0B0i cos(ξ − β) − cos(ξ − β) The correlations for χ = 3(4) can be obtained from those for χ = 1(2) by making ξ → ξ + π, and so it’s suﬃcient to consider χ = 1, 2. Thus the vector of correlations has a LHV-LHS model if and only if they can be written as a convex combination of the vectors given by the columns on Eq. (2.7). Let C1 be the convex hull of the χ = 1 column of Eq. (2.7) and C2 that for χ = 2. Then the set C of all vectors of correlations in which each correlation is of the form Eq. (2.4) is the convex hull of the union of C1 and C2. In the basis:

e1 = (1, 1, 0, 0)

e2 = (0, 0, 1, 1)

e3 = (1, −1, 0, 0)

e4 = (0, 0, 1, −1), (2.8) the vectors making up the boundaries of C1 and C2 have form cos(ξ + β)e1 + cos(ξ − β)e2 and cos(ξ + β)e3 + cos(ξ − β)e4 respectively. Now the curve (x,y) = (cos(ξ + β), cos(ξ − β)) is an ellipse which can also be expressed as x2 + y2 − 2xy cos(2β) = sin2(2β) (2.9)

This leads to the conjecture that for v = (v1, v2, v3, v4) in the basis {ei}, we have v ∈ C if and only if

1 q q v2 + v2 − 2v v cos(2β) + v2 + v2 − 2v v cos(2β) ≤ 1 (2.10) sin(2β) 1 2 1 2 3 4 1 2 In the original basis this is 1 √ √ ( u + u ) ≤ 2 (2.11) sin(2β) 1 2 where

0 2 0 0 2 0 0 0 u1 = h(A + A )Bi + h(A + A )B i − 2 cos(2β) h(A + A )Bi h(A + A )B i (2.12) 0 2 0 0 2 0 0 0 u2 = h(A − A )Bi + h(A − A )B i − 2 cos(2β) h(A − A )Bi h(A − A )B i (2.13) In other words, Eq. (2.11) is the necessary and suﬃcient inequality for the four correlations considered to have a LHV-LHS models, for arbitrary measurements on Bob’s side. It is thus an analog of the CHSH inequality for EPR steering. It reduces to Eq. (21) in [74] for β = π/4 which corresponds to µ = 0.5. The full proof of this conjecture is in Appendix A. 34 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

2.2 Equivalence Classes of Measurements

- Whilst arbitrary dichotomic projective measurements can be made on Bob’s side, there are actually equivalence classes of measurements B0, for ﬁxed A, A0, and B, for which all B0 in the same class result in the same left hand side of Eq. (2.11), as we will now show. Each equivalence class can be associated with a measurement mutually unbiased to B. Then optimising the inequality over measurements, for a given state, only requires optimising over mutually unbiased measurements by Bob. Bob’s measurements are trusted and thus in accordance with quantum mechanics they are Hermitian operators, and by convention have ±1 eigenvalues. B0 can then be expressed as √ B0 = (2µ − 1)B + 2 µp1 − µB00 (2.14) where B” is an operator mutually unbiased to B. Since cos(2β) = 2µ−1 and sin(2β) = √ √ 2 µ 1 − µ, we can rewrite u1 and u2 in Eq. (2.11) in terms of A, A’, B, B” as:

2 0 2 0 00 2 u1 = sin (2β) h(A + A )Bi + h(A + A )B i (2.15) 2 0 2 0 00 2 u2 = sin (2β) h(A − A )Bi + h(A − A )B i . (2.16)

Substituting in Eq. (2.11) we obtain:

q q h(A + A0)Bi2 + h(A + A0)B00i2 + h(A − A0)Bi2 + h(A − A0)B00i2 ≤ 2 (2.17)

This is equivalent to Eq. (2.11) for measurements A, A0,B,B00 with β = π/4, as should be for mutually unbiased measurements B,B00. Hence, if an arbitrary set of dichotomic variables {A, A0} by Alice and {B,B0} by Bob is measured and Bob’s measurements are trusted, the four correlations between variables {A, A0} and {B,B0} are consistent with a LHV-LHS model if and only if the four correlations between variables {A, A0} and {B,B00} are consistent with a LHV-LHS model, where B” is the mutually unbiased measurement to B determined √ √ by B00 = B0 − (2µ − 1)B/2 µ 1 − µ. So the demonstration of steering in this scenario implies violation of Eq. (2.11) for some pair of mutually unbiased measurements by Bob. Equation (2.11) implicitly contains µ as a variable, which depends on B and B’ set by the experimentalist, but the equivalent inequality (2.17) does not depend on µ. Given a B, each B’ is mapped to a particular B” (mutually unbiased to B) and the independence of µ means that each B” defines an equivalence class containing an infinity of B’ observables each mapped to B”. Then the inequality Eq. (2.11) for a particular B’ is not only equivalent to Eq. (2.17) but is equivalent to an infinity of inequalities involving the same A, A’,B and some B’ from the equivalence class to which B’ belongs. 2.3. STATES STEERABLE VIA CHSH-TYPE MEASUREMENTS ARE NON-LOCAL35

2.3 States Steerable via CHSH-type measurements are Non-local

- We now show that if a two-qubit quantum state violates the steering inequality (2.17) for some set of measurements then it also violates the CHSH inequality, possibly with another set of measurements. It was shown above that Eq. (2.17) is violated by some quantum state if and only if the steering inequality (2.11), for general measurements of the type in the CHSH scenario, is also violated. This means, since the inequality is necessary and suﬃcient, that a state demonstrates steering via general CHSH-type correlations if and only if it violates the CHSH inequality. Therefore all states that demonstrate steering via CHSH-type correlations are Bell nonlocal. Every bipartite state involving two qubits can be written in the form:

3 ! 1 X ρ = I ⊗ I + r · σ ⊗ I + I ⊗ s · σ + t σ ⊗ σ (2.18) 4 mn n m n,m=1

3 where, in the notation of [75], I is the identity operator, {σn}n=1 are Pauli matrices, 3 P3 r and s are vectors in R , and r · σ = i=1 riσi, tmn = Tr(ρσn ⊗ σm) forms a matrix denoted Tρ. We seek to find the maximum value of the steering inequality (2.17) for this state. Unlike the CHSH inequality this steering inequality is nonlinear so its left hand side can not be replaced by the expectation value of a single operator. Defining A = ba·σ, 0 0 0 0 0 0 3 A = ba · σ, B = b · σ, B = b · σ, where ba, ba , b, b are unit vectors in R , the left hand side of Eq. (2.17) can be written in the form: r r h i2 h i2 h i2 h i2 0 0 0 0 0 0 ESteer = b,Tρ (ba + ba ) + b ,Tρ (ba + ba ) + b,Tρ (ba − ba ) + b ,Tρ (ba − ba ) (2.19) 0 Defining orthonormal vectors bc, cb by:

0 ba + ba = 2 cos θbc 0 0 ba − ba = 2 sin θbc (2.20) where θ ∈ [0, π/2] we can express Eq. (2.19) as: q q 2 0 2 ESteer = 2 cos θ kTρbck + sin θ kTρbc k , (2.21) where Pythagoras’ Theorem has been used on the orthogonal components of Tρ (c) b 0 0 0 and Tρ cb in the b and bb directions. The disappearance of b, bb in the expression (2.21) shows that the left hand side of Eq. (2.17) is independent of measurements on Bob’s side, assuming the inequality applies to two qubits. Ultimately this means that verifying steering using two ﬁxed measurements on Alice’s side only requires choosing any pair of diﬀerent measurements on Bob’s side. 36 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

Maximising ESteer we ﬁnd: q q 2 0 2 max (ESteer) = max 2 cos θ kTρck + sin θ kTρc k 0 b b bc,bc ,θ q 2 0 2 = max 2 kTρck + kTρc k . (2.22) 0 b b bc,bc

0 In the last step above we maximise over angle θ keeping fixed bc, cb, and the optimal angle is 0 −1 kTρbcmaxk θmax = tan (2.23) kTρbcmaxk 0 2 0 2 where bcmax and bcmax are the vectors that maximise kTρbck + kTρbc k . This is exactly the maximum of the CHSH inequality calculated in [75]. Since both inequalities have a right hand side of 2, a state ρ violates our CHSH-type steering inequality if and only if it also violates the CHSH inequality, possibly for different sets of measurements. Explicitly, this violation occurs iff the sum of the squares of the largest eigenvalues η of Tρ is greater than 1 [75]. As√ an example, a Werner state W violates the CHSH inequality for η > ηCHSH = 1/ 2 [75], and therefore this result implies that it is steerable under CHSH correlations above this same threshold for η, confirming the result shown in [76] (however there it was not demonstrated that on Bob’s side only mutually unbiased bases need to be considered). This “equivalence” between steering and Bell nonlocality applies generally to dichotomic POVM measurements by Bob since the optimal measurements to show steering are some projective measurements (as a dichotomic POVM can be regarded as being a classically post-processed projective measurement [77]).

2.4 Discussion

The connection between steering and Bell nonlocality shown above is surprising since it was established in the seminal papers on the subject [21], [23] that steerable states are a strict subset of Bell nonlocal states. While all pure entangled states are Bell nonlocal, and hence also steerable, a strict hierarchy exists between entangled, steerable and Bell nonlocal mixed states in general. But we see that if CHSH- type correlations demonstrate that a state is steerable then that state must also be Bell nonlocal so in a sense the hierarchy is collapsed for these types of correlations. Furthermore the independence of the left-hand side of the steering inequality on Bob’s measurements means it is only necessary to vary over Alice’s measurements to verify that the state is Bell nonlocal using this technique. We also note that if we can demonstrate steering in one direction, then the state is Bell nonlocal, and therefore steering can also be demonstrated in the other direction. Thus there is no one-way steering in this scenario, in the sense of [24]. It was shown in [30] both theoretically and via an optical experiment that there exist Bell local Werner states which violate a steering inequality involving three dichotomic 2.4. DISCUSSION 37 measurements on either side. More recently, Bowles et al. [78] have shown that some Bell local states are one-way steerable with two projective measurements at the untrusted site and tomographic measurements on the trusted site. The present results would suggest that indeed at least three measurements at the trusted site are required for one-way steering. However, our inequality is necessary and sufficient when using the correlation data only—it is known that for some two-qubit states which are not detected by this inequality, steering can be detected using also the information about the marginals, not available from only the correlations [79]. The question then becomes whether some of those steerable states are also Bell local and one-way steerable. It would be interesting to derive an inequality that is necessary and sufficient for this general case where the marginals are also taken into account. Further work could explore necessary and sufficient steering inequalities that involve more than two measurement variables on each party or more than two outcomes for each measurement. The results above provide a partial answer to this fundamental question: for a given measurement scenario (i.e. number of parties, settings and outcomes) what are the optimal measurements to verify if an arbitrary quantum state is steerable? For the CHSH-type scenario the answer to the question above is: any pair of distinct arbitrarily chosen measurements on Bob’s side and the measurements made 0 by Alice corresponding to bamax and ab max constructed from θmax and the bcmax and 2 0 2 0 cb max that maximise kTρbck + Tρcb (eigenvectors of TρT [75]). Recently in [76] a computational optimisation over measurements on both sides was performed to find the maximum violation of Eq. (2.17) over the space of bipartite pure entangled qubits and amount of violation of the inequality over a class of Werner states, but we now see that it would have been sufficient to keep measurements on Bob’s side fixed. This independence on Bob’s measurements removes a major challenge in achieving practical nonlocality witnesses; for example in [32] the demonstration of steering inequality violation required substantial steps to account for non-mutually unbiased measurements. For pure states the connection between joint measurability (compatibility) of Al- ice’s observables and Bell nonlocality was examined in [80] and recently extended to steering in [81, 82]. The latter works suggest that two measurements at Alice are incompatible if and only if they can be used to demonstrate steering, while the former work suggests that two incompatible measurements enable CHSH inequality violation. The findings in our paper also show a kind of independence from measurements on Bob’s side (they are only required to be incompatible) and it is interesting that our approach via steering inequalities gives similar insights to their derivations using a reduced-state/“assemblage” picture. But as we allow the state to be mixed, so we include entangled unsteerable states, the criterion on Alice’s measurements to demonstrate steering with CHSH correlations is stronger than incompatibility, specifically that Eq. (2.22) must be larger than 2. In conclusion we have produced a general necessary and sufficient steering inequality for CHSH-type correlations on two qubits. The violation of the inequality for a given set of measurements implies the violation of an inequality with mutually 38 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO unbiased measurements on the trusted side hence only mutually unbiased measurements need to be examined, as in [25]. Interestingly, we are then able to prove that if any bipartite state is shown to be steerable via such measurements then it is also Bell nonlocal. Future work in this direction would find necessary and sufficient inequalities for more than two parties and several POVM’s on each site to further illuminate the differences between steerable and Bell nonlocal states. Can the distinction be shown with two d-outcome measurements, or do we need three measurements? What minimum measurements are required to demonstrate this distinction for higher-dimensional bipartite systems? Note added in proof. Recently [83] found a related result in the restricted context where Bob’s measurements used to test steering are mutually unbiased. This, however, leaves open the possibility that arbitrary qubit measurements by Bob can discriminate steering from Bell nonlocality. We establish the equivalence between steering and Bell nonlocality for the most general CHSH scenario, that is, two dichotomic measurements by both parties. More recently, the authors of [84] also showed this steering-Bell nonlocality equivalence but only for T states, i.e. states in which r = s = 0 in equation Eq. (2.18), so that Alice and Bob’s reduced states are mixed states. They also gave a geometric meaning of the maximum inequality violation in terms of the steering ellipsoid. Similarly to our paper they extended it to arbitrary states in [85], but without deriving a generalised steering inequality.

ACKNOWLEDGMENTS

The authors acknowledge Curtis Broadbent for prompting the question on connecting Bell nonlocality and steering, and useful discussions and feedback from Cyril Branciard, Andrew Doherty, Michael Hall and Howard Wiseman. We also thank an anonymous referee for suggesting further research directions based on this work. P.G. acknowledges support from the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), Project No. CE110001013.

2.5 Appendix A

We prove here a theorem that results in the necessary and suﬃcient EPR-steering inequality for the CHSH scenario, that is Eq. (2.11). Theorem: Let C1, C2 be convex sets in four dimensions and in separate planes spanned by the axes i.e. C1 ⊆ span(e1, e2) and C2 ⊆ span(e3, e4) where e1, e2, e3, e4 are basis vectors of four-dimensional space. If the boundaries of the sets are conic sections represented by Cartesian equations that only contain quadratic terms i.e. 2 2 C1 can be described as f(x1, x2) = ax1 + bx2 + cx1x2 ≤ r1 and C2 as g(x3, x4) = 0 2 0 2 0 a x3 + b x4 + c x3x4 ≤ r2, then the convex hull C of C1 and C2 has the form p p √ √ f(v1, v2) + f(v3, v4) ≤ max[ r1, r2] (2.24) Proof: 2.5. APPENDIX A 39

Let h1 = (v1, v2, 0, 0), h2 = (0, 0, v3, v4) and v = (v1, v2, v3, v4) = h1 + h2. If v is in the convex hull of C1 and C2 then v = p1w1 + p2w2 where p1 + p2 = 1 and w1 lies in C1 and w2 lies in C2. Hence, with the assumption that C1 ⊆ span(e1, e2) and C2 ⊆ span(e3, e4) where e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), e4 = (0, 0, 0, 1):

h1 w1 = (2.25) p1

h2 w2 = (2.26) p2

And as w1 lies in C1 and w2 lies in C2: v1 v2 f , ≤ r1 (2.27) p1 p1 v3 v4 g , ≤ r2 (2.28) p2 p2

Since f(v1, v2) and g(v3, v4) only contain quadratic terms this implies: 1 2 f(v1, v2) ≤ r1 (2.29) p1

1 2 g(v3, v4) ≤ r2 (2.30) p2 Putting these together we get: p p √ √ f(v1, v2) + f(v3, v4) ≤ p1 r1 + p2 r2 (2.31) √ √ ≤ max[ r1, r2] (2.32) where p1 +p2 = 1 has been used in the last line. Applying this to our situation where the boundaries for both C1 and C2 have the form of Eq. (2.9) we obtain equation Eq. (2.10) as desired. In Appendix B of [74] a proof is provided that only points in C satisfy the LHV- LHS inequality in that paper. The proof can be applied in whole to inequality (2.10) since the inequality satisﬁes the properties crucial to the proof: it is of the form f(v) ≤ 1 where f(v) is a convex function and its upper bound of 1 is obtained for points v ∈ C that can be expressed as a convex combination of a point on the boundary ∂C1 of C1 with a point on the boundary ∂C2 of C2. The latter statement is seen from the derivation above since the inequality (2.10) achieves the bound 1 if and only if f (v1/p1, v2/p1) = r1 and g (v3/p2, v4/p2) = r2, i.e., w1, w2 lie on the boundaries of C1 and C2 respectively. Hence only points in C satisfy Eq. (2.10), which in the measurement basis is Eq. (2.11). Thus Eq. (2.11) is indeed the necessary and suﬃcient EPR-steering inequality for arbitrary measurements in the CHSH scenario. 40 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

2.6 Appendix B

We examine here the case where B and B’ are dichotomic POVM’s measured by Bob. An arbitrary dichotomic POVM element associated with outcome 1 for observable B can be expressed as:

E1|B = λ1|B|1ih1| + λ2|B|2ih2| (2.33)

= kB|1ih1| + λ2|BI (2.34) where 0 ≤ λ2|B ≤ λ1|B ≤ 1 are eigenvalues of E1|B, |1i and |2i are corresponding orthonormal eigenstates and kB = (λ1|B − λ2|B). Likewise for observable B’:

0 0 0 E1|B0 = kB|1 ih1 | + λ2|B0 I (2.35)

|ψi = pµ0|1i + p1 − µ0eiφ0 |2i (2.37)

Then the probability p(1|B) of outcome 1 when measuring B is:

B p1 = ψ|E1|B|ψ 0 = kBµ + λ2|B (2.38) so that: p(1|B) − λ µ0 = 2|B (2.39) kB Then for B’ the probability of outcome 1 is:

B0 p1 = ψ|E1|B0 |ψ 0 0 2 = kB| hψ|1 i | + λ2|B0 h i 0 0 0 p 0 0 0 = kB µ µ + (1 − µ ) (1 − µ) + 2 µ (1 − µ )µ(1 − µ) cos(φ − φ) + λ2|B0 0 kB B B = λ2|B0 + p1 − λ2|B µ + kB − p1 − λ2|B (1 − µ) kB q B B 0 +2 p1 − λ2|B kB − p1 − λ2|B µ (1 − µ) cos (φ − φ) (2.40)

B0 B 0 Now let y = p1 , x = p1 , α = λ2|B0 , β = kB/kB, γ = λ2|B, δ = kB + λ2|B. The B0 B 0 boundary of the curve p1 versus p1 according to Eq. (2.40) is achieved with cos(φ − φ) = ±1 i.e. cos(φ0 − φ)2 = 1. Then the boundary has the form:

y = α + β [(x − γ)µ + (δ − x)(1 − µ) 2.6. APPENDIX B 41

i ±2p(x − γ)(δ − x)µ(1 − µ) cos(φ0 − φ) = α + r(x − γ) + s(δ − x) ± tp(x − γ)(δ − x) (2.41) where r = βµ, s = β(1 − µ), t = 2βpµ(1 − µ). Then rearranging:

(s − r)2 + t2 x2 + 2(s − r)xy + y2 + 2(s − r)(rγ − sδ − α) − t2(δ + γ) x + 2 [rγ − sδ − α] y + (rγ − sδ − α)2 + t2γδ = 0 (2.42) let:

A = (s − r)2 + t2 = β2(1 − 2µ)2 + 4β2µ(1 − µ) = β2 (2.43) B = (s − r) = β(1 − 2µ) (2.44) C = 1 (2.45) t2(δ + γ) D = (s − r)(rγ − sδ − α) − (2.46) 2 F = rγ − sδ − α (2.47) G = (rγ − sδ − α)2 + t2γδ (2.48)

So Eq. (2.42) is in the form:

Ax2 + 2Bxy + Cy2 + 2Dx + 2F y + G = 0 (2.49)

B B0 Equation (2.49) describes an ellipse in terms of the variables x, y i.e., p1 , p1 . We will now calculate its semi-axis lengths, centre and counterclockwise angle of rotation from the x axis to the major axis based on the formulas in [86]. XC , the x coordinate of the ellipse centre, is: CD − BF X = C B2 − AC −t2(δ+γ) = 2 (s − r)2 − [(s − r)2 + t2] 1 = (δ + γ) 2 k = B + λ (2.50) 2 2|B

For projective measurements kB = 1, λ2|B = 0 which implies XC = 1/2 as expected. YC , the x coordinate of the ellipse centre, is: AF − BD Y = C B2 − AC 42 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

2 t2(rγ − sδ − α) + (s−r)(δ+γ)t = 2 (s − r)2 − ((s − r)2 + t2) (r + s)(γ − δ) = − − α 2 0 kB = + λ 0 (2.51) 2 2|B

0 For projective measurements kB = 1, λ2|B0 = 0 which implies YC = 1/2 as expected. The semi-axis lengths are given by: v u 2(AF 2 + CD2 + GB2 − 2BDF − ACG) u a± = t h i (2.52) (B2 − AC) ±p(A − C)2 + 4B2 − (A + C)

Now we can write:

A = B2 + t2 (2.53) t2l D = BF − (2.54) 2 G = F 2 + t2l0 (2.55) where l = δ + γ, l0 = δγ. The numerator under the square root of Eq. (2.52) is then: " t2l2 2 B2 + t2 F 2 + BF − + F 2 + t2l0 B2 2 t2l −2BF BF − − B2 + t2 F 2 + t2l0 2 l2 = 2t4 − l0 4 " # (δ + γ)2 = 2t4 − δγ 4 4 2 = 2t kB (2.56) And the denominator under the square root of Eq. (2.52) is: q B2 − B2 + t2 ± (B2 + t2 − 1)2 + 4B2 − B2 + t2 + 1 q = −t2 ± (β2 + 1)2 + 16β2µ (µ − 1) − β2 + 1  v  u" 2 #2 2 " 2 #  u k0 k0 k0  = −t2 ±t B + 1 + 16 B µ (µ − 1) − B + 1 (2.57) k k k  B B B  2.6. APPENDIX B 43

Hence, substituting the value for t:

2 r kB p −2 a± = 2 0 µ(1 − µ) (2.58) kB S where: v u" 2 # 2 " 2 # u k0 k0 k0 S = ±t B + 1 a2 + 16 B µ (µ − 1) − B + 1 (2.59) kB kB kB

0 For projective measurements kB = kB = 1 so: s p −2 a± = 2 µ(1 − µ) ±2p1 + 4µ(µ − 1) − 2 s −1 = 2pµ(1 − µ) (2.60) ± |2µ − 1| − 1 p If µ = 0.5 then a± = −1/ − 1 = 1 as expected. The counterclockwise angle of rotation from the x axis to the major axis is: 1 A − C φ = cot−1 2 2B 1 β2 − 1 = cot−1 2 2β(1 − 2µ) 02 2 1 −1 kB − kB = cot 0 (2.61) 2 2kBkB(1 − 2µ) For projective measurements and µ 6= 0.5 we get φ = cot−1(0)/2 = π/4. The general parametric form of the ellipse in terms of the above is:

x = XC + a cos(ξ) cos(φ) − b sin(ξ) sin(φ)

= XC + T cos(ξ + κ) (2.62)

y = YC + a cos(ξ) sin(φ) + b sin(ξ) cos(φ) 0 0 = YC + T cos(ξ + κ ) (2.63) where q T = a2 cos2(φ) + b2 sin2(φ) (2.64) b κ = tan−1 tan(φ) (2.65) a q T 0 = a2 sin2(φ) + b2 cos2(φ) (2.66) ha i κ0 = tan−1 tan(φ) (2.67) b 44 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO

This implies: 1 2pB(ξ) − 1 = 2 X + T cos(ξ + κ) − (2.68) 1 C 2 0 1 2pB (ξ) − 1 = 2 Y + T 0 cos(ξ + κ0) − (2.69) 1 C 2

Then the vectors making up the boundaries of C1 to C4 in the basis (2.8) have the form: 1 1 C : 2 X + T cos(ξ + κ) − e + 2 Y + T 0 cos(ξ + κ0) − e (2.70) 1 C 2 1 C 2 2 1 1 C : 2 X + T cos(ξ + κ) − e + 2 Y + T 0 cos(ξ + κ0) − e (2.71) 2 C 2 3 C 2 4 1 1 C : − 2 X + T cos(ξ + κ) − e + 2 Y + T 0 cos(ξ + κ0) − e (2.72) 3 C 2 1 C 2 2 1 1 C : − 2 X + T cos(ξ + κ) − e + 2 Y + T 0 cos(ξ + κ0) − e (2.73) 4 C 2 3 C 2 4 Each of these correlation boundaries are elliptical and for the projective case (A = XC = 1/2), C1 and C3 reduce to cos(ξ + κ)e1 + cos(ξ − κ)e2 and C2 and C4 to cos(ξ + κ)e3 + cos(ξ − κ)e4 as we have seen before. The equation of the boundary of B C1 and C3 can be found as follows: Let m = 2p1 (ξ) − 1 = 2x − 1 i.e., x = m + 1/2 B0 and n = 2p1 (ξ) − 1 = 2y − 1, i.e., y = n + 1/2. Then from Eq. (2.49) we get m + 12 m + 1 n + 1 n + 12 A + 2B + C 2 2 2 2 m + 1 n + 1 + 2D + 2F + G = 0 (2.74) 2 2 So, A C B A + B B + C m2 + n2 + mn + + D m + + F n 4 4 2 2 2 A + C B = − + + D + F + G (2.75) 4 2

For projective measurements this reduces to Eq. (2.9). The equation for C2 and C4 involves replacing m by −m and n by −n. For the general POVM case C1 and C3 lie in the same plane but are distinct sets as with C2 and C4. The convex hull C of the sets is the convex hull of C5 and C6 where C5 is the convex hull of C1 and C3 and C6 is the convex hull of C2 and C4. The boundary of C5 consists of the two outer common tangents to the ellipses C1 and C3 and the outer arcs of the ellipses that connect with the tangents, and C6 has the same equation for its boundary as C5 (but in an orthogonal plane). The boundary is piecewise deﬁned so that C cannot be expressed as a simple inequality for the general POVM scenario. 2.6. APPENDIX B 45 46 2. EPR-STEERING AND BELL NONLOCALITY IN THE CHSH SCENARIO Chapter 3

Probing Modiﬁed Commutators with Quantum Noise

3.1 Introduction

Quantum mechanics and the general theory of relativity have both been successful at explaining most of our observations of the universe, the former is an accurate framework for predictions of phenomena at the microscopic scale while the latter explains gravity at the macroscopic scale. However quantising gravity in the standard way has well-known difficulties due to the non-renormalizability of the gravitational interaction [2]. Furthermore the combined strong quantum and gravitational effects expected at the Planck scale, and also around black hole horizons, have brought the postulates of these theories into question[87][88] [89]. One attempt to resolve this dilemma is to introduce the concept of a ‘minimal length’ of space, challenging the paradigm from relativity that space time is contin- ~ uous [63, 90]. According to quantum mechanics the uncertainty principle ∆x∆p ≥ 2 provides no restriction on measuring either the position or momentum of a particle with arbitrary precision (where precision is based on the spread of outcomes over many repeated measurements), though not both. But if there is an effective minimal length then it may mean that there is a fundamental limit to the precision of a position measurement. This is distinguished from the limit on precision measurements of position and momentum which follows from special relativity applied to single-particle quantum mechanics[91]. An effective minimal length can be accounted for by adding terms dependent on the variance of momentum on the right hand side of the standard uncertainty principle. Such generalised uncertainty principles (GUP’s) have been argued to exist in string theory[62, 92] and the finite bandwidth approach to quantum gravity[93, 64] but also follow from thought experiments involving quantum physics applied to black holes[94], relative locality[95] and double special relativity[65]. But how does one find evidence for minimal length? Or rule out specific pos- sibilities, such as modified commutators, based on experimental data? On top of the theoretical difficulties in quantising gravity it is generally thought that experiments that can probe quantum gravity effects directly require energies much higher

47 48 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE than currently available. Most effort has been devoted to find indirect signatures of quantum gravity in high energy particle collisions or astrophysics e.g. supersymmetry [96, 97] and modified gamma ray burst dispersion relations [98]. Though no direct evidence for quantum gravity has been found, constraints on modifications of the standard position-momentum commutation relation (equivalent to GUP’s), have been derived from a variety of sources and offer a novel route to test quantum gravity since they may be probed via precision measurements in low-energy experiments. Such modified relations are also motivated by the fact that they gen- eralise quantum mechanics without compromising certain physical principles. Other theories like extra space dimensions, axionic dark matter, dark energy models and semi-classical gravity, can also been constrained by table top experiments involving torsion balances[99], induced optomechanical interactions[100, 101, 102] and atom interferometry[103, 104]. The possibility of using optomechanics to constrain GUP’s was considered in [67]. There they showed that controlled interactions of optical pulses with a mechanical oscillator could lead to dependence of the output optical field on the mechanical canonical commutation relation. It was suggested that the ability to measure the average value of this field to high precision can lead to significant improvements in probing the GUP. Other proposals to probe GUP’s in the dynamics of oscillators were considered in [68] and [69]. In this spirit, in this chapter we investigate how modifying the standard commutation relation, which gives rise to a GUP, affects macroscopic oscillator motion in a noise bath environment that is typical of sensitive tabletop and interferometric experiments. We find to first order the explicit modified noise spectrum associated with Brownian motion and quantum radiation pressure noise if the oscillator is driven by an optical source. This results in constraints on the GUP via the overall noise spectrum observed in Advanced LIGO (aLIGO) and recent experiments that have reported observation of quantum radiation pressure noise on mechanical membranes. We find that current constraints from the spectra close to oscillator resonance frequency or at frequencies in a free-mass limit are comparable to the best available to date. These and related experimental scenarios are optimised to show that by adjusting optomechanical parameters by a few orders of magnitude or with additional external driving of the high quality factor oscillator new bounds are feasible.

3.2 Background

The uncertainty principle can be generalised in several ways to incorporate minimal length. If this is obtained from the Robertson uncertainty relation ∆x∆p ≥ 1 2 | h[x, p]i | then the existence of a minimal length can be loosely translated into the question: is the canonical commutation relation modified? Note that the existence of an absolute minimal length is in apparent conflict with Lorentz invariance. This poses very serious challenges to a consistent theory of quantum gravity, see for example the discussion here [105]. There are also very strong experimental bounds on violations of Lorentz invariance in many physical 3.2. BACKGROUND 49 systems, see for example [106, 107, 108]. In this chapter we are just interested in how strongly we can hope to bound the modified commutators that arise in these theories. However, see [109] for a recent discussion on reconciling minimal length with relativity via covariant band-limitation. Several modified commutation relations have been considered in the literature. For example, the simplest modified isotropic (direction symmetric) translational invariant commutator is [61]:

2! p 2 [x, p] = i~ 1 + β0 = i~ 1 + Ap (3.1) MP c

2 2 where ~ = h/2π and A = β0/MP c . A closely related uncertainty principle has been discussed as a caricature of the situation in string theory [62]:

~ β0 2 ∆x∆p ≥ 1 + 2 ∆p . (3.2) 2 (MP c) √ This inequality implies that ∆x ≥ LP β0. So if β0 = 1 then the minimal length is the Planck length. This GUP is closely related to the modified commutator model (3.1), at least when hpi = 0 after a straightforward application of the Robertson uncertainty relation. Reference [62] gives a qualitative explanation of how this arises from T- duality and a consideration of the expected behaviour of a Heisenberg microscope in the framework of string theory which, in short, implies that an effective minimal length emerges from the equivalence of high energy (small radius) and low energy (large radius) string physics. This uncertainty principle may also be arrived via a theory involving intrinsic uncertainty of spatial translations[110]. Maggiore [111] considered the problem of deriving the most general commutation relations provided that the angular momenta satisfy usual SU(2) commutation relations and satisfy usual commutation relations with the position and momenta operators, momenta commute between themselves so the translation group is not deformed, and the commutation relations depend on a parameter with dimensions of mass so that in a limit the standard relations result. The one-dimensional canonical commutator then has a unique form: s p 2 2 c + m [x, p] = i~ 1 + 2µ0 2 (3.3) MP where µ0 is a free numerical parameter, c is the speed of light, m is the mass of the particle and MP is the Planck mass. This shows that given some well-motivated constraints a consistent modification of quantum mechanics of this type is possible. Being a non-Lie algebra it also avoids certain no-go theorems [112]. In the limit of sufficiently small µ0, m, and momentum this model for a modified commutator reduces to the previous one. 50 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

Another model for modiﬁed commutators arose in an approach to quantum gravity known as ‘doubly special relativity’ theory [65]. There it is proposed that the Lorentz transformations are modiﬁed in order to keep an invariant energy-momentum scale and maximal possible momentum. In [113] the following commutator was given consistent with doubly special relativity:

2! p 2 p 2 2 [x, p] = i~ 1 − δ0 + δ0 = i~ 1 − Cp + C p (3.4) MP c MP c where Mp is Planck mass, c is speed of light, δ0 is a numerical parameter that quantifies interaction, C = δ0/MP c. In this work we will focus mainly on the β-modified commutator or (3.1), although our techniques are relevant to all these examples. This is to streamline the discussion and because bounds on β0 are typically much weaker than in the other scenarios. Constraints on β0 have been inferred from a variety of experimental observations. Some of the best constraints on β0 discussed in the literature are shown in table 3.1. The general reason for the large order of magnitudes of the upper bounds is that the modifications are applied in these experiments to particles with momenta much smaller than the Planck momentum MP c that appears in the right hand side of each commutator and the parameters being probed constrain the size of the minimal length in Planck length units.

Table 3.1: Experimental bounds on β0

Experiment Max β0 Lamb shift [114] 1036 High energy particle collisions [114] 1032 AURIGA detector [115] 1033 Scanning tunnelling microscope [114, 67] 1033 Harmonic oscillators (frequency shifts) [68, 69] 106 − 1020

One important feature of these various bounds on β0 is whether the experiments concern the modification of the commutators of macroscopic systems, or of elementary particles. The standard canonical commutator [x, p] = i~ is component-wise scale invariant in the sense that if it applies to components of an object then it also applies to the object’s centre of mass. However this is not true in general for modified commutators [67, 116]. If we assume that the modified commutator applies to elementary particles then it is necessary to make some further assumptions about the model in order to determine how the commutation relations of a composite system should be determined. The relationship between the modification parameter e.g. β0 for the centre of mass of N identical systems is derived in [66] to have approximately 2 1/N scaling with respect to the corresponding effective modification parameter βe for a single system obeying (3.1), up to a term involving the relative momenta of the systems. We presume from now on that the system of interest associated with βe is 3.2. BACKGROUND 51 an elementary particle. If this scaling is applied, bounds on β for individual systems are greatly weakened. For example, this was applied to significantly weaken a constraint on β reported in [117] from analysis of planetary motion. The constraints in [115], [68] and [69] also apply to macroscopic systems and are thus also weakened by this component-dependent factor. However in 3.1 the bounds from measurements of Lamb Shift, high energy particle collisions (we use the updated value of 13 TeV as in the Large Hadron Collider) as well as electron tunnelling directly refer to elementary particles. Even some heuristic calculations of βe from gravitational phenomena have been attempted, given values of order 1 [118, 119]. The possibility of other scaling behaviour of the bounds as a function of number of components was recently studied [120]. As we discuss in more detail in the following we will report bounds on both β0 and on βe. As mentioned earlier, [67] studied how quantum optical control and readout of a mechanical oscillator, which would have a relatively large momentum compared to experiments on subatomic particles, can be used to significantly improve the constraints on these parameters. They show that the challenge is to perform a large number of measurement runs in which each run involves measuring a large photon number coherent state interacting with an optically cooled macro-oscillator in a cavity of high finesse, but in principle if the commutators are applied to the centre of mass the parameters can be constrained to the Planck scale i.e. β0, µ0, δ0 ∼ 1. In recent experiments it was shown how other techniques probing the mean or variance of the position of harmonic oscillators lead to new bounds on the parameters. In [68] and more recently [69] the commutator (3.1) was associated with perturbation of the resonance frequency of a harmonic oscillator by an amount dependent on the initial amplitude of motion (‘amplitude-frequency effect’). Also the existence of third-harmonics was discussed in [68]. From these techniques the bounds at the bottom of table 3.1 were inferred. In this work we similarly explore the potential of macroscopic oscillators to constrain the parameters in these modified commutator theories with a focus on the modified quantum noise that an oscillator would experience due to its interaction with an environment. The simplest scenario is interaction with a thermal bath but by adding a radiation field an optomechanical interaction arises and so the oscillator ex- periences radiation pressure noise (on top of a mean classical radiation pressure)[121]. The cavity light field can be used to read out the position of the oscillator and the noise on this signal could potentially provide bounds on β0. This is a promising route to probe the canonical commutator as highly sensitive position measurements of mechanical oscillators can be performed experimentally for a very wide range of masses. For example, radiation pressure noise has recently been observed on an oscillator [122] and is expected to have a significant presence in LIGO interferometers in the near future [123]. A somewhat related earlier idea from [124] and [125] was to exploit the sensitivity of gravitational wave interferometers to probe heuristic models of quantum foam, see also [126, 127]. Also an experiment has been conducted attempting to find position variation correlations, associated with other theories of quantum gravity, between 52 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE two interferometers[128]. A closely related idea was discussed in [129] where the commutation relations of canonical operators of optical fields were modified to infer perturbed radiation pressure and shot noise in interferometers. Strong bounds on the commutators could be derived in this case. In this chapter we will only focus on the modification of commutation relations of massive particles and not on modified theories of electrodynamics. So in our setup only the commutators of the mechanical oscillator are modified.

3.3 Modiﬁed Noise Spectrum

In this section we use a perturbation expansion to derive the mechanical noise spectrum for an oscillator driven by thermal and radiation pressure noise with the modi- ﬁed canonical commutator (3.1). We proceed by an analysis of the quantum Langevin equations for this system.

3.3.1 Setup We consider a typical optomechanical system, as discussed in [121] for example. The system is composed of an optical field interacting with a mechanical oscillator inside a cavity and can be associated with the Hamiltonian: 1 p2 H = ω a†a + mΩ2x2 − Ga†ax + (3.5) ~ c 2 ~ 2m Here a, a† are creation and annihilation operators for the optical field, and x, p are the position and momentum operators for the mechanical oscillator. ωc is the optical cavity angular frequency, m is the mass of the oscillator, Ω is its resonance angular frequency, and G = ωc/L is a optomechanical coupling constant between oscillator and field. In this model we assume that only the mechanical commutators are modified, for the optical commutators we have [a, a†] = 1 as usual. Let the canonical commutation relation be that in equation (3.1). We wish to repeat a typical analysis of this optomechanical system in the limit that A is small. While there are various approaches to this, we will proceed by rewriting the Hamil- tonian (3.5) in terms of a modified momentum operatorp ˜ that satisfies the usual commutation relations [x, p˜] = i~. This will allow us to perform the calculation using standard perturbation theory. Since it is anticipated that the modification to the commutator is very small we will calculate only to first order in A. The modified momentum operatorp ˜ is defined by the following equation

p˜ = p − Ap3/3. (3.6)

It is easy to see that [x, p˜] = i~ up to first order in A [66]. Rewriting the Hamiltonian (3.5) in terms ofp ˜ and retaining terms up to first order in A we find

1 p˜2 Ap˜4 H ' ω a†a + mΩ2x2 − Ga†ax + + (3.7) ~ c 2 ~ 2m 3m 3.3. MODIFIED NOISE SPECTRUM 53

Thus the modification of the momentum operator effectively results in a potential term V = Ap˜4/3m. This bears similarity with the Duffing oscillator, which has a potential quartic in position, and assuming the dynamical equations of quantum mechanics are unaffected in a theory with a GUP it implies the equation of motion will be non-linear in position. We will assume that the mechanical oscillator is also coupled to a thermal bath with damping rate γ and temperature T . We adopt the standard approach of [121], linearising the optical field around the mean field α: a = α + δa. α describes the steady-state field amplitude in the cavity and depends on the drive power P and the optical decay rate κ. We will assume throughout that the cavity is driven on resonance. Without loss of generality we can choose α to be real. We keep only terms linear in δa, (δa)†. Furthermore we apply linear perturbation theory in A to position and momentum: x ' x0 + δx where x0 is the solution for x from the Hamiltonian with standard commutator (A = 0) and δx is the perturbation to first-order in A, and similarlyp ˜ ' p0 + δp. Writing the Langevin equations for the mechanics alone we have

˙ 3 x˙ ' (x0 + δx) = (p0 + δp) /m + 4Ap0/3m (3.8) ˙ 2 ˙ p˜˙ ' (p0 + δp) = −mΩ (x0 + δx) − γm(x0 + δx) + f (3.9)

The total driving force is f = fT + fD where fT is the stochastic thermal force and fD describes the other forces on the oscillator, including the fluctuations in the † optomechanical force ~Gα δa + δa . ˙ Note that the form of the damping term γm(x0 + δx), called ‘viscous damping’, follows from the usual derivation as in [130] that involves Heisenberg equations of motion under a Hamiltonian that contains kinetic and potential terms for the system oscillator and a thermal bath of harmonic oscillators weakly coupled to the system, provided that the modification to commutation relations of bath oscillators has negligible influence compared to that on the system. Unlike the standard case this term is inequivalent to γp. Another common model of damping for mechanical modes, known as ‘structural damping’, can be handled heuristically by making γ frequency-dependent, see 3.6.4 in the Appendix for details. Thus for x0, p0 we obtain the system of equations:

−1 x˙0 0 m x0 0 = 2 + (3.10) p˙0 −mΩ −γ p0 f and for the perturbative terms:

˙ −1 3 δx 0 m δx 4Ap0/3m ˙ = 2 + 3 (3.11) δp −mΩ −γ δp −4Aγp0/3

In the following we will be integrating these equations of motion starting from a thermal state at t = 0 and we will make the choice δx(0) = 0 = δp(0). We will assume the oscillator is driven only by noise arising from thermal and radi- 0 00 2 2 2 −κ|t0−t00|/2 ation pressure. The radiation pressure force obeys hfrad(t )frad(t )i = ~ G α e . 54 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

See 3.6.5 in the Appendix for details on the derivation. We are working in the regime kBT ~Ω in which the thermal force is well approximated by white noise 0 00 0 00 [130] hfT (t ) fT (t )i = 2kBT γmδ (t − t ), where kB is Boltzmann’s constant. The analysis can be simplified further if the optical cavity decay rate is sufficiently large κ Gα, Ω, γ. Then the radiation contribution may also be approximated well by 0 00 2 2 2 0 00 a white noise hfrad (t ) frad (t )i = (4~ G α /κ) δ (t − t ). The average number of 2 photons in the cavity is ncav = α = 4P/κ~ωc. We can also express the optomechanical coupling for a Fabry-Perot cavity as G = ωc/L where L is the length of the cavity. Thus we can re-express the radiation pressure white noise coefficient as 2 2 16~ωcP/L κ . When the mechanical system is in steady state we wish to study the power spectrum of position fluctuations Z ∞ iωt Sxx(ω) = hx(t)x(0)ie dt. (3.12) −∞ Since we are studying a quantum mechanical theory, [x(t), x(0)] 6= 0 and consequently Sxx(ω) 6= Sxx(−ω). We will mainly focus on the symmetrised noise spectrum S(ω) = [Sxx(ω) + Sxx(−ω)]/2 [131, 121]. This is because the noise spectrum of an optomechanical position is just S(ω) plus a contribution due to the optical shot noise, see for example [121]. As a result we need to calculate

hx (τ) x (0)i = h[x0 (τ) + δx (τ)] [x0 (0) + δx (0)]i

' hx0 (τ) x0 (0)i + hδx (τ) x0 (0)i (3.13)

The second expression arises as the initial condition is x0 = x0(0) = 0. The term hδx (τ) δx (0)i is neglected since it is second order in A. We can use solutions of the above systems of equations (3.10) and (3.11) to compute the required correlation functions. Detailed calculations are in the Appendix. In both terms the correlators are averaged over the steady state, which does depend on the modified Hamiltonian. However since δx (τ) is itself first order in A it is sufficient to compute hδx (τ) x0 (0)i over the unmodified steady state. In the next sections we analyse the noise spectrum of the oscillator to find constraints on β0.

3.3.2 Noise spectrum We find that the first-order modification to the oscillator noise spectrum may have significant contributions from both terms in equation (3.13). As shown in the Ap- pendix the first term in (3.13), a correlation function calculated over a steady state with Hamiltonian (3.7), is the sum of the correlation function with the standard com- 2 0 mutation relation and a contribution that is proportional to ~ when kBT ~Ω. Here T 0 is the effective temperature of fluctuations that drive the cavity with contributions from the true thermal noise and the radiation pressure noise (such an 3.3. MODIFIED NOISE SPECTRUM 55 equilibrium temperature exists provided that the relation common to most optomechanical experiments κ γ holds; see 3.6.5 in the Appendix). This higher order dependence on ~ means that this correction is smaller by a factor proportional 2 0 2 to (pZPF/mkBT ) Ω/γ than the correction due to the second term in (3.13). Here 2 0 pZPF/mkBT is just the ratio of the variance of the momentum in the oscillator ground state to the momentum variance in the thermal state, thus it is small at high temperature. But due to the presence of the quality factor Ω/γ the overall factor may be significant. This fact is not unique to this specific commutator. In the Appendix P †m n we derive, in the presence of a potential V = B m,n amna a (which may emerge from a modification to the commutator) to first order in B the modification to the correlation function due to the modified steady state ensemble alone. With this we find an expression for the modification of the symmetrized spectrum of mechanical fluctuations S(ω), (the sum of the first two terms of equation 3.85 in the Appendix). We are mainly interested in the case of high-Q oscillators γ Ω and in this case the expressions simplify considerably. We will be interested in two experimentally relevant regimes of frequency. Firstly ω Ω the so-called free-mass limit where frequencies of interest are far above the resonance frequency as is relevant for example in aLIGO. Secondly we consider frequencies close to resonance ω ∼ Ω which is where high sensitivity optomechanical experiments are usually probed and the rotating wave approximation is valid. For simplicity, for all the equations in this section we assume κ Gα, Ω, γ so that radiation pressure noise can be approximated by a white noise. In the plots in later sections of the chapter, which refer to specific experiments, we use the general spectrum (3.77) in the Appendix. In the free-mass limit the modification to the noise spectrum for the optomechanical system is (through (3.75) in the Appendix): " # 2Aγ 2 k T 0 2 Ω2 1 δS (ω) ' ~ 8 B + (3.14) ω2 ~Ω ω 3 8hνF 2P k T 0 ' + k T (3.15) B π2c2γm B where F = ∆ωFSR/κ = πc/κL is the finesse of the cavity. The dependence of δS on the various parameters changes depending on which of the noise sources contributing 0 to kBT dominates. The most important aspect here is that the spectrum has a different overall shape to the standard mechanical white noise spectrum in this frequency regime: S (ω) ∝ ω−4, for viscous/constant damping. The standard noise is inversely proportional to fourth power of driving frequency, as in the first term of the additional noise here (at lower frequencies outside the free-mass limit this takes a different shape to standard noise), but the second term from quantum zero-point energy is proportional to ω−2. This sort of effect could provide strong bounds on the minimal length. Even for the frequency-dependent ‘structural’ damping of aLIGO (explored in the next section) which can be modelled by γ = Ω2/ωQ as explained in 3.6.4 in the Appendix the spectral frequency dependence is different to the standard expression. 56 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

We will now comment on how δS(ω) depends on the various parameters of the system. Specifically we will work in terms of the oscillator mass m, frequency Ω, Q-factor, which for viscous damping is Q = Ω/γ, temperature T , and optical power P . 0 At frequencies ω/Ω kBT /~Ω the second term of the additional spectrum is dominant. In this regime the second term is inversely proportional to quality factor (keeping the resonance frequency constant). The damping model significantly affects the dependence on resonance frequency: for viscous damping the second term is proportional to Ω but for structural damping it is proportional to Ω2. In contrast to the behaviour of S(ω), δS(ω) is independent of mass. However, as we have noted earlier, in the version of the modified commutator theory in which β0 is proportional to βe for the elementary components of the system and A also depends on the number of constituent particles in the system, the noise expressed in terms of βe will depend on mass. We will explore this topic in more detail in the next section. None of the other parameters affect the noise appreciably. 0 In the other extreme, ω/Ω kBT /~Ω, we may distinguish between dominant thermal or radiation pressure noise scenarios. If thermal noise dominates radiation pressure noise the additional spectrum is proportional to the square of the temperature, the dependence on all other parameters is the same as the case where the second term of additional spectrum is dominant. On the other hand if radiation pressure noise dominates thermal noise then the additional noise increases quadratically with the power P of the input laser and its frequency ν, unlike the standard commutator radiation pressure noise which has a linear scaling with these parameters. But as in the standard expression the additional noise is inversely proportional to the square of mass in this frequency regime. The total noise is approximately proportional to the quality factor when radiation pressure noise is dominant. For viscous damping δS(ω) is proportional to Ω−1 but for structural damping it is proportional to Ω−2. Explicitly the relative contribution of this noise to the standard thermal and radiation pressure noise is:

2 2 δS (ω) m 0 ~ ω ' β0 2 8kBT + 0 (3.16) S (ω) (MP c) 3kBT

0 When the first term is dominant (again if ω/Ω kBT /~Ω) this ratio is independent of frequency, except in the case of dominant radiation pressure noise and structural damping when the ratio becomes proportional to ω. Otherwise it rises as ω2, so to obtain a better bound on β0 it is favourable to have a probe frequency away from the resonance frequency. However at ω . ωSQL, where ωSQL is the angular frequency where the standard quantum limit noise in ordinary quantum mechanics is reached , the measurement shot-noise (which we assume to be unaffected by the modified commutation relation studied here) dominates the mechanical noise. We account for this in the next section. An assessment can be made of the relevance of the second term of equation (3.16). From the functional dependence of the equation it is clear that the optimal temperature that can practically be achieved to bound β0, via a bound on the relative 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 57

noise, for a system given ω is either T+ (the maximum achievable temperature for the experiment) or T− (the minimum achievable temperature consistent with the high 0 temperature condition kBT ~Ω which was used to derive the equation). Then if the first term at T+ is much larger than the second term at T− the optimal bound will be at T+ and only the first term is relevant for the experiment. This will be p the case provided that ω/Ω (kBT+/~Ω)(kBT−/~Ω), which is usually true in the high temperature limit. We now turn to consider the second, near resonance, regime where ω ' Ω. It is instructive to look specifically at ω = Ω where the additional spectrum takes the form: 2A 2 δS (Ω) = ~ (3.17) 3γ δS (Ω) ~2Ω2m = β0 2 0 (3.18) S (Ω) 3(MP c) kBT We see that the additional noise near resonance, up to the order we have considered in perturbation theory, comes purely from zero-point energy of the oscillator. The additional noise proportional to the quality factor and to Ω−1. The relative noise however is independent of the quality factor and favours a low temperature again. It might be expected that the best place to probe δS(ω) is on the side of the resonance ω ' Ω ± γ00/2 where γ00 = Ω/Q. Close to this frequency the absolute gradient of the standard power spectrum is highest so it is to be expected that the sensitivity to perturbation of spectrum is also maximum here. Indeed for high Q the ratio between the first term of the perturbed spectrum 3.71 in the Appendix and the sum of the standard thermal and radiation pressure spectra is maximum here. We have: γ00 4 δS Ω ± ' ±A(k T 0)2 (3.19) 2 B γ2Ω γ00 0 δS Ω ± 2 4mΩk T ' ±β B (3.20) γ00 0 2 S Ω ± 2 γ (MP c) 00 0 2 The main observation here is that δS (Ω ± γ /2) ' 6Q(kBT /~Ω) δS (Ω). This relative noise always dominates the first term of the relative noise in the free-mass 0 1/2 limit and also the second term if ~ω/kBT Q . In the regime where ω is comparable to κ it is required to use the general formula 3.85 in the Appendix for the perturbed spectrum, provided that ω is not close to other resonances of the cavity. In subsequent sections where these formulae are referred this can be assumed.

3.4 Constraints on the Modiﬁed Commutator

In order to obtain constraints on the modiﬁed commutator for the oscillator’s elementary components via experimental constraints on the noise spectra such as in 58 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

previous section, parametrised in terms of β0, we must rely on a model that relates the commutator for the components with that for its centre of mass. Though an elementary component may have a simple commutation relation like (3.1) this may not be the case for the composite system. In [66] a direct approach is taken, showing that for a many-component system where each component satisfies (3.1) with modification parameter β0 it is possible to write a Hamiltonian for the centre of mass position and momentum where the modification parameter β0 is rescaled. In general there are additional terms, not present without the modified commutator, that couple the total momentum to the relative momenta of the components. Indeed, the terms involving relative momenta may dominate e.g. in ultra-hot bodies [116]. This means generally the commutator structure is not universal; it requires information about microscopic degrees of freedom. A similar conclusion was drawn in [67] and [116]. To clarify the implication for the minimal length, a β0 bound on the commutator obtained for the centre of mass of a system implies that the possible position variance of systems of that scale has a minimum value determined by β0. An underlying assumption for associating a β with a scale is that the minimum variance does not depend on the nature of the system apart from a concept of system size. If there is a minimum position variance associated to β of systems of one scale it is smaller on larger systems that are a combination of systems of the original scale, though the exact relation also depends on the interrelations of the original system e.g. relative velocities. This is in contrast to standard quantum mechanics according to which the minimum position variance is exactly zero at all scales. In our scenario we are considering a rigid body at sufficiently low temperature and small internal motion that we neglect all relative momentum terms, which al- 2 lows us to interpret that β0 scales in proportion to 1/N if all N-components are identical. This implies that for an oscillator made of a single chemical element i.e. 2 2 2 it is composed of Na atoms with mass ma then β0 ∼ βa/Na = βama/m where βa is the modification parameter for the individual atoms. But if the components are non-identical e.g. components are the nucleons and electrons of the oscillator then the relationship will be different. In the (uncharged) oscillator the number of electrons Nelec is less than the total number of nucleons Nnuc, there are 3 quarks (of 2 similar mass) per nucleon so βnuc ∼ βquark/3 , and assuming that the modification parameters of quarks and electrons are the same, being the most elementary particles currently known, βquark = βelec = βe. From these assumptions and the fact that the −27 mass of each nucleon mnuc ' 1.67 × 10 kg' 2000melec it follows that, as shown in [66]: m 3 m 3 β = N β nuc + N β elec (3.21) 0 nuc nuc m elec elec m βnuc ' 2 (3.22) Nnuc β m2 ' e nuc (3.23) 9m2 Then a constraint on β0 can be used to constrain βe and compared with previous 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 59 constraints on it e.g. via measurements of Lamb shift in hydrogen atoms. Due to the model dependence of parametric relationship between different scales it is relevant in our analysis to optimise both bounds on β0 and βe inferred from a given experiment. Consequently we will consider how bounds on both these parameters 2 depend on the properties of the system, assuming that β0 ∝ βe/m .

3.4.1 Estimating current constraints In this section we will consider how direct observations of optomechanical systems could place bounds on the modified commutator. We will consider two distinct regimes. Firstly the observation of noise in high precision interferometers like the Advanced Laser Interferometer Gravitational-Wave Observatory (aLIGO) in which the position fluctuations in the free mass limit are relevant. Secondly optomechanical experiments with nanomechanical systems where the position fluctuations of oscillators close to resonance can be observed that are due to both thermal and radiation pressure noise. These two example systems also operate in very different regimes for the mass of the oscillator. We will study the behaviour of δS(ω) as interpreted in the previous section using parameters drawn from recent experiments in these two regimes in order to get a quantitative idea of how big these bounds could be expected to be. There would be numerous additional technical issues in achieving these bounds in practice, not least because a realistic system would have more complex dynamics than the pure simple harmonic motion we consider in this chapter. The aLIGO is a Michelson interferometer in which a high power laser beam is split into two 4km long arms of the interferometer and reflected back to interfere at the output port. Extremely small relative displacements of the arms, due to gravitational radiation for example, can be detected by shifts in the light intensity at the output but the sensitivity of this measurement is constrained by radiation pressure noise on the reflecting mirrors and shot noise at the output as well as the thermal and seismic noise on the mirrors and their suspending fibres [123, 132, 133]. An accurate model of aLIGO is quite a lot more complicated than the one we have considered in this chapter. Nevertheless it is of interest to understand whether there are potentially interesting bounds on β0 arising from the measured aLIGO noise spectrum, by matching effective parameters for aLIGO to our simpler model. In the appendix 3.6.4 we provide a translation between parameters of the multi-mirror apparatus of aLIGO to our single-oscillator single-cavity model; effective parameters of our model are displayed in table 3.2. Some of the main issues in determining a set of parameters for our model that correspond to aLIGO are as follows. In aLIGO the mirrors of the optical cavities in each arm are designed such that κL ∼ ωg Ω where κL is the decay rate of each cavity, ωg is the order of magnitude of gravitational wave frequencies probed and Ω is the pendulum resonance frequency of the mirror-suspension system. Recall that in section 3.3.2 we wrote simplified expressions for the perturbed spectrum in which the cavity dynamics are implicitly adiabatically eliminated. These expressions are valid only for frequencies ω κ and when ω ∼ κ we need to use the full expressions in the 60 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

Appendix. In this regime, the radiation pressure noise spectrum is approximately inversely proportional to order sixth power of frequency[123, 133, 134], as in equation 3.85. Moreover a ‘signal recycling’ mirror in the output port of the aLIGO interferometer produces correlations between radiation pressure and shot noise [132]. Finally, leading contributions to thermal noise are internal to the suspension fibres and are usually associated with ‘structural’ damping. As discussed in section 3.3.2 a simple model for this is in terms of frequency-dependent damping of the form γ = Ω2/Qω. This results in a spectrum inversely proportional to order fifth power of frequency unlike Brownian white noise [135, 136]. The details of our model for structural damping are described in the Appendix. Thermal noise in the mirror coatings is also significant. Seismic noise is significant at frequencies below ∼ 10 Hz, outside the frequency band of gravitational waves probed by aLIGO, and shot noise is dominant at frequencies above ∼ 100 Hz. Finally, the noise spectrum for the more complicated optical system of aLIGO leads to a different balance of shot noise and radiation pressure noise contributions which we mimic by an effective detection inefficiency η2. The most recent observation of the total noise spectrum, shown in figure 7.3 of [137] (also figure 5 in [123]), has a shape and order of magnitude consistent with a theoretical description that assumes standard canonical commutation relation and that these noise sources are independent, except between 20 and 100Hz there is a gap of order one magnitude between observed and expected noise which happens to be within the range of frequencies of gravitational waves probed as well as the segment where thermal noise dominates over shot noise. Via the conservative expression δS/Sobs (ω) ≤ 1, where Sobs (ω) is the observed spectrum, a tight constraint on β0 is imposed which we can estimate by assuming that β0 must be less than the value it would take for there to be some frequency at which δS(ω) = Sobs(ω). Inserting the aLIGO parameters displayed in table 3.2 in the expression for δS(ω) 3.77 in the Appendix and comparing this to the measured value Sobs(ω) from figure 21 7.3 in [137] results in the bound β0 . 10 if we are to satisfy δS(ω) ≤ Sobs(ω). Figure 3.1 shows this rough bound as a function of probe frequency relative to resonance frequency: As per table 3.1 this is comparable to the bound obtained in [68] through a tabletop experiment on macroscopic harmonic oscillators. It would appear both of these are amongst the tightest experimental bounds obtained so far but as discussed earlier it is only unambiguous to compare bounds that refer to the same types of system. The model discussed in the previous section associated with equation (3.21) 76 implies a bound for elementary particles βe . 10 which is a much weaker constraint than that from Lamb shift measurements. Alternatively we can estimate the bounds that arise from optomechanical and electromechanical experiments with nanomechanical oscillators in which it has recently become possible to observe the effect of radiation pressure fluctuations. In the experiment of [122] a silicon nitride membrane oscillator is placed in a Fabry- Perot cavity and driven at resonance by the radiation pressure fluctuations of a ‘signal’ laser beam. A second much weaker ‘meter’ laser beam at the same frequency 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 61

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Figure 3.1: Estimate of the bounds on β arising from the measured noise spectrum in aLIGO experiments as a function of probe frequency. Here the upper bound on β0 is derived at a given frequency by requiring that the predicted additional noise δS(ω) from equation (3.77) in the Appendix is less than the observed noise power spectrum at aLIGO, see main text for details. The βe parameter is the β associated with elementary particles, inferred from β0 according to equation (3.21).

Table 3.2: Eﬀective experimental parameters and β bounds Experiment aLIGO (2016) [123, 137] Purdy et.al (2013)[122] Teufel et. al (2016)[138] T (K) 3.00E+02 1.70E-03 4.00E-02 m (kg) 1.00E+01 (reduced) 7.00E-12 8.50E-14 Ω (Hz) 4.15E+00 9.75E+06 5.88E+07 γ (Hz) 1.00E-06 (@ ω = Ω) 8.98E+03 1.53E+02 Q 1.33E+09 1.08E+03 3.83E+05 ν (Hz) 2.82E+14 2.82E+14 6.71E+09 L (m) 4.00E+03 5.10E-03 4.00E-08 κ (Hz) 4.78E+03 5.59E+06 6.64E+07 P (W) 3.60E+03 9.40E-05 7.80E-09 F 4.92E+01 3.30E+04 3.55E+08 Smallest S(ω)(m2/Hz) 9.00E-40 4.00E-32 1.00E-26 β0 upper bound @ ω = Ω - 1E+41 1E+42 βe upper bound @ ω = Ω - 1E+73 1E+70 γ00 β0 upper bound @ ω = Ω ± 2 - 1E+31 1E+28 γ00 βe upper bound @ ω = Ω ± 2 - 1E+64 1E+57

is also passed through the cavity and its intensity photocurrent is measured. This photocurrent contains an imprint of the position fluctuations of the oscillator, and for sufficient measurement strength the fluctuations due to radiation pressure noise of the signal beam override those from the thermal Brownian motion of the oscillator. As a result the radiation pressure noise power spectrum can be measured cleanly. 62 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

Another experiment [138] involves a microwave cavity optomechanical circuit containing a superconducting inductor resonating with a parallel-plate capacitor, i.e. a LC circuit in which the capacitor electrodes separation determines the resonance frequency of the microwave cavity. A coherent microwave drive at the cavity resonance is applied via a feed-line, the reflected beam is separated via a microwave circulator and the phase quadrature measured. The power spectrum of this signal is proportional to the mechanical spectrum of the capacitor, and again by optimising the measurement strength the radiation pressure fluctuations dominate the thermal noise. Both these experiments involve a single optomechanical cavity as in our setup so their parameters may be straightforwardly inserted in our equations. To obtain the bounds near resonance we note first that in these particular experiments κ is of the same order of magnitude as Ω. Thus we are again required to use the exact formula 3.77 in the Appendix to estimate the perturbed spectrum. In both the experiments the observed position noise spectra are consistent with the standard noise from thermal, radiation pressure and shot noise 3.85, assuming viscous damping, and so we may find rough estimates of bounds on β in the way accomplished for aLIGO. Substituting the parameters in table 3.2 and comparing with the recorded spectrum in the experiments, the optimal bounds at resonance are 41 73 42 70 β0 . 10 , βe . 10 for experiment [122] and β0 . 10 , βe . 10 for experiment γ00 [138]. Significant improvement of the bounds occur at ω = Ω + 2 , as described 31 64 in the conclusion of section 3.3.2. For experiment [122] β0 . 10 , βe . 10 is 28 57 obtained and β0 . 10 , βe . 10 for the other experiment. Even though tabletop experiment bounds of β0 are far worse than aLIGO (which we have explored in the free-mass limit) the corresponding βe are better due to the use of low-mass oscillators. The equation 3.20 suggests that observations on a system with parameters close to those of aLIGO but made for frequencies close to the mechanical resonance would improve bounds on β0, due its higher mass, high mechanical quality-factor, and room-temperature setup.

3.4.2 Future constraints We analyse how to optimise future experimental parameters to find better constraints on β0 (and inferred value βe) in the modified canonical commutation relation 3.1. In the previous section we derived constraints via bounds on the ratio between the observed noise on an oscillator and the additional mechanical noise on it due to the modified commutation relation (‘relative noise criterion’). Here we will derive potential future constraints in a similar way except instead of substituting observed noise we use the theoretical expression for the noise level, which is just the sum (3.85) in the Appendix of the standard mechanical and shot noise spectra. We also examine the experimental scenario where a fixed upper bound can be placed on the additional noise across frequencies. As in the previous section we choose to focus on two frequency regimes: ω Ω where the oscillator is treated as a free-mass, as well as the frequency just off its resonance frequency ω = Ω + γ00/2. Both large and small mass oscillators are of interest in future optomechanical 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 63 experiments. In the previous section we examined aLIGO as an example of the former scenario but for the latter we explore here the experiment of [139] as a test-bed, which first reported an indirect signature of quantum radiation pressure noise. In this experiment an ultracold rubidium atomic gas is confined by optical trapping in a Fabry-Perot cavity and continuously driven by a probe laser. The quantum radiation pressure noise of the laser heats the gas and via the observation of its evaporation rate the level of noise is inferred. This noise was found to be in agreement with standard quantum theory. If in a similar setup a direct observation of the mechanical position spectrum is possible, similar to the optomechanical experiments [122, 138] that were considered in the previous section, our equations may be applied straightforwardly to obtain constraints on β values. Some relevant parameters of this experiment are: T = 0.8 × 10−6K, m ' 10−22kg, Ω = 2π × 4.2 × 104Hz, Q = 42, P ' 5.02 × 10−13W, ν ' 3.84 × 1014Hz, κ = 2π × 6.6 × 105Hz and L = 1.94 × 10−4m. Note also the experiment of [140], not treated here, which also reported indirect observation of radiation pressure noise on an ultracold gas. The plots in figures 3.2 and 3.3 show β-constraints attainable in parameter regimes similar to those of the experiment of [139] and aLIGO respectively when probed in the free-mass limit. Here we have chosen to adjust either m, Ω (keeping Q constant) or P individually in each of the plots by up to three orders of magnitudes from nominal values taken from those experiments. At each frequency shown the corresponding value of β0(βe) is found via the condition that δS(ω)/Sstd(ω) . 1. We must use the full expression 3.77 in the Appendix for δS(ω) as the adiabatic approximation is violated at higher frequencies, and Sstd(ω) is taken to be the sum of thermal, radiation pressure noise and shot noise in equation 3.85. We express angular frequencies relative to the standard quantum limit (SQL) frequency at which the minimum value of Sstd(ω) is reached in the theory with β = 0. The plots show that decreasing the mass or resonance frequency of the oscillator, or increasing the laser input power, improves the bounds with respect to ω/ωSQL at all frequencies considered. We stress that ωSQL also varies with both these parameters: v u r ! u1 2 1024πνP ω ' t −κ2 + (κ2 + 4Ω2) + + 4Ω2 (3.24) SQL 8 L2m

If we had plotted with respect to ω then a diﬀerent parametric dependence holds which in the adiabatic regime obeys 3.16 e.g. then higher (lower) mass is preferred for bounding β0(βe). Across our plots the bounds decrease till ω ' ωSQL after which they sharply increase. Thus the optimal bounds occur at this frequency. The initial decreasing behaviour in the case of plots associated with the experiment of [139] can be attributed to the frequency dependence of the cavity susceptibility in the standard radiation pressure noise (see 3.77 in the Appendix), which makes an important contribution in the frequency regime we have considered, but in the plots associated with aLIGO this behaviour is due to the frequency dependence of structural damping. For aLIGO the plots corresponding to m × 10−3 and P × 103 64 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

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Figure 3.2: β bounds vs. relative frequency ω/ωSQL for different masses, resonance frequencies and powers with respect to parameters taken from the experiment of Murch et. al [139] (m ' 10−22kg, Ω = 2π × 4.2 × 104Hz and P ' 5.02 × 10−13W are the standard values). The bounds are derived similarly to figure 1 by the require- ment that δS(ω)/Sstd(ω) . 1 where δS(ω) is the perturbation to the mechanical noise spectrum due to the modified commutators and Sstd(ω) is the analytical sum of theoretical thermal, radiation pressure and shot noise spectra. We use the full expression for δS(ω) equation (3.77) in the Appendix. The sharp rise in bounds occurs when shot noise is dominant from ω ' ωSQL given by equation (3.24). 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 65

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Figure 3.3: β bounds vs. relative frequency ω/ωSQL for different masses, resonance frequencies and powers with respect to parameters taken from those of aLIGO [123] (m ' 10kg, Ω ' 2π × 6.6 × 10−1Hz and P ' 3.7 × 102W are the standard values). The bounds are derived in the same way as those of figure 2. Once again the best bounds are obtained at ω ' ωSQL. 66 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE have a sharper drop to their minima, again due to the frequency dependence of cavity susceptibility in this parameter regime where κ ∼ ωSQL. In both experiments the sharp increase in β bounds is due to the domination of shot noise over mechanical noise from ωSQL. Looking close to resonance at ω = Ω+γ00/2 we obtain for both parameter regimes the potential β bounds displayed in figures 3.4 and 3.5. In the first figure we vary by three orders of magnitude in both directions oscillator mass m, laser power P (which has same effect as varying ν), cavity decay rate κ and length L around the values in the original experiments. Each parameter is varied independently, so κ is varied with L fixed (and vice versa) which implies the cavity finesse is also varied, and varying L keeping ν fixed implies varying the optical mode number. Also the oscillator resonance frequency Ω is varied up to six orders of magnitude below the original experimental values (this is shown in the plots as varying by three orders of −3 magnitude in both directions around Ω0 = Ωi × 10 where Ωi is the original experimental value). While varying Ω we keeping the mechanical Q constant. We choose this range of values for Ω in order that κ γ which is required for the approxima- tions we use in deriving these equations to be valid, see the Appendix. In the other figure we show the dependence on the oscillator Q (keeping Ω = Ωi constant) and thermal temperature T . Both figures show a contrast in the dependence on variables between the two experiments. We see that for an experiment like that in [139] the greatest improvement in bounds are from individually increasing the quality factor or reducing the cavity length or decay rate (if probing βe then reducing the mass has the same effect); for every order of magnitude adjustment of these variables both β0 and βe are reduced by approximately two orders of magnitude. The apparent overlap of the plots varying cavity length and decay rate comes from the fact that over all the specific parameter space considered here the system remains in the adiabatic regime. Here the dependency of the β bounds on κ and L can be completely expressed in terms of cavity finesse F ∝ 1/κL. This is generally not the case. Increasing the laser power or frequency or reducing oscillator resonance frequency are also preferred though here the bounds change by only one order of magnitude for every order of magnitude adjustment of these variables. The high sensitivity to optical parameters is a consequence of dominant radiation pressure noise in this experiment, but as a result altering the mass or thermal temperature has negligible effect on the β0 bound. In the plots associated with aLIGO better bounds are obtained by adjusting the variables in the same way but in contrast the bounds do not vary appreciably for lower power and higher cavity length and decay rate, and better β0 bounds arise from increasing the mass and thermal temperature. This difference in behaviour is from the fact that in that parameter regime corresponding to aLIGO radiation pressure noise does not dominate thermal noise. We examine how to reach specific new bounds on the modified commutation relation. Firstly, comparison of the figures suggests that generally for low mass oscillators similar to [139] probing in the free-mass limit is favourable but for experiments in parameter regimes more similar to those of aLIGO it is better to observe close to 3.4. CONSTRAINTS ON THE MODIFIED COMMUTATOR 67

37 resonance. To reach βe . 10 , just one order of magnitude off the bound inferred from Lamb shift [114], one could for example probe an experiment like [139] in the free-mass limit whilst increasing the laser power by approximately three orders of magnitude or reducing the mass (resonant frequency) by two (four) orders of magnitude from the nominal values of the experiment. Note that tuning a combination of these parameters would require less adjustment of the magnitudes. This bound could also be reached when probing close to resonance by, for example, lowering any two of the oscillator mass, cavity length or decay rate by three orders of magnitude (increasing the power by same amount has a weaker effect) or increasing the oscillator quality factor quite significantly to Q ' 107. For an experiment like aLIGO we 14 see from the plots that β0 . 10 could be reached in the free-mass limit by lowering Ω by three orders of magnitude. But probing close to resonance it is possible, extrapolating figure 3.4 , to reach the target value β0 . 1 by also similarly lowering cavity length and decay rate. The above bounds on β values are derived via a constraint on the relative noise, but if an experiment places a fixed upper constraint on the additional noise itself then optimal parameters have a different scaling and tighter bounds could be achieved. In the adiabatic regime the equation (3.14) is the relevant additional noise in the free-mass limit. As discussed earlier, if thermal noise is dominant then larger T is preferred as well as smaller Q (though to derive the additional spectrum Q 1 is assumed). But once radiation pressure noise is dominant then to significantly improve bounds on both β values larger Q and optical parameters F, P, ν or smaller m and Ω are required. As an example, in the experiment of [139] if the additional noise across frequencies is bounded by the noise at the SQL frequency so δS(ω) . −29 2 38 Sstd (ωSQL) ∼ 10 m /Hz, then βe . 10 can be obtained at ω = 10Ω, a few orders of magnitude away from the bound from Lamb shift. For an experiment 74 like aLIGO we can similarly deduce βe . 10 . By lowering the reduced mass to m = 10−9kg and keeping the noise constrained to be below the noise at the SQL frequency (corresponding to the mass), we obtain a βe bound comparable to that −20 from Lamb Shift and also β0 . 1. Bringing the mass down further to m = 10 kg allows to potentially reach βe . 1. Probing at ω = Ω + γ00/2 equation (3.19) applies in the adiabatic regime and implies large Q is preferred particularly when radiation pressure noise is dominant. When the additional noise can be constrained by the SQL noise then for aLIGO we −14 have already have the strong bound β0 . 10 and by lowering (increasing) the mass (quality factor) by two orders of magnitude we can bound βe better than from Lamb shift. The bound βe = 1 could be reached by adjusting the mass(quality factor) by another ten (nine) orders of magnitude. For the experiment of [139] with a similar 31 noise constraint in fact we have βe . 10 , and β0 = 1(βe = 1) would be achieved by raising quality factor to Q = 106(Q = 109). 68 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

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Figure 3.4: Best bounds achievable at ω = Ω + γ00/2 by rescaling the value v of a given variable, in the low-mass parameter regime based on Murch et. al [139] (left) or the high-mass regime based on aLIGO [123] (right). The standard values v0 are as −3 in previous ﬁgures, except we set the central Ω value to Ω0 = Ωi × 10 where Ωi is the original Ω value. ν ' 3.85×1014Hz, κ = 2π×6.6×105Hz,L = 1.94×10−4m in the low-mass parameter regime and ν ' 2.82 × 1014Hz,κ ' 5.55 × 102Hz,L = 4 × 103m in the high-mass regime. Here the bounds are derived from the relative noise as in previous ﬁgures.

3.4.3 General driven oscillator The method we have focused on to place tight bounds on a modified canonical commutator involves constraints on the corresponding modified thermal Brownian noise and quantum radiation pressure noise on an oscillator, the key properties being that the modified noise relative to the standard noise is amplified with greater effective temperature and quality factor, and that the modified noise spectrum has a different scaling with respect to probe frequency. If β0 (βe) is probed a larger (smaller) mass is preferred when thermal noise is dominant, but either way a high effective temperature bath is also preferred. We can heuristically translate this dependency on effective temperature to a general out-of-equilibrium scenario by considering the oscillator driven by a stochastic approximately white noise force fdrive associated with an effective temperature Teff 2 that fulfils the equipartition relation: kBTeff = m hv i. Then the free-mass limit 2 2 2 2 2 2 −1 equation (3.16) becomes: β0 . (MP c) (8m hv i + ~ ω /2 hv i) . If the mirrors in 2 −2 2 −2 an experiment like aLIGO are moved so that hv i ∼ 5.3×10 m s then β0 . 1 can 3.5. DISCUSSION 69

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Figure 3.5: β bounds at ω = Ω + γ00/2 vs mechanical quality factor and varying thermal temperature, keeping other variables fixed at the values in the low-mass parameter regime based on Murch et. al [139] (left) and in the high-mass parameter regime based on aLIGO [123] (right). The bounds are derived similarly to those in the previous figure. The smallest quality factors considered here are the original values in the experiments. Increasing quality factor decreases the upper bounds in both parameter regimes but due to the high radiation pressure noise relative to thermal noise in the low mass parameter regime even an increase in over 8 orders of magnitude in temperature produces insignificant change in bounds in that case. be reached (neglecting the effect of probe frequency as we have assumed high effective temperature and taking the probe frequency to be sufficiently smaller than the resonance frequency, following the discussion after equation (3.16). But if a higher probe frequency can achieved it could be advantageous). At Ω + γ00/2 from equation 2 2 2 3.19 we have β0 . (MP c) /4Qm hv i so with the possibility of using high-Q ma- 2 −10 2 −2 terial like sapphire only hv i ∼ 10 m s may be required to reach β0 . 1. The bound on βe is independent of mass for a given speed. With a high-Q oscillator and 2 −4 2 31 hv i = 10 c the bound is approximately βe . 10 , a tighter constraint on minimal length than inferred from Lamb shift and experiments at the Large Hadron Collider. For driven systems approaching higher speeds we expect even better bounds can be attained though this would enter the relativistic regime, which is outside the scope of our analysis, and a non-perturbative treatment should be used to gain a more accurate picture.

3.5 Discussion

We have shown that probing the mechanical noise on harmonic oscillators with high precision, as in recent optomechanical experiments, is a viable technique to constrain small modifications of quantum mechanics found in models of quantum gravity, specifically the canonical commutation relations. Based on a conservative criterion 70 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE that constrains the perturbation of the standard mechanical spectrum by the observed spectrum, we inferred that currently a bound on β0 via aLIGO is comparable with that from other experiments. We found that the strongest constraints on both β0 and βe appear to arise from probing the spectrum close to the mechanical resonance frequency or close to where the standard quantum limit of noise is reached. Strikingly, by adjusting both optical and mechanical parameters by a few orders of magnitude it is possible to probe if β0 . 1, which could be interpreted as a glimpse of the Planck scale in the GUP framework. New bounds on βe using our proposal may be achievable in the near term, which would be impressive since it would mean a better understanding of the mechanics of elementary particles like electrons can be achieved by measurements of macroscopic systems. As noted in the previous section, for a general noise source the ideal scenario is to drive the oscillator so it reaches large variation of speeds. High quality factor oscillators are also generally desirable, though this may not be the case if the β bounds are derived from a fixed upper constraint on the additional noise. Other proposals like that in [67] and experiments in [115, 68], and more recently [69] have also studied macroscopic oscillators to probe modifications of commutation relations. It may seem odd that [67] provides a formula that generally favours small mass to constrain β0 unlike our result and the other experiments. However this is not a fundamental contradiction as the proposal there utilises multiple optical pulses in order to probe an induced geometric phase, whilst here we are concerned with the modification of the position noise spectrum/auto-correlation function. Also, in contrast to our work, in other papers the behaviours of the mean or variance of position or energy of systems were studied. For example in [68] and more recently [69] the commutator we have studied was mapped via the perturbed Hamiltonian to the classical solution for the mean position of a Duffing oscillator. The nonlinearity implies a consequent amplitude-frequency effect. This is also visible in the correlation function we have derived that incorporates, unlike previous papers, fluctuations from a noisy bath and both viscous and structural damping. We have also kept within a fully quantum formalism that shows potentially significant zero point energy contributions. The figure of merit for the amplitude-frequency effect turns out to be similar to that derived from equation (3.19) for the frequency close to resonance, after applying the equipartition theorem. But in the previous papers the uncertainty in resonance frequency was significantly enhanced by the resolution bandwidth of the spectrum analyser and the intrinsic oscillator non-linearity (consequently only small amplitudes of motion were probed). Here we expect over the whole frequency range of the spectrum that only close to resonance, as seen by application of the rotating wave approximation, may the intrinsic non-linearity obscure effects from modified commutators. However the experiments we have considered do not significantly show such limitations, so that the corresponding frequency uncertainty is taken to be the oscillator damping rate (hence the preference for a high quality factor). While this paper was in preparation [129] also analysed modifications of radiation pressure noise (and shot noise) in aLIGO but instead chose to only modify the optical commutation relations. But there it is reported that aLIGO measure- 3.5. DISCUSSION 71 ments already constrain the relevant modification parameters to far below 1. We focused on a modified commutator solely for the mechanical degrees of freedom of the system oscillator; this restriction can be justified as the standard quantum position and momentum operators usually apply to massive degrees of freedom (in the non-relativistic limit). Indeed we have shown how significant improvements could be found from mechanical effects alone especially in future tabletop experiments. A general treatment incorporating modified optical relations is outside the scope of our paper but it can easily be seen that the net modified mechanical position noise spectrum would be a linear combination of our results and the effects from solely modifying the optical relations, to first order in β0 and any similar modification parameters for the optical fields. One key property of a purely mechanical treatment is that the shot noise is unmodified but in [129] it is reported that the modified optical relations imply a term reducing this noise unless the optical field is squeezed to unrealistic amounts. Also while this paper was in preparation there have been multiple reports of observation of quantum radiation pressure noise on room temperature oscillators [141][142][143]. These new techniques are promising for future applications of our results, in particular the experiment of [143] consists of observations across broad- band and off-resonance frequencies relevant to our analysis of the free-mass limit. Here we mainly used a conservative bound on relative noise as the basis for our results, but in future studies other measures could give significant improvements. The relative noise approach is equivalent to assuming that the error on the noise spectrum is proportional to the total observed noise (where the factor is up to an order of magnitude). But as we have noted it could be that the precision measurements are associated with a fixed error that is independent of the observed noise, at least over a certain frequency band. Precise knowledge of the experimental error at individual frequencies would lead the way for tighter β bounds. This could account for statistical features like the number of data samples taken in the experiment. There will realistically be a variety of unaccounted noise sources that add to the error, this was discussed in the case of aLIGO in section 3.6.4 and included to calculate the current bounds A related point is that use of tailored measurement methods and/or squeezed light fields [132, 133, 144], enhanced even more by quantum entanglement[145], in new interferometers can increase position measurement sensitivity beyond the SQL limit. This is particularly important for our work which shows that in the free-mass limit the optimal bounds are inferred at approximately the ordinary SQL frequency when accounting for shot noise. This paves the way for application of fundamental limits in parameter estimation like the quantum Cramer-Rao bound which has recently been studied for gravitational wave detectors [146, 147]. Note that the modified commutation relation implies a shift of both the frequency and the noise level at the noise spectrum minimum. Our study can also be extended to other interpretations of the minimal length. As mentioned earlier there are several other proposed modified commutation relations associated with quantum gravity, like equations 3.3 and 3.4. It turns out that in previous experiments tighter bounds are generally obtained for the parameters in these 72 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE equations, so it can be expected that following our methods the same will be achieved and spectral features amplified where there is higher non-linearity. The commutator we have studied is however well-motivated from the point of view of maintaining a UV momentum cut-off [61]. It can be associated with several variance-based uncertainty principles and for some measures of ‘minimal length’ can be expressed via entropic uncertainty relations as recently shown [148]. Here we have performed a perturbative treatment in β0. This breaks down for 2 β0(p/MP c) 1, so related effects for high-energy oscillators remain to be examined which would need a fully relativistic treatment.

3.6 Appendix

In this appendix we derive the modiﬁed spectrum of mechanical ﬂuctuations from the solutions of the equations (3.10) and (3.11). Equation (3.10) describes a damped driven oscillator with frequency Ω and damping rate γ. We will consider only the underdamped regime γ < 2Ω, typical for optomechanical experiments, in which case eigenvalues of the linear system are λ± = −γ0 ± iω0 with: γ p γ = , ω = 4Ω2 − γ2/2. (3.25) 0 2 0 The solutions are

λ+t x0 (t) 1 ie [x0 (0) λ− − p0 (0) /m] + ia/m + h.c. = λ+t ∗ (3.26) p0 (t) 2ω0 iλ+e [x0 (0) λ−m − p0 (0)] − iλ+a + h.c. and

λ+t δx (t) 1 ie [δx (0) λ− − δp (0) /m] + ic/m + h.c. = λ+t ∗ (3.27) δp (t) 2ω0 iλ+e [δx (0) λ−m − δp (0)] − iλ+c + h.c. where: Z t 4 Z t λ−(t−s) 3 λ−(t−s) a = f (s) e ds, c = Aλ− p0(s) e ds (3.28) 0 3 0 The force term f is a sum of thermal and radiation pressure noise. The thermal noise is, in the high temperature limit, treated as a white noise obeying the corre- 0 00 0 00 lation function hf (s ) f (s )i = 2kBT γmδ (s − s ). For radiation pressure noise the correlation function is:

0 00 2 2 2 −κ|s0−s00|/2 hfrad(s )frad(s )i = ~ G α e . (3.29) See 3.6.5 later in the Appendix for details on the derivation. The white noise approximation of this corresponds to adiabatic elimination of the cavity ﬁeld, given by 0 00 2 2 2 0 00 hfrad(s )frad(s )i = (4~ G α /κ)δ (s − s ). 3.6. APPENDIX 73

3.6.1 First term of (3.13)

Here we ﬁnd the ﬁrst term hx0 (τ) x0 (0)i of the correlation function in (3.13). Assume the oscillator at initial time is in thermal equilibrium associated with quantum state: ρ = Z−1e−βH = Z−1e−β(H0+V ) (3.30)

2 2 2 4 0 where H0 = mΩ x /2 +p ˜ /2m, V = Ap˜ /3m and β = 1/kBT . Now ρ satisfies the Bloch equation: ∂ ρ = −Hρ (3.31) ∂β with ρ (β = 0) = I. We can use time-dependent Hamiltonian perturbation theory to compute the modified thermal state and its correlation functions since this equation is equivalent, via a Wick rotation, to the evolution of a unitary operator in the Schrödingerpicture for a system evolving under a time-dependent Hamiltonian. We keep only the first perturbation as we only keep terms first order in A. For any observable O we then have: −1 −βH hOiH = Tr ZH e O β Z 0 0 ' Z Z −1 Tr Z−1e−βH0 O − Tr Z−1e−βH0 eβ H0 V e−β H0 dβ0 O H0 H H0 H0 0 (3.32) We use the subscript H to emphasize that the expectation value is taken using the thermal state of the full Hamiltonian. Substituting the identity operator for O above we get: −1 β ! Z 0 0 −1 β H0 −β H0 0 ZH0 ZH = 1 − e V e dβ 0 H0 β Z 0 0 ' 1 + eβ H0 V e−β H0 dβ0 (3.33) 0 H0 Here we have introduced the notation hOi = Tr Z−1e−βH0 O for expectation H0 H0 values according to the thermal state of the unperturbed Hamiltonian H0. Therefore to first order in A: β β Z 0 0 Z 0 0 hOi = hOi + hOi eβ H0 V (β0) e−β H0 dβ0 − eβ H0 V e−β H0 dβ0 O H H0 H0 0 H0 0 H0 (3.34) With these perturbation theory expressions in hand we are able to compute the first term in the correlation function in (3.13): β Z 0 0 hx (τ) x (0)i = hx (τ) x (0)i + hx (τ) x (0)i eβ H0 V e−β H0 dβ0 0 0 H 0 0 H0 0 0 H0 0 H0 74 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

β Z 0 0 β H0 −β H0 0 − e V e x0 (τ) x0 (0) dβ . (3.35) 0 H0

At this point we can use the solutions for x0(τ) and p0(τ). From now on we use the notation x0 and p0 for x0 (0) and p0 (0) respectively. We will also only compute the real part of hx0 (τ) x0 (0)iH since this is all that will be required for the symmetrised spectrum which is our main interest and use the subscript R to indicate the real part of a correlation. For a standard thermal state hp x i = 0 = hax i . Then: 0 0 H0,R 0 H0 1 hx (τ) x i = x2 iλ eλ+τ − iλ eλ−τ . (3.36) 0 0 H0,R 0 H − + 2ω0 0

0 0 0 0 † β H0 −β H0 −β ~Ω † β ~Ω In our scenario H0 = ~Ω b b + 1/2 so e pe = −ipZPF be − b e p where pZPF = ~Ωm/2 is the zero-point momentum. Since we are averaging over a thermal state, which is a Gaussian state, we can apply Wick’s theorem [130]. For D 0 0 E R β β H0 −β H0 0 thermal states the only non-zero terms in 0 e V e dβ are those with H0 equal powers of b and b† and we have b†nbn = n! b†b n [149]. This implies:

β 4 4 Z 0 0 Ap Ap β H0 −β H0 0 ZPF †2 2 † ZPF 2 2 e V e dβ = 3β 2 b b + 4 b b + 1 = x0 3m mk T 0x4 H0 0 H0 B ZPF (3.37) p where xZPF = ~/2mΩ is the zero-point position. We can proceed in the same way to show

β β Z 0 0 Z 0 0 β H0 −β H0 2 0 β H0 −β H0 2 2 †2 † 0 e V e x0dβ = e V e xZPF b + b + 2b b + 1 dβ 0 H0 0 H0 ( " 2 Ap4 1 hx2i = ZPF x2 0 H0 + 1 e−2β~Ω − 1 0 H0 2 m 2~Ω xZPF 2 # 2 !) hx2i hx2i 0 H0 2β~Ω 0 H0 − 2 − 1 e − 1 + β 3 2 − 2 xZPF xZPF (3.38)

0 We will mainly be interested in the high temperature limit ~Ω/kBT 1 in which case we can simplify this expression as follows

β 4 2 2 ! Z 0 0 Ap hx i 4 β H0 −β H0 2 0 ZPF 2 0 H0 e V e x0dβ ' x0 − . (3.39) mk T 0 H0 x4 3 0 H0 B ZPF

Finally we can also show

β ! Z 0 0 β H0 0 −β H0 0 Re e V (β ) e p0x0dβ = 0 (3.40) 0 H0 3.6. APPENDIX 75

So, substituting in (3.35) we get 2Ap4 hx (τ) x i ' hx (τ) x i + ZPF x2 iλ eλ+τ − iλ eλ−τ , (3.41) 0 0 H,R 0 0 H0,R 0 0 H − + 3ω0mkBT 0 when τ ≥ 0. In the following sections we will compute these correlation functions for τ ≥ 0 only and infer the case of negative τ from general properties of the correlation functions. This expression implies that the perturbation to the standard correlation function in equation (3.41) is to leading order quadratic in ~, arising from the zero- point energy of the oscillator. Let us briefly consider how this calculation will proceed for other modified commutation relations. For other commutation relations for which [x, p] depends on p it will be possible to define a momentum-like observablep ˜ satisfying [x, p˜] = i~ to first order. Writing the Hamiltonian H in terms ofp ˜ we would find that it is perturbed by a “potential” V . A similar approach would work for those commutation relations for which [x, p] depends on x. Any perturbation V can be expressed in the form:

X †m n V = B amna a (3.42) m,n where B is a small parameter and amn are complex numbers. If B is real then as V ∗ is Hermitian we have anm = amn. The interaction picture representation of V , VI , is: X i(m−n)Ωt †m n VI = B amne a a (3.43) m,n Performing a Wick rotation it → ~β we can ﬁnd these expressions for arbitrary operator O:

Z β 1 0 0 X −1 ~Ωβ(m−n) †m n hOi VI (β ) dβ =B amn (m − n) e − 1 a a H0 Ω 0 H0 ~ m,n:m6=n ! X + β a a†nan hOi (3.44) nn H0 n X =Bβ a a†nan hOi (3.45) nn H0,th n where the last line is true for a thermal state, and

Z β 1 0 0 X −1 ~Ωβ(m−n) †m n VI (β ) dβ (O) =B amn (m − n) e − 1 a a O Ω H0 0 H0 ~ m,n:m6=n ! X †n n + β ann a a O (3.46) H0 n From equations (3.36) and (3.40), which is valid for general V averaged over a thermal state, the contribution to equation 3.35 from the perturbed Hamiltonian is:

1 Z β λ+τ λ−τ 2 0 0 δ hx0 (τ) x0i = iλ−e − iλ+e x0 VI (β ) dβ H,R 2ω H0,th 0 0 H0,th 76 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

Z β ! 0 2 0 − VI (β ) x0dβ 0 H0,th

B λ+τ λ−τ X †n n 2 †n n 2 = iλ−e − iλ+e β ann a a x0 − a a x0 2ω H0,th H0,th 0 n ! 1 X −1 ~Ωβ(m−n) †m n 2 − amn (m − n) e − 1 a a x0 Ω H0,th ~ m,n:m6=n (3.47)

Here amn can be chosen to be any non-zero complex number so for a generic potential that may be associated with a modiﬁcation to quantum mechanics this part of the modiﬁcation to the correlation function does not vanish. This holds true even in the high temperature limit where:

Bβ λ+τ λ−τ X †n n 2 †n n 2 δ hx0 (τ) x0i = iλ−e − iλ+e ann a a x0 − a a x0 H,R 2ω H0,th H0,th 0 n ! X †m n 2 − amn a a x (3.48) 0 H0,th m,n:m6=n

3.6.2 Second term of (3.13) In the previous section we calculated, at t = 0, the perturbation to the thermal state due to the modified commutation relation. Working in the Heisenberg picture we take at this time that δx (0) = δp (0) = 0 and now calculate the influence of the modified evolution of the position operators over time. Since all of these corrections will be proportional to A we can evaluate expectation values using the thermal state of the unperturbed harmonic oscillator, so now hx2i will mean hx2i . The second 0 0 H0 term of (3.13), from (3.27), is:

1 ∗ hδx (τ) x0iR = h[ic(τ) − ic (τ)] x0iR 2mω0 2A Z τ 3 λ+(τ−s) λ−(τ−s) = − p0 (s) x0 R iλ+e − iλ−e ds. (3.49) 3mω0 0 And

p0 (s) = x0j (s) + p0g (s) + h (s) , (3.50) where

λ+λ−m j (s) = ieλ+s − ieλ−s (3.51) 2ω0 1 λ−s λ+s g (s) = iλ−e − iλ+e (3.52) 2ω0 3.6. APPENDIX 77

1 ∗ h (s) = [iλ−a (s) − iλ+a (s)] . (3.53) 2ω0 Now since the unperturbed oscillator is in a Gaussian thermal state we can use Wick’s theorem to obtain symmetrically ordered moments (denoted by subscript sym) to 2 2 2 2 2 get, for example hx0p0isym = hx0i hp0i + 2 hx0p0isym. Consequently,

3 3 4 2 2 2 2 2 p0 (s) x0 R = j (s) x0 + 3g (s) j (s) x0p0 sym + 3 h (s) j (s) x0 (3.54) 2 2 2 2 2 2 = 3 x0 j(s) j(s) x0 + g(s) p0 + h(s) (3.55)

So then: 2A Z τ 2 2 3 λ+(τ−s) λ−(τ−s) hδx (τ) x0iR = − x0 x0 j(s) iλ+e − iλ−e ds mω0 0 Z τ 2 2 λ+(τ−s) λ−(τ−s) + p0 j(s)g(s) iλ+e − iλ−e ds 0 Z τ 2 λ+(τ−s) λ−(τ−s) + j(s) h(s) iλ+e − iλ−e ds (3.56) 0 The ﬁnal term is an average over the sum of thermal and radiation pressure noise. As a reminder the former is approximated to be white noise but the latter takes the more complicated form in (3.29) After the integrals that appear in the above expression can be evaluated and after some straightforward manipulations we arrive at:

hδx (τ) x0 (0)iR = λ λ −3 2 + − 3λ+τ 2 2 −4A x0 2 e x0 (λ+λ−m) (λ+ − λ−) 2 (λ− − λ+) (3λ+ − λ−) 2 2 −3λ+ −3λ+ + p0 + U + E1 2 (3λ+ − λ−)(λ− − λ+) 4 (3λ+ − λ−)(λ− − λ+) 3 2λ + λ (2λ++λ−)τ 2 2 + − + e x0 (λ+λ−m) 2 λ+ (λ+ + λ−)(λ− − λ+) 2 ! 2 2 (λ+ + λ−) + λ+λ− + p0 2 (λ+ + λ−)(λ− − λ+) (2λ+ + λ−)(λ+ + 5λ−) + U 2 + E2 4 (λ+ + λ−) (λ− − λ+) 3 λ − λ λ−τ 2 2 − + + e x0 (λ+λ−m) 2 λ+ (3λ+ − λ−)(λ+ + λ−) 2 λ− (λ− − λ+) + p0 2 (λ+ + λ−) (3λ+ − λ−) 3 2 2 3 3λ+ + 7λ−λ+ − 3λ−λ+ + λ− + U 2 + E3 2 (3λ+ − λ−)(λ+ + λ−) (λ− − λ+) 78 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

λ κ κ λ−τ − (2λ+− )τ (λ++λ−− )τ − τe U + E4 − e 2 E5 + e 2 E6 + c.c. (3.57) 2 (λ+ + λ−) where U = 2kBT γm and:

2 2 2 3α G ~ λ+ E1 = (3.58) (λ+ − λ−) (3λ+ − λ−)(κ + 2λ+) 2 2 2 α G ~ (2λ+ + λ−)(λ+ (κ + 6λ−) + λ− (5κ + 6λ−)) E2 = − 2 (3.59) (λ+ − λ−)(λ+ + λ−) (κ + 2λ+)(κ + 2λ−) 2 2 2 2 3 E3 = [2α G ~ −3κ λ+ (κ − 2λ+)(κ − 4λ+) 4 3 2 2 3 +λ− 5κ − 50κ λ+ + 44κλ+ − 24λ+ 2 3 2 +λ+λ− (κ − 3λ+) 3κ − 2λ+ 7κ + 8λ+ (2κ + λ+) 2 3 2 2 3 +κλ+λ− −7κ + 47κ λ+ − 114κλ+ + 72λ+ 3 4 3 2 2 3 4 6 −λ− κ − κ λ+ − 82κ λ+ + 140κλ+ + 72λ+ + 4λ− (κ − 6λ+) 5 2 +4λ+λ− (κ + 18λ+) ][(λ− − 3λ+)(λ+ + λ−) (λ− − λ+) 2 2 −1 (κ − 2λ+) (κ − 2λ−) (κ − 4λ+ + 2λ−)] (3.60) 2 2 2 2α G ~ κλ− E4 = (3.61) (λ+ + λ−)(κ − 2λ−)(κ − 2λ+) 2 2 2 16α G ~ κλ+ (κ − 4λ+) E5 = 2 (3.62) (κ − 2λ+) (κ + 2λ+)(κ + 2λ−)(κ − 4λ+ + 2λ−) 2 2 2 16α G ~ κλ+ (κ − 2λ+ − 2λ−) E6 = 2 2 2 (3.63) (κ − 2λ+) (κ + 2λ+)(κ − 4λ−) Recall that this expression is valid only when τ ≥ 0. We can infer the value for negative τ through the identities hx(τ)x(0)i∗ = hx(0)x(τ)i = hx(−τ)x(0)i. The first equality follows directly from the definition of the correlator in terms of the steady state density matrix and the time evolution operator. The second equality results from the fact that the system is assumed to be in steady state. Notice that these equalities imply that hx(τ)x(0)iR = hx(−τ)x(0)iR. We can compare the terms arising here with the contribution due to the perturbation of the steady state given in equation (3.41). Using thermal expectation 2 0 2 values such as hx0iH0 = kBT /mΩ we see from equation (3.41) a perturbation to the 4 2 2 correlation function that scales like ApZPF/m Ω in the limit that the oscillator is high-Q. Studying the terms in equation (3.57) that have frequencies close to ω0 it is 0 2 3 possible to see that in the high-Q regime these contributions scale like A(kBT ) γ/Ω and other contributions that are smaller by some power of γ/Ω. The earlier contri- 2 0 2 2 0 butions are therefore different by a factor (pZPF/mkBT ) Ω/γ. Since pZPF/mkBT is the ratio between the momentum variance of the oscillator in the ground state and the thermal state we expect that this term is only relevant very close to zero temperature. However even at high temperatures if the quality factor Ω/γ is sufficiently large the overall term may be significant, and indeed in the tabletop experiments we consider this is the case. Therefore we must keep both contributions in our analysis of the perturbed correlation function. 3.6. APPENDIX 79

3.6.3 Perturbed spectrum Now we can combine the results of the previous sections to obtain the perturbed spectrum. From the expressions for eigenvalues in (3.25) we can classify the terms in (3.57) according to their exponents in ω0. The assumption κ γ, typical for optomechanical experiments, allows us to treat the unperturbed oscillator as being in a steady state with an approximate effective temperature T 0 resulting from the effect of the thermal bath and the radiation pressure fluctuations. The equipartition 2 2 2 0 2 2 2 2 2 relations hp0i /2m = mΩ hx0i /2 = kBT /2 = kBT/2 + ~ α G κ/2γ0m (κ + 4ω0) then approximately hold (see section 3.6.5). Decomposing the coefficients into real and imaginary parts and adding the quantity in (3.41) the perturbed correlation function becomes (for τ ≥ 0):

γ2 + ω2 2 0 0 (−3γ0+i3ω0)τ δ hx0 (τ) x0 (0)iR ' A x0 2 e W ω0 (−γ0−iω0)τ (−γ0−iω0)τ +e (I1 + I2 + (J1 + J2)i) − τe (M + Ni) κ κ (− +i2ω0)τ (−2γ0− )τ +e 2 (Y + Zi) + e 2 R + c.c. (3.64) where:

2 2 2 2 3α G ~ ω0 W = − 2 2 2 2 (3.65) (γ0 + 4ω0)(κ + 4ω0) 2 2 2 2 4 39α G ~ ω0κ I1 + J1i = 2 2 2 2 2 2 2 (γ0 + 4ω0)(κ + 4ω0) (κ + 36ω0) 0 2 2 2 2 2 2 2 kBmT γ0 8α G ~ ω0κ(5κ + 18ω0)(γ0 + 3ω0) +i − 2 2 2 2 2 2 2 (3.66) ω0 (γ0 + 4ω0)(κ + 4ω0) (κ + 36ω0) 2 2 2 ~ ω0m ~ ω0γ0m I2 + J2i = 0 + i − 0 (3.67) 6kBT 6kBT 0 0 M + Ni = γ0kBmT + ikBmT ω0 (3.68) 2 2 2 4 3 2 4 16α G ~ κ (γ0κ + 6κ ω0 + 88κω0) Y + Zi = 2 2 3 2 2 (κ + 4ω0) (κ + 36ω0) 2 2 2 4 2 2 4 16α G ~ κω0 (κ + 12κ ω0 − 96ω0) +i − 2 2 3 2 2 (3.69) (κ + 4ω0) (κ + 36ω0) 2 2 2 2 2 16α G ~ κ (γ0κ + 2ω0) R = − 2 2 3 (3.70) (κ + 4ω0)

Here I1 and J1 comes from the second term of (3.13) and I2 and J2 arise from the first term, due to the zero-point energy of the oscillator. The existence of a time-weighted sinusoidal term in the correlation function is a property of an effectively amplitude-dependent resonance frequency, a hallmark of non-linearity, in a perturbative approximation. In the frequency regime where radiation pressure noise can modelled by white noise we can take κ Ω, γ keeping 0 terms of order kBT . In this case we find W = I1 = Y = Z = R = 0, J1 = 80 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

0 kBmT γ0/ω0 and the other terms are the same as before. The perturbed spectrum in this frequency regime is given by:

Z ∞ iωτ δS (ω) = δ hx0 (τ) x0 (0)iR e dτ −∞ 2 02 2 2 2 2 2 16AγkBT ω (ω − Ω ) 2Aγω ~ = 2 + 2 (3.71) γ2ω2 + (ω2 − Ω2)2 3 γ2ω2 + (ω2 − Ω2)

We are mainly interested in the case of high-Q oscillators (Q = Ω/γ for viscous damping) and in speciﬁc regimes of frequency in which it is possible to obtain simpler expressions. So for example, at resonance ω = Ω we obtain:

2A 2 δS (ω) = ~ (3.72) 3γ

Here only the contribution from oscillator quantum zero-point energy remains (up to ﬁrst order in A), and in fact this√ is the optimal frequency for this term. Near resonance at ω = Ω±γ0/ 3 the magnitude of the ﬁrst term of the perturbed spectrum (3.71) is approximately at maximum: " # A 2 √ k T 0 2 Ω 1 δS (ω) ' ± ~ 3 3 B + (3.73) γ ~Ω γ 2

The approximation sign indicates we have taken the limit of high Q. Another relevant frequency is ω = Ω ± γ0 where for high Q the ratio between the ﬁrst term of the perturbed spectrum and the sum of the standard thermal and radiation pressure spectra is maximum. Here: " # A 2 k T 0 2 Ω 1 δS (ω) ' ± ~ 4 B + (3.74) γ ~Ω γ 3

In the free-mass limit ω Ω we have: " # 2Aγ 2 k T 0 2 Ω2 1 δS (ω) ' ~ 8 B + (3.75) ω2 ~Ω ω 3 and at low frequencies in the range Ω/Q ω Ω: " # 2A 2γω2 k T 0 2 1 δS (ω) ' ~ −8 B + (3.76) Ω4 ~Ω 3

Over general frequencies the perturbed spectrum is given by:

δS (ω) 3.6. APPENDIX 81

2 2 2 ω ((J1 + J2)ω0 − γ0(I1 + I2)) − (γ0 + ω0)(γ0(I1 + I2) + (J1 + J2)ω0) = 2 2 2 2 4 2ω (γ0 − ω0)(γ0 + ω0) + (γ0 + ω0) + ω 1 4 2 2 + 2 2 2 2 4 2 ω −γ0 M + Mω0 − 6γ0Nω0 (2ω (γ0 − ω0)(γ0 + ω0) + (γ0 + ω0) + ω ) 2 2 2 4 4 +ω M 10γ0 ω0 + γ0 + ω0 − 4γ0Nω0 (γ0 − ω0)(γ0 + ω0) 2Rκ + γ2 + ω2 2 γ2M − Mω2 + 2γ Nω − Mω6 − 0 0 0 0 0 0 κ2 + 4ω2 2 2 2 3W γ0 (ω + 9γ0 + 9ω0) − 2 2 2 2 4 6ω (γ0 + ω0)(γ0 − ω0) + 81 (γ0 + ω0) + ω 2 Y κ κ2 2 Y κ  ω − − 2ω0Z − + 4ω − 2ω0Z 0 2 4 0 2 −4AkBT + 2  (3.77) 2 κ κ κ 2 2 4 mω2 2ω 2 + 2ω0 2 − 2ω0 + 4 + 4ω0 + ω 0

3.6.4 aLIGO modelling In the main text we analysed the modified commutator signal to get the spectrum. Doing this requires a translation of parameters between the setup of aLIGO involving multiple coupled cavities and mirrors to our setup that has a single cavity, oscillator and optical drive. In this section we explain our approach to obtaining parameters of our model pertaining to aLIGO. We can model the four identical mirrors in aLIGO as a single mirror in our setup with a reduced mass m/4 where m is the mass of each mirror in aLIGO [133]. This comes from the fact that aLIGO measures the difference between the differences in positions of centre of mass of the mirrors in each arm individually. The γ parameter above is commonly modelled in mechanical experiments as a constant over frequency, in which case the damping term in the equation of motion is termed ‘viscous damping’, proportional to the momentum. But for aLIGO the viscous damping of the mirrors, caused by collisions with surrounding gas molecules and control electromag- nets, is small as the mirrors sit in an ultra-high vacuum. A form of non-viscous damping known as ‘structural damping’ is dominant within the suspension fibres of aLIGO and visible in other experiments with low viscous noise. In the frequency domain the equation of motion is modelled in [135] via a complex spring constant that is frequency-independent:

mx¨ = −k(1 + iφ)(x − xg) + F (3.78)

2 2 2 2 where x/xg = Ω (1 + φ)/(Ω − ω + iφΩ ) is the vibration transfer function from motion at a point xg to x. This frequency-independence is a good approximation at frequencies away from zero for realistic structural damping (since close to zero frequency the true loss must drop to zero). The equation of motion can be expressed in the time domain via a Hamiltonian as follows so that its associated equation of motion is equivalent to (3.78). Consider a mirror system oscillator that is in contact with a bath modelled as consisting of many oscillators with weak coupling to the oscillator. Solving the 82 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

Heisenberg equations of bath and oscillator simultaneously, using the entire system and bath Hamiltonian, we obtain the following equation of motion for the oscillator [130] (with initial time t0 = −∞): Z t d(x − xg) 0 0 mx¨ = −k(x − xg) + Fmodiﬁed − 0 f(t − t)dt + Fnoise (3.79) −∞ dt

Here Fnoise is the total fluctuating noise force and Fmodified is the effective force due to the modification of the canonical commutator. Fnoise possesses approximately Gaussian statistics of an equilibrium thermal ensemble with a symmetrised time correlation function proportional to f(t0 −t). The third term in (3.79) is the effective dissipation force on the system. For viscous damping f(t0 − t) = 2γmδ(t0 − t). For 0 0 structural damping we set f(t − t) = −2γmΩ(γEM + ln |t − t|)), where γEM is the Euler-Mascheroni constant, to match the frequency domain expression (3.78) as in [135] associated with frequency-independent loss factor φ. Now in the regime of high frequencies, which is of interest in the free-mass limit, structural damping deviates significantly from viscous damping. Here we present a heuristic argument that in this regime structural damping may be viewed as a frequency dependent version of viscous damping. The dominant contribution to the spectrum at high frequencies is from motion in which the interval of time |t0 − t| in 0 (3.79) is small so here we may make the approximation d(x − xg)/dt ∼ d(x − xg)/dt and move it out of the integral. We can impose cutoffs of the remaining integral as we care about high frequencies/small times. We choose the cutoffs such that we recover from the equation of motion, in the case of standard canonical commutator, the position spectrum associated with structural damping as defined in [135]. The remaining integral then equals γm (Ω/ω). We may thus interpret the dissipation term as involving viscous damping but with an effectively renormalised γ that is now frequency dependent: γ0 = Ω/Q (Ω/ω). By using γ0 instead of γ throughout our derivation of the perturbed spectrum we may arrive at the approximate perturbed spectrum with structural damping. We also have to translate the optical parameters in aLIGO, which we can do by matching the radiation pressure and shot noise spectra of our setup to that of aLIGO. In our setup the standard radiation pressure noise spectrum is 2 4~2α2G2/κ (4ω2/κ2 + 1) m2 γ2ω2 + (ω2 − Ω2) ' 16hνP F 2/π2c2m2 (4ω2/κ2 + 1) ω4 where the latter is valid for ω Ω γ. In aLIGO it is in the same frequency 2 2 2 2 2 regime given by 64hνG−Parm (2πf−) /ω + 1 /c M ω where G− = 31.4 is the cavity build-up factor for the differential mode in aLIGO (difference between positions of mirrors in each arm; the degree of freedom aLIGO measures) and f− is the differential coupled cavity pole [123]. The laser frequency in our setup is taken to be the same as used in aLIGO. We now take m = M/4 as explained above, and assuming the power in each cavity of aLIGO’s arms and our setup are the same then the cavity build-up factor in our setup is Parm/P = 2F/π where Parm is the power in each cavity of aLIGO. The spectra are then equal to each other provided that F = πG−/2 which implies P = Parm/G−, and, keeping the same arm-length L as in aLIGO, that the optical decay rate is κ = 4πf− = 2c/G−L. 3.6. APPENDIX 83

To translate the shot noise spectrum we first have that in our setup 2 2 2 2 2 2 2 2 κ/16α G η2 (4ω /κ + 1) = hνλ /256P F η2 (4ω /κ + 1). Here we have introduced a parameter η2, effectively a detection efficiency factor, in order that our shot noise spectrum matches that in aLIGO. In aLIGO the shot noise spectrum is given by 2 2 2 2 2 2hνλ Gsrc/GprcPinηGarm (4π) (2πf−) /ω + 1 where Pin is the input power to the interferometer, Gprc is the power recycling cavity gain, η = 0.75 is the detection inefficiency of aLIGO (fraction of the output power that is transmitted to the pho- todiode detectors), Garm is the buildup factor of each arm cavity of aLIGO and Gsrc is the signal recycling cavity gain of aLIGO (these are expressed in rounded off form in [123]). Substituting F = πG−/2, P = PinGarmGprc/2G− which equates power in our setup cavity to that in aLIGO’s arm cavities, and G− = Garm/Gsrc we arrive at η2 = η/4. The relevant translated parameters are contained in table 3.2.

3.6.5 Steady-state expectation values without the adiabatic approximation Here we sketch the calculation of the steady-state expectation values of position/momentum variances, if the adiabatic approximation for the cavity field is not made. Recall that the radiation pressure force driving of the cavity is: † frad = ~Gα δa + δa . If the cavity is driven on resonance the required quadrature of the cavity field can be used to show that frad satisfies the equation [121] √ ˙ † frad = −κfrad/2 + κ~Gα fcav + fcav , (3.80) where fcav is the vacuum field input to the cavity and can be modelled by white noise † 0 0 with hfcav(t)fcav(t )i = δ(t − t ). Notice that this equation holds even for non-zero A. Consequently we have

Z t −κt/2 √ −κ(t−t0)/2 0 † 0 0 frad(t) = e frad(0) + κ~Gα e fcav(t ) + fcav(t ) dt . (3.81) 0 Consequently in the steady state we have

0 2 2 2 −κ|t−t0|/2 hfrad(t)frad(t )i = ~ G α e . (3.82) When the assumption is made that the oscillator dynamics are slow compared to κ this exact correlator can be approximated well by the appropriate delta-function. 2 We also need to determine the contribution of the radiation pressure force to hx0i and the other steady state expectation values of the unperturbed system. These can easily be found using equation (3.26). So we have

2 2 1 ∗ 2 hx0i = lim hx0(t) i = 2 2 lim h(ia − ia ) i t→∞ 4m ω0 t→∞ 2 2 2 ~ α G (κ + 4γ0) = 2 2 2 2 (3.83) γ0m Ω ((κ + 2γ0) + 4ω0) 84 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE

On the other hand

2 2 1 ∗ 2 hp0i = lim hp0(t) i = 2 lim h(−iλ−a + iλ+a ) i t→∞ 4ω0 t→∞ ~2α2G2κ = 2 2 (3.84) γ0 ((κ + 2γ0) + 4ω0)

3.6.6 Standard spectrum We remind the reader that for the standard canonical commutator the total noise spectrum, modelled as the sum of thermal, radiation pressure noise and shot noise spectra that we have used above, is given by [121, 131]:

4ω2 2 2 2 κ + 1 2γkT 4~ α G κ2 Sstd(ω) = + + 2 2 2 2 2 4ω2 2 2 2 2 2 2 16α2G2 m γ ω + (ω − Ω ) κ κ2 + 1 m γ ω + (ω − Ω ) (3.85) 3.6. APPENDIX 85 86 3. PROBING MODIFIED COMMUTATORS WITH QUANTUM NOISE Chapter 4

Modiﬁed Commutators and Continuous Position Measurements

Modifications of the canonical commutation relations of quantum mechanics have several motivations as explored in the previous chapter. In particular they are an expression of a minimal possible spatial length in line with theories of quantum gravity. Probing these relations experimentally has been a subject of interest, most recently due to the improved sensitivity of precision measurement experiments. In the previous chapter it was shown that bounds on such modified relations can be infered from measurements of quantum optomechanical noise, in particular according to this model a minimal spatial length associated with elementary particles like electrons could be constrained towards the Planck scale. Here I will extend this picture by providing a general quantum-mechanical treatment of continuous position measurement in the presence of modifications of the canonical commutation relation, which can be applied to a wide spectrum of experimental scenarios including that described in the previous chapter. The standard quantum mechanical description of continuous position measurement can be traced to Barchielli and collaborators [150] and further developments in particular by Caves and Milburn [70] and also Diosi [151]. We follow the conventional master equation approach though equally valid Langevin and path integral approaches also exist. A comparison of results is made with the specific scenario of optomechanical measurements studied previously.

4.1 Quantum Trajectory Equation

The evolution of an arbitrary system under continuous series of measurements according to the standard quantum theory has been studied extensively since the mid 1970’s [152]. Prior to this, studies were mostly restricted to isolated individual systems subject to few sequential measurements in well localised time intervals. But in a general scenario the system(s) may be interacting with measuring apparatus(es)

87 88 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS

(which we call meters) sufficiently frequently that we may treat the phenomenon in a limit as a continuous series of measurements. A simple example is: if light is shone on an object and the reflecting light observed over a period of time then in that period of time the object has been continuously measured. Some of the first studies of continuous measurement led to discovery of the ‘Quantum Zeno’s paradox’ [153, 154]: the quantum state of a continously projectively measured system does not evolve in time. This idea was actually understood decades earlier by Alan Turing, according to a letter at the time by his friend Robin Gandy [155]). However in the realistic scenario that we explore here of indirect measurement, involving elements of a POVM (positive operator valued measure), the evolution is non-trivial and described by a master equation. It has been shown that the presence of continuous measurement decoheres the reduced quantum state of the system in a similar manner to decoherence due to presence of a bath coupled to the system [156, 157, 158]. This has brought the theory of continuous measurements into an attempt to bridge the classical and quantum descriptions of the world, at the foundations of physics. In the standard quantum measurement framework [4] the following occurs: after each quantum measurement the meter becomes entangled with the system and every projective ‘ordinary’ measurement of the meter results in a more general operation in which for each measurement outcome a completely-positive map acts on the system. Between measurements the state undergoes unitary evolution under the Schrodinger equation (for general states the von Neumann equation). The resulting conditional master equation for the reduced system state is called a quantum trajectory equation. Here by conditional it is meant that it accounts for the result of each measurement. For a scenario obeying this model for measurement of position Caves and Milburn [70] derived the change in the moments of the position and momentum of the system under conditional evolution as well as a master equation for the non-conditional evolution. Derivations of the quantum trajectory equation partially based on this model can be found in [151, 159, 160]. It has also been deduced for specific measurement contexts e.g. optical homodyne and heterodyne measurements [161]. I will now derive the quantum trajectory equation of a system under a continuous position measurement model as in [70] but independent of the system’s canonical commutation relations. As in [70] contributions from the thermal bath are neglected. It turns out that this equation has the same form as the quantum trajectory equation with the standard canonical commutator, which is widely used in the literature. Nevertheless I will explicitly derive it for completeness. A continuous position measurement can be quantitatively modelled (following [70]) as a series of interactions between a system and meter (or identical meters) in which each interaction occurs for an infinitesimally small amount of time. The time interval τ between consecutive interactions is taken in a limit to be infinitesimally small. The position variance σ associated with the meter (modelled as a Gaussian state) is taken in a limit to be infinitely large such that τσ = 1/4ζ is a finite number (each measurement of this sort is called a ‘weak measurement’). This number characterises the rate of ‘diffusion’ of the system’s mean momentum as it is repeatedly measured. 4.1. QUANTUM TRAJECTORY EQUATION 89

Prior to the rth measurement the meter is in a pure state |Υri with Gaussian wave function: 2 − 1 x¯r hx¯ |Υ i = Υ(¯x ) = (πσ) 4 exp − (4.1) r r r 2σ

2 σ for which hx¯ri = 2 . Here the bar notation indicates that the variable refers to the meter rather than the system. The key assumption in [70] is that during measurement the meter becomes entangled with the system so that the position of the system is correlated with the displacement of the pointer of the meter. This means the interaction Hamiltonian involves coupling the system position with the generator of position of the meter i.e. the standard quantum mechanical momentum operatorp ˆr. The total system and meter Hamiltonian is then, assuming that the meter free Hamiltonian is approximately proportional to the identity and thus ignored: n X H = HS + δ(t − rτ)ˆxpˆr (4.2) r=1 In this work we assume that although the system’s position may not be directly measured (according to the projection postulate this would leave it in a quantum state with infinitely small position variance, which would violate our interpretation of a minimal length) the meter’s pointer position can be projectively measured to arbitrary precision, so this assumption of minimal length is not made for the meter. In the previous chapter, in which an optical field is used as a meter to probe a mechanical harmonic oscillator, this assumption is also not made as the standard optical commutation relations are preserved. Further studies may make this assumption, but we are following the aspect of standard quantum measurement theory [4] that holds there exists a ‘Heisenberg cut’, the end of a chain of meters entangled with a system, where a projective position measurement is made. So in our treatment the POVM associated with measurement is unaffected by modifications of the system canonical commutation relations. The (unnormalised) quantum state ρr after the rth measurement is related to the state ρr−1 after the (r − 1)th measurement by evolution under Hamiltonian HS followed by a ‘jump’ associated with the rth measurement, desribed via the system superoperator R as a function of the measured meter positionx ¯r:

2 − 1 x¯r −iHSτ iHSτ † ρr−1 → ρr = (πσ) 2 exp − R exp ρr−1 exp R (4.3) σ ~ ~ where p 2 R = exp 2ζ∆Qrxˆ exp −2ζτxˆ (4.4) and the pre-measurement result ‘current’ Qr has the properties: x¯ ∆Q = √r (4.5) r σ 2ζ 90 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS

2 ∆Qr = τ (4.6) It is convenient for our calculations to deﬁne: −iHSτ iHSτ † ρ˜r = R exp ρ˜r−1 exp R (4.7) ~ ~ Note that the probability density of each outcome is: p(¯xr) = Tr(ρr) 2 2 − 1 x¯r 2¯xrxˆ − xˆ −iHSτ iHSτ = (πσ) 2 exp − Tr exp exp ρr−1 exp σ σ ~ ~ (4.8)

Taylor expandingρ ˜r to ﬁrst order in τ:

∆˜ρr =ρ ˜r − ρ˜r−1 p 2 2 2 2 = 2ζ∆Qr (ˆxρ˜r−1 +ρ ˜r−1xˆ) + ζ∆Qr xˆ ρ˜r−1 +ρ ˜r−1xˆ + 2ζ∆Qrxˆρ˜r−1xˆ 2 2 i i −2ζ xˆ ρ˜r−1 +ρ ˜r−1xˆ − HSρ˜r−1τ + ρ˜r−1HSτ (4.9) ~ ~

Taking the limit τ → 0 and summing over an interval from t0 to δt (thereby performing an integration):

Z t0+δt 0 ρ˜(t0 + δt) − ρ˜(t0) = dρ˜(t ) t0 X = ∆˜ρr τ→0 X hp = 2ζ∆Qr (ˆxρ˜r−1 +ρ ˜r−1xˆ) τ→0 2 2 2 2 +ζ∆Qr xˆ ρ˜r−1 +ρ ˜r−1xˆ + 2ζ∆Qrxˆρ˜r−1xˆ 2 2 i i −2ζτ xˆ ρ˜r−1 +ρ ˜r−1xˆ − HSρ˜r−1τ + ρ˜r−1HSτ ~ ~ X hp = 2ζ∆Qr (ˆxρ˜r−1 +ρ ˜r−1xˆ) τ→0 2 2 +ζτ xˆ ρ˜r−1 +ρ ˜r−1xˆ + 2ζτxˆρ˜r−1xˆ 2 2 i i −2ζτ xˆ ρ˜r−1 +ρ ˜r−1xˆ − HSρ˜r−1τ + ρ˜r−1HSτ ~ ~ Z t0+δt Z t0+δt = a(t0)dQ(t0) + b(t0)dt0 (4.10) t0 t0 where

0 p a(t ) = 2ζ (ˆxρ˜r−1 +ρ ˜r−1xˆ) (4.11) 4.1. QUANTUM TRAJECTORY EQUATION 91

0 i 1 2 2 2 2 b(t ) = [˜ρr−1,HS] + 2ζ xˆ ρ˜r−1 +ρ ˜r−1xˆ +x ˆρ˜r−1xˆ − xˆ ρ˜r−1 +ρ ˜r−1xˆ ~ 2 (4.12) The second-last equation of (4.10) uses the identity dQ2 = dt valid for Ito rule integration. This identity is derived for ordinary stochastic differential equations in [130] using equation (4.5). We assume the derivation extends to operator-valued a, b. An alternative analysis is given in [151]. Q obeys the properties of a Wiener process. In the limit δt → 0 we obtain the differential change dρ˜. We first have from equation (4.10), using the cyclic property of the trace: dTr(˜ρ) = Tr(dρ˜) = Tr(˜ρ)p8ζ hxi dQ (4.13) To get dρ then: ρ˜(t + dt) ρ(t + dt) = Tr [˜ρ(t + dt)] dρ˜ ' ρ + 1 − p8ζ hxi dQ + 8ζ hxi2 dt ρ˜ = ρ − p8ζ hxi ρ(dQ − p8ζ hxi dt) +p2ζ(xρ + ρx)(dQ − p8ζ hxi dt) i +2ζD[x]ρdt − [HS, ρ] dt (4.14) ~ obtained from the expression for dρ˜ (substituting dQ2 = dt) and here D[z]ρ = † 1 † 1 † zρz − 2 z zρ − 2 ρz z . † † Let H[z]ρ = zρ+ρz −Tr(√zρ+ρz )ρ. Then applying√ generalised transformations H [x] → H e−iφx and dQ − 8ζ hxi dt → dQ − 8ζ cos φ hxi dt = dW , which often appear in literature (here dW is also a Wiener increment), we finally arrive at a master equation

i p −iφ dρ = − [HS, ρ] dt + 2ζD[x]ρdt + 2ζH e x ρdW (4.15) ~ Thus the master equation has an unmodified structure though the commutation relations affect each term implicitly when calculation moments as we show in the next section. This equation can be used to model a variety of measurement settings but also other scenarios e.g. spontaneous localisation in the GRW interpretation of quantum mechanics (see [162] for a review), quantum state diffusion for simulating coupling to an environment [163, 164], the decoherent histories interpretation of quantum mechanics [165, 166, 167] (see [168] and [169] for a comparison to quantum state diffusion and quantum jumps respectively). We are now interested in the moments of variables of the system that arise from the trajectory equation, for a given system canonical commutation relation. If the state is approximately Gaussian then we can derive all moments expressed in terms of the position and momentum p from the means and covariance matrix components 2 2 2 2 2 2 1 i.e. hxi , hpi , h(∆x) i = hx i − hxi , h(∆p) i = hp i − hpi , h∆x∆pi = 2 (hxp + pxi − 2 hxi hpi). 92 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS

4.2 Diﬀerentials of general mean & covariance matrix

In this section we calculate, for arbitrary canonical commutation relation, the differential change of the mean of any variable under the quantum trajectory equation derived above. We use this to extract the diﬀerentials of covariance matrix of the position with any other variable. The symbol hKi indicates expectation value Tr(Kρ) of an operator K on the state ρ after conditional evolution. We also denote the Hamiltonian of the system by H (replacing HS in the previous section). From equation (4.15), applying the cyclic property of the trace, we ﬁnd:

i d hKi = − h[K,H]i + ζ h[[x, K] , x]i dt+p2ζ(2 cos φ h∆x∆Ki+i sin φ h[x, K]i)dW ~ (4.16) Here the notation (different to that used in the previous chapter) is : ∆x = x − hxi, ∆K = K − hKi and: h∆x∆Ki = h(x − hxi)(K − hKi) + (K − hKi)(x − hxi)i /2 = (hxK + Kxi−2 hxi hKi)/2. The latter expression is real-valued as it is a symmetrised quantum correlator. The commutator [x, K] appears explicitly in contributions to both the deterministic and stochastic evolution. Both these contributions are explicitly independent of the Hamiltonian (though they are averaged over a state that may have been evolved under the Hamiltonian) and require non-vanishing ζ. Thus they are purely a consequence of the continuous measurement process. The deterministic contribution also requires that the commutator does not commute with position, whilst the stochastic contribution requires φ 6= mπ, m ∈ Z. From this equation we can calculate the differentials of the means and covariance matrix of position and any other operator e.g. momentum. Firstly: i d hxi = − h[x, H]i dt + 2p2ζ cos φ (∆x)2 dW (4.17) ~ To find d h(∆K)2i we note:

d (∆K)2 = d K2 − (d hKi)2 − 2 hKi d hKi (4.18)

Here equation (4.16) implies:

i d K2 = − K2,H + ζ x, K2 , x dt ~ +p2ζ 2 cos φ ∆x∆ K2 + i sin φ x, K2 dW (4.19)

This can be expressed in terms of the commutator [x, K] via:

x, K2 , x = {K, [[x, K] , x]} − 2 [x, K]2 (4.20) 4.3. MODIFIED CANONICAL COMMUTATION RELATIONS 93

For the second term: (d hKi)2 = 2ζ − sin2 φ h[x, K]i2 + 4 h∆x∆Ki2 cos2 φ + i2 sin 2φ h[x, K]i h∆x∆Ki dt (4.21) (up to O(dt) and applying dW 2 = dt) These imply that: i 2i d (∆K)2 = − K2,H + hKi h[K,H]i ~ ~ +ζ h∆K∆ [[x, K] , x]i − 2 cos2 φ h[x, K]i2 −4i sin 2φ h∆x∆Ki h[x, K]i − 8 cos2 φ h∆x∆Ki2 dt +2p2ζ cos φ ∆x∆ K2 − 2 hKi h∆x∆Ki +i sin φ h∆K∆ [x, K]i] dW (4.22) Importantly here the term −2ζ cos2 φ h[x, K]i2 depends directly on the commutator and has the unique property of being (explicitly) independent of the system Hamil- tonian or any other function of x or K (apart from that of the commutator). We will analyse this in more detail later. The diﬀerential of position variance is therefore: i 2i d (∆x)2 = − x2,H + hxi h[x, H]i − 8ζ cos2 φ (∆x)2 2 dt ~ ~ +p2ζ 2 cos φ ∆x∆ x2 − 2 hxi (∆x)2 dW (4.23) The ﬁnal element of the covariance matrix can be shown to be, applying dW 2 = dt and only keeping terms of order O(dt): 1 d h∆x∆Ki = [d hxK + Kxi − 2d(hxi hKi)] 2 i i i = − h[xK + Kx, H]i + hKi h[x, H]i + hxi h[K,H]i 2~ ~ ~ + ζ [h∆x∆ [[x, K] , x]i − (2i sin 2φ h[x, K]i +8 cos2 φ h∆x∆Ki (∆x)2 dt n + p2ζ [cos φ (h∆x∆(xK + Kx)i − 2 hxi h∆x∆Ki −2 hKi (∆x)2 + i sin φ h∆x∆ [x, K]i dW (4.24) Here there are also terms that depend explicitly on the commutator, but unlike equation 4.22 they are all weighted by extra position-dependent terms.

4.3 Modiﬁed canonical commutation relations

We may now evaluate the expressions derived in the previous section for a variety of canonical commutation relations that are found in the quantum gravity literature, 94 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS including the relation that was discussed in the previous chapter in the specific context of optomechanics. In these expressions we substitute for K the momentum p that exactly obeys a canonical commutation relation (instead of introducing a modified momentum that obeys the standard commutator to first order as in the previous chapter). At first we do not substitute a specific system Hamiltonian or state for the sake of generality. Later we use the Hamiltonian for a simple harmonic oscillator and apply the Gaussian variational ansatz, which also allows the application of Wick’s theorem. In [170] this ansatz was applied successfully to an oscillator whose position is continuously measured in a non-linear potential. The position mean (4.17) and variance (4.23) appear to be independent of the commutator though there is an implicit dependence as the Hamiltonian is a function of the momentum. For the other moments, if the momentum obeys the standard commutator [x, p] = i~ we find: i p d hpi = − h[p, H]i dt + 2ζ (2 cos φ h∆x∆pi − ~ sin φ) dW (4.25) ~

2 i 2 2i 2 2 d (∆p) = − p ,H + hpi h[p, H]i + 2ζ ~ cos φ ~ ~ 2 2 −4 cos φ h∆x∆pi + 2~ sin 2φ h∆x∆pi dt h i + 2p2ζ cos φ ∆x∆ p2 − 2 hpi h∆x∆pi dW (4.26)

i i i d h∆x∆pi = − h[xp + px, H]i + hxi h[p, H]i + hpi h[x, H]i 2~ ~ ~ 2 2 +2ζ ~ sin 2φ − 4 cos φ h∆x∆pi (∆x) dt h + p2ζ cos φ (h∆x∆(xp + px)i − 2 hxi h∆x∆pi −2 hpi (∆x)2 dW (4.27)

Notice here in equation (4.26) which measures the change in the variance of momentum the term 2~2ζ cos2 φ is completely independent of position or momentum. This term is often called the ‘backaction’ noise as it is the stochastic disturbance of the momentum after position measurement. The fact that it contains ~ and that there are no other terms to cancel this out indicates its quantum-mechanical origin. Now we look at canonical commutators from quantum gravity. Let the commutation relation be that in [61]:

2! p 2 [x, p] = i~ 1 + β0 = i~ 1 + Ap (4.28) MP c where Mp is Planck mass, c is speed of light, β0 is a numerical parameter that 2 2 quantiﬁes interaction , A = β0/MP c . Qualitatively similar commutation relations 4.3. MODIFIED CANONICAL COMMUTATION RELATIONS 95 that imply a maximum momentum together with minimal length were studied in [171] and [172]. In this case we have: i 2 3 2 3 d hpi = − h[p, H]i + 2ζA~ hpi + A (∆p) + 3 hpi (∆p) + hpi dt ~ p 2 2 + 2ζ 2 cos φ h∆x∆pi − ~ sin φ 1 + A hpi + (∆p) dW (4.29)

i 2i d (∆p)2 = − p2,H + hpi h[p, H]i ~ ~ 2 2 2 2 +ζ(2~ cos φ + 4~ sin 2φ h∆x∆pi − 8 cos φ h∆x∆pi ) 2 2 +2ζ~A 2 (∆p) + hpi h∆x∆pi sin 2φ 2 2 2 2 +~ (∆p) + 2 (∆p) + hpi cos φ 4 3 2 2 +A~ (∆p) + 2 (∆p) hpi + 3 (∆p) hpi 2 io + cos2 φ (∆p)2 + hpi4 + 2 (∆p)2 hpi2 dt n + 2p2ζ cos φ( ∆x∆p2 − 2 hpi h∆x∆pi) 3 2 −A~ sin φ (∆p) + 2 (∆p) hpi dW (4.30)

i i i d h∆x∆pi = − h[xp + px, H]i + hxi h[p, H]i + hpi h[x, H]i 2~ ~ ~ 2 2 +2ζ ~ sin 2φ − 4 cos φ h∆x∆pi (∆x) 3 2 +2ζA~ ~ h∆x∆pi + A ∆x (∆p) + 3 ∆x (∆p) hpi +3 h∆x∆pi hpi2 + (∆x)2 sin 2φ hpi2 + (∆p)2 dt n + p2ζ cos φ h∆x∆(xp + px)i − 2 hxi h∆x∆pi − 2 hpi (∆x)2 2 −A~ sin φ ∆x (∆p) + 2 hpi h∆x∆pi dW (4.31) The first line of each equation has a part that is dependent on the Hamiltonian of the system. This part appears to have a form independent of the canonical commutator, but if the Hamiltonian is dependent on position then implicitly there is a dependence. If this is not the case then still the first line of equation 4.31 is dependent on the commutator, since the Hamiltonian will be some function of the momentum. The growth of mean momentum 4.29, unlike the standard case, is non-linear. The term 2ζA~2 hpi, when taken in isolation, suggests that in this scenario continuous measurement adds an exponential growth of mean momentum. A similar effect can be seen for the other moments (in the terms that are first order in A). We also see that the mean of higher powers of momentum up to p4 are involved in the terms that are second order in A. Whilst in the standard case a contribution to the stochastic 96 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS term proportional to sin φ is only present for the mean momentum we see that for the modified commutator it is also present in the other equations for φ 6= mπ, m ∈ Z. We now substitute the Hamiltonian of a simple harmonic oscillator H = mΩ2x2/2+ p2/2m and apply Gaussian variational ansatz for the state. The use of a Gaussian variational ansatz may be justified by the assumption that the modification to the standard canonical commutator is small, together with the fact that in standard quantum mechanics general pure states evolve on fast timescales to approximately Gaussian states under the stochastic Schrodinger equation and Hamiltonian we consider [173, 174]. If the intial state is Gaussian then the evolved state is also exactly Gaussian [175]. Note that the Gaussian nature of the state was not assumed in the derivation of the trajectory equation. For the commutator (4.28), applying Wick’s theorem we can show that:

" # hpi 3 hpi (∆p)2 + hpi3 d hxi = + A dt + 2p2ζ cos φ (∆x)2 dW (4.32) m m The first term here is the standard mean velocity, but in addition to it is an effective velocity term that is non-linear as a function of mean momentum: ! 3 hpi (∆p)2 + hpi3 δ (d hxi) = A dt (4.33) m Furthermore unlike the standard case there is a dependence on a second moment (momentum variance). We emphasise that these are properties of the Hamiltonian evolution, as also seen in the previous chapter, rather than the continuous measurement process. Similarly we can show through equation (4.23) that the extra change in motion to d h(∆x)2i due to the modified commutator is: 6 δ d (∆x)2 = A h∆x∆pi hpi2 + (∆p)2 dt (4.34) m where under the Gaussian variational ansatz the stochastic term in equation (4.22) has vanished since in this case the coefficient of cos φ there is equal to a third cumulant of the state. Again this change in motion is purely due to Hamiltonian evolution. With the modified commutator if either the mean or variance of momentum is large it will mean a larger rate of change of position variance. Here unlike the standard case there is a dependence on a first moment (mean momentum). Also from equation (4.16), to first order in A: 2 2 2 2 δ (d hpi) ' −AmΩ hxi (∆p) + hpi + 2 hpi h∆x∆pi + 2ζ~ hpi dt n p 2 2 o − A~ 2ζ sin φ hpi + (∆p) dW (4.35) As for the differential of mean position (4.32) there is a dependence on the momentum variance, but also on the other second moment h∆x∆pi. We also see an additional stochastic contribution, as noted earlier. 4.3. MODIFIED CANONICAL COMMUTATION RELATIONS 97

For the variance of momentum we ﬁnd, to ﬁrst order in A:

2 2 2 2 d (∆p) ' −2mΩ h∆x∆pi + 2ζ ~ cos φ + 2~ sin 2φ h∆x∆pi −4 cos2 φ h∆x∆pi2 − 2AmΩ2 3 (∆p)2 h∆x∆pi + hpi 2 hxi (∆p)2 + hpi h∆x∆pi 2 2 2 2 +4ζ~A ~ (∆p) + hpi cos φ + (∆p) + (∆p)2 + hpi2 h∆x∆pi sin 2φ dt p 2 +4 2ζ −A~ sin φ hpi (∆p) dW (4.36) The ﬁnal term of the diﬀerential of the covariance matrix is: ( (∆p)2 d h∆x∆pi = − mΩ2 (∆x)2 + 2ζ sin 2φ − 4 cos2 φ h∆x∆pi (∆x)2 m ~ 3A + (∆p)2 (∆p)2 + hpi2 m −AmΩ2 (∆x)2 (∆p)2 + hpi2 + 2 h∆x∆pi (h∆x∆pi + hxi hpi) 2 2 2 +2Aζ~ ~ h∆x∆pi + (∆x) sin 2φ (∆p) + hpi dt n p o − 2A 2ζ~ sin φ (hpi h∆x∆pi) dW (4.37)

Under the Gaussian variational ansatz the cos φ terms that generally appear in the stochastic parts of both of these equations vanish, but the other terms remain if φ 6= mπ, m ∈ Z. This highlights a crucial difference: for the standard case the evolution of both of these moments is deterministic whilst that for the modified commutator is stochastic. Both the first moments hxi , hpi are now involved in these equations (for all values of φ), unlike in the standard case. For the change in momentum variance there is a contribution which we may interpret as an effective additional backaction, 2 2 2 2 2 4ζ~ A (∆p) + hpi cos φ + (∆p) , distinguished by the presence of ζ, ~ and that it is strictly greater than or equal to zero. Note the term here independent of φ, which is not found in the standard case. This effective back action grows with both increasing momentum mean and variance, the latter in isolation having the effect of an exponential growth of (∆p)2 . This may be called runaway backaction. For the scenario where φ = 0 the additional terms to the case of standard canonical commutator are:

δ d (∆p)2 = A −2mΩ2 3 (∆p)2 h∆x∆pi + hpi 2 hxi (∆p)2 + hpi h∆x∆pi 2 2 2 +4ζ~ 2 (∆p) + hpi dt (4.38)

3 δ (d h∆x∆pi) = A 2ζ 2 h∆x∆pi + (∆p)2 (∆p)2 + hpi2 ~ m −mΩ2 (∆x)2 (∆p)2 + hpi2 98 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS

+2 h∆x∆pi (h∆x∆pi + hxi hpi)]} dt (4.39)

In this case the additional back-action term is maximum and not diminished by a term that depends on ζ and the position-momentum correlator (which is present for other values of φ). The terms involving means of position and momentum in both equations are stochastic and so make the overall evolution stochastic, even though the equations are ‘explicitly’ deterministic. As another example we look at the canonical commutation relation:

2! p 2 p 2 2 [x, p] = i~ 1 − γ0 + γ0 = i~ 1 − Cp + C p (4.40) MP c MP c where γ0 is a dimensionless parameter C = γ0/MP c. This is considered in [113] as a part of the theory of ‘doubly special relativity’. We choose to focus on the scenario where φ = 0. First we have here:

2 2 2 2 2 2 3 2 4 2 2 h[x, p]i = −~ 1 − 2C hpi + C hpi + 2C p − 2C hpi p + C p (4.41)

2 2 2 3 3 h[[x, p] , x]i = −~ C 1 − 3C hpi + 3C p − 2C p (4.42) These imply that the additional terms produced by this commutation relation to the differentials for the standard commutation relation are to first order in C, for a harmonic oscillator and gaussian state: " # (∆p)2 + hpi2 δ(d hxi) = C − dt (4.43) m 2 2 δ(d hpi) = C mΩ (h∆x∆pi + hxi hpi) − ζ~ dt (4.44) 4 hpi h∆x∆pi δ(d (∆x)2 ) = C − dt (4.45) m 2 2 2 2 δ(d (∆p) ) = C mΩ 2 hxi (∆p) + hpi h∆x∆pi − 4ζ~ hpi dt (4.46) " # −2 hpi (∆p)2 δ(d h∆x∆pi) = C + mΩ2 hpi (∆x)2 + hxi h∆x∆pi dt m 2 2 + 3C ζ~ h∆x∆pi dt (4.47) The most striking difference between the dynamics under this model compared to that under the previous commutation relation is that the continuous measurement does not give rise to an exponential growth of the momentum moments. In particular there is now a constant mean ‘force’ −Cζ~2 on the particle (the sign here is due to the negative sign in the modified commutator). This is a strong constraint on the model since even with zero mean momentum this force exists. It can be interpreted as a shift of the oscillator origin, though note it exists even if the resonance frequency is zero. The momentum variance is now coupled to the mean momentum through the term 4.4. DISCUSSION 99

−4Cζ~2 hpi, which implies that when the mean momentum becomes negative it can result in a positive growth in momentum variance. This growth would at long times be faster than the standard backaction force but slower than the runaway backaction that we encountered earlier. In the correlator between position and momentum it can be observed that the lowest order contribution from the continuous measurement is second order in C, which suppresses its eﬀect compared to the other terms. However when treated in isolation it causes an exponential growth in the correlator, as with the previous commutator. A commutation relation derived in [111] via a set of principles is:

s p 2 2 ( c ) + m [x, p] = i~ 1 + 2µ0 2 (4.48) MP where m is the mass of the particle and µ is again a dimensionless numerical pa- √ 0 rameter. For m p/c M / µ and for µ = β , this commutator reduces to P 0 0√ 0 that of [61]. In the other limit p/c m MP / µ0 the commutator reduces to:

m2 [x, p] = i~ 1 + µ0 2 (4.49) MP which can be seen as a mass-dependent rescaling of ~. Then in this approximation of the commutator, and if we set φ = 0, the expressions for d(h∆x∆pi) and d hpi are the same as with the standard commutator. However the back-action term is 2 2 2 rescaled by a factor (1 + µ0m /MP ) .

4.4 Discussion

Now we will discuss implications of the additional terms in the dynamics due to the modified canonical commutation relation (4.28). We will focus on the behaviour that is a specific to continuous measurement. Firstly we examine the change in the mean momentum, equation (4.35). As we noted earlier there is a term 2ζA~2 hpi which gives rise to an exponential drift when taken in isolation. However in the presence of dissipation (which we have not included in our equations) there would be another similar term but with opposite sign. If the coupling of the meter and system is sufficiently large it would be possible to probe the exponential divergence. This would be a strong signature for the modified commutator as standard quantum mechanics implies the mean momentum diffuses linearly with time. Alternatively if the dissipation is significant the net result would be a renormalised dissipation dependent on the coupling. More specifically, the drift term affects the existence of steady-state solutions to the equations. In the absence of dissipation, according to standard quantum mechanics, the mean position and momentum do not reach a steady-state. However the equations for the second order moments are uncoupled from the first-order moments and have a steady-state solution provided that φ 6= π/2, 3π/2. With dissipation or feedback all moments 100 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS can reach a steady-state, as demonstrated in [70]. However with the modified commutator considered here the equations of all moments are coupled. This implies that without dissipation there is no steady-state solution even for the second-order moments. Furthermore, even in the presence of dissipation, for sufficiently large coupling meter-system coupling or β0 the drift-term can overtake the dissipation term so that none of the moments reach a steady-state. This behaviour would be expected to manifest in long-time measurements of such systems. Another significant contribution that can be probed in an experiment is the size of the backaction force. The additional backaction force in equation (4.36) depends on the momentum-dependent moments of the covariance matrix as well as φ. If an experiment is able to show that the additional backaction is smaller than the 2 2 2 2 −2 standard backaction then: β0 < (MP c) /2 (∆p) + hpi + (∆p) cos φ . We illustrate in figure 4.1 the dependence of this bound as a function of φ for some sample values of the moments . Note that at φ = π/2 the bound reaches zero, since for that scenario the standard back-action vanishes but the additional backaction remains as it contains a φ-independent term. Generally to obtain a tighter bound it is optimal to increase the value of each of the momentum moments (of the two it is preferable to increase the momentum variance). Perhaps a stronger bound can be achieved from the fact that evolution of the momentum variance is stochastic in the modified theory. For this it would be appropriate to examine the deviation of momentum-variance over several trajectories in a constant time interval. We can compare the analysis here with that of the previous chapter. There the β0 bound is derived from comparison of the standard and modified position noise spectra of an optomechanical system, involving an oscillator in a thermal state. Even though this is in general a different setup to here, under certain conditions it can be regarded as an approximately particular case. Firstly the cavity field should adiabatically eliminated and the probe frequency much larger than the oscillator’s resonant frequency (the free-mass limit). This amounts to considering only the fast-time dynamics. Also contributions from quantum zero-point energy should be neglected. 2 Then for φ = 0 = hpi and applying the equipartition theorem (∆p) = mkBT the expression here for the bound matches, up to an O(1) factor, that obtainable from the ratio of position noise spectra 3.16 in the previous chapter. But, as noted there, probing close to resonance could improve the bound by approximately the quality factor of the oscillator. 4.4. DISCUSSION 101

1019 18

0 0 /2

Figure 4.1: Upper bound on β0 as a function of φ, deduced from setting the backaction noise under the modiﬁed commutator (4.28) to be smaller than the standard backaction noise. Here as a reference for macroscopic oscillators we set m = 10kg, T = 300K, which for example corresponds to the reduced mass of aLIGO mirrors 2 2 [123]. For each blue curve hpi is set to mkBT and (∆p) is set to the number shown above, whilst for the red curves vice versa holds. 102 4. MODIFIED COMMUTATORS AND CONTINUOUS POSITION MEASUREMENTS Chapter 5

Conclusion

In this thesis we have explored ways of probing the foundations of quantum mechanics from a broad perspective. We set out with the question of how the statistics of quantum mechanics can constrain the possibility of a more complete framework and a consistent theory of quantum gravity. In relation to the former idea, the EPR-steering phenomenon shows that completeness of the theory entails the capacity to affect arbitrarily remote systems i.e. nonlocality. At the same time, Bell’s theorem shows that a more complete theory would also seem to have nonlocality. So it has been of interest to find inequalities that can experimentally detect these notions of nonlocality, and also to shed light on the relationship between these strange aspects of quantum mechanics. Here an inequality was shown to be a necessary and sufficient witness of EPR- steering using the correlations between two projective measurements on each side of a two qubit system, as in the CHSH inequality. In other words, an analog of the CHSH inequality for EPR-steering was constructed. The set of unsteerable correlations was seen to form a ellipsoid structure unlike the convex polytope structure of Bell inequalities. Furthermore the violation of this inequality does not depend upon the measurements made on the trusted party, for a given entangled state. For the scenario involving POVM’s the set of unsteerable correlations is also a convex set but of piecewise arcs of several ellipses and their tangents, so a simple inequality of this type does not hold. This intertwined with another direction, the violation of Bell inequalities. Specifically it was shown that if this witness shows EPR-steering on a quantum state then the CHSH inequality must be violated. Whilst it is known that for general states and measurements these concepts of nonlocality are strictly separate, the results here have shown that the gap closes if restricted to the CHSH- type correlations. In future work it is natural to seek necessary and sufficient inequalities for more than two parties and several POVMs on each site. This could further illuminate the correspondence with Bell nonlocality that has been uncovered here. It would also have applications in quantum networks involving both trusted and untrusted parties. As seen in section 1.4.1 witnessing Bell nonlocality is computationally hard for higher dimensional systems, but it could be simpler for EPR-steering guided by the study into the geometry of unsteerable correlations here. The problem can also be cast in

103 104 5. CONCLUSION the language of semi-definite-programming. The exact computational complexity of the problem (from a computer science perspective) has yet to be uncovered. Constraints on quantum gravity theories have been difficult to find, due to generally high energies required in experiments. But it has been promising to look in low energy experiments for modifications of the canonical commutation relations. These commutators imply the existence of an effective minimal length in nature and have been posited in theories like string theory and doubly special relativity. Notably new commutator bounds could be found on such commutators by utilising the sensitivity of position measurements, for example through the field of optomechanics. We saw here that the modified commutator found in [61] can be probed via the position noise spectrum of a simple harmonic oscillator interacting with an optical field. Two frequency regimes were seen to be optimal to detect possible deviations of the spectrum; probing just off the oscillator resonance, and probing in the free- mass limit. Recent observations of radiation pressure shot noise as well as the noise spectrum of advanced LIGO enabled to derive bounds on the commutator for the centre of mass of an oscillator, as well as its elementary components. Indeed it was demonstrated that controlling certain parameters of such experiments to increase the signal to noise ratio can make it possible for bounds that surpass those from sub-atomic measurements. To extend the work here the first priority would be to reach the target parameter values defining an optomechanical cavity or a general precision measurement apparatus, to gain advantage over previous experiments. In addition advanced statistical and quantum metrological techniques to probe noise will assist in improving commutator bounds. Expanding on the differences between various modified commutators on tabletop experiments can form a passage to distinguish theories of quantum gravity at an empirical level. We have also examined the properties of continuous position measurement of a system which obeys this modified commutator. From the associated quantum trajectory equation the moments of the system covariance matrix were derived for general system states and Hamiltonians, as well as the specific case of Gaussian state and simple harmonic oscillator. Key features that separate behaviour from the standard commutator include the exponential growth of mean momentum and backaction force, which can be constrained strongly by a variety of experiments, and stochastic evolution of the momentum variance. A large momentum variance is ideal for the ratio between modified and standard backaction, consistent with the results for the optomechanical scenario. Interestingly, this exponential growth is not present for a modified commutator that is associated with theory of doubly special relativity, but instead a constant negative force emerges that implies a shift in the origin of a harmonic oscillator . These results can be applied to any system that involves continuous position measurement, and it may be easier to probe the runaway backaction in experiments other than optomechanics. Long-time measurements on such systems should give strong constraints on the commutators. Furthermore it could also be relevant in other scenarios modelled by the quantum trajectory equation we have considered, for example 105 quantum state diffusion to simulate a system coupled to a bath of oscillators, the stochastic collapse of wavefunctions according to the GRW interpretation of quantum mechanics, and even the evolution of the universe according to the decoherent histories formalism. A consideration of other models of continuous measurement, for example by having a modified meter commutator at some level, will in general lead to different dynamics. 106 5. CONCLUSION Bibliography

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