Gravity As a Classical Information Channel: Consequences and Proposals Towards Detecting Wave Function Collapse

Gravity as a classical information channel: Consequences and proposals towards detecting wave function collapse

Kiran E. Khosla BSc (Hons), BA

A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2018, School of Mathematics and Physics. Abstract

A consistent theory including gravitational effects in quantum mechanics has not been found. Although many different approaches have been proposed for theory of quantum gravity, several authors have suggested that gravity may be fundamentally classical. As there is no experimental evidence for a fundamentally quantum vs. fundamentally classical theory of gravity, it is hoped that investigating the consequences of semiclassical gravity, one may propose an experiment to distinguish between the quantum or classical nature of gravity. As opto- and electromechanical devices allow quantum limited measurement and control over massive, mechanical degrees of freedom, they are ideal systems to test violations of standard quantum mechanics arising from gravitational effects. After a brief introduction to the background theory and literature review, we begin by introducing a semiclassical model of gravity based on quantum measurement and control. We show under which conditions the theory behaves like a collapse model, and how a Newtonian potential is recovered in the multi particle case. We then go on to analyse our semiclassical model in two scenarios: firstly in a cosmological setting, and secondly by considering quantum clocks on the surface of the earth. In the cosmological description, we show that a violation of energy conservation, resulting from our model, can be reinterpreted as a dark energy fluid in the spacetime, suggesting that perhaps local violations of conservations of energy may be resolved in a cosmological framework. Secondly, we derive a minimum, gravitational dephasing rate for quantum clocks under the semiclassical model, which unsurprisingly, is is well below the resolution of current state of the art clocks. The next chapters diverge from analysing the gravitational model, and instead focus on novel methods of manipulating and measuring micromechanical devices, that may be used to experimentally probe the results of the semi-classical theory. Firstly we show that a sequence of coherently controlled pulses in an optomechanical system can be used to directly probe the momentum of an oscillator - opening up the possibility for direct detection of momentum diffusion, which is a consequence of the semiclassical gravity model. Secondly, we introduce an electro-mechanical device, and sequence of control operations, that may be understood as an interferometer for the wave function of a mechanical oscillator coupled to a qubit. We show how the

i PREFACE ii resulting interference pattern can distinguish between unitary evolution, thermalisation/decoherence, and a purely classical description of the mechanical oscillator. We also show how the qubit-oscillator interaction can be used generate arbitrary wave functions of the mechanical oscillator, using only classical qubit driving. iii PREFACE iv Declaration by author

This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly tated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, signiﬁcant technical procedures, professional editorial advice, ﬁnancial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary in- stitution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the Univer- sity Library and, subject to the policy and procedures of The University of Queens- land, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright (manuscripts in preparation) of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis.

Kiran E. Khosla PREFACE v Publications included in this thesis

In order of appearance in this thesis

[1] Natacha Altamirano, Paulina Corona-Ugalde, Kiran E. Khosla, Gerard J. Milburn, and Robert B. Mann, Emergent dark energy via decoherence in quantum interactions, Class. Quant. Grav, 34 115007, (2017).

Contributor Statement of Contribution % K. E. Khosla Theoretical derivations 50 Numerical calculations 10 Preparation of manuscript 30 Conceptual analysis 40 N. Altamirano Theoretical derivations 25 Numerical calculations 40 Preparation of manuscript 30 Conceptual analysis 40 P. Carona-Ugalde Theoretical derivations 25 Numerical calculations 40 Preparation of manuscript 25 Conceptual analysis 20 R. B. Mann Conception and design 50 Numerical calculations 10 Preparation of manuscript 10 Supervision and guidance 50 G. J. Milburn Conception and design 50 Preparation of manuscript 5 Supervision and guidance 50 PREFACE vi

[2] Kiran E. Khosla, Natacha Altamirano, Detecting gravitational decoherence with clocks: Limits on temporal resolution from a classical-channel model of gravity, Phys. Rev. A, 95, 052116 (2017).

Contributor Statement of Contribution % K. E. Khosla Conception and design 55 Theoretical derivations 50 Numerical calculations 50 Preparation of manuscript 50 N. Altamirano Conception and design 45 Theoretical derivations 50 Numerical calculations 50 Preparation of manuscript 50

[3] Kiran E. Khosla, George A. Brawley, Michael R. Vanner and Warwick P. Bowen, Quantum optomechanics beyond the quantum coherent oscillation regime, Optica 11 1382, (2017).

Contributor Statement of Contribution % K. E. Khosla Theoretical derivations 90 Numerical calculations 100 Preparation of manuscript 80 G. A. Brawley Theoretical derivations 5 Preparation of manuscript 5 Supervision and guidance 5 M. R. Vanner Theoretical derivations 5 Preparation of manuscript 5 Conception and design 10 Supervision and guidance 10 W. Warwick P. Bowen Conception and design 90 Preparation of manuscript 10 Supervision and guidance 85 PREFACE vii

[4] Kiran E. Khosla, Michael R. Vanner, Natalia Ares and Edward A. Laird, Dis- placemon electromechanics: how to detect quantum interference in a nanomechanical resonator, Phys. Rev. X., 8–021052, (2018).

Contributor Statement of Contribution % K. E. Khosla Theoretical derivations 70 Numerical calculations 80 Analysis of results 40 Preparation of manuscript 40 M. R. Vanner Supervision and Guidance 10 Concept and design 10 Analysis of results 10 Preparation of manuscript 10 N. Ares Supervision and Guidance 10 Concept and design 10 Analysis of results 20 Preparation of manuscript 10 E. A. Laird Theoretical derivations 30 Numerical calculations 20 Supervision and Guidance 80 Concept and design 80 Analysis of results 30 Preparation of manuscript 40 PREFACE viii Submitted manuscripts included in this thesis

[5] Kiran E. Khosla, Arbitrary state preparation of a mechanical resonator via controlled pulse shaping and projective measurement in a qubit-resonator interaction, arXiv:1808.05666 (2018).

Contributor Statement of Contribution % K. E. Khosla Sole authorship 100

[6] Kiran E. Khosla, Stefan Nimmrichter, Classical Channel Gravity in the New- tonian Limit, arXiv:1812.03118, (2018).

Contributor Statement of Contribution % K. E. Khosla Conception and design 50 Theoretical derivations 50 Numerical calculations 60 Preparation of manuscript 20 Stefan Nimmrichter Conception and design 50 Theoretical derivations 50 Numerical calculations 40 Preparation of manuscript 80

Other publications during candidature

1. James S Bennett, Kiran Khosla, Lars S Madsen, Michael R Vanner, Halina Rubinsztein-Dunlop and Warwick P Bowen, A quantum optomechanical interface beyond the resolved sideband limit, New J. Phys, 18–053030 (2016). PREFACE ix Contributions by others to the thesis

Four chapters of this thesis are published works, as such significant contributions to this thesis have been made by other authors. Natacha Altamirano, Paulina Corona- Ugalde, Gerard Milburn and Robert Mann, each made significant contributions to the results and text of the published version of Ref. [1] (Chapter 3). The work in this chapter was initially developed by Gerard Milburn and Robert Mann. The published work in Ref. [2] (Chapter 4) was developed and written entirely in collaboration with Natacha Altamirano. Warwick Bowen, George Brawley and Michael Vanner each made contributions to the editing and wording in the published version of Ref. [3] (Chapter 5), with Warwick Bowen coming up with the original idea. Finally Edward Laird thought of the initial idea for Ref. [4] (Chapter 6) and along with Natalia Areas wrote most of the first two sections of and did the modeling for the displacemon architecture. Edward, Natalia, and Michael Vanner contributed heavily to the editing and polishing of the final publication. I would also like to thank James Bennett, Marco Ho, Amie Khosla, and Carolyn Wood for their help in editing this thesis.

Statement of parts of the thesis submitted to qualify for the award of another degree

No works submitted towards another degree have been included in this thesis.

Research Involving Human or Animal Subjects

No animal or human subjects were involved in this research. Acknowledgments

This thesis would not exist without the help and support of my supervisors, colleagues, friends and family. I’d first like to thank my supervisors Gerard Milburn, and Warwick Bowen for their fantastic support during my candidature. I feel privi- leged to have had your time, guidance and especially your support during my candidature, and especially your encouragement in sending me to conferences, and to work with, and introduce me to, overseas collaborators. On that note I’d like to thank Robb Mann, who acted as my supervisor during my time in Waterloo, and showed a level of hospitality and interest above and beyond any expectation. All administrative support is thanks to the efforts of Angie Bird, Ruth Forrest, Kaerin Gardner, Danette Peachey, Tara Robertson, Jenny Robinson, Lisa Walker, and Sam Zammit. In particular I’d like to extend my gratitude to Murray Kane, who was the best postgrad administration officer one could have asked for. My co-authos: Natacha Altamirano, Natalia Ares, James Bennett, George Braw- ley, Paulina Corona-Ugladem, Edward Laird, Lars Madsen, and Michael Vanner, thank-you for all your contributions and for making each project so enjoyable to work on. The work in this thesis could not have happened without all of your help. To all my friends and colleagues, Natacha Altamirano, Ardalan Armin, Sahar Basiri-Esfahani, James Bennett, George Brawley, Rob Chapman, Paulina Corona- Ugladem, Andrew Cramer, Emma Firkins-Barriere, Stefan Forstner, Glen Har- ris, Marco Ho, Sammy Hood, Susanne Kaufmann, Anatoly Kulikov, Juan Loredo, Nicholas Mauranyapin, Subhayan Moulik, Anna Pearson, Robbie Pierce, Tim Puck- ering, Eric Romero, Eugene Sachkou, Jon Swaim and Carolyn Wood. Thanks for all our stimulating talk on physics, science, and life, and for reminding me of the social side of a PhD. George, our chats in the EQuS tea room didn’t help finishing any projects, but I did learn a lot of physics. My family have been incredibly supportive and always made sure I didn’t get buried by work. Thanks Mum, Dad, Liz, Peter, and Amie for all you advice and encouragement over the past few years. Finally Franciska, thanks for getting me though some difficult patches, and know- ing how to lovingly deal with a exhausted and grumpy PhD student. I would not have finished if you didn’t continuously reminded me that I love physics.

x ACKNOWLEDGMENTS xi

Thanks all! ACKNOWLEDGMENTS xii Financial support

This research was supported by the Australian Government Australian Postgraduate Award (APA) and UQ Advantage scholarships. Travel was supported by ARC Centre of Excellence for Engineered Quantum Systems. I would like to thank The University of Waterloo, The Perimeter institute, The University of Oxford, and The National University of Singapore for their kind hospitality and support during visits, and where some of the following research was conceived and conducted.

Keywords

Gravitational decoherence, quantum collapse models, open quantum systems, quantum optomechanics, displacemon, quantum information

Australian and New Zealand Standard Research Classiﬁcations (ANZSRC)

ANZSRC code: 020604, Quantum Optics, 30% ANZSRC code: 020699, Quantum Physics not elsewhere classiﬁed, 40% ANZSRC code: 020603, Quantum Information, Computation and Communications, 30%

Fields of Research (FoR) Classiﬁcation

FoR code: 0206, Quantum Physics, 100% Contents

Preface i

Acknowledgments x

List of Figures xvii

List of Tables xviii

List of Common Abbreviations and Symbols xix

1 Introduction 1 1.1 Quantum mechanics and General Relativity ...... 1 1.2 Thesis outline ...... 2 1.3 Cavity optomechanics ...... 3 1.3.1 Optomechanical Hamiltonian ...... 4 1.3.2 Linearised optomechanical Hamiltonian ...... 8 1.3.3 Pulsed optomechanics ...... 11 1.3.4 The Displacemon ...... 12 1.4 Quantum control ...... 14 1.4.1 Quantum measurement ...... 15 1.4.2 Quantum feedback control ...... 22 1.5 Friedmann Robertson Walker space time ...... 25 1.5.1 The metric ...... 25 1.5.2 Equations of motion ...... 26 1.5.3 Energy-momentum in the FRW metric ...... 28 1.6 Summary ...... 28

List of Common Symbols 1

2 State of the Art 30 2.1 Literature Review ...... 30 2.1.1 Collapse theories and gravitational decoherence ...... 30

xiii CONTENTS xiv

2.1.2 Opto and nano-mechanical systems ...... 40 2.2 CCG: Non-linear description ...... 45 2.2.1 The Lindblad operator ...... 45 2.2.2 Systematic dynamics of the measure-feedback model . . . . . 49 2.2.3 Particle noise scaling ...... 51 2.2.4 Heating rate ...... 53 2.2.5 Summary ...... 53 2.3 Arbitrary quantum state engineering ...... 54 2.3.1 Model ...... 54 2.3.2 Generating arbitrary measurement operators ...... 57 2.3.3 Generating arbitrary quantum states ...... 61 2.3.4 Summary ...... 63

3 Emergent dark energy in a FRW cosmology 64 3.1 Foreword ...... 64 3.2 Emergent dark energy ...... 65 3.2.1 Introduction ...... 65 3.2.2 Relativistic Classical Channel Gravity ...... 67 3.2.3 Model ...... 71 3.2.4 Dark Energy from Decoherence ...... 74 3.2.5 Conclusions ...... 79 3.3 Chapter appendices ...... 82 3.3.1 Newtonian Classical Channel Model ...... 82 3.3.2 Master Equation ...... 84 3.3.3 Solutions to the evolution equations ...... 86

4 Gravitational decoherence of quantum clocks 88 4.1 Foreword ...... 88 4.2 CCG decoherence of clocks ...... 89 4.2.1 Introduction ...... 90 4.2.2 Coupled clocks ...... 91 4.2.3 Multiparticle interaction ...... 93 4.2.4 Clocks in earths gravitational ﬁeld ...... 95 4.2.5 Conclusions ...... 98 4.3 Chapter appendices ...... 99 4.3.1 Summary of CCG ...... 99 4.3.2 Dephasing Rate Minimization ...... 101 CONTENTS xv

5 Quantum optomechanics beyond the QCO regime 103 5.1 Foreword ...... 103 5.2 Quantum optomechanics beyond the QCO regime ...... 104 5.2.1 Introduction ...... 104 5.2.2 Model ...... 105 5.2.3 Nonunitary evolution ...... 109 5.2.4 Measurement and tomography ...... 111 5.2.5 Squeezed state preparation ...... 112 5.2.6 Summary ...... 114

6 The displacemon 116 6.1 Foreword ...... 116 6.2 Displacemon electromechanics ...... 117 6.2.1 Introduction ...... 117 6.2.2 Model ...... 118 6.2.3 Generating and measuring mechanical quantum interference . 121 6.2.4 Conclusion ...... 131 6.2.5 Acknowledgements ...... 132 6.3 Chapter appendices ...... 132 6.3.1 Device parameters ...... 132 6.3.2 Cooling via qubit measurement ...... 133 6.3.3 Decoherence ...... 135 6.3.4 Classical interference patterns ...... 140 6.3.5 Other device implementations ...... 141

7 Conclusion 142 7.1 Classical channel gravity ...... 142 7.2 Opto and Electromechanics ...... 143

Bibliography 146 List of Figures

1.1 Canonical optomechanical system ...... 4 1.2 Optical phase space and optomechanical correlations ...... 9 1.3 Canonical displacemon architecture ...... 13 1.4 Weak quantum measurement ...... 16

2.1 Measurement protocol for arbitrary state preperation ...... 57 2.2 Realization of arbitrary measurement operators ...... 60

3.1 Comparison of CCG model to unitary dynamics, for FRW universe and Newtonian gravity ...... 68 3.2 Scale factor dynamics in a classical channel dynamics in a flat, open and closed FRW universe ...... 75 3.3 Scale factor dynamics in a classical channel FRW universe, for different localization rates ...... 76 3.4 Scale factor evolution for different initial states ...... 78 3.5 Equation of state for the effective matter components ...... 80

4.1 Quantum circuit diagram for diﬀerent implementations of the CCG model ...... 94

5.1 Schematic for implementation of backaction evading measurements of an arbitrary mechanical quadrature with pulsed optomechanics . . 107 5.2 Conditional variance results for diﬀerent quadrature measurements . . 113 5.3 Conditional momentum quadrature variance in the ﬁnal mechanical state ...... 114

6.1 Concentric transmon design for a displacemon ...... 126 6.2 Cooling of a mechanical resonator using a displacemon architecture . 127 6.3 Ramsey sequence implementing a diﬀraction grating for the wave function of a mechanical oscillator ...... 127 6.4 Schematic for a Mach-Zender interferometer using the displacemon architecture ...... 128

xvi LIST OF FIGURES xvii

6.5 Wigner functions of the mechanical state at the output of the interferometer ...... 130 6.6 Using the probability of qubit measurement to detect interference in the wave mechanical function ...... 131 6.7 Signature of classical behavior of the mechanical resonator in the displacemon system ...... 140 List of Tables

4.1 Dephasing rate scalings for diﬀerent models of classical channel gravity 95

6.1 Displacemon parameters using for experimental realizations ...... 139

xviii List of Common Abbreviations and Symbols

a ...... Optical annihilation operator, or scale factor ain, aout ...... Input and output optical ﬁelds α, β ...... Classical amplitudes b ...... Mechanical annihilation operator c ...... Speed of light CCG ...... Classical Channel Gravity COM ...... Center of Mass CSL ...... Continuous Spontaneous Localisation dW ...... Wiener increment ∆ ...... Optomechanical detuning, ωd − ωc E(·) ...... Classical/ensemble average of · φ(x) ...... Newtonian potential FRW ...... Friedmann-Robertson-Walker g0 ...... Single photon optomechanical coupling rate g Boosted optomechanical coupling rate or metric determinant G ...... Newtons gravitational constant GRW ...... Ghirardi-Rimini-Weber γ ...... Mechanical decay rate, or qubit dephasing rate H(I0) ...... Hamiltonian (interaction or free) H.C...... Hermitian Conjugate ~ ...... Plancks constant kB ...... Boltzmans constant κi,e ...... Intrinsic/extrinsic cavity coupling rates P κ ...... Total optical (or microwave) cavity decay rate, j κj λ ...... Wavelength or interaction strength L[ρ] ...... Lindblad operator on ρ m ...... Mass µ(x) ...... Mass density operator n,¯ N¯ ...... mean phonon, and photon numbers (respectively) N [0, 1] ...... Gaussian random variable of mean zero, variance one. O ...... Arbitrary operator

xix LIST OF COMMON ABBREVIATIONS AND SYMBOLS xx

PM(L) . . . . Momentum (phase) quadrature of mechanical (optical) mode ρ(msl) Density operator, with subscript for mechanical, system, light etc. SSE ...... Stochastic Schroedinger equation SNE ...... Schroedinger-Newton equation σ . . . . . Spatial resolution (measurement model), standard deviation σx(yz+−) ...... Pauli spin projection, raising and lowering operators t, τ ...... Time intervals T ...... Temperature U(t) ...... Unitary operator Ω ...... Angular frequency ωc(lmq) ...... Angular frequency, with subscripts denoting cavity, light, mechanics, qubit XM(L) . . . Position (amplitude) quadrature of mechanical (optical) mode Υ(·) ...... Measurement operator with measurement result (·) 1 ξ(t) ...... Thermal white noise process with variancen ¯ + 2 Chapter 1

Introduction

1.1 Quantum mechanics and General Relativity

Quantum mechanics and general relativity are the two cornerstones of modern physics. Each theory has been extensively tested over a century and no discrep- ancy between theory and experiment has been observed [7, 8, 9, 10, 11]. However, it is well known that these two theories are incompatible due to their contradictory conclusions about the underlying structure of the universe [12, 13, 14, 15]. Many theories and methods have been proposed to resolve this disparity, the most well known of which are string theory [16] loop quantum gravity [17, 18, 19, 20] and holography [21, 22, 23]. Although these approaches have made significant progress in developing the theory, a consistent approach to unifying quantum mechanics and gravity remains elusive. There has been a great deal of recent attention focusing on gravitational effects in quantum mechanics [24, 25, 26, 27, 27, 28, 29], and in this direction a recent model proposed and extrapolated on by Kafri, Taylor and Milburn [30, 31, 32] proposes a path to introduce gravity into quantum mechanics without requiring a theory of quantum gravity. Their recipe is similar to previous work by Diosi [33, 34, 35, 36] which proposes gravity is a fundamentally classical theory, and the unification of the two requires a modification to quantum mechanics. The theory proposes that gravitational interactions are mediated by a classical information channel. It goes beyond the theory of quantum mechanics on curved space-time [37, 38], in which gravity plays a passive role, and allows – in principle – a dynamical interaction between a classical space-time, and quantum particles/fields. A striking feature of the model is that it does not allow macroscopic quantum superpositions. The notion that gravity may be responsible for the emergence of classicality at the macroscopic level dates back to the seminal work of Diosi [34] and Penrose [39], which built on previous ideas of wave function collapse theories [40,

1 1.2. THESIS OUTLINE 2

41]. The recent model of Kafri et al. naturally includes a gravitationally induced decoherence rate, with a less heuristic derivation than the previous work of Diosi and Penrose, and with a clear generalization to a relativistic description (although not entirely without ambiguities – as discussed in this thesis). This thesis does not aim to rewrite the standard model in this new framework, but instead has two main aims. The first: to properly formulate the model for experimentally relevant systems, and consider wider implications of the idea that gravity requires a classical information channel. The second: to propose experiments that may contribute towards falsifying or confirming its modification to unitary quantum mechanics.

1.2 Thesis outline

The rest of this chapter is dedicated to introducing the basic concepts and tools necessary to understand the central results of this thesis. The following sections give a brief introduction to cavity optomechanics, the displacemon, a new term introduced in chapter. 6, quantum control and the Friedmann-Robertson-Walker (FRW) space time. No original work (with the exception of the displacemon) is presented in these sections and the reader may choose to skip any or all of them if they are familiar with the field. The first section of chapter 2 is a literature review of approaches to coupling quantum and classical degrees of freedom (Sec. 2.1.1), and optomechanical systems (Sec. 2.1.2). The first original work is in chapter 2 (manuscripts in preparation) and is on further developing the model of Ref. [31] (Sec. 2.2) and a scheme for generating an arbitrary quantum state of a mechanical oscillator (Sec. 2.3). Chapters 3-6 each contain peer-reviewed publications (or submitted for publication) that have been an outcome of this doctoral research. Appendices from these publications appear within the chapters themselves and not in the appendices of the thesis. Finally, chapter 7 summarises the work contained within this thesis. The work in this thesis generally falls into two categories. The first category deals with the model of Classical Channel Gravity (CCG) itself and how it can be understood in different contexts or physical systems. Chapters 2.2, 3, and 4 fall under this first category, where the main focus is developing the model in more detail and understanding the phenomenological consequences. The second category, (Chapters 2.3, 5, and 6) is more concerned about how the consequences of CCG may be experimentally observed in table top experiments. CCG is a collapse theory in quantum mechanics, introducing stochastic dynamics to the usual Schrodinger or Heisenberg evolution, and there is an extensive body of work on how to detect experimental signatures of collapse theories [42, 43, 44, 45, 1.3. CAVITY OPTOMECHANICS 3

46, 47, 48]. Instead of contributing directly to this ﬁeld, the second category of this thesis contributes indirectly by proposing novel opto/nano-mechanical experiments. Even though the narrative of the ﬁnal publications focus on experimental novelty, this thesis does comment on how the proposals can be used to probe CCG.

1.3 Cavity optomechanics

The origin of modern optomechanics dates from the seminal papers of Braginsky and Manukin in 1967 [49] and Braginsky et al. in 1980 [50]. Braginsky noted as light reflects perpendicularly off a surface its momentum changes by 2~|k|, where k is the wave number of the light, and conservation of momentum requires that the surface from which the light reflected must also change by exactly 2~|k|. While the momentum transfer from a single quantum of light is negligible, modern optomechanical experiments use a variety of techniques to enhance the interaction strength between light and a mechanical element [51, 52, 53, 54, 55, 56, 57, 58]. Despite the wide variety of systems, they can almost all be understood with a relatively simple theory, which we will discuss now. State-of-the-art experiments and techniques are discussed in more detail in Chapter 2. In the following we will only discuss the so-called dispersive coupling regime in cavity optomechanics, where the motion of the reflective surface (the mirror) changes the energy in an optical field (i.e. the resonance frequency of a cavity). The model optomechanical system consists of a Fabry-Perot cavity where one of the mirrors is mechanically compliant in a harmonic trap [56, 59] (for example on a spring), Fig. 1.1. For simplicity we will assume this movable mirror has a reflectivity of unity, and the fixed mirror is semi-silvered allowing light to enter and leave the cavity (the input-output optical fields are discussed in a subsequent section). The mirror boundary conditions force the optical field inside the cavity to have nodes at the mirror surfaces, hence only certain wavelengths of light (satisfying nλ/2 = L for some integer n) are resonant in the cavity. If light enters the cavity, it will impart a momentum kick to the compliant mirror every time it is reflected. Over time1 the mirror moves, changing the length of the cavity from L to L + δL (we will assume δL L). If δL becomes too large then the light in the cavity no longer satisfies the resonance condition, nλ/2 = L, and will destructively interfere with itself reducing the amplitude of the field in the cavity. As the optical field decreases, it reduces the momentum kick imparted on the mirror, and the potential energy stored in the spring can now do work on the mirror, pushing it back to its equilibrium position. As the resonance condition is satisfied once more, light with wavelength nλ/2 = L may

1We won’t yet worry about the cavity decay times etc. 1.3. CAVITY OPTOMECHANICS 4 re-enter the cavity again imparting a momentum kick. This dynamical interplay between light in the cavity and mechanical motion is the core of optomechanical experiments.

Figure 1.1: A schematic of a canonical optomechanical system, consisting of a Fabry- Perot cavity with intracavity field a and a mechanically compliant mirror. The cavity can be driven via an input field ain though a semi-silvered mirror (with high reflectivity), with extrinsic input coupling rate κe. The reflected field aout carries information about the position of the mirror. A realistic cavity has some intrinsic loss rate κi giving a total cavity loss rate κ = κe + κi. The mechanically compliant mirror is centred at x = 0 and if displaced |x| > 0 the mirror resonates with angular frequency ωm and a decay rate γ. The mirror’s motion is actuated by the radiation pressure force of photons in the cavity.

1.3.1 Optomechanical Hamiltonian

The Hamiltonian for dispersive coupling can be intuitively obtained by considering a single mode of the canonical optomechanical system Fig. 1.1. Writing the Hamiltonian for both the optical ﬁeld and mechanical oscillator gives

p2 mω2 x2 H = ω (x)a†a + + m (1.1) ~ L 2m 2 where x and p are the position and momentum of the mirror of mass m, ωm the q mirror’s resonance frequency (ωm = k/m if the mirror is attached to a spring with spring constant k) and a the annihilation operator for the intracavity optical ﬁeld with resonance frequency ωL. When the mirror is in its equilibrium position the resonant the cavity admits a resonance at frequency ωL = 2πnc/L with c the speed of light. A single cavity mode is given by a particular choice of n, which will be left as arbitrary, with the assumption n 1. As the mirror moves from its equilibrium x = 0, the optical frequency becomes position dependent2

2πnc 2πnc 2πnc " x 2# ω (x) = = + x + O L L − x L L2 L " x 2# = ω + Gx + O (1.2) c L

2Here we will assume x > 0 decreases the length of the cavity 1.3. CAVITY OPTOMECHANICS 5

where ωc = 2πnc/L is the resonance frequency at zero displacement, and G = 2πnc/L2 is the coupling rate in units of Hz m−1. To ﬁrst order in x/L, the optomechanical Hamiltonian is given by,

s † † ~ † † H/~ = ωca a + ωmb b + G a a(b + b ) (1.3) 2mωm where the mirror degrees of freedom are expressed in annihilation (and creation) q mωm i ~ operators, b = (x+ p). By convention the bare coupling rate, g0 = G 2~ mωm 2mωm in units of Hz, is used to describe an optomechanical system. The bare coupling rate can be intuitively understood as the change in the cavity frequency per single phonon displacement of the mirror. The Hamiltonian, Eq. 1.3, results in a non-linear equation of motion and cannot be solved in general analytically, however there are a few helpful approximations that can be exploited to better grasp the underlying physics of the system (as will be seen in later sections).

The coupling rate G = ωc/L explicitly assumes a Fabry-Perot cavity where there is a well deﬁned length, but does not hold in general, for example, in evanescent coupled systems [53], a modulation of the refractive index around an optical cavity (e.g. a ring, disc or toroid) changes the eﬀective path length and therefore the resonant frequency. The basic property common to all dispersive optomechanical systems is a optical frequency dependent on the position of some mechanical device,

ωL(x). The bare coupling rate can always been written as the ﬁrst order Taylor expansion of this dependence,

s ~ dωL g0 ≡ (1.4) 2mωm dx

Some experimental designs result in a vanishing ﬁrst derivative [60, 61], leaving the lowest non-trivial expansion a†a(b + b†)2 with a coupling rate proportional to 2 d ωL dx2 . This coupling is not used in this thesis although some of the experiments and proposals are discussed in chapter 2.

Driving the cavity

So far only the interaction between light in the cavity and the oscillating mirror has been discussed. However, in order to properly understand an optomechanical system, we must be able to understand how light couples into and out-of the cavity. The following is a quick summary of the input-output formalism developed by Collett and Gardiner [62] which describes how light couples into and out of the cavity. Recently a more advanced framework has been developed [63], however this is beyond the scope of this work. The Collett and Gardiner formalism is generally written as 1.3. CAVITY OPTOMECHANICS 6 equation of motion in the Heisenberg picture and relates the propagating field that enters the cavity ain to the propagating field that leaves the cavity aout. Propagating − 1 † fields have units of Hz 2 so that hainaini is understood as the mean photon flux at the input mirror of the cavity, which is in general time dependent. The equations of motion derived in Ref. [62] are,

−i √ a˙ = [a, H] − κa + 2κain (1.5) √~ aout = 2κa − ain. (1.6)

The first term in Eq. 1.5 is the usual Heisenberg evolution for an operator, the second term causes an exponential decay from photons leaving the cavity and the third describes photons entering the cavity3. The convention used here (and the rest of the thesis) is that κ is the decay rate of the field amplitude (instead of the energy √ decay rate). The decay term, −κa, and the input term 2κ must necessarily be defined by the same rate κ, in order to satisfy the fluctuation dissipation theorem [64]. Eq. 1.6 describes the field leaving the cavity (reflected from the semi-silvered mirror in Fig. 1.1) and is intuitively understood as the interference between the reflected input field (hence the negative sign) and the intra-cavity field leaking out. In an experiment there is always a loss channel that the experimenter does not have access too. Eq. 1.5 can be generalized to include multiple input/output channels with different coupling rates,

i X q (j) a˙ = − [a, H] − κja + 2κjain (1.7) ~ j

(j) where ain denotes a diﬀerent input ﬁeld for each input with coupling rate κj. In practice the experimenter only has access to one (maybe two) of the input channels, leaving vacuum (in the case of optical or infra-red frequencies) or thermal (in the case of micro or radio wave frequencies) noise entering in the other ports.

Let us now consider a two port cavity, κ1 and κ2, where port one is driven by a coherent state of light and port two is driven by thermal noise (on top of quantum vacuum noise). Eq. 1.5 can be linearized about the coherent state by

a = α + δa (1.8) (1) (1) ain = αin + δain (1.9) (2) (2) ain = δain (1.10) where α (αin) is a complex number describing the coherent amplitude of of the cavity

3The derivation in Ref. [62] is for a general bosonic system weakly and linearly coupled to a larger number of environmental degrees of freedom. 1.3. CAVITY OPTOMECHANICS 7

(input) field. The operators marked with δ in Eqs. (1.8)-(1.10) are understood as (2) fluctuations on top of coherent amplitudes. Here ain is a thermal field and has no coherent amplitude. A thermal field is is a white noise process that satisfies [64],

D (2) E ain (t) = 0 (1.11) 1 D E 1 a†(2)(t)a(2)(t0) + H.C. = n¯ + δ(t − t0) (1.12) 2 in in 2

−1 wheren ¯ is the thermal occupation of the field (¯n = [exp(~ω/kBT ) − 1] for a photon field at temperature T and carrier frequency ω). Here δ(t − t0) is the Dirac- 1 (1) delta function. The factor of 2 is from quantum vacuum fluctuations (δain satisfies a similar equation withn ¯ = 0). Splitting Eq. 1.7 into the c-number and operator valued parts gives, √ α˙ = −iωcα − (κ1 + κ2)α + 2κ1αin (1.13) √ √ ˙ (1) (2) δa = −iωcδa − (κ1 + κ2)δa + 2κ1δain + 2κ2δain (1.14) where we have assumed a single mode cavity with no optomechanical interaction, † H = ~ωca a. The solutions to the above equations are most easily analysed in the frequency domain, and in the following we will focus on the coherent amplitude, √ 2κ a˜(ω) = 1 α˜ (ω) (1.15) κ + κ − i(ω − ω) in √1 2 c √ 2κ δa˜ (1)(ω) + 2κ δa˜ (2)(ω) δa˜(ω) = 1 in 2 in . (1.16) κ1 + κ2 − i(ωc − ω)

One can now immediately see the effect of the input field on the intra-cavity field, Eq. (1.15). The intra cavity field (in frequency space) is proportional the the input field with the pre-factor intuitively understood as a gain and phase shift. Consider

−iωdt for the moment a sinusoidal input ﬁeld with frequency ωd (i.e. αin = αe ); in

Fourier-space this ﬁeld is sharply peaked around ωd so we only consider the case 1 i arctan(y/x) ω ≈ ωd. Using x−iy ∝ e the coherent intra-cavity ﬁeld can be seen to have a phase change dependent on the detuning ∆ = ωd − ωc,

" −∆!# α˜(ω ) ∝ exp i arctan α˜ (ω ) (1.17) d κ in d where we have introduced κ = κ1 + κ2 as the total cavity loss rate. For |∆| κ the input and intra-cavity fields interfere (almost) in phase at the output, (see Eq. 1.6). If however, for |∆| κ there is a arctan(−∆/κ) ≈ ±π/2 phase difference between interfering fields. (i) As Eq. (1.7) is a linear ODE, multiple input fields at different frequencies ωL 1.3. CAVITY OPTOMECHANICS 8 simply result in a sum of phase shifted intra-cavity fields with phase shifts deter- (i) (i) mined by the individual detuning ∆ = ωd − ωc. This holds for two tone driving, (1) (2) where αin(ω) is sharple peaked around two frequencies, ωL and ωL (as analysed in Ref. [65]), or for a short pulse of input light where αin(ω) is broadly peaked at a single frequency (as in Ref. [52]).

1.3.2 Linearised optomechanical Hamiltonian

Coherent amplitude

The non-linear Hamiltonian in Eq. (1.3) is difficult to work with directly. In the following we discuss some approximations that simplify the interaction Hamiltonian. Suppose there is a large coherent amplitude in the intra-cavity field α 1, in this case the interaction term in the optomechanical Hamiltonian in Eq. (1.3) can be linearised (in the optical field operators) around α [59]. Once again, the cavity field is written rewritten as a = α + δa, and

∗ † † HI/~ = g0 α + δa (α + δa) b + b 2 † ∗ † † † † = g0|α |(b + b ) + g0 α δa + αδa b + b + g0δa δa b + b 2 † ∗ † † ≈ g0|α |(b + b ) + g0 α δa + αδa b + b (1.18) where the cubic term is dropped as it is smaller than the quadratic term by a factor of α. The first term in Eq. (1.18) produces a linear force on the mirror, shifting its equilibrium position and can be neglected by redefining the mechanical creation and annihilation operators about this shifted equilibrium. This leaves the interaction Hamiltonian quadratic in the field operators, and analytically solvable (in principle). The second term is the linearised optomechanical Hamiltonian. The linearisation may be understood as using a large coherent amplitude to convert a small rotation of the optical field to a substantial displacement Fig. 1.2 (a). Without loss of generality α may be chosen to be real(with appropriate phase of the driving field in the good cavity limit), and the optomechanical coupling is then increased from g0, the bare coupling rate, to the g = g0α the linearised coupling rate and an interaction Hamiltonian,

† † HI/~ = g(δa + δa )(b + b ). (1.19) 1.3. CAVITY OPTOMECHANICS 9

Figure 1.2: (a) A small optomechanical phase space rotation is well approximated by a displacement. The displacement is boosted by the coherent amplitude ∆PL = α sin(θ). (b) The rotating frame: (left) the coordinate system rotates with the free evolution of the optical state so the free evolution of the field is factored out. (right) Any changes to the quantum state are then solely due to the interaction Hamiltonian. In this case the rotating frame is the interaction picture with the change of basis determined by the free Hamiltonian. (c) The resonant interaction Hamiltonian for driving the cavity at ωd = ωc + ωm (blue detuned), the coherent creation and annihilation of phonons/intracavity photons leads to number state correlations between the optical field and mechanical element. (d) The resonant interaction for ωd = ωc − ωm (red detuning), the driving photon mediates a coherent exchange of quanta between the phonons and intracavity photons. Both cases (c) and (d) require the creation/annihilation of a drive photon ωd to mediate the interaction and conserve energy, however as the drive is assumed to be a large classical field (as required for the linearization in Eq. (1.18)), we do not consider the annihilation/creation operators forthe field at frequency ωd. This is the undepleted pump approximation for the large coherent field [66].

The Rotating Frame

The linearized Hamiltonian can be further simpliﬁed under certain assumptions to give Hamiltonians familiar in linear quantum optics. Moving into optical and mechanical rotating frames (see Fig. 1.2 (b)), i.e. δa → eiωdtδa and b → eiωmtb, the optomechanical Hamiltonian, the interaction Hamiltonian becomes,

† −i(ωm+∆)t † i(ωm+∆)t † † i(ωm−∆)t −i(ωm−∆)t HI/~ = g0α bδa e + b δae + b δa e + bδae (1.20) where ∆ = ωd −ωc is still the detuning between the input ﬁeld carrier frequency and the cavity resonance. Two special cases occur when the time dependent Hamiltonian above admits a stationary component, i.e. when ∆ = ±ωm. When ∆ = ωm, i.e. the input ﬁeld frequency is ωm above the cavity resonance frequency (blue detuning), 1.3. CAVITY OPTOMECHANICS 10 the interaction Hamiltonian is,

† † † −2iωm † 2iωm HI/~ = g0α b δa + bδa + bδa e + b δae (1.21) † † HI/~ ≈ g0α b δa + bδa . (1.22)

Ignoring the oscillating terms is the rotating wave approximation [59] and is valid −1 for time scales longer than the mechanical period t ωm as the second two terms average out and (almost) vanish4. This interaction generates a two mode squeezed state between the optical ﬁeld and mechanical oscillator and can be used to entangle the light and the mechanical motion [67, 68, 69, 70]. The two mode squeezer generates entangled pairs of photons and phonons (See Fig. 1.2 (c)), and although it looks like it violates energy conservation (for example it creates energy if acting on the optical and mechanical ground states), this is a consequence of the undepleted pump approximation [66] of the intra-cavity ﬁeld.

The other stationary case occurs when the input ﬁeld frequency is ωm below the cavity resonance, (red detuning) and,

† † HI/~ ≈ g0α bδa + b δa (1.23)

(where the rotating terms have been dropped). This is a beam splitter Hamil- tonian [66], coherently removing a photon and adding a phonon (and vice versa) (Fig. 1.2 (d)). In the beam splitter limit, the linearized optomechanical Hamilto- nian can be used for quantum state swaps [71] or to passively cool the mechanical oscillator [72, 73, 74, 75, 76].

Strong Coupling and Cooperativity

A goal of cavity optomechanics is to reach the single-photon strong coupling regime.

This is achieved when the photon-phonon interaction rate g0 exceeds both the optical and mechanical decay rates, g0 > κ, γ. In this regime a single quantum of energy can be exchanged between photons and phonons several times before leaving the cavity, or being dissipated in the mechanical environment. The less stringent linearised strong coupling regime g > κ, γ is a requirement to see normal mode splitting (i.e. hybridised opto-mechanical modes) [51]. Two other ﬁgures of merit are the ‘single-photon’ and ‘boosted’ optomechanical cooperativity, which is deﬁned as the dimensionless ratio of the coupling rate to 2 2 the decay rates C0 ≡ 4g0/κγ (single photon cooperativity), and C ≡ 4g /κγ = 4For example consider the time translation operator generated by this Hamiltonian, U = −i R H dt/ T [e I ~], the time dependent phase means the e−2iωm bδa† + H.C. terms oscillate between destructive and constructive contributions, while the constant terms are always constructive 1.3. CAVITY OPTOMECHANICS 11

2 α C0 (boosted cooperativity). The boosted cooperativity is the ﬁgure of merit for quantum state transfer [71], optomechanically induced transparency [65], and optomechanical cooling [76], each of which requires C > n¯, the thermal phonon occupation. Large single photon cooperativity C0 > 1 has been observed in atomic gas systems [77], but not in any other optomechanical system.

1.3.3 Pulsed optomechanics

In pulsed optomechanics, short optical pulses are sent into a cavity to interact with a mechanical oscillator [52]. The key assumption is that the duration of the optical pulse is much shorter than a single period of mirror’s motion such that the mirror is eﬀectively frozen over the span of the interaction. We are not interested in the quantum state during the evolution, but rather in the net eﬀect of the entire pulse on the system. In the following we will consider a coherent state input pulse with a real Gaussian envelope with duration τ √ Ne¯ −t2/4τ 2 αin(t) = √ . (1.24) 2πτ 2

The amplitude envelope is the square root of a Gaussian, ensuring the pulse is R † ¯ normalized to the mean number of input photons dthαinαini = N. A Gaussian pulse remains Gaussian in Fourier space with frequency width σω = 1/2τ. If σω is larger than the cavity line-width κ (i.e. a very short pulse) then most of the pulse will simply be reﬂected and not enter the cavity. In order for most of the incident photons to enter the cavity, the envelope of the coherent pulse must change over a longer time scale than the response time of the cavity κ−1, i.e. κ−1 τ so the entire pulse will enter the cavity. This along with the assumption that the −1 mechanical oscillator is frozen over the interaction τ ωm gives the conditions for −1 −1 a general pulsed optomechanical scheme, κ τ ωm . The κ−1 τ ensures that the cavity has a negligible impact on the shape of q the pulse envelope, i.e. α ≈ αin/ 2/κ. This conditions is required for the pulsed optomechanical protocol in this thesis. The κ−1 τ condition can be relaxed to κ−1 ∼ τ if an impedance matched pulse shape is used [52]. The shortest optimal5 −κ|t| pulse shape is a double exponential pulse αin(t) ∝ e . The double exponential shape (in time) exactly matches the Lorentzian cavity spectrum, but in this case the temporal envelope of aout is changed by the transfer function of the cavity, which is inconvenient for multi-pulse interactions [78, 79] The optomechanical Hamiltonian is again linearised about this time dependent

5Here optimal means all the photons enter the cavity 1.3. CAVITY OPTOMECHANICS 12 coherent amplitude,

2 † † † HI/~ ≈ α (t)(b + b ) + g0α(t)(δa + δa )(b + b ) (1.25) where we have implicitly assumed the coherent drive in on resonance ∆ = 0, and the mechanical oscillator is frozen over the entire interaction (no oscillations in b). The linearisation is valid even as α(t) → 0, as the non-linear interaction is negligible compared to the Brownian motion of the oscillator; it is not just negligible compared to α, it is absolutely negligible compared to the other dynamics in the system. The unitary operator can be easily integrated (no explicit time ordering necessary as b is constant),

Z 0 h 2 0 0 † i † U = exp −i dt α (t ) + g0α(t )(δa + δa ) (b + b ) (1.26) h i = exp −iχ(b + b†) − iλ(δa + δa†)(b + b†) (1.27)

R 2 R where χ = dtα (t) and λ = g0 dtα(t) and the integral is over the duration of the pulse (with assumed compact support). This unitary can be understood as a map that acts on the joint optomechanical Hilbert space, and describes the entire pulsed interaction. The unitary correlates the optical and mechanical operators,

b → U †bU = b + iλ(δa + δa†) + iχ (1.28) a → U †aU = a + iλ(b + b†). (1.29)

This mapping of initial to ﬁnal optical and mechanical operators (perhaps followed by measurement for further interaction) is the basis of pulsed optomechanical proposals. This interaction is a Quantum Non Demolition interaction [80]. The optical phase quadrature a + a† becomes correlated with, i.e. gains some information about the position b+b†, while the mechanical position remains unchanged b+b† → b+b†. I.e any quantum correlations in the position of the oscillator remain undisturbed (non-demolished).

1.3.4 The Displacemon

In the following we introduce the ‘displacemon’. Although the architecture for the displacemon system is quite different to optomechanics, it can be thought of as the Fermionic analogue of an optomechanical system, where the Bosonic optical mode 1 is replaced by a spin 2 degree of freedom. The canonical system for a displacemon is a superconducting flux qubit, where one of the Josephson junctions is replaced by 1.3. CAVITY OPTOMECHANICS 13 a vibrating mechanical element see Fig. 1.36. The motion of the mechanical element changes the magnetic flux though the area enclosed by the qubit and therefore changes the qubit transition frequency. This is discussed in detail in chapter 6. In the following we introduce the displacemon Hamiltonian and give some insight into the physical origin of the interaction.

Figure 1.3: (a) Simple displacemon architecture: a superconducting flux qubit has one Josephson junction replaced by an insulating mechanical resonator (a carbon nano-tube for example) which itself acts as a Josephson junction. If a transverse magnetic field is applied, the motion of the resonator periodically changes the net flux thought the qubit, and therefore the energy splitting between the clockwise and counter clockwise persistent currents. This energy splitting is responsible for the transition frequency of the qubit, hence the motion of the resonator modulates the transition frequency. (b) When the oscillator has a positive displacement (top), the effective orientation of the area of the flux qubit changes, increasing the magnetic flux. The total change in flux is proportional to the area swept out by the motion of the resonator. The change in orientation of the super conducting loop has been greatly exaggerated for illustrative purposes.

The position dependent modulation of the qubit’s resonance frequency can be can be understood by the Hamiltonian,

p2 1 H/ = ω (x)σ + + mω2 x2 (1.30) ~ q z 2m 2 m where σz is the standard Pauli-z matrix. Expanding the position dependence to lowest order in x gives the displacemon interaction,

s dωq(x) dωq(x) ~ † HI /~ = σzx = σz(b + b ) (1.31) dx dx 2mωm where we have again factored out the zero-point motion of the oscillator. Eq. (1.31) is the displacemon interaction Hamiltonian. It is a non-linear interaction, as σz =

σ+σ– − σ–σ+—with σ+(−) the qubit raising (lowering) operator—is a generator of

6This is different to the design discussed later in the thesis, however it gives a more intuitive understanding of the system 1.4. QUANTUM CONTROL 14 rotations on the Bloch-sphere, just as a†a generates rotations in phase space. q As discussed in chapter 6, the coupling rate dωq(x) ~ can be made to ap- dx 2 mωm proach the frequency of the mechanical oscillator. Hence, the device operates well q inside the strong coupling regime dωq(x) ~ > γ, κ where γ and κ the mechanical dx 2 mωm and qubit decay (inverse T2 time) rates respectively. The qubit can be probed and manipulated via coupling to a Superconducting Quantum Interference Device (SQUID) [81] (or a flux coupled transmission line for a transmon qubit [82]), and in principle one has complete control over the (classical) field used to drive the qubit, similar to the control over the cavity input field in an optomechanical system. The effect of the interaction on the mechanical oscillator is easily understood by considering an infinitesimal time evolution in the interaction picture,

dω (x) p(t + dt) = p(t) + dt q σ , (1.32) ~ dx z hence if the qubit is in the excited state (σz = 1), there is a positive momentum kick, while if the qubit is in the ground state (σz = 0) there is a negative momentum kick. If the qubit is in a superposition state, then the mechanical oscillator will receive a superposition of momentum kicks. The origin of the force is the electromagnetic Lorentz force. If the qubit is in the excited state (net current travelling anticlockwise in Fig. 1.3) the Lorentz force on the electrons in the oscillator F = I × B is directed upwards. On the other hand, if the qubit is in the ground state with the net current travelling clockwise, the net force is directed downwards. If the qubit is in a superposition state then the oscillator feels a coherent superposition of positive and negative momentum kicks. This feature discussed in chapter 6 to prepare and measure cat states of the mechanical oscillator. Given control over the state of the qubit (i.e. direction of currents and relative phase) from an external drive, one can engineer a sequence of momentum kicks to manipulate the state of the mechanical oscillator. This is discussed further in chapter 2.3, where we show arbitrary state preparation of the mechanical oscillator in the displacemon.

1.4 Quantum control

The classical channel model of gravity [30] was initially formulated in the language of quantum measurement and control [83]. Although this is the most natural formulation of the model, there are other interpretations [33]. In the CCG model, gravitational interactions may be understood as a an eﬀective direct measurement 1.4. QUANTUM CONTROL 15 and feedback protocol. To begin we will focus on weak quantum measurement, and move on to include feedback. In the following, we will focus on the particular example of measuring an arbitrary system variable X using a Harmonic oscillator as an apparatus, followed by general description. We will work exclusively in the interaction picture where the free Hamiltonian evolution is included in the basis states.

1.4.1 Quantum measurement

Among the six postulates of quantum mechanics is the measurement postulate which states: when a quantum system is measured, the quantum state immediately jumps into an Eigenstate of the measured observable. This description has some incon- sistencies when, for example, applied to the measurement of a particles’ position. The measurement postulate suggests that the post measurement wave function is in a position eigenstate, with infinite momentum variance7 – and therefore infinite Kinetic energy. However, one would still like to think about realistic measurement scenarios without introducing extra postulates. The kind of direct measurement of a particles position, as considered above, is somewhat contrived as it ignores that apparatus used to measure the particle. For example, just looking at the particles requires light to be reflected or emitted (coupling to light), or for it to reflect/emit sound (coupling to surrounding matter). By considering the coupling of the particle of interest (henceforth called the system) to a second quantum object (henceforth called the apparatus), one can infer information about the state of system given projective measurement on the apparatus [84]. While this still relies on the measurement postulate for the apparatus8, this form of measurement is known as Von Neumann measurement, or weak measurement (not to be confused with weak value) of the system. A pedagogical discussion of this type of quantum measurement can be found in Ref. [85, 86, 87]. To begin, we will consider a simple illustration of weak measurement before moving onto the general structure. Take for example an arbitrary system, and a harmonic oscillator, in the interaction picture with a simple interaction HI/~ = gxaX, where the subscript a is for apparatus and X is an arbitrary system operator. We will also assume the apparatus is in the ground state. The interaction Hamiltonian applies a force (i.e. a change in the momentum) on the apparatus depending on the system variable X. A projective measurement of the momentum of the apparatus after some interaction time will reveal some information about the system operator

7Assuming here no minimal length scale 8Strictly speaking one can relax the measurement postulate and add additional classical noise to the apparatus. The key point is that the apparatus may be measured far more accurately than the system. 1.4. QUANTUM CONTROL 16

Figure 1.4: (a) Circuit diagram depicting weak quantum measurement. The system interacts with a measurement apparatus followed by a projective measurement of the apparatus. Different interaction Hamiltonians or measurement basis of result in different measurement operators for the system. (b) Phase space picture of for the harmonic oscillator apparatus. The apparatus is initially in the ground state (left), after some interaction time the apparatus momentum is displaced by an amount proportional to the expectation value of the measured system observable (middle). The orange noise represents quantum fluctuations of the system observable about the mean displacement. As time increases (to the right), the mean displacement and quantum noise of the system observable increase, but the apparatus quantum noise remains constant, hence a single shot measurement of the apparatus momentum is strongly correlated with the measured observable. As the interaction time (or interaction rate g) increases, the quantum noise of the apparatus becomes negligible and a projective measurement of the apparatus tends towards a projective measurement of the system observable.

X. The momentum of the apparatus is called the ‘pointer variable’ as it contains information about the particles position (like the pointer needle on an analog volt- meter). The apparatus is also a quantum system, and there will be some intrinsic quantum noise in the measurement result, which limits the resolution of the position measurement (see Fig. 1.4).

The Schrodinger Picture

The process discussed above is intuitively seen by considering the evolution of the quantum state in the interaction picture,

Z 0 −igtxaX 0 0 0 0 −igtx0X hp |a e |0ia |φis = dx hp |x i hx |0i e |φis (1.33) " (X − p0/gt)2 # = (2πg2t2)−1/4 exp − |φi (1.34) 4g2t2 s 0 = Υ(p ) |φis (1.35)

0 0 where p is the outcome of a projective measurement of pa. For each p there is a different positive operator valued measure (POVM) Υ(p0), also called the measurement operator, or Kraus operator [52, 88]. Eq (1.35) is called the conditional (unnormalized) quantum state, and is the quantum state of the system if the measurement result is known. In the first line, we inserted the identity I = R dx |xi hx| in the 1.4. QUANTUM CONTROL 17 apparatus Hilbert space, and in the second line made use of ground state wave 0 − 1 −x02/4 function hx |0i = (2π) 4 e , and the position-momentum eigenstates overlap √ hp0|x0i = e−ip0x0 / 2π in dimensionless variables. The effect of the projective measurement on the system can now be seen: it narrows the wave function of the system (in the X basis) around the measurement result p0. The Gaussian nature of the measurement operator is a result of the choice of the initial state of the apparatus, and the linear interaction in the Hamiltonian. Unless the interaction strength diverges, or the measurement takes an infinitely long time, the system will never be in a singular state. During the measurement process, some, but not complete information has been gained about the system. The POVM maps pure state to pure states, and the process trivially generalises to mixed states via

0 −igtxaxs igtxaxs 0 0 † 0 hp |s e {|0ia h0|a ⊗ ρs} e |p i = Υ(p )ρsΥ (p ). (1.36)

The conditional states in Eqs. (1.35) and (1.36) are not normalized to unity, which is unsurprising as they are the result of a projective measurement — a highly non-unitary process. The normalization reﬂects the fact that the state is post selected on the result apparatus state |p0i. The norm of the conditional state gives the probability of obtaining that particular measurement outcome. This is easily derived by recalling that the probability of obtaining a particular measurement outcome is proportional to the square amplitude of the corresponding state in the wave function (or diagonal elements of the density matrix in the measurement basis). In terms of the reduced density matrix for the apparatus ρa = Trs[ρ], the probability of obtaining the measurement result p0 is given by,

Z Z h 0 † 0 i 0 0 0 Trs Υ(p )ρsΥ (p ) dp = Pr(p )dp = 1 (1.41) Z ⇔ Υ†(p0)Υ(p0)dp0 = I (1.42)

as ρs is arbitrary and using the cyclic rule of traces. Of course one could ask why not treat the apparatus projection as a weak mea- 1.4. QUANTUM CONTROL 18 surement as well? Well the projective measurement cut (sometimes called the von Neumann or Heisenberg cut) must be made somewhere and the first apparatus suffices to avoid singular quantum states [89]. If one wants to include a secondary measurement device, then it will be the joint first-apparatus–system Hilbert space that is left in a pure state, and ignorance of the first apparatus will (in general) lead to a mixture of pure state of the system.

The Heisenberg Picture

Weak measurement can also be understood by considering the Heisenberg equations of motion. We will assume that the system is also a harmonic oscillator with dimensionless canonical variables xs and ps, the Heisenberg equation of motion are,

xa(t) = xa(0) (1.43)

xs(t) = xs(0) (1.44)

pa(t) = pa(0) + gtxs(0) (1.45)

ps(t) = ps(0) + gtxa(0). (1.46) where we are still in the interaction picture. A single shot measurement of pa(t), 0 with measurement result p , is the sum of two single shot measurements of pa(0) and xs(0) (with gain gt). The apparatus is initially in the ground state, so the single shot 1 measurement of pa(0) must be a random Gaussian variable with variance 2 (from the Measurement postulate). The single shot measurement of xs(0) is also drawn from a distribution deﬁned by the initial state, and can be written hxs(0)i + δx, where δx is a classical random variable drawn from the mod-square wave function of of the system (shifted to have zero mean). We will now denotex ¯ as the scaled measurement result and p0 1 1 x¯ = = hx (0)i + δx + N 0, (1.47) gt s gt 2 where N [µ, s2] is a Gaussian variable with mean µ and variance s2. The apparatus returned the measurement result p0 and it is impossible to decouple the zero point noise from the single shot measurement result. After the measurement, the uncertainty in the particles position is given by the conditional variance,

Cov[x (0), x¯] Var [x (0)|x¯] = Var[x (0)] − a . (1.48) a a Var[¯x]

It is clear that any covariance (correlation) between the system variable and the measurement result will reduce the variance of xa after the measurement, analogous to Eq. (1.35). 1.4. QUANTUM CONTROL 19

General Structure

The general structure of weak measurement is to consider a system Hamiltonian Ia ⊗

Hs, an apparatus Hamiltonian Ha⊗Is, and an interaction Hamiltonian Oa⊗Os where

Os(a) is an arbitrary system (apparatus) operator, and Is(a) is the system (apparatus) P (i) (i) identity operator. More general interactions of the form HI = i Oa ⊗Os [90, 91], but this is not needed for this thesis. The apparatus is prepared in some initial state

|ψii a which is separable from the system state. The joint system/apparatus is then allowed to interact for some time under the full Hamiltonian, followed by projective measurement of the apparatus state. If the apparatus is found in a state |ψf ia, the conditional state of the system is found to be

Z i 0 hψf | T exp − dt (Ha ⊗ Is + Ia ⊗ Hs + Oa ⊗ Os) |ψiia ⊗ |φis = Υ (ψf ) |φis a ~ (1.49) where the terms in the square brackets deﬁne the POVM acting only on the reduced Hilbert space of the system. For a general unbiased9 Gaussian measurement (quadratic from Hamiltonain and Gaussian apparatus states) the POVM is,

" 2 # 2 − 1 (Os − x¯) Υ(¯x) = (2πσ ) 4 exp − (1.50) 4σ2 wherex ¯ is the (scaled) measurement result and the normalization ensures the identity in Eq. (1.42). Henceforce we will only consider Gaussian measurements with the exception of section 2.3, where we will require more general measurement operators.

Unconditional quantum state

If the measurement result is known, the the POVM will map pure states to pure states. However if the measurement result is unknown, there exists some classical ignorance about the quantum system and resulting system state must be mixed, regardless of the purity of the initial state. The mixed state is called the unconditional state. A general mixed can be written as a convex sum of pure states,

Z ρ = dx¯ Pr(ψx¯) |ψx¯i hψx¯| (1.51) where Pr(ψx¯) is the classical probability of ﬁnding the system in the state |ψx¯i (the integral is replaced by a sum for ﬁnite dimensional Hilbert spaces and we have dropped the a and s subscripts as we will only consider the system from now on.

9i.e. no measurement oﬀset 1.4. QUANTUM CONTROL 20

Post measurement, the set of possible (normalized) states available to the system are

( † ) Υ(¯x)ρ0Υ (¯x) {ρx¯}x¯ = † (1.52) Tr [Υ(¯x)ρ0Υ (¯x)] x¯

h † i with the probability for obtained the (normalized)x ¯ state is Pr(¯x) = Tr Υ(¯x)ρ0Υ (¯x) . The unconditional state of the post measurement quantum system is therefore given by,

Z † Z Υ(¯x)ρ0Υ (¯x) † ρu = dx¯ Pr(¯x) † = dx¯Υ(¯x)ρ0Υ (¯x) (1.53) Tr [Υ(¯x)ρ0Υ (¯x)] which is trivially normalized by Eq. (1.42). Hence the set of all measurement operators are exactly the Krauss operators for the CP map that deﬁnes a measurement with an unknown measurement results.

Continuous weak measurement

There are a number of ways to derive the stochastic Schrodinger equation for weak measurement [83, 92], however here we will generally follow the derivation (in different notation) from Ref. [85]. Consider two position measurements separated by a time interval δt (ignoring free evolution),

" 2 2 # 2 − 1 (x − x¯1) (x − x¯2) Υ(¯x )Υ(¯x ) = (2πσ ) 2 exp − − (1.54) 1 2 4σ2 4σ2 " 2 # 2 − 1 −[x − (¯x1 − x¯2)/2] (¯x +¯x )2/4 = (2πσ ) 2 exp e 1 2 , (1.55) 2σ2 the net result is to localize the wave function to within a variance of σ2/2. Extrapo- σ2 lating this results in a localization of T/δt in some ﬁnite time T . Since the continuous limit requires δt → 0, the only way to keep the wave function non-singular after some ﬁnite T is to let σ → ∞ in the continuous limit, with

σ2δt = Γ (1.56) held constant . The constant Γ has units m2 Hz−1 and is a measurement sensitivity [83]. It is understood as the increase in knowledge of position as the bandwidth decreases10 (decreasing bandwidth implies longer measurement time). The divergent measurement resolution means that in each single measurement, only an inﬁnitesi-

10For example, after a 10 s measurement (bandwidth of 0.1 Hz) the position of a particle would be known to within pΓ × (0.1 Hz) meters. Clearly as the measurement time increase, the measurement bandwidth decrease, and so too does the uncertainty of the particle. 1.4. QUANTUM CONTROL 21 mal amount of information is gained from the measurement result. Consider the statistics of the measurement resultx ¯ in this ‘ultra’ weak, i.e. continuous limit. From Eq. (1.47), it can be seen as σ diverges, Gaussian noise term dominates over the noise ﬂuctuations from the wave function. Hence as σ2 → ∞, the measurement result is well approximated by a Gaussian random variable,

2 x¯ = hxic + N [0, σ ] (1.57) s Γ = hxi + N [0, 1] (1.58) c δt √ Γ = hxi + N [0, δt] (1.59) c δt where hxic is the conditional mean of the operator x, i.e. the single shot measurement readout. Above we have used the identity N [0, s2] = sN [0, 1] and the deﬁnition of Γ, Eq. (1.56). The Gaussian random variable can also be seen by directly calculating the probability distribution ofx ¯ using Eq. (1.40),

h i Pr(¯x) = Tr Υ†(¯x)Υ(¯x)ρ (1.60) Z " (¯x − x0)2 # = dx0(2πσ2)−1/2 exp − hx0| ρ |x0i (1.61) 2σ2 h −(x0−hxi)2 i exp 2 ≈ √ 2σ (1.62) 2πσ2 where in the last line it is assumed the probability distribution Pr(x0) = hx0| ρ |x0i is sharply peaked (around hxi) compared to the with of the Gaussian. This approximation is always valid in the σ → ∞ limit, even if P (x0) has many lobes. In this case the probability distribution forx ¯ is a Gaussian random variable centred at hxi with variance σ2, in agreement with Eq. (1.59). A single measurement realizes a particular value of the random normal variable, and a sequence of continuous measurements must be conditioned on a sequence of measurement results {x¯}. The (unnormalized) post measurement wave function can now be expanded to leading order in δt (by writing σ2 = Γ/δt),

√  Γ 2  δt(x − hxi − δt N [0, δt]) Υδt(¯x) |ψi ∝ exp −  |ψi (1.63) 4Γ " δt(x − hxi)2 (x − hxi)N [0, δt] (x − hxi)2N 2[0, δt]# ' 1 − + √ + |ψi . 4Γ 2 Γ 8Γ (1.64)

Note, leading order in δt is second order in N [0, δt] as the random variable has √ variance δt and is therefore on the order δt. The continuous limit can now be 1.4. QUANTUM CONTROL 22 taken by δt → dt. In this limit N 2[0, dt] = dt, and is no longer a ﬂuctuating variable11,

" (x − hxi)2 (x − hxi) # lim Υδt(¯x) |ψi = 1 − dt + √ dW |ψi (1.65) δt→dt 8Γ 2 Γ where we have introduced dW = N [0, dt] as the standard Weiner increment, i.e. a zero-mean Gaussian random variable E(dW ) = 0 with dW 2 = dt. Although previously we neglected the normalisation of the measurement operator, Eq. (1.65) does in fact preserve the norm. The last two term in Eq. (1.65) are the inﬁnitesimal change in the state, d |ψi, from an ‘ultra weak’ measurement during dt. The normalization of the state in Eq. (1.65) can be recovered by demanding that d(hψ|ψi) = d(hψ|) |ψi + hψ| d(|ψi) + d(hψ|)d(|ψi) = 0, and although the normalization was initially disregarded, Eq. (1.65) is in fact properly normalized. Theconditional quantum state is conditioned on the measurement record which is now also written using the Weinerincrement,

√ dW x¯ = hxi + Γ . (1.66) c dt The derivative dW/dt does not formally exist but does (informally) have the statistics of a white noise process. Although the derivative in Eq. (1.66) is undefined, a well defined equation can be written in (Ito) differential form √ dJx = hxic dt + ΓdW (1.67) where the measurement current dJx ≡ xdt¯ . An observer monitoring the measurement current see the Brownian trajectory Jx, and as the observer integrates the current over time, the quantum state traces the corresponding trajectory though Hilbert space, Eq. (1.65).

1.4.2 Quantum feedback control

Once a quantum system is measured, and a measurement result obtained, the classical estimate of a quantum observable may then be used as a control parameter for some later interaction [93, 94]. For example, after measuring the position of a particle and obtaining a measurement resultx ¯, one may apply a linear potential 2 back onto the particle that dependsx ¯, i.e Hfb = mω xx¯ . The ω is suggestive as the potential looks similar to a harmonic potential. Such a feedback scheme has

11This is understood by considering the distribution of N 2[0, δt]: it is a χ2 distribution with mean δt and variance 2δt2, therefore the variance of the stochastic variable N 2[0, dt] is dt2 = 0. 1.4. QUANTUM CONTROL 23 been realised in optomechanical systems [95, 96, 97] and quantum error correcting codes [98], and in the following the exact systematic eﬀect will be derived. There are two formulations of feedback control used in this thesis. The ﬁrst is directly from the stochastic Schrodinger equation, and the second from the unconditional quantum state. We will consider exactly the case of measurement of a particles position, immediately followed by the unitary evolution generated by the Hamiltonian,

H = H0 + Hfb 2 = H0 + mω xx¯ (1.68)

1 where the usual factor of 2 is omitted for reasons that will become clear later and H0 is left as arbitrary.

Stochastic Schrodinger equation

In the SSE formulation, the measurement and feedback process is broken into two infinitesimal time steps of dt. In the fist time step, the conditional quantum state infinitesimally changes according to Eq. (1.65) (with no Hamiltonian evolution), and in the second step the state is propagated forward in time by the Hamilaontian Eq. (1.68). As this is explicitly written in the case of continuous weak measurement, √ dW the measurement result isx ¯ = hxi + Γ dt , and although the derivative does not formally exist, the unitary generates a well defined infinitesimal time translation

H " H mω2x # U(dt) = exp −i dt = exp −i 0 dt − i xdt¯ (1.69) ~ ~ ~ iH dt imω2x √ m2ω4x2Γ = 1 − 0 − hxi dt + ΓdW − dt. (1.70) ~ ~ ~2 Again, the exponential must be expanded to second order in dW , giving the ﬁnal term above, a non-unitary (i.e. no i) systematic change in the evolution This time translation operator propagates the post measurement quantum state forward an inﬁnitesimal step dt,

|ψ(t + dt)i = U(dt)(|ψi + d |ψi) (1.71) " 2 # idt 1 2 2 dt(x − hxi) dW (x − hxi) = 1 − H0 + mω x + + + √ |ψ(t)i ~ 2 8Γ0 2 Γ0 (1.72) where d |ψi is from Eq. (1.65), and the state is expanded to ﬁrst order in dt. This describes the new state after a dt step of measurement and feedback. The feedback 2 term gives an eﬀective x potential on top of H0, but at the expense of adding 1.4. QUANTUM CONTROL 24

0−1 −1 m2ω4Γ additional noise Γ = Γ + 2 . 8xzp~

Density matrix formulation

The previous formalism describes the situation where an experimenter has complete knowledge of the measurement current. However, one may be interested in the unconditional state, where the measurement and feedback is being carried out, but the experimenter has no knowledge of the measurement currents. The following description generally follows the derivation in Ref. [99]. We will again work in the interaction picture where the free Hamiltonian vanishes. Consider a system which is measured instantaneously with ﬁnite measurement resolutions σ2 at Poissonian-random intervals at a rate γ. In a short time step dt, the probability of a measurement event is γdt, and the probability that the state remains unchanged is 1−γdt. These are classical probabilities so the density matrix is simply the sum of each possible state with their corresponding probability,

Z ρ(t + dt) = γdt dx¯Υ(¯x)ρ(t)Υ†(¯x) + (1 − γdt)ρ(t) (1.73) which is doubly random. The experimenter does not know if a measurement was made, and even if it was, they do not know the measurement result (hence the unconditional state from Eq. (1.53)). To include feedback, a unitary term is added immediately after a measurement is made. If a measurement resultx ¯ is obtained, in the time step dt, then the feedback unitary, dependent on the corresponding measurement outcome is

igf xx¯ Ufb(¯x) = exp − . (1.74) ~ Recall the measurements are finite jumps, with finite resolution σ2 so the resultx ¯ is well defined. This unitary is applied immediately after the measurement, where gf is the feedback gain. Eq. (1.73) becomes,

dρ Z = γ dxU¯ (¯x)Υ(¯x)ρΥ†(¯x)U † (¯x) − γρ. (1.75) dt fb fb For Gaussian measurement and linear feedback, thex ¯ integral is solvable, the master equation becomes,

2 ! dρ i h 2 i gf Γ 1 = − gf x , ρ − + [x, [x, ρ]] (1.76) dt 2~ ~2 8Γ √ where the identity exp [−(x0 − x00)2/2s] s = R dx¯ exp [−(x0 − x¯)2/s − (x00 − x¯)2/s] has been used in the last line. Note the double commutator structure can be written 1.5. FRIEDMANN ROBERTSON WALKER SPACE TIME 25

1 1 2 2 in Lindblad from, − 2 [x, [x, ρ]] = xρx − 2 (x ρ + ρx ), where x is the Lindblad operator. The continuous limit has been taken by choosing σ, γ → ∞, holding γσ2 = Γ fixed. The Taylor expansion of the Gaussian for small (x0 − x00)/σ is valid when σ is much larger than the characteristic length scale of the wave function, as the wave function amplitudes, hx0| ρ |x00i, will vanish for x0 − x00 > σ. The linearized, i.e. continuous, master equation is exactly the same as Eq. (1.65), as it describes exactly the same process. The systematic unitary term is recovered only in the case of continuous measurement. For a finite measurement process the dhpi dhpi effective force can be understood by computing dt and recalling that dt = −∇V , as we shall do in section 2.2.

1.5 Friedmann Robertson Walker space time

In chapter 3, we investigate the consequences of a classical channel model of gravity in the relativistic regime. The details are left to the chapter, however in the following we give a brief overview of the Friedmann-Robertson-Walker (FRW) metric. In the following we will follow the convention of setting the speed of light to unity c = 1.

1.5.1 The metric

In cosmology one is often only concerned with the large scale structure of the universe. On a large enough scale, the ﬂuctuations of energy density become negligible and the space time, is isotropic and homogeneous, for all practical purposes. In this case the metric can be written with only two degrees of freedom [100],

" dr2 # ds2 = −N 2(t)dt2 + a2(t) + r2dθ2 + r2 sin2 θdφ2 (1.77) 1 + kr2 = −N 2(t)dt2 + a2(t)dx2 (1.78) as is characterised by a single parameter k which defines the intrinsic curvature of the space time, for hyperbolic, k > 0, flat k = 0 and closed, k < 0 space times. Without loss of generality, k = 0, ±1 suffices to define the curvature where radial coordinate can be rescaled. The lapse function N(t) describes how the time coordinate evolves between two spatial hyper-surfaces, and a particular choice of N(t) leads to a particular choice for the time coordinate in the metric. The scale factor a(t) defines the expansion or contraction of the spatial hyper-surfaces over the coordinate time t. Although this metric looks like a special coordinate system where space and time distinction has been chosen, it can be derived from a general metric using Weyl’s postulate and the homogeneous/isotropic symmetries [100]. 1.5. FRIEDMANN ROBERTSON WALKER SPACE TIME 26

The scale factor is understood as the scaling between coordinate spatial variable, and physical distances. For example, consider two test particles at coordinates x and y on the spatial hyper-surface at time t. The coordinate distance between these points is simply D = |x − y| (using the usual Cartesian norm), while the physical distance between these points – as measured by a physical ruler – is dt = a(t)|x − y|. After some (coordinate time) δt, and assuming the particles remain at their coordinate points, the physical distance between them is now dt+δt = a(t +

δt)|x − y| 6= dt. This apparent movement of the particles is the Hubble ﬂow and is not due to any forces perturbing the particles from their local position, but rather from the expansion or contraction of the space in-between them.

Hence there are two types of motion possible, dt+δt = a(t + δt)|x − y + dxy|. The change in the coordinate of the particle, dxy, is motion generated by forces acting on the particle (over the time δt), moving it on top of the background space time, while the a(t + δt) is the apparent motion generated by the expansion/contraction of the background.

The Hubble parameter is defined as HH =a/a ˙ , where by convention the derivative is in a coordinate system with N(t) = 1, and describes the apparent motion of the Hubble flow. Strictly speaking the Hubble parameter has units of time−1, as a is dimentionless. In order to make sense of the parameter is intuitive to consider ˙ the Hubble parameter as HH = dt/dt with coordinates points fixed, dxy = 0. This expression doesn’t change the previous definition as the |x − y| factors will cancel. ˙ Clearly dt is the speed at which the points are moving relative to each other, so length/time HH is understood as velocity per distance, or length . That is the further two (coordinate) points are from each other, the faster their relative relative motion. This is exactly Hubble’s law, discovered by Hubble in 1929 [101].

1.5.2 Equations of motion

The simplicity of the FRW metric means that it is one of the few exactly solvable space-times in GR. The equations of motion for an FRW universe may be obtained directly from the Einstein equation, which will result in the well known Freidmann equations, or from the least action principle12. In chapter 3 we require a Hamiltonian formulation for the space time dynamics, so will begin directly from the Einstein- Hilbert action,

1 Z √ S = d4x −g (R + L + L ) (1.79) 8πG M I 12A least action formulation, varying with respect to N(t) and a(t) will not exactly result in the δSg 2 2 Einstein equations as Gµν ≡ δgµν etc, which is a variation with respect to a and N not a and N. However the equations of motion give the same trajectories for the same initial conditions. 1.5. FRIEDMANN ROBERTSON WALKER SPACE TIME 27

where g = Det[gµν] is the metric determinant, LM matter Lagrangian density, and

LI space time-matter coupling Lagrangian and R is the Ricci scalar. For this thesis, it suﬃces to consider an empty universe LM = LI = 0, so we will drop these terms henceforth. The Ricci scalar depends only on time, as it must for an isotropic homogeneous universe, and the metric determinant g (can be read oﬀ the metric in Eq. (1.77)). The resulting Einstein-Hilbert action for an empty FRW space time is given by

1 Z " a˙ 2(t)# S = d4x ka(t)2N(t) − (1.80) 8πG N(t)

The integral above does not converge for an inﬁnite FRW universe, however the standard way to proceed is by separating out the temporal integral from the spatial hyper-surface, and taking the hyper-surface domain to be some ﬁnite, but arbitrarily large volume V . From the spacial independence of the metric,

V Z " a˙ 2(t)# S = dt ka(t)2N(t) − . (1.81) 8πG N(t)

This looks exactly like a standard single-particle action with generalised coordinate √ −gRV a and Lagrangian L = 8πG . The lapse function N(t) as no kinetic term hence it’s canonical momentum vanishes p = ∂L = 0. Lagrange’s equations of motion N ∂N˙ for N(t) are trivially satisfied, hence N(t) only acts as a Lagrange multiplier and is inconsequential for the evolution of the scale factor, and N(t) can be chosen to be an arbitrary function. This is not simply a consequence of the empty FRW universe here, but is a general feature of the lapse function [102], and arises because general relativity is a totally constrained theory. The standard choice is N(t) = 1, in which case the coordinate time is the comoving time, and inertial clocks in the space time tick at a constant rate in coordinate time. Another common choice is to set N(τ) = a(τ), using τ instead of t to differentiate from comoving time coordinates; the coordinate time τ measures conformal time13, and physical clocks tick at a rate dτ = dt/a(τ) in conformal time. Although these are common choices the lapse function is entirely arbitrary in the Hamiltonian. The Hamiltonian can now simply be obtained by the standard Legarandre transform defining the canonical momentum as

∂L V a˙(t) p = = − (1.82) ∂a˙ 4πG N(t)

13A ﬂat space time in conformal coordinates is conformal to a Minkowski space time. In fact formally one can choose N(t) → N(τ)a(τ) in the metric, and then later choose N(τ) = 1 for conformal coordinates. 1.6. SUMMARY 28 and 1 H ≡ pa˙ − L = − p2 − ka2. (1.83) 4 where we have now set 8πG = 1 and N(t) = 1 (as in chapter 3). Equations of motion for the space time can be readily obtained from Hamilton equations.

1.5.3 Energy-momentum in the FRW metric

So far we have only consider an empty FRW space time. We will now move on to characterizing the type of matter in the space time. The isotropic and homogeneous symmetries imply the energy-momentum tensor Tµν must be that of a perfect ﬂuid with energy density ρ and pressure P ,

T µν = diag[ρ, P, P, P ], (1.84) and the type of perfect fluid is defined by the equation of state P = wρ. The equation of state is sufficient to completely describe the solution of a(t). In comoving time (N(t) = 1), the scale factor has the power law solution a(t) = 2 a0t 3(w+1) (for w 6= −1), hence once the equation of state is specified the solution to the space time evolution is known. There are several notable equations of state in cosmology [103], these include: (i) dust: w = 0, energy density with no pressure, µ µ (ii) radiation: w = 1/3, coming from T µ = −ρ + 3P from Eq. (1.84) and T µ = 0 when expanded in the field strength tensor of electromagnetism, (iii) dark energy: w = −1, where Tµν = Λgµν from Einstein’s equations, which gives p = −ρ = Λ (by αν ν noting gµαg = δµ ). 1 There is another special case when w = − 3 . In this case the scale factor evolves 1 as a(t) = a0t. If w > − 3 , thena ¨(t) > 0 and the space time expansion will accelerate 1 for all time. However if w < − 3 thena ¨(t) < 0 the space time expansion will 1 decelerate indefinitely. The equation of state w = − 3 is the boundary between an accelerating (run away) and decelerating (collapsing) universe.

1.6 Summary

In this section we have covered some of the tools used in the rest of this thesis. The following chapters all use one or several of the concepts introduced here. In the next chapter we will cover some of the modern results in quantum opto and nano-mechanics, and wave function collapse theories. We will also introduce original results in a generalised model of CCG (using the measurement and feedback 1.6. SUMMARY 29 framework developed here), and arbitrary wave function generation using the displacemon architecture. Chapter 2

State of the Art

The ﬁrst section of this chapter contains an overview of the recent history and state- of-the-art experiments and theory discussed in this thesis. As there are already comprehensive literature reviews for each of the topics discussed in the following chapters, (gravitational decoherence [104], collapse models [46] and optomechanics [51]), the following literature review is not exhaustive, but provides context for this thesis. The two sections after the literature review each contain unpublished original research undertaken during my PhD candidature. In Section 2.2, we introduce a general formulation of the classical channel model, beyond the linearized two particle description. Finally in section 2.3, we describe a protocol for generating an arbitrary state of a mechanical oscillator in a displacemon system.

2.1 Literature Review

2.1.1 Collapse theories and gravitational decoherence

The emergence of classicality from underlying quantum dynamics remains an open problem in quantum foundations. Several authors have proposed that gravity may play a role in the resolution of this question [41, 105, 24, 106]. Early work in this direction was pioneered by Karolyhazy in 1966 [107], although the ﬁeld markedly expanded after the seminal papers of Diosi [34, 108] and Penrose [39]. Before proceeding, we should separate gravitational decoherence from the usual description of open quantum systems decoherence. Decoherence in an open quantum system can be understood via unitary evolution between a system and the environment (sometimes called the bath). Ignorance of the environment’s quantum state leads to decoherence of the system [109], even though the joint system-bath state remains coherent. This does not require any modiﬁcations to quantum theory

30 2.1. LITERATURE REVIEW 31 and is a result of classical uncertainty in the full quantum state1. As opposed to open system decoherence, gravitational decoherence may be understood in the context of a closed system. That is, intrinsic decoherence to a system in the absence of any quantum environment. Here the role of the environment is replaced by a classically fluctuating field, e.g. Refs. [31, 35]. Such models propose a fundamental modification from unitary evolution, and a change to the postulates of quantum mechanics. Some gravitational decoherence models do admit the interpretation that the system is interacting with quantum metric degrees of freedom, with the metric playing the role of an environment, e.g. Refs. [39, 110]. However, as there is no theory of quantum gravity, gravitational decoherence from ‘metric as the environment’ models (including classically fluctuating metrics) are usually discussed in the context of modifications to unitary evolution, as opposed to explicit system-environment coupling. In Ref. [34], Diosi — motivated by a minimum uncertainty bound for acceleration measurement [108] — postulates a universal white noise process in the (classical) Newtonian potential, leading to decoherence in the mass density of a quantum particle. Penrose [39] on the other hand outlines a heuristic argument noting a superposition of space times requires a superposition of time translation operators. Such a state dependent notion of time is incompatible with the single time parameter required for evolution of the Schrodinger equation. The uncertainty in time translation operators is related to the lifetime uncertainty of the quantum superposition. It has separately been shown that uncertainty of the time translation operator results in decoherence of the quantum state [111]. The early Penrose and Diosi models result in a deviation from the unitary evolution of quantum mechanics. The Ghirardi-Rimini-Weber (GRW) model [41], developed slightly earlier than the Penrose-Diosi models, also postulates this deviation from unitary evolution, but without a direct gravitational argument, although gravity is invoked as a possible source. The GRW model has since been extended to the more robust Continuous Spontaneous Localization (CSL) model [112, 113], and has recently been further extended to include gravitational effects [114, 115]. This thesis focuses on the model of Kafri, Milburn and Taylor [30, 31, 32], where the gravitational interaction between two massive particles is mediated via a classical, as opposed to a quantum information channel. Henceforth we will refer to this model as Classical Channel Gravity (CCG). The model builds on similar ideas by Tilloy and Diosi, who considered the implications of a quantum field dynami-

1In order to derive a master equation for an open quantum system we often make approximations that the environment state remains unchanged, for example the Born-Markovian approximation, in order to simply the analysis. However, the physical description is the system becoming entangled with the environment, and the decoherence arises from tracing out the environment degrees of freedom. 2.1. LITERATURE REVIEW 32 cally interacting with a classical Newtonian potential [114]. Diosi has expanded on this framework of coupling quantum and classical degrees of freedom in a series of notable papers [116, 36, 117, 35, 118].

Diosi-Penrose model

The central notion of the Diosi-Penrose model is that there exists a gravitational mechanism which destroys macroscopic quantum fluctuations in the energy of a system, preventing the existence of macroscopic quantum states. The distinguishing feature of the Diosi-Penrose collapse model is that the origin of the collapse is classical, as opposed to quantum in nature. The fact that gravity is a non-linear theory, and it has yet to be quantized, is a prime reason that one looks to gravity to destroy macroscopic superpositions. The other fundamental forces admit a well defined quantum theory and are therefore compatible with the possibility of macroscopic superposition states, hence one must look beyond these forces to explain an emergent classical theory. Furthermore, the weak and strong nuclear forces are negligible, or non-existent on macroscopic length scales, and the electromagnetic force can (in principle) be shielded against. This leaves gravity, the ever present, unshieldable, macroscopic interaction, as the natural candidate to suppress macroscopic energy fluctuations. In fact for these reasons, even a quantum theory of gravity would lead to decoherence via the usual open quantum system derivation. We will discuss the seminal papers of Penrose and Diosi individually, before introducing the Diosi-Penrose model. Penrose’s central idea is that a mass in a superposition state has some intrinsic energy uncertainty from a super position of space time geometries. The energy uncertainty results in a temporal uncertainty in the lifetime of the superposition state; from the time-energy uncertainty principle, and hence such a state cannot be an energy eigenstate. The temporal uncertainty implies a mass in a spatial superposition is unstable, and will ‘decay’ into one of the branches wave function. Underlying this, is the assumption of a superposition of the gravitational field, and therefore it implicitly assumes a quantum theory of gravity. In order to derive what the gravitational self energy is, Penrose considers the ambiguity of time translations in a superposition of space time. The ambiguity of a time translation operator, means there are no well defined stationary states (i.e. no energy eigenstate), and therefore the superposition principle breaks down2. Using

2In principle the argument also holds for a single atom in a superposition of energy states, which from mass energy equivalence, results in a superposition of space times. However the energy scales for a superposition of electron states is eV (for optical frequency transitions), however the energy scale for even a single neutron in superposition is ∼ 100 MeV 2.1. LITERATURE REVIEW 33

Lorentz invariance arguments, Penrose postulates,

Z [µ (x0) − µ (x0)][µ (x00) − µ (x00)] ∆E = −4πG 1 2 1 2 d3x0d3x00 (2.1) |x0 − x00| as a measure of the energy uncertainty between two mass densities µ1(2). The energy uncertainty is related to the decoherence time via ∆t = ~/∆E, which can be understood as the characteristic lifetime of a single particle in an equal superposition of two mass densities. Diosi’s model proposes classical, white noise fluctuations in the Newtonian potential. The variance of the classical noise is motivated by the minimum uncertainty in the estimate of gravitational acceleration if measured by a quantum particle or field [108]. The fluctuations in the Newtonian potential δφ(x0) couple directly to R 3 0 0 0 the mass density HI = d x δφ(x )M(x ). Here we have introduced a regularized mass density M(x) = R dyg(y − x)µ(y), with regularization g(x) being a smooth function (e.g. a Gaussian), to exclude the possibility of singular mass density. The fluctuations in the potential result in the stochastic Schrodinger equation (SSE),

" √ Z −iHdt 3 0 0 0 0 d |ψi = + Γ d x [M(x ) − hM(x )i] dWt(x ) ~ Γ Z # + d3x0d3x00G(x0 − x00)[M(x0) − hM(x0)i][M(x00) − hM(x00)i] |ψi . 2 (2.2)

0 00 0 00 h dWt(x )dWt(x ) i where G(x −x ) = E dt is the two point correlation function of the noise field δφ(x0) and Γ characterizes the magnitude of the fluctuations in the Newtonian potential. Diosi proposes Γ = G and G(y) = 1 . Although the SSE is non-linear in ~ |y| the pure quantum state |ψi (as it includes hM(x00)i = hψ| M(x00) |ψi), the unconditional quantum state is linear in ρ. If one assumes that there is no physical way to measure the particular trajectory (i.e. no physical access to dW (x0)), then for all practical purposes, the model is linear in ρ. The two models discussed so far look exactly nothing alike, the Penrose model invokes relativistic principles, requiring relativistic time dilation, while the Diosi model is in the Newtonian limit. Diosi assumes the potential (and by extension the metric) is classical, while Penrose writes down a superposition of space times (assuming a quantum metric), although he does admit that the interpretation, or “correct quantum mechanical description” for this state “does not greatly matter for present purposes”. Although the models appear very different, Ref. [119] shows that for a single free particle, Penroses’s idea can be understood from Diosi’s SSE Eq. (2.2), and both models result in the same decoherence rate. Although the formulation of the two models look quite different, this similarity is why the model 2.1. LITERATURE REVIEW 34 is called the Diosi-Penrose model.

The GRW model

The GRW model was constructed to ‘classicalize’ quantum mechanics, by leaving quantum scale evolution intact while requiring macroscopic objects to behave classically [41]. In the model the wavefunction of a massive particle is assumed to be localized to within some characteristic length scale σ, centered at some random point x¯ at random times, t1, t2 . . . , tn ... . The model assumes a Gaussian localization process and a Poission rate γ and the evolution of the wave function is probabilistic,

 iHdt  1 − |ψ(t)i with probability 1 − γdt ~ |ψ(t + dt)i ∝ 2 2 (2.3)  1 − iHdt e−(x−x¯) /4σ |ψ(t)i with probability γdt ~ where x is the position operator. Note this is exactly the type of localization process considered in section 1.4.2, where most of the time there is unitary evolution, interrupted by discrete jumps/localization events. The generalization to the multi particle case is straight forward with x (¯x) replaced by xi (¯xi) and γ replaced by γi. In the case of two rigidly bound particles, once one particle is localized, the other is also localized to within the same length scale. Consequently, the bipartite composite particle is localized at twice the rate of each constituent particle, giving a natural scaling: the more rigidly bound constituent particles in a composite particle, the faster the composite particle is localized. This particle scaling is a general feature of all localization models [117, 107, 120, 110], although the scaling is not always linear. The localization process is usually assumed to apply to nucleons, with γ ' 10−16 Hz, and σ ' 10−7 m. These parameters imply a Rubidium85 atom is localized to within 100 nm every 108 years, and one would not expect to see any experimental deviation from unitary quantum mechanics in Rubidium experiments. On the other hand, a grain of salt (≈ 10 µg) is localized every 1 ms and therefore behaves macroscopically3. The pure state in Eq. (2.3) is only pure to an agent with knowledge ofx ¯. However, like Diosi’s model, it is assumed that there is no physical way for any agent to obtain this information. Thus an agent without knowledge ofx ¯ must average over all possiblex ¯ and will therefore have a mixed state description for the state (see section 1.4.2). The evolution of the matrix elements (from Eq. (2.3)) are given by,

d 0 00 2 2 hx0| ρ |x00i = γ 1 − e−(x −x ) /8σ hx0| ρ |x00i (2.4) dt 3I.e. a grain of slat in a quantum superposition would have a fundamental maximum lifetime on the order of several ms, even in the absence of any other decoherence channel. A grain of salt strongly interacting with its surroundings would decohere much faster. 2.1. LITERATURE REVIEW 35 that is an exponential decay in the oﬀ diagonal elements proportional to γ, with Gaussian suppression of coherence for (x0 − x00) > σ, and quadratic suppression for (x0 −x00) σ. The GRW model does not explicitly include any gravitational eﬀects, although gravity has been invoked as a possible source for particle localization.

The CSL Model

The CSL model [112, 113, 110] was developed in the same spirit as GRW. Again it does not explicitly include gravity, however like GRW, gravitational effects are proposed to be the origin on the localization. In CSL it is the mass density operator that undergoes spontaneous localization. This fixes two issues with the GRW model. Firstly, localizing a scalar field (such as the mass density) enables a spatial noise distribution as well as a temporal noise term, enabling a Lorentz invariant description for the noise process — unlike the GRW model which only has temporal noise. Secondly the mass density is blind to any labeling of different particles and hence does not require distinguishable particles. CSL therefore admits a sensible second quantized description. In the CSL model, the usual unitary evolution of the Schrodinger equations is replaced by a SSE, written in Ito form as, √ " Z iHdt Γ 3 0 0 0 0 d |ψi = − + d x [M(x ) − hM(x )i] dWt(x ) ~ m0 Z # Γdt 3 0 0 0 2 − 2 d x [M(x ) − hM(x )i] |ψi (2.5) 2m0 where M(x0) is again the regularized mass density

Z 1 0 2 2 M(x0) = d3y e−(x −y) /2σ µ(y) (2.6) (2πσ2)3/2

0 0 and the ﬁeld dWt(x ) is an independent Wiener increment at each point x . In 2 3/2 the literature the quantities σ and Γ/(4πσ ) are often referred to as rCSL (the

CSL length scale) and λCSL (he collapse rate) respectively. However for notational uniformity we will keep the two independent parameters as σ and Γ in this thesis.

The reference mass m0 is arbitrary, but is often taken to be the neutron mass. Factoring the mass dependence out of Γ means the Γ here has different units to the Γ in Diosi’s SSE Eq. (2.2). Nevertheless there is a striking similarity to Diosi’s equations, with the exception of one less integral. In fact the two equations are actually equivalent, in structure at least. If one removes the Gaussian smearing in the mass density, and adds the same Gaussian smear to the two point correlation 0 00 function E[dWt(x )dWt(x )], the double integral of Eq. (2.2) is recovered exactly. In 2.1. LITERATURE REVIEW 36 this case the key difference between Diosi’s equation and the CSL model is the form of the correlation function of the noise field dW (x0). The regularization of the mass density plays the role of the spatial resolution in GRW, but no longer characterizes the localization strength. Instead Γ, having units m3s−1 kg−2) describes the mass density localization strength: for example in a √ −3 volume V and time t, M(x) will be localized to within the order (m0/ ΓV t) kgm . Another key difference between the GRW and CSL models is that the Wiener 0 increment dWt(x ) is now a field, and may have some spatial correlations, i.e. 0 00 0 0 00 E[dWt(x )dWt0 (x )] = G(t−t )F (x −x ), thereby allowing the possibility for Lorentz invariance. The form of Eq. (2.5) is similar to the single particle SSE in Eq. (1.65), where the single particle operator x is replaced by a field operator M(x). The difference of course is that mass density localization requires localizing an infinite number of degrees of freedom, hence the integral in Eq. (2.5). The similar structure of Eqs. (1.65) and (2.5) mean that CSL can be understood as a continuous measurement of the mass density, however the interpretation is subtly different. Continuous measurement, Eq. (1.65) is derived from standard postulates of quantum mechanics, i.e. unitary evolution and projective measurement. An agent with complete knowledge of the environment (i.e the apparatus discussed in section 1.4.1), will therefore prescribe a pure state to the system. In CSL however, the state evolution is fundamentally non-unitary (no notion of projective measurement has been invoked), and there is no way, even in principle with complete knowledge of the environment, to know the measurement result. This non-unitary evolution breaks the time translation symmetry of the wave function, and therefore, violates energy conservation [121]. The violation of energy conservation manifests as a net heating rate which scales as the decoherence rate of a composite system. The exact heating rate depend non-trivially on the distri- 0 0 bution of particles (xi − xj etc.), but has been calculated for some high symmetry geometries [122, 123, 42, 47]. For single particles the CSL and GRW models predict the same decoherence rate (for appropriately chosen Γ’s). However for the multi particle case, the two models predict different scaling with the number of individual particles in a composite object. The scaling is different depending on the mean separation of constituent particles. The simplified decoherence rate for CSL (obtained in Ref. [124]) shows a scaling of with n2N, where n is the number of particles in a volume σ3 and N is the number of such volumes in the composite particle. (see Refs. [46, 104] for a detailed discussion of decoherence scaling in different regimes). The CSL commonly quoted CSL parameters are Γ = 10−30 cm3s−1 and σ = 100 nm [113], although others parameters have been suggested [124, 41, 125]. 2.1. LITERATURE REVIEW 37

The classical channel model

The CCG model postulates the gravitational force between two massive objects is mediated by a classical information channel [31, 32]. It is instructive to compare this to the quantum channel of electromagnetism (EM). In electromagnetism, a charged particle interacts locally with, and causes a disturbance in, the EM field. The disturbance then propagates outwards, and may (locally) apply forces on distant charges as it passes, resulting in the long range Coulomb force (after tracing out the EM field and taking the c → ∞ limit). If the charged particle is in a coherent spatial superposition, the EM field will be coherently perturbed in superposition, and will apply a coherent superposition of forces to distant charges. The quantum description of the EM field unambiguously allows the superposition and propagation of the quantum state of the particle. The exchange of quantum information is readily seen by considering the Coulomb interaction, between two charged particles (q1(2)) with positions operators x1(2),

1 q q H = 1 2 . (2.7) 4π0 |x1 − x2|

If particle one is in a superposition state √1 (|x0 i + |x00i), then the eﬀective potential 2 1 1 1 1 for particle two is proportional to hHi ∝ 0 + 00 , and particle two changes 1 x1−x2 x1 −x2 by the inﬁnitesimal amount

" # 0 −iq1q2 1 0 1 0 d |x2i = √ 0 0 |x2i + 00 0 |x2i dt (2.8) 4 2π~0 |x1 − x2| |x1 − x2| where one can clearly see the two branches in the wave function of x2 evolving 4 coherently . This is because x2 can see the full quantum information of the state of x1 (and vice versa). The Hamiltonian implicitly assumes as quantized force carrier (as discussed above), and there is no classical uncertainty about the potential that acts on x2. Let us contrast this to a classical force carrier proposed by the CCG model. If the force carrier can only transmit classical information, i.e. particle one has a 0 1 well deﬁned position, in this case, the potential felt by x2 must be either H ∝ 0 |x1−x2| 00 1 OR H ∝ 00 , but cannot be a superposition of the two. In the CCG model the |x1 −x2| exact classical potential felt by x2 is unknowable, even in principle, and the state

4 0 0 00 00 The two branches are actually entangled with x1, i.e. |x1i ⊗ |x2i + |x1 i ⊗ |x2 i, although we can loosely think (but don’t think too hard) of x2 as a test particle, whos evolution does not eﬀect x1. Although this is slightly ambiguous, it does not eﬀect the this gedanken experiment, and the same conclusions may be drawn using the joint quantum state 2.1. LITERATURE REVIEW 38 changes by,

 1  |x0 i OR  0 0 2 0 −iq1q2 |x1 − x2| d |x2i = dt 1 (2.9) 4π~0  |x0 i  00 0 2 |x1 − x2| but not both in superposition. The end result is classical uncertainty, and therefore a mixed state description for particle two. Of course there is a quantum theory of electromagnetism and no need to postulate a classical channel theory. In fact such a theory would already be contradicted by experiments which have observed entanglement though the Coulomb interaction [126], which is forbidden by a classical channel theory. However, the absence of a quantum theory of gravity leaves some scope to entertain the model for gravitational interaction. The original formulation of the theory was in a quantum control framework, where a massive particle is measured, and the classical measurement result is used to apply a force onto a second massive particle (and vice versa). This guarantees that the information sent between the particles is classical as it is a real number estimate of the particle’s position. The interaction Hamiltonian was the linearized Newtonian gravitation potential, and the result is similar to the unconditional coherent feedback example in Section 1.4.2. In Section 2.3, we will extend this model to the multi particle case, with the full non-linear Newtonian potential. The force carrier in the CCG model is implicitly assumed to be a classical potential. This requires a quantum-classical interaction between a massive quantum particle, and a classical ﬁeld — the Newtonian potential. Such quantum-classical interactions have been extensively studied by Diosi [118, 36, 116, 33], although have remain largely unnoticed in the literature. Diosi has also considered this quantum- classical interaction in the context of quantum particles-classical gravity [35], with similar conclusions as the CCG model. The similarities between CCG and the GRW model is striking. The position measurement of constituent particles in CCG gives the same SSE as the GRW model. However when feedback is included in CCG, to reproduce the gravitational interaction the minimum decoherence rate is bounded below by,

GM 2 Dmin = (2.10) ~d3 for two particles of mass M separated by a distance d. In the GRW model, there is no minimum decoherence rate if one is willing to allow for larger and larger macroscopic superposition states. The authors of Ref. [31], extended the CCG model to a lattice ﬁeld description [32], in an approach similar to [114], by measuring and feeding back onto the 2.1. LITERATURE REVIEW 39 mass density operators. In this case the measurement part of the model generates a similar evolution as the CSL model, as it is an eﬀective measurement of a regularized mass density operator. Again, once the feedback is taken into account, there is a minimum decoherence rate dependent on the mass distribution of the system.

Schrodinger-Newton Equation

An overview of collapse models would be incomplete without at least mentioning the Schrodinger-Newton equation (SNE). The SNE is a simple intuitive model to include Newtonian gravity in quantum mechanics. The equation can be simply obtained by assuming the mass density of a quantum state is proportional to its probability distribution, i.e.

µ(x0) = m hψ| δ(x − x0) |ψi = m|ψ(x0)|2. (2.11)

0 R 3 0 µ(y0) The mass density generates a potential φ(x ) = −G d y |x0−y0| for wave function. This solution for the potential is only valid in three dimensions, otherwise a diﬀerent Greens function must be used, hence interpreting one dimensional solutions of the SNE can be ambiguous. The SNE is,

d i ψ(x0) = [H + mφ(x0)] ψ(x0) ~dt " Z |ψ(y0)|2 # = H − Gm2 d3y0 ψ(x0). (2.12) |x0 − y0|

Although the above equations was derived by assuming the mass density is the probability density, it can also be derived as the Newtonian limit for semi-classical 8πG gravity, Gµν = c4 hTµνi [127]. Several authors have pointed out interpretational problems with the above equation. The nonlinearity violates the superposition principle for stationary states, the equation is known to have physical solutions that allow superluminal signaling, and the solutions may have a divergent energy spectrum [128]. It is also a somewhat problematic interpretation of writing the mass density in this way. For example, consider a simple gedanken experiment of passing a test mass equidistant between a macroscopic superposition of a large mass. Standard quantum mechanics would suggest the test particle’s wave function would branch, with equal superpositions deflected left and right, entangling the test and large masses. However a Schrodinger-Newton particle would pass straight though, without deflection — it sees an equal mass distribution either side and no net force. Although the latter case does not sit well with our understanding of standard quantum mechanics, it cannot be dismissed outright without experimental measurements of a quantum 2.1. LITERATURE REVIEW 40 mass density. Note that the CCG model resolves this issue: the measurement forces the large mass to be in a definite location, and the test particle deflects accordingly. Before an observer measures the deflection of the test particle, they do not know the location, the joint quantum state is an incoherent mixture, but nevertheless the observer still the see test particle deflect on a shot-by-shot basis. Despite these problems, the SNE may be modified to avert these problems. In- troducing a regularized mass density (using a Gaussian instead of δ(x − x0)) avoids the problem of a singular mass density and divergent solutions, and introducing a stochastic term can be added to avoid superluminal signaling (on average) [129, 130]. The stochastic term in the single particle SNE behaves similarly to the GRW model and connects the SNE to other collapse models. Although the stochastic SNE is both a collapse model and includes the effect of gravity, the semi-classical interpretation slightly unsettling. CCG makes some progress to address this as mentioned in the gedanken experiment above.

2.1.2 Opto and nano-mechanical systems

Two of the key publications in this thesis involve opto- and nano-mechanical systems. This section is dedicated to a short review of the types of micromechanical resonators. The canonical optomechanical system, and their key parameters were introduced in Section 1.3. In the following we will introduce several experimental realizations of optomechanical systems. A major goal of modern physics is to demonstrate a macroscopic coherent quantum superposition of matter. Optomechanical systems are well placed to realize this target as they can (in principle) be built with macroscopic mechanical oscillators. For example, the Laser Interferometer Gravitational-Wave Observatory (LIGO) is an optomechanical system with two forty kg mechanical elements. While there is no proposal (serious or otherwise) to demonstrate a macroscopic quantum state with LIGO mirrors, there is a phenomenally wide range of mass scales available to optomechanical systems. Another obvious advantage for optomechanical systems is the ability to engineer exotic quantum states of the optical ﬁeld, and use this to observe quantum behavior of the mechanical oscillator. An example advantage is considered in Ref. [131], where a hypothetical mechanical oscillator (a mirror in a Michaelson interferometer) is fabricated with such a thin support that the momentum kick from a single photon is enough to break the oscillator. Sending a single photon though the input port puts the photon in a superposition state of interaction and not-interacting with the fragile mirror. After the photon leaves the interferometer and is detected5, there is no which

5In fact one requires number resolving detection; if one had access to the frequency/phase of the 2.1. LITERATURE REVIEW 41 way information, that can be used to infer the state of the oscillator. Hence the oscillator is in a superposition of kicked/not-kicked; broken/not-broken. Of course this example requires the breaking nonlinearity of the oscillator, but nevertheless fully illustrates the ability to use the quantum state of the optical field as a useful resource for optomechanics. There have been a number of proposals to generate exotic quantum states of a mechanical oscillator [78, 79, 132, 68, 133, 134, 135]. Despite the number of promising avenues to achieve a mechanical state superposition, observation of Wigner negativity has yet to be conclusively observed6. Recent experiments involving membranes have observed interference fringes in the position probability distribution [136], but this can be explained by classical dynamics [137]. Entanglement between an oscillator, and microwave field [70, 67], or optical field [138] has been observed, thus demonstrating quantum behavior for m = 48 pg, and m = 5 ng mechanical oscillators. In these experiments the entanglement was generated from a two-mode-squeezing interaction (blue detuned drive - see Sec- tion 1.3), resulting in a Gaussian probability distribution of the mechanical oscillator, and not a center of mass superposition state of the mechanical element.

Fabry-Perot cavities

The Fabry-Perot cavity is the canonical optomechaincal system and were the first to be experimentally realized in LIGO [139], almost a century after the first demonstration of the radiation pressure force in 1901 [140, 141], and several decades after Bra- ginski theoretically analyzed the system [49]. Due to the restriction on the minimum size of the mirror, Fabry-Perot optomechanical designs are often the largest mass systems. There are several different methods to realize the mechanically compliant mirror including suspended mirrors [142, 143, 144, 145], cantilevers [146, 147, 148], and proposed magnetically levitated mirrors [149]. Since Fabry-Perot cavities are the most obvious cavity optomechanical system, it is unsurprising that they were the first to demonstrate optical cooling [148], the optical spring effect7 [150] and the optomechanical bistability [151]. The mechanical resonance frequency range from sub- Hz for the suspended truly photon then it may, in principle reveal if it had interacted with the oscillator thereby destroying the superposition. The fact that changing the basis of measurement changes the conditional state of the oscillator comes as no surprise after the discussion of weak measurement in Section 1.4.1, and this interferometer set up can be thought of as a weak measurement of the oscillator where the apparatus measurement basis doesn’t reveal any information about the system! 6Wigner negativity is a sufficient indicator for non-classical Gaussian states. 7 Where the cavity responds quickly to the change in resonance frequency (κ ωm) such that the intracavity photon number depends only on the instantaneous position of the mirror a†a ∝ x, 2 and the interaction Hamiltonian is HI/~ ∝ g0x , i.e. a spring. The sign of the proportionally constant depends on the detuning between the drive field and the cavity resonance 2.1. LITERATURE REVIEW 42 macroscopic LOGO mirrors, to MHz, for a trampoline micromirror. Regardless of the method of suspension, there have been significant advances in reducing the coupling between the mirror and the environment. By engineering the supports and structure around the mirror, energy exchange between the environment and oscillator can be decreased, resulting in a higher Q-factor, and longer coherence times [152].

Optomechanical crystals

An optomechanical crystal consists of a dielectric slab with periodic micrometer scale structures etched out [153]. The period structure act as a pair of Bragg mirrors (i.e. mirrors of a Fabry-Perot cavity) tightly confining light in a localized region in the dielectric medium. Light can be coupled into and out of the cavity from the evanescent field of a nearby tapered fiber. The periodic structure also supports mechanical modes, which microscopically alter the effective dielectric constant in the region of optical confinement, and therefore alter the modeshape and frequency of the optical mode, resulting in an optomechanical coupling. The mechanical restoring force (i.e. spring constant) in an optomechanial crystal depends on Young’s modulus of the dielectric medium, hence the optomechanical crystals have some of the highest resonance frequencies, at ∼ GHz. This high resonance frequency means that the mechanical mode is close to its ground state at cryogenic temperatures:n ¯ ≈ kBT ≈ 102 thermal phonons for a 1 GHz oscillator ~ωm at 1 K. The low initial thermal occupation, combined with laser cooling enabled optomechanical crystals to be the first system to directly observe the ground state fluctuations of a mechanical oscillator [76]. Since their inception, several generations of optomechanical crystals have been fabricated [154, 155, 156, 157, 158]. Nano-engineering techniques such as phonic shields (a phononic analogue of Bragg mirrors), spatial dependent etching patterns, and double or triple layer crystal designs have resulted in a significant improvement in device parameters. Optical and mechanical Q-factors of up to 106, bare coupling rates of g0/2π = 90 kHz, have been demonstrated. However, the devices are much smaller than other optomechanical systems, and the effective mass of the mechanical mode is among the lightest systems.

Microtoroids and disks

Like optomechanical crystals, microtoroid [54, 159] and microdisk [160, 161] resonators also combine the optical cavity and mechanical motion into a single device, fabricated on chip. The light is conﬁned to travel around near the perimeter of the toroid (or disk) by total internal reﬂection from the toroid-air boundary. The optical 2.1. LITERATURE REVIEW 43

ﬁeld is evanescently coupled into and out of the toroid from a tapered optical ﬁber.

Adjusting the toroid-fibre distance changes the cavity input coupling rate κe, giving an extra parameter of control in the system. The mechanical modes are the eigenmodes of the toroid/disk structure [162], and the motion of each mode is enough to change the effective path length of the optical mode, giving optomechanical coupling rates of up to O(100) Hz. The mechanical modes that couple most strongly to the optical field are those that most modify the optical path length, and the highest coupling rate is the so called radial breathing mode. As toroidal resonators are intrinsically high Q, and the mechanical frequency is ∼ 100 MHz, they often operate well into the resolved side-band regime, where the mechanical frequency is much larger than the cavity decay rate, ωm κ, and therefore easily allow sideband cooling of the oscillator [163, 164]. Disk and toroidal resonators are intrinsically a degenerate multi-mode optical system: the devices support both clockwise and counter clockwise modes [165]. For an ideal device these modes are degenerate, and an input optical field will only couple into one with the same direction momentum vector at the coupling interface. However imperfections on the surface/interior of the device act as scattering centers, scattering light from one mode into the other resulting in normal mode splitting (hybridization of the modes in the clockwise-counter and clockwise basis). In this case there are two optical modes, and two possible inputs/outputs for the cavity8, and light leaving the cavity can travel in each direction along the fiber. Toroidal cavities are significantly larger than optomechanical crystals, with a ∼ 100 µmdiameter, and their size has been exploited to develop magnetometers, among other sensors [166]. By depositing magnetostrictive material on the toroid, local magnetic fields apply stresses and strains to the toroid, and this motion (above the thermal fluctuations of the eigenmodes) is detected in the optical field leaving the cavity. The optomechanical toroid is used as a transducer to convert a magnetic field into an optical signal. More recently, this effect has been used to realize an optomechanical microphone [167]. Several engineering approaches have been implemented to improve the device performance of toroids and disk resonators. Double layer devices [168] and cen- 8 ter etched (spoke) toroids, have result in mechanical quality factors of Qm = 10 , and improved coupling rates of up to G/2π = 27 GHz nm−1. Strong single photon cooperativity remains elusive, however strong linearized cooperativity has been demonstrated, and been used for both side-band and feedback cooling.

8In general three input/outputs — two from the normal mode splitting, and the third from intrinsic loss in the cavity. 2.1. LITERATURE REVIEW 44

Evanescent coupling

In a whispering gallery mode most of the optical field in an microtoroidal resonator is confined inside the dielectric material, however a portion of the mode exists in an evanescent field in the free space outside the toroid [169]. Indeed this is how light is coupled into the cavity. If a dielectric material is placed in this evanescent field, it alters the optical mode shape, and can therefore be used as a mechanical element for the optomechanical system. Such as system is called an evanescently coupled optomechanical system, or near field optomechanics [53]. The advantage of near field optomechanics is one has complete control over the choice of resonator [55, 135]. This enables one to independently optimize the fabri- cation process of the cavity and resonator [152] and combine them in a joint system, or fabricate them both on chip [170]. Silicon nitride nano-strings evanescently coupled to a toroid have demonstrated an optomechanical coupling rate of G/2π = 100 MHz nm-1 , with optical and mechanical quality factors of 107 and 105 respectively.

Microwave mechanics

There is no requirement for the light in an optomechanical system to be at optical frequencies. The key feature outlined Section 1.3 is a conﬁned electromagnetic mode with the resonance frequency dependent on the motion of a mechanical element. In the microwave system, the electromagnetic mode can be realized by an LC circuit, 2 which has a resonance at ωc = 1/LC in the microwave domain. The mechanical element is one of the capacitor plates. The motion of the capacitor plate changes the capacitance, and therefore the resonant microwave frequency of the circuit [171, 172].

Using such devices, the strong coupling domain (g0 > γ, κ) has been reached [173]. The Hamiltonian for microwave mechanics is exactly the same as optomechanics, and the same equations yield the same solutions. Hence cooling [174, 175, 176], entangling [177, 70], quantum state preparation [178, 67], have all been proposed in the microwave domain. Working in the microwave domain has advantages and disadvantages over the optical domain. For example, is advantageous for interfacing with existing superconducting devices, and one has excellent control over microwave driving fields, however this comes at the cost of good single photon detectors, and the difficulty of thermal noise in the microwaves themselves9. Microwave systems have successfully been coupled to superconducting qubits. Using this microwave-mechanical-qubit coupling, vacuum fluctuation of the optomechanical system have been detected [179]. Similar devices have successfully demonstrated coherent storage of a microwave field in the motional state of a vibrating capacitor [180], as a precursor to a realizable long lived quantum memory.

9Although recent advances in technology mean these drawbacks are becoming less problematic. 2.2. CCG: NON-LINEAR DESCRIPTION 45

Microwave-mechanics have been proposed for coherent single photon frequency conversion including optical-to-microwave conversion, using the mechanical element as a medium [181, 182]

2.2 Classical Channel Gravity: Non-linear multi particle description

Following on from the discussion of gravitational collapse models in section 2.1.1, here we introduce a multi-particle description of the classical channel model of gravity. Furthermore the full non-linear Newtonian potential is used. Before continuing however, I would like to acknowledge Stefan Nimmrichter (currently atUniversity of Erlangen-Nuremberg) for his substantial contributions to this project.A preprint of this chapter is given in Ref. [6]. In the classical channel model of gravity introduced in Ref. [31], the Newtonian gravitational interaction between two masses was linearized, and understood as a measurement and feedback model. Some authors have since discussed the consequences of such a model in the multi-particle regime (with some assumptions that we will highlight here), but always using a single directional linearized potential. Such a linearizion of the Newtonian force is not an issue if one assumes, as in Ref. [31], that the particles are tightly constrained in the transverse directions to motion. This constraint is not probing 1-Dimensional (1D) gravity as the potential still has a 1/r dependence, from the 3D Greens function solution to the potential10. However, for the general case, particles may move in any dimension, and a multi polar expansion of the 1/r potential is required to analyze a general system. In the spirit of Ref. [31], we will consider a measurement and feedback approach between N non-interacting particles and try to recover the Newtonian potential as a systematic term in the unconditional evolution.

2.2.1 The Lindblad operator

Gaussian Measurement

Instead of directly considering weak continuous measurement, we will consider a sequence of measurements at Poisson random times, at a rate γ — exactly as outlined in the section 1.4.2. The measurement operators are unbiased Gaussian POVM

10In 1D gravity the Newtonian potential has |x| scaling, as a consequence of the Greens function ∂2 for ∂x2 2.2. CCG: NON-LINEAR DESCRIPTION 46 operators acting on the jth particle,

" 2 # 1 −|rj − z¯j| M(¯zj) = 2 3/4 exp 2 . (2.13) (2πσj ) 4σj

Here rj is the 3-dimensional vector position operator of the jth particle (with canon- 2 2 ical momentum pj). The spatial resolution is mass dependent σj = (m0/mj)σ with m0 some arbitrary reference mass, and σ the reference standard deviation. The classical quantityz ¯j is a vector of measurement results replacing the scalar measurement resultx ¯ from section 1.4.2. Note the difference of units between the 3D and 1D measurement operators, (the σ−6/4 instead of σ−1/4 in the denominator), D † E this is because MJ MJ is now a probability density in three dimensions instead of one. Here σ2 still characterizes the uncertainty in the measurement, in principle one 2 2 could consider a vector of {σµ}µ=123, or matrix {σµν}µν=123 which would correspond to different spatial resolutions in different directions (vector case), or correlations 11 2 between measurement results (matrix case) . For Gaussian measurements σµν must be positive definite, and there always exists a spatial rotation — the one that diag- 2 onalizes σµν — where the same measurement operator is represented with a vector 2 2 2 2 σµ. The only choice of σ , σµ, σµν that preserves rotational symmetry is a scalar value σ, and hence we will take this to be the case.

Coherent feedback

To recover the Newtonian potential, we include a unitary term in the measurement operator (exactly analogous to the example in section 1.4.2). After measurements of all particles we have individual estimates for the positions of each particle,z ¯j. The unitary acting on the jth particle is chosen to be

i Uj = exp − rj · ∆¯pj (2.14) ~ where

X ∆¯pj = −τ∇zj Φ(|zj − zn|) (2.15) n6=j X z¯j − z¯n = − Gτmnmj 3 (2.16) n6=j |z¯j − z¯n| is a classical momentum kick felt by particle j from the feedback terms of all other particles, for a the Newtonian potential Φ(ri − rj). Here τ should be thought of

11Here we are using Greek indices to denote the possible tensor nature of the reference standard deviation σ as opposed j denoting the spatial resolution for diﬀerent particles j 2.2. CCG: NON-LINEAR DESCRIPTION 47 as O(γ−1), and is the characteristic time scale of free evolution between subsequent measurements. The unitary in Eq. (2.14) is a momentum displacement that changes the momentum depending on the position estimates of where each of the other n 6= j ixα −ixα ~ particles (recall e pe = p + α and δp = F δt = −∇V δt for small δt). The fact that only classical estimates of position (¯zj) appear in the potential mean that this unitary can be thought of as implementing the local operation part of a LOCC description. The unitary does not exactly describe a particle moving in the full 1/r potential, but rather a particle moving in a linear potential over the duration τ, after which time the gradient of the linear potential changes. We shall see that this simpliﬁed (linear) Hamiltonian can indeed generate a systematic interaction that behaves as a full 1/r Newtonian potential. There are other possibilities for the choice of unitary which will be addressed in an upcoming manuscript, the linear potential is attractive for its analytical simplicity, and results in the correct systematic term. Furthermore, as shown in section 1.4.2, if a single dimension linearized potential is used (∆¯pj ∝ zn) then this description results give exactly the same master equation as in Ref. [31]. The unitary term is included after a measurement of the jth particle resulting in the Lindblad operator,

 2  1 (rj − z¯j) i X z¯j − z¯n Lj = UjMj = 2 3/4 exp − 2 + rj · Gτmnmj  . (2.17) (2πσj ) 4σj ~ n6=j |z¯j − z¯n|

Lets now consider how the quantum state evolves in a time step dt. The conditional state after every particle is measured and the feedback applied is, ρ ∼ LρL† where

Y L = Lj (2.18) j      2  Y − 3N 3N X (rj − z¯j) i X z¯j − z¯n  2 − 4 =  σj  (2π) exp − 2 + rj · Gτmnmj  j  j 4σj ~ n6=j |z¯j − z¯n|  (2.19)

explicitly depends on the set of all measurement results {z¯j}. Here L is the multi particle Lindblad operator that acts on the joint Hilbert space of the entire N particle system. We can immediately write down the evolution of ρ in Lindblad form dρ Z = L(ρ) = γ d3N zLρL† − γρ (2.20) dt     " 2 # Y 3N Z X (rj − z¯j) i  −3N − 2 3N † =  σj  (2π) d z exp − 2 − rj · ∆¯pj ρL − γρ j  j 4σj ~  (2.21) 2.2. CCG: NON-LINEAR DESCRIPTION 48

3N 3 3 3 R 3N † where d z = d z¯1d z¯2 . . . d z¯N . The Lindblad form is obvious if one notices d zL L = I, then

Z 1 1 L[ρ] = γ d3N z LρL† − L†Lρ − ρL†L . (2.22) 2 2

Being in Lindblad form, the free Hamiltonian H0 (particle kinetic energy and any other interactions) can be intuitively included by the addition of a − i [H, ρ] term. ~ Note the similarity with Eq. 2.3 when the gravitation component is removed (i.e. set G = 0), therefore this model recovers exactly the GRW spontaneous localization model for the case of many non-interacting particle.

Comments on the Master equation

The first thing to note is that the model is invariant under Galilean boosts and translations. In quantum mechanics Galilean transformations are generated by G = P j (rjmjV + pjX), where V and X are respectively the boosted velocity and spatial translation of the transformation. The unitary e−iG/~ shifts the position coordinate of every particle by X, and the momentum coordinate by mjV , and is equivalent to e−irj P/~e−ipj X/~ up to a global phase (which plays no physical role if X and P are real numbers). The invariance is easily seen by the fact that a momentum shift has no effect on Eq. (2.21), and a position shift can be absorbed in the dummy variable 3 3 d zj → d (zj −X) (we need not consider the unitary part as rj → rj +X adds only a classical phase which will cancel in all LL† terms). Galilean invariance implies that there are no residual effects on the center of mass system, and Galilean symmetry is preserved. Equation (2.21) does not easily generalize to a second quantization picture as the particles must be distinguishable in the sum n 6= j. One could in principle let this sum run over all n, which would be equivalent to including a self interaction term (each particle feels its own gravitational field). However even if this self interaction term is included, making L symmetric in all rj (up to a relabelling ofz ¯j, the number of integration variables still depends on the total number of particles, which is problematic for second quantization. Despite the difficult path to a field theory picture, Eq. (2.21) is a self consistent model for a classical channel theory of gravity for free particles in the low energy (i.e. particle number is conserved) limit. We will consider the effect of collections particles under the evolution of Eq. (2.21). 2.2. CCG: NON-LINEAR DESCRIPTION 49

2.2.2 Systematic dynamics of the measure-feedback model

First let us consider the systematic dynamics of particles under the evolution of Eq. (2.21). The momentum of the jth particle changes as,

D † E † L (pj) = Tr [pjL(ρ)] = Tr[L (pj)ρ] Z 3N † D 3N † E = γ d zL pjL − pj = d zL [pj,L] (2.23) Z 3N † = γ d z(−i~)L ∇rj L (2.24) *Z " 2 2 #+ X γGτmjmn zj − zn 3 3 (rj − zj) (rn − zn) = − 3 3 d znd zj exp − 2 − 2 n6=j (2πσjσn) |zj − zn| 4σj 4σn (2.25)

R 3N † ∂f where we have used the fact d zL L = I (line two), and [p, f(r)] = −i~ ∂r for an analytic function in r (line 3). The integral can be computed by changing coordinates into the centre of mass and relative frame, pairwise for each n, j, recalling 2 2 σj = (m0/mj)σ ,

5/2 D † E X (mnmn) L (pj) = −Gγτ 2 3 n6=j (2πm0σ )

* h 2 mj mn 2 i 1 + Z z + r − r −(mj +mn)z − z 3 3 − j n + mj +mn − 2σ2m × d z−d z+ e 0 |z− + rj − rn| m m µ Z z D 2 2 E X n j jn 3 − −µjn(z−−rj −rn) /(2m0σ ) = −γGτ 2 3/2 d z− 3 e n6=j (2πσ m0) |z−| (2.26)

where µjn = mjmn/(mj + mn) is the reduced mass of the relative motion. We can write the Gaussian term as an expansion in Fourier space12,

D E Z exp [−m σ2k2/(2µ )] Z z D E † X 3 0 jn 3 ik·z ik·(−rj +rn) L (pj) = −γGτ mnmj d k 3 d z 3 e e n6=j (2π) |z| Z 2 2 ! X 3 exp [−m0σ k /(2µjn)] −4πik = −γGτ mnmj d k 3 2 n6=j (2π) k * 3 2 + Z d k − m0σ k2 X 2µjn −ik·(rj −rn) = γGτ mjmn ∇rj 2 2 e e (2.27) n6=j 2π k

z 1 where we have used the second line we used |z|3 = −∇z |z| and the Fourier transform of a derivative can be understood as ik. In the last line we noted that ik ≡ ∇rj in

2 2 2 2 12For a Gaussian (2πσ2)−3/2e−x /(2σ ) = R d3ke−k σ /2+ix·k, and the Newtonian potential Fourier transform R d3reik·r/r = 4π/k2 2.2. CCG: NON-LINEAR DESCRIPTION 50 the Fourier integral. The convolution theorem can be used to evaluate the integral on the RHS of Eq. (2.27). The result is a convolution between 1/r (form the 1/k2 term) and a

Gaussian (from the Gaussian term in k), the ik · rn term is included using the shifting theorem, giving

* " #+ D † E X 1 |rj − rn| L (pj) = γτG mjmn ∇rj erf √ (2.28) n6=j |rj − rn| 2σjn

2 2 where σjn = σ m0/µjn. This is exactly the evolution of pj moving under a Gaussian D E D dpj E blurred Newtonian potential, this is easily seen by recalling −∇rj V (rj) = dt . We may therefore identify the eﬀective potential as

|r −r | erf √j n X 2σjn V (rj) = −γτG mnmj . (2.29) n6=j |rj − rn|

Note that this is exactly the expression for the potential of a Gaussian smeared mass distribution,

−(r−r )2/(2σ2 ) Z Z n jn 3 µ(r) 3 e 1 Φ(x) = −Gmn d r = −Gmn d r 2 3/2 (2.30) |x − r| (2πσjn) |x − r|

(which is exactly the convolution resulting from Eq. (2.27)). The systematic eﬀect of the measurement and feedback process for a classical channel model of gravity therefore results in a Gaussian smeared interaction, even for point particles.

If the COM separation between two particles is large enough, h|rj − rn|i

σjn, the error function tends to unity and the potential is just the 1/r potential of Newtonian gravity (with the right coeﬃcients if γτ = 1). This can also be understood in terms of the mass distribution, where the Gaussian blurred mass becomes well approximated by a point mass.

On the other hand if the particles are extremely close, h|rj − rn|i σjn, then the lowest order interaction is still attractive, but is now a quadratic potential,

* 2 + D E h|r −rn|iσ (r − r ) L†(p ) −−−−−−−−→j jn γτG X m m −∇ √j n . (2.31) j j n rj 3 n6=j 3 2πσjn

This puts an experimental upper bound on a physical value of σ, which must be smaller than the length scale where the 1/r potential has been observed between to particles. We shall examine this in more detail in later subsections when considering the interaction between two composite particles. 2.2. CCG: NON-LINEAR DESCRIPTION 51

2.2.3 Particle noise scaling

Single particle

First lets consider the eﬀect of the master equation on a single particle, rj. This is a slightly unphysical situation, however we can think of a limiting case where all other particles are very far away — i.e. no other particles have wave function support nearby. In this case the unitary component of L vanishes13, and the single particle evolution is

3/2 Z m 2 2 2 2 3 −mj (r−z) /4m0σ −mj (r−z) /4m0σ Lρ = γ d z 2 3/2 e ρe − γρ (2.32) (2π m0σ ) 2 !3/2 2m0σ Z 2 2 = γ d3ke−2m0σ k /mj eik·rρe−ik·r − γρ (2.33) πmj where in the second line we have written the master equation as Fourier transform of the dummy variable z. This form is illuminating as it gives another interpretation for the random measurement process; it can be understood as a random momentum kicks of (p = ~k), with a Gaussian high energy cut off. The random kick process increases the momentum variance ensuring the uncertainty principle is obeyed. Clearly as σ → 0 (projective measurement), the particle sees a flat probability distribution of momentum kids ensuring hδp2i → ∞. This form of random kick process was considered in a general class of macrorealistic modifications studied inff Refs. [42, 47]. The introduction of other particles close by explicitly changes this. The evolution no longer factories into random kicks for each particle, and the evolution of a single particle now explicitly depends on surrounding particles (as it should for a theory of gravity). This is a major difference to previously studied macrorealistic master equations.

Composite particle

Lets consider the reduced COM motion of a composite particle. We will write the 1 P COM position operator as R = N j rj, and reparameterize the particle positions P into N−1 independent coordinates rj = R+δrj with the linear constraint j δrj = 0. In this case the master equation for the reduced center of mass is obtained by simply

13 Recall form the previous section that the probability distribution of zn is a Gaussian convolved wave function. For two particles with separable wave functions and separated by some large distance d, |zn − zj| ≈ d, and the unitary term is therefore vanishingly small for large d. 2.2. CCG: NON-LINEAR DESCRIPTION 52 tracing out the relative degrees of freedom from the center of mass,

2    P (R+δrj −zj ) i Z − + ∆pj ·(R+δrj ) Y 3  j 4σ2 ~ 2 − 2 3N j LρCOM =  (2πσj )  Trrel d ze ρ j  2 P (R+δrj −zj ) i  − − ∆pj ·(R+δrj ) j 4σ2 ~  ×e j − ρCOM  (2.34)

This is actually quite a simple expression, the phase factors ∆pj · δrj vanish in the P P trace. The other phase factor j ∆pj · R must also vanish as j ∆pj = 0, from the P cancellation of action-reaction pairs in the double sum j,k6=j ∇zj Φ(|zj − zk|) (see the deﬁnition of ∆pj in Eq. (2.15)). A change of dummy variable zj → zj − δrj, greatly simpliﬁes the integration in the trace,

  ( m (R−z )2 m (R−z )2 ) Y 3 Z P − j j P − j j 2 − 2 3N j 4σ2m j 4σ2m LρCOM =  (2πσj )  Trrel d ze 0 ρe 0 − ρCOM j (2.35)

The integral can now be evaluated in terms of the matrix elements of ρCOM in 0 00 14 hR | ρCOM |R i basis .

 h M 0 00 2i  exp − 2 (R − R ) 0 00  8σ m0  0 00 hR | LρCOM |R i = 2 3/2 − 1 γ hR | ρCOM |R i (2.36)  (2πσ m0/M)  ⇔ Z 2 2 2 2 2 −3/2 3 −(R−z) /(4σM ) −(R−z) /(4σM ) LρCOM = γ(2πσM ) d ze ρCOMe − γρCOM (2.37) which is the stochastic localization process at rate γ and localization length scale q σM = σ m0/M (compare with Eq. (2.32)). The factor of the total mass M intuitively arises from the sum over constituent particle masses in the exponential of Eq.(2.35). The localization length scale of a composite particle is enhanced by a factor of √ √ σ ∝ M −1 ∝ N −1. This can be though of as N independent weak measurements M √ of the center of mass, giving the usual signal to noise enhancement of 1/ N for independent Gaussian measurements. This mass scaling is lower than those predicted in the mass proportional CSL and GRW models, and is not suﬃcient to amplify the decoherence for macroscopic bodies. Replacing the scaling with σj = σm0/mj,

0 2 00 2 14 R 3 − (R −z) − (R −z) 3/2 (R0−R00)/(8a) Using the result d ze 4a 4a = a e . 2.2. CCG: NON-LINEAR DESCRIPTION 53 changes the scaling of the localization length scale15 to ∼ N.

2.2.4 Heating rate

The fact that pj no longer commutes with the generator of time translations (i.e. P 2 L), means the kinetic energy j pj /2mj is no longer conserved. Such energy violations are common in emergent macrorealistic theories in quantum mechanics. Even the mass proportional CSL model violates conservation of kinetic energy, and such energy violations have been exploited for proposals to test macrorealistic models. The heating rate is proportional to the sum of increases in kinetic energy, hence 2 dhpj i dt is suﬃcient to understand the heating rate,

Z * " 2 2 #+ D † 2E 3N † ~ mj 2 2 L pj = γ d z L L 4 (rj − zj) + ∆pj (2.38) v 4m0σ where the subscript v (for violation) denotes the fact that systematic terms (i.e. the rj ·∆qj terms) have been excluded on the right and side. The first term is the heating rate due the measurement (increasing the momentum variance, see Eq. (2.33)), and the second is from applying feedback with a fluctuating force. This integral is well defined, but has not been analytically solved yet. It would be of interest to consider the total heating rate for the earth (for example), to see if the model can be ruled out by simple observations.

2.2.5 Summary

This section has considered the classical channel model of gravity beyond the linearized gravitational interaction regime. We have shown that one must include a ﬁnite resolution for the measurement to preserve the Newtonian 1/r potential at the macro-scale. This is in contrast to the previously considered linearized gravitational interaction where a continuous measurement was assumed. Furthermore we have derived the systematic evolution, and the noise scaling for the center of mass motion of single/composite particles from this model. We have also written the master equation in the framework of generalized macroscopic models [43].

Acknowledgment

I would like to acknowledged and thank Stefan Nimmrichter, with whom I have worked closely with on this project, and has been very helpful in solving some diﬃcult integrals.

15 2 2 2 2 P 2 −1 −2 For example, replacing σj ∝ 1/mj with σj ∝ 1/mj gives σM ∝ ( j mj ) ≈ M if each mj are the same order of magnitude. 2.3. ARBITRARY QUANTUM STATE ENGINEERING 54 2.3 Arbitrary quantum state engineering of a nanomechanical oscillator

This section outlines how one can generate arbitrary quantum states of a massive mechanical oscillator. The model investigates the Displacemon architecture and requires a classical drive of the qubit. In the following sections a one-to-one map is given that relates the target quantum wave function to the classical qubit drive required to generate the wave function (from an initial thermal state). A preprint manuscript for this section can be found in Ref. [5].

2.3.1 Model

The model system consists of a qubit with bare frequency ωq coupled to a mechanical resonator of frequency ωm, where the frequency of the qubit depends on the displacement of the resonator [133, 183, 184, 185, 4]. The free Hamiltonian for the † 1 system is H/~ = ωmb b + 2 ωq(XM)σz, but we will approximate this by linearizing the ωq(XM) dependence. The linearization of the qubit frequency gives

(0) dωq(XM) ωq(x) ≈ ωq + XM dXM (0) † = ωq + λ0(b + b ) (2.39)

† where XM = b+b is the dimensionless position of the mechanical oscillator. The coupling rate λ0 is analogous to the single photon coupling rate g0 in an optomechanical system. For the architecture considered in Ref. [4] (i.e. chapter. 6), λ0/ωm ≈ 0.06, hence in the following we will assume strong coupling. We will move into the in- † 1 (0) teraction picture with H0 = ωmb b + 2 ωq σz. In addition to the qubit-resonator coupling we consider coherently driving the qubit with a complex drive amplitude β(t). The resulting interaction Hamiltonian (including a qubit driving term) is

† iδt ∗ −iδt HI /~ = λ0(t)(b + b )σz + e β (t)σ+ + e β(t)σ–. (2.40)

Here λ0(t) the time-dependent coupling rate between the qubit and resonator. A similar time dependent coupling was considered in Ref. [4] by using an external magnetic field. From Fig. 1.3, one can easily see that if B = 0, the change in flux though the qubit does not depend on the position of the oscillator. Hence the position dependence of the qubit frequency vanishes and there is no coupling. By changing the B field as a function of time, one may introduce a time dependent, 2.3. ARBITRARY QUANTUM STATE ENGINEERING 55

16 qubit-oscillator coupling rate, λ0(t) . The operators σ– and σ+ are the standard two level lowering and raising operators respectively. The amplitude β(t) is the time dependent envelope of the ﬁeld17 used to control the qubit, and δ the detuning between drive the carrier frequency and the bare qubit frequency. Henceforth we will choose δ = 0, as assume the driving frequency is on resonance with the the qubit frequency. The annihilation operator b is in † the resonators rotating frame so that XM = b + b , i.e. the position quadrature, is time-independent in the interaction picture ([XM,HI] = 0). Let us now consider an arbitrary (interaction picture) qubit-resonator state,

Z |ψi = ce(x) |e, xi + cg(x) |g, xi dx, (2.41) where g and e denote the ground and excited state of the qubit respectively, and ce(g)(x) is the probability amplitude of ﬁnding the qubit in the excited (ground) state, and the oscillator at position x. The interaction Hamiltonian, (Eq. (2.40)) generates the following equations of motion for the probability amplitudes

∗ c˙e(x, t) = −iλ0(t)xce(x, t) − iβ (t)cg(x, t)

c˙g(x, t) = iλ0(t)xcg(x, t) − iβ(t)ce(x, t). (2.42)

For a general time dependent coupling and drive, this coupled diﬀerential equation has no analytic solution, and must be solve numerically. We now consider a measurement protocol as follows: Fig. 2.1,

1. Prepare the qubit in the ground state, i.e. ρ = |gi hg| ⊗ ρm, with the qubit-

resonator interaction switched oﬀ λ0(t) = 0. ¯ 2. Switch on the qubit-resonator interaction λ(t) = λ0, and start the β(t) driving. During this evolution, the drive tone will populate the excites qubit state, and

σz term will rotate the qubit state around the Bloch sphere at a rate that ¯ † depends on λ0(b + b ).

3. After the β(t) pulse is complete, the qubit resonator interaction is switched

oﬀ λ0(t) = 0 and a projective σz measurement is made on the qubit.

If the qubit is found to be in the excited state, the conditional state of the mechanical

16In a practical experiment one would use flux tuning to move to a region where dωq = 0 in dXM order to avoid fast switching of a large DC field, however this is a technical point, and at this stage, only a time dependent λ0(t) is important. √ 17Here β(t) has units of Hz, while an input field should have units of Hz. This is because we have absorbed the input coupling rate κe into the definition of β(t), as is conventional in the Hamiltonian description of classical driving fields. 2.3. ARBITRARY QUANTUM STATE ENGINEERING 56 oscillator must be

Z |ψ(τ)iM ∼ ce(x, τ) |xi dx (2.43) where ∼ is used to denote the fact that the right hand side is unnormalized; which is unsurprising since it is the remaining state after a projective measurement. The time τ is the time between the switching on and off of the coupling λ0. The final wavefunction of the mechanical resonator is therefore proportional to ce(x, τ), and herein lies the key to the protocol; different choices of β(t) can be used to engineer different wave functions of the mechanical resonator. This is an entirely different mechanism of state engineering compared to Refs. [67, 180, 178] which use entirely different coupling Hamiltonians to generate quantum states, or require one to engineer a quantum state of the microwave field. The drive β(t) is classical, therefore this mechanism for wave function engineering does not require one to engineer a quantum state of the microwave field and is instead relying on the interference generated by the non-linear Hamiltonian to generate the mechanical quantum state. The measurement procedure discussed above is defined by the measurement operator:

R τ 0 0 −i HI(t )dt /~ Υe = he| T e 0 |gi (2.44)

= he| [cg(τ) |gi + ce(τ) |ei] (2.45)

= ce(τ) he|ei (2.46)

= ce(τ), (2.47) where in the second line, the time evolution has been expanded in the σz basis.

The amplitude ce is now operator valued acting on the oscillators Hilbert space. For example, the probability of ﬁnding the oscillator in the ground state depends, 2 in general, on the state of the oscillator Pr(g) = Tr[|cg(t)| ρm]. Solving for this measurement operator is equivalent to solving the ODE,

¯ ∗ c˙e(t) = −iλ0XMce(t) − iβ (t)cg(t) (2.48) ¯ c˙g(t) = iλ0XMcg(t) − iβ(t)ce(t) (2.49) for the (now operator valued) amplitudes cg(e), with initial conditions [cg(0), ce(0)] = 2 [I, 0]. These initial conditions imply Pr(g) = Tr[ρm|cg(0)| ] = Tr[ρm] = 1, is independent of the state of the resonator state, equivalent to Eq. (2.44). Hence the measurement operator can be obtained from the ce(τ) solution from equations Eq. (2.49), with initial conditions [cg(0), ce(0)] = [I, 0]. The (unnormalized) state of the oscillator after the measurement is |ψ(τ)i = 2.3. ARBITRARY QUANTUM STATE ENGINEERING 57

(a) Prepare Switch on Apply qubit Switch oﬀ Measure qubit interaction drive interaction Qubit D

(b)

Figure 2.1: Proposed measurement protocol. (a) Time line of the protocol: the qubit is initially prepared in the σz ground state with the qubit-resonator interaction switched off. The qubit-resonator interaction is switched on and remains constant over the duration of the drive tone. At the completion of the drive tone, the interaction is once again switched off and the qubit measured in the σz basis, thereby conditioning the quantum state of the resonator. (b) The corresponding state of the qubit at each point in the measurement procedure (with a final unknown measurement result.)

Υe |ψ(0)i. The generalization to mixed states is straight forward with, ρ(τ) ∼ † Υeρ(0)Υe. Note this has exactly the same structure as the weak measurement discussed in chapter 1, with the oscillator (system) coupled to a qubit (apparatus) in a known state, followed by projective measurement of the qubit (apparatus). In the following section we show how Υe depends on the drive amplitude β(t).

2.3.2 Generating arbitrary measurement operators

To understand how the time dependent drive changes the wave function of the oscillator, it’s helpful to first consider the simple case of time independent drive, β(t) = β¯. In this case the ODE Eq. (2.42) is exactly solvable and results in the well known Rabi oscillations of the qubit [81]. Rabi oscillations are coherent oscillations in the probability amplitude of finding the qubit in the excited state and depend on the detuning between the driving frequency and the qubit resonance frequency. Although the detuning δ has been set to zero, this is only true when the oscillator is at its equilibrium point. If the oscillator is displaced by a classical quantity δx, ¯ the quit appears to have frequency ωq + λ0δx, not ωq. This results in an effective ¯ ¯ detuning of δeff = λ0δx, even though δ = 0. We may therefore identify λ0XM as the effective detuning, and different XM eigenstates in superposition give a superposition of detunings. The time dependent probability amplitudes in Eq. (2.42) are exactly solvable and result in,

β¯ q ¯2 ¯2 2 Υe(τ) = q sin τ β + 4λ0XM . (2.50) ¯2 ¯2 2 β + 4λ0XM 2.3. ARBITRARY QUANTUM STATE ENGINEERING 58

¯ (note if λ0XM → δ, then the above expression is exactly the Rabi oscillation solution with drive detuning δ). Note here, as in the measurement protocol outlined above, we have chosen [cg(0), ce(0)] = [I, 0]. For reasons that will soon become clear, we restrict the drive pulse to have unsigned area π/2. If β(t) > 0, then this would exactly be a π-pulse, however in this work the π(like)-pulse requires R dt|β(t)| = π/2, which for the constant drive gives gives βτ¯ = π/2. At this time, τ = π/(2β¯), the measurement operator can be rewritten as,

 v  u ¯2 2 π π u 4λ0XM Υe = sinc  t1 +  2  2 β¯2  " ¯ # π 2πλ0 ≈ sinc XM , (2.51) 2 β¯

¯ ¯ where in the last line we have assumed 4λ0XM β. Strictly speaking as XM is an operator, the approximation is only valid for parts of the wave function at position 0 ¯ 0 ¯ 0 x , where 4λ0|x | β where x is in dimensionless units. There will always be some region x0 ≈ 0 where this approximation breaks down. However, as we shall see, this only has a minimal eﬀect for practical applications. We can characterize a general measurement operator by its spectral decomposition resolved in the position basis, i.e. a complex function f(x0) where,

Z 0 0 0 0 Υe = dx fe(x ) |x i hx | , (2.52)

0 hence to know fe(x ), is to known Υe. The functional form of the measurement operator in Eq. (2.51) is (almost) a ¯ sinc(·) function in λ0XM, ie. the Fourier transform of a top-hat function which was the functional form of β(t). The time-frequency relation of the Fourier transform is understood by considering the resulting measurement operator as a function of ¯ ¯ 0 ¯ λ0XM (or λ0x ), which has units of frequency. Since λ0 is a known parameter, it can be dropped and we can simply consider the functional shape of the measurement operator (in the |x0i basis) to be given by the shape of the Fourier transform of β(t). With this Fourier transform similarity in mind, we postulate the following:

Z 0 0 0 The functional form of the measurement operator Υe = dx f(x ) |x i hx |, is approximately a scaled Fourier transform of the drive amplitude β(t).

There is no analytical solution to Eq. (2.42) for a general drive β(t), hence this statement is diﬃcult to prove. Nevertheless this should not prevent us from interesting physics. For a given pulse shape β(t), the measurement operator can be 2.3. ARBITRARY QUANTUM STATE ENGINEERING 59 numerically solved from Eq. (2.49) and compared to the Fourier transform of the pulse shape. For the rest of this section, all numerical solutions are obtained via solving Eq. (2.42) using a numerical solver (Matlab-ode45, Mathematica-NIntegrate or python-scipy integrator) and discrete values of x with initial conditions (cg, ce) = (1, 0). For technical reasons we will introduce a pulse-shape dependent, dimensionless scaling parameter χ, that is needed to scale the bandwidth of the drive, β(t) →

βχ(t) = β(χt). The scaling χ is diﬀerent for each pulse shape and, as we shall see, the particular value of χ can be chosen in a straight forward manner for any implementation of this scheme. Generation of any target measurement operator can now be done via a straight- 0 forward procedure; once a target ΥT is chosen (i.e a choose fT (x ) — subscript R 0 0 0 0 ‘Target’ — with ΥT = dx fT (x ) |x i hx |), it can be Fourier transformed to obtain the pulse shape required to realize the operator. For example to obtain a measure- 0 ment operator with fT (x ), the drive amplitude is set to be

Z ∞ 0 0 −iχtλ¯0x 0 dx e fT (x ) π −∞ βχ(t) = . (2.53) Z ∞ Z ∞ 0 2 0 −iχtλ¯0x 0 dt dx e fT (x ) −∞ −∞

R Note the amplitude of βχ(t) must be scaled to satisfy the π-like pulse ansatz dt|βχ(t)| = π 0 2 . The realized measurement operator (subscript R) fR(x ), is obtained by solving Eqs. (2.49) given βχ(t). One may then optimize over χ to maximize the fidelity between the target and realized measurement operators. 0 18 The choice of fT (x ) is arbitrary, and any function that decays slow enough will have well defined Fourier transform with compact support, (i.e. βχ(t) being non-zero only for a finite time is a good approximation). Hence any non-linear measurement 0 operator fT (x ) can be constructed in finite time with deterministic parameters. For 2 example quadratic measurement operator discuss in Ref. [135] Υe ∝ exp(−(¯x − 2 2 XM) )) can be constructed with a deterministically chosenx ¯. The deterministic parameters come at a cost, the measurement requires conditioning on the excited qubit state, and is therefore probabilistic with the probability dependent on the † initial state of the resonator, Psuccess = Tr[ΥRΥRρm] [52] (and see Eq. (1.40)). We will deal with the details of the success probability shortly. Figure 2.2 shows the fidelity between target measurement operators, and the measurement operators (obtained numerically and optimized over χ) realized by 0 the protocol. For each plot, the target operator fT (x ) was fixed. Eq. (2.53) was 0 then used to give the shape of the required drive βχ(t) to generate fT (x ). Given

18i.e. no faster than a Gaussian 2.3. ARBITRARY QUANTUM STATE ENGINEERING 60

Real Imag

1 (a) Fidelity = 0.99808

0 target f target f t t solution f solution f e e -1

1 (b) Fidelity = 0.9952

0 target f target f t t solution f solution f e e f(x) -1

1 (c) Fidelity = 0.98546

0 target f target f t t solution f solution f e e -1

1 (d) Fidelity = 0.98711

0 target f target f t t solution f solution f e e -1 -5 0 5 -5 0 5

(e)

Figure 2.2: Fidelity between target operator and obtained operator; left and right plots show the real and imaginary part of f(x) respectively. Target operators are 1 i 1 2 1 −(x− 2 ) /4+ 2 ix (a) Coherent state α = 2 + 2 , with f(x) ∝ e , (b) n = 3 number state, −x2/4 f(x) ∝ e H3(x) where Hn(x) is the nth Hermite polynomial, (c) Superposition −(x−1.5)2/4 iπ −(x+1.5)2/4 state with relative phase f(x) ∝ e + e 2 e , (d) Plane wave f(x) ∝ e−2ix truncated by a top-hat function between [−3, 3]. (e) Qualitatively showing how the bandwidth scaling of the drive tone (χ) eﬀects the realized solution fe. R ∗ The ﬁdelity is calculated as | dxft(x)fe (x)| where ft and fe have both been L2 normalized to unity. 2.3. ARBITRARY QUANTUM STATE ENGINEERING 61

0 βχ(t), the realized measurement (fR(x )) operator was found by solving Eqs. 2.49 2 R 0 ∗ 0 0 numerically. Finally χ is fixed by optimizing the fidelity F = dx fT (x )fR(x ), between the target and realized measurement operators. 0 In particular, Fig. 2.2 shows that the protocol works for complex targets fT (x ), and even for first order discontinuities in the wave function (Fig. (2.2) (d)). Fig- ure 2.2 (e) shows the effect of scaling χ for a Gaussian measurement operator: it 0 simply scales the characteristic width of the realized function fR(x ), this not sur- prising given the relationship between χ and x0 in Eq. (2.53). In this section we have shown that one can generate any target measurement operator, but have not discussed the probabilities of obtaining a successful outcome, or the purity of the final state. Both these points are addressed in the next subsection.

2.3.3 Generating arbitrary quantum states

Once a target measurement operator is chosen, the probability of a successful outcome, and the purity of the ﬁnal quantum state will explicitly depend on the initial state of the resonator. The probability of a successful measurement is proportional to the overlap between the realized measurement operator, and the initial probability distribution. This is easily seen by explicitly computing the expectation value of † ΥeΥe in the position basis, Z Z 0 † 0 0 0 2 0 0 Psuccess = hx | ΥeΥeρ(0) |x i dx = |fR(x )| P (x , 0)dx (2.54) and the normalized ﬁnal state of the resonator is

Υ ρ(0)Υ† ρ(τ) = e e . (2.55) Psuccess

0 0 Since fR(x ) ' fT (x ) with fidelity & 0.98 (Fig. 2.2), all calculations will be done with fT (x) since it has an analytic form, and we would expect the results to be correct to within a relative error of ≈ 0.02. For generating an arbitrary quantum state we will consider a two step preparation procedure. Beginning in a thermal (Gaussian) state ρth with initial thermal 0 −x02s2/4 occupationn ¯, we apply a Gaussian measurement operator, fT (x ) ∝ e thereby squeezing the position quadrature. We can then move into another interaction pic- 1 ture time translated 4 of a period. The position quadrature squeezing can now be understood as momentum quadrature squeezing. The net effect of this first measurement is to generate a momentum squeezed state, written in position quadrature 2.3. ARBITRARY QUANTUM STATE ENGINEERING 62 matrix elements as,

h 0 00 2 2 02 002 i exp − n¯(x −x ) +s (x +x ) 0 00 4s2(2¯n+s2) hx | ρs |x i = q (2.56) 2π(2¯n + s2) where s ≥ 1 is the squeezing parameter in the momentum basis (s = 1 corresponds to ground state variance in the momentum quadrature). We now treat the state in Eq. (2.56) as ρ(0) for the second step where a second measurement operator 0 fT (x ) is applied. This second measurement operator is simply chosen to be the 0 0 wave function of the (pure) state that one wishes to generate fT (x ) = ψ(x ). If this second measurement procedure is successful in obtaining the excited state outcome, † R 0 0 0 0 19 the conditional state is ρ(τ) = ΥeρsΥe (with Υe ≈ dx fT (x ) |x i hx |) . The ﬁdelity between the target and realized quantum state is

F 2 = hψ| ρ(τ) |ψi (2.57) R 0 00 0 ∗ 00 0 ∗ 00 0 00 dx dx ψ(x )ψ (x )fR(x )fR(x ) hx | ρs |x i = R 0 0 2 0 0 (2.58) dx |fR(x )| hx | ρs |x i R 0 00 0 2 00 2 0 00 dx dx |ψ(x )| |ψ(x )| hx | ρs |x i = R 0 0 2 0 0 (2.59) dx |ψ(x )| hx | ρs |x i where in the third line we have again assumed that the target measurement operator is very close to the numerically obtained measurement operator, i.e. ψ(x0) = 0 0 fT (x ) ≈ fR(x ) in order to get analytic estimates of the fidelity. 2 0 00 In the case of strong squeezing s n¯, the matrix elements hx | ρs |x i define a broad Gaussian, centered at x0 = x00 = 0 (albeit slightly different variances). If the variance of this Gaussian is also large compared to the spatial extent of the 20 0 00 wave function , then hx | ρi |x i does not change considerably in the integral over 0 00 the wave function, and hx | ρs |x i ≈ h0| ρs |0i. In this case

R 0 0 2 R 00 00 2 2 dx |ψ(x )| dx |ψ(x )| h0| ρs |0i F ≈ R 0 0 2 = 1 (2.60) dx |ψ(x )| h0| ρs |0i and the target wavefunction can be achieved with near unit fidelity. To calculate 0 the actual fidelity one should use the realised measurement operator fR(x ), and take into account the Gaussian nature of ρs. The actual fidelity would be expected 2 2 hδXMi hδXMi to be on the order 1 − − s2 , where ≈ 0.02 is from Fig. 2.2 and the s2 term is an estimate of the “broad Gaussian” approximation (i.e. the ratio of the position variance of the target wave function to the width of the Gaussian). The

19 0 0 Recall we have assumed fR(x ) ≈ fT (x ) 2 20i.e. s2 R dx0x02|ψ(x0)|2 − R dx0x0|ψ(x0)|2 2.3. ARBITRARY QUANTUM STATE ENGINEERING 63 above expression simply highlights the fact that near unit fidelity is in principle possible for any desired wave function ψ(x0), provided s is sufficiently large. Increasing the squeezing parameter s will increase the fidelity at the expense of lowering Psuccess. The increase fidelity is a consequence of a broader Gaussian state in Eq. (2.58) and Eq. (2.60) being a better approximation. The lower success rate is also a consequence of the broader Gaussian in, resulting is a lower overlap integral 0 0 0 between the wave function and P (x ) = hx | ρs |x i (see Eq. (2.54)).

2.3.4 Summary

We have introduced a method to generate an arbitrary position basis measurement operator in a coupled qubit-mechanical system. Although the equations of motion do not have an analytic solution, we have shown that a numerical procedure can generate the drive pulse required to realize any target measurement operator and achieve ≈ 0.98 fidelity or better. The protocol requires parameters well into the strong coupling regime, λ0 κ, γ so that one can ignore decoherence over the time scale of the driving pulse. This strong coupling regime can be achieved using current state of the art qubits [186, 187] and resonators [188]. Using this measurement operator, we demonstrated that it is possible to construct any pure quantum state with ∼ 1 fidelity via a two step measurement procedure. The first step required so partially purifying a Gaussian state, and the second projecting the desired wavefuncion, enabling generation of an arbitrary wavefunction of the mechanical oscillator. Since the protocol can realize any measurement operator, one can perform full quantum state tomography via standard techniques [189, 190, 191]. Chapter 3

Emergent dark energy in a FRW cosmology

3.1 Foreword

In this chapter, (published as Ref. [1]), we investigate the classical channel model of gravity in the simplest relativistic case — that is, an empty FRW space time. Along with Ref. [192], this paper was featured on the cover of New Scientist [193]. The description of CCG in this paper is different to all previous interpretations, here we are postulating that there is a quantum description of the metric degrees of freedom, but any particles moving in the space time may only obtain classical information about the metric. This is in contrast to the previous CCG model where it was explicitly or implicitly that the background metric is fundamentally classical [31, 32, 35], and is more in line with the notion of a quantum theory of gravity. There are two main aims of this paper. The first is to give a pedagogical formulation of the CCG model in this new context. To this end, a significant portion of this paper (sec. 3.2.2) is devoted to comparing and contrasting the new CCG context to Newtonian gravity, standard approaches to quantum gravity (the Wheeler-de-Witt equation in this case), and previous models of CCG. After introducing the model in section 3.2.3, the second aim of this paper is to go through and what the effect of the model is, and analyse the resulting space time, section 3.2.4. We do not claim that this model is the physical origin of dark energy, indeed the type of dark energy found in the paper does not match observation. However, the approach is exploratory in nature, and a more detailed calculation, perhaps breaking the FRW symmetries may be informative. Ref. [194] moves toward this direction and builds upon the model in this paper by including a perfect fluid in the initial action, and obtains similar conclusions (i.e. that of a dark energy fluid filled space time at late times).

64 3.2. EMERGENT DARK ENERGY 65 3.2 Emergent dark energy via decoherence in quantum interactions

Natacha Altamirano1,2∗, Paulina Corona-Ugalde1,3, Kiran E. Khosla4,5, Gerard J. Milburn4,5, Robert B. Mann1 1Perimeter Institute, 31 Caroline St. N. Waterloo Ontario, N2L 2Y5, Canada, 2Department of Physics and Astronomy, University of Waterloo, Waterloo, On- tario, Canada, N2L 3G1, 3Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1, 4Department of Physics, University of Queensland, St Lucia, QLD 4072, Aus- tralia, 5Centre for Engineered Quantum Systems, University of Queensland, St Lucia, QLD 4072, Australia.

Received 27 February 2017, revised 11 April 2017

Accepted for publication 13 April 2017

Published 18 May 2017

In this work we consider a recent proposal that gravitational interactions are mediated via classical information and apply it to a relativistic context. We study a toy model of a quantized Friedman-Robertson-Walker (FRW) universe with the assumption that any test particles must feel a classical metric. We show that such a model results in decoherence in the FRW state that manifests itself as a dark energy fluid that fills the spacetime. Analysis of the resulting fluid, shows the equation of state asymptotically oscillates around the value w = −1/3, regardless of the spatial curvature, which provides the bound between accelerating and decelerating expanding FRW cosmologies. Motivated with quantum-classical interactions this model is yet another example of theories with violation of energy-momentum conservation whose signature could have significant consequences for the observable universe.

Keywords: dark energy, decoherence, quantum-classical dynamics

3.2.1 Introduction

Since the beginning of the 20th century researchers have tried to understand the quantum description of the different interactions that describe nature. All forces in the standard model are currently understood in terms of local quantum interactions and long range forces emerge as fluctuations of underlying gauge fields in the low energy limit, such as photons for the Coulomb force [195]. Interactions in quantum 3.2. EMERGENT DARK ENERGY 66

field theory are described by quantum gauge fields that act as force carriers and, as they admit a quantum description, they can carry quantum information. Gravita- tion, however, remains stubbornly resistant to quantization. The two most popular approaches, string theory [16] and loop quantum gravity [196], have yet to attain their goals. Other approaches to quantum gravity [197, 198, 199, 200, 201] suffer from non-local interactions or are non-renormalizable. A number of authors have questioned if gravity needs to be quantized [15, 202, 203] raising well known problems in consistently combining quantum and classical mechanics [204, 205]. In particular, gravitational decoherence models proposed by Diosi and Penrose [34, 39] use principles of relativity to limit the lifetime of spatial quantum superpositions and, as a result, breaking the unitary evolution of the wavefunction. One new approach along these lines [31] is the suggestion that gravity is fundamentally classical and therefore cannot carry quantum information [206]. This approach is motivated by the fact that gravity cannot be shielded and therefore any observer can in principle gain information about the quantum state sourcing gravity. The process of gaining (partial) information about a quantum state is equivalent to making weak measurements [207], and is consistent with the standard approach for describing open quantum systems [62]. For example, a test particle in a quantum potential will become entangled with the source of the potential, and an observer who is not aware of the test particle (i.e. traces over the test particle degrees of freedom) will necessarily see decoherence in the evolution of the source particle. This decoherence mechanism is present for any quantum potential, for example the Coulomb interaction, and not limited to gravity. The distinction between the electric and gravitational potential is the ability to, in principle, shield this effect: a superconducting shell around a source charge eliminates the test-source interaction thereby shielding the decoherence; however there is no such shield for gravity. This form of decoherence, perfectly compatible with the unitary evolution of standard quantum mechanics, motivated consideration of a classical channel model for gravitational (CCG) interactions [31]. In the CCG model, the gravitational potential is assumed to be fundamentally classical even though it can be sourced by quantum states. This quantum-classical interaction induces unavoidable decoherence on the quantum systems [116, 36]. This form of decoherence is not a consequence of tracing over an entangled state (as in the case of quantum potentials) but rather a modification of unitary evolution as a consequence of quantum-classical interactions. We will discuss the difference between CCG and standard unitary evolution in detail in Sec. 3.2.2. Previous models of CCG have only considered Newtonian gravity, [31, 32, 2, 208]. In this paper we take the first steps towards a CCG model in the relativistic regime. By ‘relativistic regime’ we mean the application of CCG in context that would oth- 3.2. EMERGENT DARK ENERGY 67 erwise be described by classical General Relativity. The question we want to answer is how to apply the CCG model when one has quantum metric degrees of freedom, and what are its consequences. We do this by considering a gravitational system with the fewest number of degrees of freedom possible, namely a canonically quantized empty Friedmann Robertson Walker (FRW) universe. In the Newtonian case, the trajectory of a test particle depends on the masses and configuration of the source, whereas in the GR description the dynamics are given solely by the metric components, i.e. the scale factor in an FRW spacetime. In CCG, the source necessarily experiences decoherence, and we will show that in the FRW context, CCG introduces decoherence of the spacetime. We will show that for an observer in such a universe this decoherence is manifested as a time dependent dark fluid.

Our paper is organized as follows: In section 3.2.2 we compare and contrast the Newtonian formulation of CCG with our relativistic extension, and highlight the distinctions between decoherence predicted by CCG and the standard notion of decoherence obtained from unitary quantum mechanics. In section 3.2.3 we will introduce the canonical spacetime Hamiltonian, and derive the equations of motion for the expectation values of the quantum observables. We analyze the solutions of the equations of motion in section 3.2.4 and compute the effective energy momentum tensor arising from this model. We then analyze the effective form of dark energy that emerges and consider how it affects the evolution of the spacetime. We close with some final remarks in section 3.2.5 and discuss future directions for this approach. Through all this work we are considering units G = 1 = c.

3.2.2 Relativistic Classical Channel Gravity

The goal of this section is to describe in detail the relativistic description of CCG. We do this by comparing unitary Newtonian gravity with the Wheeler de Witt equation, and show how the CCG model fundamentally diﬀers from unitary dynamics. In particular we explain how the presence of a test particle in the Newtonian model of CCG results in decoherence of any object that sources a (gravitational) potential for that test particle. We then argue analogously that in relativistic CCG the presence of a test particle in an FRW universe leads to decoherence in the scale factor, and therefore non-unitary evolution of the universe.

Unitary Newtonian Interaction. Consider a quantum Newtonian interaction between a source and a test particle (Fig. 3.1 top left). Under unitary evolution the two (perhaps distant) particles may become entangled, where such entanglement implicitly assumes a quantum “force carrier” (potential) — analogous to the photon 3.2. EMERGENT DARK ENERGY 68

UNITARY CCG

entanglement

source test source test potential potential particle particle particle particle WM [WM] NEWTONIAN decoherence observer intrinsic decoherence

WM [WM]

scale metric test scale metric test factor function particle factor function particle FRW

entanglement

Figure 3.1: GRAVITATIONAL MODELS. Cartoon of the four models presented in Sec.3.2.2. We describe them by considering a source particle, a potential (top – Newtonian) or metric (bottom – cosmological) and a test particle reacting to the potential or metric. The circles represent quantum degrees of freedom whereas the squares represent classical degrees of freedom. For the unitary cases (left) the joint system source-potential (metric)-test particle evolve unitarily. In this case the source/test particle may become entangled and an observer making measurements on the test particle results in a weak measurement (WM) of the source, including the associated decohere. On the other hand, the CCG model (right) assumes that the only way a test particle can respond to the source is through classical information, which is mathematically equivalent to weak measurement and feedback control [31]. This process results in decoherence for the source and classical ﬂuctuations on the potential (metric) and therefore is fundamentally a non-unitary evolution for either quantum systems. in electrodynamics; i.e. if the source particle is in a quantum superposition, the test particle will feel a coherent superposition of potentials, and thus follow two trajectories in superposition. Any observer who makes a projective measurement of the position of the test particle is eﬀectively making a weak measurement of the source particle, and this weak measurement induces decoherence of the source particle [207, 85]. This is an example of how two fundamental postulates of quantum mechanics (unitary evolution and the Born rule) lead to decoherence of a quantum state. In particular a projective measurement, i.e. an observer, is required for this type of decoherence and the fundamental evolution is unitary. Newtonian Classical Channel Gravity. The CCG model postulates that there is no quantum description for gravity and that the non-local interactions emerge from local interactions between a quantum particle and a classical potential. In this case, the gravitational interaction between a source and a test particle is mediated by a classical (as opposed to quantum) information channel (Fig. 3.1 top right). 3.2. EMERGENT DARK ENERGY 69

The question is now how a quantum particle can source a classical potential and how a test particle responds to it. In Ref. [31] it was shown that such a classical information channel is equivalent to an agent performing weak measurements of the position of the source and using the measurement outcome to control a potential for the test particle. Consequently the test particle does not respond to a quantum potential generated by the source but rather to a classical estimate of this potential. In CCG the existence of a test particle responding to the potential necessarily results in decoherence and subsequent non-unitary evolution of the source, even in the absence of an observer making any measurements of the test particle. Finally, we note that this model goes beyond the standard postulates of quantum mechanics. We devote 3.3.1 to a mathematical description of CCG as presented in [31]. The discussion so far has focused on the Newtonian description. Our goal is to understand how this same procedure can be carried out in a relativistic context. In the following we give a relativistic formulation of CCG in the cosmological context and compare it to the Wheeler-DeWitt equation. Wheeler-deWitt. The Wheeler-DeWitt equation, Hˆ |ψi = 0 where Hˆ is the Hamiltonian operator of the spacetime including any matter and |ψi is the quantum state of the universe, is the standard approach to quantum cosmology. In general relativity the least action principle always forces the classical Hamiltonian to vanish, and the Wheeler-DeWitt equation is the quantum implementation of this constraint. We will restrict the following discussion to an empty FRW universe so the only spacetime observables are the scale factora ˆ and its canonical conjugate mo- mentumπ ˆ. If we now consider a quantum test particle moving in such a universe, we would expect the particle to become entangled with the state of the universe, exactly analogous to how a test particle becomes entangled with a source particle in unitary quantum mechanics (Fig. 3.1 bottom left). By test particle we mean a particle whose contribution to the mass/energy of spacetime can be neglected, but is (in principle) able to become entangled with the spacetime state. In this context, we can view the scale factor as acting like a source that influences the dynamics of a test particle via the metric. Analogous to the unitary Newtonian case, in this scenario the presence of an observer making a measurement on the test particle is needed in order to get decoherence in the quantum state of the universe, but oth- werwise the evolution is entirely unitary. In the following we will use the analogy: source → scale factor and potential → metric to explain the main idea of this paper: the relativistic version of CCG. Relativistic Classical Channel Gravity. So far we have shown the interpretational similarity between unitary Newtonian gravity and the Wheeler-DeWitt equation: both are mediated by a quantum potential (metric). We are interested in understanding how CCG applies to the relativistic gravity: how a quantum spacetime 3.2. EMERGENT DARK ENERGY 70 can influence the dynamics of a test particle via a classical metric (Fig. 3.1 bottom right). Our interpretation in the empty FRW case is to view the quantum scale factor as a ‘source’ of the classical metric, analogous to the way that a quantum particle sources a classical potential in Newtonian CCG. In this interpretation, the quantum scale factor is equivalent to the quantum source particle in the Newtonian case, and the classical metric function is equivalent to Newtonian potential (Fig. 13.1 right hand side). A test particle in an FRW spacetime will follow a trajectory that solely depends on the scale factor, and thus the scale factor generates an effective potential for the test particle. Therefore, in analogy with the Newtonian CCG description, the scale factor-test particle interaction can be understood in terms of weak measurements and feedback control, and therefore there must be intrinsically non-unitary evolution of the quantum state. Using the same language of measurement and feedback from Ref. [31] we posit that the quantum state of the universe is subject to weak continuous measurement of the variablea ˆ2. The measurement is ofa ˆ2, as opposed toa ˆ, since classically it is the factor a2 that appears in the metric function, and therefore the trajectory of any test particle can only depend explicitly on a2. The measurement process forces the gravitational influence of spacetime on the test particle to be mediated by classical information. In other words, the test particle responds to a classical estimate of the scale factor (the measurement results) analogous to the way that a test particle responds to the Newtonian potential in CCG as described in appendix A. The presence of the weak measurement on the quantum scale factor changes the evolution of the quantum state of the universe, resulting in a master equation for the ensemble averaged state that we shall describe in detail in the next section. An observer who tries to recover the dynamics of the scale factor, must measure the trajectories of many test particles and so cannot distinguish which sequence of measurement histories took place [209, 210]; hence they observe an ensemble averaged (i.e. averaged over all possible measurement histories) spacetime. We shall denote a2 as the classical scale factor experienced by any observer in the Universe. The relationship between a2 anda ˆ2 is described in 3.3.2 and in the following section. In our model, the evolution of a is different from the standard Friedmann evolution and we will show that this is consistent with a dark energy fluid. Note that the effective measurement process avoids defining the classical scale 2 ˆ factor as haî or Tµν = hTµνi where the expectation value is calculated with the quantum state given by the Wheeler-DeWitt equation. In this section we have described the relevant properties of unitary evolution in both the Newtonian regime and in the cosmological scenario. These two models share the feature that interactions are mediated by quantum potentials that are able to entangle the interacting constituents. On the other hand, CCG postulates 3.2. EMERGENT DARK ENERGY 71 that the gravitational interaction should be mediated by a classical potential, i.e. the interacting constituents (assumed to be quantum) will communicate with each other by exchanging classical information. Such an interaction induces noise in the dynamics of the quantum constituents, and such noise can be modelled by a measurement feedback channel. The fundamental distinction between the unitary approach and CCG is studied in the next section for the cosmological case.

3.2.3 Model

In this section we present the details of cosmological CCG. We begin with a classical FRW metric describing an empty, isotropic, homogenous universe. In conformal time, the line element is

1 ds2 = a2(τ)[−N(τ)2dτ 2 + dr2 + r2dΩ2] , (3.1) 1 − kr2 where a(τ) and N(τ) are the time dependent scale factor and lapse function respectively. The curvature term k = −1, 0, 1 describes open, ﬂat, and closed spatial slices of the universe respectively. The Einstein-Hilbert action for an empty spacetime is

1 Z √ S = d4x −gR , (3.2) 16π where g is the determinant of the metric gµν and R is the Ricci scalar. From the action, the Lagrangian and Hamiltonian density can be found to be

a˙ 2 L = ka2N − , (3.3) N π2 H = − − ka2 , (3.4) 4 where π = −2˙a/N is the canonical momentum conjugate to a, with dots denoting conformal time derivatives. Note that the absence of any spatial dependence in (3.4) means the Hamiltonian density proportional to the full Hamiltonian, and so we shall take H to be the Hamiltonian (up to a factor with units of volume). The lapse function N(τ) acts as a Lagrange multiplier, ensuring the classical constraint H = 0 and it can be chosen to be unity without loss of generality. The quadratic form of the Hamiltonian density in (3.4) is due to the choice of conformal time in the deﬁnition of the line element in equation (3.1). Hamilton’s equations of motion are

a˙ = −π/2 , (3.5) π˙ = 2ka. (3.6) 3.2. EMERGENT DARK ENERGY 72

In the standard approach to quantum cosmology, the Hamiltonain becomes a quantum degree of freedom H = H(ˆa, pˆ), and along with the state of the universe |ψ(a)i are required to obey the Wheeler-DeWitt equation

∂2ψ H|ˆ ψ(a)i = 0 ⇒ − ka2ψ = 0 , (3.7) ∂a2 which is the quantum implementation of the Hamiltonian constraint1. The Wheeler De-Witt equation, (3.7), can also be written in terms of the von-Neumann equation for the quantum density matrixρ ˆ = |ψ(a)ihψ(a)|

dρˆ i = − [Hˆ, ρˆ] = 0 , (3.8) dτ ~ where τ is the time flow used to define the operator π in the Legendre transformation. Different interpretations of the meaning of |ψi lead either to single patch models (where the whole universe is thought of as a collection of interacting homogeneous patches) or to minisuperspace models (where the patch is the whole universe at early times when no inhomogeneities had formed). Either approach leads to the well-known problems of time and interpretation of the wavefunction in quantum cosmology [211], and efforts to solve them have led to a range of different models [212, 213, 214, 215]. In either case one then is left with the problem of both computing the wavefunction (or density matrix) of the universe and of interpreting it in such a manner that admits a reasonable classical limit. Here we seek to avoid the complication of a universal wave function and its associated interpretive issues by considering the relativistic extension of CCG in which the fundamental degrees of freedom of the spacetime remain quantum, but particles can only react to classical estimates of the underlying quantum observables. In the FRW context this implies that quantum scale factor acts as source of the metric potential whose evolution (by the postulate of CCG) cannot become entangled with any matter, such as a test particle. An observer, wanting to describe the spacetime dynamics will make measurements on many test particles and thus the metric this observer perceives is

1 ds2 = a2(τ)[−dτ 2 + dr2 + r2dΩ2] , (3.9) 1 − kr2 where we have set N = 1 and a2 ≡ haˆ2i = E(¯a2) is the classical information (estimate) gained from the quantum state of the spacetime experienced by each test particle (described in 3.3.2). Note that while the scale factor and its canonical

1If matter degrees of freedom are introduced they can be modelled by introducing an additional term proportional to a3µψ on the left-hand side of (3.7), where µ can be either a matter field or a phenomenological perfect fluid. 3.2. EMERGENT DARK ENERGY 73 momentum are quantized, all gravitational effects are governed by a. At early times (as we shall see) this avoids a singular universe due to the uncertainty principle. The gain in classical information of a2 implies that eq. (3.8) is no longer applicable since any gain in classical information about a quantum state, necessarily perturbs the state (see 3.3.2). Sincea ˆ2 is the observable from which a2 is estimated, we posit that

dρˆ i γ = − [Hˆ, ρˆ] − [â2, [â2, ρˆ]] , (3.10) dτ ~ 8~ governs the quantum evolution of the universe instead of (3.8). In contrast to the Wheeler De-Witt equation, which suggests [H, ρˆ] = 0, in CCG the presence of the decoherence term in general perturbs the state from a Hamiltonian eigenstate, affecting the dynamics of the quantum system. The latter term in (3.10) takes into account both the non-unitary evolution of the scale factor due to the presence of test particle(s) and the ensemble average of an observer when making measurements on multiple test particles. It can be derived from a collisional model in which the quantum degrees of freedom are continuously interacting with the external test particles. This process introduces a new fundamental constant γ that emerges as a consequence of the interaction between the test particle and the scale factor via the metric function. We give a full mathematical derivation of this equation in the 3.3.2, and also show that the uncertainty principle and positivity ofρ ˆ holds at all times provided γ > 0 [216, 217]. The form of equation (3.10) have been previously used in different scenarios, in particular to account for modifications of quantum mechanics such as collapse models [46] and with a focus on cosmological consequences of metric theories with non conservation of energy momentum tensor [192]. Here we go a step further, and analyze the consequences in the cosmological scenario of (3.10) when emergent form Quantum-classical interactions as the master equation for observables measured by a classical observer. We emphasize that the interpretation of this model is fundamentally different from that of the standard Wheeler-DeWitt approach in (3.8). Here the evolution of the universe is obtained by solving the master equation (3.10), for the time dependence of a2 = haˆ2(τ)i as in equation (3.31). The resulting spacetime will behave very differently compared to that of an empty universe, particularly at early times. As we shall demonstrate, the universe described by (3.9) will evolve as though there were a form of time-dependent dark energy present. The evolution of a2 is solved by using the master equation (3.10) and computing time derivatives of the first and second order moments of the quantum operators (â, πˆ). In particular, we note that by construction that d a2/dτ = Tr[â2dρ/dτˆ ] and 3.2. EMERGENT DARK ENERGY 74 thus we need to solve the following coupled equations

d haî = −hπî/2 , (3.11) dτ d hπî = 2khaî , (3.12) dτ d haˆ2i = −haˆπˆ +π âî/2 , (3.13) dτ d hπˆ2i = 2khaˆπˆ +π âî + γhaˆ2i , (3.14) dτ d haˆπˆ +π âî = −hπˆ2i + 4khaˆ2i , (3.15) dτ where the time derivative of an expectation value hXî is given by dhXî/dτ = Tr[Xdˆ ρ/dτˆ ] for any operator Xˆ. We see from equations (3.11)–(3.15) that only the second order moments are required to obtain the evolution of (3.9). Solving for these yields the evolution of the spacetime metric, and our subsequent task is to solve this set of equations for a variety of initial conditions for different values of the curvature constant k. We find there is always one exponentially growing mode, which makes the universe expand and two modes that either yield exponential decay or decaying oscillation of the scale factor. A general solution is a linear combination of these eigen-solutions, and so will in general asymptote to one that grows exponentially with time. We now proceed to interpret these solutions from the perspective of an observer who only has access to to the trajectories of test particles to back out the metric (3.9).

3.2.4 Dark Energy from Decoherence

As noted above, an observer can in principle determine the temporal evolution of the observable a2. Having no direct access to the underlying quantum observables, this observer can compute the Einstein tensor associated with the metric (3.9) and then use Einstein’s equations to determine the effective stress-energy tensor governing the observed evolution of spacetime. 2 (2) (2) The solution for a (τ) depends on the six variables {τ, k, γ, a0 , π0 , ζ0}, where (2) (2) τ is the conformal time and {a0 , p0 , ζ0} are the second order moments of the quantum state {haˆ2i, hpˆ2i, haˆpˆ+p âî} at τ = 0. These quantities can be constrained using a variety of physical criteria, as we shall discuss in the next section. 2 (2) (2) To see the general dependence of a on γ, we set a0 = 1 + 1/4, π0 = 1 , ζ0 = 0, describing displaced ground state of the quantum state and plot the results in comoving time in Fig. 3.2 for each value of k. To better understand the dynamics 3.2. EMERGENT DARK ENERGY 75

20 k = +1 k = 0 15 k = -1 L t H

a 10

0 0 5 10 15 20 t

Figure 3.2: Scale factor behaviour for γ = 0.1 and diﬀerent values of k as a function of comoving time. The quantum is system initially in a coherent state centered at a0 = 1. The relative behavior remains qualitatively the same as the decoherence parameter γ is varied. of the scale factor, and indeed the general physics of our model (anticipating a comparison with data), it is useful to analyze relevant physical quantities in terms of the comoving time t

Z τ t(τ) = a(τ 0)dτ 0 , (3.16) 0 with τ being the conformal time. We see that in the case k > 0 for small γ the scale factor undergoes damped oscillations. Furthermore, there exists a time scale that is half the e-folding time associated with the positive root λ+, after which the scale factor grows linearly in comoving time (exponentially in conformal time) without oscillations as noted above. This growth, while present, is not visible for the most oscillatory curve in ﬁgure 3.2. For the k ≤ 0 cases there is exponential growth but no oscillations for this choice of parameters. We also note that the growth of the scale factor is faster for k > 0 and slower for k = 0. As described in the introduction, from the point of view of an observers will infer from the motion of test particles the metric (3.9), and describe the resultant spacetime dynamics with the Einstein equations

2 Gµν(a ) = 8πTµν , (3.17)

1 where Gµν = Rµν − 2 Rgµν is the Einstein tensor. The Ricci tensor Rµν and Ricci scalar R are constructed with second derivatives of the metric tensor gµν which is given by (3.9). Such an observer will infer that the expansion is driven by a form of dark energy, whose eﬀective stress-energy is Tµν, and which we shall now compute. The symmetries of the FRW metric (homogeneity and isotropy) tell us that the 3.2. EMERGENT DARK ENERGY 76

Figure 3.3: Evolution of the scale factor in comoving time for initial thermal (top) and coherent (bottom) states, for k = 1 (left), k = 0 (middle) and k = −1 (right). In each case the steady state gradient is determined by γ. The oscillation in the k = 1 case can be understood by considering the equations of motion (eq 3.34-3.36) as γ → 0. The equations are then that of a Harmonic oscillator, with the amplitude of the oscillations defined by the Euclidean sum of the coherent amplitude (a0, p0), −1 where tan (a0/p0) defines the initial phase. energy-momentum tensor must have the form of a perfect fluid

 −ρ 0 0 0     0 P 0 0  ν   Tµ =   , (3.18)  0 0 P 0    0 0 0 P where ρ and P are the energy density and pressure of our perfect ﬂuid. The type of matter is characterized by w in the equation of state P = wρ. (2) (2) We must also choose the free parameters {a0, p0, a0 , π0 , ζ0, γ} where a0 and π0 are the initial means ofa ˆ andp ˆ respectively. This is a rather large parameter space, and constraining it to obtain useful information is a bit of a challenge. We will consider two kinds of Gaussian states: coherent states saturating the uncertainty inequality, characterized by their mean amplitude (a0, p0) (2) (2) (with ζ0 = a0π0), and thermal states for which 4a0 = π0 with ζ0 = a0 = π0 = 0. The factor of four diﬀerence comes from the transformation of equation (3.4) into natural units. Note that the spacetime only depends on a2 = haˆ2i, which in turn is governed by equations (3.34)-(3.36). Consequently, initial values of a0 = π0 = 0 result in a spacetime driven by noise from either a quantum (for minimum uncertainty states), or quantum and statistical (for mixed states) source. Additionally we can impose the following physical conditions on the spacetime 3.2. EMERGENT DARK ENERGY 77

• Strong Energy Condition: ρ > 0 and w(t) > −1 .

• Weak Energy Condition: ρ > 0 and |w(t)| < 1 .

• A non-singular spacetime: K < ∞, where K is the Kretschmann scalar K = abcd R Rabcd.

Finally, a more sophisticated model would have to be observationally constrained by early-universe data from the CMB, as well as from information on structure formation, but this is beyond the scope of this toy model. We have not imposed the Hamiltonian constraint H = 0, required in the standard picture of classical cosmology. The noise induced by the test particles measurements will break unitary evolution (as given by the Wheeler-DeWitt equation) as discuss in Sec.3.2.3; consequently the spacetime Hamiltonian (3.4) becomes time dependent

dH γ = − a2(τ) . (3.19) dτ 4 As outlined in section 3.2.1, it is unsurprising that the spacetime Hamiltonian alone is not conserved; indeed it is exactly this energy that gives rise to the gravitational source whose effective stress-energy tensor is given by (3.18). Note that the non conservation of the Hamiltonian of an empty universe, is governed by the decoherence parameter γ, and thus is deeply connected to the noise introduced by the test particles. Despite the metric being purely classical, its components are affected by the intrinsic quantum noise introduced by the interaction with test particle. When we compute the Hamiltonian using a2 and its conjugate momentum we thus find there is an excess of energy in comparison with the classical case. We now concentrate in the description of the perfect fluid and an analysis of singularities. We find that the Kretschmann scalar

12" # K = (k + a02)2 + a2a002 , (3.20) a4 does not diverge, since we must have a > 0 to have a physical quantum state. The expression for the eﬀective density is

3 a02 + k ρ(t) = , (3.21) 8π a2 and the equation of state is described by the function w(t), deﬁned via

P (t) = w(t)ρ(t) , (3.22) 3.2. EMERGENT DARK ENERGY 78

Figure 3.4: Evolution of the scale factor for k = 1. The large initial amplitude causes many oscillations before the transient behavior becomes dominated by the decoherence. The initial coherent state begins with an amplitude of (3, 0) and is (2) compared to a thermal state of a0 = 9. Without decoherence, the thermal state would remain constant in time. where we ﬁnd

2 a00a 1 w(t) = − − , (3.23) 3 a02 + k 3 where we have used Einstein equations (3.17) for a2 assuming a perfect fluid (3.18), 0 da and where a = dt . We plot the behavior of the scale factor as a function of comoving time for different parameters. The results are depicted in figure 3.3. We see that the cosmic evolution has low sensitivity to the initial conditions, indeed at late times the different curves become indistinguishable for all values of k. However there is rather high sensitivity to the choice of γ, particularly at early times, as is clear from figure 3.3. For small γ the coherent behavior is resolvable for a longer time. This is clearly visible in the case of a large coherent amplitude, shown in figure 3.4, where the large initial coherent amplitude causes many visible oscillations before the spacetime is dominated by the decoherence. a0 The behaviour of the Hubble parameter H = a as a function of comoving time (not plotted) displays considerable sensitivity for various choices of π and γ at early times, but convergence at late times. We also find that the effective energy density ρ is always positive, and that at early times its behavior can be quite oscillatory for small values γ when k = +1, but for all values of k it tends to grow initially for various choices of π and γ. At late times, regardless of these choices and values of k, the energy density is a monotonically decreasing function of time. An interesting feature of the model is that for large times, depicted in Fig. 3.5, 1 the system asymptotes to the relation P (t) = − 3 ρ(t). Although at early times the strong energy condition is generally (but not always) violated, at late times it is 3.2. EMERGENT DARK ENERGY 79 satisfied, with the equation of state settling down to the zero-acceleration case of w = −1/3. This is a rather striking feature of our model that is robust to any changes in initial conditions as long as γ > 0, and occurs for all values of k. It is a consequence of the existence of the growing mode (that always exists for any value of k – 3.3.3), which ensures at late conformal times exponential growth of the scale factor, which translates into asymptotic linear growth in comoving time. From the Friedmann equations, if w(t) > −1/3 then the universe undergoes an decelerating expansion whereas if w(t) < −1/3 we have an accelerated expansion which is a necessary condition for inflating universes. A closer analysis of the behaviour of the different quantities involved in Eq. (3.23) shows that a0(t) asymptotes to a constant finite value and, as shown in Fig. 3.3, the scale factor as a function of comoving time grows linearly with time. We thus conclude that the asymptotic behavior of the equation of state is governed by the acceleration a00(t) which goes to zero for large times. This is nothing more than the domination of the exponentially growing solution in conformal time for late times, which implies a linear growth of the scale factor as a function of comoving time and thus a00(t) = 0, yielding w → −1/3 according to Eq. (3.23). This crucially depends on the positivity of γ, whose effects are most pronounced in the k = 1 case. In the limit γ → 0, this case becomes purely oscillatory. Increasingly large values of γ both damp the oscillations and cause a more rapid growth in the scale factor, which asymptotes to a linear function of conformal time for all values of k. The fact that there are times for which the universe is expanding in an accelerated fashion suggests that our model can be used as an alternative to inflationary models, but the complete investigation of this aspect is beyond the scope of the present work.

3.2.5 Conclusions

We have explored the first implementation of CCG in a relativistic setting, showing how it can be implemented in an FRW spacetime. We found that this yields an alternative model for quantum cosmology, one in which the dynamical variables are quantum, and source a classical metric that influences test particles. By construction the evolution of the spacetime in the presence of such test particles is fundamentally non-unitary and results in an unavoidable decoherence of the quantum system and an arrow of time. The non-unitarity is required in order for a test particle to be influenced by the scale factor in the CCG model. This results in an arrow of time and unavoidable decoherence of the quantum system. Furthermore, the big-bang singularity is removed, since the scale factor is now interpreted as the mean of a positive quantum variable which is constrained by the uncertainty relations. The net effect of this interaction is manifest in a form of time-dependent dark 3.2. EMERGENT DARK ENERGY 80

Figure 3.5: Behaviour of the equation of state parameter in comoving time for thermal initial states (top) and coherent initial states (bottom), for each value of k = 1 (left), k = 0 middle, k = −1 (right). In each case w(t) → −1/3 regardless of the initial conditions or curvature. For late times t > 5 both the strong and weak energy conditions hold. Furthermore in all case w(t) approaches the asymptote faster for large γ. energy as our subsequent investigation of the evolution of the metric (as seen by an observer that measures the trajectories of test particles) indicated. For positive curvature k > 0 we found that the cosmological evolution is generally characterized by oscillatory behaviour of the scale factor (consistent with the Friedmann solutions) that is eventually dominated by exponential growth in conformal time. Transforming to comoving coordinates, the equation of state parameter w(t) initially undergoes oscillations that damp out, with this parameter reaching the asymptotic value of −1/3 at late times. This condition, present for all values of k, is robust to initial conditions and is a consequence of the aforementioned exponential growth, which in turn is driven by the constraint of the decoherence rate of the quantum system. We have thus shown than an observer in the universe will see the presence of a dark fluid as a consequence of test particles interacting with the metric. This form of fluid is characterized by γ, the fundamental parameter in our model. However the energy conditions are generically not satisfied at early times, with |w(t)| > 1, although two notable exceptions are illustrated in the upper left of figure 3.5 (the solid and dashed curves) for thermal initial states. This suggests that a more sophisticated model could generically satisfy the energy conditions. Furthermore ρ > 0 for k ≤ 0. Of course the model we have presented here is overly simplistic, ignoring matter contributions and possible spatial inhomogeneities and anisotropies. A more realistic cosmology must take such factors into account. We close by commenting on some special features of our model and future per- spectives. From Eq. (3.10) we notice that the new parameter γ has units of time. In cosmology the Hubble parameter H gives the time scale of the universe, its age, 3.2. EMERGENT DARK ENERGY 81 and the size of the Hubble horizon. Our model introduces a new time scale that sets the scale of fluctuations of the different quantities and, as discussed in section 3.2.4, these two scales are not completely independent. This parameter γ is responsible (in this simplified version of the model) of the presence of a dark fluid making the universe expand. The model discussed in this work can be extended to consider perhaps more realistic scenarios. In particular we are interested in how a matter source interacting with the scale factor will modify the equations. In this scenario, there is no need to introduce the notion of a test particle to account for non unitary evolution. In fact, the CCG model states that in order for two quantum systems to interact gravitational (in this case scale factor and matter) both subsystems need to continuously have knowledge of the other subsystem properties, and this can be achieved by a classical communication channel or weak measurements. This (effective) weak measurements will break unitary evolution and an observer in that universe that measure the matter field in order to describe its dynamics will induce decoherence on the joint system scale factor- matter therefore changing the dynamics as of a universe that behaves according to the Wheeler-deWitt equation. Let us comment on the covariance properties of the CCG model as presented in this work. As formulated here, the model explicitly brakes unitarity evolution in the frame where the weak measurements performed by the test particles are held and therefore the master equation for the scale factor was computed in the proper frame of the test particles. In this exploratory work we decided to work in conformal time, and thus both the test particles and the observers have the conformal time as their proper time. A more careful extension of the model should have this feature taken into account. For example, when matter is introduced one should look at its associated energy momentum tensor and in particular to its proper time. The proper time of the matter is then the frame in which the matter will interact with the metric and thus is in this frame where unitarity is broken. One should in principle write the master equation as a function of a the proper time of the matter. A similar description that we presented in this work will therefore hold for an observer whose proper time is the proper time of matter. For any other observer, that does not share the same proper time as the matter, one will need to perform a change of reference to described the emergent dark fluid. We postpone all this extensions to future work.

Acknowledgements

The authors would like to thank Jake Taylor for his substantial contributions in motivating this work by proposing that dark energy may arise from in the classical 3.3. CHAPTER APPENDICES 82 channel formalism. His insight into the physical grounds for the particular measurement choice were helpful in understanding the model in a wider context. The authors are also thankful to Niayesh Afshordi and Romain Pascalie for useful comments of the manuscript. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and The Australian Research Coun- cil grant CE110001013. P. C-U. gratefully acknowledges funding from CONACyT. N.A., P. C-U., and R.B.M. would like to thank the University of Queensland for their hospitality during the initial stages of this work. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research,

3.3 Chapter appendices

3.3.1 Newtonian Classical Channel Model

In the following we show the diﬀerence between decoherence in unitary quantum mechanics and the CCG model. We will use angle brackets h·i to denote expectation value of a quantum observable, and E(·) to denote average over the classical noise.

Consider two massive particles innitially separated by a mean distance d = hxˆ1(0) − xˆ2(0)i, interacting under a Newtonian gravitational potential. The potential can then be linearized about the mean separation,

2 ! Gm1m2 Gm1m2 δxˆ1 − δxˆ2 (δxˆ1 − δxˆ2) − ≈ − 1 − + 2 , (3.24) |xˆ1 − xˆ2| d d d

th where δxî is the fluctuation about the mean seperation of the i particle and has zero mean. The cross term in the second order expansion is the first non-trivial quantum interaction between the two particles. Therefore, the lowest order Newto- 3 nian interaction is HI = Kxˆ1xˆ2 where K = 2Gm1m2/d , using the notation from

Ref. [31]. Note that in general, the interaction Hamiltonian HI , may result in entanglement between the separated particles. Working in the interaction picture and beginning with a separable, pure initial stateρ ˆ(0) =ρ ˆ1 ⊗ ρˆ2, the joint system will unitarily evolve into

−iHI δt/~ −iHI δt/~ ρˆ(δt) = e ρˆ1 ⊗ ρˆ2e , (3.25) after a time δt. The time δt is assumed to be short enough such that the linearisation of HI is valid over the full duration. This is standard unitary evolution, and the joint 3.3. CHAPTER APPENDICES 83

2 2 system remains pure, Tr[(ˆρ(δt)) ] = Tr[(ˆρ1 ⊗ ρˆ2) ] = 1. However an observer who is unaware of particle two will see decoherence in the reduced state of particle one,

ρˆ1(δt) = Tr2[ˆρ(δt)]. In particular, note that even though the global evolution is non- dissipative, the observer sees decoherence in the description of their local quantum state. The decoherence is thus a consequence of thinking about the reduced evolution from the point of view of the observer, therefore necessarily requires the presence of an observer to make sense (see Fig. 3.1 top-left).

In contrast, the CCG model postulates that the interaction HI is equivalent to a measurement and feedback process. Here we outline the relevant details presented in Ref. [31] to highlight that the non-unitary dynamics in CCG is fundamental and independent of the existence of any observer. In Newtonian CCG the interaction Hamiltonian is replaced by a feedback control Hamiltonian

HI = Kxˆ1xˆ2 → Hfb = Kx¯1xˆ2 + Kx¯2xˆ1 , (3.26) wherex ¯i is the classical measurement outcome of a weak continuous measurement of xˆi. The measurement itself alters the unitary dynamics of the joint density matrix (ˆρ) to the stochastic master equation,

idt Γ1dt Γ2dt dρˆc = − [H, ρˆ] − [ˆx2, [ˆx1, ρˆ]] − [ˆx2, [ˆx2, ρˆ]] ~ 2~ 2~ s s Γ1 Γ2 + dW1H[ˆx1]ˆρc + dW2H[ˆx2]ˆρc , (3.27) ~ ~ where dWi is a standard Wiener increment with E(dWi) = 0 and E(dWidWj) = dtδij, and H[Aˆ]ˆρ = Aˆρˆ +ρ ˆAˆ − 2hAˆi for any operator Aˆ. The joint state of the system is conditioned (subscript c) on the knowledge of the measurement outcome

q x¯i = hxîic + ~/2ΓidWi/dt , (3.28) and Γi describes the strength of the measurement. While the derivative dW/dt is not formally defined, it can be understood as a white noise process, dW/dt = ξ(t) where E[ξ(t)ξ(t0)] = δ(t − t0). This modification from unitary dynamics is from the postulate that gravity is mediated by a classical information channel and has noting to do with the existence of an observer describing a reduced quantum state. After the instantaneous weak measurement is made, the joint system evolves under a unitary generated by feedback Hamiltonian (3.26), Ufb = exp (−idtHfb/~), i.e.

† ρˆ(t + dt)c = Ufb[ˆρ(t) + dρˆc]Ufb , (3.29) and the systematic interaction HI is recovered in the unitary part of the evolution. 3.3. CHAPTER APPENDICES 84

The non-unitary components of Eq. (3.27), along with the noise in the feedback unitary (i.e. the dW term in Hfbdt) result in decoherence in the joint quantum state when averaging over all possible outcomes (equivalent to an observer making an ensemble average of all possible measurement outcomes, or simply being unaware that the measurement happened). At this point we diverge slightly from the discussion in Ref. [31], and instead of treating each particle symmetrically, we considerx ˆ1 as a ‘source’ particle andx ˆ2 as a ‘test’ particle, although the distinction is made arbitrarily. Since we now have a ‘test’ particle, we are not concerned with the back reaction from the test onto the source, and therefore the eﬀective feedback Hamiltonian to consider is Hfb = Kx¯1xˆ2 to generate the dynamics of the test particle. In this case Hfb does not aﬀect the

Hilbert space of the source, and the measurement of the source (required for Hfb), does not aﬀect the Hilbert space of the test particle. Therefore if the joint state is initially separable,ρ ˆ0 =ρ ˆ1 ⊗ ρˆ2, and there is no other interaction present, then the joint state will remain separable at all times,ρ ˆ(t) =ρ ˆ1(t) ⊗ ρˆ2(t). However, the introduction of the measurement of the source implies fundamental decoherence in its quantum state and is required by the simple existence of the test particle (see Fig. 3.1 top-right). In this asymmetric treatment between the two particles, there is no way to minimize the decoherence rate, but we can conclude that there must be a non-zero decoherence rate, Γ1 > 0, of the source particle to determine the dynamics of the test particle. This one sided description is analogous to what we consider in the cosmological case. The dynamics of the test particle depends on classical information from the scale factor state. We therefore suggest, that in CCG there is some fundamental, observer independent decoherence. Since we do not consider the back reaction from the presence of a test particle on the scale factor the decoherence rate cannot be minimized, and is left as a free, but strictly positive parameter. Further exploration should include the back reaction and thus introduce a noise minimization procedure.

3.3.2 Master Equation

Following 3.3.1 we derive the master equation for the cosmological system. We propose that the state is subject to weak continuous measurement of the variable a22. The way a test particle responds to the inﬂuence of the quantum scale factor through a classical metric function is via the result of a weak measurement. In 2 2 2 2 2 2 this case the metric function is given by, ds =a ¯ (−dt + dx ) wherea ¯ = haˆ ic + q ~ γ dW/dt, relabeling Γ → γ from (3.28). This is analogous to the way a test particle responds to the Newtonian potential though Hfb = Kx¯1xˆ2. The presence

2This is because (classically) it is exactly the factor a2 that appears in the metric function, and therefore the trajectory of any test particle explicitly can only depend on a2. 3.3. CHAPTER APPENDICES 85 of the weak measurement on the quantum scale factor changes the evolution of the quantum stateρ ˆ, resulting in the stochastic master equation [91, 90] √ i ˆ γ 2 2 2 dρˆc = − [H, ρˆ]dτ − [â , [â , ρˆ]dτ − 2~dW H[â ]ˆρc , (3.30) ~ 8~ where we have assumed a continuous Gaussian measurement ofa ˆ2, and the subscript c refers to fact that the change inρ ˆ is conditioned on the measurement resulta ¯2. Any observer who is unaware of the measurement outcomea ¯2, will describe the state as an ensemble average over the measurement process, dρˆc → E(dρˆc) = dρˆ, with the corresponding ensemble averaged metric,a ¯2 → E(¯a2) = haˆ2i ≡ a2 . Consequently the corresponding spacetime is given by

ds2 = haˆ2(τ)i(−dt2 + dx2) = a2(τ)(−dt2 + dx2) , (3.31)

˙ where the evolution of haˆ2i is given by Eq. (3.30) using hAî = Tr[Aˆρˆ˙] for any operator Aˆ. The condition γ 6= 0 is required in order for the test particle to feel the presence of the scale factor in the classical metric function, and γ ≥ 0 is required preserve the positivity ofρ ˆ [216, 217]. Therefore γ > 0 is a requirement for the model to q be physical. Fluctuations in the measurement record (of order dW ~/γ) induce fluctuations in the metric function. However, any observer making multiple measurements would average over all fluctuations, and only see the ensemble averaged dynamics [209], i.e. Eq. (3.30) for the quantum system. The equations of motion for the first and second order moments are found from equation (3.30) to be

d haî = −hπî/2 , (3.32) dτ d hπî = 2khaî , (3.33) dτ d haˆ2i = −haˆπˆ +π âî/2 , (3.34) dτ d hπˆ2i = 2khaˆπˆ +π âî + γhaˆ2i , (3.35) dτ d haˆπˆ +π âî = −hπˆ2i + 4khaˆ2i . (3.36) dτ where we have assumed the Hamiltonian is given by the canonical Hamiltonian for an empty FRW space 3.4. Note that setting γ = 0 the standard Friedmann solutions are recovered. The modification γ > 0 in the CCG model is a result of postulating that a test particle responds to the scale factor though a classical metric function, avoiding the complications of considering a quantized manifold. The decoherence is only on the scale factor - this is because we have only considered the presence of 3.3. CHAPTER APPENDICES 86

‘test’ particles in the universe, similar to the source-test description in the Newtonian case in section 3.3.1. Note that our approach is equivalent to the scale factor (and more generally, all metric degrees of freedom) repeatedly interacting with additional matter degrees of freedom whose net eﬀect is to repeatedly ‘measure’ the scale factor. These additional degrees of freedom may be thought of as quantum “test particles” in the spacetime that naturally measure the scale factor along their trajectory.

3.3.3 Solutions to the evolution equations

The coupled system of differential equations (3.13)–(3.15) can be straightforwardly solved to find haˆ2i(τ), and thus the resulting spacetime. This system can be written d~x(τ) 2 2 T on the form dτ = A~x(τ), with ~x(τ) = (haˆ i, hπˆ i, haˆπˆ +π âî) and

 1  0 0 − 2         A =  γ 0 2k  . (3.37)         4k −1 0

Assuming a solution ~x(t) = ~ηeλτ , then ~η and λ are the eigenvectors and eigenvalues of A respectively where the eigenvalues are solutions to the characteristic equation γ λ3 + 4kλ − 4 = 0 , (3.38) 8 yielding

ς2e2mπi/3 − 12ke−2mπi/3 λ = , m 3ς γ √ 1/3 ς = 3 + 2 ∆ , (3.39) 4 where m = 1, 2, 3. It is straightforward to show that the characteristic equation always has one positive real root if γ > 0, which must be the case for physically sensible measurements. The general nature of the solutions is determined by the sign of ∆, where

16 γ 2 ∆ = k3 + , (3.40) 27 8 yielding distinct real roots ( ∆ < 0), multiple real roots (∆ = 0), or one real and two complex conjugate roots (∆ > 0). The former two cases occur only for 2 16|k|3 k < 0; clearly ∆ < 0 if and only if (γ/8) < 27 for k < 0. The eigenvectors are 3.3. CHAPTER APPENDICES 87

2 1 −2k+λi T ~ηi = (− , , 1) where the λi are solutions to (3.38). 2λi λi We ﬁnd there is always one positive real solution to eq. (3.38). This means that there is always one exponentially growing mode, which makes the universe expand exponentially in conformal time yielding a linear growth in comoving (or proper) time. Furthermore the real parts of the other two roots are always negative, and so the other modes will exponentially decay (∆ ≤ 0), or oscillate with an exponentially decaying envelope (∆ > 0). Since a general solution will be a linear combination of the eigen-solutions

X λiτ ~x(t) = ci ~ηie , (3.41) i the solution will in general asymptote to one that grows exponentially with time. We note that if initial conditions are chosen such that the coeﬃcient c+ of the positive 2 2 real root λ+ vanishes then haˆ i and hπˆ i will become arbitrarily small, violating the Heisenberg uncertainty principle. All valid initial quantum states must have c+ nonzero. Chapter 4

Gravitational decoherence of quantum clocks

4.1 Foreword

In thischapter (published as Ref. [2]), we investigate the consequences of CCG on the dephasing rate of optical clocks (two level systems). Unlike the previous chapter, we are working under the framework of the standard CCG model of Ref. [31], with a measurement and feedback framework in mind. The high working precision of atomic and ionic clocks make them ideal candidates for potentially testing deviations from standard quantum mechanics. The central idea to this work is that a mass, e.g. the earth, will cause a red shift in the precession frequency of a two level system. If this interaction is replaced by a measurement and feedback process (i.e. an LOCC interaction), the noise introduced necessarily leads to a dephasing term for the qubit. We also consider mass-energy equivalence with a Newtonian potential between individual two level systems as a possible source of decoherence. This work is also the first to consider different LOCC models that may be used to implement the CCG model, i.e. Fig. 4.1. For example, several LOCC models may obtain the same systematic effect, but differ in their equivalent quantum circuit. Each LOCC model discussed below is equivalent for bipartite systems, but results in different scaling of decoherence for a multiparticle systems. As each model gives the same systematic dynamics, they cannot be distinguished by looking at systematic effects. We point out that there is a preferred model (i.e. local measurement and global feedback) that conforms best with the notion of a fundamentally classical gravitational potential, and that this model also achieves the lowest dephasing rate as it most efficiently uses the available classical information. In this work we found that deviations from standard quantum mechanics are far too small to observe with quantum clocks. This was somewhat unsurprising

88 4.2. CCG DECOHERENCE OF CLOCKS 89 due to the fact that clock transitions are very weakly gravitationally interacting. However, it was not immediately obvious that the interaction between a single clock, and the many constituent particles of a gravitating mass would be undetectable in experimental observations.

4.2 Detecting gravitational decoherence with clocks: Limits on temporal resolution from a classical channel model of gravity

Kiran Khosla1,2, Natacha Altamirano3,4

1Center for Engineered Quantum Systems, The University of Queensland, Bris- bane, QLD, 4067, Australia. 2Department of Physics, The University of Queensland, Brisbane, QLD, 4067, Australia. 3Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada 4Department of Physics and Astronomy, University of Waterloo, ON, N2L 3G1, Canada

The notion of time is given a different footing in Quantum Mechanics and General Relativity, treated as a parameter in the former and being an observer dependent property in the later. From a operational point of view time is simply the correlation between a system and a clock, where an idealized clock can be modelled as a two level systems. We investigate the dynamics of clocks interacting gravitationally by treating the gravitational interaction as a classical information channel. This model, known as the Classical Channel Gravity (CCG) postulates that gravity is mediated by a fundamentally classical force carrier, and is therefore unable entangle particles gravitationally. In particular, we focus on the decoherence rates and temporal resolution of arrays of N clocks showing how the minimum dephasing rate scales with N, and the spatial configuration. Furthermore, we consider the gravitational redshift between a clock and massive particle and show that a classical channel model of gravity predicts a finite dephasing rate from the non-local interaction. In our model we obtain a fundamental limitation in time accuracy that is intrinsic to each clock. 4.2. CCG DECOHERENCE OF CLOCKS 90

4.2.1 Introduction

Despite the success of General Relativity (GR) and Quantum Mechanics (QM) to describe nature at large and small scales respectively there is still an open question as to what the interplay is between these two theories. This question becomes fundamental when treating the nature of time operationally, specifically when considering how an observer measures time in GR and QM. Operationally, a clock is a reference and the notion of time emerges as a correlation between the clock and a system [218, 219]. Even with a fundamental flow of time, any observer limited to only measurements of quantum systems will not be able to access this fundamental flow [209] with zero uncertainty. Recently, Castro et.al. [220] proposed a physically motivated quantum mechanism that produces fundamental uncertainty in measurements of coupled two level systems (clocks). The key idea in their model is that the mass energy equivalence in quantum clocks leads to a Newtonian coupling between them. This interaction entangles the clock states, and therefore a measurement of any single clock necessarily decoheres distant clocks, limiting the temporal resolution of distant observers. In this case the decoherence is entirely a consequence of mass energy equivalence with unitary quantum mechanics, similar to Ref. [24]. In this letter we take a different approach by treating the gravitational interaction between clocks in the context of classical channel gravity (CCG): a recent proposal that treats gravity as a fundamentally classical interaction [31]. This model describes the gravitational interaction between quantum systems and results in noisy dynamics with decoherence rates similar those predicted by Diósi[108, 34] and Penrose [39]. The unitary quantum interaction considered in Ref. [220] is replaced by the master equation derived in Ref. [31], resulting in non-unitary dynamics for all particles that interact gravitationally. We will show that the key difference between the two proposals resides in the ability of the gravitational interaction to entangle the clocks: in Ref. [220] the decoherence is a result of tracing out parts of an entangled state generated by standard unitary quantum mechanics, whereas in our model the decoherence is a consequence of the postulated quantum-classical interaction. Consequently, the limited temporal resolution is fundamental to each clock and we will discuss this in the context of operational time. There are several proposals to probe relativistic behaviour of quantum mechanics in the lab [24, 26, 221, 222], which focus on including standard principles of relativity within the framework of quantum mechanics. However, since CCG is fundamentally a modification of the equations of motion for quantum systems interacting gravitationally, we focus on potentially detectable deviations from standard quantum mechanics [223, 44] in a post-Newtonian regime – by allowing energy-mass equivalence. This letter is organized as follows. Firstly we show that the master equation 4.2. CCG DECOHERENCE OF CLOCKS 91

1 derived in CCG results in a fundamental phase diﬀusion for spin 2 systems, and the coherence time — inverse dephasing rate — is given by the gravitational inter- 1 action rate. We then extend the model to consider multiple spin 2 systems, and characterize how the dephasing rate depends on the number of clocks, as well as their geometric arrangement comparing our results with current experiments. Fi- 1 nally we show that CCG implies a non-zero dephasing in spin 2 clocks from earth’s gravitational ﬁeld. We conclude with a discussion of the implications of our model and its testability.

4.2.2 Coupled clocks

In the following we consider a two level system clock with its spin processing around the z-axes of the Bloch sphere [224]. The free clock Hamiltonian is H = ~ωσz where ω is the clock frequency and σz is the Pauli-z matrix. From Einstein’s mass energy 2 equivalence the clock has an eﬀective mass m = m0 +H/c where m0 is the rest mass of the clock and c the speed of light. Note that this mass operator does not violate Bargmann’s super selection rules [29, 225], and is in a similar spirit to Refs. [28, 27].

From the quantum correction to the mass, two clocks with rest masses m1 and m2, separated by a distance d12 experience a Newtonian interaction

Gm1m2 G~ (1) (2) HI = − − 2 m2ω1σz + m1ω2σz d12 d12c 2 G~ ω1ω2 (1) (2) − 4 σz σz , (4.1) d12c that couples their internal energy states. The ﬁrst term in Eq. (4.1) is a constant potential, and the second term is the gravitational redshift on clock 1 (clock 2) from the rest mass of clock 2 (clock 1) which can be absorbed into the frequencies ω1,(2) and therefore both terms are neglected. The last term is a coherent quantum interaction between the clocks that arises from the mass energy equivalence. We now examine this non-local gravitational interaction as if it were mediated by a classical information channel. That is, only classical information can be exchanged between the two separated quantum systems in a way that preserves the interaction Hamilto- nian. Kafri et.al. [31] constructed the classical channel model where the Newtonian gravitational interaction between two quantum observablesx ˆ andy ˆ emerge as a measurement and feedback process – We review this model in App. 4.3.1. The operators xˆ andy ˆ are both measured with resultsx ¯ andy ¯ respectively [85], and a feedback control Hamiltonian Hfb = ~g(¯xyˆ+x ˆy¯) replaces the unitary evolution generated by HI = ~gxˆyˆ. The net result of this process is to preserve the systematic dynamics generated by the interaction Hamiltonian HI [217]. However, the measurement and 4.2. CCG DECOHERENCE OF CLOCKS 92 feedback process leads to dissipative evolution that cannot be avoided. This interaction is explicitly a local operation and classical communication (LOCC) process and therefore cannot lead to entanglement[226, 206]. However, note that non-local entanglement is still possible through other quantum interactions present in the system such us the Coulomb interaction. Even though CCG can be considered in the framework of quantum measurement and control, this is only a convenient mathematical tool to derive a master equation, and CCG does not require the existence of an agent to perform any such measurements or feedback: the decoherence is a natural consequence of coupling quantum and classical degrees of freedom as considered in [30, 32], without violating quantum state positivity and Heisenberg’s uncertainty principle [118, 116]. However, for convenience we adopt the language of quantum control throughout this paper.

The natural measurement basis for the coupled clock system in CCG is the σz basis and following the derivation in [31], we ﬁnd the master equation that describes the interaction in Eq. (4.1) in CCG is

2 ! i (1) (2) Γ1 g12 (1) (1) ρ˙ = − [H0 + ~g12σz σz , ρ] − + [σz , [σz , ρ]] ~ 2 8Γ2 2 ! Γ2 g12 (2) (2) − + [σz , [σz , ρ]] , (4.2) 2 8Γ1

G~ω1ω2 where g12 = 4 is the Newtonian interaction rate, Γi is the measurement rate of d12c th 2 the i clock, and ρ is the density matrix. The factor g12 in the decoherence rate is due to the feedback from clock 1 onto clock 2 (and visa versa), and is required to (1) (2) get the correct magnitude of the σz σz interaction. The double commutator term in Eq. (4.2) prevents entanglement of the clocks thought the σz − σz interaction and leads to phase diffusion in the σz basis at a rate 2g12 [217]. This phase diffusion induces a fundamental limit on the time resolution of each clock that can not be avoided. Note that pure unitary evolution under the Hamiltonian in Eq. (4.1) will also result in apparent dephasing if only a single clock is measured. Indeed this is exactly the type of decoherence considered in Ref. [220]. However, measurements of the full bipartite system will be able to show violations of a Bell inequality as the apparent decoherence is due to the two clocks becoming maximally entangled, and therefore the dephasing on a single clock is a second order (in time) effect [91, 90]. In contrast, CCG results in first order (in time) dephasing, and the entanglement for- bidding LOCC nature of CCG means that Bell inequalities will always be satisfied. This difference between quantum and classical interactions, and the 1/d scaling of the dephasing rate may be used to distinguish CCG dephasing from other sources of quantum noise. 4.2. CCG DECOHERENCE OF CLOCKS 93

For two clocks of equal frequency, where one would expect the measurement rates to be equal by symmetry, the dephasing rate is minimized when Γ1 = Γ2 = g12/2; 15 for petahertz clocks (ω1 = ω2 = 2π × 10 Hz) as used in [227, 228] separated by −42 300 nm, the dephasing rate is g12/2 ≈ 10 Hz. Such a small rate would require a clock with fractional uncertainty below 10−57 to observe, and therefore cannot be ruled out by current state of the art atomic and ion clocks which have achieved a fractional uncertainty of 10−18 [227, 229, 230].

4.2.3 Multiparticle interaction

We now extend the analysis to N interacting clocks, and we investigate the enhancement of the dephasing rate due to the multiple (order N 2 − N) interactions. Before preceding we have to consider how information propagates in the classical channel model. There are two possibilities of information propagation; pairwise measurement and feedback, Fig.4.1(left), or single measurement with global feedback, Fig.4.1(right). Note that both of these models are equivalent for N = 2 clocks and hence were not discussed in Ref. [31]. The dissipative evolution of the master equation for the pairwise measurement and feedback is

2 ! X X Γij gij (i) (i) ρ˙ = − + [σz , [σz , ρ]] (4.3) i j6=i 2 8Γji

G~ωiωj where gij = gji = 4 is the interaction rate between clocks i and j and Γij > 0 dij c is the decoherence (dephasing) rate from the measurement of clock i to generate the interaction between clocks i and j. Note that Γij is related to the measurement strength, with Γij = 0 corresponding to no measurement and Γij → ∞ corresponds to projective measurement of σz. For the moment each of the Γij’s are still free parameters; later we show that there is a non-zero set of Γij’s that minimize the decoherence, and thus the minimum decoherence rate has no free parameters. The decoherence terms in Eq. (4.3) can be intuitively understood. Each clock is measured N − 1 times, and each of these measurements are used to apply a independent feedback on the other N − 1 clocks in the system. The measurements therefore P th P 2 contribute to a total dephasing rate of j6=i Γij/2 on the i clock. The j6=i gij/8Γji term for the ith clock is from the feedback of the N − 1 noisy measurements form each of the other clocks in the system. Note that similar to the two clocks case, 2 the presence of gij in the feedback term is necessary in order to recover the correct magnitude of the systematic gravitational interaction. In contrast to the pairwise measurement and feedback, the global feedback only th requires a single measurement of the i clock (single dephasing rate Γi), and the 4.2. CCG DECOHERENCE OF CLOCKS 94

Figure 4.1: Diﬀerent interpretations of the classical channel structure in CCG. We have shown only the measurements of a single clock for simplicity, but it is understood that each clock is treated equally. (left) A unique channel between each of the pairwise coupled quantum degrees of freedom as considered in [208]. (right) A single channel used for each particle used for global feedback on all other particles. single measurement result is used to apply feedback on each of the other N − 1 clocks. For global feedback, the dissipative evolution is given by

 2  X Γi X gij (i) (i) ρ˙ = −  +  [σz , [σz , ρ]]. (4.4) i 2 j6=i 8Γj

The dependence of gij (which in turn depends on dij) in the dephasing rate of Eq. (4.3) and Eq. (4.4) means that the dephasing rate of the ith clock depends on the spatial arrangement of the other N − 1 clocks. We consider N 1 clocks in 1, 2 and 3D lattice configurations with a lattice constant Lc. In this case we can find the minimum dephasing rate on a single clock (see Appendix). Our results are presented in the second column of Table 4.1. Note that there are two different scenarios we can consider for minimization.

The ﬁrst one (case A) is a pairwise minimization: the rate Γij depends only on the separation between the clocks i and j, and the minimization is taken before including it in the general master equation. The second scenario (case B) minimizes the total noise in the master equation. One can estimate the scaling of these rates by assuming that Lc is small compared to the macroscopic length scale, R, of the lattice, Lc R. In this case, the summations in Table 4.1 are well approximated by an integral expression, Eq. (4.23), for N 1. Note that the integral approach only diﬀers from the analytical sum by a factor of order one. We show how the dephasing rate depends on N in the last column of Table 4.1. With the current 6 experiments (N = 10 , Lc ≈ 800 nm [231]) we compute a dephasing rate of order 10−40Hz (similar for all arrangements). Note that in order to have a dephasing of the order of mHz, which can be detected in the lab, we need to have either a large number of clocks or small separation between them. For example, as considered in 23 26 [220] taking N = 10 , Lc = 1 fm, and a 10 GeV clock transition (10 Hz), results in a dephasing rate of order 1 Hz. Note however, the ∼ 1 Hz dephasing rate here includes the spatial distribution of the clocks, whereas the the dephasing rate quoted 4.2. CCG DECOHERENCE OF CLOCKS 95

SCENARIO SCALING

A.i Pairwise (1D) log(N)

2 √ (i) G~ω P −1 Dpw = 2c4 j6=i dij (2D) N (3D) N 2/3 q A.ii Global (1D) 1 − 2/N

2 q q (i) G~ω P −2 Dgl = 2c4 j6=i dij (2D) log(N) (3D) N 1/6 √ B.i Pairwise (1D) N √ 2 q q (i) G~ω N−1 P −2 Gpw = 2c4 j6=i dij (2D) N log(N) (3D) N 2/3

Table 4.1: Minimum dephasing rates. The first column presents our results on minimization for the cases where Γij is pairwise defined (case A) and it is a fundamental constant (case B) as outlined in the Appendix. The second column shows the scaling with the number of clocks in the array for different dimensions, assuming N 1 2 4 by using Eq. (4.23). The coefficient of the scaling is G~ω /(2Lcc ), where Lc is the characteristic separation between adjacent clocks in the lattice. in [220] assumed a 1 fm distance between each of the 1023 clocks. More realistically, we consider the Mössbauereffect in 109mAg [232] which has a transition frequency of 8 × 105 THz and a linewidth of 10 mHz, with adjacent atoms separated by ≈ 1 A.˚ Using these parameters, CCG would predict a minimum γ-ray line width of 0.01 nHz per 100 g (1 mole) of 109mAg, far below current experimental precision. To observe the linewidth at the order of mHz, would require N ≈ 1036 atoms, or 1012 kg of metallic silver.

4.2.4 Clocks in earths gravitational ﬁeld

In the previous sections we were only concerned with energy-energy coupling between spatially separated clocks. However, the gravitational redshift is a relativistic effect that has been detected in quantum systems [229, 233], and therefore is a promising candidate to study the decoherence effects predicted by CCG. Again from the mass energy equivalence, a trapped two level system with position operator x will interact with any nearby object of mass m, and position operator X via the Newtonian 4.2. CCG DECOHERENCE OF CLOCKS 96 interaction Gmωσ H = − z I ~c2|X − x| Gmω Gmω ≈ − σ + σ (δX − δx) (4.5) ~ c2|d| z ~ c2d2 z where δx and δX are deviations about the mean separation d between the clock and the mass1. The first term is the mean redshift on the clock from the presence of the rest mass, and the second term is the lowest order Newtonian interaction between the quantum degrees of freedom. The σzδx is a local interaction between the external and internal degrees of freedom of a single particle and therefore does not need to be mediated by a classical information channel. The σzδX term however, is a non-local interaction and is replaced by an effective measurement and feedback process in CCG. In the following we consider the dephasing of a single clock from treating the nearby mass as both a composite and simple particle, where the simple particle case is just the N = 1 limit of the composite particle description. Note this distinction is put in by hand and is similar to the ambiguity in the treatment of the mass distribution of an extended object [39, 234]. For the composite particle description, we treat each constituent atom as individual point particle contributing to the redshift. The dissipative part of the CCG evolution is

2 ! X Γi gi ρ˙diss = − + [δXi, [δXi, ρ]] i 2 8Γz 2 ! Γz X gi − + [σz, [σz, ρ]] (4.6) 2 i 8Γi

th where Γi is now the decoherence from the measurement of the position of the i Gmiω atom, Γz is the decoherence due to measurement of the clock, and gi = 2 2 is the c di th energy-position interaction between the clock and the i atom of mass mi and has units of in Hz m−1. Here we have assumed the single measurement-global feedback interpretation of the model - figure 4.1 (right) - which was shown previously to result in a lower bound for the minimum decoherence rate. The double commutator in position leads to momentum diffusion (heating) of each atom, an effect that is investigated in [208]. This heating is not unique to CCG, and has been predicted in continuous spontaneous localization models [41, 235, 44] and stochastic extensions to the Schrodinger-Newton equation [129]. Eq. (4.6) shows that CCG predicts a non-zero dephasing rate that accompanies the redshift, and a finite heating rate to nearby massive particles.

1Note that we have not included the δXδx term considered by Ref. [31]. This term is present but appears at sub-leading orders in this description. 4.2. CCG DECOHERENCE OF CLOCKS 97

2 2 4 As gi scales as 1/d , the dephasing due to the gi term in Eq. (4.6) scales as 1/d , meaning that only the closest particles to the clock signiﬁcantly contribute to the dephasing rate. This is easily seen by considering a macroscopic homogenous body of N atoms of equal mass mi = m (for example a single species atomic crystal) close to the clock. For such a macroscopic object, one would expect by symmetry the measurement rate of each atom to be identical, Γi = Γ. By considering gravitational interactions between neighboring atoms we use the result of Ref. [31] and ﬁnd Γ = 2 3 Gm /~Lc where Lc is now the characteristic separation between adjacent atoms (e.g. lattice constant for a crystal). In this case we can use Eq. (4.23) to express the dephasing rate as an integral over the volume V of the macroscopic object,

3 2 3 2 Z G~Lc ω X 1 G~Lc ω dV 4 4 ≈ 4 3 4 (4.7) 8c i di 8c V Lc |r − r0|

2 3 where r0 is the mean location of the clock, and we have used Γ = Gm /~Lc . Note that the integral must converge as the point r = r0 cannot be in V . This integral is non-trivial for a spherical body, nevertheless there is some intuition to be gained by considering a shell of mass centered around the clock even though there is no net redshift at the center of a mass shell. For a shell with inner radius l, outer radius L, the dephasing rate due to the redshift is given by,

Γ πG ω2 D = z + ~ (l−1 − L−1). (4.8) 2 2c4 Form this expression we see that it is only close by masses in a thick (L l) shell that signiﬁcantly contribute to the dephasing rate. Thus in a laboratory experiment, the dephasing will be dominated by the immediate environment of the clock, even though all particles contribute to the systematic redshift. Alternatively, the macroscopic particle could be treated as a single degree of freedom; the dephasing on the clock is then simply given by Eq. (4.6) with a single term in the sum,

2 2 2 Γz G M ω D = + 4 4 (4.9) 2 8c d Γi where M = Nm is the total mass of the macroscopic object with a single measurement rate Γi. For M the mass of the earth and d as the mean separation between the earths and clocks center of mass, we can use atomic clock experiments [230, 227] −2 to bound Γi > 10 Hz m and Γz < 0.1 mHz. These bounds are set as such experiments have not observed anomalous dephasing. From this result we conclude that any dephasing from a classical channel model of gravity would not be identiﬁable in any gravitational redshift measurements, and despite their precision, quantum 4.2. CCG DECOHERENCE OF CLOCKS 98 clocks are not a desirable system to observe consequences of CCG.

4.2.5 Conclusions

— The Classical Channel Gravity model proposes that the gravitational interaction between quantum systems is mediated by a classical information channel that for- bids entanglement of distant particles through gravity. It has also been shown to result in decoherence that is similar to that predicted by the models of Diosi and Penrose. In this work we have studied the consequences of this model when treating time operationally, this is by using two level systems as idealized clocks than an observer must use in order to define the rate of external dynamics. Two such clocks will couple gravitationally and in the Newtonian limit this can be understood from mass energy equivalence. In this context we derive the rate at which they will decohere under CCG, and show that the minimum rate is fixed by the post-Newtonian interaction. We have also extended this analysis to optical lattice clocks in one, two and three spatial dimensions, computing how the minimum dephasing rate scales as the number of independent two level systems in the lattice. Finally we have studied a clock coupled to the earth’s gravitational field and analyzed in detail the position-spin interaction in the context of the CCG model. However, due to the asymmetry between the mass-clock system we were not able to meaningfully minimize the dephasing rate. Nevertheless, we showed that the gravitational redshift must be accompanied by some dephasing with the dominant contribution being due to close by atoms. Although the model considered in this work for clocks predict dephasing, the weakness of the gravitational interaction and the sub-linear scaling with the number of particles (Table 4.1) give a prediction thirty-seven orders of magnitude away from the current experiments. However, note that the dephasing rates computed in this work are the minimum and it is not clear that nature will saturate this bound. This shows that despite quantum clocks being the most precise measurement devices to date and therefore seem like a natural candidate to look for deviations of standard quantum mechanics, there are not the best devices to test the CCG model. Let us emphasize that the dephasing present in our model is fundamental to each clock and can not be avoided as it is a consequence of reproducing the Newtonian force using only classical information. In particular, the dephasing on one clock does not depend on the quantum state of the surrounding clocks, which is consistent with the clocks being in a separable state. Therefore, this decoherence is to be understood as a fundamental limit to temporal resolution for any clock and can not be reduced by including measurements of other clocks. For unitary evolution of a system under 4.3. CHAPTER APPENDICES 99 the Newtonian potential, as considered in Ref. [220], the decoherence appears as a result of entanglement of a single clock with a global system, and if an observer has access to the full quantum system, there is no decoherence and therefore no limit to the temporal resolution. In contrast, each clock dephasing individually in CCG means that even access to the global quantum system is not enough to resolve time with zero uncertainty.

Acknowledgments — We would like to thank Gerard J. Milburn, and Robert B. Mann for helpful discussions throughout this project. Research at Perimeter Insti- tute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. Research at The University of Queensland is supported by the Australian Research Coun- cil grant CE110001013. K.K. would like to thank the University of Waterloo and the Perimeter Institute for their hospitality during the period when this work was completed. N.A. and K.K, contributed equally to the original idea, the theoretical modeling, and preparing the manuscript.

4.3 Chapter appendices

4.3.1 Summary of CCG

In the following we summarise the relevant points and results of Ref. [31] for convenience. Consider two masses (of equal mass m), each in a harmonic trap interacting under Newtonian gravity. If the masses are well separated (by distance d), compared to their ground state extensions, the Newtonian interaction can be linearised about small ﬂuctuations in their position δxi,

2 2 2 ! Gm Gm δx1 − δx2 (δx1 − δx2) − ≈ − 1 − + 2 . (4.10) |x1 − x2| d d d

2 3 The lowest order interaction, Kx1x2, appears at second order where K = 2Gm /d and we have dropped the delta for convenience. This system can then be canonically quantised giving the Hamiltonian,

H = H0 + Kx1x2 (4.11) where H0 includes the kinetic terms and the trapping potentials. Note the other terms in the linearized potential, Eq (4.11), can be absorbed into a redeﬁnition of the trapping frequencies (for the quadratic terms) and a displacement of the centre of the trap (for the linear terms). The key idea of Ref. [31] is to understand 4.3. CHAPTER APPENDICES 100 the interaction in terms of a local operation and classical communication (LOCC) protocol — e.g. a measurement and feedback process. Such a protocol is only able to exchange classical information, and is consistent with the notion that gravity may be mediated by a fundamentally classical force carrier. The LOCC protocol is modelled in the language of quantum measurement and control. The position of each mass is continuously measured, and the measurement result is used to apply a feedback on the other mass. For example, the mass one is weakly measured with measurement resultx ¯1 where the bar denotes a classical measurement value. This classical measurement result is then sent to mass two and used to apply a conditional feedback unitary U = exp[−idtKx¯1x2/~]. The feedback Hamiltonian is chosen as to generate an x1x2 like coupling term, however, the presence of the classical estimate x¯1 in U means there is not quantum coherence in the coupling. This process the then symmetrised by measuring mass two and applying a feedback to mass one. This process can be treated mathematically as follows: continuous measurements of the position of each mass result in the stochastic master equation [85],

2  s  idt X Γi Γi dρc = − [H0, ρc] +  [xi, [xi, ρc]] + dWi(xiρc + ρcxi − 2ρchxiic)(4.12) ~ i=1 2~ ~ where Γi characterises the strength of the continuous measurement, dWi is the standard Wiener increment with mean zero and satisfying dW 2 = dt and the subscript c denotes the state is condition on the measurement outcomes. The measurement outcomes are given by

s ~ dWi x¯i = hxii + . (4.13) 2Γi dt

Given the resultsx ¯i, the feedback unitary can be expanded to ﬁrst order in dt (second order in dW ) giving

s ! i ~ U = 1 − Kx2 hx1idt + dW1 ~ 2Γ1 s ! 2 2 i ~ K 2 K 2 − Kx1 hx2idt + dW2 − x2dt − x1dt (4.14) ~ 2Γ2 4~Γ1 4~Γ2 where the terms quadratic in K arise from the second order terms in dWi. The feedback unitary is applied immediately after the measurement, giving the joint quantum state quantum state at time t + dt

† ρc(t + dt) = U[ρ(t) + dρc]U (4.15) 4.3. CHAPTER APPENDICES 101

After expanding this expression to ﬁrst order in dt, averaging over the Wiener process and minimising the decoherence terms, one arrives at the ﬁnal master equation (Eq 19 in Ref. [31])

2 dρ i K X = − [H0 + Kx1x2, ρ] − [xi, [xi, ρ]]] (4.16) dt ~ 2~ i=1

4.3.2 Dephasing Rate Minimization

The minimum dephasing rate for the multiparticle case cannot simply be obtained by considering only a single clock. For example the dephasing of a single clock i in

Eq. (4.4) can be zero for Γi → 0, and Γj6=i → ∞. However, this would result in each of the other j 6= i clocks dephasing to a maximally mixed state instantly. Therefore, the minimization procedure must minimize each dephasing rate simultaneously to give a physically sensible result. We therefore minimize the sum of dephasing rates with respect to each of the Γij’s (or Γi’s),

 2 ! d X X Γij gij  +  = 0 (4.17) dΓkl i j6=i 2 8Γji  2  d X Γi X gij or  +  = 0. (4.18) dΓk i 2 j6=i 8Γj

For the pairwise feedback, the decoherence is minimized when Γij = Γji = gij/2, 2 P 2 while for the global feedback the decoherence is minimized when Γi = j6=i gij/4 leading to minimum dephasing rates of (assuming ωi = ωj = ω)

2 (i) G~ω X 1 Dpw = 4 (4.19) 2c j6=i dij 2 s (i) G~ω X −2 Dgl = 4 dij (4.20) 2c j6=i for pairwise and global feedback respectively. Alternatively, the measurement rates

Γij’s could be considered as some fundamental measurement rate that does not depend on the spatial distribution of the physical system. In this case, each of the Γ’s lose their ij (or j) dependence. Nevertheless, there is still a dephasing on the clocks that can be bounded by current experiments. For a ﬁxed Γ, the minimum 4.3. CHAPTER APPENDICES 102 dephasing on the ith particle is

√ 2 s (i) G~ω X −2 Gpw = N − 1 4 dij (4.21) 2c j6=i 2 s (i) G~ω X −2 Ggl = 4 dij . (4.22) 2c j6=i

Although the dephasing from the measurement is assumed to be ﬁxed, the total dephasing rate still depends on the local environment of the clock due to the feedback from all other clocks. For an arbitrary spatial distribution of clocks, the summations (i) (i) in Dpw and Dgl must be computed. However, for regular arrays of clocks the summations are well approximated by integrals and can be solved to ﬁnd the dependence on the number of clocks N, and spatial distribution. If we consider a regular array of N clocks with characteristic length Lc between adjacent clocks, the sum can be written as,

Z Z R X 1 1 dVD SD D−1−α α ≈ D α = D r dr , (4.23) j dij Lc VD r Lc Lc for a clock in the center of a D-dimensional array, e.g. linear (1D), circular planar (2D), or spherical (3D) lattices. The integral is over the macroscopic volume, (area in 2D or line in 1D) of the array, and SD = 1, 2π, 4π for linear, planar and spherical geometries respectively. The integral is explicitly an approximation to the sum for th 1/D the i clock in the center of an array of radius R = N Lc. However, by using symmetry, clocks on the sides/edge of an array would be expected to have the same scaling with N (which is ﬁxed by D − 1 − α), and diﬀer only by a constant factor of order unity. For linear arrays in 1D consider the following example:

1 Z −Lc 1 dx Z NRLc 1 dx X ≈ + j6=i dij −NLLc |x| a Lc |x| a 1 = log(NLNR) (4.24) Lc where NL > 1 (NR > 1) are the number of clocks to the left (right) of the ith clock.

As NL + NR + 1 = N, the sum scales as log(N) regardless of the physical position of the clock in the array. Chapter 5

Quantum optomechanics beyond the quantum coherent oscillation regime

5.1 Foreword

Observation of quantum behavior in a macroscopic object is a central goal of modern physics. Quantum optomechanical systems offer an attractive platform to realize this goal [51]. In this chapter published in Ref. [3], we introduce a protocol that allows quantum limited measurements and state preparation of a mechanical oscillator without the requirement of so-called “quantum coherent oscillations”. That is, the quantum state of a mechanical oscillator may be entirely decohered within a fraction of a mechanical period, yet quantum behavior may still be observed. The protocol works by using pulsed optomechanical interaction with coherent control of the optical field. By displacing the sate of the optical field after the first pulsed interaction, a second pulsed optomechanical is used to negate the quantum back action of the first, and couple directly to the momentum of the oscillator. By tuning the magnitude of the optical displacement, the protocol allows one to deterministically select which mechanical quadrature is back-action free, or equivalently which quadrature couples to the optical field. The protocol allows direct, backaction evading measurements of the mechanical oscillator’s canonical momentum. In principle this allows for sensitive measurements of a fluctuating force perturbing the oscillator (for example building on Ref. [42]), such as those predicted by the CSL model or Diosi-Penrose decoheren.

103 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 104 5.2 Quantum optomechanics beyond the quantum coherent oscillation regime

Kiran E. Khosla1∗, George A. Brawley1, Michael R. Vanner1,2, Warwick P. Bowen1

1Center for Engineered Quantum Systems, University of Queensland, St Lucia 4072, Australia 2Clarendon Laboratory, Department of Physics, University of Oxford, OX1 3PU, United Kingdom

OCIS codes (120.4880) Optomechanics, (270.0270) Quantum optics, (140.3538) Lasers, pulsed. Interaction with a thermal environment decoheres the quantum state of a mechanical oscillator. When the interaction is suﬃciently strong, such that more than one thermal phonon is introduced within a period of oscillation, quantum coherent oscillations are prevented. This is generally thought to preclude a wide range of quantum protocols. Here show that the combination of pulsed optomechanics techniques with coherent control can overcome this limitation, allowing ground state cooling, general linear quantum non-demolition measurements, optomechanical state swaps, and quantum state preparation and tomography without requiring quantum coherent oscillations. Finally we show how the protocol can break the usual thermal limit for classical sensing of impulse forces.

5.2.1 Introduction

Quantum optomechanics uses an optical or microwave field to prepare, control and characterize the quantum states of a meso- to macro-scopic mechanical oscillator, typically using a cavity to enhance the interaction [56, 51, 53, 153, 61, 60]. Optomechanical systems have been proposed for quantum information applications [236, 237, 238] and tests of foundational physics [223, 131], and are currently used for state of the art sensors [239, 240, 166], with each application requiring different optomechanical regimes. The significant progress in devices and technology of the last decade [51] suggest some of these aspirational targets will be realisable in the near future. With the exception of position non-demolition measurements and their derivatives [52], it is generally considered that operation in the quantum coherent oscillation (QCO) regime — where on average less than one thermal phonon is exchanged per mechanical period, and the oscillator remains coherent for at least a single oscillation — is a minimum requirement for such experiments [241, 51, 242, 243, 244]. This notion is reflected in recent theoretical and experimental results [175, 245, 246, 247, 248, 76, 73, 72, 55, 131, 249]. 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 105

In the high temperature limit, the QCO regime is achieved when Qωm > kBT/~ with Q the quality factor of the mechanical oscillator of frequency ωm at temperature T . This places stringent constraints on both the temperature and mechanical resonance frequency for which optomechanics protocols can be implemented. Highly 13 desirable room temperature operation requires ωmQ & 4×10 [243], which has been achieved at MHz [244, 250] frequencies, but remains beyond current technology for low frequency (ωm/2π . 100 kHz) oscillators [51]. Here we show that combining pulsed optomechanics with quantum coherent control, enables ground state cooling, general linear back-action evading measurements, state swaps, non-classical state preparation and quantum tomography, outside the QCO regime. This substantially extends the parameter regime for which quantum protocols are applicable, providing a pathway towards macroscopic tests of quantum mechanics and quantum enhanced precision force sensing [239] at room temperature, among other applications. Furthermore, when applied in the classical regime, our technique can evade thermal force noise which limits current state of the art force sensors. Such sensors operate in the non-QCO regime using a “time of ﬂight” method to translate a force signal — coupled directly to momentum — into position which can be optically read out [251, 240]. The necessary time delay introduces thermal force noise, reducing the sensitivity of the measurement. Implementing the measurement over a small fraction of a period reduces the thermal noise, increasing the measurement sensitivity. Our results extend previous work on speed meters, which have been proposed for gravitational wave detectors [252, 253]. Speed meters can achieve quantum non-demolition measurements of the relative momentum of two oscillators. Unlike our approach, they have generally been studied for detecting oscillating forces in a frequency band far from the mechanical resonance [254, 255, 256, 257], and within the free mass approximation.

5.2.2 Model

Pulsed optomechanics [52, 258] generates optomechanical correlations over a short time scale compared to free mechanical evolution. These short interactions can be used to manipulate the state of a mechanical oscillator, greatly reducing mechanical thermalisation for a given protocol. The interaction Hamiltonian for such a system † † is given by HI = ~g0a a(b + b ), where a (b) is the annihilation operator for the optical (mechanical) mode and g0 is the bare optomechanical interaction rate. We restrict the analysis to the linearized interaction where the optical field is linearised, a → α + a, about a time dependent amplitude α(t) = ha(t)i where, without loss of generality we define α to be real [59]. The pulse envelope α(t) is chosen to be 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 106 a Gaussian with pulse width τ, that may have negative amplitude. This can be achieved, for instance, thought appropriate displacements of the pulses as discussed later. Expanding to first order in a, the interaction Hamiltonian is given by

† † 2 † HI/~ = g0α(a + a )(b + b ) + g0α (b + b ). (5.1)

The first term is the linearised optomechanical interaction, and the second term is a coherent momentum displacement. This displacement is deterministic and can be cancelled by applying an opposite classical displacement to the oscillator; we therefore neglect it henceforth. Outside the single photon strong coupling regime † † the second order term, g0a a(b + b ) that has also been neglected, remains negligibly small, even when the envelope α(t) → 0, as the mechanical dynamics is dominated by Brownian motion [78]. The cavity is modeled as a single sided cavity with decay rate κ which is large enough for the optical pulse to adiabatically interact with the cavity τ κ−1. In q this regime the intracavity field is proportional to the input field α(t) = αin(t) 2/κ R 2 ¯ ¯ where the input field is normalized |αin| dt = N where N is the mean photon number in the pulse. Other pulsed optomechanics protocols [146, 78, 79] assume the oscillator is frozen during a single pulsed interaction, so that τ 1/ωm with the optomechanical system necessarily operating in the unresolved sideband limit,

κ ωm. However in this work, we require the stricter condition κ nγ¯ (this condition is only stricter outside the QCO regime) in order to approximate a single pulsed optomechanical interaction as unitary. The short duration of the pulse means that any addition mechanical modes that to couple to the optical ﬁeld cannot be spectrally resolved. This is an issue faced by many pulsed protocols, and can be avoided by using speciﬁcally designed mechanical oscillators where only a single mode is optically coupled, such as those used in Ref. [146]. Under these conditions the unitary describing the total pulsed interaction generated by the Hamiltonina in (5.1) is given by U(X X ) = exp[−iλX X ], where √ √ M L M L † † 2XM = b + b and 2XL = a + a are the mechanical position and optical am- R plitude quadratures, respectively. The dimensionless constant λ = g0 α(t)dt = q 1/4 ¯ sign(α)4(2π) g0 τN/κ is the optomechanical interaction strength [52]. Using a pulse shape other than a Gaussian simply changes the proportionality constant be- q ¯ tween λ and g0 τN/κ (where τ is now some characteristic time of the pulse shape). In the adiabatic limit, the pulse shape remains constant throughout the protocol (up to order O(κτ)−1), and the optimal pulse shape is one that maximises R |α(t)|dt subject to the constraints R |α(t)|2dt = 1 and τ κ−1. The unitary correlates the † optical phase quadrature (PL) and mechanical position as U PLU = PL − λXM at the expense of adding back action to the momentum (PM) of the oscillator 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 107

Optical Homodyne delay measurement

SBS 4

HWP Interaction pulse Displacement pulse 2 1 QWP

RBS 99:1 PBS 3

Figure 5.1: Schematic of the protocol. Half and quarter wave plates (HWP and QWP respectively), a polarizing beam splitter (PBS) and switchable beam splitter (SBS) are used to initially direct the interaction pulse along the paths 1-4, along which the first optomechanical interaction takes place. When the interaction pulse reaches the top port of the highly reflective beam splitter (RBS), its coherent amplitude is changed by the displacement pulse. Since almost all of the interaction pulse reflects from the RBS, any quantum correlations between it and the oscillator remain, as the pulse interacts with the mechanical oscillator a second time. After the second optomechanical interaction the SBS switches out the pulse to a homodyne measurement device, instead of directing it along path four a second time.

† U PMU = PM − λXL. We will now show that a sequence of such pulsed interactions, allows arbitrary mechanical quadrature measurements which are sufficient for quantum state tomography, squeezed state preparation, and pulsed optomechanical state swaps [79]. Outside the QCO regime, a single pulsed measurement scheme cannot achieve these important goals. In our back action evading protocol a single optical pulse interacts with the mechanical element twice (Fig. 5.1). This process is represented by an initial optomechanical interaction, free mechanical evolution θ = ωmt, and finally the second optomechanical interaction with the same optical field but different interaction strength. Neglecting decoherence (which will be included in the next section), the protocol is described by the overall unitary

−iθb†b U = exp[−iλ2XMXL]e exp[iλ1XMXL] φ = R(θ) χ(λ1λ2) U(XLXM) (5.2) where, without loss of generality, we have assumed a negative envelope α(t) for the ﬁrst interaction, and a positive envelope for the second. This unitary, acting φ from right to left, can be understood as follows. The ﬁrst term, U(XLXM) = φ exp(−iGXLXM), is the optomechanical interaction between a rotated mechanical φ quadrature XM = XM cos φ + PM sin φ and the amplitude quadrature, XL of the 2 2 1/2 light. The (dimensionless) interaction strength is G = (λ1 + λ2 − 2λ1λ2 cos θ) > 0, 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 108

−1 and the rotated quadrature angle is tan φ = λ2 sin θ(λ2 cos θ−λ1) . We therefore see φ that by applying two interactions it is possible to generate a single eﬀective XLXM interaction for an arbitrary mechanical quadrature φ. This interaction contains terms of the form ab + b†a† and will therefore result in entanglement between the optical and mechanical states. For a given φ, the natural choice of free parameters

(λ1, λ2, θ) are the set that maximizes G. The second unitary in Eq. (5.2), χ(λ1λ2) = i 2 exp[− 2 λ1λ2XL sin θ] is an optical Kerr nonlinearity, the pulsed analogue of the effects reported in Refs [259, 260]. The final term R(θ) = e−iθb†b is the mechanical rotation due to the delay between pulses, and must be accounted for in the final mechanical state. Different interaction strengths λ1 and λ2, with positive or negative sign, may be chosen by using a displacement pulse to change the mean photon number in-between the two interactions. A displacement pulse is realised by the highly reflective beam splitter (Fig. 5.1). After the beam splitter, almost all of the light incident on the top port is reflected, and therefore any correlations between this light and oscillator remain as the optical pulse follows the path 1-3, interacting with the oscillator a second time. This recycling of the pulse is essential to negate the quantum noise on the oscillator. The coherent amplitude of the displacement pulse is chosen to control α(t) (from (5.1)), and since it is predominantly reflected, contributes negligible quantum noise to the recycled pulse1. The phase between the displacement pulse and interaction pulse may be chosen such that the coherent amplitude of the correlated light is complete negated or reversed sign. The total unitary couples the optical and mechanical states as

XM → XM cos θ + PM sin θ − XLλ1 sin θ (5.3a)

PM → PM cos θ − XM sin θ − XL [λ2 − λ1 cos θ] (5.3b)

XL → XL (5.3c) φ PL → PL − GXM + XLλ2λ1 sin θ. (5.3d)

The cancellation of noise in Eq. (5.3b) can be intuitively understood by observing that the backaction noise arises from amplitude fluctuations in the light. The coherent amplitude acts as a gain in the linearised Hamiltonian ((5.1)), and changing its sign changes the sign of the amplitude fluctuations that kick the oscillators momentum. The cancellation is only made possible by the fact that the first pulse is recycled so that the same amplitude fluctuations kick the resonator with the reversed sign. √ √ 1 The optical annihilation operator after the beam splitter is af = 0.99ai + 0.01(di + β) ≈ ai + 0.1β where ai is the annihilation operator of the light in the top port, di is the annihilation operator of the displacement point with coherent amplitude β. We have neglected the input vacuum noise (di) since it contributes only 0.01 vacuum noise to the final operator af . 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 109

For a single optomechanical interaction (λ2 = θ = 0) the back action noise

(XL) is necessarily imparted onto the momentum of the oscillator. However, if the interaction strengths are chosen such that λ2 = λ1 cos θ the back action on the momentum exactly cancels and a homodyne measurement of the optical ﬁeld can be used to conditionally prepare a momentum squeezed state. This can be done in an arbitrarily short time, θ → 0 at the expense of increasing the overall interaction strength G. It can therefore introduce arbitrarily low levels of the thermal noise that precludes momentum squeezing with other protocols outside the QCO regime. By varying the interaction strengths for the two pulses (λ1, λ2), one may deterministically choose which mechanical quadrature the back-action is added to, and which quadrature is back-action free. For the unitary case, or indeed, in the QCO regime where a negligible amount of thermal noise is introduced after a single oscillation, there is no need to implement the two pulse protocol. A momentum measurement can be achieved with

θ = π/2, λ1 = 0 (wait then measure), correlating the phase quadrature with the initial momentum of the oscillator. Alternatively, momentum state preparation can be achieved with θ = π/2, λ2 = 0 (measurement then wait), preparing a conditional position squeezed state that rotates into momentum a quarter cycle later. However, outside the QCO regime phonon exchange during the necessary θ = π/2 delay thermalizes the mechanical state, resulting in additional measurement noise and degrading conditional state preparation, even in the limit of arbitrarily large optomechanical interaction strengths. Equations (5.3a)-(5.3d) illustrate how, with our protocol, one can reduce the duration of the protocol θ — and thus the thermalisation — while still generating the optomechanical correlations necessary to prepare and measure an arbitrary mechanical quadrature. Note that the unitary in (5.2) can be used to extend the applicability of the state swap protocol in Ref. [79] and cooling protocol of Ref. [52], which are currently limited to the QCO regime. Both of these protocols require a pulsed momentum measurement, which the authors propose to achieved via a quarter period rotation of the mechanical state, followed by a position measurement. However, by replacing these rotations and position interactions with the direct momentum interaction in (5.2) (with φ = π/2), one can implement the state swap and cooling protocols within a small fraction of one mechanical period, thereby avoiding the requirement for coherent quantum evolution of the quarter period timescale.

5.2.3 Nonunitary evolution

In this section, the description of the protocol is extended to include mechanical dissipation and optical losses [261, 262]. During each pulsed interaction, it is still 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 110 assumed that the mechanical oscillator remains eﬀectively frozen only being free to evolve during the wait time between pulses. The cavity is treated as highly over coupled such that κ ≈ κex κ0 where κex and κ0 are the extrinsic and intrinsic cavity decay rates respectively. In this case the only signiﬁcant source of optical loss occurs during the storage time between the two interactions and each optomechanical interaction is well described by the unitary operator in equation (5.2). Thermalisation is included during the free evolution of the oscillator by the Langevin equation of motion [263]

˙ XM = ωmPM (5.4a) √ ˙ PM = −ωmXM − γPM + 2γξ (5.4b)

where γ is the oscillator decay rate and ξ(t) is a zero mean white noise operator with 0 1 0 correlation function hξ(t)ξ(t )i = (¯n + )δ(t − t ) wheren ¯ ≈ kBT/~ωm in the high 2 q temperature limit. The thermal force satisﬁes [XM(t), ξ(t)] = γ/2i to preserve the commutation relations. Optical losses are modeled as a single beam splitter unitary Bη with intensity loss η after each optomechanical interaction. The protocol is then given by BηU2BηMθU1, where Ui is the ith optomechanical unitary, and Mθ is a non-unitary map that describes the dissipative mechanical evolution given by Eqs (5.4a) and (5.4b) over the rotation angle θ. After the full protocol the optical and mechanical quadratures are given by,

XM,out = XM cos θ + PM sin θ + ξX − λ1XL sin θ (5.5a)

PM,out = PM cos θ − XM sin θ + ξP − √ √ XL( ηλ2 − λ1 cos θ) − λ2 1 − ηδX1 (5.5b) q √ 2 XL,out = ηXL + η − η δX1 + 1 − ηδX2 (5.5c) q √ 2 PL,out = ηPL + η − η δP1 + 1 − ηδP2 √ φ √ + ηλ1λ2 sin θXL − GXM − ηλ2ξX (5.5d) where ξX(P ) is the thermal noise added to the mechanical position (momentum) during the non-unitary evolution and δXi (δPi) is the amplitude (phase) quadrature vacuum noise entering from the ith optical loss channel. With decoherence included, the new measured quadrature and measurement strength are given by √ −1 2 2 2 3/2 1/2 φ = arctan[λ2 sin θ(λ2 cos θ − ηλ1) ] and G = (η λ1 + ηλ2 − 2λ1λ2η cos θ) respectively. Eq (5.5d) highlights how the scheme works in the presence of mechanical φ thermalisation. A measurement of PL,out provides information about GXM with all other terms contributing zero mean Gaussian noise. For small θ the thermal noise in 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 111 the position increases as θ3 scaling (see Ref. [79] for details). Consequently, reducing the duration of the protocol, cubicly reduces this noise term, which is generally the dominant noise source outside the QCO regime wheren/Q ¯ 1. Due to correlations √ between the phase and amplitude quadratures, i.e. the Kerr term η2λ1λ2 sin θXL in Eq. (5.5d), extra noise is added to the phase quadrature. Since this noise is correlated to the optical amplitude quadrature, XL, it can be reduced by measuring a ϕ rotated optical quadrature XL = XL cos ϕ + PL sin ϕ where the optimal angle ϕ is determined numerically. The effect of optical loss is detrimental in two ways; firstly, φ it reduces the effective interaction strength G, degrading the signal of XM in the phase quadrature, and secondly adding an extra irreversible noise term (δX1) to

PM,out. In the following section, we show how the correlations in Eqs. (5.5a)-(5.5d) can be used for state preparation, measurement and force sensing.

5.2.4 Measurement and tomography

For quantum measurement and tomography, the aim is to measure the statistics φ of the a priori mechanical state, XM for each φ ∈ [0, π), without making any as- φ sumptions on the statistics of XM. From Eq. (5.5d), a measurement of PL,out (or ϕ φ indeed any optical quadrature XL 6= XL) is a measurement of XM with all other terms contributing zero mean Gaussian noise. The uncertainty in a Gaussian variable A, given a measurement of B is given in general by the conditional variance, 2 1 V (A|B) = V (A) − C(A, B) /V (B), where C(A, B) ≡ 2 hAB + BAi − hAihBi is the correlation between A and B. When the conditional variance of the measured φ ϕ quadrature V (XM|XL ) is below the ground state variance (of 1/2), full quantum state tomography can be performed efficiently, and Wigner negativity can be directly observed [264, 265]. Outside the QCO regime direct observation of Wigner negativity is not possible, using either single pulse, or stroboscopic measurement; while state tomography becomes increasingly challenging due to the convolution of the thermal noise in the measurement results. Both can be achieved, in principle, to an arbitrary degree of accuracy here. To the authors knowledge, this is the first protocol capable of efficient state tomography for oscillators outside the QCO regime. Figure 5.2(a) compares the two approaches for realistic experimental parameters. 5 The parameters are chosen as ωm/2π = 100 kHz, γ/2π = 1 Hz (Q = 10 ), similar to the silicon carbide resonators in Ref [266]. At a temperature of 1 K, 2πn/Q¯ ≈ 13 phonons enter per oscillation, residing well outside of the QCO regime. The bare optomechanical coupling rate is conservatively set at g0/2π = 1 Hz with a cavity decay rate κ/2π = 1 GHz (optical Q ≈ 2.5 × 105 at 1064 nm). To ensure a fair comparison, we choose to relate the single pulse interaction strength λ to the two 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 112

q 2 2 pulse interaction strengths via λ = λ1 + λ2 such that the mean photon number in the single pulse is equal to the sum of mean photon numbers in the two pulse scheme. As shown in ﬁgure (5.2a), there is lower bound for the conditional variance using a single pulsed interaction, independent of the interaction strength. This lower bound is removed using our protocol, allowing sub-ground state resolution outside the QCO regime. Figures (5.2b)-(5.2d) shows that sub-ground state resolution is possible for all quadrature angles allowing for full quantum state tomography for λ & 10. In order to achieve this sub-ground state resolution, the two pulses must interact with the mechanical oscillator within the thermal decoherence time. 2 ~ωκλ√ An interaction strength of λ = 10 requires a high pulse energy, E = 2 ≈ 14 g0 τ16 2π mJ in a 5 ns pulse (assuming a temperature of 100 K with τ nγ¯ ). However this is a consequence of our conservative choice of coupling strength g0/2π = 1 Hz. Coupling rates of kHz [267] and even up to 20 kHz [268] have been observed, and taking g0/2π = 2 kHz, signiﬁcantly reduces the pulse energy to 3.7 nJ for λ = 10, or 370 nJ for λ = 100. Combining this higher coupling rate with cryogenic (1 K) operation will enable longer optical pulses (increasing τ to 500 ns) further reducing the pulse energy 37 fJ for λ = 10, as is required for ground state resolution at 1 K (Fig. 5.2). We note that a 5 ns, 3.7 nJ pulse, corresponds to 0.74 W of peak power, or 3.7 µW average power at 1 kHz rep-rate, which can be routinely achieved in optical micro cavities [269, 270].

5.2.5 Squeezed state preparation

We now turn to how the protocol can be used to prepare a squeezed mechanical state in the presence of thermalization. Mechanical squeezing such as ponderomotive squeezing [271] or reservoir engineering [272] requires many oscillations of the mechanical oscillator and therefore cannot be implemented within the QCO regime. A single pulsed interaction can be used to generate squeezing outside the QCO regime but is best suited to position squeezing [52]. Our scheme allows squeezing of an arbitrary mechanical quadrature outside the QCO regime. Due to the finite rotation during the protocol there is a subtle distinction between squeezed state preparation and quantum measurement. For state preparation the aim is to φ condition the variance of the a posteriori quantum state, V(XM,out|PL) instead of φ the a priori state, V(XM|PL). Figure (5.3a) shows the a posteriori conditional variance for a momentum measurement at different temperatures, comparing the double interaction protocol introduced here with a single pulsed interaction. It shows arbitrary quadrature squeezing is in principle possible if the interaction strength is high enough, and is necessarily achievable within a fraction of an oscillation. Our scheme can achieve significantly lower variance at higher temperatures, even for 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 113

Figure 5.2: (a) Conditional variance for measurement of different quadratures (φ = π/2, π/4, π/8), using the double (solid lines) or single (dashed lines) interaction protocols. (b)-(d) Conditional variance as a function of mechanical quadrature angle, φ, for the double (solid lines) and single (dashed lines) interaction protocols for λ = 10, λ = 1 and λ = 0.1 respectively. (a)-(d) assume a bath temperature of 1 K, and ωm/2π = 100 kHz, γ/2π = 1 Hz. Grey dashed lines indicate the ground state variance. small-moderate interaction strengths. As with quantum tomography, our scheme is able to prepare states with variance below the thermalization line that bounds the single pulsed scheme, meaning the conditional variance can be arbitrarily reduced by increasing the interaction strength. This is due to the finite time taken for the position squeezed state to rotate to a momentum squeezed state. From Eqs. (5.5a)- (5.5d), it can be seen that the optimal measured quadrature is not orthogonal to the back action quadrature, as a result any squeezing will be accompanied by anti- squeezing larger than the lower bound set by the Heisenberg limit. Combined with optical losses and the mechanical thermalization, this effect reduces the purity of the final mechanical state. Note that direct measurement and manipulation of the momentum of an oscillator over a short time scale (θ/ωm) may be used to directly measure impulse forces — which couple directly to momentum through dp = F dt — at the thermal limit. Using our pulsed protocol, the momentum state preparation (measurement) can be made immediately before (after) the impulse force, therefore, the only thermal force in the signal is that which enters during the duration of the impulse. In contrast, a position measurement must necessarily wait an additional time period while the force evolves into a displacement, during which, extra thermal noise is added to the signal. For example taking the detectable momentum change to be on the order √ q of the conditional momentum standard deviation of 200 ~mω/2 at 100 K (from 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 114

Figure 5.3: Conditional variance of the ﬁnal oscillator momentum using the double (solid lines) and single (dashed lines) interaction protocols at diﬀerent temperatures. A temperature of 100 K corresponds to 1.3 × 103 phonons exchanged per cycle for ωm/2π = 100 kHz, γ/2π = 1 Hz.

ﬁgure 5.3 with λ = 1), corresponds to a thermally limited impulsed force sensitivity 1 th th of 8 pN over 50 of an oscillation period (200 ns), compared with 30 pN over 1/4 of an oscillation period (2.5 µs) for the single pulsed protocol.

5.2.6 Summary

We have introduced a protocol to realize quantum optomechanics beyond the QCO regime, where quantum state preparation and direct tomography are possible within a fraction of a mechanical period. While it may seem that a fraction of a mechanical period is an extremely short time scale, the low frequency oscillations mean that in absolute terms, it is comparable to the state life time of some of the best cryogenic quantum optomechanics experiments. For example, the resonator in the mechanical Fock state preparation and measurement experiment of Ref. [249], has a decoherence rate of 14 kHz, only two orders of magnitude smaller than the single- phonon thermalisation timenγ ¯ = 1.3 MHz (at 1 K) for the oscillators considered here. By using a high Q low frequency oscillator, the life time of a quantum state

1/nγ¯ = ~Q/kBT can be made comparable to the high frequency oscillators operating at cryogenic temperatures, while still remaining outside the QCO regime. This allows for the possible observation of quantum eﬀects in a new realm of mechanical resonators, such as low frequency, high temperature or high mass systems by removing the requirement of coherent oscillations. Furthermore, even in the classical regime, the reduction of thermal noise in our measurement protocol results in signiﬁcantly less measurement uncertainty in pulsed measurements of arbitrary mechanical quadratures. 5.2. QUANTUM OPTOMECHANICS BEYOND THE QCO REGIME 115

Funding Information

This research was jointly funded by Australian Research Council Centre of Ex- cellence for Engineered Quantum Systems (EQuS CE110001013), Discovery Project (DP140100734), Future Fellowship (FT140100650), Air Force Oﬃce of Scientiﬁc Re- search (AFOSR) grant number FA2386-14-1-4046, and the Engineering and Physical Science Research Council (EP/N014995/1).

Acknowledgments

The authors would like to thank Gerard J. Milburn for helpful discussions throughout this project. Chapter 6

The displacemon

6.1 Foreword

In thischapter (published in Ref.[4]), we introduce a novel design to realize a non- linear coupling between a two level system and a mechanical oscillator. Although there are several experimental proposals to realize the same interaction Hamilto- nian [133, 273, 274, 275], our work proposes an entirely different architecture, which allows strong coupling using existing technology. Furthermore, we analyze the proposed system in detail to optimize the design, and reduce sources of noise. The central result of the paper is to show how one can perform an interferometery experiment with the wave function of the mechanical resonator. The interferometer is realized by coherent manipulation and measurement of the qubit state, which can be understood to act as an effective diffraction grating for the mechanical wave function. We give an intuitive description of how the grating is analogous to standing wave optical gratings in atomic interferometry, and discuss the quantitative difference between interferometry with a harmonic oscillator wave function, compared to that of a free particle. The proposal allows one to directly probe the quantum nature of the mechanical oscillator consisting of up to 106 atoms, potentially demonstrating the largest mass interferometric experiment to date. We also discuss how one can differentiate between quantum and classical descriptions for the mechanical oscillator, showing an unambiguous signature of quantum behavior. The carbon nano-tube oscillator was chosen for its potential for strong coupling, however, if a larger, lower frequency oscillator were to be used, the architecture introduced here may be able to test collapse models, such as classical channel gravity, CLS, or Diosi-Penrose.

116 6.2. DISPLACEMON ELECTROMECHANICS 117 6.2 Displacemon electromechanics: how to detect quantum interference in a nanomechanical resonator

K. E. Khosla1,2, M. R. Vanner3,4, N. Ares5, E. A. Laird5. 1Center for Engineered Quantum Systems, The University of Queensland, Bris- bane, Queensland 4067, Australia 2School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4067, Australia 3QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW 4Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom 5Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom

6.2.1 Introduction

The superposition principle is a fundamental tenet of quantum mechanics and essential for understanding a wide range of quantum phenomena. As the scale of quantum objects increases, the experimental consequences of this principle become increasingly hard to isolate. Is there a scale at which this tenet begins to break down? The strongest tests of superposition come from matter-wave interferometry between trajectories of large molecules. Remarkably, interference can be measured using molecules of mass as large as 104 atomic mass units (amu) [276, 277]. The ability to create unambiguous superpositions on a mesoscopic scale would allow tests of quantum collapse theories and gravitational decoherence [34, 39, 24, 31], ulti- mately addressing experimentally the question of why we fail to see superpositions in everyday life [46]. This has inspired numerous challenging proposals to detect interference of larger particles [131, 278, 279] via optomechanical coupling [51], or using levitated nanodiamonds [280, 281]. Nanomechanical resonators span this mesoscopic mass scale from ∼ 106 to ∼ 1016 amu and therefore provide an attractive route to extend the scale over which quantum effects can be observed. Recently, cooling to the ground state [282, 76, 73], and such elements of quantum behaviour as state squeezing [283, 284, 285] and coherent qubit coupling [282, 286, 179] have become accessible with mechanical resonators of this scale. Moreover, significant progress towards observing mechanical superposition states has been made in both opto- and electromechanics, and mechanical interference fringes have recently been observed at a classical level [136]. The ob- 6.2. DISPLACEMON ELECTROMECHANICS 118 servation of quantum interference, however, remains outstanding and is a key goal of this paper. Here we introduce the “displacemon”, a device that enables strong coupling between a nanomechanical resonator and a superconducting qubit. We show how to create an effective diffraction grating that leads to an interference pattern in the resonator’s displacement. The scheme works using a sequence of manipulations on the qubit to create an effective grating with a fine pitch and therefore a large momentum displacement. In molecular interference experiments, the diffraction grating is typically an etched membrane; however, van der Waals interactions with the slits mean this is hard to extend to large particles [276]. More advanced implementations use optically defined gratings; the pitch, which sets the momentum separation of the diffracted beams, is then limited by the optical wavelength [287]. In our scheme, the pitch is limited neither by an optical wavelength nor by the size of the resonator, but by the qubit-resonator coupling strength. As we will show, this allows for diffraction gratings with a pitch narrower than the ground state wave function. Our proposed device uses a vibrating nanobeam flux-coupled to a superconducting qubit, through which all manipulations and measurements are performed. As a nanobeam that optimally combines high mechanical frequency, low dissipation, and the ability to couple strongly to superconducting quantum devices, we propose a suspended carbon nanotube. Previous proposals for quantum motion in nanotubes [288, 289] have been based on coupling to a spin qubit; however, the coherence requirement on the qubit is stringent [290]. Alternatively, a superconducting charge qubit could be used to generate and detect mechanical superposition [274], although obtaining sufficiently strong electrostatic coupling, approaching the qubit decoherence rate, remains a challenge [291, 292]. Here, using realistic parameters derived from experiments, we show how to construct an effective mechanical diffraction grating and measure quantum interference in a moving object of > 106 amu. This proposal extends by nearly three orders of magnitude the mass scale of interference between spatially distinct superpositions. Note that entanglement has been measured involving nanomechanical resonators up to a mass scale of ∼ 1014 amu [282, 70], allowing spatial superposition to be inferred but not demonstrated.

6.2.2 Model

In general, strongly coupling a mechanical resonator to a qubit is challenging because the best qubits are engineered to be insensitive to their environment [293]. We propose a design that is robust against electrical and magnetic noise, while still achieving strong mechanical coupling. We envisage a superconducting qubit of the concentric transmon design [186] in which at least one of the junctions is a vibrating 6.2. DISPLACEMON ELECTROMECHANICS 119 nanotube (Fig. 6.1). Nanotube resonators offer unique advantages for studying quantum motion [294]: (i) the zero-point amplitude is typically greater than 1 pm, much larger than other mechanical resonators; (ii) the resonant frequency is sufficiently large to allow near-ground-state thermal occupation, suppressing thermal decoherence [295, 296]; (iii) a nanotube can act as a Josephson junction [297, 185, 184]; and (iv) ultraclean devices offer quality factors greater than 106, which provide long-lived mechanical states [188]. In this design (Fig. 6.1(a)), the two junctions form a gradiometric superconducting quantum interference device (SQUID), so that the qubit frequency is set by the flux difference ∆Φ between the two loops. This flux difference is tuned primarily by means of a variable perpendicular field ∆B⊥(t), while mechanical coupling is

flux-mediated using a static in-plane field B|| [298, 299, 300]. This type of concentric transmon is insensitive to uniform magnetic fields, which has two advantages for this proposal: the qubit can be operated coherently away from a flux sweet spot [186], and any misaligned static flux does not perturb the energy levels. Both these facts are favourable for strong nanomechanical coupling. Because this variant of the transmon is designed for strong coupling to nanomechanical displacement, we refer to it as a ‘displacemon’. In this section we derive the displacemon Hamiltonian, and estimate the parameters for a feasible device.

The mechanical resonator

We consider the nanotube as a beam of length l and diameter D and focus our studies on its fundamental vibrational mode [294, 301, 302, 296, 295, 188, 58]. The mechani- † † cal resonator Hamiltonian is Hm = ~Ωa a, where a (a) is the creation (annihilation) operator for the resonator. Typically, the restoring force for a clamped nanotube is dominated by the beam’s tension T [303], so that the mechanical angular frequency q q q is ω = π T and the zero-point amplitude is X = ~ = ~ (µT )−1/4, where m l µ ZP 2mωm 2π −7 −2 µ = πρSD is the mass per unit length and ρS = 8 × 10 kg m is the sheet density of graphene. The displacement profile as a function of axial coordinate Z is √ ˜ πZ † X(Z) = X 2 sin l , where X ≡ (a + a )XZP is the displacement coordinate. This profile is normalized so that the root mean square displacement is equal to X [303]. The flux coupling is proportional to the area swept out by the nanotube, which √ 1 R l ˜ 2 2 is equal to β0lX, where β0 ≡ lX 0 X(Z) dZ = π is a geometric coupling coefficient [299]. Nanotube resonators can also be fabricated without tension, so that the restoring force is dominated by the beam’s rigidity [301, 302]. In this limit, the mechanical q 22.4 2 frequency is ωm = l2 ED /8µ and the coupling coefficient is β0 = 0.831, where E ≈ D × 1.09 × 103 Pa m is the extensional rigidity. 6.2. DISPLACEMON ELECTROMECHANICS 120

The qubit

The qubit consists of a pair of superconducting electrodes coupled through the SQUID junctions. The qubit Hamiltonian is [82]

2 Hq = 4EC(ˆn − ng) − EJ cosϕ, ˆ (6.1) where EC is the charging energy, EJ is the SQUID Josephson energy, andn ˆ andϕ ˆ are the overall charge (expressed in Cooper pairs) and phase across the junctions, with ng being the oﬀset charge. Here we have neglected the qubit inductance, which makes a small contribution on the energy levels [186]. In the transmon limit EJ EC, we can approximate Eq. (6.1) by an eﬀective Hamiltonian H ≈ 1 ω σ , where ω = √ q 2 ~ q z q 8EJEC/~ is the qubit frequency and σz is the standard Pauli matrix, acting on the qubit ground state |−i and the excited state |+i. Qubit rotations, initialization, and projective measurement are now well-established through capacitive coupling to a microwave cavity in a circuit quantum electrodynamics architecture [187, 186].

Strong and ultrastrong coupling

Strong and tunable coupling between the qubit and the mechanical resonator is achieved by ﬂux coupling to the SQUID loops, which tunes the qubit Josephson energy. Assuming equal critical current Ic in the two junctions, this Josephson energy is

π∆Φ 0 EJ = EJ cos , (6.2) 2Φ0 0 where ∆Φ is the ﬂux diﬀerence between the two loops, EJ = IcΦ0/π is the maximum

Josephson energy, and Φ0 = h/2e is the flux quantum. The flux difference can be tuned both directly, via a perpendicular magnetic field

B⊥, and via the displacement using a static in-plane ﬁeld B||. We have

∆Φ = A∆B⊥ + 2β0lB||X, (6.3) where A is the area of one SQUID loop. Since quite small perpendicular fields suffice to tune the qubit frequency over its full range, we envisage an on-chip coil to modulate ∆B⊥(t) as a function of time t [304, 186]. Substituting Eq. (6.2) into the definition of ωq gives:

dω πβ lB sin π∆Φ/2Φ q = −ω0 0 || 0 , (6.4) dX q 2Φ q 0 | cos π∆Φ/2Φ0|

0 q 0 where ωq = 8EJ EC/~ is the maximal qubit frequency. The dependence of ωq on X 6.2. DISPLACEMON ELECTROMECHANICS 121 gives rise to an electromechanical coupling, resulting in the Hamiltonian [274, 293]:

ω λ(t) H = ω a†a + q σ + (a + a†)σ , (6.5) ~ m ~ 2 z ~ 2 z where λ(t) = XZP dωq/dX (from Eq. (6.4)) is the qubit-mechanical coupling strength, dynamically controlled through the ﬁeld ∆B⊥(t). To achieve coherent interaction between the qubit and the resonator requires the strong coupling regime, where the maximum accessible coupling λ0 exceeds the thermal decay rate κth = kBT/(~Qm) of the resonator, where Qm is the quality factor, and the decoherence rate of the qubit γ = 1/T2, where T2 is the coherence time. The large zero-point motion makes nanotube resonators particularly favourable for achieving this regime. Taking device parameters from simulation and experiment (Appendix 6.3.1) leads to Ω/2π = 125 MHz, ωq/2π = 2.19 GHz, and

λ0/2π = 8.5 MHz, with the flux dependence shown in Fig. 6.1. This is favourable for achieving the strong coupling regime, since both κth/2π and γ/2π are typically less than 1 MHz. To create well-separated mechanical superpositions, a stronger condition is desirable; the qubit should precess appreciably within an interval during which the resonator can be considered stationary. This is the ultrastrong coupling regime, where λ0 > ωm [300]. It is possible that a device similar to that of Fig. 6.1 could access this regime (see Appendix 6.3.1). However, here we instead suppose that effective ultrastrong coupling is engineered by modulating ∆B⊥(t) at the mechanical frequency (see Section 6.2.3). In this modulated frame (similar to the toggled frame obtained by repeatedly flipping the qubit [305]), the resonator is effectively frozen and the ultrastrong coupling condition is relaxed to λ0 > κth, γ.

6.2.3 Generating and measuring mechanical quantum interference

To realise the nanomechanical interferometer, we propose a series of operations and measurements on the qubit. The qubit provides the necessary non-linearity to generate mechanical superposition states. The core idea is that the state of the resonator is constrained by the qubit measurement outcome in the same way that the state of a particle is constrained by passing through a diﬀraction screen. By concatenating a series of qubit rotations and measurements, the resonator can be cooled, diﬀracted and measured. 6.2. DISPLACEMON ELECTROMECHANICS 122

Cooling the resonator

The first step is to prepare the resonator close to its ground state. A mechanical frequency of 125 MHz requires a bath temperature below ∼ 5 mK for a thermal occupation less than unity. Such temperatures are achievable but challenging with cryogenic cooling [306]. At a more accessible cryostat temperature of 33 mK, the initial thermal occupation is n = 5. To approach the ground state from a thermal state, we propose here an active cooling scheme utilising the qubit as a thermal filter (Fig. 6.2). Following initialization to the |+i state, the scheme consists of applying a π burst at the bare qubit frequency ωq (Fig. 6.2(a)). If the resonator is near its equilibrium position, this results in a qubit flip. By conditioning on this outcome (i.e., utilising only those runs of the experiment where this qubit outcome is measured), the resonator state is constrained to a narrow window (Fig. 6.2(b)). A single operation of this type cools only one quadrature of the motion, because resonator states with high momentum may still pass the window. To cool the orthogonal quadrature, the same selection should be applied a quarter of a mechanical period later [52], which filters out high-energy states that pass the first selection step (Fig. 6.2(c)). The combination of these two pulse sequences therefore prepares the resonator close to its ground state, at the price of accepting only a fraction of the measurement runs.

Diﬀracting the resonator

The effective diffraction grating for the resonator (Fig. 6.3) is implemented using Ramsey interferometry to generate a periodic spatial filter [307, 133]. To understand how the grating arises, consider the time evolution operator U(t) generated from Eq. (6.5). As shown in Ref. [133], this time-ordered unitary is

iωqt − σz † U(t) = R(ωmt)e 2 D(α)|−ih−| + D (α)|+ih+| , (6.6) where D(α) = eαa†−α∗a and R(θ) = e−iθa†a are resonator displacement and rotation operators respectively, and

t i Z 0 α = eiωmt λ(t0)dt0 (6.7) 2 −∞ is the amplitude of the coherent displacement. The superposition of displacement operators in Eq. (6.6) applies equal and opposite momentum kicks (assuming Re(α) = 0 from here on 1) to the resonator depending on the state of the qubit. This is analo-

1If α is real, Eq. (6.6) leads to a position basis superposition, which is a π/2 rotation of the momentum superposition. In this way, any complex α can be understood in terms of a purely imaginary α, followed by a rotation. Without loss of generality, and to keep the analogy to optical 6.2. DISPLACEMON ELECTROMECHANICS 123 gous to standing wave gratings in atom interferometry, which when decomposed into left and right propagating beams can be understood to impart superposed positive and negative impulses to atoms. We now describe the protocol that realises this superposition of momentum kicks (Fig. 6.3(a-b)). Following initialisation of the qubit in the excited state |+i, a microwave burst applied to the qubit generates a π/2 rotation, preparing a super- √ position (|+i + |−i)/ 2. The mechanical interaction then causes the qubit state to X(t) precess at the displacement-dependent rate ∆ωq(t) = λ(t). After an interval τR, XZP a second π/2 burst is applied, followed by a σz measurement. Conditioning on the qubit outcome gives a measurement operator that acts on the mechanical system,

† Υ±(φ) ≡ h±| ΠφU(t)Π0 |+i R(ω t) = m [D†(α) ± eiφ+iωqtD(α)]. (6.8) 2

Here Πφ denotes a π/2 qubit rotation with phase φ, and ± is the result of the σz measurement. The (un-normalized) state of the resonator after the interaction is,

cos X φ! |ψiM → Υ± |ψiM = R(ωmt) |α| + |ψiM , (6.9) XZP 2 sin where the cos(·) or sin(·) correspond to finding the qubit in the excited or ground state respectively 2. The resonator wavefunction is thus projected onto an effective diffraction grating with pitch πXZP/|α| (Fig. 6.3(c)). Since the only difference between conditioning on the |±i outcomes is a relative change in phase of the effective grating, either measurement outcome may be used to define the grating. We refer to the Ramsey sequence followed by conditioning on the qubit measurement outcome as a grating operation. Its effect is to split the resonator wavefunction into a superposition of left-moving and right-moving branches. A well-separated superposition, with both branches displaced by more than the zero-point amplitude, requires |α| & 1. To achieve this with our parameters, we require that λ(t) is modulated at the mechanical frequency:

λ(t) = λ0g(t) cos Ωt, (6.10) where g(t) is a Gaussian envelope function with a maximum of unity and a full standing wave gratings, we therefore consider Re(α) = 0. 2Equation (6.9) can be intuitively understood from Eq. (6.8) by noting that for Re(α = 0), † D(α) = ei|α|(a+a ) and decomposing the exponentials in Eq. (6.8) into trigonometric functions. 6.2. DISPLACEMON ELECTROMECHANICS 124

width at half maximum of τλ 1/Ω. Equation (6.7) then gives √ π α = i √ λ0τλ. (6.11) 8 ln 2

In the following, we take α ≈ 1.9i, thus achieving the desired momentum separation. A price to pay for this modulation is that the qubit and resonator are susceptible to decoherence over the full duration of the envelope. With our parameters, we require

τλ ≈ 130 ns, corresponding to N ≈ 17 mechanical periods. This interaction time is short enough that the evolution of the resonator-qubit system is well approximated as unitary. (See Appendix 6.3.3 for modelling of qubit dephasing and mechanical decoherence). With the parameters considered in appendix 6.3.1, a 3 × 106 amu resonator achieves a maximum spatial superposition of 12 pm. Using the macroscopicity measure introduced in Ref. [308] the nano-tube has a macroscopicity measure of µ = 14.0 (assuming a fringe visibility of 0.8 and coherence time of 100 oscillator periods). The state of the resonator has a higher measure than state-of-the-art near ﬁeld atom interferometry [309] (µ = 12.1) or a micro-membrane in the ground state [73] (µ = 11.5), and is comparable other proposed experiments including a Satellite atom (Cs) interferometer [310] (µ = 14.5) or a 105 amu Talbot-Lau interferometer [43] (µ = 14.5). We do note however, that a carbon nano-tube is among the smallest mechanical resonators, and therefore has a lower measure than proposals for superpositions of a micro-mirror [131] (µ = 19.0) or nano-spere interference [278] (µ = 20.5) 3.

Nanomechanical interferometry

We now show how a sequence of grating operations can be combined to create an interferometer (Fig. 6.4). The eﬀect of a single grating operation (Eq. 6.9) with phase φ = 0 and amplitude α = α1 is to divide the resonator’s wavefunction into two components with added momentum ±|α1|~/XZP. A second grating operation with the same amplitude and phase, applied after a duration τ1 = π/2Ω corresponding to a quarter period of free evolution, further splits the branches of the superposition, allowing quantum interference between recombined branches to be observed. After a second evolution time τ2, the interference can be detected using a third Ramsey sequence. In this step, there is no conditioning; the probability p+(φ, α3) for the qubit to return to state |+i is measured as a function of the phase φ and amplitude

3The macroscopicties quoted here are obtained from Refs. [308, 287] 6.2. DISPLACEMON ELECTROMECHANICS 125

α3 of the Ramsey sequence. This probability is

† p+(φ, α3) = Tr[Υ+Υ+ρm] Z 0 ! 0 2 X φ 0 = dX cos |α3| + P (X ), (6.12) XZP 2 where ρm is the density matrix describing the state of the resonator immediately before the third grating, with position probability distribution P (X). Scanning the phase of the third Ramsey sequence is analogous to scanning the position of the third grating in a molecular interferometer [277], and the signature of interference is a sinusoidal dependence on φ. In fact, Eq. (6.12) can be understood as a Fourier decomposition in which each choice of |α3| probes the component of † P (X) with wave number 2|α3|. From here on we will use x ≡ X/XZP = (a + a ) as a dimensionless position operator.

Our goal now is to use p+ to distinguish quantum interference from classical fringes that might appear in the resonator’s probability distribution P (x). Classical fringes might arise, for example, from the shadow of the diﬀraction gratings, or from Moir´epatterns. To recognise the quantum interference, we plot the resonator

Wigner distribution at diﬀerent times τ2 after the second grating operation, choosing

|α1| = |α2| ≈ 1.9 (Fig. 6.5). The effect of applying the first grating operation is to vertically slice the distribution with cos2(|α|x), and to prepare a superposition of two momentum states (see second inset in Fig. 6.4 plotting the state after the first grating, rotated by one quarter period). Since the second grating operation is applied a quarter period after the first, it slices along an orthogonal axis, leading to the “compass-like” distribution of Fig. 6.5(a) [311]. 6.2. DISPLACEMON ELECTROMECHANICS 126

(a) (b) ΔΦ

B||

Nanotube

Flux tuning ΔΦ

Qubit

B|| Qubit drive/readout

2 8 λ 4 /2 π (MHz) 0 0 (GHz) π )/dX (MHz/pm) /2 π q /2 ω q -2 -8 0 -1 ΔΦ Φ 1 ω -1 ΔΦ Φ 1 /2 0 d( /2 0

Figure 6.1: Device for strong qubit-mechanical coupling. (a) Electrical schematic. The device is a gradiometric transmon qubit, biased by flux difference ∆Φ between the SQUID loops. With a suspended nanotube acting as at least one junction (shown here for the right junction), the displacement modulates ∆Φ and therefore the qubit levels. (b) Arrangement of the qubit, vibrating nanotube, flux tuning coil, and drive/readout cavity antenna. The in-plane magnetic field B|| introduces strong coupling between the vibrations and ∆Φ. (c) Qubit frequency as a function of flux difference, with parameters as in the text. Solid lines assume equal Joseph- son coupling in the two SQUID junctions, dotted line assumes 30 % asymmetry (see Appendix 6.3.1). Curves are plotted as a function of flux (bottom axis) and equivalently of displacement (top axis). (d) Qubit displacement sensitivity (left axis) and mechanical coupling rate (right axis) as a function of flux. The bias point that achieves the assumed coupling is indicated by a vertical dashed line. 6.2. DISPLACEMON ELECTROMECHANICS 127

(a) (b)

| Before ψ | After Initialize Measure t -40 40

Qubit drive X (pm) (c) X

Figure 6.2: Cooling the resonator using the qubit. (a) Pulse sequence (see text). Gaussian pulse shaping is chosen for near-optimal filtering. (b) Effect of this sequence, conditioned on the qubit outcome |−i, on an initial state with a thermal distribution n = 5. (Burst duration is τπ = 100 ns with λ0/2π = 800 kHz.) The pulse sequence has the effect of passing the wavefunction through a narrow slit. The filter function is obtained by numerically solving the time evolution generated by Eq. (6.5) with an additional π-pulse term g(t)σx/2. (c) Cartoon showing the effect of two pulse sequences, offset by a quarter-integer number of mechanical periods. Only the lowest-energy trajectories (solid line) survive both measurements, effectively cooling the resonator. For the purpose of this cartoon, the duration of the two pulse sequences is compressed; in fact, each pulse sequence lasts for several mechanical periods.

(a) (c)

| Before ψ | (i) (ii) (iii) After Initialize Measure

t -20 X (pm) 20 Qubit drive (b)

(i) (ii) (iii)

Figure 6.3: Using pulsed qubit operations to construct a diffraction grating. (a) Pulse sequence. (b) Evolution of the qubit on the Bloch sphere, during (i) preparation, (ii) precession at a rate set by the displacement-dependent qubit frequency shift ∆ωq, and (iii) projection. (c) Effect of this pulse sequence on the ground-state mechanical wavefunction. Conditioning on the qubit measurement outcome |+i, the resonator wavefunction is multiplied by a periodic filter (Eq. (6.9)), analogous to a grating. Here we have taken |α| = 1.9. 6.2. DISPLACEMON ELECTROMECHANICS 128

Condition on Condition on Record result

Figure 6.4: Protocol for detecting nanomechanical interference. Following cooling (not shown), the first grating operation, with amplitude α1, diffracts the resonator wavefunction into a superposition of left-moving and right-moving components. Af- ter an interval τ1 of free evolution, a second grating operation, with amplitude α2, leads to recombination of the two components a quarter mechanical period later. To measure the resulting interference, a third (unconditioned) Ramsey sequence is applied after time τ2. The resulting qubit return probability p+(α3, φ) probes the mechanical interference fringes. The main panel shows a simulated resonator spatial density |ψ(X)|2, beginning from the ground state, plotted as a function of displacement and time, with the grating operations and the final Ramsey measurement operation indicated schematically as filters. The resonator Wigner distributions and marginals (see Fig. 6.5) are shown as insets just before each filter. To illustrate the continuing periodic evolution of the resonator wavefunction, the spatial density beyond the final Ramsey measurement is plotted as it would be probed by applying the measurement instead at a later time. 6.2. DISPLACEMON ELECTROMECHANICS 129

The compass distribution is intuitively understood: the quarter-period rotation after the ﬁrst grating turns the momentum superposition state into a position superposition. Each branch of the superposition then passes the grating, generating its own momentum superpositions and resulting in the four-lobe compass state. The compass state is clearly visible if the resonator is initially prepared in the ground state,n ¯ = 0 (Fig. 6.5(a-c)), but is washed out if the resonator is initially in a thermal state, leaving only the orthogonally sliced pattern visible (Fig. 6.5(d-f)). During the evolution time τ2, the Wigner distribution rotates (Fig. 6.5(b-c) and (e-f)), so that the interference fringes oscillate between the position and momentum marginals (plotted in blue and green traces respectively). Interference patterns arise when two lobes of a coherent quantum superposition overlap in position space, for example the north-east and south-east lobes in Fig. 6.5(a) interfering around x ≈ 3, or the north and south lobes in Fig. 6.5(c) interfering around x ≈ 0. The wave numbers present in the position marginal (which is measured by the third grating) are proportional to the momentum separation of lobes in phase space, geometrically illustrating the √ 2 ratio between wave number components present in Fig. 6.5(a) and Fig. 6.5(c). The interference fringes arising when the resonator is initially prepared in its ground state can be compared with those arising from an initial thermal state (¯n = 5). If the width of the initial thermal state (as in Fig. 6.5(d)-(f)) is larger than the √ superposition size ( n¯ > |α|), then the vertical slicing of the grating is no longer accompanied by a distinct momentum superposition, but rather an increase in the momentum variance (as seen by the broader than Gaussian position and momentum distributions in Figure. 6.5(e)), and results in a checker-board pattern. We can now see the distinction between quantum and classical fringes appearing in the marginal distributions. The shadow of the gratings is dominated by components close to the wave number 2|α1| (Fig. 6.5(a),(d)). The Moir´epatterns arising from the checkerboard have components close to at most two wave numbers, 2|α | √ 1 and 2 2|α1| (see Appendix 6.3.4). By contrast, the quantum interference pattern

(Fig. 6.5(a-c)) has multiple wave numbers components at each ωmτ2, as seen in the position marginals. Furthermore, quantum interference appears for all evolution times τ2, whereas classical fringes are washed out (Fig. 6.5(e)) except at particular fractions of the mechanical period. Hence for this protocol, a marginal P (x) with multiple wave number components, observed at all rotation angles ωmτ2, indicates quantum interference. 6.2. DISPLACEMON ELECTROMECHANICS 130

-1 0 1

6 (a) (d) (g)

-6

6 (b) (e) (h)

-6

6 (c) (f) (i)

-6 -6 0 6 -6 0 6 -6 0 6

Figure 6.5: Wigner distributions of the resonator’s state at diﬀerent values of τ2 showing: (a)-(c) maximum interference from an initial pure state with unitary evolution, (d)-(f) loss of interference due to initial thermal phonon occupationn ¯ = 5 (with unitary evolution), and (g)-(i) loss of interference with non-unitary evolution equivalent to adding 0.05 (g) or 0.1 (h)-(i) thermal phonons. The initial state for (g) and (h) is the vacuum state, and the initial state of (i) is a thermal state with n¯ = 5 phonons. Blue and green traces show the position and momentum marginals respectively, normalized to the same maximum. Each plot uses |α| ≈ 1.9 and the color scale has been normalised to a maximum of unity.

We now show that this interferometer is a sensitive probe for quantum decoherence, which damps the interference fringes in P (x), and therefore destroys the signature of quantum coherence in p+. To model decoherence following the second grating operation, we consider weak thermalisation of the state, resulting in a decohered state (superscript d)

Z e−|β|2/n0 ρ(d) = d2β D(β)ρ D†(β) (6.13) m πn0 m where n0 is the number of thermal phonons eﬀectively added to the resonator, causing loss of quantum coherence. The loss of coherence is equivalent to convolving P (x) √ with a Gaussian of width n0, thereby exponentially damping oscillations of wave √ number |α| > n0 (Appendix 6.3.3). Figure 6.5 (g)-(i) plots the eﬀect of loss of coherence between the second and third gratings, assuming an initial ground state (g) and (h), and an initial (¯n = 5) thermal state (i). The decoherence is modelled 6.2. DISPLACEMON ELECTROMECHANICS 131

/2 1 (a) (b) (c)

0.5

0 0 0 1 2 0 1 2 0 1 2

Figure 6.6: The probability to find the qubit in its excited state, p+(α3), can be used to probe different wave number components in the position probability distribution P (x). As the resonator evolves (increasing ωmτ2), different wave numbers are present in P (x) (see Fig. 6.5). The probability p+(α3) is plotted for the resonator initially in (a) a pure state, (b) a thermal state ofn ¯ = 5 phonons, and (c) a pure state, but with decoherence (represented by n0 added phonons) after the second grating. only after the second grating, so that without thermalisation plots 6.5(g)-(h) would coincide with 6.5(c), and 6.5(i) would coincide with 6.5(f). The plots show that even the addition of a fraction of a phonon drastically suppresses the interference pattern in P (x) and the corresponding signature in p+. Finally in Fig. 6.6 we show explicitly how these two effects - thermal occupation before the inteferometry, and decoherence during the interferometry - degrade the quantum signatures in p+. In the ideal case (Fig. 6.6(a)), there are several values of

α3 at each time τ2 that give non-trivial probabilities of p+. In contrast, beginning in a thermal state withn ¯ = 5 phonons (Fig. 6.6(b)), all fringes are washed out, with the exception of the shadow of the grating (at α /α = 1 and ω τ = 0, π/2) √ 3 2 m 2 and the Moirépattern (at α3/α2 = 2 and ωmτ2 = π/4). Loss of coherence during the interferometry (Fig. 6.6(c)) leads to a qualitatively different behavior, namely √ 0 damping of all features in p+, including the classical fringes, for |α3| > n . This mechanical interferometery scheme could thus be used as a specific probe of quantum decoherence during the mechanical evolution.

6.2.4 Conclusion

We have introduced the displacemon electromechanical system that provides sufficiently strong coupling to generate and detect quantum interference of a massive object containing a quarter of a million atoms. By considering device parameters based on current technology, our qubit-resonator displacemon can achieve effective ultrastrong coupling using a modulated coupling scheme. A similar scheme could also be applied to other kinds of solid-state qubits coupled to high-quality resonators, such as spin qubits coupled to nanotubes [289, 288], diamond defects coupled to cantilevers [305], or piezoelectric resonators coupled to superconduct- 6.3. CHAPTER APPENDICES 132 ing qubits [312, 313]. However, the parameters estimated for the proposed device of Fig. 6.1 may be particularly experimentally favourable for this implementation, because they imply that the coupling exceeds both the thermal decay rate of the resonator and the typical dephasing rate of a qubit (see Appendix 6.3.5). Importantly, our scheme does not rely on degeneracy between the qubit and the resonator [282], nor on qubit coherence over the lifetime of the mechanical superposition [314]. Using the effective ultrastrong coupling we have shown that it is possible to extend matter wave interferometry to nano-mechanical resonators, opening up a range of new devices that can be used to study quantum physics at the meso-scale. Fur- thermore, interferometry performed on the wavefunction of a mechanically bound resonator is qualitatively distinct from existing free-particle interferometry techniques. An important advantage of nanomechanical resonators for quantum tests is that they can readily be extended to probe much larger masses than can be ac- cessed in molecular vapours or even levitated nanoparticles. Although the nanotube resonator considered here does not have enough mass to seriously challenge the interesting parameter regime of extrinsic collapse theories, a similar protocol could be extended to more massive objects still well within the range of nanomechanics. This research direction could allow for testing specific theories of quantum collapse [46], as an alternative to proposals based on single-photon optomechanics [131], or levitated nanoparticles [279, 314]. Finally, multiple resonators coupled to the same qubit (such as the pair of nanotube junctions in Fig. 6.1) could allow for implementation of entanglement generation between massive objects, leading to Bell tests of mechanical resonators [315].

6.2.5 Acknowledgements

We thank P. J. Leek and E. M. Gauger for discussions. We acknowledge FQXi, EPSRC (EP/N014995/1), and the Royal Academy of Engineering. KK would like to thank the Department of Materials, University of Oxford for their hospitality during the initial stages of this work.

6.3 Chapter appendices

6.3.1 Device parameters

We assume device parameters based on a mixture of experiment and simulation as follows. We take the parameters of the nanotube and the junctions from the nanotube SQUID device of Ref. [184]. For a resonator with length l = 800 nm, the frequency was measured as ωm/2π = 125 MHz, which with estimated diameter −21 6 D = 2.5 nm and mass m = 5×10 kg = 3×10 amu leads to XZP = 3.7 pm, typical 6.3. CHAPTER APPENDICES 133

of nanotube resonators. In each SQUID junction, a critical current Ic ≈ 12 nA was 0 achieved, implying EJ /h = 12 GHz. For the qubit, the charging energy is set by the electrode geometry, which is a design choice. Finite-element capacitance simulation for the device of Fig. 6.1(b), with qubit diameter 340 µm, gives EC/h = 0.2 GHz, typical of qubit devices and well into the transmon regime E0 E [316]. The maximal qubit frequency is then √ J C 0 ωq/2π = 8EJEC/h = 4.4 GHz, and the calculated qubit energy levels are shown in Fig. 6.1(c).

For the in-plane magnetic field we assume B|| = 0.5 T, which nanotube SQUIDs can withstand [184]. The operating flux point should then be chosen to maximise λ, while still maintaining a qubit frequency compatible with microwave resonators. We assume flux bias ∆Φ/Φ0 = −0.84, leading to a qubit frequency ωq = 2π×2.19 GHz = 0 ωq/2 (dashed vertical line in Fig. 6.1(c-d)). With symmetric junctions, and assuming that the restoring force on the nanotube is dominated by tension, the coupling is then λ/2π = 8.5 MHz. Since the coupling can be reduced by tuning ∆Φ towards zero, we take this as the maximum coupling strength λ0. In a realistic device, we must take account of asymmetry between the junctions.

Denoting the two critical currents by Ic1 and Ic2, with δ ≡ 2(Ic1−Ic2)/(Ic1+Ic2) being q 0 2 δ2 2 the asymmetry parameter, we have EJ = EJ cos (π∆Φ/2Φ0) + 4 sin (π∆Φ/2Φ0) with a corresponding modiﬁcation to Eq. (6.4) [82, 300]. This asymmetry leads to a small reduction in λ0 (Fig. 6.1(c)-(d)).

For these parameters, the device would be in the strong coupling regime (λ0 > kBT/~Qm, 1/T2) for comparatively modest resonator quality factor Qm & 15 and T2 & 120 ns. To access the ultrastrong coupling regime (λ0 > ωm) is more challenging, but may be possible [300]: if the suspended length could be increased to l ≈ 1.6 µm and the tension reduced to zero while keeping other parameters unchanged, the coupling would be λ0/2π ≈ ωm/2π ≈ 25 MHz. However, in the simulations we do not make this assumption, but instead assume that eﬀective ultrastrong coupling is engineered by toggling λ(t) as in Eq. (6.10).

6.3.2 Cooling via qubit measurement

† Here we show that the σz(a + a ) interaction Hamiltonian can be used to conditionally cool the resonator. Consider the qubit-resonator interaction in the interaction picture, with a (real) coherent qubit-drive g(t) with zero detuning to the qubit frequency,

λ(t) H / = σ (a + a†) + g(t)σ . (6.14) I ~ 2 z x 6.3. CHAPTER APPENDICES 134

The cooling protocol works as follows: first the qubit is prepared in the |−i eigenstate, then the interaction is switched on λ(t) → λ0, and remains time independent for the duration of the π-pulse g(t). At the completion of the π-pulse, the interaction is switched off and the qubit projectively measured in the σz-basis. The π-pulse cooling can be intuitively understood. If the resonator remains at a fixed position over the duration of the π-pulse (as it must in the interaction picture † since a + a commutes with the Hamiltonian) then the σz term in Eq. (6.14) looks like a detuning of the π-pulse drive from the qubit frequency. The probability of a qubit transition from the ground to excited state under a π-pulse depends on the detuning, and is maximised for zero detuning. Therefore, if the resonator is close to its equilibrium position, then there is a high probability of the qubit transitioning to the |+i state. However, if the resonator is far displaced from equilibrium (i.e. in a high potential energy state) then the qubit has a low transition probability. In the following we describe how the cooling envelope in Fig. 6.2 is obtained. Consider the state of a pure qubit-resonator system,

Z 0 0 0 0 0 |ψ(t)i = c−(x , t) |−, x i + c+(x ) |+, x , ti dx (6.15)

0 where c−(x , t) is the probability amplitude of ﬁnding the qubit in the |−i state, and the resonator at x0. The Hamiltonian Eq. (6.14) generates the equations of motion,

c˙+(x, t) = −iλ0xcˆ+(x, t) − ig(t)ˆc−(x, t) (6.16)

c˙−(x, t) = iλ0xcˆ−(x, t) − ig(t)ˆc+(x, t). (6.17)

The conditional state of the resonator after the cooling measurement (with out- R 0 0 0 come |+i) is therefore |ψim ∝ dx c+(x , tf ) |xi where c+(x , tf ) is the solution to Eq. (6.16) and (6.17) after the application of the π-pulse, with initial conditions c−(x, 0) = hx|ψ(0)im, and c+(x, 0) = 0 (i.e. the qubit is initially in the |−i state). The ∝ is used since the ﬁnal state is not normalized. This generalises to an initial mixed state of the resonator by

X ρ(0) = |−i h−| ⊗ pi |ψiim hψi|m → i Z X 0 00 0 0 00 ∗ 00 ρm(tf ) ∝ pi dx dx c+,i(x ) |x i hx | c+,i(x ), i (6.18)

where pi is the classical probability of finding the quantum system in the state |ψiim. The calculation can be greatly simplified by noting that the system of differential equations governing c± is linear, and therefore the solution for the initial conditions 0 0 0 c− = 1, c+ = 0 can be used for any c±,i(x , tf ) via c±,i(x , tf ) = hx |ψ(0)i c±(tf ). 6.3. CHAPTER APPENDICES 135

The cooling envelope in Fig. 6.2 is obtained by numerically solving, Eq. (6.16) and (6.17) with g(t) a Gaussian envelope π-pulse, and then calculating the new position marginal,

X 2 hx| ρm(tf ) |xi ∝ pi|c+,i(x)| , (6.19) i where ρm(0) is a thermal state.

6.3.3 Decoherence

Here we model the effect of decoherence on the interference. For our chosen parameters, the effect is estimated to be weak, because the interaction time τλ ≈ 130 ns is short compared with other timescales. For a superconducting qubit, the decoherence time is typically T2 > 1 µs τλ, so we may neglect qubit dephasing. For the resonator, the high quality factor Qm suppresses thermal decoherence; assuming 5 −5 Qm = 10 , there aren/Q ¯ m ≈ 5 × 10 phonons exchanged with the thermal environment every resonator period, or 1 phonon exchanged every ∼ 103 realisations of the interaction. Below, we model the effect of decoherence in detail.

Qubit dephasing

We model dephasing by adding a stochastic frequency shift to the qubit, changing the Hamiltonian (Eq. (6.5)) to

1 1 q H = ω a†a + ω σ + λ(t)(a + a†)σ + γ/2 ξ(t)σ , (6.20) ~ m 2~ q z 2~ z ~ z where γ is the qubit dephasing rate and ξ(t) is a delta-correlated white noise term satisfying E(ξ(t)) = 0 and E(ξ(t)ξ(t0)) = δ(t − t0). Here E(·) denotes an average over realisations of this stochastic process. Moving into the interaction picture, the unitary generated by this Hamiltonian is √ U(t) = e−iW (t) γ/2σz D(α)|−ih−| + D†(α)|+ih+| , (6.21) where W (t) = R t ξ(t0)dt0 is a stochastic variable with a mean of zero and a variance t0 of t. Since there is classical uncertainty in the realisation of W (t), the joint state of the resonator-qubit system will be mixed. Due to this classical uncertainty, the measurement operator cannot be understood as mapping pure states to pure states as assumed in Eq. (6.8). We must therefore consider the measurement procedure (used to impose the grating) in the density matrix description. Before switching on the interaction, i.e while λ(t) = 0, the π/2 pulse changes the |+i h+| state of the qubit to ρq = 6.3. CHAPTER APPENDICES 136

1 4 (|+i + |−i)(h+| + h−|). The state of the mechanical resonator is left unchanged in an arbitrary state ρm. This joint state must be separable, because the initialization of the qubit state at the beginning of the grating operation has the eﬀect of destroying any qubit-resonator entanglement. As the interaction is switched on, the joint state of the system evolves according to

† ρ(t, W (t)) = U(t)ρq ⊗ ρmU (t) 1 √ = e2i γ/2W (t) |−i h+| ⊗ D(α)ρ D(α) 2 m 1 √ + e−2i γ/2W (t) |+i h−| ⊗ D†(α)ρ D†(α) 2 m 1 + |+i h+| ⊗ D†(α)ρ D(α) 2 m 1 + |−i h−| ⊗ D(α)ρ D†(α), (6.22) 2 m where U(t) is the unitary operator in Eq. (6.21). Since W (t) is unknown, the resulting quantum state at time t must be weighted by the probability of obtaining √ a particular realisation of W (t), where P (W (t)) = exp[−W 2(t)/2t]/ 2πt,

Z ∞ ρ(t) = ρ(t, W (t))P (W (t))dW (t). (6.23) −∞ √ Projecting the qubit onto the state (|+i + e−iφ |−i)/ 2 gives the unnormalised conditional state of the mechanical resonator 1 ρ ∼ (h+| ± eiφ h−|)ρ(t)(|+i ± e−iφ |−i) (6.24) m,± 2 1 h ∼ D†(α)ρ D(α) + D(α)ρ D†(α) 4 m m −γt −iφ † † iφ i ±e e D (α)ρmD (α) + e D(α)ρmD(α) (6.25) where we have used ∼ because the right hand side is unnormalised. Separating this into coherent and incoherent terms, we ﬁnd

−γt e † iφ −iφ † ρm,± ∼ [D (α) ± e D(α)]ρm[D(α) ± e D (α)] 4 (6.26) 1 − e−γt h i + D†(α)ρ D(α) + D(α)ρ D†(α) . 4 m m

† We notice that the ﬁrst term is proportional to Υ±ρmΥ± where Υ± is given in

Eq. (6.8) (with ωq = 0). The first term in Eq. (6.26) is exactly the state that one 6.3. CHAPTER APPENDICES 137 would expect if the grating protocol worked perfectly, while the second term is an incoherent mixture of displacements. We can therefore understand qubit dephasing as some classical probability that the resonator coherently passed the grating, and some probability that we ended up with an incoherent mixture. Since the normalisation is state dependent, we cannot simply relate the coefficients in Eq. (6.26) with direct probabilities. However we can say the relative probability of introducing an incoherent mixture is (1−e−γt)/e−γt = eγt −1 ≈ γt for short times, or low dephasing rate. The trace of ρm,± is the probability of finding the qubit in the |±i state and using Eq. (6.25) we may read off,

1 e−γt p (α, φ) = ± [eiφχ(−2α) + e−iφχ(2α)], (6.27) ± 2 4 where χ(·) = Tr[D(·)ρ] is the characteristic function of the mechanical state (using Tr[D(α)ρD(α)] = Tr[D(α)D(α)ρ] = Tr[D(2α)ρ], etc.). The characteristic function is related to the Wigner distribution via a symplectic Fourier transform,

Z χ(α) = dxdpW (x, p)eixαi−ipαr (6.28)

where αr(i) is the real (imaginary) part of α. If Re(α) = 0, then

Z 2ixαi χ(2αi) = dxdpW (x, p)e Z = dxP (x)[cos(2αix) + i sin(2αix)], (6.29) where the complex part must vanish as χ(·) is a real function (for states with π rotational symmetry). This is simply the overlap integral between the position probability distribution P (x) and a diﬀraction grating with a pitch |α|. Therefore

1 e−γt R dxP (x) cos(2x|α|) p (α, φ) = ± cos(φ), (6.30) ± 2 2 which is exactly probing the 2|α| wave number components in P (x), with a reduced amplitude from the qubit dephasing. Thus we have seen the effect of qubit dephasing is to introduce some probability of having an incoherent mixture of different momentum kicks, thus suppressing any signatures of interference in the outcomes of qubit measurements. Since the duration of the protocol is on the order ∼ N mechanical oscillations, t ≈ 2π×N/ωm, to neglect qubit dephasing requires γ/ωm 1/N. For the parameters discussed in the main text, this requires γ/2π < 1 MHz. 6.3. CHAPTER APPENDICES 138 6 1 6 6 4 6 6 15 − − − − − 10 10 10 10 10 10 10 10 × × × × × × 7 2 1 3 7 2 6 1 7 6 2 5 5 3 2 17 − − − − − 10 10 10 10 10 10 10 10 × × × × × × × × 4 1 23 2 8 6 3 2 . . 1 1 6 4 1 5 1 6 1 2 12 − − − − 10 10 10 10 10 10 10 10 10 × × × × × × × × × 6 0 2 . 7 2 3 1 6 1 . 1 7 2 4 7 5 9 6 8 7 10 . − − − − 1 10 10 10 10 10 10 10 10 × × × × × × × × 3 1 3 19 2 4 3 2 . 9 4 3 1 6 1 6 1 2 15 − − − 10 10 10 10 10 10 10 10 10 × × × × [305] [317] [186, 318] [186, 305] [186, 176] × × 2 2 1 7 1 2 1 4 2 2 3 4 4 15 10 − − − − † 10 10 10 10 10 10 10 10 10 × × × × × × × × 6 3 0 × . . [305] 7 6 2 1 8 . 1 - 2 - 7 4 4 3 9 5 1 − − − † 10 10 10 10 10 10 × × × × 2 8 1 2 6 6 2 5 6 3 8 2 9 2 3 − − − 10 10 10 10 10 10 10 10 × × × × × × × 5 1 . 25 2 3 7 . . th (Hz) 8 m π λ/κ π 2 2 λ/ω λ/ ∗ λ/ 2 (Hz) 1 T m π (K) 3 2 Q / T m ω (amu) 3 ∗ 2 T m Coupling constant Qubit Temperature Quality factor Qubit parameter Mass Mechanical parameter SystemReferenceParameters Frequency This work 1 [281] 2Dimensionless figures of merit 3Coupling parameter 4 5 6 7 8 6.3. CHAPTER APPENDICES 139

Table 6.1: Parameters for a selection of implemented or proposed qubit-resonator coupled systems for carrying out the experiment in the main text. Systems are: (1) This proposed device; (2) Proposed levitated NV center in a magnetic field gradient (3) Implemented magnetic cantilever coupled to an NV center; (4) Pro- posed optimisation of device in (3); (5) Implemented NV center strain-coupled to diamond cantilever; (6) Proposed combination of the Al beam from [318] with the qubit of [186], in a field of 10 mT; (7) Proposed combination of the SiN membrane from [176] with the qubit of [186], in a field of 10 mT. The membrane is taken as supporting one arm of the qubit with length l = 300 µm. (8) Proposed combination of the optimized cantilever from [305] with the qubit of [186]. The field from the cantilever is taken as coupling to an enclosed area of 0.1 µm−2. None of these systems enters the bare ultrastrong coupling regime where λ/ωm > 1. However, it is possible to enter the toggled ultrastrong coupling regime where λ exceeds both the qubit and mechanical dephasing rates. In some of these works, marked †, the ∗ dephasing time T2 is not directly reported and we have therefore optimistically used the decoherence time T2 instead.

Loss of resonator coherence

To see that oscillations in p+(|α|, φ) are a quantum eﬀect, we consider the eﬀect of adding n0 thermal phonons to the state of the resonator immediately before the third grating (Eq. (6.13), restated here for convenience),

Z e−|β|2/n0 ρ(d) = d2β D[β]ρ D†[β]. (6.31) m πn0 m In this case

† (d) p+(|α|, φ) = Trm[Υ+Υ+ρm ] Z e−|β|2/n0 = d2β Tr [D†[β]Υ† Υ D[β]ρ ] πn0 m + + m 2 0 Z −βr /n ! 0 e 0 2 0 φ = dx dβr √ P (x ) cos |α|(x + 2βr) + πn0 2 1 e−4n0|α|2 R dxP (x) cos(2x|α|) = + cos(φ), 2 2 (6.32)

where Trm denotes a trace over the mechanical degrees of freedom. We therefore see that any loss of coherence between the second and third grating reduces the −4n0|α2| amplitude of the oscillations in p+ by a factor e . 6.3. CHAPTER APPENDICES 140

/2 1 (a) (b) (c)

0.5

0 0 0 1 2 0 1 2 0 1 2

Figure 6.7: Probability p+ of ﬁnding the qubit in the excited state if the resonator were described by a classical probability distribution using width σ = 0.5, 1, 5 from (a) to (c).

6.3.4 Classical interference patterns

To verify that oscillations are in fact due to quantum interference, we will consider what types of interference patterns can be understood using a classically description of the resonator. Consider a classical (superscript cl) checkerboard phase space probability density of dimensionless variables (x, p),

1 " x2 + p2 # P cl(x, p) = exp − cos2 (αx) cos2 (αp) , (6.33) N 2σ2 where N is a normalization factor and σ and α are the width and wave number of the checkerboard pattern respectively. This is the conditional state after the resonator has passed the ﬁrst two gratings (of α = α1 = α2) separated by a quarter period rotation. As the distribution rotates in phase space, we are interested in the wave number components present in the reduced probability of P cl(x) where

Z P cl(x) = P cl(x cos θ + p sin θ, p cos θ − x sin θ)dp. (6.34) and θ is the phase space rotation angle. If this probability distribution is now probed by a third grating, then the probability of ﬁnding the qubit in the |+i state is (from Eq. (6.12))

Z φ! pcl(α , φ) = dx0 cos2 |α |x0 + P cl(x0) + 3 3 2 2 2 2 ∝ cos(φ)e−2σ (|α3| +2|α3||α|(sin(θ)+cos(θ))+2|α| ) h 2 2 i × e8|α3||α|σ cos(θ) + 2e2|α|σ (2|α3| cos(θ)+|α|) + 1

h 2 2 i × e8|α3||α|σ sin(θ) + 2e2|α|σ (2|α3| sin(θ)+|α|) + 1 (6.35) √ 1 1 1 the amplitude of which peaks at {ωmτ2, |α3/α|} = { 2 nπ, 1} and {( 2 n + 4 )π, 2} for σ ≥ |α|. A plot of this function (Fig. 6.7) looks qualitatively the same as Fig. 6.6(b), with the difference being attributable to the superposition of momentum kicks that 6.3. CHAPTER APPENDICES 141 accompanies the measurement when the resonator is quantized. This confirms that the probability of finding the resonator in the |+i state can therefore be used to distinguish quantum interference patterns (Fig. 6.6(a) and(c)) from classical Moiré patterns that arise from a classical probability distribution (Fig. 6.7).

6.3.5 Other device implementations

To assess the experimental feasibility of our scheme, Table 6.1 presents parameters of the resonator, the qubit, and the coupling strength for various devices that could be used to implement it. The challenge is to achieve ultrastrong coupling between qubit and resonator without introducing either rapid dephasing of the qubit or thermal decoherence of the resonator. Assuming a toggled coupling, this requires that ∗ the coupling constant λ/2π exceeds both the qubit dephasing rate 1/T2 and the resonator thermal dephasing rate κth, as tabulated in the last two rows of the table. No existing device achieves this, although an optimized magnetic cantilever coupled to an NV center in diamond would be promising. Thus the device of Fig.6.1 is particularly attractive for investigating mesoscopic quantum interference in nanomechanics. Chapter 7

Conclusion

In this thesis we have derived new physical consequences from the model of classical channel gravity, as well as new experimental techniques that may eventually be used to test such consequences. After a simple introduction to quantum optomechanics, feedback control, and FRW cosmology, we gave a short literature review of optomechanical systems and gravitational collapse models. The following summarizes all original work presented in this thesis, and ties together the preceding ﬁve chapters.

7.1 Classical channel gravity

Although highly speculative, the semiclassical model introduced in chapter 2 allows quantum systems to dynamically interact with a classical space time while pre- serving our conventional interpretation of the wave function. Furthermore we have linked the CCG model to existing models of wave function collapse by invoking a “measurement” (i.e. collapse) and “feedback” (i.e. gravitational systematic effects) framework. We showed that there is a parameter independent, minimum collapse rate for bipartite systems in contrast with the many other wave function collapse models. However, for isolated single particles, the decoherence rate is equivalent to the GRW and CSL models. Following the introduction of the model, we then analyzed the first relativistic description for a classical channel model of gravity (chapter 3) showing that the violation of energy conservation may be understood as a matter source in an otherwise empty space time. We compared our cosmological description of CCG with unitary quantum mechanics, the Wheeler-de Witt equation, and previous models of Newtonian CCG, highlighting the physical and conceptual difference between the approaches. Understanding the violation of energy conservation as a spurious cosmological energy density hints that energy nonconservation from collapse models may be less

142 7.2. OPTO AND ELECTROMECHANICS 143 problematic when metric degrees of freedom are taken into account. This statement is left unproven in this thesis, and clearly requires further research to substantiate. However, the fact that metric degrees of freedom can contribute negative energy, (e.g. see Eq. (3.4)) to the overall Hamiltonian, and the results in chapter 3, are highly suggestive. If such such a theory if developed further, it has the potential to explain the emergence of classicality, non-baryonic energy density, allow quantum degrees of freedom to source classical space time curvature, and be verified via table top or space based experiments. However, each of these claims are very much speculative and a significant amount of further work is needed so support these statements. In chapter 4 we showed that quantum clocks are nonideal systems to detect gravitational decoherence in the energy basis. The decoherence rate from CCG scales with the Newtonian interaction rate, hence the low effective mass of the clock transition results in an undetectable decoherence rate, despite their extraordinary temporal precision. We also introduced, for the first time, the different possible information-flow possibilities (i.e quantum circuit diagrams, Fig. 4.1) for a CCG model. We then analyze how the dephasing rate scales with the number of constituent particles making the atomic clocks — i.e. size and dimensionality of atomic lattice clocks — for each of the possible quantum circuits, Table. 4.1. In each case the scaling in sub-linear in the number of clocks which is due to the 1/r scaling of the Newtonian potential. We also consider the dephasing due to the gravitational redshift under the CCG model. Even with the increased mass of the earth, and the precision of an atomic clock, the dephasing rate of the clock is still well below the detectable threshold of current (and even future) technology. Although the massive particle(s) sourcing the redshift greatly increases the total dephaing rate, the massive particle(s) decohere faster, leaving the clock relatively undisturbed. This result suggests that experiments involving massive quantum systems, such as those in opto/electromechanics, are likely to be more sensitive to gravitational decoherence.

7.2 Opto and Electromechanics

In chapters 5 and 6 we switched from directly analyzing the CCG model, and instead focus on techniques for measuring and manipulating opto/electromechanical oscillators mentioned above. In chapter 5 we developed a protocol to generate an effective φ XLXM coupling from a fixed XLXM coupling. This allows one to directly probe the momentum of a mechanical oscillator in a back-action evading measurement and opens the avenue for quantum experiments — e.g. squeezing, tomography, and quantum state swaps — even if the mechanical oscillator does not remain coherent for a single period of oscillation. This is typically the regime for low frequency, high 7.2. OPTO AND ELECTROMECHANICS 144 mass mechanical oscillators, which would have a larger decoherence rate from the CCG model. The results in chapter 5 can be viewed as an extension of speed meters; proposed devices that directly measure the relative velocity of two arms in an interferometer. Speed meters explicitly operate in the free mass approximation, however, by using a coherent control pulse to displace the optical state, we were able to correct for the finite mechanical phase space rotation due to the presence of the confining potential. Furthermore, by controlling the control pulse’s amplitude, we were able to show how one can engineer a back action evading interaction with an arbitrary mechanical quadrature. Finally in chapter 6 we introduced the “diaplacemon”; a new architecture that gives a nonlinear qubit-oscillator coupling and is capable of strong, or ultra strong coupling with current technology. Our proposed device combines a state of the art gradient transmon qubit, and a carbon nanotube mechanical oscillator. We showed how a series of manipulations and measurements of the qubit can be understood as defining an effective diffraction grating for the mechanical oscillator. A grating pitch of picometers was shown to be achievable, which, being far below the optical diffraction limit and Bohr radius, would be practically impossible with any other technology. The displacemon interferometer allows one to perform an interference experiment with the wave function of a bound harmonic oscillator. Aside from being qualitatively different to interference of free particles, the nanotube has an effective mass many orders of magnitude larger than current molecular interferometry setups. Hence a displacemon interferometer experiment would probe the superposition principle of quantum mechanics at new mass scales. Gravitational decoherence of the nanotube is unlikely to be detectable, however, one could envisage building a device with a larger mechanical oscillator (e.g. a cantilever or double clamped beam) that may seriously probe collapse models. The larger mass device would be more susceptible to gravitational decoherence, which would manifest as a loss of visibility in the interference fringes. A large mass displacemon device seems beyond current technology, however, as interference is severely susceptible to decoherence such a device would be an ideal system with which to propose a protocol to detect gravitational decoherence e.g. from the CCG models. Using the strong quibt-oscillator coupling from the displacemon, we also introduced a protocol (chapter 2.3) to engineer an arbitrary quantum state of the mechanical oscillator. The technique uses the non-linear interaction to generate the quantum state, requiring only a classical drive tone applied to the qubit. This protocol is maximally efficient for measurement based quantum state generation, with the probability of success determined by the overlap between the initial and target 7.2. OPTO AND ELECTROMECHANICS 145 states. Bibliography

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