A Guide to Experiments in Quantum Optics, 3rd edition

A Guide to Experiments in Quantum Optics

A Guide to Experiments in Quantum Optics Hans-A. Bachor and Timothy C. Ralph

Third Edition

Authors Hans-A. Bachor

The Australian Nat. University Research School of Physics Science Road Canberra ACT 0200 Australia

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Dr. Timothy C. Ralph

University of Queensland Department of Physics St. Lucia QLD 4072 Australia

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The artwork was prepared by B. Hage and is based on an experimental demonstration of photon number states published in: Chrzanowski H.M., Assad S.M., Bernu J., and Hage B. et al. (2013). Reconstruction of Photon Number Conditioned States Using Phase Randomized Homodyne Measurements, J. Phys. B At. Mol. Opt. Phys. 46: 104009.

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Contents Preface xv Acknowledgments xix 1 1.1 1.2 1.3 1.4 1.5

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2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.2.1 2.2.3 2.3 2.3.1 2.3.1.1 2.3.2 2.3.3 2.3.3.1 2.4 2.5

Introduction 1

Optics in Modern Life 1 The Origin and Progress of Quantum Optics 3 Motivation Through Simple and Direct Teaching Experiments 7 Consequences of Photon Correlations 12 How to Use This Guide 14 References 16 Classical Models of Light 19 Classical Waves 20 Mathematical Description of Waves 20 The Gaussian Beam 21 Quadrature Amplitudes 24 Field Energy, Intensity, and Power 25 A Classical Mode of Light 26 Light Carries Information 28 Modulations 30 Optical Modes and Degrees of Freedom 32 Lasers with Single and Multiple Modes 32 Polarization 33 Poincaré Sphere and Stokes Vectors 35 Multimode Systems 36 Statistical Properties of Classical Light 37 The Origin of Fluctuations 37 Gaussian Noise Approximation 38 Noise Spectra 39 Coherence 40 Correlation Functions 44 An Example: Light from a Chaotic Source as the Idealized Classical Case 46 Spatial Information and Imaging 50

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2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.5.8 2.5.9 2.5.10 2.6

State-of-the-Art Imaging 50 Classical Imaging 52 Image Detection 55 Scanning 56 Quantifying Noise and Contrast 58 Coincidence Imaging 59 Imaging with Coherent Light 60 Image Reconstruction with Structured Illumination 60 Image Analysis and Modes 61 Detection Modes and Displacement 61 Summary 62 References 63 Further Reading 64

3

65 Detecting Light 65 The Concept of Photons 68 Light from a Thermal Source 70 Interference Experiments 73 Modelling Single-Photon Experiments 78 Polarization of a Single Photon 79 Some Mathematics 80 Polarization States 81 The Single-Photon Interferometer 83 Intensity Correlation, Bunching, and Anti-bunching 84 Observing Photons in Cavities 88 Summary 90 References 90 Further Reading 92

3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.1.1 3.5.2 3.5.3 3.6 3.7 3.8

Photons: The Motivation to Go Beyond Classical Optics

4

Quantum Models of Light 93

4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.4.1

Quantization of Light 93 Some General Comments on Quantum Mechanics 93 Quantization of Cavity Modes 94 Quantized Energy 95 The Creation and Annihilation Operators 97 Quantum States of Light 97 Number or Fock States 97 Coherent States 99 Mixed States 101 Quantum Optical Representations 102 Quadrature Amplitude Operators 102 Probability and Quasi-probability Distributions 104 Photon Number Distributions 108 Covariance Matrix 111 Summary of Different Representations of Quantum States and Quantum Noise 112

Contents

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 4.7.4

Propagation and Detection of Quantum Optical Fields 113 Quantum Optical Modes in Free Space 114 Propagation in Quantum Optics 115 Detection in Quantum Optics 117 An Example: The Beamsplitter 118 Quantum Transfer Functions 120 A Linearized Quantum Noise Description 121 An Example: The Propagating Coherent State 123 Real Laser Beams 123 The Transfer of Operators, Signals, and Noise 124 Sideband Modes as Quantum States 126 Another Example: A Coherent State Pulse Through a Frequency Filter 129 Transformation of the Covariance Matrix 130 Quantum Correlations 131 Photon Correlations 131 Quadrature Correlations 132 Two-Mode Covariance Matrix 133 Summary 134 The Photon Number Basis 134 Quadrature Representations 135 Quantum Operators 135 The Quantum Noise Limit 136 References 136 Further Reading 137

5

Basic Optical Components 139

5.1 5.1.1 5.1.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.5.1 5.1.6

Beamsplitters 140 Classical Description of a Beamsplitter 140 Polarization Properties of Beamsplitters 142 The Beamsplitter in the Quantum Operator Model 143 The Beamsplitter with Single Photons 144 The Beamsplitter and the Photon Statistics 146 The Beamsplitter with Coherent States 149 Transfer Function for a Beamsplitter 149 Comparison Between a Beamsplitter and a Classical Current Junction 151 The Beamsplitter as a Model of Loss 152 Interferometers 153 Classical Description of an Interferometer 154 Quantum Model of the Interferometer 155 The Single-Photon Interferometer 156 Transfer of Intensity Noise Through the Interferometer 156 Sensitivity Limit of an Interferometer 157 Effect of Mode Mismatch on an Interferometer 160 Optical Cavities 162 Classical Description of a Linear Cavity 164

5.1.7 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.3 5.3.1

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5.3.2 5.3.3 5.3.4 5.3.4.1 5.3.4.2 5.3.4.3 5.3.5 5.3.6 5.3.7 5.3.8 5.3.9 5.3.10 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.5.1 5.4.6 5.4.7 5.4.8

The Special Case of High Reflectivities 169 The Phase Response 170 Spatial Properties of Cavities 172 Mode Matching 172 Polarization 174 Tunable Mirrors 175 Equations of Motion for the Cavity Mode 175 The Quantum Equations of Motion for a Cavity 176 The Propagation of Fluctuations Through the Cavity 177 Single Photons Through a Cavity 180 Multimode Cavities 181 Engineering Beamsplitters, Interferometers, and Resonators 182 Other Optical Components 184 Lenses 184 Holograms and Metasurfaces 185 Crystals and Polarizers 187 Optical Fibres and Waveguides 188 Modulators 189 Phase and Amplitude Modulators 191 Spatial Light Modulators 193 Optical Noise Sources 195 Non-linear Processes 195 References 196

6

Lasers and Amplifiers 199

6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.4.1 6.1.4.2 6.1.4.3 6.1.4.4 6.1.4.5 6.1.5 6.1.6 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.3.1 6.3.4 6.4 6.4.1 6.4.2

The Laser Concept 199 Technical Specifications of a Laser 201 Rate Equations 203 Quantum Model of a Laser 207 Examples of Lasers 209 Classes of Lasers 209 Dye Lasers and Argon Ion Lasers 209 The CW Nd:YAG Laser 210 Diode Lasers 213 Limits of the Single-Mode Approximation in Diode Lasers Laser Phase Noise 214 Pulsed Lasers 215 Amplification of Optical Signals 215 Parametric Amplifiers and Oscillators 218 The Second-Order Non-linearity 219 Parametric Amplification 220 Optical Parametric Oscillator 221 Noise Spectrum of the Parametric Oscillator 222 Pair Production 223 Measurement-Based Amplifiers 224 Deterministic Measurement-Based Amplifiers 225 Heralded Measurement-Based Amplifiers 228

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6.5

Summary 230 References 231

7

Photon Generation and Detection 233

7.1 7.1.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.1.3 7.2.1.4 7.2.2 7.3 7.3.1 7.3.1.1 7.3.1.2 7.3.1.3 7.3.1.4 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.3 7.3.4 7.3.4.1 7.3.4.2 7.3.4.3 7.3.4.4 7.3.4.5 7.3.4.6 7.3.4.7 7.4

Photon Sources 236 Deterministic Photon Sources 239 Photon Detection 240 Detecting Individual Photons 240 Photochemical Detectors 241 Photoelectric Detectors 241 Photo-thermal Detectors 243 Multipixel and Imaging Devices 243 Recording Electrical Signals from Individual Photons 245 Generating, Detecting, and Analysing Photocurrents 247 Properties of Photocurrents 247 Beat Measurements 247 Intensity Noise and the Shot Noise Level 248 Quantum Efficiency 249 Photodetector Materials 250 Generating Photocurrents 251 Photodiodes and Detector Circuit 251 Amplifiers and Electronic Noise 252 Detector Saturation 254 Recording of Photocurrents 255 Spectral Analysis of Photocurrents 257 Digital Fourier Transform 257 Analogue Fourier Transform 258 From Optical Sidebands to the Current Spectrum 258 The Operation of an Electronic Spectrum Analyser 259 Detecting Signal and Noise Independently 260 The Decibel Scale 261 Adding Electronic AC Signals 262 Imaging with Photons 263 References 264 Further Reading 267

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Quantum Noise: Basic Measurements and Techniques 269

8.1 8.1.1 8.1.1.1 8.1.2 8.1.3 8.1.4 8.1.4.1 8.1.5 8.1.5.1 8.2

Detection and Calibration of Quantum Noise 269 Direct Detection and Calibration 269 White Light Calibration 273 Balanced Detection 273 Detection of Intensity Modulation and SNR 275 Homodyne Detection 275 The Homodyne Detector for Classical Waves 275 Heterodyne Detection 279 Measuring Other Properties 280 Intensity Noise 281

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8.2.1 8.3 8.3.1 8.3.2 8.3.2.1 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5

Laser Noise 281 The Intensity Noise Eater 282 Classical Intensity Control 282 Quantum Noise Control 285 Practical Consequences 289 Frequency Stabilization and Locking of Cavities 290 Pound–Drever–Hall Locking 292 Tilt Locking 293 The PID Controller 294 How to Mount a Mirror 295 The Extremes of Mirror Suspension: GW Detectors 296 Injection Locking 296 References 299

9

Squeezed Light 303

9.1 9.1.1 9.1.1.1 9.1.1.2 9.1.2 9.1.2.1 9.2 9.2.1 9.2.2 9.2.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.4.4.1 9.4.4.2 9.4.4.3 9.4.4.4 9.4.4.5 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5

The Concept of Squeezing 303 Tools for Squeezing: Two Simple Examples 303 The Kerr Effect 304 Four-Wave Mixing 307 Properties of Squeezed States 310 What Are the Uses of These Various Types of Squeezed Light? 312 Quantum Model of Squeezed States 314 The Formal Definition of a Squeezed State 314 The Generation of Squeezed States 317 Squeezing as Correlations Between Noise Sidebands 319 Detecting Squeezed Light 322 Detecting Amplitude Squeezed Light 322 Detecting Quadrature Squeezed Light 322 Using a Cavity to Measure Quadrature Squeezing 324 Summary of Different Representations of Squeezed States 325 Propagation of Squeezed Light 325 Early Demonstrations of Squeezed Light 330 Four Wave Mixing 330 Optical Parametric Processes 333 Second Harmonic Generation 339 The Kerr Effect 343 The Response of the Kerr Medium 343 Optimizing the Kerr Effect 345 Fibre Kerr Squeezing 346 Atomic Kerr Squeezing 348 Atomic Polarization Self-Rotation 349 Pulsed Squeezing 349 Quantum Noise of Optical Pulses 349 Pulsed Squeezing Experiments with Kerr Media 352 Pulsed SHG and OPO Experiments 353 Soliton Squeezing 354 Spectral Filtering 355

Contents

9.5.6 9.6 9.7 9.8 9.9 9.9.1 9.9.2 9.9.3 9.10

Non-linear Interferometers 356 Amplitude Squeezed Light from Diode Lasers 358 Quantum State Tomography 360 State of the Art of CW Squeezing 363 Squeezing of Multiple Modes 365 Twin-Photon Beams 365 Polarization Squeezing 367 Degenerate Multimode Squeezers 368 Summary: Quantum Limits and Enhancement 370 References 371 Further Reading 376

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Applications of Quantum Light 377

10.1 10.1.1 10.1.2 10.1.3 10.1.4 10.2 10.3 10.3.1 10.3.1.1 10.3.2 10.3.2.1 10.3.2.2 10.3.2.3 10.3.3 10.3.3.1 10.3.4 10.3.4.1 10.4 10.4.1 10.4.2 10.5 10.5.1 10.6

Quantum Enhanced Sensors 377 Coherent Sensors and Sensitivity Scaling 377 Practical Examples of Sensors 380 Ultimate Sensing Limits 382 Adaptive Phase Estimation 384 Optical Communication 384 Gravitational Wave Detection 389 The Origin and Properties of GW 389 Concept and Design of an Optical GW Detector 392 Quantum Properties of the Ideal Interferometer 393 Configurations of Interferometers 396 Recycling 397 Modulation Techniques 398 The Sensitivity of GW Observatories 400 Enhancement Below the SQL 402 Interferometry with Squeezed Light 405 Quantum Enhancement Beyond the SQL 410 Quantum Enhanced Imaging 411 Imaging with Photons on Demand 411 Quantum Enhanced Coincidence Imaging 412 Multimode Squeezing Enhancing Sensors 414 Spatial Multimode Squeezing 414 Summary and Outlook 419 References 419

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QND 425

11.1 11.2 11.3 11.4 11.4.1

QND Measurements of Quadrature Amplitudes 425 Classification of QND Measurements 427 Experimental Results 430 Single-Photon QND 432 Measurement-Based QND 434 References 437

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Fundamental Tests of Quantum Mechanics 441

12.1 12.2 12.3 12.3.1 12.3.2 12.3.2.1 12.3.2.2 12.3.3 12.3.3.1 12.3.3.2 12.4

Wave–Particle Duality 441 Indistinguishability 446 Non-locality 453 Einstein–Podolsky–Rosen Paradox 453 Characterization of Entangled Beams via Homodyne Detection 458 Logarithmic Negativity and Two-Mode Squeezing 459 Entanglement of Formation 460 Bell Inequalities 461 Long-Distance Bell Inequality Violations 466 Loophole-Free Bell Inequality Violations 466 Summary 468 References 468

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Quantum Information 473

13.1 13.1.1 13.2 13.3 13.3.1 13.3.2 13.4 13.5 13.5.1 13.5.2 13.5.3 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.7 13.7.1 13.7.1.1 13.7.1.2 13.7.1.3 13.7.2 13.7.2.1 13.7.2.2 13.7.3 13.7.3.1 13.7.3.2 13.7.4 13.8

Photons as Qubits 473 Other Quantum Encodings 475 Post-selection and Coincidence Counting 475 True Single-Photon Sources 477 Heralded Single Photons 477 Single Photons on Demand 480 Characterizing Photonic Qubits 482 Quantum Key Distribution 484 QKD Using Single Photons 485 QKD Using Continuous Variables 489 No Cloning 492 Teleportation 492 Teleportation of Photon Qubits 493 Continuous Variable Teleportation 495 Entanglement Swapping 502 Entanglement Distillation 502 Quantum Computation 505 Dual-Rail Quantum Computing 506 Quantum Circuits with Linear Optics 507 Cluster States 511 Quantum Gates with Non-linear Optics 513 Single-Rail Quantum Computation 514 Quantum Random Walks 515 Boson Sampling 516 Continuous Variable Quantum Computation 518 Cat State Quantum Computing 519 Continuous Variable Cluster States 521 Large-Scale Quantum Computation 522 Summary 525 References 526 Further Reading 531

Contents

14

The Future: From Q-demonstrations to Q-technologies 533

14.1 14.2 14.3 14.3.1 14.3.2 14.3.3 14.3.4

Demonstrating Quantum Effects 533 Matter Waves and Atoms 535 Q-Technology Based on Optics 537 Applications of Squeezed Light 537 Quantum Communication and Logic with Photons 539 Cavity QED 542 Extending to Other Wavelengths: Microwaves and Cryogenic Circuits 542 Quantum Optomechanics 542 Transfer of Quantum Information Between Different Physical Systems 543 Transferring and Storing Quantum States 544 Outlook 544 References 545 Further Reading 547

14.3.5 14.3.6 14.3.7 14.4

Appendices 549

Appendix A: List of Quantum Operators, States, and Functions 549 Appendix B: Calculation of the Quantum Properties of a Feedback Loop 551 Appendix C: Detection of Signal and Noise with an ESA 552 Reference 554 Appendix D: An Example of Analogue Processing of Photocurrents 554 Appendix E: Symbols and Abbreviations 556 Index 559

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Preface The Idea Behind This Guide Some of the most interesting and sometimes puzzling phenomena in optics are those where the quantum mechanical nature of light is apparent. Recent years have seen a rapid expansion of quantum optics. Beautiful demonstrations and applications of the quantum nature of light are now common, and optics has been shown to be one of the key areas of physics to build quantum technologies based on the concepts behind quantum mechanics. The field of photonics has already spawned major industries and changed the way we communicate, gather data, and handle information. Future applications of quantum information will enhance this trend and will introduce even more quantum aspects into our everyday activities. Why should we still write and read books at the age of search engines, blogs, and Wikipedia? This question did not arise when the first edition of this book was published in 1998, which is a short step back in history but a long time ago in terms of the changes in information technology and the way we live, learn, and do research. You are likely to read this on a screen and can follow up links within seconds, tapping into an amazing wealth of information. This calls even more on this Guide to Experiment in Quantum Optics to explain the concepts, introduce reliable techniques, and help to select from the enormous amount of information that is available. We intend to guide you through the many experiments that have been published and to present and interpret them in one common style. This book provides a practical basis for exploring the ever-expanding technology in optics and electronics and provides a practical link to the rapidly expanding field of quantum information, which is now being used in many different platforms. Finally, this book is an attempt to communicate our personal insights and experiences in this field of science. Science is driven by creative people, and some insight in the way researchers advance the field should find interest with you the reader. Many excellent textbooks are available on this topic. In the same way as this field of research has been initiated by theoretical ideas, most of these books are written from a theoretical point of view. They provide fascinating, solid, and reliable descriptions of the field. However, we found the literature frequently leaves out some of the important simpler pictures and intuitive interpretations of the experiments. In this guide we focus on both the first and the most recent

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experiments. The guide explains the underlying physics in the most intuitive way we can find. It addresses questions such as the following: What are the limitations of the equipment? What can be measured and what remains a goal for the future? The answers will prepare the readers to form their own mind map of quantum optics. It gives them the tools to make independent predictions of the outcome of their own experiments. For those who want to use quantum optics in practical ways, it tries to give an understanding of the design concepts of the machines that exploit the quantum nature of light. We assume that most of our readers will already have a fairly good understanding of optical phenomena and the way light propagates and how it interacts with components such as lenses, mirrors, detectors, etc. They can imagine the processes in action in an optical instrument. It is useful in the everyday work of a scientist to have practical pictures of what is going on inside the experiment or the machine. Such pictures have been shaped over many years of systematic study and through one’s own experimentation. Our guide provides such pictures, which can be filled in with many more details from theory and technology. In quantum optics three different pictures, or interpretations, are used simultaneously: light as waves, light as photons, and light as the solution to operator equations. To these we add the picture of fluctuations propagating through an optical system, a description using noise transfer functions. All of them are useful; all of them are correct within their limits. They are based on specific interpretations of the one formalism, and they are mathematically equivalent. In this guide all these four descriptions are introduced rigorously and are used to discuss a series of experiments. We start with simple demonstrations, gradually getting more complex and include most of the landmark experiments in single-photon and quantum noise detection, photon pair generation, squeezed states, quantum non-demolition, and entanglement measurements published to date. All four pictures – waves, photons, operators, and noise spectra – are used in parallel. In each case the most appropriate and intuitive interpretation is highlighted, and the limitations of all four interpretations are stated. In this way we hope to maintain and present a lot of the fascination of quantum optics that has motivated us and many of our colleagues, young and old, over so many years.

The Expansion of Quantum Optics and the Third Edition of This Guide The development of the field has been rapid in recent years, and many new technical advances have been reported: vastly improved sources for photon pairs, squeezed and entangled states. We have better detectors and nonlinear media. With the ability to record and process much more information the era of analogue technology is gone. Post-processing is now the norm, quantum gates and quantum memories have been reported. We can generate and process multiple optical modes in frequency, space and time, demonstrate multimode entanglement and design and build reliable complex optical systems.

Preface

The field of quantum information is growing rapidly, and new concepts for quantum communication, quantum logic, and quantum internet are emerging. Optics will play a major role in this. The field has broadened to include both more fundamental concepts about the quantum nature of information and new ideas for the engineering of complex optical designs. Measurements using non-classical light are now well established and have become a routine operation in some select fields such as gravitational-wave detectors and sensors. Entanglement has grown from an abstract concept to a technical tool. At the same time commercial companies using quantum techniques for secure communication have already carved out a large market for themselves. The ideas in this guide, and the examples chosen, draw attention to both the techniques of single-photon detection and single-event logic and the techniques of continuous variable logic and the use of continuous coherent and squeezed beams. There are now far more practitioners of the former, and the latter has become a specialized niche, mainly used for metrology. However, both will play a role in the development of future applications and in the understanding the concepts of the quantum world. In addition, quantum optics based on pairs of photons has been combined with the technology of strong laser beams and continuous variables in a series of exciting hybrid experiments. This third edition focusses on the latest landmark experiments, and examples that are now outdated have been deleted. The coverage of quantum information concepts and protocols has been expanded to keep pace with the evolution of the field. Reference is made to the state-of-the-art technologies, hinting at potential future developments. This guide follows the well-tested idea of providing independent building blocks in theory and experimental techniques that are used in later chapters to discuss complete experiments and applications. We found that most of them were still valid and thus remain unchanged. Optics is no longer dominant as the medium of choice for quantum technologies. Technical advances have allowed the same concepts now to be used in areas such as cryogenic electric circuits, superconducting systems, trapped ions and neutral atoms, Bose–Einstein condensates (BECs) and atom laser beams, and optomechanical systems, and other examples yet to be tested. This has posed a challenge: how to define the limits of this guide? To keep the book compact, we decided to focus entirely on optical systems and to point to links with other systems where required, but not to venture into the technical details of these fields. It is clear that future quantum technology will be using many complementary platforms. Optics will remain a major contributor and quite likely the most direct approach to the fascinating and sometimes counterintuitive effects of the quantum world. Australia, 2018

Hans Bachor and Timothy Ralph

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Acknowledgments This book would not have been possible without the teaching we received from many of the creative and motivating pioneers of the field of quantum optics. We have named them in their context throughout the book. HAB is particularly in debt to Marc Levenson, who challenged him first with ideas about quantum noise. From him he learned many of the tricks of the trade and the art of checking one's sanity when the experiment produces results that seem to be impossible. TCR wishes particularly to thank Craig Savage for instilling the importance of doing theory that is relevant to experimentalists. Both of us owe a great deal to Dan Walls and Gerard Milburn who provided such beautiful vision of what might be possible. Over the years we learned from and received help for this book from a large international team of colleagues and students. Too many to name individually. All of you made important contributions to the our understanding and the way we communicate our knowledge and sight. We enjoy the feedback and are keen to discuss the mysteries and the past and future of quantum optics. We like to thank Aska Dolinska for the creative graphics work. Most importantly this Third Edition would not have been possible without the support of our loved ones. TCR wishes to thank his wife Barbara for her support and son Isaac for putting up with a distracted father. And HAB wishes to thank his wife Cornelia for the patience she showed with an absent-minded and many times frustrated writer. Hans Bachor Timothy Ralph 2018

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1 Introduction 1.1 Optics in Modern Life The invention of the laser and the use of the special properties of laser light created one of the great technical revolutions of the twentieth century and has impact on many aspects of our modern lives. It allows us to manufacture the gadgets we love to use, including the computer with which this has been written. It allows us to communicate literally at the speed of light and with an information bandwidth that is unprecedented. Optics has been for a while the technology of choice for data storage, with DVD and Blu-ray still being used as a medium for data and 3D optical storage that might emerge in future. The modern alternatives, downloading and streaming, depend in many ways on photonics. In addition, there is a plethora of other applications of light, from laser welding in manufacturing to laser fusion for power generation and from laser-based microscopes to laser surgery. Imaging and display technologies are in many gadgets. The developments of optical technology have been relentless and will continue to be so for many more years. This has allowed us to access new levels of power, controllability, accuracy, complexity, and low cost that were deemed impossible only a few years ago. This is a convincing reason to study and teach quantum optics, as a scientist or engineer. Whilst all of the techniques in photonics use the quantum concept of stimulated emission and the laser [1], most of these applications still use quantum ideas in a quite simple and peripheral way. Most of the designs are based on conventional optics and can be understood on the basis of wave optics and traditional physical optics [2]. An interpretation of light as a classical wave is perfectly adequate for the understanding of effects such as diffraction, interference, or image formation. Non-linear optics, such as frequency doubling or wave mixing, is well described by classical theory. Even most of the properties of coherent laser light can be understood and modelled in this way. There is a wealth or new ideas and applications emerging, exploiting the regimes of wave optics that were previously not accessible, with new theories and devices. This includes the many fascinating new devices made possible through new engineered materials, waveguide structures, photonic bandgap materials, linear and non-linear metamaterials. They all hold the promise of even higher performance, better specifications, and even new applications, such as cloaking or 3D storage, optical data processing at unheard A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Introduction

speeds, or optical data processing. Conferences and research reviews all document that in 2018 a bright future is ahead in these areas of optics. However, which role will optics play in a future that will use quantum technologies? Firstly, the question of what ultimately limits the performance of optical devices lies outside the classical theory of light. We have to focus on a different aspect of optics. We follow the ideas that come directly from the quantum nature of light, the full properties of photons, and the subtle effects of quantum optics [3]. Initially, optics was recognized as a pretty ideal medium to study the finer effects of the quantum world. Photons in the visible spectrum were easily generated, detected with high efficiency, and not masked by thermal noise at room temperature, and any correlations could be recorded very well. Non-linear effects allowed the generation of a variety of different forms of light. This established optics as the medium of choice to explore the quantum world, and the early improvements in the technology in the 1980s led to great demonstrations, such as the Bell inequalities, signalling the beginning of experimental quantum optics. This exploration of the ‘weirdness’ of the quantum world is still today one of the frontiers in quantum optics and discussed in this guide. It was recognized that quantum effects pose a limit to the sensitivity of many measurements, with the now successful detectors for gravitational waves the pre-eminent example. A closer analysis showed that there are ways of, at least partially, circumventing such limitations. Optical quantum metrology was born and is a major topic in this guide. It will continue to evolve – and we now see the first routine applications of quantum-enhanced metrology, with few photons and with laser beams. The ideas are spreading and can be realized in other systems such as coherent groups of atoms or ions or in circuits. Quantum-enhanced metrology in its various forms is becoming more widespread, from the definition of the SI units to very practical situations. Optics technology and our knowledge of protocols and devices have steadily grown in time. Almost gone are the analogue devices of the 1990s, and we have direct access to digital data. For example, we can now detect, record, process, and store information in our experiments in much better ways, with A/D converters, multichannel analysers, memories of Terabit size, etc. This means that we can access much more information and process and analyse data more thoroughly. The next step might well include the use of artificial intelligence to optimize data analysis and also the performance of devices. The detection of quantum correlations are a case in hand: we can now directly explore the correlation properties, use different approaches and measures to evaluate the quality of entanglement, and post-select data, or use heralded information, to select the best quantum data. This provides a path to optical quantum gates and more elaborate quantum logic. We can post-process the information and make it conditional on other measurements. This allows us to not only describe the state with tomography in great detail but also provide a path for looking inside the statistical properties of a beam of light. An example is the Wigner function shown on the front cover, which shows strong negative values that are contained in one beam from an entangled pair of beam – after

1.2 The Origin and Progress of Quantum Optics

conditional detection and letting information from one beam influence the reading from the other entangled beam [4]. Furthermore, so-called hybrid detection uses entangled beams, where the information from a single photon detector in one beam is used to post-select the most relevant data from stream of photon recorded from the other beam. Using this form of hybrid detection, it is possible to filter out states of light with specific quantum properties, for example, Schrödinger cat states [5]. They represent a state that is a quantum superposition of two orthogonal states and is one of the icons and hallmarks of quantum physics, as recognized in the 2012 Nobel Prize of Physics [6]. On the other hand, the last few years has seen an ever-increasing interest in the information that is contained within a beam of light, whether it be images, analogue pulses, or digital data encoded in the light. The emergence of the concept of quantum information has created a rising interest in the ability to communicate, store, and process more complex information than available in classical digital electronics, all by using beams of light or individual photons. This has created a great deal of interest, an expanding research field, and a demand for practical quantum optics-based devices. Optics overlaps more and more with condensed matter and soft matter physics, materials science, and mechanical engineering. Ever-improving technology drives these fields, as well as now insights into quantum logic and quantum control. Optics is in many cases a precursor, sometimes a testing ground, and increasingly a component of such quantum devices.

1.2 The Origin and Progress of Quantum Optics Ever since the quantum interpretation of the black body radiation by M. Planck [7], the discovery of the photoelectric effect by W. Hallwachs [8] and P. Lennard [9] and its interpretation by A. Einstein [10] has the idea of photons been used to describe the origin of light. For the description of the generation of light, a quantum model is essential. The emission of light by atoms, and equally the absorption of light, requires the assumption that light of a certain wavelength 𝜆, or frequency 𝜈, is made up of discrete units of energy each with the same energy hc∕𝜆 = h𝜈. This concept is closely linked to the quantum theory of the atoms themselves. Atoms are best described as quantum systems; their properties can be derived from the wave functions of the atoms. The energy eigenstates of atoms lead directly to the spectra of light which they emit or absorb. The quantum theory of atoms and their interaction with light has been developed extensively; it led to many practical applications [11] and has played crucial role in the formulation of quantum mechanics itself. Spectroscopy relies almost completely on the quantum nature of atoms. However, in this guide we will not be concerned with the quantum theory of atoms, but concentrate on the properties of the light, its propagation, and its applications in optical measurements. Historically, special experiments with extremely low intensities and those that are based on detecting individual photons raised new questions, for instance, the famous interference experiments by G.I. Taylor [12] where the energy flux corresponds to the transit of individual photons. He repeated Th. Young’s double-slit

3

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1 Introduction

experiment [13] with typically less than one photon in the experiment at any one time. In this case, the classical explanation of interference based on electromagnetic waves and the quantum explanation based on the interference of probability amplitudes can be made to coincide by the simple expedient of making the probability of counting a photon proportional to the intensity of the classical field. The experiment cannot distinguish between the two explanations. Similarly, the modern versions of these experiments, using the technologies of low noise current detection or, alternatively, photon counting, show the identity of the two interpretations. The search for uniquely quantum optical effects continued through experiments concerned with intensity, rather than amplitude interferometry. The focus shifted to the measurement of fluctuations and the statistical analysis of light. Correlations between the arrival of photons were considered. It started with the famous experiment by R. Hanbury-Brown and R. Twiss [14] who studied the correlation between the fluctuations of the two photocurrents from two different detectors illuminated by the same light source. They observed with a thermal light source an enhancement in the two-time intensity correlation function for short time delays. This was a consequence of the large intensity fluctuations of the thermal light and was called photon bunching. This phenomenon can be adequately explained using a classical theory that includes a fluctuating electromagnetic field. Once laser light was available, it was found that a strong laser beam well above threshold shows no photon bunching; instead the light has a Poissonian counting statistics. One consequence is that laser beams cannot be perfectly quiet; they show noise in the intensity that is called shot noise, a name that reminds us of the arrival of many particles of light. This result can be derived from both classical and quantum models. Next, it was shown by R.J. Glauber [15] that additional, unique predictions could be made from his quantum formulation of optical coherence. One such prediction is photon anti-bunching, where the initial slope of the two-time correlation function is positive. This corresponds to greater than average separations between the arrival times of photons; the photon counting statistics may be sub-Poissonian, and the fluctuations of the resulting photocurrent would be smaller than the shot noise. It was shown that a classical theory based on fluctuating field amplitudes would require negative probabilities in order to predict anti-bunching, and this is clearly not a classical concept but a key feature of a quantum model. It was not until 1976, when H.J. Carmichael and D.F. Walls [16] predicted that light generated by resonance fluorescence from a two-level atom would exhibit anti-bunching, that a physically accessible system was identified that exhibits non-classical behaviour. This phenomenon was observed in experiments by H.J. Kimble et al. [17], where individual atoms were observed. This opened the era of quantum optics. It is now possible to track and directly display the evolution of the photon number inside a cavity, thanks to the innovation introduced by S. Haroche and coworkers. They have designed better and better superconducting cavities to store microwave photons and can carry out non-demolition experiments in a cavity with very long decay time. They can now directly observe the evolution of the photon number in an individual mode [18]. See

1.2 The Origin and Progress of Quantum Optics

for details Figure 3.7. This is a most beautiful demonstration of the theoretical concepts. The concept of modes will be a central feature in this guide. This concept has evolved in time; it now has several distinct meanings and is playing a pivotal role in quantum optics. For an experimentalist, a single mode can mean a perfect beam shape without any distortion. A single-mode laser means a device that is emitting a beam with exactly one frequency, in one polarization, and with a perfect beam shape. In the theory, a mode is the fundamental entity that can be quantized and be described by simple operators. We have a reliable and direct model for the behaviour of such a single mode, and up to recently most quantum optics were formulated in this single-mode model. At the same time, in optical technology, such as communication or imaging, the concept of a mode is associated with an individual channel of information. This could be a specific modulation frequency, as we are familiar with from radio transmission, or it could be a specific spatial image. In the world of optical sensing, we wish to detect and transmit changes of various parameters, such as lengths or position, magnetic or electric fields, pressure or temperature, to name a few. The evolution of these parameters are then encoded into the light and transmitted to the receiver. In this case, each parameter, and each degree of freedom for the change of the parameter, corresponds to one specific mode. It has now been recognized that a sensor can be optimized and will show the best possible performance, when the optical mode is fully matched to the information. This can be a simple mode, as mentioned in the case of the traditional single-mode laser, or it can actually be a much more complicated spatial and temporal distribution. The first extension in quantum optics has been to multiple beams of light, each containing one mode each. We can have many separate beams, all originating from one complex optical apparatus. However, a single optical beam can also contain multiple modes, for example, spatial modes, temporal modes, or frequency modes. They correspond to separate degrees of freedom or separate information channels within the beam. Such a multimode beam might contain only a few, or the occasional, photon, or it might have the full flux of a laser beam. In either case, the underlying physics is the same. The information transmitted is best described in terms of a set of independent orthogonal modes. Each can be quantized individually and has their own independent statistics. The signals, and the fluctuations, transmitted by these modes do not interact with each other, and the full quantum description of these devices is straightforward using this concept. There has been great progress in the technology to generate individual photons in a controlled way. The technology has evolved to a stage where many of the initial thought experiments dealing with individual atoms can now be performed with real machines. In addition, a whole range of new types of single photon emitters has been developed and studied by many groups around the world. They include quantum dots that show the required quantum properties and are continuously improving in their robustness and reliability. They are now being used regularly in a variety of applications where small size matters. This includes applications for quantum communication with quantum dots placed in specific places or in the form of nanodiamonds imbedded as emitters in biological samples that respond to the local conditions in living cells.

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Initially, these single photon effects were a pure curiosity. They showed the difference between the quantum and the classical world and were used to test quantum mechanics. In particular, the critique by A. Einstein et al. and their EPR paradox [19] required further investigation. Through the work of J.S. Bell [20], it became possible to carry out quantitative tests. They are all based on the quantum concept of entanglement. In the simplest case, this means pairs of quantum systems that have a common origin and have linked properties, such that the detection of one allows the perfect prediction of the other. Such systems include not only individual photons or atoms, or coherent states of light, such as laser beams but also physical matter, such as was demonstrated with Bose–Einstein condensates. See Chapter 14. Starting with the experiments by A. Aspect et al. [21] with atomic sources, and later with the development of sources of twin photon pairs and experiments by A. Zeilinger [22] and many others, the quantum predictions have been extremely well tested. The concept of entanglement has now been demonstrated and tested with single photons and also with intense beams of light, over large distances of many kilometres and through optical fibres. There has been a long series of experiments with the aim to demonstrate the validity and necessity of quantum physics beyond any doubt, closing step by step every loophole. It has even been possible to extend the concept of entanglement from two entities, such as two particles, to many particles, most prominently to many trapped ions. Such multipartite entanglement is at the core of plans to build quantum simulators and quantum logic devices. Similarly there has been an extension from two modes of light to several modes. This leads to the idea of multimode entanglement, which is a new topic of research, with both interesting physics and the potential for new applications. In summary, the progress in quantum optics, which initially was concerned with the quality of the quantum demonstrations, has recently led to the conquering of complexity and the study of the robustness, or decoherence, of such complex systems. The peculiarities of single photon quantum processes have led to many new technical applications, mainly in the area of communication. For example, it was found that information sent by single photons can be made secure against listening in; the unique quantum properties do not allow the copying of the information. Any eavesdropping will introduce noise and can therefore be detected. Quantum cryptography and quantum key distribution (QKD) [23] are now viable technologies, based on the inability to clone single photons or quantum modes. There is great anticipation of other future technological applications of photons. The superposition of different states of photons can be used to send more complex information, allowing the transfer not only of classical bits of information, such as 0 and 1, but also of qubits based on the superposition of states 0 and 1. The coherent evolution of several such quantum systems could lead to the development of quantum logic connections, quantum gates, and eventually whole quantum computers that promise to solve certain classes of mathematical problems much more efficiently and rapidly [24]. Just as astounding is the concept of teleportation [25], the disembodied communication of the complete quantum information from one place to another. This has already been tested in experiments (see Chapter 13).

1.3 Motivation Through Simple and Direct Teaching Experiments

Optics is now a testing ground for future quantum information technologies and for data communication and processing. At the same time, the concepts and ideas of quantum optics are emerging in other related quantum technologies, for example, based on quantum circuits or mechanical systems, which are all quantized and show in principle the same physics. We can see the equivalence of the physics of photons, phonons, polaritons, and other particles that describe the excitation of quantum systems. Technology is moving very fast, and we can expect over the next decade to see machines that utilize any of these individual quantum systems or even combine several of them. The rules of quantum optics, as described in this guide, will become important in many different other types of instruments. Optics could become one of the best ways to visualize the broader quantum concepts that apply to many other types of instruments.

1.3 Motivation Through Simple and Direct Teaching Experiments The best starting point for the demonstration of photons in current technology is to start with images that are generated by CCD (charge-coupled device) cameras. It is worthwhile to note that basically all information we have about light and optical effects is based on either some form of photodetector or the human eye. The first generates an electric signal, either pulses or a continuous photon current. Whilst the eye produces, in a clever and complex way, a signal in the neurones in the brain, even they can be interpreted as a form of electric signal. It is the characteristics of these signals and their statistics that provides access to the quantum properties of light. In the extreme case we would have a detector that is capable of detecting one individual photon at a time. This is now possible with special CCD cameras, where the sensitivity has been lifted to such a high level and thermal noise has been reduced so much that it hardly interferes with the observation. The number of dark counts, that is, clicks that are generated at random without photons present, can be neglected. In this way we can actually detect and record images at the single photon level as shown in Figure 1.1. The human eye is actually capable of seeing at this level, and many animals exceed our capability. At this level of illumination, the world looks very grainy. We can see individual events. The image is composed out of individual local detection events. An example, taken with a special CCD, is shown in Figure 1.1. At some level this can be regarded as trivial – obviously it will be like this. We have photons. On the other this image demonstrates several subtle points. We can see that the concept of intensity, bright and dark, is directly related to the concept of probability. Areas with a high local photon flux, or high probability, correspond to high intensities, that is, a high amplitude of the electromagnetic field. Conversely, dark areas of the image correspond to a low intensity. The central concept in quantum science of a probability amplitude become directly obvious. Two, or many, waves with different amplitudes and relative phases can interfere, and this is a central concept of the classical wave picture of light. The example

7

1 Introduction

(a) 300 Number of photons

8

250 200 150 100 50 0 0

(b)

50

100

150

200

Pixel

Figure 1.1 Image of the fringes from a double-slit experiment recorded at the single photon level. (a) Distribution of individual photons. (b) Histogram of photons in each vertical line. The full movie can be seen at https://www.lcf.institutoptique.fr/Groupes-de-recherche/Gazquantiques/Membres/Permanents/Alain-Aspect. Source: After https://www.lcf.institutoptique .fr/Groupes-de-recherche/Gaz-quantiques/Membres/Permanents/Alain-Aspect.

shown in Figure 1.1 is a real interference fringe system, generated with coherent light and a double-slit apparatus. This is one of the simplest arrangements that shows the wave nature of light – and yet in this case it is detected by a number of individual photons. There is no contradiction here – the two explanations, waves and particles, are fully complementary. The shape of the fringes, the overall intensity function, is best described with the wave model. At the same time, the statistics of the light, the visibility that is observed, and thus the quality of the fringes can best be described with the photon model. This very simple example immediately shows the strengths and weaknesses of both models. It demonstrates how we want to use these two models for different aspects of the experiment. We should choose the pictures in our mind that is most suitable to understand the specific aspect of the experiment we wish to investigate. The next level would be to quantify the statistics. What would be an appropriate statistical model for the fluctuations, detected in one specific area of the image and averaged over many time intervals? Equivalently we could ask: what is the statistics of the number of photons detected in many separate areas of equal size, all receiving the same intensity? In the first case one could imagine listening to the number of clicks being generated. It would sound like a random sequence of clicks, as we know them from a Geiger counter detecting 𝛼 or 𝛾 rays.

1.3 Motivation Through Simple and Direct Teaching Experiments

A proper analysis would reveal that we have a Poissonian distribution for the different time intervals. The same result applies to the analysis of the number of photons in many image areas of equal size. The graininess we see in Figure 1.1 is a direct representation of the counting statistics, which can be derived from simple statistical arguments, assuming random processes. The randomness in the arrival times of the photons in a weak beam of light is so perfect that it is now used as the gold standard for randomness, exceeding in quality the widely used random number generators based on electronic processes or imbedded in computers. This has led to commercial activities where optically generated random numbers now can be accessed via the internet and the generators are for profit. The demand is substantial – and already a small niche market for quantum random numbers has emerged. The 1960s saw a rapid development of new laser light sources and improvements in light detection techniques. This allowed the distinction between incoherent (thermal) and coherent (laser) light on the basis of photon statistics. The groups of F.T. Arecchi [26], L. Mandel, and R.E. Pike all demonstrated in their experiments that the photon counting statistic goes from super-Poissonian at the lasing threshold to Poissonian far above threshold. The corresponding theoretical work by R.J. Glauber [15] was based on the concept that both the atomic variables and the light are quantized and showed that light can be described by a coherent state, the closest quantum counterpart to a classical field. The results are essentially equivalent to a quantum treatment of a classical oscillator. However, it is an important consequence of the quantum model that any measurement of the properties of this state, intensity, amplitude, or phase of the light, will be limited by these fluctuations. This is the optical manifestation of Heisenberg’s uncertainty principle. Quantum noise can impose a limit to the performance of lasers, sensors, and communication systems, and near the quantum limit the performance can be quite different. To illustrate this, consider the result of a very simple and practical experiment: use a laser, such as a laser pointer with a few milliwatt of power, and detect all of the light with a photodiode. The average number of detection events is now very large. Listening to the output, we can no longer distinguish between the individual clicks; we will hear a continuous noise. The level of noise will increase with the intensity, as the width of the Poissonian distribution increases with the average number of photons detected. This quantum noise is the direct experimental evidence of the quantum statistics of light. If we record the current from the photodetector on an oscilloscope, we would see a trace with random fluctuations, above and below a mean level, representing the average intensity. The excursions have a Gaussian distribution around the mean value, which is fully consistent with the transition from a Poissonian to a Gaussian distribution. Relatively speaking this effect gets smaller if we average the signal over more photons, that means using a brighter laser or measuring for a longer time. As we will see, this effect cannot be avoided completely. Quantum noise is not an effect that can be eliminated by good traditional engineering. Decades ago quantum noise was called shot noise and was regarded by many as a consequence of the photodetection process, as a randomness of the stream of electrons produced in the photodetector. This view prevailed for a long time,

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Modulator A Laser B

X2

X1

Figure 1.2 Experiment to investigate the correlation between two parts of a laser beam, using a laser pointer, modulator, two detectors, and an x–y oscilloscope.

particularly in engineering textbooks. However, this view is misleading as we find in the squeezing experiments described in this book. It cannot account for situations where non-linear optical processes modify the quantum noise, whilst the detector remains unchanged. In the squeezing experiments the quantum noise is changed optically, and consequently we have to assume it is a property of the light, not of the detector. Today it is straightforward to see these effects of quantum noise with simple equipment, accessible to undergraduate students. The light source can be a laser pointer, carefully selected to operate in a single mode. The laser beam is split by a beamsplitter into two beams of equal power that are detected by two individual detectors, A and B, at different locations. Each photocurrent is amplified, and they are both displayed via the x and y axes of one oscilloscope. We want to see both the average (DC) value of the photocurrent, which represents the time averaged intensity of the beams A and B, and the fluctuations (AC) of the two photocurrents, as they represent any modulation and the fluctuations, or noise, of the intensity of the two beams. This can easily be done by looking at the signals at a few megahertz, which is typically the bandwidth of the RF amplifiers used. This is shown in Figure 1.2. The laser can be modulated, through periodic variations of the laser current. Classical optics predicts that the intensity of both beams A and B will be modulated, and this will show up clearly as a line at 45∘ on the oscilloscope. That means the two photocurrents change synchronously in time, an increase in one is accompanied by an increase in the other etc. In practice some care is required to achieve a clear 45∘ line: there should be no delay in the signals from the detector to the oscilloscope and the size of the signals has been matched. This result shows a strong correlation between the two beams. A correlation of the photocurrents means a correlation of the flux of photons. We can also look at the fluctuations in more detail. If we look at beams A and B individually, we see fluctuations of the current above and below the long-term average value. Closer inspection would show a Gaussian distribution of the excursions below and above, as we expect it for random noise in the beam. The magnitude of the fluctuations will follow the gain curve of the RF amplifiers. With careful calibration of the frequency response of our equipment, we find that the noise in the light is not frequency dependent; it is white noise. The magnitude of the fluctuations is the same for a laser beam with or without a small modulation.

1.3 Motivation Through Simple and Direct Teaching Experiments

A

A

C

C

X2

X2

X1

X1

(a)

(b) A C

X2

X1

(c)

Figure 1.3 The noise of beams A and B displayed on an oscilloscope. (a) No beam, technical background noise. (b) The laser beam has no modulation, quantum noise. (c) The laser beam with modulation. The signal from the two detectors is displayed in the coordinates X1 and X2 , correlations appear in the direction C, and uncorrelated signal appears in both directions A and C.

The noise increases with intensity, and careful measurements show that the variance of the noise is proportional to the intensity. Is the noise from A correlated with the noise from B? The oscilloscope image immediately provides the answer. The noise from A and B results in a fuzzy blob on the screen. See Figure 1.3b. The fluctuations in the two photocurrents are independent. The point on the screen of the oscilloscope traces out a random walk independently in both directions. This is in stark contrast to the modulation where the photocurrent fluctuates synchronously, as shown in Figure 1.3c. The lack of correlation is immediately evident. This confirms the special properties of quantum noise; it does not obey the classical rules. One simple explanation is that for small fluctuations we would notice the effect that one photon cannot be split into halves at the beamsplitter. It will appear in one or the other beam, with a random choice which beam it will join. We will see in Chapter 3 how this simple statement leads to an explanation of the properties of quantum noise in this situation. If we have no modulation, the blob is right in the middle of the screen. The blob is circular, given there is no additional noise in the system and the photon noise dominates over the electronic noise generated in the apparatus. For a laser beam with modulation, we will see a fuzzy area tilted at 45∘ , as shown in Figure 1.3c. This is a combination of the two effects, a line

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in direction C for the modulation and a fuzzy blob for the laser noise. Classical modulations and quantum noise have different properties, even when they have similar magnitudes, and this simple experiment can show the difference.

1.4 Consequences of Photon Correlations It is worthwhile to explore the properties of quantum noise somewhat further. It was found that, in clear distinction to classical noise, no technical trick can eliminate quantum noise. This becomes evident from our experiment. A common conventional idea for noise suppression would be to make two copies of the signal, both containing the noise, and to subtract them for each other. We have all the necessary components in the apparatus used in Figure 1.2. We have made two copies of the beam. We can use one beam, say, A, for our experiment, record it with detector A, and then subtract the electronic fluctuations generated by detector B. Based on our observations we now see that we can subtract the classical modulation, which is common to both beams. However, we cannot subtract the quantum noise, which is independent and uncorrelated. The subtraction works for any modulation, for harmonic modulation demonstrated here, and for random intensity modulations, such as those generated by a low-quality, noisy current source for the laser. That means we can distinguish between classical laser noise and quantum noise that is a consequence of the quantum properties of the photons. The result remains unchanged if the subtraction is replaced by a sum or if the two currents, or if the two currents are added with an arbitrary short time delay, such as introduced be an extra length of cable. The resulting noise is always the quadrature sum of the two-input signal. It is independent of the sign, or phase, of the summation. This is equivalent to the statement that the noise in the two photocurrents is not correlated. At this stage it is not easy to identify the point in the experiment where this uncorrelated noise is generated. One interpretation assumes that the noise in the photocurrents is generated in the photodetectors. A different interpretation assumes that the beamsplitter is a random selector for photons and consequently the intensities of the two beams are random and thus uncorrelated. A distinction can only be made by further experiments with squeezed light, which are discussed later in this book. An alternative scheme for noise suppression is the use of feedback control, as shown in Figure 1.4. It achieves equivalent results to difference detection. The Modulator

Laser

Feedback control

Figure 1.4 The second attempt to eliminate laser noise: an improved apparatus using a feedback controller, or ‘noise eater’.

1.4 Consequences of Photon Correlations

intensity of the light can be controlled with a modulator, such as an acousto-optic modulator or an electro-optic modulator. Using a feedback amplifier with appropriately chosen gain and phase lag, the intensity noise can be reduced, and all the technical noise can be eliminated [27]. It is possible to get very close to the quantum noise limit, but the quantum noise itself cannot be suppressed [28, 29]. This phenomenon can be understood by considering the properties of photons. As mentioned above, the quantum noise measured by the two detectors is not correlated; thus the feedback control, when operating only on quantum noise on one detector, will not be able to control the noise in the beam that reaches the other detector. This can be explained using a full quantum theory. The role of the photon generation process can be explored further. It was found that the noise of the light may be below the standard quantum limit if the pumping process exhibits sub-Poissonian statistics. This is particularly easy to achieve for LEDs, which are high efficiency light sources driven directly by electric currents, or with semiconductor lasers. The currents driving these devices are classical, at the level of the fluctuations we are concerned with, and the fluctuations can be controlled with ease to levels well below the shot noise level. For sources with high quantum efficiencies, the sub-Poissonian statistics of the drive current is transferred directly to the statistics of the light emitted. Such experiments were pioneered for the case of diode lasers by the group of Y. Yamamoto [30]. They showed that intensity fluctuations can be suppressed in a high impedance semiconductor laser driven by a constant current. Similar work had earlier been carried out with LEDs by several groups. If we use this type of source in our experiment shown in Figure 1.3, the fuzzy blob would be smaller, in both directions, than the blob for a normal laser with the same output intensity. To explain the quantum noise and the correlations fully, we should not restrict ourselves to fluctuations of the intensity of the light or the statistics of the photon arrival time. We will find that there is noise both in the magnitude and phase of the electromagnetic wave. Consequently our quantum model requires a two-dimensional description of the light, with properties which we will call quadratures. Fluctuations, or noise, can be characterized by the variance in both the amplitude and phase quadrature. Almost a decade after the observation of photon anti-bunching in atomic fluorescence, another quantum phenomenon of light was observed – the suppression, or squeezing, of quantum fluctuations [31]. For a coherent state the uncertainties in the quadratures are equal and minimize the product in Heisenberg’s uncertainty principle. A consequence is that measurements of both the amplitude or the phase quadrature of the light show quantum noise. In a squeezed state, the fluctuations in the quadratures are no longer identical. One quadrature may have reduced quantum fluctuations at the expense of increased fluctuations in the other quadrature. Such squeezed light could be used to beat the standard quantum limit. After the initial theoretical predictions, the race was on to find such a process. A number of non-linear processes were tried simultaneously by several competing groups. The observation of a squeezed state of light was achieved in 1985 first by the group of R.E. Slusher at Bell Labs in four-wave mixing in sodium atomic beam [32], soon followed by the group of M.D. Levenson and R. Shelby at IBM [33] the group of H.J. Kimble with an optical parametric oscillator [34]. In recent years a number of other non-linear processes have been used to

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Non-linear medium Spectrum analyser

Laser

Squeezed light

Local oscillator + _ Homodyne detector

Figure 1.5 A typical squeezing experiment. The non-linear medium generates the squeezed light that is detected by a homodyne detection scheme.

demonstrate the quantum noise suppression based on squeezing [35]. A generic layout is shown in Figure 1.5. The experiments are now reliable, and practical applications are feasible. Quantum information is a key to many applications of quantum optics. Plans for using the complexity of quantum states to code information, to teleport it, to use it for secure communication and cryptography, and to store quantum information and possibly use it for complex logical processes and quantum computing have all been widely discussed and demonstrated to a greater or lesser degree. The concept of entanglement has emerged as one of the key qualities of quantum optics. As it will be shown in this guide, it is now possible to create entangled beams of light either from pairs of individual photons, which is now possible in technical applications and in teaching laboratories [37], or from the combination of two squeezed beams, and the demand and interest in non-classical states of light has sharply risen. We can see quantum optics playing a large role in future communication and computing technologies [23, 36]. For this reason, the guide covers both single photon and CW beam experiments parallel to each other. It provides a unified description and compares the achievements and tries to predict the future potential of these experiments.

1.5 How to Use This Guide This guide leads us through experiments in quantum optics, experiments that deal with light and demonstrate, or use, the quantum nature of light. It shows the practicalities and challenges of these experiments and gives an interpretation of their results. One of the current difficulties in understanding the field of quantum optics is the diversity of the models used. On the one hand, the theory and most of the publications in quantum optics are based on a rigorous quantum model that is rather abstract. On the other hand, the teaching of physical optics and the experimental training in using devices such as modulators, detectors, and data acquisition systems are based on classical wave ideas. This training is extremely

1.5 How to Use This Guide

Experiment versus theory | a1>, | v > States

c1| a1> + c2| a2> + c3| v >

H Operators

(a)

|0> Operator and vacuum state

Complex state Prob. of det and correlation

Info SNR

(b)

Beams and modulation Experiment linear and non-linear intensity, phase components Signals and noise at

Loss

α2in, Φin, δX1(Ω), δX2(Ω), V1in (Ω), V2in (Ω) Quantum transfer function

Output beams

Detectors electronics

2

αout, Φout, V1out (Ω), V2out (Ω) Vv = 1 Vacuum beam

Coherent states variances at Ω

(c)

Figure 1.6 Comparison between an experiment (b) and the two theory descriptions for few photon states (a) and laser beams (c).

useful, but frequently does not include the quantum processes. Actually, the language used by these two approaches can be very different, and it is not always obvious how to relate a result from a theoretical model to a technical device and vice versa. As an example compare the schematic representation of a squeezing experiment, shown in Figure 1.6, both in terms of the theoretical treatments for photons and laser beams and in an experimental description. The purpose of this guide is to bridge the gap between theory and experiment. This is done by describing the different building blocks in separate chapters and combining them into complete experiments as described in recent literature. This guide starts with a classical model of light in Chapter 2. Experiments reveal that we require a concept of photons, Chapter 3, which is expanded into a quantum model of light in Chapter 4. The properties of optical components and devices are given in Chapter 5, followed by a detailed description of lasers and amplifiers in Chapter 6. Next is a detailed discussion of photodetection for single photons and beams in Chapter 7. On this basis we build with the discussion of complete experiments. The technical details required for reliable experimentation with quantum noise, including techniques such as cavity locking and feedback controller, are given in Chapter 8. The concept of squeezing is central to most attempts to improve optical devices beyond the quantum noise limit and is introduced and discussed in Chapter 9. It also describes the various squeezing experiments and their results are discussed; the different interpretations are compared. Finally, the applications of squeezed light are described in Chapter 10 with a detailed description of the largest application,

15

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1 Introduction

the detection of gravitational waves. Similar to squeezing Chapter 11 discusses quantum non-demolition experiments. In Chapter 12 more and more complete experiments to test the fundamental concepts of quantum mechanics are discussed. Finally, the concepts of quantum information and the rapidly evolving state of art in using quantum optics, both single photons or CW beams, in quantum information devices are presented in Chapter 13. Quantum optics has helped to build the foundations of many other form of quantum technology, and the links to these many parallel lines of research is briefly outlined in Chapter 14, which tries to look a little into the future. This guide can be used in different ways. A reader who is primarily interested in learning about the ideas and concepts of quantum optics would best concentrate on Chapters 2–4, 9, 12, and 13 but may leave out many of the technical details. For these readers Chapter 5 would provide a useful exercise in applying the concepts introduced in Chapter 4. In contrast, a reader who wishes to find out the limitations of optical engineering or wants to learn about the intricacies of experimentation would concentrate more on Chapters 2, 5, 6, and 8, and for an extension into experiments involving squeezed light, Chapters 9 and 10 can be added. A quick overview of the possibilities opened by quantum optics can be gained by reading Chapters 3, 4, 6, 9, 11–14. We hope that in this way our book provides a useful guide to the fascinating world of quantum optics.

References 1 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 2 Born, M. and Wolf, E. (1959). The Principles of Optics. Pergamon Press. 3 Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford

University Press. 4 Hage, B., Janousek, J., Armstrong, S. et al. (2011). Demonstrating various

quantum effects with two entangled laser beams. Eur. Phys. J. D 63: 457. 5 Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., and Grangier, P. (2007). Gener-

6 7 8

9 10 11

ation of optical “Schrödinger cats” from photon number states. Nature 488: 784. https://doi.org/10.1038/nature06054. Haroche, S. (2012) Nobel Lecture 2012. http://www.nobelprize.org/ nobelprizes/physics/laureates/2012/haroche-lecture.html. Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verh. Dtsch. Phys. Ges. 2: 237. Hallwachs, W. (1889). Ueber den Zusammenhang des Elektrizitätzverlustes durch Beleuchtung mit der Lichtabsorption. Ann. Phys. 273 (8): 666. https:// doi.org/10.1102/andp.18892730811. Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 313 (5): 149. https://doi.org/10.1002/andp.19023130510. Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Ann. Phys. 17: 132. Thorne, A., Litzen, U., and Johansson, S. (1999). Spectrophysics, Principles and Applications, 3e. Springer-Verlag.

References

12 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 13 Young, Th. (1807). Course of Lectures on Natural Philosophy and Mechanical

Arts. London: Taylor and Walton. 14 Hanbury-Brown, R. and Twiss, R.Q. (1956). Correlation between photons in

two coherent beams of light. Nature 177: 27–29. 15 Glauber, R.J. (1963). The quantum theory of optical coherence. Phys. Rev.

Lett. 130: 2529. 16 Carmichael, H.J. and Walls, D.F. (1976). A quantum-mechanical master

equation treatment of the dynamical stark effect. J. Phys. B 9: 1199. 17 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance fluorescence. Phys. Rev. Lett. 39: 691. 18 Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive field state collapse

and quantum non-demolition photon counting. Nature 448: 889. 19 Einstein, A., Podolsky, B., and Rosen, N. (1935). Can a quantum-mechanical

description of physical reality be considered complete?. Phys. Rev. A 47: 777. 20 Bell, J.S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1: 195. 21 Aspect, A., Grangier, P., and Roger, G. (1982). Experimental realisation of

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31 32

Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49: 91. Zeilinger, A. (2000). Quantum cryptography with entangled photons. Phys. Rev. Lett. 84: 4729. Kok, P. and Lovett, B.W. (2010). Introduction to Optical Quantum Information Processing. Cambridge University Press. Nielsen, M.A. and Chuang, I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. Bennett, C.H., Brassard, G., Crepau, C. et al. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70: 1895. Arecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of the time evolution of the radiation field by joint photocurrent distributions. Phys. Rev. Lett. 17: 260. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilisation of an Argon laser using an electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Taubman, M.S., Wiseman, H., McClelland, D.E., and Bachor, H.-A. (1995). Quantum effects of intensity feedback. J. Opt. Soc. Am. B 12: 1792. Machida, S. and Yamamoto, Y. (1989). Observation of amplitude squeezing from semiconductor lasers by balanced direct detectors with a delay line. Opt. Lett. 14: 1045. Walls, D. (1983). Squeezed states of light. Nature 306: 141. Slusher, R.E., Hollberg, L.W., Yurke, B. et al. (1985). Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett. 55: 2409.

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33 Shelby, R.M., Levenson, M.D., Walls, D.F. et al. (1986). Generation of

34 35

36 37

squeezed states of light with a fiber-optics ring interferometer. Phys. Rev. A33: 4008. Wu, L.-A., Xiao, M., and Kimble, H.J. (1987). Squeezed states of light from an optical parametric oscillator. J. Opt. Soc. Am. B 4: 1465. (a) Kimble, H.J. and Walls, D.F. (ed.) (1987). Squeezed states of the electromagnetic field. J. Opt. Soc. Am. B 4 (10 Special Issue); (b) Loudon, R. and Knight, P.L. (ed.) (1987). Squeezed light. J. Mod. Phys. Opt. 34 (Special Issue); (c) Yamamoto, Y. and Haus, H.A. (1986). Preparation, measurement and information capacity of optical quantum states. Rev. Mod. Phys. 58: 1001. Kok, P., Munro, W.J., Nemoto, K. et al. (2007). Linear optical quantum computing. Rev. Mod. Phys. 79: 135. Galvez, E.J. (2014). Resource Letter SPE-1: Single-photon experiments in the undergraduate laboratory. Am. J. Phys. 82 (11): 1019. https://doi.org/10.1119/1 .4872135.

19

2 Classical Models of Light The development of optics was driven by the desire to understand and interpret the many optical phenomena that were observed. Over the centuries more and more refined instruments were invented to make observations, revealing ever more fascinating details. All the optical phenomena investigated during the nineteenth century were observed either directly or with photographic plates. The spatial distribution of light was the main attraction. The experiments measured long-term averages of the intensity; no time-resolved information was available. The formation of images and the interference patterns were the main topics of interest. In this context the classical optical models, namely, geometrical optics, dating back to the Greeks and developed ever since, and wave optics, introduced through Huygens’ ideas and Fresnel’s mathematical models, were sufficient. Excellent technology emerged, largely in the form of imaging devices of any kind, from microscopes through photography to telescopes. In parallel spectroscopy emerged as a tool for atomic and molecular physics. A revolution came with the invention of the laser. This changed optics in several ways. It allowed the use of coherent light, bringing the possibilities of wave optics to their full potential, for example, through holography. Lasers allowed spectroscopic investigations with unprecedented resolution, testing the quantum mechanical description of the atom in even more detail. The fast light pulses generated by lasers allowed the study of the dynamics of chemical and biological processes. In addition, a range of unforeseen effects emerged, for example, the high intensities resulted in non-linear optical effects. One fundamental assumption of the wave model, namely, that the frequency of the light remained unchanged, was found to be seriously violated in this new regime. The sum, difference, and higher harmonic frequencies were generated. Finally, optics was found to be one of the best testing grounds of quantum mechanics. The combination of sensitive photoelectric detection with reliable, stable laser sources meant not only that interesting quantum mechanical states could be produced but also that quantum mechanical effects had to be accounted for in technical applications. Light is now almost exclusively observed using photoelectric detectors, which can register individual events or produce photocurrents. A description of these effects requires the field of statistical optics. We need a description of the beam of light that emerges from a laser. A laser beam is a well-confined, coherent electromagnetic field oscillating at a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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2 Classical Models of Light

frequency 𝜈 or angular frequency 𝜔 = 2π𝜈. This frequency is constant for all times and locations and can only be changed by the interaction of light with non-linear materials. Furthermore, the beam propagates and maintains its shape unperturbed by diffraction. The beam carries information in the form of modulation, spatial in the form of images or temporal in the form of signals or pulses. Transmitting information in various forms, such as intensity, phase, polarization, and spatial changes, is the biggest use of optics. In this chapter we develop classical models suited to describing light. In the following chapters we will see how these must be modified to account for quantum mechanical effects.

2.1 Classical Waves 2.1.1

Mathematical Description of Waves

An electromagnetic wave in an isotropic insulating medium is described by the wave equation 1 𝜕2 E(r, t) = 0 c 𝜕t 2 with a solution for the electric field vector E(r, t), given by ∇2 E(r, t) −

(2.1)

E(r, t) = E0 [𝛼(r, t) exp(i2π𝜈t) + 𝛼 ∗ (r, t) exp(−i2π𝜈t)] p(r, t)

(2.2)

The wave oscillates at a frequency 𝜈 in Hz. For now we consider only a single-frequency component, but in general the solution can be an arbitrary superposition of such components. The electric field vector is perpendicular to the local propagation direction. The orientation of the electric field on the plane tangential to the local wave front is known as the polarization and is described by the vector p(r, t). The dimensionless complex amplitude 𝛼(r, t) of the wave can be written as a magnitude 𝛼0 (r, t) and a phase 𝜙(r, t): 𝛼(r, t) = 𝛼0 (r, t) exp{i𝜙(r, t)}

(2.3)

The magnitude 𝛼0 (r, t) will change in time when the light is modulated or pulsed. It will also vary in space, describing the extent of the wave. The wave will typically be constant in space for dimensions of the order of several wavelengths. The direction and shape of the wave front are determined by the phase term 𝜙(r, t), and the spatial distribution of the 𝜙(r, t) describes the curvature of the waves. Simple examples of specific solutions are monochromatic plane waves and spherical waves. These represent the extreme cases in terms of angular and spatial confinement. The monochromatic plane wave is the ideal case in which the complex amplitude magnitude 𝛼0 is a constant and the phase is given by 𝜙(r) = −k ⋅ r

(2.4)

The wave propagates in a direction and with a wavelength 𝜆 represented by the wave vector k, where k∕k is a unit vector that provides the direction of the local

2.1 Classical Waves

wave front and k = 2π∕𝜆. The wavelength is related to the frequency via 𝜆 = c∕n𝜈, where n is the refractive index of the medium in which the wave propagates and c is the speed of light in the vacuum. Thus we can write 𝛼(z) = 𝛼0 exp{−ikz}

(2.5)

where we have assumed that the wave propagates in the z direction. The wave has a constant amplitude in any plane perpendicular to this optical axis. Substituting Eq. (2.5) into Eq. (2.2) and assuming 𝛼0 as real, we obtain the familiar solution E(z, t) = E0 cos(𝜈t − kz) p(z, t)

(2.6)

Linear polarization means that p(z, t) has a fixed direction perpendicular to k. Circular polarization means that in any one location p(z, t) the direction is rotating in time in the plane perpendicular to k. Equation (2.6) represents a wave that has no angular spread. In contrast the spherical wave originates from one singular point. All its properties are described by one parameter r, the distance to its point of origin. The amplitude magnitude reduces with 1∕r, and the phase is constant on spheres around the origin: 𝛼 (2.7) 𝛼(r, t) = 𝛼(r) = 0 exp{ikr} r 2.1.2

The Gaussian Beam

Both of these models are not physical, they are only crude approximations to a beam of light emitted by a real laser. Such a beam is confined to a well-defined region and should be as close as possible to the ideal of a spatially localized but non-divergent wave. It has to be immune to spreading by diffraction and therefore has to be a stable solution of the wave equation. An example of such solutions is the Gaussian modes. The simplest of these is circularly symmetric around the optical axis, thus depending only on the radial distance 𝜌 from the beam axis: 𝜌 2 = x2 + y 2 This special solution is the Gaussian beam or fundamental mode, with a complex amplitude distribution of ( ) i𝛼 −ik𝜌2 𝛼(r, t) = 0 exp (2.8) exp(ikz), q(z) = z + iz0 q(z) 2q(z) where z0 is a constant. The Gaussian beam is a good spatial representation of a laser beam. Laser light is generated inside a cavity, constructed of mirrors, by multiple interference of waves and simultaneous amplification. These processes result in a well-defined, stable field distribution inside the cavity, the so-called cavity modes [1–3]. The freely propagating beam is the extension of this internal field. It has the remarkable property that it does not change its shape during propagation. Both the internal and external fields are described by Eq. (2.8). By taking

21

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the square of the complex amplitude 𝛼(r, t), the intensity is found, resulting in a Gaussian transverse intensity distribution: ( ( ) ) W0 2 −2𝜌2 exp I(𝜌, z) = I0 W (z) W 2 (z) √ ( )2 z (2.9) and W (z) = W0 1 + z0 The properties of this Gaussian beam are the following: The shape of the intensity distribution remains unchanged by diffraction. The size of the beam is given by W (z). The beam has its narrowest width at z = 0, the so-called waist of the beam. It corresponds to the classical focus, but rather than pointlike, the Gaussian beam has a size of W0 . This value gives the radial distance where the intensity has dropped from I0 = I(0, 0) at the centre to I(W0 , 0) = I0 ∕e. At the waist the phase front is a plane, and 𝜙(𝜌) is a constant. Moving away from the waist, the beam radius W (z) expands, the intensity at the axis reduces, and, at the same time, the wave front curves. Along the optical axis the Gaussian beam scales with the parameter z0 , which is known as the Rayleigh length: z0 =

πW02

𝜆 It describes the point where the intensity on axis has dropped to half the value √ at the waist, I(0, z0 ) = 12 I0 , and the beam radius has expanded by a factor of 2. This can be regarded as the spatial limit to the focus region, and the value 2z0 is frequently described as the depth of focus or as the confocal parameter. Far away from the waist, for z ≫ z0 , the Gaussian beam has essentially the wave front of a spherical wave originating at the waist. Obviously, the Gaussian beam is still centred on the beam axis and has only a finite extent W (z). At these large distances, the size of the beam W (z) depends linearly on z; the beam has a constant divergence angle of Θ=

2𝜆 2πW0

(2.10)

In addition to the wave front curvature, a gradual phase shift occurs on the beam axis between a uniform plane wave with identical phase at z = 0 and the actual Gaussian beam. The phase difference increases to π∕4 at z = z0 and asymptotically approaches π∕2 (Table 2.1). Table 2.1 Properties of a Gaussian beam. z=0

z = z0

Beam waist

W = W0

√ W = 2W0

W (z) = z∕z0 W0

Intensity

I0

I0 ∕2

I0 (z0 ∕z)2

Wave front

Plane wave

Curved wave front

Spherical wave

Phase retardation

0

π∕4

π∕2

Location

z ≫ z0

2.1 Classical Waves

Figure 2.1 Gaussian beam propagation along the z axis.

I(y) I(x)

z W(z) W0 = W(z = 0) x x

As known from experiments with laser beams, the Gaussian beam is very robust in regard to propagation through optical components. Any component that is axially symmetric and the effect of which can be described by a paraboloidal wave front will keep the Gaussian characteristic of the beam. This means the emerging beam will be Gaussian again, however, now with a new position z = z′ of the waist. All spherical lenses and spherical and parabolic mirrors transform a Gaussian beam into another Gaussian beam. Another important consequence of this transformation is the fact that Gaussian beams are self-reproducing under reflection with spherical mirrors (Figure 2.1). However, the Gaussian beam (Eq. (2.8)) is not the only stable solution of the paraxial wave equation. There are many other spatial solutions with non-Gaussian intensity distributions that can be generated by a laser and propagate without change in shape. One particular set of solutions is the Hermite–Gaussian beams. These beams have the same underlying phase, except for an excess term 𝜁l,m (z), which varies slowly with z. They have the same paraboloidal wave front curvature as the Gaussian beam, which matches the curvature of spherical optical components aligned on the axis. Their intensity distribution is fixed and scales with W (z). The distribution is two-dimensional (2D), with symmetry in the x and y directions and usually not with a maximum on the beam axis. The intensity on the beam axis decreases with the same factor W (z) as the Gaussian beam along the z axis. The various solutions can be represented by a product of two Hermite–Gaussian functions, one in the x and the other in the y direction, which directly leads to the following notation: (√ ) ( ) (√ ) W0 2y 2x Gm Gl 𝛼l,m (x, y, z) = 𝛼0 Al,m W (z) W (z) W (z) )} { ( kr2 for l, m = 0, 1, 2, … − 𝜁l,m (z) × exp −i kz + 2R(z) (2.11) Here Al,m is a constant and Gl (u) and Gm (u) are standard mathematical functions, the Hermite polynomials of order l, m, respectively, which can be found in tables [2, 4]. In particular, G0 (u) = exp{−u2 ∕2}, and thus the solution 𝛼0,0 (x, y, z) is identical to the Gaussian beam described above. This set of solutions forms a complete basis for all functions with paraboloidal wave fronts. Any such beam can be described by a superposition of Hermite–Gaussian beams. All these components propagate together, maintaining the amplitude distribution. The only

23

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2 Classical Models of Light

00

10

20

30

01

11

21

31

02

12

22

33

Figure 2.2 Intensity distributions 𝛼l,m (x, y)2 of Hermite–Gaussian beams. These modes are labelled transverse excited mode TEMlm with the parameters l and m describing the horizontal and vertical order of the modes. Source: https://www.wikipedia.org/.Licensed Under CC BY SA 3.0.

component with circular symmetry is the Gaussian beam 𝛼0,0 (x, y, z). However, it is possible to describe other circularly symmetric intensity distributions. Consider, for example, the superposition of two independent beams with 𝛼1,0 (x, y, z) and 𝛼0,1 (x, y, z) and equal intensity; if these beams have random phase, their interference can be neglected and the sum of the intensities results in a donut-shaped circularly symmetric function, occasionally observed in lasers with high internal gain (Figure 2.2). 2.1.3

Quadrature Amplitudes

The description of a wave as a complex amplitude (Eq. (2.3)), with amplitude 𝛼0 (r, t) and phase distribution 𝜙(r, t), is useful for the description of interference and diffraction. As shown above, the phase distribution 𝜙(r, t) describes the shape of the wave front as well as the absolute phase, in respect to a reference wave. However, it is useful to separate the two to consider the absolute phase independently from the phase distribution or wave front curvature. An equivalent form of Eq. (2.2) can be given in terms of the quadrature amplitudes, X1 and X2, the amplitudes of the associated sine and cosine waves: E(r, t) = E0 [X1(r, t) cos(2π𝜈t) + X2(r, t) sin(2π𝜈t)] p(r, t)

(2.12)

The quadrature amplitudes are proportional to the real and imaginary parts of the complex amplitude: X1(r, t) = 𝛼(r, t) + 𝛼 ∗ (r, t) X2(r, t) = −i[𝛼(r, t) − 𝛼 ∗ (r, t)]

(2.13)

2.1 Classical Waves

Figure 2.3 Phasor diagram (a) for a constant single wave and (b) the representation of interference of several coherent waves.

X2

X2

E

α total

ϕ (a)

E

X1

(b)

X1

In this notation the absolute phase 𝜙0 of the wave is associated with the distribution of the amplitude between the quadratures X1 and X2: ( ) X2 (2.14) 𝜙0 = tan−1 X1 For X2 = 0 the wave is simply a cosine wave (Eq. (2.6)). We can regard this wave arbitrarily as the phase reference. A common description of classical waves whenever interference or diffraction is described is phasor diagram, a 2D diagram of the values X1 and X2 as shown in Figure 2.3a. In such a diagram, which represents the field at one point in space and time, each wave corresponds to one particular vector of length 𝛼 from the origin to the point (X1, X2), and this vector could be measured as the complex amplitude averaged over a few optical cycles of length 1∕𝜈. The magnitude of the wave is given by the distance 𝛼 of the point from the origin, and the relative phase 𝜙0 is given by the angle with the X1 axis, since the absolute phase is arbitrary. It is useful to choose 𝛼 as real (𝜙 = 0), which means the vector is parallel to the X1 axis. If several waves are present at one point at the same time, they will interfere. Each individual wave is represented by one vector; the total field is given by the sum of the vectors, as shown in Figure 2.3b. These types of diagrams are a useful visualization of the interference effects. After the total amplitude 𝛼total has been determined, we can again select a phase for this total wave and choose 𝛼total as being real. All forms of interference can be treated this way and we obtain a value for the average, or DC, value of the optical amplitude. In addition, the wave can have fluctuations or modulations in both amplitude and phase. Such variations correspond to fluctuations in the quadrature values. We can describe the time-dependent wave with the complex amplitude 𝛼(t) = 𝛼 + 𝛿X1(t) + i 𝛿X2(t)

(2.15)

In the limit 𝛼 ≫ 𝛿X1(t), which means for waves that change or fluctuate only very little around a fixed value 𝛼, we see that 𝛿X1(t), 𝛿X2(t) describe the changes in the amplitude and phase, respectively. This is the regime that we use to describe beams of light detected by detectors that produce a photocurrent. We will call them the amplitude and phase quadratures. The direct link between variations in the quadrature values and those of the field magnitude and phase is shown diagrammatically in Figure 2.4. 2.1.4

Field Energy, Intensity, and Power

The energy passing through a small surface area element dx, dy around the point x, y in a short time interval dt around the time t is obtained by integrating the

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2 Classical Models of Light

Δϕ

X1 ΔX1

E

ΔE

ΔX2

t

X2

Figure 2.4 Fluctuations shown in phase space: the field changes in time and moves from point to point in the phasor diagram. This corresponds to waves with different amplitude and phase.

square of Eq. (2.2). Note that because the field is propagating in the z direction, this is equivalent to the energy in the volume element dx, dy, c dt. We get (x, y, t) = E02 2 𝛼 ∗ 𝛼 dt dx dy

(2.16)

where we have assumed that dx dy dt is sufficiently small that 𝛼 does not vary over the interval, but that dt ≫ 1∕Ω is satisfied and the time average will be over a large number of optical cycles. As a result only the phase-independent terms survive. The intensity of the beam is I(x, y) =

∫one second

E02 2 𝛼 ∗ 𝛼 dt

(2.17)

What is normally measured by a photodetector is the power P of the light, the total flux of energy reaching the detector in a second, integrated over the area of the detector: P=

∫

I(x, y) dx dy

(2.18)

The energy in a small volume element (Eq. (2.16)) can also be expressed in terms of the quadratures as (x, y, t) = E02 (X12 + X22 ) dt dx dy where we have used Eq. (2.13). The parameter E02

(2.19) is proportional to the frequency

of the light, 𝜈, specifically

ℏ2π𝜈 (2.20) 2 where ℏ is Planck’s constant. It is the optical power P(t) in Eq. (2.18) that is measured in most experiments, and from this we infer the amplitude, the phase, and the fluctuations of the beam. E02 =

2.1.5

A Classical Mode of Light

The idea of a phasor diagram can now be combined with the concept of spatial modes as described in Eq. (2.3). Let us concentrate on the simplest type of beam, a Gaussian beam. Equation (2.8) relates the relative amplitude and phase at any one point back to a point at z = 0 at the centre of the waist. Once the size W0 is given, the entire spatial distribution is well defined. We can assign one overall optical

2.1 Classical Waves

P

W0

z ν, ϕ0

I0

Figure 2.5 Graphical representation of the parameters of a Gaussian mode of light.

power, or peak intensity I0 , or one overall phase to the entire spatial distribution. Thus the entire laser beam can be described by just six parameters. The spatial shape of the laser beam with its very narrow focus, see Figure 2.5 is the key for many applications of laser light. It allows us to deliver very high power to a very small volume, which is the key to most applications for cutting, ablating, or ionizing materials and many of the medical applications. The small size of the focus, limited by diffraction to a about half of a wavelength, allows high spatial resolution in microscopy or the coupling of light into a single-mode optical fibre. Even more magically the high gradients in intensity in a laser focus can be used to exert forces onto particles that move inside the laser beam, and this can be used to trap particles, alas individual atoms or ions inside the focus, the physics behind optical tweezers invented by 2018 Nobel laureate A. Ashkin [5, 6]. This itself is the foundation of many applications in biology and also of many new quantum experiments with laser-cooled atoms, single atoms, and ions. Such a mode of light is strictly speaking constant in time, at least for short times. It corresponds to an idealized laser that has a perfectly stable, time-independent output, with an intensity distribution that is not changing at all in time. More realistically the laser output will not be perfectly constant, but there will be fluctuations of the average intensity, the amplitude and phase, and all parameters listed in Table 2.2. Experiments confirm that there are always fluctuations (see Figure 1.2). They can be stabilized to some degree by technical means, but ultimately quantum mechanics limits how small the fluctuations can be. This description of the light as a mode, which can be quantized, will be the key to an elegant quantum model. Table 2.2 The parameters of a mode of light. Parameters required to describe one mode of light

Intensity of the mode

I0

Frequency of optical field

𝜈

Absolute phase of this mode

𝜙0

Direction of propagation

z

Location of the waist (z = 0)

r0

Waist diameter of the beam

W0

Direction of polarization

P

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2 Classical Models of Light

2.1.6

Light Carries Information

Light is used to carry information. In many cases we think about images, where the information is encoded in the spatial domain, transverse to the propagation of light. In this case the magnitude and the phase of the light are a function of location r, the image is in the transverse x − −y plane, and the image can change along the z axis of propagation. Quite frequently this is constant in time – until the input image is changed. In many other situations we want to use light to measure some parameter or to communicate information from one place to another. In both cases we need a beam of light that can carry information, a simple constant beam of light that can be modulated in time. The physical characteristics of a freely propagating laser beam define such a mode of light. The shape of this mode is well determined. In all these cases the propagation of the light, be it in free space, through a complex apparatus, through an optical fibre, etc., can be described by the mode of light, the simplest case being the Gaussian beam. We will select the appropriate degree of freedom that is being modulated, and it is useful to describe the entire beam, including all its time and frequency components, as one entity, as one mode. The beam can change in its appearance while it propagates through the apparatus – but it remains one entity. We will find in Chapter 4 that this is the most convenient and practical way to quantize the light. However, the temporal dependence is most important, since it contains the information in the form of modulations. We describe this with a time-varying amplitude 𝛼(t) for this mode. The various aspects of the measurement and manipulation of laser beams occur at different timescales and at their corresponding frequencies. Let us introduce the nomenclature for a set of natural and technical frequencies and timescales (Figure 2.6).

Δtusp

Δtsamp

Δtmeas

One sample Integrate interval for variance and average

Δtosc = 1/ Ωmod

Period of oscillation in periodic signal

Δtrec

T

Interval between recordings

Envelope s(t)

2π νopt

Ωsamp

Ωmod, Ωsens

Ωenv

High

Sampling of photon flux frequency

Modulator and signal carrier frequencies

Signal content frequency

CW

Long

Low

Figure 2.6 Definition of different timescales and frequencies used in measuring and manipulating light.

2.1 Classical Waves

The natural starting point is the optical frequency 2π𝜈L of a laser beam. This is the highest frequency, typically 1015 Hz, and all the modulation frequencies Ωmod will be lower. We can sample the photon flux at very high rates with the sampling frequency Ωsamp , which is between 2π𝜈L and Ωmod . The shortest technical timescale is Δtsamp = 1∕Ωsamp , which in many situations is achieved through the fast gating of either a photon counter or an analogue-to-digital converter. We require a number of samples to evaluate a variance for the fluctuations or an average value for the photon flux, which requires the time Δtmeas . Many such data points are required to impose a periodic signal at Ωmod or measure a periodic signal Ωsens from an external source. This requires a time interval Δtosc = 1∕Ωmod . In a sensing application we might want to take a whole series of readings, taken at time intervals Δtread , and we will have a full evaluation of the signal and noise for a considerable period of time T. In most cases we can assume that the performance of the instrument remains constant over a long time, that the noise is stationary and we can use samples of the noise to describe the performance, and that the laser parameters, such as optical power and optical frequency, are well stabilized. In this case we can record the envelope function from many readings and analyse the behaviour of a parameter we are sensing. We might find periodic behaviour and typical envelope frequencies Ωenv . The absolute values for these timescales and frequencies can vary by several orders of magnitude and cannot be stated in general terms. The instruments can contain anything from femtosecond pulses to a constant laser. We can have short pulses of light where each pulse carries one piece of information. This will be particularly important when we think about individual photons in each pulse. Here the typical timescale Δtmeas for each measurement can be shorter than the time between two pulses, and with modern equipment this can be as short as a few picoseconds. In the other extreme we can have a continuous wave (CW) laser that is steady, and we measure for a time interval T and take many readings. The modulation and sensing frequencies can vary from single hertz for sensors such as environmental or astronomical detectors to gigahertz in communication systems. Still, there is a great deal of commonality that we will use in describing the properties of light, classical and quantum. Finally, there is the option of quasi-CW communication where we use trains of pulses with high repetition frequency, in particular to achieve high peak powers, and the measurement averages over a number of pulses. For the modulation and sensing, we can choose several degrees of freedom, such as phase, frequency, polarization, position and mode shape, etc., and in this way pack more information into one beam. Technically this is described as multiplexing. One example is the modulation of the polarization, which has 2 degrees of freedom, and we require two orthogonal polarization axes for one single beam. More degrees of freedom are possible, for example, additional optical frequencies or several additional co-propagating spatial modes. In addition we will encounter situations where we have several beams within one apparatus, for example, created by a series of beamsplitters. In all these cases we can define the available degrees of freedom, find the corresponding orthogonal functions that describe

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these optical fields and their modulations, and describe this apparatus with a set of multiple modes. 2.1.7

Modulations

Information carried by a beam of light can be described in the form of modulation sidebands. Note that the equations used here are identical to those describing radio waves where amplitude modulation (AM) and frequency modulation (FM), or technically speaking phase modulation (PM), are the familiar, standard techniques. Many of the technical expressions, such as carrier for the fundamental frequency components and sidebands, are taken directly from this much older field of technology. A beam of light that is amplitude modulated by a fraction M at one frequency Ωmod can be described by ( ) M 𝛼(t) = 𝛼0 1 − (1 − cos(2πΩmod t)) exp(i2π𝜈L t) 2) ( M exp(i2π𝜈L t) = 𝛼0 1 − 2 M + 𝛼0 [ exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t) ] (2.21) 4 We see that the effect of the modulation is to create new frequency components at 𝜈L + Ωmod and 𝜈L − Ωmod . These are known as the upper and lower sidebands, respectively, whilst the component remaining at frequency 𝜈L is known as the carrier (see Figure 2.7a). For completely modulated light, which means M = 1, α X2

νL − Ω2 νL − Ω1 νL + Ω1 (a)

X2

=

AM

νL + Ω2 ν

X2

Ω1

+

Ω2

FM

α

α X1

X1

(b)

δX2

δX1

X1 Ω1

Ω2 (c)

Ω

Figure 2.7 Three different graphical representations of amplitude (AM) and frequency (FM) modulation at frequencies Ω1 and Ω2 , respectively. Part (a) shows the sidebands of the optical frequency at 𝜈L ± Ω1 and 𝜈L ± Ω2 , (b) shows the same modulations in the time domain and in a phasor diagram, and (c) shows the complete phasor diagram in frequency space.

2.1 Classical Waves

the sidebands have exactly 12 the amplitude of the fundamental component. The sidebands are perfectly correlated, and they have the same amplitude and phase, which means the two beat signals, generated by the sidebands at 𝜈L + Ωmod and 𝜈L − Ωmod beating with the carrier, are in phase. A second representation is in the time domain through oscillating vectors 𝛿𝛼 in the phasor diagram Figure 2.7b. The two vectors, 𝛿𝛼+Ω and 𝛿𝛼−Ω , rotate in opposite directions with angular frequency 2πΩ. A third representation is in frequency space in the phasor diagram shown in Figure 2.7c. Notice that for a given modulation depth M, the sideband amplitudes scale with the amplitude of the carrier, 𝛼0 . Thus even very small modulation depths, M, can result in significant sideband amplitude if the carrier amplitude is large. The detected intensity is given by ( )2 M (2.22) I(t) = I0 |𝛼(t)|2 = I0 1 − (1 − cos(2πΩmod t)) 2 and oscillates at the frequency Ωmod around the long-term average I0 . The resulting photocurrent has two components, a long-term average iDC and a constant component at frequency Ωmod , labelled i(Ωmod ). A common (and useful) situation is when the modulation depth is very small, M ≪ 1. Then the resulting intensity modulation is linear in the modulation depth: I(t) ≈ I0 [1 − M(1 − cos(2πΩmod t)) ]

(2.23)

A light beam with modulated phase can be described in a similar way. The amplitude can be represented by 𝛼(t) = 𝛼0 exp[iM cos(2πΩmod t)] exp(i2π𝜈L t)

(2.24)

Clearly the intensity will not be modulated as |𝛼(t)|2 = |𝛼0 |2 . Expanding 𝛼t we obtain ) ( M2 2 𝛼(t) = 𝛼0 1 + iM cos(2πΩmod t) − cos (2πΩmod t) + · · · exp(i2π𝜈L t) 2 [( ) M2 = 𝛼0 1 − + · · · exp(i2π𝜈L t) 4 ) ( M + · · · (exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)) +i 2 ( 2 ) M − + · · · (exp(i2π(𝜈L + 2Ωmod )t) + exp(i2π(𝜈L − 2Ωmod )t)) 8 +· · · ]

(2.25)

Suppose we measure the quadrature amplitudes (X1 and X2) of this field relative to a cosine reference wave (i.e. we take 𝛼 to be real). The X2 quadrature will have pairs of sidebands separated from the carrier by ±Ωmod , ±3Ωmod , … The X1 quadrature on the other hand will have sidebands separated from the carrier by ±2Ωmod , ±4Ωmod , … Again a simple, and useful, situation is for the modulation depth to be very small, M ≪ 1. The amplitude is approximately [ ] M 𝛼(t) ≈ 𝛼0 exp(i2π𝜈L t) + i [exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)] 2 (2.26)

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and the sidebands will only appear on measurements of the X2 quadrature. Exact expressions for larger M involve nth-order Bessel functions [7]. The phasor diagram corresponding to this double sideband approximation is shown in Figure 2.7b. The sidebands have the same magnitude. The relative phase of the amplitudes is such that the two resulting beat signals have the opposite sign and the beat signals cancel. Thus, as we have seen, phase modulation cannot be detected with a simple intensity detector. Another way of saying this is that the sidebands for phase modulation are perfectly anti-correlated. The treatment of frequency modulation proceeds in a similar way. Any information carried by the mode can be expressed as a combination of amplitude and phase modulation. Any modulation, in turn, can be expressed in terms of the corresponding sidebands.

2.2 Optical Modes and Degrees of Freedom 2.2.1

Lasers with Single and Multiple Modes

A single mode is only the simplest option for a laser beam output. The resonator inside the laser will have several resonance conditions. Optical waves with different frequencies can be amplified, and also different spatial amplitude and phase distributions can be in resonance. Each of these waves, with their specific set of parameters, can be considered as a different mode, with distinctively different values of the parameters listed in Table 2.2, and in the case of spatial modes different amplitude and phase distributions. The details depend crucially on the geometry of the laser resonator [2] and the spectral response of the gain medium, and some of the details are summarized in Chapters 5 and 6. In many cases the laser might produce an output in several of these modes simultaneously – with all of these modes competing for gain. Examples of other laser modes are beams with the same shape but different carrier frequencies, spaced in regular intervals and all linked by a common phase. This concept, known as a frequency comb, won the inventors a Nobel prize [8] and is the basis of many new applications in optical metrology of time signals and of new forms of spectroscopy and non-linear optics. Short laser pulses can be thought of as a frequency comb with a specific envelope function determining the number and strength of the individual components. These envelope functions are shaped to be the effects inside the laser, or they can also be modified outside by instruments that have a frequency-dependent response, or dispersion, or with active means such as electro-optic pulse shapers. Frequently the aim of engineering is to produce a single-mode laser, which can be described by the simple set of parameters, with a waveform such as stated in Eq. (2.8). However, not every single-mode beam has to be this simple. A single-mode beam can be transformed, for example, with lenses or beamsplitters, or we could delete a part the beam, and during propagation diffraction would change the intensity distribution, possibly quite dramatically. The underlying idea of the mode remains the same in such a mode transformation, and the relevant question remains: can we find one single mathematical description that contains all properties of the laser beam? Clearly, yes we can.

2.2 Optical Modes and Degrees of Freedom

This is correct for the spatially transformed single-mode beam. Another example is a carefully crafted pulsed laser system where techniques such as mode locking produce trains of pulses as short as femtoseconds, with correspondingly wide frequency combs, where all components of the comb are phase and amplitude linked. This entire ensemble of spectral components can be described as a single mode with one simple set of parameters. In this case the mode looks complex, in space or frequency, but we can still speak of one mode since all the information is still contained as a modulation of this specific entity. In each case we have to consider the number of degrees of freedom that are available, not the apparent complexity of the beam of light. 2.2.2

Polarization

Let us consider beams that require a description by two or modes. One clear example is the polarization of an otherwise single-mode beam. The polarization properties can be described by separately considering the amplitude of the field oscillating in two transverse directions; let us call them horizontal H and vertical V . These are two orthogonal coordinates that could be in any physical directions. Mathematically we require two orthogonal basis vectors, H and V. A 2D complex vector 𝛼 p(r, t) with magnitudes |𝛼H |, |𝛼V | and phases 𝜙H , 𝜙V for the two directions provides the full description: 𝛼 p(r, t) = |𝛼H (r, t)| ei𝜙H H + |𝛼V (r, t)| ei𝜙V V = (𝛼H (r, t), 𝛼V (r, t))

(2.27)

The important point is that the properties, and also the information, in the two coordinates are completely independent. For example, a beam can have all its intensity in one or the other coordinate V and H, or it can have a shared amplitude in both. We would call a beam with 𝛼V = 0 horizontally polarized and with 𝛼H = 0 vertically polarized. A beam with 𝛼V = 𝛼H is diagonally polarized, D, while a beam with 𝛼V = −𝛼H is anti-diagonally polarized, A, orthogonal to D. We can make beams with the amplitude 𝛼V = i𝛼H , which we call right-hand circularly polarized (R), and with 𝛼V = −i𝛼H , which are left-hand circularly polarized (L). All of these descriptions, or projections on different basis vectors, are equivalent, and we can transform the projection between any of them without gaining or losing information. We can choose the basis vectors that match best to the experiment, resulting in the simplest description. In all cases the projection is based on a 2D vector, and we have two orthogonal modes, both travelling within the same laser beam. Any transformation of the polarization in the apparatus, for example, through polarizers, wave plates as shown in Figure 2.8, through filters, glass plates, etc., can be described by 2D matrices Ξ, also known as Jones matrices, described in most textbooks about optics, such as [3]: 𝛼 p (r, t)out = Ξ 𝛼 p (r, t)in = Ξ1 Ξ2 Ξ3 … 𝛼 p (r, t)in

(2.28)

Here each component in the apparatus is described by a specific matrix Ξi , i = 1, 2, …, and light propagating through a sequence of such components can be described by multiplication of the individual matrices Ξ1 ,Ξ2 ,Ξ3 , … Examples for such Jones matrices include Ξprop for simple propagation, ΞH for a linear

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λ 2

V

PBS

R L

D

H

(a)

λ 4

A

(b)

PBS

(c)

PBS

Figure 2.8 Three examples of the polarization state of a single laser beam. The different projections used are (a) horizontal H and vertical V, (b) diagonal D and anti-diagonal A, and (c) left-hand circular L and right-hand circular R polarization. In each case two detectors are used, each detecting one of the polarization components. The transformation from one basis to another is achieved by a combination of polarizing beamsplitter (PBS) and either quarter-wave or half-wave plate.

polarizer in direction H, ΞV for a linear polarizer in direction V , and Ξ𝜙 for a wave retarder by a phase angle 𝜙: ( ( ( ( ) ) ) ) 1 0 1 0 0 0 1 0 , ΞH = , ΞV = , Ξ𝜙 = Ξprop = 0 1 0 0 0 1 0 ei𝜙 (2.29) Note that the order of the matrices is important and has to be the same as the order of components in the experiment. Interchanging components, such as placing a polarizer in a different section of the apparatus, can have dramatic effects. A very important point is that the beam can be modulated differently in the coordinates H and V , and as long as we detect the light in this basis, the information streams that we receive in these coordinates are completely independent. The mathematical statement of orthogonality between H and V is fully equivalent to having no crosstalk between the coordinates. We can use this fact to optimize our experiment. For example, by modulating only 𝛼H and not 𝛼V and by using two detectors, set for the H and V polarization, respectively, we can finely tune the optics until we find no signal coming from the V detector. We then change the modulation to V and should find no signal coming from the H detector. We now have established two independent communication channels. The laser beam can carry twice as much information using polarization modulation than in amplitude modulation of an unpolarized beam. This shows the distinction of a two-mode beam from a single-mode beam: it has 2 degrees of freedom, it contains more information, and we can consider the question of correlation between the two modes. For example, if we take a beam with pure horizontal modulation, 𝛼V = 0. That means we only use one single mode and we have no correlation between the information in H and V. If we detect the same beam in the D and A coordinates rotated at 45∘ , we share the modulation in the two components and we have a strong correlation between 𝛼D and 𝛼A . However, there is no additional information, and the beam can still be described by a single mode. No information is added in the choice of a different projection. In contrast, if we have independent modulations in both H and V, and that means no correlation between 𝛼H and 𝛼V , then we require a two-mode description. Detection in H and V would give us two independent channels and no

2.2 Optical Modes and Degrees of Freedom

crosstalk. In this case the detection in the basis D and A would create a correlation and would also mix up the information from the two channels. We still have a two-mode case, just but not an optimal description. 2.2.2.1

Poincaré Sphere and Stokes Vectors

It is useful to mention an alternative description of polarization. With the complex Jones vector, we have four components, and we can describe the state of the polarization by two real parameters: the magnitude ratio 𝛼H ∕𝛼V and the phase difference 𝜙H − 𝜙V . Alternatively, we might characterize the state of polarization by two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse, see Figure 2.9. We can now describe the polarization in the form of a three-dimensional (3D) diagram, the Poincaré sphere where each state is given by a point on the surface of the sphere of unit radius, with the spherical coordinates given by (r = 1, Θ = 90 − 2𝜒, Φ = 2𝜓). Each point on the sphere is a different polarization state. For example, points on the equator, that is with 𝜒 = 0, represent linear polarization. The two points on the equator, u1 with 2𝜓 = 0, and u2 with 2𝜓 = 90 degree, represent linear polarization in the H and V axis. The north pole u3 with 2𝜒 = 90 and south pole with 2𝜒 = −90 degree represent left-handed and right-handed circular polarized light respectively. Other states on the sphere represent the various forms of elliptical polarization. The coordinates u1 , u2 , u3 of a point on the Poincaré sphere in Cartesian coordinates are also known as the first three Stokes parameters (S1 , S2 , S3 ). The fourth parameter S0 is simply proportional to the intensity of the beam. For the full details, see Ref. [3]. This four-component vector, introduced by Stokes, is the original description of polarization. It contains the same information as the Jones vector with its two complex components. We find that the latter is mathematically more elegant and fits to the two-mode description, which is so powerful in describing quantum optics systems. However, the Poincaré sphere is a practical way to illustrate the effects of polarizing components. It will appear explicitly when we discuss the effect of polarization squeezing, where this description u3

y χ ψ x

1 1

2χ Polarization ellipse (a)

2ψ u1 (b)

u2

Poincaré sphere

Figure 2.9 Graphical representation of polarization (a) through the two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse and (b) the Poincaré sphere, in both spherical and Cartesian coordinates. Source: From B.E.A. Saleh and M.C. Teich 2007 [3].

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provides the most direct insight in the quantum noise. In addition it allows us to see the correspondence between polarization and other two-mode systems such as the spin of an atom or the time evolution of a two-level system. This is a beautiful generalized graphical representation that has its place in any description of atomic physics or non-linear optics. Finally, it is worthwhile to see the link between the two modes, which exist within the one laser beam, and the two separate beams, which are normally required for the simultaneous detection of the modes. A common example in polarization is the use of the polarizing beamsplitter, which sends all the light including modulation in the H polarization to one output beam and all the light and modulation in the orthogonal V polarization into a separate beam, as shown in left-hand side of Figure 2.8. In this case we would associate the beam travelling to one detector with one mode and the other beam with the other mode. Similarly we could have a beam with two different carrier frequencies and then use a grating to separate the two modes into separate beams and modes.

2.2.3

Multimode Systems

Is it strictly necessary to physically separate the modes to detect them? Not necessarily. A counterexample would be a laser beam that has two separate spatial modes, like two orthogonal Hermite–Gaussian modes (see Figure 2.2), which are detected by a photodetector with many pixels. We can detect the two spatial modes independently by recording the photocurrents from all pixels of the detector and then doing some clever recombination of the currents to represent the individual modes. At no point are the modes spatially separated, but they propagate together. However, the mathematical description is the same as for a set of modes, and beams, travelling completely separate through space. This example shows the importance and need for a multimode description. And there are situations where the number of modes can be much larger than two. It is possible to create beams of light that have N independent modes, or information channels. For example, this could be beams with N different carrier frequencies, with N different spatial modes, or with N separate beams created by an apparatus with a complex components that are sensitive to polarization, frequency, spatial shape, etc. In all these situations we can create a basis of N orthogonal modes, which we can modulate and detect individually. For the case of spatial modes, two modes described by 𝛼i (r, t) and 𝛼j (r, t) are orthogonal if ∫

𝛼i (r, t)𝛼j∗ (r, t) dr = 0

(2.30)

We can have N independent information channels with no crosstalk between any of them, or we can create, through mode transformation, correlations between the modes. The analysis of a beam using a large set of orthogonal basis vectors allows us to determine the smallest possible number N of modes that is required for a specific situation: ( 𝛼1 (r, t), 𝛼2 (r, t), … , 𝛼(N−1) (r, t), 𝛼N (r, t) )

(2.31)

2.3 Statistical Properties of Classical Light

In practice this leads to the mathematical challenge of finding the lowest possible value of N or the lowest possible number of independent degrees of freedom for any given situation. There are many applications of this type of multimode optical physics. In optical communication it allows us to increase the information flux that is communicated. The commonly used terminology is multiplexing. In optical sensing it allows us to isolate exactly the degrees of freedom of the laser beam we wish to detect and suppress other effects on the beam we do not want to detect. In coherent image detection and analysis, we can define an object that we wish to detect as an optical mode and the changes to this object as orthogonal modes. That means each of the higher-order modes contains information about one specific degree of freedom, for example, a translation, rotation, or stretching of the object in the image. These changes can be tracked by detecting only the specific higher-order mode. One interesting example is the translation of a Gaussian beam in the x and y coordinates. The information about this change is stored in the mode that represents the derivative of TEM00 in these two directions, respectively, and corresponds directly to the modes TEM01 and TEM10 , which are shown in Figure 2.2. By recording the intensity changes in exactly these two modes, the position of the centre of the beam can be isolated and tracked with high precision [9]. Many other examples are being developed for precision sensing.

2.3 Statistical Properties of Classical Light 2.3.1

The Origin of Fluctuations

The classical model so far assumes that light consists of a continuous train of electromagnetic waves. This is unrealistic: atoms emitting light have a finite lifetime; they are emitting bursts of light. Since these lifetimes are generally short compared to the detection interval td , these bursts will not be detected individually. The result will be a well-defined average intensity, which will fluctuate on timescales > td . Similarly in most classical, or thermal, light sources, there will be a large number of atoms, each emitting light independently. In addition, the atoms will move, which leads to frequency shifts, and they can collide, which disrupts the emission process and leads to phase shifts. All these effects have two major consequences: 1. Any light source has a spectral line shape. The frequency of the real light is not perfectly constant; it is changing rapidly in time. The description of any light requires a spectral distribution that is an extended spectrum, not a delta function. 2. Any light source will have some intensity noise. The intensity will not be constant. The amplitude of the light will change in time. There will be changes in the amplitude due to the fluctuations of the number of emitting atoms but also due to the jumps, or discontinuities, in the phase of the light emitted by the individual atoms.

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The first point leads to the study of spectral line shapes, an area of study that can reveal in great detail the condition of the atoms. The study of spectra is a well-established and well-documented area of research. However, in the context of this guide, we will not expand on this subject. The relevant results are cited, where required. For the analysis of our quantum optics experiments, we are not concerned as much with the spectral distribution of the light. We are more interested in the noise, and thus the statistical properties, of intensity, amplitude, and phase. Only the intensity fluctuations can be directly measured in the experiments. However, with the homodyne detector, which will be explained in Section 8.1.4, we have a tool to investigate both the phase and amplitude of the light. The measurement of the statistical properties of light can be carried out in two alternative ways: either through a measurement of the fluctuations themselves in the form of temporal noise traces or noise spectra or, alternatively, through the measurement of coherence or correlation functions. While requiring entirely different equipment, the two concepts are related, and in this chapter predictions for both types of measurement will be derived. The emphasis here is on the results and the underlying assumptions and not on the details of the derivation. Many of the details can be found in the excellent monographs by Loudon [7], Louisell [10], and Gardiner [11]. In Chapter 8 it will be shown how these statistical properties can be measured and how they have been used to distinguish between classical and quantum properties of the light. 2.3.1.1

Gaussian Noise Approximation

In many of our discussions, we will be assuming that the fluctuations have a Gaussian distribution over time. This assumption is certainly true for classical random systems that obey the central limit theorem. They are generally applicable for systems where the individual fluctuations are small and the overall field evolves smoothly in time. For example, the amplitude varies in small steps, and the phase evolves smoothly at the timescales at which we are observing. Thermal noise is a typical example. We will also find that the quantum noise of laser beams does obey this limit. In these cases the variance, Eq. (2.34), is the only parameter of importance, and it gives a complete description of the fluctuations. In many applications the Gaussian approximation is valid. However, there are situations in classical and quantum optics where non-Gaussian processes are important. One simple example would be a beam where the amplitude, or phase, is digitally modulated through an on–off switch or controlled by a function with several discrete values. Here the switching can be fast and abrupt. Or consider an apparatus that contains a component with a non-linear transmission function, such that the fluctuations are clipped or limited at a certain level. In these cases we would see non-Gaussian fluctuations. They require more complex mathematical descriptions, and the statistics would include higher-order moments, not just the first-order variance. Non-Gaussian systems are not only interesting but also increasingly important in applications. While they are not common in metrology, we will find that they play a major role in digital communication, in optical logic systems, and in digital data storage and data processing. We will have to consider the validity of the

2.3 Statistical Properties of Classical Light

Gaussian approximation when any of these applications are the motivation for our experiments in quantum optics. 2.3.2

Noise Spectra

A very direct way of measuring the fluctuations, or the noise, of the light is to record the time history of the intensity I(t) and to evaluate the variance ΔI 2 (t) for a certain detection time. The variance of a classical value is defined as ΔI 2 (t) = ⟨(I(t) − ⟨I⟩)2 ⟩ = ⟨I 2 ⟩ − ⟨I⟩2 = Var(I(t))

(2.32)

where ⟨I⟩ is the average intensity and the brackets ⟨· · ·⟩ denote averaging over a detection interval of length td . Similarly, we define the root mean square (RMS) value as √ √ RMS(I(t)) = Var(I(t)) = ΔI 2 (t) (2.33) As we have seen in Section 2.1.7, information can be carried on a light beam as small amplitude or phase modulations. The variance gives limited information about the level of noise at a particular modulation frequency and thus how successfully information can be carried. More useful is the spectral variance or noise spectrum, which gives the noise power as a function of frequency from the carrier. The spectral variance of a variable y is given by the Fourier transform of the autocorrelation function for that variable, ∞

V (Ω) =

1 ei2πΩ𝜏 ⟨y(t)y(t + 𝜏)⟩d𝜏 2π ∫−∞

(2.34)

and can be evaluated experimentally using a spectrum analyser (see Chapter 7). Most commonly we will be evaluating the spectral variances of the intensity, y = 𝛼 ∗ 𝛼, and the quadrature amplitudes, y = 𝛼 + 𝛼 ∗ and y = i(𝛼 − 𝛼 ∗ ). In many practical situations we can write our variable in the form of y(t) = y0 + 𝛿y(t)

(2.35)

where y0 is the average value of the variable and all the signals and noise are fluctuations around the average, carried by the term 𝛿y(t), which is assumed to have zero mean (i.e. ⟨𝛿y(t)⟩ = 0). Under these conditions the spectrum is simply given by V (Ω) = ⟨|𝛿̃y(Ω)|2 ⟩

(2.36)

where the tilde indicates a Fourier transform. A simple example is illustrative at this point. Suppose the fluctuations in our variable are made up of two parts: a deterministic signal, 𝛿ys = g cos(2πΩt), and randomly varying noise that we represent by the stochastic function 𝛿yn = f 𝜁 (t) (f and g are positive constants). The signal and noise are uncorrelated, so we can treat them separately. For the signal we use Eq. (2.36). The Fourier transform of a cosine is a pair of delta functions at ±Ω. Squaring and averaging over a small range of frequencies (say, 𝛿Ω) around ±Ω, we end up with two signals of size g 2 . For the noise it is easier to work from Eq. (2.34). We can take the

39

2 Classical Models of Light

log (ΔI2(Ω))

40

M(Ω2)

Signal SNR

Noise

Ω=0

Ω1

Ω2

Detection frequency, Ω

Figure 2.10 Example of a noise spectrum showing noise at many frequencies and two modulations at Ω1 and Ω2 . This plot has a logarithmic scale and the signal-to-noise ratio can be read off it directly if M(Ω) ≫ Var(I(Ω)).

autocorrelation of the noise to be ⟨𝜁 (t)𝜁 (t + 𝜏)⟩ = 𝛿(𝜏). That is, the noise is random on all timescales; there is no correlation between how it behaves from instant to instant. The Fourier transform of a delta function is a constant. Thus, after integration over the small bandwidth, the noise contributes 𝛿Ωf 2 to the spectrum at all frequencies. The signal-to-noise ratio (SNR) is defined by SNR + 1 =

g2 𝛿Ωf 2

(2.37)

In Figure 2.10 an actual intensity spectrum with AM modulation is plotted. In this case the coefficient g corresponds to the modulation M𝛼0 . The relative intensity noise RIN is the ratio of the variance to the average intensity. When stated as a function of the detection frequency Ω, the integration time td , or bandwidth, has to be taken into account. For a bandwidth of 1 Hz, it is given by RIN(Ω) = Var(I(Ω))∕⟨I⟩2

(2.38)

In classical theory it is in principle possible to make any noise arbitrarily small in the amplitude and phase such that the amount of information that can be carried by a beam of light is limited only by technical considerations. We shall find that this is not true in the quantum theory of light. Strict bounds apply to the amount of information that can be encoded on light of a particular power. 2.3.3

Coherence

Light emitted by a pointlike source has properties closely resembling one single wave. The amplitudes at two closely spaced points are linked with each other, as part of the same wave. In contrast, the amplitudes at two very different points or at two very different times will not be identical. The question how far the detectors can be separated in space, or in time, before the amplitudes will differ depends on an additional property of the light: its coherence. Any source of light has a coherence length and a coherence time. The amplitude at two points within the coherence length and coherence time is in phase and has the same fluctuations. Interference can be observed. Light at two points separated by more than the

2.3 Statistical Properties of Classical Light

Figure 2.11 Stellar interferometer as proposed and developed by A. Michelson and F.G. Pease [12].

δθ

M3

M4 M2

M1 Lens

Detector

coherence length, or the coherence time, has independent amplitudes that will not interfere. One clear example of this concept is the technique of measuring the size of a star, which has a small angular size but is not a point source, using a stellar interferometer. The size of a distant star can usually not be resolved with a normal optical telescope. The starlight forms an image in the focal plane of the telescope, which is given by the diffraction of the light. The image depends on the geometrical shape and the size of the aperture of the telescope and has little to do with the actual shape and size of the object, the star. Diffraction limits the resolution. Only a bigger aperture of the telescope, larger than technically feasible, would increase the resolution. To overcome this limitation, A.H. Fizeau proposed the idea of a stellar interferometer, which was first used by A.A. Michelson and F.G. Pease [12]. The concept of the interferometer is shown in Figure 2.11. In this apparatus the starlight is collected by two mirrors, M1 and M2 , which direct the light via the mirrors M3 and M4 to a telescope. There the two rays of light are combined in the focal plane. In addition we have optical filters in the instrument that select only a limited range of optical wavelengths. The mirrors M1 and M2 are separated by a distance d, and the apparatus is carefully aligned such that the distances from the focal plane to M1 and to M2 are identical. As a consequence we have interference of the two partial waves in the focal plane. An interference system will appear due to the fact that the mirrors M1 and M2 are not exactly at 45∘ to the axis of the telescope. This arrangement is equivalent to a Michelson interferometer with an optical wedge in one arm, and it measures the first-order coherence of the light. For an extended source we get light from different parts of the source reaching the telescope under slightly different directions – let us say with a very small angular difference of 𝛿𝜃. Each wave will produce an interference system, but at a slightly different location in the focal plane. This is due to the additional optical path Δl = d sin(𝛿𝜃) ≈ d 𝛿𝜃 inside the interferometer. We assume that the light from the different parts of the star is independent. Consequently, it will interfere with light from the same part of the star, but not with light from other parts, and the intensity of the interference patterns from the different directions is added. The interference pattern in the focal plane will be visible as long as the path

41

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2 Classical Models of Light

difference is small compared to an optical wavelength. If the separation d between M1 and M2 is increased, the total interference pattern will eventually disappear, since the individual patterns are now shifted in respect to each other. An estimate for the separation at which fringes are no longer visible is d 𝛿𝜃 = 𝜆. In this situation the two individual patterns from the edges of the star are shifted by one fringe period, and all other patterns lie in between, washing out the total pattern. A more detailed analysis predicts the change in the visibility of the interference patterns as a function of the separation d. The interferometer measures the coherence of the light. In this case the coherence is limited due to the independence of the different parts of the star, and the coherence length is the path difference for the light travelling from different edges of the star to the Earth. The corresponding coherence time for the entire starlight is the difference in travel time. We find that interferometry can be used to measure the coherence length and coherence time. Michelson and Pease built an instrument based on this idea and was able to measure star diameters down to about 0.02 arc seconds [12, 13]. There are improved versions of this instrument in operation, in particular an instrument called SUSI (Sydney University Stellar Interferometer), which have achieved resolutions as good as 0.004 arc seconds. However, there are technical limitations. It is very easy to lose the visibility of the fringes, already the wavelength is restricted to an interval of about 5 nm, and even then the path difference inside the interferometer has to be kept to less than 0.01 mm. A wider spectrum, containing more light, would require an even better tolerance. It is very difficult to maintain such a small path difference for the entire time of the observation, while the mirrors track the star and the separation between the mirrors is changed by up to several metres. To avoid these complications, R. Hanbury Brown & R.Q. Twiss proposed an entirely different scheme to measure coherence. Theirs is a very elegant approach that uses intensity fluctuations rather than interference, an idea that had shortly before been used in radio astronomy. In this way they founded a new discipline based on the statistical properties of light. They set out to measure the correlation between the intensities measured at different locations rather than the amplitude and therefore avoided all the complications associated with building an optical interferometer. In their experiment they replaced the mirrors M1 and M2 (Figure 2.11) with curved mirrors, which collected the light onto two different photomultipliers, as shown in Figure 2.12 [14]. The photocurrents from these two detectors are amplified independently and then multiplied with each other, and the time average of the product is recorded. This signal reflects the correlation between the intensities. This apparatus does not require the mechanical stability of an interferometer. However, some questions arise: In which way is this experiment related to an interferometer? How can it provide similar information if we have disbanded the crucial phase measurement? To answer these questions, we remind ourselves that stellar interferometers measure the spatial coherence of light, transverse to the direction of observation. At any one point the phase of the light fluctuates wildly, and so does the amplitude of light from a thermal source. Generally the amplitude and the phase are not directly linked, and both fluctuate independently. However, for two points within

2.3 Statistical Properties of Classical Light

d

x Multiplier

Figure 2.12 Intensity interferometer proposed and developed by Hanbury Brown & Twiss. Source: Brown and Twiss 1956 [14]. Reproduced with the permission of Springer Nature.

a coherence length, the phase and the amplitude are linked to each other. The light will interfere and the intensity fluctuations are correlated to each other. For two points separated by more than a coherence length, the interference will disappear, and so will the intensity correlation. Fortunately for Hanbury Brown and Twiss, and for us, it is a characteristic property of thermal light that the coherence length over which the phase is constant is approximately the same length in which the amplitude changes very little. Thus a measurement of the path length over which the intensity correlation disappears is in most cases identical to a measurement of the phase coherence length. Hanbury Brown and Twiss measured this distance and therefore were able to determine the size of the source. Already in 1956 they measured the diameter of the star Sirius. An improved apparatus was installed in Narrabri, Australia, with an extended baseline. The detectors are moved on a circular track with a diameter of 396 m [15], and the star diameters can be determined down to 0.0005 arc seconds. How does a simple multiplier reveal the correlation between the signal? We can describe the intensity at one detector as the sum of a constant average value, ⟨I⟩, and a zero mean fluctuating term, 𝛿I1 (t) by I1 (t) = ⟨I⟩ + 𝛿I1 (t). Similarly the intensity at the other detector can be written as I2 (t) = ⟨I⟩ + 𝛿I2 (t). While both ⟨𝛿I1 (t)⟩ and ⟨𝛿I2 (t)⟩ are equal to zero, for correlated waves, we get ⟨𝛿I1 (t)𝛿I2 (t)⟩ ≠ 0. If the intensities are perfectly correlated, then 𝛿I1 (t) = 𝛿I2 (t) and the time average of the product will be ⟨I1 (t)I2 (t)⟩ = ⟨I 2 + 𝛿I1 (t)2 ⟩. If the separation of the sources exceeds the coherence length, then the intensities are not correlated and the product 𝛿I1 (t)𝛿I2 (t) randomly changes sign and the average ⟨𝛿I1 (t)𝛿I2 (t)⟩ vanishes. That means ⟨I1 (t)I2 (t)⟩ = ⟨I⟩2 , which is distinctly different to the correlated case. A reduction in the correlation signal indicates that the coherence length has been reached. A study of the correlation signal for different receiver separations allows a measurement of the second-order coherence of the light.

43

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2 Classical Models of Light

2.3.3.1

Correlation Functions

After this introduction and motivation, we will now examine first- and secondorder coherence, and the correlation functions that describe them, quantitatively. First-order coherence is important when two light beams are physically superimposed. As a generic example let us consider the case of two beams being combined on a 50 : 50 beamsplitter, e.g. a semi-transparent mirror, followed by direct detection of the intensity of one output beam. We allow a time delay, 𝜏, to be imposed on one of the beams before the beamsplitter. We can describe the input fields to the beamsplitter by their complex amplitudes 𝛼1 (t) and 𝛼2 (t + 𝜏). We assume the beams are in single identical spatial modes that are completely detected; thus we can ignore the spatial dependence of the amplitudes. For our purposes here we can assume that √ the action of the beamsplitter is simply to produce the output 𝛼out (t) = (1∕ 2)(𝛼1 (t) + 𝛼2 (t + 𝜏)), an equal linear superposition of the inputs. The detector produces a photocurrent proportional to the intensity of the output field, which in turn is proportional to the absolute square of the amplitude, |𝛼out |2 . Our measurement result will then be proportional to |𝛼out |2 averaged over some measurement interval, say, td , such that 1 |𝛼out |2 dt td ∫ 1 = ⟨|𝛼1 (t)|2 + |𝛼2 (t + 𝜏)|2 + 𝛼1 (t)𝛼2∗ (t + 𝜏) + 𝛼1∗ (t)𝛼2 (t + 𝜏)⟩ 2 (2.39)

⟨|𝛼out |2 ⟩ =

where we have used ⟨· · ·⟩ to indicate the time average. Suppose that 𝛼1 and 𝛼2 both derive from the same continuous source oscillating at frequency 𝜈 such that 𝛼1 (t) = 𝛼2 (t) = 𝛼(t) exp(i2π𝜈t). Then ⟨|𝛼out |2 ⟩ = ⟨|𝛼|2 ⟩ + cos(2π𝜈𝜏 + 𝜙) ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩

(2.40)

There is interference between the waves resulting in a modulation of the photocurrent as a function of the path difference. This basic effect is the source of fringes in all first-order interferometry including double-slit experiments. The temporal coherence of the light is quantified by the first-order correlation function, g (1) (𝜏), defined via g (1) (𝜏) =

⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩ ⟨|𝛼|2 ⟩

(2.41)

To see that g (1) (𝜏) measures the coherence of the light, consider the depth of the modulation in the above example. This is given by the visibility , defined by =

Imax − Imin Imax + Imin

(2.42)

Maximum modulation results in  = 1, whilst no modulation results in  = 0. For the output of Eq. (2.40), we find |⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩| ⟨|𝛼|2 ⟩ = |g (1) (𝜏)|

(𝜏) =

(2.43)

2.3 Statistical Properties of Classical Light

Thus, for this simple example, the first-order correlation is equal to the visibility. If the source was a perfect sinusoid, then 𝛼 would be a constant in time and we would have g (1) (𝜏) = 1 regardless of the time delay. This corresponds to an infinite coherence length. More realistically the source will have some bandwidth. Suppose that the source has a Gaussian spread of frequencies Δ𝜈 around 𝜈. This can be represented in Fourier (or frequency) space by 𝛼(Ω) ̃ = 𝛼c exp(−2π(𝜈 − Ω)2 ∕Δ𝜈 2 )

(2.44)

where 𝛼(Ω) ̃ is the Fourier transform of 𝛼(t) and 𝛼c is a constant. Taking the inverse transform gives Δ𝜈 𝛼(t) = 𝛼c √ exp(−πΔ𝜈 2 t 2 ) 2

(2.45)

for our amplitude in time space. Thus our first-order correlation function becomes g (1) (𝜏) =

Δ𝜈 2 1 |𝛼c |2 exp(−πΔ𝜈 2 𝜏 2 ) exp(−πΔ𝜈 2 t 2 )dt td 2 ∫

1 Δ𝜈 2 |𝛼c |2 exp(−πΔ𝜈 2 t 2 )dt td 2 ∫ ) ( π (2.46) = exp − Δ𝜈 2 𝜏 2 2 a Gaussian function of 𝜏 with a width inversely proportional to the frequency spread. For a narrow frequency spread, Δ𝜈 small, we have a large coherence length, whilst for a broad spectrum, Δ𝜈 large, the coherence length becomes very short. It is this first-order coherence length that is measured by the stellar interferometer. Now let us consider intensity correlation measurements. As a generic example we consider direct detection of the intensity of two beams that have not been optically mixed, but one of which has a variable time delay imposed on it before detection. The two photocurrents from the detectors are then sent into a correlator, which multiplies the photocurrents together. Suppose firstly that, as in our previous discussion, the two beams originate from the same source. The output of the correlator will then be proportional to the second-order correlation function, g (2) , which is defined by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨𝛼 ∗ (t)𝛼(t)𝛼 ∗ (t + 𝜏)𝛼(t + 𝜏)⟩ = ⟨I(t)⟩2 ⟨𝛼 ∗ (t)𝛼(t)⟩2

(2.47)

where the I are the detected intensities. From the symmetry of this definition follows the property g (2) (−𝜏) = g (2) (𝜏), and measurements need only to be made for positive values of 𝜏. Whilst the degree of first-order coherence g (1) (𝜏) takes on values between 0 and 1 and is unity for 𝜏 = 0, the second-order correlation is not bounded above. Remembering that the variance of I is given by ΔI 2 = ⟨I 2 ⟩ − ⟨I⟩2 , we can write the second-order correlation at 𝜏 = 0 as g (2) (0) =

ΔI 2 +1 ⟨I⟩2

(2.48)

45

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2 Classical Models of Light

Given that the variance obeys ΔI 2 ≥ 0, we find that the value of the second-order coherence at zero time delay is g (2) (0) ≥ 1

(2.49)

and it is not possible to derive an upper limit. This proof is only valid for 𝜏 = 0 and cannot be applied to other delay times. The only restriction stems from the positive nature of the intensity, resulting in ∞ ≥ g (2) (𝜏) ≥ 0

for all

𝜏≠0

(2.50)

It is possible to relate the degree of second-order coherence at later times to the degree of coherence at zero delay time by using the Schwarz inequality in the form ⟨I(t1 )I(t1 + 𝜏)⟩2 ≤ ⟨I(t1 )2 ⟩⟨I(t1 + 𝜏)2 ⟩

(2.51)

For ergodic or stationary systems, two time averages depend only on the time difference, not the absolute time. For such systems we then have ⟨I(t1 )2 ⟩ = ⟨I(t1 + 𝜏)2 ⟩. Consequently we have ⟨I(t1 )I(t1 + 𝜏)⟩ ≤ ⟨I(t1 )2 ⟩ yielding g (2) (𝜏) ≤ g (2) (0)

(2.52)

For a perfectly stable wave (i.e. zero variance), the equal sign applies and g (2) (𝜏) = 1 for all delay times. Equation (2.52) means that the degree of coherence for any classical wave should always be less than g (2) (0). This is a result that has been contradicted for quantum states of light and has been used to demonstrate experimentally the quantum nature of light as shown in detail in Chapter 3.

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case In most practical cases classical light will be noisy. Only a perfect oscillator would emit an electromagnetic wave with perfectly constant amplitude and phase. There are some relatively simple models of realistic light sources. Chaotic light is the idealized approximation for the light generated by independent sources emitting resonance radiation. A practical example is a spectral lamp. In contrast, thermal light is an approximation of the light emitted by many interacting atoms that are thermally excited and together emit a broad, non-resonant spectrum of light. A practical example is a hot, glowing filament. Thermal light will be discussed in Chapter 3. The source of chaotic light is a large ensemble of atoms that are independent and thermally excited. They all emit light at the same resonance frequency. For simplicity, we assume they are sufficiently slow so that the effect of Doppler shifts can be neglected. The atoms will collide with each other; thus the waves of radiation emitted by each atom will be perturbed by collisions. The noise properties are best derived by analysing the total complex amplitude 𝛼(t) of the wave, which is the sum of the amplitudes emitted by the individual atoms. We can evaluate the coherence function of the total amplitude, and from there we can determine the

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

intensity noise spectrum. For a large number M of atoms, the total wave amplitude is 𝛼(t) =

M ∑

𝛼j (t) = 𝛼0 exp(i2π𝜈0 t)

j=0

M ∑

(exp(i𝜙j )) = |𝛼(t)| exp(i𝜙(t))

(2.53)

j=0

where 𝜙j represents the phase of the individual contributions. For simplicity we assume that all atoms emit into the same polarization. For a large number of atoms, |𝛼(t)| and 𝜙(t) are effectively randomly varying functions that describe the magnitude and the phase of the total amplitude at any time t. If the average time between collisions is 𝜏0 , one can derive [7] a probability (𝜏) that an atom is still in free flight after a time 𝜏 since its last collision which is given by ) ( 𝜏 (2.54) (𝜏)d𝜏 = exp − 𝜏0 The time interval 𝜏0 is usually very short. If we assume that a single measurement takes a time larger than 𝜏0 , the coherence function can be evaluated as ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩ = 𝛼02 exp(i2π𝜈0 𝜏) ⟨[exp(−i𝜙1 (t)) + · · · + exp(−i𝜙M (t))] [exp(i𝜙1 (t + 𝜏)) + · · · + exp(i𝜙M (t + 𝜏))]⟩ In multiplying out the large brackets, many cross terms appear, which correspond to waves emitted by different atoms. Since the phase between the light from different atoms is random, they cancel and give no contribution to the total average. Thus ⟨𝛼 (t)𝛼(t + 𝜏)⟩ = |𝛼0 | exp(i2π𝜈0 𝜏) ∗

2

M ∑

exp[i(𝜙j (t + 𝜏) − 𝜙j (t))]

j=1

= M⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ where 𝛼j (t) is the amplitude emitted by a single atom and thus the entire correlation is completely described by the single-atom correlation function. A single atom emits a steady train of waves until a random phase jump occurs due to a collision. If one neglects the small amount of light that is emitted during the collision (impact approximation), one gets the correlation terms ⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ = |𝛼0 |2 exp(−i2π𝜈0 𝜏)P(𝜏) ( ) 𝜏 = |𝛼0 |2 exp −i2π𝜈0 𝜏 − 𝜏0 and the first-order correlation function ( ) 𝜏 g (1) (𝜏) = exp −i2π𝜈0 𝜏 − 𝜏0

(2.55)

For all types of chaotic light, it is possible to derive the following relationship between the first- and second-order correlation coefficient [7]: g (2) (𝜏) = 1 + |g (1) (𝜏)|2

(2.56)

47

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2 Classical Models of Light

The exact form of both g (2) (𝜏) and g (1) (𝜏) depends on the type and the statistical distribution of collisions. For illustration a numerical example is given in Figure 2.13 showing the various properties and descriptions of a fluctuating light beam. Here the chaotic light is simulated by calculating explicitly the emission from 20 independent sources, all with the same amplitude. The initial phase of these sources is selected at random. The initial conditions are given in Figure 2.13a: all twenty partial waves 𝛼i (t = 0) and the total amplitude 𝛼(t = 0) are shown. The phase evolution of the emitters is simulated by adding to each emitter a change in phase that can randomly vary in the range −0.13π to 0.13π for each time step. Thus each emitter has a random development of its phase, occasionally with a phase jump. For each time interval the total amplitude can be described as one point in a polar diagram, which leads to the distribution of points shown in Figure 2.13b. The vector describes a convoluted path; it moves from one point to another seemingly at random. The density of points can be interpreted as a probability of finding a certain amplitude. The points are scattered around the origin. While the amplitudes would average out, the most probable intensity is not zero. The full time development of the magnitude of the total amplitude |𝛼(t)|, the total phase 𝜙(t), and the scaled intensity I(t) = I0 𝛼 ∗ 𝛼 are all shown in Figure 2.13c–e. The magnitude |𝛼(t)| and the intensity I(t) show dramatic fluctuations, occasionally dropping close to zero. The intensity can be described by the long-term average ⟨I⟩ and the RMS value, both included in Figure 2.13d. For this numerical example, the correlation functions can be calculated directly. Equations (2.41) and (2.47) have been evaluated, with the results shown in Figure 2.13f,g. The results are very close to the predictions for a chaotic source. Over the time interval shown, the first-order correlation functions change smoothly from g (1) (0) = 1 down to 0.15. In the same time interval, g (2) (𝜏) declines from g (2) (0) = 1.8–1.0. There are some oscillations in g (2) (𝜏), which is typical for such a short statistical sample. The analysis of the longer time sample would approach the theoretically predicted smooth decline of the values of g (2) (𝜏). The link between g (2) and g (1) given in Eq. (2.56) is clearly reproduced in this example. In order to predict the intensity noise of the chaotic source of M atoms, we consider the instantaneous intensity I(t) and the averaged intensity ⟨I⟩. We find 1 ⟨I⟩ = ⟨I(t)⟩ = E02 𝜖0 c|𝛼0 |2 2

⟨| M |2 ⟩ |∑ | 1 | exp(i𝜙 (t))| = E02 𝜖0 c|𝛼0 |2 M j | | 2 | j=1 | | |

(2.57)

since all the cross terms average out due to the randomness of the phase functions. The total average intensity is simply the sum of the average intensity emitted by a single atom. In order to evaluate the variance, we require a value for (

1 2 ⟨I(t) ⟩ = E 𝜖 c|𝛼 |2 2 0 0 0 2

)2

⟨| M |4 ⟩ |∑ | | exp(i𝜙j (t))| | | | j=1 | | |

(2.58)

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

5 (a) Vector sum at one time (magnified)

(b) Time development of vector

10

0

0

10 5

5 10

0

5

10

0

10

(c) Magnitude 5

0 100

(d) Intensity

50 Average Int. 0 π

(e) Phase

0 −π 1

2

(g) g(2)(τ)

(f) g(1)(τ)

0.5

1

0 0

τmax

0

0

τmax

Figure 2.13 An example of the statistical properties of chaotic light. The light is simulated as the total field emitted by 20 independent emitters, each with constant amplitude but randomly evolving phase. Part (a) shows the individual vectors at t = 0 and (b) the resulting total phase vector for various times. The time history is shown for the magnitude (c), the intensity (d), and the phase of the field (e). This leads to the first-order (f ) and second-order (g) correlation function.

49

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2 Classical Models of Light

The only terms that will contribute to this value are those where each factor is multiplied by its own complex conjugate. These terms result in ⟨( M )2 ( M )2 ⟩ )2 ( ∑ ∑ 1 ei𝜙j e−i𝜙j E2 𝜖 c|𝛼 |2 ⟨I(t)2 ⟩ = 2 0 0 0 j j ⟨M ⟩ M )2 ∑ ( ∑ 1 2 2 i(2𝜙j −2𝜙j ) i(𝜙j +𝜙k −𝜙j −𝜙k ) = e +2 e +··· E 𝜖 c|𝛼 | 2 0 0 0 j j≠k )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 ⟨M + 2M(M − 1) + · · ·⟩ = 2 )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 (M + 2M(M − 1)) = 2 where · · · represents terms that average to zero. With Eq. (2.57) we obtain ( )2 ) ( ⟨I⟩ 1 (2M2 − M) = ⟨I⟩2 2 − ⟨I(t)2 ⟩ = M M and using ⟨I(t)2 ⟩ = 2⟨I⟩2 for large M, the variance is ΔI(t)2 = (2⟨I⟩2 − ⟨I⟩2 ) = ⟨I⟩2

(2.59)

This is a simple and important result: for a chaotic source the variance of the time-resolved intensity is equal to the square of the average intensity. Calculations for more practical situations, which include Doppler broadening and realistic models of the line broadening, show basically the same result. The intensity noise spectrum will be confined to the spectral linewidth. Outside the linewidth this classical description predicts an arbitrarily low intensity noise spectrum.

2.5 Spatial Information and Imaging 2.5.1

State-of-the-Art Imaging

Imaging is a widely used term. It transfers the spatial properties of an object, typically a 2D cross section, into an image that we either see directly or record in the form of paper or canvas and now more likely in an electronic format as a dataset that can be displayed or printed. Viewing something by eye or assisted with a lens, telescope, or microscope involves imaging, so is any form of recording with a camera. The quality and range of applications of imaging is improving and expanding all the time, with better spatial and temporal resolution, higher sensitivity, and the ability to transform a wider array of physical properties into images. Technology is improving at a rapid pace and with lower costs is making it much more widely accessible. A smartphone is an amazing imaging device for more and more purposes. We are surrounded by images. Figure 2.14 shows four examples from very different situations. Figure 2.14a is a demonstration of the quantum nature of light using Young’s double-slit fringes recorded with increasing photon flux. This is a teaching demonstration by T.L. Dimitrova and A. Weis for the classroom [16].

2.5 Spatial Information and Imaging

(a)

(b)

(d)

(c)

Figure 2.14 A variety of new imaging techniques: (a) teaching demonstration of Young’s double-slit experiment; (b) 3D image taken with short pulses, photon packets, and a depth-resolving fast detector; (c) scan of a brain tumour using PET-MRI, single-photon detection reconstruction, and false colour display; and (d) image of a live neurone in a rat brain created with scanning two-photon microscopy. Source: From [16–19].

Figure 2.14b shows a 3D reconstruction of an image where not only the position of the photons but also their arrival time is recorded using short light pulses, photon packets, and an optical time-of-flight (TOF) technique. In this example from 2005, the depth resolution is less than a millimetre, and the resolution is 64 × 64 pixels [17]. From the data a 3D image is reconstructed, and this technology is constantly improving. Systematic progress is being made to add more pixels and enhance the specifications of the single-photon detector arrays, reaching 256 × 64 pixels by 2012 [20] and more since, by creating new applications such as TOF LIDAR imaging systems for autonomous devices. Figure 2.14c shows a fictional slice through the brain of a person who has tumour. The image has been generated by multimodal PET-MRI [18]. This medical diagnostic technique uses time-resolved imaging of individual photons generated in scintillating crystals when hit by gamma rays, which are created in a nuclear decay. The image shows false colour coding, which scales with the local production rate of gamma rays and conversion into detectable photons. This signal is interpreted as an unusual activity in the brain. Finally, Figure 2.14d shows an in vivo scan of a live neurone in a rat brain. The neurone has been filled with a fluorescent dye, and the dye is stimulated in a two-photon (2P) process with a focussed laser beam. The focal region is scanned in the three spatial dimensions,

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many slices were taken, and a 3D image was reconstructed. This well-known confocal microscopy technique can be used to visualize the neurone, to modify the neurone with lasers beams [19], and in the future to record the electrical activity and the information flow between neurones optically. It is clear from Figure 2.14 that the generation and detection of individual photons plays a major role in these state-of-the-art examples of imaging. Photons are a concept that we require to understand how the image is created, and in particular Figure 2.14a shows explicitly that the statistics of the photons influences the quality of the images. When the intensity, and thus the photon flux, is low, we can clearly see the graininess of the images. The contrast improves with more light and more photons being detected. This is barely noticeable with our own eye in everyday situations but is rather pronounced when modern detectors with high sensitivity are used.

2.5.2

Classical Imaging

Imaging refers in general to the creation of a 2D intensity and colour distribution I(X, Y , 𝜆), a function that we see with our eyes or record for posterity and for data analysis. Most of these images involve colour, which means a dependence on the spectrum of wavelengths 𝜆. With our well-trained vision system, we can see images in three dimensions (3D) I(X, Y , Z, 𝜆), and most people can distinguish more than 100 000 hues of colour, which are mixtures of wavelengths. And we do this with a reasonable time resolution of a few tens of milliseconds, which can be described by a function with a total of 5 degrees of freedom, I(X, Y , Z, t, 𝜆). We all have the capacity to deal with such a complex situation with our vision system. And we all operate devices that record and process images, nowadays in digital and pixelated format, and that extend the capability of the human eye. Central to the description is an object distribution of complex optical amplitude 𝛼(x, y) = 𝛼0 (x, y) exp(i𝜙(x, t)) with 𝛼0 (x, y) the magnitude of the optical field and 𝜙(x, y) the corresponding optical phase across the object plane. These are the important components if we deal with coherent imaging where the phase of the wave front is stable, propagating in well-defined ways, and detectable. In practice this might require working with a very narrow bandwidth Δ𝜆 or a laser beam. Note that it is possible to build special imaging devices using interferometric tricks, which can make the phase distribution visible even for light with low coherence, for example, through phase contrast imaging and microscopy or white light interferometry. The object distribution is created by the illumination of the object plane from a light source, and various effects create the light distribution, including straight transmission of the light, specular reflection or scattering of the object, fluorescence of a concentration of dye, etc. We can treat each point in the object plane as a source of radiation, which follows through to the image system according to the rules of wave propagation. For incoherent illumination imaging we only require the intensity distribution Iobject (x, y). In the detection plane we normally form the intensity distribution Iimage (X, Y ) = |𝛼image (X, Y )|2 , the intensity distribution we are going to see. To have a reasonable intensity, we can integrate the intensity

2.5 Spatial Information and Imaging

over a small area in the object plane of the size 𝛿x times 𝛿y: 𝛿x,𝛿y

Iobject (x, y) =

∫−𝛿x,−𝛿y

|𝛼object (xi , yj )|2 dx, dy

(2.60)

Optics is generally concerned with the transfer of the light to see how the amplitude 𝛼object (x, y) in the object plane is transformed into the image intensity Iimage (X, Y ). For the propagation of this image through an optical system, we should keep track of both amplitudes and the optical phase terms. This will allow us to consider both coherent imaging, including speckle, holography, and spatial filtering. It is common to describe the properties of an imaging system by the mathematical link between the object and the image, in particular the image of a single point, or pixel in the object plane, a delta function 𝛿(x0 , y0 ). The mathematical link between object and image is conventionally done in two steps – first the scaling function s(x, y) of the image, which includes magnification, stretching, and distortion, and second a function that images each object point at location (x, y) in the object plane into a point at (X, Y ) in the image plane: Iimage (X, Y ) = Iobject [s(x), s(y)]

(2.61)

In the simplest case this could be a magnification by a factor M: (X − X0 ) = M(x − x0 ) and (Y − Y0 ) = M(y − y0 )

(2.62)

where (x0 , y0 ) and (X0 , Y0 ) are the origin of the coordinate systems in the two planes and are connected by the central ray, pointing in the direction of the propagation of the wave and perpendicular to the wave front. In many cases there will be distortions, which means that s(x, y) takes on a far more complex form. This can be simulated by ray tracing where we distribute the intensity at the point Iobject (x, y) into several components emitted in different directions and each is traced through the imaging system, see Figure 2.15. We now have access to very powerful software routines that can do this for complex lens systems. In the ideal case the rays would meet again in a point. In reality they will form a distributed function, in the simplest form a symmetric Gaussian-like distribution. The image of a single point, or delta function, leads to a point spread function psf(X,Y ) for the amplitude or PSF(X,Y ) for the intensity distribution in the image plane around the point (X, Y ). In most cases we can assume that the point spread function is constant across the image plane and just one function PSF(0,0) (dX, dY ). The complete image can then be evaluated as a convolution of the object with one constant PSF: 𝛿X

Iimage (X, Y ) =

𝛿Y

∫−𝛿X ∫−𝛿Y

Iimage (X, Y )PSF(0,0) (dX, dY )dX, dY

(2.63)

This has the effect of reduced spatial resolution. A great deal of engineering and ingenuity has gone into optimizing the PSF, and this is what we gain when we purchase a superior and expensive lens system, camera, or microscope. A very powerful method to describe the imaging process has been Fourier optics. It is very practical to describe the image quality and in particular the resolution of the image by analysing the object in terms of a series of orthogonal

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y Light source Image = detection plane

x

Image system Object amplitude

Object intensity

a (x, y) Φ (x, y)

I (x, y)

Coherent

Y Yk Xj

Incoherent

X

I (X,Y, Ω)

Wave propagation Ray approximation

Detector pixels digitization N (X j,Y k)

Point spread function Resolution

Figure 2.15 Schematic of classical imaging.

spatial harmonic functions. These are normally given in dimensionless units of oscillation per full frame of the image. The spatial frequencies describe the modulation of the amplitude across the image frame, starting with a constant average amplitude F0 and then contributions Fobj,k and Fobj,k at k = 1, 2, 3, … and l = 1, 2, 3, … oscillations per image frame in both directions. This reaches up to a maximum at the Nyquist limit of k = N∕2, l = M∕2 oscillations per image where N and M are the number of pixels considered. This series of harmonics, added up in both directions and all spatial frequencies, is equivalent description to the object amplitude: 𝛼object (x, y) =

N∕2 M∕2 ∑ ∑ k=0 l=0

( Fobj,l cos

2πk(x − x0 ) xmax

(

) Fobj,l cos

2πl(y − y0 ) ymax

)

(2.64) Alternatively, we can use a continuous Fourier transform. One of the reasons for the usefulness of this transformation in optics is that the Fourier analysis can be done directly by optical means. It is the distribution in far-field diffraction plane and also in the plane of a perfect lens system. For collimated input beams in the focal plane of the lens, this Fourier description provides direct access to the understanding of the limits of resolution and the impact of diffraction on the imaging system. This concept, made popular by scientists such as Ernst Abbe in Germany and A. J. Fresnel in France, in the nineteenth century allowed the progress of modern lens design. They showed that the high spatial frequencies are essential to maintaining the resolution. An imaging system needs large numerical aperture in order to achieve high resolution. The notion of diffraction-limited resolution means that the actual PSF is very close to the

2.5 Spatial Information and Imaging

Figure 2.16 Phase distribution described by the low-order Zernike polynomials. 0

Z0

–1

1

Z1

–2

Z2

Z1

0

Z2

2

Z2

PSF given by the aperture size and geometry of the system. This theory was first developed for incoherent light where the intensity spread function is considered. There is an extensive area of optics design that describes the propagation of the wave front with a different set of functions, known as the Zernike polynomials. This is a different set of orthogonal spatial functions that are used to describe the wave front when it propagates through the lens system. It is specifically chosen to describe the properties of the imaging process in terms of the most common distortions that can be observed. The first six terms are shown in Figure 2.16. These polynomials are both a mathematical basis to describe the optics and a tool for the experienced expert to describe the quality and details of the imaging system, and one by one they describe tip in x, tilt in y, defocus, stigmatism, coma, etc. A quantitative description of the Zernike polynomials can be found in many optics textbooks, such as the classic text written by Born and Wolf. There is widespread literature about the use of these mode functions in lens design and vision correction. More recently these modes of light are playing an increasing role in the dynamic compensation for wave front distortions. With the use of fast spatial light modulators, wave front detection methods, and feedback algorithms, it is now possible to compensate dynamically for the fluctuations that occur in the atmosphere above ground-based telescopes. The quality of the image and optical focussing can be improved using adaptive optics techniques. They can also be used to correct distortions that occur when imaging structures inside biological samples, particular for in vivo studies of brain tissue. These techniques have become much more accessible and powerful with the advent of fast multi-pixel spatial light modulators and more and more powerful control algorithms. 2.5.3

Image Detection

Finally, the intensity in the image plane is sampled by the active components of the eye or the electronic detector. In the case of the human eye, these are the receptors, cones, and rods, which are sampling the intensity and convert them into neurological signals to be processed by the optical nerve and the brain. In technical applications we now have multi-pixel detectors, commonly now with thousands of pixels in both directions, N > 1000, M > 1000. In each pixel the amount of light is integrated across a pixel and also in time due to the finite response time of the detector, and this is converted into a photocurrent, ideally proportional to the intensity or averaged over time and stored as an electric

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charge. The abundance of photodiodes, with fast response time, charge-coupled devices (CCD), or complementary metal–oxide–semiconductor (CMOS) detectors now allows optimization for every imaging task. More than 10 megapixels and less than 10 ms exposure time are now easily achieved. For practical reasons there is a strong demand for image compression techniques; otherwise the information content could not be easily handled, in particular for video recordings. There are several algorithms, and those most widely used, JPEG for still images and MPEG for video images, are based on a projection on a set of 2D basis functions. In the case of JPEG, this is a set of wavelet functions that are related but not identical to the harmonic functions in Eq.(2.64). Normally, only the magnitude of these contributions is stored. The resolution of the stored image depends on the order to which this projection is extended. In most cases a trade-off is made between the resolution limits due to data storage capacity and the properties of the image system. The technical capacities are increasing, and with this the quality is improving, leading to the developments of new standards such as ultra high definition (UHD) and 4K. There is now a convergence of the standards that define detection and displays, resulting in very realistic images. 2.5.4

Scanning

An alternative imaging scheme is to scan a small well-defined spot, for example, to project an image or to record absorption or fluorescence as shown in Figure 2.17. Many different projection displays have been developed over the last decade, initially incoherent, using various forms of modulators, lamps, or more recently LEDs. The data from the recorded image are used to control the spatial intensity modulators. As an alternative, coherent laser beam can be used, which are scanned either with mechanical mirror scanning systems, such as galvanometers or spinning prisms as used in some laser printers, or alternatively with spatial phase modulators and optical arrangements using concepts similar to holography. These types of laser projection imaging systems are now becoming widespread in commercial cinema displays with the UHD standard or even higher pixel count. An equivalent imaging technique is laser scanning for imaging in LIDAR, the optical equivalent of RADAR that uses the detection of the reflected light synchronized to the scanning to recreate an image. Here the scanning is over a wide angular section, and the detector might contain just one pixel, a bulk detector, or alternatively a chip with many pixels. Short pulses, fast scanning speed, and excellent synchronization are important in order to measure distances with a resolution of centimetres or even millimetres. Sufficient contrast is achieved by developing detectors with a high dynamic range. LIDAR used to be a scientific speciality for atmospheric research or specific defence applications. However, since 2015, the need for vision systems in autonomous vehicles has created a rapid development and expansion of this imaging technique. At the microscopic scale the equivalent scanning techniques are used in scanning beam microscopy and confocal microscopy, where the ability to focus a laser down to the size of about one wavelength with a high-quality high numerical aperture lens system is used to produce high-resolution images. The basic idea is

2.5 Spatial Information and Imaging

Laser Scanning unit

Laser beam focussed at the mask

Beamsplitter

Time recording, synchronized with scanner

Object plane, with mask

I1 ( t)

or fluorescent dye

Bulk detector with one element

Figure 2.17 Schematic of a scanning imaging system. In this example a mask is illuminated with a scanned focussed laser beam, and all the intensity is recorded with a large single-element bulk detector. Alternatively a structure filled with a fluorescent dye could be placed in the object plane, and all the light emitted at the focus is recorded by this detector. A time series is recorded and stored as a stack of slices from which a 2D or 3D image can be reconstructed.

that the sample under investigation contains a fluorescent dye that is excited by the laser beam. One dye molecule requires a single photon with matching energy for excitation and emits one other photon, with lower energy, into an arbitrary direction. This fluorescent light is detected with a large numerical aperture lens, in order to increase both sensitivity and resolution. The spatial resolution is given by the size of the laser beam, typically a Gaussian beam with a focus inside the sample. However, the fluorescence can come from many parts of the beam, and this technique works well with a thin sample. In order to produce more localized images, 2P excitation is used. Here the probability of excitation is proportional to the square of the beam intensity, or local photon flux, and thus the fluorescence is strongly concentrated in the location of the laser beam focus. In this way spatial resolution close to the diffraction limit can be achieved in the x and y direction, with typically 10 times lesser resolution in the z direction. One elegant technique to scan the beam is holographic projection with a spatial phase modulator that allows fast scanning in all three dimensions. After recording and storing multiple x, y slices, each with a specific value of z, it is possible to reconstruct a complete 3D representation of the fluorescence in the sample, which can be stored and displayed from any point of view. This technique has widespread use in biology and in particular neuroscience, as shown in figure 2.14d where one individual neurone is being imaged in a living

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sample of brain tissue. This is very powerful and in future can be extended to functional imaging of the activities inside an organ or a cell. For example, with the help of a dye or other substances, which are sensitive to the local electric fields, we can not only display the spatial structure of the neurone but also its electrical activity. The action potential and information flow can be imaged. Considerable progress is being made in developing more and more advanced image scanning techniques. 2.5.5

Quantifying Noise and Contrast

Spatial and temporal resolution is one specification of image quality. How can we describe the quality of the object and image distributions in terms of clarity, noise, and contrast? One universal approach is to analyse the SNR. Let us consider this first for a scanned image, where the entire information is contained in a long time series. The total time signal can be described, after Fourier analysis, by a set of signal components at modulation frequencies Ωj with amplitudes 𝛼j for each component. In order to describe the quality, we also need to consider the random fluctuations or noise in the time series. This could be due to statistical effects in the light intensity or due to noise generated in the detection device. To make the effect of noise visible, we compare recordings of the image, which includes both signal and noise, [S + N], and of the noise alone. Assuming that the noise is Gaussian in its statistical properties, we can quantify the noise by evaluating the variance VN for a long time interval. We might find noise that is stronger at certain frequencies Ωj , or we might find white noise with the same strength at all frequencies. Then we can apply the same process to the signal and determine [S + N](Ωj ). For a strong high contrast image, each simple harmonic signal component at Ωj contributes a signal that is proportional to the square of the modulation amplitude. For such a large modulation amplitude 𝛼j (Ωj ), we can approximate the SNR as SNR(Ωj ) ≈ Var([S + N](Ωj ))∕Var(N(Ωj )) ∝ |𝛼j (Ωj )|2

(2.65)

This is used as a description of the quality of the time series, and once we place the scans together in an image, it can also describe the quality of the 2D images. For a very simple example, such as the fringes shown in Figure 1.1a, which have a cosine intensity distribution, we can obviously use this process, now replacing the time with the spatial coordinate x and the time frequency Ω with a spatial frequency 𝜈. The SNR(𝜈) for such a single harmonic spatial function describes the quality, or contrast, of this particular image of the modulation. Let us expand this to all other spatial frequencies and consider the SNR separately for each spatial Fourier component 𝜈j resulting in a spectrum SNR(𝜈, 𝜇), where 𝜈 and 𝜇 describe the spatial frequencies along the x and y axes. We can describe the quality of the object function by such a 2D contrast function and is linked directly to the contrast or visibility of the object. The formal process includes an extension of Eq. (2.65) into two dimensions. The equivalent of white noise required is now independent of the position anywhere in the image plane. Towards high spatial frequencies 𝜈, 𝜇, the signal will get smaller, and the contrast will drop off due to diffraction effects. Setting a practical minimum value

2.5 Spatial Information and Imaging

for the SNR(𝜈, 𝜇), we can define the spatial resolution of the object function. In practice this is tested with a variety of object functions, for example, in the various forms of test charts, which are widely used in photography or optical testing. They include features of smaller and smaller size, corresponding to a series of spatial frequencies. An important specification in the design, manufacture, and testing of imaging systems is the transmission of the SNR as a function of spatial frequencies. Low-quality optics lowers the contrast and the resolution. 2.5.6

Coincidence Imaging

There is also an alternative process of imaging where two beams of light correlated in both intensity and direction of propagation are used and the image is formed through coincidence detection. Each beam is directed onto separate detectors, in this case by the use of a beamsplitter and mirror (see Figure 2.18). One detector (2) has spatial resolution. The other is a bulk detector with only one spatial element and records the total intensity of the beam (1). The object to be imaged is inserted into the beam (1). Is it possible to obtain spatial resolution despite the use of a bulk detector and the imaging system being on the wrong beam? By recording only the correlated component of the photocurrents between the two photodetectors, a clear image can be created. By recording the coincidence counts for all outputs, simultaneously or in a scanning sequence, the information from the bulk detector (1) will provide the required information of how much light has been absorbed in any specific location of the mask. This might appear to be a complex arrangement. However, it has the major technical advantages that it might not be possible to build the imaging system Mirror

Imaging system Detector 2 with spatial resolution Y Yk

Beamsplitter

ed lat re m and r Co bea ns s tio itie ec ns dir inte

Xj

X

I2 (X,Y, Λ1)

Object plane, with mask

I1 ( Λ2)

Detector 1 with one element

Correlation processing

Figure 2.18 Schematic of coincidence imaging using pairs of optical rays with correlated directions, one single-element bulk detector, and one detector with spatial resolution.

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around the mask, for example, because it is inside a small container, in a vacuum system, or an environment that does not allow direct optical access. This type of imaging has been given the popular name of ghost imaging, presumably because there is nothing tangible in front of the conventional imaging system. It has been shown that this system works with coherent light such as laser beams, as well as with light that has low coherence and with light that has thermal fluctuations. The real strength comes once we manage to have beams with different wavelengths fully correlated. For example, beam 1 could be at a wavelength, such as infrared, where no detector with spatial resolution is available, while beam 2 can be conveniently imaged. In all these situations we can make use of the rules of classical optics to synthesize an image from the coincidence counts. First, demonstrations have been reported [21], and the goal is to create images of otherwise invisible objects, for example, clouds of gas such as methane or carbon dioxide that have absorption lines only in the infrared where it is difficult to build tunable light sources. 2.5.7

Imaging with Coherent Light

Note that in the case of coherent imaging, for example, with a monochromatic laser beam, the propagation is slightly more complex due to the diffraction processes than the predictions from ray optics. This would require full analytic solutions and can also be summarized in the form of propagating closed forms of laser modes, such as the TEM modes shown in Figure 2.2. One frequent situation is the focussing of a laser beam into a second waist at locations that do not simply follow the normal lens equations used in incoherent illumination. A second area where coherence will show dramatic effects is the illumination and imaging of irregular objects, such as rough surfaces or liquids with scatter centres, which will create a very complex diffraction pattern known as speckle. This pattern is predictable but also responds to the slightest changes in alignment and thus appears unsteady due to the influence of mechanical vibrations. Thus speckle is normally seen as an unwanted side effect of coherent illumination. 2.5.8

Image Reconstruction with Structured Illumination

Images can be formed using a single-pixel photodetector by illuminating the object with a range of specifically chosen patterns and recording the reflection from each of these patterns. The general principle is that the total reflection will be different for each pattern and that with a clever algorithm it should be possible to reconstruct the spatial details of the object. Clearly great care has to be taken in selecting the set of patterns, which should be orthogonal in the sense that the information created by each individual pattern is as independent from the others as possible. Various sets of patterns have been tested over time; starting in the 1970s [22], the algorithms have been optimized, and it is now possible to create useful images. The system can be portable and fast. An impressive demonstration comes from the team led by Miles Padgett at Glasgow of the real-time detection of methane gas leaks using a single-pixel camera [23]. In this case 256 different masks are used, and imaging with an equivalent of

2.5 Spatial Information and Imaging

16 × 16 pixels at 25 frames per seconds is demonstrated, all with a single-pixel detector. With improved computing power, many more such applications of image reconstruction will emerge [24]. 2.5.9

Image Analysis and Modes

In many situations the imaging process is not completed when the intensity distribution, or photon flux, has been recorded. The image is analysed for its content. For our own vision we have a very sophisticated processor for this in our optical nerve and the brain. This can analyse images with remarkable speed and accuracy. In technical terms we can analyse one image through its characteristic features, for example, we can enhance or suppress the contrast or certain colours or make edges more visible. This can be done by processing the photocurrents from the various pixels of the detector. It is also possible to filter out certain shapes in the image, which we could achieve by adding the photocurrents from a group of pixels. The analogue equivalent is that we consider a certain mode of light, which correspond to intensity distribution and wave front shape. We can expand the actual image at the detector plane over a whole set of such modes. A mode can describe a specific feature we are looking for. It can be a local feature, for example, a face or a feature in a medical image, which can be represented by a specific weighted sum of pixels. Or it could be a recurrent feature, such as regular set of windows in the image of a high-rise building or a specific set of tracks in the image of an electronic circuit. The detection mode is a very versatile concept. Once defined we can track it from image to image. Face recognition, fingerprint recognition, and similar biometric schemes are one form of image analysis. It can be implemented directly by combining the outputs of many pixels in a clever way. 2.5.10

Detection Modes and Displacement

This approach allows us to detect spatial changes within the detection plane. We can determine the movement of the mode, how far and how fast. We can measure how it rotates, expands, or shrinks. That means we can detect changes in specifically designed degrees of freedom by comparing images or analysing a whole string of images recorded like a video stream. In this way spatial measurement with beams of light can be seen as a special form of imaging by defining a specific mode u(X, Y ) and tracking its changes. One of the simplest examples is to use a round beam of light. This is our mode that we place on a detector with two pixels, or two groups of pixels, and we measure the difference of the photocurrents. The mode that comes from the object is round. The detection mode is a plane cut in half. If the beam moves, we will get a strong signal for the displacement as the symmetry between the top and bottom half is broken. This simple device can measure the displacement with remarkable accuracy and high intensities; this can be much less than the diffraction limit, only limited by the SNR. In the language of modes, we can define a mode that represents the displacement of the beam, with the new mode for displacement in X defined as uΔX (X, Y ) = 𝜕u(X, Y )∕𝜕X.

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The device should now have a detection mode udet , which is matched well to uΔX . The split detector mentioned above is a good but not perfect approximation. Let us use the example of a coherent laser beam to derive the optimum detection scheme. The beam produced by a perfect laser can be described by the mode that is shaped inside the laser by the resonator where a set of mirrors build up a powerful internal standing wave selectively amplified by the gain of the pumped laser medium. The resulting output beam is the TEM0,0 mode (see Figure 2.5), the lowest order of the set of Hermite–Gaussian modes described in Eq. (2.11) and shown in Figure 2.5. A beam perfectly aligned on the centre of the split detector beam is described by 𝛼0,0 (x, y, z), with the beautiful property that it is transformed into another TEM0,0 mode when imaged with a lens, in the limit of the paraxial approximation. We can easily propagate a laser beam through an imaging system and will have a TEM0,0 mode in the detection plane. When we now consider a shift of the beam, using Eq. (2.11), we find that the first derivative 𝜕𝛼0,0 ∕𝜕x = 𝛼1,0 . This means that all the information of the movement of the beam is contained in a TEM1,0 mode. Measuring this mode is the best and complete way of measuring the shift of the beam. The optimum device is a detector that has the detection mode TEM1,0 . Similarly we can detect shifts in the y direction. We can expand this idea further and find that each of the TEMl,m modes, shown in Figure 2.2, represents one independent degree of freedom, such as tilt and expansion. Generalizing this concept further, we can see that imaging can be interpreted as a transformation from one optical mode into another. We start with an illumination mode that enters the object plane and creates the mode uobj that contains the amplitude and phase distribution. This is a new mode, which is projected through the imaging system to the detector. There we have a detector mode udet , which is defined by the shape of the detector and its pixels and the wiring and combination of pixels that is used. The detection process is now a projection, or overlap, of the image of uobj onto udet . With a properly chosen detection mode, we can measure the change of the spatial properties of uobj in one well-specified degree of freedom. It is quite possible to do this for several degrees of freedom at the same time, by monitoring several detection modes, ideally all defined orthogonal to each other. A straightforward example is displacement and expansion in two orthogonal directions, or rotation in opposite directions [9].

2.6 Summary We have now a set of tools to describe the properties of a beam of light: it is a mode with a set of DC values and we can describe the power that can be measured both in space and in time. In addition, we can communicate information with the beam through modulations using the AC sidebands. Communication of data is now the most widespread use of light; it has led to a whole industry of photonics. Noise limits the quality of communication. We can describe the fluctuations of the light as a spectrum or alternatively through correlation functions. This noise is intrinsic to the way the light is generated, and we distinguish between incoherent light, such as sunlight, coherent light from a laser, and all cases in between.

References

Correlation measurements have opened new ways in determining the properties of the light, with many applications ranging from astronomy to microscopy. Finally, we have so many applications of such a diverse range of techniques in imaging. Initially in the domain of human sight and photography, this is expanding rapidly with the ever-improving technology for detection, image processing, and reconstruction. Optical engineering has found many alternative approaches to image the very large and the very small and many ways to show us the dynamics of objects and effects and processes that were invisible before [25]. The field of classical optics is wide, versatile, and beautiful. However, all of this relies on photons and their special properties.

References 1 Kogelnik, H. and Li, T. (1966). Laser beams and resonators. Appl. Opt. 5:

1550. 2 Siegman, A.E. (1986). Lasers. University Science Books. 3 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 4 Abramovitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions.

New York: Dover. 5 Askin, A. (1970). Acceleration and trapping of particles by radiation pressure.

Phys. Rev. Lett. 24: 156. 6 Askin, A. (1978). Traping of atoms by resonance radiation pressure. Phys. Rev. 7 8 9

10 11 12 13 14 15 16 17

Lett. 40: 729. Loudon, R. (1973). Quantum Theory of Light. Oxford: Oxford University Press. Hänsch, T. (2005) Nobel Prize Lecture 2005. http://www.nobelprize.org/nobel prizes/physics/laureates/2005/hansch-lecture.html. Delaubert, V., Treps, N., Harb, C.C. et al. (2006). Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam. Opt. Lett. 35: 1537. Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Gardiner, C.W. (1991). Quantum Noise. Berlin: Springer-Verlag. Michelson, A.A. and Pease, F.G. (1921). Measurement of the diameter of 𝛼 orionis with the the interferometer. Astrophys. J. 53: 249. Pease, F.G. (1931). Interferometer methods in astronomy. Ergeb. Exakten Naturwiss. 10: 84. Brown, R.H. and Twiss, R.Q. (1956). Correlation between photons in two coherent beams of light. Nature 177: 27. Brown, R.H. (1964). The stellar interferometer at Narrabri observatory. Sky Telescope 28: 64. Dimitrova, T.L. and Weis, A. (2008). The wave particle duality of light: a demonstration experiment. Am. J. Phys. 76: 137. Niclass, C. and Charbon, E. (2005). A single photon detector array with 64 × 64 resolution and millimetric time depth accuracy for 3D imaging.

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18 19 20

21

22 23 24 25

In: ISSCC. 2005 IEEE International Digest of Technical Papers. Solid-State Circuit Conference, 364. https://doi.org/10.1109/ISSCC.2005.1494020. Charbon, E. (2014). Single photon imaging in complementary metal oxide semiconductor processes. Trans. R. Soc. A 372: 20130100. Go, M.A., Choy, J.M.C., Colibaba, A.S. et al. (2016). Targeted pruning of neuron’s dendritic tree via femtosecond laser dentrotomy. Sci. Rep. 6: 19078. Niclass, C., Ito, K., Soga, M. et al. (2012). Design and characterisation of a 256×64-pixel sinle-photon-imager in CMOS for a MEMS-based laser scanning time-of-flight sensor. Opt. Express 20: 11863. Kim, C.C. and Kanner, G. (2010). Infrared two-color ghost imaging using entangled beams. Proceedings Volume 7815, Quantum Communications and Quantum Imaging VIII; 781503. Pratt, W.K., Kane, J., and Andrews, H.C. (1969). Hadamard transfer image coding. Proceedings of IEEE57, 58. https://doi.org/10.1109/PROC.1969.6869. Gibson, G.M., Sun, B., Edgar, M.P. et al. (2017). Real-time imaging of methane gas leaks using a single-pixel camera. Opt. Express 25: 002998. Altmann, Y. et al. (2018). Quantum-inspired computational imaging. Science 361: 2298. SPIE International Year of Light (2015). Celebrating Light.

Further Reading In this area is numerous. Highly comprehensive reference books are: Born, M., and Wolf, E. (1959). The Principles of Optics. London: Pergamon Press. Mandel, L., and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press.

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3 Photons: The Motivation to Go Beyond Classical Optics Although classical optics is very successful in describing many aspects of light, it is not all inclusive. Many fascinating effects are due to the quantum nature of light; they are the foundation of quantum optics. This chapter introduces the concept of the photon as the smallest detectable quantity of light and develops some of the basics required for modelling quantum optics experiments.

3.1 Detecting Light So far we have considered light to be just one form of electromagnetic waves, much like radio waves. The detection of radio waves and of light is fundamentally different. Radio waves are detected with an antenna, which is a macroscopic structure made out of many atoms. The waves induce an electric field in the antenna, a movement of many electrons. This leads to an electric signal that can be amplified and processed. The signal is proportional to the magnitude of the electric field. In contrast, the detection of light relies on the interaction with individual atoms. The microscopic, or atomic, properties of the material of the detector determine the detection process. A photodetector does not measure the field; it cannot follow the frequency 𝜈 of the oscillations, typically of the order of 1015 Hz. It can, however, determine a transfer of energy from the radiation field to the atoms in the detector. And from this process we can conclude properties of the light intensity. What are the experimental details of the process of photodetection? One fundamental fact is the resonance-like property of the interaction. Atoms interact only with radiation that matches their eigenfrequencies. The absorption and emission of light results in very specific spectra. For free atoms, as those in a gas, these spectra have a resonant structure; they consist of several narrow, well-defined lines. Thus, free atoms are not suitable for a general-purpose detector. Consequently, we use a material with many interacting atoms with a broad absorption spectrum. But the fact remains – individual atoms absorb the radiation that matches their individual eigenfrequencies. One well-established detector is the photographic plate. The atoms in the photosensitive emulsion can absorb light. One individual absorption process can lead to the chemical change of an entire grain in the emulsion and therefore A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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create one dark dot on the plate. This technique has one major technical problem: long exposure times. Not every photon incident leads to the change of a grain. It requires a lot of intensity to expose a plate. It is also far more convenient to deal with electric signals. Thus we use the photoelectric effect. Very early experiments with this effect showed that radiation interacts with such a detector by the absorption of multiples of the energy h𝜈 [1] (Figure 3.1). The probability det per unit time of detecting one individual photoelectron averaged over the detection time interval tmeas scales with the integral of the intensity det =

D h𝜈tmeas ∫0

tmeas

I(t ′ )dt ′

(3.1)

This equation applies to any form of the photoelectric effect. The response time is very similar. The conversion efficiency D depends on the frequency of the detected light and on the material used. Each individual detection process results in one individual electron. This is not sufficient to produce a signal that can be processed. Several techniques have been invented to turn one electron into a measurable pulse of many electrons. This has led to the development of a range of modern photodetectors, such as photomultipliers, photodiodes, and charge-coupled devices (CCDs) and complementary metal-oxide semiconductor (CMOS) sensors [2]. For technical detail see Chapter 7. Whilst different in their technology, they all produce a photocurrent that is proportional to the light intensity, and the detector circuitry averages over a time tmeas , which is much longer than a cycle of the oscillation. The link between the photocurrent i(t) and intensity I(t) is given by i(t) =

D′

tmeas

tmeas ∫0

I(t ′ )dt ′

(3.2)

where D′ describes the sensitivity and the gain of the detector. Detection times as short as 10−11 seconds have been achieved. Such a detector can only record changes of the intensity that occur on timescales longer than tmeas . The electric signal will contain a direct current (DC) term, proportional to the long-term average intensity, and alternating current (AC) components with frequencies up to 1∕tmeas . Please note that in our description so far, there has been no need for the quantization of the radiation. We need a quantized detection process, but Eq. (3.1) could also be used for a classical electromagnetic field. And indeed, at high intensities, such a classical interpretation is valid. However, at low intensities, we find experimental evidence that the light field has to be quantized and that we need the Photo-current i(t)

Beam of light

t Sequence of photons

Figure 3.1 Photoelectric detection.

3.1 Detecting Light

concept of particles of light, of photons. For example, consider a situation where a pulse of light contains energy less than h𝜈. In this case the classical equation (3.1) would predict a probability that is larger than zero, whilst the experimental evidence shows that there is no detection at all. This can be taken into account by choosing a quantized theory of the detection process [3] that replaces Eq. (3.1) with t

meas D′ ′ ̃ † A⟩dt ̃ ⟨A (3.3) h𝜈 tmeas ∫0 ̃ and A ̃ † are operators that describe the complex amplitude of the electric where A field and its conjugate. We will find that there are some peculiar features of these operators compared with the classical functions. For example, the sequence of the operators in the integral is important. The sequence in Eq. (3.3) corresponds to the normal ordering that has been chosen such that absorption of photons is the detection process and a different ordering would lead to completely unrealistic effects. This quantum model is described in Chapter 4. One motivation for introducing quantized radiation stems from the question: how quickly, or with what response time, can low intensity light pulses be detected? In a classical theory there would have to be a delay in order to accumulate sufficient energy in one atom to release an electron. The response time tmin for light with a photon flux density of N photons/m2 s and for an atom with cross section A m2 can be estimated as tmin = 1∕(NA). This is the time required for the energy equivalent to one photon to be transmitted through one atom; it would be a minimum for the response time. Is this realistic? Let us consider sunlight as an example. Green light, with a wavelength of 500 nm, is selected with an optical filter that transmits about 1% of the entire spectrum. This green light has a photon flux density of about 1019 photons/m2 s as can be calculated from Planck’s radiation law using the well-known data for the sun, with a surface temperature of 6000 K and diameter of 32 arc minutes. Assuming an atomic cross section of 10−18 m2 , the detection time is about 0.1 second. For less intense beams the time would increase to fantastic values, whilst we know that our eye can be fairly fast even at intensities much lower than full sunlight. Actually, we can almost detect individual photons with our eyes. The latest studies have shown that we can sense the presence of individual photons and that the actual sensitivity depends on the actual conditions, including what has been seen immediately beforehand [4]. In order to see detectable features, the sensitivity of the human eye for green light can be estimated as about 5 × 10−18 W, which corresponds to a photon flux of about 12 photons/s into the eye or a photon flux density of about 109 photons/m2 s . This low level is sufficient to release photoelectrons, and the detection process certainly does not take longer than one second. If the simple classical model was correct, we could not see low intensity images, such as the impressive display of star patterns in a dark night. Furthermore, the vision process would be entirely different to our experience. The main difference between bright and dim objects would be the time it takes to see them; the detected intensity would always appear to be similar. Thus we conclude that our personal experience of vision requires the concept of photons.

det =

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It was pointed out that the simplified model of the atom interaction overestimates the response time. A faster response can be estimated by assuming that atoms can draw energy from a wider area. This would be the case when the atoms are separated or when only some of the densely packed atoms contribute to the detection process. Estimates for such a situation have been made for classical oscillators by Lorentz [5] and can be combined with the assumptions of modern atomic theories [6, 7]. These calculations show the ability of atoms to draw energy from a wide area. One physical interpretation of this effect is that the incoming field induces a dipole moment in the atom, which in turn emits a wave that destructively interferes with the incoming wave. In this way the energy is concentrated in the atom. For the case of sunlight, a response time of about 10−7 seconds, and for the sensitivity limit of the eye, a response of 10−1 seconds can be calculated. Whilst these values are short enough to be comparable with the response time of the human eye, they are still orders of magnitudes too long compared with experiments. Fast response times were measured even before the advent of modern short pulse laser technology. In 1927 Lawrence and Beams [8] used an electric spark to generate a light pulse, which was detected with a photocathode that could be gated using an additional internal grid. The timing of the grid could be delayed by propagating the electric gating pulses along various lengths of cable, and it was observed that the photoelectrons from the detector appeared simultaneously with the light pulse, within an accuracy of 10−9 seconds. Forrester et al. [9] used the detection of beat signals between two monochromatic thermal sources and concluded that the detectors can follow the electromagnetic field with accumulation times less than 10−10 seconds. Nowadays even shorter pulses, femtoseconds, and higher modulation frequencies are routinely used in optical instruments. All these observations show the need for a concept of light as a quantized system – the need for photons.

3.2 The Concept of Photons We really need more than one concept of photons. So far we have taken the view of an experimentalist who would talk about detecting photons from a propagating beam of light. Photons have properties quite different from those of classical particles. They have energy h𝜈, zero rest mass, momentum ℏk, and spin ℏ, and they are non-interacting particles following the Bose statistics. A laser beam contains a large number of these particles (see Table 3.1), and the measurement of the beam intensity corresponds to a measurement of the flux of photons. The photons that were created by the emission process, that is, the de-excitation of the atoms via stimulated or spontaneous emission, are destroyed in the detection process. This in turn creates an electron–hole pair in the detector material. For the experimentalist photons are particles that can be localized, for example, at the detector. The experimentalist takes the consequence of the detection, the electron created or the grain in a photographic plate that is darkened, as the evidence of the photon. Figure 3.2 shows two examples of images that are typical for the detection of individual photons in one laser beam at very low light intensity. In this case the

3.2 The Concept of Photons

Figure 3.2 Two images from multi-pixel sensors, such as a CCD or CMOS camera sensor, with low intensity illumination. Each image has an array of individual detectors, or pixels, where the individual photons trigger electrical events that are recorded and displayed. The number of photons recorded in each pixel is colour-coded, from no photon as black to several photons as red or organ yellow to even higher numbers as green. These images show graininess or randomness, with no specific pattern. The images would be similar for thermal and laser light. The information can be used for imaging or for the measurement of photon statistics or could be added up as a total photon flux. Table 3.1 Examples of the photon flux and the photon density of typical optical fields.

Type of light

Intensity I (W/m2 )

White light (T = 6000 K)

103 4

Spectral lamp

10

5

Electric field E (V/m)

Photon density (m−3 )

Photons/mode

103

1013

10−4

3

14

10−2

15

3 × 10

4

10

CW laser

10

10

10

1010

Pulsed laser

1013

108

1023

1018

For simplicity it is assumed that all fields have the same wavelength of 𝜆 = 500 nm, and thus all photons have the same energy h𝜈 = 2.510−19 J. In the case of the pulsed laser, the photon density inside the pulse has been evaluated. The table shows that we can consider a mode to contain a large number of photons only for laser beams.

detector array has pixels that are smaller than the size of the laser beam, which is in the form a Gaussian single-mode beam. A small faction of the beam illuminates the array of pixels shown here, each creating a number of detection events during the measurement interval chosen. The two images could be representative of two samples in time at the same location with the same average intensity or alternatively of two spatial samples in the one laser beam recorded at the same time. In either case the selection of events is random, and all the photons belong to the same mode. There is one probability distribution for detection for the entire mode. It is tempting from this point of view to envisage a beam of light as a stream of photons travelling at the speed of light. But this is incorrect and, as we shall see in the following sections, cannot successfully explain many experimental results. A different concept for photons, perhaps more familiar to a theoretician, is needed to describe the propagation of light. Here we would talk about the photon number of a mode, where ‘mode’ refers to the radiation field modes described in Section 2.1. Each mode has energy states like a harmonic oscillator, and the

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state of excitation of a mode is said to be the number of photons in the mode. In this concept photons are distributed, they do not exist at any specific location, and they are a property of the entire mode. The only possible processes are the annihilation and creation of a photon in which the entire mode is affected, and to describe this we need the formal rules of quantum mechanics. The wonder of quantum mechanics is that the apparently contradictory points of view – localized, particle-like emission and detection but delocalized field-like propagation, often referred to as wave–particle duality – can coexist. A laser beam can be described by a single mode propagating from the laser, with physical properties much like the Gaussian mode in Eq. (2.8). The photon number of the laser mode is linked to the flux of photons in the external laser beam, which can be detected. Laser modes typically contain a large number of photons, and the flux is high. For thermal light, on the other hand, the number of possible modes can exceed the photon number, and on average there will be less than 1 photon in any mode (see Table 3.1). A more detailed review of the concept of photons can be found in books by H. Paul [7] and R. Loudon [10].

3.3 Light from a Thermal Source We will now look at the example of light emitted by a broadband thermal source. We will assume that the emission properties of the atoms are quantized, as first proposed by M. Planck [11], who tried to explain new observations of the black-body radiation or Schwarzkörperstrahlung, which had been made at the newly founded Physikalische-Technische Reichsanstalt (PTR) in Berlin, precursor to the modern PTB (Physikalische Technische Bundesanstalt), by O. Lummer and E. Pringsheim in 1900. These beautiful technical and theoretical achievements deserve description in some details. See also [12]. In 1885 W. Wien and O. Lummer followed an idea by G. Kirchhoff and proposed to build an isothermal resonator that should produce inside the cavity the so-called black-body radiation. The light can be observed through a small hole in the cavity and was expected to match the predictions of the new Wien’s law [13]. In 1900 the team of O. Lummer, F. Kurlbaum, and H. Rubens constructed at the PTR a unique electrically heated resonator that could reach a temperature of 1600 ∘ C, and they recorded the radiation spectrum from 1 to 6 μm, as shown in Figure 3.3), using a new bolometer design. With even more technical developments of the radiation detectors, they were able to make remarkable improvements and extended the range of measurements to 18 μm with sufficient precision. The data showed a deviation from Wien’s radiation law of up to 50% [14]. The experimentalist reported the result personally to Max Planck, who was Kirchoff ’s successor at the Universität in Berlin. On the very same day, Planck discovered an empirical formulation of the radiation law. He reported his results a few days later at a meeting of the DPG (Deutsche Physikalische Gesellschaft) following lecture of the experimentalists. Together, they found a close fit to the observed radiation data. Two months later, in December 1900, Planck presented a full theoretical deduction of his equation using an atomistic–probabilistic description of entropy introduced by L. Boltzmann,

3.3 Light from a Thermal Source

E 140 130 120 110 1646 K (1653 K)

100 90 80 70 60

1460 K

50 40 30

1259 K

(b)

20 1095 K

10 (a) 0

998 K 904 K 723 K

1

2

3

4

5

6

λ

Figure 3.3 (a) Absolute measurements of the spectral energy distribution carried out in 1900 by O. Lummer and E. Pringsheim on the electrically heated black body over a spectral range of 1–6 μm. The broken line was calculated with Wien’s radiation law. The continuous line represents the measurements that increasingly deviate from Wien’s law with higher temperature and increasing wavelength, but were in good agreement with Planck’s radiation law. (b) The apparatus is also shown. Source: From PTB historical records and summary by J. Hollandt [12].

which he had rejected until then. In an ‘act of desperation’, Planck only allowed certain discrete states of energy. Using the experimental data he was able to determine the free variables in his equation. Whilst now regarded as the hour of birth of quantum mechanics, neither Planck himself nor his audience were aware of the widespread consequences of this discovery of Planck’s constant in the new century ahead. The term photon was not used at the time. We will use it to describe the fact that the light is emitted and absorbed in ‘lumps’ or ‘quanta’ by atoms, molecules, and solid materials. Using a semi-classical approach, light emitted by a thermal source can be analysed by considering a finite volume of space with zero field boundary conditions. The region will support a discrete number of standing waves of the electromagnetic field. We first derive the number and the spectral density of these modes, and from there we determine the distribution of photons in the light emitted by the source. If the region has the physical dimension L, it can support a variety of modes with the wave vectors kx = 𝜋𝜈x∕L, ky = 𝜋𝜈y∕L, kz = 𝜋𝜈z∕L, resulting in a mode density: 𝜌𝜈 d𝜈 =

8𝜋𝜈 2 d𝜈 c3

(3.4)

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These modes are thermally excited according to a Boltzmann distribution: exp(−En ∕kB T) (En ) = ∑∞ n′ =0 exp(−En′ ∕kB T)

(3.5)

Assuming the condition En = (n + 12 )h𝜈 for the energy of a mode, we can derive a probability distribution for n photons in one mode: n (𝜈, T) =

1 − exp(−h𝜈∕kB T) exp(n h𝜈∕kB T)

(3.6)

We can now evaluate the average number n(𝜈, T) of photons in a mode: n(𝜈, T) =

∞ ∑ n=0

nn (𝜈, T) =

1 exp(h𝜈∕kB T) − 1

(3.7)

which is the well-known black-body formula. It explains the existence of the black-body spectrum of a solid at temperature T. The fluctuations in a single mode of this radiation can be rather large. For normal thermal conditions and for radiation in the visible or near infrared, the average excitation is very low. For example, at T = 300 K and for 𝜈 = 1015 Hz, the average excitation is about 7 × 10−4 photons in each mode. The probability of finding a fixed number of photons, for example, n = 0, 1, 2, …, in one mode follows a Bose–Einstein distribution that actually has a maximum at n = 0. However, this calculation is only valid if we count photons in a single spatio-temporal mode. More typically, for the detection of light from a thermal source with measurable power, we have to consider many modes at different frequencies. They all contribute to the measured intensity. We have to evaluate the probability m that m photons are emitted in the detection interval td . The interval is chosen so large that m ≫ 1. Further we assume that the average intensity of the light is not time dependent. The light source sends out a stream of photons, and at all times we measure the same average intensity and the same average number of photons. The result from classical statistics for many independent sources is t ⟨I⟩ ⟨m⟩m exp(−⟨m⟩) with ⟨m⟩ = d (3.8) m! h𝜈 which is known as the Poissonian distribution. One important property of this distribution is Δm2 = ⟨m⟩, which states that the width of the distribution, measured by the variance, is equal to the mean value of the distribution. The probability distribution in Eq. (3.8) describes the fluctuations of the intensity emitted by a thermal source. The nature of these fluctuations is such that they are independent of the time intervals from which the photon number data is collected. These time intervals can be close and regularly spaced or widely separated and intermittent; the results will be the same, provided a sufficiently large sample is taken. Noise that behaves in this way is known as stationary noise. As a consequence g (1) (𝜏) and g (2) (𝜏) are independent of 𝜏. In particular, the noise spectrum I(Ω) (averaged over a small frequency range 𝛿Ω about Ω) is independent of the detection frequency Ω, i.e. all frequency components carry the same intensity (I(Ω) is flat). This type of noise m (td ) =

3.4 Interference Experiments

is referred to as white noise. The value of the variance is fixed for a given average intensity and depends only on the length of the detection interval: Var(I(Ω)) =

⟨I⟩h𝜈 td

(3.9)

This statistical result can be applied to any other situation where events generated by independent processes are counted and the average number of such events is not dependent on time. Examples are the measurement of particles from radioactive decay or the number of raindrops, particles of rain, in a steady downpour. The spectrum of the noise for all these cases is equivalent to the sound of a heavy rain shower on a metal roof. It was this analogy, namely, the dropping of lead shot on a metal surface, that was the origin for the name of this statistical phenomenon: shot noise. The classical description provides the tools for describing the statistical properties of light. Chapter 4 gives a complete quantum mechanical description. The classical description can serve as a reference, and the interesting question will be: how do the classical predictions compare with the more complete results from quantum theory? We will find that the predictions for thermal light, that is, light emitted by many independent atoms, are identical in both the classical and the quantum models. The lesson here is that it is sometimes sufficient to quantize the emitting (and absorbing) atoms whilst still retaining a classical description of the light (this is often referred to as a semi-classical description). In contrast, the results from a quantum mechanical description of the laser will be quite different from the results for a classical oscillator. A detailed comparison between the two models will show up several features that are uniquely due to the quantum nature of the light.

3.4 Interference Experiments The measurement of interference, made accessible through the use of instruments such as a Michelson interferometer, clearly demonstrates the wave nature and the coherence properties of light. However, the same type of interference experiments carried out at very low intensities also documents some of the quantum properties. This was done in very early experiments by Taylor [15]. These experiments used photographic plates and extremely long exposure times – up to three months. The experiments showed that the visibility of the fringes remained constant, at a given total exposure, independent of how long it took and thus how many photons could have been in the instrument at any one time. It should be noted that all these interference experiments are most impressive achievements. The interferometers are very sensitive devices, and nothing would have been easier than wiping out the interference fringes through a small perturbation of the apparatus. More recent experiments [16, 17] using photoelectric detection confirmed these results. In addition, some of these experiments used interferometers with total optical paths far exceeding the coherence length of the light source. For example, in the experiment by Janossy, each arm of the Michelson interferometer

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was 14.5 m long, requiring a very quiet and temperature-stable (ΔT < 0.001 K) environment, which was achieved in an underground tunnel. Again, the visibility remained unchanged – independent of the actual intensity. Complementary results were obtained with the synchrotron radiation emitted by a storage ring. Here the number of electrons generating the radiation can be determined separately from the intensity of the radiation – and again the visibility was shown to be independent of the photon number. All these experiments support the statement by P. Dirac: The new theory, which connects the wave function with probabilities for one photon, gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs. [18] The beamsplitter in the interferometer does not divide the light into smaller parcels – any detection of the light shows quanta of the same amount of energy. This result cannot be derived if we think of photons as small local parcels of energy – miniature billiard balls – with the properties of classical particles. It reveals one of the consequences of quantum mechanics, namely, that a conclusion about the specific outcome of an experiment cannot be drawn before the experiment is carried out. This consequence of quantum mechanics becomes very clear by considering the various possible outcomes of an interference experiment where detectors and counters are used in an attempt to measure the presence of the photon in the one or the other of the interferometer arms. Any such attempt would fail, as it has been so elegantly expressed by R. Feynman et al. in his lecture notes [19]. The duality of the wave–particle model is a clear conclusion from these experiments: light is neither exclusively a wave nor exclusively a stream of particles. The type and condition of the experiment determines which one of these models will dominate and which one will give the simpler interpretation. All the time light is something more inclusive, more complicated than either of these models would suggest. The statement that ‘a photon will only interfere with itself’ should not be taken too literally in the sense that it could be tested through the observation of individual events. In the case of the photographic recording, each detection event produces one clear dark dot on the photographic plate. This dot could be part of an interference pattern or of a random distribution of dots. It is the distribution of many dots – a frequent repeat of the same measurement – that reveals the result of the experiment. Theoretically, one single dot measured exactly at the location on the plate where there should be absolutely no intensity according to the calculation would have an enormous significance. Such an event would prove the calculation wrong (Figure 3.4). But we have to note that in reality there will always be some other imperfection, such as an imbalance of the beamsplitter, finite coherence length, or background noise, which could have resulted in this particular single event. It is the distribution of the dots, or of the counts in the electronic instrument, that is crucial for the test of a model. As is common

3.4 Interference Experiments

4 1

3

2

Figure 3.4 Historical example: interferometer at the turn of the century. Source: From the notes of A.A. Michelson and E.W. Morley [20]).

for all tests of quantum mechanics, a large number of individual experiments are required to allow a valid conclusion. The nature of the light seems to depend on the type of experiment: a single beamsplitter can often be well described by the photon picture, and a complete interferometer requires the wave picture. An interesting extension of the demonstration described above are the so-called delayed choice experiments. Here we are trying to trick the light into the wrong behaviour by delaying the decision of which type of experiment is being performed until it has already passed the first beamsplitter. Will the photon be able to predict the type of experiment that is lying ahead and react in the appropriate way? Whilst this view is certainly naive, it poses a valid question: does it matter whether we have a delay built into the interferometer? One example for such an experiment [21], shown in Figure 3.5, uses pulses of light of short duration (150 ps). The intensity was chosen sufficiently low that on average only every fifth pulse contained one photon. Pulse 150 ps

n = 0.2 photons

Electro-optic shutter

Polarizing beamsplitter

5m Fibre

Det 1

5m

Det 2

Figure 3.5 Schematics of a delayed choice experiment. Source: After Hellmuth et al. [21].

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Both arms of the interferometer contain 5 m of single-mode fibre. One arm contained an electro-optical switch that allowed this arm to be blocked and the interference to be turned off through an electronic signal. In order to put the photons in the above-mentioned dilemma, the interferometer was operated such that the switch was operated after the pulse had passed through the first beamsplitter. However, the delay had no effect on the visibility of the interference pattern. The outcome of the experiment depended only on the configuration of the experiment that was present at the time of detection, independent of the earlier state of the apparatus. A more thorough discussion of these types of experiments is given in Chapter 12. This raises the question as to whether photons from two completely independent sources can interfere with each other. It is well known that light from different sources, or from different parts of the same source, does not interfere when observed for a long time. This is a consequence of the fact that the phase of the electromagnetic wave emitted by the two sources has two independent phase terms. In a normal interferometer, which has wave front splitting mirrors, this does not matter as long as the path length difference is less than the coherence length. Similarly, for independent sources, the fringes should also appear if the measurement takes less time than the coherence time tcoh . During such a short time interval, the phase will not vary enough to reduce the visibility. This has been shown in an experiment with two independent lasers [22], in this early experiment two ruby lasers. Spatial interference patterns from single pulses of the lasers were recorded, clearly showing the interference between photons from different sources. An alternative is to measure the beat between two lasers; this was first demonstrated for two independent He–Ne lasers [23]. The experiment operated continuously, and the narrow linewidths of the gas lasers made the observations straightforward. The major problem was the drifts in the laser frequency due to mechanical and temperature variations. Once this was overcome, this experiment demonstrated clearly that interference exists between beams from independent sources, provided that the detection time t1 is less than the coherence time tcoh . With modern technology, such as tunable diode lasers and radio wave spectrum analysers, this type of experiment is almost trivial. The effect is common knowledge for a laser physicist. However, the interesting question remains: should this interference effect between independent lasers persist in the limit of very small intensities? The first answer might be No, since we cannot be certain of the relative phase during the long measurement time required for the weak beams. However, if we select only data measured at times when the relative phase is within a set small range, one could expect an interference pattern to become visible. Exactly this experiment was carried out by Radloff et al. [24]. Two He–Ne lasers were used that had been built in one mechanical structure in order to minimize the technical noise and the phase fluctuations due to vibrations. The beat frequency between the beams was continuously monitored. A very small fraction of the light was selected and formed a spatial interference pattern. This was gated by an electro-optic shutter and only opened when the frequency was within a preset range. The photon flux was as low as 105 photons/s with a typical shutter time of 10−5 seconds.

3.4 Interference Experiments

After exposures of about 30 minutes, clearly recognizable interference patterns became visible. These patterns were present even at arbitrarily low intensities. The particle picture is not at all suitable for this situation. It is extremely unlikely that one photon from each laser will pass together through the shutter. The wave model on the other hand provides a very simple explanation. The only problem arises when we want to understand the quantization of the energy together with interference. These results for the interference of photons from independent sources are in some contradiction to the statement by P. Dirac, to repeat: ‘Each photon interferes with itself. Interference between two different photons does not occur’. Dirac made this statement whilst considering only interference experiments known at his time. It should not be applied to the experiments involving independent sources. Nevertheless, attempts have been made to rescue Dirac’s statement by evoking additional processes that guarantee that photons are emitted simultaneously by both lasers. But these attempts are really not all that helpful in understanding the quantum nature of light. In summary, we can say that the only conditions under which interference can be observed are those where it is impossible to detect which path the photon has taken. Any attempt to measure the path, or even to reconstruct the path after the event, would wipe out the interference pattern. A discussion of these types of experiments has led to the postulate of Heisenberg’s uncertainty principle. For this particular case the principle states that ΔxΔp ≥ h∕4𝜋. It enforces that we can measure either the interference pattern, by determining the position x of the photon, or alternatively the path of the photon, as measured by its momentum p. But we cannot measure both simultaneously [7]. Discussion about these results has played a prominent role in the development of quantum mechanics as a whole. A related idea was behind the design by J.D. Franson, who designed an interferometer that can be used to distinguish between single- and two-photon interference [25]. His design is shown in Figure 3.6. In this apparatus let us consider two coincident photons, emitted by a source, which individually travel in opposite directions towards two identical interferometers. Each interferometer contains θ2

θ1

γ2

D2

Source

D′2

γ1

D1

D′1

Figure 3.6 The Franson interferometer. Source: Franson 1991 [25]. Reproduced with permission of APS.

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a shorter and a longer path, and the difference Δttrans in transit times over the two paths is taken to be much larger than the coherence time of the photons. In addition, it has individual phase settings Θ1 and Θ2 . That means each interferometer by itself would not produce fringes individually and the visibility of a fringe due to a change of the phase Θ1 and Θ2 would be extremely low for such a large path length imbalance. However, interference between the quantum mechanical amplitudes for the photons to have both travelled the shorter paths and the longer paths produces a modulation in the coincidence counting rate R of photons from the two sets of detectors, given by cos2 [(Θ1 + Θ2 + 2𝜋𝜈 Δttrans )∕2]. This type of interferometer would only show a modulation due to the quantum mechanical wave function of the photons, which is sensitive to indistinguishability of the paths. That means an interference can occur for the quantum mechanical amplitudes and thus for the counting rates. This would be a clear distinction between two types of interference. Moreover, if we illuminate this interferometer with an entangled pair of photons, there is a link between the probability amplitudes on both sides of this interferometer, and we can measure the visibility of the fringes due to the settings Θ1 and Θ2 . This should allow us to learn about the quantum properties of the photon pairs and will be discussed in Chapter 13.

3.5 Modelling Single-Photon Experiments We now turn to a more idealized description of single-photon experiments in order to see how they may be modelled. We will introduce a number of basic optical elements as ‘black boxes’, describing and contrasting their operation for classical light waves and single photons. Descriptions of how these optical elements can be constructed will be found in later chapters. Similarly, the simple quantum mechanical models introduced here will also later be expanded. The average number of photons per second in a beam of light is proportional to its intensity. However, the quantization of the field introduces a degree of freedom not present in the classical theory, the photon statistics. Suppose we had a ‘photon gun’, that is, a light source that emits just one photon when we pull the trigger. Such a photon gun can be produced through pulsed excitation of a single molecule, though presently not conveniently. Such a source would share some characteristics with the highly attenuated sources described in Section 3.4. However, whilst an attenuated laser may produce one photon on average in some time interval, there will be fluctuations: sometimes there will be two or more photons, and sometimes there will be none. On the other hand, we will always detect just one photon in the measurement interval coming from our photon gun. There is no way to describe this difference in the photon statistics with a classical description of light. We must introduce a new concept, that of the quantum state of the light. The quantum state describes all that is worth knowing about a light beam. The photon gun produces light in a single-photon number state, which we will write as |1⟩. Studying the behaviour of such a light state in simple situations illustrates some of the basics of the quantum description of light.

3.5 Modelling Single-Photon Experiments

3.5.1

Polarization of a Single Photon

Let us first consider how a single polarized photon behaves. We will describe this behaviour using single-photon polarization states. We will see how it is possible to build a consistent description of light phenomena in spite of wave–particle duality. To do so, however, we will need to introduce some quite different concepts to those found in classical theory. Recall that in our classical description the polarization of the field was described by the vector p(t), which gave the direction of the electric field tangential to the wave front. For linearly polarized light p will be fixed. For circularly polarized light p will rotate at the optical frequency. It is possible to design optics that will decompose a light field into two orthogonal modes. The amplitudes of the two modes will be equal to the projection of p onto the chosen axes. For example, a horizontal/vertical polarizing beamsplitter will project the field onto a horizontally polarized mode and a vertically polarized mode. These two modes will exit through different ports of the beamsplitter. Similarly, one can use a diagonal/anti-diagonal polarizing beamsplitter to decompose the field into linearly polarized fields at 45∘ and −45∘ away from horizontal. With slightly more effort it is also possible to create a left/right circular polarizing beamsplitter that decomposes the field into clockwise and anticlockwise rotating circular polarizations. Suppose we fire a string of single photons at a horizontal/vertical polarizing beamsplitter. As we have noted, classically such a device will direct horizontally polarized light in one direction and vertically polarized light in the other. Thus we will define photons that exit through the horizontal port of this beamsplitter as being either in the horizontal state or the vertical state as |H⟩

or |V ⟩

(3.10)

By ‘exit through the horizontal port’, we mean that a photon detector placed at the output of the horizontal port will detect a photon, or equally a photon detector placed at the vertical port does not detect a photon. These situations are illustrated in Figure 2.8. The symbol for the state, |−⟩, is referred to as a ket. These definitions are sensible because a photon that exits through the horizontal port of the beamsplitter and is passed through another polarizing beamsplitter will certainly exit through the horizontal port. A similar definition applies to vertical photons. The diagonal single-photon state, |D⟩, and the anti-diagonal state, |A⟩, can be defined in a similar way by analysing the beams with a diagonal/anti-diagonal polarizing beamsplitter. So far this is straightforward. The physics of classical polarized beams and single photons looks the same. Things become more interesting when we start to mix up the polarizations. If we send a diagonally polarized classical beam of light into a horizontal/vertical polarizing beamsplitter, then half the beam will exit through the horizontal port and half will exit through the vertical port. What will happen if we send single photons in the state |D⟩ through this beamsplitter? As we have discussed in Section 3.4, the photons cannot be divided, in the sense that a

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detector placed at one of the output ports will either detect a whole photon or no photon. Instead they must go one way or the other. To be consistent with the classical result for many photons, it must be that they go one way 50% of the time and the other way the other 50% of the time. But can we tell in which direction an individual photon will go? To answer this first remember how |D⟩ photons behave at a diagonal/anti-diagonal polarizing beamsplitter: they all exit through the diagonal port. That is, they all behave perfectly predictably and identically. Yet when these identically behaving photons are sent into a horizontal/vertical beamsplitter, we have argued some must take one path and others take the other. Thus the path an individual photon in the state |D⟩ takes through the horizontal/vertical beamsplitter is not specified. All that is specified is that on average half the photons will go one way and half will go the other. This example illustrates the fact that in quantum mechanics it is probabilities of outcomes, not particular outcomes, that are predicted. Even though we can say with certainty that a photon in the state |D⟩ will emerge from the diagonal port of a diagonal/anti-diagonal beamsplitter, its reaction to a horizontal/vertical beamsplitter is as random as a flip of a coin. It is tempting to think that maybe there are other variables (perhaps hidden to us) that do determine in a precise way the polarization behaviour of individual photons under all conditions. However such a possibility can be ruled out experimentally (see Chapter 12). We are forced to accept the intrinsic indeterminacy of the quantum world. Continuing our discussion of polarization states, we can also introduce the right circular single-photon polarization state, |R⟩, and the left circular state, |L⟩, in an analogous way to the other states. A photon in state |R⟩ will randomly take one port or the other when sent into either a horizontal/vertical or a diagonal/anti-diagonal beamsplitter. A photon in state |L⟩ will behave in the same way. Similarly a photon in state |H⟩ will give a random result for both diagonal/anti-diagonal and right/left circular polarizing beamsplitters and so on for the other states. We will now outline the mathematical formalism for treating these situations. 3.5.1.1

Some Mathematics

The state kets we have introduced are vectors. They live in a vector space called Hilbert space. Hilbert space is a complex vector space: when we add vectors, we are allowed to do so using complex coefficients. The dimensions of the vector space are equal to the number of degrees of freedom of the property being described. In the polarization example our vector space is discrete and has dimension 2. That is, in all the experiments we have described, there are only two possible outcomes: the photon comes out from one port or the other. It can only take two discrete values. This is in stark contrast to the classical case in which the amount of the field that comes out of one or other port can vary continuously between all and nothing. In general, discrete systems can have any number of dimensions, and continuous systems can also be described. A continuous variable can give any outcome in some range, and their vector space will have infinite dimensions. We will look at such examples in optics in Chapter 4. As for other vector spaces, we can reach any vector in our space through linear superpositions of n orthogonal basis kets, where n is the dimension of the system.

3.5 Modelling Single-Photon Experiments

To define orthogonality we introduce bra vectors, which are written ⟨x|. Bras and kets are related by the Hermitian conjugate, an operation similar to complex conjugation and indicated by the ‘dagger’ symbol, †. Thus we can write z∗ ⟨x| = (z|x⟩)† .

(3.11)

where z is just a complex number. Vectors |x⟩ and |y⟩ are orthogonal if ⟨x| × |y⟩ ≡ ⟨x|y⟩ = 0

(3.12)

In general the object ⟨x|y⟩ is called a bracket and is similar to the dot product in real vector spaces. A ket is said to be normalized if ⟨x|x⟩ = 1. We will generally assume this property. Simple physical questions can be asked in the following way. If we prepare our system in the state |𝜓⟩ and then use an analyser to ask the question ‘Is my system in the state |𝜎⟩?’, the probability, P, that we get the answer ‘Yes’ is given by P = |⟨𝜓|𝜎⟩|2

(3.13)

The bracket ⟨𝜓|𝜎⟩ is referred to as the probability amplitude and will in general be a complex number. The orthogonality condition then says that if we prepare our system in |𝜎⟩, our analyser will never find it in |𝜓⟩ if these two states are orthogonal. Normalization reflects the fact that if we prepare our state in |𝜓⟩, we will certainly (P = 1) find it there with our analyser. We will try in this book to minimize the mathematics by introducing concepts on a need-to-use basis. However, a deep understanding of the workings of quantum optics requires a good understanding of the mathematical structure of quantum mechanics. Readers unfamiliar with the basics of quantum mechanics are encouraged to study other texts such as Refs. [26] and [27].

3.5.2

Polarization States

The concepts introduced above allow us to identify our polarization states |H⟩ and |V ⟩ as orthogonal states. That is, ⟨H|V ⟩ = ⟨V |H⟩ = 0, which means that a photon in state |H⟩ never comes out the vertical port and vice versa. We will assume also that our states are normalized such that ⟨H|H⟩ = ⟨V |V ⟩ = 1. This means that we should be able to express all our polarization states as linear superpositions of the horizontal and vertical states. Consider first the diagonal states. From our discussion in Section 3.5.1, we know that they must have the properties ⟨D|A⟩ = ⟨A|D⟩ = 0 |⟨D|H⟩|2 = |⟨D|V ⟩|2 = 0.5 |⟨A|H⟩|2 = |⟨A|V ⟩|2 = 0.5

(3.14)

or the diagonal states are orthogonal and a diagonal state placed through a horizontal/vertical polarizing beamsplitter has a 50% probability of exiting through

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either port. It is straightforward to check that 1 |D⟩ = √ (|H⟩ + |V ⟩) 2 1 |A⟩ = √ (|H⟩ − |V ⟩) 2

(3.15)

satisfy the conditions of Eq. (3.14). Now consider the circularly polarized states. We need to satisfy ⟨R|L⟩ = ⟨L|R⟩ = 0 along with |⟨R|H⟩|2 = |⟨R|V ⟩|2 = 0.5 |⟨L|H⟩|2 = |⟨L|V ⟩|2 = 0.5 and |⟨R|D⟩| = |⟨R|A⟩|2 = 0.5 2

|⟨L|D⟩|2 = |⟨L|A⟩|2 = 0.5

(3.16)

This looks impossible until we remember that Hilbert space is a complex vector space. Thus we are allowed to form the complex coefficient superpositions 1 |R⟩ = √ (|H⟩ + i|V ⟩) 2 1 |L⟩ = √ (|H⟩ − i|V ⟩) 2

(3.17)

which satisfy the above requirements. It follows that an arbitrary polarization state can be represented by |𝜔⟩ = x|H⟩ + ei𝜙 y|V ⟩

(3.18)

where x2 + y2 = 1 and 𝜙, x, and y are real numbers (absolute phase factors have no physical significance). Notice that such a representation is not unique. We could equally well write all our states as superpositions of diagonal and anti-diagonal states or circular states. Any two orthogonal states can span the Hilbert space. We have seen that in order to describe the behaviour of a quantized light field, we have had to introduce two new concepts: (i) the state of the light and (ii) that predictions are made of probabilities of events, not actual events. The fact that these concepts lead us to represent, for example, a diagonal photon as an equal superposition of horizontal and vertical photon states may seem relatively benign at this point. That such a superposition has no analogue for classical particles will be demonstrated more graphically in Section 3.5.3. In Chapters 12 and 13, we will discuss in detail photon experiments that use polarization as their degree of freedom. In particular, we will investigate the unique communications and information processing abilities these quantum states can exhibit.

3.5 Modelling Single-Photon Experiments

3.5.3

The Single-Photon Interferometer

We will now describe a simple single-photon interference experiment using a Mach–Zehnder interferometer, similar in concept to the experiments described in Section 3.4. A more general description of this device can be found in Section 5.2. Here we will illustrate the basic physics of the arrangement in the extreme situation of only a single photon present in the interferometer at a time and how the mathematics introduced in the previous sections can describe it. A Mach–Zehnder interferometer consists of a 50/50 beamsplitter, which divides the input light into two equal parts, followed by a series of mirrors, which direct the two beams back together again on a second 50/50 beamsplitter that recombines them. The path length difference between the two arms of the interferometer can be adjusted. This then corresponds to the situation discussed in Section 5.2, and thus classically we expect the amount of light exiting one port to change as a function of the path length difference. In particular, if our path length difference remains well within the coherence length, we will be able to pick a path length such that all the light exits from a single port. This set-up is depicted schematically in Figure 3.7. Let us now describe the behaviour of this interferometer when single photons pass through it. We will assume that the path length difference is held in the situation described above such that for classical beams all the light enters and exits through a single port. We will also assume that our beamsplitters and mirrors are polarization independent so that the polarization of our single photons is irrelevant, and we can simply label a single photon in a beam by the number state ket |1⟩. There are however two ways in which the photon can enter the interferometer: via the beam path labelled a in Figure 3.7 or by the beam path labelled b. Suppose we in fact send our photon in by beam path a. We can describe this initial situation by the state ket |𝜎⟩ = |1⟩a |0⟩b

(3.19)

meaning there is one photon in beam a and no photon in beam b. There is an even probability of the photon exiting from either port of the first beamsplitter. The output state from the beamsplitter can thus be written as 1 |𝜎 ′ ⟩ = √ (|1⟩a |0⟩b + |0⟩a |1⟩b ) 2 Figure 3.7 Schematic of single-photon interferometer.

50/50

(3.20)

Pathlength adjustment

a

b 50/50

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The path length condition means that, apart from an overall phase, this state remains the same as it propagates through the interferometer. The time symmetric nature of light propagation through a beamsplitter means that when the beams reach the second beamsplitter, they must return to their original state of a photon in a and no photon in b (see Eq. (3.19)). Thus we just get the photon equivalent of the classical result – the photons always exit through a single port. Now let us consider the physical significance of our description. In particular, what does Eq. (3.20) mean? It says that the photon is in a superposition of being in each path. It is tempting to think that this just means the photon is in one or the other path with a 50% probability. Certainly the probability of finding the photon in path a Pa = |⟨1|𝜎 ′ ⟩a |2 = 0.5 is 50%. But is the interference effect, i.e. the fact that the photons always exit from the a port (see Figure 3.7), consistent with the photons taking one path or the other? Well, the second beamsplitter is identical to the first one so our first puzzle is: why should the photon only exit through one port? We might imagine that it picks up some kind of ‘label’ at the first beamsplitter that tells it what to do at the second. So suppose we find our photon in beam a in the middle of the interferometer, but then we ‘release’ it and let it finish its journey to the second beamsplitter. Experimentally we find that now there is no interference and the photon is just as likely to come out from either of the output ports. Quantum mechanically, because we now know that the photon is in beam a, the state inside the interferometer becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b , and so the photon just behaves as it did at the first beamsplitter and has a 50/50 chance of emerging from either port. We might argue that when we ‘caught’ the photon, it forgot its label and so just behaved randomly at the second beamsplitter. So consider another possibility. Suppose we look for the photon in beam b but we do not find it. Now quantum mechanically the situation is the same as above; we know that the photon must be in beam a so the state again becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b and the interference should disappear. But any common- sense description would say this cannot be. We have not interacted with the photon, so any label it has must be unaffected. Thus we would expect the interference to still appear. Experiments, however, confirm the quantum mechanical result (see Chapter 12). We are forced to conclude that the superposition state of Eq. (3.20) represents in some objective sense the presence of the photon in both paths of the interferometer. Any experiment that causes the photon to manifest itself in one or the other path, whether an actual detection occurs or not, destroys this superposition and in turn wipes out the interference. This is wave–particle duality and we will encounter it in many guises throughout this book.

3.6 Intensity Correlation, Bunching, and Anti-bunching The quantum properties of light can be investigated by using the techniques of photon counting. We will find that the statistics of photons are considerably different from the statistics of independent particles. Again, we have to combine

3.6 Intensity Correlation, Bunching, and Anti-bunching

the particle picture with the wave picture. For these statistical experiments it is necessary to measure quantitatively the correlation between different beams of light. This approach has already been discussed in Section 2.3.3. In particular the discussion of the Hanbury Brown and Twiss experiment for thermal light, in their case starlight, showed a way for determining the correlation between fluctuating currents ia (t) and ib (t) by recording and analysing the product ia (t)ib (t) of these currents. This technique can be used for any type of light, including laser light, and it also applies to any intensity of light. At very low intensities the photodetector becomes a photon counter. For a photon counter the probability of a count is proportional to the intensity of the light. Now the number n(t, tdec ) of counts recorded at time t within a time interval tdec represents the intensity of the light. The detection time interval tdec should be shorter than the coherence time tcoh of the light source; otherwise the fluctuations are simply averaged out. As before we determine the intensity correlation by using detectors Deta and Detb , measuring the photon counts na (t, tdec ) and nb (t, tdec ) independently, and then forming the product na (t, tdec )nb (t, tdec ). Actually, such medium intensities, ranging from a few to several tens of counts per detection interval, are very challenging. No particularly satisfactory technology for this range has yet emerged, though progress is steadily being made (see Chapter 12). On the other hand, at very low intensities, where the probability of more than a single count per detection interval can be neglected, good and reliable technology exists (see Chapter 7). A moment’s thought shows that the probability of coincidences, which means events where both detectors register a photon within a short time interval, is proportional to the value of na (t, tdec )nb (t, tdec ). It is technically very attractive to build a circuit that only responds to such coincidences. This technique shows an increase of the coincidence rate under the same conditions where big correlations are measured. Going back to the Hanbury Brown and Twiss experiment, the coincidence rate is high if the separation between the detectors is sufficiently low. This is an excess of coincidences – compared to the numbers one would expect from completely random counting events. In other words, whenever a photon has reached detector Deta , the probability for another photon to be at Detb is enhanced. This could not be understood by regarding photons naively as independent particles, thereby ignoring the wave properties of the light. The photons emitted from the different part of the source would not know of each other. Their arrival time would not be correlated. In contrast, the wave picture predicts that the instantaneous intensity on the detector is influenced by all atoms in the source. Each part of the source is making a small contribution. Thus both detectors see related contributions where light is originated from one atom in the source. Again, this means we cannot consider individual photons, each with one well-defined birthplace. Atoms cannot be localized in that way. These observations are concerned with spatial effects. Intensity information can also be gained from temporal effects within a single beam. To measure the time correlation of a beam of light, we can use a trick that was used by Hanbury Brown and Twiss as a precursor to the stellar observations. We can use a beamsplitter to generate two light beams with each one being detected individually, as

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na (t, tdec)

Deta

nb (t, tdec)

Detb

X

Figure 3.8 Schematic layout of an optical intensity correlator.

Correlator (product)

shown in Figure 3.8. The coincidence rate can now be directly measured using detectors that are gated. Only events that happen within the gate time at both detectors are recorded. By moving one of the detectors either further away or closer to the beamsplitter, we introduce a delay time 𝜏 and can determine the coincidence rate as a function of the delay time. Alternatively, we can leave the detectors stationary and use electronic delays. For thermal light the coincidence rate for 𝜏 < tcoh should be higher than for 𝜏 > tcoh . This is a direct analogy to the properties of the second-order correlation coefficient. It corresponds to g (2) (𝜏 < tcoh ) > g (2) (𝜏 > tcoh )

(3.21)

as shown in Eq. (2.47). Since the coincidence rate for 𝜏 ≫ tcoh can only be those events that are completely random, we simply have to look for excess coincidences. The long delay time measurements serve as a normalization. One practical difficulty is that the detection time tmeas has to be less than the coherence time if we are to be within the single-mode approximation. For normal thermal light the linewidth is very large, the coherence time is extremely short, and the response of the detectors is too slow. A direct measurement of the coincidences for thermal light is not possible. In contrast, laser light has a small linewidth and long coherence time, and the experiment is feasible. One clever experimental trick is to simulate thermal light through scattering of laser light on rapidly and randomly moving objects. This was first done by Arrecchi et al. [28]. The experiments used light from a gas laser with long coherence time, which was scattered by polystyrene particles of various sizes suspended in water and detected by two detectors. The coincidence rate, or second-order correlation coefficient, was measured. For comparison the laser light itself was investigated, and the results show a clear excess of coincidences. The shape of g (2) (𝜏) fits properly to the prediction of a thermal light source with a Gaussian linewidth, and the difference between thermal light and coherent laser light is clearly demonstrated. An alternative interpretation of these results is to say that the photons from the thermal light arrive in bunches and the probability for photons to arrive together, in coincidence, is large for short delay time intervals. This phenomenon is known as photon bunching. It is typical for thermal light that has naturally large fluctuations. In contrast, for coherent light there is no preferred time interval between photons. Coherent light is not bunched. The coincidences show random Poissonian statistics on all timescales. Can there be anti-bunched light, where the photons keep a distance, in time, to each other? This would be light that clearly violates the classical conditions. The second-order correlation function would be less than 1, which is not possible for classical statistics, as

3.6 Intensity Correlation, Bunching, and Anti-bunching

Microscope objective

g2 (τ) 3 Optical excitation 2

Timer Start

Phototube Stop

1 Computer

25 (a)

(b)

50 Delay τ (ns)

75

100

Figure 3.9 The experiment by Kimble et al. [29]. (a) Schematic outline. (b) Measured values for g(2) (𝜏) derived from the data. The broken curve shows the theoretically expected form for a single atom, arbitrarily normalized to the same peak.

already shown in Section 2.3. A measurement of anti-bunched light would be a clear demonstration of quantum behaviour, of non-classical light. Well in fact the single-photon quantum state we discussed in the last section would be just such a state. Recall from Eq. (3.20) that such a beam will send a photon in one or other direction at a beamsplitter. Thus, at zero time delay, it is impossible to record a coincidence from such a state in the set-up of Figure 3.8. Hence we have g (2) (0) = 0. A pioneering experiment by Kimble et al. succeeded in 1977 in demonstrating the generation of non-classical light [29]. They used the fluorescence of single atoms to generate light with the statistical properties of anti-bunching. The experiment was carried out with Na atoms, which were first prepared into a two-level system and then excited by two resonant continuous-wave (CW) laser beams, as shown in the schematic layout in Figure 3.9a. The fluorescence light from a very small volume, containing on average not more than a single atom, was imaged through a 50/50 beamsplitter on to two photodetectors, D1 and D2 . Upon arrival of a photon, the detectors generated pulses that were processed, via counting logic, and produced the probability function (t, t + 𝜏) for the arrival of photons at different detectors with a time delay 𝜏. This information was converted into the second-order correlation function g (2) (0) (see Section 2.3). The excitation of individual atoms is determined by the quantum mechanical evolution of the atomic wave function. After the single atom has been excited by a laser beam, it will emit one photon, let us say at time t = 0. The atom cannot immediately emit another photon; it requires time for re-excitation. Consequently the probability for the generation of pairs of photons with small time separation is very small. The fluorescence light has a value of g (2) (0) close to zero, as shown in Figure 3.9b, which is not possible for classical light. This light clearly shows quantum properties – it is anti-bunched. Since this first demonstration, several experiments, for example, reported by J. Cresser et al. [30], J.G. Walker and E. Jakeman [31], and P. Grangier et al. [32], have been able to reveal the properties of similar quantum systems in more and more detail.

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More recently, the observation of anti-bunching has become the method of choice for identifying single quantum emitters. For example, fluorescent light emitted by nitrogen vacancy (NV) centres in diamond can be analysed, and when strong anti-bunching is observed, one can be sure that the observed fluorescence comes from a single NV centre in diamond [33]. See Chapter 13 for details.

3.7 Observing Photons in Cavities Many researchers pointed out that the most direct evidence of the quantization of the radiation field, namely, the discrete nature of the photocurrent, could also be explained by a classical description of the radiation field, provided that the detector is a linear quantum system. Do we have better evidence of the quantum properties of the radiation field? Researchers set out to find a more direct proof for the discreteness of the field and analysed the light interacting with atoms contained inside a cavity. The atom can be approximated as a system with just two energy levels, a two-level system. The coupling of this system with a single mode of the radiation field leads to a dynamical evolution of the atomic excitation in the form of nutations at the Rabi frequency. This simple model for the interaction is referred to as the Jaynes–Cummings model [34], well described in the literature [35], and extensively used in coherent optics with light or microwaves (nuclear magnetic resonance, NMR). In a strong coherent field, which means a beam with many photons, the coupling leads to a nutation at a frequency associated with the mean photon number n𝛼 , which is a continuously variable number. However, for a weak field (n𝛼 ≈ 1), the Rabi frequencies should be discrete and linked directly to the photon numbers in the field. By placing the atoms inside a resonator, interesting properties of the coupled atom–light system can be studied. This idea led to a series of beautiful experiments with the micromaser built at Garching in the group around H. Walther. A similar technology of cavity atom coupling has been perfected by a team in Paris around S. Haroche and coworkers [36]. In these systems so-called Rydberg atoms, that is, atoms in excited states with very high quantum numbers (n > 40) and that have very simple, long-lived states, are placed inside a microwave cavity of extremely high quality. The cavity encloses the atoms almost completely, and it can be cooled to exclude thermal noise, and the coupling between the internal field, the photons, and the probe in the form of the atoms can be made to dominate over all other effects. The atoms couple to the microwave field, and the coupled systems evolve in an elegant way determined by quantum mechanics. By observing the atoms that leave from the resonator, conclusions can be drawn about the state of the light–atom field inside. These systems can show features such as the collapse and revival of the wave function of the atom, as reviewed, for example, early by P. Meystre and M. Sargent III [26] and more recently by S. Haroche and coworkers [36]. We will come back to these experiments when we discuss different ways of generating strong non-classical states of light in Chapter 12 (Figure 3.7). Early experiments with Rydberg atoms by G. Rempe et al. [38] in a micromaser built in Garching have revealed an oscillation of the atomic population in a

0.5

76 Ph 5 4 oto nn 3 2 um be 1 0 r, n

0

Probability

7 6 5 4 3 2 1 0

5.0 〈n〉

1.0

Photon number, 〈n〉

3.7 Observing Photons in Cavities

4.5 4.0 0.06 0.07 0.08 Time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 100 Time (s) 80 60 r, N e 40 mb Direct observation of 20 nu m dynamics of photons in a cavity o t A

Figure 3.10 Direct observation of photons in a cavity via QND detection by S. Haroche team. The left side shows the progressive collapse of the field into a photon number state in the cavity by observing more and more atoms. The photon number probabilities are plotted versus the photon number n and the atom number N. The histograms evolve, as N increases from 0 to 110 atoms, from a flat distribution into n = 5. The right-hand side shows the dynamics. The mean photon number ⟨n⟩ is continuously monitored via QND over 0.7 s. The inset zooms into the n = 5 to 4 jump, showing that it is detected with a time resolution of about 0.01 s. Source: Haroche and coworkers 2007 [37]. Reproduced with permission of Springer Nature.

thermal field with (1.5 < n𝛼 < 3.8) and in the micromaser field. Similarly, the series of experiments in Paris led by S. Haroche and coworkers has achieved extreme conditions [36]. Here the Rydberg atoms traverse a cavity where both atom–cavity interaction time and cavity damping time are very long. This requires a cryogenic cavity (T = 0.8 K) of very high quality (Q = 7 × 107 ) and a very stable source of slow atoms. The average delay between successive atoms was set to 2.5 ms, much longer than the cavity decay time. In the cavity the radiation field and the atoms are coupled together, resulting in a time evolution that is dominated by oscillations at the Rabi frequency, and this determines the state of the atom at the end of its travel through the field in the resonator. After leaving the cavity the atoms were detected by state-selective field ionization. From these data, accumulated for many atoms, the dynamics of the atom–cavity evolution was reconstructed, as shown in Fig. 3.10. This showed a mixture of oscillations in time. Taking the Fourier transform of the traces at low average photon numbers, the experiments clearly showed discrete frequency components that correspond to the discrete Rabi oscillations associated with the photon numbers n = 1, 2, 3, 4, …. The discreteness of the radiation field is directly evident. The technology has been systematically improved. More recent experiments by the Paris group have refined this approach to the point where it is now possible to probe the field inside the cavity repeatedly – in this way first preparing and then analysing discrete photon number states in the cavity [37, 39]. It is now possible to carry out continuous quantum non-demolition (QND) measurements by using the coupling of the light field to the excitation of atoms in the cavity, and we can

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directly observe the establishment of the photon number. More details of this approach are given in Chapter 11. The major improvement over the earlier experiments is an increase of about 2 orders of magnitude in the quality of the cavity, pushing the damping time up to t = 0.13 s. This means that an average photon will travel over 10 000 km, back and forth within the cavity, before escaping. Given that successive atoms can now be sent across the cavity and measured every 0.5 ms, the trapped cavity field can now be probed with thousands of individual atoms before it decays. Quantum optics predicts that if a near stationary cavity field is measured to have a particular number of photons, then successive measurements will find the same number of photons in the field, as described in Chapter 4. This is exactly what is observed, with hundreds of successive atomic measurements returning the same discrete photon number. These experiments unambiguously show that the discreteness of optical energy measurements is a property of the field, not of the atoms in the measurement device. S. Haroche shared the Nobel Prize for physics in 2012 for this work, and his lecture provides a fascinating overview of the search for the evidence of photons.

3.8 Summary We have compelling experimental evidence that photons exist and that the classical wave picture has to be extended. There is no doubt that we require quantum description of light for many current experiments and even technologies. Some of these observations are linked directly to the detection process, whilst others show in detail the quantum nature of light. This is introduced formally in Chapter 4. For many years optical experiments in the visible spectrum have had the strong advantage that they combine high efficiency light generation and detection, can be operated at room temperature without much impact from thermal noise, and, importantly, are directly convincing to the observer. Optical experiments continue to play an important role of testing and verifying the quantum theory and its many predictions. At the same time optical techniques have been a driver of many, many technologies such as communication, sensing, data storage, and even entertainment. We are now busy in creating new applications based on quantum ideas.

References 1 Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 8: 149. 2 Rieke, G. (2003). Detection of Light: From the Ultraviolet to the Submillimeter,

2e. Cambridge: Cambridge University Press. 3 Glauber, R.J. (1962). Coherent and incoherent states of the radiation field.

Phys. Rev. 131: 2766. 4 Tinsley, J.N., Molodtsov, M.I., Prevedel, R. et al. (2016). Direct detection of a

single photon by humans. Nat. Commun. 7: 12172. 5 Lorentz, H.A. (1910). Alte und neue Fragen der Physik. Phys. Z. 11: 1234.

References

6 Paul, H. and Fischer, R. (1983). Light absorption by a dipole. Usp. Fiz. Nauk

141: 375. 7 Paul, H. (1995). Photonen, eine Einführung in die Quantenoptik. Stuttgart:

Teubner. 8 Lawrence, E.O. and Beams, J.W. (1927). The instantaneity of the photoelectric

effect. Phys. Rev. 29: 903. 9 Forrester, A.T., Gudmundsen, R.A., and Johnson, P.O. (1955). Photoelectric

mixing of incoherent light. Phys. Rev. 99: 1691. 10 Loudon, R. (2000). The Quantum Theory of Light, 3e. Oxford: Oxford Univer-

sity Press. 11 Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Nor-

malspektrum. Verh. Dtsch. Phys. Ges. 2: 237. 12 Hollandt, J. (2012). 125 Years of Research On and with the Black Body,

Metrology Throughout the Ages, vol. 3, no.3, 87. PTB Mitteilungen. 13 Wien, W. (1896). Über die Energievertheilung im Emissionsspectrum eines

schwarzen Körpers. Ann. Phys. 294: 662. 14 Lummer, O. and Pringsheim, E. (1900). Über die Strahlung des schwarzen

Körpers fur lange Wellen. Verh. Dtsch. Phys. Ges. 2: 163–180. 15 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 16 Janossy, L. and Naray, Zs. (1958). Investigation into interference phenomena

17 18 19 20 21 22 23 24 25 26 27 28

at extremely low light intensities by means of a large Michelson interferometer. Il Nuovo Cimento 9: 588. Hariharan, P., Brown, N., and Sanders, B.C. (1993). Interference of independent laser beams at the single-photon level. J. Mod. Opt. 40: 113. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4e. Oxford University Press. Feynman, R., Leighton, R.B., and Sands, M. (1963). The Feynman Lectures on Physics, vol. 3. Addison and Wesley. Michelson, A.A. and Morley E.W. (1927). On the relative motion of the earth and the luminiferous ether. Am. J. Sci. 1887 XXXIV (203). Hellmuth, T., Walther, H., Zajonc, A., and Schleich, W. (1987). Delayed-choice experiments in quantum interference. Phys. Rev. A 35: 2532. Magyar, G. and Mandel, L. (1963). Interference fringes produced by the superposition of two independent maser light beams. Nature 198: 255. Javan, A., Ballik, E.A., and Bond, W.L. (1962). Frequency characteristics of a continuous wave He-Ne optical maser. J. Opt. Soc. Am. 52: 96. Radloff, W. (1971). Zur Interferenz unabhängiger Lichtstrahlen geringer Intensit á́t. Ann. Phys. 26: 178. Franson, J.D. (1991). Violation of a simple inequality for classical fields. Phys. Rev. Lett. 67: 290. Meystre, P. and Sargent, M. III (1989). Elements of Quantum Optics, 448. Springer-Verlag. Sakurai, J.J. (1985). Modern Quantum Mechanics. Addison-Wesley. Arrecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of time evolution of the radiation field by joint photo-current distributions. Phys. Rev. Lett. 17: 260.

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29 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance fluorescence. Phys. Rev. Lett. 39: 691. 30 Cresser, J.D., Haeger, J., Leuchs, G. et al. (1982). In: Dissipative Systems in

Quantum Optics (ed. R. Bonifacio), 21. Springer-Verlag. 31 Walker, J.G. and Jakeman, E. (1985). Photon-antibunching by use of a photo-

electron event triggered optical shutter. Opt. Acta 32: 1303. 32 Grangier, P., Roger, G., Aspect, A. et al. (1986). Observation of photon anti-

bunching in phase-matched multiatom resonance. Phys. Rev. Lett. 57: 687. 33 Scgirhagle, R., Chang, K., Loretz, M., and Degen, C.L. (2014).

34

35 36 37 38 39

Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65: 83. Jaynes, E.T. and Cummings, F.W. (1963). Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51: 89. Cohen-Tannoudj, C., Diu, B., and Laloë, F. (1980). Quantum Mechanics. Wiley. Brune, M., Schmidt-Kaler, F., Maali, A. et al. (1996). Quantum Rabi oscillation: a direct test of field quantization in a cavity. Phys. Rev. Lett. 76: 1800. Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive field state collapse and quantum non-demolition photon counting. Nature 448: 889. Rempe, G., Walther, H., and Klein, N. (1987). Observation of quantum collapse and revival in a one-atom maser. Phys. Rev. Lett. 58: 353. Deleglise, S., Dotsenko, I., Sayrin, C. et al. (2008). Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature 455: 510.

Further Reading Haroche, S. (2012). Nobel Lecture 2012. www.nobelprize.org/nobel prizes/physics/laureates/2012/haroche-lecture.html. Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford University Press. Kingston, R.H. (1995). Optical Sources, Detectors and Systems. Academic Press. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization entangled photon pairs. Phys. Rev. Lett. 75: 4337. Mandel, L. and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press. Zeilinger, A., Weihs, G., Jennewein, T., and Aspelmeyer, M. (2005). Happy centenary, photon. Nature 433: 230.

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4 Quantum Models of Light 4.1 Quantization of Light 4.1.1

Some General Comments on Quantum Mechanics

In this chapter we will develop quantum mechanical models for light that will help us understand the various experimental results and techniques we will encounter in this book. Let us begin by reviewing the types of predictions one can make about experimental outcomes of quantum mechanical systems and how these are obtained from the mathematics. In classical mechanics, an observable quantity is represented by a variable such as x, the position of a particle. This is just a number that may vary as a function of time and/or other parameters. The outcome of an experiment that measures position will just be the value of x. In general, for quantum mechanical systems, the result of an experiment will not be unique. Instead, a range of possible outcomes will occur with some probabilities. Consequently, a single number is insufficient to describe all the possible outcomes of a particular experiment. Thus, in quantum mechanics, variables are ̂ the position operator, describing the type of meareplaced by operators such as x, surement to be made, and states, represented by kets, |𝜎⟩. As we saw in the last chapter, these contain the information about the possible outcomes. Each operator representing an observable quantity will have a set of eigenstates associated with it whose eigenvalues represent the spectrum of possible results that a measurement of this observable can produce. Typically a system will be in a state that is a superposition of various eigenstates of some observable. A special case is when the system is actually in an eigenstate of the observable, in which case a definite result is predicted. Most generally the system will be in a statistical mixture of various quantum states. Quantum theory makes predictions about the results of large ensembles of measurements. The average value of an observable is given by the expectation value of the observable’s operator: ̂ = ⟨𝜎|x|𝜎⟩ ̂ ⟨x⟩

(4.1)

Equation (4.1) predicts the average value of measurements performed on a large ensemble of systems all prepared in the same state, |𝜎⟩, of the observable ̂ We can also consider expectation values of higher represented by the operator x. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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moments such as ⟨x̂ 2 ⟩, which naturally leads us to define the variance of an observable as ̂ 2 ⟨Δx̂ 2 ⟩ = ⟨x̂ 2 ⟩ − ⟨x⟩

(4.2)

and similarly for higher-order properties. In classical mechanics a non-zero variance, or noise, is associated with imperfect preparation of the system with the desired property. In quantum mechanics even ideally prepared systems can exhibit non-zero variance due to the probabilistic nature of the theory. Such non-zero fluctuations are thus often referred to as quantum noise. The characteristics of a system with respect to a particular observable can be completely characterized by the expectation values of all the moments of the observables’ operator. However, in many cases of interest in this book, the systems will exhibit Gaussian statistics. In such cases, the system is completely characterized by just the first- and second-order moments. A powerful link between classical variables and quantum operators exists in the procedure of canonical quantization, first pioneered by Dirac. If a system is described classically by the canonically conjugate variables x and p (in the sense of Hamilton’s equations [1]), then the correct quantum mechanical description can be obtained by simply replacing x and p with the operators x̂ and p̂ in the Hamiltonian and replacing the classical Poisson bracket with the commutation ̂ p] ̂ = x̂ p̂ − p̂ x̂ = iℏ. In the following section we will use this procerelation [x, dure to convert the classical description of light introduced in Chapter 2 into a quantum mechanical description. 4.1.2

Quantization of Cavity Modes

The classical description of light in Section 2.1 assumed as a model for a laser beam a single frequency wave. Although an idealization, for classical light it is not an unreasonable one as it would neglect modulations and fluctuations of the light. Anticipating this we start our discussion by considering the discrete optical modes in a ring cavity. A sufficiently high finesse cavity will only support fields with wavelengths 𝜆 according to 𝜆 = L∕n,

n = 0, 1, 2, 3, …

(4.3)

where L is the cavity round-trip length. This discrete frequency approximation is valid in both the classical and quantum domains. Recall first from Section 2.1.4 that the energy in a classical light field is proportional to the sum of the squares of the quadrature amplitudes X1 and X2. Specifically for polarized modes in a high finesse cavity, the energy is given classically by ℏ𝜔k (4.4) (X12k + X22k ) 4 where k labels the different frequency modes and 𝜔k = 2π𝜈k is the angular frequency. Moreover we can identify suitably scaled versions of the quadratures H = Σk

4.1 Quantization of Light

as the canonical variables of the classical light wave. Specifically qk and pk are canonical variables where √ ℏ𝜔k qk = X1k 2 √ ℏ pk = − X2 (4.5) 2𝜔k k We can now apply canonical quantization by introducing the operators q̂ k and p̂ k with [q̂ k , p̂ l ] = iℏ𝛿kl . This immediately implies that the operator equivalents of ̂ k and X𝟐 ̂ k obey the commutator the quadrature amplitudes X𝟏 ̂ l ] = 2i𝛿kl ̂ k , X𝟐 [X𝟏 The energy operator or Hamiltonian for the system is just ∑ ℏ𝜔k ̂ 2) ̂ 2 + X𝟐 ̂ = (X𝟏 H k k 4 k 4.1.3

(4.6)

(4.7)

Quantized Energy

The Hamiltonian is composed of a sum of components with a whole range of frequencies. These different waves are independent in the sense that different frequency components commute with each other (see Eq. (4.6)) – much like we are used to considering classical waves of different frequencies to be independent. Like in a classical cavity the frequency is a discrete variable, and only frequencies 𝜔 associated with the spatial modes are possible. In fact the Hamiltonian is mathematically equivalent to a sum of uncoupled harmonic oscillators, one for each frequency 𝜔. Each term describes one mode. For each mode we can solve the operator equation. Before carrying out the calculations in detail, let us consider the overall properties of the quantized harmonic oscillator. One important consequence of the quantization of the oscillator is the discreteness of the possible energy states of this system. The system has a series of possible energies that are equally spaced as illustrated in Figure 4.1: ) ( 1 ℏ𝜔 (4.8) n = n + 2 Similarly, the quantized field has discrete energies, n . The energy of the field can only be increased by multiples of the energy quantum of ℏ𝜔. This quantization suggests that the field is made up of particles – photons. However, one has to be careful to use only those properties of particles that can be directly derived from the quantized field equations. Care has to be taken with this picture to avoid an over-interpretation and to regard photons too much as particles behaving like small classical particles. It is worthwhile to contemplate one peculiar case, the lowest possible energy 0 = 12 ℏ𝜔, known as the zero point energy. The value of 0 is not zero. There is remaining energy. However, this does not upset our measurements since it is not

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4 Quantum Models of Light

Figure 4.1 Energy levels of a harmonic oscillator.

Energy

96

(n + 1/2) ħω

possible to extract this energy from the system. It is the lowest energy that can be achieved with this particular mode of light. All measurements of the energy of the mode refer back to it. The fact that we can only measure changes in the energy, not the absolute energy, means that the ground state is not directly accessible. Further discussions on this may be found in E.A. Power [2]. It is noteworthy that the value of 0 is unique to the optical frequency. A field that consists of many modes has a total zero point energy of 12 ℏ𝜔 per mode. Since in theory there is no upper bound on the frequencies, the energy of the ground state is infinite – this creates a conceptual problem. Mathematically this can be overcome by renormalization [3], and in practice any real experiment will have a finite detection bandwidth and the infinity is avoided. It is possible to measure the effect of the zero point energy directly. One can create an experimental situation where the number of allowed modes in a multimode system can be changed, and as a consequence the total zero point energy changes. This increase in energy results in a force that tries to change the system towards the lowest total zero point energy. Such effects were investigated by H.B.G. Casimir and D. Polder [4]. Consider a cavity with length d. The cavity supports modes for all wavelengths 𝜆 = 2d∕n = L∕n. The longest supported wavelength is L. The size of the cavity determines the energy of the allowed low frequency modes and thus the total zero point energy. That means that for a smaller cavity, shorter L, the total zero point energy is smaller than for a larger cavity. The change in total zero point energy is proportional to the change in number of modes. H.B.G. Casimir and D. Polder [4] worked this out for a system of two conducting plates. In this case the total zero point energy varies with L−3 , and the resulting force Fcas is given by Aπhc [N] (4.9) 4.810−20 d4 where A is the area of the plates and d = L∕2 is the separation. This force does not depend on the electron charge, and it does not require any electric field between the plates. It is a quantum mechanical effect that applies to two plates in vacuum. The forces can be significant for very small plate separations. As an example values of A = 10−4 m2 and d = 0.01 mm result in Fcas = 1.3 × 10−11 N. This pure quantum force was demonstrated in careful measurements where all other forces were excluded [5]. Without elaborating, it is worth noting that similar forces are part of the explanation of the well-known van der Waals forces [2]. These forces are essential for the explanation of a range of intermolecular effects and include Fcas =

4.2 Quantum States of Light

such practical effects as the stability of colloids in paint suspensions. The zero point energy can play an important role. 4.1.4

The Creation and Annihilation Operators

Just as, given certain assumptions about the spatial modes, we could describe the essential characteristics of a classical light field via its complex amplitude, 𝛼 (see Chapter 2), so its quantized equivalent, â , plays a similar role in a quantum description of light. The operator â and its adjoint, â † , raise and lower the excitation of a specific mode by one increment, respectively. It follows from Eq. (4.6) ̂ = i(̂a† − â ) that they obey the Boson ̂ = â + â † and X𝟐 and the relationships X𝟏 commutation rules [̂a𝜔 , â †𝜔′ ] = (̂a𝜔 â †𝜔′ − â †𝜔 â 𝜔′ ) = 𝛿𝜔𝜔′

(4.10)

[̂a𝜔 , â 𝜔′ ] = [̂a†𝜔 , â †𝜔′ ] = 0

(4.11)

and We can rewrite the Hamiltonian operator of the radiation field of a single mode in terms of these operators as ̂ = 1 ℏ𝜔(̂a† â + â â † ) H 2 ( ) ( ) 1 1 = ℏ𝜔 n̂ + (4.12) = ℏ𝜔 â † â + 2 2 where n̂ = â † â is the photon number operator. The Hamiltonian above represents the number of photons in one mode multiplied by the energy of one photon in that mode. For a multimode field the total Hamiltonian would be the sum of several terms such as given in Eq. (4.12), one for each frequency component. Note the specific ordering of the operators â † and â . The ordering is important since the operators do not commute. ̂ p̂ and the creation Finally, the link between the conjugate variable operators q, and annihilation operators â † , â is given by √ ℏ𝜔k q̂ = (̂a + â † ) 2 √ ℏ p̂ = −i (̂a† − â ) (4.13) 2𝜔k It is again worth noting that the operator algebra of the quantized electromagnetic field is identical to that of the quantized simple harmonic oscillator.

4.2 Quantum States of Light 4.2.1

Number or Fock States

The Hamiltonian for a single mode of the radiation field (Eq. (4.12)) has the eigenvalues ℏ𝜔(n + 12 ). The corresponding eigenstates can be represented by the

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4 Quantum Models of Light

notation |n⟩. These are known as the number or Fock states and are eigenstates of the number operator ̂ = n|n⟩ â † â |n⟩ = n|n⟩

(4.14)

The ground state, also known as the vacuum state, is defined by â |0⟩ = 0

(4.15)

It cannot be lowered any further. The energy of the ground state has already been discussed in Section 4.1.3 and is given by 1∑ ̂ ⟨0|H|0⟩ = ℏ𝜔k (4.16) 2 k where we sum over all frequencies. The operators â and â † are raising and lowering operators on a series of equally spaced eigenstates: √ √ (4.17) â |n⟩ = n|n − 1⟩ and â † |n⟩ = (n + 1)|n + 1⟩ They can be used to generate all possible number states starting with the ground state |0⟩ by a successive application of the creation operator (̂a† )n |n⟩ = √ |0⟩ n!

(4.18)

They are orthogonal and complete, i.e. ∑ ⟨n|n′ ⟩ = 𝛿nn′ and |n⟩⟨n| = Î

(4.19)

n

where Î is the identity operator. The number of photons n is exactly known for a number state; hence the variance of the photon number is zero for a number state: ̂ ̂ 2=0 Vn = ⟨Δn2 ⟩ = ⟨n|n̂ n|n⟩ − ⟨n|n|n⟩

(4.20)

We encountered the single photon number state in Section 3.5 where we found that a consideration of their properties in an interferometer led us to a radically different view of light from the classical one. We can illustrate this further by contrasting the average value of the quadrature amplitudes for classical light and light in a number state. Recall that in a classical description of light, an arbitrary quadrature amplitude, X 𝜃 , can be described as a function of the phase angle, 𝜃, according to X 𝜃 = cos[𝜃]X1 + sin[𝜃]X2

(4.21)

We can convert this equation to the correct quantum mechanical form by simply replacing the quadrature variables by quadrature operators. The average value of the quadrature amplitude for light in a particular quantum state is then found by taking the expectation value over the state of interest. However what we find for a number state is that ̂ 𝜃 ⟩n = ⟨n|X ̂ 𝜃 |n⟩ = 0 ⟨X

(4.22)

4.2 Quantum States of Light

where the orthogonality of the number states ensures that all terms like ⟨n|̂a|n⟩ and ⟨n|̂a† |n⟩ are zero (see Eq. (4.17)). The results in Eqs. (4.21) and (4.22) could hardly be more different. In the classical case the quadrature amplitude oscillates as a function of the quadrature angle, 𝜃, whilst for the number state the average value is always zero, independent of the quadrature angle. It is clear that number states represent quite exotic states of light, quite dissimilar to what is produced directly by a laser. We will return to consider them again later but for now we look for light states with more familiar behaviour, which, in turn, turn out to be those most naturally produced. For now we use the number states as a basis set and find more realistic superpositions. However, one particular number state plays a major role in all quantum optical experiments: the vacuum state |0⟩. 4.2.2

Coherent States

The number states do not represent an optical state with a well-defined phase. To describe laser light we require a state that has a more precisely defined phase. This is the coherent state |𝛼⟩, which is the closest quantum approximation to the field generated by a laser [6]. Consider, in an idealized way, how a light wave is produced in the classical picture. Initially the energy in the electric and magnetic fields is zero. Then, by some means, the electric field is displaced to some non-zero value, say, |𝛼|. This in turn displaces the magnetic field and so on, producing a propagating light field. Suppose we follow the same idealized recipe in quantum mechanics. This suggests that we should displace the quantum mechanical vacuum state to some non-zero value. We might hope that this will produce an oscillating field. However, because the quantum mechanical vacuum has non-zero energy, we also expect uniquely quantum features. ̂ acting on the vacuum ket has the effect of displacThe displacement operator D ing it by an amount 𝛼, where 𝛼 is a complex number that can be directly associated with the complex amplitude in classical optics (see Chapter 2). We define ̂ |𝛼⟩ = D(𝛼)|0⟩ The displacement operator is given by ̂ D(𝛼) = exp(𝛼 â † − 𝛼 ∗ â )

(4.23)

It is a unitary operator, i.e. ̂ −1 (𝛼) = D(−𝛼) ̂ ̂ † (𝛼) = D D and has the properties ̂ ̂ † (𝛼)̂aD(𝛼) = â + 𝛼 D † † ̂ ̂ D (𝛼)̂a D(𝛼) = â † + 𝛼 ∗

(4.24)

The family of states |𝛼⟩ is called coherent states, and, as we will now show, they behave in many ways like classical laser light. Due to the unitarity of the displacement operator, they are normalized to unity: ⟨𝛼|𝛼⟩ = 1

(4.25)

99

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4 Quantum Models of Light

Also the coherent states are eigenstates of the annihilation operator â , as can be verified from ̂ † (𝛼)̂aD(𝛼)|0⟩ ̂ ̂ † (𝛼)̂a|𝛼⟩ = D = (̂a + 𝛼)|0⟩ = 𝛼|0⟩ D ̂ Multiplying both sides with D(a) results in the eigenvalue condition â |𝛼⟩ = 𝛼|𝛼⟩

(4.26)

Using Eq. (4.26) we now see that the expectation values for the quadrature amplî 𝜃 ⟩𝛼 , are identical to the classical ̂ 𝜃 |𝛼⟩ = ⟨X tude operator in a coherent state, ⟨𝛼|X variables, provided we identify the displacement 𝛼 with the classical complex amplitude. That is ̂ 𝜃 ⟩𝛼 = cos[𝜃]X1 + sin[𝜃]X2 ⟨X

(4.27)

Hence we see that the average value of the field oscillates as a function of quadrature angle. Similarly the expectation value for the photon number in a coherent state is ⟨̂a† â ⟩𝛼 = 𝛼 ∗ 𝛼

(4.28)

which again is the same as the classical value. Thus we see that the average values of the observables for a coherent state are identical to those for a classical field. However, unlike a number state or a classical field that contains a definite amount of energy, a coherent state does not contain a definite number of photons. This follows from the fact that a coherent state |𝛼⟩ is made up of a superposition of number states: ∞ ∑ |n⟩⟨n|𝛼⟩ (4.29) |𝛼⟩ = n=0

Using Eq. (4.18) and the completeness of the number states, Eq. (4.19), one obtains ) ( 1 𝛼n (4.30) ⟨n|𝛼⟩ = √ exp − |𝛼|2 2 n! which is the contribution of the various number states. Now consider the relationship between two different coherent states. Using Eqs. (4.29) and (4.30), we find that their scalar product is ) 2 ( 1 | | |⟨𝛼1 |𝛼2 ⟩|2 = | exp − (|𝛼1 |2 + |𝛼2 |2 ) + 𝛼1∗ 𝛼2 | | | 2 2 = exp −|𝛼1 − 𝛼2 | (4.31) which is not zero. That means two different coherent states are not orthogonal to each other. However, for two states with significantly different photon numbers, that is, |𝛼1 − 𝛼2 | > 1, the product is very small. Such states are approximately orthogonal. One consequence is that the coherent states form a basis that is over-complete. But despite this, the coherent states can be used as a basis on which any state |Ψ⟩ can be expanded [7]. In summary, the coherent states have properties similar to those of classical coherent light and in fact represent its closest quantum mechanical

4.2 Quantum States of Light

approximation. Just as under certain conditions laser light can be approximated as idealized classical coherent light, so in situations where a quantum mechanical description is needed, laser light can often be well approximated by a coherent state. We will find in Section 5.1 that coherent states are stable in regard to dissipation, which means one coherent state is transformed into another coherent state when attenuated. Consequently, the coherence properties of the light remain similar after attenuation. Coherent states will be used extensively in our modelling of laser experiments. Being a complex number, the range of possible values of 𝛼, and thus the range of possible coherent states, requires a two-dimensional representation. This is only natural since we can distinguish different states of light experimentally by two parameters, intensity and phase. But we note that only one of them, the intensity, has a simple quantum mechanical equivalent (i.e. the number operator). A similarly simple quantum mechanical operator for the phase is not available. Phase operators can only be approximated in a rather complex way through a limit of a series of operators [7–9], and their definition has stimulated a lively debate amongst theoreticians. However, the real and imaginary parts of the annî and X𝟐, ̂ are both hilation operator â , i.e. the quadrature amplitude operators X𝟏 well-behaved operators. This leads to a pair of Hermitian operators that describe the field. 4.2.3

Mixed States

So far we have only considered pure field states. More generally we can have fields that are in an incoherent mixture of different pure states. This is referred to as ̂ a mixed state. A mixed state is normally represented by a density operator, 𝝆, such that 𝝆̂ = 𝑤1 |𝜎1 ⟩⟨𝜎1 | + 𝑤2 |𝜎2 ⟩⟨𝜎2 | + · · · represents a mixture of the states |𝜎1 ⟩, |𝜎2 ⟩, … with fractional weightings 𝑤1 , 𝑤2 , …, respectively. It is important not to confuse mixtures of states and superpositions√ of states; they are very different. Consider the equal superposition state |𝜙⟩ = 1∕ 2(|a⟩ + |b⟩). Written as a density operator, this state is 𝝆̂ p = |𝜙⟩⟨𝜙| 1 1 1 1 = |a⟩⟨a| + |a⟩⟨b| + |b⟩⟨a| + |b⟩⟨b| 2 2 2 2 In contrast an equal mixture of the states a and b is just

(4.32)

1 1 |a⟩⟨a| + |b⟩⟨b| (4.33) 2 2 The coherences between the states that are present for the superposition are absent in the case of the mixture. Expectation values for mixed states are given by 𝝆̂ m =

̂ = Tr{x̂ 𝝆} ̂ ⟨x⟩ where the trace, Tr, is taken over a complete set of eigenstates. An important example of a mixed state in optics is the thermal state, which can be represented

101

102

4 Quantum Models of Light

as a mixture of number states by ) ( ) ( 1 G−1 G−1 2 𝝆̂ th = |2⟩⟨2| + · · · |1⟩⟨1| + |0⟩⟨0| + G G G

(4.34)

where G is a parameter that ranges from one for a zero temperature thermal field (the vacuum) up to a high temperature thermal field for G → ∞.

4.3 Quantum Optical Representations 4.3.1

Quadrature Amplitude Operators

Let us now look in some detail at the quadrature amplitude operators defined by ̂ = â + â † X𝟏

̂ = i(̂a† − â ) and X𝟐

(4.35)

We will sometimes refer to them as the amplitude quadrature and the phase quadrature, respectively, for reasons that will become apparent when we discuss bright fields (Section 4.5). A variety of nomenclature for the quadrature ̂ X𝟐} ̂ ≡ amplitude operators exist in the literature – some common ones are {X𝟏, + ̂− ̂ ̂ ̂ ̂ p}. ̂ {X , X }; {X, P}; {x, ̂ We can define eigenstates of these operators via X𝟏|x1⟩ = x1|x1⟩ and similarly ̂ X𝟐|x2⟩ = x2|x2⟩. The eigenvalues xi form an unbounded continuous set. Because ̂ and X𝟐, ̂ it is not possible to of the non-zero commutation relation between X𝟏 ̂ ̂ be in a simultaneous eigenstate of X𝟏 and X𝟐. The resulting uncertainty principle constrains the variances of the operators in any state to ̂ 2⟩ ≥ 1 ̂ 2 ⟩⟨ΔX𝟐 ⟨ΔX𝟏

(4.36)

where the quadrature fluctuation operators are defined: ̂ = X𝟏 ̂ − ⟨ΔX𝟏⟩ ̂ ΔX𝟏

̂ = X𝟐 ̂ − ⟨ΔX𝟐⟩ ̂ and ΔX𝟐

(4.37)

̂ or X𝟐 ̂ are unphysical as infinite uncertainty It follows that exact eigenstates of X𝟏 in an unbounded spectrum implies infinite energy. This is analogous to the situation for position and momentum of a free particle. We saw in the previous section that the expectation values of the quadrature operators in a coherent state were equal to their classical values. It is straightforward to show that the uncertainty product for the quadratures in a coherent state is given by ̂ 2 ⟩𝛼 = 1 ̂ 2 ⟩𝛼 ⟨ΔX𝟐 ⟨ΔX𝟏

(4.38)

which means a coherent state is a minimum uncertainty state. Equally important is the fact that the uncertainty of the coherent state is split equally between the two quadratures. The uncertainty relationship is symmetric: ̂ 2⟩ = 1 ̂ 2 ⟩ = ⟨ΔX𝟐 ⟨ΔX𝟏

(4.39)

Hence the coherent state can be thought of as the best allowed compromise for a simultaneous eigenstate of the two quadratures. The quadrature operators allow us to represent a beam of light graphically. These so-called phasor diagrams are

4.3 Quantum Optical Representations

very popular in quantum optics – ever since their introduction by C. Caves [10]. Any state of light can be represented on a phasor diagram of the operators, that ̂ versus ⟨X𝟏⟩. ̂ is, a plot of ⟨X𝟐⟩ In such a diagram a particular coherent state |𝛼⟩ is ̂ ̂ equivalent to an area centred around the point given by the value (⟨X𝟏⟩, ⟨X𝟐⟩) for 2 ̂ ̂ this state. The size of the area is given by the variances ⟨ΔX𝟏 ⟩ and ⟨ΔX𝟐2 ⟩ and ̂ 2 ⟩, and the area of the is usually represented just by a circle with diameter ⟨ΔX𝟏 circle describes the extent of the uncertainty distribution. This phasor diagram is a qualitative description that shows several important results: 1. For a coherent state the uncertainty area is symmetric. Any measurement of the fluctuations will yield the same result, independent of the projection angle 𝜃 chosen. 2. The uncertainty area is of constant size, independent of the intensity |𝛼|2 of the state. ̂ X𝟐 ̂ 3. Two states are only distinguishable if their average coordinates in the X𝟏, ̂ 2 ⟩. plane are separated by more than ⟨ΔX𝟏 The last result is significant. It gives us a practical rule when two states are measurably different. The uncertainty area sets a minimum change in amplitude or phase that we have to make before we can distinguish two states. In a classical system such an uncertainty region would be associated with noise, due to technical imperfections. However, here the uncertainty area is a consequence of the quantum rules – not technical imperfections. Consequently, the uncertainty area represents the effect of quantum noise on a measurement, whether it is amplitude, phase, or any other property of the light. Experimental measurements cannot be more precise than the quantum noise. The uncertainty area is two-dimensional. One can imagine the entire parameter space being covered, or tiled, with circular uncertainty areas of uniform size (see Figure 4.2). The number of tiles, which are not all overlapping, approximately determines the number of states that can be distinguished. As a practical example, consider the light in an optical communication system. The information is transmitted by changing either the intensity or phase of the light at one modulation frequency. We assume the intensity is limited. How many distinguishable Figure 4.2 Uncertainty areas can cover the entire parameter space like tiles.

ˆ 2 ΔX2

X2

ˆ 2 ΔX1

X1

103

104

4 Quantum Models of Light

states of light, that is, different pieces of information, could we send? The light is represented by a two-dimensional plane with a circle that indicates the largest possible intensity Imax . Inside this circle we can place our tiles. The number of tiles estimates the upper limit to the number of states we can use, that is, the number of different pieces of information that a beam of light modulated at one frequency can contain. As described in Section 10.2, in 2018 many more systems are approaching this quantum resolution, and ultimately technology will reach this limit in most cases. 4.3.2

Probability and Quasi-probability Distributions

The uncertainty area is a simple, useful tool that can, under certain circumstances, be used to completely characterize the state. To understand under what conditions this is possible, we need to look at the probability distributions ̂ and X𝟐. ̂ Such measurements can be carried out associated with measuring X𝟏 using the technique of balanced homodyne detection, which will be described in detail in Chapter 8. The probability of obtaining a result in a small interval, 𝛿x, ̂ performed on a system around a particular value x1 from a measurement of X𝟏 in the state |Ψ⟩ is given by PΨ (x1 ) = |⟨x1 |Ψ⟩|2 𝛿x

(4.40)

̂ the probability of obtaining results in a and similarly for a measurement of X𝟐 small interval around x2 is PΨ (x2 ) = |⟨x2 |Ψ⟩|2 𝛿x

(4.41)

In other words |⟨xi |Ψ⟩| is the probability density at the point xi . These distributions can be evaluated for various states using the relationships 2

d ̂ ⟨x1 |X𝟐|Ψ⟩ = −i ⟨x |Ψ⟩, dx1 1

d ̂ ⟨x2 |X𝟏|Ψ⟩ =i ⟨x |Ψ⟩ dx2 2

(4.42)

For example, for a coherent state, we can show ⟨x1 |̂a|𝛼⟩ = 𝛼⟨x1 |𝛼⟩ 1 ̂ + iX𝟐|𝛼⟩ ̂ = ⟨x1 |X𝟏 2( ) d 1 ⟨x1 |𝛼⟩ x1 + = 2 dx1

(4.43)

Solving the relevant differential equation then gives 2 1 |⟨x1 |𝛼⟩|2 = √ e−(x1 −x1 ) π

(4.44)

where x1 = (𝛼 + 𝛼 ∗ )∕2 is the real part of 𝛼. We see that our x1 results form a Gaussian distribution. The width or variance of the distribution is 1 as expected. Similarly we find 2 1 |⟨x2 |𝛼⟩|2 = √ e−(x2 −x2 ) π

(4.45)

4.3 Quantum Optical Representations

for the probability distribution of x2 results. Here x2 = i(𝛼 − 𝛼 ∗ )∕2 is the imaginary part of 𝛼. Again, we find a Gaussian distribution of width 1. Because the coherent states are Gaussian, they can be completely characterized by the first̂ and X𝟐 ̂ and thus the uncertainty area of the preand second-order moments of X𝟏 vious section. This property leads to an elegant approach based on the covariance matrix, which will be discussed in Section 4.3.4. It turns out that many experimentally accessible optical quantum states have Gaussian quadrature distributions and therefore are quite easily characterized. This is not in general true however. For example, the quadrature distributions for a single photon number state are given by |⟨x1 |1⟩|2 =

2 1 √ 2x2 e−x1 1 π

(4.46)

and similarly 2 1 |⟨x2 |1⟩|2 = √ 2x22 e−x2 π

(4.47)

These distributions are clearly non-Gaussian. A two-dimensional probability distribution for a state |Ψ⟩ is given by the Q-function defined as |⟨𝛼|Ψ⟩|2 π which is positive and bounded by QΨ (𝛼) =

(4.48)

1 π Physically the Q-function represents the probability density of obtaining the ̂ and result 𝛼 = (x1 + ix2 )∕2 from an optimal simultaneous measurement of X𝟏 ̂ ̂ ̂ X𝟐. Because X𝟏 and X𝟐 do not commute, such a measurement is necessarily nonideal. By optimal we mean that the resolution of the measurement is as good as the uncertainty principle allows. Such an optimal measurement can be achieved using the techniques of heterodyne or dual homodyne detection, which will be discussed in Chapter 8. For a coherent state |𝛽⟩, the Q-function is simply QΨ ≤

exp(−|𝛼 − 𝛽|2 ) |⟨𝛼|𝛽⟩|2 = (4.49) π π It is a Gaussian function centred around the point (e(𝛽), m(𝛽)). The width of the Gaussian is two, that is, twice that of what we found when the quadratures were measured individually. The additional noise is a result of the nonideal measurement in which one extra unit of quantum noise is added. Note that this function has a width that is independent of the average photon number of the state. A strong coherent state has the same-sized Q-function as a weak field – just displaced further from the origin. The Q-function for a number state is quite different to that of a coherent state and is given by Q𝛽 =

Qn =

|𝛼|2n exp(−|𝛼|2 ) |⟨𝛼|n⟩|2 = π πn!

(4.50)

105

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4 Quantum Models of Light

Figure 4.3 Uncertainty area of a number state.

X2

X1

This distribution has the shape of a ring (see Figure 4.3). The maximum probability is a circle around the origin, and the distribution in the radial direction is Gaussian centred at the average photon number with a width that decreases very rapidly, with n! That means for large photon numbers the radial distribution is approaching a delta function. A number state contains no phase information at all. The ring shape does not favour any particular phase. A two-dimensional distribution more closely related to our homodyne distributions is the Wigner function, which can be written as WΨ (𝛼) =

̂ D(−𝛼)|Ψ⟩ ̂ ̂ 2⟨Ψ|D(𝛼) 𝚷 π

(4.51)

̂ is the parity operator. The parity operator measures the symmetry of where 𝚷 the probability amplitude distribution of a state. The number states are parity eigenstates with eigenvalues +1 for n even and −1 for n odd. So the action of the parity operator on an arbitrary superposition of number states is ̂ 0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · ·) = (c0 |0⟩ − c1 |1⟩ + c2 |2⟩ − · · · ) 𝚷(c This is effectively a π rotation on the phasor diagram. Thus the Wigner function is the overlap between the state, displaced by −𝛼, and itself, also displaced by −𝛼 but then also rotated by 180 degrees. For a coherent state |𝛽⟩, we will obtain 2⟨𝛽 − 𝛼|𝛼 − 𝛽⟩ 2e−2|𝛼−𝛽|2 = (4.52) π π This is a symmetric Gaussian with width 1, centred at the point (e(𝛽), m(𝛽)). Its projections along the X1 and X2 directions correspond with the homodyne probability distributions of Eqs. (4.44) and (4.45). In this way the noise introduced by the quantum fluctuations can be interpreted as producing ‘classical’ noise distributions. However, the Wigner function is a quasi-probability function. Although for Gaussian states it is always positive and generally well behaved, it can become negative for non-Gaussian states, such as the number state, and thus cannot be interpreted as a true probability distribution. For such states the quantum fluctuations do not have a classical’ noise interpretation. Producing and analysing exotic states with negative Wigner functions is technically challenging but is now W𝛽 (𝛼) =

4.3 Quantum Optical Representations

Figure 4.4 The randomness of the quadrature values represented by a weighted random distribution of dots. Each dot represents one particular instantaneous value. The histogram of all ̂ and X𝟐 ̂ dots both in the X𝟏 directions corresponds to the projection of the Wigner function.

ˆ X2

ˆ X1

achieved routinely by several groups. We will look at some examples of negative Wigner functions in Chapter 12. The uncertainty areas that are shown in the phasor diagrams are simplified representations of the Wigner function. They are contours drawn at the half maximum value of the full two-dimensional Wigner function (see Figures 4.4 and 4.5). This line is the edge of the uncertainty area. It is described by only two values, the size of the minimum and the maximum axes. From an experimental point of view, we use the measured variance, for a certain angle 𝜃, as a value for the uncertainty of this particular quadrature. That means we select one 𝜃, measure the variance of this quadrature, and obtain two points. We can repeat this for many angles, and in a real experiment we would actually do this to check the theory and reduce the experimental errors. All these points together describe the edge of the uncertainty area. Whilst the Wigner function of a coherent state is circularly symmetric, this is not the only option. The width of the distribution in orthogonal directions could be different, and it is possible for one of the variances to fall below 1. The only requirement is that the product of the two widths, that is, the product of the variances, satisfies the uncertainty relationship. We shall see in Chapter 9 that such states can be generated in optical experiments, which are known as squeezed states. It follows from the special properties of Gaussian functions that for such Figure 4.5 A two-dimensional Gaussian distribution and its contour. This is the origin of the circular uncertainty area.

X2

X1

107

108

4 Quantum Models of Light

a state the contour at the half maximum value is always an ellipse. The diameter along the minor and major axes corresponds to the minimum and maximum quantum noise. The various distributions we have looked at can be generalized to mixed states such that PΨ (xi ) = ⟨xi |𝝆̂ 𝚿 |xi ⟩ QΨ (𝛼) =

⟨𝛼|𝝆̂ 𝚿 |𝛼⟩ π

and 2 ̂ ̂ ̂ Tr{D(−𝛼) 𝝆̂ 𝚿 D(𝛼) 𝚷} (4.53) π where in each case 𝝆̂ 𝚿 is the density operator of the state to be characterized. For example, the Wigner function for a thermal state (see Eq. (4.34)) is given by WΨ (𝛼) =

2

2e−2|𝛼| ∕𝜎 (4.54) π which is again a symmetric Gaussian with width 𝜎 = 2G − 1. Finally, let us make a comment on other possible representations. Another obvious representation would generate a delta function for any particular coherent state. A representation with this property is the P-representation. A plot of the P-function would immediately show which coherent state is present. However, there are a number of mathematical problems that arise with the P-function once, for example, squeezed states are considered. Since for a squeezed state the variance of one of the quadratures is smaller than for a coherent state, and since a delta function cannot be made any narrower, the P-function will get negative values and even singularities. However, regularized versions of the P- function have been developed, which avoid some of these mathematical problems, and these can form a useful way of theoretically analysing quantum optical systems. It can actually be shown that all three representations (Q, P, and W ) are only limiting cases of a whole continuum of possible representations. Wth (𝛼) =

4.3.3

Photon Number Distributions

In the previous section we looked at results of measurements of the quadrature amplitudes. We now consider photon number measurements of a single optical mode. These are modelled as measurements of the observable n̂ = â † â . In general the probability distribution, 𝜌 (n), of a state described by the density operator 𝝆̂ is given by its diagonal elements in the photon number basis, i.e. ̂ 𝜌 (n) = ⟨n|𝝆|n⟩

(4.55)

For a photon number state, there is a unique photon number; it will be exactly the same in every measurement, and hence Eq. (4.55) has only one non-zero element. However, for a coherent state, one expects to find a spread in the results of many measurements of the photon number. We can use the expansion in Eq. (4.29) to determine the probability 𝛼 (n) of finding n photons in the coherent state |𝛼⟩: 𝛼 (n) = |⟨n|𝛼⟩|2 =

|𝛼|2n exp(−|𝛼|2 ) n!

(4.56)

4.3 Quantum Optical Representations

0.2 nα = 4

0.1

nα = 1000

nα = 36

0.05

2Δn = 64

P(n)

P(n)

0.15

0.035 0.03 0.025 0.02 0.015 0.01 0.005

0 0

10

20

30

40

50

60 n

950

1000

1050

1100

Figure 4.6 Poissonian distribution of photons for three coherent states: (i) n𝛼 = 4, (ii) n𝛼 = 36, and (iii) n𝛼 = 1000.

This is a Poissonian distribution with a mean value of |𝛼|2 , which represents the average number n𝛼 of photons in this mode. The intensity fluctuates according to this distribution. The noise is not a result of technical imperfections of the experiment. This is a quantum property of the light. Mathematically speaking it is a direct consequence of the quantum mechanical commutation rules. One important property of any Poissonian distribution is that the variance of the distribution is equal to the√mean value. Consequently, the width of the distribution Δn is equal to Δn = n𝛼 . Examples of three such distributions for n𝛼 = 4, n𝛼 = 36, and n𝛼 = 1000 are shown in Figure 4.6. Evaluating the relative uncertainty of√the mean photon number of a coherent state results in Δn∕n = √ n𝛼 ∕n𝛼 = 1∕ n𝛼 , which decreases with n𝛼 . A state with a high photon number is better defined than that of a state with a small number of photons. We can interpret this result in the photon picture: a photodetector will not see a regular stream of photons, but it will see a somewhat randomized stream such that the measurement of the absorption events – for a perfect detector equal to the number of photons incident – has a distribution given by 𝛼 (n). Each event creates a fixed number of electrons, and the stream of electrons leaving the detector forms the photocurrent. The distribution 𝛼 (n) is directly represented by the variation of the photocurrent. The intensity noise properties of the light could be expressed in terms of the distribution function 𝛼 (n). Thinking in the time domain, this means that the photons arrive in unequal time intervals as shown in Figure 4.7d. For large intensities 𝛼 (n) tends to a normal Gaussian function. In contrast, for low intensities, the Poissonian distribution differs markedly from a Gaussian function. A convenient way to classify the photon number distribution, particularly for bright fields, is via its width relative to Poissonian. In many practical situations light has more fluctuations than a coherent state, and its photon distribution is wider than a Poissonian function. In this case we would call it super-Poissonian. On the other hand, we will find that non-linear processes can be used to make the distribution narrower. This we will call sub-Poissonian light. In order to quantify the noise one can define the Fano factor, which is given by the ratio of the actual width of the distribution to the width of a Poissonian distribution with the same mean photon number. For some arbitrary state 𝜌𝛽 with

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δX2

δX2

δX2 α2 δX1

α1

(a)

X

δX1

(d) I(t)

P(n)

I2

[I2]

[I1]

Vnα1

n nα2

nα1

(b)

t (e) I(Ω)

Counts

α2 1

α1 t

(c)

0

Ω (f)

Im(α(Ω)) Re(α(Ω))

(g)

Ω

Figure 4.7 Different representations of a coherent state of light, shown for states |𝛼1 ⟩, |𝛼2 ⟩ with different intensities. (a) Phase space diagram, (b) photon number distribution, (c) arrival times of photons, (d) Wigner function, (e) time evolution of the intensities, (f ) variance of the intensities, and (g) the noise at different frequencies.

4.3 Quantum Optical Representations

mean photon number n𝛽 and distribution V n𝛽 , we define the Fano factor as f𝛽 =

V n𝛽 n𝛽

(4.57)

which classifies the states in the following way: f f f f

=0 <1 =1 >1

number state sub-Poissonian state coherent state super-Poissonian state

Note that the Fano factor is determined only by the variance. In fact the Fano factor is identical to the normalized variance of the photon flux, which, for many practical situations, is equivalent to the variance V 1 of the quadrature amplitude, as we shall see in Section 4.5. In much of this guide, we will use V 1 interchangeably with the Fano factor. Another important way to classify photon number distributions, particularly for low intensity fields, is the second-order coherence function, g (2) , previously introduced for classical fields, which we will discuss in detail in Section 4.6.1. For zero time delay, we have the following relationship between the second-order coherence function and the Fano factor: 1 g (2) (0) = f + 1 − (4.58) n 4.3.4

Covariance Matrix

The second-order quadrature moments can be arranged into a 2 × 2 matrix V, referred to as the covariance matrix [11]: ( ) 1 ̂ X𝟐 ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨Δ X𝟏Δ ⟨ΔX𝟏 2 V= 1 ̂ (4.59) ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨ΔX𝟏ΔX𝟐 ⟨ΔX𝟐 2 where the quadrature fluctuation operators are defined in Eq. (4.37). Similarly, the displacement vector, d, can be defined via the first-order moments: ( ) ̂ ⟨X𝟏⟩ d= (4.60) ̂ ⟨X𝟐⟩ For Gaussian states the covariance matrix plus the displacement vector completely characterizes the state. For simplicity, the covariance matrix is often presented in its standard form: ( ) ̂ ′2 ⟩ 0 ⟨ΔX𝟏 ′ V = (4.61) ̂ ′2 ⟩ 0 ⟨ΔX𝟐 ̂ + sin 𝜃 X𝟐 ̂ and X𝟐 ̂ ′ = cos 𝜃 X𝟐 ̂ − sin 𝜃 X𝟏 ̂ are rotated ver̂ ′ = cos 𝜃 X𝟏 where X𝟏 sions of the quadratures with 𝜃 specifically chosen to make the off-diagonal

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terms vanish. Such a 𝜃 can always be found. Physically this corresponds to rotating the quadratures until they line up with the major and minor axes of the uncertainty ellipse (see Section 4.3.2). In other words, in the standard form, ̂ ′2 ⟩ is minimized, or vice versa. ̂ ′2 ⟩ is maximized and V 2 = ⟨ΔX𝟐 V 1 = ⟨ΔX𝟏 Many useful properties about Gaussian states can easily be extracted from the covariance matrix. This will be particularly important when we consider multimode states and their correlations. A simple single-mode example is that the uncertainty principle can be expressed as DetV ≥ 1. Here DetM is the determinant of the matrix M, reducing to the familiar Heisenberg form where the covariance matrix is in its standard form (Eq. (4.61)). 4.3.4.1 Summary of Different Representations of Quantum States and Quantum Noise

We have examined a number of representations of quantum optical states, several based on quadrature amplitudes, plus the photon number distribution. All representations are complete, so the choice of which to use for a particular problem may depend on mathematical tractability, the physics one wishes to highlight, or simply taste. Being quasi-probability distributions, these representations often suggest interpretations in terms of noise. Quantum noise is the most easily observable consequence of the quantization of light. It is a direct result of the use of operators for describing the amplitude of the electromagnetic field and of the commutation rules applied to these operators. In other words, the minimum uncertainty principle applied to light leads to the quantum noise that affects many optical measurements. For Gaussian states a noise interpretation is always possible for the quadrature amplitudes; however in general such an interpretation is at odds with photon number measurements. For non-Gaussian states this fails even for quadrature measurements as negative probabilities’ can result. A useful quantum state, sometimes seen as lying on the boundary between classical and quantum states, is the coherent state. It is an idealized state of light representing the output of a perfect laser source. In mathematical terms it is the eigenstate of the annihilation operator. A physical consequence of this is that attenuation changes one coherent state into another coherent state, just with lower energy. The characteristic properties of the coherent state are preserved. In many situations the coherent state can be interpreted as a classical light field with the addition of stochastic noise. The coherent states can be used as a basis set, and all types of light can be described as a superposition of these basic states. This expansion can be represented by a probability distribution for measuring particular 𝛼, the Q-function, which includes the intrinsic measurement penalty. The characteristic properties of a coherent state of light can be summarized in Table 4.1. The corresponding graphical representations of quantum noise are shown in Figure 4.7.

4.4 Propagation and Detection of Quantum Optical Fields

Table 4.1 Properties of coherent states. Properties of coherent states

Average complex amplitude

𝛼 = |𝛼|ei𝜙

Average photon number

n𝛼 = |𝛼|2

Quadrature amplitude operators

̂ X𝟐 ̂ corresponding to a complex Operators X𝟏, amplitude 𝛼

Wigner function

Quasi-probability distribution function is given by a two-dimensional Gaussian function of unity width. Contour of the Wigner function is a circle

Fluctuations of the quad. amplitude

Variance of the operator is given by V𝛼 (𝜃) = V 1𝛼 = 1, independent of quadrature and detection frequency. Standard quantum limit (SQL)

Intensity noise

V n𝛼 = n𝛼 , which is equivalent to the shot noise formula. The intensity noise spectrum is flat, independent of detection frequency

Photon number distribution

Poissonian: 𝛼 (n) = |⟨n|𝛼⟩|2 =

Photon statistics

Time sequence of photon arrival is irregular as shown in Figure 4.7c. The statistics of the number of photons arriving in a time interval times follows a Poissonian distribution

|𝛼|2n n!

exp(−|𝛼|2 )

Fano factor

f𝛼 = 1 for all detection frequencies

Classical field

Carrier component has an amplitude |𝛼|, with optical phase arg(𝛼). It has completely uncorrelated noise sidebands

4.4 Propagation and Detection of Quantum Optical Fields Up till now we have been implicitly considering a stationary single frequency mode in a cavity and the behaviour of observables of that mode without specifying how they would be measured. Now we wish to address the experimentally relevant situation of propagating fields and photodetection. We begin with a quite general discussion of the problem whilst still limiting our discussion to that of single-mode fields. In free space there is an infinite continuum of frequency modes that must all be quantized. The approach described in this section can be a satisfactory approximation in situations where, over the detection bandwidth, the frequency spectrum of the source and the frequency response of the optical elements are both flat. That is, although there are many modes present, they are all doing the same thing’. A single mode can also be constructed from a superposition of frequency modes as we shall describe in the first part of this section. The approach of this

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section is also appropriate if the optical elements act coherently on a single mode of any kind to map it to another single mode. In the following section we will discuss a more specific approach in which the multimode effects of non-uniform spectral responses are included. 4.4.1

Quantum Optical Modes in Free Space

A single propagating mode is defined quantum mechanically in an analogous way to the classical approach (see Section 2.2): ̃k dk e−i𝜔t F(k, x)A (4.62) ∫ ̃ k are continuum annihilation operators. They obey the commuThe operators A tation relation ̃ † ] = 𝛿(k′ − k) ̃ k′ , A (4.63) [A â (x, t) =

k

and the mode function is normalized such that ∫

dk|F(k, x)|2 = 1

(4.64)

As a result it is easy to show that [̂a(x, t), â † (x, t)] = 1

(4.65)

as required for Boson creation and annihilation operators. Two such modes, say, ̂ are said to be independent or orthogonal if they commute, i.e. â and b, [̂a(x, t), b̂ † (x, t)] = 0

(4.66)

Notice that if the mode function for the mode b̂ is G(k, x), then this implies ∫

dkF(k, x)G(k, x)∗ = 0

(4.67)

which is identical to the definition of independent classical modes in Section 2.2.3. Indeed all the properties of classical modes discussed there transfer across to the quantum realm – a feature exploited in many of the experiments we will discuss later in the book. An idealized quantum optical mode that is nonetheless a good approximation in many situations is the plane wave Gaussian pulse: â (z, t) = where

√ ̃𝜔 = A

∫

̃𝜔 d𝜔e−i(𝜔t−kz) F(𝜔)A

−2zo k⃗ 2 ko ⃗ ko A ̃k dke √ 4zo π ∫

(4.68)

(4.69)

and k⃗ = (kx , ky ), k ≡ kz is the direction of propagation and 𝜔 = 2π𝜈 = |k|∕c. Also, ko is the average wave number of the beam. The parameter zo is defined in Section 2.1.5. Note that ̃ † ′ ] = 𝛿(𝜔 − 𝜔′ ) ̃ 𝜔, A (4.70) [A 𝜔

4.4 Propagation and Detection of Quantum Optical Fields

Physically, Eq. (4.68) describes a pulse of light in a transverse Gaussian mode (see Sections 2.1.5 and 2.2.3) on axis and close to its waist. The pulse shape as a function of frequency is given by F(𝜔). The relatively simple form of Eq. (4.68) comes from the fact that the wave fronts of a Gaussian mode are approximately plane waves close to their waist such that the transverse and longitudinal degrees of freedom decouple. It may seem very restrictive to have the requirement that we are close to the waist of the beam; however, lenses can be used to focus a Gaussian beam down to its waist at any spatial location. Typically in experiments it is arranged that optical beams interact with optical elements at their waists. A simple case of a pulse shape that is easy to work with is if the pulse shape is also Gaussian: √ 1 −(𝜔−𝜔o )2 (4.71) F(𝜔) = √ e 2𝑣 𝑣π where 𝜔o = ko ∕c is the average frequency of the beam and 𝑣 is the variance around this mean. With this pulse shape, the following conditions on the parameters for Eq. (4.68) to be a good approximation close to the waist: √ are required √ z, 1∕𝑣 ≪ zo and 𝑣 ≪ 𝜔o . Now consider the quantum mode ̂ t) = b(z, where

√ B̃ 𝜔 =

d𝜔e−i(𝜔t−kz) F(𝜔)B̃ 𝜔

(4.72)

−2zo k⃗ 2 ko ⃗ ko eixo kx A ̃k dke √ 4zo π ∫

(4.73)

∫

Equations (4.68) and (5.47) describe two pulses with identical spectral characteristics co-propagating in parallel Gaussian modes separated by a distance xo . √ ̃ 𝜔 , B̃ † ′ ] = 0 and the two modes will be indepenProvided xo ≫ ko ∕zo , then [A 𝜔 dent. Often when we talk about independent modes, we will be thinking of this type of situation with spatially separated optical beams; however, as discussed in Section 2.2, there are many other degrees of freedom that can be used to create independent modes. In the rest of this section, we will work at the level of the mode operators without being specific about the type of modes involved. 4.4.2

Propagation in Quantum Optics

The propagation of quantum optical modes through optical elements is modelled ̂ † . Two distinct pictures ̂ where U ̂ −1 = U via unitary transformation operators, U, can be employed: (i) the Schrödinger picture, in which the evolution of the state of the mode |Ψ⟩ → |Ψ′ ⟩ is calculated via ̂ |Ψ′ ⟩ = U|Ψ⟩ and the system observables remain stationary, or (ii) the Heisenberg picture, in which the evolution of a mode observable â → â ′ is calculated via ̂ ̂ † â U (4.74) â ′ = U

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4 Quantum Models of Light

and the state of the system is stationary. The two pictures are equivalent in the sense that all expectation values are equal in the two pictures because ̂ ̂ † â U|Ψ⟩ = ⟨Ψ|̂a′ |Ψ⟩ ⟨Ψ′ |̂a|Ψ′ ⟩ = ⟨Ψ|U

(4.75)

However, depending on the particular situation, the two pictures offer different levels of computational complexity and different physical insights. For propagation through stationary elements (that do not change with time), we can write ̂ = ei Ĥ 𝜏∕ℏ U

(4.76)

̂ is the interaction Hamiltonian characterizing the particular optical elewhere H ment and 𝜏 characterizes the strength of the interaction or the time for which it acts. For the latter case it is easy to show that the equation of motion for an arbitrary Heisenberg operator is then 1 dâ ̂ = [̂a, H] (4.77) dt iℏ The Heisenberg picture most resembles the classical treatment thanks to canonical quantization that requires that the equations of motion of the annihilation operator â are identical to those of the classical complex amplitude 𝛼. Thus in many situations it is straightforward to write down the propagation transformations in the Heisenberg picture, and one is able to retain a considerable amount of classical intuition in the quantum treatment. This is particularly true for Gaussian states as will be discussed in the next section. However, this is not always the case, and in certain situations the higher-order moments of the annihilation and creation operators may have equations of motion quite dissimilar from their classical counterparts. The Schrödinger picture on the other hand gives a uniquely quantum mechanical view of the transformations and as such tends to be most useful when quantum effects dominate, such as in photon counting experiments. In general it is not trivial to convert from a potentially straightforward analysis of an optical circuit in the Heisenberg picture to the corresponding Schrödinger state transformation. An exception is propagation through linear optical elements, that is, passive elements that do not change the photon number in the modes. Clearly the action on the vacuum mode of unitary operators representing such elements will be the identity, that is, ̂ lin |0⟩ = |0⟩ U

(4.78)

Now suppose we write out the state we wish to transform in the number basis: |𝜓⟩ = c0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · · ) ( 1 †2 † = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.79)

4.4 Propagation and Detection of Quantum Optical Fields

where we have used Eq. (4.18). Now we find ( ) 1 † † 2 ̂ lin c0 + c1 â + c2 √ (̂a ) + · · · U ̂ |0⟩ ̂ lin |𝜓⟩ = U ̂† U U lin lin 2 ( ) 1 ′′† ′′† 2 = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.80)

̂ aU ̂ † is the inverse Heisenberg picture annihilation operator, where â ′′ = Û obtained by inverting the standard Heisenberg equations of motion. This technique is easily generalized to optical elements that mix more than one mode and, where applicable, is a convenient tool for moving between the two pictures. 4.4.3

Detection in Quantum Optics

The basic detection tool in quantum optics is photodetection. One way or another almost all measurements of optical observables rely on photodetection. For bright fields photodetection measures the intensity of the field; for weak fields it measures the photon number of the field. Although quite distinct technologies must be employed in the bright and weak field limits, the process can be modelled in both cases as the measurement of the photon number observable â † â . More sophisticated detection models will be discussed in later chapters. As for propagation, detection in quantum optics is modelled differently in the Heisenberg and Schrödinger pictures. In the Heisenberg picture photodetection can be viewed as the transformation of the quantum mode â ′ into the classî a represents the photocurrent or, more gen̂ a = â ′† â ′ . The mode N cal mode N erally, the information generated by the detector and is classical in the sense that ̂ †a ] = 0. Nevertheless N ̂ a is an operator and must be so to correctly predict ̂ a, N [N ̂ a can be investigated by calculating expectation all behaviour. The statistics of N values of its various moments: ̂ a ⟩ = ⟨Ψ|N ̂ a |Ψ⟩ ⟨N j

j

(4.81)

where |Ψ⟩ is the initial state of the mode â . Using the technique of homodyne detection (see Section 8.1.4), photodetection can also be used to measure the quadrature observable â ′ ei𝜃 + â ′† e−i𝜃 . This then can be modelled as a transfor̂ 𝜃a = â ′ ei𝜃 + â ′† e−i𝜃 . mation of the quantum mode â ′ into the classical mode X 𝜃 ̂ Once again, although Xa represents a classical mode, it must be modelled as an ̂ a the statistics of the quadrature amplitude can be calculated operator. As for N by taking expectation values over the initial state of mode â . The operators representing the photocurrents can be manipulated as they might be in an experiment. For example, correlation measurements can be modelled by adding, subtracting, or multiplying photocurrent operators representing the measurement of different modes.

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In the Schrödinger picture photodetection can be viewed as projection onto the eigenstates of â † â , i.e. the number states |n⟩. Thus we can predict the probability that a particular result n will occur via Pn = |⟨n|Ψ′ ⟩|2

(4.82)

Average values, variances, etc. can be calculated from the probability distribution thus generated. Similarly if the experiment is configured such that a quadrature observable is measured, then the projection is onto the eigenstates of â ei𝜃 + â † e−i𝜃 , |x𝜃 ⟩, and the probabilities are given, as was used in Section 4.3.2, by PΨ (x𝜃 ) = |⟨x𝜃 |Ψ⟩|2 𝛿x

(4.83)

It is important to note that the predictions about the photocurrent generated in the Schrödinger picture are just numbers, unlike the operators generated in the Heisenberg picture. A major conceptual difference arises between the two approaches when multimode systems are considered. Suppose a system initially describable by a pure state in a single mode interacts with a number of other modes, also initially in pure states, during propagation. In the Heisenberg picture there will now be a number of output modes that are all a function of the input modes. Now suppose we have lost’ all but one of these modes, â ′ . Nevertheless we proceed as before by ̂ a = â ′† â ′ , just as we would with classical modes. This analysing the properties of N scenario looks quite different in the Schrödinger picture however. After propagation we are left with a state that describes the global properties of all the modes. If we have lost the other modes, we must describe that explicitly by averaging over all the possible detection results the other modes may have given. This is achieved by tracing over all the lost modes such that 𝝆̂ a = Trc≠a {𝝆̂ ′ }

(4.84)

where 𝜌 is the global state of the system, whilst 𝝆̂ a is the reduced density operator that describes the state of only mode â . The subscript c ≠ a indicates that the trace is over all modes except â . Under certain conditions the trace will not change the state of mode â . The state is then said to be separable as it indicates that the portion describing mode â can be factored out. More generally though the state will be inseparable or entangled, meaning no factorization is possible. Then inevitably 𝝆̂ a will be a mixed state, and ′

Pn = ⟨n|𝝆̂ a |n⟩

(4.85)

will give the probability of a particular detection result n. If the results of measurements on the other modes are known, then we must insert them explicitly so |Ψa ⟩ = {⟨nb |⟨nc |⟨nd | …} |Ψ′ ⟩

(4.86)

where now |Ψa ⟩ is the conditional state of the system given that the results nb , nc , nd , … were obtained for the other modes. 4.4.4

An Example: The Beamsplitter

To illustrate some of the ideas introduced in this section, we consider a simple example: a single photon number state passing through a beamsplitter. A more

4.4 Propagation and Detection of Quantum Optical Fields

general discussion of beamsplitters will be given in Chapter 5. The beamsplitter is assumed to have transmittance 𝜂. The other mode entering the beamsplitter is unoccupied. We label the mode containing the single photon state â and the unoĉ Thus the initial state can be written as |1⟩a |0⟩b . Let us consider the cupied mode b. Heisenberg approach first. As described in Section 5.1, the classical equations for mixing two modes with amplitudes 𝛼 and 𝛽 on a beamsplitter can be written as √ √ 𝛼out = 𝜂 𝛼 + 1 − 𝜂 𝛽 √ √ 𝛽out = 1 − 𝜂 𝛼 − 𝜂 𝛽 (4.87) Canonical quantization then allows us to immediately write √ √ â ′ = 𝜂 â + 1 − 𝜂 b̂ √ √ b̂ ′ = 1 − 𝜂 â − 𝜂 b̂

(4.88)

Hence, if we photodetect â , we obtain ′

̂ a = â ′† â ′ N = 𝜂 â † â + (1 − 𝜂)b̂ † b̂ +

√ 𝜂(1 − 𝜂)(̂a† b̂ + b̂ † â )

(4.89)

The average photon number detected is then ̂ a ⟩ = ⟨1|a ⟨0|b â ′† â ′ |0⟩b |1⟩a n = ⟨N ̂ b+ = 𝜂⟨1|a â † â |1⟩a + (1 − 𝜂)⟨0|b b̂ † b|0⟩ √ ̂ b + ⟨0|b b̂ † |0⟩b ⟨1|a â |1⟩a ) 𝜂(1 − 𝜂)(⟨1|a â † |1⟩a ⟨0|b b|0⟩ =𝜂

(4.90)

where we have used the relationships of Eq. (4.17). In a similar way the variance is ̂ 2a ⟩ − ⟨N ̂ a ⟩2 Vn = ⟨N

= 𝜂(1 − 𝜂)⟨1|a â † â |1⟩a ⟨0|b b̂ b̂ † |0⟩b = 𝜂(1 − 𝜂)

(4.91)

The result for the average photon number can be understood from a classical field point of view: the intensity is reduced according to the transmission of the beamsplitter; the non-zero variance after some loss at the beamsplitter reflects the particle aspect of light. More on this is said in Chapter 5. Now let us consider the same problem in the Schrödinger picture. Using the technique of Eq. (4.80), we find the state evolution through the beamsplitter is given by |Ψ′ ⟩ = â ′′ |0⟩a′ |0⟩b′ √ √ = ( 𝜂 â ′ + 1 − 𝜂 b̂ ′ )|0⟩a′ |0⟩b′ √ √ = 𝜂|1⟩a′ |0⟩b′ + 1 − 𝜂|0⟩a′ |1⟩b′

(4.92)

In order to consider only detection of the â mode, we need to trace over the b̂ mode as per Eq. (4.84). Writing 𝝆̂ ′ = |Ψ′ ⟩⟨Ψ′ |, we find 𝝆̂ a′ = ⟨0|b′ 𝝆̂ ′ |0⟩b′ + ⟨1|b′ 𝝆̂ ′ |1⟩b′ = 𝜂|1⟩a′ ⟨1|a′ + (1 − 𝜂)|0⟩a′ ⟨0|a′

(4.93)

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Using Eq. (4.85), we find that detection of this state gives the probabilities P1 = ⟨1|𝝆̂ a′ |1⟩ = 𝜂 P0 = ⟨0|𝝆̂ a′ |0⟩ = 1 − 𝜂

(4.94)

which leads to the same average value and variance as predicted from the Heisenberg result. Now, instead of losing’ the b̂ mode, let us suppose that we have photodetected it. If the photodetection of the b̂ mode gave the result 0’, then the conditional state of the â mode will be √ (4.95) |Ψ⟩a′ = {⟨0|b′ }|Ψ′ ⟩ = 𝜂|1⟩ √ indicating that a photon will be found in mode â with certainty. The factor of 𝜂 indicates that the probability of this state being prepared √ (i.e. the probability of the measurement of the b̂ mode giving the 0’ result) is ( 𝜂)2 = 𝜂. Similarly, if the measurement of the b̂ mode gave the result 1’, then the conditional state of the â mode would be √ |Ψ⟩a′ = {⟨1|b′ }|Ψ′ ⟩ = 1 − 𝜂|0⟩ indicating that it is certain that no photon will be found in the â mode. How could we predict this latter result in the Heisenberg picture? Well, we found that ̂ one was not found in â and vice versa. whenever a photon appeared in mode b, Another way of saying this is that the two modes are anti-correlated. To test for anti-correlation in an experiment, we might add the photocurrents obtained from the photodetection of the two modes and look for reduced noise (variance). Adding the photocurrents, we would obtain (using Eq. (4.88)) ̂ b = â ′† â ′ + b̂ ′† b̂ ′ ̂a+N N = â † â + b̂ † b̂

(4.96)

which has zero variance indicating perfect anti-correlation as expected. Although the results match in the two pictures, as they must, the calculational techniques and the physical pictures suggested are quite different.

4.5 Quantum Transfer Functions As we saw in Section 4.3, in general a complete description of a state of light, including its quantum properties, can be carried out through various probability and quasi-probability distributions [7]. For the description inside an optical cavity, a single function will suffice. In contrast, for a freely propagating beam, the situation is more involved [12]. We can expand our beam in a basis of single frequency modes. We will then find that we need a whole continuum of distribution functions to describe these modes. These functions can be derived in the Schrödinger picture as the solutions of complex stochastic differential equations, in many cases master equations [7, 13, 14]. Whilst complete, this approach is often more general than required in practice. Here we present an alternative route based on the Heisenberg picture in which we obtain operator equations that describe how the individual frequency modes transform – the

4.5 Quantum Transfer Functions

quantum transfer functions. We restrict our attention to systems that are linear or can be linearized. Linearization can be done if the modulations and the fluctuations we are concerned with are small compared with the amplitude of the laser beam. This is usually correct for situations with optical powers larger than microwatts. A continuous-wave (CW) laser beam is composed of many independent single frequency modes and so is inherently a multimode situation. In contrast a single pulse from a pulsed laser may be in a single mode described by a coherent superposition of many frequency modes. In either case a complete description of the quantum evolution of the beam is available via the quantum transfer functions. Working in the Heisenberg picture means the operator equations are very similar to the classical equations encountered in Section 2.3.2, and similar interpretations can be made. This approach is not directly applicable to experiments that resolve single photons using photon counting technology. However, in Chapter 12, we will adapt this approach to treat such experiments. 4.5.1

A Linearized Quantum Noise Description

Consider a propagating CW laser beam in a single spatial mode. We will represent ̂ where we use capitals to remind us that the beam by the continuum operator A, this is a continuum operator with units s−1∕2 . We can break this operator up into two contributions: (i) the mean value of the amplitude, given by one complex number 𝛼, and (ii) the fluctuations and modulations, described by the operator ̂ 𝜹A(t), representing small fluctuations around the mean. We thus write ̂ = 𝛼 + 𝜹A(t) ̂ A

(4.97)

Note the similarity here with the classical treatment of small fluctuations, Eq. (2.15), the difference being that now the fluctuating term is represented as an operator. Note also that Eq. (4.97) is really just a special case of a fluctuation ̂ = 𝛼. We use small 𝛿 here to emphasize that operator (see Section 4.3) where ⟨A⟩ these fluctuation operators are assumed small compared to their expectation ̂ is values. If the beam is detected by a photodetector, then the photocurrent N given by ̂ ̂ † (t)A(t) ̂ =A N ̂ ̂ † (t))(𝛼 + 𝜹A(t)) = (𝛼 + 𝜹A ̂ † (t) + 𝛼 𝜹A(t) ̂ + 𝜹A ̂ † (t) 𝜹A(t) ̂ = 𝛼 2 + 𝛼 𝜹A ̂ = 𝛼 2 + 𝛼 𝜹X𝟏(t)

(4.98)

where we have taken the arbitrary phase of 𝛼 real and linearized by assuming the fluctuations are small and hence neglecting higher than first-order terms in the fluctuations. By small’, we mean that ultimately the contribution to the calculated expectation values due to these terms will be negligible. We see that from the quantum point of view, we are effectively measuring the in-phase ̂ is often referred ̂ = 𝜹A ̂ ̂ † (t) + 𝜹A(t). For this reason 𝜹X𝟏 quadrature operator 𝜹X𝟏 to as the amplitude quadrature operator. By introducing another bright beam as a reference, it is also possible to measure the fluctuations in the phase of the

121

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4 Quantum Models of Light

beam (see Chapter 8). Given the linear approximation, these turn out to be ̂ = i(𝜹A ̂ † (t) − 𝜹A(t)), ̂ proportional to the out-of-phase quadrature, 𝜹X𝟐 which is thus often referred to as the phase quadrature. This link is illustrated in ̂ and 𝜹X𝟐 ̂ have continuous eigenvalue spectra. This Figure 4.7a. Recall that 𝜹X𝟏 implies that under such measurement conditions, discrete single-photon events cannot be observed, regardless of the resolution of the detectors. The discussion is more naturally continued in frequency space. The Fourier transform of the operator in Eq. (4.97) is ̃ ̃ A(Ω) = 𝛼 𝛿(Ω) + 𝜹A(Ω)

(4.99)

where 𝛿(Ω) is the Dirac delta function. Recall that 𝛼 is defined in a frame rotating at the mean optical frequency 𝜈o (see Section 2.1). Thus Ω = 2π(𝜈o − 𝜈)

(4.100)

is the frequency difference from the mean optical frequency. The mean coherent amplitude of the field appears as a delta function spike, the carrier, 𝛼 𝛿(Ω), which is only non-zero for Ω = 0, i.e. at the mean optical frequency. We can consider the Fourier transformed fluctuation operator, in some small frequency interval around radio frequency Ω, as approximately describing a single quantum mode at this frequency. Such modes obey the commutation relations ̃ ̃ † (Ω′ )] = 𝛿(Ω − Ω′ ), [A(Ω), A

̃ ̃ ′ )] = 0 [A(Ω), A(Ω

(4.101)

This is a useful and practical picture because experimentally we can perform Fourier analysis on the photocurrent produced by photodetection using a spectrum analyser, discussed in Chapter 8. We can also place information on the laser beams in the form of modulation at various different frequencies Ωmod , which is measured in Hz, and each frequency can be thought of as an independent information channel. From Eq. (4.98), we find that the operator describing the photocurrent in Fourier space is ̃ out (Ω) ̃ out = 𝛼 2 𝛿(Ω) + 𝛼 𝜹X𝟏 N

(4.102)

Now the first term describes the average power, concentrated in a DC spike, and the second term describes the measurement of the amplitude quadrature at frequency Ω. The variance of the photocurrent at frequency Ω is given by 2 ̃ out (Ω)|2 |𝛼⟩ = 𝛼 2 ⟨𝛼||𝜹X𝟏 V Nout (Ω) = 𝛼out out

V 1out (Ω)

(4.103)

We have a direct link between the measured values V Nout (Ω) and the property V 1out (Ω) of the state. In particular we see that ultimately the intensity noise of a bright laser is determined by the quantum states of its frequency sidebands. If the laser mode is subjected to amplitude or phase modulations (either deterministic signals or stochastic noise), these can be incorporated into the description as c-number variables. Thus we can modify Eq. (4.99) to read ̃ ̃ ̃ A(Ω) = 𝛼𝛿(Ω) + 𝜹A(Ω) + 𝛿 S(Ω)

(4.104)

where the ̃ indicates the Fourier transform of a classical variable. If the power ̃ spectra of 𝛿 S(Ω) are much larger than those associated with the quantum part

4.5 Quantum Transfer Functions

̃ (𝜹A(Ω)), then the beam will behave classically. However, once again, if all classical noise is suppressed, we will still be left with fluctuations due to the quantum noise. If we are trying to measure or send information via small modulations of the beam, then the signal to noise associated with this measurement or communication will likewise be determined by the quantum noise. 4.5.2

An Example: The Propagating Coherent State

Let us illustrate the preceding discussion with an idealized and then more realistic example. In Section 4.2.2 we described a coherent state as the displacement of a single cavity mode initially in a vacuum state. We may think of a propagating coherent state in a similar way as a beam with all its power at a single frequency. Notice that this implies a CW laser, as a single frequency corresponds ̃ to a constant output in time. This also means that the quantum state of the 𝛿 A(Ω) sideband modes is the vacuum. From Eq. (4.103) and using Eqs. (4.15) and (4.101), we immediately find 2 ̃ out |2 |0⟩ ⟨0||𝜹X𝟏 V Nout (Ω) = 𝛼out ̃ 2 + (𝜹A ̃ † )2 + 𝜹 A ̃ † 𝜹A ̃ + 𝜹A𝜹 ̃ A ̃ † |0⟩ = 𝛼 2 ⟨0|(𝜹A out

2 = 𝛼out

(4.105)

where the absolute square is taken because we are in Fourier space. The laser intensity exhibits Poissonian statistics at all frequencies. In a similar way it follows that the quadrature variances are at the quantum noise limit (QNL) for all angles and all frequencies. These results are as might be expected for a travelling field generalization of a coherent state. Of course this is an idealization. Real lasers have a finite linewidth and may have structure in their fluctuation spectra (see Chapter 6). Nevertheless, for lasers oscillating on single longitudinal and spatial cavity modes, so-called single-mode lasers, at sufficiently high frequencies, the power in the sidebands will become negligible. At or above such frequencies, the laser will behave like a coherent state, and it is in this sense that lasers are often referred to as approximately producing a coherent state. A typical solid-state laser will be coherent’ at frequencies above about 10 MHz. Another way of saying this is that if we sample the field on short timescales, the laser can be thought of as producing a coherent state. On the other hand, as we go to low sideband frequencies (long timescales), eventually the power in the sidebands will become an appreciable fraction of the central peak power and our linearization assumption will break down. 4.5.3

Real Laser Beams

We will find in Chapter 6 that a real laser beam contains much more noise than a coherent state. Figure 4.8 shows an example of a typical laser that has an intensity noise spectrum V 1(Ω) well above the QNL V 1(Ω) = 1 of the coherent ̃ state. This figure shows the link between the values for the quadratures 𝜹X𝟏, ̃ 𝜹X𝟐 and the actual measured intensity noise. In addition such a laser beam will contain modulation signals at certain frequencies. We need a technique to derive

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4 Quantum Models of Light

ˆ (Ω) δX1

ˆ (Ω) δX2

V1(Ω)

its 1

r un

e oov

H

0 Ω

Ω

Figure 4.8 The Fourier components of the noise spectrum for a typical real laser. On the right the fluctuations for six different detection frequencies Ω are represented by their uncertainty areas. On the left is the corresponding continuous intensity noise spectrum V1(Ω).

such spectra and to describe how the noise and the signals will change whilst the light propagates through the apparatus. 4.5.4

The Transfer of Operators, Signals, and Noise

We are now in a position to describe a complete optical system and to evaluate the propagation of the noise and the signals through the system. Each optical component of the system can be thought of as having several inputs, described ̃ in1 , by their coherent amplitudes 𝛼in1 , 𝛼in2 , … etc., their fluctuation operators 𝜹A ̃ in2 , … , etc., and their states |Φin1 ⟩, |Φin2 ⟩, … , etc. The corresponding spectral 𝜹A variances for each input are ̃ inj (Ω)|2 |Φinj ⟩ V 1inj (Ω) = ⟨Φinj ||X𝟏 ̃ inj (Ω)|2 |Φinj ⟩ V 2inj (Ω) = ⟨Φinj ||X𝟐

(4.106)

where j = 1, 2, … Inside the optical components the modes are combined. The algebraic equations for the coherent amplitudes are solved to give the output coherent amplitude. We usually choose the arbitrary overall phase of the input fields such that the output coherent amplitude is real. After linearization the equations of motion for the fluctuation operators can be solved in frequency space. The fluctuation operators in the frequency domain for one output mode

4.5 Quantum Transfer Functions

can then be described by equations of the form ̃ in1 (Ω) + c1c (Ω) 𝜹A ̃ † (Ω) ̃ out (Ω) = c1a (Ω) 𝜹A 𝜹A in1 ̃ ̃ † (Ω) + · · · + c2a (Ω) 𝜹Ain2 (Ω) + c2c (Ω) 𝜹A in2

̃ in1 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) ̃ † (Ω) = c∗ (−Ω) 𝜹A 𝜹A 1c 1a out in1 ̃ in2 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) + · · · + c∗ (−Ω) 𝜹A 2c

2a

in2

(4.107)

and similarly for all other output modes. The coefficients ckl (Ω) are complex numbers that describe the frequency response of the component. Note that the response is not necessarily symmetric about the carrier. The cka and ckc are linked via commutation requirements. For linear systems the ckl do not depend on the beam intensities and all ckc = 0. We can also treat the case of propagation through non-linear media, such as 𝜒 (2) and 𝜒 (3) media. This includes non-linear processes such as frequency doubling, parametric amplification, the Kerr effect, or non-linear absorption. Equation (4.123) remains linear, but in these cases the coefficients c1a , c1c , c2a , c2c , … are dependent on the average optical power of the beams. In various examples throughout this book, we will show how the coefficients cij can be derived from the physical properties of the component, ̃ out (Ω) for such as reflectivities, conversion efficiencies, etc. Once the operator 𝜹A the output mode in question has been found, the variances can be determined using Eq. (4.106). Let us consider some general cases: (1) The simplest case is where all the inputs are independent: ̃ inl (Ω)⟩ = 𝛿kl . This will be true if all the input beams represent ̃ ink (Ω)𝜹X𝟏 ⟨𝜹X𝟏 beams from independent sources or different vacuum inputs. Secondly, we require that the coefficients obey c∗ij (−Ω) = cij (Ω), i.e. magnitude and phase of the frequency response are symmetric. This will be true if there are no detunings or strong dispersion in the optical system. In this case we obtain the simplest type of transfer function: V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 V 1in1 (Ω) + |c2a (Ω) + c2c (Ω)|2 V 1in2 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 V 2in1 (Ω) + |c2a (Ω) − c2c (Ω)|2 V 2in2 (Ω) + · · ·

(4.108)

Notice that for a linear system (c1c = c2c = · · · = 0), the transfer functions for both quadratures are identical. (2) The coefficients still obey c∗ij (−Ω) = cij (Ω), but we allow for inputs that are not independent. Now there can be correlations between the quadrature variances of different inputs. However, we can always regain the form of Eq. (4.108) by breaking down each of the inputs into a function of its own inputs until we end up with all the inputs being independent. (3) The coefficients obey c∗ij (−Ω) = c∗ij (Ω), i.e. the magnitude of the frequency response is symmetric but not the phase. This will occur if the fluctuations

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4 Quantum Models of Light

suffer a different phase rotation to that of the coherent amplitude. In this case we get cross-coupling between the transfer functions of V 1out and V 2out such that now V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + · · ·

(4.109)

where 𝜃 is defined by cij (Ω) = |cij (Ω)|e . We have assumed that V 1 and V 2 are the quadratures of minimum and maximum fluctuations. (4) The coefficients obey c∗ij (−Ω) ≠ c∗ij (Ω). This will occur when the frequency response of the optical components is asymmetric around the carrier frequency, for example, due to detunings. It is not possible to represent the output as a transfer function of quadrature variances in this case. i𝜃

The simplest example of a transfer function describes the influence of loss terms only and has the form V 1out (Ω) = |c1a |2 V 1in1 (Ω) + |c2a |2 + |c3a |2 + · · ·, where all the inputs apart from V 1in1 (Ω) are vacuum inputs that have the numerical value of 1. Such a transfer function has identical effects on both quadratures. An example of case (2) would be where either a modulation is common or two or more of the input modes have correlated noise. This is the case when the beams originate from one source, such as two beams formed by a beamsplitter, or where two beams are generated by an above threshold optical parametric oscillation (OPO). Examples of case (3) can be as benign as mixing beams on a beamsplitter where the coherent amplitudes of the inputs have different phases, or components with strong dispersion where the refractive index changes within a frequency interval of Ωdet , or as complex as single-ended cavities with optical detuning. The simplest example of case (4) is a double-ended optical cavity with detuning. An appropriate name for Eq. (4.108) is quantum noise transfer function. This equation is very similar to the classical transfer functions used by communication engineers. The classical function can be derived using a description of the various inputs as classical waves with modulation at Ω. Noise behaves exactly in the same way as the modulations. Quantum noise is not included. The propagation and interference of these waves are determined, and the result is the size of the modulation at the output. In contrast, Eq. (4.108) includes all the quantum effects of the light, in particular the effects of quantum noise. We can directly see how the signals vary in comparison to the quantum noise and can evaluate the coefficient for the transfer of the signal-to-noise ratio from input to output. This approach has been developed since the 1980s by a number of research groups, very prominently by the groups in France who pioneered squeezing [12]. 4.5.5

Sideband Modes as Quantum States

We will now outline how we can relate the quantum transfer functions to the detailed quantum state of the field at different frequencies. In particular we will

4.5 Quantum Transfer Functions

consider the example of modulation of a quantum noise limited beam and see how this can be viewed as producing a coherent state at the modulation frequency. In Section 2.1.7 we saw how small modulations of a classical light beam resulted in coherent power being injected symmetrically into upper and lower sidebands around the carrier. In Section 4.5.1 we described classical modulations in the Heisenberg picture as evolution of the quantum fluctuation operator ̃ ̃ ̃ 𝜹A(Ω) → 𝜹A(Ω) + 𝛿 S(Ω)

(4.110)

This Heisenberg evolution is equivalent to the Schrödinger evolution of displacement (see Section 4.2.2), i.e. ̃ ̃ A(Ω))|𝜓⟩ |𝜓⟩Ω → D(𝜹 Ω

(4.111)

Now consider the amplitude quadrature operator in Fourier space (Figure 4.9). It ̃ † (Ω), is is easy to show that in general the Fourier transform of a conjugate, e.g. A † ̃ , via a change in sign related to the conjugate of the Fourier transform, e.g. A(Ω) † ̃ ̃ † (Ω) = A(−Ω) . Using this property of the Fourier transform, we can of Ω, e.g. A observe that ̃ † (Ω) ̃ ̃ 𝜹X𝟏(Ω) = 𝜹A(Ω) + 𝜹A † ̃ ̃ = 𝜹A(Ω) + 𝜹A(−Ω)

(4.112)

Thus we see that a quadrature measurement actually probes both the upper and lower sidebands. Suppose the initial state of the sidebands is a vacuum. Using Eq. (4.111) we find that modulation produces the coherent sideband state ̃ ̃ |𝜓 ′ ⟩ = |𝛿 S(Ω)⟩ Ω |𝛿 S(−Ω)⟩−Ω

(4.113)

where, in response to Eq. (4.112), we have written the state explicitly in terms of an upper and lower sideband part. We are ignoring the low frequency contributions from the carrier. For an arbitrary mix of ideal amplitude and phase modulation, we have ̃ 𝛿 S(Ω) = 𝛽(𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod ))

(4.114)

The complex number 𝛽 = Ma ∕4 + iMp ∕4, where Ma and Mp are the amplitude and phase modulation depths, respectively, and Ωmod is the modulation X2

E

vL–Ω

vL+Ω

X1 Random distribution in time (a)

vL

(b)

Random components at all frequencies

v

Figure 4.9 Comparison between (a) the phase space diagram and (b) the sideband picture. The quantum properties are encapsulated in noise sidebands at all modulation frequencies above and below the laser frequency 𝜈L .

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4 Quantum Models of Light

frequency (see Section 2.1.7). Averaging over a small range of frequencies around Ωmod , we can write our state as |𝜓 ′ ⟩ = |𝛽⟩Ωmod |𝛽⟩−Ωmod |0⟩Ω≠±Ωmod

(4.115)

A pair of identical coherent states has been created at the upper and lower sideband frequencies. As quadrature measurements probe the sidebands symmetrically (see Eq. (4.112)), the results of quadrature measurements are identical to those for a single frequency coherent state. For example, the expectation value of X1 in the Schrödinger picture is given by † ̃ ̃ ̃ + ⟨𝛽|𝜹A(−Ω ⟨𝜹X𝟏(Ω mod )⟩ = ⟨𝛽|𝜹A(Ωmod )|𝛽⟩Ω mod ) |𝛽⟩−Ω mod

mod

= 𝛽 + 𝛽∗

(4.116)

This is of course the same as the result that would be obtained in the Heisenberg picture. This result generalizes to other states provided the transfer functions remain symmetric about the carrier. Most interesting are states for which the upper and lower sideband modes become entangled as this has no classical analogue. Squeezed states, which are described in Chapter 9, are an example of such states. Using a homodyne detector it is possible to measure the variances VΘ (Ω) at all frequencies Ω. The properties of the light in a small interval around each frequency can then be described by one Wigner function W (Ω). The interval is assumed small enough such that the frequency response is flat. In this way the entire beam can be described as a series of quantum states, one for each small frequency interval. This is a Schrödinger picture view of the travelling field, which complements very well the actual technology used in optical communication. Here each of these frequency intervals is used as a separate communication channel. The variance VΘ (Ω) can be determined mathematically by first projecting all parts of the Wigner function onto one axis at angle Θ. This produces a series of one-dimensional distribution functions DF(Θ) that describe in the form of a histogram the time-varying photocurrent in a small frequency range. In a second step the variances of these functions are evaluated, which provides information about the width of DF(Θ). In general we would need to evaluate the moments of DF(Θ) to all orders. However, as we have seen, for the restricted class of Gaussian functions, a direct link exists between the Wigner functions and the variances, and a complete description is given by Vmin (Ω), Vmax (Ω), and Θmin (Ω), as shown in Section 4.3.2. It is also possible to obtain the average photon number in the sidebands from the quadrature phase measurements. Consider the sum of the amplitude and phase quadrature variances written in terms of annihilation and creation operators. We have † ̃ † ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω)𝜹 A(−Ω) ⟩ V 1(Ω) + V 2(Ω) = 2⟨𝜹A(Ω) † † ̃ ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω) 𝜹A(−Ω)⟩ +2 = 2⟨𝜹A(Ω)

(4.117)

Hence we can write 1 † ̃ ̃ 𝛿N(|Ω|) = ⟨𝜹A(|Ω|) 𝜹A(|Ω|)⟩ = (V 1(Ω) + V 2(Ω) − 2) 4

(4.118)

4.5 Quantum Transfer Functions

For a coherent state (of the entire beam), we expect V 1 = V 2 = 1 at all frequencies, and so from Eq. (4.118) there are no photons in the sidebands as discussed in Section 4.5.2. This discussion requires a caveat: we have assumed throughout that the sidebands are symmetric around the carrier, i.e. our transfer function is not of type (4) as described in the previous section. Asymmetric transfer functions can occur. For example, it is possible to produce single-sideband modulation. The state produced in such a situation is |𝜓 ′ ⟩ = |𝛽⟩Ωmod |0⟩Ω≠Ωmod

(4.119)

Such states are not fully characterized by homodyne quadrature measurements. 4.5.6 Another Example: A Coherent State Pulse Through a Frequency Filter So far we have only considered CW or quasi-CW situations where our beam can be approximated as many independent single frequency modes (or mode pairs as in the case of the sidebands). Now we wish to illustrate how the quantum transfer functions can also be used to describe the propagation of pulses through optical elements with non-uniform spectral response. The example we will discuss is a pulse of light in a coherent state transmitted through a frequency filter. We consider a plane wave Gaussian mode as described by Eq. (4.68). Written in the nomenclature of this section, we have â =

∫

̃ dΩF(Ω)A(Ω)

(4.120)

Suppose we ask: what is the expectation value of an amplitude quadrature measurement that projects onto this mode when the mode is prepared in a coherent state of amplitude 𝛼? Using the techniques of Section 4.4 and Eq. (4.24), we find ̂ † (𝛼)(̂a + â † )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂a + â † |𝛼⟩â = ⟨0|D â = ⟨0|̂a + â † + (𝛼 + 𝛼 ∗ )

dΩF(Ω)F(Ω)∗ |0⟩ ∫ = ⟨0|̂a + â † + 𝛼 + 𝛼 ∗ |0⟩ = 𝛼 + 𝛼 ∗

(4.121)

As expected we obtain 2 e 𝛼. The intermediate step of the calculation shown in the second line is included to emphasize that when generalizing from the displacement of a single frequency mode (Eq. (4.24)) to that of a pulse, the overlap between the mode that is displaced and that which is detected appears. In this case the detected mode is identical to the displaced mode, and so the integral evaluates to unity via normalization (Eq. (4.64)). Now suppose that a frequency filter is placed in front of the detector. The detector mode is now â o =

∫

̃ o (Ω) dΩF(Ω)A

(4.122)

̃ o is determined by the transfer function of the frequency filter. where A A frequency filter can be modelled as the transmission through an optical cavity,

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as described in Section 5.3. Here we just quote its transfer functions: 𝜅 iΩ ̃ ̃ B(Ω) (4.123) A(Ω) + 𝜅 − iΩ 𝜅 − iΩ ̃ where 𝜅 is the linewidth of the filter and B(Ω) is a second mode, initially in the vacuum state, coupled in by the filter. Now the expectation value of the amplitude quadrature becomes ̃ o (Ω) = A

̂ † (𝛼)(̂ao + â †o )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂ao + â †o |𝛼⟩â = ⟨0|D â ) ( 𝜅 𝜅 F(Ω)F(Ω)∗ |0⟩ = ⟨0|(̂ao + â †o ) + dΩ 𝛼 + 𝛼∗ ∫ 𝜅 − iΩ 𝜅 + iΩ ( ) 𝜅2 Ω𝜅 ∗ = dΩ|F(Ω)|2 (𝛼 + 𝛼 ∗ ) 2 − i(𝛼 − 𝛼 ) ∫ 𝜅 + Ω2 𝜅 2 + Ω2 (4.124) In general the filter produces two effects: a rotation of the quadratures such that a component of the imaginary part of 𝛼 now appears in the expectation value and attenuation by a Lorentzian factor. Notice that if |F(Ω)| = |F(−Ω)|, then the m 𝛼 term goes to zero as it is an odd function of Ω. Hence, if the signal is symmetric and the centre frequency is on resonance with the filter, there will be no rotation. Restricting to this case we can get some insight √about the attenuation by 2

−(Ω) 1 √ e 2𝑣 (c.f. Eq. 𝑣 π ⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 𝛼 +

considering two limits. Suppose F(Ω) is the Gaussian

(4.68).

𝛼 ∗ , i.e. If 𝑣 ≪ 𝜅, then we can approximate F(Ω) ≈ 𝛿(Ω) and if the frequency spread of the pulse is narrow compared to the linewidth of the filter, almost all the power of the pulse is transmitted to the detector. On√the other 1 √ and hand, in the opposite limit where 𝑣 ≫ 𝜅, we can approximate F(Ω) ≈ 𝑣 π

⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 0, i.e. if the frequency spread of the pulse is broad compared to the linewidth of the filter, almost none of the power of the pulse is transmitted to the detector. 4.5.7

Transformation of the Covariance Matrix

Throughout this section we have been considering linear transfer functions arising from linear, or linearized, optical interactions. Such transfer functions always map one set of Gaussian states to another set of Gaussian states and as such are known as Gaussian operations. Thus, if our system is initially in a Gaussian state and it is transformed according to linear transfer functions, then the final state will also be Gaussian. This means that in such a situation all the information required to completely describe the output state can be obtained by finding the transfer functions of the elements of the input displacement vector and covariance matrix (see Section 4.3.4). For transfer function cases (1)–(3), a covariance matrix initially in standard form, Eq. (4.61), will remain in standard form, and it is only necessary to calculate how the variances transform. See also Section 2.1.5. However, for case (4), a covariance matrix initially in standard form will transform to one not in standard form, and hence the transformation of all matrix elements must be calculated.

4.6 Quantum Correlations

4.6 Quantum Correlations A key difference between classical optical fields and their quantum counterparts is that the correlations that can exist between spatially separated beams in quantum mechanics can be stronger than is allowed by classical theory. We now look at examples of how this difference can be quantified both in the single-photon and linearized domains. 4.6.1

Photon Correlations

In Section 2.3.3 we saw that the intensity correlations between different beams could be characterized by the second-order correlation function, g (2) , which was defined by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨I(t)2 ⟩

(4.125)

We also saw that for classical fields g (2) (0) ≥ 1. In contrast we argued in Section 3.6 that g (2) (0) < 1 should be possible for quantum mechanical fields. We will now treat this question with the full quantum formalism. The calculation that led to the restriction on g (2) in the classical case relied on the fact that ⟨A2 ⟩ ≥ ⟨A⟩2 . This relationship still holds if A is an operator, so how is it that we can have g (2) (0) < 1 for a quantum mechanical field? To answer this we must look more carefully at what the actual measurement of g (2) involves. As outlined previously we need to divide the field in half at a beamsplitter and then separately measure the intensities of the two halves with a variable time delay, 𝜏, introduced. For a classical field the action of the beamsplitter is I(t) → 1∕2I(t). Thus by finding the average of the product of the intensities of the two beams and normalizing by their individual averages, we arrive at the above expression for g (2) . Now consider the quantum mechanical case. As we have seen in Section 4.4.4 (see Chapter 5 for more details), the correct quantum mechanical expressions for the outputs of a 50: 50 beamsplitter are √ √ â ′ = 0.5 â + 0.5 b̂ √ √ (4.126) b̂ ′ = 0.5 â − 0.5 b̂ where â represents the beam whose intensity we are measuring, whilst b̂ is initially in the vacuum state. If we now form the ratio of the average of the product of the intensities of the two beams normalized to their individual averages, we obtain g (2) (𝜏) =

̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))(̂ a† (t + 𝜏) − b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t ̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))⟩⟨(̂ a† (t + 𝜏) + b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t (4.127)

Using the field commutation relations and the fact that the b̂ mode is initially in the vacuum state (see Section 4.2.1), this expression can be significantly

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simplified to g (2) (𝜏) =

⟨̂a† (t)̂a† (t + 𝜏)̂a(t)̂a(t + 𝜏)⟩ ⟨̂a† (t)̂a(t)⟩2

(4.128)

where we have also assumed the individual beam averages are not a function of time (the system is ergodic). Equation (4.128) is then the correct quantum mechanical expression for the second-order correlation function. Notice that the ordering of the operators under the averages is such that all creation operators stand to the left whilst all annihilation operators stand to the right. This is referred to as normal ordering and is often a convenient way of expressing results. We can now consider two simple examples. First, suppose the mode â was initially in a coherent state. Using Eq. (4.128) we immediately find that for a coherent state, g (2) (0) = 1, as expected for an ideal classical field. However, if the mode â is in a number state, then the numerator is immediately zero, as two applications of the annihilation operator to the one photon state produces zero. Thus, for a single photon number state g (2) (0) = 0, in stark contrast to the classical limit. In Chapters 12 and 13, we will see other examples of quantum intensity correlations as functions of different degrees of freedom such as polarization. 4.6.2

Quadrature Correlations

We can also consider correlations between the quadrature amplitude fluctuations of spatially separated beams, recorded by separate detectors. These correlations can be quantified by measuring the quadrature difference and sum variances defined by 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟏 V 1s = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟐 V 2s = ⟨(𝜹X𝟐 2 for the sum variances of the two quadratures and

(4.129)

1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟏 V 1d = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟐 V 2d = ⟨(𝜹X𝟐 (4.130) 2 for the difference quadrature variances. Here the subscripts label the beams 1 and 2. If the beams are correlated, say, in the amplitude quadrature, we would expect V 1d < V 1s . Conversely, if the beams were anti-correlated, we would find V 1d > V 1s . Two coherent beams are found to be uncorrelated, with their noise at the QNL; that is, for two coherent beams, we have V 1d = V 1s = V 2d = V 2s = 1. Beams are said to be quantum correlated if they have sum or difference quadrature variances that fall below 1. Such correlations cannot be consistently accommodated in a classical theory. In particular, if, for example, V 1d < 1 and V 2s < 1, then the quantum state of the beams is necessarily said to be entangled (see Section 12.3) and no consistent local realistic description of the correlations is possible.

4.6 Quantum Correlations

More generally we might consider the conditional variance of, say, beam 1’s amplitude quadrature, given that we have measured beam 2’s amplitude quadrature. This tells us how accurately we can infer the value of beam 1’s amplitude quadrature given the measurement of beam 2 and is defined as ̃ 1 + g 𝜹X𝟏 ̃ 2 )2 ⟩ V 11|2 = ⟨(𝜹X𝟏

(4.131)

Here g is a gain that is chosen such as to minimize the value of V 11|2 . By calculus we can find that the optimum value of g is g=−

̃ 2 𝜹X𝟏 ̃ 1 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ ⟨𝜹X𝟏 2 V 12

(4.132)

Substituting back into Eq. (4.131), we obtain V 11|2 = V 11 −

̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1 ⟩2 ̃ 1 𝜹X𝟏 ⟨𝜹X𝟏 2 V 12

(4.133)

Similarly for the phase quadrature conditional variance, we have V 21|2 = V 21 −

̃ 2 𝜹X𝟐 ̃ 2 + 𝜹X𝟐 ̃ 1 ⟩2 ̃ 1 𝜹X𝟐 ⟨𝜹X𝟐 2 V 22

(4.134)

For coherent states we find V 11|2 = V 21|2 = 1, that is, the conditional variance is no better than the single beam variance indicating no correlation. Quantum correlation occurs when a conditional variance falls below 1. A special situation, which can only happen when the beams are entangled, is when both V 11|2 and V 21|2 fall below 1, and this will be discussed in Chapter 13. It is convenient to define the expectation value of the product of the quadratures of the two beams as the amplitude quadrature correlation coefficient, C1, i.e. 1 ̃ ̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ (4.135) ⟨𝜹X𝟏1 𝜹X𝟏 2 Experimentally we can obtain C1 by measuring the sum and difference quadrature variances, i.e. 1 C1 = (V 1s − V 1d ) (4.136) 2 In terms of C1 we then have that uncorrelated beams have C1 = 0, correlated beams have C1 > 0, and anti-correlated beams have C1 < 0. Similarly for the phase quadrature correlation coefficient, we have C1 =

C2 = 4.6.3

1 (V 2s − V 2d ) 2

(4.137)

Two-Mode Covariance Matrix

If the state of our two-mode system is Gaussian, then a complete description is provided by the covariance matrix and displacement vector. The covariance matrix of a two-mode system can be written in the following way: ( ) V1 C V1,2 = (4.138) CT V 2

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where Vi is the covariance matrix of the ith mode as defined in Eq. (4.59). The matrix C is defined as ) (1 ̂ 1 ΔX𝟏 ̂ 2 + ΔX𝟏 ̂ 2 ΔX𝟏 ̂ 1 ⟩ 1 ⟨ΔX𝟏 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟏 ̂ 1⟩ ⟨ΔX𝟏 2 2 C= 1 ̂ 2 ΔX𝟐 ̂ 1 + ΔX𝟐 ̂ 1 ΔX𝟏 ̂ 2 ⟩ 1 ⟨ΔX𝟐 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟐 ̂ 1⟩ ⟨ΔX𝟏 2 2 (4.139) and describes the correlations between the modes. The matrix CT is the transpose of C. In standard form the two-mode covariance matrix reduces to

V1,2

⎛ V 11 ⎜ ⎜ 0 =⎜ ⎜ C1 ⎜ ⎝ 0

0 V 21 0 C2

C1 0 V 12 0

0 ⎞ ⎟ C2 ⎟ ⎟ 0 ⎟ ⎟ V 22 ⎠

(4.140)

where the subscripts on the variances indicate the mode and C1 and C2 are defined in the previous section. The displacement vector of the two-mode state is ̃ 1⟩ ⎞ ⎛ ⟨𝜹X𝟏 ⎜ ⎟ ̃ 1⟩ ⎟ ⎜ ⟨𝜹X𝟐 d=⎜ (4.141) ̃ 2 ⟩ ⎟⎟ ⎜ ⟨𝜹X𝟏 ⎜ ̃ ⎟ ⎝ ⟨𝜹X𝟐2 ⟩ ⎠ The covariance matrix approach will turn out to be very useful when analysing quantum information protocols with Gaussian states.

4.7 Summary We have introduced quantum optical theory and have discussed various quantum models of light. As is generally true in quantum mechanics, there are different pictures and/or representations that can be used in calculations that seem very different but lead to the same solutions. Which one to use can depend on the application. Here we have focused mainly on single photon states and coherent states as examples of the most quantum and most classical states, respectively; however the approaches described will be applied more generally in the coming chapters. 4.7.1

The Photon Number Basis

The photon picture is intuitive and obvious for the explanation of emission and detection processes. However, care has to be taken since photons do not behave like classical particles. Their quantum properties have to be taken into account. The description of state evolution in the Schrödinger picture using the photon number basis is optimized for situations with low photon numbers and photon

4.7 Summary

counting. Technology for generating and detecting single photons and photon coincidences has been developed that matches this representation well. The special quantum properties of particles are directly accessible in this regime, and they are used extensively for quantum applications such as cryptography and quantum information processing, described in Chapter 13. We will find elsewhere in this guide, Chapter 9, that the photon number representation is very useful in explaining the properties of non-linear processes of CW laser beams, such as second harmonic generation (SHG) or OPO. Mathematically the discrete form of the number basis can have calculational advantages and is particularly useful for numerical solutions. 4.7.2

Quadrature Representations

We can also expand our quantum optical state in terms of the eigenstates of the quadrature observables in various ways, leading to continuous probability and quasi-probability distributions in phase space. The most popular probability distribution is the Q-function, which can be interpreted as the probability density of measuring the state to be in the coherent state |𝛼⟩. The most popular quasi-probability distribution is the Wigner function whose marginals are the probability densities for measuring particular quadrature outcomes, but which can take on negative values for non-Gaussian states. For the special but ubiquitous case of Gaussian states, the first and second quadrature moments are sufficient to completely characterize the state. It is useful then to tabulate these moments into a displacement vector and covariance matrix that can be used to calculate various properties of the state. This is a compact representation particularly when one is considering multimode states. 4.7.3

Quantum Operators

A useful quantum optical approach is to calculate how the quantum mode operators evolve in the Heisenberg picture. Starting from some fiducial initial state, often the vacuum state, the transformation of the annihilation operators of the various modes involved completely characterizes the final quantum state of the system. We have described how propagation and detection are modelled in the Heisenberg and Schrödinger pictures and highlighted the differing physical interpretations associated with each. For many practical applications a simplified linearized model is fully sufficient. In this case we can completely describe the beam by the evolution of the operators ̃ ̃ 𝜹X𝟏(Ω) and 𝜹X𝟐(Ω). This model can be used to derive the noise transfer functions for V 1(Ω) and V 2(Ω), describing how the quantum (and classical) noise propa2 ̃ gates through different devices, where V (Ω) = ⟨|𝜹X(Ω)| ⟩. In this way we can analyse complete optical systems from source to detector by evaluating the operators at the various locations. This model of transfer functions will be extremely useful to derive the properties of light beams that have deeper quantum properties, such as squeezed light (Chapter 9), entanglement between the quadratures, and CW quantum processing such as optical teleportation (Chapter 13).

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If all the interactions are linear or linearizable and our initial state is the vacuum, then all output states will be Gaussian, and so the transfer function approach can conveniently be used to calculate the covariance matrix elements and their evolution. 4.7.4

The Quantum Noise Limit

We have found that the QNL defines the quietest light that can be produced with a conventional source, such as thermal light from a lamp, or coherent light from a laser. Quantum noise is the way the quantum nature of light manifests itself in CW laser beams and in individual pulses of laser light. However, there is more to come, and other types of light states are possible. Within the quantum model it is possible to define light that has reduced fluctuations in one of its quadratures. Such squeezed states of light and their generation are the topic of Chapter 9. We find that for all devices that are sensitive enough to detect quantum noise, and these are getting more and more common due to the advances in technology, it is important to understand the quantum properties of the light. They will influence the performance of the devices. In many cases the quantum effects will set a fundamental limit to the performance, the QNL. We will find tricks to avoid these limitations. In other cases the performance will be quite different near the QNL compared with the classical regime well above the QNL. Finally we will show applications and devices that are only possible because of the existence of these quantum effects.

References 1 Goldstein, H. (1980). Classical Mechanics. Singapore: Addison-Wesley. 2 Power, E.A. (1964). Introductory Quantum Electrodynamics. Longmans. 3 Vogel, W. and Welsch, D.-G. (1994). Lectures on Quantum Optics. Berlin:

Akademie Verlag. 4 Casimir, H.B.G. and Polder, D. (1948). The influence of retardation on the

London - van der Waals forces. Phys. Rev. 73: 360. 5 Kitchener, J.A. and Prosser, A.P. (1957). Direct measurement of the

long-range van der Waals forces. Proc. R. Soc. A 242: 403. 6 Glauber, R.J. (1962). Coherent and incoherent states of the radiation field.

Phys. Rev. 131: 2766. 7 Walls, D.F. and Milburn, G.J. (1994). Quantum Optics. Springer-Verlag. 8 Pegg, D.T. and Barnett, S.M. (1988). Unitary phase operator in quantum

mechanics. Europhys. Lett. 6: 483. 9 Freyberger, M., Heni, M., and Schleich, W.P. (1995). Two mode quantum

phase. Quantum Semiclass. Opt. 7: 187. 10 Caves, C.M. (1981). Quantum-mechanical noise in an interferometer. Phys.

Rev. D 23: 1693. https:doi.org/10.1103/PhysRevD.23.1693. 11 Weedbrook, C., Pirandola, S., Garca-Patron, R. et al. (2012). Gaussian quan-

tum information. Rev. Mod. Phys. 84: 621. 12 Fabre, C. (1992). Squeezed states of light. Phys. Rep. 219: 215.

Further Reading

13 Scully, M.O. and Zubairy, M.S. (1997). Quantum Optics. Cambridge Univer-

sity Press. 14 Gardiner, C.W. and Zoller, P. (2000). Quantum Noise, 2e. Berlin:

Springer-Verlag.

Further Reading Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Mandel, L. and Wolf, E. (1996). Optical Coherence and Quantum Optics. Cambridge University Press. Schleich, W.P. (2001). Quantum Optics in Phase Space. Wiley-VCH. Grynberg, G., Aspect, A., and Fabre, C. (2010). Introduction to Quantum Optics. Cambridge University Press.

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5 Basic Optical Components Real experiments consist of many components – at first sight, any quantum optics experiment looks like a bewildering array of lenses, mirrors, etc. filling an optical table in a more or less random fashion. However, they all serve a purpose, and their individual characteristics are described in this chapter. In quantum optics experiments we consider light as laser beams or modes of radiation. Using optical components, we want to change the parameters of the mode, such as beam size, direction, polarization, or frequency. This means we wish to transform one mode into another mode. This can be achieved by a whole range of optical components, all of which are linear, which means that their output is proportional to their input, independent of the intensity of the beam. One of the simplest optical processes is attenuation by a beamsplitter, and we will see how it affects both the intensity as well as the fluctuations of the light. Whilst the first is simple, that is, the intensity is reduced, the latter requires more detailed modelling. The complementary process, gain, is similar but requires much more elaborate equipment and theory, which will be covered in Chapter 6, describing lasers. Combining several mirrors allows us to build interferometers and cavities. Several other optical components are used to manipulate the light: spatial parameters can be controlled with lenses and curved mirrors; polarization can be influenced with various crystals and surface effects; and frequency can be shifted using electro-optic modulators (EOM). In this chapter we will first describe the effect of the various components on a single spatial and temporal mode. The quantum description in Chapter 4 will be used to demonstrate and model the distinctive quantum effects. The concept of linear systems, beamsplitters, interferometers, and resonators is immensely useful and can be used to describe many devices. Any machine that is based on the propagation of waves and modes will contain these components. This could be as simple as the unavoidable losses that take power out of the mode, any coherent splitting and recombination that leads to interference, and any resonators that have tuned transmission functions. Finally, we summarize the many practical ways of building beamsplitters, interferometers, and resonators that have been developed to create devices that use the principles of quantum physics. These go well beyond optics.

A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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5.1 Beamsplitters One of the most common and simplest optical components is a mirror. This is made either of thin layers of metal or multilayer dielectric films deposited on glass. A mirror has two output beams for every input beam, one transmitted and the other reflected. Thus it is referred to as a beamsplitter. Let us analyse the properties of a beamsplitter. We can use the different models described in Chapter 4; they will obviously lead to the same conclusion but will provide different interpretations of the effect of a beamsplitter. 5.1.1

Classical Description of a Beamsplitter

Consider a beamsplitter as shown in Figure 5.1 with an incoming beam described classically by the complex amplitude 𝛼in exp(i𝜙in ). There are two waves leaving the beamsplitter: the transmitted beam, 𝛼t exp(i𝜙t ), and the reflected beam, 𝛼r exp(i𝜙r ). We describe the properties of the beamsplitter by the intensity reflection coefficient 𝜖: |𝛼r |2 = 𝜖|𝛼in |2 |𝛼t |2 = (1 − 𝜖)|𝛼in |2

(5.1)

The special case 𝜖 = 12 describes a balanced 50/50 beamsplitter. However, there is also a second input axis defined such that a wave 𝛼u exp(i𝜙u ) would produce at the output a wave in the same place and propagating in the same direction as 𝛼t and 𝛼r . For a simple beamsplitter only 𝛼in exists and 𝛼u = 0. In other applications two waves 𝛼in and 𝛼u are superimposed and, in this way, are made to interfere. The optical frequencies of all input and output beams are identical. All the frequencies, including all the sidebands, are preserved. Assuming it has no losses, the mirror will preserve energy. This means |𝛼in |2 + |𝛼u |2 = |𝛼r |2 + |𝛼t |2

(5.2)

The intensity response to the other input beam can be written similarly to Eq. (5.1). The response for amplitude transmission and reflection is not that simple, since the effect of the beamsplitter on the phase of the light has to be considered. Let us arbitrarily set the phase of the two incoming waves to zero, which means 𝜙in = 𝜙u = 0. The relative phase between the two output beams is of interest: √ √ 𝛼t = (1 − 𝜖) 𝛼in exp(i𝜙in,t ) + 𝜖 𝛼u exp(i𝜙u,t ) √ √ 𝛼r = 𝜖 𝛼in exp(i𝜙in,r ) + (1 − 𝜖) 𝛼u exp(i𝜙u,r ) Φu

αu

ε

αin

Figure 5.1 The classical beamsplitter.

αt

Φt

Φin Φr

αr

5.1 Beamsplitters

There is not sufficient information to determine all four relative phase terms 𝜙in , t; 𝜙n , r; 𝜙u,t ; and 𝜙u,r directly. This could be done by solving the electromagnetic boundary conditions explicitly. The solutions, known as Fresnel equations, can be found in textbooks on physical optics. These equations are different for metal and dielectric mirrors, and they depend on the polarization of the input wave. However, we can apply energy conservation and find a general condition for the four waves: |𝛼in |2 + |𝛼u |2 = |𝛼t |2 + |𝛼r |2 = (1 − 𝜖)|𝛼in |2 + 𝜖|𝛼u |2 + 𝜖|𝛼in |2 + (1 − 𝜖)|𝛼u |2 √ + 𝜖(1 − 𝜖) 𝛼u∗ 𝛼in exp(i(𝜙in,t − 𝜙u,t )) + c.c. √ + 𝜖(1 − 𝜖) 𝛼in 𝛼u∗ exp(i(𝜙in,r − 𝜙u,r )) + c.c.

(5.3)

which leads to exp(i(𝜙in,t − 𝜙u,t )) + exp(i(𝜙in,r − 𝜙u,r )) = 0

(5.4)

The usual choice is to set three of the relative phases to zero: 𝜙in,t = 𝜙in,r = 𝜙u,r = 0, and 𝜙u,t = π, which means √ √ 𝛼r = 𝜖𝛼in + 1 − 𝜖𝛼u √ √ 𝛼t = 1 − 𝜖𝛼in − 𝜖𝛼u (5.5) The normal explanation is that one of the reflected waves has a phase shift of 180∘ in respect to all other waves. For a simple dielectric layer, this would be the reflection of the layer with the higher refractive index. But this is only one possible choice; various mirrors have different properties. All mirrors can be represented by Eq. (5.5). The above analysis had been discussed as early as 1852 by Stokes [1, 2] and leads to a more general matrix format: ( ) ( ) ( )( ) 𝛼r 𝛼in Mr,in Mr,u 𝛼in =M = (5.6) 𝛼t 𝛼u Mt,in Mt,u 𝛼u If there is no loss, the matrix M has to be a unitary transformation, which means the following conditions apply: |Mr,in | = |Mt,u | and |Mt,in | = |Mr,u | |Mr,in |2 + |Mt,in |2 = 1 Mr,in M∗t,in + Mr,u M∗t,u = 0 and our choice of the beamsplitter matrix is ( √ ) √ 𝜖 1 − 𝜖 √ √ 1−𝜖 − 𝜖

(5.7)

We should note that any real beamsplitter will introduce losses due to absorption and internal scattering. This can be described by a loss term (1 − A) such that |𝛼t |2 + |𝛼r |2 = (1 − A)(|𝛼in |2 + |𝛼u |2 )

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In most cases the transmitted and the reflected waves are affected by the same amount. For metal mirrors typical values are A = 0.05; for dielectric mirrors, A of 10−2 –10−4 is common, but losses as low as 5 × 10−6 have been achieved. 5.1.1.1

Polarization Properties of Beamsplitters

In some cases we might want a beamsplitter that acts equally on all polarizations. In the other extreme case one could imagine a polarizing beamsplitter (PBS) that reflects only one polarization and transmits the orthogonal polarization. Both these cases can be approximately achieved with different types of beamsplitters. Another possibility would be a partially polarizing beamsplitter (PPBS) that reflects all of one polarization but only partially transmits the orthogonal polarization. In general, the polarization response is not trivial, and the reflectivity 𝜖 is polarization dependent. For this purpose we define two orthogonal polarization states: S and P linear polarization. For P polarization the electric field vector is parallel to the surface of the mirror. This is out of the plane of the drawing in Figure 5.1. For S polarization, the electric field is perpendicular (or senkrecht in German) to the P polarization. This is in the plane of Figure 5.1. The S polarization contains a component that is perpendicular to the surface of the mirror, and the boundary condition on the mirror surface for the parallel and perpendicular fields is different. Consequently the S and P components of the light are reflected differently, both in amplitude and in phase, and in general 𝜖P and 𝜖S differ from each other. They depend on the way the mirror is manufactured and changes with the input angle and wavelength. For a simple dielectric material, such as uncoated glass, the problem can be solved easily, resulting in the Fresnel reflection coefficients [3]. For real multilayer mirrors such calculations are rather involved. One special solution is 𝜖S = 0 for a dielectric surface under Brewster’s angle. Stacks of tilted glass plates, near Brewster’s angle, are used as polarizers. As long as we consider only light linearly polarized in the S or P direction, the polarization is maintained, and the relative phase shift between the S and the P wave is of no concern. For any other light the polarization is modified by the mirror. The mirror acts like a combination of a non-PBS and a wave plate. In summary, to describe a mirror it is not sufficient to state just the reflectivity 𝜖; the following properties have to be specified as well: input angle, wavelength, polarization, and losses. Finally, the following practical hints are useful for the selection of mirrors: there is a choice of materials for mirrors. For non-PBS, that means ideally 𝜖P = 𝜖S . Metal surfaces are very useful, and 𝜖S = 𝜖P = 0.48 can be readily achieved, but the losses are considerable. With dielectric mirrors the difference between 𝜖P and 𝜖S is usually significant, unless they are specifically designed for the specific application. Dielectric mirrors should preferably be used with linearly polarized light aligned in the S or P direction. The losses can be small (<0.001% = 10 ppm). Polarizing beamsplitters with 𝜖P = 0.001 and 𝜖S = 0.99 are available. Note that this property degrades with a change in wavelength or input angle, and these polarizers have to be used for specified parameters only. In addition, there is a range of wavelength-independent components available that rely on internal total reflection rather than on metallic or dielectric films.

5.1 Beamsplitters

5.1.2

The Beamsplitter in the Quantum Operator Model

The properties of a beamsplitter can be described in a quantum mechanical model. We assume that the input beam is in a mode described by the operator â in . A beamsplitter will maintain the mode properties, which means the frequency of the light, the size of the beam, and the curvature of the wave front are all preserved. The direction of propagation is given by geometrical considerations. The polarizing properties of the mirror can be neglected by assuming that the incoming light is either P or S polarized and the calculation for a different polarization is treated as a linear combination of these special cases. Thus a beamsplitter can be simply described by a transformation from two input mode operators, â in and â u , into two output mode operators, â r and â t [4] (Figure 5.2). Thanks to canonical quantization (see Section 4.1) the transformation rules for the quantum operators are equivalent to those of classical complex amplitudes, and we can use again the matrix definition of Eq. (5.7) and obtain )( √ ( ) ( ) ( √ ) â r â in â in 𝜖 1 − 𝜖 √ =M = √ (5.8) â t â u â u 1−𝜖 − 𝜖 with the definition of the matrix as given in Eq. (5.7). Please note that in many publications in quantum optics the definition of the matrix in Eq. (5.7) is frequently replaced by alternative definitions. Two examples commonly used are ( √ ) ( √ ) √ √ i 𝜖 1 − 𝜖 𝜖 i 1 − 𝜖 √ √ √ √ or (5.9) 1−𝜖 i 𝜖 1 − 𝜖 −i 𝜖 They are equivalent, and preference to either definition is normally given purely on aesthetic grounds. For consistency this book uses only one definition, as given in Eq. (5.8). Suppose there is no light entering the second input. Do we really need to specify both input modes, â in and â u ? We find a reason for having to include â u by checking the properties of the output operators. The operators for all modes obey the Boson commutation relations [̂aj , â †j ] = 1

with j = in, r, t

and [̂ar , â †t ] = 0

The last equation describes the fact that both outputs can be measured independently, without affecting each other. If we simply ignored the input mode â u , we âu

Figure 5.2 The beamsplitter in the quantum operator model. âin

ε

âr

ât

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obtain on the basis of Eq. (5.8): √ â r = 𝜖 â in √ â t = 1 − 𝜖 â in

(5.10)

and we find that the commutator equations do not hold. Instead we find [ ] [ ] â r , â †r = 𝜖 â in , â †in =𝜖 [ ] [ ] † † â t , â t = (1 − 𝜖) â in , â in = (1 − 𝜖) [ ] √ ] √ [ â r , â †t = 𝜖(1 − 𝜖) â in , â †in = 𝜖(1 − 𝜖) The reason for this discrepancy is that we have ignored the second input port. This was justified in the classical model, since no light enters that way. However, even if no energy is flowing through the second port, in the full quantum model, there is zero point energy in the vacuum mode that enters here and contributes to the two output modes. Accordingly, we should use the full description of Eq. (5.8) with both input beams and find [̂ar , â †r ] = 𝜖[̂ain , â †in ] + (1 − 𝜖)[̂au , â †u ] = 𝜖 + (1 − 𝜖) = 1

(5.11)

and similarly for â t . This is in agreement with the commutation rules. We see that the vacuum mode plays an important role and is required to satisfy the rules that apply to any quantized field. 5.1.3

The Beamsplitter with Single Photons

On the one hand, the beamsplitter is not splitting individual photons; it is influencing the probabilities for a photon to leave at one or the other of the two outputs. On the other hand, the beamsplitter is splitting the quadratures, dividing the field between the two ports. This ambiguity that makes the action of macroscopic objects on the quantum world dependent on the type of measurements subsequently made is a unique feature of quantum mechanics. This can be described through the quantum states of light (see Chapter 4). From the photon point of view, the beamsplitter acts like a random selector that reflects the photons with a probability 𝜖 and transmits them with a probability (1 − 𝜖). The effect on any particular photon is unpredictable. If we had an input that always contains a photon, or consider only those intervals when there is a photon, then the input can be approximately described by |in⟩ = |1⟩. We have the output states |r⟩ = |1⟩ with the probability 𝜖 and |r⟩ = |0⟩ with the probability (1 − 𝜖). This means that after the beamsplitter the certainty of finding the input photon in one particular beam is lost. However, for the lossless beamsplitter, we have certainty for the correlation between the results for the two outputs. For example, with |r⟩ = |1⟩ we have simultaneously |t⟩ = |0⟩ in the same time interval. Since we have two input ports, we can write the total input state in the format of combined states as |1⟩in |0⟩u for the ports in and u and obtain the two possible output states |1⟩r |0⟩t and |0⟩r |1⟩t , meaning the photon appears in one or the

5.1 Beamsplitters

other output beam. Since the photon number is preserved, the states |0⟩r |0⟩t and |1⟩r |1⟩t cannot occur. We see that the beamsplitter expands the basis of states that we have to use. For each beamsplitting process we have to include an extra input and output channel, and that means we increase the basis for the state vectors. More rigorously we can use the techniques of Section 4.4. Let the unitary ̂ We know that operator representing evolution through the beamsplitter be U. † † ̂ ̂ ̂ ̂ â r = U â in U and â t = U â u U. Writing the initial state as |1⟩in |0⟩u = â †in |0⟩in |0⟩u , we then have ̂ a† U ̂ †U ̂ |0⟩in |0⟩u ̂ a† |0⟩in |0⟩u = Û Û in in † † ̂a U ̂ |0⟩r |0⟩t = Û in

(5.12)

̂ a† U ̂ † represents time-reversed evolution through the beamsplitter and is Now Û in obtained by inverting Eq. (5.8) and taking the Hermitian conjugate. Explicitly √ √ ̂ † = 𝜖 â †r + 1 − 𝜖 â † ̂ a† U Û t in Therefore

√ √ ̂ a† |0⟩in |0⟩u = ( 𝜖 â †r + 1 − 𝜖 â † )|0⟩r |0⟩t Û t in √ √ = 𝜖 |1⟩r |0⟩t + 1 − 𝜖 |0⟩r |1⟩t

(5.13)

As expected we obtain a superposition of the two possibilities, but notice that the ‘plus’ sign between them is not arbitrary but comes directly from the phase convention of the beamsplitter (Eq. (5.5)). To illustrate this, consider the situation in which the photon arrives at the other port of the beamsplitter. Now our input state is |0⟩in |1⟩u = â †u |0⟩in |0⟩u , and our output state is √ √ ̂ a†u |0⟩in |0⟩u = ( 1 − 𝜖 â †r − 𝜖 â † ) |0⟩r |0⟩t Û t √ √ = 1 − 𝜖 |1⟩r |0⟩t − 𝜖 |0⟩r |1⟩t (5.14) Keeping track of the sign or, more generally, the phase of the superposition is crucial in understanding interferometers at the single-photon level (see Section 5.2). So far we have assumed only a single photon arrives at the beamsplitter. Now, let us consider a beamsplitter that mixes two single-photon inputs. This can produce some surprising results. The combined single-photon states can be written as |1⟩in |1⟩u = â †in â †u |0⟩in |0⟩u . A particularly interesting case is the balanced beamsplitter with 𝜖 = (1 − 𝜖) = 0.5. We obtain the result )( ) ( ̂ a† â †u |0⟩in |0⟩u = 1 â †r + â † â †r − â † |0⟩r |0⟩t Û t t in 2 ) 1( † † = â r â r − â †t â †t |0⟩r |0⟩t 2 ) 1 ( = √ |2⟩r |0⟩t − |0⟩r |2⟩t (5.15) 2 if we illuminate the beamsplitter with the states |1⟩in |1⟩u , which means we know with certainty that single photons are present at both inputs in the same time interval. The key result from Eq. (5.15) is that we will have two photons either in

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the reflected or the transmitted beam, and the terms |1⟩r |1⟩t and |1⟩t |1⟩r , with single photons in each output beam, cancel since the input photons are indistinguishable. This means that for the right condition, namely, a well-balanced beamsplitter and good beam overlap, the events of detecting pairs of single photons, one in each arm, should be rare. This is a quantum interference effect, not a classical interference effect. It can be detected by a coincidence counter, a device that is triggered only when both output beams contain one photon. An experimental demonstration of this effect is discussed in Section 12.2. For a single input beam, |1⟩in |0⟩u , or for ordinary laser beams at the input, with a mixture of photon states, there will be no such interference and reduction in coincidence counts. To produce the required light source, which generates many time intervals that are well described by |1⟩in |1⟩u , is not trivial (see also Section 3.5). However, with these sources, which can produce pairs of input photons, we can use the beamsplitter and correlator both as a demonstration of quantum interference and as a test of the quality of our photon source (Figure 5.3). 5.1.4

The Beamsplitter and the Photon Statistics

Let us now consider a beamsplitter with a single input and a single output. That is, we consider the other input to be vacuum and assume that the second output is ‘lost’. Such a situation can be used as a generic model for loss due to absorption or scattering. The output state of the single beam will in general be mixed. Using the techniques of Section 4.4 to average (trace) over the possible states of the lost field, we find that for a single-photon input, |1⟩in , the output state is given by the density operator 𝝆̃t = 𝜖|1⟩⟨1| + (1 − 𝜖)|0⟩⟨0|

(5.16)

This is just a classical mixture of the possibility that a photon was transmitted or not, weighted by the probability of transmission. Thus in this situation a beamsplitter can be thought of as a random selector of photons, and classical statistics can be applied. In the case of many photons per counting interval, we can measure the average photon flux and the width of the counting distribution function. The average photon flux is fully equivalent to the classical intensity. If we have only one input with flux Nin and Nu = 0, we obtain, based on the probability of the selection of individual photons, Nr = 𝜖 Nin and Nt = (1 − 𝜖) Nin . The photon statistic, however, will be affected differently. We describe the input statistics by the Fano factor fin defined in Eq. (4.57), where fin = 1 corresponds to a Poissonian input statistic and fin = 0 to a perfectly regulated beam. The effect of a random selection process is illustrated in Figure 5.4. Here the time sequence Figure 5.3 The beamsplitter with two input beams with exactly one photon per time interval. In this hypothetical case we should have only pairs of photons leaving each output and no coincidences between the two output beams.

|1〉in |1〉u

|2〉r |0〉t – |0〉r 2〉t

5.1 Beamsplitters

P(n)

Coherent or Poissonian input NAin

t P(n)

ε = 0.9

t P(n)

n

NAin

ε NAin

n

ε = 0.5 t

n P(n)

ε = 0.1

t

n

(b) Sub-Poissonian input

P(n)

NAin t P(n) ε = 0.9

t P(n)

NAin

ε NAin

n

n

ε = 0.5 t

n P(n)

ε = 0.1

t

n

(b)

Figure 5.4 The effect of a beamsplitter on a stream of photons for (a) a Poissonian and (b) a sub-Poissonian input. The left part shows the arrival sequence of photons, and the right side the corresponding distribution functions.

of photons in the input and output beams and their counting distribution functions is sketched. The effects of three beamsplitters with different reflectivities 𝜖 = 0.9, 0.5, and 0.1 are shown. In the first case, part (a) of Figure 5.4, we assume that the input beam is a coherent state, with a Poissonian distribution function. We should note that the general property of any Poissonian distribution is that a random selection yields again a Poissonian distribution. Consequently, the output beam has a Poissonian distribution, with a peak that has moved from NAin to NAr = 𝜖NAin . Similarly,

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the transmitted beam has a Poissonian distribution peaked at NAt = (1 − 𝜖)NAin . The normalized width of the distributions, the Fano factors, of all these beams is constant. We can generalize these results and conclude that a coherent state transmitted or reflected by a mirror remains a coherent state (see next section). The situation is quite different for states with strongly non-Poissonian distributions. In part (b) of Figure 5.4, the extreme case of an absolutely regular stream of photons is shown. The counting distribution for this input beam is a 𝛿-function, which means during each counting interval, the same number of photons is available, as shown in the top line of Figure 5.4. Since only a random selection of these photons is transmitted, this regularity is no longer correct for the output beam. There is an uncertainty in the number of photons present in a counting interval. The distribution function is centred at a lower average photon number and is wider. If only a very small fraction is selected, the statistical properties of the selection process will dominate. In this case (𝜖 ≪ 1) we will have a Poissonian distribution again, as shown in the bottom line of Figure 5.4. This effect of a beamsplitter can be quantified by applying the rules of counting statistics (4.57). The Fano factor for the reflected light, which is a random selection of a fraction 𝜖 of the incoming light, is given by fr = 1 − 𝜖(1 − fin )

(5.17)

The mathematical details are given in Appendix D. For the transmitted, or reflected, beams, the distribution functions are closer to the Poissonian limit than for the input beam. Using Eq. (5.17) the Fano factors can easily be calculated: fr = 1 and ft = 1 for

fin = 1 (Poissonian input)

This means that for the case of a Poissonian input, both outputs are Poissonian. However, for a completely noise-free input, one would obtain fr = 1 − 𝜖 for

and ft = 𝜖

fin = 0 (completely noise free input)

When the input is sub-Poissonian, and the second port is not used, the output is always noisier than the input. It appears that the beamsplitter adds noise. However, as we will see in Section 5.1.5, it is identical to the effect of the vacuum fluctuations entering through the unused port. If the input beam is super-Poissonian, Eq. (5.17) shows that 1 < fr < fin and 1 < ft < fin . The output beams are closer to the Poissonian distribution than the input beam. This matches the expectation that a beam with a fixed modulation depth, which corresponds to a very regular change in the number of photons, will have a reduced signal-to-noise ratio (SNR) after it passed through the beamsplitter. The photon model, which means the stochastic interpretation of a beamsplitter, allows us to predict the properties of the intensity fluctuations. The effect on the quadrature fluctuations requires a full quantum model.

5.1 Beamsplitters

5.1.5

The Beamsplitter with Coherent States

The view of the beamsplitter as a random photon selector is in stark contrast to the classical picture of the division and superposition of fields by a beamsplitter. We saw in Section 4.2.2 that coherent states are the closest quantum analogues to classical light, so it is of interest to examine the effect of the beamsplitter on coherent inputs. ̂ in (𝛼in )D ̂ u (𝛼u )|0⟩in |0⟩u . Using the We can write our input state as |𝛼in ⟩in |𝛼u ⟩u = D techniques of Section 4.4.2, evolution through the beamsplitter is described by transforming the displacement operator ̂ in (𝛼in ) = exp(𝛼in â † − 𝛼 ∗ â in ) D ) (√ )) ( in(√ in √ √ † ̂ ̂ ̂ 𝜖 â †r + 1 − 𝜖 â †t − 𝛼∈∗ 𝜖 â r + 1 − 𝜖 â t UDin (𝛼in )U = exp 𝛼in (√ ) (√ ) ̂r ̂t (5.18) 𝜖 𝛼in D 1 − 𝜖 𝛼in =D where we have simply transformed the operators in the argument of the exponential in accordance with Eq. (5.8). Similarly we have (√ ) ( √ ) ( ) † ̂D ̂ u 𝛼u U ̂r ̂ =D ̂ t − 𝜖𝛼u (5.19) U 1 − 𝜖𝛼u D Putting these together we find the output state of the fields is given by ) ) √ √ |(√ |(√ | 𝜖 𝛼in + 1 − 𝜖 𝛼u ⟩r || 1 − 𝜖 𝛼in − 𝜖 𝛼u ⟩t (5.20) | | | The output state remains a pair of coherent states. Their amplitudes are transformed in exactly the same way as the amplitudes of a classical field are transformed by the beamsplitter. As a result the expectation values of the quadrature observables will be identical to the classical predictions. Note also that, in contrast to the single-photon situation, the output fields are not entangled. That is, the output state of Eq. (5.20) can be factored into a product of a beam r part and a beam t part. Although the average values of the quadratures are the same as in the classical case, we must consider quantum noise in the quantum case. To do this we next discuss the transfer function of the beamsplitter. 5.1.5.1

Transfer Function for a Beamsplitter

Now we calculate the effect of a beamsplitter on fluctuations and modulations. This is given by a transfer function that transforms V 1in (Ω), V 2in (Ω) and V 1u (Ω), V 2u (Ω) into V 1r (Ω), V 2r (Ω) and V 1t (Ω), V 2t (Ω). The beamsplitter is one of the simplest cases of a transfer function. Let us consider a single input beam, which means a photon flux N in and no flux in the other beam, N u = 0. The beamsplitter transformation is linear, so the fluctuations transform in the same way as the full operators (Eq. (5.8)). We wish to calculate the variances using the recipe V 1r (Ω) = ̃ r (Ω)|2 ⟩ derived in Section 4.5.1, and ̃ r (Ω)|2 ⟩ and similarly V 2r (Ω) = ⟨|𝜹X𝟐 ⟨|𝜹X𝟏 ̃ = thus we convert to the description in terms of the quadrature operators 𝜹X𝟏 † † ̃ ̃ ̃ ̃ ̃ 𝜹A + 𝜹A and 𝜹X𝟐 = i(𝜹A − 𝜹A ). We describe the input beams by the spec̃ in (Ω) and 𝜹X𝟐 ̃ u (Ω) and calculate 𝜹X𝟏 ̃ r (Ω) and 𝜹X𝟐 ̃ r (Ω). Assuming the tra 𝜹X𝟏

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beamsplitter has a flat frequency response (an excellent approximation over radio frequency ranges), the quadrature operators in frequency space transform in the same way as the operators in the time domain, and we can use )( √ ) ( √ ) ( ̃ in (Ω) ̃ r (Ω) 𝜖 1 − 𝜖 𝜹X𝟏 𝜹X𝟏 √ √ = (5.21) ̃ t (Ω) ̃ u (Ω) 𝜹X𝟏 𝜹X𝟏 1−𝜖 − 𝜖 ̃ r (Ω)|2 , we find that for independent inputs When we calculate the product |𝜹X𝟏 ̃ u (Ω) will average to zero. Only the terms ̃ in (Ω)𝜹X𝟏 all terms of the type 𝜹X𝟏 ̃ u (Ω)|2 contribute. We obtain ̃ in (Ω)|2 and |𝜹X𝟏 |𝜹X𝟏 V 1r (Ω) = 𝜖 V 1in (Ω) + (1 − 𝜖) V 1u (Ω) V 1t (Ω) = (1 − 𝜖) V 1in (Ω) + 𝜖 V 1u (Ω)

(5.22)

and the same result for V 2r (Ω) and V 2t (Ω). In all these quantum calculations, we ̃ u for the second have to keep track of the fluctuation properties of the operator 𝜹A input beam that is incident on the beamsplitter. Despite the fact that this mode contains no energy, the fluctuations have to be taken into account as V 1u (Ω) = V 2u (Ω) = 1. This is a direct consequence of the commutation rules and is the essential difference between the classical and quantum calculations. If we have only one input beam with intensity, then V 1u (Ω) can be replaced by 1, and we get V 1r (Ω) = 𝜖 V 1in (Ω) + (1 − 𝜖). As a rule the variance of the output light will be moved closer to the quantum noise limit (QNL). In the limiting case 𝜖 → 1, which means a totally reflecting mirror, the output beam is identical to the input. For a balanced 50/50 beamsplitter, the output variance of each beam is an equal mixture of the fluctuations from the input state and the empty input port. This result is identical to that for the attenuation of a beam of light (see Figure 5.4). As mentioned earlier a beamsplitter is fully equivalent to attenuation. Each beamsplitter, or in practical terms each loss mechanism in the experiment, opens one new input channel, and this is affecting the properties of the light for fluctuations or modulation close to the quantum noise level. The beamsplitter reduces the intensity but retains the relative intensity noise (RIN). Consequently the amplitude of the carrier √ and the modulation sidebands are both reduced by the amplitude reflectivity 𝜖. In contrast the quantum noise sidebands remain unchanged. That means that the SNR of a quantum noise limited signal is reduced. Finally, we have the general result again that in the situation where all input beams are coherent, which means V 1in (Ω) = V 2in (Ω) = 1, all output beams are coherent as well, and we have V 1r (Ω) = V 2r (Ω) = V 1t (Ω) = V 2t (Ω) = 1. There is a school of thought that wishes to avoid the rigour of quantum mechanical calculations and prefers to use a classical analogue [5]. In simple situations it is possible to interpret a beamsplitter as a beam divider with a noise source built into it. This source generates just the right amount of partition noise and mimics the properties of the stochastic device. All calculations are carried out classically. Obviously, with the correctly chosen noise source, the results for intensity noise are identical to those from a full quantum model. However, complications arise when other quadratures of the light have to be taken into account. Examples are interferometers and cavities, where the phase noise of the light is as important as the amplitude noise, and non-linear media, where phase and amplitude are linked

5.1 Beamsplitters

to each other. In these cases the quantum transfer functions provide a complete and simple description. 5.1.6 Comparison Between a Beamsplitter and a Classical Current Junction It is worthwhile to compare an optical beamsplitter with a junction for electrical currents. These two components have very different, complementary properties. The photons traversing a beamsplitter are non-interacting particles. The beamsplitter acts like a random selector for the photons. It thereby determines the statistical properties of the stream of photons emerging from the beamsplitter and leads to the quantum fluctuations that have been discussed in detail. In the case of electrical currents, a strong interaction exists between the particles, a consequence of their electrical charge. The stream of electrons is divided in a completely deterministic way, in accordance with Kirchoff ’s law. Let us describe the input current iin (t) as a time-averaged current iin plus some time-dependent fluctuations 𝛿iin (t). The junction splits off a fraction 𝜁 of the current and produces two output currents it (t) = it + 𝛿it (t) and ir = ir + 𝛿ir (t). Both the average current and the fluctuations are affected in the same way: ir = 𝜁 iin it = (1 − 𝜁 )iin 𝛿ir (t) = 𝜁 𝛿iin (t) 𝛿it (t) = (1 − 𝜁 )𝛿iin (t)

(5.23)

The statistical properties of the currents can be described by a normalized variance, defined in the same way as for photons (Eq. (4.57)). Again Vi = 1 represents a Poissonian distribution. The statistics of the output current can be evaluated as 𝛿ir (t)E2 𝜁 𝛿iin (t)E2 = = 𝜁 Viin ir 𝜁 iin Vit = (1 − 𝜁 )Viin

Vi r =

(5.24)

This means that, relatively speaking, the two output currents are quieter than the input current and the variance is always reduced. For a Poissonian input current, both output currents are sub-Poissonian. The only distribution that remains unchanged would be a delta function, the distribution for a perfectly quiet current without any fluctuations, Vi = 0. The junction does not introduce any noise. It actually reduces it, as measured by the Fano factor. This is a consequence of the way the Fano factor scales. The actual fluctuations have not disappeared; they were redistributed in the same way as the average current. The other consequence of such a fully deterministic junction is that the fluctuations of the two outputs are completely correlated to each other and they are correlated to the input. In this case, which can be referred to as the classical case, we could detect ir (t) and it (t) separately and could cancel the fluctuations in one of the signals with the other. Just by measuring one output, all three currents, the input and the two outputs, can be determined with perfect accuracy. The difference between the properties of a random selector (optical beamsplitter)

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and the deterministic divider (electrical junction) is striking. The beamsplitter will preserve the Poissonian distribution. The output beams generated by a QNL input beam are completely uncorrelated. In contrast, the junction always reduces the Fano factor and creates perfect correlations between the input and output beams. We clearly have to apply different rules to the two cases. Optical beamsplitters cannot be treated like electrical junctions. Does the simple description of a junction hold under all conditions? The interaction between the particles (electrons) dominates at the level of fluctuations discussed here, namely, relative changes in the current of the order of one part on 104 –109 . This is the regime where measurements of optical noise are carried out. At the level of much smaller fluctuations, the quantum nature of the electrons will contribute some randomness, and at such low level of fluctuations, Kirchoff ’s law (Eq. (5.24)) will no longer be correct. In general, quantum effects can be ignored in macroscopic systems such as wires, resistors, and other normal electronic components. However, quantum effects have to be considered where the physical dimensions of the components approach the size of the wave function for the electron. This field is sometimes referred to as electron quantum optics and displays some quite unique features. It is now becoming possible to build and operate such systems, resulting in new and interesting ways to control currents at the microscopic scale [6]. 5.1.7

The Beamsplitter as a Model of Loss

Any component that takes power out of the mode is a beamsplitter, and thus any loss in power can be modelled as a beamsplitter. The difference is that this is not the conventional four-port device described in Eq. 5.5 that has two coherent outputs. Rather we need only one output port that is coherent and a second output port that couples into one or a large number of modes, all of which are never returning. This power is simply lost and will not interfere again with the remaining wave. Practical examples could be scattering, absorption, apertures, losses inside fibres or on surfaces, or inefficiencies of components such as sources or detectors. Distributed loss can often be lumped together as a single beamsplitter. Consider two sources of loss in series. Now model them as two beamsplitters of transmissions 𝜂1 and 𝜂2 , one following the other. In the Heisenberg picture an input mode, â , which passes√through the beamsplitters is trans√ √ formed to an output mode 𝜂1 𝜂2 â + 𝜂2 (1 − 𝜂1 )̂v1 + 1 − 𝜂2 v̂ 2 , where v̂ 1 and v̂ 2 are the independent vacuum mode inputs. However, it is easy to show that √ all expectation √ values will √remain the same if we make the substitution 𝜂2 (1 − 𝜂1 )̂v1 + 1 − 𝜂2 v̂ 2 → 1 − 𝜂1 𝜂2 v̂ T where v̂ T also is vacuum mode input. Thus the pair of beamsplitters is formally equivalent to the single beamsplitter √ √ transformation: 𝜂T â + 1 − 𝜂T v̂ T , with 𝜂T = 𝜂1 𝜂2 . This type of equivalence generalizes to multiple beamsplitters. When we engineer real devices, we will find loss at many places within one device, and we will make repeated use of this simple approach.

5.2 Interferometers

This is a universal model for any loss or in theory terms a model for an open system with a coupling to a thermal bath. We can use the normal beamsplitter model, with all its consequences, to describe the effect of loss in open quantum systems. The quantum effect is basically the coupling of power out of the system to be replaced by quantum mechanical vacuum (or thermal) fluctuations that are coupled back in. The effect applies both for simple beamsplitters as well as complex devices. This simple trick allows us to reliably predict both the classical and quantum performances of many practical devices, all of which have some less than perfect efficiency, and that means some losses. We will find that this is a universal tool for describing the quality of the apparatus and for optimizing the technology.

5.2 Interferometers One of the most useful and interesting applications of a beamsplitter is to split and later recombine the electromagnetic waves associated with a beam of light. In the process of recombination the waves interfere, the result is an output beam with an intensity that depends on the relative phase between the two partial waves. Mirrors do not dissipate energy, apart from the (usually very small) scattering losses. Thus the total energy in an interferometer is preserved. For any occurrence of destructive interference in the output, there must be a corresponding part of this beam or of another beam leaving the interferometer that shows constructive interference. The physical layout of interferometers can vary tremendously. But they can all be traced back to one fundamental system, the basic four-mirror arrangement, shown in Figure 5.5, known as the Mach–Zehnder interferometer. This arrangement contains two beamsplitters M1 and M4 . The other two mirrors (M2 , M3 ) are used to redirect the beams and overlap them at the output. For a balanced interferometer reflectivities of the beamsplitters are 𝜖1 = 𝜖4 = 1∕2 and 𝜖2 = 𝜖3 = 1. The first beamsplitter produces two beams of equal amplitude. These beams travel on different paths. They experience a differential phase shift Δ𝜙, which is indicated by the wedge of transparent material with refractive index n > 1. The second beamsplitter recombines the beams. Figure 5.5 The components of a classical interferometer.

M1

ε1

α|

αin

M3

M2

M4

α||

∆ϕ

α1

ε4 α2

153

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5 Basic Optical Components

5.2.1

Classical Description of an Interferometer

In the classical wave model, a balanced interferometer with 50/50 beamsplitters can be described by the following evolution of the complex amplitudes: √ 𝛼I (t) = 1∕2 𝛼in (t) √ 𝛼II (t) = 1∕2 𝛼in (t) √ √ 𝛼1 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) − 1∕2 𝛼II (t) √ √ (5.25) 𝛼2 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) + 1∕2 𝛼II (t) √ √ where the amplitude reflectivity 𝜖 has been set to 1∕2. The calculation can easily be generalized to other values of 𝜖. Here the convention of Eq. (5.7) is used. The minus sign in the third equation represents the 180∘ phase shift imposed by one, but not the other, of the reflections. Strictly speaking, this set of equations applies only to plane waves, but it can also be applied to spatial Gaussian beams. The output intensity of the interferometer is given by I1 (Δ𝜙) = Iin cos2 (Δ𝜙∕2) I2 (Δ𝜙) = Iin sin2 (Δ𝜙∕2)

(5.26)

and I1 + I2 = Iin . There are many applications for interferometers: they can be used to measure physical processes that affect the relative phase difference Δ𝜙. One should think about an interferometer as a device that transforms changes in the optical phase into changes of intensity. Equation (5.26) holds for any interferometer that is based on the interference of two waves – independent on the layout and the details of the apparatus. This includes Michelson interferometers, Lloyd’s mirrors, fibre-optic interferometers, polarimeters, etc. The main task in describing a specific configuration is to evaluate Δ𝜙; once this is done the intensity distribution can be directly evaluated from Eq. (5.26). Under practical conditions the interference will not be perfect. The overlap of the two wave fronts will not be perfect, or the two waves might not be completely coherent. The spatial or temporal coherence between the two waves, which means the correlation between 𝛼I and 𝛼II , will determine the strength of the interference. In the case of perfect correlation, the result in Eq. (5.26) is valid. In the case of no correlation between 𝛼II and 𝛼I , or if they are incoherent, the resulting intensity will be I1 (Δ𝜙) = (𝜖4 |𝛼II |2 + t4 |𝛼I |2 )hΩ = constant I2 (Δ𝜙) = (t4 |𝛼II |2 + 𝜖4 |𝛼I |2 )hΩ = constant I1 (Δ𝜙) + I2 (Δ𝜙) = Iin

(5.27)

where we are now considering an interferometer that is not necessarily balanced. Thus the visibility, or contrast, of the interference system can be used to measure the degree of coherence. We can measure the visibility VIS by comparing the extrema Imax and Imin of the intensity in one output beam and use the definition VIS =

Imax − Imin Imax + Imin

(5.28)

5.2 Interferometers

We can then compare the measured value of VIS with the best possible value VISperfect that we would have achieved for completely correlated beams, where 𝛼II and 𝛼I completely interfere: √ 2 III II (𝛼II + 𝛼I )2 − (𝛼II − 𝛼I )2 VISperfect = = (5.29) (𝛼II + 𝛼I )2 + (𝛼II − 𝛼I )2 III + II VISperfect can be determined separately by individually blocking the beams and by recording the intensity of the beams independently. Combining these four measurements of Imax , Imin , III , and II in Eq. (5.29) yields the value =

VIS VISperfect

(5.30)

This is the value for the coherence of the beams, assuming a perfectly aligned interferometer, which is also the value of the correlation between the beams. This measurement cannot produce easily the sign of the correlation since correlated and anti-correlated lights produce fringe systems of identical visibility – only shifted with respect to each other. A knowledge of the relative phase between 𝛼II and 𝛼I at the maximum interference would allow us to determine the sign. Please note that this technique can be used for any situation; the intensity of the interfering beams do not have to be balanced. However, a simple, direct evaluation is only possible for the balanced situation |III | = |II | since in this case VISperfect = 1 and  = VIS. In all cases we have  = |C(𝛼II , 𝛼I )|. One application of this procedure is to determine the degree of spatial coherence of a beam of light using a double slit experiment. In this particular case 𝛼II and 𝛼I are parts of the same wave but from spatially different sections of the beam. The phase difference Δ𝜙 is a purely geometrical term that is given by Δ𝜙 = d sin(Θ) where d is the separation of the two slits and Θ the angle of detection measured from the axis normal to the plane of the two slits. If the detection occurs at a distance much larger than the slit separation, the fringes are intensity minima and maxima that are parallel and equally spaced. The visibility of these fringes is a quantitative measure of the spatial coherence of the light. Similarly, a measurement of the visibility can be used to ascertain the correlation between two electronic signals, for example, the currents from two photon detectors. By adding and subtracting the signals, or by adding them with a variable time delay, interference signals can be generated. The visibility, and thus the correlation coefficient, can be determined. In Section 7.3.4 this application is discussed in detail. 5.2.2

Quantum Model of the Interferometer

We now consider a quantum model of the interferometer. As in the classical case, we will assume that the instrument is perfectly aligned and that the spatial overlap of the two beams at the recombining beamsplitter is perfect. The effect of spatial and temporal mode mismatch will be discussed in Section 5.2.6 (Figure 5.6). The interferometer is illuminated by modes with the operators â in and â u ; inside we have the operators â I and â II . After recombination we have the two output operators â 1 and â 2 , which are detected individually. As with the classical model, the case of a Mach–Zehnder interferometer with a phase difference Δ𝜙 is treated

155

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5 Basic Optical Components

Avac Ain

ε1

Figure 5.6 Description of an interferometer in the quantum model.

A|

A||

A1

∆ϕ

ε4 A2

first. The results can be generalized to any other interferometer and to applications where Δ𝜙 is a function of time or space. By canonical quantization the internal operators are linked to the input operators via √ √ â I = 1 − 𝜖1 â in − 𝜖1 â u √ √ â II = 𝜖1 â in + 1 − 𝜖1 â u (5.31) The output operators are linked to the internal states via √ √ â 1 = exp(iΔ𝜙) 1 − 𝜖4 â II − 𝜖4 â I √ √ â 2 = exp(iΔ𝜙) 𝜖4 â II + 1 − 𝜖4 â I

(5.32)

For the balanced interferometer, we can set 𝜖1 = 𝜖4 = 12 , and the equations can be combined to give â 1 = exp(iΔ𝜙∕2)(cos(Δ𝜙∕2) â in + i sin(Δ𝜙∕2) â u ) â 2 = exp(iΔ𝜙∕2)(i sin(Δ𝜙∕2) â in + cos(Δ𝜙∕2) â u ) 5.2.3

(5.33)

The Single-Photon Interferometer

The output state of a single-photon incident on a balanced interferometer can be calculated by inverting Eq. (5.33). For the input state |1⟩in |0⟩u = â †in |0⟩in |0⟩u , we obtain the output state cos(Δ𝜙∕2) |1⟩1 |0⟩2 + i sin(Δ𝜙∕2) |0⟩1 |1⟩2

(5.34)

Depending on the phase shift in the interferometer, the photons can be made to exit from one or other output ports with certainty or be in a superposition of both ports. This effect is clearly a wave effect and cannot be understood by thinking of the beamsplitters as random photon partitioners. The behaviour of the photons depends on path length differences between the arms of the interferometer on the order of fractions of the optical wavelength. 5.2.4

Transfer of Intensity Noise Through the Interferometer

We now consider the intensity noise observed at the interferometer output when a bright beam is incident. In order to determine the photon flux, for example, at

5.2 Interferometers

output A2 , we have to evaluate the operator n̂ 2 = â †2 â 2 . However, for a bright beam we can consider this as a classical part 𝛼 for the average values and the operator 𝛿 â for the fluctuations, e.g. â 2 = 𝛼2 + 𝛿 â 2 . To calculate the average intensities, we only have to consider the terms in 𝛼; all terms with 𝛿 â make no contribution, and we obtain N 1 = N in cos2 (Δ𝜙∕2) N 2 = N in sin2 (Δ𝜙∕2)

(5.35)

Not surprisingly, this is the same result as for the classical calculation. The interferometer is like a variable beamsplitter that distributes the energy between the two output ports. The ratio of the distribution can be adjusted by the differential phase Δ𝜙. Now we come to the transfer function for the fluctuations. The direct calculation of the fluctuations would require the evaluation of V N1 = ⟨𝛿 â †1 𝛿 â 1 𝛿 â †1 𝛿 â 1 ⟩ − ⟨𝛿 â †1 𝛿 â 1 ⟩2 . This would be very laborious, but for bright beams we can linearize (see ̃ 1 (Ω)|2 ⟩. Consequently, turning Section 4.5) and use V N1 = 𝛼12 V 11 (Ω) = 𝛼12 ⟨|𝜹X1 Eq. (5.33) into a transfer function for the amplitude quadratures, the result is V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 2u (Ω) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2) V 2u (Ω)

(5.36)

Again, this is not a surprising result. The fluctuations of the two outputs are proportional to the average photon fluxes. The size of the fluctuations reflects the division of the flux between the two outputs. However, it is important to note that the fluctuations of the phase quadrature of the unused port enter these calculations, not the phase fluctuations of the input laser beam, which cancel completely. It was first pointed out by C.M. Caves [7] that the noise in the output can be interpreted as a beat between the coherent input beam and the vacuum state with V 2u = 1. In this balanced interferometer the phase noise of the input beam cancels; it does not contribute. If we have only one input beam, we can make the assumptions that (i) the second port is not used, but the other input beam is the vacuum mode and V 1u = 1, and (ii) the photon flux of the input beam is dominant (N in ≫ 1). Now we obtain V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2)

(5.37)

This is identical to the transfer function of a single beam with a beamsplitter of variable reflectivity, determined by the phase shift difference Δ𝜙 in the interferometer. Again, the spectrum of the fluctuations remains unchanged.

5.2.5

Sensitivity Limit of an Interferometer

How should we operate the interferometer to get the best sensitivity? What is the limit in sensitivity? It is necessary to determine the signal that can be created by modulating the phase difference. For a small change 𝛿𝜙 of the phase

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5 Basic Optical Components

shift (𝛿𝜙 ≪ π), the change in the optical flux at the detector is equal to the first derivative of the output photon flux in regard to the phase multiplied with 𝛿𝜙: 𝛿N2 = 𝛿𝜙

d(N2 ) 𝛿𝜙 1 = 𝛿𝜙2 cos(Δ𝜙∕2) sin(Δ𝜙∕2) Nin ≈ sin(Δ𝜙)Nin d𝜙 2 2 (5.38)

An AC detector measures the signal power psig that is proportional to the square of the change in optical power (see Section 2.1) and includes 𝜏, the inverse of the detection bandwidth: ( )2 𝛿𝜙 (5.39) sin(Δ𝜙) Nin 𝜏 psig = 2 As illustrated in Figure 5.7, the signal is largest when the change in power is most rapid, that is, at Δ𝜙 = π∕2, in the middle between the maximum and minimum of a fringe. The signal can be compared with the variance of the photon number or the quantum noise on the detector: pqnoise = V N2 = sin2 (Δ𝜙∕2) Nin 𝜏 and we can evaluate the SNR: sin2 (Δ𝜙) 𝛿𝜙2 SNR = Nin 𝜏 2 4 sin (Δ𝜙∕2)

(5.40)

(5.41)

which has the largest value when Δ𝜙 = 0, 2π, where the output optical power is at a minimum. In the jargon of the trade, we operate the interferometer at a dark fringe. This is a very different operating point than one would use for a classical interferometer with intensity-independent noise. The reason for this result is that whilst both signal and noise disappear at Δ𝜙 = 0, the quantum noise disappears more rapidly than the signal and the ratio of the two reaches a maximum (see the SNR trace (c) in Figure 5.7). Note that this is a somewhat academic result. In practice the perfect dark fringe is not an optimum since there are always some additional intensity-independent noise terms such as electronic noise. The optimum will be at a point where the quantum noise component in the current fluctuations is larger than the additional 20

Linear units (QNL)

158

15

(b) Quantum noise

(c) SNR

10

(a) Signal

5

0 0

π/2

π

3π/2

Phase offset (∆ϕ)

2π

Figure 5.7 Results for an interferometer that is limited by quantum noise. Shown as a function of the detuning Δ𝜙 are (a) the optical power that also corresponds to a signal with fixed modulation depth, (b) the quantum noise, and (c) the signal-to-noise ratio.

5.2 Interferometers

noise. Details are given in Section 10.3. We define a QNL instrument as a device where the quantum noise is the dominant noise source. The minimum detectable phase change 𝛿𝜙QNL in a QNL interferometer corresponds to an SNR of 1. Near the optimum dark fringe, we can approximate sin2 (Δ𝜙) (Δ𝜙)2 =4 ≈ sin2 (Δ𝜙∕2) (Δ𝜙∕2)2 1 𝛿𝜙QNL = √ Nin 𝜏

(5.42)

for a coherent input state. This is the standard QNL for interferometry. It is the smallest achievable measurement of a change in phase using coherent light. A comment on units is that 𝛿𝜙QNL is measured in rad. Nin is the number of photons travelling from the interferometer to the front face of a detector in one measurement interval. Thus Nin is a number with no dimensions. The value of 𝛿𝜙QNL is not fixed; it depends on the integration time 𝜏. If we spend more time on the measurement, integrate for longer, the value would improve. But the integration time cannot be longer than the period of the signal. As an example the QNL for an interferometer operated with 1 (mW) of light at 𝜆 = 1(μm) and a detection interval of 1 (ms) is 4.4 × 10−7 (rad). This quantum description is not complete. Equation (5.42) shows that the effect of the quantum noise decreases with more optical power. It seems that it would be possible to reach any arbitrary precision; this must be impossible. It is one of the deep truths of quantum mechanics that we need to complement (using an expression coined by Niels Bohr) the concept of quantum noise of the light with a description of the measurement process. We have to consider both the system we investigate and the process of measurement. In straightforward analogy to the Gedanken experiment of Heisenberg’s microscope, there is a back effect of the measurement process onto the object we measure. In an interferometer we investigate the position of the mirrors, which defines the optical path length of the interferometer. The measurement process is the interference of the two reflected modes of light. The reflection of a beam of light with power Prefl exerts a force on the mirrors: 2Prefl c The fluctuations of this force are due to the quantum noise fluctuations in the photon flux, as discussed above, and the amplitude spectral density of the force will be 2 ℏ Prefl 2 (Ω) = (5.43) 𝛿FRP c𝜆 which is independent of the detection frequency. Such a stochastic force will produce fluctuations in the mirror position and results in a position noise spectrum [8]: √ ℏPrefl 1 1 2 2 𝛿F (Ω) = (5.44) 𝛿x (Ω) = m(2πΩ)2 RP mΩ2 8π3 c𝜆 FRP =

159

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5 Basic Optical Components

where m is the mass of the mirror. This displacement occurs in both arms of the interferometer but with opposite sign. The total effect gives a spectrum of the angular fluctuations: ℏ √ Nin 𝜏 𝛿𝜙RP (Ω) = mΩ2 𝜆2 This noise contribution increases with the input power. It also rises at lower detection frequencies. The total noise will be 𝛿𝜙2QM = 𝛿𝜙2QNL + 𝛿𝜙2RP

(5.45)

and the balance between the two terms depends on detection frequency and the power level chosen. Only at low frequencies and high powers does the radiation pressure dominate. At higher frequencies and lower powers, the system is limited by intensity quantum noise alone (QNL). Very special arrangements, such as high-power lasers and very lightweight mirrors, are required to make the light pressure visible. An example for this balance is shown in Figure 10.10 in Chapter 10. For each frequency there is a minimum noise level, also called the standard quantum limit (SQL). This limit can be reached for an optimum power of Popt = πc𝜆mΩ2 , which is typically quite large. For example, with m = 1 kg, 𝜆 = 1 μm, and Ω = 100 Hz, one obtains Popt = 9.4 MW. At the optimum condition the system reaches its limit in sensitivity as dictated by the quantum physics of the measurement process. The QNL described in Eq. (5.45) cannot be avoided with any measurement process based on normal coherent light, but it can be surpassed by using correlated photon states and quantum non-demolition techniques that are described in Chapters 9–11.

5.2.6

Effect of Mode Mismatch on an Interferometer

So far we have assumed that the modes aI and aII that meet at the second beamsplitter in the interferometer are perfectly mode-matched. In general this will not be true, leading to the visibility of the interferometer being reduced. There are many ways in which mode mismatch can occur, for example, the transverse modes might not perfectly overlap, either because they are displaced from each other or because their spot sizes are unequal; their polarizations might not perfectly match due to some birefringence in one arm; or their longitudinal mode functions might not perfectly overlap. Qualitatively, all these different possibilities lead to similar effects. As a particular example we will consider the latter case for a single photon through an interferometer. We consider a single-photon input state as in Section 5.2.3 but now show its mode decomposition explicitly using the techniques of Section 4.4.1: â †in (ti ) |0⟩in |0⟩u =

∫

̃ † |0⟩in |0⟩u d𝜔e−i𝜔ti F(𝜔)A in,𝜔

(5.46)

where ti is some initial time. We model propagation through an interferometer as shown in Figure 5.6 but now allow for an arbitrary time delay, td , in the lower arm.

5.2 Interferometers

Transforming the frequency operators and assuming a balanced interferometer, we obtain the output state |𝜙⟩1,2 =

∫

1 ̃ † + ((e−i𝜔(t+td ) + e−i𝜔t )A ̃ † ) |0⟩1 |0⟩2 d𝜔F(𝜔) ((e−i𝜔(t+td ) − e−i𝜔t )A 1,𝜔 2,𝜔 2 (5.47)

This is a spectrally entangled state. To evaluate the visibility of the interferometer, we need to introduce a particular detection model. We model broadband photon counting of the output modes of the interferometer by the operator n̂ i =

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.48)

̃ represent ̃† A Here it is assumed that the although the individual operators A i,𝜔 i,𝜔 number detection with strong frequency resolution, the output photocurrent of the detector averages over all frequencies. The photon counting rates are then given by 𝜔t 1 d𝜔|F(𝜔)|2 |1 − ei𝜔td |2 = d𝜔|F(𝜔)|2 sin2 d ∫ ∫ 2 2 𝜔 t 1 d𝜔|F(𝜔)|2 |1 + ei𝜔td |2 = d𝜔|F(𝜔)|2 cos2 d ⟨n̂ 2 ⟩ = ∫ ∫ 2 2 ⟨n̂ 1 ⟩ =

(5.49)

We can identify two limits to this expression. If 𝜔 td is sufficiently small, the sine and cosine terms will be approximately constant over the width of the mode function and so can be taken outside the integrals giving 𝜔o td 𝜔t d𝜔|F(𝜔)|2 = sin2 o d 2 ∫ 2 𝜔 t 𝜔 t 2 o d ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = cos2 o d →0 = cos 2 ∫ 2

2 ⟨n̂ 1 ⟩𝜔tdlim →0 = sin

(5.50)

where 𝜔o is the central frequency of the single-photon pulse. In this case the time delay is so small that it only imparts a phase shift. The result agrees with those of Section 5.2.3. In this limit there is perfect mode matching and the visibility is unity. Alternatively, if 𝜔td is sufficiently large, the sine and cosine terms will oscillate very rapidly. As a result, over intervals for which the mode function is approximately constant, both the sine and cosine terms will average to one-half. We obtain 1 1 d𝜔|F(𝜔)|2 = 2∫ 2 1 1 ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = →∞ = 2∫ 2 ⟨n̂ 1 ⟩𝜔tdlim →∞ =

(5.51)

Now the time delay in the lower arm is so long that there is no overlap at the final beamsplitter between pulses travelling along the upper arm and the lower arm. In this limit there is complete mode mismatch, and the visibility is zero. For delay times in between these limits, there will be partial mode match, and the visibility will be between zero and one.

161

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From the point of view of the broadband detection model (Eq. (5.48)), it is clear that the visibility is reduced because, although individual frequency components still display interference, the fringes shift their position with frequency and hence are washed out when we average over many frequencies. Interestingly, Eq. (5.48) can also arise from a completely different detection model in which our individual detection operators have excellent temporal resolution, but we then integrate the photocurrent over time. Specifically 1 ̂† A ̂ d𝜏 A i,𝜏 i,𝜏 2π ∫ ′ 1 ̃† ̃ i,𝜔′ d𝜔′ ei𝜔 𝜏 A = d𝜏 d𝜔e−i𝜔𝜏 A i,𝜔 ∫ ∫ 2π ∫

n̂ i =

=

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.52)

̂† A ̂ represent number Now it is assumed that the individual operators A i,𝜏 i,𝜏 detection with strong temporal resolution. The expansion in moving from lines one to two uses the Fourier transform relation between the time and frequency domains. The simplification in moving from the second to third line results from recognizing that the integral over the detection time, 𝜏, creates the Dirac delta function 𝛿(𝜔′ − 𝜔). We arrive at the same measurement operator as for the broadband case, and so the visibility also decreases with increasing time delay for this detection model. However, now the reduction of visibility can be attributed to which-path information in the interferometer. In principle the time-resolved detectors can resolve the arrival times of the photons at the two outputs with high precision. When the time delay is very small, the arrival time of the photons reveals no information about which path they took through the interferometer. However, when the time delay is large, photons that take the lower arm will arrive noticeably later than those that take the upper arm, thus revealing the path taken through the interferometer and erasing the interference. Even though we integrate over different times and hence do not reveal the arrival times to the experimenter, the in-principle existence of which-path information erases the interference.

5.3 Optical Cavities One component of widespread use in quantum optics experiments is the optical cavity. In the simplest case this consists of two mirrors that are aligned on one optical axis such that they exactly retro-reflect the incident light beam. In the more general case, it is a group of mirrors aligned in such a way that the beam of light is reflected in a closed path, such that after one round trip, it interferes perfectly with the incident wave. The curvature of the mirrors is chosen such that they match the curvature of the internal wave fronts at all reflection points. The optical path length for one round trip through the cavity is the perimeter p, which takes into account all the refractive indices inside the cavity. The light is reflected

5.3 Optical Cavities

(a)

(b)

Figure 5.8 Schematic diagrams of linear and ring cavities.

many times around the cavity. Each round trip generates one partial wave; all these waves will interfere (Figure 5.8). For the case that p is an integer multiple of the wavelength of the incident light, a strong field will be built up inside the cavity due to constructive interference; the cavity is on resonance. The resonance frequencies satisfy the condition 𝜈cav =

jc ; p

j = integer

(5.53)

Note that for a linear cavity the optical length is p = 2nd12 , where d12 is the physical separation between the two mirrors and n is the refractive index. If either the cavity length or the wavelength of the light is scanned, the cavity will go in and out of resonance. The cavity resonance frequencies are equally spaced, and the difference between two adjacent resonance frequencies is called the free spectral range (FSR): FSR = c∕p

(5.54)

The magnitude of the enhancement of the field and the width of the resonance depend on the reflectivities of the mirror and losses inside the cavity. Cavities are used for a number of purposes: the laser itself is a cavity with an active medium inside that provides the gain. The cavity provides the optical feedback required to achieve oscillation, hence the name laser resonator. The cavity determines the spatial properties of the laser beam. In addition, it selects the frequencies of the laser modes. A cavity can be used to investigate the frequency distribution of the incident light. For this purpose, the length of the cavity is scanned, and the transmission is plotted as a function of the length. In this mode of operation, a cavity is called an optical spectrum analyser. On the other hand, a cavity can be used as a very stable reference for the frequency of the light. The detuning of the incident light from the resonance frequency can be measured, and the frequency of the laser light can be controlled via electronic feedback. In this way the laser is locked to the cavity, and the frequency of the laser light is stabilized. For details see Section 8.4. The light reflected from the cavity provides optical feedback. This is very undesirable for many laser systems and sometimes even destructive. However, feedback can be used with some lasers, such as diode lasers, to form an external cavity and thereby control the laser frequency. The high optical field inside the cavity can be used to enhance non-linear optical effects. Such build-up cavities are used in many of the CW squeezing experiments described in Chapter 9.

163

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On resonance, the cavity will set up internal spatial modes. In the case of a laser resonator, this determines the shape of the laser mode and its waist position and waist size. Only certain combinations of mirror curvatures will lead to a stable resonance condition [9]. A passive cavity will only be resonant with that component of the incident field that corresponds to its internal spatial mode. All other light will be directly reflected. In other words, the incident light has to be mode-matched to the cavity in order to get into the cavity. In return, the transmitted light has a very well-defined, clean spatial mode. For this reason cavities can be used as spatial mode cleaners. There are several applications of cavities for influencing the fluctuations of the light. The build-up of the internal field is not instantaneous. It requires the interference of many waves that have to travel through the entire cavity. A steady state can be achieved after a time tcav . This is also the time it takes for the internal field to decay in the absence of an incident light field. Thus the cavity acts like an integrator on the fluctuations with a time constant tcav . The high frequency components of the incident light, at frequencies larger than 𝛾 = 1∕tcav , will not enter the cavity; they will be reflected. These frequency components will be strongly attenuated in the transmitted light. On resonance, the cavity acts like a low-pass filter on the intensity fluctuations. Finally, there are interesting applications of a cavity that is close to, but not exactly on, resonance with the input field. The phase of the reflected amplitude, compared with the phase of the incident light, varies strongly, by about π, when scanned across one cavity linewidth. Thus the sidebands of the light, which have different cavity detunings to the carrier frequency, are reflected with different optical phases. As a consequence, the quadrature of the reflected light is rotated in respect to that of the incident light. We will find (in Section 9.3) that cavities can be used to analyse the quadratures of light, coherent, or squeezed and as a device to change the squeezing quadrature. 5.3.1

Classical Description of a Linear Cavity

The intensity of the internal, reflected, and transmitted light and the relative phase of these beams can be evaluated with a purely classical model. Let us consider a simple cavity with input mirror M1 and output mirror M2 . This could be a standing wave cavity with two mirrors only or a ring cavity where all other mirrors are near-perfect reflectors. The cavity can be described by the parameters and the corresponding total cavity properties in Figure 5.9 and Table 5.1. These properties can be derived using the classical equations for multiple interference of waves. Here we will use the standard textbook approach of adding partial waves and combine it with the more elegant approach of solving d12

Figure 5.9 Properties of a linear cavity.

Loss: e–α2d R1 T1

R2 T2

5.3 Optical Cavities

Table 5.1 Parameters and properties of a linear cavity. Parameters of a linear cavity

T1

= intensity transmission of the input mirror

T2

= intensity transmission of the output mirror

R1

= intensity reflectivity of the input mirror

R2

= intensity reflectivity of the output mirror

p

= optical path length for one round-trip

d12

= distance from input to output mirror

𝛽

= the loss coefficient for internal absorption or scattering

exp(−𝛽p)

= round-trip intensity loss of the cavity

gm

= loss parameter

Pin

= power of the incident beam

Pout

= power of the transmitted beam

Prefl

= power of the reflected beam

𝛾

= cavity linewidth

Δ

= normalized cavity detuning = (𝜈L − 𝜈cav )∕𝛾

Properties of a linear cavity

FSR

= free spectral range = c∕p

F

= finesse = FSR∕𝛾

self-consistency equations at the boundaries of the cavity. The amplitude 𝛼in of the incident field will be partially transmitted through the √ first mirror, and the internal field just past the first mirror field will be 𝛼0 = T1 𝛼in . It will circulate inside the cavity, and during each round trip it will be reduced in amplitude by a factor [9]: √ (5.55) gm = R1 R2 exp(−𝛽p) which includes the amplitude reflectivities at the mirrors and the internal losses. The wave will also pick up a phase difference 𝛿𝜙 = 2π(𝜈L − 𝜈cav )p∕c for every round trip (Figure 5.10). T1 R1 A1 Ein

Pin

T2 R2 A2

E0 e–αp ei δϕ E1 E2 E3 E4

Erefl

Prefl

Ecav

Pcav

Eout

Pout

Figure 5.10 Multiple amplitude interference inside a linear cavity.

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Let us describe the field amplitude 𝛼ca𝑣 inside the cavity as a sum of all circulating terms: 𝛼cav = 𝛼0 + 𝛼1 + 𝛼2 + 𝛼3 + 𝛼4 + · · · =

∞ ∑

𝛼j

j=0

where 𝛼j+1 = g(𝜈)𝛼j and g(𝜈) = gm exp(−i𝛿𝜙(𝜈))

(5.56)

is a complex number that describes the total effect of the cavity during one round trip. The amplitude of 𝛼2 is related to the amplitude of 𝛼1 by the same factor. The total field is described by 𝛼0 (5.57) 𝛼cav = 𝛼0 (1 + g(𝜈) + g(𝜈)2 + g(𝜈)3 + g(𝜈)4 + · · · ) = 1 − g(𝜈) Note that this is equivalent to the effect of a closed feedback system that adds a component g(𝜈)𝛼0 inside the feedback loop. The intensity, and thus the power of the light inside the cavity, is built up to |𝛼0 |2 |1 − g(𝜈)|2 |𝛼0 |2 = |1 − gm exp(−i𝜙)|2 |𝛼0 |2 = (1 − gm )2 + 4gm sin2 (𝛿𝜙∕2)

Pcav ∝ |𝛼cav |2 =

(5.58)

Thus we get 1 1 + (2F∕π)2 sin2 (𝛿𝜙∕2) )2 ( 𝛼 with Pmax = 1−g0 and m √ π gm F = Finesse = (1 − gm ) Pcav = Pmax

(5.59)

The finesse is generally used as a measure for the quality of cavity. It is the number most frequently quoted in experimental publications. We can now link the amplitudes of the various fields through the following boundary conditions, where we take 𝛼cav as the internal field just on the inside of mirror M1 : √ 𝛼0 = T1 𝛼in √ 𝛼out = T2 exp(−𝛽d12 + i𝛿𝜙∕2)𝛼cav √ √ − R1 𝛼in + T1 𝛼refl = (5.60) √ R1 𝛼cav

5.3 Optical Cavities

The second condition takes into account the internal losses on the path d12 from mirror M1 to mirror M2 and the transmission through mirror M2 . The last condition takes into account that the wave at the outside of mirror M1 is out of phase (minus sign) with the wave transmitted through this mirror from the inside. Combining Eq. (5.60) with Eq. (5.57) leads to the link between the transmitted and reflected powers with the input power Pout exp(−2𝛽d12 ) = T1 T2 Pin |1 − g(𝜈)|2 Prefl |R − (R1 + T1 )g(𝜈)|2 = 1 Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.61)

Consider the special case of a linear cavity with no losses in the mirrors, which means p = 2d12 and R1 + T1 = 1, R2 + T2 = 1. This is commonly discussed in textbooks as the case for an optical spectrum analyser or a Fabry–Perot etalon. In this case we obtain the result Pout T1 T2 |g(𝜈)|2 = Pin R1 R2 |1 − g(𝜈)|2 Prefl |R1 − g(𝜈)|2 = Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.62)

The cavity build-up can be dramatic, and the transmission of a cavity can be much larger than the individual transmissions of the mirrors. Note that these equations hold for all detunings; on resonance, simply replace g(𝜈) by gm . For illustration it is worthwhile to consider some special cases: (i) A symmetric, lossless cavity. That means T1 = T2 = T and 𝛽 = 0. The loss parameter for this case is gm = R = (1 − T), and we obtain from Eq. (5.61) Pout ∕Pin = T 2 R∕R(1 − R)2 = 1 and Prefl ∕Pin = (1 − R)2 ∕R(1 − R)2 = 0. Such a cavity is perfectly transmitting and has no reflected power at all. This clearly illustrates the dramatic effect of the constructive interference. The internal power is Pcav = Pin T∕(1 − R)2 = Pin ∕T, which means that mirrors of transmission T = 0.01, corresponding to R = 0.99, are required for a power build-up by a factor of 100. The finesse of such a cavity can be approximated as F = π∕T = 300. (ii) A cavity with loss. All cavities will have some losses. An interesting case is a cavity where the input transmission T1 just balances the sum of all losses. That means R1 = gm , which is equivalent to R1 = R2 exp(−𝛽p). For this case we get Prefl = 0; there is no reflection. The leakage of the intra-cavity power out through the input mirror is equal to the directly reflected amplitude. They cancel each other. This case is known as the impedance matched case,

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in analogy to the matching of electronic transmission lines: Under-coupled:

R1 < R2 exp(−𝛽p)

Impedance matched:

R1 = R2 exp(−𝛽p)

Over-coupled:

R1 > R2 exp(−𝛽p)

The two other cases, under-coupled and over-coupled, lead to some reflection, and the transmission is not optimum. Whilst both cases result in similar reflected power, the sign of the reflected wave changes from the under-coupled to the over-coupled case. However, in most experiments this sign cannot be detected. All these cases are illustrated in Figure 5.11 by the traces (i), (ii), and (iii). 1 (iii) A single ported cavity (T2 = 0). This obviously has no transmission. However, it is still useful. It shows internal power build-up, and all the leakage back out is through the input mirror. The cavity will have a partial power reflection. This light will contain information about the internal field. The reflected beam can be separated and detected by using a linear cavity and an optical isolator or alternatively a ring cavity where the light is reflected into a different direction. The loss parameter gm describes the reduction in the intensity of the internal field on resonance with the cavity after one round trip. The fraction (1 − gm ) leaves the cavity during each round trip. One round trip takes the time p∕c. Thus the energy loss from the cavity is (1 − gm )Pcav c∕p measured in J/s. Since this loss is proportional to the energy stored in the cavity, the power Pcav will decay exponentially in time in the absence of an incident field: Pcav (t) = Pcav (0) exp(−t∕tcav ) c 1 = (1 − gm ) tcav p The amplitude of the field decays with twice the time constant, and we can define a coupling rate 𝜅 for the amplitude 𝛼cav (t) = 𝛼cav (t = 0) exp(−t∕2tcav ) = 𝛼cav (t = 0) exp(−𝜅t) with 𝜅 = (c∕p)(1 − gm ). This decay constant is the appropriate term for the temporal description of the cavity response. It can be linked to the cavity linewidth 𝛾. The Fourier transform of the intensity decay gives the spectral response function of the cavity. It results in a function proportional to 1∕(1 + i4πtcav ). The resonance of the spectral response has a full width half maximum (FWHM) of 𝛾 = 1∕(2πtcav ) and is measured in units of Hz. The same result could have been obtained by determining the FWHM of the cavity response function given in Eq. (5.59). 1 Note that the definition of under-coupled and over-coupled used here is the opposite of the conventional definition used, for example, in the book Lasers by A.E. Siegman [9].

5.3 Optical Cavities 0.4 π/2

(ii)

Pout

Φout (Rad)

0.3 0.2

Pin

0.1

(i)

0 0.064

(a)

(iii) 0

0

−π/2 0.064

0.064

0

0.064

0

0.064

0 Detuning (∆ν)

0.064

40 π/2

Φrefl (Rad)

30

Pcav 20

Pin

0

10 −π/2

0 0.064

(b)

0

0.064

0.064

1

π

Φcav (Rad)

0.75

Prefl ___ 0.5 Pin

0

0.25

−π 0 0.64

(c)

0

0.64

0.064

Detuning (∆ν)

Figure 5.11 Illustration of the cavity response. On the left side is the power response, and on the right side the phase response. Shown are the three results: (a) transmitted light, (b) the internal field, and (c) the reflected light. For each situation three different cases are shown. (i) The largely under-coupled case (long dashes), (ii) the impedance matched case (solid line), and (iii) the over-coupled case (dotted line).

5.3.2

The Special Case of High Reflectivities

It is useful to treat the transmission through the mirrors and the loss inside the cavity in a similar way. They are all processes for the leakage of the field out of the cavity. For this purpose one can describe the properties of the mirrors and the losses with coupling constants 𝛿 [9]. These replace the exponential terms that are required for cavities with large losses. We convert the mirror reflectivity into

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a mirror coupling constant via Ri = exp(−𝛿mi )

or 𝛿mi = ln(1∕Ri ) = − ln(Ri )

and define an equivalent coupling constant for the intra-cavity losses as 𝛿lo = 𝛽p. In this way the loss parameter gm , defined in Eq. (5.55) for a cavity with two partially reflecting mirrors, turns into gm = exp(−(𝛿m1 + 𝛿m2 + 𝛿lo )) = exp(−𝛿) An important case in quantum optics is the case of a high finesse cavity. Here the round-trip losses are small. That means 𝛿mi ≪ 1, 𝛿lo ≪ 1, and thus 𝛿 ≪ 1. This effectively means that the amplitude is changing so little per round trip that spatial effects along the propagation axis can be neglected. We assume a uniform internal amplitude. We also assume that losses inside the mirrors can be neglected, which means (Ri + Ti ) = 1. This leads to the following approximations: gm = exp(−𝛿) ≈ 1 − 𝛿 𝛿mi = − ln(Ri ) ≈ 1 − Ri = Ti (1 − gm )c (𝛿 + 𝛿m2 + 𝛿lo )c 𝛿c 𝜅= ≈ = m1 2p 2p 2p

(5.63)

We can define the loss rates for the total cavity and for the individual processes as 𝜅 = 𝜅m1 + 𝜅m2 + 𝜅lo

with 𝜅mi =

Ti c 2p

and 𝜅lo =

𝛽c 2

(5.64)

These definitions are commonly used in describing high finesse cavities. We find them, for example, in laser models (Chapter 6) and many papers on non-linear processes inside cavities. They are very useful for most quantitative models. However, the limitations of this approximation should not be forgotten. There are many experiments where intra-cavity losses as large as several percent are unavoidable. Or we might have a strong non-linear conversion out of the internal mode into another mode. One example would be efficient second harmonic generation (SHG). In these cases the approximations of the high finesse limit might not hold, and the exact equations should be used.

5.3.3

The Phase Response

The cavity introduces a phase shift between the incoming, the intra-cavity, the reflected, and the transmitted fields. The boundary conditions for the electric field, given in Eq. (5.60), can be used to evaluate these phase shifts. Let us define all phase shifts in respect to the phase of the incoming amplitude. That means 𝜙cav = arg(𝛼cav ) − arg(𝛼in ) = arg(𝛼cav ∕𝛼in ), where the function arg(z) determines the phase angle 𝜙 of the complex number z. Using the results of Eq. (5.60) and

5.3 Optical Cavities

some algebraic manipulation, the following results are obtained: ( ) gm sin(𝛿𝜙) 𝜙cav = arctan 1 − gm cos(𝛿𝜙) ( ) − sin(𝛿𝜙) 𝜙out = arctan g − cos(𝛿𝜙) ) ( m √ −T1 R1 sin(𝛿𝜙) 𝜙refl = arctan √ √ R1 + T1 R1 (1 − gm cos(𝛿𝜙))

(5.65)

A graphical evaluation of these equations for three special cases of under-coupled, over-coupled, and impedance matched is shown in Figure 5.11 (i)–(iii) alongside the results for the transmitted powers. On resonance, which means 𝛿𝜙 = 0, all three phase terms are equal to zero, modulus π. The electric fields of the incoming, intra-cavity, and transmitted light are all in phase. The reflected light is either in phase or out of phase. Around the resonance the phase terms show a dispersive behaviour. The width of the dispersion curve (right-hand side) is comparable with the cavity linewidth 𝛾 (left-hand side). There is a 2π phase change right across the cavity resonance. The intra-cavity field and the transmitted field have a phase advance over the incoming field that increases with detuning. For the reflected light the phase can be advanced (+ sign) or retarded (− sign), depending on the parameters of the cavity. If the directly reflected light dominates (under-coupled case), the phase difference is 180∘ on resonance, and the phase is retarded. If the light from the cavity dominates (over-coupled case), the phase is advanced. Note that sidebands of the incoming wave can experience different attenuation and phase shifts. For the limiting case where the modulation frequency Ωmod of the sidebands is less than the cavity linewidth (Ωmod ≪ 𝛾), this effect is negligible. In this case the sidebands remain in phase with the cavity. For modulation frequencies much larger than the cavity linewidth (Ωmod ≫ 𝛾), the two sidebands are shifted by +π∕2 and −π∕2, respectively, and again they appear to be in phase with the central component. If, however, the modulation frequency is of the order of the cavity linewidth, the two sidebands at ±Ωmod will experience very different phase shifts. This effect can be exploited to measure the detuning of the cavity. The incoming light is modulated, and the transmitted or reflected light intensity is detected. At Ωmod we obtain a signal that is the sum of two beat signals between the sidebands and the central laser frequency. Commonly, phase modulation is used with the consequence that on resonance these two beat signals are out of phase and cancel each other. On resonance, there is no detectable intensity modulation on either the incoming, the transmitted, or the reflected light. However, for a detuned cavity the difference in the phase shifts for the two sidebands will prevent this cancellation. A modulation signal at frequency Ωmod appears, and phase modulation is partially turned into intensity modulation. The signal strength can be measured by demodulating the photocurrent, for example, with a mixer driven at Ωmod . The shape of this response as a function of detuning is closely linked to the dispersion curve for the individual phase terms. It can be derived analytically, and details are given in Section 8.4.

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5.3.4

Spatial Properties of Cavities

So far we have treated the cavity and the light in the cavity as one-dimensional systems, and we have lost all spatial information. However, not all cavities with the same mirror reflectivity and length are the same. The curvature of the mirrors and the shape of the wave front have to be taken into account. There are certain combinations of mirror separation and mirror curvature where a resonance is possible. In this case the wave front, returning after one round trip, is keeping its size and is not expanding significantly. A spatial resonance will build up, and as a consequence the curvature of the wave front of the resulting resonant mode will be identical to the curvature of the mirror. As the simplest example let us consider a linear two-mirror cavity. Here the resonator is defined by the separation d12 of the mirrors and the mirror curvatures 1 and 2 . The focal length of the mirrors is given by f1 = 1 ∕2 and f2 = 2 ∕2. In the standard notation we define the mirror parameters [9] g1 = 1 −

d12 2f1

and g2 = 1 −

d12 2f2

(5.66)

The condition for a stable resonator, which means a resonator that builds up a resonant wave, can be derived [9] and results in the stability criterion 0 ≤ g1 g 2 ≤ 1

(5.67)

Based on this criterion we find that for a given set of mirror curvatures (focal lengths), resonance only occurs for a restricted range of cavity lengths. We have to choose the length and the curvatures together. Some examples of stable resonators are: • The (near) concentric cavity: d12 = 2 1 = 2 2 , which means g1 = g2 = −1 and g1 g2 = 1. • The (near) planar cavity: 1 = 2 = ∞, which means g1 = g2 = 1 and g1 g2 = 1. • The confocal cavity: d12 = 1 = 2 , which means g1 = g2 = 0 and g1 g2 = 0. The concentric and planar cavities are right at the limit of the stability criterion. They can be stable, but even minor changes in length or small misalignments tend to affect the performance. These cases are not recommended. It is better to satisfy the stability criterion with some margin. The confocal cavity is in the middle of the stability region and thus very useful as a robust design for a resonator. Cavities can support higher-order TEMi,j spatial modes, and if the cavity is not confocal, this will occur at cavity lengths different to the resonance length for the TEM0,0 mode. This feature can be used to devise techniques for locking a cavity to the frequency of a laser beam [10] by analysing the spatial properties of the transmitted and/or reflected beam (see Section 8.4). 5.3.4.1

Mode Matching

The cavity has its own internal geometric mode, determined by the curvature of the mirrors. The wave front of the mode of light illuminating the cavity might have a different wave front than the internal mode. The incoming wave can only fully interact with the cavity if the incoming wave front and the internal wave front are

5.3 Optical Cavities

matched. This is known as the mode-matched case. The spatial properties of the incoming wave are matched, the intensity distribution (mode size) is identical to the internal mode, and the wave front curvature is identical to the curvature of the mirror, which in turn is the curvature of the internal mode. In this particular case, which is highly desirable but in practice requires extensive alignment work, the description of the cavity and the coupled light is simple, and the one-dimensional model in the previous section is correct. All the light interacts with the cavity. Most theoretical models assume mode matching. What will happen to light incident on a cavity that has a different geometric mode? The cavity will support one fundamental mode and a whole series of higher-order spatial modes. In general, these modes will not be resonant simultaneously – different detunings, which means small shifts in the mirror separations, will be required for the resonance condition of the higher-order modes. For any given mirror separation, the cavity will only support one mode. The one special case, where all spatial modes are resonant simultaneously, is the confocal cavity. For all other cases the resonators are non-degenerate – the resonances occur at different cavity detunings. On resonance, the cavity will select the part of the external field that corresponds to the resonant internal mode, an eigenmode of the cavity. In order to calculate the reflection and transmission of a cavity, it is useful to describe the external field in terms of all the internal eigenmodes of the cavity. These spatial eigenmodes are a useful basis set for a mathematical expansion of the external field. Let us classify the spatial modes as Hermite–Gaussian modes (see Section 2.1) with the classification TEMmn describing the possible modes. We can calculate the fraction Pmn of the total power that corresponds to the resonant internal mode TEMmn . In most cases we align the cavity to resonate in the fundamental mode TEM00 . Only this light will enter the cavity and contribute to the build-up of the internal power. The remaining power will be reflected, assuming that the front mirror has a high reflectivity. The equations for the transmitted and reflected power (Eq. (5.61)) have to be modified to include the fraction Pmm . For example, the transmission will then be Pmn × Ptrans ∕Pin , and the reflectivity will be (1 − Pmn ) + Pmn × Prefl ∕Pin . Obviously, P00 = 1 corresponds to the perfectly mode-matched situation. To achieve P00 > 0.95 requires considerable patience and good-quality optical components. Once aligned to the fundamental mode, both the internal and the transmitted fields will have the spatial intensity distribution of a Gaussian beam. The cavity selects the Gaussian component from the incident field; it acts like a spatial filter. Such a cavity is therefore also referred to as a spatial mode cleaner. How can the mode-matched condition be achieved, and how can the mode mismatch be quantified? By detuning the cavity, which means by changing the mirror separation, a small fraction of a wavelength, the cavity can be set on resonance with the higher-order spatial modes. The exact resonance frequencies can be evaluated by taking into account the phase differences between the various higher-order modes and the fundamental mode, known as Gouy phase shifts. These phase √ shifts depend on the order numbers m and n and can be expanded as cos−1 (± g1 g2 ) where g1 and g2 are the mirror parameters defined in Eq. (5.66). The changes in the resonance frequencies Δ𝜈mn for the modes TEMmn away from

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5 Basic Optical Components

the resonance of the fundamental mode are given by [9] √ (n + m + 1)cos−1 (± g1 g2 ) c Δ𝜈mn = π2d12

(5.68)

with the following special cases: =0

for confocal cavity

1 FSR for near planar cavity 2 = FSR for near concentric cavity

Δ𝜈mn =

If the cavity is not confocal, the higher-order spatial modes map out a whole series of resonances as shown in Figure 5.12b. These can be made visible by scanning the cavity length or the frequency of the laser. At each side peak the cavity is in resonance with one of the higher-order spatial modes, and the transmitted light will show the corresponding spatial patterns of this mode. In most practical situations we want to achieve the mode-matched case. That means we want to minimize the higher-order modes, as shown in Figure 5.12. By using lenses to change the wave front curvature and by optimizing the position of the lenses systematically, mode matching can be achieved. In the same way the alignment of the beam to the axis of the cavity is improved. This is largely an iterative process, since in most situations, there are six degrees of freedom that have to be optimized: two degrees for the tilt of the input beam in regard to the cavity, two degrees for a lateral shift, and two degrees for the wave front curvature. Measuring the size of the high-order peaks Pmn is far more sensitive than measuring the reflected power Prefl . Clever strategies have been devised to detect misalignment and mode matching separately. There have been several approaches how alignment can be automized [11–13], but the problem of automatic mode matching is still largely unresolved. 5.3.4.2

Polarization

Polarization was not included in the above discussion. For a completely isotropic cavity, the polarization does not matter; the description using a single, arbitrary polarization state of light would be correct. However, in most practical cases at least one of the components of the cavity, or inside the cavity, is anisotropic and induces some birefringence or dichroism. In that case one has to find the polarization eigenstates of the cavity. Light in these states will keep its polarization whilst P00 Pmn

FSR

(a)

νcav

(b)

νcav

Figure 5.12 The frequency response of a scanning cavity that is longer than confocal to (a) a mode-matched input beam and (b) a mismatched beam. Improved alignment will get us from case (b) to the optimum case (a).

5.3 Optical Cavities

propagating. It will return after one complete round trip with the same polarization. Note that at any individual point inside the cavity, these eigenstates can be quite different. There are always two such orthogonal states. Losses, detuning, and thus the finesse can be different for the two states, and the response of the cavity is described by two independent sets of equations, one for each polarization eigenstate. The best-known example is the unidirectional device inside a ring laser that uses polarization and losses to suppress laser oscillation in one of the two counter-propagating directions. 5.3.4.3

Tunable Mirrors

In many practical situations it would be very desirable to be able to tune the reflectivity of a mirror in a cavity. This can be achieved by making one of the mirrors a cavity itself or by using a trick called frustrated internal reflection. Here the two uncoated surfaces are used with a small gap to form a mirror. The light is aligned such that each surface would totally reflect the light. However, when the surfaces have a separation of a wavelength or less, the evanescent field, on the outside of the totally reflecting surfaces, reaches to the next surface, and light is transmitted. The magnitude of this effect can be controlled by changing the separation, thus a mirror of tunable reflectivity can be constructed [3, 14]. It should be noted that such mirrors are polarization dependent. 5.3.5

Equations of Motion for the Cavity Mode

For the rest of our discussion of cavities, we will assume that the technical requirements for obtaining high finesse operation have been satisfied. Under such conditions the equations of motion of a single cavity mode can be obtained in a straightforward way. Conceptually it is easiest to consider the case of a ring cavity as depicted in Figure 5.13. The cavity has two partially transmitting mirrors with transmissions T1 and T2 and internal loss with a transmission of Tl . External fields 𝛼in1 and 𝛼in2 enter at the first and second mirrors, respectively. We calculate the change in the circulating mode amplitude, 𝛼c , incurred in making one round trip of the cavity under the assumption that that change is small. We have √ √ √ 𝛼c (t + 𝜏) = 𝛼c (t)ei𝜙 1 − T1 1 − T2 1 − Tl √ √ √ + 𝛼in1 (t)ei𝜙 T1 1 − T2 + 𝛼in2 (t)ei𝜙 T2 (5.69) where 𝜏 is the cavity round-trip time and 𝜙 is the differential phase shift acquired by the fields in that time. If T1 , T2 , 𝜙 ≪ 1, then the change in the field in one round trip will be small. As such we expand to first order the LHS of Eq. (5.69) in a Taylor Figure 5.13 A cavity described in the quantum model.

Aout1

Ain2 acav

Ain1

κ1

κloss

Aout2

κ2

175

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5 Basic Optical Components

series and make small T approximations on the RHS. We obtain ) ( ( √ √ ) 1 1 1 𝛼c (t) + 𝜏 𝛼̇ c (t) = 𝛼c (t) 1 − T1 − T2 − Tl + 𝛼in1 T1 + 𝛼in2 T2 ei𝜙 2 2 2 Making a small 𝜙 (modulo 2π) approximation, i.e. assuming we are close to a cavity resonance, and rearranging, we obtain ( ) √ √ i𝜙 − 12 T1 − 12 T2 − 12 Tl √ √ T1 T2 𝜏 𝛼̇ c = 𝜏𝛼c + 𝛼in1 + 𝛼in2 (5.70) 𝜏 𝜏 𝜏 √ Equation (5.70) is normally presented with the following substitutions: 𝛼 = 𝜏𝛼c is the amplitude of the standing mode of the cavity; Δ = 𝜙∕𝜏 is the detuning of the cavity; and 𝜅i = Ti ∕(2𝜏) are the decay rates of the cavity. With these substitutions we obtain the following equation of motion for the cavity mode: √ √ (5.71) 𝛼̇ = 𝛼(iΔ − 𝜅1 − 𝜅2 − 𝜅l ) + 𝛼in1 2𝜅1 + 𝛼in2 2𝜅2 This equation of motion will be the starting point for our quantum description. The field exiting from the first mirror is √ √ 𝛼out1 = T1 𝛼c − 1 − T1 𝛼in1 √ ≈ 2𝜅1 𝛼 − 𝛼in1 (5.72) and similarly the field exiting from the second mirror is √ √ 𝛼out2 = T2 𝛼c − 1 − T2 𝛼in2 √ ≈ 2𝜅2 𝛼 − 𝛼in2

(5.73)

Equations (5.72) and (5.73) are referred to as the boundary conditions for the cavity. 5.3.6

The Quantum Equations of Motion for a Cavity

The fundamental object of the quantum model of light is the cavity mode. The cavity is the experimental equivalent to the abstract concept of a single frequency mode and its subsequent identification with the quantum model of a harmonic oscillator. In general, a description of the quantum evolution of a cavity requires the specification of the Hamiltonian describing the interactions within the cavity as well as those between the cavity mode and the continuum of field modes outside the cavity. Equations of motion for the field can be obtained either in the Heisenberg picture (operator Langevin equations) or in the Schrödinger picture (master equation) [15]. Combining this with quantum boundary conditions [16] allows calculations of the properties of fields leaving the cavity. It was techniques such as these that were used to rigorously establish the behaviour of quantum fields that we have described in this and the previous chapters. In later chapters we will use such approaches. However, here we proceed via the simpler approach of canonical quantization of the classical equations of motion (Eq. (5.71)). This leads directly to √ √ √ ̂ in1 + 2𝜅2 A ̂ in2 + 2𝜅l A ̂l (5.74) â̇ = −(𝜅 − iΔ)̂a + 2𝜅1 A

5.3 Optical Cavities

̂ j are travelling modes where â is the standing cavity mode, whilst the A and 𝜅 = 𝜅1 + 𝜅2 + 𝜅l is the total decay rate of the cavity. As for the beamsplitter and the interferometer, the essential differences between the quantum and classical equations are the transformation to operators and the explicit inclusion of an operator representing the vacuum mode introduced by loss. Equation (5.74) is an example of an operator quantum Langevin equation. The boundary condition for the transmitted field √ ̂ in2 ̂ out2 = 2𝜅2 â − A (5.75) A and the reflected field √ ̂ out1 = 2𝜅1 â − A ̂ in1 A

(5.76)

can also be obtained directly from the classical boundary conditions (see Eqs. (5.72) and (5.73)). 5.3.7

The Propagation of Fluctuations Through the Cavity

In order to analyse the effect of a cavity on the noise of an incident propagating mode of light, we consider the transfer functions for the cavity. We use the model described in Chapter 4. We approach the problem step by step: 1. The fluctuations of the input field are described as the spectrum of the amplitude variance V 1Ain (Ω). 2. The cavity is described through the equation of motion. The resulting fluctuations are analysed. We first consider the amplitude quadrature of the fluctuations. 3. The variances V 1out of the various output fields are determined. They are given by a linear combination, the transfer function, of the variances of all the input fields. The variances of the fluctuations of the internal field are of academic interest only. They cannot be measured by conventional means. We find that, for a cavity on resonance, the output amplitude fluctuations are linked directly to the input amplitude fluctuations. For a cavity with detuning, both quadrature amplitudes of the input contribute to the output amplitude fluctuations. The output noise spectrum can be written as a linear function of the variance of the input mode and of contributions from the various unused input ports, which is the major difference to the classical model. Whilst the unused ports do not contribute to the spectrum of the output power, they contribute to the noise spectra of the light. In a way, the effect of the cavity on the fluctuations becomes similar to that of an electronic filter. The response is frequency dependent; it is linear, and it can be described by a transfer function. We consider the system shown in Figure 5.13 and include the fluctuations. As usual each operator is written as a classical complex value for the amplitude and ̂ in1 , etc. The classical ̂ in1 = 𝛼in1 + 𝜹A ̂ A an operator for the fluctuations: â = 𝛼 + 𝜹a, parts obey Eqs. (5.71)–(5.73), whilst the operator parts obey Eqs. (5.74)–(5.76). Considering just the quantum fluctuations, we find √ √ √ ̂̇ = −𝜅 𝜹a ̂ + 2𝜅 𝜹A ̂ + 2𝜅 𝜹A ̂ ̂ + 2𝜅 𝜹A (5.77) 𝜹a 1

in1

2

in2

l

l

177

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5 Basic Optical Components

By adding the complex conjugate, an equation for the amplitude quadrature can be formed: √ √ √ ̂̇ = −𝜅 𝜹X𝟏 ̂ + 2𝜅1 𝜹X𝟏 ̂ in1 + 2𝜅2 𝜹X𝟏 ̂ in2 + 2𝜅l 𝜹X𝟏 ̂ l 𝜹X𝟏 (5.78) The boundary conditions of Eqs. (5.75) and (5.76) are rewritten for the fluctuations √ ̂ out2 = 2𝜅2 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in2 𝜹X𝟏 √ ̂ out1 = 2𝜅1 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in1 (5.79) 𝜹X𝟏 The next step is to consider the Fourier components of the fluctuation. Using the property of Fourier transforms  (df (t)∕dt) = −i2πΩ (f (t)), we can combine Eq. (5.78) with Eq. (5.79) and obtain √ √ ̃ in1 + (2𝜅2 − 𝜅 + i2πΩ) 𝜹X𝟏 ̃ l ̃ in2 + 4𝜅1 𝜅l 𝜹X𝟏 4𝜅1 𝜅2 𝜹X𝟏 ̃ out2 = 𝜹X𝟏 𝜅 − i(2πΩ) (5.80) In order to obtain the spectrum of the fluctuations transmitted through the cavity, we determine the variance. This is the desired set of transfer functions. It shows how the different input variances are linked and how the fluctuations propagate through the system. We can repeat the above calculations and obtain for the reflected beam V 1out1 (Ω) = {((2𝜅1 − 𝜅)2 + (2πΩ)2 ) V 1in1 (Ω) + 4𝜅2 𝜅1 V 1in2 (Ω) + 4𝜅1 𝜅l V 1l (Ω)} 1 × 2 (5.81) 𝜅 + (2πΩ)2 and for the transmitted beam 4𝜅 𝜅 V 1in1 + ((2𝜅2 − 𝜅)2 + (2πΩ)2 ) V 1in2 (Ω) + 4𝜅2 𝜅l V 1l (Ω) V 1out2 (Ω) = 2 1 𝜅 2 + (2πΩ)2 (5.82) The first useful check is to consider the special case where all inputs are QNL, which means they are not either used or illuminated by a coherent state. We find V 1out2 (Ω) = V 1out1 (Ω) = 1 As expected, the two output noise spectra for this special case are also QNL. This is an important result that can be applied to all passive systems no matter how complicated the instrument is: if all the input beams are at the QNL, then also all the output states are at the QNL. But note that even when all output variances have the variances V 1out = 1, the origin of this noise can still vary with the frequency since the components of the transfer function itself are frequency dependent. This becomes important when at least one of the input beams has fluctuations above or below the QNL, and either noise or squeezing is propagating through the cavity. In general, the transfer function is frequency dependent.

5.3 Optical Cavities

Consider the special case of a perfect ring cavity, with 𝜅1 = 𝜅2 = 𝜅∕2. Then the set of transfer functions will become 𝜅 2 V 1in1 + (2πΩ)2 V 1in2 𝜅 2 + (2πΩ)2 2 (2πΩ) V 1in1 + 𝜅 2 V 1in2 V 1out1 (Ω) = 𝜅 2 + (2πΩ)2 V 1out2 (Ω) =

We can now examine the low and high frequency performances of the cavity, which are distinctly different. For low frequencies, we see that in the limit Ω → 0 the spectra approach V 1out2 (0) = V 1in1 V 1out1 (0) = V 1in2 In other words, the input noise passes directly through to the output, and the noise of the unused port, normally vacuum noise, passes straight through into the reflected beam. The cavity is transparent to noise in either direction. Interestingly, at high frequencies, we see that in the limit of Ω → ∞, these spectra approach V 1out2 (∞) = V 1in2

(5.83)

V 1out1 (∞) = V 1in1

(5.84)

This is the opposite situation to that above. The cavity is completely opaque to noise at high frequencies. The input noise is reflected into the reflected field, and the noise of the unused port is reflected into the output field. This result is true for all cavities, as described by Eqs. (5.82) and (5.81). For high frequencies ((2πΩ)2 ≫ 𝜅 2 ), the noise from the input source will not be able to propagate through the cavity. Only low frequency noise, within the linewidth of the cavity, is transmitted. The cavity acts like an integrator and smoothes out fast fluctuations. However, these fluctuations are not lost; they appear in the reflected light. For low frequencies, (2πΩ)2 ≪ 𝜅 2 , the noise is transmitted. For frequencies around (2πΩ) = 𝜅, there is a smooth transition. This response is shown in Figure 5.14 for the two cases of a noisy input (V 1in1 = 2) and a squeezed input (V 1in1 = 0.1). 2 Vrefl = Vout2

Vin = 2

1.5 Vtrans = Vout1 V

Figure 5.14 Noise response of a cavity as a function of the detection frequency. Both the transmitted and the reflected noise are shown. Two cases: (i) a cavity with noisy input (V1in1 = 2) and (ii) a cavity with an input noise below the QNL (V1in1 = 0.1).

1 Vtrans = Vout1 Vin = 0.1

0.5 Vrefl = Vout2

0 0

2

4

Ω/∆ν

6

8

179

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5 Basic Optical Components

In practice, it means that for a laser beam with modulation at Ωmod incident on a cavity, the transmission function for the laser carrier frequency 𝜈L and for the modulation sidebands is not the same (compare Eqs. (5.61) and (5.82)). In particular, for high frequencies the modulation sidebands cannot be transmitted; they are reflected. A cavity can be used as a filter for noise. The cavity is placed after the output coupler of the laser, all high frequency classical noise is rejected, and the output beam is QNL at these frequencies. Please note that all of the above equations have treated the resonant case. The transfer functions for the phase quadrature, V 2, or indeed any arbitrary quadrature angle, are equivalent to those we have calculated for the amplitude quadrature, V 1, for this case. It is straightforward to include detunings from cavity resonance, but this generates significantly more complex equations. The main difference is that in the detuned case both quadrature amplitudes, X1 and X2, have to be taken into account. A detuned cavity couples the fluctuations of the two quadratures – it links amplitude and phase fluctuations. This effect can be exploited for very sensitive measurements of phase fluctuations, for cavity locking (see Section 8.4), and for the rotation of the squeezing axis (see Section 9.3).

5.3.8

Single Photons Through a Cavity

We now look at the effect of the cavity on a single-photon field. As for previous elements we will use the techniques of Section 4.4 to generate the state evolution. However, in contrast to the beamsplitter and interferometer, here we need to explicitly work in the frequency picture. Let us consider the case of a resonant, lossless, two-sided cavity with the input and output mirrors of equal reflectivity. Using Eqs. (5.74), (5.75), and (5.76) with 𝜅1 = 𝜅2 = 𝜅∕2 and 𝜅l , Δ = 0, we obtain the following operator evolutions through the cavity in frequency space: ̃ ̃ ̃ out2 = 𝜅 Ain1 + i𝜔 Ain2 A 𝜅 − i𝜔

(5.85)

̃ ̃ ̃ out1 = 𝜅 Ain2 + i𝜔 Ain1 A 𝜅 − i𝜔

(5.86)

and

where 𝜔 = 𝜔o − 𝜔cav is the difference between the optical frequency of the mode and the cavity resonant frequency in radians per second. Let us consider a single-photon input at the first input port and vacuum at the other. We can write the input state as ∞

|1⟩f |0⟩ =

∫−∞

̃ † |0⟩in1 |0⟩in2 d𝜔f (𝜔)A in1,𝜔

̃ † is the photon creation operator at frequency 𝜔 in the input mode to where A in1,𝜔 ̃ † |0⟩in1 = |1⟩in1,𝜔 . Also |f (𝜔)|2 d𝜔 is the probability to the first port, such that A in1,𝜔 find the photon in the small frequency interval d𝜔 around 𝜔.

5.3 Optical Cavities

Using the inverted forms of Eqs. (5.85) and (5.86), we can evolve the state through the cavity giving ̃ U(𝜔)|1⟩ f |0⟩ =

∞

∫−∞

d𝜔f (𝜔)

̃† ̃ † + i𝜔 A 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.87)

As expected, the effect of the cavity in frequency space is to act like a frequency-dependent beamsplitter, transmitting the low frequency components of the single-photon state but reflecting the high frequency components. For example suppose the input photon has a very narrow frequency range centred at 𝜔 = 0. We can write approximately f (𝜔) = 𝛿(𝜔), and so the input state is just |1⟩in1,0 |0⟩in2 , whilst the output state is |0⟩out1 |1⟩out2,0 , that is, the photon is definitely transmitted through the cavity. On the other hand if the photon has a narrow frequency range centred on some frequency 𝜔 ≫ 𝜅, then the output state will be |1⟩out1,𝜔 |0⟩out2 , that is, the photon will be completely reflected by the cavity. For an input state with a very broad frequency distribution, which we can approximate by supposing f (𝜔) = K, where K is a constant, the output state will be in a superposition of the two output modes: ̃ U(𝜔)|1⟩ f |0⟩ = K

∞

∫−∞

d𝜔

̃ † + i𝜔 A ̃† 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.88)

If we consider the reflected field to be ‘lost’, then the transmitted field will be a mixed state of the vacuum and a single photon state with the Lorentzian frequency distribution 𝜅2 (5.89) + 𝜔2 In this way cavities select specific components of the radiation field, even in the limit of single photons. 𝜅2

5.3.9

Multimode Cavities

In our quantum treatment, so far we have focused on a single high finesse cavity mode. This occurred because we made several approximations that assumed high reflectivity mirrors and low round-trip loss and that we were close to a single cavity resonance, to obtain Eq. (5.71), before applying canonical quantization to obtain Eq. (5.74). The advantage of doing this was it gave a simple equation of motion, useful in many situations, which corresponds to the quantum Langevin equation one obtains from a quantum Hamiltonian approach. However, it is also possible to simply quantize the exact expression (Eq. (5.69)) in order to obtain the more general quantum expression √ √ √ ̂ c (t + 𝜏) = A ̂ c (t)ei𝜙 1 − T1 1 − T2 1 − Tl A √ √ √ ̂ in1 (t)ei𝜙 T1 1 − T2 + A ̂ in2 (t)ei𝜙 T2 +A (5.90) In principle this expression can be used to describe multiple resonances of a cavity of arbitrary finesse. Here we will still consider a high finesse cavity but

181

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retain the multimode feature. Taking T1 , T2 ≪ 1 and using the definitions in Section 5.3.5, we obtain √ √ √ ̂ in1 (t) + 2𝜅2 A ̂ in2 (t) + 2𝜅l A ̂ l (t)) â (t + 𝜏) = ei𝜙 (𝜅 â (t) + 2𝜅1 A (5.91) i𝜏𝜔 ̃ Using the Fourier transform relationship  â (t + 𝜏) = a(𝜔)e and the input–output relations (Eqs. (5.75) and (5.76)), we can solve for the output field in Fourier space to obtain √ √ ̃ in2 + 4𝜅1 𝜅l A ̃ in1 + (2𝜅2 − 𝜅 + Γ(𝜔)) A ̃l 4𝜅1 𝜅2 A ̃ out2 = (5.92) A 𝜅 − Γ(𝜔)

where 1 − ei(𝜔−Δ)𝜏 (5.93) 𝜏 This now describes a multimode cavity in which cavity resonances explicitly occur whenever (𝜔 − Δ)𝜏 = 2πn (with n as an integer). Γ(𝜔) =

5.3.10

Engineering Beamsplitters, Interferometers, and Resonators

Whilst in Section 5.1 we focussed on one specific type of beamsplitter, the well-known device made of a plain piece of glass with specifically prepared surface, many other devices have the same effect. Some of them transfer the power into one or several specific other modes; this could, for example, be done by a grating that reflects into different orders. Or we can use surface or evanescent coupling when two modes approach each other by bringing two surfaces close to each other, or we can achieve this inside an optical fibre coupler. We can set up periodic nanostructured materials that couple different propagating modes. The list includes various forms of active modulators that produce gratings or other spatial modes that couple into other spatial modes and those that create different frequency components. We can combine several beamsplitters into interferometers and cavities and resonators. This creates many technological options for coherent linear devices. This is illustrated in Figure 5.15, which shows the diverse range of optical resonators that are used today. All are using different approaches and techniques, ranging from simple optical cavities made out of two mirrors to multi-mirror ring cavities, fibre couplers, and fibre rings, evanescent waves in discs, toroids and spheres, Bragg reflectors, planar optical devices, and photonics crystals. The first generation of devices shown in the top row of Figure 5.15 was designed as resonators for lasers. One of the main design criteria is to make the resonator stable, which means a large build-up after many round trips is possible and minor mechanical perturbations have only a small effect. A second design goal is to minimize losses such that the components can handle high internal fields. All of these resonators have several spatial eigenmodes, which are the different stable solutions for the resonance. For example a long linear resonator can support the family of Hermite–Gaussian spatial modes, shown in Chapter 2 in Figure 2.2, and these modes can be generated in a laser. For some of these configurations, in particular the case of the confocal resonator, the spatial modes

Planar mirror

Spherical mirror

Microtoroid

Figure 5.15 Various optical resonator designs.

DBR

Microdisk

Microsphere

Rectangular cavity

Integrated optic ring

DBR

Fibre ring

Ring mirror

Micropillar

Photonic crystal

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5 Basic Optical Components

exist at the same resonance frequency; they are degenerate and will appear at the same time. However, for most geometries the spatial modes have different optical resonance frequencies, the modes are non-degenerate, and the laser either operates on several modes with different frequencies or in one spatial mode at a time. There is a considerable knowledge about the engineering of such resonators to make them insensitive against mechanical vibration and also to make their resonance frequencies tunable. A good starting point for the design techniques can be found in the widely used book by A.E. Siegman [9]. The lower part of the diagram shows a variety of resonators that have emerged in the last few years and have been optimized to create resonators that are stable, have low losses, and more recently are designed to be very small and mass produced as building blocks for complex systems. Each technology has its own design limits, but in general they are improving quite dramatically to achieve almost perfect resonators. The ever-increasing quality of these devices is well described in a series of review articles and textbooks [17].

5.4 Other Optical Components 5.4.1

Lenses

We require optical components to change the spatial properties of a laser beam or mode of light. Actually, in most of optics imaging is the main concern. The ability of lenses and mirrors to form and reproduce images is the reason for using these components. The effect of lenses is a change of the wave front curvature and consequently a transformation from one Gaussian input beam to another Gaussian beam with different waist position and size. In practice, lenses are used to transform the spatial mode coming from the laser into the mode required in the experiment, to match it to any cavity that exists in the experiment, and finally, to focus the mode onto the detector (Figure 5.16). The quality of a lens or mirror is determined by the shape of the lens, its surface finish, and the uniformity of the material. Most imaging lenses are multi-element lenses, and the details of the lens design are of crucial importance to the quality of the image, the spatial resolution that can be achieved. For practical details consult one of the handbooks on optical instruments [18]. For most cases the paraxial transformation will hold, and we can use the results discussed in Section 2.1. It

Lens Gaussian beam

Figure 5.16 Spatial mode transformation by a lens.

Gaussian beam

5.4 Other Optical Components

assumes that the input is in a single spatial mode and that the lens is well represented by a parabolic phase term. Consequently, the output will be in a single spatial mode. Imperfections of the lens will result in deviations from a Gaussian beam, and aspherical aberrations are fairly common. In this case the light has no longer a parabolic wave front; it is distributed over a variety of spatial modes. This will make any interference experiment, including homodyne detection, very difficult. It will reduce the visibility of the interference fringes and therefore make it an inefficient process. The loss of light due to reflection on the lens surfaces is a major experimental problem. As an example, for glass lenses, n = 1.5, the intensity reflection per surface is typically 4% for every surface. This can quickly add up in a whole experiment. Thus most surfaces have an anti-reflection coating that can reduce the reflection to less than 0.5%. Scattering losses inside the components are usually even smaller than 0.1% and will only be of concern if the optical component is placed inside a cavity or is extremely sensitive to scattered light. Whilst losses are one concern in quantum experiments, the other is the need to maintain one single mode. Assume we have a mode of light with an interesting intensity noise statistics that is sub-Poissonian. It passes a lens with distortions and turns into a non-Gaussian mode. If we detect all of this light, we still find a sub-Poissonian result. If, however, the detector is sensitive to only the Gaussian mode, because it contains a cavity or it is a homodyne detector that requires the interference and beating with an optical local oscillator, the results will be different. If the partition between the modes is random, then the other modes will appear like a power loss, and the statistics will be closer to Poissonian. If the partition is somehow deterministic, then additional noise could be introduced. Fortunately all passive components, such as mirrors and lenses, result in a random partition. The other modes can be treated as loss terms. In contrast, active systems such as lasers are different and can introduce additional partition noise, as shown in Chapter 6. A problem that exists with high intensity experiments, and in particular pulsed experiments, is thermal lensing. Glasses will absorb some small fraction (typically 0.1 [%/cm]) of the light, and the absorbed power can lead to local heating of the glass that results in local changes of the refractive index. At such high intensities, light becomes non-linear. In this case self-focussing can occur since the intensity-dependent refractive index effectively forms the profile equivalent to a lens, formed by the beam itself. Self-focussing leads to high higher intensities at the core of the beam increasing the effect event more. This results in positive feedback, which can have disastrous effects. Many lenses have been destroyed with holes and channels burned into the glass. With clever design, and the right materials and pulse length, this can be used to create a localized bubble in the glass, which can be seen from the outside and can serve as on voxel for a 3D structure, very popular with artists and tourists. 5.4.2

Holograms and Metasurfaces

The concept of using structured surfaces to alter the spatial wave front and to manipulate the shape of beams of light has a very long tradition. Fresnel lenses,

185

186

5 Basic Optical Components

which use circular phase shifts or even absorption, were a very early invention that could take on the task handled by lenses. Early versions, such as those used in light houses, had structures that were much larger than the wavelength of light, frequently several millimetres, and can be understood in the same way as lenses. Later it was possible to make small structures using photographic processes, and the understanding of coherent light with known phase distributions allowed the concept of holograms by D. Gabor [19] even lasers were available and in many versions since. Holograms can be used to record and play back complex waveforms that result, for example, in the reconstruction of 3D images. Fourier optics, which means the spatial frequencies of the light field, created a whole new field of understanding imaging systems, designing instruments, and shaping light. Holograms have been refined, and it is not possible to create computergenerated hologram using the high-resolution spatial light modulators (SLM) available today. That means we can now design optical components that can be manipulated from the outside. This allows the shaping of light for many purposes, such as displays, microscopy, optical traps, and optical tweezers for atoms and particles. Practical examples are devices for the investigation of biological processes or the functional imaging of brain cells. The external control also allows the use of the modulators, and phase structures inside feedback loops, which allows for intelligent alignment and imaging processes. Recent development in the design and manufacture of structured surfaces allow the direct control of the wave fronts by very thin components. New nanoscale building techniques can create the required three dimensional structures and have resulted in a completely new class of materials and of passive and active components for the manipulation of light. The properties of metamaterials were explored in 1990s, first as a curiosity, and initially tested at longer wavelengths, such as microwaves. Now we have optically thin structured materials made from sub-wavelength nanostructures, sometimes referred to as meta-atoms, which allow us to influence the properties of the light interacting with them. Metasurfaces enable the manipulation of the wave front at the surface. They can be used as optical elements for beam shaping and holography. At the same time, metasurfaces enable the control of light in the intermediate and near-field zones. By tuning the phase properties of waves in the intermediate regime, also known as the Fresnel zone, it is possible, for example, to focus beams or engineer-desired point spread functions. At shorter distances, and straight after the surface where the behaviour of light is governed by its near-field characteristics, carefully crafted components allow the concentration of the local intensity as required for the non-linear generation of new frequencies, surface-enhanced Raman scattering, or enhancement of the interaction with atoms via the Purcell effect. For an overview of the design and use of metasurfaces, see, for example, S. Keren-Zu et al. [20]. As long as the metasurfaces, and thicker strictures made of metamaterials, behave in a linear way, they can be seen as the equivalent of the bulkier conventional optical components, such as lenses, interferometers, and optical cavities. They will not influence the photon statistics or add noise. Increasingly these will be the technique of choice for designing optical instruments, for focussing, and for the generation of multiple spatial modes.

5.4 Other Optical Components

5.4.3

Crystals and Polarizers

Glass and crystals vary in two very important points: crystals are non-isotropic, and they can show an electro-optics response and they can show non-linear effects. The anisotropy of the crystals results in properties such as birefringence, optical activity, and dichroism. The details obviously vary with different types of crystals and their composition and geometrical structure. It also varies with the alignment of the light beam with the crystal axis. All these properties can be used to control the polarization properties of the light. The idea here is to introduce a relative phase change between two different polarization components of the field. For example, introduce a π∕2 phase change of one linear polarization and not the other, and thereby turn linearly polarized light, aligned with the axis of the crystal, into circularly polarized light. This would be called a quarter wave plate. The results of any such component can be analysed using the rules of classical optics, in particular the formalism of Jones matrixes [3]. Birefringent crystals can be manufactured into wave plates and polarizers. In this regard quantum optics experiments are no different to classical optics experiments. All tricks available to control and influence the polarization will be used in the various experiments. One particularly useful component is the optical isolator. This device is equivalent to a diode in electronics; it provides a different effect in the forward and backward direction. The light propagating in the reverse direction is redirected and does not go back to the source. This avoids interference between the incoming beam and the retro-reflected light. This effect is described as optical feedback that can lead to very noisy or even unstable performance of a laser. At the same time an optical isolator will give access to the retro-reflected beam, which is important in experiments where all of the light has to be detected. It is also useful for situations where locking of a cavity to a laser is important. In this case the redirected light contains the information about the cavity detuning. Two types of optical isolators are commonly used: the Faraday isolator and wave plates (see Figure 5.17). The Faraday isolator uses a material, usually a special glass or crystal, with a high Verdet constant. This means a large rotation of the plane of polarization for linearly polarized light can be induced by a static magnetic field along

Laser

PBS

B Mirror Faraday isolator

(a)

Laser

PBS

4 Mirror Quarter wave plate

(b)

Figure 5.17 Optical isolators using either a Faraday rotator (a) or a quarter wave plate (b).

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the crystal. The isolator is tuned to the wavelength used in the particular experiment by adjusting the magnetic field strength to give a 45∘ rotation. The return beam is rotated again by 45∘ in the same direction, which means both beams are linearly polarized at 90∘ to each other and can be separated by a PBS. The second design uses wave plates (see Figure 5.17b). The source generates linearly polarized light. A quarter wave plate is used to turn this into left-hand circularly polarized light (𝜎+ light). Upon reflection in the experiment, the returning light is right-hand circularly polarized (𝜎− light), and the wave plate turns this into a counter-propagating beam with orthogonal linear polarization. This light can be separated with a PBS. Crystals can also contribute entirely new effects. They can have a non-linear response that allows us to generate new frequencies, such as twice the frequency (SHG), half the frequency (optical parametric gain), or sums and different frequencies (four-wave mixing (4WM)). Crystals can show a Kerr effect, which means an intensity-dependent refractive index, or non-linear absorption. These effects will influence not only the intensities but also the fluctuations of the light, and consequently they are discussed in Chapter 9 on squeezing experiments. 5.4.4

Optical Fibres and Waveguides

Optical fibres are a convenient component to deliver light from one place to another. The idea is simple: the fibres have a refractive index structure that guides the light. For sufficiently small core sizes, a few times the optical wavelength, the field inside the fibre has one single mode with low loss, and all other modes are highly attenuated over length. In addition, larger diameter fibres show many modes. The mode properties of fibres are complex and are discussed in great detail in the specialized literature [21]. It is possible to design a lens that will couple very efficiently a free-space Gaussian laser beam into a single-mode fibre. This requires a lens of short focal length, such as a microscope objective, which has a numerical aperture matched to the fibre geometry. When the beam parameters are correctly chosen, at least 80% of the power can be coupled into the fibre. The losses along the fibre itself are so exceedingly small that the fibre length is normally of no consequence. Fibres can be spliced together with low losses, and efficiencies of 95% are feasible. In this way whole fibre networks, including couplers, can be built. In recent years it has also been possible to create other optical waveguides, frequently through direct writing optical circuits inside blocks of glass and other transmitting materials. Materials that are structured on nanometre scale have been used as so-called photonics crystals that are used as waveguides and optical circuits. All of this has extended the options for guiding and distributing light tremendously. However one should note that in communication applications an overall loss of 3 dB or 50% is fully acceptable. This is also the case for some single-photon experiments, where the overall quantum efficiency may not be critical. However, in CW applications these devices are not yet suitable since the high losses reduce the precious quantum properties. It should be noted that simple symmetric fibres completely scramble the polarization of the light. Any polarization dependence of the optics after the fibre

5.4 Other Optical Components

can lead to enormous intensity fluctuations introduced by the fibre. There are special non-symmetric polarization-preserving fibres, which have to be carefully aligned. And even these are very sensitive to mechanical movements of the fibre. The output mode shape from a single-mode fibre is very close to a Gaussian beam. As a result fibres can be used as spatial mode cleaners, and they will transform an input beam with a complex wave front into a very nice Gaussian beam. The light that is not matched will be lost by scattering. The higher losses of a fibre mode cleaner are frequently compensated by its simplicity and the fact that, unlike a mode cleaner cavity, it requires no active locking. Modern technology now allows the writing of waveguides inside solid materials, and the advantages of fibres can be extended to more complex structures that can contain splitters and combiners and other components all integrated into one chip. One of the main attractions is the high internal intensity inside the fibre or waveguide, which make it very attractive for non-linear interactions. Some of the first and some of the best squeezing experiments have been carried out in fibres (see Section 9.4.4). Apart from producing quantum effects, the non-linearities in waveguides can be exploited for future active control and the operation of quantum logic operations, all firmly integrated into solid components [22].

5.4.5

Modulators

Modulators are an essential component of most electro-optical systems. They allow us to encode information onto laser beams. The most common application is to modulate a single beam, which means one single optical mode, without noticeably changing the mode but simply adding frequency sidebands that contain the information. The classical description of modulation (see Section 2.1.7) can be transferred into the quantum mechanical description (see Section 4.5.5) by introducing a time-dependent component to the model of the beamsplitter. The properties of the classical transfer equation remain the same. However, a quantum mechanical vacuum operator appears that represents the coupling of vacuum fluctuations back into the beam. This term is responsible for the special quantum noise properties that are so important in any system operating close to the QNL. For example an intensity modulator is essentially a time-dependent beamsplitter, where an external modulation function controls the strength of the beam splitter. In general a time-dependent beamsplitter that mixes two input modes ̂ transforms them such that ̂ and b(t) represented by the operators a(t) √ √ ̂ â o (t) = 𝜂(t)̂a(t) + 1 − 𝜂(t)b(t) √ √ ̂ − 1 − 𝜂(t)̂a(t) (5.94) b̂ o (t) = 𝜂(t)b(t) where 𝜂(t) is the time-dependent transmitivity of the beamsplitter. Transforming into the frequency domain, we obtain ̃ ̃ o (Ω) = 𝜈(Ω) ̃ ̃ ∗ A(Ω) + 𝜇(Ω) ̃ ∗ B(Ω) A ̃ ̃ ̃ ∗ B(Ω) − 𝜇(Ω) ̃ ∗ A(Ω) B̃ o (Ω) = 𝜈(Ω)

(5.95)

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√ where 𝜈(Ω) ̃ is the Fourier transform of 𝜂(t) and 𝜇(Ω) ̃ is the Fourier transform √ of 1 − 𝜂(t). The ∗ symbol indicates a convolution, that is, M(Ω) ∗ L(Ω) =

∫

dΩ′ M(Ω′ )L(Ω − Ω′ )

(5.96)

A good model for the modulator is a beamsplitter that takes off a small fraction of the light 𝜖o and then adds an even smaller modulation term with amplitude 𝜖(t). Hence we can write 𝜂(t) = 1 − 𝜖o + 𝜖(t) cos(2πΩmod t)

(5.97)

where Ωmod is the modulation frequency and 𝜖o > 𝜖(t). If we assume the modulation amplitude is very small such that 1 − 𝜖o > 𝜖o ≫ 𝜖(t), then we can make the approximations √ √ 𝜖(t) 𝜂 ≈ 1 − 𝜖o + √ cos(2πΩmod t) 2 1 − 𝜖o √ √ 𝜖(t) 1 − 𝜂 ≈ 𝜖o − √ cos(2πΩmod t) (5.98) 2 𝜖o and hence we can write √ 1 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) 𝜈(Ω) ̃ = 1 − 𝜖o 𝛿(Ω) + √ 4 1 − 𝜖o √ 1 𝜇(Ω) ̃ = 𝜖o 𝛿(Ω) + √ 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) (5.99) 4 𝜖o where 𝜖(Ω) ̃ is the Fourier transform of 𝜖(t). Let us assume that the modulation is turned on at some time −T∕2 and turned off at T∕2 and has the value 𝜖 when on. Hence 𝜖(Ω) ̃ = 𝜖T sinc(πΩT) where sinc(x) = sin(x)∕x. Further we assume that ̃ ̃ the {b(t) mode is initially in the vacuum state and write B(Ω) ≡ V(Ω). As a result, in the expectation values, non-zero modulated terms coming from the vacuum input will only appear to second order or higher in 𝜖 and so can be neglected. Thus, inserting the convolutions, we obtain the following expression for the output mode from the intensity modulator: √ √ ̃ o (Ω) = 1 − 𝜖o A(Ω) ̃ ̃ A + 𝜖o V(Ω) 𝜖T ̃ ′ + Ωmod ) + A(Ω ̃ ′ − Ωmod ))sinc(π(Ω − Ω′ )T) dΩ′ (A(Ω + √ 4 1 − 𝜖o ∫ (5.100) In the limit that the modulation is on for a long time (T ≫ Δtmod ), the sinc function approaches a 𝛿 function, and the sidebands become very sharp (as assumed in Section 4.5.5). In contrast, if the modulation is only on for a short time, as a burst, the sidebands are spread out over a range of frequencies around the sideband frequencies. Similarly the information about a slowly varying time evolution of the modulation will be contained in the spectral components close to the pair of sidebands.

5.4 Other Optical Components

5.4.5.1

Phase and Amplitude Modulators

1

1

FM

n

atio

cal

ti Ver (a)

riz pola

Fast axis

Modulators normally operate across a frequency bandwidth: some are broadband, and others are more resonant. By feeding a signal at a frequency Ωmod into the modulator, we create the specific sidebands, which contain the information we want to impose. Within the bandwidth complete modulation spectra can be used, for example, in communication systems. The modulation can be either a phase or an amplitude modulation, or even a combination of both, depending on the design of the modulator. An intensity modulator controls only the magnitude of the transmitted beam and should leave all other properties unchanged. A phase modulator introduces a time-dependent phase shift and should leave the overall power constant. However, we should note that many instruments, for example, interferometers, convert phase modulation into amplitude modulation, and this can make it difficult to build a pure phase modulator. Reducing crosstalk between amplitude and phase modulation is a technical challenge (Figure 5.18). Many crystals have significant electro-optic coefficients. That means the refractive index can be manipulated with an external electric field. The direction of the electric field that affects the refractive index along a particular optical axis depends on the type of crystal used. In some cases the transverse effect is large, which means the electric field is applied orthogonal to the optical beam. In other crystals a longitudinal field, parallel to the optical beam, is required. The refractive index will follow the field up to very high frequencies; more than 20 GHz are possible. Thus these devices can be used as EOM. Large changes in refractive index will require electric fields of the order of 100–1000 V/cm. The limit in speed is normally set by the electric power requirements. Fast response and large degrees of modulation can be achieved by using resonant electronic circuits that provide a large electric field in a very narrow frequency range. This effect can be used to make a phase modulator. Either a crystal with isotropic response is chosen, or the axis of linear polarization is carefully aligned

ted dula mo ization s t ron polar se f n Pha ange i PBS ch o N

n

atio

ariz

du mo

m n A latio u d mo

tion

riza

ola ar p

l

u Circ (b)

n

latio

de

Pol

axis

1

plitu

1

AM

Slow

Figure 5.18 Electro-optical modulators for phase or amplitude.

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5 Basic Optical Components

parallel to the crystal axis. For a single modulation frequency Ωmod , the output light will get one set of sidebands, Ωmod and its higher harmonics, as described in Eq. (2.21). In the ideal case phase-modulated light does not produce any modulation of the photocurrent. In practice, considerable care has to be taken to avoid amplitude modulation. FM sidebands are frequently used to probe the detuning of cavities and to lock lasers to cavities. Alternatively, a birefringent crystal can be used and aligned such that the linear input polarization is at 45∘ to the optical axis of the crystal. In this case the two orthogonal components of the linear polarization are affected differently. The voltage induces a difference in the change of the refractive index, and the modulator is acting like the AC equivalent of a wave plate. The polarization is rotated synchronously with the applied electric field. After transmission through a polarizer, the light has a mixture of amplitude modulation, from the polarization rotation, and phase modulation. This is a common configuration for intensity modulation. Normally a small reduction in intensity is chosen, and the intensity can be increased and decreased around this offset point, as per the example in Eq. (5.97). An entirely different effect is used in acousto-optical modulators (AOM). These utilize the fact that an acoustic standing wave can be generated inside crystals through an RF electric field. Light transmitted through such a standing wave sees a periodic density grating and is diffracted. The diffracted beam is shifted in frequency by ΩRF and is also shifted in its output direction, proportional to ΩRF (Figure 5.19). Since the diffraction angle is linked to the modulation frequency, we can redirect a beam by simply changing the frequency ΩRF of the RF driving voltage, which is typically in the range of 10–1000 MHz as a function of time. We can change the intensity of the diffracted beam by changing the RF power. Since both RF and power can be controlled rapidly, the output beam is thus scanned in one dimension. One common application of an AOM is as fast scanners, such as laser printers and displays. The AOM can be used as an intensity modulator by keeping the RF fixed and modulating the RF intensity at frequencies Ωmod ≪ ΩRF . In quantum optics experiments the main application is the shift in optical frequency. For an ideal AOM the diffracted beam has the same mode properties as the input beam, apart from a change in frequency and direction. All the fluctuations of the light are simply transmitted, and when carefully aligned there should be no additional noise. There are many applications, such as tuning to Crystal with internal acoustic standing wave vL

Figure 5.19 Acousto-optical modulator. vL + Ωmod vL

~

Ωmod

Modulation frequency

5.4 Other Optical Components

an atomic or cavity resonance or the probing of atoms with a frequency offset. AO modulators can also be used as isolators. The return beam from the experiment, for example, a cavity, will have twice the frequency shift 2ΩRF imposed by the modulator and is less likely to interfere with the laser. This retroflection trick can also be used to produce a beam with high frequency modulation and no directional shift and intensity modulation. 5.4.6

Spatial Light Modulators

SLMs allow the modification and control of the optical wave front through many pixels by electronic means. They are a fairly new technology that is steadily improving in quality and are now almost omnipresent through their applications in laser scanners, data projectors, and adaptive optics. They are part of much of the optics that is being developed. SLMs allow completely new applications, such as laser movie projection [23], in the form of spatial intensity modulators, or they can modify the shape of the optical wave front in the form of spatial phase modulators, for example, to produce dynamically controlled optical tweezers and traps [24] or neurostimulators [25]. At the same time they can be used to optimize the shape of optical beams, which means they can be used as an alternative to optical phase plates to create different spatial modes, for example, beams with orbital angular momentum, or they can dynamically improve a wave front that has been distorted by the atmosphere in the case of astronomy or by propagating through biological samples or tissue in the case of medical applications. In quantum optics laboratories SLMs are used to optimize mode matching and have been used for quantum logic and entanglement measurements. The main advantage of the SLM is the high speed and the ability to control through suitable software very complex wave fronts and intensity distributions. The commonly quoted characteristics are the number of pixels controlled, now up to several megapixels; the contrast ratio, which is important for intensity modulators and can now reach up to 100 000 or higher; and the dynamic range, in multiples of full wavelength in the case of phase modulators. The other factor is the speed, which is linked to the ability to switch one pixel, typically on the timescale of 1–100 μs. The second aspect of speed is the ability to control so many pixels simultaneously, which means transmission rates of 10 Mbit/s to 1 Gbit/s and is rapidly progressing through the use of dedicated graphics cards and optimized software. Soon speed and resolution will not be a limiting factor in the applications. There are several different technologies: the SLMs can be used in transmission and in reflection. The transmission mode with very small mirrors, known as digital light projection (DLP) is widely used in display technology such as video and cinema projectors. However, it does not have the characteristics required in quantum optics due to the considerable losses. More useful are phase modulators that work either in transmission mode or deformable mirrors that operate in reflective mode. The liquid crystal display (LCD)-type phase modulator is the most common; it combines large pixel numbers with good access speed. Such a phase modulator can be effectively programmed as a hologram and can be used

193

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5 Basic Optical Components

as variable lens, changing the focal length, compensating for beam distortions, or transforming the mode, for example, from one TEM mode to another TEM mode or from one Gauss Leguerre LG mode to another LG mode, or to create completely different type of modes, for example, special modes for optical tweezers. This can be done with reasonable efficiency, and in particular when used as a grating, where the transmission into first order or higher order is used, the contrast can be very high. For high efficiency applications, transformable mirrors perform far better. In particular those mirrors that have a continuous mirror surface can reach efficiencies above 90%. These SLMs have a low number of pixels. For reasonable cost one can get about 1000 pixels, which is sufficient for a number of mode matching and entanglement applications. The question arises: do SLMs add technical noise that will reduce their usefulness in quantum optics? The LCD-type modulators have considerable phase jitter that is commonly very noticeable due to the strong phase modulations created by the electronic drivers. In many normal display applications, the image on the SLM is refreshed frequently, for example, every millisecond, which means a rapid reorientation of the liquid crystals at that rate. The crystals then tend to oscillate around their equilibrium position, and this produces a strong driving of the phase setting for each pixel, leading to a large phase noise. This effect can be avoided either by using very different electronic drivers that have a very fast refresh rate that cuts the oscillation out of the frequency spectrum that is of interest or by using the SLM as a grating. The phase noise will be visible in the zero order beam. However, by observing the first-order transmission or reflection, the phase noise will appear as translation noise only and thus not be detectable with the normal optical detectors. This is the reason why transmission SLMs have been used so successfully in combination with single-photon detection where optical losses are not so important [26]. For CW applications with squeezed light, the deformable mirror appears to be promising. A systematic test has been performed to show that no noticeable phase noise is present. This experiment shows that the mirror can be used to transform the mode shape of a beam of squeezed light with an efficiency of larger than 95% and that no detectable noise is present that would add to the quantum noise and destroy the quantum correlations [27]. That means no fundamental limits have to be considered – the right type of SLM will exist for every experiment. The future generation of SLMs will have further improved specifications, and this opens up the field of quantum optics experiments to electronic mode matching that enhances the reliability of experiment. The controller of the SLM can use learning algorithms to iterate the many pixel settings with the single goal to optimize a chosen global parameter. An example would be to optimize fringe visibility between two interfering beams to get the best possible mode matching. In this way the combination of SLM and intelligent multidimensional controller will tune itself, and it can be an alternative to the manual alignment process performed now. Future students will have at least one SLM in each experiment and will not need to worry about frequent mirror realignment. The SLMs will also allow more complex experiments with multiple beams. They will allow the design

5.4 Other Optical Components

of reconfigurable layouts, for example, for quantum logic, and alternative modulation techniques for quantum communication. 5.4.7

Optical Noise Sources

Optical materials can contribute noise. This is equivalent to saying that additional light can be scattered into the mode under investigation. One example is atoms that are excited by one of the laser beams in the experiment and that spontaneously emit at the frequency and into the spatial mode that is detected. In a similar way stimulated processes can add light to the mode. These could be Raman transitions or other non-linear processes, such as 4WM or SHG. We have to be very careful in experiments that contain strong optical fields; basically any process that can generate light at the frequency of the mode we wish to detect is potentially a source of competing noise that could swamp an effect as small as quantum noise. These types of noise add to the intensity fluctuations in the mode. In addition, the refractive index of the optical material can be affected by fluctuations that will generate noise in the phase quadrature of the transmitted light. For example, a common source of noise in optical fibres is guided acoustic wave Brillouin scattering, abbreviated as GAWBS. This noise occurs in any system where the wave is guided by geometrical structures such as a fibre or planar waveguide. The noise spectrum shows very pronounced maxima, which can be related to the resonances of acoustic waves that are reflected by the boundaries of the fibre. It covers such a wide spectrum that the detection of QNL light becomes very difficult. Non-linear crystals can be the source of additional noise [29]. In all experiments a careful characterization of the noise spectrum has to be carried out, before we can assume that quantum noise is actually observed. Chapter 8 describes these techniques in detail. 5.4.8

Non-linear Processes

In summary, all linear components have a very simple effect on the fluctuations: a coherent state of light will remain a coherent state. Even for light that is not a coherent state, the effects are simple; we will find that any losses make the fluctuations look more like those of a coherent state. In the limit of large losses, any input light will approach a coherent state (Figure 5.20). Quantum optics would not be much fun if there were only linear processes, but fortunately there is more. Gain combined with saturation allows us to make a laser, the all-important light source (see Chapter 6). One of the main features of non-linear processes is that they can generate additional new frequencies. In particular, SHG and degenerate optical parametric oscillation (OPO) are used to generate beams at twice or half the original frequency. The non-linear Kerr effect transfers changes in the intensity to the refractive index. 4WM links modulation sidebands. The non-linear processes are too numerous to list here explicitly, and instead the extensive literature should be consulted, for example, the textbooks by M.D. Levenson and S.S. Kano [30], and R.W. Boyd [31] as the current reference. For a recent review of non-linear fibre effect, see the review by

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5 Basic Optical Components

40 Noise power (rel. units)

196

Figure 5.20 An example of optical noise. The noise introduced in a single-mode optical fibre by guided acoustic wave Brillouin scattering (GAWBS) measured with a CW laser. Source: Courtesy of Shelby and Levenson [28].

Hitachi fibre GAWBS

20 10 8 4 2

Quantum noise

1 0

500 MHz

1000

S.M. Hendrickson et al. [22], and for non-linear metamaterials, see the colloquium review by M. Lapine and the team lead by Y.S. Kivshar and coworkers [32]. These non-linear effects allow us to change the wave front of the light, leading to self-focussing or defocussing; they can also turn a single spatial mode, like a TEM modes, into a high-order mode or generate even multimode situations, all as a consequence of the interplay between high local intensities and non-linear materials. Important for quantum optics is the fact that non-linear effects can change the modulation and information carried by the beam. And it can change the statistics of the light and can be used to manipulate quantum noise and thus allows to create light for measurements with better SNR than allowed by the QNL. Non-linear media are essential for the generation of photon pair, sub-Poissonian light, and squeezed and entangled light, and such experiments will be discussed, one by one, in Chapter 9.

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metasurfaces. Rev. Mod. Phys. 86: 1093.

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6 Lasers and Amplifiers 6.1 The Laser Concept The laser is the key component in any quantum optics experiment. It was the invention of the laser (see Figure 6.1) that gave us access to the world of coherent optics, non-linear optics, and the quantum features of light. Without the laser we would be restricted to thermal light – missing out on most modern applications and on almost all the experiments described in this guide. In quantum optics we are concerned with the characteristics of the light generated by a laser. In particular, we will investigate the fluctuations in the intensity of the light. In theoretical terms the laser is the system of choice for the realization of many quantum optics ideas: the laser light itself is close to a pure coherent state. In many cases, the laser can be regarded as the source for a single mode of radiation as described in the theory. If we consider a continuous-wave (CW) laser, the mode will have constant properties, and even pulsed lasers can be considered as constant in the sense that we describe them as a steady stream of pulses – each pulse like any other. Many books have been written about the laser while the technology constantly evolves and makes the devices more powerful and more accessible. One of the most comprehensive introductions remains the masterpiece by A.E. Siegman [1] in which he provides all the technical details required to understand and operate lasers. In simple technical terms, the laser is an optical oscillator (see Figure 6.2) that relies on feedback, much like an electronic circuit. It consists of a narrow band optical amplifier, which generates a well-defined output beam of light and an optical arrangement that feeds part of the light, in phase, back to the input. The amplification is provided by a gain medium, atoms or molecules that are excited by an external source of energy, the pump. The realization that such a situation is possible was one of the early results of atomic quantization [2]. The feedback is provided by a cavity surrounding the active medium. The mirrors of the cavity generate a large internal optical field, which is amplified during each round trip. Of course there are also losses inside the cavity, and in order to achieve sufficient feedback for oscillation to occur, the gain has to be larger than the losses. Without any input there would be no output, no optical feedback signal, and no laser beam. However, the presence of even a small amount of light, some small optical amplitude within the bandwidth of the resonant amplifier system, A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Figure 6.1 The original laser demonstrated by Theodore Maiman in 1960. The ruby crystal with coatings is the gain medium, the flash lamp surrounding it the pump source. The pulses wore out the metal coating of this version of the laser. Source: Courtesy Welling (2015).

Pump Output beam

Feedback Output Amplifier Power (a)

Cavity with gain medium

(b)

Figure 6.2 Comparison between an (a) electronic amplifier/oscillator and (b) a laser.

will initiate the process and lead to a rapid growth of the output. Spontaneous emission from atoms in their excited state can provide such a seed. The amplitude increases until the gain of the amplifier saturates at a level that just balances the losses due to the mirrors and other sources. The output power will stabilize at a steady-state value, and the laser achieves constant CW operation. The conditions for lasing are twofold. Firstly, the amplifier gain must be greater than the loss in the feedback system so that a net gain is achieved in each round trip. All lasers show a threshold, which means they require a minimum pump power Pthr to start steady oscillation and generate an output beam. Secondly, the total phase accumulated in a single round trip must be an integer multiple of 2π to provide phase matching to the feedback signal, i.e. the laser frequency must correspond to a cavity mode. Ignoring small frequency pulling effects, which can be somewhat larger in diode lasers, the laser mode(s) will correspond with the cavity mode(s) closest to the peak of the gain profile. In the simplest case this is a single fixed frequency.

6.1 The Laser Concept

Above threshold the laser produces a steady output beam. The geometrical properties of the output beam are those of the spatial modes of the cavity, in the simplest case the fundamental Gaussian TEM00 beam (see Eq. (2.8)). The combination of these two cases is known as single-mode operation, both transversely (spatially) as well as longitudinally (spectrally). This is the ideal case for the study of the quantum properties of light. The intra-cavity field se well described as one quantum mechanical mode of the radiation field that extends throughout the cavity. The output beam can be described by a propagating mode or coherent state as described in Section 4.5. As we will see, the present technology gets us extremely close to the realization of such a perfect laser. 6.1.1

Technical Specifications of a Laser

For CW lasers we can use a wide range of lasing materials, excitation mechanisms, and cavity designs. CW lasers include gas lasers, such as the once widely used He–Ne lasers and the powerful Ar ion and CO2 lasers, to the complex tunable dye lasers. We do not have space here to describe all the technical details and numerous engineering tricks that are used in the design of these lasers. A range of specialized literature is available to provide this information [1, 3]. Nowadays, the majority are the more robust, technically almost simplistic solid-state diode lasers. The trend is to fibre lasers, and various other solid-state systems all pumped by diode lasers [4]. These lasers are starting to dominate the technological applications. In the area of atomic spectroscopy, where tunability and narrow bandwidth is required, diode lasers and Ti/sapphire rapidly replace the traditional dye lasers. For short pulses solid-state lasers have allowed much shorter pulses. For complex experiments and for applications, the compactness and robustness of the diode laser are the main advantage. As a consequence most of the practical examples of laser performance in this book refer to solid-state lasers. The performance of a laser can be described by the following parameters: optical power, spatial mode structure, output wavelength, frequency spectrum, and noise spectrum. The power is quoted as the total optical power Pout of the output beam. Of practical interest is the total efficiency of the laser 𝜂total = Pout ∕Ppump , where Ppump is the pump power required to achieve an output power Pout . Well above threshold the power scales linearly, and the slope efficiency 𝜂slope = dPout ∕dPpump is constant. This is shown in Figure 6.3. Note that operation near threshold is not reliable. At that point the power is not stable; it tends to be noisy, and Pthr may not be easy to measure. The spatial properties of the laser beams are determined by the properties of the optical resonator. In most cases a stable resonator is used, and the mode analysis in terms of Hermite–Gaussian TEMm,n modes, as discussed in Chapter 2, can be used. There are a few exceptional cases, particularly for the design of high-power lasers where more complicated unstable resonators are used. However, many lasers operate in a single spatial mode, the lowest order TEM0,0 mode, which is particularly simple to describe. In practice the resonator is never a perfect optical cavity, there will always be some spatial distortions in the output beam, and a parameter Mm,n is normally quoted that describes how closely the beam is

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6 Lasers and Amplifiers

Figure 6.3 Optical output power versus pump power. The laser has a threshold above which the output increases linearly with the pump power.

Pout Output power

202

Pthr

Ppump Pump power

approximated by a TEMm,n mode. The perfect match corresponds to Mm,n = 1. Some lasers produce beams that have elliptical rather than circular beam profiles. For these lasers it is important to know the aspect ratio of the ellipse and whether the spatial mode is simply a stretched version of a Gaussian beam or a more complex wave front. The output wavelength is determined by the properties of the gain medium and can vary from the UV (excimer laser 169 nm) to the far-infrared NH3 laser (440 μm). Many laser materials have gain for several output wavelengths. For example, the Ar ion laser has several green/blue laser transitions. The desired wavelength can be selected by optimizing the mirror coatings and by using selective intra-cavity components, such as a prism or a grating. Some lasers are tunable, that is, a range of laser wavelengths can be chosen. Some materials with a tuning range have a broad gain profile – prominent examples are dye lasers and the Ti/sapphire laser; in other materials the gain profile is temperature dependent. For example, in a diode laser the wavelength can be tuned by changing the temperature of the device, controlling the current, and using an extended cavity for precise control of the laser frequency. The frequency distribution of the laser output is determined by the cavity properties and the gain profile. In addition, it is important whether the gain profile is an inhomogeneous or homogeneous profile. In the first case different cavity modes obtain their gain from different active atoms, and the various modes can oscillate independently. The result is a series of spectral output modes, separated by the free spectral range of the laser resonator. In the latter case all active atoms in the laser contribute to the laser in the same way. There will be competition between the spectral modes, and normally only one will win and the output spectrum will be dominated by one single spectral mode. Such a single-mode laser still has a frequency bandwidth. The absolute frequency can change with time. The bandwidth is given as the range of frequencies that are present in a set observation time. The bandwidth is a measure of the size of the frequency fluctuations. For example, suppose only frequencies within a 1 MHz interval are measured within one second interval. This information tells us that the laser could be used to resolve a detailed feature of a given spectrum with a resolution of 1 MHz and no less. However, short-term laser bandwidth

6.1 The Laser Concept

and long-term frequency drift can vary widely. The same laser might drift by 100 MHZ in 10 minutes. We would have to carry out the experiment very quickly. A more complete description of the frequency properties is given by the Allan variance, a statistical analysis of the frequency changes for a range of different observation times [5]. We will find that the frequency fluctuations are commonly a consequence of technical imperfections, such as vibrations and electrical and acoustical noise. The quantum mechanical fluctuations play hardly a role in the frequency noise. The phase of a laser is unconstrained and will drift. This is known as phase diffusion. This means that the variance of the phase quadrature (see Chapter 4) will grow unlimited over long time periods. On the other hand for short times, or high frequencies, the phase can be quite stable and will tend to the coherent state level at sufficiently high frequencies. Finally, there are fluctuations of the intensity of the output beam. These are described as intensity noise spectra, as the relative changes of the laser power, 𝛿Pout ∕Pout , measured with a bandwidth RBW at a frequency Ω. The measured variance of the optical power fluctuations can be directly compared with the modelled variance of the amplitude quadrature V 1las (Ω) (see Chapter 4), which is normalized by the quantum noise associated with a coherent state of the same optical power. In summary, lasers are available that produce output beams, which are very close to the theoretical limit. Their light can be considered as the real-world analogue of a coherent state. But not all lasers do this. Some lasers are not built well enough and produce beams that are distorted or noisy. Some other lasers have noise that is intrinsic to the lasing process. Finally, we will find some lasers that actually perform better than a coherent state. 6.1.2

Rate Equations

Although different laser systems differ widely in their details, it is possible to model many aspects of laser behaviour from a quite general rate equation approach. The main assumption in such an approach is that coherent atomic effects are negligible. Coherent effects can be neglected in the presence of strong dephasing effects, such as interatomic collisions in a gas or lattice vibrations in a solid. This is an excellent approximation for all but a few lasers, such as far-infrared gas lasers, which we will not consider in this discussion. In a rate equation model, we consider the gain medium to be made up of M atoms. These atoms have internal states of excitation, with energies E1 , E2 , and E3 as shown in Figure 6.4. We assume lasing occurs between the states with energies E3 and E2 . The frequency of the laser light is centred around 𝜈L = (E3 − E2 )∕h. The actual laser frequency will be determined by the cavity resonance, taking into account all the non-linear effects inside the laser resonator. The excitation of the medium is described by the populations M1 , M2 , M3 of the individual states. These populations should really be interpreted as the probabilities J1 , J2 , J3 of finding any individual atom in one of the three states. The total population is preserved, thus M = M1 + M2 + M3 = M(J1 + J2 + J3 )

(6.1)

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6 Lasers and Amplifiers

Figure 6.4 The semi-classical model of a three-level laser system.

E3 , J3 γ32

G Energy

204

E2 , J2 Γ

γ31 γ21

E1 , J1

A number of processes link the states. Pumping is modelled as a uni-directional transfer of population from state |E1 ⟩ to state |E3 ⟩. Normally any pumping process will also produce de-excitation of the upper pump level at a rate proportional to its population. The uni-directional pumping is to be understood as the limit of pumping to a higher state, which then decays very rapidly to state |E3 ⟩. The population of all excited states decay through spontaneous emission. Stimulated emission and absorption of photons in the laser field couples the states |E3 ⟩ and |E2 ⟩. All these processes are described by rates. As an example, the spontaneous emission from state |E2 ⟩ to state |E1 ⟩ results in M2 𝛾21 = MJ2 𝛾21 photons emitted per second, which are leaving the laser in random directions, and a corresponding reduction in the population of state |E2 ⟩ and increase of population of state |E1 ⟩. Similarly the rate of spontaneous emission from state |E3 ⟩ is described by rate constant 𝛾32 . The pumping rate depends on the pump power, the excitation cross section, and the pumping efficiency. It is described by one rate M1 Γ. For simplicity, all we neglect spatial effects and consider one spatial mode only. Similarly only one spectral mode, at frequency 𝜈L , is considered. All optical effects are described by the gain and loss of light from this single mode. For a high finesse cavity, the total loss of the resonator is just the sum of the individual losses (see Section 5.1.7). We write the total loss 𝜅 as the sum of losses due to the output mirror 𝜅m and those due to all other mechanisms 𝜅l . If the mode inside the laser contains nt photons, then the number of photons leaving the cavity per second will be 2𝜅nt . Of these only the fraction 𝜅m ∕𝜅 exits in the output beam; the remainder are dissipated internally. The energy difference between the atomic states |E3 ⟩ and |E2 ⟩ corresponds to a frequency that is resonant with the mode. Stimulated emission occurs at a rate G32 J3 nt , and stimulated absorption at a rate G32 J2 nt . The rate constant G32 is defined as G32 = 𝜎s 𝜌c∕nr

(6.2)

where 𝜎s is the stimulated emission cross section for the |E3 ⟩ to |E2 ⟩ transition, 𝜌 is the density of lasing atoms in the medium, c is the speed of light, and nr is

6.1 The Laser Concept

the refractive index. The two processes produce an overall change of the photon number of G32 (J3 − J2 )nt (photons per second). The inversion (J3 − J2 ) drives this amplification process. At this point we ignore the small contribution to the photon number due to spontaneous emission into the lasing mode. Combining all these individual rates, we can write a system of four coupled rate equations that describe the dynamics of the laser: dJ1 = −ΓJ1 + 𝛾21 J2 dt dJ2 = G32 (J3 − J2 )n + 𝛾32 J3 − 𝛾21 J2 dt dJ3 = −G32 (J3 − J2 )n − 𝛾32 J3 + ΓJ1 dt dn (6.3) = G32 (J3 − J2 )n − 2𝜅n dt where n is the photon number per atom such that nM = nt . We can also write the equation of motion of the cavity field (per atom) as d𝛼c (6.4) = G32 (J3 − J2 )𝛼c − 𝜅𝛼c dt where 𝛼c∗ 𝛼c = n. The steady state solution for the photon number per atom, n, can be obtained by setting the time derivatives to zero. We find either n = 0 or n=

Γ(𝛾21 − 𝛾32 ) 𝜅 Γ𝛾21 + 𝛾21 𝛾32 + Γ𝛾32 − 2𝜅(2Γ + 𝛾21 ) G32 2𝜅(2Γ + 𝛾21 )

Provided we have 𝛾21 > 𝛾32 , there is a threshold pump rate, Γt , above which the non-zero solution for n will be positive and stable. This represents laser oscillation while the zero solution is unstable. For pump rates below the threshold, the non-zero solution is negative, and it is the zero solution that is stable. That is, the laser does not oscillate. Physically that means the losses in the resonator, per round trip, are larger than the gain so that any light introduced into the cavity is quickly damped away. Generally the lifetime of the lower lasing level will be very short so we can set 𝛾21 ≫ 𝛾32 , Γ. Under this condition the population of the second level can be neglected, and the ground state can be considered undepleted (J2 = 0, J1 = 1). Our equations of motion reduce to dJ3 = −G32 J3 𝛼c∗ 𝛼c − 𝛾32 J3 + Γ dt d𝛼c (6.5) = G32 J3 𝛼c − 𝜅𝛼c dt and we obtain the following simpler solution for the output photon flux per atom of the laser, nout : ( ) 𝜅m 2𝜅 nout = 2𝜅m n = Γ − 𝛾32 (6.6) G32 𝜅

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The pump threshold is Γt = steady state, 2𝜅 J3 = G32

2𝜅 𝛾 . G32 32

We also have that above threshold, in the (6.7)

and the following alternative form of Eq. (6.6) can be written 𝜅 (6.8) nout = (Γ − 𝛾32 J3 ) m 𝜅 That is, the rate at which laser photons leave the cavity is simply the rate at which atoms are placed in the upper lasing level minus the rate at which photons are lost through spontaneous emission. Threshold occurs when pumping exceeds the loss due to spontaneous emission. The useful output is scaled by the fraction of photons that leave via the cavity mirror and hence depends on the internal losses. The total efficiency of the laser, 𝜂total , depends on this fraction, the relative pump power Ppump /Pthr , as well as the pumping efficiency. Very close to threshold the performance is obviously very dependent on fluctuations. The laser can switch on and off, and our simple theoretical model would have to be extended to include such effects. For reliable operation the actual pump rate Γ should be considerably higher than Γt , which means Ppump > Pthr . Above the threshold the output power increases linearly with the pump rate (see Figure 6.3). Hence the slope efficiency, 𝜂slope , is constant above threshold, depending only on the internal losses and pump efficiency. At even higher pump powers, the output power can level off. Theoretically this occurs when the pump rate begins to deplete the ground state. In practice this tends to happen at lower pump powers than a simple estimate would suggest, since a number of additional non-linear effects might occur that can have power limiting back effects. For example, laser materials show thermal expansion, due to absorption, which can lead to a non-uniform refractive index and thus thermal lensing. This and other power-induced wave front distortions introduce additional losses in the resonator, which limit the internal power. An alternative possible configuration of the laser is where the lasing transition occurs between state |E2 ⟩ and the ground state, state |E1 ⟩, with a stimulated emission cross section proportional to G21 . Important examples of this type of laser are the solid-state Ruby and Erbium lasers. The analysis proceeds as before. Making the assumption now that state |E3 ⟩ decays very rapidly, we get the following solution for the output photon flux per atom: ) ( 𝜅m 1 2𝜅 (Γ − 𝛾21 ) nout = 2𝜅m n = (6.9) Γ − 𝛾21 − 2 G21 𝜅 The threshold pump rate is given by Γt =

1+

2𝜅 G21

1−

2𝜅 G21

𝛾21

(6.10)

The major difference between this laser system and the previous one is the threshold behaviours. Whilst in the previous case, the pump threshold could in principle be made arbitrarily small by reducing the ratio G2𝜅 , that is, increasing the 32

6.1 The Laser Concept

cavity finesse or active atom density, in this case the threshold cannot become less than the spontaneous emission rate between the laser levels, 𝛾21 . Physically this is because we have to at least half empty the ground level before we can create an inversion. 6.1.3

Quantum Model of a Laser

In order to determine the dynamics of the fluctuations of the laser System, a quantum model is required. In the quantum description the laser consists of a laser medium of M atoms with internal states |E1 ⟩, |E2 ⟩, |E3 ⟩, a pump source with pump rate Γ and noise spectrum V 1P (Ω), and an internal optical mode â with an expectation value of 𝛼c . Operators Ĵ1 , Ĵ2 , Ĵ3 describe the collective excitation of the atoms. As in the semi-classical model, given in the previous section, the expectation values of these operators, J1 , J2 , J3 , respectively, give the probability of finding any one atom in the respective state. Figure 6.5 shows the laser system in this notation. Making the simplifying assumptions that led to Eq. (6.5) (i.e. negligible second-level population and an undepleted ground state), the quantum operator Langevin equations of motion can be written as √ √ √ dĴ3 ̂ p − Γ 𝜹X ̂ 𝛾 + G32 â † â 𝜹X ̂P = −G32 Ĵ3 â † â − 𝛾32 Ĵ3 + Γ + 𝛾32 𝜹X dt √ √ √ dâ G32 ̂ ̂ l + 2𝜅m 𝜹A ̂ m − G32 Ĵ3 𝜹A ̂ †p (6.11) = J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 where all terms with ‘𝛿’ prefixes are vacuum fields, with zero expectation values. The equations of motion of the expectation values correspond to the semi-classical rate equations (Eq. (6.5)). All the processes are described in a similar way – now by using operators to describe the states of the atoms and the single mode of the light. In addition, the full quantum model has to contain operators describing all possible noise sources. These include the pump ̂ P , and with an intensity noise spectrum noise, characterized by the operator 𝜹X 2 ̂ m , and ̃ V 1P (Ω) = ⟨|𝜹XP | ⟩, the vacuum input through the cavity mirror, 𝜹A ̂ the vacuum input due to other intra-cavity losses, 𝜹Al . The rapid phase decay

|3〉, σ3

Energy

G

V1las |2〉, σ2

Γ

V1vac

V1pump

γ32

γ31 γ21

|1〉, σ1

Figure 6.5 The quantum model of a three-level laser.

V1spont V1dipole V1losses

207

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6 Lasers and Amplifiers

of the atomic coherence due to collisions with atoms or, in the case of solids, lattice-induced disturbances introduces dipole fluctuations that can be charaĉ p + 𝜹A ̂ †p . Finally, noise in the populations due ̂ p = 𝜹A terized by the operator 𝜹X ̂ 𝛾 . The effects of to spontaneous emission has to be included via the operator 𝜹X all these noise sources appear explicitly in the fluctuations of the output operator ̃ las . The derivation of Eq. (6.11) is quite involved and beyond the scope of this A book. Details of the derivation and the identification of the noise operators with their various physical origins can be found in Ref. [6]. The steady-state solutions for the probabilities J3 and photon number per atom, n, in the quantum model, obtained when all the time derivatives are set to zero, are identical to those obtained from the semi-classical Eq. (6.5). We proceed by ̂ linearizing around these stable solutions (see Section 4.5.1). Writing â = 𝛼c + 𝜹a and Ĵ3 = J3 + 𝜹Ĵ3 and only keeping terms to first order in the vacuum fields, we obtain the following linearized equations for the quantum fluctuations: √ d𝜹Ĵ3 ̂ a − (𝛾32 + G32 𝛼c2 )𝜹Ĵ3 + 𝛾32 𝜹X ̂𝛾 = −G32 J3 𝛼c 𝜹X dt √ √ ̂ p − Γ 𝜹X ̂P + G32 𝛼c2 𝜹X √ √ √ ̂ G d𝜹a ̂ l + 2𝜅m 𝜹A ̂ m − G32 J3 𝜹A ̂ †p (6.12) = 32 𝛼c 𝜹Ĵ3 + 2𝜅l 𝜹A dt 2 √ where we have taken 𝛼c = n real without loss of generality (see Section 6.1.5). These equations can be solved in Fourier space. The internal mode is coupled to the external √ field (see Section 5.3 – resulting in the frequency space ̃ las = 2𝜅m ã − 𝜹A ̃ m , describing the fluctuations of the output field. operator, A The result of interest is the noise spectrum V 1las (Ω) of the photon number, or intensity, of this mode. Under our linearization condition this is given by the ̃ A |2 ⟩ where amplitude quadrature spectrum of the output mode: V 1las (Ω) = ⟨|𝜹X † ̃ ̃ ̃ 𝜹XA = 𝜹Alas + 𝜹Alas . We obtain the following solution for the system of transfer functions: ) ( 4𝜅m2 ((2πΩ)2 + 𝛾L2 ) − 8𝜅m 𝜅G32 n𝛾L Vm V 1las (Ω) = 1 + ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 + + + +

2 2𝜅m G32 nΓ

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2 (2𝜅m G32 n𝛾32 J3 )

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2𝜅m G32 ((𝛾32 + Γ)2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 4𝜅m 𝜅l (𝛾L2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2

VP (Ω) V𝛾 Vp Vl

(6.13)

where 𝛾L = G32 n + 𝛾32 + Γ, (2π ΩRRO )2 = 2G32 n𝜅, and the solutions for n and J3 are given by Eqs. (6.6) and (6.7). It should be noted that the parameters have to be chosen separately for each laser. The material constants 𝜎s , 𝜌, nr , 𝛾32 , and

6.1 The Laser Concept

𝛾21 are specific to the type of laser and reasonably well known. In particular the rapid decays from the highly excited states and the lower lasing level for most lasers justify the use of this simplified three-level model. Other parameters such as the resonator parameters 𝜅l and 𝜅m and the total number of lasing atoms vary with the individual laser. How close to a coherent state with V 1las (Ω) = 1 can this solution be? The general answer is: many lasers approach this limit if they are operated well above threshold. However, there are important differences for the various types of lasers. We will now discuss a number of typical cases focusing especially on solid-state and diode lasers. 6.1.4 6.1.4.1

Examples of Lasers Classes of Lasers

The finesse of the laser resonator, determined by the magnitude of 𝜅, is generally chosen to achieve optimum laser efficiency. The finesse is controlled by changing the transmission of the output mirror (𝜅m ). Reducing the mirror transmission lowers the laser threshold and hence acts to increase efficiency. However, it also 𝜅 reduces the ratio 𝜅m due to the presence of internal losses (𝜅l ), which decreases the efficiency. Thus for a particular laser material, there will be an optimum value of 𝜅. Three classes of lasers with distinctly different dynamical behaviours can be identified on the basis of the relationship between the sizes of 𝜅 and the decay rate of the upper lasing level (𝛾32 ). They are: • Class 1: 𝛾32 ≫ 𝜅 – e.g. dye lasers. • Class 2: 𝛾32 ≈ 𝜅 – e.g. visible gas lasers. • Class 3: 𝛾32 ≪ 𝜅 – e.g. solid-state and semiconductor lasers. We will now examine the predicted behaviour of examples from each of these classes, focusing particularly on the class 3 solid-state and semiconductor lasers. A fourth class of lasers, mentioned earlier, in which the photon decay rate is more rapid than the coherence dephasing rate cannot be described via rate equations and will not be discussed here. 6.1.4.2

Dye Lasers and Argon Ion Lasers

There is a wide range of possible lasers that are used in quantum optics experiments. In particular when the non-linearity is a resonant atomic or molecular medium, it is important to have a laser that can be tuned on or arbitrarily close to the resonance. While even that can be achieved nowadays with diode lasers, using external cavities, the traditional work horse in these situations is the argon ion laser pumped dye laser. In these devices the active medium is a dye solution, made up of large, complex dye molecules that can provide gain continuously over a wide range of optical frequencies. The actual laser frequency is selected with a number of frequency-dependent components inside the laser resonator. Typically a dye laser will have parameters of 𝜅 ≈ 1.5 × 107 1/s and 𝛾32 ≈ 3 × 108 1/s and hence is a class 1 laser. Provided we are operating more than about five times above threshold, we can approximate the noise spectrum (6.13) by V 1las (Ω) = 1 +

(2𝜅)2 (V 1P − 1) (2𝜅)2 + (2πΩ)2

(6.14)

209

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This is a Lorentzian spectrum with a half width equal to the cavity decay rate. At low frequencies the spectrum will be dominated by the noise of the pump laser. At high frequencies (2πΩ ≫ 2𝜅), the spectrum will be quantum noise limited (QNL). In order to obtain QNL operation from a dye laser at frequencies in the tens of MHz, a QNL pump laser is needed. A dye laser is pumped by an Ar ion laser, providing large optical powers (more than 10 W) at wavelengths in the blue end of the visible spectrum. Typically an argon laser will have 𝜅 ≈ 1.0 × 106 1/s and 𝛾32 ≈ 2 × 106 1/s and hence is a class 2 laser. Sufficiently far above threshold a class 2 laser has a noise spectrum very similar to that of a class 1 laser (see Eq. (6.14)). Due to the slower cavity decay rate and hence narrower pump (electric discharge) noise region, we would expect the noise spectrum of the argon laser to be QNL at frequencies of tens of MHz. However, due to the small gain that the Ar plasma provides, these lasers have long cavities and thus in the presence of inhomogeneous broadening frequently run on several longitudinal modes. The output spectrum of such a pump laser would hence be dominated by strong spectral features with a spacing of typically 100 MHz. This noise from an Ar ion laser will appear on the intensity noise spectrum of the dye laser. It is possible to suppress the higher-order modes using intra-cavity etalons but only with a large power penalty. For quantum optics experiments, which require a laser beam that is as close as possible to the quantum limit, the noise properties of the Ar ion and dye lasers have posed considerable problems. Most experiments were carried out at specific detection frequencies, in between the large noise spikes, where the noise spectrum happened to be QNL. In recent years the progress in the development of high-power, low noise diode lasers and diode laser pumped solid-state lasers has been so remarkable that now dye lasers are avoided as much as possible. 6.1.4.3

The CW Nd:YAG Laser

We will now discuss the most popular solid-state laser, the Nd:YAG laser. Gain is provided by excited neodymium ions in a yttrium aluminium garnet glass substrate. Traditionally the neodymium ions were excited by a flash lamp, but it is becoming more common these days to use diode laser arrays as the pump source. In a miniature monolithic ring configuration, these lasers can be made to perform very close to the ideal single-mode laser described by our rate equation (6.3). Thus a careful quantitative comparison between theory and experimental results is warranted here. Typically solid-state lasers have 𝜅 ≈ 107 − 108 1/s depending on their size and 𝛾32 ≈ 102 − 103 1/s and hence are class 3 lasers. For a class 3 laser, all the terms in the noise spectrum (Eq. (6.13)) remain important even when operating up to 10 times above threshold and the spectrum is quite complex. It contains contributions from the pump noise V 1P (Ω), spontaneous emission, dipole fluctuations, and vacuum fluctuations. The mixture of these contributions varies with the detection frequency. It is instructive to plot these various contributions separately. This is done in Figure 6.6a, where the various contributions are described in the figure caption. There are some very clear features, and the spectrum is dominated by a resonant feature, called the resonant relaxation oscillation that occurs at the frequency √ 2π ΩRRO = 2G23 n𝜅 (6.15)

Noise power relative to QNL (dB)

6.1 The Laser Concept

80 60 (v)

40 (iv) 20

(vi)

(i)

QNL

0

0.01

0.1

(a)

Noise power relative to QNL (dB)

(ii)

(iii) –20 0.001

1

10

100

500

10

30

Frequency (MHz)

75 (i) 55

(ii)

35

15

(iv)

(iii) QNL

–5 0.001 (b)

0.01

0.1

1

Frequency (MHz)

Figure 6.6 Noise spectrum of a free-running Nd:YAG laser. (a) The different contributions to Eq. (6.3) are drawn separately. Trace (i) is the contribution of the vacuum noise, (ii) is the effect of the pump noise, (iii) is the noise due to spontaneous emission, (iv) is the noise associated with the dipole fluctuations, (v) is the noise due to intra-cavity losses, and (vi) is the total noise for a QNL pump. Part (b) shows experimental results for such a Nd:YAG laser. In this realistic case (i) the pump noise level was about 40 dB above the QNL. For comparison the predicted noise performance is shown in (ii). Trace (iii) was obtained with a pump noise of 20 dB. Trace (iv) is the corresponding theoretical prediction. Source: Adapted from Harb et al. [7].

In this particular case ΩRRO = 400 kHz. This oscillation is due to the interaction of the atomic state inversion (J3 − J2 ) ≈ J3 with the excitation of the internal optical mode. In physical terms, the energy in the laser is moving back and forth between the laser medium and the cavity mode, resulting in a strong modulation of the output power. This oscillation is a prominent feature and corresponds to an underdamped oscillation that is continually excited by the noise sources within the laser [8]. Note that this oscillation does not occur in class 1 and 2 lasers because for them 𝛾L ≫ ΩRRO , making the oscillation overdamped, as can

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be seen from the denominator in Eq. (6.13). An alternative expression for ΩRRO in terms of the pump rate is √ √ Γ − Γt (6.16) 2π ΩRRO = 2𝜅𝛾32 Γt That means the harder the laser is pumped, increasing Γ, the higher is the frequency of the relaxation oscillation and the less pronounced it is. This can be seen by comparing the traces in Figure 6.7 that are plotted for the same laser but with different pump powers. In some situations fitting the measured frequencies ΩRRO for a range of pump powers is a more reliable way to determine Γt than an attempt to measure the power at which the laser turns on. Above ΩRRO the laser has a long, drawn-out noise tail. For very high frequencies the laser approaches the quantum noise limit. The origin of the quantum noise can be traced back to the various vacuum inputs in the model. It should be noted that the noise of the pump source has no influence on the laser noise for frequencies Ω ≫ ΩRRO . A noisy pump will not affect the laser at such high frequencies. However, it also means that in this frequency regime the laser output power cannot be controlled by modulating the pump power. This is certainly a disadvantage for communication systems, since pump power modulation is normally the simplest and cheapest modulation scheme. At low frequencies, Ω ≪ ΩRRO , the noise spectrum of the laser is dominated by the noise of the pump source. There is a strong transfer from the pump to the laser output. By carefully choosing the noise properties of the pump, the laser output can be quietened down. Figure 6.6b shows two such cases. The first case, traces (i) and (ii), is realistic for a laser with large pump noise. Here V 1p (Ω) is a factor 10 000, or 40 dB, above the standard quantum limit. The second case, –20

129

170

205 266

–40 Power (dBm)

318 360 413

78 kHz

–60 455 –80

1

0.1

–100 0.01

212

Frequency (MHz)

Figure 6.7 Experimental noise trace for a Nd:YAG laser with various pump powers. The relaxation oscillation frequency ΩR is stated for each case in units of kHz. It increases with pump power as stated in Eq. (6.16). Source: Adapted from Harb et al. [7].

6.1 The Laser Concept

traces (iii) and (iv), was measured with a quieter pump, V 1p (Ω) = 20 dB. Note that even for the case of a quantum-limited pump (V 1p (Ω) = 0 dB), the resonant relaxation oscillation would survive. Only for the impossible case of a completely noise-free pump source (V 1P (Ω) = 0) and no cavity losses or dipole fluctuations would the relaxation oscillation completely disappear. 6.1.4.4

Diode Lasers

Perhaps the simplest and most efficient lasers are semiconductor or diode lasers. Diode lasers use the band gap in doped semiconductor materials to produce optical gain directly from an electrical current. Under suitable conditions diode lasers can be described by our rate equation (6.3) as class 3 lasers. Major differences with solid-state lasers arise from their small size and high gain. Typically a diode laser will have 𝜅 ≈ 1011 1/s and will be run 5–10 times above threshold. Under these conditions the RRO will be pushed out to frequencies of tens of GHz and will be quite small. Most detection schemes are only sensitive up to frequencies of a few GHz; thus for all practical purposes the spectrum can be written as 𝜅 (6.17) V 1las (Ω) = m [1 + 𝜁 (V 1P (Ω) − 1)] 𝜅 where it is understood that 2πΩ ≪ 𝜅 and 𝜁 is the pump efficiency. Moreover, unlike the situation of the dye laser where the pump source is a potentially very noisy Ar ion laser, the pump source of the diode laser is an electric current, which can be made very quiet. Indeed we can have V 1las (Ω) = 1 if the pump current is given by V 1P (Ω) = 1. This means that for a quiet Poissonian noise-limited pump current, a diode laser has a QNL output. Equation (6.17) also means the laser output responds directly to modulation of the pump source. This is important for communication applications where we want to use the current driving the diode laser as the input of modulation. In fact, there is no fundamental reason why we cannot have V 1P (Ω) < 1, that is, a sub-Poissonian current. Unlike light, electric currents can easily be made sub-Poissonian. This would imply V 1las (Ω) < 1; the laser output would be sub-Poissonian. Diode lasers have successfully been made to produce squeezed light using this method, and technical details are discussed in Chapter 9. From Eq. (6.17) we see that, at least in theory, the squeezing is limited only by the internal losses and the pump efficiency 𝜁 . 6.1.4.5

Limits of the Single-Mode Approximation in Diode Lasers

From the previous discussion of diode lasers, it would appear that, provided sufficient care was taken to suppress noise in the pump current, these lasers would always run at or below the QNL. For a number of reasons, in practice this is not the case. In particular, multimode behaviour can lead to increased noise. Normally one expects single-mode operation from a homogeneously broadened medium, such as that of a diode laser, due to gain competition. However, this is not the whole story; if there is strong spatial dependence in the laser mode, some regions of the gain medium can become gain depleted, whilst others do not, allowing modes with different spatial dependence to access the undepleted gain and get above the lasing threshold. Competition between these modes can lead to elevated noise levels. An example is the high-power diode array, consisting

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of a strip of diode gain medium sharing a common cavity. The cavity can support a number of transverse modes, many of which lase simultaneously due to spatial hole burning, thus introducing excess noise. Further, the far-field spatial patterns of the different modes are different, such that different spatial regions of the beam can have different noise properties. Even in a purely homogeneously broadened medium, spontaneous emission into the lasing mode, which is neglected in our rate equation (6.3), can allow for the existence of weakly lasing longitudinal side modes [9]. The homogeneity of the medium means that the production of a photon in the main mode implies no photon is concurrently emitted in the side modes and vice versa. Hence there is a strong anti-correlation between the side modes and the main mode. If just the main mode is observed, it is found to be very noisy due to the deletion of photons to the side modes [10]. However, the anti-correlation means that the noise tends to cancel when all the light is observed. Unfortunately, differences in the losses experienced by the main mode and the side modes can lead to a small amount of excess noise, which does not cancel. This can be sufficient to destroy squeezing. This problem has been solved using injection locking or external cavities to suppress the side modes. In this way reliable squeezing has been obtained – the details of these experiments are given in Chapter 9. In summary we find that lasers themselves are interesting but in many cases complex light sources. Only the single-mode laser operating well above threshold will be QNL at all detection frequencies. For all other cases we have to explicitly derive the intensity noise spectrum or measure it directly. 6.1.5

Laser Phase Noise

In Section 6.1.2 we solved for the steady-state values of the intra-cavity photon number, n, and the upper laser level population, J3 . We also introduced the intra-cavity complex amplitude 𝛼c . Notice however that it is not possible to solve for the phase of 𝛼c . Physically this indicates that the laser phase is metastable and will tend to drift in time. This effect is known as phase diffusion. When we linearized the quantum equations of motion to obtain the equations of motion for the fluctuation operators (Eq. (6.12)), we took the phase of 𝛼c to be constant and real. This is of no consequence for the intensity spectrum because ̂ are phase insensitive. However, we need to be ̂ † A) intensity measurements (A more careful if considering phase-sensitive measurements. We can calculate the phase fluctuation spectrum of the laser output from ̃ las |2 ⟩ (6.18) V 2las (Ω) = ⟨|𝜹X𝟐 ̃ las = i(𝜹A ̃ las − 𝜹A ̃ † ). The phase fluctuation spectrum can be where as usual 𝜹X𝟐 las obtained by making a phase-sensitive measurement of the field with respect to a stable phase reference, which means using homodyne detection described in Section 8.1.4. Our assumption of constant classical phase (𝛼c real) will only be valid for the phase spectrum if the fluctuations around that constant phase are small. The solution is given by V 2las (Ω) =

(2πΩ)2 + 2(2𝜅)2 (2πΩ)2

(6.19)

6.2 Amplification of Optical Signals

At very high frequencies the phase spectrum tends to the quantum noise level, V 2las (Ω) = 1. As we move to lower and lower frequencies, the phase noise becomes larger and larger, eventually diverging at zero frequency. The divergence of the phase noise spectrum at zero frequency indicates that our linearization breaks down for sufficiently small Fourier frequencies. On long enough timescales, or low enough frequencies, the phase drifts significantly, thus invalidating the linear approximation. This is the signature of laser phase diffusion. In the absence of technical noise, this phase diffusion determines the laser linewidth. In practice phase diffusion does not present a major problem as generally the same laser is used as a phase reference as is used to produce the signal. The phase drift is then common to both modes and so is not observed. Thus provided we stay within the coherence length of our laser (see Section 2.3.3 and work at frequencies for which the laser’s intensity noise is at the QNL, a laser beam can represent an excellent realization of an ideal coherent state. 6.1.6

Pulsed Lasers

The focus in this section is primarily CW lasers since they are very common in quantum optics experiments, in precision metrology. The steady optical power is necessary to avoid unwanted non-linear effects and the related laser noise. They are also easy to explain and comprehend. However, the first laser demonstrated was pulse, and much of the research work and many applications use pulsed lasers, ranging from nanosecond (ns) down to femtosecond (fs) pulses for commercial systems and attosecond (as) pulse lengths for the highness optical intensities required in research applications. This advance of laser technology uses non-linear processes and the elegant concept of beam pulse compression and expansion, as first shown in the late 1980s [11]. These techniques have revolutionized the use of lasers and were honoured by the award of the Nobel prize to Donna Strickland and Gerard Mourou in 2018. In all of these systems, the optical amplification is similar, and the optical resonator shares some commonalities. Most of the pulsed lasers have a high repetition frequency Ωrep , frequently, around 100 MHz for femtosecond lasers, and thus a typical applications uses many of these pulses to detect the desired effect. Each pulse has a high peak power, which is useful to drive non-linear effects, and this is key reason to use them in applications such as two- and three-photon imaging, materials processing, ionization, etc. The signal-to-noise ratio however of a measurement is normally measured at signal frequencies that are much lower than the repetition frequency Ωsig < Ωrep , and the influence of the noise is at low frequencies as well and basically identical to the case of a CW laser. Clear examples of this for the use of pulsed lasers in quantum optics are shown in Section 9.5.

6.2 Amplification of Optical Signals It is possible to use an optical medium to amplify the power of a laser beam. For example, we can imagine an input power of 1 mW and an output power

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of 1 W, which means a power gain G of a factor of 1000. A medium that can achieve this purpose is called an active optical medium. Normally it is just a laser medium without optical feedback from a cavity. Such a device would obviously be very useful for practical applications where high power is required. Some of the biggest and most expensive optical systems are the Nd:glass amplifiers that have been built to generate powerful laser pulses for fusion studies using inertial confinement [12]. Another application is in optical communications where the beam is attenuated over distance, for example, by losses inside an optical fibre. In this case it is necessary to amplify not only the power but also the modulation sidebands, which contain the information that is communicated. In a fibre system an in-line amplifier, such as an erbium-doped fibre amplifier, is a very useful and attractive device. The gain bandwidth should extend to the largest modulation sideband that is used. Typically this is a few GHz, which is not difficult to achieve. The next question is: how does an optical amplifier affect both the signal and the noise? Does the signal-to-noise ratio vary? It might be easiest to understand the properties of optical amplifiers by first describing the simpler and more obvious case of an electronic amplifier. Consider a circuit with a linear amplifier that has a power gain Gel (Ω). The gain is frequency dependent; usually Gel (Ω) drops to less than unity at high frequencies. The input into the amplifier is a current with modulation at Ωm . In linear systems we can consider each Fourier component, each modulation frequency, separately and independently of all the others. Ideally the input and output are related by √ 𝛿iout (Ω) = Gel (Ω)𝛿iin (Ω) The variances of modulations, or fluctuations, of the input and the output are linked by Var(iout ) = Gel

Var(iin )

(6.20)

This amplifier affects both the signal and the noise in the same way. In principle, the signal-to-noise ratio will be maintained. In practice a small additional noise component will be added by the amplifier. However there is no fundamental limit to how small this added noise can be. As long as the added noise is much smaller than the noise floor of the input, the signal-to-noise ratio is maintained. The performance of this system could be improved further if the amplifier is given a reference signal, which is at the same frequency and phase locked to the modulation signal. In this case a significant part of the noise, which has random phase, can be suppressed by this apparatus, normally called a lock-in amplifier. The performance of an optical amplifier is quite different [13–15]. The input and the output modes can be described in the Heisenberg picture by the operators â in and â out . Suppose our input state is a coherent state, |𝛼⟩. We expect the coherent amplitude to be amplified such that √ 𝛼out = ⟨̂aout ⟩𝛼 = G 𝛼in This suggests the following operator description: √ â out = G â in

6.2 Amplification of Optical Signals

However, this violates the commutation rule [̂aout , â †out ] = G [̂ain , â †in ] = G ≠ 1 This problem is similar to the quantum theory of damping, where a careful consideration of the role of quantum noise has to be made. The vacuum state entering through any unused port has to be taken into account. See the description of a beamsplitter in Section 5.1 for details. In a similar way we have to include the vacuum state in this amplification process. The correct description is [16] √ √ â out = G â in + G − 1 v̂ † (6.21) where v̂ is an operator describing an input mode originally in the vacuum state that is mixed in by the amplifier. It is completely independent of â in . The result is that the light, after amplification, is considerably noisier than before. The amplifier is increasing not only the power but also the noise level. The variance of the intensity of the amplified light is V 1out = G V 1in + (G − 1) V 10

(6.22)

Here V 10 = 1 is at the QNL. For large gains (G ≫ 1), the second term results in an increase of an extra 3 dB compared to the classical case (Eq. (6.20)). As shown in Figure 6.8, both the average amplitude and the fluctuations increase. To see that this is indeed the effect of amplification by a laser medium, consider again the quantum mechanical equations of motion for the laser (Eq. (6.11)). Suppose that we are below threshold such that 𝜅 > G32 J3 ∕2 but that instead of ̂ in , is injected. Further let us assume vacuum entering the output mirror, a field, A that we are sufficiently below threshold, that there is very little depletion of the upper lasing level, and that we can therefore replace the operator Ĵ3 with its expectation value J3 . We obtain the following equation of motion: √ √ √ G dâ ̂ l + 2𝜅m A ̂ †p ̂ in − G32 J3 𝜹A (6.23) = 32 J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 Equation (6.23) is now linear and can be solved exactly in Fourier space. Neglecting internal losses (𝜅l = 0) leads to the following solution for the output from the amplifier: √ 2𝜅G32 J3 𝜅 + G32 J3 ∕2 + i2πΩ ̃ in − ̃ †p ̃ out = (6.24) A 𝜹A A 𝜅 − G32 J3 ∕2 + i2πΩ 𝜅 − G32 J3 ∕2 + i2πΩ Figure 6.8 Effect of optical gain on a quantum state.

X2

X1

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The spectrum for either quadrature is then V iout (Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 2𝜅G32 J3 V iin + (𝜅 − G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

which is of the same form as Eq. (6.22) with G(Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

We see that in this case the additional quantum noise is introduced via the dipole fluctuations of the gain medium. The 3 dB penalty is imposed by the optical amplifier when it acts equally on all quadratures. Such an amplifier is called phase insensitive. This penalty can be avoided by using an amplifier that is phase sensitive, as will be discussed in the next section.

6.3 Parametric Amplifiers and Oscillators Phase-insensitive amplification is of limited utility when dealing with quantum fields because of the excess noise it introduces. We would really like to be able to amplify our quantum-limited light beam without adding additional noise. Let us consider the fundamental commutation problem again. If we write the annihilation operator in terms of the amplitude and phase quadratures, we see the problem clearly: 1 ̂ ̂ + iX𝟐) (6.25) â = (X𝟏 2 If both quadratures are amplified equally, then â is just multiplied by some gain factor, and via the arguments of the last section, it is clear that an additional field, inevitably introducing noise, must be mixed in by the amplifier. However, consider √ the situation in which the amplitude quadrature is amplified by some amount, K, but the phase quadrature is de-amplified by the same amount. The output operator would now look like ( ) i ̂ 1 √ ̂ K X𝟏 + √ X𝟐 (6.26) â out = 2 K It is straightforward to confirm that the commutation relations still hold in this case and that this is a legitimate transformation and no additional noise operators are required. Thus, provided that the product of the gains of the two quadratures is unity, noiseless amplification of one of the quadratures can be achieved. This is a very important quantum optical interaction. In analogy with the linear amplifier, we can also write Eq. (6.26) in the form √ √ (6.27) â out = G â in + G − 1 â †in where the gain is G=

(K + 1)2 4K

6.3 Parametric Amplifiers and Oscillators

For historical reasons a phase-sensitive amplifier is referred to as a parametric amplifier. 6.3.1

The Second-Order Non-linearity

Classically the response of an atomic medium to an incident optical beam can be described by the induced dipole polarization P(E) caused by the beam’s electric field E: P(E) = 𝜒E + 𝜒 (2) E2 + 𝜒 (3) E3 + · · ·

(6.28)

A non-linear optical material is one in which, at the available field strength, higher than linear terms in the expansion become significant. With the invention of the laser, incident field strengths became sufficiently high that many materials could be made to show non-linear behaviour. We will find that phase-sensitive amplification and other interesting quantum optical phenomena can be produced by materials exhibiting a second-order 𝜒 (2) non-linearity. The key feature of the second-order non-linearity is its ability to couple a fundamental field (oscillating at 𝜈) to a harmonic field (oscillating at 2𝜈). The Hamiltonian, or energy function, for such an interaction can be written as H = iℏ𝜒 (2) (𝛽 ∗ 𝛼 2 − 𝛼 ∗ 2 𝛽) where 𝛼 is the fundamental whilst 𝛽 is the harmonic field. Canonical quantization then leads to the quantum mechanical Hamiltonian ̂ ̂ = iℏ𝜒 (2) (b̂ † â 2 − â †2 b) H This Hamiltonian has an elegant interpretation in terms of photons. The first term represents the annihilation of two photons at the fundamental frequency and the creation of one at the harmonic, whilst the second represents the reverse process of a single harmonic photon being annihilated and two fundamental photons being created. The Heisenberg equations of motion (see Section 4.4) are then given by 1 ̂ ̂ H] = 𝜒 (2) â 2 b̂̇ = [b, iℏ â̇ = −2 𝜒 (2) â † b̂

(6.29)

Notice that because the process is energy conserving (and coherent), i.e. two photons of energy h𝜈 equals one of energy h 2𝜈, then no additional vacuum modes are required to be coupled into the fields. In order that either the up-conversion (fundamental to harmonic) or the down-conversion (harmonic to fundamental) process takes place with some reasonable efficiency, we require not only energy conservation but also that momentum is conserved for the converted beam in some particular direction(s). In this way the emission from many dipoles can coherently add to produce a single output beam. This condition is known as phase matching. Let us consider the situation shown schematically in Figure 6.9. A non-linear crystal is placed inside a pair of cavities, one resonant at the fundamental and the other resonant at the harmonic, which are oriented such that there is good phase

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M1 Ain

vA

M2 ζ

αA

Detunings

αB

Aout

κA κB

Δa

Δb

vB =

1 v 2 A

κℓa κℓb

κa = total loss at vA

κb = total loss at vB

Figure 6.9 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive amplifier of the vacuum modes and above threshold as an oscillator. The subscript A refers to the fundamental frequency, and the subscript B refers to the subharmonic frequency.

matching between the two spatial modes defined by the cavities. By combining the cavity equations of motion derived in Section 5.3.6 with those of Eq. (6.29), we obtain the equations of motion for the internal cavity modes as √ √ (6.30) b̂̇ = 𝜒 (2) â 2 − 𝜅b b̂ + 2 𝜅B B̂ in + 2𝜅lb 𝜹B̂ l â̇ = −2 𝜒 (2) â † b̂ − 𝜅a â +

√ √ ̂l ̂ in + 2𝜅la 𝜹A 2𝜅A A

(6.31)

where 𝜅A and 𝜅B are the loss rates of the input/output mirrors for the â and b̂ modes, respectively. Similarly 𝜅la and 𝜅lb are the respective internal loss rates for the two modes and 𝜅a = 𝜅A + 𝜅la , 𝜅b = 𝜅B + 𝜅lb . We also have the respective input ̂ in and B̂ in and the respective vacuum modes that couple in due to the modes A ̂ l and 𝜹B̂ l . Equations (6.30) and (6.31) can describe a large range internal losses 𝜹A of important quantum optical experiments. 6.3.2

Parametric Amplification

Let us now consider parametric amplification. We will assume that the harmonic field is an intense field, which is virtually undepleted by its interaction with the non-linear crystal, whilst the fundamental field is ‘small’. We expect down-conversion under these conditions with energy being transferred from the harmonic to the fundamental field. We shall also assume that 𝜅a ≪ 𝜅b such that the dynamics of the harmonic mode can be neglected. Under these conditions we can replace the operator b̂ with the c-number 𝛽. We are then left with just a single, linear equation for the â mode: √ √ ̂l ̂ in + 2𝜅la 𝜹A (6.32) â̇ = −𝜒 â † − 𝜅a â + 2𝜅A A where

√ 𝜒 = 2𝜒 (2) 𝛽 = 2𝜒 (2)

2 𝛽 𝜅b in

(6.33)

6.3 Parametric Amplifiers and Oscillators

and 𝛽in = ⟨B̂ in ⟩. The parameter 𝜒 is intensity dependent. Equation (6.32) can be solved in Fourier space to give the output mode at the fundamental frequency ̃ out = A

𝜒 2 + 𝜅a2 + (2πΩ)2 2𝜒𝜅a ̃ in + ̃† A A 2 2 (𝜅a + i2πΩ) − 𝜒 (𝜅a + i2πΩ)2 − 𝜒 2 in

(6.34)

Here we have neglected internal losses to the fundamental mode (𝜅la = 0) for simplicity. Equation (6.34) is formally equivalent to Eq. (6.27). The variance of the amplitude quadrature is given by V 1out (Ω) =

(𝜅a2 + 2𝜒𝜅a + 𝜒 2 + (2πΩ)2 )2 𝜒 4 + (𝜅a2 + (2πΩ)2 )2 − 2𝜒 2 (𝜅a2 − (2πΩ)2 )

V 1in (Ω)

(6.35)

where V 1in is the amplitude quadrature variance of the input at the fundamental. The output is an amplified version of the input amplitude quadrature with no added noise as hoped. This can be seen more clearly at zero frequency where the expression reduces to V 1out (0) =

(𝜅a + 𝜒)2 V 1in (0) (𝜅a − 𝜒)2

(6.36)

As the harmonic intensity is increased 𝜒 becomes larger, the denominator of Eq. (6.36) becomes smaller, and hence the amplification increases. On the other hand the phase quadrature (at zero frequency) is given by V 2out (0) =

(𝜅a − 𝜒)2 V 2in (0) (𝜅a + 𝜒)2

(6.37)

which suffers increasing de-amplification as 𝜒 increases. Notice that if V 2in is at the QNL, then V 2out will be below the QNL, which means V 2out < 1. This is termed squeezing and is discussed in detail in Chapter 9. 6.3.3

Optical Parametric Oscillator

Examining Eq. (6.36) we note that the spectrum diverges as the value of 𝜒 approaches that of 𝜅a . This indicates that an approximation is failing, in this case that of negligible pump depletion. As the amplification of the harmonic field becomes larger and larger, eventually it starts to significantly deplete the harmonic. To see what happens in this case, we must go back to Eqs. (6.30) and (6.31) and linearize. We continue to consider the experimental situation where the cavity for the harmonic mode is either only weakly resonant or even non-existent, that is, we effectively have 𝜅a ≪ 𝜅b . As a result the evolution of the mode b̂ occurs on a much faster timescale than that of the â mode, and we can adiabatically eliminate it by setting its time derivative to zero. We also continue to assume that 𝜅lb ≪ 𝜅b . That means the losses for the mode b̂ can be neglected and Eqs. (6.30) and (6.31) reduce to √ (2)2 √ √ 2 ̂ †̂ ̂l ̂ in + 2𝜅la 𝜹A ̂ † â 2 − 2𝜒 (2) ̂ + 2𝜅A A ̂̇ = − 2𝜒 𝜹a 𝛿a 𝜹a Bin − 𝜅a 𝜹a 𝜅b 𝜅b

221

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6 Lasers and Amplifiers

To proceed we first consider the semi-classical steady-state solution of this equation. By taking expectation values, and making the approximation that they may be factored, we obtain √ 2 √ 2 𝜒 (2) ∗ 2 2 ∗ 𝛼̇ = − 𝛼 𝛼 − 2𝜒 (2) 𝛼 𝛽in − 𝜅a 𝛼 + 2𝜅A 𝛼in (6.38) 𝜅b 𝜅b The steady-state solution is obtained by setting the time derivative to zero. If we assume that the first term on the RHS of Eq. (6.38) is small and can be neglected, then we obtain the parametric amplification considered in the previous section. Instead we now retain that term but consider the situation in which there is no input field at the fundamental (𝛼in = 0). The equation that needs to be solved is √ 2 2𝜒 (2) ∗ 2 2 ∗ (2) − 𝛼 𝛼 − 2𝜒 𝛼 𝛽in − 𝜅a 𝛼 (6.39) 𝜅b 𝜅b Equation (6.39) has two solutions: either 𝛼 = 0 or 𝛼 ≠ 0. We recognize this as threshold behaviour similar to that of the laser. At a sufficiently high value of the harmonic pump field, the system will start to oscillate and produce coherent output at the fundamental. Under these conditions the device is called an optical parametric oscillator. If 𝛼 ≠ 0 we have √ 2𝜅b 𝜅 𝜅 ∗ 𝛼 𝛼 = − (2) 𝛽in e−i2𝜙𝛼 − a (2)b 2 (6.40) 𝜒 2𝜒 where 𝜙𝛼 is the phase of 𝛼. To be a valid stable solution, 𝛼 ∗ 𝛼 must be real and positive. This is only possible if 𝜙𝛽 − 2𝜙𝛼 = π. For simplicity we take 𝛽in real and negative (𝜙𝛽 = π); then 𝛼 will be real, and the solution for the output photon number is 𝜅 𝜅 nout = 2𝜅a 𝛼 2 = a(2)b2 (𝜒 − 𝜅a ) (6.41) 𝜒 where we have used the definition of 𝜒 from Eq. (6.33). When 𝜒 < 𝜅a the 𝛼 = 0 solution will be stable; when 𝜒 > 𝜅a then Eq. (6.41) gives the photon flux of the output field at the fundamental. Notice that the threshold condition is the same as the point of divergence of the noise spectrum in the previous section. 6.3.3.1

Noise Spectrum of the Parametric Oscillator

We can now investigate the fluctuations of the above threshold fundamental output by linearizing around the semi-classical solution. By linearizing Eq. (6.34) we obtain the following equation of motion for the fluctuations: √ (2)2 2 ̂ † ̇ = − 2 𝜒 𝛼 2 (𝜹a ̂ † + 2𝜹a) ̂ − 2𝜒 (2) ̂𝛿a (𝜹a 𝛽in + 𝛼 𝜹B̂ in ) 𝜅b 𝜅b √ √ ̂ in + 2𝜅la 𝜹A ̂l ̂ + 2𝜅A 𝜹A (6.42) −𝜅a 𝜹a This equation can be solved in Fourier space to give the quadrature fluctuations of the output field: √ 2 2𝜅a (𝜒 − 𝜅a ) 4𝜅a − 2𝜒 − i2πΩ ̃ ̃ ̃ Bin 𝜹X𝟏out = 𝜹X𝟏Ain + 𝜹X𝟏 2𝜒 − 2𝜅a + i2πΩ 2𝜒 − 2𝜅a + i2πΩ √ 2 2𝜅a (𝜒 − 𝜅a ) 2𝜅a − 2𝜒 − i2πΩ ̃ out = ̃ Ain + ̃ Bin 𝜹X𝟐 (6.43) 𝜹X𝟐 𝜹X𝟐 2𝜒 + i2πΩ 2𝜒 + i2πΩ

6.3 Parametric Amplifiers and Oscillators

where we have neglected internal losses of the fundamental for simplicity ̃ Ain and 𝜹X𝟐 ̃ Ain are the amplitude and phase quadrature fluctuations and 𝜹X𝟏 entering at the fundamental input/output mirrors, respectively. Notice also that non-negligible pump depletion now couples fluctuations from the pump ̃ Bin and 𝜹X𝟐 ̃ Bin , into the fundamental field. The quadrature harmonic field, 𝜹X𝟏 variances at zero frequency are V 1(0) = 1 +

𝜅a2 (𝜒 − 𝜅a )2

V 2(0) = 1 −

𝜅a2 𝜒2

(6.44)

where we have assumed that the harmonic pump fluctuations are at the QNL. The coupling of the harmonic fluctuations means that the amplification of the amplitude quadrature is no longer noiseless above threshold. Strong noise reduction (or squeezing), that is, V 2 ≪ 1, is still present on the phase quadrature close to threshold. However, it becomes negligible far above threshold where the output tends towards a coherent state.

6.3.4

Pair Production

Finally let us return to the parametric amplifier operating far below threshold for another important application of the 𝜒2 interaction. Far below threshold, with no input at the fundamental, we may form the following physical picture: the harmonic beam is mostly transmitted without change through the non-linear medium; however every now and then a harmonic photon is converted into a pair of fundamental photons. This picture suggests that if we counted photons at the fundamental output under these conditions, we should only ever see them arrive as pairs. To test this proposition, we consider the Schrödinger evolution of the fundamental far below threshold where its initial state is the vacuum. The evolution of the fundamental mode is not passive (energy conserving) as energy is being added; however it is unitary (reversible) as no noise is being added. This means we cannot directly use the techniques described in Chapter 4 but we can use a modified version. Let us work in the Fourier domain and use the fact that we can ̃A ̃ † |0⟩ that is clearly still the vacwrite the vacuum state in the following way: A uum state as the action of creation and then annihilation nullify each other. This motivates us to write the output state as ̃A ̃ † |0⟩ ̃ ̃A |𝜙⟩ = U|0⟩ =U ̃ †U ̃U ̃A ̃ ̃A ̃ † )(U ̃ † )(U|0⟩) = (U ̃ ′A ̃ ′† |𝜙⟩ =A

(6.45)

Equation (6.45) shows that the output state is the eigenstate of the operator ̃ ′A ̃ ′† with eigenvalue ‘one’. Here the dashes indicate reverse time HeisenỸ = A berg evolution (see Section 4.4). Solving this eigenvalue equation leads to (at worst) a second-order recurrence relation for linear or linearized systems.

223

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6 Lasers and Amplifiers

Inverting Eq. (6.34) and its corresponding conjugate equation, we obtain † ̃ ̃ ̃ ̃ ̃ − pq∗ A(Ω) Y(Ω) = |p|2 A(Ω) A(Ω) A(−Ω) ∗ ̃ †̃ † 2̃ ̃ −p qA(−Ω) A(Ω) + |q| A(−Ω)† A(−Ω)

(6.46)

where 𝜒 2 + 𝜅a2 + (2πΩ)2 (𝜅a + i2πΩ)2 − 𝜒 2 2𝜒𝜅a q= (𝜅a + i2πΩ)2 − 𝜒 2

p=

(6.47)

Note the presence of creation and annihilation operators for both upper and lower sidebands in the result (see Section 4.5.5). Solving the eigenvalue equation under the assumption that 𝜒 ≪ 𝜅a leads to the following output state for a particular frequency component: |𝜙⟩Ω = |0⟩ +

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.48)

As expected, most of the time ‘vacuum in’ results in ‘vacuum out’. However, occasionally a pair of photons is produced: one in the upper sideband and one in the lower sideband, as required for energy conservation. Given that the input state is a superposition of all frequencies, we may write the total output state as |𝜙⟩T = |0⟩ +

∫

dΩ

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.49)

Operating in this configuration the system is often referred to as a down-converter or photon pair generator. The technical specifications have been improved steadily via a variety of new non-linear media, engineering of the non-linear crystals and novel optical arrangements. The main emphasis is on improving the brightness of the source, measured as the number of pairs per second, the suppression of individual photons and other noise sources, and the quality of the optical modes of the pairs. Optical down-converters and related systems are presently the workhorses of single-photon experiments, as described in Chapters 12 and 13. The quality of the entanglement that has been measured is a direct indication of the quality of the source of photon pairs.

6.4 Measurement-Based Amplifiers Amplification of quantum states can also be implemented via techniques that employ partial measurement of the state followed by feedforward of the measurement results. We will refer to such devices as measurement-based amplifiers. These can be divided into two distinct classes: (i) deterministic and (ii) heralded. Deterministic measurement-based amplifiers implement amplification techniques like those discussed in the previous section. However, they avoid the need for in-line, non-linear crystals or other gain media. This can be quite an advantage as coupling into and out of non-linear media can be inefficient and imperfections in the media can lead to unwanted technical noise. In contrast,

6.4 Measurement-Based Amplifiers

heralded measurement-based amplifiers can beat the quantum limits discussed in the previous section. This is allowed because these devices are probabilistic in their operation. Nevertheless successful operation is heralded by particular measurement results. Hence such devices can be useful for certain applications.

6.4.1

Deterministic Measurement-Based Amplifiers

We consider first a noiseless, phase-sensitive amplifier. The idea here is to amplify one particular quadrature whilst maintaining the signal-to-noise ratio of any signals on that quadrature (i.e. not adding additional noise). Initially we are not interested in what happens to the other quadrature. A brute force measurement-based technique is the complete detection and re-emission of the light using high efficiency detectors and LEDs [17, 18]. The disadvantage here is that no phase reference or coherence exists between the input and output beams. An alternative, which does not require any non-linear process whilst its output still retains optical coherence, was developed and demonstrated by Lam et al. using electro-optic feedforward [19]. This scheme is based on partial detection of the light with a beamsplitter and detector (see Figure 6.10A). The light reflected from the beamsplitter is detected, and the photocurrent is amplified and fed forward to the transmitted beam via an electro-optic modulator (EOM). The electronic gain and phase can be chosen such that the intensity modulation signals carried by the input light are amplified, whilst the vacuum fluctuations that enter through the empty port of the beamsplitter are cancelled, thus making the amplification noiseless. Since not all of the input light is destroyed, the output is still coherent with the input beam. Here a half-wave plate controlled the transmittivity, 𝜀1 , of the polarizing beamsplitter. On the in-loop beam, a high efficiency (𝜀2 = 0.92 ± 0.02) photodetector was used. The photocurrent is then passed through multiple stages of RF amplification and filtering to ensure sufficient gain. An amplitude modulator is formed by using the EOM in conjunction with a polarizer, and finally the fluctuation spectrum of the output is measured using detector Dout and a spectrum analyser. This system can be modelled with the linearized operator approach as used for the electro-optic noise eater (see Section 8.3). We obtain the following transfer functions for the spectrum of the output beam normalized to the QNL: √ |√ |2 Vout (Ω) = 𝜀3 | 𝜀1 + 𝜆 (1 − 𝜀1 )𝜀2 | Vin (Ω) | | √ |√ |2 + 𝜀3 | (1 − 𝜀1 ) − 𝜆 𝜀1 𝜀2 | V1 | | |2 | √ + 𝜀3 |𝜆 (1 − 𝜀2 )| V2 | | + (1 − 𝜀3 )V3 (6.50) where the electronic gain 𝜆(𝜔) is in general a complex number. Vin (Ω) is the amplitude noise spectrum of the input field. The vacuum noise spectra due to the beamsplitter V1 , the in-loop detector efficiency V2 , and the out-of-loop losses V3 are shown separately to emphasize their origins. All vacuum inputs are QNL, i.e.

225

6 Lasers and Amplifiers

ε2

Dil

Filter

Amplifier Dout

PBS Signal

ε1 λ/2

EOM AM

δv1

(A)

ε3

–73 Input signal

2.8 dB (i)

–78 Measured noise power (dBm)

226

(ii)

–83 –75 Output signal

(c)

2.6 dB

–80 (a)

–85 (b)

–90 5 (B)

10

15

20

25

Frequency, f (MHz)

Figure 6.10 (A) Schematic of the feedforward experiment. Dil , in-loop detector; Dout , out-of-loop detector; PBS, polarizing beamsplitter; 𝜆∕2, half-wave plate; AM, amplitude modulator. (B) Top: noise spectra of the squeezed input beam. Bottom: noise spectra of the output beam. (a) Direct detection of the input light after lossy (86%) transmission. (b) Output noise spectrum without feedforward (effectively 98.6% loss). (c) With optimum feedforward gain the signal and noise are amplified with little loss of SNR. Source: After Lam et al. [19].

V1 = V2 = V3 = 1. Due to the opposite signs accompanying the feedback parameter 𝜆 in Eq. (6.50), it is possible to amplify the input noise (first term) while cancelling the vacuum noise from the feedforward beamsplitter (second term). The third and fourth terms of Eq. (6.50) unavoidable experimental √ represent √ losses. In particular if we choose 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , the vacuum fluctuations from the beamsplitter, V1 , are exactly cancelled. Then, under the optimum condition of unit efficiency detection and negligible out-of-loop losses (𝜀2 = 𝜀3 = 1),

6.4 Measurement-Based Amplifiers

we find Vout (Ω) =

1 V (Ω) 𝜀1 in

(6.51)

That is, the fluctuations are noiselessly amplified by the inverse of the beamsplitter transmittivity. Hence the system ideally can retain the SNR for a signal gain of G = 1∕𝜀1 . This system was used to demonstrate low noise amplification of amplitude squeezed light (see Chapter 9). The second harmonic output from a singly resonant frequency doubling crystal (Section 9.4.3) provided the input beam. The top half of Figure (6.10B) shows the input noise spectra. Trace (i) shows the QNL, and trace (ii) is the noise spectrum of the input light. Regions where (ii) is below (i) are amplitude squeezed. The maximum measured squeezing of 1.6 dB is observed in the region of 8–10 MHz on a 26 mW beam. A small input modulation signal is introduced at 10 MHz that has SNRin = 1.10 ± 0.03. Other features of the spectra include the residual 17.5 MHz locking signals of the frequency doubling system (see Section 9.4.3 and the low-frequency roll-off of the photodetector, introduced to avoid saturation due to the relaxation oscillation of the laser. The bottom half of Figure 6.10B shows the noise spectra obtained from the single output detector. The input beam is made to experience 86% downstream loss. As trace (a) shows, the SNR is greatly reduced by the attenuation. Signal amplification can be performed by first introducing a beamsplitter of reflectivity 90%. This further attenuates the output beam to 𝜀tot = 𝜀1 𝜀3 = 0.014. With no feedforward gain, as trace (b) shows, the modulation signal is completely lost, and the trace itself is QNL to within 0.1 dB over most of the spectrum. Finally, by choosing the optimum feedforward gain, trace (c) shows the amplified input signal with SNRout = 0.82 ± 0.03 and G = 9.3 ±0.2 dB. Note that both (b) and (c) are of the same intensity; hence the output signal is now significantly above the QNL. This means the amplified output is far more robust to losses than the input. The results are best described by the transfer coefficient Ts = SNRout ∕SNRin . The spectra in Figure 6.10B corresponds to a value of Ts = 0.75 ± 0.02, in good agreement with the theoretical result of Ts = 0.77. This is to be compared with the performance of an ideal phase-insensitive amplifier, with similarly squeezed light, of Ts ≈ 0.4. Signal amplification via feedforward is a versatile technique that can be adapted to various different applications. For example, by replacing the homodyne detector in the feedforward loop with a dual homodyne detection set-up, one can implement phase-insensitive linear amplification. In particular, now the light reflected off the beamsplitter is split again on a 50 : 50 beamsplitter and sent to two homodyne detectors, one of which measures the amplitude quadrature and the other the phase quadrature. The signals are fed forward to amplitude and phase modulators, respectively, beam. By choosing the √ in the transmitted √ feedforward gains to both be 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , one can obtain (in the unit efficiency limit) the input/output relation √ â out =

1 â + 𝜖1 in

√

1 − 1 v̂ † 𝜖1

(6.52)

227

228

6 Lasers and Amplifiers

Given the substitution G = 𝜖1 , Eq. (6.52) becomes formally equivalent to Eq. 1 (6.21). Hence quantum-limited phase-insensitive amplification can be implemented in this way without needing a laser gain medium (and avoiding various technical noise sources associated with gain media). The vacuum mode, v̂ , leading to the quantum amplification noise, is coupled in via the dual homodyne beamsplitter in this case. An experimental demonstration of this technique was carried out by Josse et al. at the University of Erlangen [20]. Coherent state sideband states at 14.3 MHz on a 1064 nm Nd:YAG laser were amplified using this method, achieving transfer coefficients within about 10% of the theoretical limit (see Eq. (6.22)) and in good agreement with the theory when finite detection efficiency is included. For many applications we are only interested in the amplification of the quantum state carried on the sidebands (see Section 4.5.5), i.e. the signal. However, if we are considering an optical beam with a bright carrier, an unavoidable consequence of this signal amplification technique is the reduction in intensity of the carrier beam. It turns out that injection locking can be used to amplify the intensity of the output beam without affecting the signal or noise (see Section 8.5). When the signal frequency is much larger than the linewidth of the laser cavity, the output spectrum of the injection-locked field is the same as the input spectrum (Vout = Vin ). Because the injection-locked output is of a higher optical intensity, we can thus regain the optical intensity lost by the feedforward amplification scheme. This was demonstrated by E. H. Huntington et al. [21] using the output of the feedforward set-up to injection lock a Nd:YAG nonplanar ring oscillator. A transfer coefficient of Ts = 0.88 ± 0.05 with signal gain of G = 9.3 ± 0.2 dB was obtained for a range of output powers from 14 to 350 mW. Hence with this set-up it is possible to independently vary the amount of signal and carrier amplification with minimal loss of signal-to-noise ratio. 6.4.2

Heralded Measurement-Based Amplifiers

So far we have seen how measurement-based amplifiers can mimic the performance of quantum-limited amplifiers but without the technical difficulties associated with non-linear optical media. We now consider a situation in which the measurement result obtained heralds whether the amplification has been successful or not. Because such an amplifier is nondeterministic, i.e. only certain results herald success, it is able to exceed the quantum limits that apply to deterministic amplifiers. Consider a device that performs the transformation |𝛼⟩⟨𝛼| → 𝜌(𝛼) = P|g𝛼⟩⟨g𝛼| + (1 − P)|0⟩⟨0|

(6.53)

where g is a real number obeying |g| > 1, |𝛼⟩ is an optical coherent state with complex amplitude 𝛼, and P is a probability. The RHS of Eq. (6.53) is a mixed state, but we assume a heralding signal identifies which term in the output density operator has been produced by any particular run of the device: the first term being produced by a successful run and the second term by an unsuccessful run. Thus with probability P the coherent amplitude of the input is amplified

6.4 Measurement-Based Amplifiers

whilst keeping the heralded state a pure coherent state. The quantum limits discussed in the previous sections would clearly be violated if this occurred with P = 1. What probability of success is allowed? The linearity of quantum mechanics requires that the distinguishability of two quantum states cannot be increased by any transformation. This condition is equivalent to requiring that the magnitude of their overlap does not decrease under any transformation. Consider the input states |0⟩ and |𝛼⟩. We require |⟨0|𝛼⟩|2 ≤ ⟨0|𝜌(𝛼)|0⟩ = P|⟨0|g𝛼⟩|2 + 1 − P. ′ 2 Inserting the values of the overlaps using |⟨0|𝛼 ′ ⟩|2 = e−|𝛼 | and taking the limit of small 𝛼, we find 1 (6.54) P≤ 2 g We conclude that provided P is bounded according to Eq. (6.54), then the transformation of Eq. (6.53) is physically allowed for small coherent states. The transformation of Eq. (6.53) was discussed by Ralph and Lund who referred to it as heralded noiseless linear amplification (NLA) [22]. They proposed a method for realising NLA based on single-photon sources, single-photon detection, and linear optics. The simplest NLA circuit is shown schematically in Figure 6.11a. The optical mode to be amplified is interacted with a single-photon ancilla as shown. This interaction is a generalization of the quantum scissors introduced by D. Pegg et al. [23]. The interaction is successful if one and only one photon is counted at one or other of the indicated ports. |ψ〉 |0〉

|0〉

Ns p l i t t e r

Tˆ |ψ〉

Ns p l i t t e r

A A

A

〈0|

〈0|

(a)

|in〉 A

=

a″

a′

〈1|〈0| or 〈0|〈1|

a′″

|0〉

|out〉

η |1〉 (b)

Figure 6.11 Schematic of the noiseless linear amplifier. (a) The interaction labelled ‘A’ interacts a single-photon ancilla with an input beam as shown. The upper beamsplitter is 50 : 50, whilst the lower beamsplitter has transmission 𝜂 as shown. The interaction succeeds if a single count is recorded at a′ and no count at a′′ , or vice versa. (b) The N-splitter is an array of beamsplitters that evenly divides the input beam. The second N-splitter coherently recombines the beams and is considered to have succeeded if no light exits through the other ports, as determined by photon counters. Each arm contains the interaction shown in (a).

229

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6 Lasers and Amplifiers

Consider the effect of this device on a small input coherent state, i.e. |𝜓⟩ = |𝛼⟩, where we assume 𝛼 ≪ 1. The action of the generalized quantum scissor is to truncate the coherent state to first order and simultaneously amplify it. Specifically, detection of a single photon at output port a′ and zero photons at output port a′′ , or detection of a single photon at output port a′′ and zero photons at output port a′ , produces the transformation ) √ √ ( 2 𝜂 1−𝜂 † − |𝛼|2 |𝛼⟩a′ → e (6.55) â 𝛼 |0⟩a′′′ 1± 2 𝜂 where the plus sign corresponds to the former case and the minus sign to the latter case. If the latter case occurs, the phase flip can be corrected by feeding forward to a phase shifter. Equation (6.55) is approximately equivalent to 1

(1−g 2 )|𝛼|2

(6.56) |𝛼⟩ → 𝜂 2 e− 2 |g 𝛼⟩ √ where g = (1 − 𝜂)∕𝜂, provided that g𝛼 ≪ 1 holds. Thus for 𝜂 < 1∕2 we have g > 1, and hence the device in Figure 6.11a achieves NLA as per Eq. (6.53). The probability of success is given by the norm of the state P = 𝜂e−(1−g

2

)|𝛼|2

≈

1 g2 + 1

(6.57)

in agreement with the condition of Eq. (6.54). This device can be scaled up such that it can amplify larger coherent states as depicted in Figure 6.11b; however the complexity of the set-up increases rapidly, and the probability of success decreases rapidly [22]. The basic operation of the NLA in the low photon number regime was first demonstrated by G.Y. Xiang et al. [24], F. Ferreyrol et al. [25], and A. Zavatta et al. [26]. The first two publications implemented the quantum scissor approach described above. However, Zavatta et al. used a different approach based on single-photon subtraction and addition that was able to noiselessly amplify somewhat larger photon number states with high fidelity. The NLA has been proposed as a tool for improving various quantum information protocols, and such demonstrations are discussed in Chapter 13.

6.5 Summary In this chapter we have examined the theory of laser oscillators, the conditions under which real lasers behave like the simplified theory, and the characteristics of the laser light under those conditions. We conclude that well-built lasers can produce output beams that behave approximately like coherent states. We call these QNL lasers or beams. We considered the theoretical limits to amplification of quantum limited light and discussed the theory of the non-linear 𝜒2 interaction. We found a general description of this process and considered the specific examples of parametric amplification, parametric oscillation, and pair production. We also discussed measurement-based quantum amplifiers based on electro-optic feedforward and heralded devices based on photon counting. The latter type could circumvent

References

the quantum limits to amplification because they are nondeterministic. Experiments based on examples discussed in this chapter and many variations on these themes will feature strongly in the following experiments in quantum optics.

References 1 2 3 4 5

6 7

8 9

10 11 12

13 14 15 16 17

18

19

Siegman, A.E. (1986). Lasers. University Science Books, and many reprints. Einstein, A. (1917). Zur Quantentheorie der Strahlung. Phys. Z. 18: 121. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics. Wiley. Koechner, W. (1999). Solid State Laser Engineering, 5e. Springer-Verlag. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Ralph, T.C. (2003). Squeezing from lasers. In: Quantum Squeezing (ed. P. Drummond and Z. Ficek), 141. Springer-Verlag. Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1997). Intensity-noise dependence of Nd:YAG lasers on their diode-laser pump source. J. Opt. Soc. Am. B 14: 2936. Yariv, A. (1989). Quantum Electronics. Wiley. Inoue, S., Ohzu, H., Machida, S., and Yamamoto, Y. (1992). Quantum Correlations between longitudinal-mode intensities in a multi-mode squeezed semi-conductor laser. Phys. Rev. A 46: 2757. Marin, F., Bramati, A., Giacobino, E. et al. (1995). Squeezing and inter-mode correlations in laser diodes. Phys. Rev. Lett. 75: 4606. Strickland, D. and Mourou, G. (1985). Compression of amplified chirped optical pulses. Opt. Commun. 56: 219. Di Nicola, J.M., Bond, T., Bowers, M. et al. (2018). The national ignition facility: laser performance status and performance quad results at elevated energy. Nucl. Fusion 59, 3. Haus, H.A. and Mullen, J.A. (1962). Quantum noise in linear amplifiers. Phys. Rev. 128: 2407. Caves, C. (1982). Quantum limits on noise in linear amplifier. Phys. Rev. D 26: 1817. Haus, H.A. (2000). Electromagnetic Noise and Quantum Optical Measurements. Springer-Verlag. Vogel, W. and Welsch, D.-G. (1994). Lectures on quantum optics. Berlin: Akademie Verlag. Goobar, E., Karlsson, A., and Björk, G. (1993). Experimental realization of a semiconductor photon number amplifier and a quantum optical tap. Phys. Rev. Lett. 71: 2002. Roch, J.-F., Poizat, J.-Ph., and Grangier, P. (1993). Sub-shot-noise manipulation of light using semiconductor emitters and receivers. Phys. Rev. Lett. 71: 2006. Lam, P.K., Ralph, T.C., Huntington, E.H., and Bachor, H.-A. (1997). Noiseless signal amplification using positive electro-optic feedforward. Phys. Rev. Lett. 79: 1471.

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20 Josse, V., Sabuncu, M., Cerf, N.J. et al. (2006). Universal optical amplification

without nonlinearity. Phys. Rev. Lett. 96: 163602. 21 Huntington, E.H., Lam, P.K., Ralph, T.C. et al. (1998). Independent noiseless

signal and power amplification. Opt. Lett. 23: 540. 22 Ralph, T.C. and Lund, A.P. (2009). Nondeterministic noiseless linear amplifi-

cation of quantum systems. AIP Conf. Proc. 1110: 155. 23 Pegg, D.T., Phillips, L.S., and Barnett, S.M. (1998). Optical state truncation by

projection synthesis. Phys. Rev. Lett. 81: 1604. 24 Xiang, G.Y., Ralph, T.C., Lund, A.P. et al. (2010). Noiseless linear amplification

and distillation of entanglement. Nat. Photon. 4: 316. 25 Ferreyrol, F., Barbieri, M., Blandino, R. et al. (2010). Implementation of a non-

deterministic optical noiseless amplifier. Phys. Rev. Lett. 104: 123603. 26 Zavatta, A., Fiurášek, J., and Bellini, M. (2011). A high-fidelity noiseless ampli-

fier for quantum light states. Nat. Photon. 5: 52.

233

7 Photon Generation and Detection What are photons? In the simplest language, they are what make a photon counter click. A practical working definition of a photon links directly with the process of detection. Photons are not directly detected in the laboratory: they are nearly always converted to electrical signals, and it is these events that are detected and analysed. In this way we can recover the information that is encoded on the light and is carried by the photons. Light propagates as a high frequency electromagnetic wave, and, as with all such fields, there are two methods of detection: field detection, where both amplitude and phase information can be measured, and intensity detection, which yields no phase information. In the radio, microwave, and mid-infrared (mid-IR) regions of the spectrum, field detection can be done directly. In the near-infrared (near-IR), visible, and ultraviolet (UV) regions, field detection is achieved indirectly via homodyne or heterodyne techniques: the light field is interfered with a reference field of known amplitude and phase of either the same frequency (homodyne) or shifted frequency (heterodyne), and the resultant field is detected with intensity detectors, as described in Section 8.1.4. Thus in the optical region of the spectrum, intensity detectors are used for both field and intensity detection. Broadly speaking there are two different ways of recording light intensity: we can detect individual photons or we can measure currents generated by a stream of photons. The time response of the photodetector and the magnitude of the photon flux determine which mode of operation is suitable. Both technologies exist in parallel and are used for different purposes.

Figure 7.1 The origin of quantum noise. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Table 7.1 Comparison of different detection modes, the type of detection, the output results, the processing required, and the applications made possible with this type of detection. Type

Detection

Result

CW weak

Fast, single photon

|0⟩, |1⟩, …

Many samples Pulsed

Fast, synchronized

weak

Single photons

|1⟩ on demand

Processing

Application

Store

Q-info, Q-logic

each event

QKD

Store

Q-info, Q-logic

each event

Many samples CW strong

Slow

Modulation

Cont. homodyne

Sensors, anologue +

averages

Noise variance

Record + FFT

dig. communication

Wigner function

Tomography

QKD

Pulsed

Slow

Modulation

Cont. homodyne

Sensors, analogue +

strong

averages

Noise variance

Record + FFT

dig. communication

Δtmeas ≫ Δtrep

Wigner function

Tomography

Hybrid

Fast, CW, and

Negative

Cont. homodyne

single photon

Wigner function

and post-selection

Many samples

Cat states

Tomography

Q-info, Q-logic

Table 7.1 provides an overview of the types of detection that are required for the different applications. All of these are now possible with the different photodetectors and processing electronics that are described in this chapter (Figure 7.1). If the photon flux is low enough and the detector is fast enough, with a recording time Δtmeas shorter than the time intervals between the photons, we can record the individual events, resulting in a sequence of numbers, typically 0 for no photon and 1 for at least one photon in each of the counting intervals. This would be the most fundamental way of sending information using this elementary quantum state of light. In addition photons can carry more complex information that is encoded in the quantum state using additional degrees of freedom such as polarization, momentum, or frequency. In these cases we use photons in beams that have several independent modes, for example, a beam with orthogonal polarizations, with multiple spatial modes or optical frequencies. Photons are ideal for these types of applications; since they travel at the speed of light (flying qubits), they interact only weakly with the environment, even over long distances, and photons can be manipulated very efficiently and with linear and non-linear optics. By directing the modes to separate detectors, we can record the state of the qubit. Many quantum communication protocols make use and demand exactly one single-photon travelling in a mode, or channel, and more than one photon recorded in one click would compromise the results. For this reason increasing effort is made to both create single photons on demand and to have detectors that can alert to the presence of additional photons that could compromise the security of the communication. Ideally, we would have detectors

Photon Generation and Detection

that can distinguish the number of photons in every counting interval and could then work with multiphoton states. As an alternative to photon counting, we can integrate the individual events and continuously record a photocurrent i(t) that varies in time. We average during the measurement interval Δtmeas over a large number of photons and analyse the optical power P(t) that contains information in the form of a slowly varying modulation of the light, up to a maximum modulation frequency Ωmod < 1∕2Δtmeas . There will always be some fast fluctuations, or noise, and the more accurate the information measured in the photocurrent will get, the more photons have been detected. We can characterize the statistics of the fluctuations by recording and analysing large sets of these short measurement intervals. Depending on the nature of the light source, the photons arrive with quite different temporal distributions: for example, thermal (a natural light source), Poissonian (a coherent light source such as a laser), or regularly spaced in time (a Fock state or single-photon source). The information of the recorded power in a particular beam of light is frequently not dependent on the underlying statistics of the light source. Indeed, many quantum optics phenomena based on self-interference are insensitive to the photon arrival statistics with the general caveat that the light is sufficiently attenuated such that the chance of a multiple photon count in any given measurement interval is very low. Examples include quantum key distribution (QKD) and ‘interaction-free’ measurements, which are discussed in Chapters 12 and 13. In principle, photons are detected as discrete events, and the experiment is about registering the place and time of arrival of these events. In arguably the first quantum optics experiment, G.I. Taylor attenuated a thermal light source down to the single-photon level, passed the light through a double slit, and recorded the photon arrival positions as an image on photographic film, as discussed in Chapter 3. It was the race to develop television that led to the development of the photomultiplier in the 1930s, which allowed the multiplication of the charge created by a single photon into a useful electronic pulse that could be made audible as a click or as a dot on a screen [1, 2]. In the 1960s the Geiger-mode avalanche photodiode (APD) made solid-state optical single-photon detection possible [3]. Many more innovations, some in the last decade, have extended our detector technology. We can now combine specifically designed light sources with the corresponding detectors to cater for the various applications. This includes imaging, sensing with strong laser beams or a few photons, and reading optical memories. It could be classical communication with either analogue or digital signals or quantum information applications such as secure communication with QKD, or it could be quantum logic, quantum memories, or potentially quantum computation. For each case there are now sources and detectors available. Many sophisticated applications involve the interference of several modes, which are all individually in a single photon state. These include non-classical interference, teleportation, and quantum computation as they are discussed in Chapter 13 on single-photon experiments and quantum information. In these experiments it is critical that the various modes are indistinguishable. That is, they have identical distributions in energy, which means photon number and frequency, momentum (spatial modes), and time. In general the modes must

235

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also have states of definite photon number, i.e. Fock states. In the case of pulsed modes, this requires pulses that are identical from pulse to pulse. Finally we want to know when such a pulse arrives, providing us with a source for photons on demand, which would be the ultimate quantum light source [4].

7.1 Photon Sources In practical terms, it is not easy to build a source that meets all these requirements. Approaches tried to date have been numerous and include single atoms [5], colour centres [6], molecules [7], atomic ensembles [8], quantum wells, and quantum dots [9]. We can describe and classify photon sources by a number of critical parameters and compare them with the requirements for the ideal single-photon source. Are they deterministic or probabilistic? Probabilistic sources, such as spontaneous down-converters, produce pairs of photons that are entangled but appear at random times. They can be used as heralded sources, where one photon is used to determine the time when the other photon can be observed. Many of the more recent approaches hold the promise of a deterministic photon source where the photons arrive at predetermined times or can be triggered from the outside [4], allowing the use of a range of advanced quantum information protocols. To date in 2018 we are approaching the goal of building such an idealized photon gun [10, 11]. At what temperature do they operate? Some materials work well close to room temperature (300 K), and some require precise temperature control at an elevated temperature in order to optimize the non-linear processes through phase matching. Others, in particular those designed for photon number resolution, work with ultra-cold atomic samples; individual atoms or ions trapped in a vacuum might require cryogenic conditions. In which wavelength region can they be used, and are they broadband or restricted to an atomic transition? Whilst the non-linear crystalline materials have broadband characteristics and work in the visible regime, some require near-IR wavelengths or near-UV wavelengths. Some sources are tunable and can be used in systems that include optical materials with specific spectral properties, in particular atomic samples or specific organic materials such as dyes or even biological samples. What is the efficiency of extracting the photon from the source and sending it into the detector, including any spectral filtering that would be necessary for a typical quantum information application? This property determines the choice of protocol to be used. A low efficiency makes the apparatus slow. Whilst we can only process a small subset of the pulses, the results otherwise do not change. In deterministic sources we know that we will have only one photon at a time, but we do not know exactly when it works. The ideal photon on-demand source has close to 100% efficiency. Do they produce a good single spatial mode? This is an important ability because if the probability of finding the single photon is spread across incoherently related modes, the photon state will be mixed. In many situations, we want

7.1 Photon Sources

to overlap the different modes at the output to create a quantum interference at the output of the apparatus. The interference will be maximized if each source operates in a single spatial and frequency mode. One measure of the quality of the single-photon source is the g (2) function. Ideally the value should be zero, which indicates that no more than one photon is contained in each of the outputs from the source and the apparatus is perfect. Values as low as 4 × 10−4 have been achieved, which is extremely useful in practical terms and almost closes potential loopholes for quantum demonstrations. A faint laser would be the simplest approach. It allows to direct all the light on to the detector and can create a single mode. However, photon arrival times are random and thus g (2) = 1. Many sophisticated sources are now available, as the review written by Eisaman et al. shows very clearly [12]. Table 7.2 lists the most common and advanced sources and presents their typical specification. However, note that g (2) says nothing about the efficiency or modal purity of the source. The most common technology used in quantum optics experiments to date is still spontaneous parametric down-conversion (spontaneous PDC) and is probabilistic. PDC sources are bright with high photon count rates, can produce clean optical beams with well-defined frequency and spatial characteristics [13–16], and can produce tunable entangled modes [17, 18]. In a spontaneous PDC, high frequency photons are passed through a non-linear crystal, where they are converted down into two lower frequency daughter photons, born within 10s of Table 7.2 Comparison of different photon sources and their specifications.

Source

Type

Temperature (K)

Wavelength

Efficiency

Mode

g(2)

Faint laser

Prob

300

vis-IR

1

Single

1

PDC bulk

Prob

300

vis-IR

0.6–0.8

Multi

0.0014

PDC pp

Prob

350 ctr

vis-IR

0.85

Multi

—

PDC waveguide pp

Prob

350 ctr

vis-IR

0.07

Single

0.0007

FWM in fibre

Prob

300

vis-IR

0.02

Single

0.02

PDC multiplexed

Prob/Det

300

vis-IR

0.1

Single

0.08

Q-dot (GaN)

Det

300

340–370 nm

<0.1

Multi

0.4

Q-dot (InAs) cav

Det

5

920–950 nm

0.1

Single

0.02

Single-ion cav

Det

0

Atomic line

0.08

Single

0.015

Single-atom cav

Det

0

Atomic line

0.05

Single

0.06

Colour Centre (NV)

Det

300

640–800 nm

0.022

Multi

0.07

a) This table describes the state of the art in about 2012 as summarized by Eisaman et al. [12]. The properties are the operating temperature, the wavelength regime that can be covered, the mode matching efficiency into the detector, whether it can produce a single or multiple spatial mode, and the observed two-photon correlation g (2) . Source: After Eisaman et al. [10].

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femtoseconds of one another, and so are tightly correlated in time. Their energy and spatial distributions are a common function of the pump mode and crystal mode matching characteristics and are also highly correlated. In fact, they are entangled. The tight time correlation means that if a daughter photon is detected in one mode, then any photon detected within a short time interval in the other arm is likely to be its twin – given the source brightness, it is unlikely that there are unrelated photons in that time. Thus the information gathered by one mode, the heralded photon, can be used to condition the other, so that it is a good approximation of a true single-photon source – albeit one with probabilistic arrival times. This conditioning can be a feedforward circuit that unblocks the optical path on detection of a photon in the twin mode or more commonly a coincidence circuit that post-selects events from one mode based on results from the other. Indeed, if two single photons are required in an experiment, it is not necessary to set up two PDC sources. Instead both modes of one source can be used, where only events detected in coincidence are post-selected. Such photon coincidences can be registered with commercial photon coincidence counters, which are typically limited to a 5 ns coincidence window, or with fast counting instrumentation used in particle physics, where the coincidence window can be sub-nanosecond. The advantage of a short coincidence window is that it allows screening of background counts due to extraneous light sources, particularly absorption-induced fluorescence, which can be a problem at short wavelengths. Currently, the lower limit to coincidence window length is the combined jitter of the photodetector digitization circuit and the coincidence circuit, which is in the order of a few nanosecond. Coincidence windows lower than this can result in lost coincidence counts. With sufficient control of the timing of the photodetector through time gating, it is possible to enhance the detection of individual photons and to suppress the probability of detecting two photons in one interval. The PDC can be built from different materials. The most common approach was to use bulk crystals such as BBO (beta-barium borate), KD*P (potassium dideuterium phosphate), LiNbO3 (lithium niobate), and LiIO3 (lithium iodate). Phase matching is frequently achieved through angular tuning. Such devices are very robust and achieve high photon rates. However, they tend to generate complex output modes. Using periodically poled materials, such as periodically poled potassium titanyl phosphate (PPKTP), increases the efficiency and allows a large fraction of the photons to be captured. Here the phase matching is achieved through careful temperature control [19]. By configuring the periodically poled material in the form of a waveguide, single spatial mode operation can be achieved. Whilst this reduces the efficiency of generating photon pairs and the output rate is relatively low, the quality of the correlation can be extremely good, and the output mode is well matched to other components of the experiment. In addition, many experiments have demonstrated the generation of photon pairs using the process of four wave mixing (FWM) in fibres, using non-linear materials such as birefringent single-mode fibres [20] or photonic crystal [16] with inbuilt filters. One limitation of these sources is a considerable single-photon background due to Raman scattering that has to be actively suppressed.

7.1 Photon Sources

7.1.1

Deterministic Photon Sources

A promising concept is to multiplex several spontaneous sources, such as several PDCs, by operating them in parallel in a regime of reasonably low emission probability and by emitting into several modes. Following a pulse it is likely that at least one of the PDCs will release a pair. A set of heralding detectors identifies the successful PDC, and the heralded photon can be guided to the output. As a consequence, we will have a single photon and obtain it with a higher probability than achievable with a single device. In principle it is possible to integrate the entire group of converters using waveguide technology, and the plan is to have one self-contained device with one output with characteristics that are close to deterministic [21–23]. There is also a whole range of techniques to build truly deterministic single emitter sources, each using a different material. They all rely on some external control that is used to put the system into an excited state that will emit a single photon, with short random delay, into a lower energy state. Often optical cavities are used to enhance the coupling between the atoms in the sample and the pump light. This also helps to engineer the source such that the photon emerges in a well-specified spatial mode of the cavity. Semiconductor quantum dots have been developed and studied for a considerable time, and their specifications are steadily improving. They are essentially small 3D islands of a semiconductor material with a smaller band gap imbedded in a larger band-gap semiconductor. The small size of the dot results in a discrete energy structure for the electrons and holes, resembling the energy level structure of an atom. In the weak excitation regime, an exciton (electron–hole pair) can be produced on demand, and the radiative recombination results in the emission of a single photon. Q-dots can be pumped electrically or optically. In the optical case the saturation of the transition provides the required control of event. In the electric case the effect of Coulomb blockage allows the temporal control of the movement of the charges, creating one electron at a time [24]. Q-dots can be placed between distributed Bragg reflection (DBR) mirrors, or inside micro-cavities and photonic crystal cavities, and the rate of spontaneous emission can be significantly enhanced. The main disadvantage so far is the need for cryogenic operation and a fairly low g (2) value. In addition, different Q-dots have different spatial properties. Even for a single Q-dot, its spectral properties tend to drift in time, making the output effectively multimode since the photons are distinguishable. Single neutral atom photon emitters make use of the strong coupling regime in quantum electrodynamics. Here the cavity and the atom form a coupled system, and the emission of a single photon by the atom has a profound effect and enhances the emission of the photon into a Gaussian spatial mode defined by the cavity. Atoms such as caesium or rubidium are captured and cooled in a magneto-optical trap (MOT) and then transferred, normally through free fall, into a high finesse optical cavity, where a single atom is trapped. With a pump laser beam of appropriate intensity and temporal control, the combined atom–photon state and thus the emission of photons can be controlled. The efficiency of single-photon generation inside the cavity can be close to unity,

239

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7 Photon Generation and Detection

although the losses for exiting the cavity are significant. Such a system can produce g (2) values of no less than 0.06 [25]. Single ions have similar energy level system to neutral atoms, and excitation by Raman scattering is possibly the most efficient path to single-photon generation. Radio frequency (RF) traps can be used to localize one ion inside an optical cavity, and thus the likelihood of multiphoton emission is avoided. g (2) values as low as 0.015 have been achieved. One disadvantage is the presence of strong spontaneous emission channels, which compete with the coherent control of the ion. There has been significant progress in positioning several individual ions in compact structures [26–28] to design scalable multi-ion systems for quantum information processing. The technical limitation seems to be in the low emission efficiency and the fact that the strong coupling regime is not accessible with ions. Colour centres are individual ions within crystals that have the appropriate modified level structure for single-photon emission. Most common are nitrogen-vacancy (NV) colour centres in diamond, which are locations where a nitrogen atom and a vacancy are substituted at adjacent lattice positions [29]. These ions can have long-living shelving states and control of the emission. The NV centres have the advantage that they are photostable and do not show bleaching. So far their g (2) values of about 0.1 are less impressive, and the disadvantage is that they, like Q-dots, are not identical [30]. However, with so many different types of colour centres available, significant improvements can be expected in the future.

7.2 Photon Detection In this field considerable progress has been made in the last 10 years, and the technologies continue to develop rapidly. Here we discuss the principles of detection, the different technologies for individual photons and for photocurrents. We want to provide guidance to answer the question: how can we avoid as much of the technical imperfections as possible and get close to detecting the actual electromagnetic field with all its quantum features? 7.2.1

Detecting Individual Photons

The photons in the field carry an energy proportional to the field frequency, 𝜈, specifically, E = h𝜈, where h is Planck’s constant. For visible light, frequencies are on the order of 1015 Hz, meaning the energies of photons in this region are on the order of 10−19 J – a very small amount of energy indeed. To detect single photons a mechanism is needed to convert this small amount of energy into a macroscopic signal that can be detected in the laboratory. Three mechanisms are used in current photon counters: photochemical, where the incoming photon triggers a measurable chemical change; photoelectric, where the photon is converted to an electron, and this then produces a measurable current or voltage; and photo-thermal, where the photon is converted into heat and vibrational excitation, and this produces a measurable temperature change that in turn influences

7.2 Photon Detection

secondary characteristics such as electrical properties. A fourth mechanism has been proposed but not yet demonstrated as a photon counter: photo-optical, where a single photon is converted to a many-photon cascade, which is then detected via one of the above mechanisms [31]. 7.2.1.1

Photochemical Detectors

The earliest single-photon detectors were photochemical: the energy of a single photon is sufficient to lead to a chemical change in a single grain of photographic film [32]. Film has the advantages of high sensitivity, extremely low noise (spontaneous changes in the grains are very rare), and good spatial resolution. The chief disadvantage of film is that it does not produce an electronic signal output – it must be chemically and mechanically processed, and this does not allow for real-time monitoring of an experiment. Thus it is rarely used in modern quantum optics experiments except as an inexpensive imaging device, e.g. to measure the spatial variation of a photon source [33]. 7.2.1.2

Photoelectric Detectors

In contrast to film, fast electrical signals are provided by a variety of photoelectric detectors, all built using different techniques and with different nonideal specifications that make them suitable for different applications. The technology is still improving, and improved components will appear on the market. A good overview up to 2011 was given by Eisaman et al. [12] that summarizes the technology and specifications we have available. There are a number of technical solutions and associated acronyms. The most ubiquitous is the photomultiplier tube (PMT). A PMT works via the photoelectric effect, where light incident on a metal surface in a vacuum liberates electrons from the surface (as long as the energy of the photon is greater than the binding energy of electrons to the metal – the work function). Typically a single electron is liberated by a single photon and would be equally difficult to detect. However, in a PMT there is a series of metal plates placed by the photosensitive surface. Each plate is at an increasingly positive voltage with respect to the one before it, such that when a single electron strikes a plate, it liberates several electrons. A relatively small cascade of plates thus leads to a large number of electrons, which can be detected as a macroscopic current pulse (this is the origin of the multiplier part of the name PMT). This mode of operation is sometimes known as Geiger mode, as early Geiger counters worked in a similar fashion. Current commercial PMTs can operate quickly (with linear outputs up to 10 million counts per second), suffer moderate dark noise (>10 dark counts per second with moderate Peltier cooling), and have a high spectral bandwidth (detecting light from 185 to 900 nm). Unfortunately, they are not very efficient at turning photons into electrons: 𝜂 ranges from fractions of a percent at high wavelengths (600–900 nm) to a few percent at low wavelengths (185–300 nm) to tens of percent near the peak wavelengths. Currently the best commercial PMT achieves 𝜂 ∼ 40% at 550 nm. A photon counter with higher efficiencies than the PMT is the APD used in a regime where it operates as single-photon avalanche diode (SPAD). An APD works via the internal photoelectric effect, where light incident on a semiconductor surface promotes an electron from the valence to the conduction band,

241

7 Photon Generation and Detection

Electrons

Energy

242

Ebg

hν

Figure 7.2 Semiconductor and band gap.

Photon

Holes

as illustrated in Figure 7.2. Of course, this requires a material with a band gap smaller than the photon energy, h𝜈 – a difficult task in certain regions of the spectrum. As with a PMT, a single electron is typically liberated by a single photon, so SPADs are also run in Geiger mode to amplify the single electrons to macroscopic current pulses. Commercially available APDs are often packaged with an amplifier, a discriminator, and a transistor–transistor logic (TTL) output driver, which turns the current pulses into pulses with well-defined height and rise time, so-called TTL voltage pulses (defined as Logic 1 = 2–5 V and Logic 0 = 0–0.8 V), which make them very user-friendly. These detectors are also reasonably fast, with maximum counting rates of ∼10 × 106 photons/s, linear to ∼ 106 photons/s, and suffer moderate dark noise (>100 dark counts per second). The wavelength range is smaller than PMTs but decidedly more efficient, depending on the way they are designed. Examples of the possible peak efficiencies are for silicon SPADs are 50% at 450 nm, 70% at 600 nm, and 50% at 830 nm and for InGaAs SPADs, up to 33% at 1060 nm. An even more efficient device is the solid-state photomultiplier (SSPM). This is a semiconductor material doped with neutral atoms where the energy levels of the dopant atoms are very close to the conduction band of the semiconductor (e.g. Si doped with As has a 54 meV gap). Briefly, at an operating temperature of 6 ∼ 7 K, the electrons are frozen onto the impurity atoms, as they lack enough thermal energy to be excited into the conduction band. When an incoming photon is absorbed, it generates an electron–hole pair: the electron is accelerated to the anode, impact ionizing other impurities as atoms on the way and so generating an electron cascade within the device. Since little energy is needed to photoexcite an impurity atom so close to the conduction band, SSPMs are sensitive well into the infrared (IR): from 30 μm down to the visible, except for wavelengths in the silicon absorption band, which unfortunately includes the common telecom wavelengths. The broad IR sensitivity means that SSPMs must be carefully screened so that only light at the desired wavelength is coupled to the detector – or else ambient IR will quickly lead to detector saturation. So screened, and corrected for optical losses, the quantum efficiency of an SSPM in the visible has been measured to be 96±3% [17]. In an SSPM, photoabsorption and gain occur throughout the bulk of the detector: in a variant known as the visible light photon counter (VLPC), the absorption and gain regions are separated. This reduces

7.2 Photon Detection

the sensitivity to 1–30 μm radiation to ∼2% whilst maintaining the visible spectral response. At 694 nm, uncorrected quantum efficiencies of 88±5% have been measured [34], corresponding to corrected efficiencies of 95±5%. Unlike an APD or a PMT, the SSPM and VLPC can distinguish between one and several photons hitting the detector at the same time: an SSPM has been shown to distinguish two photons with 47% quantum efficiency [35]. Apart from requiring cryogenic operation, typically at 7 K, the other disadvantages to these systems are their cost and relatively high dark noise, typically 2 × 104 dark counts per second. They also have a remarkable wide wavelength range and thus require shielding from IR photons. 7.2.1.3

Photo-thermal Detectors

The superconducting transition-edge sensor (TES) detector operates as a bolometer where the energy of the absorbed photon is detected as a rise in temperature. In order to achieve the sensitivity required to detect a single photon, the heat capacity of the absorber has to be very small, and the sensor must create a large response to a very small temperature change. This is achieved by using a thin layer of superconducting material, deposited onto an insulating material. The device is operated at a temperature between superconducting and normal resistances, such that a minimal change in temperature yields a large change in resistance. The device is kept at the operating temperature by electrothermal control, which also produces the output current. The current sensitivity can be enhanced by using an array of superconducting quantum interference devices (SQIDs). Single-photon sensitivity was first demonstrated in the visible and IR [36]. To date TES detectors have recorded the highest efficiency so far [37, 38] and a very low dark count rate. However, they suffer so far from several drawbacks: the need to operate at about 100 mK and, until recently, a slow response of typically 100 ns, resulting in low maximum count rates of typically 100 kHz. Major technical advances can be expected soon. There are several further approaches that are being followed. They include heat-to-voltage conversion with digital readout (QVD), superconducting nanowire single-photon detectors, quantum dot field-effect transistor-based detectors, superconducting tunnel junction (STJ)-based detectors, up-conversion single-photon detectors, and the use of atomic vapours with lambda three-level systems as reviewed by Eisaman et al. [12]. Even quantum non-demolition sensing of photon number using materials with giant Kerr non-linearities [39] is being pursued. Research into all these devices is very active, and it seems rather likely that the quality of detectors, their applicability, and price will improve steadily, and thus we should assume that in the future, almost perfect single number receiving detection with small pixel size and imaging capability will be achievable and practical (Table 7.3). 7.2.1.4

Multipixel and Imaging Devices

Spatial images can be obtained photoelectrically, although typically the response time is orders of magnitude slower than that possible with a single detection element device like an APD. There are several technologies for achieving

243

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7 Photon Generation and Detection

Table 7.3 Comparison and typical specifications of different single-photon detector technologies: PMT, SPAD, SSPM, VLPC, and TES. Detector

Efficiency (%)

Temperature (K)

Dark counts (1/s)

Max counts (1/s)

PNR

PMT

40 at 500 nm

300

100

10 × 106

Some

SPAD

76 at 600 nm

300

200

16 × 106

None

SSPM

76 at 700 nm

50

7 000

10 × 106

Full

VLPC

88 at 694 nm

7

20 000

10

Some

TES

95 at 1556 nm

0.1

3

0.1

Full

Source: From Eisaman et al. [12].

this, perhaps the most common being the photon counting charge-coupled device (CCD). A traditional configuration is the intensified charge-coupled device (ICCD), where an image intensifier (e.g. a photoelectric photocathode, followed by an array of multichannel plates for electronic amplification, followed by a phosphor screen) is imaged, via free space or fibre bundle, to a low noise CCD camera. More recent devices dispense with the image intensifier, utilizing high voltage to produce an on-chip electron cascade, and thus gain. Current CCDs have a quantum efficiency in excess of 92% (with resolutions up to 1024 × 1024 pixels and an associated readout rate of 30 frames per second) and will doubtlessly improve. Figure 7.3 shows the output of a spontaneous down- conversion photon source imaged with an image- intensified photon counting CCD. The photoabsorbing surface in most photoelectric devices is a metal or semiconductor: high refractive index materials that intrinsically reflect a significant proportion of incident light (e.g. see Table 7.4). The quantum efficiency of photoelectric detectors can be substantially improved via the application of anti-reflection (AR) coatings that allow more light to be coupled into the bulk of the material, where the intrinsic quantum efficiency may be quite high, e.g. 𝜂 = 99% for Si [41]. In commercial single element devices, the AR coatings are relatively simple (e.g. a single layer of TiO2 ), and the improvement in quantum efficiency is modest. In commercial CCDs the situation is somewhat better: mid-band visible AR coatings are available that improve the efficiency from

Figure 7.3 Images acquired with a photon counting CCD camera: output of a type II spontaneous down-converter. Source: After [40].

7.2 Photon Detection

Table 7.4 Properties of photodiode materials. Si

Ge

InGaAs

Band-gap Ebg [eV]

1.11

0.66

≈ 1.0

Cut-off wavelength [nm]

1150

1880

≈ 1300

Peak detection wavelength 𝜆peak [nm]

800

1600

≈ 1000

Material quantum efficiency

0.99

0.88

0.98

𝜆min and 𝜆max for 𝜂mat = 0.5

600–900

900–1600

≈ 900–1200

Refractive index n at 𝜆peak

3.5

4.0

3.7

Reflectivity at normal incidence (%) Brewster’s angle (∘ )

31

36

33

74

75

76

𝜂mat at peak wavelength [nm]

∼61% to ∼92%. It is worth remembering that for a specific wavelength application, a custom coating can lead to a significant improvement, up to the intrinsic efficiency of the device. 7.2.2

Recording Electrical Signals from Individual Photons

The improvements of electronics and computing in the last decade have resulted in a shift of approach to photodetection. The availability of very fast detectors and digitizing boards, large processing capacities, and data storage has led to a paradigm shift in the way experiments are designed and data are analysed. The technique that most people now consider is to store the entire output from the photodetectors directly and to process these digitally, either in real time or in post-processing. This has many advantages: it allows the implementation of a much wider range of processes; it has capabilities for both single-photon experiments and continuous-wave experiments; the data are stored and can be analysed again and again; and the processing tools can be refined after the data were taken. This allows the experimentalist to concentrate on the experiment for the purpose of getting the best data and separately, at other times, on the analysis. In practice this is very conducive to daytime experimentation and automated repeats of the experiments. But it is still followed by long evenings, in front of computer displays. The first and very common approach is the direct counting of photons within regular time intervals. The arrival of one or more photons will result in an electric pulse or click. Only PNR detectors will allow us to classify the pulses according to the number of photons. We store such a pulse above a preset threshold as binary 1, whilst we ignore all small events below the threshold and record them as 0, meaning no photon. The processing is now a simple matter of storing the string of 1s and 0s, which can then be used to plot histograms and analyse trends as a function of various parameters. The average value can be associated with the average photon number ⟨N⟩ in this mode. The next level will be to look for correlations of the output of two photodetectors. The most direct way is to set the system

245

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7 Photon Generation and Detection

such that it is only triggered by coincidences, that is, when both detectors have produced a pulse above the threshold and thus show a 1. There are many important experiments, such as entanglement, tests of Bell’s inequality, etc., which rely directly on coincidence counts. The results can be used to determine a two-mode correlation function g (2) quantitatively. The alternative to coincidence counting is to store the binary files and evaluate the sequences during or after the recording. This would yield the same results for coincidences but allows other types of protocols. We can consider conditional recording via post-selection where the pulse in one channel is recorded only when a pulse was present in the other channel. This is used in many protocols where one channel is designed to produce photons that herald a particular event in another channel. It has been used to demonstrate logic gates and describe their properties. Or it can be used to evaluate the conditional variance between the two channels. By recording the sequences of events from several detectors, more complex situations have been investigated, for example, the properties of a system with more than two modes. This could be systems that have entanglement between several modes, or degrees of freedom, or even systems where the modes are linked together to form a logic system. That means a large number of correlation functions between different modes can be measured, the result leading to a full analysis of the covariance matrix (see Chapter 4). All of this is possible with the straightforward recording of binary sequences, typically with a sampling time of nanosecond, resulting in datasets with a few megabits. Analysis in these experiments is relatively straightforward, since the important parameter is the rate of photons detected at a given output port, measured over some integration time. If film is used, the integration time is set by the length of the exposure. For photodetectors, the integration time is set by the counting time, which in turn is set by the electronics, including capacitors as low-pass filters, and the frequency of the trigger pulses. The electric current is digitized, e.g. TTL pulses, converted into binary values and counted with a commercial pulse counter; otherwise it can be digitized using a commercial discriminator, fast oscilloscope, or similar device and then counted. In some protocols, such as quantum cryptography, it is important that the photons arrive in a well-defined time period. Even in these cases, any source, thermal, coherent, or Fock state source, can be used, as long as some combination of the source, the optical path, or the photodetector detection time is appropriately modulated. Such pulses will have regular time intervals. Alternatively, photon time-of-arrival information is a useful diagnostic in these experiments and may be obtained by using a time-to-amplitude converter (TAC) to measure the difference between the expected time, as provided by the modulator, and the actual time, as provided by the photodetector. The modulator provides a periodic start signal, for a measurement, which is ended by a photodetection event if one occurs. The time difference between the start and stop pulses then sets the amplitude of a regular pulse. The amplitude distribution of these pulses can be measured with a multichannel analyser (MCA), providing information on the distribution of photon arrivals.

7.3 Generating, Detecting, and Analysing Photocurrents

7.3 Generating, Detecting, and Analysing Photocurrents 7.3.1

Properties of Photocurrents

The technology used with CW laser beams is quite different from that used with single photons, as described in Section 7.2. Here we analyse the full-time history of the photocurrent i(t), and the results we obtain for a laser beam are a value for the average optical power, which is obtained with a power meter, and in parallel a spectrum of the variance of the optical power, such as shown in Figure 7.9, which is obtained with an electronic spectrum analyser (ESA). For CW detection the electronic signal from the photodetector is integrated over many detection events for a time interval Δt, and this generates a current i(t), which is proportional to the power P(t), measured in Watt, of the light beam on the detector or, in other words, the rate at which photons reach the detector: i(t) =

ne (t)e P(t) e 𝜂det = Δt h𝜈

(7.1)

where ne (t) is the number of electrons released by the detector, e is the charge of a single electron, and 𝜂det is the efficiency of the conversion process (see for comparison Section 4.4.3). We will be interested in the average value of the current i = ⟨i(t)⟩. This value is proportional to the average optical power Popt ; thus a device that measures i is generally referred to as a power meter. Each such apparatus will have to be calibrated, and care should be taken that the response is linear, since the device will saturate if the optical power is too high.

7.3.1.1

Beat Measurements

The photodetector provides not only an average current i, the DC current, but also a whole spectrum of time-varying signals, the AC. Any AC component at detection frequency Ω can be traced back to a beat between two components of the optical spectrum with a separation Ω. These beat signals are very useful to identify the mode structure of the light source. Multimode laser operation will lead to strong beat signals, which can be detected and processed with high accuracy and sensitivity using conventional electronic RF equipment. The measurement of beat signals between two lasers, for example, will establish the frequency difference and the convolution of the linewidth of both lasers. Such observations are extremely useful for the establishment of optical frequency standards and also the measurement of laser linewidth, as discussed in Section 8.1.5. In addition, the AC signal also contains information about the noise and the signal carried by the laser beam, since the fluctuations and modulation of the light intensity corresponds directly to the size of the amplitude sidebands. The presence of these sidebands leads to beat signals, which are similar but much weaker and noisier than the beat signals from two lasers.

247

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7 Photon Generation and Detection

7.3.1.2

Intensity Noise and the Shot Noise Level

For the quantitative measurement of the fluctuations, we are interested in the variance Δi(t)2 of the photocurrent. This is linked to the variance of the number of single-photon detection events via ( ) ne (t)e 2 Δ(ne (t))2 e2 2 Δi(t) = Δ = (7.2) Δt Δt 2 If we consider the generation of all the photoelectrons as independent processes, we can apply the statistics for independent particles. The counting statistics for the electrons should be a Poissonian distribution with variance Δ(ne (t))2 = ne = iΔt∕e. For large numbers of electrons in each counting interval, this should be a Gaussian distribution, resulting in ie Δt We will be investigating the spectrum of the current fluctuations. In this case the detection interval Δt is related to the bandwidth B of the apparatus via B = 1∕2Δt, which leads to Δi(t)2 =

Δi(t)2 = 2 i e B

(7.3)

assuming that i does not vary during the measurement. This is the well-known shot noise formula in electronics that also applies to photodetection. It should be noted that Eq. (7.3) can be derived for any stream of electrons that are generated independently. However, this is a rather special case for a current, and generally a current can be noisier or quieter. There are many practical situations, such as the use of electronic current controllers, which can generate currents with less noise than the shot noise limit. It is important to note that shot noise is not a property of the detector. The noise reflects the fluctuations of the light intensity, and we can use detectors to probe the statistics of the light fluctuations. This will be correct as long as the quantum efficiency of the detector is high. For 𝜂det = 1 the statistics of the photons and electrons will be identical, the current fluctuations are directly linked to those of the light, and we can investigate the statistical properties of the light by studying the current fluctuations. It should be noted that after the detection of the light, any quantum properties of the fluctuations no longer exist. Current noise can be treated classically, as already discussed in Section 5.1. For any practical purpose there is no quantum limit for the noise of a current. However, at the same time the current fluctuations are a reliable measure of the quantum properties of the light. This distinction is important in the context of measurement theory, the generation of non-classical light (Chapter 9), and the applications of electro-optic feedback described in Section 8.3. How closely the current noise reflects the light fluctuations depends on the efficiency 𝜂det . A quantum efficiency much less than unity corresponds to a random selection of a fraction 𝜂det of the light. This is fully equivalent to having a perfect detector behind a beamsplitter, which selects a small fraction of the photons. This results in a variance of the current that is closer to the shot noise limit than to the variance of the light intensity. To describe this quantitatively, it is useful to

7.3 Generating, Detecting, and Analysing Photocurrents

consider normalized variances for individual Fourier components of the current fluctuations: ) ( Δi(t)2 Vi (Ω) =  (7.4) 2ieB The corresponding statistics of the light is described by the normalized spectrum V 1out (Ω) of the photon flux, as was shown in Chapter 4. The effect of the detection efficiency 𝜂det is given by Vi (Ω) = 1 + 𝜂det (V 1out (Ω) − 1) This is identical to treating the inefficient detector as a perfect detector that sees only the light reflected by a beamsplitter with reflectivity 𝜖 = 𝜂det as calculated in Section 5.1. The measured spectrum Vi (Ω) varies with the fraction of the light detected. For a very inefficient detector, the normalized variance of the current approaches the shot noise limit Vi (Ω) = 1. 7.3.1.3

Quantum Efficiency

The detection efficiency 𝜂det contains all processes that reduce the conversion from light into a current. In practice, there are a number of such processes, as shown in Figure 7.4: only a fraction 𝜂r of the light reaches the photodetector material, and the rest is reflected or scattered at the surface of the detector. The detector absorbs a fraction 𝜂a of the light, and the rest is scattered or transmitted through the detector. The absorption scales exponentially with the thickness of the material and depends on the doping of the material. The material has a quantum efficiency 𝜂mat , which depends strongly on the wavelength of the light. This efficiency can be very close to unity as long as the wavelength of the light is close to but shorter than the cut-off wavelength set by the band-gap energy of the material. Finally, there is a chance for the electron–hole pair to combine before it can leave the detector. This recombination is more likely at the surface of the material than inside. At shorter wavelengths the light tends to be absorbed more strongly; it does not penetrate as deeply. Thus more detection processes occur close to a

N(t)

(1 – ηcom) recombination

(1 – ηr) reflection

ηdet =

N(t) i(t)

(1 – ηa) transmission

ηmat conversion

i(t)

Figure 7.4 The different loss mechanisms present in a photodetector.

249

7 Photon Generation and Detection

surface, and the losses due to recombination are larger. Only the fraction 𝜂com will actually contribute to the photocurrent. All these losses are independent and can be combined to the total detector efficiency 𝜂det : 𝜂det = 𝜂mat × 𝜂a × 𝜂r × 𝜂com The efficiency 𝜂det is generally independent of the intensity and the detection frequency Ω. However, it has been noted that 𝜂com will get worse at higher intensities and that some photodetectors get slower at higher intensities when either the detector material is getting hot or there are insufficient fast charge carriers to drive the AC. The efficiency 𝜂det has to be determined for the parameters of the light used in the specific experiment, in particular its wavelength, total power, and focusing geometry. Techniques for increasing 𝜂det in order to approach the ideal case of 𝜂det = 1 are very important in all experiments with non-classical light. 7.3.1.4

Photodetector Materials

Photodetection relies on the interaction of light with materials, which results in the absorption of photons and the generation of electron–hole pairs. These pairs change the conductance of the material, and they contribute to the photocurrent. The material for the photodetector has to be selected for a given frequency of the light. The energy of the photons h𝜈 has to be matched to the band-gap separation Ebg of the material. Semiconducting solids, such as germanium (Ge), silicon (Si), or indium–gallium–arsenide (InGaAs) are well suited for this application. These materials work in the visible and near IR. The properties of these materials are listed in Table 7.4, and the wavelength dependence of the efficiency is shown in Figure 7.5. Note that the properties of primary semiconductors (Si, Ge) are fixed but that the properties of tertiary materials (such as InGaAs) depend on the relative composition, which means the concentration of the individual elements. The material quantum efficiency 𝜂mat of a very pure and selected material can be very high, up to 𝜂mat = 0.98. This is the limit of the efficiency of the detector and can be achieved in practice provided the sample of material used is thick enough to absorb all the light and reflection losses are avoided. Since most photodetector 1 Si

Ge

InGaAs η

250

0.5 InSb InGaAsP

Ag–ZnS 0 0.1

0.2

0.4

0.6 1 2 Wavelength λ (μm)

4

Figure 7.5 Quantum efficiencies of various photodiode materials.

6

7.3 Generating, Detecting, and Analysing Photocurrents

materials have very high refractive indices, their reflectivity is also very high; for example, for Si with a refractive index of 3.5 at the peak detection wavelength of 800 nm, the reflectivity under normal incidence is 31% of the intensity. To this the reflection loss at the cover glass (typically 8% from both surfaces of the glass) has to be added. In order to increase the quantum efficiency, it is common to remove the cover glass and to either tilt the detector closer to Brewster’s angle, which requires large detectors, or use an AR coating on the detector. In addition, the light reflected at the surface can be retro-reflected on to the same detector. Using these techniques typical quantum efficiencies that were achieved by 2005 were 65% at 553 nm (EG&G FND 100 Si photodiode) and 95% at 1064 nm (Epitax 500 InGaAs photodiode). Similarly high efficiencies around 1600 nm have also been achieved, which is important due to the interest in optical communication applications and the optimal performance of silicon fibres in this wavelength range. In most cases one would expect technology to simply improve. However, there have been situations where certain optimized components are no longer produced, for example, the Epitax diodes, and it can be costly and time consuming to find replacements. Alternatively, several detectors can be cascaded, each subsequent detector receiving the light reflected by the previous detector, and all the individual photocurrents are combined. Here electronic delays of the different currents have to be carefully avoided in order to keep the signals synchronized with an accuracy better than the inverse of the detection frequency. Using cascaded detectors efficiencies close to 100% have been reported for detection frequencies of up to 100 kHz using Si detectors [42] and up to 10 MHz using InGaAs detectors. 7.3.2 7.3.2.1

Generating Photocurrents Photodiodes and Detector Circuit

The photodetection process leads to a change in the conductivity of the semiconductor material. A simple photoresistor could be used to produce a photocurrent. However, the most common configuration is a PIN photodiode, which is a composite structure made out of P and N semiconductors with an intermediate layer. In this device an external voltage, the bias voltage, Ubias , creates an electrical field in the region between the P layer and the N layer. Here electron–hole pairs can be separated, and they contribute to the photocurrent. There are three modes of operation for a photodiode. The diode can be operated with an open circuit. This will produce a voltage proportional to the photon flux; no current will flow. Alternatively, a low resistance is switched across the diode, and the circuit is effectively a short circuit. In this configuration a DC current is generated and the photodiode operates like a solar cell. Both these modes have slow responses. It is advantageous to apply a voltage, the reverse bias voltage Ubias , to the detector that accelerates the charge carriers. In this mode the photodetector is used like a diode to generate a voltage signal across an external resistor. The diode current is proportional to the photon flux N(t). For a sufficiently large bias voltage, the photocurrent is proportional to N(t). A schematic circuit and a typical response curve are shown in Figure 7.6. It is common to

251

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7 Photon Generation and Detection

i N(t)

DC Ubias

–Ubias

R i(t)

U

N(t) AC –Ubias / R

Figure 7.6 Circuit of a photodiode with reverse bias voltage Ubias and the typical diagram that shows the photocurrent i(t) as a function of the bias voltage for various values of the photon flux N(t).

use the highest possible bias voltage (Ubias = 5–80 V depending on the diode) since this will increase the drift velocity of the charge carriers and thereby the speed of the detector. The photocurrent contains both a DC component and an AC component that can be directed to different receivers using simple capacitors or more complicated electronic networks. The AC components contain all the information about modulation and the fluctuations. 7.3.2.2

Amplifiers and Electronic Noise

The typical AC noise component is of the order of 1 part in 107 of the DC current. It requires amplification. This can be achieved by various types of amplifiers, as described in [43]. The most obvious solution is to terminate the photodiode into a 50 Ω resistor and to use a commercial RF amplifier. Such an amplifier can be obtained with wide bandwidth (1–200 MHz), large gain (10–60 dB), and low noise performance. Please note that the gain is normally stated as the power amplification in dB, where dB is 10 log Pout ∕Pin (see Eq. (7.9)). The amplifier will increase both signal and noise at the input. Figure 7.7 shows the combination of noise and signal that is generated. The different noise contributions are: • Dark noise of the detector Δi(t)2dark . Note that this is not the shot noise; it is only the noise due to false detection events. For most of our measurements with optical powers of a few milliwatt, this can be neglected. • Thermal, or Johnson, noise of the circuit given by Δi(t)2th ≈ 4kB TB∕R where T is the temperature, B is the bandwidth, and R is the impedance of the circuit. • Amplifier noise Δi(t)2amp ≈ 2eGBF. Here G is the voltage gain and B the bandwidth. F is the excess noise factor, a composite of different contributions inside the amplifier. The excess noise factor is normally quoted in the technical specifications. Note that a large bandwidth detector contributes more noise than a narrowband amplifier. All these noise contributions are independent; thus their variances can be added. Note that the noise Δi(t)2optical created by the light is the actual information

7.3 Generating, Detecting, and Analysing Photocurrents

Light

Photodiode

Gain

Output Thermal noise Gain noise

Noise

(a)

Dark noise Quantum noise

Signal

(b)

Modulation and classical noise

Figure 7.7 Signal and noise contributions in a photodetector. Noise terms (a) and signals (b) propagate differently through the stages of the detectors circuit.

we wish to detect. It corresponds to the signal in a communication application, and in order to achieve a clear detection of the signal, the total noise has to be less than the optical noise: Δi(t)2el = Δi(t)2dark + Δi(t)2amp + Δi(t)2th ≪ Δi(t)2optical

(7.5)

One of the largest contributions is normally the thermal noise. Consider the following example: for a DC current of 1 [mA], equivalent to 1 mW of power at 𝜆 = 1000 nm, the thermal noise at room temperature for an impedance 2 2 of 50 Ω and a detection bandwidth of 100 kHz is Δi(t)2thermal ∕i = 4kB TB∕Ri = 4 ∗ 1.38 × 10−23 ∗ 300 × 105 ∕(50 × 10−6 ) = 3.3 × 10−11 . The relative noise that corresponds to the shot noise for this bandwidth and average power is 2 Δi(t)2 ∕i = 2eB∕i = 2 ∗ 1.6 × 10−19 ∗ 105 ∕10−3 = 3.2 × 10−11 . This means that for the given conditions both quantum noise and electrical noise produce comparable relative fluctuation Δi(t) of about 5 × 10−6 . In this example, the thermal noise is as large as the shot noise at this optical power, and such a detector could only show quantum noise at much higher powers. The importance of this so-called front end of the circuit cannot be overestimated; it will determine the noise characteristics of the detector circuit. Several technical tricks exist to optimize, which is summarized, for example, by P.C.D. Hobbs [44]. Most custom-made photodetectors contain a so-called transimpedance amplifier, a circuit that converts the AC photocurrent into an AC voltage, which is fast and still has a high impedance. In such a circuit the thermal noise will be much lower, about 10−8 at 1 mA DC current, and we can clearly resolve quantum noise. State-of-the-art experiments provide a combined electronic noise of typically −10 dB and in some cases −30 dB below the quantum noise limit (QNL). This is the reason why photodetectors for quantum optics experiments are frequently custom made. For illustration, an example of a complete circuit diagram of such an amplifier is shown in Figure 7.8. It shows the photodiode,

253

7 Photon Generation and Detection

3

C1 22u

C2 100n

C3 100n

D1 ETX 500

7

I2 10u

2 – 3 +

5

1N4007 D2

+15

~4PF R1 2K 6

IC1 R2 CLC420 200R

U2 79L05 3 GND 2 IN –5V 1

–15

1N4007 D3

I3 10u C8 22u

C6 100n

C4 100n C10

GND 2

7

U1 78L05 Vin +5V

3 + 2 –

8

1

IC2 CLC430 6

R5

Vout

51R

4

I1 10u

4

254

R3 750R R4 300R

C5 100n

C7 100n

C9 100n

Figure 7.8 Example of an actual circuit diagram of a photodetector. For other examples, see M.B. Gray et al. [43].

the transimpedance amplifier (IC1), the driver amplifier (IC2), and all the filter components required for a practical low noise circuit. Whilst the development of electronics has continued constantly, the design of photodetectors still follows the same pattern. A good reference in 2017 is still the article by P.C.D. Hobbs [44], which summarizes the concept of the electronic design and the references given in the textbook by the same author [45]. 7.3.2.3

Detector Saturation

In many cases the light has strong modulations at specific detection frequencies. This could be due to laser noise, such as beats with other laser modes, or due to the sidebands introduced for the purpose of locking the laser to resonators, as discussed in Section 8.4. We expect the photodetector to handle both the very small fluctuations and the modulation without distortions, and consequently we require a very large dynamic range for some applications as large as 60 dB. If the dynamic range of the amplifiers is not chosen large enough, the circuit will be saturated. This frequently leads to a decrease of all AC components, even those at other than the modulation frequency that originally caused the saturation. This saturation can be confusing and can mimic false results. It is particularly worrisome if several amplifiers are used, a first amplification stage with wide bandwidth and a later stage with narrow bandwidth. In this situation the first stage can saturate from a modulation at frequencies outside the bandwidth of the final stage. Thus the modulation is not visible at the output, and the noise level decreases when the signal is turned on – mimicking a noise suppression. In more than one case, this malfunctioning of the detector has fooled researchers and was taken as the demonstration of some quantum optical effect. As a precaution one should always record spectra for the entire bandwidth of the detector, and any

7.3 Generating, Detecting, and Analysing Photocurrents

amplifier and low-pass filter should best be placed directly after the photodiode. In addition, the scaling of the noise traces with the input optical power should be checked, as outlined in Section 8.1.1, to verify that no saturation exists. 7.3.3

Recording of Photocurrents

In most situations experiments are conducted by recording the photocurrents directly via fast analogue-to-digital (A/D) converters storing the entire information with a large bandwidth and analysing the physical properties, such as histograms and correlation functions. Historically, this is a form of post-processing that was typical for the analysis of single-photon events. However, due to the vastly increased speed of the processors, this can also be used now for all types of data analysis of continuous photon currents. For example, the modulation spectrum of the photocurrent can be determined with a numerical fast Fourier transform (FFT) providing similar results as the direct analogue ESA that were used in the beginning of the quantum experiments. Provided that the speed of recording, the dynamic range of the AD converter, and the size of the memory provides no limitation, we can assume that the Fourier transform will produce a true spectrum. In 2017, A/D converters with 18–20 bit conversion and sample speeds of 100 megasamples/s are available. In addition the size of the memory is now almost no issue, say, a few terabits, and the size is only noticeable in the time it takes to process a trace. However, there are limitations in the digital data recording that have to be considered and tested thoroughly to avoid misinterpretations: the main constraints are digitization speed or sampling frequency (samples per second), the dynamic range (bit per sample), storage capacity (total number of bits), and sensitivity to aliasing, which means the interference between the fixed sample frequency and higher harmonics present in the input, leading to spurious low frequency effects. For a simple practical example, we might consider a technical test where we record a communication channel that transmits signals with a carrier frequency of 100 MHz and modulation sidebands of around 100 kHz. We want to record an experiment for about 10 seconds. And we want to see a signal-to-noise ratio of 40 db. All this seems quite reasonable. Using the Nyquist theorem as a guide, we would set the sampling frequency to about 200 megasamples/s. To achieve a dynamic range, or contrast, of 10 000, which is necessary in order to see the fluctuations and signal in reasonable detail, we would have to use a 16-bit A/D converter. This would generate 2 × 108 ∗ 65 536 ∗ 10 = 1.31 × 1014 = 130 terabits. That is, in excess of 2017 technology and such a brute force approach would simply create an unmanageable amount of data. However, the real information content is actually much smaller. We actually have only a maximum of 1 million possible signal values per second, we can average out the noise to a slowly varying value with an accuracy of one part in 10 000 of the biggest signal. So with little distortion a 12-bit A/D converter would be sufficient for simple communication applications. After mixing down the signal from 100 MHz and recording at a speed of 0.1 megasamples/s, only about 106 ∗ 4000 ∗ 10 = 4 gigabits of recording would be generated. This volume of data is comparable with the recoding of a

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high-definition video signal for that length of time and would be sufficient for most applications. In comparison, this could have been an experiment recorded with analogue instrumentation, such as an ESA that creates a printout of a spectrum centred at 100 MHz and a width of 200 kHz with a 50 dB vertical scale with a resolution of 0.5. Whilst such a spectrum contains only an average of the information over the entire 10 seconds scan, it would give us excellent information about the signal-to-noise ratio, and any potential problems or non-linearities within the system bandwidth would be recognized. The printout contains only a hundred values between 0 and 120 db, but this represents the complete spectrum. In order to describe the technical specifications of the apparatus, such a simple graph or even just a few numbers are sufficient in most cases. This example shows that the recording of an arbitrary signal with very fine detail requires extraordinary number of samples. This not only exceeds our storage capacity, but it also makes processing practically impossible, and it is only necessary if we have no predictor of the signal that we wish to find. That means that we should tailor our system to exactly what we need; there is no universal solution and just digitizing everything is not efficient. In most technical applications we have a specific goal in mind. Let us consider two extreme examples. We can store and show with a smartphone or tablet any imaginable sequence of video images. These could potentially change completely from frame to frame, say, 30 times per second. However, in most cases the change from frame to frame might be small. In order to make this practical, the information has to be cleverly compressed, for example, in one of the current standards as MP4. This allows us to store only a fraction of the total information contained in all the pixels at all times. Clever algorithms project the images unto mathematical base functions and select a small amount of data that represents the video. The spatial resolution can be selected. The information can be stored and can be analysed quite readily. This technique requires online compression and storage of a medium number of data points. At the other end of the range of applications is the investigation of something unknown, for example, of gravitational waves (GW) detected as the correlation between the outputs of several optical interferometers placed at different locations. Since late 2015 this has been dramatically demonstrated by the global LIGO collaboration. The GW have frequencies from megahertz to kilohertz. The detectors have good sensitivity above about 10 Hz and are being sampled at 40 kilosample/s continuously for 24 hours per day for months on end, since we do not know when the next event will happen and what it will look like. This leads to many terabits of data. Such large datasets have been recorded by LIGO since 2011 and have been constantly analysed by many groups around the world. In quantum optics experiments we typically lie between these extremes, and we have to design our experiments carefully to make use of the existing digital recording and processing technology. We can now record 100 megasamples/s with 16-bit resolution. In 2017 we can handle 1–100 gigabit files quite readily. However, we should also consider any analogue steps as part of the recording process, such as mixing the RF signal before recording, including low-pass filters or band filter to isolate the frequencies we are interested in and to protect

7.3 Generating, Detecting, and Analysing Photocurrents

us from aliasing. Thus the optimum system will be a combination of analogue preprocessing and digital recording and post-processing. 7.3.4 7.3.4.1

Spectral Analysis of Photocurrents Digital Fourier Transform

The time spectrum of the photon current contains a wealth of information that describes the properties of the apparatus. It is thus a widely used test procedure and the key to demonstrating many optical techniques. Figure 7.9 shows a typical spectrum and all the settings of the analyser being used. The most direct approach for creating a spectrum from digitized data is to use an FFT. This commonly available software routine is frequently directly built into the data acquisition system, for example, LabVIEW . The obvious requirements are as follows: a sufficiently large number of samples, according to the Nyquist limit; the dynamical range, which is limited by the sampling resolution; and protection against interference with other periodic events, for example, at higher frequencies. The latter is an unavoidable consequence of recording with strictly periodic samples and can only be avoided by optimizing the recording electronics. Since FFT deals with stored data, the central RF, resolution bandwidth (RBW), and the averaging, or in traditional terms, video bandwidth (VBW) can all be selected after the recording. The recording should satisfy the following

Figure 7.9 A typical intensity spectrum as displayed by an electronic spectrum analyser. The modulation is about 8.9 MHz. The smaller signal at about 5.2 MHz is a radio interference from another source, affecting both the signal and the dark trace. Here the settings of the instrument are shown together with the data.

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requirements: a sampling frequency must be at least twice the central RF; the number of samples should be at least larger than the ratio of RF to RBW; and the VBW is set by the inverse of the recording length. For slow long-term recordings, the choice is to either process all frequencies simultaneously and display long-term average values or alternatively analyse every section of the spectrum independently one at a time, which provides less averaging but allows to catch transient phenomena. It is worth noting that some of the commercially available ESA are actually a combination of A/D converter, storage, and FFT, rather than the full analogue version of using RF mixers. In these devices all the sample settings are automatically adjusted by selecting the required frequency and bandwidth settings. These machines are limited in their specifications but avoid the complication of having to individually optimize the digitization and sampling settings. 7.3.4.2

Analogue Fourier Transform

An ESA is used in some experiments to record the Fourier spectrum of the photocurrent. A typical spectrum displayed by such an instrument is shown in Figure 7.9. The principle function of an ESA is to determine and display the power spectrum of the electrical input signal. We consider in detail the transfer function from the spectrum of the input signal to the display. It is important that we understand the many options and settings of the analyser in order to obtain reliable data that can be interpreted as the properties of the light beam. A state-of-the-art analyser is a very powerful instrument with many options, menus, and knobs. An ESA contains a combination of analogue and digital electronics that produce a string of data that will be recorded alongside its actual settings. These machines are designed for the recording of signals, such as radio signals. In this section we will discuss the technical details of the ESA, such as the various bandwidth settings, sensitivity scales, sweep speeds, etc. and the tricks for using the analyser for the recording of noise and signals. Finally, we will discuss alternatives such as electronic mixers, which are useful for more advanced experiments such as tomography. 7.3.4.3

From Optical Sidebands to the Current Spectrum

We are interested in the sidebands of the optical field that contain all the information about the fluctuations and the statistics as well as all the information that is transmitted with the light in the form of modulation. Typical sideband frequencies range from kilohertz to gigahertz. These frequency intervals are too small for ordinary spectroscopic instruments, such as grating spectrometers or optical spectrum analysers (OSA), and are just about accessible with optical cavities. However, the most reliable and best separation of the sidebands is by the Fourier analysis of the photocurrents. The information is transferred from the optical domain (𝜈 ≈ 1015 Hz) into the domain of electronic instruments, which means detection frequencies Ω < 10 GHz. For this purpose several light beams are combined on a photodetector. The resulting beat signals contain information about the quadrature of the light. These techniques, homodyne and heterodyne detection, are described in

7.3 Generating, Detecting, and Analysing Photocurrents

Section 8.1.4. In all cases the information is contained in the temporal development of the current i(t) generated by the photodetector. The purpose of an ESA is to measure and display the spectral composition pi (Ω) of the electrical power of an electrical signal i(t). Mathematically this corresponds to a determination of the power spectrum, or the square of the Fourier transform, of the input voltage. The instrument measures the power pi (Ω) of the oscillating field in a spectral interval around the detection frequency Ω. The spectral response of the analyser is given by the function res(Ω), which has a width given by the RBW. For a fixed input resistance R, typically 50 Ω, the measured power is ( )2 pi (Ω) = time average res(Ω) i(Ω) R dΩ (7.6) ∫ where i(Ω) is the Fourier transform of the current i(t). We are measuring the square of the AC photocurrent integrated over the detection interval. The ESA is not sensitive to the phase of the electrical input signal. It will display only positive frequencies; there is no difference in the result of a measurement of signals with positive or negative frequencies, of sine or cosine modulations, of the current. 7.3.4.4

The Operation of an Electronic Spectrum Analyser

Technically, the ESA consists of a mixer that selects a certain frequency; an envelope detector, in most cases a logarithmic detector; and circuitry for averaging the signal, as shown in Figure 7.10. In most modern analysers the signal is digitized and displayed with a wide choice of scales and other options. In the following we analyse the transfer function between the electronic input signal and the display. The ESA is built around the concept of electronic heterodyne detection. A reference signal with fixed amplitude and variable frequency, the local oscillator, is generated internally. The frequency Ωdet of the local oscillator determines the frequency under detection. The input signal is amplified and mixed with the local oscillator. The output from the mixer is filtered by a narrow band pass filter with bandwidth RBW. The result of the measurement, at a given Ω, can still fluctuate in time, and we can describe the output of the ESA by a distribution function with the mean value 𝜇disp , and it is this value from which we infer the spectral density pi (Ω) of the photocurrent. In our optical experiments we make relative measurements, which means we are not interested in the absolute values of 𝜇disp ,

Mixer Voltagein

Filter RBW filter

Envelope detector

Log amplifier

Filter VBW filter

Local oscillator

Display

Scale dB power

Figure 7.10 Block diagram of a typical spectrum analyser.

AD sample

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but in relative values, such as the ratio of the 𝜇disp to a calibrated value that represents quantum noise. The output signal is integrated using a low-pass filter with a cut-off frequency known as the VBW. The VBW is the inverse of the time constant Δtave with which the measured signals is averaged, Δtave = 1∕VBW. In the limit of small VBW, large integration times, the output is one single, steady value 𝜇disp (Ω) for each detection frequency. In many practical situations we want to see the spectrum rapidly, and a lower value for the VBW is chosen; this does not vary 𝜇disp (Ω) but produces a more noisy trace. Modern RF spectrum analysers cover a wide range of input frequencies, from about 1 kHz to 100 GHz. They can achieve this wide span through a sophisticated combination of several internal mixing processes. Nevertheless, the idea outlined above of a narrow band scanning filter followed by an integrating power detector is appropriate. The instrument covers a total frequency interval, or span, SPA, and scans it with a set scan speed, measured in Hz/s. The instrument requires at least a time of 1/RBW for each measurement; after that it moves on by a frequency interval RBW. Consequently the maximum scan speed is RBW2 . For typical values of RBW of 104 Hz, the fastest possible speed is 108 Hz/s = 100 MHz/s. Faster scans would lose detailed information. The highest possible VBW is equal to the RBW, but usually the VBW is chosen lower. In particular for noise measurements, it is necessary to select a low VBW in order to obtain a well-defined result. Frequently, measurements with fixed detection frequency Ωdet are carried out. In this case the scan represents a time history of the power of the single Fourier components. 7.3.4.5

Detecting Signal and Noise Independently

It is not possible to distinguish between a pure noise source and a mixture of signal and noise by just looking at a trace recorded at one frequency. In this case we could not say if we have a laser beam with quantum noise alone, a laser beam with quantum and classical noise, or laser beam with noise and modulation. We need additional information. For example, a scan of frequencies would reveal a peak if the signal is narrowband and the noise is broadband. Or a measurement of the orthogonal quadrature, using a homodyne detector, would show the characteristics of a signal and of noise or squeezing and anti-squeezing. We can isolate signal and noise, as measured by the traces on the spectrum, and we can determine the SNR as SNR =

2 2 𝜎(signal+ − 𝜎noise noise) 2 𝜎noise

(7.7)

What would happen if we had more than one noise source and more than one signal present? As long as the noise terms have Gaussian distributions with 2 2 2 widths 𝜎n1 and 𝜎n2 , we can combine them into one value 𝜎noise = 𝜎n1 + 𝜎n2 . In a similar way we can combine two signals S1 and S2 with a random phase relationship. However, if the two signals have a fixed phase relationship, the amplitudes 2 will have to be added coherently, before the variance 𝜎signal is formed. In either

7.3 Generating, Detecting, and Analysing Photocurrents

case, one combined signal intensity Ssignal can be found. Consequently, the calculation in Eq. (7.7) for one noise and one signal term is generally sufficient for all cases that we will encounter. 2 for a given noise source by decreasing Note that it is possible to reduce 𝜎noise the RBW, effectively detecting less noise and the same amount of signal. This will improve the balance in Eq. (7.7) and allow smaller signals to be measured. However, this will make the detection process slower. We can also determine the measured value better by averaging the fluctuations for a sufficiently long time, which means using a low VBW. In the limit of VBW → 0, or extremely long integration times, the response of the analyser to a noise input with constant RMS will be a constant reading. The entire noise distribution would now be reduced to one piece of information. In most practical situations the integration time will be limited to about 0.1 s (VBW = 10 Hz) since systematic drifts influence the performance of the experiment and do not allow excessively long scans. When the noise is white, which means the fluctuations are independent of the frequency, the spectrum will be a flat horizontal line on the display. 7.3.4.6

The Decibel Scale

Some examples of actual measurements of laser intensity noise are given in Chapter 8. The display of the spectrum analyser is normally logarithmic. The units used are logarithmic power units, called dBm. The conversion of Watt into dBm is given by ( ) P[W] (7.8) pi [dBm] = 10 log 1[mW] The unit dBm is an absolute unit of power, for example, the RF power of 1 mW corresponds to 0 dBm, 0.001 mW = 1 μW corresponds to −30 dBm, etc. Whilst these numbers appear to be rather small, it should be noted that they refer to the power of the AC components only, not the optical power of the beam that generated the photocurrent. It would require a strong modulation of the intensity of the beam to get a significant fraction of the energy into the sidebands. For example, a beam with average optical power Popt and single frequency harmonic modulation of the intensity by 1% has both a positive and a negative sideband at the modulation frequency, ±Ωmod . Each contains 1% of the amplitude of the optical field, which corresponds to i(Ωmod ) = 0.02 of i and pi (Ωmod ) = 4 × 10−4 Popt . In most situations the relative change of the signal, or the noise, is of interest, and a relative unit of power, the decibel or dB, is used. It describes the ratio of two power levels. On a logarithmic scale this corresponds to a difference ΔP(Ω) given by ( ) P1 (Ω) (7.9) ΔP1,2 (Ω)[dB] = 10 log P2 (Ω) An expression such as ‘the signal was amplified by 3 dB’ means that the power of the signal increased by a factor of 103∕10 = 1.995 ≈ 2. Note √ that for such an increase the actual AC current would grow by only a factor 2. Similarly 6 dB corresponds to a factor 4 increase in power. The ratios in Eq. (7.9) are independent of the units; we record them from the display as ratios between values of 𝜇disp , and

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we think of them as the ratio of two noise powers P and interpret them as the ratio of quadrature variances V 1. All results of noise increase or noise suppression are quoted in dB, which gives directly the factor of change in the quadrature variance V 1 of the light. 7.3.4.7

Adding Electronic AC Signals

In many situations more than one signal or one noise component are present at the same detection frequency. Only the total power of all the contributions will be displayed. It is important to be able to determine the contributions from the individual terms, and we need rules for the adding of several AC. These rules depend on the nature and the correlation of the individual currents. Clearly, there is a distinction between currents that are coming from the same source or from different independent sources. First consider the extreme case of two currents, i1 (Ω) and i2 (Ω), which are generated by the same source. They are perfectly correlated, and their cross-correlation coefficient as defined in Chapter 2 is equal to unity: C(i1 , i2 ) = 1. In practical terms that means that the two signals have identical time histories. For single harmonic functions, this means that they have the same frequency and are locked in phase to each other. For these signals the combined power is simply the sum of the amplitudes: Pi1 +i2 (Ω) =

∫

res(Ω)[i1 (Ω′ ) + i2 (Ω′ )]2 R dΩ′

(7.10)

If the ratio of the amplitudes of the two currents is given by the number c = i2 (Ω)∕i1 (Ω), the measured power will change by a factor Pi1 +i2 = (1 + c)2 (7.11) Pi1 or in dB by ΔPi1 +i2 [dB] = 10 log((1 + c)2 )

(7.12)

For the case of two identical signals, i1 (Ω) = i2 (Ω) or c = 1, the power increase is 6 dB. The other extreme case is that of two signals that have a random phase difference to each other, which means they are uncorrelated. In this case the two signals add in quadrature, which means ΔPi1 +i2 [dB] = 10 log(1 + c2 )

(7.13)

and for the case of two signals with equal amplitude, the power increase is 3 dB. The same technique can be applied to noise terms, which are only different from signals insofar as the power of the noise at detection frequency Ω varies in time whilst the power of a signal remains fixed. In particular, signal and noise terms are always uncorrelated to each other. Two noise terms are uncorrelated when they come from different sources. They are correlated when they can be traced back to the same source. Correspondingly, these noise signals are added in the respective way, using either Eq. (7.12) or Eq. (7.13). In the first case, uncorrelated terms, the variances of the two noise terms are added in quadrature. Note that this procedure is strictly speaking only correct if both noise signals have the same statistical distribution functions, e.g. Gaussian distributions, which we can assume for many practical situations.

7.4 Imaging with Photons

7.4 Imaging with Photons All of the images shown in Figure 2.14 can be described by classical optics. At the same time, they use low intensities and thus photon flux, and the limiting effects of photon statistics are clearly visible. The first step is to consider the effects of photons in the generation of light in the light source. This could be due to the various processes that generate the light, such as thermal processes for an incandescent light, excitation of atoms in a plasma in a discharge lamp, excitation of atoms in a semiconductor that operates as a LED, or amplification of light in a laser resonator. In each of these situations, the clearest and most elegant description of the process is on the basis of photons. There are well-described probabilities for an atom to emit a photon, and the wavelength of the light is determined by the energy of each photon via Ephoton = hc∕𝜆, and this is controlled by the energy level structure of the atoms or the semiconductor being employed. The amplification of light inside a laser resonator is best described in terms of stimulated emission. Starting with the object that is being illuminated and imaged, the major processes are the transmission and absorption of the light from the illumination, where some of the photons are absorbed and the flux is reduced in proportion. In addition photons can be generated in the object plane with new propagation directions through elastic scattering. These photons have the same energy per photon, which means they carry the same wavelength information. There could also be inelastic scattering, or fluorescence, where the emitted photon has less energy than the photon required for the excitation of the atom involved in the process. This leads to new wavelengths being created in the object. There can be time delays between the excitation and fluorescence, which can be used through precise time gating to distinguish between the photons from the illumination and the sample and enhance contrast and sensitivity. The improvement in detectors and gating has allowed imaging and spectroscopy to reach the level of detecting single molecules. This in turn has led to a major expansion of imaging techniques in biological and medical applications [46]. Another option is the use of multiphoton excitation where the presence of several low energy photons is required to created one high energy photon. For example, Figure 2.14d was created using 2-photon excitation of a dye in the brain sample. This has the advantage that the excitation has a much larger probability in the focus of a laser beam since the probability scales with the square to the local photon flux. Therefore superior spatial resolution is achieved in the x, y plane and to some degree in the z direction. It is possible to reach with high precision individual or many points simultaneously for photostimulation and to image light from multiple spots. Higher-order three-photon excitation is possible. Such non-linear processes, which use multiphoton processes, are also at the core of so-called super-resolution techniques, including stimulated emission depletion (STED) microscopy proposed by Stefan Hell in 1994 and demonstrated in 1998 [47, 48], fluorescence photoactivation localization microscopy (FPALM), and other techniques in microscopy that bypass the conventional diffraction limits. For a summary of the techniques and some of the wide impact they have had, see, for example, [49].

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As we have seen in Chapter 4, the time-averaged photon flux of the quantized field is proportional to the intensity of the classical field, N(X, Y ) ∝ I(X, Y ), and the losses, redistribution of light, and change in flux can be described by one standard model. The quantum efficiency 𝜂 is the one single parameter that is used to model the change in photon flux. This includes all the finer details of Fourier optics and any of the imaging techniques. As a consequence we can easily translate the intensity distribution to determine the photon flux captured by each of the pixels of the photon detector. This applies independently to each pixel. For all these linear processes, the photon statistics will be either a Poissonian distribution or approach this distribution for lower and lower values of 𝜂. Consequently, we have a universal description of the noise in each part of the imaging system and for each pixel. Moreover, as with multiple beamsplitters, there are no correlations between the fluctuations in each of the pixels; the photon counting noise in different pixels is independent and only scales with the photon flux. It is easy to visualize an image that has been taken in very dim light by just a few photons. It is grainy and noisy and is obviously not of the best quality. With modern technology, and cooled CCDs, it is now quite common to work with just a few photons/pixel and time interval. More photons, more intensity, would obviously help until we have a beautiful image at high intensities. Many situations where this is not possible come to mind, in night viewing, astronomy, etc. The human eye is close to the limit of being able to detect individual photons. Whether this is possible is still a topic of active research, and it seems that this is possible for some situations and some people but not for all, as discussed in Section 3.1. Detectors have steadily improved, with improvements to the sensitivity and reduction of other noise sources, as listed in Section 7.2. Thus it is now more common to image with a few photons per pixel and to be photon counting and shot noise limited. This means imaging has in many cases reached the limits set by the quantization of the light. Examples are given in Figure 2.14 where parts (a) and (d) were made using specialized CMOS detectors. In addition, there has been rapid progress with the development of arrays of single-photon detectors based on SPADs as used in part (d) where both fast timing and high sensitivity are important. All these types of single-photon detectors are now employed in more and more low light imaging situations in astronomical, medical, biological, and engineering applications.

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of heralded single-photon sources. Nat. Commun.. 4. https://doi.org/10.1038/ ncomms3582. Kim, J., Benson, O., Kan, H., and Yamamoto, Y. (1999). A single-photon turnstile device. Nature 397: 500. McKeever, J., Boca, A., Boozer, A.D. et al. (2004). Deterministic generation of single photons from one atom trapped in a cavity. Science 303: 1992. Riebe, M., Monz, T., Kim, K. et al. (2008). Deterministic entanglement swapping with an ion-trap quantum computer. Nat. Phys. 4: 839. Home, J.P., Hanneke, D., Jost, J.D. et al. (2009). Complete methods set for scalable ion trap quantum information processing. Science 325: 1227. Zhang, J., Pagano, G., Hess, P.W. et al. (2017). Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551: 601. Brouri, R., Beveratos, A., Poizat, J.-Ph., and Grangier, P. (2000). Photon antibunching in the fluorescence of individual color centres in diamond. Opt. Lett. 25: 1294. Beveratos, A., Kuhn, S., Brouri, R. et al. (2002). Room temperature stable single-photon source. Eur. Phys. J. D 18: 191. James, D.F.V. and Kwiat, P.G. (2002). Atomic vapor-based high efficiency optical detectors with photon number resolution. Phys. Rev. Lett. 89: 183601. Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Philos. Soc. 15: 114. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization-entangled photons. Phys. Rev. Lett. 75: 4337. Takeuchi, S., Kim, J., Yamamoto, Y., and Hogue, H.H. (1999). Development of a high-quantum-efficiency single-photon counting system. Appl. Phys. Lett. 74: 1063. Kim, J., Takeuchi, S., Yamamoto, Y., and Hogue, H.H. (1999). Multiphoton detection using visible light photon counter. Appl. Phys. Lett. 74: 902. Cabrera, B., Clarke, R.M., Colling, P. et al. (1998). Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors. Appl. Phys. Lett. 73: 735. Lita, A.E., Miller, A.J., and Nam, S.W. (2008). Counting near-infrared single-photons with 95% efficiency. Opt. Express 16: 3032. Fukuda, D., Fujii, G., Numata, T. et al. (2009). Ti/Au TES to discriminate single photons. Metrologica 46: S288. Munro, W.J., Nemoto, K., Beausoleil, R.G., and Spiller, T.P. (2005). High-efficiency quantum-nondemoltion single-photon-number resolving detector. Phys. Rev. A 71: 033819. Takeuchi, S. (2001). Beamlike twin-photon generation by use of type II parametric downconversion. Opt. Lett. 26: 843. Zalewski, E.P. and Geist, J. (1980). Silicon photodiode absolute spectral response self calibration. Appl. Opt. 19: 1214. Gardner, J.M. (1994). Transmission trap detectors. Appl. Opt. 33: 5914.

Further Reading

43 Gray, M.B., Shaddock, D.A., Harb, C.C., and Bachor, H.-A. (1998). Photode-

44 45 46 47

48 49

tector designs for low-noise, broadband and high power applications. Rev. Sci. Instrum. 69: 3755. Hobbs, P.C.D. (2001). Photodiode front needs: the real story. Opt. Photon. News, OSA, April. 42. Hobbs, P.C.D. (2001). Building Electro-Optical Systems: Making it All Work. Wiley. Michalet, x., Siegmund, O.H.W., Vallerga, J.V. et al. (2007). Detectors for single-molecule fluorescence imaging and spectroscopy. J. Mod. Opt. 54: 239. Hell, S.W. and Wichmann, J. (1994). Breaking the diffraction resolution limit by stimulated emission: stimulated emission depletion microscopy. Opt. Lett. 19: 780. Hell, S.W. (2009). Microscopy and its focal switch. Nat. Methods 6: 24. Chi, K.R. (2009). Super-resolution microscopy: breaking the limits. Nat. Methods 6: 15. https://doi.org/10.1038/nmeth.f.234.

Further Reading Rieke, G. (2002). Detection of Light: From the Ultraviolet to the Submillimeter. Cambridge University Press. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley.

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8 Quantum Noise: Basic Measurements and Techniques We now turn our attention to the measurement of quantum noise – that is the fundamental noise we observe on the photocurrent when we detect optical beams with a large coherent amplitude (|𝛼|2 ≫ 1) using photodetectors. Several theoretical models have already been described in Chapters 2 and 4, and the experimental components have been discussed in Chapters 5 and 7. Based on this let us build up, step by step, the experiments for measuring and manipulating quantum noise.

8.1 Detection and Calibration of Quantum Noise We will concentrate on CW laser beams and the noise associated with the intensity, amplitude, and phase of the light. We have to devise ways of interpreting the electrical noise spectra. How do we distinguish quantum noise from ordinary technical noise? How can we can calibrate our measurement? Three different techniques are commonly used: direct detection (Section 8.1.1), balanced detection (Section 8.1.2), and homodyne detection (Section 8.1.4). The choice depends on the properties of laser beam. 8.1.1

Direct Detection and Calibration

The obvious way of detecting quantum noise is to directly detect the light with a single photodetector and to analyse the resulting photocurrent. As long as we can assume that the noise is Gaussian, we will have the full information in the form of the variance of the noise, and we can use an electronic spectrum analyser or we can perform a spectral decomposition of the digitized data using a Fourier transform. This is the same arrangement as one would use to measure an intensity modulation of the laser beam, and both of these effects will be present in the resulting spectrum. In most applications we will be interested in the quantum noise as the ultimately limiting effect in the signal-to-noise ratio (SNR) (Figure 8.1). Direct detection gives us an electrical noise power pi (Ω), which for a perfect laser mode and detection system should be entirely due to the quantum noise of the light. There are obviously some technical pitfalls that need to be avoided, and we need a reliable procedure that convinces us, and anybody else, that we A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

8 Quantum Noise: Basic Measurements and Techniques

V1las(Ω) Laser

Photodetector efficiency η

Amplifier gain gel Noise pel

Spectrum Analyser Vi (Ω) pi (Ω)

Figure 8.1 Schematic diagram of direct detection.

are really measuring quantum noise. The noise power pi (Ω) displayed by the spectrum analyser is a measure of fluctuations of the photocurrent, the amplification, and the noise contributions of the electronics. For a quantum noise limited source, we obtain the value pi,QNL (Ω). All measurements will be normalized to this value, resulting in the normalized variance Vi (Ω) =

pi (Ω) pi,QNL (Ω)

(8.1)

Once the recorded noise traces have been calibrated, it is obvious that a new unit should be used. A creative suggestion was to label the standard quantum noise limit (QNL) as 1 Hoover, in recognition of the role of the vacuum noise [1]. All noise diagrams in this guide with linear scales, or a logarithmic scale in decibels (dB) defined on page 261, use this unit. We need to know the total quantum efficiency of the detection system to link the variance Vi (Ω) to the fluctuations of the light V 1(Ω) (Eq. (4.57)). All values for the noise power are long-term averages, as shown on the display depicted in Figure 8.2. In this case a quantum noise limited light source with modulation at about 9 MHz illuminates the detector. The left-hand side traces show the display for a spectrum from 6 to 10 MHz with constant resolution bandwidth (RBW) of 100 kHz and variable video bandwidths (VBWs) of 30, 300, and 3000 Hz. Note that the average noise remains constant whilst the actual time fluctuations of the noise increase with VBW. The signal remains unchanged, and the width of the signal is given by RBW. The right-hand side traces show the results for the RBW –86 Noise power (dbm)

270

–90

–94

–98

VBW: 10 Hz RBW: 100 kHz

VBW: 100 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 300 kHz

VBW: 300 Hz RBW: 1 MHz

Figure 8.2 Experimental noise and signal spectra for different combinations of resolution and video bandwidth. These are four separate experimental traces, with settings as shown in Figure 7.9. The increases in the noise floor with larger resolution bandwidth are clearly visible.

8.1 Detection and Calibration of Quantum Noise

of 300 kHz. The average noise level increases with RBW. In contrast, the signal height remains the same, since the signal is much larger than the noise and the signal width increases with RBW. Clearly we gain by selecting a larger RBW and get further away from the electronic noise. However, there are limitations, in particular if we have not a flat but a structured noise spectrum, and there are a number of technical complications: 1. Electronic noise. The photodetector generates independent electronic noise, pel (Ω), which adds to the optical noise spectrum. We can avoid this complication by designing special detectors with low electronic noise (Section 7.3.2) and by independently measuring the dark noise of the detector. This is usually done by simply blocking the detector. If the dark noise is significant, we can correct the measured signal by subtracting the dark noise after the measurement. It can be assumed that the electronic noise and the optical noise are independent and are added in quadrature. It should be noted that some exotic amplifiers have the unpleasant property that the noise depends on the actual photon current. In these cases a normal dark current measurement is not sufficient, and several measurements at various photocurrents, or intensities, are required to investigate the scaling of the electronic noise. The performance of a given detector is commonly described by the noise equivalent power (NEP), which is the optical power on the detector at which quantum noise and electronic noise are equal (pi,QNL (Ω) = pel (Ω)). That means the recombined trace is 3 dB above the dark noise trace. Values of NEP = 1 mW are typical. Values as low as 10 μW have been achieved. 2. AC response. Most detector circuits will have a frequency-dependent gain, given by gel (Ω). For all detectors there will be a high frequency roll-off, depending on the type of photodiode and amplifier used. Frequently, there is also some electronic mismatch between the components leading to a sinusoidal frequency response curve. This is more a nuisance than a complication. It is taken into account by normalizing the traces (Eq. (8.1)), and gel (Ω) does not affect Vi (Ω) directly. 3. Efficiencies. The detector efficiency is not perfect. A low efficiency has the same effect as a beamsplitter that diverts only a fraction of the light into the detector. For measurements of the quantum noise alone, this is not crucial. For example, we can measure the QNL even from only a fraction of the beam and compensate with more electronic gain. However, knowledge of the efficiency is crucial to calculate the SNR of a signal or the degree of squeezing in the light, which means differences to the QNL. These effects disappear with high quantum efficiency. Note that electronic gain and efficiency play different roles. The golden rule for any squeezing experiment is that all of the light should be detected and that quantum efficiencies have to be close to unity. 4. Power saturation. Detectors can only handle a limited amount of optical power. The actual limit varies from mW for small fast photodiodes up to W for large detectors. Once the limit in power is exceeded, the detector can actually be damaged due to heating processes. The absorbed light will heat the diode material, and, more importantly, the bias current will lead to further ohmic heating. Even below this damage limit, the performance can become

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non-linear. The number of available carriers inside the detection material is limited, and the response drops if the local intensity is too high. Thus the detector has to be uniformly illuminated even at intensities below the recommended power limit. 5. Saturation of the amplifiers. Several cases have been reported where a detector showed a peculiar response and the reason was the presence of a strong modulation signal that saturated the entire circuit. Saturation at one radio frequency (RF) can affect the performance of the detector at many other frequencies. Care should be taken that no such strong signal is present by recording large bandwidth spectra up to the roll-off frequency of the detector. A particular pitfall is to use a series of amplifiers where the bandwidth of the earlier stages is broader than that of later stages, for example, by using a low-pass filter at the output of an amplifier. In this case the large high frequency signal would not be visible in the output, but it is saturating the amplifiers ahead of the filter. Saturation can change the noise spectrum over a wide bandwidth, and it can even lower the noise level, and in some experiments saturated amplifiers have generated noise traces that mimicked, and were mistaken for, squeezing. After all precautions have been taken, we can trust the detector. But in order to measure the QNL, we still need to calibrate it. One obvious technique is to calculate the expected noise power pi,QNL (Ω) from the formula for shot noise (Eq. (3.9)), and using the appropriate constants, we get pi,QNL (Ω) (dBm) = −138.0 + 10 log(i × RBW)(dBm) + pel (Ω)(dBm) + gel (Ω)(dB) where RBW is measured in Hz and i in A and an input resistance of 50 Ω is assumed. pi,QNL (Ω) can be determined once the average photocurrent iave is known. This in turn should be done best by a direct measurement of the DC from the photodiode. It also requires a knowledge of the detection bandwidth, which is normally the RBW of the spectrum analyser. The AC gain of the various amplifying stages has to be included. Finally, the actual response characteristics of the spectrum analyser to noise, rather than to a signal, have to be taken into account (Section 7.3.4), which could require corrections as large as 2 dB. After all these measurements and calculations, one could expect to get a value for the noise level with an accuracy of typically 20% (1 dB). This is good enough to test the apparatus for inconsistencies but insufficient to calibrate the traces accurately. A useful check is to investigate the scaling of the detected electrical noise power with the optical power, as shown in Figure 8.3. Quantum noise, electronic noise, and classical photon noise, or modulations, all scale differently. Electronic or dark noise is usually independent of the optical power Popt . The noise power 0.5 . The noise due to classical optical due to quantum noise is proportional to Popt noise is proportional to Popt . Figure 8.3 shows four cases of different mixtures of noise and their scaling. A reliable measurement of the quantum noise should only be attempted for situations that show a linear dependence of the measured noise level on Popt , in this case near Popt = 1.

8.1 Detection and Calibration of Quantum Noise

Log noise power, Pi (Ω)

100 (d) 10 (c) 1 (b) (a) 0.1 0.1

1 10 Log optical power, Popt

100

Figure 8.3 Scaling of different types of noise for a beam of light with fixed classical relative intensity noise (RIN = constant) and variable optical power Popt . The noise terms scale differently with Popt : (a) electronic noise, (b) quantum noise, (c) classical noise, and (d) total noise as seen in an experiment. For convenience all terms are normalized by the quantum noise present at Popt = 1.

8.1.1.1

White Light Calibration

An alternative is to calibrate the detector with a light source that we know is quantum noise limited. For example, we could use a thermal white light source, which has Poissonian photon statistics (Section 2.3). In principle all that is required is to illuminate the detector with white light that generates the same DC photocurrent and to record the RF noise spectrum. However, care has to be taken: the wavelength should be filtered to the same wavelength (about ±20–50 nm) as used in the actual experiment since the response of photodiodes is known to be wavelength dependent. The absorption depth depends on the wavelength. In addition, the illumination pattern has to be similar since localized illumination can lead to charge carrier depletion. Finally, the diode, particularly the very small InGaAs diodes, can easily be geometrically overfilled. Once a large fraction of the light misses the photosensitive area, it can heat up the entire chip and distort the results. From experience, large photodetectors can be calibrated using white light to about ±0.1 dB, whilst small InGaAs detectors can be calibrated to an accuracy of about ±0.4 dB. 8.1.2

Balanced Detection

An alternative technique for calibration is to use a balanced detection system. Here the incoming light is divided into two beams of equal optical power, which are detected by two matched photodetectors (Figure 8.4). The two photocurrents are either added or subtracted. Two noise spectra are created by the electronic spectrum analyser and are recorded as the sum and difference spectrum. This technique can distinguish between optical quantum noise and classical optical noise. The beamsplitter sends classical noise onto both detectors – the classical optical noise in both beams, and thus the photocurrents are correlated to each other. That means they can be subtracted and cancelled. When added, the full

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Vvac (Ω)

η

Vlas (Ω)

η

+

– gel (Ω) pel (Ω)

Vi (Ω)

Figure 8.4 Balanced detector.

noise power is recovered. For quantum noise the effect of the beamsplitter is different – the two resulting photocurrents are not correlated. That means the two channels add in quadrature for both the sum and the difference. In the quantum noise limited case, the two spectra are identical. This argument is based purely on the correlation properties of the sidebands containing the information about the noise, as described in Section 4.5. A more formal derivation of these results is given in Appendix B, which describes the balanced detector in terms of the propagation of coherent states through a beamsplitter. Electronic noise from the two detectors is obviously not correlated and adds in quadrature. It cannot easily be distinguished from quantum noise. Consequently, the apparatus can suppress all classical technical noise on this mode of light and leaves only the optical quantum and the electronic noise. The description is p+ (Ω) = gel (Ω) pi (Ω) Vi (Ω) + pel (Ω) p− (Ω) = gel (Ω) pi (Ω) + pel (Ω) Vi (Ω) =

p+ (Ω) − pel (Ω) p− (Ω) − pel (Ω)

(8.2)

In the case of sub-Poissonian intensity noise, the plus trace displays the noise suppression, and the minus trace shows quantum noise only. The technical difficulties with this detection scheme is the balancing of the detectors. The two photodetectors have to have the same electronic RF response. We have effectively built an electronic interferometer, and both the gain and the phase have to be matched at all RF that are used. This can be extraordinarily time consuming since the response frequently depends not only on the notional values of the components used but also on the actual physical layout of the circuit. Obviously cable lengths and delay times in the circuits have to be matched to achieve a proper balance. Passive splitters and combiners are used as adders and subtractors at high frequencies (≥ 50 MHz), whilst active circuits are useful for low and medium frequencies. The optical balancing can be tricky since it ideally requires a precise 50/50 beamsplitter. This can be achieved by careful adjustment since the splitting ratio for most dielectric beamsplitters depends on the input angle. Alternatively, a polarizing beamsplitter (PBS) can be used. For a linearly polarized output beam, the ratio can be continuously adjusted by rotating the polarization direction ahead of the beamsplitter with a half-wave plate.

8.1 Detection and Calibration of Quantum Noise

In all these arrangements optical losses have to be avoided if squeezing is to be detected since they would effectively decrease the quantum efficiency. It is tempting to compensate a mismatch in the electronic gain with an adjustment in the optical power. But again, this would decrease the quantum efficiency. Thus many of the common tricks used in detecting ordinary coherent light are not allowed for squeezed light. 8.1.3

Detection of Intensity Modulation and SNR

The size of a modulation can be measured directly. After all, electronic spectrum analysers were designed for this purpose. The height of the spectral peak is proportional to the power of the Fourier component. The relative intensity noise is linked to the measured power by pi (Ω) = (i RIN(Ω))2 × RBW 1

RIN(Ω) = 10 2 pi (Ω) dBm

(8.3)

For example, a beam that generates an average photocurrent of 1 mA with a modulation of one part in 104 (RIN = 10−4 ) produces a noise peak of −10 dBm when measured with a RBW of 100 kHz. For large modulations, the photocurrent SNR, as defined in Eq. (7.7), can be measured directly as the difference, in dB, between the signal peak and the noise floor. For small modulations we have to take into account that the signal peak contains both the signal and the noise and have to calculate the value SNR = (signal peak − noise)/noise. For very small SNR (< 2), the exact response of the spectrum analyser has to be considered (Appendix E). Electronic gain will not affect the SNR. Finally, the total quantum efficiency, including any form of attenuation of the light, has to be taken into account when using Eq. (7.7) to evaluate the optical SNR from the measurements of the photon current. 8.1.4

Homodyne Detection

The direct detection schemes described in Section 8.1 are used to measure the intensity as well as the intensity noise of a laser beam. For a single-mode field, the combination of two detectors with a beamsplitter can be used to evaluate the contribution of classical modulation and quantum noise. However, the direct detection cannot differentiate between the quadratures, effectively only measuring the in-phase (or amplitude) quadrature. To measure different quadrature angles, we need a reference beam, commonly called the local oscillator. This beam has to be phase-locked to the input; otherwise it cannot provide a phase reference to distinguish between the quadratures. If the local oscillator has the same frequency as the input, we have a homodyne detector. Alternatively, if the local oscillator is frequency shifted, the system is known as a heterodyne detector. 8.1.4.1

The Homodyne Detector for Classical Waves

For a homodyne detector we can assume that the local oscillator is created by the same source as the input beam we want to measure. In practice the link

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Unknown beam αin Laser

M1 M4 Local oscillator αlo

D1

ϕlo Phase shifter

D2

_

i_(t)

Figure 8.5 Homodyne detector.

could be much more indirect. The entire system can be represented by one large interferometer (Figure 8.5). The input beam 𝛼in and the local oscillator beam 𝛼lo are combined by the mirror M4 that has a balanced 50/50 reflectivity. The whole apparatus has to be aligned in the same way and with the same precision as a conventional interferometer. The two detectors D1 and D2 will receive the interference signal. Firstly, let us classically analyse the performance of the system. The two beams can be represented by the amplitudes 𝛼in (t) = 𝛼in + 𝛿X1in (t) + i𝛿X2in (t) 𝛼lo (t) = [𝛼lo + 𝛿X1lo (t) + i𝛿X2lo (t)] ei𝜙lo

(8.4)

i𝜙lo

Here e represents the relative phase difference between the input amplitude and the local oscillator amplitude. It can be controlled by changing the length of the one arm of the interferometer designated as the local oscillator. In practice, this involves the fine control of a mirror position by a piezo or the rotation of a glass plate in the local oscillator beam. The local oscillator should dominate the output signal. We consider the case that the local oscillator beam is far more 2 . In this case we can assume that all the intense than the input beam: 𝛼lo2 ≫ 𝛼in intensity is from the local oscillator and that the intensities on both detectors are the same: 1 |𝛼D1 |2 = |𝛼D2 |2 = |𝛼lo |2 (8.5) 2 All terms proportional to 𝛼in can be dropped in comparison with those proportional to 𝛼lo . The signal at the two detectors can be described using the standard convention for the phase shift in a single beamsplitter (Eq. (5.1)), including a 180∘ phase shift for one of the reflected beams (Figure 8.6): √ √ 𝛼D1 = 1∕2 𝛼lo (t) + 1∕2 𝛼in (t) √ √ (8.6) 𝛼D2 = 1∕2 𝛼lo (t) − 1∕2 𝛼in (t) The intensities and the photocurrents from the two detectors are proportional to |𝛼D1 |2 and |𝛼D2 |2 , respectively, ∗ |𝛼D1 |2 = 1∕2 [ |𝛼lo (t)|2 + 𝛼lo (t)𝛼in (t) + 𝛼in (t)𝛼lo∗ (t) + |𝛼in (t)|2 ]

8.1 Detection and Calibration of Quantum Noise

Figure 8.6 Projection through the uncertainty area.

δX2

θ = ϕlo δX1

which can be approximated as |𝛼D1 |2 = 1∕2 [|𝛼lo |2 + 2𝛼lo 𝛿X1lo (t) + 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo )) ]

(8.7)

∗ compared with |𝛼lo |2 and 𝛼lo 𝛿X ∗ compared by neglecting terms such as 𝛼lo 𝛼in ∗ with 𝛼in 𝛿X . In addition all higher-order terms in 𝛿X have been left out. Similarly, the result for the other detector can be found. We can now define the difference current i− (t) from the two detectors:

i− (t) ≈ 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo ) )

(8.8)

This is a remarkable result. The current i− (t) scales with the amplitude of the local oscillator, but the fluctuations, or noise, of the local oscillator are completely suppressed. Its noise level could be well above the QNL, and the result would not be affected. On the other hand, the power of the input beam has no influence, as long as it is small compared to the power of the local oscillator. This is an extremely useful and somewhat magic device. The difference current i− (t) contains only higher frequency, or AC, terms. Obviously the DC cancel each other. In the experiment we could use a splitter/combiner or a differential AC amplifier to generate this fluctuating current. Finally, the variance of this current is evaluated. We should note that in this calculation all cross terms of the type 𝛿X1𝛿X2 cancel out in the averaging process. We obtain Δi2− ≈ 4𝛼lo2 (𝛿X12in cos2 (𝜙lo ) + 𝛿X22in sin2 (𝜙lo ))

(8.9)

This means the measured variance is a weighted combination of the variance of the fluctuations in the two quadratures. In particular for the phase 𝜙lo = 0, the variance in the quadrature amplitude is measured, whilst for 𝜙lo = 90 = 𝜋∕2 the variance in the quadrature phase is measured. An alternative interpretation of Eq. (8.9) is that the homodyne detector measures the variance of the fluctuating state under a variable projection angle 𝜃 = 𝜙lo . By scanning the local oscillator phase, the projection angle can be selected. The sideband interpretation says that the different types of modulation differ by the relative phase 𝜃 between the two beats between the two sidebands and the carrier. The homodyne detector selects one such phase. The modulation with the other phase (𝜃 + 90) is not detected at all. The homodyne detector filters out specific quadrature of the modulation sideband. With the fixed phase shift 𝜙lo for the local oscillator, the detector will have best sensitivity for one particular type of modulation. For example, for 𝜙lo = 0, only amplitude modulation is detected, and

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for 𝜙lo = 90, only phase modulation is detected. The sensitivity of the detector to this quadrature changes with cos2 (𝜙lo ). The homodyne detector is very popular in radio applications and is a key technology in our present communication systems. Signals are detected in phase with a local oscillator, in the case of radio waves an electronic oscillator inside the receiver, which can be controlled in frequency very well, for example, using quartz crystal oscillators, or could even be locked to the incoming RF (known as phase-locked loop (PLL)). Radio homodyne and all the later heterodyne detection techniques have allowed us to place different carriers very close to each other in the radio spectrum and to detect very weak signals. This increases dramatically the carrying capacity compared to a simple intensity-coded system and allowed applications such as the mobile phone, satellite communications, etc. Quantum mechanical calculations are required to check the validity of the above classical results for the quantum properties of the light. The balanced homodyne detector is a special case of a Mach–Zehnder interferometer (Section 5.2) with a 50/50 output mirror. The two incident fields on the beamsplitter M4 can be described by the operators ̂ in (t) + i 𝜹X𝟐 ̂ in (t) â in (t) = 𝛼in + 𝜹X𝟏 ̂ lo (t) + i 𝜹X𝟐 ̂ lo (t)] ei𝜙lo â lo (t) = [𝛼lo + 𝜹X𝟏

(8.10)

where we are using the nomenclature of Section 4.5.1. In complete analogy to the classical derivation, we can now derive the equivalent to Eq. (8.9) but now for the full quantum operators: ) ( ̂ 2 ⟩sin2 (𝜙lo ) ̂ 2 ⟩cos2 (𝜙lo ) + ⟨𝜹X𝟐 Δi2− ≈ 4𝛼lo2 ⟨𝜹X𝟏 in in = 4𝛼lo2 (V 1in cos2 (𝜙lo ) + V 2in sin2 (𝜙lo ))

(8.11)

The derivation of Eq. (8.11) holds for the approximation that 𝛼lo ≫ 𝛼in . In this simple interferometer we have a beautiful device for the measurement of the variance for a single quadrature, including all quantum properties. We can select the quadrature by simply varying the phase angle of the local oscillator. For a coherent state, where all quadratures are identical, this technique has no particular advantage over direct detection. However, once the variances of the quadratures change, as in a squeezed state, the homodyne detector is the essential measuring system. It was first proposed by H.P. Yuen and J.H. Shapiro [2] and D.F. Walls and P. Zoller [3] and has since been a key component in all experiments with quadrature squeezed states, as discussed in Chapter 9. Finally, a few practical hints are given: It should be noted that the homodyne detector can be operated even when the intensity of the input is small. As long as |𝛼in |2 is smaller than the intensity |𝛼lo |2 of the local oscillator, all the results given above are correct. Thus the detector can be used to measure squeezed vacuum states as well as bright squeezed light. If |𝛼in |2 is comparable with |𝛼lo |2 , the output intensity will change periodically with 𝜙lo , and the variance will be a mixture of variances from both orthogonal quadratures. Balancing the homodyne detector is the most difficult part of this experiment. Both the amplitude and the wave front curvature have to be matched for both outputs. This generally means that

8.1 Detection and Calibration of Quantum Noise

both the local oscillator and the input beam should travel comparable distances and should experience similar optical components such as lenses and mirrors. In squeezing experiments we cannot afford any losses; otherwise some vacuum noise would be added, and the noise suppression would deteriorate. That means some of the tricks quite common with normal classical interferometers cannot be used here. For example, we have to detect all of the beam, and using an aperture is not allowed. We cannot balance the two beamsplitter outputs by attenuating one of the beams. It is also important that all the electrical components are balanced and they should have the same gain and phase difference at all detection frequencies used. The quality of the alignment can be judged by measuring the fringe visibility VIS of the system (definition in Eq. (5.28)). This can be done in some experiments by turning up the power of the input beam to be the same as that of the local oscillator. As long as some input intensity is available, the normalized visibility  = VIS∕VISperfect can be measured, where VIS is the directly measured value and VISperfect is the visibility predicted from the measurement of the intensities. For  = 1 the homodyne detector records all the correlations accurately. The measurement of the extrema is reduced when  < 1, and the correlations between the input and the local oscillator beams appear to be smaller than they really are. The quality of the homodyne detector is described by an overall efficiency: 𝜂hd = 𝜂det  2

(8.12)

where 𝜂det is the normal efficiency of the photodetector. In most situations 𝜂det is smaller than  2 . Values as good as  2 = 0.98 have been achieved. These details will be discussed in the description of the individual experiments. The homodyne detector is the most common technique to measure the noise and signal properties of both quadratures. It is more complex and requires more patience than direct detection, but it provides the full information. 8.1.5

Heterodyne Detection

Whilst in homodyne detection we are mixing two beams with the same optical frequency on one detector, with a beamsplitter we can also mix beams with different optical frequencies 𝜈1 and 𝜈2 . This is known as heterodyne detection. As long as the difference frequency ΩHT = 𝜈1 − 𝜈2 is within the detection range of the photodetector, and in practice that is limited typically to about 1–2 GHz, we will record a strong, narrow signal at ΩHT , which is surrounded by beat signals created by all the sidebands of either laser beam beating with the respective carriers of the other beam. Note that a spectrum analyser cannot distinguish between positive and negative beat frequencies and these two will be folded together (Figure 8.7). Heterodyne detection is frequently used to determine the linewidth Δ𝜈2 of an unknown laser. The spectrum around the detection frequency ΩHT = 𝜈1 − 𝜈2 contains information about both beams. The lineshape of the heterodyne signal is a convolution of the lineshapes of the two lasers. If one of the laser beams has a narrow linewidth compared with the other, e.g. Δ𝜈1 ≪ Δ𝜈2 , the width ΔΩHT of

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Spectrum analyser Laser 1

Laser 2 Mode matching optics

Figure 8.7 Example of heterodyne detection of the bandwidth of two lasers.

the heterodyne signal corresponds to the broader linewidth, in this case Δ𝜈2 . The exact link between ΔΩHT and Δ𝜈 depends on the lineshape. For example, if both lasers have Lorentzian lineshapes, which are quite common, and Δ𝜈1 ≈ Δ𝜈2 , the link is ΩHT = Δ𝜈1 + Δ𝜈2 . Furthermore, the heterodyne spectrum will show any additional laser modes in the input beams. These will appear as strong, narrow spectral features around ΩHT . In heterodyne detection we are measuring signals at high detection frequencies, and outside the prevailing technical noise at low frequencies, the heterodyne technique is very sensitive in the amplitude quadrature. Ideally it is only limited by the QNL of the stronger of the two input beams, and in this case the detection is at the QNL, or commonly called shot noise limited. This technique has been used to detect with very high sensitivity the fluorescence spectrum of individual ions in traps or individual atoms and to determine such properties as the ion trap frequency (as low as hundreds of Hz) and the excitation temperature. In this regime heterodyne detection competes with single-photon detection in sensitivity. 8.1.5.1

Measuring Other Properties

The detection schemes discussed so far concentrate on the measurement of the intensity and the quadratures of the laser light. However, there are other properties of the light, in particular the position and the polarization of the light, that are worth a consideration. Many applications rely on the modulation and detection of these properties, as can be seen from Figure 8.8. Detector (c) measures the beam position by comparing the difference in the intensity falling onto the two halves of a split detector. This could also be done in both transverse directions using a quadrant detector. Detector (d) uses a PBS to analyse the degree of polarization, in this case horizontal and vertical linear polarization. Other states of polarization can be determined by including a wave plate ahead of the PBS. This comparison shows that in all cases we are using two photodetectors and we are processing the difference of the photocurrents. In all these cases we will find that for a coherent and single-mode beam, the perfect laser beam, each detector produces a photocurrent that includes quantum noise and that simply adds to the quantum noise from the other detector. This means we have a quantum noise for the detection of all these properties: intensity, quadrature position, and

8.2 Intensity Noise

D1 Laser

Laser D2

D1

+/–

(a)

Phase shifter

(b)

D2

_

D1

Laser (c)

Laser (d)

PBS

D2

Figure 8.8 Comparison of detection schemes: (a) balanced detector for V1(Ω), (b) homodyne detector for the quadratures, (c) split detector for the position of the beam, and (d) polarization detector.

polarization. The last one is normalized differently, and the uncertainty can be expressed in terms of the Stokes parameters that have a cyclic commutation relationship [4, 5], but is still a QNL. We will find in Chapters 9 and 10 that for all these properties we can generate non-classical states of light with reduced noise below the QNL.

8.2 Intensity Noise 8.2.1

Laser Noise

Lasers are the light sources in all quantum optics experiments, but they are far from ideal; most lasers have intensity noise well above the QNL as discussed in Chapter 6, page 210. With careful preparation some lasers can be QNL in restricted parts of the spectrum. Good examples are the diode-pumped monolithic Nd:YAG laser, which is one of the quietest and most reliable lasers available (https://www.coherent.com/lasers/main/mephisto-lasers) [6, 7]. These lasers are QNL for frequencies well above 5 MHz, as shown in Figure 6.6, and thus are rather useful light sources for quantum optics experiments. Diode lasers themselves can be excellent low noise light sources at frequencies up to several hundred MHz, as discussed on page 213. In particular the low-power (<200 mW) single-element diode lasers can be QNL at a wide range of frequencies. The cavities are very short (<1 mm), but the gain bandwidth can be very large, and thus single-mode operation is usually only achieved with either an extended cavity design or an internal distributed Bragg reflector (DBR). Even without these additional features, the total photon flux can indeed be very quiet, basically representing the fluctuations of the input current that can easily be controlled to the QNL. However, in practice these lasers tend to excite other modes near threshold, which results in large intensity noise, as discussed on page 213.

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Modulator

η2

V1las (Ω)

Laser PID h (Ω)

η1

V1out (Ω)

V1il (Ω)

Figure 8.9 Schematic diagram of a noise eater.

8.3 The Intensity Noise Eater It is an obvious idea that the intensity noise of a laser can be reduced using an electro-optic modulator and a feedback controller. In the jargon of the experimentalists, such a device is called a noise eater. Practical applications are the suppression of the large modulations associated with the relaxation oscillation, as shown in Figure 6.6, and the suppression of technical noise due to mechanical imperfections of the laser. The schematic layout of a noise eater is shown in Figure 8.9. It consists of the intensity modulator, a beamsplitter to generate a beam that is detected by the in-loop photodetector Dil , and control electronics, with a combination of proportional, integral, and differential (PID) gain that drives the modulator. The laser has a noise spectrum V 1las (Ω) that is reduced by the feedback system to a much quieter spectrum V 1out (Ω). For large intensity noise, well above the QNL, the feedback system is entirely classical, and we can use conventional control theory to describe it. However, once we approach the QNL, the properties change. We will find that for a QNL laser input, the noise actually increases, completely reversing the classical predictions. For the control of the intensity, any one of the different intensity modulators, described in Section 5.4.5, can be used. A fixed voltage is applied to the modulator that reduces the power Plas of the laser to a working point (1 − D)Plas . We assume D ≪ 1, but even the largest fluctuation of the laser is less than DPlas . The voltage to the modulator is controlled by the feedback circuit to counteract any changes in the intensity coming from the laser. In principle this could be achieved for all frequencies Ω. As a result, the noise at the output, V 1out (Ω), is quiet. We want to derive the transfer function for V 1out (Ω). 8.3.1

Classical Intensity Control

Using standard control theory [8], the feedback system can be described by the open-loop amplitude gain h(Ω). This gain can be visualized as the action of the entire feedback system, including beamsplitter, detector, amplifier, and modulator, on an intensity modulation. Here h(Ω) = 1 means the effect of the modulator would be equal to the signal on the laser beam, and h(Ω) = 0 means that there is no feedback. For good noise suppression we need |h(Ω)| > 1. The complex gain h(Ω) changes the magnitude and the phase of the modulation. The phase

8.3 The Intensity Noise Eater

Bode plot 10 (i) Magnitude 1 |h|

Nyquist plot

0.1 |h| cos (arg(h))

4

(ii) Phase

10

0

100

10 000 1000

–2

500

arg(h)

–4 –4 (b)

(a)

2

1

10

100

1000

104

–2 0 2 |h| sin (arg(h))

4

105

Figure 8.10 (a) Bode plot of a typical feedback loop round-trip gain. (b) Nyquist plot of the same gain.

shifts are due to the delay in the loop, introduced by the physical layout of the system, and the phase shifts inside the electronic components. The magnitude |h(Ω)| is affected by the conversion efficiency of the detector, the gain of the electronic amplifier, and the characteristics of the modulator. Figure 8.10 shows one typical example of the gain spectrum h(Ω). This is usually plotted in separate diagrams for the magnitude |h(Ω)|, Figure 8.10a (trace i), and the phase arg(h(Ω)) as functions of Ω, Figure 8.10a (trace ii). This pair of diagrams is known as the Bode plot. A more useful alternative is the Nyquist plot (Figure 8.10b), in which h(Ω) is shown in a polar plot. The individual frequencies are marked along this plot. It should be noted that in all practical cases the gain will be low, |h(Ω)| < 1 at very low frequencies and very high frequencies. In addition any such system has to have a phase shift from DC to the highest frequency of at least 2𝜋. We can derive the effect of the feedback system by tracing the amplitude of the modulation at different places within the system (Figure 8.9). For a classical system we can ignore the effect of the vacuum fluctuations. We have the amplitude ̃ f after the beamsplitter, 𝜹A ̃ el inside the elec̃ il directly after the modulator, 𝜹A 𝛿A ̃ tronic feedback system, and 𝜹Aout for the laser beam leaving the noise eater. For the classical system the different amplitudes are linked via the equations ̃ las + h(Ω) 𝜹A ̃ el ̃ il = 𝜹A 𝜹A ̃ il = 𝜹A ̃ f = 𝜹A ̃ el 𝜹A

(8.13)

This results in ̃ las + h(Ω)𝜹A ̃ il ̃ il = 𝜹A 𝜹A

(8.14)

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8 Quantum Noise: Basic Measurements and Techniques

And consequently, still ignoring the effect of the vacuum fluctuations and with h(Ω) compensating the reduction in amplitude caused by the beamsplitter, we obtain ̃ las 𝜹A ̃ il = 𝜹A (8.15) |1 − h(Ω)| and we get the well-known result from classical control theory: V 1il (Ω) =

V 1las (Ω) |1 − h(Ω)|2

(8.16)

which is also detected in the beam leaving the noise eater. Without feedback the noise of the light in front of the in-loop detectors is V 1il (Ω) = V 1las (Ω). Once the feedback loop is closed, the fluctuations will be suppressed, and the tendency is to make the noise of the in-loop current V 1e (Ω) as low as possible. This noise suppression is best for large gain. Naively one might expect that the noise suppression depends simply on the sign of ℜe h(Ω): for negative feedback, ℜe h(Ω) < 0, the noise is suppressed, and for positive feedback, ℜe h(Ω) > 0, the noise is increased. This is not correct. The suppression depends on the value |1 − h(Ω)|2 . This is best explained with a Nyquist diagram (Figure 8.10b). Equation (8.16) means that the feedback loop decreases the variance at all those frequencies where |1 − h(Ω)| > 1, that is, where h(Ω) is outside a circle of radius 1 drawn around the point (1, 0) on the Nyquist diagram. The reduction in the variance is proportional to the square of the distance between h(Ω) and (1, 0). If the line that traces h(Ω) is inside the circle, the variance is actually increased, not reduced. If h(Ω) gets close to (1, 0), the resulting noise is extremely large. The stability of the systems can be described by one simple rule: if the line that traces h(Ω) in the Nyquist diagram encloses the point (1, 0), the system is unstable. It will oscillate madly. This includes the case where h(Ω) = 1. All other systems, where h(Ω) does not enclose (1, 0), are stable. But note that even in a stable system, there can still be frequencies where the noise is increased, as demonstrated in the example shown in Figure 8.11. Finally, we find that the variance of the modulation of the beam leaving the classical system is scaled by the transmission 1 − 𝜖 of the beamsplitter and the quantum efficiency 𝜂 of the detector, in the same way as the intensity of the output beam, resulting in the transfer Figure 8.11 The action of the noise eater on a large classical signal using the feedback gain shown in the previous diagram. Input noise is V1las (Ω) = 1000, very large compared to the QNL. The resulting output V1out (Ω) remains well above the QNL of V1out (Ω) = 1.

10 000 Vin = 1000 1000 Vout

284

100

10

1

10

100

1000 Ω

10 000

100 000

8.3 The Intensity Noise Eater

function V 1out (Ω) = 𝜂2 (1 − 𝜖) V 1il (Ω) = 𝜂2 (1 − 𝜖)

V 1las (Ω) |1 − h(Ω)|2

(8.17)

where 𝜂2 is the quantum efficiency of the output detector. To illustrate Equation (8.17) consider the output noise spectrum shown in Figure 8.11, which is calculated for a large constant laser noise V 1las (Ω) = 1000, very much larger than the QNL. The feedback system has the gain h(Ω) shown in Figure 8.10. We see the effect of a feedback amplifier with a maximum gain at 300 kHz and unity gain points at 6 and 1000 kHz. The gain drops below 1 at frequencies below 6 kHz and above 1000 kHz. This feedback loop has an internal delay time of about 4 μs. The phase response is determined by the somewhat resonant nature of this circuit. In the Nyquist diagram this corresponds to a line that starts with straight negative feedback until h(Ω) changes sign at about 400 kHz. However, the noise is reduced even at higher frequencies up to 1000 kHz since the noise reduction is proportional to the square of the distance to the point (1, 0). The complete spectrum of the output variance is shown in Figure 8.11. The noise is suppressed from very low frequencies to 1000 kHz. The maximum gain results in the best noise suppression at 300 kHz. But when the line h(Ω) enters the circle around (1, 0), the noise increases. A clear hump is visible from 1000 kHz to 40 MHz. This feedback loop functions well at medium frequencies with high gain. But at higher frequencies the phase has increased too much, and excess noise is created. This result is not atypical for a realistic feedback system. In practice one has to work very hard to find a design that reduces the gain quickly enough and simultaneously avoids too much electronic phase change. 8.3.2

Quantum Noise Control

The behaviour of the feedback system near the standard quantum limit (QNL) is quite different. In this case we have to include the vacuum fluctuations from the unused input port of the beamsplitter. We can also no longer assume that the variance inside and outside the feedback loop will remain the same. Figure 8.12 shows the apparatus including all vacuum fields. The input into Modulator

Avac

η2

V1las (Ω) Avac

Laser PID h(Ω)

η1

Avac V1out (Ω)

V1il (Ω)

Figure 8.12 Schematic diagram of a noise eater with all vacuum terms.

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8 Quantum Noise: Basic Measurements and Techniques

̃ las (Ω). We distinguish between the in-loop the noise eater is described by 𝜹A ̃ il in front of the beamsplitter, the field in front of the in-loop detector field 𝜹A ̃ f , the field that is leaving the apparatus 𝜹A ̃ o and the two electrical signals, one 𝜹A ̃ el that is fed back to the modulator, and 𝜹A ̃ out from inside the feedback loop 𝜹A the detector that measures the beam leaving the apparatus. We recall that we are using nomenclature introduced in Section 4.5.1 whereby capitals and tildes indicate continuum operators in the frequency domain with commutators: ̃ ̃ † (Ω′ )] = 𝜹(Ω − Ω′ ). [𝜹A(Ω), 𝜹A It is now necessary to define a quantum description of the feedback. This problem was addressed as early as 1980 and was approached in various ways. The most obvious and direct approach was pioneered by Wiseman et al. [9]. The main feature is that the current signal at time t is not only related to the instantaneous value of the photon flux Nil but is the convolution with a response function that takes the history of the signal into account and simulates the gain of the feedback loop. For frequencies that are smaller than the inverse of the round-trip time of the feedback loop (Ω < 1∕tloop ), the effect is similar to the effect in a classical feedback system, and we can describe the action of the modulator simply by √ ̃ il = 𝜹A ̃ las + g(Ω) 𝜹A ̃ el 𝜹A (8.18) which describes the action of the intensity modulator that combines the input field and the electrical signal. The very small absorption of this device, which also corresponds to a vacuum input, has been ignored. The gain of the in-loop electrical amplifier is given by the value g. The various fields are linked by √ √ ̃ il + 1 − 𝜖 𝜹A ̃ vac ̃ f = 𝜖𝜹A 𝜹A √ √ ̃ el = 𝜂𝜹A ̃ f − 1 − 𝜂 𝜹A ̃ vac 𝜹A √ √ ̃ o = 1 − 𝜖𝜹A ̃ il + 𝜖 𝜹A ̃ vac 𝜹A √ √ ̃ o − 1 − 𝜂 𝜹A ̃ vac ̃ out = 𝜂𝜹A 𝜹A (8.19) where we have assumed that the two detectors have the same quantum efficiency 𝜂 and recall that 𝜖 is the reflectivity of the feedback beamsplitter. We can now ̃ out , ̃ el and 𝜹A combine the equations in (8.19) to provide explicit expressions for 𝜹A ̃ ̃ convert these into the quadrature operators 𝜹X𝟏el and 𝜹X𝟏out , and use our standard recipe to calculate the noise spectra V 1el (Ω) for the current inside the loop and V 1out (Ω) for the current from the external detector. The details of the calculation are given in Appendix D, and the resulting transfer functions are V 1el (Ω) =

𝜂𝜖(V 1las (Ω) − 1) + 1 |1 − h(Ω)|2

(8.20)

and V 1out (Ω) = 1 +

2 1 − 𝜖 𝜂𝜖(V 1las (Ω) − 1) + |h(Ω)| 𝜖 |1 − h(Ω)|2

(8.21)

√ where h(Ω) = 𝜂𝜖 g(Ω) is the open-loop round-trip gain, which includes both the electronic gain g(Ω) and the optical attenuation. Note that our model

8.3 The Intensity Noise Eater

of the modulator requires the intensity attenuation to be combined with the voltage gain. These results show that the properties of the light inside the controller and the beam leaving the controller are entirely different. Inside (Eq. (8.20)), the modulation and the noise are strongly suppressed, and for a large gain the variance V 1el (Ω) can be well below the QNL. The noise eater works well inside, which is possible, since the laser beam inside is a conditional field. It only exists inside the feedback system, and the conditions for a freely propagating beam do not apply. The standard commutation rules do not apply and allow this violation of the normal quantum noise rule. Such light is sometimes referred to as squashed light [10]. For high gain, |h(Ω)| > 1, and a low level of laser noise, V 1las (Ω) ≈ 1, the variance V 1el (Ω) of the control current can be substantially less than the QNL. This was observed by A.V. Masalov et al. [11, 12] and others. It was initially concluded that the light inside the loop was sub-Poissonian and could be used for measurements with a better SNR than the QNL would allow. However, it was overlooked that any signal, for example, a modulated absorption, would be controlled by the feedback loop in a similar way as the classical noise. A detailed analysis has shown that even with sub-Poissonian control currents, it is not possible to improve the SNR. Any apparent advantage of the low noise of the control current cannot be used [13]. The properties of the freely propagating output beam (Eq. (8.21)) are quite different. This beam obeys the standard commutation rules and conditions for the quantum noise, and the spectrum V 1out (Ω) in most cases is above the QNL. We find that this light has properties that are quite different from the predictions of the classical model (see Eq. (8.17)) and contain unique quantum features. In many situations the effect of the noise eater is actually counterproductive, and the machine increases the noise level. This is one of the striking examples where the properties of an optical instrument with small signals, near the QNL, are considerably different to the conventional classical predictions, and standard engineering solutions will actually fail. In the area of laser noise control, this lesson had to be learned, and quantum optics rules are important for the technical progress. It is worthwhile to consider the following special cases: 1. For very low gain (h(Ω) → 0), this result shows simply the effect of attenuation by the combination of beamsplitter transmission (1 − 𝜖) and quantum efficiency 𝜂 with V 1out (Ω) = 1 + (1 − 𝜖)𝜂 (V 1las (Ω) − 1). This is identical to a simple beamsplitter. Note that the effect of the quantum noise contribution is included. 2. For a QNL input laser (V 1las (Ω) = 1), the feedback loop creates extra noise |h(Ω)|2 that increases with the gain h(Ω) with V 1out (Ω) = 1 + 1−𝜖 . This case 𝜖 |1−h(Ω)|2 is interesting for engineering applications since it means that in systems with very small signals, close to the QNL, the real feedback system shows a behaviour that is quite different from the classical approximation. The higher the gain and the smaller the fraction 𝜖 of the light that is used for control, the more excess noise is generated. The physical explanation is that the vacuum noise at the beamsplitter is added and amplified. For example, if only

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8 Quantum Noise: Basic Measurements and Techniques

10 8 Vlas = 10 6 V1out

288

Vlas = 5 4 Vlas = 1 2 0

0

–1

–2

–3

–4

–5

Round-trip gain, h

Figure 8.13 The effect of the feedback system with 𝜖 = 0.1, for different modulations V1las = 10, 5, 1, as a function of the open-loop gain h. At small gains the larger signals are suppressed, and at larger gains (h = −5) all signals approach the same value, in this case V1out = 7.3.

10% of the light is used (𝜖 = 0.1) and the open-loop gain is h = −5 (negative feedback), the output variance will be 1 + ((1 − 0.1)∕0.1) 52 ∕(1 + 5)2 = 7.25. It could be significantly larger if the feedback phase is not optimum. In order to avoid this noise penalty completely, all the light would have to be detected (𝜖 = 1), which is clearly impractical. One can minimize the excess noise by detecting most of the light, leaving only a small intensity at the output. In . the limit of very large gain, we obtain the limiting value V 1out (Ω) = 1 + 1−𝜖 𝜖 This value is independent of the gain, a kind of saturation effect producing a fixed noise level, as shown in Figure 8.13. ), the system shows the classical proper3. For very large signals (V 1out (Ω) > h(Ω) 𝜂𝜖 ties with the noise suppression given by V 1out (Ω) = no longer play any role.

𝜂𝜖V 1las . |1−h(Ω)|2

Quantum effects

To illustrate this quantum effect, the performance of the noise eater previously described in Figure 8.10b has been evaluated for a wide range of input laser noise levels. They range from V 1las = 1000 to V 1las = 1. All traces in Figure 8.14 are normalized to the QNL. At the highest frequencies where the gain is much less than one, the noise spectrum is unaffected. The top trace, with the large noise level, V 1las = 1000, clearly shows the same response as the classical feedback loop. Biggest suppression is achieved close to the maximum gain 300 kHz and some excess noise at higher frequencies, 2000 kHz. However, the lowest trace for a QNL laser clearly demonstrates the noise penalty. In this particular case the feedback system produces about 8 dB of excess noise. At laser noise levels of about 10 times the QNL, the feedback loop has very little effect indeed since in this case noise suppression and noise penalty balance each other. One explanation for the noise penalty is the fact that the beamsplitter acts as a random selector of photons, not a deterministic junction (see the discussion in Section 5.1). The noise measured by the in-loop detector is a random selection

8.3 The Intensity Noise Eater

104 Vin = 1000

1000

300 100

Vout

100

30 10

10

1

3 1

0.1

1

10

100

1000

104

105

Ω

Figure 8.14 The effect of a noise eater on a laser. Various input noise levels (V1las = 1, 3, 10, … , 1000) are shown. At low noise levels this device creates rather than eats noise.

of the laser noise that is not correlated to the fluctuations transmitted to the output. This small fraction of the noise is detected, converted into a current, amplified, and added by the modulator to the remaining fluctuations of the light from the laser. Obviously all this has to happen within a time less than 1∕Ω. Unlike in a classical feedback circuit, the noise that was detected cannot cancel the remaining noise since there is no correlation between them. Instead, they are added in quadrature, leading to the excess noise. The more gain we apply, the worse it gets. This phenomenon has been observed and described by a number of researchers. One particularly instructive description was given by Mertz and Heidmann who used a Monte Carlo technique to describe the properties of the beamsplitter [14]. 8.3.2.1

Practical Consequences

The consequence of this performance at the QNL is that we have to reduce the gain of the feedback loop to much less than unity at all detection frequencies where the laser should be quantum noise limited. Most lasers are QNL at large frequencies. Consequently, the task is to develop special electronic amplifiers that combine high gain at low frequencies, where classical noise dominates, with low gain (|h| < 10−2 ) at the frequencies where we need QNL performance. The trick to achieve this is a very steep roll-off in the gain without too much phase lag being incorporated [15], and this requires specialized electronic designs. There are different options for intensity control. As discussed so far, an external intensity modulator can be added to the laser. It is also possible to control the pump power to the laser by feedback. In the case of diode lasers, the injection current can be directly controlled, which is a very elegant approach. The performance of this system can be predicted by incorporating the laser noise transfer function for pump noise into the description of the feedback loop. The solution is not trivial. The practical difficulty is that most current control units are not

289

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8 Quantum Noise: Basic Measurements and Techniques

suitable for this since their external inputs include filters designed to avoid noise transients. This produces a phase lag that is far too large for a good feedback performance. In many cases it is better to leave the current source untouched and to add a small control current. This was demonstrated successfully by C.C. Harb et al. [16] for a diode laser-pumped CW Nd:YAG ring laser. The relaxation oscillation was suppressed by more than 40 dB, and the noise at low frequencies was largely eliminated. Similar circuits are available in commercial lasers [17]. In conclusion, a feedback intensity controller, or noise eater, is extremely useful to reduce technical noise in the intensity of a laser. With careful design the QNL can be achieved. However, it cannot be used to either generate an output beam that is sub-Poissonian (squeezed) or measure any effect with an SNR better than the standard quantum limit that can already be reached with a coherent state. A complementary design, electro-optic feedforward, has quite different properties that can be utilized for noiseless amplification (Chapter 10) or quantum information processing (Chapter 13).

8.4 Frequency Stabilization and Locking of Cavities In many experiments it is very important to operate lasers and cavities that are locked to each other, which means they are tracking each other in the frequency domain. Active feedback techniques are required to achieve this aim. The length of either the cavity or the laser resonator is constantly adjusted to keep both of them in resonance at one frequency. Applications for such a system are build-up cavities for non-linear processes, mode cleaner cavities, and the stabilization of the laser frequency. In the latter case, a very quiet reference cavity is chosen, which is protected as much as possible from external influences, such as mechanical vibrations and temperature fluctuations. The laser is locked to this reference and will have a much narrower linewidth than a free-running laser. The performance of lasers can best be described using the concept of the Allan variance and quoting the timescale on which the smallest variance is observed [18]. Frequency stabilization has been used since the very early days of laser design. Many of the milestones in this technology have been achieved by J. Hall and his colleagues in a continuous series of technical developments since the 1970s. Two current examples illustrate the degree of improvement that can be achieved. A free-running dye laser has a linewidth of many MHz, measured over an interval of 1 second. Using active control techniques, the linewidth can be reduced to about 1 kHz. A free-running diode laser-pumped CW Nd:YAG laser has already a narrow linewidth, 10 kHz measured over 100 seconds, due to its simple mechanical structure. Using active control techniques, linewidths of less than 1 Hz measured in 100 seconds can be achieved. And this is not a fundamental limit [19]. Many tricks are used in the design of the special reference cavities, such as low expansion materials, special mounting systems, and cryogenic control. Extraordinary precision has been achieved through painstaking engineering and some ingenious design. Here only the principles of the locking technique

8.4 Frequency Stabilization and Locking of Cavities

will be outlined, as an introduction to the techniques commonly used. In most CW quantum experiments, at least one cavity locking system is required. Frequently several are used simultaneously. In most cases the locking system can be described by the schematic block diagram shown in Figure 8.15a. The aim of Modulator PBS

Piezo element

Faraday isolator

Reference cavity

Laser

RF signal generator

~ Error signal Mixer

(a)

PBS

Piezo element

Faraday isolator

Reference cavity

Laser

Error signal (b)

Transmitted power (V)

2 TEM00

1.5

TEM10

TEM20

TEM30

(i)

1 0.5

PDH error signal (V)

0 0.05 (ii) 0

Tilt error signal (V)

–0.05 1 0.5

–0.5 –1

(c)

(ii)

0

0

1

2

3 4 5 Scanning time (a.u.)

6

7

Figure 8.15 Frequency locking system. (a) Schematic layout for PDH locking and phase response of the cavity. (b) Layout for tilt locking. (c) Experimental error signals as demonstrated by D.A. Shadock et al. (i) Transmitted power, (ii) error signal generated via PDH locking, and (iii) error signal generated via tilt locking technique. Source: Courtesy of D.A. Shadock et al. [20].

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the device is to dynamically control the detuning Δ𝜈, the difference between the cavity and the laser frequency, and to maintain resonance (Δ𝜈 = 0). The light from the laser is mode-matched, using a lens system, into the reference cavity. An error signal is generated, which is a measure of the detuning Δ𝜈. This signal is processed using an amplifier of suitable gain, bandwidth, and phase response. Finally, an actuator is used, such as a piezo crystal mounted on a mirror, to control either the laser frequency or the cavity frequency. Let us discuss the most important features of these components. The cavity parameters are chosen to produce a suitable control range. Typically the control range is as wide as the cavity linewidth. For lasers with large frequency excursions, such as dye lasers, a low finesse cavity is required. For narrowband lasers, such as CW Nd:YAG lasers, a high finesse (up to 20 000 or more) cavity is chosen to provide a very steep frequency response. The cavities can be confocal, which reduces the difficulties of mode matching, or can have a conventional stable resonator configuration.

8.4.1

Pound–Drever–Hall Locking

The error signal has to be responsive to very small changes in Δ𝜈. We require a technique that measures the detuning accurately and with low noise. The most popular technique is to measure the response of the cavity to frequency modulation sidebands. This technique was first used by Pound for microwave systems and later refined by several scientists for the optical domain. It is frequently known as the Pound–Drever–Hall (PDH) technique. In this case a phase modulator, driven by a frequency Ωpm , is placed between the laser and the cavity. The light reflected from the cavity is detected. Since optical feedback has to be avoided, an optical isolator (Section 5.4) is normally placed between the cavity and the laser, with the reflected beam directly accessible. The reflected light has modulation sidebands at frequency Ωpm . These are detected using a mixer and, as local oscillator, a part of the modulator driving signal. The amplitude of the resulting low frequency signal provides a suitable error signal. If the cavity is on resonance with the laser beam, which means Δ𝜈 = 0, the reflected light will be purely phase modulated. After the mixing process, no signal is available since the two sidebands balance each other exactly. The error signal for Δ𝜈 = 0 is zero. However, once the cavity is detuned, both PM sidebands will experience a different phase shift, and the two beat signals will have a phase difference Δ𝜙out (Δ𝜈) with respect to each other. They will no longer cancel, and there will be some AM, a detectable error signal after the mixer. The shape of the error signal, as a function of cavity detuning, is easily explained for the case that the modulation frequency Ωpm is much larger than the cavity linewidth Γcav . The relative phase of the two beat signals is determined by the phase difference between the central component 𝜙c and the two sidebands 𝜙− , 𝜙+ . The three components have a distinctively different phase, as discussed in Section 5.3. A typical phase response is shown in Figure 8.15c(ii). The corresponding power response, in this case for an impedance-matched cavity, is given in Figure 8.15c(i). We can arbitrarily define 𝜙(0) = 0 at resonance. That means

8.4 Frequency Stabilization and Locking of Cavities

for large modulation frequencies, we obtain 𝜙(−Ωpm ) = 𝜙(+Ωpm ) = 0 The phase difference at the output is Δ𝜙out (Δ𝜈) = (𝜙+ − 𝜙c ) − (𝜙c − 𝜙− ) = 𝜙(Δ𝜈 + Ωpm ) − 𝜙(Δ𝜈) − (𝜙(Δ𝜈) − 𝜙(Δ𝜈 − Ωpm ))

(8.22)

This output phase difference is shown in Figure 8.15c(ii). On resonance both terms are equal, 𝜙(Δ𝜈) = 0, and the two beat signals keep their original phase, Δ𝜙out (0) = Δ𝜙in (0). The modulation remains pure phase modulation. The error signal is zero. For any other detuning we have to evaluate the phase response explicitly. It crosses zero on resonance, Δ𝜈 = 0, and rises steeply on either side. Close to resonance it has a linear slope, which can be measured in rad/MHZ. The steeper the slope, the higher the gain in the feedback loop. At a detuning less than one cavity linewidth, the phase difference peaks and then rolls off gently. At such large detunings the feedback system still has the correct sign. However, it has a small gain and the system is only slowly pushed backed to the resonance. The regime of constant feedback sign is called the capture range. As long as the error signal stays within the capture range, the detuning will be reduced and will eventually arrive at resonance; outside it the system moves away from resonance. The capture range of this technique is large, as large as the modulation frequency. A different interpretation of the result given in Eq. (8.22) is to say that the vector describing the reflected beam is rotated in the phasor diagram in regard to that of the input beam. However, the demodulated signal after the mixer corresponds to a measurement along the quadrature amplitude axis. Phase modulation of the input beam is orthogonal to this axis. The reflected light, with a rotated phasor diagram, now has some component of modulation in this direction. Phase modulation is partially converted to amplitude modulation; an error signal appears. The rotation angle 𝜃(Δ𝜈) is the half phase difference Δ𝜙out (Δ𝜈), and the maximum rotation is ±𝜋∕2. This rotation obviously applies to all types of modulations and also to fluctuations. Error signals can be generated using AM or mixtures of AM and PM. The rotation of the quadrature is also used as a self-homodyning detector for quadrature squeezed light (Section 9.3). 8.4.2

Tilt Locking

An alternative technique for the generation of the error signal is tilt locking. This is based on the interference of spatial modes reflected by the cavity. The input field is slightly misaligned with respect to the cavity, resulting in a TEM0,0 and other higher-order modes TEMi,j inside the cavity, which are all resonant at different detunings Δi,j as a result of the Gouy phase shift (Sections 5.3.3 and 2.1.2). To first order, the misaligned beam can be approximated by the sum of the TEM0,0 and TEM0,1 modes that have a phase shift Φ0,0 0,1 between them. Assuming the cavity is not confocal and that TEM0,0 is near resonance, the mode TEM0,1 will be far off resonance and will be fully reflected, like the sidebands in the PDH scheme. The mode TEM0,1 has a phase shift of 𝜋 between the two spatial components. The

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mode TEM0,0 will be partially reflected, and we now have to combine the amplitudes of the two modes and detect the resulting beam with a split detector that is aligned to resolve the two parts of the TEM0,1 field. The interference is critically dependent on Φ0,0 0,1 , which in turn depends on the detuning Δ0,0 . On resonance, Δ0,0 = 0, the two modes are exactly Φ0,0 0,1 = 𝜋∕2 out of phase, and the resulting amplitude on both sides of the split detector has equal magnitude. For a detuning to one side, Δ0,0 < 0, and the phase difference Φ0,0 0,1 is larger than 𝜋∕2, and this results in a reduction of the combined amplitude on one side of the split detector and an increase on the other side, as shown in Figure 8.15b. The amplitude 𝛼0,0 is rotated in the phasor diagram with respect to the amplitude 𝛼0,1 . The result is, for argument’s sake, a positive signal. For the opposite detuning, Δ0,0 > 0, the roles are reversed, and we will obtain a negative signal. Altogether we generate the error signal as displayed in Figure 8.15c(iii). We see that both techniques are equivalent and produce similar error signals. Tilt locking requires less equipment but is more sensitive to mechanical instabilities. PDH locking is the more commonly employed technique. A third alternative, after Hänsch and Couillaud, is based on the response of the cavity to two input polarizations. Provided the cavity has a higher finesse for one polarization than the other, the reflected light, when analysed in the appropriate polarization direction, will again produce an error signal with the required dispersion shape. Which particular technique should be chosen depends on the specific technical requirements. 8.4.3

The PID Controller

The other component is the feedback amplifier. It processes the error signal. The gain should be as large as possible, and the bandwidth should span as far as possible to control slow and fast frequency excursions. However, there are severe limitations set by the requirements for the phase of the feedback gain. As already discussed in Section 8.3, a reliable rule for stable operation is that the gain rolls off with high frequencies and the phase shift at the upper unity gain point should be less than 𝜋. A more precise definition can be obtained using a Nyquist diagram (Figure 8.10). As a consequence the delay times have to be kept short, and any resonances, mechanical or electronic, have to be avoided. The gain is usually controlled by a PID amplifier, which means a combination of proportional amplifier (middle frequencies), integrator (low frequencies), and differentiator (high frequencies). All these components contribute to the electronic round-trip phase and have to be carefully optimized. Unfortunately, most electromechanical actuators have resonances in the acoustic range kHz (Figure 8.16), severely limiting the bandwidth of the amplifier that has to reach unity gain at frequencies well below the resonance. One option is to split the control into several independent branches, for example, to use a piezo with large range (length changes of several optical wavelengths) at low frequencies to compensate for a drift of the frequencies and to use electro-optic components, such as a phase modulator inside the cavity, which has a smaller range but no resonance, for the control of the high frequency excursions. Extensive research

8.4 Frequency Stabilization and Locking of Cavities

kPZT, γPZT

Universe

(a)

Counter weight

PZT

kc,γc Universe

Counter weight

mm

mc xc(t)

x=0 Amplitude response

(b)

xm(t)

Phase response

20

0 –0.5 –1 –1.5

10 0 (i)

Amplitude (dB)

(iii)

ϕ control–π

10 0

Stability

Stability region

–10 region 20

0

–π ϕ control –0.5

10 0 Stability –10 region 20

–1 –1.5

Stability region

(iv)

–π 0

ol

ϕ contr

10 0

Stability

–10 region 10

0 –0.5 –1 –1.5

Stability region

20

30

40

50

10

Phase (rad/π)

(ii)

–10 20

20

30

40

–0.5 –1 –1.5 50

Frequency (kHz)

Figure 8.16 (a) Mirror mounted on a PZT modelled as a mass-and-spring system. The counterweight is the mass on which the PZT is pushing. This model includes the option of preloading. (b) Amplitude and phase response of a PZT (i) with small counterweight, (ii) with large counterweight, (iii) with large counterweight and low tension preloading, and (iv) with large counterweight and high tension preloading. Source: From W. Bowen [21].

has been carried out into these control systems, for example, for the applications in frequency standards and precision interferometry. 8.4.4

How to Mount a Mirror

Where the control is done using a lead zirconate titanate (PZT)-mounted mirror, the bandwidth of the feedback control system will be severely limited. The mirror, PZT, and mirror mount effectively form a system with a series of mechanical resonances Ωres,j , from the lowest j = 1 to higher frequencies. Since this system is inside the feedback loop, we can expect stable operation only at frequencies well below the lowest resonance Ωres,1 . Otherwise the phase shift induced by the resonance will interfere with feedback control. The resonance frequency depends on the mass of the mirror, the mass of the mirror mount or counterweight, and the way the PZT and the mirror are connected. The first improvement is to use the smallest possible mirror substrate and a PZT glued with a hard

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compound onto a massive backplate as the counterweight. This type of solution is found in many commercial systems, for example, many laser systems with an extended optical resonator. The next step is to modify the mount by preloading it, which means we contain the mirror and PZT in a rigid cage and exert a pressure force onto the mirror. In practical terms the cage is pushing via a rubber O-ring onto the mirror. The result is that the effective spring constants are increased and that the resonance frequencies are increased. This trick has been used by many laboratories that specialize in laser frequency control, and many different mechanical systems have been developed. One example was built and analysed by W. Bowen et al. [21] where the counterweight and the preloading tension can be systematically varied. They mounted the mirror as part of a cavity and recorded the cavity locking signal with an electronic network analyser that can record the amplitude and phase response. The results are shown in Figure 8.16b(ii). We can see that the resonances in the spectrum move towards higher frequencies. The shaded area indicates the useful locking range, and this is extended by more and more preloading. The preloading force is limited to values that cause no damage to the mirror. 8.4.5

The Extremes of Mirror Suspension: GW Detectors

This is only one of many examples of optimizing a mirror mount. Many special engineering solutions are being employed, ranging from very small and passive systems to large, complex active systems. The most extreme case is the mounting of the beamsplitter and end mirror in a gravitational-wave (GW) detector as outlined in Section 10.3. Here the mirrors are mounted on a most elaborate and optimized seismic suspension that includes both active and passive components located inside a vacuum system to suppress the vibrations in the environment by many orders of magnitude. The overall concept is to suspend the mirrors on a sequence of pendula, each with different frequencies. The combined transfer function will show a large attenuation at all frequencies. The ground vibrations are the largest frequencies due to many seismic and man-made effects from mHz to hundreds of Hz. The multi-pendula suspension in a GW detector has a very sharp cut-off and is typically used above10 Hz. Whilst very complex both mechanically and in regard to the feedback systems, such suspension systems can be extremely reliable and have been used in continuous long-term operation of GW detectors.

8.5 Injection Locking For a number of reasons, high-power lasers tend to have a much wider linewidth and not have a well-defined output mode. The mechanical design is simple, allowing more mechanical vibrations; the control of the pump source is not very sophisticated, and the electric current driving the pump source could be noisy or more than one pump source may be used. On the other hand, very high-quality lasers of low power but with excellent frequency and intensity noise

8.5 Injection Locking

quality and perfect spatial mode can be built. The trick is to combine the high output power of one laser with the high beam quality of the other. This can be achieved through the technique of injection locking. The basic idea is to seed the high-power laser, referred to as the slave laser, with some of the output from the well-controlled, high-quality laser, the master laser, and force the high-power laser to take on most of the mode qualities of the master laser. The slave laser will change its output frequency and phase to that of the master, and it will remain locked to this frequency. As a consequence the laser linewidth of the slave will be reduced to that of the master. If the slave laser was already operating in a single spatial mode, no changes would occur, and the spatial mode is determined by the geometry of the resonator of the slave laser. However, if it was operating in several modes, one mode will get an advantage, through the injection of light, and the laser can change over to this specific spatial mode. The output power of the slave laser is basically unaffected by the injection locking. Injection locking is the optical equivalent of the locking of two oscillators, an effect that was known for a long time for mechanical systems. It was reported that in 1865 Christiaan Huygens, whilst confined to his bed by illness, observed that the pendula of two clocks in his room moved in synchrony if the clocks were hung close to each other, but became free running when hung apart. He eventually traced the coupling to mechanical vibrations transmitted through the wall from one clock to the other. This observation was later confirmed using electrical oscillators and has developed into a widespread technique in electronic signal processing and microwave technology. Thus it was well known and established before the laser was invented. Optical injection locking will occur simply when the output beam of the master laser is matched into the resonator of the slave laser, both in the spatial waveform and by having the same polarization state. In addition, the optical frequency 𝜈m of the master laser has to be within the locking range 𝜈s ± Δl of the slave laser. Inside this range locking will happen completely. Outside this range no locking will occur at all, and the slave laser will operate as a free-running laser. The change from one state to another is abrupt. In practice this means that the slave laser will track the master laser as long as their independent free-running frequencies are within the locking range, |𝜈s − 𝜈m | = Δms < Δl . It is actually not easy to detect where the slave laser is within the locking range since the free-running frequency cannot be observed. The only indication would be a phase difference between the injection locked output beam and the master beam. To maintain locking, it might be necessary to add a control system that suppresses frequency drifts of the slave to keep the detuning Δms within the locking range. This can be achieved by a much simpler control system than would be necessary for a full frequency stabilization of the slave laser (Figure 8.17). An expression for the locking range can be derived by comparing the the power Ps of the free-running slave laser with the power of the master laser after amplification inside the slave resonator. This later value is given by the product of the amplitude gain |G(𝜈)|2 and the power Pm injected from the master into the slave laser Pm . The gain |G(𝜈)|2 can be estimated as (2𝜅)2 ∕(2𝜋(𝜈 − 𝜈s ))2 where 𝜅 is the amplitude decay rate of the slave resonator that, in many small coupling cases, can be estimated as 𝜅 = (1 − R)∕t, R the reflectivity of the output coupler, and t

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V1pm

Slave pump

Master pump V1master

V1slave

Master laser Slave laser

Figure 8.17 Schematic layout for injection locking.

the round-trip time. Setting the two values equal leads to the locking condition 2𝜅 √ |G(𝜈)|2 Pm ≥ Ps or Δl ≤ Pm ∕Ps (8.23) 2𝜋 As a practical example, if the slave laser has a power of 10 W, the laser resonator has a round-trip length L of 1 m, the cavity loss rate is 10% per round trip, and the √ master laser has a power of 0.1 W, the locking range will be Δl = 2(1 − R)c∕2𝜋L Pm ∕Ps = 0.5 MHz. The first observation of optical injection locking was carried out in 1966 by Stover and Steier who used two He–Ne lasers for this purpose [22]. They controlled the frequencies of the two lasers and were able to confirm the predicted locking range. The technique is now widespread, and it is used to control the properties of the light emitted by high-power lasers, to improve the mode quality, and to build powerful tunable lasers by locking a low cost powerful diode laser to well-controlled low-power diode lasers. On the other hand, frequency locking can also be an undesirable process. It prevents the measurement of very small frequency differences. For example, in laser gyroscopes, which are used in navigational applications, a beat signal between the two counter-propagating modes in a ring laser is measured. The modes experience different cavity lengths, if the ring laser rotates in an inertial frame. They will beat and the frequency of the beat is a measure of the rate of rotation. These frequencies fbeat are very small, and they are given by 2𝜋fbeat = Ωrot A∕p where A is the area and p the perimeter of the gyroscope. Ωrot is the angular frequency of the rotation. For the Earth’s rotation the value, at the pole, is 2𝜋 in 24 hours or 75 × 10−6 rad/s. For a 1 m2 ring gyroscope located at the pole, the beat frequency is about 20 Hz. Locking of the two modes is very likely, but it obviously has to be avoided; otherwise the beat signal is zero, and it would appear that the Earth stood still. Extraordinarily small scatter losses are sufficient to provide enough light to cause locking. As a consequence the demand for laser gyroscopes has been the motivation for most of the recent improvements in low-loss optics and the advances in the technology of mirror manufacture. Quantum optics experiments have gained from these improvements. It is interesting to ask what the noise properties of the injection-locked laser system could be. Does the master laser dominate the system completely? Is it possible to build in this way a high-power quantum noise limited laser? This question was addressed by several researchers in the regime of classical fluctuations [23]. More recently, complete laser models have been developed, and using the

References

65 (iii) Power (dB above QNL)

55 45 35

(ii)

25 15 (i)

5 QNL –5 0.01

0.1

1

10

Frequency (MHz)

Figure 8.18 The spectral intensity noise of an injection-locked laser. (i) A polynomial fit to the experimental data. (ii) Experimental data. (iii) Excess noise due to amplification of the master laser’s phase noise by the injection-locked slave laser. Source: From C.C. Harb et al. [25].

linearized quantum theory, it is possible to derive complete noise spectra for the injection locked systems. In these models the laser is literally described as an oscillator with two distinct internal modes, at the frequency of the master and the slave laser, and the combined system is solved within one laser model, similar to that described in Chapter 6 [24]. The outcome of these models and of experiments [25] is that the characteristic of the laser intensity noise varies with the detection frequency. There are clearly distinguishable regions of frequencies that show different properties. A practical example is the locking of two Nd:YAG lasers. One region is centred roughly around the relaxation oscillation frequency of the slave laser (Figure 8.18). Here the slave laser operates effectively like an amplifier for the noise of the master laser. Note that if the master laser is quantum noise limited, this means output is considerably noisier than a QNL laser with the same power as the slave laser. At very high frequencies the output can reach the quantum limit, and it maintains the properties of the slave laser. At very low frequencies the dominant source of noise is the pump source of the slave laser. Different strategies have to be followed to optimize the system at the different frequencies.

References 1 Levenson, M.D., Shelby, R.M., and Perlmutter, S.H. (1985). Squeezing of clas-

sical noise by nondegenerate four-wave mixing in an optical fiber. Opt. Lett. 10: 514. 2 Yuen, H.P. and Shapiro, J.H. (1980). Optical communication with two-photon coherent states. I. Quantum-state propagation and quantum-noise reduction. IEEE Trans. Inf. Theory IT-26: 78.

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3 Walls, D.F. and Zoller, P. (1981). Reduced quantum fluctuations in resonance

fluorescence. Phys. Rev. Lett. 47: 709. 4 Robson, B.A. (1974). The Theory of Polarisation Phenomena. Oxford: Claren-

don. 5 Schnabel, R., Bowen, W.P., Treps, N. et al. (2003). Stokes operator-squeezed

continuous-variable polarization states. Phys. Rev. A 67: 012316. 6 Kane, T.J. and Byer, R.L. (1985). Monolithic, unidirectional single-mode

Nd:YAG ring laser. Opt. Lett. 10: 65. 7 Köchner, W. (1996). Solid-State Laser Engineering, 4e. Springer-Verlag. 8 Dorf, R.C. (1992). Modern Control Systems, 6e. Addison Wesly. 9 Wiseman, H.M., Taubman, M.S., and Bachor, H.-A. (1995).

10 11

12 13 14 15

16

17

18

19

20

21 22

Feedback-enhanced squeezing in second-harmonic generation. Phys. Rev. A 51: 3227. Wiseman, H.M. (1999). Squashed states of light: theory and applications to quantum spectroscopy. J. Opt. B: Quantum Semiclassical Opt. 1: 459. Masalov, A.V., Putilin, A.A., and Vasilyev, M.V. (1994). Sub-Poissonian light and photocurrent shot-noise suppression in a closed optoelectronic loop. J. Mod. Opt. 41: 1941. Troshin, A.S. (1991). Photon statistics in a negative-feedback optical system. Opt. Spektrosk. 70: 389. Taubman, M.S., Wiseman, H.M., McClelland, D.E., and Bachor, H.-A. (1995). Effects of intensity feedback on quantum noise. J. Opt. Soc. Am. B 12: 1792. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilization of an Argon laser using electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Harb, C.C., Gray, M.B., Bachor, H.-A. et al. (1994). Suppression of the intensity noise in a diode pumped neodymium YAG nonplanar ring laser. IEEE J. Quantum Electron. 30: 2907. Kane, T.J. (1990). Intensity noise in a diode pumped single frequency Nd:YAG laser and its control by electronic feedback. IEEE Photon. Technol. Lett. 2: 244. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Day, T., Gustafson, E.K., and Byer, R.L. (1992). Sub-Hertz relative frequency stabilisation of two diode laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer. Opt. Lett. 17: 1204. Shaddock, D.A., Gray, M.B., and McClelland, D.E. (1999). Frequency locking a laser to an optical cavity by use of spatial mode interference. Opt. Lett. 24: 1499. Bowen, W. (2003). Experiments towards a quantum information network with squeezed light and entanglement. PhD thesis. Australian National University. Stover, H.L. and Staier, W.H. (1966). Locking of laser oscillators by light injection. Appl. Phys. Lett. 8: 91.

References

23 Farinas, A.D., Gustafson, E.K., and Byer, R.L. (1995). Frequency and intensity

noise in an injection locked, solid state laser. J. Opt. Soc. Am. B 12: 328. 24 Ralph, T.C., Harb, C.C., and Bachor, H.-A. (1996). Intensity noise of

injection-locked lasers: quantum theory using linearized input-output method. Phys. Rev. A 54: 4359. 25 Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1996). Intensity-noise properties of injection locked lasers. Phys. Rev. A 54: 4370.

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9 Squeezed Light 9.1 The Concept of Squeezing So far we have worked with coherent states of light and the limits imposed by quantum noise that affect all quadratures equally. However, it is possible to generate light that has less noise in one selected quadrature than the quantum noise limit (QNL) dictates. Light with this property is called squeezed light. This form of light is clearly the first milestone in quantum optics with many photons. Squeezing technology made many applications of quantum optics possible and lets us create many improvements unique to quantum optics. In optical measurements and metrology, we can achieve a better signal-to-noise ratio than is possible with coherent light, and this is now state of the art in interferometry. Squeezed light is the tool for entanglement of laser beams and for techniques such as teleportation that play a role in communication and signal processing. Even quantum logic might be feasible. Combined with optimized detection techniques, squeezed light can be the source of specific effects, such as Schrödinger’s cat states of light. In this chapter, we start by describing the effect of non-linear optical processes on the photon statistics and the fluctuations of the light. This is first done in a classical approach – since it is possible to see some of the key results immediately. We use two examples that are intuitive and instructive. Next we introduce the concept of squeezed states of light rigorously. This will provide the mathematical tools to analyse the experiments. Finally, we discuss the first experiments that have been performed since 1985 to generate squeezed light and describe the state of the art in 2018. 9.1.1

Tools for Squeezing: Two Simple Examples

The properties of quantum noise have already been introduced in detail in Chapter 4, and now we wish to reduce the fluctuations of the light. We cannot suppress the fluctuations altogether that would violate the uncertainty principle. But we can rearrange them between the quadratures. We can build experiments, and protocols, where we measure only one quadrature at a time and only the noise in this one quadrature contributes to the final result. Consequently, it will be sufficient to reduce the fluctuations in this one quadrature alone. The challenge is: What are the tools that would reduce the noise in one quadrature? How can we generate states of light that have asymmetric quadrature noise A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Figure 9.1 Tools for squeezing. X2

X1

distributions that are squeezed in one quadrature? We will see in this section that the trick is to start with a normal coherent state and to introduce correlations ̂ and X𝟐 ̂ through non-linear between the fluctuations of the two quadratures X𝟏 interactions (Figure 9.1). 9.1.1.1

The Kerr Effect

To illustrate how correlations can reduce the quantum noise, the case of the non-linear Kerr effect is instructive [1]. Consider a beam of light travelling through a medium, which has an intensity-dependent refractive index. The beam can be described as an optical field Ein with both amplitude and phase fluctuations: 𝛼(t) = 𝛼in + 𝛿X1in (t) + 𝛿X2in (t)

(9.1)

This could be a QNL laser beam, in which case 𝛿X1in (t) and 𝛿X2in (t) represent the quadratures of the quantum noise. The non-linear refractive index n(𝛼) can be described by n(𝛼) = n0 (1 + n2 𝛼 2 )

(9.2)

As a consequence fluctuations in the intensity will modulate the refractive index. The refractive index, in turn, will modulate the phase of the transmitted light. For a medium of length L and light at the wavelength 𝜆, the output phase is given by 2πL n 2πL = 𝜙in (t) + (n0 + n2 (𝛼in + 𝛿X1in (t))2 ) 𝜆 𝜆 For small fluctuations this can be approximated by 𝜙out (t) = 𝜙in (t) +

𝜙out (t) = 𝜙Kerr +

4πL n (𝛼 𝛿X1in (t) 𝜆 2 in

(9.3)

(9.4)

9.1 The Concept of Squeezing

ϕin – ϕout αout

(a)

αin

(b)

αin

Figure 9.2 Effect of a Kerr medium on the electric field vector: (a) input field and (b) output field. Note that the field illustrated here has a large amplitude, compared to the fluctuations. Thus the origin of the diagram is far away and not shown. Consequently any changes in the phase are orthogonal to changes in the amplitude. The main result is that the change in the phase fluctuations depends on the intensity proportional to 𝛼 2 .

where 𝜙Kerr is the time averaged phase added by the medium. This is illustrated in Figure 9.2. There is no change to the intensity of the light: 2 2 = 𝛼in = 𝛼2 𝛼out

and 𝛿X1out (t) = 𝛿X1in (t)

(9.5)

At the output of the medium, the phase fluctuations are those of the input light plus a component that is linked to the input intensity fluctuations. Using the relationship between the quadratures 𝛼𝛿𝜙 = 𝛿X2, which is valid for small angles, this can be written in terms of the quadrature amplitudes: 4πLn0 n2 2 𝛼 𝛿X1in (t) 𝜆 = 𝛿X2in (t) + 2rKerr 𝛿X1in (t) 2πLn0 n2 𝛼 2 (9.6) with rKerr = 𝜆 where rKerr is a parameter that includes the non-linearity and the average amplitude 𝛼 of the input field. Higher-order terms in 𝛿X1 have been neglected since they are exceedingly small, and only the terms linear in 𝛿X1 have been kept. We see from Eq. (9.6) that the non-linear medium couples the fluctuations in amplitude and phase quadratures. The amplitude 𝛼 drives the non-linear refractive index. This in turn controls the quadrature phase X2. At the output of the medium, the fluctuations are no longer independent. The two quadratures are now correlated. What is the effect of this process on the statistics of the fluctuations? On the one hand, we have made the light noisier by adding fluctuations. On the other hand, we now have correlated fluctuations. We can estimate the consequences by plotting the effect not just for one point but for a whole distribution of classical amplitudes. This is done in Figure 9.3. The input and the output fields are represented by density plots, already discussed in Chapter 4 and shown in Figure 2.13. The axes of the plot for Kerr effect (Figure 9.3A(a)) are the input quadratures X1in and X2in . The mean value is given by the point (𝛼in , 0). The fluctuating amplitude is represented by a group of dots, wherein each dot gives one instantaneous value of (𝛿X1in (t), 𝛿X2in (t)). The density of the dots indicates the probability of finding a specific value of the quadrature amplitude fluctuation, as introduced in 𝛿X2out (t) = 𝛿X2in (t) +

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9 Squeezed Light

Output E2

Input δX2

δY1

δY2 (a)

(a) δX1in δX1

(b)

δX1 δX2

(b)

Histogram

δY1 Histogram

306

δY2

(A)

(B)

Figure 9.3 Simulation of the effect of a Kerr medium on the fluctuations of an electromagnetic field. Left-hand side (A) is the input field and right-hand side (B) the output field. The top part (a) shows the probability distribution of the field, and the bottom part (b) shows the histograms of the samples above.

Section 4.1. This simulation uses a random distribution, with two-dimensional (2D) Gaussian weighting, as described in Appendix A. The probability of measuring a certain value X1(t) is given by a histogram of the projection of all dots onto one of the axes. Two such histograms, for both the X1 and X2 coordinates, are shown in Figure 9.3A(b). Both of them are Gaussian curves with the same width. Figure 9.3B shows the distribution at the output of a non-linear Kerr medium for a parameter rKerr = 0.375. Note that the total phase shift 𝜙out has been ignored. The axes are again the quadrature amplitudes X1out and X2out . The distribution is no longer circularly symmetric. The area covered by the distribution is actually larger since we have added fluctuations ((Figure 9.3B(a)). But note that the distribution is narrower in one special direction. The histogram of this distribution, taken for a projection axis Y 1 rotated by the angle 𝜃s to the X1 axis (Figure 9.3B(b)) is narrower than the histogram for the input state (Figure 9.3A(b)). In this particular case the full width at half maximum (FWHM), or standard deviation, is reduced from 1 to 0.79. The histogram for the projection Y 2 (Figure 9.3B(b)), orthogonal to Y 1, is wider, by a factor 1.23, than at the input. The width of the distribution in the squeezing direction is given by √ 2 2 VY 1 = 1 − 2 rKerr 1 − rKerr + 2 rKerr (9.7)

9.1 The Concept of Squeezing

well below the QNL. This result is derived on page 343. This concept can also be used for a QNL laser beam or coherent state. The dot diagram represents the ̂ and X𝟐. ̂ The variance of the rotated measured probabilities of the observables X𝟏 ̂ ̂ quadrature Y𝟏, rotated by 𝜃s with respect to X𝟏, is smaller than that for a coherent state. Just by introducing this correlation between the quadratures, we have created light with noise reduced below the QNL in one specific quadrature. Note that in this example the uncertainty area has been increased. The output is no longer a minimum uncertainty state, but this is not of great importance for squeezing. ̂ The details of the We are only interested in measurements of the quadrature Y𝟏. size of the noise suppression and of the squeezing angle in a real experiment are described in Section 9.4.4. The Kerr effect acts on all fluctuations, independent of their frequency. The model can be applied to each individual Fourier component of the fluctuations. For each detection frequency the process correlates the quadratures of the modes. The correlations are introduced separately for each Fourier component. In this way, the Kerr effect correlates the amplitude and the phase quadrature of the light, and this applies to modulations as well as the quantum noise. 9.1.1.2

Four-Wave Mixing

A second example of a squeezing tool is four-wave mixing (4WM). This is a well-known non-linear process that occurs in many resonant media, such as atoms. Imagine three laser beams that are interacting with the medium and are tuned close to resonance with one absorption line, as illustrated in Figure 9.4a. This process is best described in the frequency domain. Two of the laser beams, the pump beams, have the optical frequency 𝜈L . They enter from different angles. The third beam is at the frequency 𝜈L + Ω. The medium responds with the generation of a new beam at 𝜈L − Ω. This process is energetically allowed; the energy comes from the abundance of photons at 𝜈L . The strength or probability, of this process depends on three factors: (i) a non-linear coefficient 𝜒 (3) that varies from material to material, (ii) the density of the sample, and (iii) a condition that describes how well the momentum of the photons is preserved. The latter is also known as phase matching and determines the geometry of the input and output beams [1]. Conventionally, this process is used as a technique to produce a new beam that emerges at a different frequency or direction. This new beam can be used to investigate the properties of the non-linear medium or as a source of light at the new wavelength. This process also works if we have only one single input beam that has a single modulation sideband at 𝜈L + Ω. In this case all frequency components are co-linear, and 4WM produces a new sideband at 𝜈L − Ω, which is phase-locked to the original sideband. The efficiency of the process is given by the relative size of the new sideband. It can be quantified by the ratio C between the amplitude of the new and the original sideband. This applies to any sideband. Now consider the effect of 4WM on an amplitude-modulated laser beam. That means we have pairs of sidebands, each pair in phase with each other, i.e. their respective beat signals are in phase and constructively interfere. Each sideband is partially transferred to the other sideband respectively by 4WM (Section 2.1.7). This will increase the sidebands and thus the modulation by a factor (1 + C).

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4WM vL

Non-linear medium

vL – Ω

vL + Ω

C

vL

vL – Ω (b)

(a)

δX1in

vL

vL + Ω

δX1in vL

vL More AM Less FM

δX1out (c)

δX1out vL

(d)

vL

Figure 9.4 The effect of a 4WM medium. (a) The geometrical layout: a new beam is generated. (b) The effect of 4WM in the frequency domain is a transfer from one sideband to another. (c) As a consequence amplitude modulation (AM) is increased. (d) Phase and frequency modulation (FM) is reduced.

The medium provides gain for amplitude modulation, as shown in Figure 9.4c. In contrast, phase modulation consists of two sidebands 180∘ out of phase. Again, 4WM creates additional components at the opposite sideband frequency. The transfer from one sideband to the other will decrease the phase modulation by a factor (1 − C), which means 4WM reduces phase modulation (Figure 9.4d). In conclusion, the non-linear process of 4WM provides a selective gain to one quadrature and reduces the other. 4WM is an example of a phase-sensitive amplifier (Section 6.2), which has different gain for each quadrature. This process applies both to modulation and to noise sidebands. Any amplitude modulation noise is amplified; any phase modulation noise is reduced. Consequently, a beam that initially has equal amounts of both types of noise will turn into light with enhanced amplitude modulation and reduced phase modulation noise. This classical argument shows us that an input beam that is in a coherent state will turn into a state with suppressed phase noise. This result is confirmed by a full quantum treatment [2]. It is useful to consider the 4WM process in terms of a phasor diagram, much like we have done for the Kerr effect. Again, we will consider one Fourier component of the fluctuations at a time. Any periodic excursion from the long-term average value corresponds, in the phasor diagram, to an oscillation at the modulation frequency Ω. It can be a mixture of amplitude and phase modulation. This can be drawn, as shown in Figure 9.4c, as an additional vector, the sum of the vectors 𝛿X1(Ω) and 𝛿X2(Ω). The 4WM process will increase 𝛿X1(Ω) by a factor (1 + C) and reduce 𝛿X2(Ω) by a factor (1 − C) (Figure 9.4d). This is called phase-sensitive gain. We can test the effect of 4WM on a truly noisy state, which

9.1 The Concept of Squeezing

Input

Output

δX2

δY2

δY1 (a)

(a)

δX1 δX2

(b)

Histogram

(b)

Histogram

δX1

δY2

δY1 (A)

(B)

Figure 9.5 Simulation of the effect of a 4WM medium on the fluctuations of an electromagnetic field. Part (A) shows the input state. The distribution is circularly symmetric. Part (B) shows the distribution after the 4WM medium. It is stretched. The top part (a) shows the probability distribution of the field, and the bottom part (b) shows the histograms of the samples above.

is represented by a 2D random Gaussian distribution in time. The transformation we apply point by point is 𝛿X1out (t) = (1 + C) 𝛿X1in (t) and 𝛿X2out (t) = (1 − C) 𝛿X2in (t)

(9.8)

Figure 9.5 shows an example of both input and output distributions, Figure 9.5A andB, respectively, based on the random sample from Figure 9.3 and a value of C = 0.3. Clearly the output state has increased amplitude fluctuations and reduced phase fluctuations. The size of the uncertainty area is preserved. This phasor diagram represents a QNL input; clearly the output has phase fluctuations below the QNL. It is worth noting that the arguments used here are exactly the same as those applied in phase conjugation. The only difference is that there the spatial phase contributions are reduced by the generation of a reflected wave with opposite phase whilst in 4WM we consider temporal fluctuations. These two examples, Kerr and 4WM, are only illustrations of a concept. They show that correlations introduced by non-linear processes can be used to generate a new type of light with smaller fluctuations, in one specific quadrature, as compared with the input light. The above classical examples can be used as a guide for a complete quantum model of this new type of light. In Chapter 4, it was shown that the quadrature probability distribution for a quantum state can be

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described by the Wigner function. The coherent light has a Gaussian distribution of the fluctuations around the time averaged value, and this will not change for the strength of non-linearities that are available. In particular, for a coherent state, the Wigner function is a 2D Gaussian function with circular symmetry. The new state has a Wigner function that is compressed in one direction and expanded in the other. Certainly squeezed state is a fitting description. The first step for a complete description of this light is a formal and rigorous definition. 9.1.2

Properties of Squeezed States

With squeezed states we mean states that have the following property: the distribution function of the quadratures is not symmetric, that is, the uncertainty area is no longer a circle, and the width of the distribution function in one particular direction Y 1 is narrower than the QNL. The direction Y 1 can be at any angle with respect to the quadratures X1 and X2. Measurements that are designed to be sensitive only to Y 1 and not the orthogonal quadrature Y 2 contain noise less than the QNL. For this reason we call it non-classical light, meaning that it cannot be described by the normal theory of coherent light, given in Chapter 4, which prescribes a minimum uncertainty area that is independent of the quadratures. Quantum mechanics and non-linear optics have to be combined to avoid the symmetry and to go below the standard uncertainty limit (Figure 9.6). X2

X2 δX2

δX1

δX2

X1

Coherent state

(a)

δX1

(b)

X2

Minimum uncertainty state X2

δX2

δX1

(c)

X1

δX2

X1

Squeezed and excess noise

δX1

(d)

X1

Noisy and not squeezed

Figure 9.6 Phasor diagram comparing squeezed and other states. (a) Coherent state. (b) Minimum uncertainty squeezed state that is narrower than the coherent state in one direction. (c) Squeezed state with excess noise. (d) An asymmetric, noisy, but not squeezed state – it is described by an ellipse, but no projection is narrower than the coherent state.

9.1 The Concept of Squeezing

δX2

δX2

(a)

δX1

(b)

δX1

Figure 9.7 Wigner function of (a) coherent state and (b) squeezed state.

The uncertainty area for a squeezed state is generally drawn as an ellipse. Why does it have this particular shape? The Wigner function for any coherent state is a Gaussian distribution in every direction. The contour line for this state is a circle. We found that this corresponds to fluctuations in the measurements of the variable such that V (Ω, 0) = V (Ω, 90) = 1 for all detection frequencies. It was already shown by W. Pauli [3] that any minimum uncertainty state, even squeezed, will have 2D Gaussian distribution functions. The only requirement is that the widths of these functions satisfy the uncertainty principle V (Ω, 𝜃) V (Ω, 𝜃 + 90) = 1 for any direction 𝜃. One of the two variances can be less than 1. The contour line for such a 2D Gaussian function is always an ellipse. An ellipse can be described by exactly two parameters: the direction of the narrowest axis Y 1 and the ellipticity. Note that the area inside the contour can never be smaller than the area inside the uncertainty circle of a coherent state (Figure 9.7). There are states with excess noise, that is, states with V (Ω, 𝜃) V (Ω, 𝜃 + 90) > 1 for some or even all frequencies. Such states need not be described by 2D Gaussian with elliptical contours. However, we find that most realistic optical systems can be described by theoretical models that are linearized in the fluctuations (Section 4.5). These systems can have non-linear properties that are essential for the generation of squeezed light, but its description requires only first-order terms in regard to the fluctuations. All such models result in Gaussian distribution functions – and the contours are ellipses. It would be of great interest to generate light with more exotic contours, with parts of the Wigner functions that have negative values and strong non-classical properties – but this has so far only been achieved in the single photon counting domain. Further details are given in Section 9.7. For the remainder we shall restrict ourselves to squeezed states that are described by elliptical contours. Squeezed states require a 2D description. The quadrature amplitudes Y 1 and Y 2 provide the most useful coordinate system – the theoretical model is

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X2

X2

δY2 θs = 0

X1

θs =

δX1

(a)

X1

π 2

δX1

(b)

X2

X2

δY2

δY1

δY2 δY1

θs

X1 (c)

X1 (d)

Figure 9.8 Different types of squeezed states. (a) Amplitude. (b) Phase squeezed state. (c) Quadrature squeezed state. (d) Vacuum quadrature squeezed state.

best derived using these parameters. Direct intensity measurements will only determine the intensity fluctuations. Whilst this determines the width of the variance V 1(Ω) of the amplitude, measured along the X1 axis, it does not measure the quadrature fluctuations in any other direction. We require experiments that can measure rotated quadratures, in particular V (Ω, 𝜃s ) and V (Ω, 𝜃s + 90). This is usually achieved by using a homodyne detector (Section 8.1.4). There are different types of squeezed states, as shown in Figure 9.8. Each squeezed state ellipse has a narrow axis (Y 1) and, at right angles, the axis with the widest distribution (Y 2). If the narrow axis is aligned with the amplitude axis, i.e. (Y 1) parallel to (X1), we call the light amplitude squeezed. If it is aligned with the quadrature phase axis, i.e. (Y 1) parallel to (X2), it is called phase squeezed light. In general, (Y 1) could be rotated at some arbitrary angle 𝜃s relative to the axis X1. This is called quadrature squeezed light. 9.1.2.1

What Are the Uses of These Various Types of Squeezed Light?

Amplitude squeezed light is useful in absorption or modulation measurements. It can be detected simply by direct detection. Due to the reduced amplitude noise, the signal-to-noise ratio for amplitude modulation is improved. The fluctuations in the other quadrature, the phase quadrature, are above the QNL. There could even be excess noise in the phase quadrature. None of this would affect the direct intensity measurement. Amplitude squeezed light is obviously related to light with sub-Poissonian counting statistics. It has a Fano factor of less than unity. The detailed connection between squeezing and photon statistics has to be determined in each individual case. It depends on the actual photon probability distribution that could be very different to a

9.1 The Concept of Squeezing

Poissonian distribution. However, for the generally observed moderate degree of squeezing, the link is simple: F𝛼 = V 1𝛼 . This light is frequently called bright squeezed light. The generation of bright squeezed light using second harmonic generators (SHGs) [4–7] or diode lasers is now a reliable experimental technique and is described in Section 9.4.3. Quadrature phase squeezed light can be used to improve the signal-to-noise ratio of optical measurements. Care has to be taken that only the phase quadrature is measured. This can be achieved in interferometric arrangements where amplitude fluctuations are cancelled out and phase modulations are turned into modulation of the intensity at the output. In such an arrangement only the noise of the light in the phase quadrature will contribute to the final signal-to-noise ratio. Finally, there is a peculiar state, centred at the origin of the phasor diagram, that has a squeezing ellipse. This state has an axis of reduced fluctuations. But for each fluctuation there is an equally likely fluctuation in the opposite direction. This state is given the confusing name squeezed vacuum state. We will see that it is closely related to the coherent vacuum state |0⟩. An attempt to detect squeezed vacuum state directly would not result in any measurable intensity. To be precise, this state contains a rather small number of photon pairs (Eq. (9.13)), and the number of such pairs increases the more the state is squeezed. However, this effect is far too small to be detected directly with photodiode technology. Only with photon counting could such small intensities be seen. In any other regard this state has normal properties: it is a beam with a well-defined mode, direction, size, frequency, and polarization. However, once this beam is aligned with a homodyne detector and beats with a strong local oscillator beam, we can see its properties: the quantum fluctuations vary with the detection phase. The fluctuations corresponding to the Y 1 quadrature are smaller than those of the Y 2 quadrature. This squeezed vacuum state can have significant practical applications, as shown in Section 5.2. All interferometers, and other instruments based on beamsplitters, contain input ports that are not used. That means there are locations in the experiment that could be used as the input for a real laser beam that would propagate right through the instrument, probably with some attenuation, and finally reach the detector. Such an input port has the properties of a real mode of light. In practice this port is not illuminated, and it can be ignored in classical calculations. However, in a quantum model of such an instrument, we cannot ignore this port, as shown in Chapter 5, and we have to represent it with a vacuum mode, the coherent state |0⟩. These vacuum input modes are the origin of the quantum noise in the output, as was shown in Section 5.1. Consequently the performance of the instrument can be improved by illuminating the unused input port by a squeezed vacuum mode. The squeezing quadrature Y 1 has to be aligned such that it is now the driver of the quantum fluctuations that we measure in our apparatus. The result will be reduced quantum noise and a better signal-to-noise ratio. This was proposed by C.M. Caves [8] for the case of interferometers and has been demonstrated in an impressive landmark experiment by M. Xiao et al. [9] who achieved an improvement of the signal-to-noise ratio of 3 dB. The details and limitations of this technique will be discussed in the Chapter 10.

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9.2 Quantum Model of Squeezed States In Chapter 4 we described pure states of light with Gaussian quadrature statistics as coherent states. We now want to expand this to include a much larger class of states, the squeezed states. They have to be based on a wider definition and can be derived from the lowest energy state, the vacuum state. 9.2.1

The Formal Definition of a Squeezed State

Squeezed states are represented by |𝛼, 𝜉⟩

and 𝜉 = rs exp(i2𝜃s )

(9.9)

where |𝛼|2 is the intensity of the state, 𝜃 the orientation of the squeezing axis, and rs the degree of squeezing. The squeezed states can be generated from the vacuum state: ̂ ̂ S(𝜉)|0⟩ |𝛼, 𝜉⟩ = D(𝛼) ̂ where the squeezing operator S(𝜉) is defined as ) ( 1 1 ̂ S(𝜉) = exp 𝜉 ∗ â 2 − 𝜉 â †2 2 2 The properties of the squeeze operator are

(9.10)

(9.11)

̂ Ŝ † (𝜉) = Ŝ −1 (𝜉) = S(−𝜉) ̂ = â cosh(rs ) − â † exp(−2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂aS(𝜉) ̂ = â † cosh(rs ) − â exp(2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂a† S(𝜉)

(9.12)

̂ in Eq. (9.10) is possible. This results ̂ Note that the reverse order of D(𝛼) and S(𝜉) in the so-called two-photon correlated state, which was introduced by H.P. Yuen even before the name squeezed state was coined [10]. However, this distinction has little relevance for experiments. It makes no detectable difference for any realistic state. The formal description allows us to derive the noise properties of a squeezed state exactly. The squeezed states have the following expectation values for the annihilation and creation operators: ⟨𝛼, 𝜉|̂a|𝛼, 𝜉⟩ = 𝛼 ⟨𝛼, 𝜉|̂a2 |𝛼, 𝜉⟩ = 𝛼 2 − cosh(rs ) sinh(rs ) exp(2i𝜃s ) ⟨𝛼, 𝜉|̂a† â |𝛼, 𝜉⟩ = |𝛼|2 + sinh2 (rs )

(9.13)

The last equation means that the intensity of a squeezed state is slightly larger than that of the coherent state with the same value |𝛼|. However, this is a minute effect for any laser beam with detectable intensity. Using these properties we can ̂ derive the variances of the generalized quadrature X(𝜃) = exp(−i𝜃)̂a + exp(i𝜃)̂a† , the quadrature that would be measured at the rotation angle 𝜃. This variance is given by ̂ Var(X(𝜃)) = cosh(2 rs ) − sinh(2 rs ) cos(2 (𝜃 − 𝜃s ))

(9.14)

9.2 Quantum Model of Squeezed States

10

V(θ) (dB)

5

0

–5

–10

(a)

π/2

0

π

3π/2

2π

1

0.75 >7 [5, 7] [3, 5] [1, 3] [−1, 1] [−3, −1] [−5, −3] [−7, −3] <−7

0.5

0.25

0 0 (b)

π 2

π

3π 2

2π

Figure 9.9 (a) A plot on a logarithmic scale in dB above QNL of the variance of a squeezed state with 𝜃s = π∕3 as a function of the squeezing angle and for different squeezing parameters rs = 0.25 → 1.00. (b) Contour plot of the same data.

This quite complex expression shows that the variance is a periodic function of the rotation angle, as one would expect from the concept of an ellipse being rotated. It has a minimum when 𝜃 = −𝜃s and a maximum in the orthogonal direction 𝜃 = −𝜃s + π∕2. This variance is plotted in Figure 9.9 for one fixed squeezing angle 𝜃s = −π∕3 and for the range of squeezing parameters rs = 0.25, 0.5, 0.75, and 1.00. Figure 9.9a shows a linear plot of the variance, whilst (b) shows the same data on a logarithmic scale, which means the vertical scale is the normalized variance in dB above and below the QNL. Here the quantum limit corresponds to a circle with radius 1 (bottom of the plot), and the orientation of the suppressed quadrature Y 1, the squeezing angle 𝜃s , is immediately visible. It can be seen that as the squeeze parameter is increased, the minimum variance decreases and the maximum increases. This agrees with the concept of stretching the ellipse. What is not immediately obvious is the fact that the range of rotation

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angles for which the variance is below the QNL gets narrower with increasing squeezing. The points where the variance is equal to the QNL get closer to 𝜃s . This is particularly obvious from the logarithmic plot (Figure 9.9b). This means that the rotation angle has to be very well specified in order to obtain the best noise suppression. The noise properties are most easily described by introducing the ̂ and Y𝟐, ̂ which are rotated in regard to the standard quadrature quadratures Y𝟏 ̂ ̂ amplitudes X𝟏 and X𝟐 by the angle 𝜃s : ̂ + iY𝟐 ̂ = (X𝟏 ̂ + iX𝟐) ̂ exp(−i𝜃s ) Y𝟏

(9.15)

The variances for these two quadratures, that is, the minimum and maximum of Eq. (9.14), have particularly simple expressions: ̂ 2 ⟩ = V𝛼,𝜉 (𝜃s ) = exp(−2rs ) ⟨𝚫Y𝟏 ) ( ̂ 2 ⟩ = V𝛼,𝜉 𝜃s + π = exp(2rs ) ⟨𝚫Y𝟐 (9.16) 2 Thus the minor and major axes of the contour ellipse are of the size exp(rs ) and exp(−rs ). Note that the contour has a value that corresponds to the square root of the variance. The details of the link between the variance and the Wigner function are discussed in Section 4.1. Coherent states are eigenstates of the annihilation operator. This is not true of squeezed states. The squeezed states are eigenstates of b̂ = cosh(rs )̂a + sinh(rs )̂a† and are thus not eigenstates of the annihilation operator. Notice that in the limit ̂ Thus the squeezed states tend towards rs → ∞, the operator b̂ approximates X𝟏. quadrature eigenstates. Notice also in the limit rs → 0 that the operator b̂ goes to â and we reduce to coherent states. At the photon detector we will have one single beam, and in our calculations we can arbitrarily set the amplitude 𝛼 as real. The average number of photons in a squeezed state is evaluated in Eq. (9.13), and the variance in the photon number is given by V n𝛼,𝜉 = |𝛼|2 [cosh(2 rs ) − exp(−2i𝜃s ) sinh(2 rs )]2 + 1∕2 sinh2 (2 rs ) This complex expression is the variance measured by direct intensity detection. For quadrature squeezed states, there is no longer a simple link between V n𝛼,𝜉 and the variance V𝛼,𝜉 (𝜃s ). The variance measured for the intensity behaves differently to the quadrature amplitude variance. There are two limiting cases that have simple properties: for bright squeezed light, (𝛼 ≫ rs ) and (𝛼 ≫ 1), the average photon number is close to the coherent state |𝛼⟩. Thus intensity measurements of |𝛼, 𝜉⟩ will provide the same average results as those of a coherent state and the intensity fluctuations will be proportional to V 1. The noise in the quadratures will range from exp(−2rs ) to exp(2rs ) depending on the detection angle. In general the noise will depend on the detection frequency, and we define the noise spectra VY 1𝛼,𝜉 (Ω) =  (V𝛼,𝜉 (𝜃s )) at frequency Ω VY 2𝛼,𝜉 (Ω) =  (V𝛼,𝜉 (𝜃s + 90)) at frequency Ω

9.2 Quantum Model of Squeezed States

Note that in many situations we are only interested in the best possible squeezing. This is expressed by the squeezing spectrum S(Ω). This is the value of V𝛼,𝜉 (Ω, 𝜃) for the optimized squeezing angle 𝜃s , which can vary with the detection frequency. Thus, any rotation of the ellipse is simply ignored. In most theoretical papers the final result quoted is S(Ω). The other extreme case is a squeezed vacuum state |0, 𝜉⟩. This state contains a small number of photons: n0,𝜉 = ⟨0, 𝜉|̂a† â |0, 𝜉⟩ = sinh2 (rs )

(9.17)

Any squeezed vacuum state cannot be perfectly empty of photons. It is not the minimum energy state and therefore not a vacuum at all. However, for realistic degrees of squeezing (rs < 1), the number of photons is so small compared to the coherent state used as the local oscillator in a homodyne detector that it can be neglected. As mentioned already, the squeezed vacuum state has many practical applications. It is of interest to consider the case of a squeezed state with a few photons, typically |𝛼|2 < 10. In these cases even more unusual effects can occur. Take, for example, photon statistics. A squeezed state has larger probabilities for even than for odd photon numbers. A strongly squeezed state contains mainly pairs of photons. For these cases the photon number distribution is not at all simple, and for completeness it is stated here: 1 (tanh(rs ))n exp[−𝛼 2 (1 + tanh(rs ))] 2n n! cosh(rs ) )|2 | ( | 𝛼 exp(rs ) || | (9.18) × |Hn √ | | | sinh(2 r ) s | | where 𝛼 is taken real and Hn are Hermite polynomials. Please note that this formula is frequently misprinted. For illustration the photon probability functions for amplitude squeezed states (𝜃s = 0) with different average photon numbers (𝛼 2 = 4, 36, 1000) and a degree of squeezing (rs = 1) are shown in Figure 9.10. We see that for moderate squeezing, or for large photon numbers, the distributions are similar to a Poissonian with a narrower width. If the squeezing parameter rs approaches the value of 𝛼, the distribution develops extra oscillations. This is a sign that photon pairs are predominant in strongly squeezed states. (n) =

9.2.2

The Generation of Squeezed States

Conventional light, from either thermal sources or from lasers, is not squeezed. Laser light can be approximated by coherent states at sufficiently high detection frequencies. How can a coherent state be transformed into a squeezed state? In the introduction to this chapter, examples were given for this process: the Kerr effect and 4WM. The conclusion was that light that contains correlated quadrature fluctuations can show a reduction in the variance of a specific quadrature. This result can be generalized. The feature that distinguishes any squeezed state from a coherent state is these correlations. Whilst for coherent states all quadrâ ̂ + 90) are equal, the correlations will single out a particular ture pairs X(𝜃), X(𝜃 ̂ Y𝟐, ̂ and the fluctuations in Y𝟏 ̂ will be below the QNL. set of quadratures Y𝟏,

317

9 Squeezed Light

0.3 0.2

2

α =4

0.15 0.1 0.05 0 0

(a)

10

20

30

40

50

60 n

950

1000

1050

0.3 rs = 1.0

0.25

2

α = 1000

0.2 0.1 0.05 0 (b)

0

10

20

30

40

50

60 n

950

1000

1050

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1100

P(n)

2

α =4

0.15

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 1100

P(n)

P(n)

2

α = 1000

rs = 0

0.25

P(n)

318

Figure 9.10 Photon probability functions for three different average photon numbers 𝛼 2 = 4, 36, 1000 for (a) a coherent state (rs = 0) and (b) a squeezed state (rs = 1.0).

We need a non-linear process to achieve this correlation between the quadratures. Obviously there are several such interactions that can be used. Two classes of interactions are described by the generic Hamiltonian operators: ̂ = iℏ [𝛼𝜒 (2) â 2 − 𝛼𝜒 (2) â †2 ] H

(9.19)

̂ = iℏ [𝛼 2 𝜒 (3) â 2 − 𝛼 2 𝜒 (3) â †2 ] H

(9.20)

and where 𝜒 (2) and 𝜒 (3) are the second- and third-order non-linear susceptibilities of the medium and 𝛼 is the amplitude of the pump field that drives the non-linear process, which is assumed to be a large classical field. The first Hamiltonian describes a degenerate parametric amplifier, the second 4WM, or a Kerr medium. There are other more complex situations that also lead to squeezing. These will contain higher-order terms or other combinations of â and â † . In his landmark paper in 1983, D.F. Walls pointed out the general effects of these operators and coined the name squeezed states [11]. The effect of these Hamiltonians on the fluctuations will be discussed in detail in Sections 9.4.2 and 9.4.3. In all practical cases, the linearization technique discussed in Section 4.5 can be used. These Hamiltonians are frequently oversimplifications, not a complete description of a realistic system where other effects will take place at the same time, but the Hamiltonians, Eqs. (9.19) and (9.20), are essential for the generation of the correlations. In general, calculations based only on this part of the total Hamiltonian would overestimate the size of the correlations and the degree of squeezing. Additional processes will degrade the

9.2 Quantum Model of Squeezed States

correlations and thus the noise suppression. A different type of Hamiltonian that generates the required correlation is of the form ̂ a1 â † ) ̂ = iℏ𝜒 (2) (b̂ † â † â 2 − b̂ H 1 2 between the modes â 1 , â 2 , b̂ and that converts, in a non-linear process, the photons in the mode b̂ into photons in the modes â 1 , â 2 such that the energy is conserved. Examples are SHG from pairs of photons in â to b̂ and the optical parametric oscillator (OPO) above threshold operating in the reverse direction. In this case the correlation is not created between the quadratures of one state but between the quadratures of two different states, for example, the fundamental and the second harmonic (SH) beam. This process results in cross correlations between beams of different optical frequencies as well as autocorrelations of the individual beams. In the following chapters experiments will be described that use a whole range of different processes and materials. In each case the noise spectrum is derived, and the practical limitations and the relevant competing noise processes are discussed in detail.

9.2.3

Squeezing as Correlations Between Noise Sidebands

A very useful interpretation can be given for the properties of the squeezed states using the engineering point of view described in Section 4.5.5. The quantum noise of a coherent state at a specific detection frequency Ω can be interpreted as the contributions to the beat signal by the sidebands of the complex amplitude. There are two contributing sidebands, and these have frequency-independent fluctuations (white noise). The two sidebands 𝛿E+Ω , 𝛿E−Ω , at the frequency +Ω and −Ω, are not correlated. In contrast, squeezed states have correlated sidebands. All squeezing experiments are based on non-linear effects and work due to such a correlation. The observed noise suppression is a consequence of the addition of both sidebands in the detection process, whether it is direct or homodyne detection. For this reason most theoretical books call this two-mode squeezing. However, it is worth noting that in our nomenclature the central amplitude component and all sidebands within the bandwidth of the detector are part of the same single mode. When we talk in this book about two or multiple modes, we mean orthogonal modes that can be separately detected, without crosstalk, by two or more separate detectors. For a coherent state the noise is generated by these two fluctuating sidebands added in quadrature. The root mean square (RMS) value of this noise corresponds to the quantum noise. The fluctuations at the sidebands can have a random phase to each other. They are described by complex numbers. Graphically they can be represented by a vector located on the frequency axis at 𝜈L + Ω and 𝜈L − Ω with a random orientation, shown in Figure 9.11a. Only the RMS size is fixed, which is indicated by the circles at 𝜈L + Ω and 𝜈L − Ω. The amplitude of the field is described by 𝛼(t) = 𝛼0 exp(2πi𝜈L ) + 𝛿E+Ω (t) exp(2πi𝜈L + Ω) + 𝛿E−Ω (t) exp(2πi𝜈L − Ω)

319

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9 Squeezed Light

X2

E

Coherent state

No correlation X1

Ω (a) Partially correlated noise

X2

E

Phase squeezed state

X1

Ω (b) Completely correlated signals

X2

E

AM

X1

Ω (c) Partially anti-correlated noise

X2

E

Amplitude squeezed state

X1

Ω (d) X2

E Completely anti-correlated signals

Phase modulation

X1

Ω (e) X2

E Partially correlated with phase angle θ

Quadrature squeezed state Y2

Ω

Y1

X1

(f)

Figure 9.11 The link between correlated sidebands, left-hand side, and phase diagrams for squeezed states, right-hand side.

9.2 Quantum Model of Squeezed States

Squeezed states have correlated sidebands, and the observed noise suppression is a consequence of both sidebands. For this reason theoretical books refer to two-mode squeezing. Consider the following extreme cases: If the sidebands are completely correlated, the wave would be modulated. A complete positive correlation 𝛿E+Ω (t) = 𝛿E−Ω (t) means the left and the right sidebands fluctuate in phase with each other. This corresponds to an amplitude-modulated beam. Since the fluctuations at one frequency are random in time, this will be a noisy amplitude. The beam has intensity noise. It corresponds to the modulated light introduced in Section 4.5.5. The corresponding quadrature diagram is a line in the X1 directions shown in Figure 9.11c. Similarly, a complete anti-correlation, 𝛿E+Ω (t) = −𝛿E−Ω (t), leads to a vector that oscillates in the X2 direction (Figure 9.11e). The two components are completely out of phase with each other. This is closely related to a phase-modulated light beam that is equivalent to a beam with a whole series of sidebands (Eq. (2.25)). The oscillating term 𝛿E+Ω (t) corresponds exactly to the first sidebands. The higher-order sidebands (at ±3Ω, ±5Ω, …) can be neglected since we deal with extremely small modulations. This is light with random phase modulation but with no amplitude noise and direct detection will show no extra noise at all. More generally we can have complete correlation with a fixed phase angle 𝜃. That means any contribution of a fluctuation 𝛿E+Ω (t) is accompanied by a fluctuation 𝛿E−Ω (t) = exp(i𝜃)𝛿E+Ω (t). The result is again a straight line in the phase diagram (Figure 9.11f ). The light is randomly modulated in a quadrature Y 1. When detected with a homodyne detector with local oscillator phase 𝜃∕2, the full noise is detected; at local oscillator phase (𝜃∕2 + π∕2), no noise is detected. Now let us turn our attention to partially correlated sidebands. That means that 𝛿E+Ω (t) and 𝛿E−Ω (t) contain both fluctuations that are completely correlated and some other components that are completely random and independent. If we add the contributions in a quadrature phase diagram, the correlated part will again be represented by a line. The random, uncorrelated part corresponds to a circle, with a diameter smaller than the QNL. The total result is a convolution of the two. The vector is a sum of the instantaneous components of both. If the partial correlation is positive, we obtain an extended elliptically shaped area along the X1 direction (Figure 9.11b). This light is tending towards amplitude modulation. Its quadrature phase is squeezed. Similarly, a partial anti-correlation leads to an uncertainty area along the X2 direction (Figure 9.11d). This light is tending towards phase modulation. Thus the light is quadrature amplitude squeezed. If the correlation is of a fixed phase angle 𝜃s ,the uncertainty area is drawn out along the Y 1 axis (Figure 9.11f ). This light is quadrature squeezed. The question is: can this semi-classical interpretation of squeezed light be used for any quantitative calculations? The answer is yes. This model will produce a result as reliably as the full quantum model. The degree of correlation between the quadratures and the correlation angle is equivalent to the squeezing spectrum V (Ω, 𝜃) and the squeezing angle 𝜃s . This is because the Wigner function for Gaussian states is always positive and can be interpreted as a classical probability distribution for the quadrature measurement.

321

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9 Squeezed Light

The only assumption in this interpretation that is not classical is the fixed size of the noise sidebands that correspond to the QNL, as discussed in Section 4.5.5. This is based entirely on the quantum nature of light and is mathematically equivalent to the inclusion of the vacuum state in the full quantum calculations. As stated in Section 4.5.5, the sidebands |𝛿E+Ω (t)| and |𝛿E−Ω (t)| can be associated ̃ and 𝜹a ̃ † . Once the variances of the fluctuations |𝛿E+Ω (t)| with the operators 𝜹a and |𝛿E−Ω (t)| have been set, everything else can be calculated classically. More importantly, the effect of a non-linear medium on the quantum noise can be clearly visualized.

9.3 Detecting Squeezed Light There is a great similarity between all squeezing experiments. They all are concerned with the measurement of the reduction of the noise in one specific quadrature of the light. A non-linear medium is used to introduce the correlation that generates the squeezed light inside the experiment. In many situations the non-linearity is enhanced by placing the medium inside an optical resonator. This not only increases the squeezing but also influences the spectrum and the orientation of the squeezing. Detecting squeezed light means detecting noise fluctuations. These fluctuations are very small, and we need to have a minimum detectable energy to overcome the limitations imposed by electronic noise in our detection system. All detection techniques rely on the detection of the beat signals between a strong mode, the local oscillator or the carrier, and the correlated noise sidebands at ±Ω around this mode. In the case of bright amplitude squeezed light, we can use the same balanced detector as we used to measure the noise of a laser. For bright quadrature squeezed light, we can use a cavity to rotate the angle of detection. For weak squeezed light, or for a squeezed vacuum state, a balanced homodyne detector is required. In all these schemes loss has to be rigorously avoided, and this is one of the chief challenges in squeezing experiments. 9.3.1

Detecting Amplitude Squeezed Light

There is no difference between the detection technique for sub-Poissonian light or bright quadrature amplitude squeezed light. The details of the balanced detector have already been given in Section 8.1 where it is used to measure classical noise. This technique allows us to measure the noise suppression reliably, and a comparison between the (+) and the (−) current gives a direct calibration of the apparatus (Figure 9.12). If the light is QNL, the variance of the (+) and the (−) currents will be identical, when the squeezing process is turned on and the difference between the (+) and the (−) channels directly displays the noise reduction. 9.3.2

Detecting Quadrature Squeezed Light

For quadrature squeezed light we use the full homodyne detection scheme (discussed in Section 8.1.4). The experiment looks like a Mach–Zehnder interferometer with the non-linear medium in one of the arms. A typical squeezing

9.3 Detecting Squeezed Light

Balanced direct detector

Squeezing spectrum

Laser Non-linear medium

+

Spectrum analyser

Figure 9.12 Detection scheme for bright squeezed light: self-homodyne balanced detection. The energy comes from the bright beam itself and only amplitude fluctuations are measured. Non-linear medium

Laser

θ Phase shifter

– V(θ)

The result

θ

1 Hoover

θs

Fixed frequency Ω

Figure 9.13 A typical squeezing experiment. The fluctuations of the quadratures are selected by tuning the phase of the local oscillator. One single detection frequency is selected. V𝛼,𝜉 (Ω) is displayed as a function of the detection angle 𝜃.

experiment is shown in Figure 9.13. The light emerges from the squeezing medium with the correlated fluctuations in one quadrature. A strong local oscillator is generated and sent around the squeezing experiment. Its phase is controlled and the two beams are recombined. The alignment procedure is identical to that of an interferometer; in particular the spatial modes of the two interfering beams have to be matched. They have to overlap and contain identical wave fronts and essentially form one large interference fringe. Both output beams are detected, and the difference (−) of the two photocurrents is formed. The optical power of the local oscillator is chosen to be far higher than that of the squeezed light. An analysis of this apparatus shows that the noise level of the photocurrent is proportional to the optical power of the local oscillator and contains only the fluctuations of one quadrature of the squeezed light. Any local oscillator noise is suppressed.

323

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9 Squeezed Light

The detection frequency is determined by the spectrum analyser (SA) that operates with zero span, which means it detects only one single frequency. The quadrature is chosen by scanning the phase of the local oscillator. In this way all quadratures can be investigated, one at a time. The result is an oscillating trace, showing the variances for all quadratures 𝜃, as derived in Eq. (9.14) and plotted in Figure 9.9. The apparatus has to be calibrated with an independent Poissonian light source. This arrangement is the essential scheme for the detection of squeezed vacuum states. The whole experiment is one large interferometer and thus has to be protected from vibrations. A change in the local oscillator phase during the measurement will smear out the trace, and this tends to reduce the best noise suppression. This is particularly noticeable when the squeezing is large. 9.3.3

Using a Cavity to Measure Quadrature Squeezing

For bright squeezed light there is an alternative of using a cavity to analyse the degree and the quadrature of the squeezing. In the conventional homodyne detector, the rotation of the quadratures is achieved by controlling a phase shift between the sidebands and the local oscillator. We could use the squeezed light itself as the local oscillator if we had a device that distinguishes between the two noise sidebands and the centre component of the output beam and gives one a phase shift in respect to the other. This can be achieved by reflecting the light from a cavity that has a linewidth narrower than the detection frequency (Figure 9.14). Once the cavity is detuned, a phase shift 𝜃(Δ𝜈) is introduced. The details of this technique have already been described in Section 8.4, where we found that the effect of a detuned cavity is to rotate the quadratures in the reflected light. The rotation is almost linear with the detuning particularly if the detuning is less than one cavity linewidth. With a phase shift of 𝜃, the noise sidebands that are measured correspond to the fluctuations of the light in the quadrature X(𝜃). By varying the detuning Δ𝜈, the angle 𝜃 can be selected.

Non-linear medium

δX2

Input δX1

Laser

δX2

+

Detuned cavity

Output

δX1

Figure 9.14 The detection of quadrature squeezed light using a phase shifting cavity that rotates the squeezing at a specific frequency.

9.3 Detecting Squeezed Light

In practice, the cavity has to be locked to a fixed detuning Δ𝜈 during each measurement. This can be achieved by a servo control system and by using the Pound–Drever–Hall technique described in Section 8.4. The phase modulation for the locking should be at frequencies far away from the detection frequency for the squeezing (Figure 9.14) to avoid contamination of the signal. The actual detuning, and thus 𝜃, has to be determined separately. Note that the noise level for the QNL, which is proportional to the optical power on the detector, also varies with the detuning. We have to record the DC photocurrent for each detuning and calibrate each measurement individually. Typical examples of the power attenuation and the phase shift of a detuned cavity are given in Figure 5.11, and we notice that it is quite possible to achieve rotations of ±π∕2. This detection technique was first used in the fibre Kerr experiment by M. Levenson and co-workers [12, 13] and has been described in detail by P. Galatola et al. [14]. A more complete approach for characterizing the light is tomography, as discussed in Section 9.7. 9.3.4

Summary of Different Representations of Squeezed States

A coherent state can be represented by just one parameter, |𝛼⟩. However, we need more parameters to define a squeezed state. Any minimum uncertainty squeezed state could be represented by |𝛼, 𝜉⟩ and could be described by three parameters (𝛼, rs , 𝜃). A squeezed state with excess noise is mixed and requires a fourth parameter, which describes the ratio of the minimum variance to the maximum variance. The properties of the pure squeezed states can be summarized by the following expressions: VY 1 = exp(−2rs ) and VY 2 = exp(2rs ) n𝛼,𝜉 = |𝛼|2 + sinh2 (rs ) V n𝛼,𝜉 = (𝛼 cosh(rs ) − 𝛼 ∗ rs e2i𝜃s sinh(rs ))2 + 2cosh2 (rs ) sinh(rs ) Both rs and 𝜃s can be functions of the detection frequency. The best noise suppression, for given frequency Ω, is quoted as the squeezing spectrum: S(Ω) = exp(−2rs (Ω)) with optimized 𝜃s (Ω) We have seen that there are a number of graphical representations for a coherent state, each concentrating on one particular aspect. These are compiled in Figure 4.7. Similarly, there are a number of graphical representations for squeezed states, which are summarized in Figure 9.15. 9.3.5

Propagation of Squeezed Light

A major concern in all experiments is the sensitivity of the squeezed light to losses. The light will have to propagate through various optical components; it needs to be detected and has to overlap with other beams. Each such loss, mismatch, or inefficiency will reduce the degree of squeezing. This is very different from conventional optics experiments where losses can be tolerated and can always be compensated by an increase in input intensity. The quantitative effect

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9 Squeezed Light

X2

Pn δY2

δY1

 α, ξ >

θs X1

n δX1(Ω)

δX2(Ω)

V1(Ω) 1

Ω

0

δX2

δX1

Ω S(Ω)

V(θ)

θ

1

1

0 θs

–Ω

0

Ω

Figure 9.15 Summary of the various graphical representations of a squeezed state.

of losses is easy to predict since all passive losses behave in the same way and can be simulated by beamsplitters. Only non-linear processes are excluded from this model. In terms of the sideband picture, the effect of the beamsplitter is easy to describe. The squeezed light corresponds to light with partially correlated sidebands 𝛿E+Ω and 𝛿E−Ω . For a beamsplitter with reflection 𝜖, transmission √ 𝜂 = (1 − 𝜖), the correlated part of the noise sidebands is reduced by 𝜂, in the same way as modulation sidebands are reduced (Section 2.1), whilst simultaneously the uncorrelated part is increased to keep the magnitude of the quantum noise sidebands constant. This is equivalent to including the quantum noise sidebands of the vacuum state from the empty port of the beamsplitter. The combined sidebands 𝛿E+Ω,out , 𝛿E−Ω,out at the output are less correlated than those at the input, and the degree of squeezing is reduced (Figure 9.16a). In terms of the phasor diagrams, the beamsplitter replaces the squeezed Wigner function with one that is less squeezed (Figure 9.16b), which means the contour ellipse

9.3 Detecting Squeezed Light

X2

Partially αin correlated noise

δY2 Ω

δY1 θs

V(θ) θ

1

X1 θs X2

αout Less correlated noise

V(θ) δY2

Ω (a)

δY1 θs

θ

1

X1 (b)

(c)

θs

Figure 9.16 The effect of a beamsplitter on squeezed light, input at the top and output at the bottom. Three representations are used: (a) correlated sidebands, (b) phasor diagrams, and (c) quadrature variances. The top row is representative of the light at the input, and the bottom row at the output of the beamsplitter.

is not as narrow. The corresponding variance V (𝜃) at the output is to the QNL (Figure 9.16c). Loss transforms the contour from an elongated ellipse towards a circle, as shown in Figure 9.17. This might not be obvious if one considers that the fluctuations of the input of the beamsplitter simply add. Just adding the two Wigner functions would suggest a rather more complicated result, as indicated in Figure 9.17b, for the case of adding two squeezed states with different squeezing angle 𝜃s . However, all features of the Wigner function have to be taken into account, including the phase terms – and the answer is actually simple: the result is another ellipse with the new orientation 𝜃s and smaller ellipticity. It has to be an ellipse since we know that any solution of the linearized model will result in a 2D Gaussian quasi-probability distribution (Section 9.2). Thus the only parameter that can vary is the variance VY 1 of the squeezed state after the beamsplitter. If the input state is a squeezed state with the general parameters (𝛼, rs , 𝜃s ), the output state, after the beamsplitter, is (𝛼 ′ , rs′ , 𝜃s′ ). We know that the intensity is reduced such that |𝛼 ′ |2 = 𝜂|𝛼|2 . The beamsplitter has a second input port for vacuum fluctuations and is described in the standard way: √ √ ̃ in + 1 − 𝜂 𝜹X ̃ vac ̃ out = 𝜂 𝜹X(𝜃) (9.21) 𝜹X(𝜃) The calculation can be carried out independently for each angle 𝜃. As a result, the minimum will remain at 𝜃s since |0⟩ has no angular dependence. Thus the squeezing angle is unchanged, 𝜃s′ = 𝜃s . We have to evaluate the variances and this is a linear operation √ that leads to the sum of two contributions. The second term simply results in 1 − 𝜂 since any contribution from the vacuum state results in 2 ̃ ⟩ = 1. As a result we obtain Vvac = ⟨|𝜹X(𝜃)| VY 1out = 𝜂VY 1in + (1 − 𝜂) or (1 − VY 1out ) = 𝜂(1 − VY 1in )

(9.22)

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9 Squeezed Light

(a)

No

Yes (b)

Figure 9.17 Wigner functions cannot be added directly; the phase terms must be taken into account. (a) Attenuation of a squeezed state by a beamsplitter results in a state with reduced squeezing. (b) The combination of two squeezed states with different 𝜃s values results in a normal squeezed state, not a Wigner function with four contributions.

This means the noise suppression (1 − VY 1out ) is reduced linearly with loss 𝜂. The noise suppression is affected in the same way as the intensity. Note that the noise suppression is commonly quoted on a logarithmic scale in dB for the normalized variance and this value is not reduced linearly. The effect of 𝜂 is shown in Figure 9.18 where the output variance VY 1out is plotted in dB versus the input variance VY 1in . The effect of losses is more severe on strongly squeezed light. A loss of 50%, (𝜂 = 0.5) transforms VY 1in = −3 dB into −1.3 dB, whilst for the much better squeezing of VY 1in = −10 dB, the output is very similar, namely, −2 dB. Strong squeezing is easily lost in a system with limited efficiencies. The minimum variance of a squeezed state is VY 1𝛼,𝜉 = exp(−2rs ). One could formally try to link the squeezing parameter rs′ to rs . The link is exp(−2rs′ ) = exp(−2rs )𝜂 + (𝜂 − 1). This is not a simple relationship, and the squeezing parameter is not practical for this type of calculation. Since the conversion of the squeezing due to losses is constantly required in the evaluation

9.3 Detecting Squeezed Light

Output log (VY1(α′,ξ′)) (dB)

0

2

0.5 0.6

4

0.7 0.8

6

0.9 η = 1.0

8

10

0

2

4

6

8

10

Input log (VY1(α, ξ)) (dB)

Figure 9.18 The quantitative effect of different efficiencies 𝜂 on the minimum variance of the squeezed state.

of squeezing data, the squeezing parameter is rarely used by experimentalists. The most common parameter cited is (1 − VY 1) either on a linear or logarithmic scale, and this scales easily, as stated in Eq. (9.22). Note that a beamsplitter, or any other phase-independent loss, does not change the spectrum of the squeezing. Equation (9.22) holds equally well for all detection frequencies. Both the shape of the spectrum and the squeezing angle are not affected by losses, and only the degree of squeezing is reduced. The squeezed light produced by some devices, for example, a below threshold OPO, can be minimum uncertainty state, a pure state with VY 1 VY 2 = 1. Does it remain a minimum uncertainty state after losses? The beamsplitter will mix in a fraction of the vacuum state, itself a minimum uncertainty state, and the combined output will be a mixed state, above the minimum uncertainty value. How much above? We can directly calculate VY 1out VY 2out = (𝜂VY 1in + (1 − 𝜂)) (𝜂VY 2in + (1 − 𝜂)) but do not obtain a simple answer. The result is easiest interpreted from a diagram. Figure 9.19 shows the values for the product of the variances as a function of 𝜂 for three degrees of squeezing: VY 1in = 0.5, 0.2, and 0.1. We see that the product rises quickly and for 𝜂 ≈ 0.5 reaches a maximum that depends on the initial degree of squeezing. For large attenuation all these states approach again a minimum uncertainty, the coherent state. If the light is too intense for the detector, we should not simply attenuate the light but reflect it off a cavity. If a cavity with smaller linewidth than the detection frequency is chosen, the cavity will fully reflect the noise sidebands whilst the carrier component is partially transmitted. In this way the optical power can be attenuated without losing the squeezing. An interesting extension would be to incorporate an active gain medium into the cavity [6]. This now becomes

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9 Squeezed Light

3 VY1in = 0.1 VY1out * VY2out

330

2.5

2

VY1in = 0.2

1.5 VY1in = 0.5 1 0

0.2

0.4

0.6

0.8

1

Loss η

Figure 9.19 The uncertainty product for a squeezed state after loss. Three different degrees of squeezing are shown.

a noiseless optical power amplifier – it amplifies the energy but leaves the noise sidebands outside the cavity linewidth unaffected. Alternatively we could detune the cavity and therefore rotate the squeezing ellipse, as discussed in the Section 9.4.4. Amplitude quadrature squeezing can be turned into phase quadrature squeezing and vice versa. We have all the components available to modify the squeezing parameters, but the only thing we cannot do is to increase the degree of squeezing.

9.4 Early Demonstrations of Squeezed Light 9.4.1

Four Wave Mixing

We are now discussing the pioneering squeezing experiments in detail. By the middle of 1984 several research groups were competing to measure the first non-classical noise suppression – to detect the first squeezed state. Each group favoured a different non-linear process, and eventually all these attempts were successful. The first group to report experimental observations of squeezing was led by R.E. Slusher at the AT&T Bell Laboratories in Murray Hill. Their experiment used 4WM in a Na atomic beam [15]. This is a resonant medium. The non-linearity was enhanced by an optical cavity. The first proposal of squeezing with a 4WM mixing medium had been published as early as 1974 by H.P. Yuen and J.H. Shapiro [16]. The difficulty was to achieve sufficient 4WM gain whilst avoiding all other sources of noise. The analysis of squeezing by 4WM confirmed that we require a non-degenerate situation [2]. That means the pump beam, which provided the energy for the process, and the signal and idler beams should have separate frequencies. This led to the classical interpretation of 4WM as correlating two sidebands, each detuned by ±Ω from the pump frequency, as discussed in Section 9.2.3

9.4 Early Demonstrations of Squeezed Light

One important noise source that could swamp any squeezing was spontaneous emission. Light from the pump beam, which was absorbed by the atoms, was re-emitted either at the same frequency or at the atomic resonance frequency. Such light produced fluctuations in the photocurrent that raise the noise level above the QNL. It was necessary to detune the pump wave from the atomic resonance by many atomic linewidths. However, this reduced the non-linearity and a compromise between non-linearity and spontaneous emission noise had to be achieved. One technique to reduce the spontaneous emission, at a given detuning, was to reduce the Doppler broadening. Consequently most experiments were performed with atomic beams that are essentially Doppler-free as long as all optical beams are aligned at right angles to the propagation axis of the atoms. The atomic beams had to be optimized for high densities. In Slusher’s experiment an atomic absorption of exp(−0.5) on line centre was measured. The experiment consisted of a beam of Na atoms that was illuminated, pumped by an optical pump beam, and detuned by about 2 GHz away from the hyperfine components of the Na D1 transition at 590 nm. Figure 9.20 shows the experimental layout and Figure 9.21 the frequencies of all the optical beams in the experiment. A second beam, the probe, was aligned through the same atoms at

ϕLO

Dye laser

BS1 EO2 FB2

AO3,4

LO

AO1,2

vL+3vSC

vL+2vSC

PM1

BS2

SM2

DB

EO1 S1

SM1 Na

PM2

DA –

FB1

SA

Figure 9.20 Layout of the atom 4WM experiment. Source: After R.E. Slusher et al. [15].

vatom

vsignal

vprobe

vidler

vlock

Figure 9.21 The different frequencies involved in the 4WM experiment. The cavity modes are shown as thin solid lines. They are shifted by the dispersion of the atoms.

331

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9 Squeezed Light

a small angle to the pump beam. This probe beam was aligned into a heterodyne detection system together with a local oscillator beam that was split off from the pump beam. Two photodiodes detected the beams and generated a beat signal at the difference frequency between probe and pump, in this case at 595 MHz. In the absence of the atoms, and with the probe beam blocked, the noise level recorded was the QNL (1 Hoover). It can be interpreted as the noise due to the vacuum state entering the beamsplitter of the heterodyne detector. With the atoms present and the probe beam turned on, the 4WM medium generated a second sideband with exactly the opposite detuning to that of the probe. Now two beams were present and were detected together. In this way the probe beam was used to measure the classical 4WM gain and to align the detection system. The 4WM process can be interpreted as phase-sensitive amplification: it increases amplitude modulation and reduces phase modulation sidebands. This gain correlates the two sidebands, as discussed in Section 9.2. Even with the probe beam switched off, this correlation exists. The ever-present quantum noise sidebands, which enter the medium on the axis that was defined by the probe beam, will be correlated, resulting in a squeezed vacuum state. For a single transit of the Na beam, this effect was far too small to be detectable. The 4WM process, and the resulting correlation, could be built up by a cavity that is tuned to both sidebands simultaneously. Each sideband was on resonance with a different cavity mode. In this experiment they were not even adjacent modes, but they were separated by nine free spectral ranges (FSRs) (Figure 9.21). Again, the probe beam was used for the alignment of the cavity. A locking system kept the cavity on resonance. This used an optical beam, which probed the cavity at yet another optical frequency, avoiding the pump and the probe frequency. One practical problem was the fact that the atoms produced a strongly dispersive refractive index that shifts all cavity resonances. The cavity modes were asymmetric to the probe frequency, but by using acousto-optical modulators (AOMs), the locking beam could be shifted until both the probe and the newly generated beam were in resonance. The pump intensity was raised until it approached the level of the saturation intensity of the sodium vapour at a detuning of 2 GHz, equivalent to 400 atomic linewidths. and the pump beam was retro-reflected to provide an even higher pump intensity. Finally, the probe beam is blocked. The vacuum mode, which is collinear with the probe beam, experiences the correlation at the sidebands at +Ω and −Ω induced by the atoms inside the single-sided cavity. A squeezed beam is created that travels towards the detectors. The light is squeezed in a band of frequencies around the cavity resonance frequency. The strength and the quadrature of the squeezing are measured by scanning the phase of the local oscillator. This first squeezing result is shown in Figure 9.22a. On the right-hand side, Figure 9.22b shows the result after continuous improvements 15 months after the first observation [15]. The dotted line in Figure 9.22b gives the QNL, obtained with the cavity blocked. The measured noise increases 1.3 dB above and decreases −0.7 dB below the SQL. The electronic noise (amplifier noise) is −10 dB below the SQL and contributes about 10% to the noise level. The detection frequency is 594.6 MHz, resolution bandwidth 300 kHz, and video bandwidth 100 Hz. The theoretical model predicts ±2 dB (solid curve) for an ideal measurement. The efficiency of the photodetectors is 𝜂det = 0.8, and the efficiency of the

9.4 Early Demonstrations of Squeezed Light

–58

Noise level (dBm)

–59

–60

–61

–62 0 (a)

(b)

π/2 ϕLO

π

3π/2

Figure 9.22 The first squeezing results. (a) Photograph of the first trace [15]. (b) Quantitative comparison between experimental results and theoretical predictions after months of further work. Source: After R.E. Slusher et al. [2].

interferometer is 𝜂vis = 0.81, resulting in a total efficiency of 0.65. Taking this and the electronic noise into account, the theoretically predicted noise is given as the dashed-dotted line in Figure 9.22b. It shows good agreement with the enhanced noise and moderate agreement with the squeezed noise quadrature, namely, −0.7 dB observed versus −1.05 dB predicted. One practical limitation is the mechanical stability of the interferometer required for the homodyne detection. Any mechanical vibration can wash out the noise suppression. This effect, described as phase jitter, is cited as the reason for the disagreement with the prediction. It can explain this particular set of results. However, at higher intensities, the disagreement gets even larger, and other processes have to be considered. Non-linear coupling between the light used for the locking of the cavity to the laser frequency and the mode of the squeezed light is considered by the authors as a possible source of excess noise. This experiment clearly demonstrates the existence of a squeezed state of light, in this case a squeezed vacuum state. All predictions about the properties of the squeezed light and the detection systems were confirmed qualitatively. The observed noise reduction is modest. Despite the predictions of squeezing of up to −10 dB, 4WM has never been really successful as a squeezing process. The technical difficulties with the atomic beam were frustrating, and this type of experiment has not been repeated by any other group. It remains a milestone in the early exploration of squeezed light. 9.4.2

Optical Parametric Processes

Some of the most successful results in the early period of squeezing came from the group led by H.J. Kimble at the University of Texas [17]. He used an

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9 Squeezed Light

M2

M1 αp

A1 vp

Detunings

ζ

αd κp1 κd1 κp = Total loss at vp

Aout

Δp Δd

vd = 1/2 vp

κp2 κd2 κd = Total loss at vd

Figure 9.23 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive amplifier of the vacuum modes and above threshold as an oscillator. The subscript p refers to the fundamental frequency, and the subscript d refers to the subharmonic frequency.

optical parametric amplifier (OPA) to generate squeezing. We have already seen in Chapter 6 that the OPA is ideally suited since it is phase sensitive and amplifies one quadrature whilst it attenuates the other as shown in Figure 9.24. The trick to enhance the process is to build a cavity around the OPA as shown in Figure 9.23. The experiments with the non-linear OPO are carried out with a cavity containing a 𝜒 2 material, usually a crystal such as LiNbO3 . A single-mode laser illuminates the cavity. Following the derivations in Section 6.3.3, we know that a parametric oscillator below threshold will produce quite different gain for the two quadratures, similar to 4WM, and therefore suppresses the noise selectively. Figure 9.24 symbolically shows how the quadratures are affected and the conversion of a vacuum input state (circle) into a squeezed vacuum (ellipse). From Eq. (6.43) we can derive a noise spectrum and find that the effect is greatest at low frequencies (Ω ≪ cavity linewidth). For such an oscillator the crucial c , experimental parameter is given by the normalized pump amplitude d = 𝛼in ∕𝛼in c where 𝛼in is the input amplitude required to reach threshold. We wish to operate with d < 1 and can optimize the performance by approaching d → 1 by changing the input intensity. In terms of the cavity parameters, losses and non-linear gain, Figure 9.24 Symbolic representation of parametric gain in the phase diagram.

δX2

δX1

9.4 Early Demonstrations of Squeezed Light

1

S(Ω)

0.8 0.6 0.4 d = 0.5

0.2 0

4

0

2

2

4

Ω

Figure 9.25 The squeezing spectrum for an OPO below threshold for a fixed value of d = 0.5. The modulation frequencies are scaled in units of the cavity bandwidth.

we find d = 𝜒∕𝜅a , and consequently we can derive the variance from Eq. (6.35): V 2(Ω) = 1 −

4d (1 + d)2 + (2πΩ)2

(9.23)

which is identical to the squeezing spectrum, S(Ω) = V 2(Ω), since the noise suppression always occurs in the phase quadrature. An example is shown in Figure 9.25. At very low frequencies, Ω → 0, we get the optimum value V 2(Ω) =

(1 − d)2 (1 + d)2

(9.24)

and at the same time the noise enhancement in the orthogonal quadrature V 1(Ω) =

(1 + d)2 (1 − d)2

(9.25)

We see that the fluctuations in the X2 quadrature tend to zero. On the other hand, the fluctuations in the X1 quadrature diverge. The effect for the two quadratures is shown in Figure 9.26. The system has a critical point at the 40

100

VY1

30 20 V (dB)

Gain

10 Amplitude

0

1 Phase 0.1 (a)

10

20 0

0.2

0.6

0.4 d2

0.8

VY2

10 1 (b)

0

0.2

0.6

0.4 d

0.8

1

2

Figure 9.26 Phase-dependent gain and resulting variances V1 and V2 for the two quadratures of a OPO below threshold as a function of the normalized input power d2 . This calculation is for very low detection frequencies less than the linewidth of the cavity Ω = 0 and for a finite quantum efficiency 𝜂 ≤ 1.

335

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9 Squeezed Light

Ba2NaNb5O15 SHG

MgO:LiNbO3 0.53 μm OPO

1.06 μm Laser Nd:YAG

θlo +_

Figure 9.27 The schematic layout of the experiment by L.-A. Wu et al. [17] generating a squeezed vacuum state with a sub-threshold OPO.

threshold. Perfect squeezing is possible at threshold, at least in theory, but in practice the operation gets unstable and in most situations values no larger than d = 0.8 are chosen. Interestingly the noise suppression improves only slowly for intensities larger than half the threshold value (d2 = 0.5). The efficiency 𝜂 of the apparatus has to be taken into account, and the complete equation for the noise suppression is V 2(Ω) = 1 −

𝜂 4d (1 + d)2 + (2πΩ)2

(9.26)

The first realization of this experiment was carried out by Wu et al. in 1986. The layout of their experiment is shown schematically in Figure 9.27. The light source is a continuous-wave (CW) Nd:YAG ring laser, in this case lamp pumped. The light is frequency doubled using an intra-cavity Ba2 NaNb5 O15 crystal. The laser cavity has two output beams, at 1064 and 532 nm, with orthogonal polarizations. The green light is sent to the parametric oscillator, which is a MgO-doped LiNbO3 crystal inside a linear cavity. When operated above threshold as a degenerate OPO, the cavity sends out a beam at exactly the laser frequency. This beam is recombined in a Mach–Zehnder interferometer with the infrared output from the laser cavity. This arrangement forms the homodyne detector for the output of the parametric oscillator. The laser provides the local oscillator. When operated below threshold, the output ceases, but the apparatus is still aligned. Now the OPO emits a squeezed vacuum state that is detected by the homodyne detector. All measurements were carried out at detection frequencies less than the cavity linewidth. The phase angle 𝜃 of the local oscillator is scanned. In this way the quadratures can be probed. The vacuum noise level is determined by blocking the beam from the OPO. The results are shown in Figure 9.28a. The minimum of the trace gives the optimum noise suppression. It was difficult to stabilize the phase of the local oscillator; thus mainly scans of the projection angle 𝜃 were published by L.-A. Wu et al. [17]. We can investigate the properties of the OPA in different ways. The noise suppression will improve when the threshold is approached. This has been tested and results are given in Figure 9.28a. The agreement with the theoretical predictions (Eq. (9.26)) is excellent. The best noise suppression observed is

9.4 Early Demonstrations of Squeezed Light

(ii) 2.0 (iv) V (θ) 1.0 (i) (iii) 0 (a)

θ0

θ0 + π

θ0

θ

θ0 + π

θ0 + 2π

θ0 + 3π

d2 = P2 /P0 0

0

0.5

1.0

S_(r) –0.5

–1.0

(b)

Figure 9.28 Experimental results. (a) Dependence of noise power on the local oscillator phase 𝜃. With the output of the OPO blocked, the vacuum field entering the detector produces trace (i). With the OPO input present, trace (ii) exhibits phase-sensitive deviations both below and above the SQL. Trace (iii) is the amplifier noise. (b) Noise suppression for different normalized intensities d2 . Comparison of the inferred noise suppression V2(d2 ) = 1 + S− (r) after correction for the quantum efficiencies derived from measurements such as shown in (a) with optimized phase angle with those given by theory (solid curve). Detection frequency Ω is fixed. Perfect squeezing corresponds to V2 = 0 and is indicated by the dashed line. Source: After L.-A. Wu et al. [17].

V1 = 0.33. Once the detector efficiencies and the losses are taken into account, the best inferred squeezing corresponds to a factor of 10, or −10 dB, for the light emitted by the cavity. Finally, this experiment is also a most impressive demonstration of the concept of a minimum uncertainty state. In Figure 9.28b both the minimum and the maximum variances for each input amplitude are plotted.

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10.0

Squeezed states

Minimum uncertainty (1 + S+ )(1 + S_ ) = 1

(1 + S+ )

338

5.0

Vacuum state 1.0 0

0

0.5

1.0

1.5

(1 + S_ )

Figure 9.29 Variances V1 and V2 determined from measurements. The solid curve is the hyperbola V1 V2 = 1, which defines the class of minimum uncertainty states. Squeezed states are those states for which either V1 or V2 < 1 and that lie in the region bounded by the hyperbola and the dashed lines. Source: From L.-A. Wu et al. [17].

According to the theory, the product of the two variances should be unity, since the parametric amplification preserves the minimum uncertainty area. The data points should lie on a hyperbola, and this is indeed the case. Note that this result can only to be expected for data describing directly the OPO, with perfect efficiency, since any loss would have turned these minimum uncertainty states into mixed states. The data for Figure 9.29 are inferred after removing the effect of the detector loss. This was a landmark experiment. Some of the most striking features of squeezed light have been demonstrated in a single experiment: noise suppression, noise enhancement in the opposite quadrature, and preservation of the minimum uncertainty. And the light generated, a squeezed vacuum beam, is certainly not classical in any regard. The work of Liang Wu et al. is a remarkable achievement and remained unsurpassed for many years. The main advance has been in the reduction of losses inside the OPO. The limitations are now the efficiencies of the detector and technical fluctuations in the phase of the local oscillator 𝜃 that wash out the very sharp minima. Work at Caltech, Konstanz, ANU, Shanxi University, Tokyo, Paris, and Hannover have established the OPO/OPA as the most reliable source for squeezed light.

9.4 Early Demonstrations of Squeezed Light

9.4.3

Second Harmonic Generation

The process of frequency doubling or SHG has been used to generate amplitude squeezed light. It is presently the most reliable and efficient process to generate light with high intensity and good noise suppression. A qualitative description why SHG leads to amplitude squeezed light is obtained from the photon model. A coherent input beam can be thought of as a stream of photons with Poissonian distribution. The time separation between photons is random. This beam illuminates a non-linear crystal, which converts a fraction of the light into a new beam with twice the optical frequency. Two beams emerge, the remaining fundamental (f ) beam and an SH beam. The crystal converts pairs of photons from the incoming fundamental beam into single photons of the SH beam. If the probability for the conversion were constant, the statistics of the two emerging beams would be identical to that of the input beam. However, the conversion probability is proportional to 𝛼 2 𝜒 (2) , that is, the instantaneous intensity of the fundamental beam multiplied by the second-order non-linearity. This means the conversion is more likely when the pump intensity is high, when the photons in the pump are closely spaced. Closely spaced pairs of photons are most likely to be converted into a single SH photon. As a consequence the statistics of the SH photons is closely linked to the statistics of pairs in the fundamental beam. The SH photons come in a more regular stream than the pump photons. If the pump beam has a Poissonian photon statistics, the SH beam has a sub-Poissonian distribution. Similarly, the remaining fundamental light from which the pairs have been deleted has a more regular stream of photons; it also has a sub-Poissonian photon statistics. Similar arguments apply to non-linear absorption, which could also produce squeezing. This simple interpretation is suitable only for a single pass through the non-linear SHG material. In CW experiments the non-linearity is not large enough, and the crystal is placed inside a cavity to build up a mode with high intensity at the fundamental frequency. The first successful observations were carried out by S.F. Pereira et al. [18] who detected a noise suppression of −0.6 dB in the fundamental beam. These early experiments were plagued by additional technical noise sources inside the build-up cavity. A breakthrough occurred when it became possible to polish and coat the non-linear crystal directly. In such a monolithic design, the internal losses can be kept to a minimum, the fundamental light is built up to increase the non-linearity, and the SH field is reflected at one end to have all of the SH field escape on one side. Care has to be taken that the fundamental and SH waves stay in phase, which is achieved by selecting the correct phase matching temperature. The coatings are designed such that phase shifts upon reflection preserve the optimum phase relation for a large number of round trips. Such a monolith was first designed and operated by the group in München [4]. A. Sizmann et al. measured a noise suppression of −1.0 dB in the fundamental light in the presence of significant electronic noise from which they inferred a noise suppression of at least −2.2 dB. This was one of the earliest applications of the balanced detector system (see Section 8.1). Unfortunately the system was not stable, but squeezing was observed only for milliseconds at a time. Systematic

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1064 nm

532 nm

High Refl 1064 High Refl 532

High Refl 1064 Low Refl 532

Figure 9.30 The monolithic design for second harmonic generation (SHG) using CW lasers. The fundamental field (FF) is enhanced by the cavity, and the second harmonic field is reflected at one end and forms one single output beam. Source: From R. Paschotta et al. [5].

PD1 + –

SA

BS PD2 532 nm Oven 110 °C Laser

F1

1064 nm

M1

1064 nm Lock-in 12 MHz

Figure 9.31 Typical layout of an SHG squeezing experiment. The monolithic resonator is pumped through an isolator (FI) by a single-mode ND:YAG laser at 1064 nm. The reflected second harmonic light is detected by a balanced detector, and the sum and difference currents from the two detectors (D1 and D2) are measured with a spectrum analyser (SA). The parameters used by R. Paschotta et al. are MgO:LiNbO3 , cavity length 7.5 mm, radius of curvature 10 mm, input reflectivities of RFH = 99.9%, RSH < 1%, and output reflectivities close to perfect for both fields. Source: After R. Paschotta et al. [5].

improvements were carried out by the group in Konstanz who achieved very reliable squeezing of considerable magnitude. They used a monolithic cavity (Figure 9.30). R. Paschotta et al. [5] measured −0.6 dB squeezing (−2.0 dB inferred) on the SH beam. This system showed great reliability (Figure 9.31). A quantitative analysis of the fluctuations can be obtained from a description ̂ in at frequency of the cavity using quantum operators. The input mode A 𝜈L enters the cavity through a mirror with decay constant 𝜅f 1 . It builds up an intra-cavity mode â f . A mode â sh at frequency 2 𝜈L is generated via the second-order non-linearity of strength 𝜒 (2) and leaves the cavity via a mirror of decay constant 𝜅sh1 . The total decay of the internal fundamental mode is described by 𝜅f = (𝜅f 1 + 𝜅fl ), which includes losses due to scattering, whilst that of the SH cavity is 𝜅sh = (𝜅sh1 + 𝜅shl ). A corresponding amount of vacuum noise is coupled in to each cavity. The quantum Langevin equations describing the

9.4 Early Demonstrations of Squeezed Light

–22

Noise power (dBm)

–24

Signal Shot noise

–26

–28

–30 0

10

20

30

Noise frequency (MHz)

Figure 9.32 Squeezing spectrum obtained by R. Paschotta et al. Source: From R. Paschotta et al. [5].

generation of the SH field are given by √ dâ f √ ̂fl ̂ in + 2𝜅fl 𝜹A = −𝜅f â f − 2𝜒 (2) â †f â sh + 2𝜅f 1 A dt √ √ dâ sh ̂ sh + 2𝜅shl 𝜹A ̂ shl (9.27) = −𝜅sh â sh + 𝜒 (2) â 2f + 2𝜅sh1 𝜹A dt ̂ i are all vacuum fields. Equations like these were used by M. Collett where the 𝜹A and later R. Paschotta and co-workers [5, 19]. They concluded that the squeezing was best in the limit of low detection frequencies (Ω → 0) and that the variance of the SH beam could ideally reach the value of V 1SH = 0.11 or −9.5 dB. The squeezing extended up to frequencies of the order of the linewidth of the cavity. This model assumed that the laser was driven by a coherent state. However, in the experiment, the laser noise swamped the noise suppression at very low frequencies. The best noise suppression observed (Figure 9.32) was only V 1SH = 0.8, or −1.0 dB, at a frequency of 16 MHz. The full spectrum can be calculated using the formalism of linearization. It was shown that the spectrum of the SH light (assuming low scattering losses) is given by a noise transfer function [6]: V 1SH (Ω) = 1 −

8𝜅nl2 − 8𝜅nl 𝜅f (V 1las − 1) (3𝜅nl + 𝜅f )2 + (2πΩ)2

2

where 𝜅nl =

2𝜒 (2) |𝛼f |2 𝜅sh (9.28)

̂ in , the driving field where 𝛼f = ⟨̂af ⟩ and V 1las is the amplitude noise spectrum of A of the laser at the fundamental. At low frequencies the large excess noise of the laser V 1las , derived in Chapter 6, dominates the measurement. An optimum detection frequency exists. The noise that masks the squeezing is not a technical imperfection, but it is intrinsic to the laser. It is possible to reduce the laser noise by placing a mode cleaner cavity between the laser and the squeezing cavity. This cavity will reflect the high

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9 Squeezed Light

Piezo A

PID

Mode cleaner (optional)

±

Lens

Mixer

Dichroic

Photodiode

Signal generator

Power monitor

λ 2

λ 4

Adder/subtracter

Monolithic frequency doubler

λ Faraday 2 rotator

Transmitted 1064 nm detector

Laser Slow control

Fast control

PBS

PBS PBS Reflected pump beam from doubler

28 MHz

45 MHz

BS

BS Reflected 1064 nm self-homodyne detector Highpass filter

Combiner

Laser controller

A

±

532 nm self-homodyne detector

±

PID

Figure 9.33 Schematic layout of the improved SHG experiment with mode cleaner. Source: After A.G. White et al. [20].

frequency noise for frequencies larger than its linewidth. The system can now be modelled by two sequential transfer functions, one for each cavity. This theoretical model was tested by A.G. White et al. [20]. The experimental layout shown in Figure 9.33 is similar to that used by R. Paschotta (see Figure 9.31). The light source is a diode laser-pumped monolithic Nd:YAG laser. The non-linear material is MgO-doped LiNbO3 . The crystal is phase-matched by tuning the temperature. The operating temperature is at about 110 ∘ C and has to be optimized to within ±2 mK. The SH light is separated by a dichroic mirror and detected by a balanced detector. The detectors receive optical powers of up to 15 mW each. The efficiency of the photodetectors (EG&G FND100) is low (𝜂det = 0.55). In a repeat of the experiment, a mode cleaner cavity, locked to the laser frequency, is placed between the laser and the crystal. The experimental noise spectra are shown in Figure 9.34 for the frequency range 2–34 MHz. The QNL was determined by measuring the difference of the currents from the two detectors. The electronic noise, typically −6 dB, has been subtracted. The experimental trace (a) shows the noise suppression for the monolithic cavity. The optimum of −0.6 dB is measured at 13 MHz. A theoretical prediction, trace (b), based on Eq. (9.28) and using measured values for the non-linearity, losses, intensities, and cavity reflectivities agrees very well. In a second experiment the mode cleaner cavity is inserted. As shown in trace (c), the high frequency noise from the laser (2–15 MHz) is substantially rejected. The best observed squeezing is increased to −1.6 dB at 10 MHz (3 dB inferred). Trace (d) shows the theoretical prediction obtained by including the transfer function for the mode cleaner cavity, which means replacing V 1las in Eq. (9.28) with the output variance from the cavity. The good agreement shows that all

10

8.36

7.5

6.03

5

3.81

(a)

1.78

2.5 (b) Quantum noise limit

0 (d)

–2.5

–1.45

(c)

(e)

0

Normalized noise power (dB) (measured)

Normalized noise power (dB) (inferred)

9.4 Early Demonstrations of Squeezed Light

–2.55

–5 2

7

12

17

22

27

32

Frequency, f (MHz)

Figure 9.34 Comparison between theory and experiment for the squeezing generated by second harmonic generation as shown in Figure 9.33. The linearized model describes the complete system in a modular approach through the use of transfer functions. (a, b) No cavity. (c, d) With mode cleaner cavity. (e) Model with no laser noise. Source: After A.G. White et al. [20].

effects are included in the combined transfer function. How can these results be improved? Can we get closer to the limit of V 1SH = 1∕9? A systematic study has shown that the main obstacles are the limited ratio of non-linearity to absorption in available non-linear materials and competing non-linearities [21]. One appealing, and technically simple, alternative to the external cavity frequency doubler would be to incorporate the crystal into the laser cavity. Such internal, or active, systems show very high conversion efficiencies for SHG. Early theoretical studies predicted large squeezing in these systems, but a careful analysis by White et al. [20], including the intensity and dephasing noise of the laser, shows that in practice no squeezing can be expected. This prediction applies to a wide a range of lasers, and thus active doubling is not worth considering for experiments. 9.4.4

The Kerr Effect

A third process for squeezing that had been proposed very early is the Kerr effect. It was favoured by the group of M.D. Levenson and R.M. Shelby at IBM Almaden Research Laboratories. It led to some of the earliest published squeezing results and was later extended by H.A. Haus and K. Bergman at MIT to the pulsed regime where it achieved impressive −3.5 dB noise reduction. The responses of the Kerr medium was briefly discussed in Section 9.1. Here we analyse it in detail. 9.4.4.1

The Response of the Kerr Medium

The classical description of light propagating through a non-linear Kerr medium has already been given in Eqs. (9.1) and (9.6). The fluctuations of the quadrature

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phase of the output field can be described by quadrature operators, either in the time or as here the frequency domain: ̃ in ̃ out = X𝟏 X𝟏 2πn0 n2 L𝛼 2 (9.29) 𝜆 This output field is detected using a homodyne detector by beating the output signal against a local oscillator. The detected quadrature is a linear superposition of 𝛿X1 and 𝛿X2 given by ̃ in + 2rKerr X𝟏 ̃ out = X𝟐 ̃ in X𝟐

with rKerr =

̃ ̃ cos(𝜃) + X𝟐 ̃ sin(𝜃) X(𝜃) = X𝟏

(9.30)

The variance is a measure of the noise in the quadrature and is calculated as shown in Section 4.1.1. It depends on the Kerr parameter rKerr and is normalized to the variance of a coherent state. Note that the amplitude remains unchanged and that for 𝜃 = 0, the amplitude quadrature variance is always equal to the QNL. This particular projection is independent of the non-linearity: 2 sin2 (𝜃) V (𝜃, rKerr ) = 1 + 2rKerr sin(2𝜃) + 4rKerr

(9.31)

where V (𝜃 = 0, rKerr ) = 1 is the QNL, and for V (𝜃, rKerr ) < 1 we have achieved squeezing. The angle 𝜃s for the best squeezing, the minimum of V (𝜃, rKerr ), can be found by evaluating d∕d𝜃(V (𝜃, rKerr )) = 0, which leads to ) ( 1 −1 (9.32) cot(2𝜃s ) = −rKerr or 𝜃s = arctan 2 rKerr In the limit of a small non-linear parameter (rKerr → 0), the best squeezing is observed for 𝜃s = −0.25π. For large squeezing parameters the angle approaches asymptotically the value −0.5π. Figure 9.35 shows an example of the normalized variance V (𝜃, rKerr ) for three values of the non-linearity, namely, rKerr = 0.25, 0.5, 0.75, and 1.0. It can be seen that the noise suppression is improving with larger non-linearity and that the angle of best squeezing is rotated. The noise suppression increases steadily with the non-linearity, and the optimum squeezing is given by √ 2 2 V (𝜃s ) = 1 − 2 rKerr 1 + rKerr + 2 rKerr (9.33) 4

1

3

rKerr = 0.5

2

rKerr = 0.25

rKerr = 0.75

0.8 V(θs)

V(θ)

344

1 0 (a)

0.6 0.4 0.2

0

π/4

π/2 θ (rad)

3π/4

0

π

(b)

0

0.5

1

1.5

2

rKerr

Figure 9.35 (a) V(𝜃) for three values of rKerr . (b) Dependence of optimum value V(𝜃) on rKerr .

9.4 Early Demonstrations of Squeezed Light

The calculation did not assume any specific detection frequency Ω. This is correct for a situation where the non-linearity does not depend on Ω. In practice, this applies only to the case of a travelling wave through a non-resonant Kerr medium. The two different cases of (i) a broadband Kerr medium inside a cavity and (ii) of a resonant Kerr medium, such as atoms close to an absorption line, will require a description of n(𝛼 2 ) as a function of Ω and have to be evaluated for each detection frequency individually. It is useful to consider the Kerr effect in view of quantum noise sidebands. At the core of the above calculation is the correlation between the quadratures stated in Eq. (9.29). This correlation enforces a link between the noise sidebands. Without such a correlation 𝛿EΩ and 𝛿E−Ω would be completely random. There would be independent contributions from 𝛿EΩ and 𝛿E−Ω to both 𝛿X1 and 𝛿X2. The correlation requires that for each component of 𝛿EΩ , there is also an additional component −2 rKerr 𝛿EΩ at the other sideband, and vice versa. The sidebands at the output are larger than at the input, as can be seen in Figure 9.4. In detection the correlation leads to a reduction of the noise in one quadrature. 9.4.4.2

Optimizing the Kerr Effect

The Kerr effect has been used in both CW and pulsed systems. It was actually one of the first successful demonstrations of squeezing. The main practical problem is to achieve a large Kerr parameter rKerr , and this requires the largest possible values for the non-linear coefficient 𝜒 (3) , interaction length L, and pump intensity I. There are four strategies for obtaining a large value for rKerr : (i) Optical fibre. The coefficient for the non-linearity is small, but the length of the medium and the intensity can be increased by using several hundred metres of single-mode fibre. Values for rs in the range 0.1–0.5 have been obtained for the case of a CW squeezing and in the range 0.5–2.0 have been achieved using pulses (see Section 9.5). (ii) Non-resonant medium inside a cavity. The same material (quartz) as used in the optical fibre can be used in a high finesse resonator. Whilst the physical length is short (typically 10 mm), the effective length and the high intra-cavity intensity in a high finesse cavity could compensate this shortcoming. No results have yet been published. (iii) Resonant media. The non-linear coefficient 𝜒 (3) can be enhanced several orders of magnitudes by operating close to an atomic resonance. Every absorption line shows a strong dispersive effect, which means a strong linear refractive index. The spectral lineshape of the refractive index is affected by the driving field. The transition can be saturated and power broadened, which in turn provides a non-linear refractive index. In the experiments this effect is again enhanced by a cavity around the medium. (iv) The use of cascaded 𝜒 (2) processes. It is possible to create a process that has the properties of the Kerr effect by combining two 𝜒 (2) processes. It was shown [22] that the simultaneous occurrence of OPO and SHG is equivalent to one single process. Provided the system is degenerate, the excitation of the pump and down-converted mode will be coupled in a way that resembles a Kerr effect for the pump mode. This was demonstrated in single-pass experiments [23] and for the CW case with a non-linear crystal in a cavity [24].

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These strategies suggest that large Kerr parameters should be achievable. However, when we analyse the experiments in detail, we will find that the reality of squeezing experiments is not that simple. All four approaches have severe limitations since, in all experiments, there are extra sources of noise and competing non-linear effects that will degrade the squeezing. In the actual experiments special tricks are used to suppress the extraneous noise, and a maximum of the non-linearity is traded off against a minimum of degradation due to competing effects. 9.4.4.3

Fibre Kerr Squeezing

Fibre-optic quantum noise reduction was eventually developed to reliable and strong photon number noise reduction through three following generations of experiments [25]. After the pioneering experiments with CW laser light described here, the next generations used ultrashort pulses to achieve a larger non-linear phase shift with less Brillouin scattering noise in shorter fibres. With short pulses the distortion of pulse shape and phase becomes an issue in coherent detection schemes. The elegant solution is to employ solitons that propagate free of chirp. For solitons, the local oscillator phase can easily be optimized for the entire pulse in the phase-sensitive detection of quadrature amplitude squeezing. No phase-matched local oscillator was needed in the two latest generations of experiments in fibre-optic Kerr squeezing. Using the methods of spectral filtering and asymmetric non-linear interferometers with picosecond or sub-picosecond solitons, directly detectable photon number squeezing was produced. These experiments are described in Section 9.5.4 on soliton squeezing. A comprehensive summary of fibre-optic Kerr squeezing experiments is given in the tables of Ref. [25]. One of the pioneering experiments was carried out by M.D. Levenson, R.M. Shelby, and their collaborators at IBM in 1985 [13]. The idea was very straightforward: use a quartz fibre with a Kerr coefficient to turn a coherent state, generated by a krypton laser, into quadrature squeezed light. The experimental arrangement is shown in Figure 9.36. The detection system used the scheme of a detuned cavity that provides attenuation for the light without destroying the squeezing and provides a variable projection of the uncertainty area as a function of detuning. The details of this scheme are described in Section 9.3.3. Phase-dependent noise was readily observed. However, the initial squeezing results were rather disappointing. The apparatus appeared to be dominated by classical noise, frequently significantly larger than the QNL. The reason is that optical fibres show Brillouin scattering, which is driven by the thermal excitations of the solid-state material. This drives small fluctuations in the refractive index. Inside the fibre, which has a well-defined refractive index guiding geometry, acoustic waves can be set up that have a complex spectrum of resonances. A consequence of these effects is a broadband spectrum of phase noise, called guided acoustic wave Brillouin scattering (GAWBS) (Section 5.4.7). This adds to the noise in the squeezing quadrature and can mask all squeezing. The GAWBS noise can be partially suppressed by cooling the fibre; thus the experiments by Levenson and co-workers

9.4 Early Demonstrations of Squeezed Light

DC

DVM

AC

100 Ω Laser

Spectrum analyser

D

Lens Faraday rotation isolation Phase modulator ωM

Amp

Cavity ωM

LHe Fibre

Figure 9.36 The schematic layout of the pioneering Kerr squeezing experiment in an optical fibre at IBM. Source: After [26].

were performed at 4 K using liquid helium-cooled fibres [26]. The GAWBS effect also varies with the materials and the geometry used in the fibre. Improved materials have more recently reduced the effects of GAWBS [27]. It is possible to suppress many of these phase fluctuations by cancelling the classical phase noise due to GAWBS in an interferometric arrangement. Here, the local oscillator and the squeezed beam both propagate through the fibre in opposite directions, and both beams experience the same classical linear phase fluctuations that then are cancelled in the detection scheme. Finally, there is stimulated Brillouin scattering in a fibre. This is a non-linear phenomenon with a threshold above which a significant fraction of light in the fibre is backscattered. The transmitted light is extremely noisy, thereby limiting the intensity that can be used in a fibre experiment. This effect can be partially overcome using not a single-mode but a multimode input, where each mode is squeezed individually and has its own stimulated Brillouin threshold. The experiment has to be arranged such that all these squeezing features add up in phase. The results of the experiment are shown in Figure 9.37. The data clearly show squeezing at a squeezing angle of 28∘ . The best noise suppression observed is V 1(𝜃s ) = 0.85 (−0.7 dB), which corresponds to an inferred squeezing of V 1(𝜃s ) = 0.7 for the light emitted by the fibre. What are the alternatives? For the non-resonant medium in a cavity, the main obstacle is the problem associated with achieving a sufficiently high finesse and the effects of losses in the cavity. The cavity will have to be locked actively to the laser frequency using the techniques described in Section 8.4. Squeezing can be achieved only at sidebands around each cavity resonance. That means the squeezing spectrum will show maxima at multiples of the FSR of the cavity. Any dispersion effects have to be carefully avoided. Otherwise the FSR will change with the optical frequency, and the sidebands at plus and minus one FSR cannot be brought into resonance simultaneously.

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Figure 9.37 Early squeezing results from the fibre Kerr squeezing experiment at IBM. Source: After [26].

2.5 Normalized noise level V

348

2.0

1.5

Standard quantum limit = 1.0 –100

–50

0

50

100

Phase shift θ (°)

How high a finesse is required? In the fibre the interaction length is typically 100 m, the input intensity is 100 mW, and the waist size is 3 μm. In a resonator with the same input power, with a waist size of 10 μm and a material length of 10 mm, a finesse of about 100 000 would be required to achieve the same parameter rKerr for the interaction. For a single-port cavity, this requires a total loss of about 10−5 in optical power, including all absorption and scatter losses. Such low values require extremely pure and homogeneous materials that are not yet available. 9.4.4.4

Atomic Kerr Squeezing

The use of a resonant medium relies on the ability to tune the input light, which is to be squeezed, close to resonance with an absorption line of the medium. The non-linear refractive index is enhanced. However, for these small detunings, the spontaneous emission becomes an extraneous source of noise, as with 4WM (Section 9.4.1). The light absorbed and re-emitted by the atoms produces intensity fluctuations that are detected alongside the fluctuations due to the quantum noise of the input beam. This produces uncorrelated noise at the detection frequency and swamps the carefully arranged correlations between the sidebands that we want to observe. Therefore the squeezing experiment requires detuning larger than the atomic linewidth absorption. For thermal atoms this would mean the Doppler linewidth. To avoid this the experiments by A. Lambrecht et al.[28] were carried out in laser-cooled atoms. A calculation for the Kerr effect generated by a sample of cold atoms inside an atomic trap predicts significant noise suppression. In practice several complications arise: The high intensities inside the cavity lead to instabilities. The cavity with a non-linear medium displays bistability and irregular switching. In addition, real atoms have many states due to hyperfine and Zeeman splitting – the prediction of a simple two-level model can frequently not be achieved, and optical pumping has to be taken into account. After overcoming all these difficulties, good noise suppression of more than −1.8 dB below the QNL

9.5 Pulsed Squeezing

has been observed [28]. These are difficult experiments, and despite an intensive optimization, such higher atomic densities in magneto-optical traps or even the proposal of using a Bose–Einstein condensate as the non-linear sample and much smaller detunings, the atoms are not a promising Kerr medium. The best results were probably achieved by the group led by P. Grangier and co-workers [29] who optimized the medium for the demonstration of the quantum non-demolition (QND) effect, as described in Chapter 11. 9.4.4.5

Atomic Polarization Self-Rotation

A related mechanism for squeezing is the rotation of the polarization of the light induced by an atomic vapour that occurs at high intensities. Interaction of the near-resonant atoms with elliptically polarized light can cause the initially isotropic 𝜒 (3) non-linear medium to display circular birefringence: the two orthogonal circularly polarized components of the light will propagate at different velocities. The result is a rotation of the linear polarization of a light beam. This in turn corresponds to a shear in phase space and the distortion of the initial symmetric Wigner function into an ellipse [30]. The effect was demonstrated in rubidium and the experiment can be very simple: an atomic vapour cell, heated to achieve the correct atomic density and shielded from stray magnetic fields, is illuminated by carefully controlled polarized light. The output beam is separated with a polarizing beamsplitter into a squeezed vacuum beam and a local oscillator beam that are combined, after rotation of the polarization of the local oscillator, in a standard homodyne detector. Whilst a simple theory predicts squeezing of up to −6 to −8 dB [31], the experiment achieved so far a squeezing of −0.85 dB [30]. The limitations are largely due to self-focussing of the vapour, resonance fluorescence, and phase mismatch. It remains to be seen if this tantalizingly simple experiment can achieve its full potential.

9.5 Pulsed Squeezing Some of the best squeezing results have been achieved using the very high intensities and non-linearities that can be achieved with optical pulses. 9.5.1

Quantum Noise of Optical Pulses

The concepts of quantum noise and squeezing can be extended to pulsed light sources. Consider a series of optical pulses, each pulse with a length tpulse repeated at time intervals trep ≫ tpulse , as shown in Figure 9.38a. The spectrum of the photocurrent will contain frequency peaks at the repetition frequency Ωrep and its higher harmonics. The width of each peak will depend on the length of the pulse train measured or the integration time of the detection. The distribution of power amongst, or the envelope of, the frequency peaks is set by the Fourier transform of the pulse shape of the individual pulses. This is schematically shown in Figure 9.38b. For the short pulses that can be generated with mode-locked lasers, (Ωrep typically 100 MHz, tpulse 0.2–100 ps), the envelope is very wide, but only

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tpulse

Intensity

trep

Time

(a) Intensity

350

(b)

Envelope due to pulse shape Quantum noise δΩ Ω rep

2 Ω rep

Frequency Ω

Figure 9.38 Optical pulses in the time domain (a) and the frequency domain (b). Quantum noise is present at frequencies between the fundamental components associated with the pulses.

a few of the fundamental peaks fit within the detection bandwidth. In between these peaks the spectrum contains a quantum noise floor. This is equivalent to the quantum noise spectrum of a CW laser. Thus we can call this quasi-CW . One can interpret this noise as being the sum of many pairs of quantum noise sidebands, each beating with one of the fundamental peaks in the optical spectrum. Any part of the recorded spectrum at frequency Ω < Ωrep represents the Fourier component of the variation in the energy of many pulses. For example, the pulsed source could be intensity modulated at a frequency Ωmod . This would create spectral sidebands to all fundamental peaks of the optical spectrum, which means signals at nΩrep ± Ωmod . The quantum noise is the noise floor in such a sideband. In these experiments noise represents the lack of reproducibility of the pulses. An ideal classical light source would produce a sequence of perfectly reproducible pulses, and thus no noise outside the fundamental components would be detected. Coherent light cannot have exactly identical pulses, but the energy will vary from pulse to pulse. We are comparing the variation of one group of pulses with that of the next group – and observing the noise floor. This quantum noise is independent of the detection frequency, as long as it is less than the repetition frequency, and a QNL pulsed laser will have a flat, white quantum noise spectrum between the fundamental peaks. The detection scheme for pulsed light is almost identical to that for CW laser light. The same photodiodes and SA are used. A single detector can reveal intensity fluctuations. A balanced detector will be able to distinguish between classical and quantum intensity noise, and a balanced homodyne detector can detect noise of specific quadratures. However, three additional requirements have to be satisfied in the detection of pulses: (i) The detector should not respond to fundamental frequency components and their harmonics; otherwise saturation of the detector and the electronic amplifiers will occur even at very low optical powers. This can be achieved by using sophisticated filter networks with notch filters suppressing any

9.5 Pulsed Squeezing

response at Ωrep . Such detectors require extensive development. The only exception is the detection at very low frequencies, Ω ≪ Ωrep , where the detector sensitivity can simply be rolled off with low-pass filters. (ii) The pulses have to be carefully aligned in the time domain. For example, the pulses used as the signal and local oscillator beams in the homodyne detector have to arrive simultaneously, with a time difference of much less than the pulse length. This requires an accurate adjustment of all distances in the experiment. Any cavity would have to have round-trip times that are an integer of trep . (iii) The pulses contain very high intensities. Whilst this is an advantage in that the non-linear response is significantly enhanced, it can introduce at the same time other non-linear processes, such as self-focussing, which limit the beam quality and thus the detectability of the light. The fundamental advantages of pulsed experiments are the high intensities and consequently the very large non-linear effects that can be observed. The intensity during a pulse makes the build-up cavities, which are necessary in CW experiments, obsolete. Single-pass SHG or OPO operation is possible. Similarly, the Kerr effect in single pass is sufficiently strong to introduce a correlation between noise sidebands and therefore induce squeezing. However, there are some effects that limit noise suppression. The squeezing occurs only during the pulse if we assume that the response of the material is instantaneous. When calculating the degree and the quadrature of the measurable squeezing, it is necessary to take the time development of the squeezing into account and to average over the temporal development of the noise properties and the squeezing during the duration of a pulse. Figure 9.39 shows an example where the degree of squeezing and the orientation vary during the pulse. The measurement averages the noise contributions, which are weighted by the intensity at the detector at any one time. Although the squeezing at the peak intensity gets the highest weighting, the observed total squeezing will be considerably less than predicted for the peak intensity alone. In addition the squeezing angle is again a weighted average, and squeezing in different quadratures can partially cancel each other. Figure 9.39 The development of the squeezing during a light pulse, indicated by the change of the squeezing ellipse. Source: From H.A. Haus et al. [32]. Time

Squeezed axis

351

9 Squeezed Light

Input 85/15 coupler

3 dB coupler

Squeezed vacuum

Local oscillator

Balanced detector –

Fibre (non-linear)

Output

Stabilization circuit (a)

15 10 Relative noise level (dBm)

352

5 Shot noise level

0 –5 –10 –15 –20 0

(b)

1

2

3

4

5

6

Pulse non-linear phase (rad)

Figure 9.40 The pulsed Kerr experiment. Source: After H.A. Haus et al. [32]. (a) The schematic layout. (b) Noise suppression results from the pulsed Kerr experiment.

9.5.2

Pulsed Squeezing Experiments with Kerr Media

A very successful series of squeezing experiments with optical pulses was carried out by the MIT group [32, 33] Figure 9.40b. Here the optical medium is a single-mode fibre, and the non-linear process is the Kerr effect. The light source is a mode-locked Nd:YLF laser generating 100 ps pulse with a repetition frequency of 100 MHz, Ωrep = 10 ns. The wavelength is 1.3 μm, which is the zero-dispersion wavelength of the fibre and assures that the pulses maintain their pulse shape during the experiment. The experiment is shown schematically in Figure 9.40a. The experimental arrangement is equivalent to a non-linear Sagnac interferometer. The pulses from the laser enter the interferometer via a 3 dB fibre coupler, equivalent to a 50/50 beamsplitter. There they are separated into two pulses of

9.5 Pulsed Squeezing

Time (s) Noise with swept mirror

Figure 9.41 The temporal trace of the noise from the pulsed experiment. Detection frequency is fixed at Ω = 55 kHz. Source: After K. Bergman and H.A. Haus [33].

identical intensity, which move in opposite directions through the fibre. After a full traverse, the pulses arrive simultaneously back at the coupler. They interfere and form a pulse that is moving back towards the laser. A pulse of squeezed vacuum light leaves through the other port of the interferometer. This pulse is detected with a homodyne detector. A fraction of the reflected light is used as a local oscillator, generated by the 85/15 coupler. The arrival time of the local oscillator has to be synchronized with the pulse of squeezed light. The phase of the local oscillator is set and stabilized with a piezo controller. The effect is seen in different experiments in two frequency bands: at low frequencies (80–100 kHz) and at high frequencies (≥ 100 MHz). The Sagnac interferometer is essential for such measurements. All fluctuations induced by the fibre, such as phase shifts and intensity variations due to mechanical vibrations, are experienced by both counter-propagating pulses. They are cancelled when the pulses interfere after one round trip and do not appear on the output. This also applies to phase noise induced by the fibre, such as GAWBS. Equally, noise in the laser light frequency or intensity is compensated by the interferometer. In this particular example the suppression was so good that all noise with the exception of large relaxation oscillations were cancelled. What remains is the correlation between the quadratures, induced by the Kerr effect. The squeezing in this case is so strong that they did not even need a SA, but it was directly detectable with an oscilloscope in the time traces of the laser intensity. Figure 9.41 shows alternating noise traces with the squeezing process turned on and off. The difference between these traces gives directly the amount of squeezing. A noise suppression of −3.5 dB was achieved at frequencies as low as 55 kHz. 9.5.3

Pulsed SHG and OPO Experiments

The high peak intensities of short laser pulses allow the realization of direct single-pass travelling-wave second harmonic generation (TWSHG) and the

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associated squeezing. Consider that in a single pass through a crystal of length , a beam of input power P(0) generates a beam of power PSHG () = ΓP(0). We call Γ the single-pass conversion efficiency. At the same time the beam will experience an effective non-linear phase shift 𝜙eff between the SH light generated at position z =  and SH light generated at position z = 0 and propagating to z = . Best SHG is achieved with complete phase matching, or 𝜙eff = 0. It was shown by the group of P. Kumar [22] that in this case the predicted noise suppression at low frequencies is S(Ω → 0) = 1 − Γ

(9.34)

This indicates the possibility of almost perfect squeezing provided that most of the light is converted into the SH. Even with a moderate amount of phase mismatch, the noise suppression should be very strong, and detailed calculations predict squeezing as large as 10 dB [22]. However, the corresponding experiments were so far not successful, but a noise suppression of only 0.3 dB has been observed [34]. This was achieved with a Na-doped potassium titanyl phosphate (KTP) crystal. An SHG efficiency of Γ = 0.15 was obtained at a peak intensity of about 250 MW/cm2 . The main technical challenge has been in isolating the squeezed from the fundamental light. Using a type II phase-matched crystal, it was possible to separate the SHG beam from the fundamental beam using polarization optics. But the achievable extinction ratio was too small. Excessive leakage of the input field into the detector led to saturation of the detection system and limited the observed noise suppression. This type of squeezing requires further technical developments. Far more successful has been the generation of squeezed vacuum state using a degenerate optical parametric amplifier (DOPA). The group of P. Kumar used a KTP crystal in single-pass operation. The fluctuations are measured with a homodyne detector. Special attention was given to the matching of the local oscillator beam to the squeezed beam in both spectral and temporal domains. Noise suppression of 5.8 dB was observed [35], matching the best results from the corresponding CW cavity experiment. 9.5.4

Soliton Squeezing

One of the requirements for quadrature squeezing in the pulsed experiments is to maintain the temporal shape of the pulse throughout the medium. This is commonly achieved by working at the zero-dispersion wavelength of an optical fibre. However, the material shows self-phase modulation that results in a chirp of the frequencies. Consequently a non-soliton pulse does not allow the local oscillator phase to be optimized for the pulse as a whole. This limited early pulsed fibre squeezing experiments where Gaussian zero-dispersion pulses were used. A clever alternative was proposed by P. Drummond and his group [36] and in parallel by the group of H.A. Haus [37]. A special pulse form can be used to form temporal solitons that preserve their shape by a balance between dispersion and non-linear refractive effects and propagate free of chirp. Solitons have almost particle-like properties, are robust against a variety of perturbations, and preserve their shape, energy, and momentum even after numerous interactions with

9.5 Pulsed Squeezing

each other [38]. The pulse shape is a hyperbolic secant. The pulse width tFWHM is dependent on the peak power Ppeak because shorter pulses experience stronger dispersive broadening and consequently require a higher peak power to balance dispersion. A fundamental soliton is obtained for a wide range of pulse energies and wide varieties of initial pulse shapes such as Gaussian or square pulses. After some initial self-reshaping, the exact balance of pulse width and peak power is eventually attained with a smooth soliton shape. 2 , It is convenient to introduce the pulse parameter N 2 = LD ∕LNL ∝ Ppeak tFWHM which indicates whether dispersive pulse broadening (N < 1) or non-linear pulse compression (N > 1) dominates the initial pulse evolution. The N parameter is the ratio of the length scale for dispersive pulse broadening, LD ∝ (tFWHM )2 , and the length per one radian non-linear phase shift in the pulse peak, LNL ∝ (Ppeak )−1 . These length scales are given by the dispersive and non-linear coefficients of the fibre. A fundamental soliton is also called a N = 1 soliton because it exactly balances dispersion with non-linear effects. The quantum mechanical description, using a non-linear Schrödinger equation, was able to describe the soliton’s noise properties and unique quantum soliton effects were found [37, 39]. The first such effect was amplitude quadrature squeezing, predicted by Carter et al. [40]. It was observed at IBM by M. Rosenbluh and R.M. Shelby [41], who used a colour centre laser to produce soliton pulses with a pulse length of 200 fs at 1500 nm propagating through a polarization-preserving single-mode fibre. The Kerr effect in the fibre generated the squeezing. Noise suppression of −1.67 dB was measured with a homodyne detector. A second quantum effect is the ‘entanglement’ of pairs of solitons. Their quadratures get linked, which can be exploited for QND experiments (Chapter 11) and quantum information experiments (Chapter 13). 9.5.5

Spectral Filtering

A third quantum effect is soliton photon number squeezing by spectral filtering, which was unexpectedly found by S.R. Friberg et al. [42] in 1996. The experimental method was fairly simple: a pulse launched into a fibre was spectrally bandpass filtered after emerging from the fibre end and was then directly detected – an unusual idea since filtering deliberately introduces losses to achieve squeezing. The effect can be heuristically explained by considering the propagation of solitons in the fibre. As they propagate, self-phase modulation and group velocity dispersion interact to cause a periodic broadening and shrinking of the soliton spectrum. For a fixed length of fibre the soliton energy, or photon number, determines the periodicity. When the soliton, after leaving the fibre, is attenuated by a fixed bandwidth spectral filter, the losses depend on the soliton’s pulse energy. The combination of propagation and filtering results in a non-linear input–output relationship, i.e. non-linear absorption. At specific lengths, or pulse energies, the intensity noise is suppressed below the QNL – photon number squeezing is possible. These simple ideas were confirmed by full quantum models [43]. The underlying squeezing mechanism can be best understood in terms of the multimode

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quantum structure of solitons. Spectrally resolved quantum noise measurements showed that anti-correlations in photon number fluctuations emerge within the spectrum of the soliton as it propagates down the fibre [44, 45]. If these anti-correlated spectral domains are selectively transmitted through a spectral filter and then detected simultaneously and coherently, their fluctuations cancel each other in the observed photon flux. This anti-correlation produces a photocurrent noise reduction far below shot noise. The pioneering experiment was carried out with 2.7 ps pulses with a repetition rate of 100 MHz centred at 1455 nm with a bandwidth of 1.25 nm. After propagating through 1.5 km of fibre, the pulses emerge with a pulse width of 1.3 ps and a spectral bandwidth of 1.65 nm. They were filtered with a variable bandpass filter to remove the high and low cut-off frequencies, resulting in the observation of −2.3 dB squeezing with N = 1.2 solitons [42]. The method was further investigated in Erlangen by A. Sizmann and G. Leuchs et al. with sub-picosecond solitons and a variety of filter functions, fibre lengths, and pulse energies below and above the fundamental soliton energy [45]. When optimized in the experiment, up to −3.8(±0.2) dB of photocurrent noise reduction below shot noise was achieved in direct detection. If corrected for linear losses, a reduction in the photon number uncertainty by −6.4 dB can be inferred [45]. This technique is very elegant and has significant practical advantages: It requires no coherent noise measurements and thus is insensitive to phase (GAWBS) noise. It is broadband and all that is required is a pulsed soliton source and suitable fibre. By using an in-line filter, this device can be made exceedingly lossless and robust. 9.5.6

Non-linear Interferometers

The most promising and most successful method of photon number noise reduction with solitons so far are highly asymmetric non-linear interferometers (Figure 9.42a). The CW [46] and soliton-pulse analysis [47–49] show that the beam splitting ratio in the interferometer must be highly asymmetric in order to produce strong photon number noise reduction, in contrast to earlier experiments with symmetric non-linear fibre interferometers that produced a squeezed vacuum. More than 10 dB of noise reduction was predicted. After the first experimental demonstration in Erlangen by S. Schmitt et al. using 130 fs solitons, the interferometer parameters were optimized [48], and the observation of less than −5 dB in the photocurrent noise was reported (Figure 9.42c). This and parallel work reported by K. Bergman and co-workers [50] are the strongest noise reductions observed for solitons to date. When corrected for linear losses, the inferred photon number squeezing was −7.3 dB [51]. A simple model shows a steplike input–output energy transfer function. Around each minimum the transmission is non-linear, the output energy is stabilized at a fixed value, and the photon number uncertainty is reduced below the shot noise limit. However, the squeezing mechanism can be better understood in terms of number–phase correlations induced by the Kerr non-linearity [52], similar to the CW case studied by M. Kitagawa and Y. Yamamoto [46]. A Kerr squeezed state from the high-power path of the interferometer is slightly rotated by the

9.5 Pulsed Squeezing

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Figure 9.42 Intensity squeezing of solitons with the non-linear interferometer. (a) Experimental arrangement. (b) The concept of projecting the squeezing ellipse after the Kerr effect. (c) Experimental results with solitons of different orders N. The top trace shows the transmission and the bottom trace the measured noise suppression. Source: From A. Sizmann and G. Leuchs [25].

addition of the weak field from the low-power path of the interferometer, such that the amplitude quadrature of the total output field is squeezed (Figure 9.42b). This squeezing mechanism implies that noise reduction grows with the non-linear phase shift in one arm of the non-linear interferometer [52]. This is in contrast to spectral filtering where noise reduction originates from multimode photon number correlations, which does not grow with propagation distance. Today the non-linear loop interferometer is seriously considered in technical applications for stabilizing communication systems operating with Terabaud

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information rates. It is also employed in basic research as an effective squeezing source for generating entangled beams in quantum information experiments (Chapter 13). To summarize, optical solitons in fibres have provided the perspectives of building fibre-integrated sources of noise reduction and given new insight into multimode quantum correlations with a terahertz bandwidth through spectrally resolved quantum measurements.

9.6 Amplitude Squeezed Light from Diode Lasers The simplest intensity squeezing experiment is to produce amplitude squeezed light directly with a laser of high efficiency that is driven by a sub-Poissonian current. The laser can be driven inside a high impedance circuit with a very quiet current (V idr ≪ 1). The conversion efficiency of the diode laser can be as high as 𝜂las = 0.5 − 0.75. An important distinction is the fact that a diode laser has an internal cavity to build up the radiation. The laser also involves both spontaneous and stimulated emission. Consequently, it has laser dynamics and spontaneous emission noise. The earliest models for diode lasers were developed by Y. Yamamoto and co-worker [53]. More recently linearized laser models have been developed [54]. It is possible to apply them to the diode laser system (see Chapter 6) and to calculate directly the transfer function between the fluctuations of the pump source (idr ) and the light. As Eq. (6.17) shows, the noise of the laser intensity is completely dictated by the pump noise. For an electrically pumped laser, the efficiency of pumping is close to unity, and, in diode lasers, the output coupling is large. Thus the efficiency 𝜂las is basically given by 𝜅m ∕𝜅. The variances V 1 of the amplitude quadrature and VN of the intensity are equivalent, and it is common practice to use the Fano factor for this situation. With these assumptions, Eq. (6.17) turns into (1 − V 1las ) = 𝜂las (1 − V idr )

(9.35)

This is the same result as for any other passive device. The maximum noise reduction for a perfectly quiet drive current is V 1las = 1 − 𝜂las . Two further important assumptions have been made in this model, namely, that the laser emits all light into a single output mode and that all the current entering the laser is contributing to the lasing process. Bias currents have been neglected. In practice, however, both these assumptions are not easy to justify. An experiment by W.H. Richardson and Y. Yamamoto [55] showed a noise suppression of 6.5 dB. This is a most impressive result that was actually better than predicted by Eq. (9.35) and the overall efficiency of the laser used. No definite explanation has been given to date for this result. However, it is possible that the diode laser had significant internal parallel currents that did not participate in the lasing process. Thus the actual laser efficiency 𝜂las might have been better than measured directly. Alternatively, some unknown process might have correlated the drive current with the light. Unfortunately, later experiments were not able to reproduce these early measurements, but they showed less noise reduction and tended to confirm the prediction of Eq. (9.35). In particular Inoue observed noise reduction up to −4.5 dB in diode lasers cooled to 20 K [56].

9.6 Amplitude Squeezed Light from Diode Lasers

It has been observed that diode lasers will excite more than one laser mode. These might not contribute much to the intensity of the output beam, but they can influence the noise behaviour dramatically. A particularly systematic study was carried out by the group in France [57]. They showed that a free-running diode laser was emitting light into over 50 individual modes. The total intensity was found to be sub-Poissonian. However, by measuring only some of the modes, rejecting the others with a spectrograph, the noise increased drastically. The noise of the single central mode, which carried the vast majority of the power, was up to 40 dB above the Poissonian level. This suggests that the various modes have anti-correlated noise and the noise terms cancel each other when all modes are detected simultaneously. This behaviour can be predicted by a simple multimode laser model that includes the competition between the laser modes [57]. Only some of the modes will be operating well above threshold, whilst others will remain at threshold and only contribute fluctuations. The competition incorporates a limitation of the total gain based on the available number of excited atoms. The division between the modes is not stochastic but balances the fluctuations, similar to a conservation of the total number of available photons. The result is anti-correlation between the modes. More detailed theoretical models have been published to take mode competition properly into account [58], but further study is still required. It is possible to enforce single-mode operation, and simultaneous noise suppression, by either injection locking or optical feedback from an external cavity. When these techniques are optimized, the laser system will operate on a truly single-mode output and generate amplitude squeezed light. M.J. Freeman et al. [59] used injection locking, and H. Wang et al. [60] used external cavities to achieve single-mode operation. They achieved a noise suppression of 1.8 dB with lasers at room temperature. More recently, D.C. Kilper et al. observed a noise reduction by 4.5 dB in an injection-locked laser at 15 K. On the whole, these results fit well to Eq. (9.35). An overview of all these results is given in Figure 9.46. However, it appears that not all diode lasers show reproducibly the same degree of noise suppression. The group led by Y. Yamamoto has succeeded for the first time in designing and manufacturing diode lasers that operate in single mode due to their geometry and that reliably operate at −2.8 dB below the Poissonian limit [61]. In contrast, most commercial systems do not perform as well as earlier prototype versions of the same laser design. Large variations in the noise performance have been found even within one commercial series of lasers with otherwise identical specifications. The detailed reasons for this variation can be many. Finally, one experimental problem is that the maximum efficiency of the lasers occurs at the highest drive currents, well above threshold, where the output powers are large. For a measurement of the noise suppression, all the light has to be detected, unless complex structures are used to selectively attenuate the power but not the squeezed sidebands. Thus the beam cannot be attenuated or otherwise the noise suppression is lost. The limit of detectable power is set by the saturation properties of the detector; presently only a maximum of 50 mW can typically be detected with fast detectors capable of measuring fluctuations at radio frequencies (RF) (≥ 10 MHz). This means that some of the best noise suppression cannot actually be detected.

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In summary, the expectation of simple and reliable diode lasers as low cost sources of non-classical light has not yet come to fruition.

9.7 Quantum State Tomography So far we have concentrated only on the measurements of the variances of quantum noise. It has been shown that this is fully sufficient for a description of squeezed states, or to be more general, any states that can be described by linearizable models. Figure 9.43 shows a very clear example of the noise suppression that was available already in 1996 after optimization of the experimental parameters. This type of scan provides a good knowledge of the properties of the quantum state. We saw in Section 4.5 that these states have 2D Gaussian Wigner functions. For a quadrature squeezed state, with unknown squeezing angle, we have to measure the variance V (𝜃) for a whole range of angles, and from these data the minimum value V (𝜃s ) is determined. We actually quote only this one number, and the entire squeezed state is described in this way. At least as long as it is a minimum uncertainty state, otherwise a second number, V (𝜃s + 90), is required to describe the state completely. On the other hand, techniques have been developed to measure the Wigner function. This is of fundamental interest since some quasi probability distributions (QPDs) exhibit unusual features and it would be nice to show diagrams of completely reconstructed QPDs. The key idea to these techniques is to measure higher-order moments of the noise statistics. Rather than measuring only the variance for a certain projection angle 𝜃, the full noise trace is recorded (see Figure 9.45). This was described by U. Leonhardt and H. Paul [63] and further elaborated by U. Leonhardt [64] and others. The idea was first applied to pulsed –75 (a)

–80 Noise power (dBm)

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Figure 9.43 Experimental data for CW squeezing using a monolithic OPO: the typical scan of the variances indicating squeezing and anti-squeezing. For comparison a theoretical curve with a noise suppression of −7.1 dB is shown. Source: After P.K. Lam et al. [62].

9.7 Quantum State Tomography

light. M.G. Raymer’s group at Oregon [65] recorded histograms of the energy of many pulses and in this way the full noise characteristics of the pulses. Many such histograms were recorded for different angles 𝜃, and the underlying Wigner function can be reconstructed using tomographic techniques [65]. For CW experiments the process is similar. The full time history of the fluctuations is recorded. This can be achieved by directly measuring the noise fluctuations of the photocurrent and demodulating the signal with a fixed single detection frequency. Alternatively an electronic SA is used, as shown in Figure 9.45a. By leaving the video bandwidth sufficiently large, the full noise trace is recorded. For a squeezed state these traces vary dramatically with the projection angle, as shown Figure 9.45b. The many traces are then transformed into histograms of the fluctuations for this specific projection angle and detection frequency Ω. The restriction to specific values of Ω is particularly important for those squeezing sources that have a distinct squeezing spectrum S(Ω). Here it is important to detect only those sidebands that are strongly squeezed. For each detection frequency a function W (Ω) can be reconstructed using the tomographic technique [63, 67]. Note that all the measured Wigner functions will be centred around zero and they show the fluctuations that are independent of the mean value of the intensity. The quadrature of the squeezing is only determined as a rotation angle, which is measured either with a homodyne detector or in the case of a bright squeezed source with large intensity, in comparison with the phase of this beam. Apart from a rotation of the Wigner function, the phase and amplitude squeezed states look the same. In addition, the shape of the Wigner function is frequency dependent and can vary considerably with Ω. The same light source can simultaneously be squeezed at one frequency and noisy at another. The complete beam is described by a whole spectrum of Wigner functions (Figure 9.44).

X2 X1

Figure 9.44 The concept of quantum state tomography: complete noise traces are recorded for different projection angles, and one such example is given here. These data can be recombined to reconstruct the Wigner function of the quantum state. Source: After U. Leonhardt [64].

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0.2 Local oscillator

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Figure 9.45 Quantum tomography experiment. Part (a) shows the schematic layout. Results for a squeezed vacuum state are given: (b) the phase-dependent noise, (c) the squeezed two-dimensional Wigner function, and (d) the reconstructed photon probability distribution. Source: From S. Schiller et al. [66].

How does this compare with the pulsed situation that seems to provide one unique answer for the Wigner function? In this case the selection of the detection frequencies is made automatically; they are given by the Fourier components of the pulses, the pulse repetition frequency, and its higher harmonics. Only these specific detection frequencies contribute. Only one combined answer, a weighted average of several single frequency Wigner functions, was recorded in the measurements by M.G. Raymer’s group [65]. Using their OPO, the Konstanz group demonstrated the capability of tomography [66] very early on. They investigated a squeezed vacuum, measured the noise projections, and converted them into Wigner functions and photon distributions. One example is given in Figure 9.45. Their results, the complete squeezed Wigner function (c) and the photon statistics (d) with enhanced probabilities for even photon numbers, are a beautiful demonstration of quantum optics. This is now a routine diagnostic technique, both for the quality of the light that is generated and for the quality of the apparatus that is processing the squeezed light and using it for a variety of applications. The comprehensive review by A.I. Lvovsky and M.G. Raymer [68] provides many details of the experiments and the data analysis. It is interesting to note that we can also apply tomography usefully in situations where the light is not squeezed but at the QNL or weakly squeezed and contains

9.8 State of the Art of CW Squeezing

some well-known modulation. After testing the light at several locations in the apparatus, a comparison of the tomographic results will show quite clearly the performance and losses imposed by the apparatus. In this way the quality and capability of the apparatus can be tested and verified. This would, for example, mean that in the case of commercial production of components that should be able to handle squeezed light or entangled beams, tomography is a useful tool for quality testing and assurance of components and without the need to use non-classical light. As long as we stay within the limit of Gaussian states, tomography should in principle not reveal more than the measurement of the two orthogonal quadratures. However, can we use tomography to show even more exotic results? This depends not so much on the measurement techniques but on the types of states that we can manufacture, and this has not been easy. As pointed out in Section 4.5, we would require a system that cannot be linearized and fluctuations that do not follow Gaussian statistics. One successful approach is to consider a state with a very small photon number, which is heralded by photon counting techniques. Alternatively photon number selective post-processing can be used to select the data for such a specific state of light imbedded in the optical beam. In these cases strongly non-classical Wigner functions have been recorded, as discussed in Chapter 13. An example is shown on the front cover of this book, where the non-Gaussian Wigner functions are made visible using a modified detection scheme, where the homodyne detection is conditional on a photon counting measurement. Alternatively we would require a CW system that is extremely non-linear, for example, close to an instability point, and where the coefficients for the non-linearities (e.g. 𝜒 (2) or 𝜒 (3) ) change within the uncertainty limit of the state. These conditions are extremely difficult to achieve since we do not have such non-linearities, and this remains a challenge for the future.

9.8 State of the Art of CW Squeezing Squeezing moved from the stage of a cute theoretical idea to the first experimental demonstration in a remarkably short time. From 1980 to 1985 the details of possible squeezing were explored theoretically; then within one year several squeezing experiments were performed more or less simultaneously. Since the first demonstration of squeezed states, we have seen a steady improvement in the size and the reliability of the noise suppression. During the first few years, several unexpected new noise sources were discovered that hampered the progress. These had to be avoided, in most cases through improvements of the materials used. Examples are as follows: the new fibres used by K. Bergman et al. had significantly less GAWBS noise than the earlier fibres used by M.D. Levenson et al., the early noise problems encountered by H.J. Kimble et al. with external cavity SHG were avoided in the more recent monolithic devices, and the limitation in squeezing with the OPO was set by noise created inside the crystal, which can now be avoided by lower-loss materials. Very quiet monolithic diode-pumped Nd:YAG lasers have replaced the much noisier lamp-pumped lasers used in the earlier

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Squeezing history 3 0 Noise supression (dB)

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Figure 9.46 Best observed noise suppression in different squeezing experiments in the time 1985–2016. The symbols describe different processes: Boxes = CW vacuum squeezing using a sub-threshold OPO, diamonds = CW bright squeezed light (SHG, atoms, Kerr), triangles = diode laser noise suppression, crosses = pulsed squeezing experiments (OPO, SHG, solitons, etc.), and circles = twin-photon beam experiments.

experiments. And the detectors have been improved with a quantum efficiency of more than 0.95 now available for the visible (Si) as well as the near-infrared spectrum (InGaAs). All this technology had matured by about 2000. Some processes did not show sufficient non-linearities to produce significant and reliable squeezing, including bistability [69] and atom–cavity interactions [70] and did not stand the test of time. New processes were found, such as the non-linear interferometer for pulsed squeezing. Now hundreds of different experiments have been reported, too many to list separately. Instead, Figure 9.46 shows the historical development of the experiments. The observed noise suppression is plotted for several groups of experiments producing squeezed light. As can be seen, the OPO generating a squeezed vacuum state was the first to show large squeezing and still remains the best process. At this point in time a suppression to −15 dB has reliably been achievable, albeit only by very few groups, since it requires an amazing attention to detail. If the long-term trend holds up, one could expect a suppression to −20 dB once all the components and materials are optimized. The CW experiments with atomic systems, both 4WM and strong cavity coupling, and with Kerr media were beautiful demonstrations of concepts and test of the theory. They were essential for our understanding but are less likely to play a role in applications. The singly resonant SHG was rapidly improving and is a simple and reliable source of bright amplitude squeezed light. Noise suppression down to −5 dB is a realistic expectation. These systems can be made very robust, small, and reliable, and with the use of periodically poled materials, we might see even simpler and better devices. For the pulsed experiments the results look equally impressive. Here the Kerr effect, soliton squeezing, and the non-linear interferometer all have the

9.9 Squeezing of Multiple Modes

potential to achieve a noise suppression to −10 dB. The potential of new fibres with engineered band-gap structures has not been explored. The only exception to this positive trend has been the diode lasers. Some outstanding results have been achieved very early on, but they could not be confirmed. The best of the present results is supporting the direct theoretical prediction, and further understanding and refinements of the manufacturing process could make −3 dB of noise suppression available in commercial low cost lasers. Squeezed vacuum states can be transformed into various types of squeezed states, the squeezing axis can be rotated, and coherent amplitude can be added through reflection off a cavity. We can control the squeezing independently of the coherent amplitude. In addition we can produce polarization squeezed light and spatially squeezed light. Fibre-based squeezing and diode lasers have the potential to be readily integrated into practical systems. In parallel the other main advance is reliability. In 1984 squeezing light was regarded as a ‘Hero’ experiment, which would normally work no earlier than 2 a.m. ‘Daylight squeezing’ during normal business hours became a norm by about 1996. Now we have systems squeezers that operate continuously for many days, ‘24/7’, for example, within a gravitational-wave observatory with both very high duty cycle and reliability. In addition we have single-pass squeezers as part of integrated optics that are directly built into custom-made devices. Technically this is all possible and commercialization is a clear option. The future use is limited only by our creativity for applications and the demand for such devices.

9.9 Squeezing of Multiple Modes 9.9.1

Twin-Photon Beams

Closely related to amplitude squeezed light is the generation of twin-photon beams. In this case two optical beams are generated by one above threshold OPO (Sections 9.4.2 and 6.3.3). The single input beam is converted, in a non-linear crystal inside a resonant cavity pumped above threshold, into two output beams that have identical statistical properties. In a simple picture one can imagine that each input photon generates two identical photons, one for each output beam, but unlike the parametric down-converter we have here not correlated pairs of photons but two single-mode beams with correlated photon flux. Using type II materials, which means the two modes have orthogonal polarizations (Section 9.4.2), the beams can easily be separated and detected individually. The OPO in most cases is non-degenerate, which means the optical frequencies of the beams are different but this has no consequences since they are not recombined optically. Theoretical calculations [71, 72] show that each beam individually has intensity fluctuations that depend on the dynamics of the OPO, as discussed in Section 6.3.3. In particular, close to threshold, each beam has very large fluctuations, V 1(Ω) ≫ 1, at four times the threshold the beams are at the QNL, and far above the threshold each beam should be squeezed, V 1(Ω) = 0.5. This has not yet been observed.

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However, if we detect the difference between photocurrents, these fluctuations cancel because of their strong correlations, and we can observe a noise level well below the QNL. The QNL in this case corresponds to the noise measured on the difference between the intensities of ‘classical twin beams’, obtained by splitting any light beam on a 50∕50 beamsplitter. It is equal to the shot noise level for the combined photocurrent. This experiment was first carried out with CW lasers by the Paris group, led by E. Giacobino and co-workers [73] who recorded a noise suppression of 27 % (V 1 = 0.73). Next came an experiment with mode-locked Q-switched lasers by O. Aytür and P. Kumar [74]. C.D. Nabors and R.M. Shelby [75] used a doubly resonant OPO to generate two beams of different colours and reported a noise suppression of −3.1 dB. Systematic improvements of their apparatus by the group in Paris led to observations of noise suppression by −8.5 dB, after correction for the dark currents, which is a record that stood for a long time. It was beaten by the Taiyuan group [76] with a noise reduction of −9.2 dB and finally improved again by the Paris group to −9.5 dB [77]. Since these experiments involve only intensity measurements, and no local oscillators, they avoid the complications of mode matching and beam overlap, and the data are limited purely by the quantum efficiency of the detectors and the escape efficiency. Note also that twin-photon beams are obtained at the two outputs of a 50∕50 beamsplitter, which has at its two input ports a squeezed vacuum state and a local oscillator of appropriate phase. This is actually what is done in any homodyne measurement of squeezing. It is possible to generate one beam that has sub-Poissonian statistics by measuring one beam and using this information to control, via electro-optic feedforward, the intensity of the other. This was demonstrated by J. Mertz et al. [78] who achieved a noise suppression of −4 dB of one beam compared to the QNL of this individual beam. This new output beam can be directly used for all the applications of sub-Poissonian light, which forms a direct alternative to the quiet diode lasers mentioned previously. Twin-photon beams can also be directly used to beat the QNL in the spectroscopy of very weak absorption lines by −1.9 dB [79] and the measurement of very weak absorptions by −7.0 dB [76]. Finally, the Paris group showed that twin beams offer the opportunity to introduce post-selection processes, commonly used in the photon counting regime, to the regime of continuous variables [77]. If the fluctuations of the two photocurrents at a fixed detection frequency Ωdet are recorded directly and stored, as it was described in the section on state tomography (Section 9.7), we can now produce a sequence of recordings, each containing short intervals of the fluctuations. Thanks to the quantum correlations, we can use one beam to select intervals where V 1(Ωdet ) is within a narrow, well-defined band and select from the other beam the corresponding intervals. The selected light describes a beam with unusual properties. For example, we could select information that describes a beam with strong sub-Poissonian distribution, which was demonstrated experimentally and produced an intensity noise reduction of −4.4 dB below the QNL. Alternatively the selection could be done with an electro-optical modulator. The resulting beam is now no longer continuous but intermittent, depending on the selection process that rejects the majority of the time intervals but has the advantage of representing a strongly non-classical quantum state.

9.9 Squeezing of Multiple Modes

9.9.2

Polarization Squeezing

The polarization of a beam of light is another property that is limited by quantum noise. In order to detect the polarization, we require a combination of two detectors. Each is producing quantum noise, and consequently one expects a QNL for the polarization. The polarization is well described by the Stokes parameters S0 , S1 , S2 , S3 (Section 3.5.2), which describe the light in terms of three degrees of freedom for linear and circular components and one normalization factor. The state of polarization can be described by a three-dimensional diagram, the 1 Poincaré sphere, defined by the radius S = (S12 + S22 + S32 ) 2 . Each point on the sphere is one polarization, and the radius of the sphere is given by the intensity of the beam. An analysis of the quantum properties of polarization shows that they obey the cyclical commutation relation [Ŝ 1 , Ŝ 2 ] = 2iŜ 3 ,

[Ŝ 2 , Ŝ 3 ] = 2iŜ 1 ,

[Ŝ 3 , Ŝ 1 ] = 2iŜ 2

(9.36)

This means that the polarization has an uncertainty volume and the polarization is only defined within a small sphere around each point on the Poincaré sphere, as shown in Figure 9.47A. The possibility of achieving polarization fluctuations below the QNL was investigated extensively by the group around A.S. Chirkin in Moscow. In particular N.V. Korolkova and A.P. Alodjants investigated the possibility of using amplitude squeezed light to influence the polarization [81, 82]. It was shown that two squeezed beams can be combined on a beamsplitter and by selecting the correct polarization and phase between the two beams, the output beam will have one or more of the Stokes variables measurable below the QNL [83]. The uncertainty volume is compressed and can take the shape of a cigar, with the two Stokes parameters Ŝ 𝟏 Ŝ 𝟑 suppressed at the same time, or alternatively like a pancake, with suppression of only one parameter, Ŝ 𝟐 . The first experimental demonstrations were made by the group at Bell labs [84] and the Aarhus group [85]. The ANU group, including R. Schnabel, W. Bowen, and P.K. Lam, achieved the goal to squeeze several Stokes parameters simultaneously and below the −3 dB limit. They generated two independent squeezed beams [80] with two seeded OPAs driven by one Nd:YAG laser and frequency doubler. The squeezed beams were combined on one beamsplitter. The apparatus is shown in Figure 9.48 as an example of the actual complexity of quantum optics experiments. The multitude of components, all individually aligned and optimized, is required to produce the pump, seed, and local oscillator beams, to produce the two squeezed beams, to combine them on a beamsplitter, and to detect the properties of the output state. The main challenges are to reduce the loss of all components, to optimize the mode match between all the beams, and to maintain the phase locking between all the cavities, such as lasers, mode cleaners, and OPOs. For this purpose this experiment contains 4 temperature control systems and 7 optical locking systems. By generating two bright squeezed beams with about 1 mW power and about −4 dB of squeezing each, they were able to achieve both the cigar and pancake states, with suppression of the Stokes parameters simultaneously by better than −3 dB at detection frequencies around 8 MHz. Figure 9.47B shows the experimental results in the form of reconstructed

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9 Squeezed Light

S3

(a)

Right-circularly (b) polarized

ˆ S 3

ˆ S 1

(A)

ˆ S 2

S2

S1 Horizontally polarized

45° polarized σS

(a) σ S3

σS

(c) σ S1

3

1

σS

σS

2

ˆ S //

3

σS

σS

1

σS

2

σS

(b) σ S1

1

σS

2

1

σS 2 σS

σS

3

2

Sˆ ⊥ b

Sˆ ⊥a

σS 3 σS

(e) σS 3

3

σS

σS

σS

(d) σ S3

3

σS

2

3

σS

2

2

σS

σS

1

σS

σS 1

2

1

σS

σS 1

2

(B)

Figure 9.47 (A) Diagram of (a) the classical and (b) quantum Stokes vectors mapped on a Poincaré sphere. The ball at the end of the vector visualizes the quantum noise of the polarization states. (B) Measured quantum polarization noise at 8.5 MHz from different combinations of input beams: (a) single coherent beam, (b) coherent beam and squeezed beam, (c) bright squeezed beams, (d) two phase squeezed beams, and (e) two amplitude squeezed beams. Source: After R. Schnabel et al. [80].

uncertainty areas on the Poincaré sphere. This type of light can now be used to couple the quantum properties of the light to the quantum properties of atoms, as discussed in Chapter 14. 9.9.3

Degenerate Multimode Squeezers

Following the lead of the polarization squeezer, it is possible to use the output of two squeezers within one experiment. We will see that, for example, teleportation requires two squeezed inputs, and for quantum gates and error correction, we might need up to eight squeezed inputs. This can be achieved by building several devices, each producing a separate beam, and these can be combined using a whole array of beamsplitters, resulting in several output beams that are detected individually. This technique has been developed into a fine art by the group of A. Furusawa at Tokyo University, and many more details can be found in his book [86].

9.9 Squeezing of Multiple Modes

2011 ANU double squeezer, entangler, tweezer

1989 IBM Kerr squeezer QND

Figure 9.48 Squeezing experiments are complex. They combine low loss optical systems and nonlinear devices, high efficiency detectors and high quality opto-mechanical control. They are operated in well controlled environments, built on optical tables with minimum mechanical noise, excellent thermal and dust control. These photos shows two examples: on the right had side an experiment at IBM in 1989 for the demonstration of squeezing and QND in optical fibres. On the left hand side an apparatus at ANU in 2011 with two OPO squeezers for entanglement, spatial mode squeezing and applications to tweezers with biological samples. The design principles remained, the complexity and performance of the devices had greatly improved. The future will see integrated opto-mechanical designs with low loss wave guides, custom manufactured materials and opto-mechanical systems providing reduced size, ruggedness and higher reliability. Source: After R.Schnabel et al. [80].

Alternatively, one could search for one device that can generate squeezing in several modes simultaneously. This is in principle possible and has been used for OPOs that operate simultaneously in both output directions and in this way produce two squeezed beams. This apparatus has considerably fewer components than the complex machine that has two separate squeezers, as shown in Figure 9.48. The goal of producing a device with many modes has for a long time been elusive. Early attempts of using degenerate OPOs in special cavity configurations to generate several spatial modes simultaneously were found to be too delicate. However, since 2010, there has been considerable progress, in particular with temporal modes, where more than 10 squeezed modes have been generated and detected by the Paris group in a femtosecond frequency comb [87] using one OPO and creating orthogonal modes where each is a different combination of a large set of individual frequencies in the comb. This is now being applied to several ideas from quantum information, communication, and sensing. At the same time the team in Virginia led by O. Pfister has produced and verified 60 modes of carefully engineered combinations of frequencies within an OPO with two non-linear crystals [88]. Each combination represents a whole entity, or mode, which combines the simple quantum states at individual frequencies of the

369

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9 Squeezed Light

optical field and their magnitudes and phases in such a way that the combined mode is part of an elaborate basis set. The apparatus is carefully designed and crafted to create and then let us test quantum correlations not only between two of these modes but a whole set of multimode correlations, following carefully set out rules for graph states. These states are the realization of complex rules designed to represent mathematical algorithms and the realization of quantum logic protocols.

9.10 Summary: Quantum Limits and Enhancement In summary, the tools for optical quantum engineering have now emerged. Squeezed light is now being used in a number of applications to improve the detection of light beyond the QNL associated with the photon statistics of the best classical light, with the details in Section 10.2. There are several other situations in which the apparatus is limited by the QNL, for example, attenuators and amplifiers. In all cases the QNL, which we experience with a coherent state of light produced by a perfect laser, represents the limit of the performance of the instrument. Fortunately there are ways in which these limits can be overcome through the use of quantum correlations of the optical fluctuations created by non-linear optical systems. Table 9.1 provides an overview of these situations. Table 9.1 Other effects that are related to squeezing. Limit of a coherent state

Quantum effect

QNL

Squeezing

V1 = V2 = 1

Vsq < 1, Vantisq > 1

Attenuation

Squeezing

Vout = 𝜂Vin + (1 − 𝜂)

V1out = 𝜂V1in

Amplification

Anti-squeezing

Vout = G Vin + 1

V2out = GV2in

Independent beams

Entangled beams

V1(A|B) V2(A|B) = 1

V1(A|B) V2(A|B) < 1

Measurement

Quantum non-demolition

Vout > Vin

V1out = V1in

Classical communication

Teleportation

V1out ≥ V1in + 2

V1in ≤ V1out < V1in + 2

V2out ≥ V2in + 2

V2in ≤ V2out < V2in + 2

Beam displacement

Quantum laser pointer

VΔx = Vdx 1

VΔx < Vdx 1

VΔ𝜃 = Vdx 2

VΔ𝜃 < Vdx 2

References

The first column states the quantum limits as set by the coherent state in terms of the variances of the fluctuations. One aim of quantum optics is to go beyond these limits. The second column of the table names the devices that use quantum correlations and states the limits that can be achieved. If only one squeezer is used, we see an improvement in one selected quadrature Vsq . By using two squeezed sources, we can arrange an improvement in both quadratures of the same mode, for example, in the case of entanglement between two beams and teleportation. The improvement applies to all aspects of the information that is encoded in these orthogonal quadratures. In the case of spatial measurements, the detection of a displacement or tilt in a transverse direction x is limited by the quantum noise in the quadratures of a separate orthogonal mode with quadratures Xdx 1 and Xdx 1. This mode contains all the information of the displacement and tilt, and squeezing this mode will yield improved measurements of this particular degree of freedom. The following chapter shows examples of the use of squeezed light for measurement applications. Presently the quality and reliability of the devices is being improved. After more creative engineering, the complexity and usefulness of the state generators will improve to a point where they are simply sources of a desired quantum correlation pattern. More and more complex tests are being devised and will be used for in-principle demonstrations and quality control. The results of today will appear to be just the beginning – and it is now time to consider real applications of muitlmode entanglement, based on squeezing of light, or an equivalent set of modes based on particles, on photons, or ions, or atoms, or other particle-like quantum effects. We shall now progress to see in Chapter 10 the effects such quantum correlations have on actual devices, real sensors, communication, and imaging and in Chapter 13 on quantum logic protocols and devices.

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66 Schiller, S., Breitenbach, G., Pereira, S.F. et al. (1996). Quantum statistics

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85 Soerernsen, J.L., Hald, J., and Polzik, E.S. (1998). Quantum noise of an atomic

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Further Reading Drummond, P. and Ficek, Z. (ed.) (2003). Quantum Squeezing. Springer. Furusawa, A. (2015). Quantum States of Light. Springer. ISBN: 978-4-431-55958-0. Schleich, W.P., Mayr, E., and Krähmer, D. (1997). Quantum Optics in Phase Space. Wiley-VCH. Walls, D.F. and Milburn, G. (1993 and 1997). Quantum Optics. Springer-Verlag.

377

10 Applications of Quantum Light Squeezed states and entangled pairs of photons allow us, in principle, to circumvent some of the quantum limits that apply to coherent states of light. Quantum measurements provide opportunities that classical systems do not have. This has practical applications and four areas stand out as the most likely applications: optical sensors benefit from squeezed light in gaining sensitivity. Optical communication benefits when information can be transmitted with non-classical light. Spatial effects and imaging as well as timing are worth discussing as multimode examples for quantum enhanced measurements. Finally, interferometric detectors for gravitational waves (GWs) are the most impressive example for the use of squeezed light to date. For all these cases we will discuss the standard quantum limits (SQLs) that apply to coherent states and ways to overcome them. A different class of applications based on quantum information concepts is discussed in Chapter 13.

10.1 Quantum Enhanced Sensors 10.1.1

Coherent Sensors and Sensitivity Scaling

An optical sensor is a device that monitors a property and translates a change of this property into an optical signal. The property under investigation could be a position, orientation, acceleration, temperature, pressure, concentration, chemical composition, light level, electric or magnetic field, or even a GW, to name the most common. The sensor could be a local device, sometimes with very high microscopic spatial resolution, or it could be a remote sensing device allowing measurements of a property at a distance. Frequently, the sensor is used to produce the data for a 2D or 3D spatial or temporal map of the property. In all these cases we want to measure the property with the best possible sensitivity, low noise, and big dynamic range. We may wish to monitor the effect continuously, and it is then normally an advantage to push the limit of sensitivity and signal-to-noise ratio (SNR). It is this goal where quantum optics makes a contribution (Figure 10.1). Technically speaking a coherent optical sensor can be regarded as a modulator (see Section 5.4.5), where the modulation of a beam of light is affected by just one external parameter; let us call it ℘. The concept is that a change in ℘ leads to a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Parameter ρ (x,y,z,t) changing in time and space

Parameter ρ (x,y,z,t) changing in time and space

DET Input beam optical power A in ^2(t)

(a)

One- or two-photon excitation

Output A out ^2(t)

DET

(b)

Figure 10.1 Coherent optical sensor (a) and incoherent sensor based on fluorescence (b).

change of the optical output power, or intensity for a fixed beam size, which can then can be detected. Given an input mode â (t), we can write the output mode as √ √ â o (t) = 𝜖(℘(t))̂a(t) + 1 − 𝜖(℘(t))̂v(t) (10.1) where the effective optical transmission, 𝜖, is a function of ℘ and v̂ (t) is the vacuum mode that inevitably couples for non-unit transmission. In a high sensitivity situation, we may typically want to estimate a small change in our parameter, 𝛿℘, away from a known value ℘o . Given that a small change in ℘ leads to a small change in the transmission, we can write ) ( √ K ′ 𝛿℘ â o (t) = â (t) 𝜖(℘o ) − √ 2 𝜖(℘o ) ( ) √ K ′ 𝛿℘ + v̂ (t) (10.2) 1 − 𝜖(℘o ) + √ 2 1 − 𝜖(℘o ) 𝜕𝜖 where K ′ = 𝜕℘ |℘o . We will discuss the ultimate limits of this type of sensor in Section 10.1.3. A related situation is where we want to use a sensor to continuously monitor small time-dependent changes 𝛿℘(t) of the parameter. The changes can be due to changes of ℘(x, y, z, t) in time at a fixed location or due to scanning the sensor in space or both. The changes in the photocurrent at the output iout (t) = â †o (t)̂ao (t) are seen as a proxy for the changes in ℘(t). The design of the sensor can vary widely – we can have very direct links, such as the measurement of concentration of a compound by absorption or a displacement measurement in an interferometer. However, we also have some quite complex sensors, for example, for the measurement of magnetic fields, temperature, or pressure. We also assume that any change 𝛿℘(t) can occur with any time dependence, which means there could be harmonic signals with fixed or variable frequencies, step functions, or bursts. Consequently we want to sample the data stream from the sensor at regular intervals, Δtsamp , fast enough to capture the dynamics of interest. On the other hand we assume that the underlying noise, from the input beam and from other sources such as the electronics, is of Gaussian nature.

10.1 Quantum Enhanced Sensors

As a concrete example let us take a simple case where our external parameter is a harmonic signal and there is a linear relationship between 𝜖 and the parameter, i.e. 𝜖 = K℘ = K(℘o + 𝛿℘o cos(2𝜋Ωsig t)), where K and 𝛿℘o ≪ ℘o are constants. ̃ = 𝛿℘o(𝛿(Ω + Ωsig ) + 𝛿(Ω − The Fourier transform of the fluctuating term is 𝛿℘ ′ Ωsig )) and K = K. We may also assume that the fluctuations and the signal are small compared to the average power such that the input mode can be writ̃ ̃ ̂ ten as â (t) = 𝛼 + 𝜹a(t) with |𝛼| ≫ 1 and hence A(Ω) = 𝛿(Ω)𝛼 + 𝜹A(Ω). Taking the Fourier transform of Eq. (10.2), substituting and neglecting small terms (see Section 5.4.5) gives √ √ ̃ o (Ω) = ℘o A(Ω) ̃ ̃ A + 1 − ℘o V(Ω) K 𝛼 𝛿℘ (10.3) + √ o (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4 ℘o where we have taken 𝛼 real. We see that the harmonically oscillating parameter maps onto frequency sidebands. The spectral power as measured via direct detection is proportional to the amplitude quadrature (see Section 4.5.1), so we find ̃ o (Ω) + A ̃ †o (Ω)|2 ⟩ VI (Ω) = ⟨|A = 1 + ℘o (V 1A (Ω) − 1) K 2 𝛼 2 𝛿℘2o + (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4℘o

(10.4)

The signal to noise of, for example, the upper sideband is SNR =

K 2 𝛼 2 𝛿℘2o 4℘o (1 + ℘o (V 1A (Ωsig ) − 1))

(10.5)

If our input beam is in a coherent state, then V 1A (Ωsig ) = 1. However, we see that in this simple example amplitude squeezing, i.e. V 1A (Ωsig ) < 1, can improve the signal to noise of the sensor and hence improve the precision of our estimate of the parameter. Notice that in the coherent state case, the SNR scales with the input power Pin ∝ |𝛼|2 = n. This scaling is sometimes referred to as the quantum noise limit (QNL) for sensors. However, if we use squeezed light at the input, we find that the quantum noise term is reduced, and, under ideal conditions, for the same parameter ℘, the signal now scales proportional to the optical power squared, 2 . This limit is known as the Heisenberg limit in the sensitivity. This can be Pin seen in the above example by setting ℘o ≈ 1 and assuming V 1A ≪ 1. We also require that the number of photons arising from the squeezing, ns ≈ 1∕(4V 1A ), is equal to the number of photons from the coherent amplitude, na = |𝛼|2 , such that na = ns = 1∕2 n ∝ 1∕2 Pin . Substituting these conditions into Eq. (10.6) gives 2

K 2 n 𝛿℘2o (10.6) 4 thus exhibiting the expected Heisenberg scaling. One test for the operation below the quantum limit is to measure the SNR as a function of the optical power Pin SNR ≈

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at the input, which would be proportional to Pin for the quantum noise lim2 for the quanited case and approach, but not exceed the proportionality to Pin tum enhanced case. The fact that this is the best possible scaling can be derived rigorously using the concept of information theory, Fisher information, and the Cramer–Rao Bound as will be discussed in Section 10.1.3. 10.1.2

Practical Examples of Sensors

In all of these cases, the sensor is measuring and tracking small changes in a property with high sensitivity. To reach its full potential, the sensor should be first optimized to operate at the QNL, commonly referred to as the shot noise limit, and then the sensitivity can be improved by increasing the laser power. Once this is no longer possible, most likely through the onset of non-linear effects, squeezing can provide an additional improvement. At the moment typically by a factor of 2 (or 3 dB) in the SNR. Potentially, this could be expanded to a factor 10 (or 10 dB) but at a tremendous cost in refinement since even the smallest optical losses have to be avoided. Here the GW detectors stand out as the shining future example [1], since already all other improvements have been made and the scientific gain of even a factor of 2 improvement allows the detection in 3D from an eight times larger volume, which is very important for cosmology. The most notable examples of quantum enhanced sensors are described in more detail in the following. It has been shown that absorption spectra can be recorded with improved SNR [2]. Interferometers can be improved, as first predicted by C.M. Caves et al. [3] and first implemented for table top instruments [4, 5]. It is now a routine component of some of the large-scale GW detectors [6], as will be discussed in more detail in Section 10.3. Squeezed light improves the measurement of the polarization of a laser beam [7] and any effect that introduces changes to the polarization, the macroscopic spin of a sample of atoms [8], and the displacement or position of a laser beam [9]. Squeezed light enhances the tracking of particles within living cells [10]. It can be used for the improvement of the uncertainty in timing pulses and frequency combs [11, 12]. Other technical applications will follow. Very early in the game, the concept was applied to the detection of absorption, see Figure 10.2(a,b). Amplitude squeezed light was used, as generated with quiet diode lasers (see page 213 or, from a second harmonic generator (SHG) experiment, Section 9.4.3. The experiments were identical to conventional absorption sensors or absorption spectroscopy with the only change being that the input beam is squeezed. The absorption has to be modulated at a frequency Ωem within the bandwidth of the squeezing spectrum. Care has to be taken that the overall efficiency of the apparatus and the detectors is very high; otherwise the degree of squeezing and the improvement in SNR rapidly disappears. One of the first demonstrations was made by E.S. Polzik et al. [2], who demonstrated an improvement of 2 dB for the SNR for saturated absorption spectroscopy of a Cs vapour. They used an optical parametric oscillation (OPO) as a source for squeezed light, and the major challenge was to make this light source tunable. Another fine demonstration was carried out by C.D. Nabors and R.M. Shelby [13] who showed that a modulation can be sensed with improved SNR using

10.1 Quantum Enhanced Sensors

SHG

Intensity squeezed laser

Absorber

Laser

OPO

Detector

Absorber

(a)

(b)

SHG

Laser PBS

Modulator

OPO Absorber

(c)

Intensity squeezed laser

Moving scattering particles

(d)

Figure 10.2 Different schemes for using squeezed light for improved SNR in optical sensors. (a) Absorption using amplitude squeezed light. (b) Absorption using tunable OPO. Source: After E.S. Polzik et al. [2]. (c) Measuring DC absorption using twin photon beams. Source: After Hai et al. [14]. (d) Detecting scattered laser light for motion detection.

the bright squeezed light from an above threshold doubly resonant monolithic OPO. One of the difficulties is to modulate the absorption at a sufficiently high frequency. The experiments by the group in Shanxi University [14] and in Paris and now many others use an elegant trick: twin photon beams generated by an above threshold OPO. One beam is used as a reference. The other is modulated at Ωmod and split into another pair of beams, probe 1 and probe 2, and only probe 1 is sent through the sample (see Figure 10.2c). By first forming the sum of the two probe photocurrents and then the difference between the total probe and the reference, a current is generated that contains an AC signal at Ωmod , which is proportional to the strength of the absorption. In this way a slowly varying absorption can be observed with high sensitivity. A similar approach can be used to detect the intensity of scattered light (see Figure10.2d). It was shown that the light scattered by fast-moving smoke particles can be detected using an optical homodyne detectors and tracking the frequency sidebands. The Doppler shift provided the necessary frequency shift, and the spectrum was used to detect the density and the velocity distribution of the sample. This technique is known as Doppler velocimetry. Using amplitude squeezed light from a diode laser, Y.Q. Li et al. were able to demonstrate an increase in the SNR of 1 dB compared to an experiment with classical light [15]. More recently this was used, for example, for tracking particles inside biological samples with quantum enhancement (see page 418).

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10.1.3

Ultimate Sensing Limits

In Section 10.1.1 we found measurement precision limits from SNRs given a particular detection strategy, i.e. direct detection. Here we discuss a more general approach that finds the ultimate precision limit for any detection strategy. We also discuss recent suggestions for non-linear quantum sensors. As discussed earlier, in general, optical sensors use some interferometric arrangement as an actuator such that changes in a parameter of interest lead to a modulation of the intensity of a probe field. Hence the beamsplitter interaction described by Eq. (10.2) can be considered a generic model of a quantum optical sensor, with different types of sensors being characterized by different functional dependences of 𝜖(℘) on the parameter of interest, ℘. Previously we used a simple signal-to-noise approach to analyse the precision of the parameter estimation. More generally the lower bound on the variance of an unbiased estimator, maximized over all possible measurements on the output probe, can be found by calculating the quantum Fisher information, H(𝜖). The variance of the estimator, V (℘), is bounded by the Cramer–Rao inequality [16], generalized to the quantum case [17]: V (℘) ≥ (

1 )2 𝜕𝜖 H(𝜖)N 𝜕℘

(10.7)

where N is the number of measurements taken. The Fisher information is the second moment of the classical logarithmic derivative of the likelihood function. The quantum Fisher information is further optimized to coincide with the best measurement strategy allowed by quantum mechanics. The quantum Fisher information can be evaluated via several different methods. One often convenient ̂ 𝝈), ̂ between the two density operators 𝝆̂ method is via the quantum fidelity,  (𝝆, ̂ where and 𝝈, | [√√ √ ]||2 |  = |Tr 𝝆̂ 𝝈̂ 𝝆̂ | | | | |

(10.8)

The quantum fidelity is a well-known quantification for the similarity of quantum states. For pure states, the fidelity reduces to the overlap between states:  = |⟨𝜓|𝜓 ′ ⟩|2 . The fidelity can be used to calculate the quantum Fisher information via √ 8(1 −  (𝝆̂ 𝜖 , 𝝆̂ 𝜖+𝛿𝜖 )) H(𝜖) = lim (10.9) 𝛿𝜖→0 𝛿𝜖 2 where 𝝆̂ 𝜖 is the output state of the probe after the interaction with the beamsplitter, whilst 𝝆̂ 𝜖+𝛿𝜖 is the output state after the interaction given a small perturbation, 𝛿𝜖, to 𝜖. For example, a coherent state probe remains a pure state after transmission through the beamsplitter, so the fidelity is easily calculated as ( 2 2 ) √ √ |𝛼| 𝛿𝜖 )  (𝜌𝜖 , 𝜌𝜖+𝛿𝜖 ) = |⟨ 𝜖𝛼| 𝜖 + 𝛿𝜖𝛼⟩|2 = exp − (10.10) 8𝜖

10.1 Quantum Enhanced Sensors

Substituting Eq. (10.10) into Eq. (10.9) and hence into Eq. (10.7) gives the precision bound for a coherent probe: V (℘) ≥ (

𝜕𝜖 𝜕℘

1 )2 4|𝛼|2 N

(10.11)

Given that the variance of the estimator is inversely proportional to the SNR, we can note that Eq. (10.11) has the same 1∕n scaling as obtained from the signal-to-noise analysis. As we observed earlier a squeezed probe can do better, 2 leading ideally to a 1∕n scaling [18]. By using the general expression for fidelity between Gaussian states (see Section 13.6.2), we can arrive at the same conclusion from the Fisher information analysis. Under more practical conditions, the addition of squeezing to a coherent probe can give a constant improvement factor as can also be shown from the Fisher information analysis. Determining the measurement basis that will saturate the quantum Cramer–Rao bound is more involved, but for Gaussian states it has been shown in general to be a displaced, squeezed number basis [19]. Recently it has been observed that if there is a strong non-linear coupling to 2 the probe, then variance scalings better than the 1∕n can be achieved [20], and an experimental example has been demonstrated [21]. For example, consider the following non-linear beamsplitter relationship: â o =

† † † † 1 1 (1 + ei𝜒∕2(̂a −̂v )(̂a−̂v) ) â + (1 − ei𝜒∕2(̂a −̂v )(̂a−̂v) ) v̂ 2 2

(10.12)

where, as in Eq. (10.1), â represents the probe mode and v̂ a vacuum mode. Here we wish to estimate the parameter 𝜒; however the transmission depends both on 𝜒 and the intensity of the probe in a non-linear way. A signal-to-noise analysis can be carried out by assuming that the fluctuations and the signal are small compared with the average power such that the input mode can be written as ̂ with 𝛼 ≫ 1 and 𝜒 = 𝜒o + 𝛿𝜒 with 𝛿𝜒 ≪ 1 and linearizing the fluctuâ = 𝛼 + 𝜹a ations. This leads to an approximate power spectrum for the fluctuations of the output field of 𝜒o 𝜒o 2 2 1 1 Vo = V𝜒 |𝛼|6 + Va |1 + ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 + |1 − ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 4 4 (10.13)

Now suppose we are looking for fluctuations around the point 𝜒o = 0. Under these conditions, with a coherent probe, we find SNR = V𝜒 |𝛼|6 ∕16, and so the 3 estimation precision scales as 1∕n , exceeding the Heisenberg scaling. These claims have generated some controversy. In particular it should be noted that: 1. The enhanced precision typically only applies locally, i.e. around the operating point (𝜒o = 0 in our example), and the range in which an enhancement is obtained may be too narrow to be useful [22]. 2. In order to have practical applications, an efficient way to map some physical parameter of interest onto a non-linear parameter, such as 𝜒2 or 𝜒3 , is required. This is a technical challenge.

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Nevertheless, such non-linear devices are an interesting field of future research in quantum optical sensors. 10.1.4

Adaptive Phase Estimation

Finally, we note that so far we have only discussed sensors that are measuring small variations starting with a known value. That means we do not need to estimate the full value of the parameter ℘. All we need to record in each time step is the change 𝛿℘(t). In contrast it is possible to consider sensors that estimate parameter values for which we have no previous knowledge, and thus we have to start in each time interval from the same point, e.g. ℘ = 0. For example, in an interferometric sensor we normally consider situations where we track the small changes in the phase 𝛿𝜙(t) of the light and compare it to the value of 𝜙(t − 𝜏) at the previous time step. In contrast one could start with no knowledge of 𝜙 in the range 0 to 2𝜋 in each time step. This task requires an adaptive approach and more resources and photons per time interval than a sensor, which constantly tracks small changes of the phase. A considerable amount of theory work has been done in this field, in particular by H. Wiseman, G. Milburn, and H. Mabuchi and their respective teams [23, 24]. In more recent times experiments by a combined Tokyo and Canberra team, led by A. Furusawa and E. Huntington, have explored adaptive optical phase estimation using time-symmetric quantum smoothing [25] and combined with squeezed light [26], and they have verified the theoretical predictions. It is important to establish the correct quantum limit and minimum number of quantum resources, which are required to achieve the phase estimation. This team later studied the quantum-limited estimation of a mirror position. A detailed study was made to optimize which data should be used in post-processing. The authors concluded that using only data recorded before the measurement was not as efficient as using the same number of data points recorded before and after the timing of the signal [27]. These studies are the beginning of more complex analysis and the design of devices that can be used in more sophisticated optical control situations at and below the QNL. This work has been successful with squeezed light and also with the corresponding experiments using single photons and the corresponding phase estimation algorithms. See, for example, the work at Griffiths University by G.J. Pryde and co-workers [28], which summarizes the approaches to measurements and estimation and identifies which are the most appropriate quantum states and how best to use them.

10.2 Optical Communication Optical communication has been one of the motivations for the development of quantum optics technology. In the last three decades, there has been a tremendous advance in optical communication technology: low loss fibres, high quantum efficiency photodiodes, diode lasers, and laser amplifiers have all improved the system performance. The speed has increased relentlessly over the decades by a factor of 10 about every 4 years, as seen in Figure 10.3.

10.2 Optical Communication

1017 y ver 0e 1 × :

1016 1015 Capacity (bit/s)

1014

Capacity limit for current technology

nd Tre

1013 1012

s ear 4y

Space division multiplexing

WDM

1011 High spectral efficiency coding

1010 109 108 107 106 105 1980

EDFA Improved transmission fibres 1990

2000 Year

2010

Figure 10.3 Communication speed and multiplexing. Source: From Richardson et al. [29].

In optical systems the information is encoded on one or more channels or modes of light that are propagated via free space or more commonly via an optical fibre and decoded at the receiver. The attraction of optical systems is the wide modulation bandwidth and the ability to multiplex the wavelengths, which means carrier frequencies, the polarization, and, very soon, spatial modes in the fibre. The communication speed can be measured by bits per second (bits/s), and an overview of the communication speed can be seen in Figure 10.3. We have now systems approaching a speed of 1014 bit/s, which was hard to imagine only a few years ago, and this requires the simultaneous use of many advanced techniques. One trick is multiplexing, which means in our language that information is encoded in a series of orthogonal modes, with minimum crosstalk between the modes. Multiplexing technology is currently one of the major topics of international research and development. Combined with all the tricks used in coherent detection, using a known detection phase reference via a local oscillator, and homodyne detection, the information density and thus also communication bandwidth can be increased further. The technology of the fastest communication systems and quantum optics experiments is remarkably similar [29, 30], and we find increasingly an overlap and a demand from the communication industry and engineers. It can be foreseen that we will soon reach the situation where the quantum properties of light become an important limit for the performance of optical communication systems. At that stage squeezed light might play a role in overcoming these limits and in enhancing the performance. In this chapter we want to estimate the size of such possible improvements. In principle, a communication system contains the components shown in Figure 10.4. Different encoding and detection schemes are in use, with the simplest and most commonly used one direct incoherent modulation. In this case the transmitter is a diode laser, the

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Data in

Laser

Transmission line Demodulator

Fibre laser amplifier

Modulator

Data out

Coupler

Figure 10.4 Components of an optical communication system without coherent detection.

information is encoded by an amplitude modulator, and the receiver is a simple photodetector. More advanced systems use coherent schemes such as phase shift keying (PSK). This requires a phase modulator in the transmitter and a homodyne, or heterodyne, detector with a local oscillator beam that is phase-locked to the input mode. These systems can operate within a few decibels of the QNL. The question of how non-classical light can improve the performance of the system was the motivation for the early work by H. Takahashi [31] and H.P. Yuen and J.H. Shapiro [32]. The techniques for the encoding, propagating, and decoding of information on a quantum system were reviewed by Y. Yamamoto and H.A. Haus [33] and by C.M. Caves and P.D. Drummond [34]. Non-classical light can be used in several ways to improve components of the system: (i) the laser can produce sub-Poissonian or quadrature squeezed light and thus possibly allow a better SNR. This is particularly attractive and practical when diode lasers are used. (ii) The phase-insensitive laser amplifier can be replaced by a phase-sensitive parametric amplifier. (iii) The beamsplitter, or coupler, can be equipped with a squeezed vacuum input that reduces the noise introduced at this point or alternatively a quantum non-demolition (QND) device can be used [35]. In all cases the same detection system can be used. The overall improvement can be illustrated by considering the amount of information that can be encoded on a mode of light with given maximum optical power. For an incoherent system, where only intensity information is processed, this information content can be estimated by evaluating the number of states with X2

α

δX1

X1

X2

δX1

X1

Figure 10.5 Illustration of the number of distinguishable states for incoherent detection. Top, coherent states; below, squeezed states.

10.2 Optical Communication

fGW

Two rotating neutron stars or similar

Time

Figure 10.6 A typical source for gravitational waves and its effect on space and time. The effects observed by the end of 2017 include colliding black holes and spinning neutron stars.

different intensities that can be distinguished. This is illustrated in Figure 10.5, where the sequence of distinguishable coherent states are shown, all with one phase = zero. Each different state is represented by an uncertainty area, symmetric circles in the case of coherent states. An incoherent system recognizes different intensities; the phase information is ignored. With sub-Poissonian light the uncertainty in the amplitude is smaller, and for given maximum power a larger number of states can be distinguished with the same SNR. Hence more information can be encoded Figure 10.6. A quantitative figure of merit for a communication system is the channel capacity, a concept that describes the maximum amount of information that can be transmitted based on statistical arguments. The Shannon capacity [36] of a communication channel with Gaussian noise of power (variance) N and Gaussian distributed signal power S operating at the bandwidth limit is ] [ S 1 (10.14) C = log2 1 + 2 N where C is in units of bits per symbol. Equation (10.14) can be used to calculate the channel capacities of quantum states with Gaussian probability distributions such as coherent states and squeezed states. Consider first a signal composed of a Gaussian distribution of coherent state amplitudes all with the same quadrature angle. The signal power Vs is given by the variance of the distribution. The noise is given by the intrinsic quantum noise of the coherent states, Vn = 1. Because the quadrature angle of the signal is known, homodyne detection can in principle detect the signal without further penalty, and thus the measured signal to noise is S∕N = Vs ∕Vn = Vs . In general the average photon number per bandwidth per second of a light beam is given by 1 + 1 (V + V − ) − (10.15) 4 2 where V + (V − ) are the variances of the maximum (minimum) quadrature projections of the noise ellipse of the state. These projections are orthogonal quadratures, such as amplitude and phase, and obey the uncertainty principle V + V − ≥ 1. In the above example one quadrature is made up of signal plus n=

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quantum noise such that V + = Vs + 1, whilst the orthogonal quadrature is just quantum noise, so V − = 1. Hence n = 1∕4Vs , and so the channel capacity of a coherent state with single quadrature encoding and homodyne detection is [√ ] Cc = log2 1 + 4n (10.16) Establishing in an experiment that a particular optical mode has, this capacity would involve (i) measuring the quadrature amplitude variances of the beam, V + and V − , (ii) calibrating Alice’s signal variance, and (iii) measuring Bob’s signal to noise. If these measurement agreed with the theoretical conditions above, then Shannon’s theorem tells us that an encoding scheme exists that could realize the channel capacity of Eq. (10.16). An example of such an encoding is given in [37]. For photon numbers n > 2 improved channel capacity can be obtained by encoding symmetrically on both quadratures and detecting both quadratures simultaneously using heterodyne detection or dual homodyne detection. Because of the non-commutation of orthogonal quadratures, there is a penalty for their simultaneous detection that reduces the signal to noise of each quadrature to S∕N = 1∕2Vs . Also because there is signal on both quadratures, the average photon number of the beam is now n = 1∕2Vs . On the other hand the total channel capacity will now be the sum of the two independent channels carried by the two quadratures. Thus the channel capacity for a coherent state with dual quadrature encoding and heterodyne detection is [ ] [ ] S− 1 S+ 1 + log2 1 + Cch = log2 1 + 2 N 2 N (10.17) = log2 [1 + n] which exceeds that of the homodyne technique (Eq. (10.16)) for n > 2. The above channel capacities are the best achievable if we restrict ourselves to a semi-classical treatment of light. However the channel capacity of the homodyne technique can be improved by the use of non-classical squeezed light. With squeezed light the noise variance of the encoded quadrature can be reduced such that Vne < 1, whilst the noise of the unencoded quadrature is increased such that Vnu ≥ 1∕Vne . As a result the signal to noise is improved to S∕N = Vs ∕Vne , whilst the photon number is now given by Eq. (10.15) but with V + = Vs + Vne and V − = 1∕Vne where a pure (i.e. minimum uncertainty) squeezed state has been 2 assumed. Maximizing the signal to noise for fixed n leads to S∕N = 4(n + n ) for a squeezed quadrature variance of Vne,opt = 1∕(1 + 2n). Hence the channel capacity for a squeezed beam with homodyne detection is ] [ Csh = log2 1 + 2n (10.18) which exceeds both coherent homodyne and heterodyne for all values of n. Although squeezing gives an improvement in channel capacity, this improvement is rapidly washed out by nonidealities in the system such as propagation loss and other inefficiencies. It still seems some way off before squeezing might be considered a viable option for increasing channel capacity. In principle, a final improvement in channel capacity can be obtained by allowing non-Gaussian states. The absolute maximum channel capacity for a single mode is given by

10.3 Gravitational Wave Detection

the Holevo bound and can be realized by encoding in a maximum entropy ensemble of Fock states and using photon number detection. This ultimate channel capacity is CFock = (1 + n)log2 [(1 + n)] − nlog2 [n]

(10.19)

which is the maximal channel capacity at all values of n. Quantum optics can describe the ultimate limits of data communication, and we can find ways to approach this limit. In 2018 such quantum limits can be reached in exceptional cases. The main emphasis is still on expanding the existing technologies, to create more and better communication channels, and in some cases the quantum limit can be reached.

10.3 Gravitational Wave Detection One of the most likely and promising applications of squeezed light will be the improvement of the sensitivity of interferometers, in particular those designed to measure GWs. These instruments are already optimized and are designed to operate at the QNL and with minimum losses. The extra benefit of squeezed light could have an important scientific impact. 10.3.1

The Origin and Properties of GW

One of the most fascinating consequences of the general theory of relativity is the coupling between mass and the geometry of space, leading to the prediction that changes in the mass distribution can produce a perturbation of the geometry that propagates through space like a wave. These ripples in space-time, or GWs, propagate at the speed of light and carry with them information about the dynamics of those cosmological objects that created them. Collapsing or exploding stars will emit bursts of waves, and rotating systems will constantly emit waves. The detection of GWs is a new form of astronomy, complementing the conventional optical, infrared, radio, and X-ray astronomy. In 2017 the first observation was reported that combined all these types of observations for one single astronomical event. The GW induces a change of the geometry in a plane transverse to the direction of propagation. It causes an expansion in one direction and simultaneously a contraction in the orthogonal direction. This change will oscillate in time. A length, measured in one direction of the plane, will alternate between expansion and contraction. The frequency of this oscillation, fGW , is linked to the frequency of the movement of the space-time system and thus the mass in the object that emitted the wave. GWs are analogues to quadrupole radiation; they follow the higher-order moments of the movement of the mass. In principle there are two possible polarizations of the GW , which can be detected with different geometries of detectors. We should be able to detect the GW by continuously monitoring the length in two orthogonal directions. However, the extremely small size of the effect makes the GWs so very elusive. The properties of the sources and the waves were actively discussed in the 1970s; a systematic approach became evident that evolved more and more and

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predicted magnitudes and probabilities for these astronomical events. This was, for example, summarized by K. Thorne and his collaborators in 1987 [38]. After earlier unsuccessful attempts with solid bar antenna, a proposal was made by R. Weiss as early as 1972 to detect GW with laser interferometers [39]. The size of the effect of a GW is given by the relative change in distance or the relative strain h = Δl∕l induced. A value of h = 10−20 means a change in a distance measurement between two points in space by one part in 1020 . If we could measure the distance between two or three points in a plane perpendicular to the propagation of the GW wave, we should be able to detect a periodic signal of the expansion and contraction of these distances, driven by the GW. Since space-time is extremely stiff, the attenuation, and thus the recorded signal, is exceedingly small. The audio frequency range interferometric measurements of the distances in two orthogonal directions are the way to reach such extremely low strain sensitivities. It has occasionally been suggested that the argument above contains a logical trap. One could argue that not only the distance between the mirrors changes as a consequence of the GW but also the time it takes for the light to travel from the beamsplitter to the mirror is modified. After all, general relativity affects both distance and time. Would these two effects cancel each other? No. When the calculations are carried out, one has to make a choice which set of equations should be used, either a set with normalized time coordinate or, alternatively, a set with normalized space coordinates. The results of both calculations are identical, but they cannot be used simultaneously. A detailed description of these calculations can be found in [40]. A list of predictions of cosmological sources of GW is shown in Figure 10.7 with information from an early design study for the American LIGO project. 10–18

Strain in space

390

10–20

Black hole binary coalescence 106 M

Co

m

Black hole formation 106 M ar Black hole ies binary 105 M

pa

ct

bin

10–22 BH–BH 103 M

White dwarf binaries

10–24

10–4

Co m co pa Stellar ale ct sc bin en ar ce y

Collapse

10–2

Space-based detectors

1 Frequency (Hz)

102

104

Ground-based detectors

Figure 10.7 Early predictions of the magnitude of the strain h and the frequency fGW of the oscillations in the space coordinates induced by various sources of gravitational waves. This is a compilation of the estimates from cosmological models. Source: From S. Rowen and J. Hough in the year 2000 [41].

10.3 Gravitational Wave Detection

This simple diagram incorporates most of the astrophysical predictions available in 1994. Such predictions have been refined steadily with improvements in the cosmological models [42]. The various types of objects differ in frequency, waveform, and amplitude of the predicted GW we can expect to traverse the Earth. The likely frequencies for GWs, fGW , range from 10−6 to 103 Hz. Some objects, such as spinning pulsars, have fixed frequencies. Others will change their dynamics rapidly, over days to seconds, and move through an entire spectral evolution. Each object has its own typical signature. The details depend critically on the spatial distribution and the symmetry and size of the masses involved. More details can be calculated using the astrophysical models that are presently being developed, and over the next few years, we can expect to get even more detailed results from these models (Figure 10.8). A very talented and ambitious group of research in many countries set out to design and then build a group of detectors: first in the United States with two LIGO detectors built far apart in the NW and SE regions to gain the best spatial pointing resolution in the sky and then in Europe with detectors VIRGO near Pisa, Italy, and GEO near Hannover, Germany. The steadily improving technology was developed in various parts of the world, including Germany, Scotland, Japan, Australia, etc. Many existing technologies had to be improved by several orders of magnitude to achieve the required specifications. At the same time many new forms of data analysis, all based on fully relativistic models and simulations, had to be developed. It took over three decades to refine the instruments to make the first observation.

Hanford, Washington (H1)

Strain (10–21)

1.0 0.5 0.0 –0.5 –1.0 1.0 0.5 0.0 –0.5 –1.0 0.5 0.0 –0.5

Livingston, Louisiana (L1)

H1 observed

L1 observed H1 observed (shifted, inverted)

Numerical relativity Reconstructed (wavelet) Reconstructed (template)

Numerical relativity Reconstructed (wavelet) Reconstructed (template)

Residual

Residual

Figure 10.8 The waveform of the event detected by the two LIGO detectors located in Hanford, Washington, and Livingston, Louisiana, on 14 September 2015. The top trace shows the experimental signal in the low audio range. We can hear such a cosmic event for the first time. The middle trace is the matched prediction from full relativistic model calculation of the collision of two black holes. The bottom trace is the difference between experiment and model. Source: From [1].

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As of 14 September 2015 we know that such sources do indeed exist and theoretical predictions can be confirmed. The collision of two black holes was recorded by the LIGO interferometers, matching perfectly with a waveform that had been calculated. In other words, the waveform recorded allowed a unique identification of the source, and the calibrated measurement of the size allowed the determination of type, size, and distance of the source. This is one of the key events in astronomy, opening a completely new window of observation [1]. The observation was published in 2016 after a most extensive analysis. By the end of 2017, at least four events of colliding black holes have been published, providing the first estimate of the rate at which these events occur. Since 2017 we have combined GW and conventional astronomy, with spinning neutron stars the first object observed in this way [43]. This object was pinpointed by the three GW detectors, the two LIGOs that made direct observations and VIRGO, which was operational but did not detect the radiation, and thereby helped to improve the pointing accuracy. In addition, optical, radio, and gamma ray observations of the object provided a wealth of data. These remarkable scientific achievements were acknowledged by a Nobel Prize in Physics in 2017 for the pioneers of the field Ray Weiss, Kip Thorne, and Barry Barrish, who represent the entire global GW activity led by LIGO. The field of GW astronomy will thrive. 10.3.1.1

Concept and Design of an Optical GW Detector

The predicted strain amplitude for GW sources ranges from h = 10−25 to 10−20 [1 Hz−1∕2 ]. Over which length should we best measure the change in length? At any one time the largest effect will be at two points with a separation that corresponds to half the wavelength 𝜆GW of the GW. At any one place the biggest effect can be seen between two time intervals separated by 12 tGW = 2f1 . GW For example, let us assume a GW at fGW = 500 Hz, which means tGW = 2 ms and 𝜆GW = 6 × 105 m = 600 km. In this case the largest amplitude is a length change of Δl = 12 h𝜆GW = 3 × 10−16 m. This is ridiculously small, less than the typical size of a single electron. However, we should recall that we will measure the movement of surfaces, not individual atoms. Surfaces are typically made of more than 1020 atoms, the measurement averages over all these atoms, and the idea of such a small shift of the surface becomes meaningful again. The displacement of a mirror surface could possibly be measured with this accuracy. The idea is to use a mirror that has no internal motion, which means it is a free mass without internal oscillations and reflects light back to the source. With a Michelson interferometer we can measure the displacement in two orthogonal directions and translate the difference in the displacement of the mirrors into an optical phase shift. The biggest effect would be achieved by positioning the mirror at a distance 1 𝜆 from the beamsplitter. In this case the distance from the beamsplitter to 4 GW one mirror will be at maximum expansion when the other will have the maximum contraction. This is shown in Figure 10.9. Since we do not know the frequency in advance, a broadband detector is desirable that works over many GW frequencies. This can be achieved by placing resonators into the two arms of the interferometer with s suitable ring down, or storage time, for the light (see

10.3 Gravitational Wave Detection

δX1las δX2las

Cavity

Laser Circulator Unused port δX1u δX2u (a)

(b)

(c)

δX1out δX2out Detector

Figure 10.9 Three configurations for the detection of gravitational waves (a) Michelson interferometer, (b) Sagnac interferometer, and (c) Michelson interferometer with internal cavities.

Figure 10.9). This results in a smooth detection response declining gradually from 10 to 2000 Hz, as shown in Figure 10.11. Two optical waves leave from the beamsplitter to the output port that are displaced by 3 × 10−16 m. Using light with a wavelength of 1000 nm, we would get a maximum phase shift of 5 × 10−9 rad. This is clearly an incredibly small value. Note that this instrument has an optimum response only for GW that approach from a direction orthogonal to the plane of the interferometer. In practice we will see only a smaller effect due to the projection of the GW onto this detection plane. To measure it will require the use of all tricks known in optics and interferometry and the best materials and components, which have all been implemented step by step and refined in a succession of improved detectors. 10.3.2

Quantum Properties of the Ideal Interferometer

Before we get drawn into the intricacies of a specific configuration, we shall determine the quantum noise properties of a simple, idealized instrument, the Michelson interferometer. What is the fundamental limit in sensitivity of an interferometer? This question was analysed by V.B. Braginsky [44] and also eloquently answered by C. Caves and R. Loudon in 1980 [3, 45]. Their results actually stimulated the interest in squeezed states of light. C. Caves analysed the ideal, lossless interferometer – for the situation that the photon fluctuations and the readout of the reflected light dominate. This is an excellent assumption for all situations until recently. For an interferometer where classical fluctuations dominate, one might choose a setting where the response of the output intensity to a change in phase, or length, is largest. This is the conventional choice, with arm length difference chosen to create the working point halfway between a dark fringe and a bright fringe. However, when quantum noise dominates, the interferometer performs best when the difference in the length of the arms is chosen such that the light interferes destructively at one output port; it forms a dark fringe. The intensity at

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this port is detected with a single detector. The intensity noise depends on the quadrature fluctuations of the two input fields, the mode of light from the laser ̃ u entering the unused port. The sidebands are ̃ las and the mode A represented by A ̃ las and 𝜹X𝟐 ̃ u . It also depends on the momenrepresented by their quadratures 𝜹X𝟏 tum and position state of the mirror, described by the expectation values of the ̃ The total variance of the intensity momentum operator q̃ and position operator p. of the output light at detection frequency Ω can be summarized as [46] ̃ 2out ⟩ = A⟨𝜹X ̃ u (𝜃 + 𝜋∕2)𝜹X ̃ †u (𝜃 + 𝜋∕2)⟩ + B⟨𝜹X ̃ las (𝜃)𝜹X ̃ † (𝜃)⟩ ⟨𝜹X𝟏 las ̃ p̃ † ⟩ ̃ q̃ † ⟩ + D⟨𝜹p𝜹 + C⟨𝜹q𝜹 (10.20) ̃ ̃ + sin 𝜃⟨𝜹X𝟐⟩. ̃ where ⟨𝜹X(𝜃)⟩ = cos 𝜃⟨𝜹X𝟏⟩ The first term describes the effect of the vacuum quantum noise, or photon noise, entering at the unused port. The second term describes the quadrature fluctuations of the laser mode. The third and fourth terms are the fluctuations of the mirror position and momentum and the mechanical vibration modes inside the mirror, and these can be ignored for a GW detector with very heavy mirrors. The factor B depends on the mode matching of the interferometer and can be assumed to be zero for a well-balanced device. The angle 𝜃 is power dependent. At low powers it tends to zero such that the noise of the output is dominated by the phase quadrature fluctuations of the vacuum (see Section 5.2). Under these conditions the factor A depends only on the optical power Pin of the input light and the detection interval 𝜏. As the power increases the 𝜃 parameter becomes non-zero as a result of radiation pressure noise induced by the intensity fluctuations of the laser mode. This acts to couple amplitude fluctuations from the vacuum port into the output. The factor A now depends on the mass M of the mirror and on Pin and 𝜏. This calculation is correct for a free mass or for a mirror suspended by a pendulum with an eigenfrequency much less than the detection frequency. The photon noise itself is frequency independent; the relative sensitivity improves for lower frequencies, as shown, for example, for a real interferometer in Figure 10.13. The mirror noise scales with Ω−2 , and in the LIGO instruments the sensitivity is limited towards low frequencies by the noise transmitted through the suspension and inside the mirror. As a consequence the sensitivity of an interferometer dominated by photon noise will be best at lower frequencies, just above the noise isolation limit given by the mechanical set-up of the instrument. The coefficients A and 𝜃 in Eq. (10.20) are intensity dependent. With increasing intensity the effect of photon noise, or shot noise, is reduced, whilst that of radiation pressure noise is increased. For most current instruments this means that the maximum power is sought. The practical limitation will be the performance of the control systems and the many locking loops. In practice, various forms of instabilities or non-linearities can appear, and the servo systems have to work harder at larger intensities. Much of the engineering optimization has gone in stabilizing the instrument in order to reach the QNL. The variance of the radiation pressure noise scales with Ω−4 . At very high powers this noise could be the dominant term, and in this case higher frequency signals would have lower noise and would be easier to detect. In the ideal world

10.3 Gravitational Wave Detection

there would be a minimum of the sensitivity at a specific frequency for a given optical power. At the same time, there is an optimum intensity at which the best performance can be achieved for a given detection frequency Ωdet . Figure 10.10 shows such a noise balance, as a function of laser power. The minimum value of the combined noise is called the SQL of the interferometer, which is trace (c) in Figure 10.10 and is given by √ 2 1 2ℏ𝜏 ̃ out ⟩ = ⟨ 𝜹X𝟏 (10.21) L M This result is independent of the quantum mechanical noise of the free mass M. It represents a balance of two noise effects, the noise in the measurement of the light and the noise induced by the light onto the mass, and is a typical example of the ideas of Heisenberg applied to this optical measurement. This noise limit can be interpreted as the consequence of the back action of the measurement (see Chapter 11). For heavy masses the SQL is much larger than the zero point fluctuations in the position of a free mass that is shown as trace (d) in Figure 10.10. This fundamental limit, sometimes called the Heisenberg limit of the test mass, cannot be achieved completely. We will see that it is possible to approach it, and pass the SQL, by using correlated non-classical states of light [47]. In 1980 Carlton Caves drew several important conclusions from his analysis: (i) For any reasonable mass the first two terms in Eq. (10.20) dominate. That means the interaction with the light, the measurement process, is the origin of the largest quantum fluctuations. We have to optimize this process. (ii) At reasonable optical powers the photon counting noise dominates. With an increase in the optical power, the noise is reduced, and the sensitivity is improved. This holds for all optical powers used in GW detectors until recently. 10–12

(a) Photon noise

7×10–13 (b) Photon pressure noise

δϕ (rad)

4×10–13 2×10–13 (c) SQL (d) Zero point noise

10–13 105

106 107 108 Optical power (W)

109

Figure 10.10 The noise contributions in an ideal interferometer as a function of optical input power. (a) Photon noise (dashed line). (b) Photon pressure noise (dashed line). The solid line gives the total noise. The standard quantum limit (c) is given by the horizontal dotted line. (d) The zero point quantum fluctuations of the mass are even lower (dash dotted line). The parameters for this diagram are m = 1 kg, 𝜏 = 1 s, 𝜆 = 1 μm, and fGW = 100 Hz. This simulation shows that the optical powers to achieve the SQL are out of reach, and only case (a) should be considered.

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(iii) The noise at the output is entirely due to the fluctuation in the phase quadrature of the unused port of the interferometer. Illuminating the empty port with a squeezed vacuum state that is phase-locked to the laser can reduce ̃ 2u ⟩ = V 2 = the noise. For a squeezed state with minimum uncertainty ⟨𝜹X𝟐 exp(−2rs ), the improvement is equivalent to an increase in the optical power of a factor exp(rs ). 10.3.2.1

Configurations of Interferometers

An alternative to the Michelson interferometer is to measure the interference of two optical waves that have travelled from the beamsplitter to both mirrors and back but in the opposite sequence. One beam travels first to mirror one and then to mirror two, the other in the opposite order. The crucial point for achieving the maximum effect is the travel time. It should be 12 tGW . This can be achieved with an optical arrangement that is basically similar to a Sagnac interferometer (see Figure 10.9b). One can imagine a number of other configurations with identical travel, or storage, time and with beams travelling in orthogonal paths in opposite order. They all will produce the same result. Both systems have the optimum sensitivity at frequency fopt = 1∕tRT . At lower frequencies, fGW < fopt , the response of the Sagnac interferometer drops, proportional to 1∕f , simply because the slow changes in the length will affect both counter-propagating beams similarly. The Sagnac is insensitive to DC differences in the arm length, whilst here the Michelson interferometer has its full sensitivity. At larger frequencies, fGW > fopt , the responses are similar, and at fGW = 2∕tRT both instruments do not register the GW at all, since we are comparing the position of the mirror at time intervals separated by a full period of the GW. For even higher frequencies detection is possible again but only for frequencies between the response minima at multiples of 2∕tRT . For a broadband detector we would choose the round-trip time to be the highest GW frequency we wish to detect. These sensitivity spectra are shown in Figure 10.11. The idea of the round-trip time can be generalized. With the Michelson interferometer we do not have to send the two beams the full distance 14 𝜆GW . We could send it forwards and backwards from the location of the beamsplitter to the far mirror several times until it has covered a total travel distance of 12 𝜆GW or equivalently has spent the time 12 tGW in this arm of the interferometer before it interferes with the other beam. Any practical GW detector will use this trick since 𝜆GW is far too large for earthbound systems; it is typically several hundreds of kilometres. For this purpose, some designs include so-called delay lines, two mirrors of radius of curvature 0 placed at a separation of 0 (1 + 1∕j). This configuration has the remarkable property that an incoming beam is reflected j times between the pair of mirrors and emerges with the same beam size and wave front curvature it had at the input. The beam is well confined by this geometry. The number of reflections that one chooses depends on the amount of technical noise that is added at each reflection. In practice more than 50 reflections are not useful. For this reason today’s GW interferometers are typically 4 km long and reach frequencies below 100 Hz.

10.3 Gravitational Wave Detection

2 (c) Michelson with cavities

Relative response

1.75 1.5 1.25 1

(b) Sagnac

0.75 0.5 0.25

(a) 0

1000

2000 3000 4000 Detection frequency, Ω (Hz)

5000

Figure 10.11 Comparison of the response spectra for a small displacement of the mirror for (a) Michelson interferometer, (b) Sagnac interferometer, and (c) Michelson interferometer with internal cavities.

This arrangement is suitable for both the Michelson and Sagnac interferometers. The major difference between the two is that in the Sagnac all the components are sampled by both beams, which means that the two beams will have very similar wave fronts when they emerge and the visibility of the interference will be high. Secondly, any length changes that occur on timescales slower than the round-trip time will be sampled by both beams; the interferometer is immune to technical noise at frequencies f ≪ 1∕tRT . Both these points are advantages for the Sagnac configuration. On the other hand the Sagnac interferometer requires an exactly balanced beamsplitter, since one beam experiences two transmissions and the other two reflections. A detailed comparison is given by Sun et al. [48]. As we will see the idea of a Sagnac might have a revival once the optical powers are high enough to reach the regime where radiation pressure noise becomes accessible and interferometers as speed meters have an advantage over position measurements (see Section 10.3.3). A better alternative of obtaining the necessary storage time for the light is to place long cavities with storage time tst into the arms of the interferometer. The length changes will now be integrated. The effect is that for fGW ≤ 1∕tst the response of the instrument with cavities is similar to that of a simple Michelson interferometer. At larger frequencies, fGW > tst , the sensitivity of the instrument decreases due to the integration property of the cavity. The effects induced by the GW inside the cavity cannot be detected in the reflected light. A comparison of the spectrum of the signal response for all three types of instruments is shown in Figure 10.11. 10.3.2.2

Recycling

The interferometer that will finally be used to detect gravity waves will have a more complex structure than the Michelson or Sagnac we considered here for simplicity. The instrument will use some form of recycling. This can be power recycling, which remedies one glaring defect of the interferometer, namely, that all

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the light that is not lost by absorption is reflected back towards the laser. Whilst we actually use optical isolators to dump it effectively, this is obviously a considerable waste. Ron Drever [49], Roland Schilling and others recognized that by introducing a mirror MPR between the input laser and the interferometer, the light reflected by the instrument can be used again. MPR essentially forms a cavity with the whole interferometer acting as one compound mirror. This is an elegant and efficient way to enhance the laser power. In practice all we have to achieve is to keep the mirror locked to a resonance position. Alternatively, the sensitivity can be enhanced by placing a mirror MSR at the output and reflecting the sidebands back into the interferometer. This leads to a build-up of the sidebands that contain the information, with the consequence of increased sensitivity. The position of MSR can be chosen such that the laser frequency, the carrier, is resonantly built up or, alternatively, such that the cavity formed by MSR is detuned and only one of the signal sidebands is built up. The latter is known as tuned signal recycling and allows us to tune the entire instrument to a specific signal frequency fGW . Both signal and power recycling can be combined, dual recycling. The major penalty is loss in bandwidth and increased complexity of the instrument and the control system required to keep all components locked simultaneously. The ideas of recycling were pioneered by B.J. Meers [50, 51] and extended by Mizuno et al. [52, 53]. Recycling allows an improvement of the instrument beyond the limits set by the simple Michelson interferometer. It allows the detection of weaker GW at specific frequencies. Alternatively, they would allow more compact, and cheaper, instruments with enhanced sensitivity at specific frequency. Many of these ideas have been implemented in the German/British GEO project with 0.600 km baseline (http://www.geo600.org), which has been a technology leader and demonstrator for many new ideas, components, and devices before they were implemented in the larger 4 km LIGO interferometers. This included in particular low noise lasers, locking techniques, and squeezed light. All these teams have worked together to build the most sensitive GW detector. 10.3.2.3

Modulation Techniques

It was shown in Section 5.2 that in the absence of noise, the largest signal can be obtained with 𝜙0 = 𝜋∕2. This is also the best operating condition for a classical interferometer where the relative intensity noise is constant and independent of 𝜙0 . In contrast, for a QNL interferometer, where the noise is proportional to the square root of the intensity, the best performance is achieved very close to 𝜙0 = 0 or 𝜙0 = 𝜋, which means at a dark fringe. In practical instruments, there will always be some additional noise, 𝜀 > 0 in Eq. (10.29), and thus the optimum operating condition will be further offset from the exact dark fringe, 𝜙0 = 0. The signal could be detected directly, and laser noise would be no problem. All laser noise, in frequency and in intensity, is common to both waves in the interferometer, and the ideal interferometer, with matched arms, has common mode suppression to reject this noise. In practice, however, the common mode suppression is rather limited. Thus intensity noise has to be kept within stringent limits set by many imperfections of the real instrument.

10.3 Gravitational Wave Detection

Similarly, the length of the two interferometer arms will not be perfectly balanced, in particular when cavities are used inside the interferometer and not only the physical length but also the storage time has to be matched. In this case the instrument is sensitive to laser frequency noise. Typical specifications cannot be derived from first principles. For a large instrument they typically are RIN< 10−7 and d𝜈𝜈 < 10−17 [6], values that can be achieved with the latest control technology. Measurements at very low frequencies, such as 10 Hz, are prone to a large number of difficulties, the most severe being the laser noise. Consequently all interferometer designs use modulation techniques. By phase modulating the light, it is possible to transfer all information to frequencies Ωmod where the laser is QNL. As a result the photocurrent will have a strong Fourier component at 2Ωmod . All information is contained in the sideband of this spectral component, called the carrier. Demodulation using an electronic mixer and an electronic local oscillator signal, derived from the modulator driving signal, allows very sensitive heterodyne detection. There are three alternatives for modulation, as shown in Figure 10.12: internal modulation requires a modulator inside one of the interferometer arms. Frontal, or inline, modulation involves a phase modulation of the light before it enters the interferometer. The phase modulated light propagates through both arms. The arms have to have different lengths to convert common mode into differential mode phase modulation. External modulation uses an optical local oscillator with phase modulation, which is mixed on a beamsplitter with the output beam from the interferometer. This latter scheme is analogous to an optical heterodyne detector as discussed in Section 8.1.4. However, we have to pay a slight penalty, a reduction in the sensitivity. The SNR with internal modulation, assuming perfect optics ( = 1) and no electronic noise, is given by [54] SNRint =

J12 (Φm ) 𝜌Pin 𝜙2s 8 e RBW 1 − (1 − 𝜙2s ∕4)J0 (Φm )

(10.22)

where Φm is the modulation depth in rad. By selecting the optimum modulation depth and phase, a maximum value of SNRint,max =

8𝜌Pin 𝜙2s 2 = SNRmax 12 e RBW 3

(10.23) ΔL

Modulator

Modulator (a)

(b)

Modulator (c)

Figure 10.12 Three types of modulation schemes (a) internal modulation, (b) external modulation, and (c) frontal modulation.

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can be achieved. This means the instrument with internal modulation has an optimum performance only 2/3 as good as with direct detection (see Eq. (10.23)). This is a consequence of loss of information; modulation and demodulation are carried out by the same waveform, and in the process information is transferred to other harmonic frequencies. This effect would be particularly strong for modulation functions other than a sine function. In addition, information from other harmonic frequencies is brought in via demodulation. Uncorrelated noise is added in the process. It was suggested [55] and demonstrated [56, 57] that by using inverse pairs of functions for modulation and demodulation, one would assure that all Fourier components are fully recovered and no noise is added. However, the technical complications in generating and processing such matching functions can be prohibitive. A similar analysis can be made for an instrument with external modulation. For the optimized instrument, with Φm = 1.84 rad,  = 1, 𝜖 = 0, the limit is given by SNRext = 0.67 SNRmax , a result which is very similar to the case of internal modulation [54]. To date GW observatories have remained with direct detection techniques. 10.3.3

The Sensitivity of GW Observatories

The effect of the GW has to be measured in the presence of noise, both in the position of the mirrors as well as noise of the light. The quality of the signal is described by the SNR, as defined in Section 2.1. Consider first the signal; the GW induces a length change Δl(t), which is transformed by the interferometer into a phase change Δ𝜙s (t) between the two interfering output waves. The link between Δl(t) and Δ𝜙s (t) depends on the type of instrument used. For a simple Michelson interferometer, we obtain 4𝜋Δl(t) (10.24) Δ𝜙s (t) = 𝜆 If we consider one Fourier component fGW only, we can simulate the effect of the GW as a sinusoidal modulation of the phase term and get the phase difference between the two optical waves as Δ𝜙s (t) = 𝜙0 + 𝜙s sin(2𝜋fGW t + 𝜒s )

(10.25)

where 𝜙0 is the DC phase offset and 𝜒s is the signal phase offset at time t = 0. The optical power at the output port is Pout = Pin (1 +  cos(Δ𝜙s (t))

(10.26)

where Pin is the optical power incident on the interferometer and  is the fringe visibility. A perfect instrument would have a visibility of one, but in most cases scattering and wave front distortions will result in  ≤ 1. Combining Eqs. (10.25) and (10.26) leads to an expansion in term of Bessel functions Jn of the first kind. For small phase modulations, Δ𝜙 ≪ 1, harmonics above the fundamental modulation frequencies are small, and the higher-order terms can be neglected. Using a detector with responsitivity 𝜌(𝜆) A/W, this optical power can be converted into a photocurrent. The DC current is 1 (10.27) iDC = 𝜌Pin (1 + J0 (𝜙s ) cos(𝜙0 )) 2

10.3 Gravitational Wave Detection

whilst the variance of the AC components is Vi (fGW ) =

1 (𝜌Pin J1 (𝜙s ) sin(𝜙0 ))2 2

(10.28)

As expected (see Section 5.2) the received signal is strongest for 𝜙0 = 𝜋∕2, midway between a bright fringe and a dark fringe. No signal is received at the turning points (𝜙0 = 0, 𝜋, 2𝜋, …). However, the best SNR is obtained at a dark fringe. Realistic interferometers contain imperfections, or noise sources, which will reduce the performance of the idealized system discussed by Caves. These additional noise sources will add to the quantum noise and thus reduce the sensitivity. In order to optimize the performance of the instrument, the operating conditions will be different from those of the QNL interferometer. The important noise sources are seismic noise ⟨Δism ⟩, thermal noise in the suspension ⟨Δisp ⟩, thermal noise in the test mass ⟨Δitm ⟩, and electronic noise ⟨Δiel ⟩. All these noise sources are independent of each other and add in quadrature. The total noise budget present in the instrument is ⟨Δi2noise ⟩ = ⟨Δi2sm ⟩ + ⟨Δi2sp ⟩ + ⟨Δi2tm ⟩ + ⟨Δi2el ⟩ = ⟨Δi2qn ⟩(1 + 𝜀(fGW ))

(10.29)

The combined additional noise term 𝜖(fGW ), scaled in units of the quantum noise, has a strong and complex dependence on the detection frequency. Inserting the absolute value for the quantum noise (see Chapter 4), we obtain the SNR as a function of the offset angle 𝜙0 , the optical power Pin , and the detection bandwidth RBW. After expanding the Bessel functions for small signals, we obtain SNR =

𝜌Pin 𝜙2s  2 sin2 (𝜙0 ) 8 e RBW (1 + 2𝜀) +  cos(𝜙0 )

(10.30)

with an optimum value for the quantum noise limited interferometer (𝜀 = 0) and perfect visibility ( = 1) of SNRmax =

𝜌Pin 𝜙2s 4 e RBW

(10.31)

The value of Δ𝜙 that can be measured with an SNR of 1 corresponds to the quantum noise limited value Δ𝜙QNL that is derived in Section 5.2. The sensitivity of the GW detector is quoted as the relative strain Δh that produces an SNR of 1 measured with a bandwidth of 1 Hz. A plot of Δh as a function of the detection frequency fGW provides an overview of the frequency response of the instrument. This spectrum is influenced by both the various frequency-dependent noise contributions and the response function of the interferometer. This is used to predict the sensitivity of the large baseline LIGO interferometers. These instruments will be limited at low frequencies, fGW < 10 Hz), by seismic and man-made vibrations of the laboratory, which are strong from mHz to tens of Hz. They are attenuated by elaborate multistage pendulum and active suspension systems. As a consequence the suspension introduces a number of sharp resonances, the so-called violin modes, which are clearly visible but can be filtered out. Other technical spiked are due to EM radiation, for example, 60 Hz in the United States and its

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higher harmonics. All those contributions are well known, are being suppressed as much as possible, and are included as known signals in the data analysis. Let us have a look at the evolution of the LIGO instrument in time, starting around the year 2000 and looking into the future: Figure 10.13a shows the performance of a LIGO instrument in 2003. The residual noise (red trace) exhibits the typical response that is dominated by seismic noise at frequencies up to 40 Hz, followed by thermal noise from the mirror up to 200 Hz. The instrument is dominated by optical noise, and the impressive achievement is that it can actually be operated very close to the QNL for frequencies above 200 Hz. This shows that almost all other technical noise sources have been eliminated. At that stage the LIGO interferometer had reached a sensitivity of h = 210−22 [1∕Hz1∕2 ] at fGW = 200 Hz. Over the years many components were improved. This resulted in the so-called Advanced LIGO of 2015 shown in Figure 10.13b after almost all components were optimized or renewed. This includes more laser power, better low loss mirrors, a new mirror suspension, and a much refined feedback control systems. An overall improvement of more than a factor of 100 in sensitivity was achieved in a decade. Please compare the noise spectra from 2003 with the present sensitivity in order to appreciate the increase in performance of the LIGO detectors. In 2016 they achieved h = 8 10−23 [1∕Hz1∕2 ] at 200 Hz. These achievements are truly remarkable and show the strength of ingenuity, engineering, and international collaboration. More improvements are possible. The solid black line is based on the design specifications for 2019. Such an instrument would have a reach of about 200 megaparsecs (Mpc) compared with about 80 Mpc achieved in the year 2016. This trend in improved sensitivity is most impressive and is summarized in Figure 10.13c, which was made in 2012. The technological progress will continue, and this will require optimization of the existing facilities in the Unites States with the assistance of the full global LIGO collaboration. The instruments in Europe and LIGO India, potentially in Japan and China, will show similar advances. In parallel, even larger and more sensitive instruments are in the design phase, such as the Einstein GW telescope. In addition there will be space-based detectors – GW detectors such as LISA that will allow detection at ultra-low frequencies fGW of micro and milli hertz. A crucial first mission LISA pathfinder has successfully demonstrated the concept in 2017, and a launch is likely in the early 2020s. In the future we will see a complete network and global collaboration in GW detection, linked into all other forms of astronomy. The improvement of optical technology is at the core of all these advances in science. 10.3.3.1

Enhancement Below the SQL

In early experiments it was impossible to operate the interferometers at intensities that would make the frequency regime around and below the SQL accessible. Technical noise sources, such as seismic noise, normally dominate the performance at low frequencies, and the laser powers were not sufficiently high. One of the key features of the interferometer is the mechanical resonance Ωmech of the system and the difference response to the optical noise below and above this resonance. For simplicity it is worthwhile to step back and consider just a simple

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10 Applications of Quantum Light

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Michelson interferometer, with cavity build-up and recycling. Figure 10.14 shows such an example with Ωmech set to 1 Hz for simplicity. Here we can see how the quantum noise, or shot noise, drives up the minimal detectable displacement at the resonance for a very high laser power, set at 4000 W in this case. We can see that the radiation pressure dominates in this specific case for frequencies below 200 Hz, and the photon counting noise dominates at high frequencies. The SQL is a combination of the effects of the quantum noise of coherent states, including the vacuum state, entering through all inputs of the interferometer and in all quadratures. It is clear that we want to operate at the highest power, to lower the SQL, as long as we are in the regime where photon counting noise dominates (see Figure 10.10). This might not always be possible. Even in 2017 the GW observatories cannot be operated at the highest available power. Whilst the light source in the LIGO might be capable of delivering close to 1000 W of quantum noise limited power, which is amazing, the actual power used was called back to more like 125 W to avoid unexplained noise at low detection frequencies. This noise is not fully understood and can be due to a number of non-linearities in the instrument. One candidate is self-focussing inside the optical components. This can be counteracted by active waveform control, for example, by measuring the wave front curvature during operation and by selectively heating the components in a preselected spatial pattern using an additional heating laser operating at some different wavelength. Such a system can be controlled by a slow feedback system to balance the non-linearity during high-power operation. This is one of the many control measures that had to be developed specifically for GW observatories and that are routinely operated during the science runs. However, these urgent technical issues will be resolved through even better technical solutions, and the optical power to reach the SQL will become accessible. It is now time to consider options for optimizing the instrument in this

10.3 Gravitational Wave Detection

regime and in particular to look at techniques for combating radiation pressure noise. This effect, observable at low frequencies, depends on the intensity fluctuations of the light in the interferometer, the opposite quadrature to the phase fluctuations that dominate the performance at high frequencies. Secondly, we have a well-known concept for improving the measurement by determining the momentum, or speed, of the end mirrors in the instrument and not the position. This concept has been proposed by V.B. Braginsky and F.Ya. Khalili [59, 60] first for microwave sensors and then optical devices and is now of interest again. The trick is to combine two successive measurements, with a delay time 𝜏D , and to combine them into one measurement of the speed 𝑣(t) via 𝑣(t + 𝜏D ∕2) = (x(t) + x(t + 𝜏D )∕𝜏D

(10.32)

In such a speedmeter using coherent light, the effect of the GW can be read out with a precision better than the SQL. Such a speedometer has superior low frequency performance. Various configurations are a possible and have been analysed in recent years. The details are intricate and many proposals have been made. The work of Y. Chen [61] considers Sagnac interferometers, and S.L. Danilishin [62] and others show that instruments based on the current Michelson design with some additional components to use the same laser beam and measure the position of the end mirror again after a full round trip of the instrument could show the desired enhancement at low frequencies. For the design of future of large-scale GW observatories, it is also useful to consider triangular instruments that combine essentially three linked interferometers, which cannot only detect the GW in their two polarizations but also include the advantages of speed meters. 10.3.4

Interferometry with Squeezed Light

All instruments have basically the same QNL. As can be seen from Figure 10.13, the quantum noise is the most important effect in the present GW detector for higher detection frequencies. It should be possible to improve the sensitivity using squeezed states of light. In most instruments the laser intensity will be sufficiently low that photon counting noise, the first term in Eq. (10.20), dominates, and improvements can be achieved by using a squeezed vacuum state as the input to the second port of the beamsplitter. A reduction of the ̃ †u ⟩ would lower the total noise. As shown in Figure 10.13, ̃ u 𝜹X𝟐 noise term ⟨𝜹X𝟐 this is equivalent to increasing the laser power, or shifting the noise curve along the horizontal axis, in Figure 10.15. This experiment requires that the squeezed light is phase-locked to the laser input. It has to be generated by a beam from the same laser and carefully aligned with the axis of the empty port. The effect is rather surprising; blocking off the invisible squeezed vacuum beam will increase the noise at the detector. This effect was demonstrated in a beautiful experiment by M. Xiao and co-workers [4]. Their aim was to show that squeezed light, which they had generated with an OPO (see section 9.4.2), could improve a practical measurement by at least 3 dB. Their apparatus is shown in Figure 10.16. A Nd:YAG laser with internal frequency doubler generated a beam at 1064 nm that illuminated a

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10.3 Gravitational Wave Detection

Mach–Zehnder interferometer. The beam at 532 nm is used to drive a degenerate OPO. Inside the interferometer are two electro-optic phase modulators. One controls the DC phase shift and thereby sets the operating point, namely, the dark fringe on the output. The other is used to introduce the signal, an AC phase shift at 80 MHz. The vacuum port of the interferometer is blocked. The amplitude of the signal 𝜙s is chosen to provide an SNR of 3 dB, as shown in Figure 10.16b. Now squeezed light is injected into the interferometer, and the noise floor drops (Figure 10.16c). Note that the displayed signal trace also drops since it is the quadrature sum of signal and noise. The SNR is improved by about 3 dB. These landmark experiments showed beyond any doubt that squeezing can be a powerful tool. Almost simultaneously Slusher’s group at ATT-Bell Laboratories obtained similar results using a polarimeter and pulsed squeezed light [7] with an improvement of 2.8 dB. The light used for the empty port does not actually have to be a squeezed vacuum state. The requirement is only that the power of the beam at this port is much weaker (≤ 10 percent) than that in the other port; otherwise the uncorrelated quantum noise of the laser would appear in the output signal. The second condition is that the correct quadrature (X2) is squeezed. The group at Stanford [63] realized that this can be achieved with two phase-locked lasers, one for each input port. A high-power diode laser illuminates one input; a low-power, intensity-squeezed diode laser illuminates the other. The squeezing quadrature is selected by choosing the absolute phase difference between the two waves. If a beam, which is squeezed in the X1 quadrature, is shifted by 𝜋2 , the reduced fluctuations appear in the X2 quadrature as seen by the interferometer. The initially surprising result is that intensity squeezed light can be used to improve the sensitivity of an interferometer. So far we have assumed that the two inputs, laser and vacuum state, have independent fluctuations. What would happen if both inputs were strongly correlated? Quite early the case of two number states has been investigated by M.J. Holland and K. Burnett [64], who showed that an improvement in sensitivity can be achieved that is as good as the use of vacuum squeezed light. In √theory the sensitivity can be improved such that Δ𝜙 no longer scales with 1∕ I but with 1∕I. However, their scheme is currently not practical since it requires number states and is extremely sensitive to losses. A simplified optical layout of the squeezed light-enhanced interferometer is shown in Figure 10.17. The observatory contains the established GEO600 and the additional squeezed light source (yellow box). The detector has two singly folded arms with 600 m each resulting a total optical length of 2400 m. The instrument is operated such that almost all the light is back-reflected towards the 12 W input laser system by keeping the interferometer output on a dark fringe by a control system. A power-recycling mirror (MPR) leads to a resonant enhancement of the circulating light power of 2700 W at the beamsplitter. Similar to the power-recycling technique, a partially transmissive signalrecycling mirror (MSR) is installed to further resonantly enhance the GW-induced signal at the interferometers output. When operated with squeezed light, this instrument achieved the predicted quantum reduction of

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Figure 10.17 Simplified optical layout of the GEO600 GW observatory enhanced by squeezed light. Details of components are BS, 50/50 beamsplitter; SHG, second harmonic generator; OPA, optical parametric amplifier; DBS, dichroic beamsplitter; PLL, phase locking loop; MFe/MFn, far interferometer end mirrors (east/north); MCe/MCn, central interferometer mirrors; and T, mirror transmissivity. All interferometer optics are suspended by multistage pendula and located in a vacuum system. Source: From R. Schnabel et al. [65].

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10.3 Gravitational Wave Detection

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Figure 10.18 Non-classical reduction of the GEO600 instrumental noise using squeezed vacuum states of light, calibrated to GW-strain and differential mirror displacement, respectively. In black the observatory noise spectral density is shown without the injection of squeezed light. An injection of squeezed vacuum states into the interferometer leads to a broadband noise reduction of up to 3.5 dB (red trace) in the shot noise limited frequency band. The spectral features are caused by excited violin modes of the suspensions (600–700 Hz and harmonics) as well as by calibration (160–2500 Hz) and OMC alignment control (250–550 Hz) lines. Both traces were averaged over four minutes. The resolution bandwidth is 1 Hz for frequencies below 1000 Hz and 2 Hz at higher frequencies. Source: From R. Schnabel et al. [65].

the GEO600 instrumental noise calibrated to GW-strain and differential mirror displacement, respectively (see Figure 10.18). In black the observatory noise spectral density is shown without the injection of squeezed light. Below 700 Hz residual noise from the environment couples into the observatory. At higher frequencies GEO600 is shot noise limited; note that the slope in the kHz regime is due to the normalization and the frequency-dependent signal enhancement of GEO600. An injection of squeezed vacuum states into the interferometer leads to a broadband noise reduction of up to 3.5 dB (red trace) in the part of the spectrum that is quantum noise limited. In this first example the traces were averaged for minutes. Later it was demonstrated that the squeezing was as reliable as all the other locking systems, operating for many hours and hardly effecting the duty cycle of the observatory at all [66]. GW detectors are an ideal match for squeezed light. For the enhanced LIGO, which was successful in detecting GWs in the last year, there are many technical options to improve the performance and sensitivity, as shown in Figure 10.13 and described by the GEO and LIGO teams[67, 68]. A careful comparison of all these options, including improved light sources, beamsplitters, reduction of thermal noise by cooling the detectors etc., will lead to a strategy that include squeezing in the upgrades over the next few years. Any GW detector has an event horizon to which it can see an event with a certain strength. The volume around us that can be seen scales with the third power of the strain sensitivity of the detector.

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That means quantum enhancement by a factor of 2 or 3 dB enlarges the detection volume by a factor of 8. If the universe was populated homogeneously with such sources, this should increase the frequency of observation of events with similar strength by a similar factor. This would be a major step forward and without doubt squeezing will play an important role in future GW detection. 10.3.4.1

Quantum Enhancement Beyond the SQL

In order to find a practical way to enhance the interferometer and to go below the SQL, we have to consider the behaviour of the 𝜃 parameter in Eq. (10.20). The interferometer system, even without cavity enhancement, will have an optomechanical resonance Ωmech given by the mechanical suspension and test mass, and the observations of GW will be carried out around this frequency. The response of the interferometer will be very different at both sides of the resonance. By using squeezed light of the correct quadrature, we can reduce the noise in the output. However, we have to recognize that the optimum squeezing quadrature depends on the intensity through intensity dependence of 𝜃. Hence at low intensities I < ISQL , the light should be phase squeezed (𝜃s = 90), at high intensities where radiation pressure dominates I > ISQL the light should be amplitude squeezed (𝜃s = 0), and at the SQL the squeezing angle should be between (𝜃s = 45). Furthermore, we find that for GW detectors that contain resonant storage cavities, the squeezing quadrature, or 𝜃s , depends on the detection frequency. For such systems the value of the quantum noise equivalent detection limit will increase with higher frequencies – as shown in Figure 10.13. Consequently, we will require squeezing with detection frequency-dependent squeezing angle. This is an additional requirement that can be satisfied by using a standard squeezing source (𝜃s = 90), such as an OPA, and two cavities to rotate the squeezing quadrature. This is shown in the study by H.J. Kimble et al. [47]. This early design called for mode cleaners of the full length, e.g. for LIGO 4 km. When operated at the SQL and with Vu = 0.1, this design would allow an improvement Δ𝜙sq ∕Δ𝜙SQL = 0.3. There are alternative configurations that can be used to improve an interferometer below the SQL. The study by H.J. Kimble et al. [47] evaluated a scheme where the squeezed light with suitable frequency-dependent quadrature is used in a homodyne arrangement to suppress in the detection. An enhancement of the sensitivity follows. An alternative, which has been investigated, is to create the correlation between the photon counting noise and the radiation pressure noise inside the actual interferometer. In recycled interferometers, such as LIGO, the dynamics of the motion of the mirror is more complex than the dynamics of a free mass. The radiation pressure acts like a force, and one can consider this to act like an optical spring. This was analysed, for example, by the Caltech LIGO team and also H.J. Kimble et al. [47]. This force follows the intensity fluctuations and thus the second quadrature of the light. By appropriately engineering the frequency-dependent squeezing and optimizing the quadratures to the noise suppression, it is possible to combine the effects below and above the resonance frequency in a constructive way as described, for example, by the LIGO team [69] and by R. Schnabel in Hannover [58]. Figure 10.19 shows the simulation for the case of a simple interferometer with neither resonator nor recycling, for

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an optical power of 1 MW at the beamsplitter, a test mass of 1 kg and light at 1550 nm. The noise equivalent minimum displacement is shown for a coherent state and squeezing light. Through the right adjustment of the apparatus, the squeezing quadrature and the optical phase are possible to achieve operation below the SQL for all observation frequencies. This application of squeezed light will become a component of future GW observatories.

10.4 Quantum Enhanced Imaging 10.4.1

Imaging with Photons on Demand

In Section 7.4 we discussed the impact of quantum noise and photon statistics on imaging technology. The images in Figure 2.14 show the effects of low photon flux and the corresponding graininess of the images. The improved technology through low noise detectors operating at room temperature, for example, the latest generation of CMOS imaging chips, allow quantum noise, or shot noise limited imaging to be applied to many more situation and at moderate cost. The graininess due quantum noise is unavoidable, unless we can use light sources that would produce perfectly well-defined numbers of photons for each pixel and time interval. Such improvements are coming from light sources that produced a well-defined stream of photons, as described in Section 7.1 and summarized in Table 7.2. There has been some progress in making such photon on demand light sources, and it is possible to improve an imaging system in this way. However, in practice the complexity, low efficiency, and high cost of these devices results in a low uptake of this technology for imaging. In most cases it is easier to

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use a stronger conventional light source with Poissonian statistics and to simply increase the photon flux and in this way the SNR. 10.4.2

Quantum Enhanced Coincidence Imaging

An alternative is to use the steadily improving photon pair generators, as described in Section 6.3.4. We can arrange for the two members of each photon pair to propagate in two separate spatial modes, for example, two Gaussian modes propagating in different directions or two orthogonal modes in one direction. For each pair it is possible to use the information from the detector of one mode as a trigger to look for the detection of the photon in the complementary mode. In the literature this is described as using a heralded photon. If there is no detection event in the second counter, the measurement event is suppressed. This type coincidence imaging is a form of post selection, which creates information with much suppressed noise. In the ideal case of only pairs, this would retain the speed of detection arriving at the detectors. However, in practice there will be losses in one or the other of the two modes, and the speed of detection will decline. That means we should ideally look for lossless imaging systems with a total quantum efficiency for detection close to unity. Situations to be avoided are, for example, where one or the other of the photodetectors has a low quantum efficiency or the case of two photon fluorescence, where the generation of photons for imaging has a low efficiency. Again, averaging over longer averaging time intervals would result in the equivalent improvement of the SNR. The demand for such complex quantum enhanced imaging devices is likely to grow for applications that demand both low light flux and high speed of imaging. Many opportunities exist, for example, in biological and medical imaging. The techniques of coincidence imaging is an excellent use of photon pairs or photon entanglement. Many applications have been studied in great detail under the title of ghost imaging. Take the concept shown in Figure 2.18 of using correlated or possibly even entangled beams. Here the image information comes from those events where both detectors, one with spatial resolution and one without, record in coincidence one photon each, which means a complete pair. The first demonstration was in 1995 by T.B. Pittman et al. [70]. This was rapidly followed by many more demonstrations and an intense discussion and a large number of publications concerned with the questions: how well could ghost imaging be performed with classical light and what advances could be expected from quantum light and entangled photons [71–73]? The answer depends significantly on the type of imaging system used, either using a lens based or a lensless system. It matters whether we are considering the near- or far-field images and most importantly the photon flux involved. Whilst the whole story is complex, the main conclusion for practical purposes was that the imaging process does work with a large variety of light sources and the advantage of quantum light is essentially all contained in the improvement of the SNR of the images. We can enhance the SNR by using quantum light sources with a better than Poissonian photon statistics or with entanglement. By post-selecting the photon pairs, limitation due to quantum efficiency can be largely avoided. The most recent summary of the state of the art is given by A. Meda et al. [74].

10.4 Quantum Enhanced Imaging

In order to apply this to imaging, we could just use a single mode of light with reduced fluctuations and use it in a scanning arrangement in order to create an image. This is possible and would certainly enhance the quality of the image. However, it would also be slow. The alternative is to use several spatial modes simultaneously. As discussed in Section 10.5.1, this allows us to focus on many degrees of freedom, the same number as independent and orthogonal modes, and to handle the rich information content of an image. Since about 1995 spatial multimodal light can be efficiently generated. For example, the experiments by the team around P.D. Lett at NIST demonstrated beautiful early imaging results with quantum light generated by four-wave mixing (4WM) [75]. They showed that different shapes could clearly be distinguished. The next level of improvement is to optimize the multimode output of the down-converter to create a multimodal structure, which effectively produces very well-defined beams with multiple areas that can be imaged in a wide-field microscope as individual imaging sections [76]. By preserving the correlation between the two beams down to modes that effectively cover only micron-sized areas, N. Samantaray et al. have built a microscope that can resolve areas of about 400 μm in size [77] with a resolution of about 5 μm. This system uses a high-resolution CCD camera, and the data from the individual pixels of the detector are averaged over a number of d2 neighbouring pixels to optimize the performance. Figure 10.20 shows results for averaging for d = 6, which means averaging over 36 pixels, and compares the results for direct imaging (DR); noise improvement via classical differential measurements, where independent noise is suppressed (DC); and genuine sub-shot noise (SSN) imaging. This is the first demonstration of a quantum enhanced wide-field microscope that is suitable for dynamic imaging, including videos. There are some fundamental questions that can still be followed. One cute extension of the concept of coincidence imaging was explored by the Vienna team with Anton Zeilinger, who designed an apparatus that creates images in such a way that the photons that go through the object are never actually detected and the photons that form the image have never gone through the object [78]. This can be achieved by using two non-linear crystals for the down-conversion. The photon from the first photon pair, which contains the image information, Figure 10.20 Noise reduction in images created with quantum light. Comparison between the single shot direct image (DR), the differential classical (DC), and the sub-shot noise (SSN) images. Top right is a faithful image of the object, about 400 μm in size, after the averaging over 300 shots with the best resolution of 5 μm. Source: From N. Samantaray et al. [77].

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Figure 10.21 LHS. Schematic layout for quantum imaging with undetected photons. Source: From G.B. Lemos et al. [78].

is used as the input to one of the crystals and thus controls the generation of the second photon pair. The details are described in Figure 10.21. Laser light (green) splits at BS1 into modes a and b. Beam a pumps the non-linear crystal NL1, where collinear down-conversion may produce a pair of photons of different wavelengths called signal (yellow) and idler (red). After passing through object O reflects at dichroic mirror D2 and is aligned with the idler beam produced in NL2 such that the final emerging idler f does not contain any information about which crustal produced the photon pair. Therefore, signals c and e combined at beamsplitter BS2 interfere. Consequently, signal beams g and h reveal the spatial properties of object O. RHS: intensity images obtained (a) constructive and destructive interference are observed at the output of the beamsplitter, when the cardboard cut-out shown in (b) is place in the path D1–D2. This arrangement follows the concept of the which-path experiments, and as long as the pathways are indistinguishable, there will be interference of the final photons, as shown in Figure 10.21. For a team based in Vienna, the home town of Schrödinger, the object to be imaged obviously has to be a cat.

10.5 Multimode Squeezing Enhancing Sensors 10.5.1

Spatial Multimode Squeezing

When strong beams of light are used for imaging, questions appear such as: could there be quantum correlations within a beam in the transverse plane? Can one have local squeezing in small parts of the beam? Could there be a spatial manifestation of squeezing? The study of such questions formed a growing domain in quantum optics, and this form of quantum imaging, which was pioneered by M.D. Levenson and the St. Petersburg group, and it has been extensively studied from the mid-1990s. For an overview see, for example, the summaries by M.I. Kolobov, L.A Lugiato et al. [79, 80]. Let us consider a single transverse mode of the field. In 1989, M. Levenson et al. commented that strictly speaking a single spatial mode is not possible and that a real laser beam would always have some fluctuating contributions from other higher-order spatial modes. These do not affect the intensity of the laser, but bring fluctuations to its transverse variation.

10.5 Multimode Squeezing Enhancing Sensors

In a discussion aptly named ‘Why a laser beam can never go straight’ [81], he described the consequences of the uncertainty principle on the spatial mode distribution. Using a diode laser he showed that the presence and the noise of such higher-order spatial modes can be detected and are in agreement with simple models. One can show that in a highly squeezed single-mode beam, the local quantum fluctuations, which can be measured on a very small photodetector, are at the QNL [82], the local fluctuations at different points of the transverse plane being correlated in such a way that they cancel when all these contributions are summed up on a large area detector. It is only when the many transverse modes are simultaneously squeezed that one can get a local noise reduction below the QNL. Spatial quantum effects and multi-transverse modes are thus intimately related. M.I. Kolobov and I.V. Sokolov [83] predicted that travelling wave frequency-degenerate parametric amplification in a thin crystal pumped by a plane wave is able to produce such a multimode squeezed light. When the parametric gain is large, the non-linearity creates quantum correlations between various tilted plane wave modes at the subharmonic frequency with wave vectors k and k ′ such that k + k ′ = kpump ; in the same way as in the time domain the correlation between symmetrical sideband modes creates broadband squeezing. L.A. Lugiato and A. Gatti [82] predicted that the same property holds if one places the parametric crystal in a resonant cavity with plane mirrors, provided the system is operated just below the oscillation threshold. In both situations strong quantum correlations should appear between the symmetrical pixels in the far field, which means correlations between the k vectors. Only transverse-degenerate cavities are likely to produce these spatial intensity quantum effects. L.A. Lugiato and P. Grangier [84] have verified this and showed that frequency-degenerate OPOs with a confocal cavity produce below threshold squeezed vacuum states whatever the size and shape of the detector-sensitive area, provided that it is symmetrical with respect to the cavity axis. Almost all the temporal quantum effects of squeezing discussed in Chapter 9 can be extended into the spatial domain. For example, one can produce spatially resolved quantum-correlated beams using parametric amplification [85], and noiseless parametric amplification of images can be performed [86]. Parametric down-conversion is not the only way of producing such spatial quantum effects: second harmonic generation can produce interesting spatial quantum effects [87]. Spatial solitons propagating either in 𝜒 (2) or 𝜒 (3) media have interesting spatial quantum properties [88]. One can show, for example, that more squeezing on a spatial soliton can be measured if one takes only the central part of the pulse [88]. Such a spatial filtering is an exact analogy of the spectral filtering, which has been shown to improve the squeezing in temporal solitons (see Section 9.5.4). Possible applications are numerous, such as the detection of weak phase and amplitude images [89], noiseless amplification and teleportation of images [90], two-photon microscopy [91], microlithography [92], and detection of small changes in an image [93]. Multimode squeezed light might also improve the ultimate resolution in imaging [94] using special reconstruction techniques. One pioneering experiment is the noiseless amplification of a double slit image

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by frequency-degenerate travelling wave amplification, performed by the team led by P. Kumar and co-workers [95]. The nontrivial spatial distribution of correlations inside twin beams produced by a confocal OPO has been reported [96]. The group led by P.D. Lett at NIST has extended this work to the detection of spatial patterns [75] showing that significant improvements can be made with quantum-correlated photons and coincidence detection. A different way to produce multimode non-classical light was chosen in the experiments at the ANU, where the idea is to synthesize such a beam from single-mode squeezed beams. It aims at measuring with maximum accuracy the transverse displacement d of a light beam, using a split detector. The difference between the photocurrents measured on the two segments produces a very sensitive signal proportional to the transverse displacement from the centre. The quantum noise from the two segments cannot be subtracted and sets the QNL dQNL for the measurement of the displacement. This value is equivalent to the size Δ of the beam at the detector divided by number of photons n detected during one measurement: √ (10.33) dQNL = Δ∕ n In order to go beyond this limit, it was shown by the Paris group that a normal intensity squeezed beam is not useful; instead one must have a mixture of two transverse modes: a normal squeezed TEM00 vacuum mode and a bright coherent ‘split mode’, which has the same Gaussian intensity distribution as the TEM00 mode but with a 𝜋 phase shift between the two halves. This solution is in true analogy to the homodyne detector (Section 8.1.4), where two beams, one bright and one vacuum, are combined on a beamsplitter and one of the four beams recorded by the two detectors has a 𝜋 phase shift. Experimentally such a multimode state is synthesized from normal TEM00 modes using split phase plate with a 𝜋 retardation left to right and combining the beams either with a beamsplitter [97] or a ring cavity [9], which transmits the symmetric mode and reflects the asymmetric mode with minimum losses (see Figure 10.22a). In a first experiment a noise reduction of −1.7 dB below the QNL in the position measurements was achieved, allowing the measurement of beam displacements of a less than 1 nm. The measurement was extended to two dimensions, which requires the mixing of three spatial modes, one bright coherent and two squeezed vacuum beams. A noise reduction of −3.5 dB in the horizontal and −2.0 dB in the vertical direction was obtained. This quantum laser pointer can measure spatial modulations of the beam position with a detection limit a factor of 2 smaller than a conventional laser, which is shown in Figure 10.22b. This is a truly sub-QNL spatial multimode state of light that can be used in a sensor. One example of possible applications of this concept was to demonstrate quantum enhancement in optical force microscopy in rheology and the tracking of small particles in living cells. The work of M. Taylor et al. in Australia, using a custom made optical force microscope, was based on the concept of optical tweezers built at UQ Brisbane and transported to a squeezer at ANU Canberra [10]. The experimental layout (Figure 10.23) shows a combination of several tricks discussed in previous sections. One laser beam was used to hold a yeast cell in the microscope using optical forces. A second modulated probe beam, with a

10.5 Multimode Squeezing Enhancing Sensors

Local oscillator MC

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TEMf00

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Figure 10.22 The quantum laser pointer. (a) The experimental layout that shows the mixing of three beams, two vacuum beams, and one bright coherent beam with different spatial phase distributions using a ring cavity. (b) The measurement of a spatial oscillation with increasing amplitude (horizontal scale). The recorded value (vertical scale) is noisy due to the short recording intervals. The noise floor for the squeezed beams (II) is below the noise floor for the conventional beam (I), and the value for smallest detectable oscillation is reduced by a factor of 2. Source: After N. Treps et al. [9].

Gaussian spatial mode profile, is scattered by the moving lipid, which will generate light in both a symmetric mode for a stationary particle and an asymmetric mode from the movement of the particle. This light is detected by a homodyne detector with a squeezed local oscillator beam that has a flipped phase profile and thus is closely matched to the required asymmetric detection mode. In this way the displacement of the small particle, which in this case is the fast jittering of the lipid, around a slowly changing average position was optimally detected. The detection mode is closely matched to the degree of freedom that we want to observe. The scattered light follows the time modulation at the frequency ΩAM of the probe beam, moving the information far away in frequency from technical

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Phase plate

PBS

CCD Trapping field

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be

o Pr

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AM

Figure 10.23 Detecting the mobility of small particles inside a living cell. The cell is located in an optical trap generated by 10–500 mW of light at 1064 nm (orange). Polarizing optics isolates the trapping light from the detector. An imaging field at 532 nm and a CCD camera are used for alignment. A separate Nd:Yag laser produces the probe light at 1064 nm for a squeezed local oscillator beam (red) with flipped phase for the homodyne detector that measures the scattering of the AM modulated probe beam by the moving particle into the asymmetric mode defined by the local oscillator. Source: After M.A. Taylor et al. [10].

laser noise. With electronic spectral analysis of the sidebands around ΩAM , the movement of the particle was monitored for a signal range between a few Hz for slow drifts to about 10 kHz for the fast random dynamic movement. The local oscillator beam was squeezed to −3 dB, and it was demonstrated that the SNR of the spatial signal was improved by a factor of 2. The fluctuating signal, at higher frequencies, was used to measure the mobility of the particle by comparing and calibrating it with the standard spectrum of a particle experiencing Brownian motion. This was tested with recording the mobility of needs moving in water. The size and spectrum of the signal showed the restricted mobility of the lipid inside the cell, which is valuable information for biologists. With squeezed light the mobility was determined with 22% enhanced precision in the same time interval or a 64% increase in measurement speed with the same precision. At the same time the authors were able to track the particle in space, by integrating the slow frequency components of the signal. By combining the information from both frequency regimes, Hz and kHz, they were able to measure the change in mobility and position in one coordinate at the same time. A map of mobility, created by following the particle, in this case in one dimension, can provide information about the internal structure of the cell. Extending the technique to 3D tracking is in principle feasible. It would require multimode squeezing tailored for this application.

References

This device is an example of how optical instruments can be improved using the techniques of quantum optics. An additional feature of this experiment was that it was performed with a living biological sample, a situation where the maximum optical power is limited since an increase in power could have resulted in photobleaching and damage the cells. Such situations make good use of the enhancement provided by quantum optics techniques.

10.6 Summary and Outlook There are already very impressive examples for the use of squeezed light in practical devices and for various applications. This is one form of quantum technology, which will grow in the future. The main prerequisite is that the device is already quantum noise limited and that a simple increase in the laser intensity to boost the SNR is not possible. The ideal candidate has been the GW detector, where the increase in sensitivity due to squeezed light has been demonstrated and will be implemented eventually in the large-scale GW observatories. Spatial sensors, imaging devices, and timing sensors are the second category. Squeezing is particularly advantageous in the study of photosensitive samples, such as living organisms where photobleaching at high intensities is a concern. In all these cases it is important to identify the special mode that contains the information to be measured and the degree of freedom to be investigated and then to match the squeezed light to this detection mode. Another application of squeezed light is the calibration of the quantum efficiency 𝜂 of photodetectors that can be measured with high accuracy be recording the effect of 𝜂 on the degree of squeezing. This method is free of many of the common assumptions and limitations for calibrating photometers through measuring of the optical power and has been elegantly demonstrated by H. Vahlbruch et al. [98]. More applications will emerge from using multiple squeezed modes and the generation of beams of entangled light. The simplest and most famous case is a pair of entangled beams from two squeezed beams, followed by more complex sets of beams with high-dimensional quantum correlations and generalized version of entanglement. Such multimode entangled light has applications in many quantum information devices and protocols and is discussed in Chapter 13.

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11 QND One old question in quantum optics is whether we could measure and record a state of light and maintain it at the same time. Whilst it is clear that we could not achieve this for all the properties of the state or laser beam simultaneously, it looks possible as a goal if we restrict ourselves to one quadrature, or one property, at a time. Such an experiment would be called a quantum non-demolition (QND) measurement. In this chapter we will discuss techniques that allow non-destructive optical measurements with a quality better than possible with coherent light and conventional optics.

11.1 QND Measurements of Quadrature Amplitudes The properties of a beam of light can be described by its two quadratures X1 and X2. They are conjugate variables linked via the commutation relation, and that means, more descriptively, they are linked through the uncertainty principle. Quantum mechanics of coherent states predicts that a measurement of one of these properties will perturb the quantity that is measured. There is a back-action that adds noise to the system. A very simple example is the measurement of the modulation of the X1 amplitude quadrature of a QNL beam of light using a beamsplitter. We have seen that the remaining light, which was not used for the measurement, will still be at the QNL, but the signal-to-noise ratio (SNR) of the modulation will be reduced. Some part of the light, and with it some of the SNR, has been demolished in the experiment. Is there any chance of avoiding this back-action? Can we invent any technique to measure a property of the light without adding noise to this particular quadrature (Figure 11.1)? Historically, QND techniques were first discussed in the context of the problem of detecting the motion of a mechanical oscillator driven by a very small force. The limitations imposed by the quantum mechanical properties of the oscillator had to be avoided. V.B. Braginsky et al. [1, 2] developed these ideas first for mechanical systems and then generalized them to all harmonic oscillators. Different authors, including C.M. Caves and co-workers, developed the theoretical formalism [3, 4]. More recently, a number of experiments have demonstrated that, in principle, an optical QND measurement is feasible. The basic idea is to introduce a link between the quadratures, through a non-linear process, and to A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Figure 11.1 Cartoon of a QND.

Standard light detection T

LA

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1224 Counter

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Quantum non-demolition measurement

arrange the experiment such that all the back-action noise is transferred into the quadrature that is not interrogated. In this way the back-action is not avoided completely but is directed towards a quadrature that is not important for that particular measurement. As long as we ask only questions about one quadrature, we can, in principle, avoid the noise effects on the measurement. An idealized QND experiment of this type can be described by the interaction between two beams: one is the signal beam with quadratures X1s and X2s . Let us assume we wish to measure amplitude modulation, which means the properties of the X1s quadrature. The second beam is a probe, or meter beam, with the quadratures X1m and X2m . It will somehow interact with the signal beam. We are going to detect one quadrature of the meter beam, say, X2m , and learn from this about X1s . All the back-action induced by this measurement should influence only X2s , and it should leave X1s unchanged. The meter is distinguished from the signal through its frequency, polarization, or direction of propagation. We can describe the QND measurement by a set of equations that reflect the coupling of the quadratures: ̃ in ̃ out X𝟏 s = X𝟏s ̃ out ̃ in ̃ in X𝟐 s = X𝟐s + GX𝟏m ̃ in ̃ out X𝟏 m = X𝟏m ̃ in ̃ in ̃ out X𝟐 m = X𝟐m + GX𝟏s

(11.1)

The first equation describes the fact that there is no back-action on the quadrature we wish to measure. The second shows the back-action on the orthogonal

11.2 Classification of QND Measurements

Figure 11.2 Schematic layout of a QND device.

QND device

Signal in SNR sin

Non-linear medium

Signal out SNR sout

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Meter out

in SNR m

out SNR m

quadrature of the signal beam. The last equation shows that the information conout tained in X1in s is transferred to X2m , one of the quadratures of the meter beam (Figure 11.2). In the ideal QND experiment, as proposed by V.B. Braginsky and F.Ya. Khalili [5], the output beam has the same properties as the input beam. In particular, no change in the optical power or frequency has occurred. The quadrature under investigation is not changed at all, and the orthogonal quadrature is perturbed only as much as absolutely necessary. Such a measurement is almost impossible to carry out, but we can approach the ideal case. The question is: what type of interaction should be used? The best interactions are very similar to those used to create squeezed light. Again, the Kerr effect comes to mind. In a Kerr medium the amplitude quadrature of the signal influences the refractive index of the medium. This in turn influences the phase quadrature of the meter beam propagating through the sample, and a measurement of the meter will reveal the modulation of the signal amplitude. The amplitude quadratures of both signal and meter will not be affected since the medium is transparent. However, the phase quadrature of the signal will experience the refractive index change induced by the meter beam. This is the back-action of the measurement. What we have to avoid is any self-action of one quadrature to the other quadrature of the same beam. In this case the parametric gain induced in Eq. (11.1) is simply linked to the Kerr coefficient of the medium and the intensities of the beams. The perfect QND measurement corresponds to an infinite gain G, which is clearly not attainable in practice. A true QND experiment can be performed on one signal beam several times in a row, without introducing noise or affecting the quality of the quadrature under investigation. The key factor of QND measurements is repeatability [3]. In practice, other sources of noise accompany any system with large parametric gain. The reasons are basically the same as with squeezing experiments: all noise sources have to be suppressed, but this gets harder and harder with increasing intensity and non-linearity. Thus it is useful to characterize the degree of performance for a given QND measurement and to quantify how close a given experiment comes to the ideal QND device.

11.2 Classification of QND Measurements The characterization of a QND device can be carried out by comparing the four variances of the signal and meter beams before and after the interaction.

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Alternatively, one can perform correlation measurements between the fluctuations of the two beams, as proposed by N. Imoto and S. Saito [6]. The standard for any comparison should always be the ideal lossless beamsplitter, a device that introduces only the absolutely minimal amount of noise. Please note that the perfect beamsplitter is not a QND device – it is used as a reference. Many practical devices could actually be much worse than the ideal beamsplitter. An excellent summary of the characterization of QND measurements is given by J.-Ph. Poizat et al. [7]. One proposal to quantify the quality of a QND measurement was made by M.J. Holland et al. [8]. The performance of the QND measurement device is described by correlation coefficients between the input and the output modes in frequency space. A coefficient Cm links the signal input to the metre output, and it characterizes the fact that a measurement has been carried out: ̃ in ̃ out Cm = C(𝜹X𝟏 s , 𝜹X𝟐m ) ̃ in 𝜹X𝟐 ̃ out ⟨𝜹X𝟏 m ⟩ = √ s out V 1in s V 2m

(11.2)

The values for Cm are −1 < Cm < 1. A large correlation coefficient indicates a strong transfer from signal to meter. For a perfect measurement, Cm = 1. A second coefficient Cs describes the ability of the device to avoid back-action. It compares the input and output properties of the device: ̃ in ̃ out Cs = C(𝜹X𝟏 s , 𝜹X𝟏s ) ̃ in ̃ out ⟨𝜹X𝟏 s 𝜹X𝟏s ⟩ = √ out V 1in s V 1s

(11.3)

The values for Cs are −1 < Cs < 1. A system that introduces no back-action has Cs = 1. These values cannot easily be obtained in an experiment since they ̃ in require knowledge of the input fluctuations 𝜹X𝟏 s , which in turn would require a QND device for the unperturbed measurements. Any practical characterization will be based on readily available experimental data. This was first described by N. Imoto and S. Saito at NTT Japan [6], and later P. Grangier and co-workers showed that the measurement of SNR is both practical and can be used to quantify the performance of a QND experiment [7]. This is best done by transfer coefficients Tm and Ts that describe the transfer of the SNR from the input signal to the output of the meter and the signal, respectively: Tm = Ts =

SNRout m SNRin s SNRout s

(11.4)

SNRin s

It can be shown that for a perfect beamsplitter with reflectivity 𝜖, one obtains the values Ts,BS = 1 − 𝜖

and Tm,BS = 𝜖

(11.5)

11.2 Classification of QND Measurements

and thus the result Tm,BS + Ts,BS = 1. For the ideal QND device, we have Ts = 1, Tm = 1 and Tm + Ts = 2. In this way we have a criterion that a QND device should satisfy Tm + Ts > 1

(11.6)

1 Ts

The inverse value is known as the noise figure. Any classical device will have a noise figure > 1. To complement this criterion, we need a second independent quantity that measures the quantum correlations between the two outputs. This is the conditional variance, which is defined as 2 2 ̃ out (11.7) Vs|m = ⟨(𝜹X𝟏 s ) ⟩(1 − Cm ) For a perfect measurement, this should vanish, Vs|m = 0. A determination of Vs|m requires a measurement and comparison of the fluctuations of the photocurrent at the output. The conditional variance is already normalized to the quantum noise level. For the ideal beamsplitter, we obtain Vs|m,BS = 1. The achievement of Vs|m < 1 can be considered as conditional squeezing, it is non-classical in the same sense as squeezed light. The value of Vs|m is identical to the best noise suppression that could be obtained in a feedback control system (see Section 8.3) using the meter beam as the input for the control loop. The experimental determination of the conditional variance involves the addition and subtraction of fluctuations of Fourier components of the photocurrents. To achieve the best interference between the two terms, an optimum attenuation should be used as discussed in Section 7.3.4. This can be found empirically during the experiment and should agree with the correlation factor in Eq. (11.3) [7]. Because the classification of a QND device requires two independent criteria, it is best represented by a two-dimensional diagram with the axes Vs|m and T = Ts + Tm as shown in Figure 11.3a. The reference point is the ideal beamsplitter with Vs|m = 1 and T = 1. The extreme cases are the perfect twin beams with Vs|m = 0 and T = 1, the quantum duplicator with Vs|m = 1 and T = 2, and the perfect QND experiment with Vs|m = 0 and T = 2. The parameter space can be divided into four quadrants, each with a certain distinct meaning. The classical quadrant, top left, describes experiments with no quantum correlations, which can be performed with normal linear components. Any imperfections in the system or in the measurement, such as losses and nonideal quantum efficiencies, will move the result away from the optimum point. A value T > 1 indicates noiseless amplification as it can be achieved with a phase-sensitive amplifier, described in Section 6.2. On the other hand, Vs|m < 1 indicates quantum state preparation. The combination of the two, represented by the bottom right quadrant in Figure 11.3, indicates a true QND device. Finally, the best and most important test for a QND device is that it can be operated in sequence, which means several QND measurements can be applied to the one mode of light, each producing the same result and each adding as little noise as possible. A complete demonstration of a QND measurement would show that the state is not more perturbed than absolutely necessary in accordance with the quantum description of the process. This classification is not without dispute. It is probably not complete since it allows devices to be classified as QND that one would otherwise not regard as

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1 T = T m + Ts

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Figure 11.3 Characterization of quantum measurement devices introduced by Poizat et al. [7]. The parameters Vs|m and T = Tm + Ts classify all possible devices. The conditions for a true QND device are satisfied with Vs|m < 1 and T > 1. The best experimental results in about 2001 are shown by numbered circles, referenced in the text. This diagram is similar to the review by P. Grangier et al. [9]. 1, M.D. Levenson et al. Kerr Fibre [10]; 2, A. La Porta et al. OPA [11]; 3, J.A. Levenson et al. OPA [12]; 4, K. Bensheikh et al. OPA Fibre [13]; 5, S.F. Pereira et al. OPO [14]; 6, S.F. Schiller et al. OPO [15]; 7, J.Ph. Poizat and P. Grangier atoms [16]; 8, J.F. Roch et al. atoms [17]; 9, R. Bruckmeier et al. squeezing [18]; 10, B.C. Buchler et al. squeezing and feedforward [19]; 11, B.C. Buchler et al. squeezing and feedforward [20]; 12, S.R. Friberg et al. solitons [21]; 13, F. König et al. solitons [22]; 14, E. Goobar et al. LED [23] and J.F. Roch et al. LED [24].

such. One example is to use a combination of photodetector, amplifier/splitter, and light source to detect a beam of light, generate a new beam, and measure the intensity fluctuations electronically in between. In this experiment all the light is detected, and assuming the very good quantum efficiencies for both detection and generation, the intensity fluctuations, including the quantum noise, can be measured without back effect [23, 24], shown as point (14) in Figure 11.3. This good result is not surprising since in this case we measure current fluctuations, not quadrature fluctuations of the light. However, only information of the square of one quadrature is available, and the information about the other quadrature is completely destroyed; the only property that is preserved is the intensity. It has been argued that such a destructive device should not be called QND device and that a third parameter is required to classify the experiment. This additional parameter would measure how close the overall effect on all quadratures is to the minimum destructiveness required by quantum rules. Unfortunately, no generally accepted definition of such a third parameter has been given so far, and the review by P. Grangier et al. [9] still summarizes the results very well.

11.3 Experimental Results The fundamental difficulties in QND measurements are very similar to those of squeezing experiments. We require strong non-linear effects that couple the quadratures of the signal and meter modes, no losses or noise, and very efficient

11.3 Experimental Results

detectors. The measurements of both Vs|m and T can be achieved with the same instruments as used for squeezing measurements. The only addition is the optimization of the electronic gain in order to measure the combination of electronic voltages that corresponds to the parameters Vs|m and T. A summary of the presently achievable results is given in Figure 11.3. Early experiments, such as the work by M.D. Levenson et al. [10, 25], point (1), used the non-linear properties of optical fibres to introduce correlation between the quadratures. They obtained some small degree of quantum correlation but insufficient transfer. A whole group of experiments used optical parametric down-converters. La Porta et al. [11] carried out an experiment with a potassium titanyl phosphate (KTP) crystal as the optical parametric oscillation (OPO). This was driven by pulses at 532 nm from a frequency-doubled Nd:YAG laser. A phase-locked ring cavity was used to recirculate these green pump pulses through the crystal. Two infrared beams at 1064 nm, which are distinguished by their polarization, served as the signal and meter. The results, point (2) in Figure 11.3, show that the conditional variance is smaller than in a classical device but the transfer values are not sufficient. This experiment demonstrates quantum state preparation. More recent experiments using better materials and with smaller losses satisfy all the requirements of a QND device. J.A. Levenson et al. [12] and K. Bensheikh et al. [26], points (3) and (4), respectively, used a KTP crystal pumped by pulses from a frequency-doubled Nd:YLF laser a single-pass travelling-wave optical parametric amplifier (OPA). The signal and meter beams are at 1054 nm, and the meter beam is generated inside the OPA. Improved results were achieved by S.F. Pereira et al. [14] with a continuous-wave (CW) experiment. A KTP crystal, inside a resonant cavity, is optically pumped. Signal and meter beams, represented by orthogonally polarized beams at 1080 nm, are measured in direct detection. Their results, point (5), show a clear QND performance. S. Schiller et al. [15] utilized the properties of a LiNbO3 OPO that operated as the generator for a strongly squeezed vacuum state. This device did not only squeeze, but introduced quantum correlations on to a second beam injected into the same cavity. This is clearly a QND result, shown as point (6) in Figure 11.3. A completely different experimental approach was chosen by P. Grangier et al. [27]. They used the Kerr effect of a sodium atomic beam to correlate the quadratures. The signal beam at 589 nm was coupled into a resonant cavity, with the atoms inside. A second beam, the probe at 819 nm, tuned to a second cascaded atomic resonance, picks up a modulation of the phase introduced by the intensity fluctuations of the signal beam. Losses and competing noise sources that appear when the light is tuned close to the atomic resonance limit this particular scheme. Whilst these first results produced only a conditional variance of 1, the improved experiments by J.P. Poizat and P. Grangier [16], point (7), clearly showed sufficient transfer. More recent experiments from this group used laser-cooled atoms [17] in order to take advantage of the reduced atomic linewidth and higher densities (Figure 11.8a). This is one of the most difficult and most advanced quantum optics experiments based on atoms and produced some of the best QND results, as shown by point (8) in Figure 11.3. The QND measurement of optical solitons, which was proposed by H.A. Haus et al. [28], was demonstrated by S.R. Friberg et al. [21] (point (12) in Figure 11.3).

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The underlying effects are identical to those responsible for the generation of squeezed solitons as described in Section 9.5.4. This technology was extended by F. König and co-workers in Erlangen who used spectral filtering [22]. In a series of experiments, they showed the QND measurement of the soliton photon number and achieved, with optimized gain, the values T = 1.37, Vs|m = 0.92, point (13) in Figure 11.3, and the best value of Vs|m = 0.73. Finally, it is possible to build a very simple experiment with an LED and an optical detector that can achieve a high combined quantum efficiency. This arrangement can achieve a very good performance for intensity signals and thus a transfer function T. At the same time the value of V is limit to 1 and thus the classical limit. Such systems were demonstrated in 1997 by E. Goobar et al. [23] and J.F. Roch et al. [24] to demonstrate the capability of such simplified systems and are shown as point (14) in Figure 11.3. The crucial tests for repeatability were carried out in the groups led by K. Bencheikh and J.A. Levenson [13] and S. Schiller [15]. Both could show that whilst the second measurement is not quite as good as the first, both consecutive measurements can be classified as QND measurements. In addition, J.A. Levenson et al. had demonstrated that two cascaded optical taps [12] produce a better transfer of information than a perfect beamsplitter. These experiments open the way for the design of optical taps, devices that can be used to distribute, or eavesdrop, on information sent via optical fibres. QND promises the ability to tap into information transferred optically without reducing the SNR to a large degree. However, in practice, QND in this form is unlikely to be a useful attack in a communication system due to its high sensitivity to losses.

11.4 Single-Photon QND We have seen in Chapter 3 that the coupling between light and atoms inside a cavity can be used as a direct evidence for the existence of photons. Very clear demonstrations exist for the coupling of microwave radiation to atoms in highly excited Rydberg states. In such a micromaser the radiation is contained inside an almost lossless superconducting micro-cavity, and the advantages are the very long lifetime of the atoms, several milliseconds, the ability to avoid thermal noise by cooling the cavity, and the ability to prepare and monitor the atoms, which are injected into the cavity and are analysed, via field ionization when they leave the cavity. Such an arrangement is shown in Figure 11.4. The optical state of interest is contained inside the cavity, which interacts closely and is modified by the atoms, and we can gain information of the light by analysing the quantum state of the atoms coming from the cavity. The optical field is not detected directly, but instead the information is indirectly gained from the properties of the atoms. The atoms and light are in a combined state, sometimes referred to as a mesoscopic superposition state, and a measurement of the atoms, leaving the cavity, provides information about light inside the cavity [29], with minimum back-action. Whilst the atoms outside the cavity are destroyed, the properties of the photons inside the cavity can be inferred.

11.4 Single-Photon QND

O

L′1 S

B L1 L2 R1

C De

S′ R2

Dg

Figure 11.4 Schematic layout of a single-photon QND experiment. The components are O = oven, L1 = optical pumping and velocity selection, L2 = optical excitation to Rydberg state, R1 = state preparation by 𝜋/2 pulses from microwave source S′ , C = cooled, low-loss microwave cavity, R2 = second 𝜋/2 microwave pulse for state preparation, and D = field ionization. Source: After M. Brune et al. [29].

The measurement of the atoms is equivalent to a QND measurement of the light in the cavity. Like many single-photon experiments, the data acquisition relies on post-selection, but here that means using only those events where exactly one atom is present. In the first experiments (Section 3.7), it was possible to see the quantization of the microwave field by observing the Rabi oscillations of the atoms. By carefully selecting the velocity of the atoms, the magnitude of the Rabi frequency provides a measure of the field strength inside, and the apparatus is sufficiently sensitive to distinguish individual photons. Next we can use the interaction inside the cavity to create special states of light, and it has been shown, both in the experiment in München [30] and Paris [31], under the respective guidance of H. Walther and S. Haroche, to produce results closely resembling one- or two-photon Fock states. One early example, given in Figure 11.5, shows a remarkable agreement between theory and experiment for both measuring the Rabi frequency of the resonator and the corresponding photon probabilities. The technology has been steadily improved, and dramatic progress is being made, in particular in the laboratory in Paris where the all aspects of the cavity QND experiment have been improved. The theoretical background and technical advances are beautifully described by S. Haroche and J.-M. Raimond [32]. The technical progress includes improved control of the environment of the atoms, better superconducting mirrors, less influence from thermal noise and other sources, and perfection in the delivery and detection of the Rydberg atoms that are used to probe the microwave field via state tomography. Using data from a long uninterrupted sequence of these QND measurements, the decay of the photon state can be tracked for up to a second. This is shown in Figure 11.6 taken from an experiment published in 2008, which describes advances in the measurement technology and in particular the progress tomography being used [33].

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Figure 11.5 Results from the München micromaser experiment in (2000) with the aim of creating well-defined microwave photon number states, or Fock states, inside a cavity. The results shown are for conditions |0 >, |1 >, and |2 > for the photons inside the cavity. The left side (a, b, c) shows the dynamics of the population or inversion inside the cavity. The different Rabi frequencies are clearly visible. The right side (d, e, f ) shows the probabilities of the reconstructed photon numbers. Source: From B.T.H. Varcoe et al. [30].

11.4.1

Measurement-Based QND

One direction of the microwave experiments that was further developed is to implement quantum control back onto the photon state. In this apparatus it is possible to control the state of the photon field in both photon number and phase up to the precision given by the standard quantum limit (SQL) and then to go beyond using quantum resources. This can be seen as quantum-enabled metrology. It can also be seen as measurement-based QND. The microwave cavity field is one excellent example for this, as shown in Figure 11.7. These experiments start with the preparation of a mesoscopic quantum state superposition (MQSS) prepared by the resonant interaction of Rydberg atoms with the field stored inside the cavity. Information placed on the field by modulating one specific parameter 𝛽 of the field is measured, and the precision of the measurement, in the form of the standard deviation Δ𝛽, is recorded. Without the help of quantum resources, the measurement is limited by the SQL. The data shown in Figure 11.7 demonstrate that it is now possible to use a quantum resource; in this case a well-prepared MQSS of the field improves the precision of the measurement of the encoded information to below the SQL.

11.4 Single-Photon QND

P(n,t)

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Figure 11.6 Evolution of a coherent state. (a) Evolution for 600 ms of the photon number probabilities P(n, t) with n = 0, … , 7 given by the colour code. (b) Average photon number ⟨n⟩ versus time with exponential fit (red line). The inserts show the photon number distributions and the Poisson fit (blue line) at the three time marks shown by the arrows. Source: From M. Brune et al. [33]. T1 (μs) 1.2

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Figure 11.7 Measurement precision versus preparation time T1 and superposition size D. The circles are the variance of the state extracted from the measured data. The straight line is the predicted evolution over the measurement time. The shaded (cyan) zone is the quantum regime bounded above by the standard quantum limit (SQL) (in this publication Δ𝛽 = 0.5) and below by values from the quantum model. Source: From M. Penasa et al. [34].

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We can interpret this as a very elegant demonstration that a QND experiment is achievable through the use of specific resources outside the cavity. Returning to the physics of laser beams, what would be the equivalent process for measurement-based QND? We would definitely need quantum correlations between the quadratures of the laser beam, which means we should use squeezed light as the resource. One way to achieve a QND measurement is to improve the properties of a normal beamsplitter by filling the unused port with squeezed light. This was proposed by J.H. Shapiro [35] and was first realized by R. Bruckmeier et al. [18] who used a CW OPO to generate the squeezed light. We note that the beamsplitter will attenuate the light but will leave the information on the beam largely unperturbed. The results were already impressive, with good T and V (point (9) in Figure 11.3). This concept was extended by B.C. Buchler et al. [19] who showed that the combination of the squeezed light on the beamsplitter with electro-optic feedforward control, as described in Section 11.4.1, will enhance the QND measurement. Analogous to the noiseless amplifier, reported in Section 11.4.1, the feedforward will improve the SNR, which means the value of Ts+m in the two-dimensional TV diagram that describes the quality of the QND system, whilst at the same time retaining the correlation Vs|m . This was demonstrated by B.C. Buchler et al. [20] using a beamsplitter with a squeezed input beam. They improved QND from the original values of Ts+m = 1.12, Vs|m = 0.46 to Ts+m = 1.81, Vs|m = 0.55 through the additional use of feedforward. Figure 11.8 shows on the left the raw data, with and without squeezing, and the results in the TV diagram for beamsplitters (50/50) and (92/8) and different feedforward gain settings. The details are given in the caption of Figure 11.8. 3 = Previous QND experiments 2

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Figure 11.8 Using squeezed light to create QND. Left-hand side: data around the squeezing frequency of 4.5 MHz where (a) is the signal into the beamsplitter, (b) the output without squeezing, and (c) the output with squeezing. Right-hand side: a comparison of the best QND experiments in 2001, in particular the results (RC) by J.F. Roch et al. [17] with the results of this system reported by B.C. Buchler et al. [20]. (V) 92/8 beamsplitter no feedforward, (W) 50/50 beamsplitter no feedforward, (X) 92/8 beamsplitter feedforward with unity signal gain, (Y) 50/50 beamsplitter with maximized Ts , and (Z) 92/8 beamsplitter with maximized Ts .

References

A comparison is given with the QND data shown in Fig.11.3, in particular point (8), which are the results of the experiment by J.F. Roch et al. using atoms for QND, and points (10) and (11), which are for the beamsplitter experiment. We can see that in 2001 the two approaches are already comparable in quality. With the improved squeezing sources available, now better QND results can be achieved. This approach can be part of a bigger systems approach to quantum logic as it has been proposed by A. Furusawa and P. van Loock [36] and many others. QND and the closely related concept of teleportation become a tool for designing systems to process optical quantum information encoded on laser beams. Squeezed light can be seen as a resource for building quantum logic circuits, and with the arrival of practical optically integrated multimode squeezing sources, there might well be applications of QND and teleportation in more complex devices. For further examples, see Chapter 13.

References 1 Braginsky, V.B. and Vorontsov, Yu.I. (1975). Quantum-mechanical limitations

2 3

4 5 6 7 8 9 10

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12

in macroscopic experiments and modern experimental techniques. Sov. Phys. Usp. 17: 644. Braginsky, V.B., Vorontsov, Yu.I., and Thorne, K.S. (1980). Quantum nondemolition measurements. Science 209: 547. Caves, C.M., Thorne, K.S., Drever, R.W. et al. (1980). On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I: Issues of principle. Rev. Mod. Phys. 52: 341. Unruh, W.G. (1979). Quantum nondemolition and gravity-wave detection. Phys. Rev. D 19: 2888. Braginsky, V.B. and Khalili, F.Ya. (1992). Quantum Measurement (ed. K.S. Thorne), 106–121. Cambridge University Press. Imoto, N. and Saito, S. (1989). Quantum nondemolition measurement of photon number in a lossy optical Kerr medium. Phys. Rev. A 39: 675. Poizat, J.Ph., Roch, J.F., and Grangier, P. (1994). Characterization of quantum non-demolition measurements in optics. Ann. Phys. Fr. 19: 265. Holland, M.J., Collett, M.J., Walls, D.F., and Levenson, M.D. (1990). Nonideal quantum nondemolition measurements. Phys. Rev. A 42: 2995. Grangier, P., Levenson, J.A., and Poizat, J.Ph. (1998). Quantum-non-demolition measurements in optics. Nature 396: 537. Levenson, M.D., Shelby, R.M., Reid, M., and Walls, D.F. (1986). Quantum nondemolition detection of optical quadrature amplitudes. Phys. Rev. Lett. 57: 2473. La Porta, A., Slusher, R.E., and Yurke, B. (1989). Back-action evading measurements of an optical field using parametric down conversion. Phys. Rev. Lett. 62: 28. Levenson, J.A., Abram, I., Rivera, T., and Fayolle, P. (1993). Quantum optical cloning amplifier. Phys. Rev. Lett. 70: 267.

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13 Bencheikh, K., Simonneau, C., and Levenson, J.A. (1997). Cascaded

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amplifying quantum optical taps: a robust, noiseless optical bus. Phys. Rev. Lett. 78: 34. Pereira, S.F., Ou, Z.Y., and Kimble, H.J. (1994). Backaction evading measurements for quantum nondemolition detection and quantum optical tapping. Phys. Rev. Lett. 72: 214. Schiller, S., Bruckmeier, R., Schalke, M. et al. (1996). Quantum nondemolition measurements and generation of individually squeezed beams by a degenerate optical parametric amplifier. Europhys. Lett. 36: 361. Poizat, J.Ph. and Grangier, P. (1993). Experimental realization of a quantum optical tap. Phys. Rev. Lett. 70: 271. Roch, J.F., Vigneron, K., Grelu, Ph. et al. (1997). Quantum nondemolition measurements using cold trapped atoms. Phys. Rev. Lett. 78: 634. Bruckmeier, R., Hansen, H., Schiller, S., and Mlynek, J. (1997). Realization of a paradigm for quantum measurements: the squeezed light beam splitter. Phys. Rev. Lett. 79: 43. Buchler, B.C., Lam, P.K., and Ralph, T.C. (1999). Enhancement of quantum nondemolition measurements with an electro-optic feed-forward amplifier. Phys. Rev. A 60: 4943. Buchler, B.C., Lam, P.K., Bachor, H.-A. et al. (2001). Squeezing more from a quantum nondemolition measurement. Phys. Rev. A 65: 011803. Friberg, S.R., Machida, S., and Yamamoto, Y. (1992). Quantum-nondemolition measurements of the photon number of an optical soliton. Phys. Rev. Lett. 69: 3165. Kεonig, F., Buchler, B.C., Rechtenwald, T. et al. (2002). Soliton backaction-evading measurement using spectral filtering. Phys. Rev. A 66: 043810. Goobar, E., Karlsson, A., and Bjoerk, G. (1993). Experimental realization of a semiconductor photon number amplifier and a quantum optical tap. Phys. Rev. Lett. 71: 2002. Roch, J.F., Poizat, J.Ph., and Grangier, P. (1993). Sub-shot-noise manipulation of light using semiconductor emitters and receivers. Phys. Rev. Lett. 71: 2006. Bachor, H.-A., Levenson, M.D., Walls, D.F. et al. (1988). Quantum nondemolition measurements in an optical fiber ring resonator. Phys. Rev. A38: 180. Bencheikh, K., Levenson, J.A., Grangier, P., and Lopez, O. (1995). Quantum nondemolition demonstration via repeated backaction evading measurements. Phys. Rev. Lett. 75: 3422. Grangier, P., Roch, J.F., and Roger, G. (1991). Observation of backaction-evading measurement of an optical intensity in a three-level atomic nonlinear system. Phys. Rev. Lett. 66: 1418. Haus, H.A., Watanabe, K., and Yamamoto, Y. (1989). Quantum-nondemolition measurement of optical solitons. J. Opt. Soc. Am. B 6: 1138. Brune, M., Hagley, E., Dreyer, J. et al. (1996). Observing the progressive decoherence of the “meter” in a quantum measurement. Phys. Rev. Lett. 77: 4887. Varcoe, B.T.H., Brattke, S., Weidinger, M., and Walther, H. (2000). Preparing pure photon number states of the radiation field. Nature 403: 743.

References

31 Bertet, P., Auffeves, A., Maioli, P. et al. (2002). Direct measurement of the

32 33

34 35 36

Wigner function of a one-photon fock state in a cavity. Phys. Rev. Lett. 89: 200402. Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford University Press. Brune, M., Bernu, J., Guerlin, C. et al. (2008). Process tomography and field damping and measurements of Fock state lifetimes by quantum non demolition photon counting in a cavity. Phys. Rev. Lett. 101: 240402. Penasa, M., Gerlich, S., Rybarcyk, T. et al. (2016). Measurement of microwave field amplitude beyond the standard quantum limit. Phys. Rev. A 94: 022313. Shapiro, J.H. (1980). Optical waveguide tap with infinitesimal insertion loss. Opt. Lett. 5: 351. Furusawa, A. and van Loock, P. (2011). Quantum Teleportation and Entanglement. Wiley-VCH.

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12 Fundamental Tests of Quantum Mechanics Quantum mechanics is undoubtedly the most successful theory yet devised in terms of its ability to accurately predict physical phenomena. However, the physical picture that emerges from quantum mechanics is so foreign compared with the one experienced in the macroscopic world that many have rebelled against it and sought a more ‘sensible’ physical picture hidden beneath. For a long time, it was not technologically possible to test the fundamental tenets of quantum mechanics directly. The advent of the laser and hence the birth of quantum optics changed that situation, and now many of the Gedankenexperiments proposed in the early days of the development of quantum theory have been demonstrated experimentally. Whilst quantum optics was the leader here, the increasing control at the quantum level of atomic and solid-state systems in recent years has led to complementary fundamental demonstrations in matter systems. In this chapter we discuss experiments aimed at testing three of these concepts: complementarity, or more particularly wave–particle duality; indistinguishability; and non-locality.

12.1 Wave–Particle Duality Quantum systems can display both particle- and wave-like properties. This was first realized by A. Einstein in his description of the photoelectric effect [1] but later extended by L. de Broglie to massive particles [2]. However, any experiment that clearly delineates the particle aspects of a system will completely fail to see the wave aspect and vice versa. This is an example of the concept of complementarity first described by N. Bohr [3]. The canonical example of wave–particle duality is the interferometer, whether optical or matter, in which wave interference disappears if precise path information is obtained. In some situations wave–particle duality is upheld by a straightforward application of the uncertainty principle [4], whilst in others the connection is more subtle [5, 6]. In Chapter 3 we described experiments in which it could be postdicted that only one photon had passed through the interferometric apparatus at a time. We saw that provided we had no information about the path taken by the photon, then perfect interference fringes could be observed. On the other hand, if the path A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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taken was determined, say, by placing a detector in one arm, then no interference will occur. This latter case is trivial if the photon is detected, but notice that the absence of a detection in one arm also gives which-path information, i.e. that the photon took the other path. In this case the photon arrives at the output, but no interference takes place. This effect can be used to detect the presence of an object without depositing any energy into it. Consider a balanced interferometer set-up such that photons entering, say, input port ‘1’ always exit by output port ‘1’ due to destructive interference. Now suppose an absorber is placed in one path of the interferometer such that no interference will occur. It is now possible for a photon to exit through output port ‘2’. If the observer finds a photon at output port ‘2’, they can be sure that an absorber was present, even though there was no energy transfer to the absorber. This effect is referred to as an interaction-free measurement [7] and has been demonstrated experimentally [8]. These are extreme cases of all-or-nothing path information: what if we obtain partial information about the path taken? It turns out that a strict bound can be placed on the quality of the interference versus the which-path knowledge obtained [9]. The quality of the interference is quantified in the usual way by the fringe visibility: n − nmin (12.1)  = max nmax + nmin where nmax (nmin ) is the number of particles counted at the fringe maximum (minimum). Which-path information is quantified by the likelihood PL . This is the probability that the experimenter will correctly guess the path taken by the particle. The path knowledge, K, can be defined as K = 2PL − 1

(12.2)

which has the property of being zero when no path information is obtained and one when the information is perfect. Given these definitions, the following relation is obeyed: 2 + K2 ≤ 1

(12.3)

The inequality is saturated for pure input states and optimal read-out of which-path information. A few experiments have been carried out to probe wave–particle duality in this intermediate domain, both with massive particles [10] and in optics [11]. Perhaps the most sophisticated experiments with massive particles were the atom-interferometer experiments carried out by the group of G. Rempe [12]. Here increasing which-path information stored in the internal states of the atoms was seen to progressively erase the atomic interference effects. A quantitative test of wave–particle duality was carried out in optics by P.D.D. Schwindt et al. [13]. In this experiment a highly attenuated 670 nm laser was used to illuminate a Mach–Zehnder interferometer, and single photon counting was performed at one of the output ports. Inside the Mach–Zehnder a half-wave plate (HWP) in one of the arms was used to encode partial which-path information on the photons. The set-up is shown in Figure 12.1. The attenuation

12.1 Wave–Particle Duality

QWP

Pat h

2

(a)

From source

(b)

Laser 670 nm

V H

HWP

Beamsplitters

1 2

HWP

Path 1

HWP

PBS

PBS

HWP

Figure 12.1 Experimental set-up of Schwindt et al. [13] designed to quantify wave–particle duality.

was sufficient that the maximum count rates at the output were ≤ 50 000 s−1 . Given a transit time through the interferometer of 1 ns, the average number of photons in the interferometer at any time was ≤ 10−4 , and hence, given the Poissonian photon statistics of a coherent field, the probability of more than one photon travelling through the interferometer at one time was negligible. By post-selecting only those events that triggered the detector, an excellent approximation to a single-photon source is achieved (Section 13.2). The single-photon quantum state inside the interferometer (see Section 5.2), before the wave plate, is given by 1 i𝜙 √ (|1⟩1V |0⟩2V + e |0⟩1V |1⟩2V )|0⟩1H |0⟩2H 2

(12.4)

where the ket |n⟩jK indicates photon occupation n of the jth spatial and Kth polarization mode. The input modes to the interferometer are assumed to be vertically polarized. Here 𝜙 is the phase difference between the two arms of the Mach–Zehnder. This phase can be adjusted experimentally by varying the path length of one arm by small amounts using a piezoelectric transducer mounted on one of the mirrors. After passing through the HWP, the state becomes ( ) 1 √ 1 − 𝜖 2 |1⟩1V |0⟩2V |0⟩1H |0⟩2H + 𝜖 |0⟩1V |0⟩2V |1⟩1H |0⟩2H √ 2 1 + √ ei𝜙 |0⟩1V |1⟩2V |0⟩1H |0⟩2H (12.5) 2 √ There is now an amplitude (1∕ 2) 𝜖 to find a photon in one of the horizontal modes, but this can only occur if the photon took path (1) through the interferometer. After recombining the interferometric paths at the output beamsplitter, we have the output state ( ) √ 1 √ ( 1 − 𝜖 2 + ei𝜙 ) |1⟩1V |0⟩2V + ( 1 − 𝜖 2 − ei𝜙 ) |0⟩1V |1⟩2V |0⟩1H |0⟩2H 2 1 + 𝜖 |0⟩1V |0⟩2V ( |1⟩1H |0⟩2H − |0⟩1H |1⟩2H ) (12.6) 2

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It is clear that when the wave plate is oriented such that it produces no rotation of the polarization, 𝜖 = 0, then perfect visibility is realized. That means that if we look at one spatial output, say |0⟩1 |1⟩2 , then we see perfect extinction of counts when the phase shift is zero, 𝜙 = 0, and hence we obtain  = 1. On the other hand, no which-path information is obtained as the photons always emerge vertically polarized. Qualitatively one can see from Eq. (12.6) that as polarization rotation is introduced, the interferometric visibility is decreased whilst which-path information is increased. When the polarization of a photon taking path (1) is completely flipped, 𝜖 = 1, the visibility is reduced to zero, which means the number of counts at an output is unchanged by varying the path length. However, knowledge of the path taken is perfect as a vertically polarized photon at the output definitely took path (2), whilst a horizontally polarized photon definitely took path (1). The visibility of the probability distribution was measured by counting the total number of photons emerging from one of the outputs when the path length difference was adjusted to produce maximum counts, giving nmax , and when the path length was adjusted to produce a minimum, giving nmin . To find the which-path knowledge, the output polarization analyser, comprising a HWP, quarter-wave plate, polarizing beamsplitter, and two detectors, was employed (Figure 12.2). This combination allows measurements to be made in an arbitrary polarization basis. By blocking one of the paths through the interferometer and looking at the relative count rates on the two detectors for some particular polarization basis and repeating this process with the other path blocked, it is possible to evaluate the likelihood for that particular basis. In particular, consider the case where more counts are registered on detector 1 than detector 2 when the path (1) through the interferometer is taken and vice versa when the path (2) is taken. If we now send a single photon through the interferometer with both paths open and we find that detector 1 ‘clicks’, then we would guess that the

1

V 2 + K2

0.8 0.6

Knowledge Visibility 0.4 (Optimized basis) 0.2 (Non-optimized basis) 0 30° 37.5° 0° 7.5° 15° 22.5° Angle of internal HWP, θHWP

45°

Figure 12.2 Experimental data and theoretical curves from the experiment by Schwindt et al. The results are for a vertically polarized input as a function of the orientation of the half-wave plate in path (1). The crosses are the measured visibilities. The dotted line and the triangles correspond to K measurements in the horizontal/vertical basis, and the solid line and diamonds correspond to measurements in the optimal basis. Source: Data from Schwindt et al. [13].

12.1 Wave–Particle Duality

photon took the first path. The likelihood that this guess is correct is given by [14] PL (λ) =

n1,1 (λ) + n2,2 (λ) n1,1 (λ) + n2,2 (λ) + n1,2 (λ) + n2,1 (λ)

(12.7)

where ni,j (λ) is the count rate on detector i when the photons are made to take the jth path and the polarization analysis is in the λ basis. More generally we can write PL (λ) =

max[n1,1 (λ), n1,2 (λ)] + max[n2,2 (λ), n2,1 (λ)] n1,1 (λ) + n2,2 (λ) + n1,2 (λ) + n2,1 (λ)

(12.8)

The results obtained in the experiment by Schwindt et al. [13] are shown in Figure 12.2 along with theoretical curves. The trade-off between visibility and knowledge is clearly seen. The results from using two measurement bases are shown: horizontal/vertical and the optimized basis. The optimal basis is (𝜙1 + 𝜙2 )∕2 + 45∘ and (𝜙1 + 𝜙2 )∕2 − 45∘ where 𝜙j is the polarization angle of the photons if they traverse the jth path. When the optimal measurement basis is used, the sum  2 + K 2 is within a few per cent of 1 over the whole parameter range. This is a powerful experimental demonstration of the fundamental quantum property of complementarity. With the advent of optical quantum gates (see Section 13.7), it became possible to make quantum non-demolition measurements of photon path through an interferometer. In this way G.J. Pryde et al. [15] were able to map path information about the photon onto a second ancilla photon. Plotting the sum of visibility (of the original photon) and knowledge (carried by the ancilla) resulted in a similar graph to Figure 12.2. In the previous discussion it is quite clear that under conditions of high visibility, the path or ‘trajectory’ of individual photons through the interferometer is undefined. From Eq. (12.3) we see that if the visibility is high, then the path information must be negligible. Whilst this is true for individual photons, it turns out somewhat surprisingly that it is possible to define average trajectories for ensembles of photons even when high visibility fringes are produced. The trick is to make very weak measurements of path, such that the interference is hardly disturbed, but repeat these measurements many times. An interesting example is the well-known two-slit experiment in which single photons impinge on a screen with two slits in it. Spatial interference fringes form on the far side of the screen due to interference between the two paths the photons can take through the slits. Just as for the interferometer, if the slit through which individual photons pass is determined, then the interference fringes disappear. Now consider the following strategy: we sit at a particular distance behind the screen and make a weak measurement of the photon’s transverse momentum followed by a strong measurement of the photon’s transverse position. We assume that the momentum measurement is sufficiently weak that the back-action on the position measurement is very small. Shot by shot we gain very little information about the photon’s transverse momentum. However, if we repeat this procedure many times, only keeping those cases where the strong measurement finds the photon at a particular position, then we can determine an average transverse momentum for photons that arrive at that particular transverse position. We can

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12 Fundamental Tests of Quantum Mechanics

obtain such average momenta for many photon positions. Now we can move to a different distance behind the screen and repeat the whole procedure. Wiseman [16] showed that if a sufficient sample of such measurements were performed at many different distances behind the screen, then average trajectories of photons passing through the double slit and forming an interference pattern in the far field could be reconstructed. Interestingly, the trajectories constructed in this way correspond to those predicted by the Bohmian interpretation of quantum mechanics. An experimental demonstration of this effect was carried out by S. Kocsis et al. [17]. Single photons generated by an InGaAs quantum dot source (see Section 13.7) were coupled into an optical fibre and then split into two fibres using a 50 / 50 fibre beamsplitter. The output ends of the fibres were placed, with guiding optics, such as to mimic the geometry of the two-slit experiment. In particular, the spatial modes of the photons exiting from either fibre overlapped in the far field so as to produce a spatial interference pattern. The spatial pattern was monitored at the single photon level using a cooled charge-coupled device (CCD) (see Section 13.7) with a spatial resolution set by the 26 μm pixel width of the CCD. Imaging optics allowed the effective distance between the fibre ends and the CCD to be varied such that the spatial pattern of single-photon arrivals could be observed at various planes between the near and far field. As for P. Schwindt et al., polarization was used as the additional degree of freedom for measuring the photons. However, in this case, it is the transverse momentum that couples to the polarization. This is achieved by passing the photons through a thin calcite plate. Initially circularly polarized light becomes slightly elliptical as a function of the transverse component of the photon’s wave number. Using a polarization displacer, it was possible to monitor the extent of this very small ellipticity and hence, after many trials, extract an average transverse momenta for sub-ensembles of photons arriving at different positions, as required. In this way average trajectories of the photons in the two-slit experiment were obtained. A reconstruction of these average trajectories is shown in Figure 12.4. We see that the trajectories evolve from a particle-like situation as they emerge from the fibres (back of the figure) to a wave-like interference pattern (at the front of the figure).

12.2 Indistinguishability Another fundamental difference between classical and quantum physics is the indistinguishability of quantum particles. In classical mechanics one can always label different particles in an ensemble by their position and momentum. However, in quantum theory, the uncertainty principle makes this not possible in general. Consider the situation depicted in Figure 12.5. A collision occurs within a region smaller than the position uncertainty of the particles. It is thus not possible, even in principle, to distinguish between the two trajectories shown and hence not possible to ‘keep track’ of a particular particle. The analogous situation for photons was described in Section 5.1. If two photons that cannot be

x^

|R〉

|R〉 |A〉 y^

z^

Single photons from quantum dot

|D〉 |L〉

QWP HWP

|H〉

|H〉

|H〉 |A〉 |L〉

φx

|R〉 |A〉

|D〉

|V〉

|V〉

1 (|H〉 + |V〉) |D〉 = 2

–1 1 1 e 2 φx |H〉) e 2 φx |V〉) 2

|D〉 |L〉 |V〉

y^

50:50 fibre beamsplitter HWP QWP

State preparation

Polarizer

Calcite

Weak measurement

QWP L1

L2

Translation stage

L3

Polarization beam displacer

Post-selection

Cooled CCD

Figure 12.3 Experimental set-up for measuring the average photon trajectories. Single photons from an InGaAs quantum dot are split on a 50 / 50 beamsplitter and then out-coupled from two collimated fibre couplers that act as double slits. A polarizer projects the photons into a diagonal polarization state. Quarter-wave plates (QWP) and half-wave plates (HWP) allow the flux of photons passing through each slit to be varied. The weak measurement is performed using a 0.7 mm thick calcite crystal with its optical axis at 42∘ . A cooled CCD measures the final transverse position of the photons. Source: From S. Kocsis et al. [17].

12 Fundamental Tests of Quantum Mechanics

Probability density (mm–1)

448

0.3 0.2 0.1 0

2000 3000

c distan gation Propa

4000 5000 6000

)

e (mm

7000 8000 9000

–12

–10

–8

–6

4 2 0 –2 –4 ) Transverse coordinate (mm

6

8

Figure 12.4 The reconstructed average trajectories of an ensemble of single photons in the double-slit apparatus shown in Figure 12.3. Source: From S. Kocsis et al. [17].

Figure 12.5 Particle collision in which the uncertainty region, represented by shading, makes it impossible to distinguish alternate trajectories.

distinguished by their spatial or spectral modes collide on a beamsplitter, then it is not possible to distinguish between events in which both were reflected or both were transmitted. Because photons are bosons, the amplitudes for these two events have opposite signs and, if transmission and reflection are equally likely, will exactly cancel (Figure 12.6). The first experimental demonstration of this effect was performed by C.K. Hong et al. [18]. A parametric down-converter (Section 6.3.4), constructed from a non-linear crystal of potassium dihydrogen phosphate (KDP) and pumped by the 351 nm argon ion laser line, was used to produce pairs of visible photons. The photons were sent down paths of almost equal length and then

12.2 Indistinguishability

Pinhole

IF2 Amp. and disc.

UV filter

D2

ω2 UV

M2 Coincidence counter

BS

KDP

ω0

Counter

ω1

M1

PDP 11/23+

D1 Amp. and disc. Pinhole

Counter

IF1

Figure 12.6 Set-up of the Hong–Ou–Mandel experiment [18]. Movable mirrors are labelled M; interference filters (IFs); the beamsplitter (BS); and single photon counters (D).

No. of coincidence counts in 10 min

mixed on a 50 / 50 beamsplitter. The difference in the path lengths followed by the two photons could be varied from zero to ± a few tens of μm by shifting the position of the beamsplitter. The beams were then passed through pinholes and interference filters with a passband of order 5 × 1012 Hz before striking single photon counters. The photon counters were connected to a coincidence counter that was set to register only events that occurred within a 7.5 ns time window. The pinholes served to erase spatial information about the paths taken by the photons and were oriented to isolate two well-defined spatial modes that overlap well at the beamsplitter. Similarly the interference filters served to erase frequency information that might distinguish the photons. The experimental results obtained are shown in Figure 12.7. Plotted are the numbers of coincidences obtained over a 10 minute time interval as a function of 1000 800 600 400 200 0 260

280 300 320 340 Position of beamsplitters (μm)

360

Figure 12.7 Numbers of coincidences measured in the Hong–Ou–Mandel experiment as a function of the path length difference. This instrument has equal paths at ≈ 300 μm with photons arriving at the same time. Source: From [18].

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12 Fundamental Tests of Quantum Mechanics

Figure 12.8 HOM interference scan. The fourfold coincidences are measured directly behind the final beamsplitter versus time delay. Background has not been subtracted. The error bars indicate standard deviations. The solid line is a Gaussian fit to the data with a visibility of 96 ±1%. Source: From R. Kaltenbaek et al. [19].

140 Four-fold coincidences in 9000s

450

120 100 80 60 40 20 0

–6

–4

–2 0 2 Relative time delay (ps)

4

6

the path length difference. The rate of accidental coincidences has been measured and subtracted from the data. A strong dip in the coincidence rate is observed at a beamsplitter position corresponding to equal path lengths, experimentally demonstrating the destructive interference of the indistinguishable ‘collision’ events of double transmission and double reflection. This is now known as the Hong–Ou–Mandel (HOM) dip (Figure 12.8). The HOM dip can be modelled by considering two single-photon pulses (see Section 4.4.1): |1⟩A =

∫

̃ †𝜔 |0⟩; d𝜔H(𝜔)A

|1⟩B =

∫

d𝜔G(𝜔)B̃ †𝜔 |0⟩

(12.9)

We propagate these states through a 50 / 50 beamsplitter (see Section 4.4.2) and consider detection with broadband photon counters represented by the operators ̃ †𝜔 A ̃ 𝜔 and N ̂ B = ∫ d𝜔B̃ †𝜔 B̃ 𝜔 . The coincidence signal is found by con̂ A = ∫ d𝜔A N sidering the expectation value of the product of these operators over the evolved state: ̂ AN ̂ B ⟩ = 1 (1 − (12.10) d𝜔d𝜔′ H(𝜔)G∗ (𝜔)H ∗ (𝜔′ )G(𝜔′ )) CAB = ⟨N ∫ ∫ 2 Now suppose that the photon pulses are identical Gaussian wave packets except for temporal displacements τA and τB , i.e. H(𝜔) = F(𝜔)ei𝜔τA and G(𝜔) = F(𝜔)ei𝜔τB 2 2 with F(𝜔) = (2∕(𝜎 2 𝜋))1∕4 e−(𝜔−𝜔o ) ∕𝜎 where 𝜎 2 is the variance of the wave packets and 𝜔o is the centre frequency. The coincidence rate can then be evaluated to be ( ) (τ −τ )2 ̂ AN ̂ B ⟩ = 1 1 − e−𝜎 2 A 4 B (12.11) CAB = ⟨N 2 As expected the coincidence count rate vanishes for zero time delay (τA − τB = 0), but asymptotes to half the total count rate for large time delays, the result expected for distinguishable particles. The solid curve in Figure 12.7 is a scaled version of Eq. (12.11) with 𝜎 = 1.5 × 1013 rad/s, determined by the bandpass of the filters in the experiment, and position ≡ c(τA − τB ) + 302. The width of the

12.2 Indistinguishability

dip reflects the range of times for which the finite width of the photon wave packets leads to partial indistinguishability. Since the groundbreaking experiment of Hong, Ou, and Mandel, much work has been carried out into phenomena associated with photon indistinguishability, first by L. Mandel’s group (see [20]) and later by many others including the demonstration by the group of Y. Shih and co-workers, who showed that photon indistinguishability, and hence two-photon interference, could be produced even in situations where the photons never actually overlapped at a beamsplitter [21]. The main limit to the visibility in HOM interferometers is the quality of the spatial mode matching at the beamsplitter. A common method for enhancing this mode overlap is to first couple the outputs from the down-converter into optical fibres. This cleans up the spatial mode, with the outputs from the fibre being Gaussian modes that can more easily be interfered at the beamsplitter. By conducting the entire experiment in single-mode optical fibre and employing a fibre beamsplitter, near-perfect mode matching can be achieved with corresponding high visibilities, for example, 99.4% in the 2003 experiment of T.B. Pittman and J.D. Franson [22], where the visibility in HOM experiments is normally defined as VHOM = (Imax − Imin )∕Imax . More recently, HOM dips have been demonstrated on optical chips achieving similarly high visibilities, for example, the experiment of A. Laing et al. using silica-on-silicon waveguides [23]. In contrast to the long counting times of the original experiment, modern HOM experiments of this type can gather statistics at close to MHz rates due to significant improvements in both source brightness and coincidence counting technology. In all the experiments so far discussed, the two photons were generated by the one down-conversion crystal. However, in theory, it should be possible to interfere photons from independent sources. J.G. Rarity and P.R. Tapster were the first to show that an HOM dip could be observed between separately produced photons [24]. In their experiment they interfered a single photon, heralded from a down-conversion source, with a strongly attenuated coherent state. The coherent state was derived from the pump laser that drove the down-conversion process. The heralding was achieved by gating the coincidence electronics based on a detection pulse from the heralding detector. The HOM visibility was measured to be 63%, thus exceeding the classical bound of 50%. Soon after, separately heralded photons from the same down-conversion crystal were interfered by Bouwmeester et al. [25] and used for teleportation (see Section 13.6.1). The next step was to generate the photons in separate crystals, pumped by the same laser, as realized by H. de Riedmatten et al. [26]. Arguably, to consider the photons to be truly independently produced, one would like there to be no optical link between the photon sources. The problem is that the spectral and temporal properties of the pump pulses must be very similar in order that the photons exhibit high indistinguishability. This was first achieved by R. Kaltenbaek et al. [27] who actively synchronized their pump lasers through electronic feedback. In their experiment the two down-conversion crystals were pumped by frequency doubled IR pulses from independent Ti/sapphire femtosecond mode-locked lasers. The lasers had a 76 MHz repetition rate, the same centre wavelengths of 788.5 ± 0.4 nm, with

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12 Fundamental Tests of Quantum Mechanics

bandwidths of 2.9 ± 0.1 and 3.2 ± 0.1 nm, respectively, and rms pulse widths of 49.3 ± 0.3 and 46.8 ± 0.3 fs, respectively. The lasers were synchronized by a commercial electronic feedback loop from Coherent that was able to achieve a relative timing jitter of 260 ± 30 fs. Uniquely, in this experiment, the time delay between the photons was not introduced by adjusting a physical path length but rather was introduced by varying the time delay offset between the two pump lasers. Narrow bandwidth filters of order 1 nm were used to increase the pulse widths of the photons so they would be much larger than the residual relative timing jitter of the electronic feedback. In this way an HOM dip with 83% visibility was observed. In later work by the same group, the visibility was increased to 96% by implementing narrower 0.4 nm frequency filters on the photons before interference. The lack of perfect visibility in HOM dip experiments can be modelled by allowing the input photons to be in mixed states (see Section 4.2.3). Consider the single-photon states introduced in Eq. (12.9) and the discussion following. Suppose that shot by shot the temporal shift of the photons randomly varies around their mean values (τA and τB ) according to Gaussian distributions with means 𝜎T . The photons will now be in mixed states given by √ 2 − τ2 ′ 1 𝜎 ̃ 𝜔′ ̃ †𝜔 |0⟩⟨0|A T 𝝆̂ A = dτe d𝜔d𝜔′ F(𝜔)F ∗ (𝜔′ )ei(𝜔−𝜔 )(τA −τ) A 2 ∫ ∫ ∫ 𝜋𝜎T √ 2 − τ2 ′ 1 𝜎 T 𝝆̂ B = dτ d𝜔d𝜔′ F(𝜔)F ∗ (𝜔′ )ei(𝜔−𝜔 )(τB −τ) B̃ †𝜔 |0⟩⟨0|B̃ 𝜔′ e B 2 ∫ ∫ ∫ 𝜋𝜎T (12.12) The coincidence rate now becomes ) ( √ (τ −τ )2 −𝜎 2 A 2B 2 1 2 4+2𝜎 𝜎 ) ( ̂ ̂ T CAB = Tr{𝝆̂ AB NA NB } = e 1− 2 2 + 𝜎T2 𝜎 2

(12.13)

̂ BS 𝝆̂ A 𝝆̂ B U ̂ † is the state of the photon pulses after mixing on the where 𝝆̂ AB = U BS beamsplitter. Now the coincidence rate does not go to zero when the time difference is zero, as the time jitter in the photon arrival times introduces some level of distinguishability between the photons. The visibility is given by √ 2 VHOM = (12.14) 2 + 𝜎T2 𝜎 2 Notice that the visibility can be increased both by reducing the time jitter between the photons, 𝜎T , achieved by R. Kaltenbaek et al. by the electronic feedback loop and by reducing the intrinsic linewidth of the photon wave packets, 𝜎, through frequency filtering. Taking 𝜎T =260 fs and 𝜎 = 2𝜋c∕λ2 𝜎λ , where λ =788.5 nm and 𝜎λ is the bandpass of the frequency filters in nm, gives visibilities that are consistent with the experimental results. We note that similar effects occur for frequency jitter or any other fluctuating parameter, which can serve to distinguish the photons.

12.3 Non-locality

In 2001 E. Knill et al. [28] made the remarkable discovery that indistinguishability of photons allows massive non-linear interactions between single photons to be conditionally induced through photon number measurement. This allows the possibility of performing quantum computation tasks using photons as qubits as will be discussed in Section 13.1.

12.3 Non-locality A unique feature of quantum mechanics is its non-local character. Consider two photons in the entangled state: |H⟩A |H⟩B − |V ⟩A |V ⟩B

(12.15)

where |K⟩J indicates that a single photon occupies the Kth polarization mode and the Jth spatial mode. The meaning seems clear: the photons are in a superposition of either both being horizontally polarized or both being vertically polarized. But notice that it is not possible to write down a ket, which describes the polarization state of just one of the photons: the ket must describe both photons jointly, hence the term entangled. Now suppose the photons are moved far apart. If this is done carefully enough, Eq. (12.15) will continue to describe their state. But now we have a system described by a single ket whose individual parts may be space-like separated. If a measurement is made on one of the photons and it is found to be, say, horizontally polarized, then the state of the other photon immediately also becomes horizontal. It appears that actions on one photon immediately affect its entangled partner seemingly in contradiction with special relativity that requires any communication to be limited by the speed of light. Because of this behaviour of entangled systems, quantum mechanics is often referred to as a non-local theory. In the following we will discuss quantum optical experiments that have tested these non-local effects.

12.3.1

Einstein–Podolsky–Rosen Paradox

The first major discussion of non-local effects in quantum mechanics is due to A. Einstein and his colleagues B. Podolsky and N. Rosen (EPR) [29]. EPR’s aim was in fact to show that quantum mechanics must be flawed. They considered two particles travelling away from each other in such a way that their transverse position was always correlated. The state of the particles can be written in the position basis as |EPR⟩ =

∫

dx|x⟩A |x⟩B

(12.16)

where |x⟩K is a position eigenstate of the Kth particle. If a measurement is made of the position of particle A giving the result x′ , then we immediately know that the position of particle B is also x′ . On the other hand, we can also express Eq. (12.16)

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12 Fundamental Tests of Quantum Mechanics

in the momentum basis as |EPR⟩ =

∫

=

∫

=

∫

dp|p⟩A ⟨p|A dp|p⟩A

∫

∫

dp′ |p′ ⟩B ⟨p′ |B

dp′ |p′ ⟩B

dp|p⟩A | − p⟩B

dx|x⟩A |x⟩B

∫ ′

∫

dxei(p+p )x∕ℏ (12.17)

where |p⟩K is a momentum eigenstate of the Kth particle. We see that the momentum is exactly anti-correlated for this state, and if a measurement of the momentum of particle A gave the result p′′ , then we would immediately know that the momentum of particle B was −p′′ . EPR argued that this situation violated the spirit of the Heisenberg uncertainty relation. They held that nothing an observer did to particle A (if it was space-like separated from B) could affect the values of the position and momentum of particle B. And yet an observer in possession of particle A could choose to determine either the x or p value of B to arbitrary precision whilst only touching particle A. Because the choice of measurement was arbitrary and apparently did not affect the ‘real’ situation of the other particle, it seemed that particle B must have had well-defined values for these observables in the first place – contrary to the Heisenberg uncertainty relation.1 The quantum description implies a connection between the particles even when far apart, which Einstein characterized as spooky action at a distance. EPR’s conclusion was that quantum mechanics was incomplete and that there must be additional ‘hidden variables’ determining the ‘real’ situation of each particle. The state |EPR⟩ has infinite momentum uncertainty and so is not normalizable and hence is unphysical; however any state for which Δx2inf Δp2inf < ℏ∕4

(12.18)

is upheld satisfies the basic EPR argument. Here Δx2inf is the variance in the value of the position that can be inferred from measurement of the other particle, and similarly Δp2inf is the variance in the value of the momentum that can be inferred from measurement of the other particle. M.D. Reid and P.D. Drummond [30] pointed out that such a state can be produced in optics. Consider the Heisenberg evolution of two vacuum modes, â and ̂ such that b, √ √ â → G â + G − 1 b̂ † √ √ (12.19) b̂ → G b̂ + G − 1 â † This interaction is produced by parametric amplification ( Section 6.3.2) either directly by a non-degenerate system or alternatively via degenerate parametric amplification squeezing, followed by the out-of-phase mixing of the two modes on a 50 / 50 beamsplitter. The inferred variance of the amplitude quadrature of 1 Note that of course both the momentum and position of particle B cannot actually be ascertained precisely because only one measurement can actually be made on particle A.

12.3 Non-locality

beam â conditional on the measurement of the amplitude quadrature of beam b̂ is given by the conditional variance V 1a|b (see Section 4.6.2). From Eqs. (4.133) and (12.19), we obtain 1 (12.20) 2G − 1 Notice that as soon as we introduce some gain, that is, G > 1, the amplitude conditional variance falls below the QNL. Similarly the inferred value of the phase quadrature of mode â conditional on a measurement of the phase quadrature of mode b̂ is given by the conditional variance, V 2a|b , where V 1a|b =

1 (12.21) 2G − 1 which also falls below the quantum limit for non-unity gain. The EPR argument suggests that a paradox exists when two non-commuting observables can be inferred non-locally to better precision than the uncertainty principle would predict. This is the case here as )2 ( 1 (12.22) V 1a|b V 2a|b = 2G − 1 which will be less than the normal uncertainty bound of 1 for all non-unit gain. That this is indeed an entangled state can be seen from the state of the beams in the Schrödinger picture: V 2a|b =

(√ )2 √ ⎛ ⎞ G−1 G−1 1 ⎜ ⎟ |EPR⟩ = √ |0⟩a |0⟩b + √ |1⟩a |1⟩b + |2⟩ |2⟩ + · · · √ a b ⎟ G ⎜⎝ G G ⎠ (12.23) which is a result that can be derived using the techniques of Section 6.3.4. The first experimental demonstration of an EPR state was made by the group of H.J. Kimble at Caltech in 1992 [31]. The experimental layout is shown in Figure 12.9. A crystal of potassium titanyl phosphate (KTP) was placed in a folded ring cavity arrangement and pumped at 540 nm by a frequency-doubled Nd:YAP laser to form an optical parametric oscillator. This was run sub-threshold to give parametric amplification. The system gave a frequency and spatially degenerate output at 1080 nm. However the output was polarization non-degenerate. Such operation is known as type II parametric amplification. In practice this was achieved via precise millikelvin temperature tuning of the specially cut crystal. The cavity was actively locked to the fundamental laser line at 1080 nm via a weak counter-propagating field (Figure 12.10). A polarizing beamsplitter was used to separate the two polarization modes. The resulting pair of beams was approximately in the EPR state described by Eq. (12.23) or the transfer functions of Eq. (12.19). The beams were sent to two homodyne detection systems using tap-offs from the fundamental laser line as local oscillators, which could then simultaneously measure the amplitude fluctuations or the phase fluctuations of the two beams. The amplitude conditional variance was measured by feeding the weighted difference of the photocurrents

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12 Fundamental Tests of Quantum Mechanics

M3

KTP

M2 Pump

M1

M4

λ/2 P

LO1

LO2

i2

i1 i– ψ1(Ω, θ1)

g ψ2(Ω, θ2)

Φ1(Ω, θ1, θ2)

Figure 12.9 Layout of the experiment by Z.Y. Ou et al. designed to demonstrate entanglement between laser beams [31]. Figure 12.10 Results from the experiment by Z.Y. Ou et al. [31] from Figure 12.9 showing the product of the conditional variances V1a|b V2a|b . Here unity represents the QNL of beam a alone.

1.0

0.9 Δ2inf × Δ2inf Y

456

0.8

0.7

1.0

1.5

2.0 Gain, Gq

2.5

3.0

from the amplitude-locked homodyne systems into the spectrum analyser and finding the variance ̃ a − g1 X𝟏 ̃ b )2 ⟩ ⟨ (X𝟏

(12.24)

where g1 is the electronic gain factor in the sum. The conditional variance is then the minimum variance obtained as a function of g1. Similarly the phase conditional variance is obtained from the minimum of ̃ a + g2 X𝟐 ̃ b )2 ⟩ ⟨(X𝟐

(12.25)

V1 V2

12.3 Non-locality

History of CW beam entanglement 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1990 1995 2000 2005 2010 2015 Year

EPR quadrature EPR polarization EPR spatial

Figure 12.11 History of measurements of CW entanglement between laser beams that started with the results from Caltech in 1992. The dramatic advances in squeezing technology also led to sudden improvements in the simultaneous noise suppression V1a|b V2a|b down to 0.03 as reported by the team in Hannover. Source: From Caltech in 1992 [30] and Hannover [31].

A key consideration in the experiment was to limit the total losses in the experiment – in production, propagation, and detection of the beams. If the total losses on each beam exceed 50%, then it is not possible to observe correlations strong enough to satisfy the EPR condition V 1a|b V 2a|b < 1. This can easily be confirmed by introducing beamsplitter losses to the transfer functions of Eq. (12.19) and calculating the subsequent conditional variances. The overall efficiency of the experiment was measured to be about 63%. The experimental results are shown in Figure 12.11. A minimum product of V 1a|b V 2a|b = 0.7 is observed, clearly demonstrating the EPR effect. Very strong EPR correlations have been observed using type I parametric amplification by T. Eberle et al. [32]. Type I parametric amplification produces single-mode squeezing. In order to obtain two-mode squeezing, it is sufficient to mix two such produced single-mode squeezed beams on a 50/50 beamsplitter with a 𝜋∕2 phase shift between them such that the squeezed quadrature of one beam mixes with the anti-squeezed quadrature of the other. This technique was first demonstrated by A. Furusawa et al. [33] using propagating and counter-propagating beams in a single crystal (see Section 13.6.2) and later from two different crystals by W.P. Bowen et al. [34] (see Section 12.3.2). In the experiment reported by T. Eberle et al., both squeezed light sources were made of a ≈10 mm length periodically poled potassium titanyl phosphate (PPKTP) crystal forming a semi-monolithic cavity. One end face of the crystal was curved and coated to be highly reflective, and the other was plane and coated to be anti-reflective. The cavity was formed from the curved end face of the crystal and a curved coupling mirror. The coupling mirror had a reflectivity of 90% for 1550 nm (the squeezed light frequency) and 20% for 775 nm, which is the fibre laser pump frequency. Separate locking loops were used to lock the squeezed light source cavities and the phase between the two squeezed beams. Locking the phase between the beams before the beamsplitter at 𝜋∕2 resulted in strong and highly stable two-mode squeezing. Measuring the quadrature correlations

457

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12 Fundamental Tests of Quantum Mechanics

between the beams at 8 MHz resulted in the observation of a minimum conditional variance product of V 1a|b V 2a|b = 0.0309 ± 0.0002. A summary of the improvement of EPR correlations over the years is shown in Fig. 12.11. 12.3.2

Characterization of Entangled Beams via Homodyne Detection

As we have seen, the effect of entanglement can be observed by measuring continuous quadrature variables of the entangled beams with homodyne detection. The detection of quantum correlation is carried out via the suitable combination of the photocurrents from both beams, as discussed in Section 4.6.2. These measurements show that the information about the quadratures of one of the beams can be used to predict the corresponding quadratures of the other beam better than the QNL. In this section we look at methods for quantifying the entanglement. Also, as we have seen, one technique for generating entangled beams is to combine two squeezed beams with a phase shift such that, effectively, one has noise suppression in the X1 quadrature and the other in the X2 quadrature. The squeezed beams may be generated in independent non-linear systems pumped by one laser, such as the apparatus described in Section 9.7. The two beams are mode matched, locked in phase and combined on a beamsplitter, and provide a pair of entangled beams. Whilst the two individual output beams are not squeezed but rather very noisy (Figure 9.17), we find that information about one beam is contained in the other and vice versa. These beams can propagate and be detected separately – and one can be used to predict properties of the other beam. This is the essence of entanglement. In the Section 12.3.1 we saw that EPR correlations could be demonstrated when squeezed states produced by parametric amplification were mixed. However, other sources of squeezing will also produce entanglement. For example, EPR correlations have been demonstrated in fibre-optic systems [35], where two squeezed beams in orthogonal polarizations were produced by the Kerr effect in the fibre and then mixed. EPR correlations have also been produced from non-degenerate above threshold parametric oscillators [36] and four-wave mixing [37]. In the sub-threshold degenerate parametric oscillator experiment of W.P. Bowen et al. [34] mentioned in the Section 12.3.1, the connection between entanglement and EPR correlations was explored. For pure Gaussian states, entanglement is a necessary and sufficient condition for the observation of EPR correlations. However, for mixed states, entanglement is only necessary, not sufficient. Bowen et al. demonstrated this experimentally by introducing attenuation, and therefore mixing, progressively into their EPR state and showing that the state remained entangled even though the EPR correlations were rapidly lost. The experimental results are shown in Figure 12.12. The entanglement of the beams is evaluated using the two-mode squeezing. It was shown by L.-M. Duan et al. [38] that a pair of symmetric Gaussian beams, as used in this experiment, will be entangled if and only if √ ̃ a − X𝟏 ̃ b )2 ⟩⟨(X𝟐 ̃ a + X𝟐 ̃ b )2 ⟩ ⟨(X𝟏 <1 (12.26) I= 2

12.3 Non-locality

Degree of entanglement

1.75

I

(a)

II

III

IV

1.5 1.25 1 0.75

(b)

0.5 0.25 0

0

0.1

0.2

0.3

0.4 0.5 0.6 Total efficiency

0.7

0.8

0.9

1

Figure 12.12 Results from the experiment by Bowen et al. showing (a) the product of conditional variances V1a|b V2a|b and (b) the entanglement criteria I as a function of loss (symmetric between the beams). The solid and dashed lines are fitted to theory. Unavoidable losses in the experiment are indicated by vertical bars and were (I) detection efficiency, (II) homodyne efficiency, (III) optical loss, and (IV) the OPA escape efficiency. Source: After Bowen et al. [34].

̃ exhibits the maximal correlation and X𝟐 ̃ exhibits where we have assumed that X𝟏 the maximal anti-correlation. The experimental data shows that whilst the product of the conditional variances is above 1 for loss greater than 50% and thus no longer violates the EPR condition, entanglement, as characterized by I < 1, is present for all levels of loss. The maximum observed value for the EPR correlation was V 1a|b V 2a|b = 0.58. 12.3.2.1

Logarithmic Negativity and Two-Mode Squeezing

From quantum information theory (see Chapter 13), it can be shown that a valid measure of entanglement for continuous variable systems is the logarithmic negativity, EN . For the special case of two-mode Gaussian systems, EN = − ln 𝜈− where 𝜈− is the lowest symplectic eigenvalue of the covariance matrix of the partially transposed state. Any two-mode Gaussian state can be characterized by the following covariance matrix (see Section 4.6.2) in the standard locally symmetric form [38]: ⎛ ⎜ V=⎜ ⎜ ⎝

A 0 k1 0

0 A 0 k2

k1 0 B 0

0 k2 0 B

⎞ ⎟ ⎟ ⎟ ⎠

(12.27)

where local squeezing operations and phase rotation can convert any two-mode covariance matrix to this form. For this state we have √ √ 𝜈− =

A2 + B2 − 2k1 k2 −

(A2 − B2 )2 − 4(A2 + B2 )k1 k2 + 4AB(k12 + k22 ) √ 2 (12.28)

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12 Fundamental Tests of Quantum Mechanics

There is actually a direct relationship between the uncertainty product of the two-mode squeezing correlations and the lowest symplectic eigenvalue. We can generalize the two-mode squeezing uncertainty product by introducing correlation gains xi , similar to what is done for the conditional variance. The generalized form is then found to be equal to the lowest symmetric eigenvalue [39], specifically √ ̃ a − x1 X𝟏 ̃ a + x2 X𝟐 ̃ b )2 ⟩⟨(X𝟐 ̃ b )2 ⟩ ⟨(X𝟏 Min I = x1 ,x2 = 𝜈− (12.29) 1 + x1 x2 where the minimization is over both correlation gains. Note that for symmetric states, this expression is minimized by x1 = x2 = 1, in which case it reduces to the form of Eq. (12.26), and is equivalent to the product of the two-mode correlations introduced in Section 4.6.2. In the experiment of T. Eberle et al. [32], the value of two-mode squeezing was measured to be I = 0.09 ± 0.0003, giving a logarithmic negativity of EN = 2.4. 12.3.2.2

Entanglement of Formation

Another prominent but distinct measure of entanglement is the entanglement of formation, EF . An arbitrary mixed two-mode Gaussian state can always be formed by starting with a pure two-mode squeezed state and adding classical noise to it through random displacements. In general there will be a range of pure states that can be used as the starting point, but in that range there will be a least entangled state. The entanglement of formation of a mixed two-mode Gaussian state is the entanglement of the least squeezed pure state needed to generate it. J.S Ivan and R. Simon [40] have given a straightforward recipe for calculating EF (see also [41, 42]). Any two-mode Gaussian state can be characterized by a different form of the covariance matrix referred to as the canonical form: 2 ⎛ C + uc ⎜ 0 V=⎜ ⎜ S + ucs ⎜ ⎝ 0

0

S + ucs

C + 𝑣c2

0

0

C + us2

−S − 𝑣cs

0

0

⎞ −S − 𝑣cs ⎟ ⎟ ⎟ 0 ⎟ C + 𝑣s2 ⎠

(12.30)

where C = cosh 2r0 , S = sinh 2r0 , c = cos 𝜃0 , s = sin 𝜃0 , and u, 𝑣 ≥ 0, and transformations between the standard form (Eq. (12.27)) and the canonical form (Eq. (12.30)) can always be achieved by local squeezing operations. By comparing the covariance matrix of the state of interest to the canonical form, one can determine the value of the parameters 𝜃0 , 𝜇, 𝜈, and in particular r0 . The entanglement of formation is then given by EF = cosh2 r0 log2 (cosh2 r0 ) − sinh2 r0 log2 (sinh2 r0 )

(12.31)

We note that this procedure should only be used to quantify the entanglement of states that have already been established to be entangled (by considering, say, negativity). We also note that in general this procedure can lead to two solutions, the least entangled of which should be chosen. Finally we note that analytical solutions are not always obtained via this method, though in a number of situations of

12.3 Non-locality

physical interest simple analytic results can be found by other means [39]. Transforming the measured covariance matrix in the experiment of T. Eberle et al. [32] into the canonical form, one finds an entanglement of formation of EF = 5.0. 12.3.3

Bell Inequalities

Although of considerable philosophical interest, the EPR paradox did not lead to an experimentally testable difference between quantum mechanics and hidden variable theories. However, in 1965, J.S. Bell derived such a difference [43] based on the properties of entangled two-level systems [44]. The beauty of Bell’s argument was that it produced an inequality that must be obeyed by all local realistic theories of the type envisaged by Albert Einstein, regardless of the actual details of the theories. But, as John Bell also showed, entangled quantum systems could violate this inequality. Consider the generic correlation experiment shown in Figure 12.13. Correlated beams of particles are emitted from a source (S) in opposite directions, A and B. Two distinct ‘paths’ or modes (p and m) are available to the particles in each beam. These could be different spatial or temporal paths or orthogonal polarizations (or spins). The two paths are coherently combined and then spatially separated to form a different pair of orthogonal paths + and −. The combiners, C(𝜃), are black boxes such that it is not possible, for a general value of the mixing parameter 𝜃, to determine from measurements of + and − whether a particular particle took path p or m. Occupation measurements are then made on the + and − paths of each beam, giving results R+ (𝜃) and R− (𝜃), respectively. In the standard case these measurements are simply the presence (1) or absence (0) of a particle in a particular path in some time interval. More generally they may represent the count rate of particles in a particular path. We can form correlation functions of the following form: j

Rij (𝜃A , 𝜃B ) = RiA (𝜃A )RB (𝜃B )

(12.32)

where i, j = +, −. We then construct the normalized averages Pij (𝜃A , 𝜃B ) = ∑

⟨Rij (𝜃A , 𝜃B )⟩ kl k,l=± ⟨R (𝜃A , 𝜃B )⟩

(12.33)

It can then be shown [45] that provided the P’s have the form of probabilities (bounded between 0 and 1), then in any local realistic description the correlations will be bounded by the following Bell inequality: B = |E(𝜃A , 𝜃B ) + E(𝜃A′ , 𝜃B′ ) + E(𝜃A′ , 𝜃B ) − E(𝜃A , 𝜃B′ )| ≤ 2

(12.34)

E(𝜃A , 𝜃B ) = P++ (𝜃A , 𝜃B ) + P−− (𝜃A , 𝜃B ) − P+− (𝜃A , 𝜃B ) − P−+ (𝜃A , 𝜃B )

(12.35)

where

Figure 12.13 Schematic layout of a correlation experiment.

+

RA

A –

RA

+ –

+ C θ1

p m

p S

m

C θ2

–

+

RB

B –

RB

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12 Fundamental Tests of Quantum Mechanics

Optics offered a unique opportunity for testing the inequality of Eq. (12.34) and hence this fundamental property of quantum mechanic. Entangled photons can be produced, widely spatially separated, and measured in the required way with relative ease compared to other quantum systems. Consider the standard optical example of a state that violates the Bell inequality. Such a state is the number-polarization entangled state χ (12.36) |0⟩ + √ (|H⟩A |H⟩B − |V ⟩A |V ⟩B ) 2 which is approximately produced by a parametric down-converter operating at low conversion efficiency (χ ≪ 1) as will be described shortly. The requirement of low conversion efficiency is that higher photon number terms (which appear as products of higher powers of χ) can be neglected. In the following it will be more convenient to work in the Heisenberg picture. In this picture the action of the down-converter is to evolve a vacuum state, with input annihilation operators ci and di , where i = h, 𝑣, according to â i = ĉ i + χd̂ †i , b̂ i = d̂ i + χ̂c†i

(12.37)

where as before χ ≪ 1 has been assumed. Our two paths, p and m, in this example are the horizontal (̂ah and b̂ h ) and vertical (̂a𝑣 and b̂ 𝑣 ) polarization modes. The mixer C is then some combination of polarizing optics that decomposes our beams into a different orthogonal polarization basis set (+ and −). This corresponds to the transformation â + (𝜃A ) = cos 𝜃A â h + sin 𝜃A â 𝑣 , b̂ + (𝜃B ) = cos 𝜃B b̂ h + sin 𝜃B b̂ 𝑣 ,

â − (𝜃A ) = cos 𝜃A â 𝑣 − sin 𝜃A â h b̂ − (𝜃B ) = cos 𝜃B b̂ 𝑣 − sin 𝜃B b̂ h

(12.38)

Photon counting is then performed on the beams, and we define j

RA (𝜃A ) = â †j (𝜃A )̂aj (𝜃A ) j RB (𝜃B ) = b̂ †j (𝜃B )b̂ j (𝜃B )

(12.39)

with j = +, −. The definitions then follow as per Eqs. (12.32) and (12.33). An explicit calculation then gives the result E(𝜃A , 𝜃B ) = cos 2(𝜃A − 𝜃B )

(12.40) 2

where terms of order higher than χ have been neglected.√Choosing the angles 𝜃A = 3𝜋∕8, 𝜃A′ = 𝜋∕8, 𝜃B = 𝜋∕4, 𝜃B′ = 0, we find B = 2 2, a clear violation of Eq. (12.34). This form of the Bell inequality was developed by J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt (CHSH) [46]. The first experimental demonstrations of Bell inequality violations were actually achieved using the polarization-correlated photons produced by atomic cascades. Using the 4p2 1 S0 –4s4p 1 P1 –4s2 1 S0 cascade in calcium to produce polarization entangled visible photons, A. Aspect et al. [47] performed the first complete test in 1981. Their experimental set-up is shown in Figure 12.14. Two-photon excitation via a krypton ion laser at 406.7 nm and a dye laser at 581 nm was used to selectively populate the upper state. Large aperture aspheric lenses were used to isolate and collimate beams, which were sent to the analysers

12.3 Non-locality Atomic beam Filter ν2

Kr laser νK

Pol II

Filter ν1

Pol I

P.M.2

P.M.1 Dye laser νD

Disc

Disc

Coinc Delay Stop

Coinc T.A.C.

Start

MCA

Figure 12.14 Schematic diagram of the apparatus and electronics used in the first Bell inequality test. The laser beams are focussed onto the atomic beam perpendicular to the figure. Feedback loops from the fluorescence signal control the krypton laser power and the dye laser wavelength. The output of discriminators feed counters (not shown) and coincidence circuits. The multichannel analyser (MCA) displays the time delay spectrum. Source: After A. Aspect et al. [47].

spatially separated by up to 6.5 m. The analysers consisted of plate polarizers followed by photomultipliers connected to coincidence counting electronics with a 19 ns coincidence window around zero time delay (see Section 7.1). The coincidence counting selects those events for which one of the photons from the entangled pair travelled to and was successfully counted by one detector, whilst the other photon of the pair similarly travelled to and was successfully detected by the other detector. The post-selected state is approximately the maximally entangled state |H⟩A |H⟩B − |V ⟩A |V ⟩B . The results of the experiment by A. Aspect et al. [47] are shown in Figure 12.15. Here the normalized coincidence rates are plotted as a function of the angle difference, 𝜙, between the polarizer settings at the two analysers. The counts were Figure 12.15 Normalized coincidence rate as a function of the relative polarizer orientation. Indicated errors are ± 1 standard deviation. The solid curve is not a fit to the data but the prediction of quantum mechanics. Source: After A. Aspect et al. [47].

R(ϕ)/Ro 0.5

0.4

0.3

0.2

0.1 ϕ (°) 0

90

180

270

360

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12 Fundamental Tests of Quantum Mechanics

found to only depend on 𝜙 and not on the absolute √ value of the settings. Under such conditions, a fringe visibility in excess of 1∕ 2 is sufficient to ensure violation of the Bell inequality (Eq. (12.34)). The observed visibility is about 0.9, resulting in a value of B = 2.465 ± 0.035 violating the inequality by more than 13 standard deviations (Figure 12.16). The use of parametric down-conversion to produce entangled states, starting with Mandel’s group in the late 1980s [49], significantly reduced the complexity of the source and increased its brightness. For example, in the experiment of 1995 by the team led by P.G. Kwiat [50], 1500 coincidences per second were observed, an order of magnitude brighter than observed by A. Aspect et al. A Bell violation of over a 100 standard deviations was observed in this experiment. Novel optical arrangements are required to directly produce the polarization entangled state of Eq. (12.36) from parametric down-conversion. In the Section 12.3.2 we discussed a type II parametric oscillator. When operated at low conversion efficiency, without a cavity, a type II crystal will produce orthogonally polarized photon pairs that radiate into conical spatial modes, which appear as rings in cross section. Momentum conservation requires that if a photon is detected in a particular position on one ring, its paired photon will appear in the mirror image position on the other ring. By placing a pair of apertures at the appropriate positions on the rings, the spatial modes thus isolated will occasionally contain a pair of photons in the state |H⟩A |V ⟩B . They are not entangled in the polarization degree of freedom. However, if the angle between the pump beam and the optical axis of the crystal is increased, the rings will intersect. If apertures are placed at the intersection points, it is possible to obtain the pair |H⟩A |V ⟩B or the pair |V ⟩A |H⟩B . If the walk-off due to birefringence in the crystal is compensated, then these two possible pairs become only distinguishable by polarization, and the effective state of the spatial modes is χ (12.41) |0⟩ + √ (|H⟩A |V ⟩B + ei𝜙 |V ⟩A |H⟩B ) 2 which is a polarization entangled state like that of Eq. (12.36) [50]. It is also possible to create polarization entanglement using type I crystals (where the photon pairs are produced with the same polarization) by sandwiching two thin crystals with their polarization axes orthogonal, oriented at, say, H and V , and pumping with a diagonally polarized pump beam [51]. Pairs of horizontally polarized photons are produced in one crystal, whilst pairs of vertically polarized photons are produced by the other. If the crystals are thin enough, the conical spatial modes in which the photon pairs are produced can overlap such that they become indistinguishable, again resulting in an entangled state. More recently, periodic poling of non-linear crystals has considerably relaxed the phase matching requirements for parametric amplification, allowing for a much bigger proportion of the photon pairs to be entangled than in the conical spatial mode geometry. However, new techniques were required to spatially separate the now collinear output modes. The key idea was to interferometrically interfere the outputs of two periodically poled type II sources on a polarizing beamsplitter [52]. The orthogonally polarized photon pairs from each crystal are separated into individual beams, H, V from one crystal and V , H from the other.

Source and transmitter

Optical Ground Station Tracking laser

Tracking beam CCD

1016 mm

Transmitter Fibre

144 km DC source BB0

OGS telescope

La Palma

High-power laser

Tenerife La Gomera

Alice on La Palma

Bob on Tenerife

Polarization compensation

BS HWP

PBS

Polarization compensation

GPS clock

GPS clock

Time tagging

Time tagging

PBS

BS PBS

HWP PBS

N

Classical Internet connection E

Polarization analyser

W

Polarization analyser S

Figure 12.16 Experimental layout for the propagation of entanglement through the atmosphere and detection after length of 144 km. Source: From R. Ursin et al. [48].

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12 Fundamental Tests of Quantum Mechanics

Provided the spatial mode matching at the beamsplitter is good, the spatial states will be indistinguishable, and we will obtain the entangled state Eq. (12.36) again, but this time in a collinear arrangement. An ingenious way of constructing a practical entanglement source using this concept was developed by T. Kim et al. [53] who used a bidirectionally pumped type II PPKTP crystal in a polarization Sagnac interferometer geometry to produce a stable, high brightness CW source at 810 nm, which was subsequently optimized by A. Fedrizzi et al. [54]. R.-B. Jin et al. [55] later developed a pulsed PPKTP Sagnac source that can be operated at the telecom wavelength of 1550 nm and exhibits high spectral purity, as required for entanglement swapping and quantum processing applications (see Chapter 13). 12.3.3.1

Long-Distance Bell Inequality Violations

Seeing Bell violations over significant distances was first achieved by sending the entangled photons through optical fibre links. Preserving polarization under such conditions is difficult, and so a temporal form of the Bell inequality, first derived by J.D. Franson [56], can be used. Using this approach W. Tittel et al. were able to demonstrate a Bell violation of 16 standard deviations over a distance of almost 11 km [57]. In a tour de force experiment, R. Ursin et al. [48] were able to obtain a Bell inequality violation over a distance of 144 km using a free-space link and polarization-entangled photons. The experiment was performed in the Canary Islands between the islands of La Palma and Tenerife at a height of about 2400 m above sea level. The entanglement source was a bulk beta barium borate (BBO) crystal pumped by a picosecond pulsed Nd:vanadate laser, producing entangled photon pairs at 710 nm. In later experiments the PPKTP Sagnac source was used [54]. One photon from the entangled pair was measured locally at La Palma, whilst the second photon was sent via a single-mode fibre to a transmitter telescope. There, the beam was guided via a 150 mm diameter lens with 400 mm focal length over the 144 km long free-space link to the mirror telescope with 1 m aperture diameter of the Optical Ground Station (OGS) of the European Space Agency on Tenerife. The link was actively stabilized by analysing the direction of a tracking beam (532 nm) sent from OGS to La Palma. The attenuation of the link varied between 25 and 30 dB. In spite of the attenuation, a clear peak was observed within the coincidence window of 0.8 ns, centred around the flight time from La Palma to Tenerife of about 487 μs, and high polarization visibility was maintained for the photons that were successfully detected at Tenerife. Measuring ≈ 7000 coincidence events over typical times of ≈ 200 s, they found B = 2.508 ± 0.037, demonstrating the violation of the Bell inequality by more than 13 standard deviations without any background subtraction. This experiment showed no evidence of any fundamental decrease in entanglement with distance and set up the possibility of distributing quantum entanglement via satellites, a feat achieved in 2017 over a distance of 1200 km [58]. 12.3.3.2

Loophole-Free Bell Inequality Violations

There are a number of ways that the Bell inequalities can be ‘tricked’, and a Bell violation can be observed even if the system obeys local realism. These are referred to as loopholes. There are three main loopholes:

12.3 Non-locality

(i) The locality loophole occurs if the measurement result or choice of measurement setting on one side can be communicated to the other side in time to influence the measurement results there. This loophole is closed if each side is space-like separated from the other’s setting choice and measurement outcome. (ii) The freedom of choice loophole occurs if the measurement settings are not chosen freely and randomly. The possibility then arises that there is some interdependence between the chosen settings and the system properties. If one takes the extreme philosophical view that there is no freedom of choice, then closing this loophole becomes impossible. However, under the normal scientific worldview, this loophole will be closed if the measurement settings are generated independently at the measurement stations and are space-like separated from the entangled particle creation. (iii) The fair sampling loophole occurs if only a small sub-ensemble of the entangled particles are detected. This is because it is possible under the local realistic assumptions for a carefully chosen sub-ensemble of particles to violate the Bell inequality even though the entire ensemble does not. This loophole is closed if a sufficiently large proportion of the particles are successfully measured. In 2015, two experiments achieved the violation of a loophole-free Bell inequality in an all optical set-up. These large multi-institutional groups with significant overlap of personnel were led by A. Zeilinger [59] from the University of Vienna and S.W. Nam [60] from the National Institute of Standards and Technology in Boulder. The locality loophole had been addressed by A. Aspect et al.’s experiment [47] and improved upon with fast switching by G. Weihs et al. [61]. The freedom of choice and locality loopholes were first simultaneously closed in the experiment of T. Sheidl et al. [62]. Closing the fair sampling loophole was very challenging for optical systems due to the simultaneous problems of low production and detection efficiency. As a result the fair sampling loophole was first closed using matter systems [63] and was not closed for optical systems until the experiments by M. Giustina et al. [64]. Both of the 2015 experiments used a modified version of the Bell inequality due to P.H. Eberhard [65]. For the CHSH type of Bell inequality discussed up until now, the fair sampling loophole is not closed unless efficiencies greater than ≈ 83% are achieved. In contrast, the Eberhard–Bell inequality is implicitly free of the sampling loophole and can be violated with efficiencies greater than ≈ 67%. The inequality can be written as J ≡ P++ (𝜃A , 𝜃B ) − P+0 (𝜃A , 𝜃B′ ) − P0+ (𝜃A′ , 𝜃B ) − P++ (𝜃A′ , 𝜃B′ ) ≤ 0

(12.42)

where we only consider events in which particles are measured on the ‘+’ path or not in the ‘+’ path, ‘0’. For example, the element P+0 (𝜃A , 𝜃B′ ) is the probability of Alice measuring a particle in the ‘+’ path and Bob not finding a particle in ‘+’ path when the measurement settings are 𝜃A and 𝜃B′ , respectively. It turns out that non-maximally entangled states give the strongest violations, for example, states of the form 1 (|V ⟩A |H⟩B + r |H⟩A |V ⟩B ) (12.43) √ 1 + r2

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In the experiment performed by M. Giustina et al. [64], the PPKTP Sagnac source was used to produce this anti-correlated state conditionally, whilst L.K. Shalm et al. [60] used a PPKTP Mach–Zehnder source to produce a correlated version of the state. In both experiments the parameter r was tuned by adjusting the pump polarization. The entangled photons were launched into single-mode fibres, which transmitted them to the measurement stations, each 29 m away from the source in opposite directions in the Giustina et al. experiment, such that Alice and Bob were separated by 58 m. The geometry was that of a right angle triangle in the L.K. Shalm et al. experiment with the source at the right angle. Alice was 132 m from the source, and Bob 126 m from the source, and Alice and Bob were separated by 185 m. We focus now on the Giustina et al. experiment. Fast random number generators located at Alice and Bob generated the measurement settings in approximately 26 ns, giving estimated margins of ≈ 4 ns for the space-like separation of Alice’s setting from Bob’s measurement, and vice versa, and ≈ 7 ns for the space-like separation of each setting from the emission event. The measurements were implemented by fast electro-optical modulators followed by a polarizer and a transition-edge sensor (TES) single-photon detector at each station. The overall heralding efficiency, i.e. the ratio of twofold coincidences divided by the total events measured in one detector, was found to be 78.6% for Alice and 76.2% for Bob. Given these efficiencies are higher than 67%, we expect it to be possible to violate the Eberhard–Bell inequality. Indeed, with r ≈ −2.9, and the angles 𝜃A = 94.4∘ , 𝜃A′ = 62.4∘ , 𝜃B = −6.5∘ , and 𝜃B′ = 25.5∘ , a positive J value of 7.27 × 10−6 was measured. The statistical significance of the violation was estimated to be 11.5-sigma. Similar results were obtained by L.K. Shalm et al. [60]. The experiments discussed here, plus many others, have removed any lingering doubts about the accuracy of quantum mechanics in describing experiments with entangled states. They show that hidden variable theories of the general type suggested by EPR are not consistent with observation.

12.4 Summary In this chapter we have looked at quantum optical techniques and experiments that test fundamental principles of quantum mechanics. We have seen that quantum optics has been able to quantitatively test many of the Gedankenexperiments proposed in the early days of the formulation of quantum theory. In all cases the results have strongly supported the quantum mechanical predictions.

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13 Quantum Information We discussed in Section 10.2 communicating information using light. We saw that the quantum nature of light leads to intrinsic limits on the amount of information that can be sent optically. We looked at experiments using squeezed light that could reduce those limits as much as possible. In those discussions we imagined encoding the information on the light in much the same way as one would if sending information via a classical medium. Now we consider a quite different idea: that of encoding information in ways only possible for quantum systems. Information encoded in this way is referred to as quantum information. From this point of view, we will find communication and computation tasks that, rather than being limited by quantum mechanics, are actually enhanced by it. The technical challenges in working with quantum information are formidable. Nevertheless significant experimental demonstrations have been made, a number of which we will discuss here. As we shall see, quantum optics has some considerable advantages (and disadvantages) over alternative quantum systems from a quantum information point of view.

13.1 Photons as Qubits In Chapter 3 we introduced the idea of a single polarized photon. For example, we described a light beam as being in the state |H⟩ if in some time interval there is unit probability that one and only one photon will be detected at the horizontal output of a horizontal/vertical polarizing beamsplitter (PBS) placed in the beam path. There is zero probability a photon will be found at the vertical output in the same time interval. A beam in the state |V ⟩ will conversely only be found at the vertical output. It is clear that such light states could be used to carry bits of information. For example, we could assign the value ‘zero’ to the |H⟩ state and ‘one’ to the |V ⟩ state. A string of horizontal and vertically polarized photons could then faithfully represent an arbitrary bit string. However, being quantum objects, photons offer more possible manipulations than classical carriers of bits. In particular not only can we have zero’s and one’s, but we can √ also have superpositions of zeros and ones such as the diagonal state |D⟩ = 1∕ 2(|H⟩ + |V ⟩). Indeed bits can just as effectively be encoded in such A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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√ superposition states, for example, using |D⟩ as a zero and |A⟩ = 1∕ 2(|H⟩ − |V ⟩) as a one. Because of these extra degrees of freedom, we refer to information digitally encoded on quantum systems (such as photons) as quantum bits or qubits. One non-classical feature of encoding in this way is the fact that different bases do not in general commute. Thus simultaneous ideal measurements in both bases cannot be made. Furthermore any measurements that obtain any information about the bit values of one basis inevitably disturbs the bit values of the other basis. These features can be used to create a secure communication channel via the technique of quantum key distribution (QKD) (also referred to as quantum cryptography). A number of demonstrations of QKD have been made in optics. Another feature of qubits is their ability to span all different bit values simultaneously. This is obviously true of a single qubit where the |D⟩ state, when viewed in the horizontal/vertical basis, equally spans the two different bit values (i.e. H = 0 and V = 1). This continues to be true for multi-qubit states. For example, if we start with two qubits in the state |H⟩|H⟩

(13.1)

and we rotate both their polarizations by 45∘ , we end up with the state |H⟩|H⟩ + |H⟩|V ⟩ + |V ⟩|H⟩ + |V ⟩|V ⟩

(13.2)

which is an equal superposition of all four possible two-bit values. This generalizes to n qubits where the same operation of rotating every individual qubit leads to an equal superposition of all 2n bit values. The single-qubit rotations required here are just polarization rotations and can thus be performed easily using wave plates as described in Section 5.4.3. Although this ability to span all possible inputs simultaneously hints at the possibility of increased communication or computation power using qubits, it is not the whole story. Note in particular that analogues of the sort of superpositions represented by Eq. (13.2) can also be created in classical optical systems as superpositions of classical waves. In order to unlock the full power of quantum information, we need to create entangled states. We have seen examples of entangled states before (see Section 12.3) and observed that they have no classical analogue. In the following we will see that entanglement can be used as a resource to perform information tasks more efficiently than possible classically and discuss experimental demonstrations of these tasks using entangled optical states. Finally we could consider performing computations using qubits instead of classical bits. To do this we need to introduce quantum gates. Some of these will have classical counterparts, for example, the NOT gate takes |H⟩ to |V ⟩ and vice versa. On the other hand, some gates will have √ no classical analogue, 2(|H⟩ + |V ⟩) and |V ⟩ to such as the Hadamard gate, which takes |H⟩ to 1∕ √ 1∕ 2(|H⟩ − |V ⟩). As we have noted, these types of operations can easily be achieved in optics using wave plates. However, we also require two-qubit gates such as the controlled-NOT (CNOT), which performs the NOT operation on one qubit (the target) only if the other qubit (the control) has ‘one’ as its logical value. Such two-qubit gates can produce entanglement. Controlled gates

13.2 Post-selection and Coincidence Counting

can be implemented in optics nondeterministically. We will look at examples of such gates. However, implementing two-qubit gates deterministically is much more difficult. We will discuss experimental efforts in this direction. Eventually, if large arrays of gate operations can be implemented efficiently, one could consider performing quantum computation. Algorithms such as Shor’s algorithm [1] show that for certain problems an exponential speed-up over the best known classical programs is in principle possible with a quantum computer, leading to considerable interest. We will discuss small-scale algorithms that have been demonstrated. Whilst the realization of universal quantum computation experimentally, based on quantum optics or any other quantum platform, still remains a long way off, the prospects for nearer-term, non-universal quantum optical processors are good. 13.1.1

Other Quantum Encodings

The polarization encoding discussed so far is not the only possibility. A general class of dual-rail encodings are defined by |𝟎⟩ ≡ |0⟩|1⟩,

|𝟏⟩ ≡ |1⟩|0⟩

(13.3)

Here the boldface 𝟎 and 𝟏 indicate the corresponding logical values of the qubits. The dual-rail encoding assigns logical value depending on whether there is a single-photon excitation in the first or second of two orthogonal modes represented by the two kets. The polarization encoding is a special case of this type of encoding for which the two orthogonal modes are the vertical and horizontal polarization modes. However, there are many other possibilities for the two modes such as orthogonal spatial modes, orthogonal temporal modes, and orthogonal orbital angular momentum modes. A second general class are the single-rail encodings defined by |𝟎⟩ ≡ |𝜓⟩,

|𝟏⟩ ≡ |𝜙⟩

(13.4)

where |𝜓⟩ and |𝜙⟩ are orthogonal (or approximately orthogonal) states. Common choices are the number encoding, where |𝟎⟩ ≡ |0⟩ and |𝟏⟩ ≡ |1⟩, and the coherent state encoding, where |𝟎⟩ ≡ |𝛼⟩ and |𝟏⟩ ≡ | − 𝛼⟩. Finally, we note that higher-dimensional encodings are also possible. For example, by adding an extra rail, i.e. an additional orthogonal mode and degree of freedom, to the dual-rail scheme, we can form a three-level system, i.e. a qutrit. Similarly, adding the two-photon Fock state to the number encoding also allows a qutrit encoding. We can even consider encoding quantum information on continuous variables (CV) such as the quadrature amplitudes.

13.2 Post-selection and Coincidence Counting Producing and detecting single-photon states efficiently presents a major technological challenge. In Section 7.2 we saw that current commercial room temperature photon counting detectors are ‘off/on’ type detectors that only ‘click’ when there are one or more photons incident in a particular time window. Typical

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maximum detection efficiencies are around 70%. However, specialized cryogenic temperature detectors can now obtain detection efficiencies over 95%. Light sources that produce very low photon numbers tend to be spontaneous sources, that is, they produce photons randomly. In the next section we will examine experimental progress towards creating light sources that produce single photons on demand. Here we will discuss how in principle experimental demonstrations can be performed using on/off detectors and spontaneous sources. As discussed in Sections 3.4 and 12.1, single-photon experiments can be performed by strongly attenuating a laser source. We can represent the state of such a laser source by the state (see Section 4.2.2) 𝛼2 |2⟩ + · · · (13.5) 2! where we are ignoring frequency dependence and normalization factors for the present. As we attenuate the source more and more, 𝛼 becomes much less than one, and we can write to a good approximation |𝜓⟩ = |0⟩ + 𝛼|1⟩ +

|𝜓⟩ = |0⟩ + 𝛼|1⟩

(13.6)

We now have a source that in any particular time interval (length determined by the frequency dependence) has a high probability of producing vacuum, some small probability of producing a single-photon state and a negligible probability of producing a multi-photon state. If a photon counter is placed at the end of the experiment and we only worry about those times when the detector ‘clicks’, then we will post-select just the single-photon part of the state. If the source is polarized, then it can be manipulated as a qubit. Notice, however, that it is a rather inconvenient qubit source as it rarely works and you only know it worked after the fact by evaluating the detection record. Nevertheless this type of source has successfully been used in single-qubit-type experiments such as quantum key distribution (see Section 13.5.1). A major problem arises with an attenuated coherent source if we try to move to experiments requiring two qubits. One might assume we could use two highly attenuated coherent sources and then post-select only those events where two photons appear at the end of the experiment. However, the joint state of two equal power, attenuated lasers is ( ) ( ) 𝛼2 𝛼2 |0⟩ + 𝛼|1⟩ + |2⟩ + · · · |𝜓⟩ab = |0⟩ + 𝛼|1⟩ + |2⟩ + · · · 2! 2! a b = |0⟩a |0⟩b + 𝛼(|1⟩a |0⟩b + |0⟩a |1⟩b ) 𝛼2 (13.7) + (2!|1⟩a |1⟩b + |2⟩a |0⟩b + |0⟩a |2⟩b ) + · · · 2! where the first ket refers to one source whilst the second one to the other source. Notice that if we go to order 𝛼 2 , then there is indeed a term with a single-photon state in each beam. However, terms involving pairs of photons in one beam with vacuum in the other occur with the same probability. Post-selecting on two-photon events will not in general remove these terms. Hence it is not possible in general to perform two-qubit experiments using highly attenuated laser sources. A more sophisticated solution is required.

13.3 True Single-Photon Sources

As we saw in Section 12.2, since the late 1980s, the solution of choice has been parametric down-conversion. As was discussed in Section 6.3.4, weak parametric amplification results in the spontaneous creation of pairs of photons. If the down-conversion is spatially non-degenerate, then, in the Schrödinger picture, initial vacuum inputs are transformed according to 1 |0⟩a |0⟩b → √ (|0⟩a |0⟩b + 𝜒 ′ |1⟩a |1⟩b + 𝜒 ′2 |2⟩a |2⟩b + · · · ) G

(13.8)

where 𝜒 ′ is an effective interaction strength, related to the parameters in the discussion of Section 6.3.4 by 𝜒 ′ = 𝜒𝜅 . If we now allow 𝜒 ′ to be very small, which is not hard to arrange experimentally, then the state produced is given to an excellent approximation by |𝜓⟩ab = |0⟩a |0⟩b + 𝜒 ′ |1⟩a |1⟩b

(13.9)

In contrast to Eq. (13.7), the state in Eq. (13.9) has only the desired two-photon term to first order in 𝜒 ′ . If we post-select only those events from the detection record in which two photons are detected ‘simultaneously’ or in coincidence (within some preset time window), then we will only record the part of the state that is due to the pairs of photons. Thus, by using the combination of parametric down-conversion, the polarization degree of freedom and post-selection, we can perform, at least in principle, two-qubit experiments. Experiments carried out this way are sometimes referred to as coincidence basis experiments. In Sections 13.6 and 13.7, we will see several examples of such experiments. Note though that this source is still spontaneous, i.e. successful events are rare, and random and we do not know if they have occurred until after the fact. Progress in producing sources without these drawbacks is discussed in the next section.

13.3 True Single-Photon Sources We now discuss two distinct approaches to producing better approximations to single-photon states. The first is to create a heralded single-photon source, that is, a source that, though not always producing a single-photon state, produces a clear signal when successful. The second approach is to produce an on-demand source, which deterministically produces a single-photon state when requested. 13.3.1

Heralded Single Photons

A heralded single-photon source can be created using spatially non-degenerate down-conversion by detecting one of the output modes and only accepting the other output if a photon is detected. From an idealistic point of view, the conditional state when a single photon is detected in mode a can be obtained from Eq. (13.9) as ⟨1|a |𝜓⟩ab = 𝜒 ′ |1⟩b

(13.10)

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indicating that a single-photon state is created in mode b with probability |𝜒 ′ |2 . In reality things are not so simple. The output state of the down-converter is more realistically described by |𝜓⟩ab = |0⟩a |0⟩b + 𝜒 ′

∫

̃ † B̃ † |0⟩a |0⟩b dka dkb F(ka , kb )A k k a

b

(13.11)

where ki is the wave number of the ith beam and the function F(ka , kb ) describes the spatio-temporal structure of the modes. The photon counter selects an ensemble of distinguishable single-photon modes that can be described by the mixed state 𝝆̂ a =

∫

̃ † |0⟩a ⟨0|a A ̃k dka T(ka )A k a a

(13.12)

where T(ka ) is the spatio-temporal distribution of the detected ensemble. The output state, conditional on a photon count, is then 𝝆̂ b = Tra [|𝜓⟩ab ⟨𝜓|ab 𝝆̂ a ] = |𝜒 ′ |2

∫

dka T(ka )

∫

dkb dkb′ F(ka , kb )B̃ †k |0⟩b ⟨0|b′ B̃ k ′ F(ka , kb′ )∗ b b (13.13)

In general 𝝆̂ b is a mixed state; however, if T(ka ) is centred on but is much narrower in frequency than F(ka , kb ) (such that F(ka , kb ) is approximately constant over the range of wave numbers for which T(ka ) is non-zero), then to a good approximation the pure single-photon number state |𝜓⟩b = 𝜒 ′

∫

dkb F(kao , kb )B̃ †k |0⟩b b

(13.14)

is produced, where kao is the centre wave number of T(ka ). The state |𝜓⟩b is unnormalized with the probability of producing the single photon given by P = ⟨𝜓|𝜓⟩b = |𝜒 ′ |2

∫

dkb |F(kao , kb )|2

(13.15)

The above technique was demonstrated conclusively by A.I. Lvovsky et al. [2] at Universität Konstanz by performing homodyne tomography on the conditionally produced photon states. A beta barium borate (BBO) crystal was used in a type I arrangement to produce frequency degenerate but spatially non-degenerate photon pairs. Transform limited pulses at 790 nm from a Ti/sapphire laser were doubled and used to pump the crystal. Great care was taken to minimize any distortions of the spatial and temporal modes of the outputs due to walk-off. The trigger photons were passed through a spatial filter and a 0.3 nm frequency filter before being counted on a single-photon detector. A key issue in analysing the conditional photon state is that the signal detector only ‘looks’ at the single-photon mode; otherwise spurious vacuum modes will also couple to the analyser. This means that the local oscillator (LO) pulse used in the homodyne detection of the signal must be accurately mode-matched to the single-photon state. Mode matching with a visibility of about 80% was achieved. The homodyne data collected was then used to produce the Wigner distribution of the single-photon state (see Sections 4.3.2 and 9.7). Normalization of the

13.3 True Single-Photon Sources

0.3

W(X, P)

0.2 0.1

–2

–1

1 P

2

X

–0.1

Figure 13.1 Reconstructed Wigner function from experiment of Lvovsky et al. [2] showing negative values close to the origin. Source: From Lvovsky et al. [2].

distribution was achieved by simultaneously collecting vacuum state data. The resulting Wigner function is plotted in Figure 13.1. The plot is consistent with a mixed state composed of 55% single-photon state and 45% vacuum. The source of the vacuum contribution can be identified as the non-unit LO mode matching (an effective 35% loss) and loss due to non-unit homodyne efficiency and signal transmission of about 10%. The most salient feature of the plot is the negativity close to the origin. This demonstrates the strongly quantum mechanical nature of the detected single-photon state. The experiment by A. Lvovsky et al. clearly showed that a single-photon state can in principle be produced in this manner but it also highlights problems with this approach. For example, in order to obtain a conditional state that is as pure as possible, strong attenuation in the form of frequency filtering had to be applied to the trigger photon, resulting in low single-photon state production rates of about one photon every four seconds. This problem can be avoided by engineering the source in such a way that the spectral mode function becomes separable, i.e. F(ka , kb ) = Fa (ka )Fb (kb ). Under these conditions a pure single-photon state can be heralded without the need for spectral filtering and hence with much higher efficiency. A recipe for achieving this was first suggested by W.P. Grice and I.A. Walmsley [3] and demonstrated by P.J. Mosley et al. [4]. In addition, spatial mode matching of the single-photon state is seen to be a difficult problem. A practical solution is to mode-match the single-photon state into an optical fibre for subsequent use downstream, a technique demonstrated by C. Kurtsiefer et al. [5] who obtained about 40% single-photon contribution to the conditional state. The development of quasi-phase matched crystal waveguides allowed good spatial mode matching to fibres and spectral engineering to be achieved simultaneously. Jin et al. [6] developed a pulsed periodically poled KTP (PPKTP) Sagnac source (see Section 12.3.3) that can be operated at the telecom wavelength of 1550 [nm] and exhibits high spectral purity. This source not only allows for high efficiency heralding of pure single photons but can also produce spectrally pure entanglement, as required for entanglement swapping and quantum processing applications (as will be discussed in Section 13.7). An alternative solution, particularly suited to homodyne tomography, is to use a cavity-based down-conversion source. The cavity naturally leads to a narrowband source with a good spatial mode profile. Such sources have demonstrated

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high Wigner function negativities. For example, M. Fuwa et al. report a negativity of −0.17 and single-photon probability in their state of 75% [7] (see also Section 13.7). Further advances to the tomography of entangled beams of light can be made by using advanced reconstruction techniques for tomography. With an approach called phase randomized homodyne tomography [8, 9] it is possible to recreate Wigner functions of the quantum states that make up the entangled beams. This optimized conditional reconstruction technique is applied directly by post processing the time series data recorded with normal photo detectors, without the need for single photon detectors. This can be done with high efficiency and as a result strong negative components in the distribution function can be reconstructed. An example is shown on the front cover of this book. Finally, it is clearly inconvenient that the photons in these experiments are produced at random times. One solution to this problem, proposed and demonstrated in principle by T.B. Pittman et al. [10], is to inject the single-photon state into an optical fibre storage loop when the trigger photon is detected. The captured photon is then held there till required, when it is switched out of the loop. If the round-trip time of the loop is matched to the repetition rate of the pulsed pump laser, then a number of loops can be loaded and then released simultaneously to produce several single-photon states at the same time. The technical difficulty in obtaining an efficient source in this way is that the losses associated with the switch that takes photons in and out of the loop must be very small. Recently considerable progress has been made on this problem [11]. Alternatively, instead of temporally multiplexing, one could spatially multiplex the sources as suggested by A.L. Migdall et al. [12] and demonstrated in principle by M.J. Collins et al. [13]. In this case a large number of down-conversion sources, N ≫ 1, are simultaneously pumped, and there trigger photons detected. If N is very large, then the chance that at least one of the down-converters produces a photon pair per pump pulse can be close to one, even if 𝜒 is much less than one. A switching network is then used to send a successfully heralded photon into the output. Having 𝜒 ≪ 1 is desirable so that non-photon number resolving photon counters of finite efficiency can be used without introducing multi-photon components to the output. However, this leads to the need for a very large number of down-converters and hence a very complicated switching network. For example, to obtain a photon production probability > 96% with a g2 ≈ 0.2 requires N > 350. 13.3.2

Single Photons on Demand

Approximate single-photon states can be directly created on demand by generating light from a single isolated emitter such as a single ion or atom. The trick here is that a single emitter can only produce a single photon ‘at a time’ with some dead time between emissions whilst the source is re-excited. The effect is that the output state can be written (in an idealized fashion) as ( 2 ) 𝛼 |𝜓⟩ = |0⟩ + 𝛼|1⟩ + 𝜏 |2⟩ + · · · (13.16) 2! where 𝜏 is a number between 0 and 1 representing the suppression of higher photon number terms. If 𝜏 is very small, then 𝛼 can be made large, such that there is a high probability that a single photon will be emitted, whilst the probability of multi-photon emission remains very low.

13.3 True Single-Photon Sources

The first experiments of this kind were performed in the late 1970s (as discussed in Section 3.6). However, although they clearly displayed the photon anti-bunching expected of a single-photon source, they were very inefficient because they radiated into 4π steradians and being based on atomic beams there was little that could be done to improve matters. More recently various attempts have been made to create more efficient single emitters. These included placing single neutral atoms or ions into high finesse optical cavities [14, 15] such that the photon emission should be into a single Gaussian mode, close coupling to single solid-state emitters such as neutral vacancy (NV) centres in diamond [16], and the construction of single quantum dot emitters integrated into distributed Bragg reflector (DBR) cavities [17]. Perhaps the most promising of these is the quantum dot emitters. In the early experiments of Santorini et al. [17] at Stanford University, self-assembled InAs quantum dots embedded in GaAs were sandwiched between DBR mirrors to form tiny, high finesse monolithic cavities in the form of 5 μ high pillars. These were then cooled to 3–7 K and pumped by a pulsed Ti/sapphire laser. The quantum dot emission, at around 935 nm, was spectrally filtered (0.1 nm), and a single polarization was selected before being coupled into single-mode optical fibre. The quantum efficiency of single-photon production was estimated to be about 30%. The single-photon states thus produced were characterized by their g (2) factor (see Section 3.6), which was typically of the order of 0.06 (𝜏 2 ≈ g (2) ) showing good suppression of two-photon emission. To test the indistinguishability of the photon states, the Hong–Ou–Mandel (HOM) dip (see Section 12.2) between consecutive emissions was measured. This was achieved by forming a Michelson-type interferometer with a path length difference of 𝜏 + Δt where 𝜏 = 2 ns is the time interval between consecutive pulses. The set-up is shown in Figure 13.2 and the resulting dip is shown in Figure 13.3. The inferred visibility of the dip when measurement imperfections are taken into account is about 70%. Progress in improving both the efficiency and photon quality of quantum dot emitters was slow, but sources with qualities approaching those of down-conversion are now emerging. In the experiments of H. Wang et al. [18], Fabry–Pérot

Pols

APD g1(τ), g2(τ), and HOM set-up QD BS 4.3 K cryostat

HWP

BS

APDs

Delay lines (free space, 2 m to 3 km fibre)

Figure 13.2 Generation of single photons from a quantum dot (QD) embedded in a micro pillar. The QD is grown by molecular epitaxy and sandwiched between 25.5 (15) lower (upper) distributed Bragg reflector stacks. Pillars with 2 μm diameter are defined with electron beam lithography. The Q factor is measured to be about 5350. The photons are analysed via a Fabry–Pérot cavity or with an unbalanced Mach–Zehnder interferometer for g1 (𝜏), g2 (𝜏) and HOM measurements. Source: From Wang et al. [18].

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Photon indistinguishability

482

0.96 0.95

0.96 2 0.94 0.92

0.94

Number of consecutively emitted single photons

23 64 128 390 1120

0.1 1 10 Time separation (μs, in log scale)

0.01

0.93 0.92

0

2

4

6

8

10

12

14

Emission time separation (μs)

Figure 13.3 Results obtained from the experiment shown in Figure 13.2 showing the extracted photon indistinguishability as a function of the emission time separation.

a single InAs/GaAs quantum dot, embedded in a 2 μm diameter micro-pillar cavity and cooled to 4.3 K, was resonantly excited via a picosecond pulsed laser with a repetition rate of 76.4 MHz, producing photons at 893 nm. Photon production rates of around a million per second at the output of a single-mode fibre were achieved corresponding to an absolute brightness of 7%. Strong suppression of two-photon emission was observed (g (2) (0) = 0.007 ± 0.001). The key achievement of this work was to show that not only consecutive single-photon pulses showed good indistinguishability but that photons separated by up to 1000 pulses also showed good indistinguishability. This was achieved by introducing fibre delay lines in the experiment that could be varied between 2 m and 3 km. HOM visibilities over 0.95 were observed between photons separated by up to around 20 consecutive emissions. A slight reduction in visibility then occurred, saturating to a visibility of approximately 0.92 by about 100 consecutive emissions. This visibility was then maintained out to over 1000 consecutive emissions. The authors attribute this effect to a small spectral wandering, which slightly broadens the intrinsic lifetime-limited linewidth. As we shall see in Section 13.7, these properties are well suited to optical quantum computing using time-bin encoding protocols.

13.4 Characterizing Photonic Qubits We now return to the post-selected domain and ask the question, given that a photon is detected, what is its state? Of course this question cannot be answered for a single event because of the probabilistic interpretation of quantum states. But given a large ensemble of detection events, corresponding to identically prepared photons, a recipe can be given for determining the state of the ensemble. We consider polarization-encoded qubits. The polarization state of the photons is most generally described by the density operator 𝝆̂ (see Section 4.2.3).

13.4 Characterizing Photonic Qubits

Our observables are the Stokes operators, which correspond to the classical Stokes parameters [19]: Ŝ 1 = n̂ H − n̂ V = |H⟩⟨H| − |V ⟩⟨V | Ŝ 2 = n̂ D − n̂ A = |H⟩⟨V | + |V ⟩⟨H|

Ŝ 3 = n̂ R − n̂ L = i(|V ⟩⟨H| − |H⟩⟨V |)

(13.17)

where n̂ J is the number operator for the Jth polarization mode. The eigenstates of Ŝ 1 are |H⟩ and |V ⟩ with eigenvalues +1 and −1, respectively. Similarly the eigenstates of Ŝ 2 are |D⟩ and |A⟩ and of Ŝ 3 are |R⟩ and |L⟩. The expectation values of the Stokes operators are related to measurement by 2RH −1 RH + RV 2RD ⟨Ŝ 2 ⟩ = −1 RH + RV 2RR ⟨Ŝ 3 ⟩ = −1 RH + RV ⟨Ŝ 1 ⟩ =

(13.18)

where RH and RV are the count rates recorded at the H and V output ports of a horizontal/vertical PBS, respectively, and similarly for the diagonal/anti-diagonal and right/left bases (see Section 3.5.1). On the other hand, we can also express the expectation values in terms of the density operator as ⟨Ŝ 1 ⟩ = Tr[𝝆̂ Ŝ 1 ] = 𝜌h,h − 𝜌𝑣,𝑣 ⟨Ŝ 2 ⟩ = Tr[𝝆̂ Ŝ 2 ] = 𝜌h,𝑣 + 𝜌𝑣,h ⟨Ŝ 3 ⟩ = Tr[𝝆̂ Ŝ 3 ] = i(𝜌h,𝑣 − 𝜌𝑣,h )

(13.19)

̂ are the elements of the density matrix 𝜌 representing the denwhere 𝜌i,j = ⟨I|𝝆|J⟩ sity operator in the H∕V basis. Equations (13.19) are obtained using the ket representation of the Stokes operators given in Eq. (13.17). Combining Eqs. (13.18) and (13.19), we can obtain all the elements of the density matrix in terms of the Stokes operators’ expectation values and hence in terms of count rates via 𝜌h,h =

RH 1 + ⟨Ŝ 1 ⟩ = 2 RH + RV

𝜌𝑣,𝑣 =

RV 1 − ⟨Ŝ 1 ⟩ = 2 RH + RV

𝜌𝑣,h =

⟨Ŝ 2 ⟩ + i⟨Ŝ 3 ⟩ RD + iRR 1 + i − = 2 RH + RV 2

𝜌h,𝑣 =

⟨Ŝ 2 ⟩ − i⟨Ŝ 3 ⟩ RD − iRR 1 − i − = 2 RH + RV 2

(13.20)

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where we have used the normalization 𝜌h,h + 𝜌𝑣,𝑣 = 1. The density matrix contains all the information about the polarization state of the photons, and properties such as the purity of the state are readily extracted. A question often asked is ̂ is to some pure target state how similar the experimentally produced state, 𝝆, |𝜙⟩. A common measure of this is the fidelity, F, given by ̂ F = ⟨𝜙|𝝆|𝜙⟩

(13.21)

which is easily found in terms of the matrix elements of 𝜌. This technique can be extended to two, or more, photons by considering the expectation values of products of Stokes operators of each photon. For example, ⟨Ŝ 1,a Ŝ 1,b ⟩ = 𝜌hh,hh − 𝜌h𝑣,h𝑣 − 𝜌𝑣h,𝑣h + 𝜌𝑣𝑣,𝑣𝑣

(13.22)

̂ where a, b label the two photons and 𝜌ij,kl = ⟨i|⟨j|𝝆|k⟩|l⟩. By considering the expectation values of all the different combinations of Stokes operator products, the two-photon density matrix can be evaluated. Whilst 4 measurement settings are needed to completely characterize a single photon, 16 measurement settings are needed in general for two photons. Of course, if the two photons are known to be in a separable state, then 4 measurement settings on each individual photon will suffice. The greater number of measurements needed to characterize entangled states is a consequence of their increased complexity. The exponential increase in measurements required as a function of photon number continues with three photons requiring 64 measurement settings and so on. The above techniques were developed and applied by A.G. White et al. [20] and D.F.V. James et al. [21]. One problem that arises is that due to experimental errors, the density matrix produced from the data may be unphysical. To deal with this, maximum likelihood techniques were applied such that the physical density matrix ‘closest’ to the data is identified [21]. A large range of single- and two-photon states have now been analysed in this way with very high fidelity (≥ 99%). In Section 13.7 we will see how these techniques can be applied to the characterization of quantum gates. There is also considerable experimental progress in making the required measurements more reliable and efficient and to get results with a high fidelity from a robust set-up. A good tomographic characterization and finite number of measurements are require, and a device with a multiport output and no other losses would be a very helpful. This has recently been demonstrated for the case of polarization tomography by the team around Y. Kivshar in Australia using a specifically designed ultra-thin metasurface, which creates all the required detection channels [22]. This device has been tested satisfactory for the tomography of a two-photon entangled state using one multiport device and several single-photon detectors and achieving a fidelity of >99%. This and other similar devices have good potential for the future.

13.5 Quantum Key Distribution Perhaps the simplest application of quantum information is in the field of secure communications. It is referred to variously as QKD, quantum cryptography, or

13.5 Quantum Key Distribution

sometimes quantum key expansion and was initially proposed by C.H. Bennett and G. Brassard [23]. The idea is to set up a communication channel that is secure in the sense that any attempt to eavesdrop on the communication can be detected after the fact. The channel is used to send a random number encryption key between two parties, usually referred to as Alice (the sender) and Bob (the receiver). The parties then perform checks on a subset of their data to determine how much information could possibly have leaked to an eavesdropper, called Eve. If the amount of information that could have leaked to Eve is sufficiently small then, after some further post-processing of the data, Alice and Bob can proceed to use the remaining random number key to encrypt secret messages. If they find too much information has leaked to Eve, they scrap that attempt and try again. 13.5.1

QKD Using Single Photons

QKD’s ability to detect eavesdroppers is based on the fact that any process that acquires information about an observable of a quantum mechanical system inevitably disturbs the values of any non-commuting observables. For example, first suppose Alice sends out a ‘zero’ encoded as a horizontally polarized photon, |H⟩. Eve measures in the horizontal/vertical basis (see Section 3.5.1) and obtains the result ‘zero’ and so sends a horizontally polarized photon on to Bob who will definitely get a zero if he measures in the horizontal/vertical basis. However, now suppose Alice and Bob switch to encoding in the diagonal/anti-diagonal basis without √ Eve knowing. Alice sends a zero as a diagonally polarized photon, |D⟩ = 1∕ 2(|H⟩ + |V ⟩). Eve still measures in the horizontal/vertical basis and so has a 50/50 possibility of getting either zero or one as the result, regardless of what Alice sent. Furthermore, what Eve sends on to Bob is basically the mixed state 𝜌 = 1∕2(|H⟩⟨H| + |V ⟩⟨V |). So when Bob measures in the diagonal/anti-diagonal basis, he also gets a random result. Thus, by measuring in the wrong basis, not only does Eve potentially get the wrong result, but she also completely erases the qubit value that is sent on to Bob who then may also get the wrong result. The trick then is to arrange a situation in which Eve does not know in which basis the information on any particular photonic qubit has been encoded because then she is bound to make mistakes that Bob will be able to detect. A typical protocol would go as follows: 1. Alice sends a random number sequence to Bob, encoded on the polarization of single photons. She randomly swaps between encoding on the horizontal/vertical basis and encoding on the diagonal/anti-diagonal basis. 2. Bob measures the polarization of the incoming photons and records the results, but he also swaps randomly between measuring in the horizontal/vertical basis and measuring in the diagonal/anti-diagonal basis. 3. After the transmission is complete, Alice and Bob communicate on a public channel. First Bob announces which basis he measured in for each transmission event. Alice tells him whether or not this corresponded to the basis in which she prepared the photon. They discard all transmission events for which their bases did not correspond.

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4. Bob then reveals the bit values he measured for a randomly selected subset of the remaining data. Alice compares the values revealed by Bob with those she sent. Inevitably there are some errors in the transmission due to imperfections in the system. Alice and Bob set limits on the amount of information that Eve can have obtained based on the error rate they observe. Provided this error rate is sufficiently small, post-processing of the data using techniques called error reconciliation and privacy amplification [24] can be used to produce a shorter secret key. Eve’s information about this shorter key can be made vanishingly small. Hence Alice and Bob are free to use the remaining undisclosed data as a secret key with unconditional security. If there are too many errors, the data is discarded and they try again. An important caveat is that Alice and Bob must initially share some secret information that they can use to identify each other. Otherwise Eve can fool them by pretending to be Bob to Alice and vice versa. Given these conditions, QKD is provably secure. An alternative way for Alice to prepare the states she sends to Bob is via entanglement [25]. Suppose Alice prepares the entangled polarization state 1 √ (|H⟩a |H⟩b + |V ⟩a |V ⟩b ) and distributes mode b to Bob whilst keeping mode a. 2 Alice and Bob then make measurements on their respective halves of the state, swapping between horizontal/vertical measurements and diagonal/anti-diagonal measurements. If they both measure in the same basis, they will obtain correlated results, whilst if the bases are different, their results will be uncorrelated. The protocol can then proceed as per items 3 and 4 above. The importance of the entanglement-based QKD protocols is that very general security proofs are possible for them. This is because there is a direct quantitative relationship between the purity of the entangled state that is distributed between Alice and Bob and the amount of information that Eve could have gained about the state. Thus, if Alice and Bob can determine the purity of their joint state via their public communications, they can bound the maximum information Eve could have obtained regardless of the type of attack she has made. Furthermore, from the point of view of Bob and Eve, there is no difference between Alice sharing entangled states with Bob and Alice directly preparing and sending randomly selected polarization states for Bob to measure as per item 1 above. Thus the powerful limits on the information Eve could have obtained about the key can be determined for prepare-and-measure schemes by ‘pretending’ that Alice produced entangled states. In this way QKD can be shown to be unconditionally secure against any eavesdropper attack [26, 27]. No comparable result exists for classical communications. The first experimental demonstration of QKD was carried out by C.H. Bennett et al. in 1992 over a distance of centimetres [28]. Demonstrations over distances of up to one or two hundred kilometres in free space or fibre are now the state of the art. Typically highly attenuated lasers are used as the qubit source. Switching between the four input states may be achieved through electro-optic control or via the passive combination of four separate laser sources. In all cases it is crucial that the spatial and temporal modes of the four input states are identical so that no additional information is leaked to Eve. The receiver station can be a

13.5 Quantum Key Distribution

passive arrangement. A 50/50 beamsplitter is used to randomly send the incoming photons either to a horizontal/vertical analyser or to a diagonal/anti-diagonal analyser. A schematics of the QKD transmitter and receiver used in the free space demonstration by R.J. Hughes et al. [29] at Los Alamos National Laboratory is shown in Figure 13.4. To increase the signal to noise of the detection system, the detectors are gated, only opening for the one nanosecond or so window in which the single-photon pulse is expected. Synchronization may be arranged via bright timing pulses preceding the single-photon pulses or via more standard public communication links. Sophisticated reconciliation and privacy amplification algorithms then need to be implemented over the public channel. In the Hughes et al. experiment, secure key rates of hundreds per second were achieved over a 10 km range in broad daylight, and somewhat higher rates were achieved at night. The free-space entanglement distribution experiments of Ursin et al. (see Section 12.3.3) were used to perform entanglement-based QKD. This allowed secret key to be generated over a free-space link of 144km, albeit at the very slow rate of about 2 bits per second. The main motivation for free-space systems is to transmit secret keys between ground and satellites securely [30]. For terrestrial systems transmission through fibre-optic networks is more desirable. Although this has the advantage of less stray light, it has the problem that optic fibre is birefringent, and hence polarization-encoded qubits can become scrambled. One solution is to go to a temporal mode qubit encoding. For example, one could use the non-commuting encodings |0⟩ ≡ |T1⟩ + |T2⟩ |1⟩ ≡ |T1⟩ − |T2⟩

(13.23)

and |+⟩ ≡ |T1⟩ + i|T2⟩ |−⟩ ≡ |T1⟩ − i|T2⟩

(13.24)

where |Ti⟩ represents a single photon occupying a temporal wave packet centred at time Ti. Alice could produce the state |T1⟩ + |T2⟩ by allowing a single-photon pulse to pass through a Mach–Zehnder interferometer with unequal arm lengths, in particular where the arm length difference is T1 − T2. The other states are created in a similar way but where an additional phase of π (for the state |T1⟩ − |T2⟩) or π∕2 or 3π∕2 (for the other basis states) is added to one arm of the interferometer. Unfortunately, a read-out by Bob would require him to have an interferometer that is phase-locked to Alice’s, something that is difficult to arrange. D. Stucki et al. at the University of Geneva came up with an elegant solution to this problem by having Bob first send a bright pulse to Alice [31]. This pulse acts as a phase reference, thus avoiding the locking issue. With this idea they were able to transmit secure keys over distances of nearly 70 km at rates of about 50 per second. Dramatic increases in key rates have occurred since these early experiments. In the experiment of L.C. Comandar et al. [32] at the University of Cambridge in 2014, key rates 5 orders of magnitude higher than the Geneva experiment were

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Figure 13.4 Laser and detector arrangement used in free-space QKD experiments of R.J. Hughes et al. Source: After R.J. Hughes et al. [29].

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13.5 Quantum Key Distribution

achieved. A number of technical and theoretical advances were needed to produce this result. A limit on the secure key rates occurred in the early experiments because of the use of attenuated laser sources. The initial intensity cannot be too great; otherwise the probability of two-photon events will be too high. Eve can use two-photon events to extract information about the key without penalty. A solution to this problem, referred to as decoy states, was first suggested by Hwang [33]. The idea is to randomly switch the type of signal sent during the protocol. Because the switching is random, Eve is forced to perform the same attack on all signals. However, in the post-processing, Alice and Bob can use the knowledge of which type of signal was sent to characterize the channel parameters better. This technique was first demonstrated by Zhao et al. [34] by Alice modulating the intensity of her laser. In the Comandar et al. experiment, the use of decoy states allows them to use higher laser intensities and hence achieve higher key rates. In order to achieve low-loss transmission in fibre, one should operate in the telecom window around 1550 nm. However, high efficiency, low noise, room temperature single-photon counters based on silicon operate well in the visible, but poorly in the infrared. Developing high efficiency, low noise, room temperature single-photon counters at 1550 nm was a key technical requirement for achieving high key rates over fibre. Comandar et al. used detectors based on InGaAs APDs incorporating a self-differencing circuit that strongly suppresses after-pulsing and allows high-speed operation with low dark counts. Their QKD system was based on the temporal encoding scheme, also known as phase encoding, with decoy states. Bob’s interferometer was locked to Alice’s via real-time feedback minimizing the bit error rate, thus avoiding the need for a bright phase reference. Using this system they achieved key rates ranging from 1.26 Mbits/s at 50 km to 1.2 Kbits/s at 100 km. These rates were achieved with full composable security, taking into account finite key length effects. An alternative solution to the multi-photon problem of weak coherent states is to use a true single-photon source. Beveratos et al. [16] were the first to demonstrate such a scheme. They used the fluorescence from a single NV colour centre inside a diamond nanocrystal at room temperature as their single-photon source. The nanocrystals were attached to a dielectric mirror, and their emission was collected by a high numerical aperture microscope objective. The effective value of 𝜏 (see Eq. (13.16)) for this source was 0.07; thus the number of two-photon events is reduced by a factor of 14 over an attenuated laser source of the same intensity. The test system employing this source was able to achieve secure bit rate of 7700 per second over 50 m in free space. The very high rates achieved by laser diodes using the decoy state protocol have reduced interest in the single-photon source approach to QKD. The QKD protocol we have discussed here is called BB84. Many other protocols have been proposed and demonstrated, and new protocols and demonstrations appear regularly [27, 35]. A number of companies commercializing this technology have appeared. 13.5.2

QKD Using Continuous Variables

So far we have only considered quantum information carried on the polarization of single photons – a discrete two-state variable. It is also possible to carry

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quantum information on CV such as the amplitude and phase of a light mode. We now discuss how QKD can be implemented with CV. As we have seen the basic mechanism used in QKD schemes is the fact that the act of measurement (by Eve) inevitably disturbs the system. This measurement back-action of course also exists for continuous quantum mechanical variables. In particular let us consider the situation in which Alice sends a series of weak coherent states to Bob whose amplitudes are picked from a two-dimensional (2D) Gaussian distribution centred on zero. Bob chooses to measure either the in-phase (amplitude) or out-of-phase (phase) projections of the states onto a shared local oscillator using homodyne detection (see Section 8.1.4). Bob will effectively see a Gaussian distribution of real amplitude coherent states when he looks at amplitude and a Gaussian distribution of imaginary amplitude coherent states when he looks at the phase. Alice can encode two different random number sequences on the amplitude and phase distributions, respectively (see Section 10.2). Because the two quadrature measurements do not commute (see Section 4.3), Eve now has a similar problem as in the discrete case: any attempt to extract information about one quadrature from the beam will inevitably erase information carried on the other quadrature. If Alice and Bob compare some of the data at the end of the protocol, they will thus notice increased error rates as a result of any intervention. This protocol was developed by F. Grosshans and P. Grangier [36] based on earlier work [37, 38]. As for single-photon QKD, the most general security proofs are based on the equivalence with an entanglement-based scheme [27]. In the CV case the equivalent scheme involves Alice producing an EPR state (see Section 12.3.1), keeping one of the modes and sending the other to Bob. If Alice makes heterodyne measurements on the modes she keeps, then the modes sent to Bob will be equivalent to Gaussian-modulated coherent states, as per the prepare-and-measure scheme. Coherent state QKD can be implemented either by sending very weak coherent pulses of light or by sending bright, quantum-limited light with amplitude/phase modulation of the sidebands playing the role of the coherent states (see Section 4.5.5). Early experimental demonstrations of CV QKD were performed by F. Grosshans et al. [39] using the former technique and A.M. Lance et al. [40] using the latter technique. In the Grosshans et al. experiment, they were able to transmit secret bits at a rate of 75 kbits/s in the presence of 50% line loss, whilst in the Lance et al. experiment a secret key rate of 1 kbit/s in the presence of 90% line loss was achieved. To go beyond ‘laboratory conditions’, the local oscillator needed for Bob’s homodyne measurements should be distributed over an equivalent distance as for the quantum signal. P. Jouguet et al. [41] demonstrated such a scheme using a combination of time and polarization multiplexing to send the local oscillator along the same fibre link as the signal, as shown in Figure 13.5. The authors split 100 ns pulses into a weak signal and a strong LO with an unbalanced coupler. The signal pulse is modulated with a centred Gaussian distribution using an amplitude and phase modulator. The variance is controlled using a course, variable attenuator and an amplitude modulator. The signal pulse is 200 ns delayed with respect to the LO pulse using a 20 m delay line and a Faraday mirror. Both pulses are multiplexed with orthogonal polarization using a PBS. The time and

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Figure 13.5 Optical layout of a long-distance CVQKD prototype. Alice sends Bob 100 ns coherent light pulses generated by a 1550 nm telecom laser diode pulse with a frequency of 1 MHz. Alice and Bob are in the same laboratory and separated by fibre spools. The maximum operating distance of the experiment is 80 km. Source: From [41].

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polarization multiplexed pulses are then sent through the channel. They are demultiplexed on Bob’s side with another PBS combined with active polarization control. A second delay line on Bob’s side allows for time superposition of the signal and LO pulses. After demultiplexing, the signal and LO interfere on a balanced homodyne detector. A phase modulator in the LO path allows for random choice of the measured signal quadrature. This experiment is notable for operating in the telecom window at 1550 nm and performing parameter estimation on finite-sized data blocks. Using these techniques they achieved a maximum distance of 80 km of fibre with a secret key rate of 0.2 kbits/s. 13.5.3

No Cloning

An alternative way of understanding the basic mechanism of QKD is the fact that quantum information cannot be copied or cloned. That is, if given an unknown quantum state, it is not possible to produce an identical copy or clone of it. This was first pointed out by W.K. Wootters and W.H. Zurek [42]. The best that can be achieved is to turn the original input state into a pair of output states, which are approximate copies of the original. For qubits the best achievable fidelity of the clones with the original is 5∕6. For coherent states the best achievable fidelity is 2∕3. Experimental realizations of approximate cloning approaching the optimal bounds have been realized for single-photon qubits [43] and coherent states [44].

13.6 Teleportation When Alice and Bob share an entangled state, their ability to communicate is enhanced. This is quite remarkable given that entanglement is undirected and carries no information itself. For example, suppose Alice and Bob share a pair of qubits in the entangled state |0⟩a |0⟩b + |1⟩a |1⟩b . Alice chooses to do one of four operations on her qubit: 1. Leave the qubit unchanged. 2. Perform a bit-flip or X operation, which takes |0⟩ → |1⟩ and |1⟩ → |0⟩. 3. Perform a phase-flip or Z operation, which leaves |0⟩ unchanged but takes |1⟩ → −|1⟩. 4. Perform a bit flip and a phase flip. She then sends her qubit to Bob who now has a pair of qubits in one of the four states: 1. 2. 3. 4.

|𝜙+⟩ = |0⟩a |0⟩b + |1⟩a |1⟩b , |𝜓+⟩ = |1⟩a |0⟩b + |0⟩a |1⟩b , |𝜙−⟩ = |0⟩a |0⟩b − |1⟩a |1⟩b , |𝜓−⟩ = |1⟩a |0⟩b − |0⟩a |1⟩b ,

respectively. These states are known as the Bell states. The important point is that they are mutually orthogonal and so Bob can now, in principle, perform a measurement that unambiguously picks which of the four states he has (this is known as a Bell measurement). The outcome is that Alice has communicated

13.6 Teleportation

two bits of information to Bob whilst only sending a single qubit! This effect is called quantum dense coding, or super-dense coding [45], and has no classical analogue. Perhaps even more surprising is quantum teleportation [46]. Here the presence of entanglement enables Alice to send Bob an unknown qubit by simply sending a classical message. Consider first what Alice can do in the absence of entanglement. Her best strategy is to measure the qubit in some basis and send a message that tells Bob to make a qubit corresponding to the result she gets. Sometimes she will be lucky and will measure in a good basis, and then Bob will make a close approximation to the qubit. Other times she will measure in a bad basis and the result she gets will be completely random, and Bob will make a poor approximation to the qubit. It can be shown that on average the fidelity of Bob’s qubit with Alice’s original is 2∕3. If they share entanglement, they can perform teleportation. This works in the following way: Alice and Bob again share an entangled pair of qubits, say, |0⟩a |0⟩b + |1⟩a |1⟩b . Alice also has a qubit in the arbitrary state 𝜇|0⟩a′ + 𝜈|1⟩a′ that she wishes to send to Bob. Alice does not know the state of her qubit. If we write down the state of the three qubits and then rearrange it, we notice a remarkable feature: Bob’s qubit can be represented as an equal superposition of four states, each differing from Alice’s unknown qubit by at most a bit flip and a phase flip. What is more, Alice can tell which of these four states Bob actually has by making a Bell measurement on her two qubits. Explicitly (𝜇|0⟩a′ + 𝜈|1⟩a′ )(|0⟩a |0⟩b + |1⟩a |1⟩b ) = (|0⟩a′ |0⟩a + |1⟩a′ |1⟩a ) (𝜇|0⟩b + 𝜈|1⟩b ) + (|0⟩a′ |0⟩a − |1⟩a′ |1⟩a ) (𝜇|0⟩b − 𝜈|1⟩b ) + (|0⟩a′ |1⟩a + |1⟩a′ |0⟩a ) (𝜇|1⟩b + 𝜈|0⟩b ) + (|0⟩a′ |1⟩a − |1⟩a′ |0⟩a ) (𝜇|1⟩b − 𝜈|0⟩b )

(13.25)

If the result of Alice’s Bell measurement is |𝜙+⟩, she tells Bob not to do anything; if it is |𝜙−⟩, she tells him to do a phase flip; if it is |𝜓+⟩, a bit flip is required; and finally if she measures |𝜓−⟩, she tells him to perform both a bit flip and a phase flip. In the end Bob has turned his qubit into an exact copy of Alice’s original, but all Alice has sent is a two-bit classical message. 13.6.1

Teleportation of Photon Qubits

Optics looks a good candidate for a demonstration of teleportation. We have already seen in Section 12.3.3 that spatially separated polarization-entangled states of the required form (at least after post-selection) can be generated by down-conversion and the local operations can be implemented using wave plates. Notice though that in addition to the entangled photons we need another photon to be produced simultaneously to act as the teleportee. Also the Bell measurement represents a problem as differentiation of all four Bell states requires a non-linear Kerr interaction much stronger than is presently available. These problems were solved in the demonstration by D. Bouwmeester et al. at the University of Innsbruck [47]. Their experimental layout is shown in

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ALICE f2

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Figure 13.6 Layout of the experiment by D. Bouwmeester et al. Source: After D. Bouwmeester et al. [47].

Figure 13.6. Pulsed UV pumping (200 ps pulse length, 76 MHz repetition rate) of a non-linear crystal in a type II arrangement was used to produce pairs of polarization-entangled photons at 788 nm (2 and 3). The pump pulse was then retro-reflected through the crystal such that a second pair of counter-propagating photons (1 and 4) might be produced. Teleportation can then proceed by giving Alice entangled photon 2 and photon 1 as the teleportee, after it has been prepared in some arbitrary state, and giving Bob entangled photon 3. Although a complete Bell measurement is not practical, a partial Bell measurement can be achieved quite straightforwardly by passing photons 1 and 2 through a 50/50 beamsplitter and then photon counting at the outputs. The action of a beamsplitter on the Bell states is to make the photons bunch (i.e. both exit through the same port) in much the same way as for the HOM dip. This applies to all Bell states, but not the state |𝜓−⟩, for which the photons always exit by different ports. Thus, if Alice records a coincidence count at the output of the beamsplitter, then she has unambiguously identified the |𝜓−⟩ Bell state and teleportation has succeeded. The experiment is arranged such that the |𝜓−⟩ state is the ‘do nothing’ result. If she does not obtain a coincidence, the protocol has failed. Results from the experiment are shown in Figure 13.7. Teleportation should have succeeded when photon 4 is detected at p (indicating that photon 1 is on its way), a coincidence is detected at f1 and f2 (indicating the correct Bell state has been detected), and a photon is detected at either d1 or d2 (indicating the teleported photon has been successfully detected). When these fourfold coincidences occur, we should expect the polarization state of Bob’s photon to match that of the photon prepared for Alice. This is what is shown in Figure 13.7 for input photons in the non-orthogonal states |A⟩ and |H⟩ (−45∘ and 0∘ , respectively). At zero time delay, when photons 1 and 2 meet at their beamsplitter, the

13.6 Teleportation

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Figure 13.7 Experimental results from the experiment by D. Bouwmeester et al. Source: After D. Bouwmeester et al. [47].

original states are reproduced with a visibility of 70% ± 3%. When a time delay is introduced, photons 1 and 2 become distinguishable at the beamsplitter, and the Bell measurement fails and so too does the teleportation. This experiment was very technically challenging at the time. The probability of four-photon events was very low. To prevent any temporal distinguishability of photons 1 and 2, a frequency filtering producing a 4 nm bandwidth was applied, further reducing the counts. Finally the protocol itself only succeeded one quarter of the time. This resulted in roughly one successful event per minute. With recent improvements to sources and detectors, such experiments now run at rates of up to a few per second. A subtlety of the original experiment was that there was a significant probability for the down-converter to produce two photons each in modes 1 and 4. Then, even under perfect conditions of zero loss, a threefold coincidence on Alice’s side of the experiment does not guarantee a photon is sent to Bob. In a later manifestation of the experiment, the possibility of such errors was made negligible, and fidelities of > 80% were observed, well in excess of the 2∕3 limit [48]. 13.6.2

Continuous Variable Teleportation

So far we have considered teleportation of qubits, as carried by the polarization degree of freedom of single photons. This technique will only work for single-photon states. What if we wish to teleport a general field state with

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contributions from vacuum and higher photon number terms? The answer is to implement a teleportation protocol based on the measurement of the quadrature amplitudes of the field. Because the quadrature amplitudes are continuous, rather than discrete, variables, this is known as continuous variable teleportation. It was developed by S.L. Braunstein and H.J. Kimble [49] based on earlier work by L. Vaidman [50]. Consider the following situation. Alice wishes to teleport to Bob an unknown coherent state, |𝛼⟩, drawn from a broad Gaussian distribution. In the absence of entanglement, Alice’s best approach is to divide the field into two equal parts at a beamsplitter and then measure the amplitude quadrature of one half and the phase quadrature of the other. This is referred to as a dual homodyne measurement. The amplitude measurement gives an estimate of the real part of 𝛼, whilst the phase measurement gives an estimate of the imaginary part of 𝛼; however both estimates are imperfect due to noise from the vacuum field, which inevitably enters through the open port of the beamsplitter. Alice sends these estimates to Bob who uses them to produce a coherent state by displacing his local vacuum state by the relevant quantities. This situation is most easily described in the Heisenberg picture. Let the initial field mode be represented by the annihilation operator â and the vacuum entering at the 50/50 beamsplitter by v̂ a . The measurement results obtained by Alice are then represented by the quadrature operators (see Section 4.4.3): ̂ Va ) ̂ a = √1 (X𝟏 ̂ a − X𝟏 X𝟏 2 ̂ Va ) ̂ a = √1 (X𝟐 ̂ a + X𝟐 (13.26) X𝟐 2 These are sent to Bob who uses them to displace his vacuum field, v̂ b giving the output field 1 ̂ 1 ̂ ̂ ̂ (13.27) b̂ = v̂ b + g (X𝟏 a + iX𝟐a ) − g (X𝟏Va − iX𝟐Va ) 2 2 where g is a gain factor for the displacement. Choosing g = 1, unity gain, Bob’s output field is (13.28) b̂ = â + v̂ b − v̂ a† Notice that two vacuum fields have been added to the output, one entering through Alice’s measurement and the other through Bob’s reconstruction. As a result Bob’s state is mixed and is three times noisier than the quantum noise limit (QNL) level of the input coherent state. Now suppose Alice and Bob share an entangled state, in particular an EPR entangled state of the type discussed in Section 12.3.1. Alice again divides and measures her beam, but this time instead of allowing vacuum to enter the empty port of her beamsplitter, she sends in her half of the EPR pair. As a result her quadrature measurement results are now given by √ √ ̂ Va − G − 1 X𝟏 ̂ Vb ) ̂ a − G X𝟏 ̂ a = √1 (X𝟏 X𝟏 2 √ √ ̂ Va − G − 1 X𝟐 ̂ Vb ) ̂ a = √1 (X𝟐 ̂ a + G X𝟐 (13.29) X𝟐 2

13.6 Teleportation

where we have used Eq. (12.19) to describe the entanglement. These results are sent to Bob who now uses them to displace his half of the EPR pair, obtaining (at unity gain) √ √ √ √ (13.30) b̂ = â + ( G − G − 1)̂vb + ( G − G − 1)̂va† √ √ Now in the limit G → ∞, ( G − G − 1) → 0; hence in this limit Eq. (13.30) reduces to b̂ = â

(13.31)

Evolution through the teleporter is the identity, and so the output state is identical to the input (this is obviously true not only for the coherent input states we have been considering but for any input state). An alternative to unity √ gain√is to employ gain tuning whereby the entanglement-dependent gain

ge = ( √G+√G−1−1 )2 is used. The resulting output is then [51] G+ G−1+1 √ √ b̂ = ge â + 1 − ge v̂

(13.32)

where v̂ is a vacuum mode. Notice that now the output is simply an attenuated version of the input, with the effective transmission coefficient, ge , a function of the strength of the entanglement. In the limit of G → ∞, we have ge → 1 and we retrieve the unity gain result. Indeed, in general for all gains and entanglement strengths, the CV teleporter can be considered equivalent to a quantum optical channel with varying levels of attenuation, or amplification, plus added thermal noise [52]. Once again optics is an obvious candidate for demonstrating this type of teleportation. The required EPR states can be produced via a squeezing interaction (see Section 12.3.1), and the coherent input states can be created through sideband modulation of a quantum-limited laser (see Section 4.5.5). Bob’s displacement can also be achieved via modulation. The first demonstration of this type was by A. Furusawa et al. at the California Institute of Technology [53]. The experimental set-up is shown in Figure 13.8. EPR entanglement is produced by the mixing of two out-of-phase squeezed beams on a 50/50 beamsplitter. Both squeezed beams (at 860 nm) are generated in a single ring-cavity optical parametric oscillator (OPO) by simultaneously pumping counter-propagating cavity modes. The cavity is actively locked, and the parametric medium is temperature-tuned potassium niobate. One of the EPR beams is sent to Alice who mixes it with her signal beam and performs dual balanced homodyne measurements, actively locked to be 90∘ out of phase, such that conjugate quadrature measurements are made. The photocurrents thus generated are sent to Bob who uses them to impose phase and amplitude modulations on a bright laser beam. By mixing this bright beam with his EPR beam on a highly reflective beamsplitter, Bob can efficiently impose on it a displacement proportional to the modulations. All the beams in the experiment originate from a single Ti/sapphire master laser, including the signal beam that has a known modulation (coherent amplitude) imposed on it before being sent to Alice. The modulation on the signal beam was imposed at 2.9 MHz and had an amplitude of about 25 dB above QNL. When, based on the signal size observed on Bob’s side, unity gain was achieved, the quadrature noise floors of

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Figure 13.8 Schematic layout of the Caltech teleportation experiment. Source: From A. Furusawa et al. [53].

the teleported beam were measured by an independent balanced homodyne detector, both with and without entanglement. The quality of Bob’s reconstruction can be evaluated via the fidelity of it compared with the initial coherent state Alice sent. Provided the output is Gaussian (which it is), this fidelity is given by ] [ 2 2 2 2 exp − √ |𝛼| (1 − g) F=√ (V 1b + 1)(V 2b + 1) (V 1b + 1)(V 2b + 1) (13.33) Recalling from our earlier discussion that without entanglement the quadrature variances of the outputs are V 1b = V 2b = 3, then we find from Eq. (13.33) that for large 𝛼 the best fidelity with no entanglement is achieved at unity gain and is F = 0.5. This is confirmed by Furusawa et al. who find a best fidelity without entanglement of Fc = 0.48 ± 0.03. On the other hand, with entanglement, a fidelity of Fq = 0.58 ± 0.02 is measured, clearly exceeding the classical bound. Experimental results as a function of gain are shown in Figure 13.9. A subsequent experiment by Bowen et al. at the Australian National University made significant improvements [54]. These included higher fidelities (Fq = 0.64 ± 0.02), stable operation over long periods, and a greater range of input state amplitudes. Their experiment used two independent monolithic sub-threshold OPOs to produce twin squeezed beams at 1064 nm, which were then mixed on a beamsplitter to produce the required EPR entanglement. In Figure 13.10 spectra of the

13.6 Teleportation

0.7 0.6

Fidelity F

0.5 0.4 0.3 0.2 0.1 0.0 –4

–2

–3

–1

0 1 Gain g2 (dB)

2

3

4

Figure 13.9 Measured fidelities from the experiment by A. Furusawa et al. Results with and without entanglement are shown along with theoretical curves. Classical (red) and quantum (blue) teleportation with the theoretical results as solid lines and experimental data points. Source: From A. Furusawa et al. [53].

Time (min)

Signal (dB relative to QNL)

0 5 4 3 2 1 0

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Figure 13.10 Input and output spectra for amplitude and phase quadratures from the ANU teleportation. Time traces show noise on amplitude and phase quadratures of the output over a five minutes interval. Source: From W.P. Bowen et al. [54].

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amplitude and phase quadratures of the input and teleported signals are shown along with time traces showing the stability of the system over several minutes. In addition the performance of the Bowen et al. teleporter was characterized in terms of the teleportation T–V diagram, introduced by one of us (TCR) and P.K. Lam and co-worker [55]. The teleportation T–V diagram is in a similar vein to the QND T–V diagram (see Section 11.2). As we have seen, in the absence of entanglement, strict bounds are placed on both the accuracy of measurement and reconstruction of an unknown state. These are represented by the vacuum modes that appear in Eq. (13.28). These bounds can be expressed in such a way that, in contrast to fidelity, they are input state independent. Alice’s measurement accuracy is limited by the generalized uncertainty principle V 1M V 2M ≥ 1 [56], where V 1M , V 2M are the quadrature measurement penalties, which holds for any simultaneous measurements of conjugate quadrature amplitudes of an unknown quantum optical system. Note that in most current publications the notation has evolved into X1 = X + , X2 = X − , V 1 = V + , V 2 = V − , etc. This relationship can be rewritten in terms of quadrature signal transfer coefficients T1 = SNR1out ∕SNR1in and T2 = SNR2out ∕SNR2in as ( ) 1 Tq = T1 + T2 − T1 T2 1 − ≤1 (13.34) V 1in V 2in where SNR1 = 𝛼1 ∕V 1 and SNR2 = 𝛼2 ∕V 2 are the relevant signal-to-noise ratios. This expression reduces to Tq = T1 + T2 for minimum uncertainty input states (V 1in V 2in = 1). Without entanglement it is not possible to break the inequality given in Eq. (13.34). Bob’s reconstruction must be carried out on a mode of the E∕M field, the fluctuations of which must already obey the uncertainty principle. In the absence of entanglement, these intrinsic fluctuations remain present on any reconstructed field; thus the amplitude and phase conditional variances, V 1in|out = ̂ in 𝛿 X𝟏 ̂ out ⟩|2 ∕V 1in and V 2in|out = V 2out − |⟨𝛿 X𝟐 ̂ in 𝛿 X𝟐 ̂ out ⟩|2 ∕V 2in , V 1out − |⟨𝛿 X𝟏 respectively, which measure the noise added during the teleportation process, will satisfy V 1in|out V 2in|out ≥ 1. This can be written in terms of the signal transfer and quadrature variances of the output state as prod

Vq

= (1 − T2)(1 − T2) V 1out V 2out ≥ 1

(13.35)

The criteria of Eqs. (13.34) and (13.35) are then used to represent the quantum teleportation on the T–V graph. The Tq and Vq bounds have independent physical significance. If Bob’s state passes the Tq bound (Eq. (13.34)), then he can be sure, regardless of how it was transmitted to him, that no other party can possess a copy of the state that also passes this bound (i.e. carries as much information about the original). Surpassing the Vq bound is a necessary prerequisite for reconstruction of non-classical features of the input state such as squeezing. Clearly it is desirable that the Tq and Vq bounds are simultaneously exceeded. It was first pointed out by F. Grosshans and P. Grangier [57] that the crossover point (1, 1), corresponds to a fidelity of 2∕3, the no-cloning limit for coherent states (see Section 13.5.3). Perfect reconstruction of the input state would result in Tq = 2 and Vq = 0.

13.6 Teleportation

2.0

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57 0.5

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Figure 13.11 T–V graph of results from a sequence of experiments by A. Furusawa et al. [53], W.P. Bowen et al. [54], M. Yukawa et al. [58], and S. Takeda et al. [59]. The dashed line represents unity gain, whilst below the dot-dashed line is unphysical.

In Figure 13.11 data from a sequence of experiments are plotted on the T–V diagram. The Bowen et al. experiment was the first to (marginally) exceed both Tq and Vq simultaneously. More recent experiments have seen significant improvement in the quality of the teleportation. For example, M. Yukawa et al. at the University of Tokyo achieved a fidelity of 83% for teleportation of coherent states [58]. CV teleportation teleports the state of the optical mode. This means that it is not restricted to traditional CV states, like coherent states or squeezed states, but can also teleport number states or other more exotic states. This is in stark contrast to discrete variable teleportation, which can only teleport qubits in some particular encoding. To demonstrate this versatility, S. Takeda et al. at the University of Tokyo used CV teleportation to teleport dual-rail, time-bin qubits [59]. To achieve this, the two modes of the dual-rail qubit were independently, but coherently teleported. Single-photon states compatible with the CV teleporters were conditionally produced from a continuous-wave (CW) down-conversion source via techniques similar to those discussed in Section 13.7.3. The result was deterministic teleportation of the photonic input states with fidelities greater than the classical bound. The teleporter could also be run under gain-tuned conditions for which the teleported output appears attenuated, but otherwise undistorted, even for low levels of squeezing (see Eq. (13.6.2)). This allows a direct comparison with the post-selective discrete variable teleportation experiments discussed in the previous section. Employing post-selection for the CV qubit teleportation results in fidelities of around 80% and success probabilities around 43%. The fidelities are similar to those achieved with discrete variable teleportation; however, the efficiency of the CV technique is at least an order of magnitude higher.

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13.6.3

Entanglement Swapping

An interesting special case of teleportation is when the state to be teleported is itself part of an entangled state. In this way entanglement between two modes that are close together can be teleported to modes that are far apart [46]. The first experimental demonstration of entanglement swapping was carried out with a discrete variable system by J.-W. Pan et al. [60]. The experiment was based on the four-photon source used in the original teleportation experiment at the University of Innsbruck (see Section 13.6.1). The visibility of the correlation fringes between the teleported photons was ∼65%, exceeding the 50% bound, hence confirming the entanglement had survived the teleportation. In later experiments, R. Kaltenbaek et al. [61] were able to perform entanglement swapping between independent sources of entangled photons (see Section 12.2) with high enough quality to violate a Bell inequality (see Section 12.3.3) between the swapped photons. As for single-photon teleportation experiments, the results are post-selected on the basis of fourfold coincidences, and the Bell measurement producing the swapping is nondeterministic. Entanglement swapping can also be carried out using CV teleportation. The first experimental realization was carried out by X. Jia et al. [62]. Using two polarization non-degenerate optical parametric amplifiers operating at 1.08 μm, they were able to deterministically swap EPR entanglement, producing two-mode squeezing between the swapped modes of 1.23 and 1.12 dB below the QNL for the amplitude and phase quadratures, respectively. This corresponds to a negativity of EN = 0.27. It is also possible to teleport single-photon entanglement using CV teleportation [51]. In the experiment of S. Takeda et al., a heralded single photon was split on a beamsplitter to produce single-photon mode entanglement, one mode of which was then deterministically swapped using CV teleportation [63]. The output state had a negativity of EN = 0.28 as shown in Figure 13.12. 13.6.4

Entanglement Distillation

Teleportation can be used to communicate quantum information provided that entanglement has been established between Alice and Bob. But it seems that a 0.3 0.25 Logarithmic negativity

502

Figure 13.12 Results for entanglement swapping from S. Takeda et al. Gain dependence of the logarithmic negativity for r = 0 (blue diamonds), r = 0.71 (green triangles), and r = 1.01 (red circles). Theoretical curves are also plotted in the same colours. Source: From S. Takeda et al. [63].

r = 0.00 r = 0.71 r = 1.01

0.2 0.15 0.1 0.05 0 0.50

0.63 0.79 Gain, g

1.0

13.6 Teleportation

noise-free quantum channel is needed between Alice and Bob in order to establish the entanglement in the first place. From a practical point of view, one may then ask: why not send the quantum state directly through this noise-free channel and not bother with teleportation? It turns out that it is not necessary to have a noise-free channel to distribute high quality entanglement. This then means that teleportation can be used to communicate quantum states with high fidelity when the existing quantum channel between Alice and Bob is noisy. The distribution of high quality entanglement through a noisy channel is achieved using a technique called entanglement distillation [64]. Distillation converts an ensemble of weakly entangled states shared by Alice and Bob into a smaller sub-ensemble of more strongly entangled states using operations performed locally by Alice and Bob. Local operations cannot increase entanglement overall, so it must be true that the total entanglement of the final sub-ensemble is less than or equal to the total entanglement of the original ensemble. However, this still allows the individual members of the final sub-ensemble to be more entangled than the individual members of the original ensemble. The first experimental demonstration of distillation of Bell-type entanglement was carried out by P.G. Kwiat et al. using a simple polarization filtering scheme [65]. Consider non-maximally entangled polarization Bell states of the form √ √ 𝜖|H⟩|H⟩ + 1 − 𝜖|V ⟩|V ⟩ (13.36) where we can take 𝜖 > 1∕2. Suppose each beam is passed through a partially PBS with transmission for vertical states of unity, but transmission for horizontal states of 𝜂 (see Section 5.1.1). If we now herald on no photon emerging from the reflected port of the partially PBS, the conditional state is √ √ √ (𝜂 𝜖|H⟩|H⟩ + 1 − 𝜖|V ⟩|V ⟩)∕ 1 − 𝜖(1 − 𝜂 2 ) (13.37) √ √ If we choose 𝜂 = (1 − 𝜖)∕𝜖, then Eq. (13.37) reduces to 1∕ 2(|H⟩|H⟩ + |V ⟩|V ⟩), i.e. a maximally entangled Bell state. The experiment used downconversion through a pair of thin type I BBO crystals to produce non-maximally polarization-entangled photon pairs (see Section 12.3.3). In a typical result, starting with a non-maximally entangled state with an 𝜖 = 0.17, a photon pair with 99% fidelity with the maximally entangled Bell state was distilled. In the experiment the distilled states were not actually heralded by a zero count at the reflected ports, but rather were post-selected by a pair of photons arriving at the final detector. Single-photon modal entanglement can be distilled using the noiseless linear amplifier (NLA) (see Section 6.4). The ideal transformation enacted by the NLA on a general state, |𝜓⟩, can be written as |𝜓⟩ → g n̂ |𝜓⟩

(13.38)

This is an unphysical operation as it can, in general, cause the state to become unnormalizable for sufficiently large g. Physical NLAs approximate this operation up to some photon number cut-off, N. Consider the non-maximal single-photon modal entanglement obtained by sending a single photon through a beamsplitter, 𝜇|0⟩|1⟩ + 𝜈|1⟩|0⟩, where we assume 𝜇 > 𝜈. If the NLA is applied to the first arm

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of this state, then (according to Eq. (13.38)) the state is transformed as 𝜇|0⟩|1⟩ + 𝜈|1⟩|0⟩ →

𝜇|0⟩|1⟩ + g𝜈|1⟩|0⟩ √ |𝜇|2 + g 2 |𝜈|2

(13.39)

If we pick the gain such that g|𝜈| = |𝜇|, then the state becomes maximally entangled. This type of distillation was demonstrated by G.Y. Xiang et al. using the quantum scissor technique to implement the NLA [66]. Pairs of photons were generated by a pulsed type I down-conversion source. One photon was split on a beamsplitter to produce the modal entanglement, whilst the other photon was used as the ancilla for an NLA that operated on one arm of the modal entanglement. An interferometric arrangement was used to measure the entanglement with and without the noiseless amplification. The entanglement was shown to be increased by a factor of approximately 1.75 by the distillation step. Unlike the filtering experiments, successful entanglement distillation events were heralded by the NLA. A no-go theorem due to J. Eisert et al. shows that Gaussian entangled states cannot be distilled by Gaussian operations alone [67]. This means that the traditional elements of a CV experiment, such as displacements, squeezing operations, linear optical networks, and homodyne detection, are insufficient to distill Gaussian entangled states. A ‘loop hole’ is that Gaussian states that have been affected by non-Gaussian noise will become non-Gaussian states. In this situation distillation with Gaussian elements becomes possible [68]. However, if the noise is Gaussian, then a non-Gaussian element like the NLA is required. An example is the results by A.E. Ulanov et al. and their team in Moscow [69] who demonstrated how the NLA could undo the effects of losses on an EPR state (see Figures 13.13 and 13.14).

A Distilled state

EPR source

Homodyne detector

B Loss |1〉

Noiseless amplification

〈1| SPCM2

Homodyne detector λ/2

EOM Loss channel PPKTP1

SPCM1

λ/2 Noiseless amplification PPKTP2

Ancilla photon preparation

EPR state preparation

Figure 13.13 Experimental set-up for distillation of Gaussian noise on an EPR state. The NLA distillation event corresponds to the click in the single-photon detector placed in the beamsplitter output. Insert: conceptual scheme. Source: From A.E. Ulanov et al. [69].

13.7 Quantum Computation

〈(XA ± XB)2〉

1.4 Anti-squeezing (initial state)

1.2 Anti-squeezing after loss

1.0 0.8

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Figure 13.14 Analysis of the distilled states as a function of the NLA gain. (a) The case of the initial squeezing and anti-squeezing growing with the gain to a limiting case (dashed line). In addition, the reduction of squeezing after the introduction of losses is shown. The vertical axes are in units of the QNL. (b) The logarithmic negativity. The theoretical curves are calculated assuming an initial squeezing parameter of 𝛾 = 0.05, detection efficiencies in the undistilled and distilled channels of 𝜂A = 𝜂B = 0.5, and single-photon preparation efficiency of 𝜂 = 0.65. For the case of a loss channel, 𝛾 = 0.135 and 𝜂A = 0.45 are assumed. Source: From A.E. Ulanov et al. [69].

13.7 Quantum Computation We have seen that optics has proven an excellent testing ground for many of the principles and basic tasks in quantum information and is the system of choice for quantum communication applications. However, what about the more challenging task of quantum computation? It turns out that optics has certain advantages, as well as certain drawbacks, with respect to quantum computing. Whilst the eventual physical platform that will realize the first practical quantum computer is still undecided, it seems very likely that the winning platform will contain at least some quantum optical components. Experimental demonstrations of quantum computation with optics have been carried out over dual-rail, single-rail, and even CV quantum encodings. Indeed, the variety of different encodings available in optics is one of its advantages.

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In this section we will look at key proof-of-principle experiments performed with each of these encodings. We will finish by examining the paths towards a full-scale optical quantum computer, progress, and prospects. 13.7.1

Dual-Rail Quantum Computing

It turns out that to be able to implement arbitrary processing of information encoded on a set of qubits, it is sufficient to be able to implement arbitrary operations on single qubits as well as having at least one non-trivial two-qubit operation. Single-qubit operations on dual-rail qubits can be implemented using linear optics. For example, for polarization-encoded single-photon qubits, arbitrary single-qubit operations can be implemented using a succession of half- and quarter-wave plates. For spatial mode qubits, beamsplitters and phase shifters suffice. An example of a non-trivial two-qubit gate is the CNOT gate. In terms of logical qubits, its operation is summarized by the following truth table: |𝟎⟩c |𝟎⟩t → |𝟎⟩c |𝟎⟩t |𝟎⟩c |𝟏⟩t → |𝟎⟩c |𝟏⟩t |𝟏⟩c |𝟎⟩t → |𝟏⟩c |𝟏⟩t |𝟏⟩c |𝟏⟩t → |𝟏⟩c |𝟎⟩t

(13.40)

When the control qubit is in the zero state, |𝟎⟩c , the value of the target qubit |𝟎⟩t or |𝟏⟩t is unchanged. However, when the control is one, |𝟏⟩c , the value of the target qubit is flipped, zero to one, and vice versa. The effect of a CNOT gate on superposition states is simply a superposition of the transformations of Eq. (13.40). For example, if the control is in the diagonal basis, we get the following transformations: 1 1 √ (|𝟎⟩c + |𝟏⟩c )|𝟎⟩t → √ (|𝟎⟩c |𝟎⟩t + |𝟏⟩c |𝟏⟩t ) 2 2 1 1 √ (|𝟎⟩c + |𝟏⟩c )|𝟏⟩t → √ (|𝟎⟩c |𝟏⟩t + |𝟏⟩c |𝟎⟩t ) 2 2 1 1 √ (|𝟎⟩c − |𝟏⟩c )|𝟎⟩t → √ (|𝟎⟩c |𝟎⟩t − |𝟏⟩c |𝟏⟩t ) 2 2 1 1 √ (|𝟎⟩c − |𝟏⟩c )|𝟏⟩t → √ (|𝟎⟩c |𝟏⟩t − |𝟏⟩c |𝟎⟩t ) 2 2

(13.41)

Notice that the resulting output states are the four Bell states (see Section 13.6). How might such an interaction between two photons be implemented? The only deterministic solution is the use of a Kerr medium with a non-linearity so strong that a single photon can induce a π phase shift on another single photon, as first suggested by G.J. Milburn [70]. This is very technically challenging to achieve. However, if we are willing to accept probabilistic gates, then a dual-rail CNOT gate (and several other quantum processing gates) can be implemented via linear optics.

13.7 Quantum Computation

13.7.1.1

Quantum Circuits with Linear Optics

The simplest linear optical CNOT design was introduced by T.C. Ralph et al. [71]. The gate is shown schematically in Figure 13.15 for a polarization qubit implementation. Consider the input |H⟩c |H⟩t . The target wave plate produces the transformation 1 |H⟩c |H⟩t → √ |H⟩c (|H⟩t + |V ⟩t ) 2 which is a Hadamard transformation on the target qubit. The PBS then spatially separate the polarization modes of the two beams. An array of different possibilities are present after the middle beamsplitters; however, we are only interested in those events where photons arrive at both the target and control outputs. There are two ways for this to happen: the control photon must take the top path and reflect off beamsplitter 1, and the target photon may take its upper path and reflect off beamsplitter 2 or take the bottom path and reflect off beamsplitter 3. In both cases the effect is just to reduce the amplitude of the successful components by a factor of 1∕3. The output state is then transformed by the second target wave plate such that 1 1 1 √ |H⟩c (|H⟩t + |V ⟩t ) → |H⟩c |H⟩t 3 2 3 Similarly the input |H⟩c |V ⟩t is unchanged by passage through the circuit, other than a 1∕3 reduction in amplitude. The probability of this event happening is 1∕32 = 1∕9. Events where photons do not appear at both outputs are discarded by post-selection. Things are different when the control is in the vertical state. Consider the input state |V ⟩c |H⟩t . The target wave plate produces the transformation 1 |V ⟩c |H⟩t → √ |V ⟩c (|H⟩t + |V ⟩t ) 2 Now there are three ways for a successful detection to occur. The control takes its lower path. If the target photon takes the bottom path, then both must reflect off their respective beamsplitters as before, simply reducing the amplitude by 1∕3. However, if the target photon follows its upper path, then there are two possibilities at beamsplitter 2: either both photons may be reflected, giving an amplitude Figure 13.15 Schematic representation of a coincidence basic CNOT gate.

PBS

PBS (1)

1/3

c

co (2)

1/3

t

to (3)

HWP PBS 22.5

1/3 PBS HWP 22.5

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of 1∕3, or both may be transmitted, giving an amplitude of −2∕3. If the photons are indistinguishable, then these amplitudes are added, giving a total amplitude for that component of −1∕3! Thus, when the polarization modes are recombined, the state carries a minus sign on one target component, and the second target wave plate makes the transformation 1 1 1 √ |V ⟩c (|H⟩t − |V ⟩t ) → |V ⟩c |V ⟩t 3 2 3 and the value of the target qubit is flipped as required. Similarly the circuit does the transformation |V ⟩c |V ⟩t → 1∕3|V ⟩c |H⟩t . Hence CNOT operation is realized whenever a coincidence is recorded. The probability of success is (1∕3)2 = 1∕9. If the Hadamard transformations on the target are not implemented, then a Controlled-Sign (CZ) gate is implemented. J.L. O’Brien et al. at the University of Queensland used this technique to demonstrate CNOT operation for single-photon qubits [72]. Their experimental set-up is shown in Figure 13.16. A pair of polarization beam displacers was used to spatially separate and then recombine the polarization modes of the qubits in an interferometrically stable configuration. A single wave plate oriented at 62.5∘ and inserted between the displacers then played the role of all three one third beamsplitters in the conceptual diagram (Figure 13.15). Photons pairs at 702 nm were produced via type II down-conversion in beam-like modes. The pairs were not polarization entangled. They were spatially filtered through optical fibres before being prepared in particular qubit states via a sequence of half- and quarter-wave plates and injected into the circuit. An automated state tomography system could then be used to obtain the density operator of the output state for comparison with the expected outcome (see Section 13.4). After passing through 0.36 nm frequency filters, coincidences detected within a 5 ns time window were excepted. The truth table performance ranged from about 95% correct for ‘control-off’ operation to 73% for ‘control-on’ operation. The difference is attributed to the need for both single- and two-photon interference when the control is on. All four Bell states were produced as described in Eq. (13.41) with fidelities ranging from 77% to 87%, thus demonstrating the ability of the gate to produce entanglement. This leads to an average gate fidelity of around 81%. 1/3 (62.5°)

State preparation

X

Cout

C0 Cin

C1

Tin

T+

Automated quantum state tomography

Tout

T–

X

H (22.5°) PBS

BS

QWP

HWP

702.2 ± 0.18 nm filter

SPCM

X Dump

Figure 13.16 Experimental layout for the demonstration of a CNOT gate used by O’Brien et al. Source: After O’Brien et al. [72].

13.7 Quantum Computation

(i)

(ii)

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II IX IY IZ XI XX XY XZ YI YX YY YZ ZI ZX ZY ZZ

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II IX IY IZ XI XX XY XZ YI YX YY YZ ZI ZX ZY ZZ

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XZ YI YX

Figure 13.17 Process matrix of the CNOT gate. (a) Ideal process matrix. (b) Maximum likelihood experimentally reconstructed process matrix. (i) Real and (ii) imaginary parts are shown. A process fidelity of 96.13 ± 0.17 is observed. Source: From X.-Q. Zhou et al. [74].

Much higher fidelities were achieved in subsequent experiments. For example, in Ref. [73], an average gate fidelity of 97% was reported. More generally the process matrix representing the operation of the CNOT gate on any state can be calculated. A high fidelity example is shown in Figure 13.17. Many small-scale circuits based on the CNOT scheme of Ref. [71] have been implemented using both the original polarization encoding and spatial and temporal encodings. For example, in A. Politi et al. [75], key elements of Shor’s factoring algorithm were demonstrated using a dual-rail spatial mode encoding, implemented on an integrated waveguide silica-on-silicon chip. However, there are limitations on the quantum circuits that can be constructed in this way. Besides the reduction in probability of success as the circuit grows,

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there is the problem that cascaded CNOTs of this kind can simultaneously fail in such a way that it looks like a success – that is, each qubit is still occupied by a photon, but the evolution is wrong. Ingeniously, there are ways to mitigate both the reduction in probability of success and the occurrence of false successes by utilizing additional degrees of freedom to implement larger circuits directly, rather than decomposing them into circuits containing multiple CNOT gates. For example, in B.P. Lanyon et al. [73], the three-qubit Toffoli gate was demonstrated. The Toffoli gate implements a bit flip on the target state if both of two control qubits are in their logical one states. It is a key three-qubit gate, but requires at least six CNOT gates to implement. As such it seems impossible to implement using the gates so far discussed. Nevertheless it turns out that if one of the qubits in the circuit is temporarily extended to be a qutrit, i.e. a three-level system, then the Tofolli gate can be implemented using only two CNOT gates. The additional level is introduced by adding an extra spatial rail to one of the polarization qubits. Ideally this gate operates with a probability of success 1/72 (a low probability, but much higher than (1∕9)6 ). An average gate fidelity of around 80% was achieved for the Toffoli in this experiment. A more general version of this technique was developed by X.-Q. Zhou et al. [74] that could be applied to any Controlled-Unitary gate, i.e. a gate that applies any desired unitary evolution, to some arbitrary number of target qubits, dependent on whether a control qubit is ‘off’ or ‘on’. This technique has been used to implement a number of quantum circuits that would have been inaccessible otherwise, including the demonstration by R.B. Patel et al. of the Fredkin gate (a Controlled-SWAP) with an average gate fidelity of around 93% [76]. Ideally this gate operates with a probability of success 1/16. So far all the gates we have considered have the nice property that they only require as many photons per run as there are qubits. But this has the downside that it is not necessarily possible to concatenate these gates or circuit elements because we do not know if individual elements succeeded without determining whether the photons exited in the correct modes after each element. A way around this problem is to introduce additional ancillary photons that can herald the successful operation of each element. Gates of this kind were introduced by Knill, Laflamme, and Milburn [77]. In particular E. Knill, R. Laflamme and G. Milburn (KLM) showed how to make a CNOT gate that was nondeterministic, but heralded. That is, the gate does not always work, but an independent signal heralds successful operation. A somewhat simplified version of this gate is shown in Figure 13.18. In addition to the single-photon polarization qubits incident at ports c (control) and t (target), the gate also has ancilla inputs comprising two vacuum input ports, 𝑣1 and 𝑣2, and two single-photon √ input ports, p1 √ and p2. The beamsplitter reflectivities are given by 𝜂1 = 5 − 3 2 and 𝜂2 = (3 − 2)∕7. It can be shown that when no photons are detected at outputs 𝑣o1 and 𝑣o2 and one and only one photon is detected at each of po1 and po2, then the gate has succeeded and the photon qubits exiting through co and to have had the CNOT transformation applied to them. The probability of successful operation is 𝜂22 ≈ 0.05.

13.7 Quantum Computation

Figure 13.18 Schematic representation of nondeterministic CNOT gate. Successful operation is heralded by the detection of no photons at outputs 𝑣o1 and 𝑣o2 and the detection of one and only one photon at each of the outputs po1 and po2. Source: From Ref. [78].

PBS

PBS v1

c

vo1 p1

η1

po1

1/2

1/2 η1

t

co

η2

v2

η2 vo2 p2

HWP PBS 22.5

to

po2 PBS

HWP 22.5

This gate was experimentally realized with a spatial mode encoding on a reprogrammable optical chip by J. Carolan et al. [79]. The four photons needed to demonstrate the gate were generated through down-conversion in a BiBO crystal. An average fidelity of 94% was achieved. 13.7.1.2

Cluster States

A very different method for realizing quantum computing is via cluster states, introduced by R. Raussendorf and H.J. Briegel [80]. In this protocol a generic entangled state, called a cluster state, is produced. Measurements of the cluster state can be used to simulate arbitrary quantum computations. Roughly speaking, an n × m 2D cluster state can be used to simulate any n qubit circuit with a gate depth of m. The cluster can be created by performing CZ gates between qubits that have all been prepared in the diagonal state. The measurements that simulate the quantum evolution are single-qubit measurements in the |0⟩ ± ei𝜃 |1⟩ basis, where 𝜃 depends on both the particular single-qubit gate being applied at that time step and the result of previous measurements. It was pointed out by M. Nielsen that optics could be a good candidate for realizing cluster state quantum computation [81]. A small-scale cluster state quantum computation was performed by R. Prevedel et al. using a four-photon cluster state directly generated by down-conversion, single-photon measurements, and active feedforward [82]. Using a similar source to that described in Section 13.6.1, four folds of photons could be produced in the state: 𝛼(|H⟩1 |H⟩2 + |V ⟩1 |V ⟩2 )(|H⟩3 |H⟩4 − |V ⟩3 |V ⟩4 ) + 𝛽(|2H⟩1 |2H⟩2 + |2V ⟩1 |2V ⟩2 )|0⟩3 |0⟩4 + 𝛾|0⟩1 |0⟩2 (|2H⟩3 |2H⟩4 − |2V ⟩3 |2V ⟩4 )

(13.42)

where we have only written down the events where four photons are produced. The state components in the first line represent events where two pairs are simultaneously emitted to the left (modes 1 and 2) and to the right (modes 3 and 4), whilst the state components in the second line represent events where four photons are emitted to the left and none to the right and vice versa. All these events occur to the same order in the non-linearity though in general they are numerically different, represented by the 𝛼, 𝛽 and 𝛾 factors. By combining photons 1

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and 4 on a PBS and similarly for photons 2 and 3 and additionally introducing some carefully chosen polarization rotations to correct phases and equilibrate state components, one can arrange that the only fourfold events that survive are in the desired cluster state: 1 (|H⟩1 |H⟩2 |H⟩3 |H⟩4 + |H⟩1 |H⟩2 |V ⟩3 |V ⟩4 + |V ⟩1 |V ⟩2 |H⟩3 |H⟩4 2 − |V ⟩1 |V ⟩2 |V ⟩3 |V ⟩4 ) (13.43) By adaptively measuring the photons in different bases and sequences, they could implement single-qubit evolution, a two-qubit entangling evolution, and a small-scale version of Grover’s search algorithm [83], all from the same cluster state (see Figure 13.19). The adaption of the measurement bases was performed in real time using feedforward to polarization rotating elements and fibre delay lines of 30 and 60 m for photons 3 and 4, respectively, to enable sufficient time for the electronics to act. State fidelities of around 80% were achieved for the state evolutions and the search algorithm succeeded in 85% of cases.

EOM switching scheme Linear cluster

Outcome Outcome Error photon 2 photon 3 correction 0 0 None 0 1 x 1 0 z 1 1 xz

‘1’

Photon 1

Logic 1

PBS

‘0’

‘1’ Logic 2

Photon 2

PBS

‘0’ HWP

‘1’ ±β

Photon 3

PBS Delay 150 ns Photon 4

Delay 300 ns

EOM 1 adapts measurement basis

QWP

‘0’

X

Z

Output

EOMs 2 and 3 apply error correction

Figure 13.19 Experimental set-up for interferometric cluster state preparation. An UV laser pulse creates four photons in a non-linear BBO crystal. The computation evolves through consecutive polarization measurements on photons 1–4. Dependent on the outcomes of photons 1–3, three fast electro-optic modulators (EOMs) are employed to implement the active feedforward. One EOM adapts the measurement basis of photon 3, whilst two EOMs, aligned for 𝜎x and 𝜎z operation, apply the error correction on output photon 4. Two single-mode fibres locally delay the photons during the detection stage, logical operation and switching process of the EOMs. The polarization of the photons is measured by a PBS preceded by a HWP and a QWP in each mode. Source: From R. Prevedel et al. [82].

13.7 Quantum Computation

13.7.1.3

Quantum Gates with Non-linear Optics

Finally, we revisit the idea of a deterministic solution to implementing a CNOT via the use of a Kerr medium with a non-linearity so strong that a single photon can induce a π phase shift on another single photon [70]. Such a system would not require heralding in order to concatenate and could in principle be used repeatedly without a drastic reduction in probability of success. Unfortunately, the use of bulk crystals to implement such a gate seems to be ruled out, not only because of the weakness of available non-linear materials but also due to inevitable noise and mode distortion that is introduced due to the multimode nature of bulk crystals [84]. On the other hand, using a single quantum emitter (such as an atom or a quantum dot) as an intermediary seems to be a viable alternative. For example, L.-M. Duan and H.J. Kimble showed that consecutive interactions between two photonic polarization qubits and a single atom in a high finesse cavity could lead to a strong Kerr-like interaction and hence a CNOT gate between the photonic qubits [85]. A modified version of this gate was realized by B. Hacker et al. at Garching [86]. In their experiment a single 87 Rb atom was trapped in a three-dimensional (3D) optical lattice at the centre of a one-sided optical high finesse cavity. Three rubidium atomic levels are relevant for the scheme: |+⟩ = |F = 2, mF = 2⟩, |−⟩ = |F = 1, mF = 1⟩, and |e⟩ = |F = 3, mF = 3⟩. We can consider the |±⟩ as the atomic qubit levels. The atom cavity system is arranged such that only the |+⟩ ↔ |e⟩ transition and the right circularly polarized cavity mode at 780 nm are resonant. As a result, only a right circularly polarized photon, at 780 nm, impinging on the front face of the cavity will couple strongly to the atomic transition, and then only if the atom is in the |+⟩ state. The effect of this strong coupling is to introduce a π phase shift on the reflected photon, relative to events where the reflected photon was left circularly polarized or the atomic state was |−⟩. Thus we find that the various possible input states are mapped such that |+⟩|R⟩ → −|+⟩|R⟩, |+⟩|L⟩ → |+⟩|L⟩, |−⟩|R⟩ → |−⟩|R⟩, and |−⟩|L⟩ → |−⟩|L⟩. This truth table is equivalent to a CZ gate between the photonic and atomic qubits. The protocol requires the sequential enacting of such CZ gates between the atomic qubit and the first photonic qubit, and then the atomic qubit and the second photonic qubit, interleaved with atomic Hadamard gates. Finally the atomic qubit is measured by driving the |+⟩ ↔ |e⟩ transition with a strong classical field and looking for fluorescence. If the atom is found in the |−⟩ state, a CZ gate has been enacted between the first and second photonic qubits. If the atom is found in the |+⟩ state, then a phase correction has to be implemented before the required CZ gate is achieved (Figure 13.20). The experiment uses two highly attenuated coherent states, in co-propagating pulses separated by about 2 μs and prepared in different polarization states, as the photonic qubits. This means that the launching of the qubits into the gate is very inefficient and errors can occur due to two-photon components. However, it is important to note that two-qubit linear optical gates would not work at all with coherent state inputs (see Section 13.2), thus emphasizing the very different physics behind the working of this non-linear gate. After both pulses are reflected from the cavity, the atomic state was measured and fed forward to make the phase correction, if needed, to the first qubit. A 1.2 km fibre delay was introduced to allow time for the measurement and feedforward to occur. The photons

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r pai tion an tec de

ate St

m Ra

Level scheme |e〉 |L〉 |R〉

p2

FPGA

In p1

SPD

EOM 87

Rb

NPBS

Switch

Delay fibre

Out

Figure 13.20 Schematic for cavity CNOT. Qubit carrying weak coherent pulses p1 and p2 enter in two separate spatio-temporal modes via a non-polarizing 98.5% transmitting beamsplitter (NPBS) that acts as a circulator. The photons are reflected from a cavity containing a single atom before a switch directs them into a delay fibre for storage. Meanwhile the state of the atom is read out via fluorescence photons (blue arrows) directed by the switch towards a single-photon detector (SPD). A field programmable gate array (FPGA) applies a conditional phase feedback to p1 via an EOM. Eventually, the photons leave the gate set-up towards polarization analysers. The inset shows the atomic energy levels of 87 RB. Source: From B. Hacker et al. [86].

were then sent to polarization optics and single-photon counters to analyse the output states. The average gate fidelity was found to be about 76%. A considerable reduction in the fidelity is due to the use of coherent states as the input qubits. The authors estimate the average fidelity would rise to about 90% if true single-photon states were used. In spite of the low fidelity, entanglement production from separable input states was demonstrated. The efficiency of the experiment is also very low because of the use of coherent states to produce the qubits. Factoring out the inefficiency of the source, the gate still has a quite low efficiency of around 4.8%. This is due mainly to the loss in the fibre delay line and non-unit reflectivity of the cavity (around 67%). Thus, whilst the protocol is in principle deterministic, in practice it is strongly probabilistic. Nevertheless, significant improvements seem plausible. If the losses can be reduced sufficiently, such a gate could become the building block for much larger quantum circuits or cluster states than have so far been achieved. 13.7.2

Single-Rail Quantum Computation

Single-rail photonic qubits have advantages and disadvantages. An advantage is that deterministic entangling gates are available. For example, splitting √ a single photon on a 50/50 beamsplitter leads to the single-rail Bell state, 1∕ 2(|0⟩|1⟩ + |1⟩|0⟩) (see Section 13.6). On the other hand, basic single-qubit operations such as a bit flip are difficult in single rail because they are not energy preserving. Another disadvantage is that photon loss leads to unheralded logical errors, also known as amplitude damping errors. In contrast, the loss of a photon in dual rail is detectable by coincidence detection. Nevertheless there are certain algorithms that suit the single-rail encoding. Typically these are scattering problems in which

13.7 Quantum Computation

we start with a set number of qubits in the logical one state and the rest in logical zero. After the evolution we still have the same number of qubits in the logical one state; however they are entangled over many different configurations. Such algorithms can often be mapped onto deterministic circuits with single-photon inputs and single-photon measurements. 13.7.2.1

Quantum Random Walks

In classical computing random walks are a powerful tool used in a broad range of fields. A simple example is the ‘drunken walk’ in which the walker starts at zero on the number line and at each time step moves one unit either to the right or the left depending on the outcome of a coin toss. The probability distribution of possible end points for the walker is a normal distribution that, after a large number of steps, approaches a Gaussian centred on zero, but with a width that increases with time, analogous to a diffusive process. Now consider a ‘drunken quantum walk’ in which the walker is placed into a superposition of moving left or right at each time step. Due to interference, the resulting probability distribution is quite different from the classical case, exhibiting a symmetric twin-peak structure around the zero position. Hence, the most likely place to find the walker after many steps is not the centre of the distribution but out in either of the wings. A quantum walk like this can be demonstrated using a unary single-rail optical encoding – or put more simply by allowing a single photon to be the walker and using 50/50 beamsplitters to implement the quantum coin tosses. Several experiments have been performed along these lines. For example, M.A. Broome et al. used polarizing beam displacers and wave plates to implement a six-step quantum walk on heralded single photons from down-conversion [87]. By introducing small tilts to the beam displacers, interference between the different paths could be reduced, allowing the experiment to be tuned between implementing a quantum random walk and a classical random walk. The probability distributions clearly exhibited the expected double-peaked and Gaussian distributions, respectively. A quantitative comparison between the obtained and expected distributions showed excellent agreement. In a more sophisticated experiment, A. Schreiber et al. implemented a 2D quantum walk over 12 steps and 169 positions using an optical fibre network and a time-bin encoding [88]. The photon source in this experiment was a pulsed diode laser with wavelength 805 nm and a repetition rate of 110 kHz. Single-photon events were post-selected by photon detection with APDs. Polarization optics allowed steps forward or backward in two different spatial directions to be simulated. By looping repeatedly through the set-up, multiple steps could be implemented. Fibre delay lines separated time bins sufficiently that different walker positions could be resolved by the photon counting as different arrival times. Photons were coupled out of the loops via weak beamsplitters, separated into horizontal and vertical polarization components and counted at four APDs. The bias of the quantum coin toss at each site could be independently adjusted via a fast-switching EOM. The main physics explored in this experiment was the equivalence between a single quantum particle on a 2D walk and two interacting quantum particles on a 1D walk. Their results demonstrated the simulation of the emergence of entanglement between two interacting walkers

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and the different dynamics that emerge when the walkers interact locally versus interacting globally as they move through the position lattice. The similarity between the predicted and observed probability distributions was ≥90%. A drawback to more extensive applications of quantum walks implemented via single-rail photonic systems is the use of the unary encoding, i.e. where different position states are represented via |1⟩, |2⟩, |3⟩, … ≡ |1000 …⟩, |0100 …⟩, |0010 …⟩, … This is to be contrasted with a binary encoding where |1⟩, |2⟩, |3⟩, … ≡ |0⟩, |1⟩, |10⟩, … The key difference here is that whilst it requires N modes to represent N position states with a unary encoding, it only requires Log2 N modes using a binary encoding. This difference means that an exponential computational speed-up is not possible with a unary encoding without incurring an exponential blow-up in mode number, although it is still possible in principle to obtain a quadratic speed-up. 13.7.2.2

Boson Sampling

The boson sampling algorithm was introduced by S. Aaronson and A. Arkhipov [89]. Similar to the quantum random walk approach, we represent particle position by the corresponding qubit being in the logical one state. Hence, for the single-rail encoding, this translates to a one-to-one correspondence between particle position and modes containing a single-photon state, with other modes in the vacuum state. Also like random walks, the modes are mixed via linear optics, and the output position of particles is detected via photon detection. Unlike single-particle random walks, we now consider several particles traversing a large linear optical network. The linear optical network is not highly structured like in random walks, but highly unstructured, with the parameter values of optical elements chosen randomly across the network. The aim of the protocol is to obtain samples, i.e. particular instances, from the resulting unstructured probability distribution. The probability distributions created by boson sampling are ‘more’ quantum than those produced by quantum random walks because they involve multi-particle quantum interference effects like the HOM dip (see Section 12.2). Experimentally, a boson sampler consists of a large multipath interferometer. N single photons are injected into some subset of the paths. The total number of paths through the interferometer, M, is assumed much larger than N – typically one takes M ≈ N 2 . This condition ensures that it is unlikely that more than one photon will turn up at a particular output – sometimes referred to as the no collision output. Single-photon counters are placed at all M output modes, and the arrangement of photons at the output, shot by shot, is recorded. In the presence of loss, one can post-select on events where all N photons that were nominally launched make it to the outputs. The Heisenberg evolution between a particu∑M lar input mode â k and a particular output mode b̂ j is given by b̂ j = k=1 Ujk â k , where Ujk is the jkth element of the unitary transformation matrix, U, describing the optical network. Now suppose we inject our single photons into the first N modes of the network and ask for the probability that they emerge in the configuration n⃗ . For example, consider the small-scale case where N = 2 and M = 4 and we are interested in the output configuration where a single photon is detected at the first and third output modes. We would represent this by n⃗ = (1, 0, 1, 0).

13.7 Quantum Computation

Boson sampler Electronic stage

U

m=5

m=7

m=9

m = 13

Figure 13.21 Photonic implementation of boson sampling: n indistinguishable photons interfere in a random, linear m-mode interferometer, with photodetection at the output modes. Source: From N. Spagnolo et al. [90].

A remarkable result is that in general the probability for an event of this kind is given by p(⃗n) = |Per([U]1N ×⃗n )|2

(13.44)

where [U]1N ×⃗n is the sub-matrix of U obtained by selecting the first N columns and n⃗ rows of U and Per(A) indicates the permanent of the matrix A. The permanent of a matrix is a similar function to the determinant of a matrix; however, unlike the determinant, (in general) the permanent becomes intractable to compute as the size of the matrix grows (Figure 13.21). A number of groups have performed small-scale boson sampling experiments. In the experiments of N. Spagnolo et al. carried out at Sapienza Università di Roma, three photons were injected into 5-, 7-, 9-, and 13-mode optical networks [90]. The optical networks were multimode integrated interferometers fabricated in glass chips by femtosecond laser writing. Evanescent coupling between the waveguides when brought close to each other was used to realize beamsplitters, and careful control of path lengths realized phase shifts. The in-chip waveguides were single mode around 800 nm and exhibited losses of about 0.5 dB/cm. The photon source was down-conversion in a BBO crystal, producing photons at 785 nm. Four-photon events were identified via post-selection, where three of the photons were directed through the on-chip network and the fourth acted as a trigger. Single-photon detectors were placed at all outputs, enabling the probability distribution to be sampled. The number of possible (no collision) output configurations is given by the binomial coefficient (m n ), which grows exponentially with n (given n ≪ m). For the 13-mode experiment this leads to 286 different possible output configurations. To obtain the expected probability distributions, the permanents for the sub-matrices corresponding to all configurations were calculated. Comparing the experimentally obtained probabilities with the predictions showed excellent agreement for all the chips. Such a direct comparison would become intractable for larger systems – both because of the exponentially rising complexity of calculating the probabilities and because of the exponentially rising amount of

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data needed to experimentally characterize the distribution. N. Spagnolo et al. demonstrated an alternative approach whereby partial validation of the device can be obtained efficiently by ruling out the possibility that the distribution was simply a uniform one or that the distribution was generated by sending distinguishable particles through the device. In both cases, only small subsets of the data were needed, and the tests could be calculated efficiently. The preceding discussion suggests that solving the boson sampling problem on a classical computer may become intractable as the system grows. S. Aaronson and A. Arkhipov were able to prove that this is the case, even when the quantum device draws samples from a probability distribution that only approximates the actual distribution (modulo a couple of reasonable assumptions) [89]. This is quite a remarkable result given the limitations to the gates that can be implemented in single-rail systems with linear optics. In general, quantum computers that have an advantage over classical computers for specific calculations but do not employ a universal set of gates are referred to as intermediate quantum computers. 13.7.3

Continuous Variable Quantum Computation

So far we have considered encodings based on the discrete number state spectrum. It is also possible to consider qubits or quantum processing resource states constructed from the eigenstates of operators with a continuous spectrum. For example, one could consider qubits formed from superpositions of coherent states or squeezed states. One popular choice is to assign the logical values |𝟎⟩ ≡ |𝛼⟩ and |𝟏⟩ ≡ | − 𝛼⟩, and so consider quantum computing with qubits of the form 𝜇|𝛼⟩ + 𝜈| − 𝛼⟩

(13.45)

A curious feature of such an encoding is that the logical states are non-orthogonal (see Section 4.2.2). Nevertheless, quantum computing using such qubits has been shown to be possible in principle if equal superposition resource states (i.e. |𝛼⟩ + | − 𝛼⟩) are available [91]. These resource states are sometimes referred to as cat states. An alternative CV encoding is based on multiple superpositions of squeezed states [92], but here the resource states are much more difficult to produce. A generalization of cluster state computing to CV also exists [93]. Instead of preparing single-photon qubits in their diagonal states and then joining them together with CZ gates, optical modes are prepared in squeezed states and are joined together using beamsplitter interactions. Remarkably, in this way a cluster state that is universal for quantum computation can be produced deterministically using only linear optics. On the other hand, whilst Gaussian evolution can be simulated with such a state by making homodyne measurements on it, universal evolutions require the inclusion of cubic phase state projections that are much more difficult to implement. In general, when considering CV schemes, it must be borne in mind that any optical system comprising only Gaussian state generation and evolution, plus homodyne detection, will be classically simulatable [94].

13.7 Quantum Computation

Thus any optical quantum computing scheme must contain at least one fundamentally non-Gaussian element, such as photon counting. 13.7.3.1

Cat State Quantum Computing

The scheme of Ref. [91] and related works led to significant interest in the experimental generation of cat states and CV qubits similar to Eq. (13.45). Conditional approaches based on both pulsed systems and CW systems have been demonstrated. The pulsed approach is similar to that employed to herald single photons by A.I. Lvovsky et al. [2] (see Section 13.3). Pairs of entangled pulses are produced, one of which is measured. If the measurement result is the ‘correct’ one, then the other pulse is projected into the desired state. In contrast, in the CW approach, pairs of entangled modes are produced, one of which is measured continuously. An appropriate ‘click’ in the measured mode not only projects the desired state but also heralds the particular ‘pulse’ in which that state exists. If the other mode is continuously monitored via homodyne detection, then the projected state can be ‘cut out’ of the measurement record given the heralding signal. A sophisticated version of the pulsed approach was used by A. Ourjoumtsev et al. at the Université Paris-Sud to produce a squeezed cat state [95]. Two separate conditioning steps were used in this experiment. First, two-photon number states were heralded from pulsed down-conversion driven by ultrashort pump pulses (180 fs). The idler beam from the down-converter was split at a 50/50 beamsplitter and detected by two photon counters. Events were accepted only when both detectors fired simultaneously – thus heralding a two-photon number state in the signal pulse. The signal pulse was then sent to a second conditioning step in which it was split at a 50/50 beamsplitter and one arm was sent to a homodyne detector measuring the phase quadrature. Only when this homodyne detector registered a result close to zero was the state in the other arm successfully heralded. Ideally, given no loss or other technical imperfections, and if the homodyne result is close enough to zero, then the heralded state to ̂ ̂ an excellent approximation is S(r)(|𝛼⟩ + | − 𝛼⟩) – a squeezed cat. Here, S(r) is the squeezing operator with r such that the squeezing is in the amplitude direction and of strength 3.5 dB and |𝛼| ≈ 1.6. Whilst the effects of loss and other technical imperfections limit the fidelity with the target state, the experimentally produced state clearly exhibits twin peaks corresponding to ±𝛼 and interference between the peaks leading to negative regions of the Wigner function (see Figure 13.22). A model incorporating the loss and other technical imperfections agrees well with the observed state. This approach scales to larger amplitude cat states if higher photon number states are prepared before the homodyne conditioning. The first application of the CW approach to producing cat states was performed by J.S. Neergaard-Nielsen et al. at the Niels Bohr Institute in Copenhagen [96]. In later experiments at the National Institute of Information and Communications Technology in Tokyo, CV qubits similar to that in Eq. (13.45) were produced [97]. The basic idea is that if you subtract a single photon from a pure squeezed vacuum state, you can obtain a state that is very close to a small odd cat, i.e. |𝛼⟩ − | − √ 𝛼⟩, where 𝛼 =

3 (V 1 4

− V 2) ≤ 1. Here the squeezing is out of phase with the

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0.12 0.10

‘Dead’ state

‘Alive’ state

0.08 0.06 0.04 0.02 0 –0.02 x

p

Figure 13.22 Experimental Wigner function W(x, p) produced with n = 2 photons, corrected for the losses of the final homodyne detection. An interference between the dead and alive states with two negative regions is clearly visible. Source: From A. Ourjoumtsev et al. [95].

amplitude, 𝛼, which is taken to be real. For example, with V 2 = 0.5, we have 𝛼 ≈ 1 and a fidelity with the exact state of over 99.5%. This is referred to as an ‘odd’ cat because the state contains only odd photon number state components, i.e. |1⟩, |3⟩, etc. The photon subtraction is achieved by sending the optical mode through a highly transmissive beamsplitter and looking for single photons at the reflected output port. When a photon is detected, the photon subtraction operation is approximately enacted. That is, if the input state of the mode is |𝜙⟩, then the output from the transmission port is approximately â |𝜙⟩. The quality of the approximation improves as the transmission of the beamsplitter approaches unity; however the probability of success of the operation decreases. In the experiment of Ref. [97], an extra twist was added by displacing the reflected beam before detection. In this way it is not possible to tell if the detected photon came from the input state or from the displacement. This has the effect of creating a superposition between subtracting a photon and not subtracting a photon. In particular, in the ideal situation, the output state, conditioned on a successful detection, is a superposition of a squeezed vacuum state and a squeezed single-photon state: 𝜃̂ 𝜃̂ + ei𝜖 sin S(r)|1⟩ (13.46) |𝜓⟩ = cos S(r)|0⟩ 2 2 √n d , where 𝛽 is the amplitude of the displacewith 𝜖 = π − arg 𝛽 and cos 𝜃2 = n t

ment, nt is the average photon number of the reflected beam, and nd = |𝛽|2 is the contribution to that average photon number from the displacement. This state is approximately equal to that in Eq. (13.45) given appropriate choices of 𝜇 and 𝜈 and 𝛼 < 1. In the experiment the initial squeezed state was generated by a CW OPO operating at 860 nm and with bandwidth of 9 MHz. The photon subtraction

13.7 Quantum Computation

beamsplitter had a transmission of 95%. The transmitted beam was sent to a homodyne detector. The reflected ‘trigger’ beam was spectrally filtered via two Fabry–Pérot resonators in order to isolate the single mode detected in the homodyne detector. It was then displaced by a weak coherent beam via an unbalanced beamsplitter and then sent to a photon counter. When the photon counter clicked, this heralded the conditioned state in a specific temporal mode centred on the click time. This mode could be identified and extracted from the continuous output of the homodyne detector and subjected to tomographic analysis. CV qubits in a large range of different superposition states were produced and analysed in this way. Fidelities with the target states ranging between about 70% and 95% were obtained. These varied mainly depending on the fragility to loss of the various target states. 13.7.3.2

Continuous Variable Cluster States

Demonstrations of CV cluster states have been performed using spatial, spectral, and temporal modes. The initial experiments used bulk optics to produce CV cluster states in 4 spatially separated modes by combining squeezed states on beamsplitters [98]. Later experiments were able to generate much larger CV clusters in a single spatial mode by utilizing time-bin [99] and spectral bin [100] encodings. In the experiment of M. Chen et al. carried out at the University of Virginia, a 60-mode linear CV cluster state was produced [100]. Arguably this is the largest entangled state produced to date in any architecture in which all modes are simultaneously available and individually addressable. The system was a quantum optical frequency comb formed by the resonant frequency modes of the optical cavity of a doubly resonant optical parametric amplifier. The optical parametric amplifier contained two PPKTP non-linear crystals, pumped individually by two Nd:YAG lasers with frequency doubled to 532 nm. The crystals were oriented and pumped so as to produce distinct sets of EPR states around a wavelength of 1064 nm. Each set is composed of pairs of cavity resonances arranged symmetrically around a central frequency. The two sets are displaced from each other by one cavity free spectral range. The two sets are distinct due to having orthogonal polarizations. Rotating the polarization into the diagonal/anti-diagonal basis, an effective beamsplitter operation is enacted between the individual members of the two sets, entangling them all into a large CV cluster state. A third phase-locked laser, frequency shifted via an EOM, was used as the local oscillator in a homodyne detector to define the particular modes to be detected. The EOM’s maximum bandwidth of 14 GHz limited the number of modes detectable in this way to 60, though it was estimated that over 6000 modes were actually entangled. By probing the quantum correlations between various pairs of modes in this way, it is possible to determine the entanglement properties of the system. An observed (inferred) squeezing level 3.2 dB (3.4 dB) was found throughout the cluster, confirming the multipartite entanglement of the state. Note that a squeezing level above 3 dB implies multipartite inseparability [101]. The measurement results were consistent with the expected correlations for a dual-rail CV cluster state.

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An even larger linear CV cluster state of over 10 000 modes was generated by the time-bin experiment of S. Yokoyama et al. at the University of Tokyo [99]. Although all modes of the entangled state are not simultaneously present in this experiment, this is not a drawback with respect to quantum computation applications as cluster state quantum computation proceeds via sequential measurements of the modes. The state was produced via two CW OPOs, producing two beams of squeezed light at 860 nm. The squeezed light beams were nominally divided into time bins of time period T, where 1∕T is much narrower than the bandwidths of the two OPOs. The beams were combined out of phase at a 50/50 beamsplitter, producing a string of entangled EPR states in the time bins. One beam was then sent through 30 m of optical fibre, resulting in a delay between the beams of T = 157.5 ns, corresponding to the time-bin period. The beams were then combined again on a 50/50 beamsplitter where each time-bin EPR state interacts with the previous and successive states, producing the desired CV cluster state. The beams are then sent to two homodyne detectors where analysis of the quantum correlations between different time bins and quadratures can be carried out. The quantum correlations expected for a linear CV cluster state of this type were observed. Squeezing of around 5 dB was observed for the correlations from the first 1000 modes. It was not until around 10 000 modes that this fell below the 3 dB limit required to demonstrate true multipartite entanglement. In later experiments [102] the number of multipartite entangled modes has been pushed out to over 1 000 000! See Figure 13.23. 13.7.4

Large-Scale Quantum Computation

We now briefly examine some of the requirements needed to push the small-scale quantum processing experiments we have discussed towards large-scale universal optical quantum computation and the rather softer requirements for demonstrating a quantum advantage for intermediate quantum computing. Universal optical quantum computation requires that the quantum encoding being employed is compatible with fault-tolerant error correction; otherwise, inevitably, errors will grow until they make the results of any calculation essentially random. The field of optical quantum computation basically began when KLM proved that circuit-based, fault-tolerant quantum computation could be carried out with dual-rail encoding, photon production and detection, and linear optics [77]. A fault-tolerant architecture for linear optical dual-rail cluster states was described by C.M. Dawson et al. [103], and detailed architectures for its implementation have been proposed [104]. If non-linear optical CNOT gates become available, then fault-tolerant architectures based on 3D cluster states and dual-rail time-bin encoding can be employed as described by S.J. Devitt et al. [105]. Fault-tolerant architectures have also been described for cat state quantum computing by A.P. Lund et al. [106] and for quantum computing with CV cluster states by N.C. Menicucci [107]. Generic issues of precision, efficiency, and scale apply to all-optical quantum computation schemes (and indeed to all proposed quantum computation architectures); however there are also unique issues that can be identified for different encodings. Dual-rail linear optical schemes are challenged by the large overheads

13.7 Quantum Computation

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Figure 13.23 Experimental results for up to 800 000 modes by J.-I. Yoshikawa et al. Residual noise variances of the nullifiers’ (i) variances of X̃ k and (ii) variances of P̃ k . (iii) Corresponding reference quantum noise. (iv) A sufficient condition for full inseparability is the −3 dB bound. Horizontal axes are the temporal mode index k. Any correction of optical losses or detection noises is not applied. After J.-I. Yoshikawa et al. [102].

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Figure 13.24 State of the art in photonic engineering in 2016: universal linear optical processor demonstrated at the University of Bristol. This device uses 6 input modes and combined them with 15 Mach–Zehnder interferometers Mi,j , each containing a phase shifter and beamsplitter. The photons are arriving via polarization-maintaining fibres. The interferometers are controlled by 30 thermo-optic phase shifters, set by a digital-to-analogue converter (DAC). Output photons are sent to up to 6 fibres and up to 12 single-photon avalanche diode (SPAD) photon number resolving detectors and counted using 12-channel time-correlated single photon counting (TCSPC) module. Source: after J. Carolan et al. [79].

13.8 Summary

that come from the probabilistic nature of their entangling gates. Whilst these overheads have been lowered by orders of magnitude over those in the original KLM scheme, they still remain very high. Another problem is the current inefficiency of single-photon sources, although detection efficiencies are now close to those needed for scale-up. For CV cluster states the situation is somewhat reversed. As we have seen, deterministic sources and gates mean that scale-up to large numbers of modes is possible and squeezing levels are approaching what is required for fault tolerance. However, for universal quantum computing, detectors that can resolve the difference between, say, 50 and 51 photons are currently believed to be needed – a significant technological challenge. Cat state computing faces a combination of these challenges – difficult state preparation and challenging detection requirements that have restricted it to one- and two-qubit operations – although it should be noted that larger-scale demonstrations have been carried out in the microwave domain [108]. The boson sampling problem is interesting because there are very strong arguments to suggest that medium-scale systems, such as 50 bosons in 2500 paths, are intractable for classical computers. This suggests that interesting quantum computations can be carried out in this space without fault-tolerant error correction. Because single-rail quantum optics can implement the necessary gates deterministically, scale-up does not carry the large overhead associated with dual-rail schemes. In addition, there is a variation of the problem referred to as scattershot or Gaussian boson sampling, which can be solved efficiently by a single-rail scheme directly using the squeezed states deterministically produced by down-converters as the input (rather than single-photon states) [109]. Thus the major challenge to realizing an intermediate optical quantum computer of this kind is the ability to efficiently (i.e. with very low loss) implement a reconfigurable universal linear optical network over hundreds to thousands of modes. On-chip designs such as the 6-mode reconfigurable universal circuit demonstrated by J. Carolan et al. at the University of Bristol [79], shown in Figure 13.24, are one of several promising ways forward. Indeed integrated silicon chips with on-chip sources and over 500 photonic elements are now possible [110].

13.8 Summary Remarkable progress has been made in implementing quantum information protocols in quantum optics. The progress in QKD is sufficiently advanced that commercial applications are emerging. Teleportation, approaching unit fidelity, has been demonstrated in both the discrete and continuous domains and is beginning to see applications. Optical quantum computation has probably seen the most significant advance over the past decade, though there is still a long way to go before universal quantum computation becomes a reality. Nevertheless the prospects for intermediate optical quantum computers are good, and demonstrations of a quantum speed-up for problems such as boson sampling may not be so far off.

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Further Reading

99 Yokoyama, S., Ukai, R., Armstrong, S.C. et al. (2013). Optical generation of

ultra-large-scale continuous-variable cluster states. Nat. Photonics 7: 982. 100 Chen, M., Menicucci, N.C., and Pfister, O. (2014). Experimental realization

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of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett. 112: 120505. van Loock, P. and Furusawa, A. (2003). Detecting genuine multipartite continuous-variable entanglement. Phys. Rev. A 67: 052315. Yoshikawa, J.-I., Yokoyama, S., Kaji, T. et al. (2016). Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing. APL Photonics 1: 060801. Dawson, C.M., Haselgrove, H.L., and Nielsen, M.A. (2006). Noise thresholds for optical quantum computers. Phys. Rev. Lett. 96: 020501. Gimeno-Segovia, M., Shadbolt, P., Browne, D.E., and Rudolph, T. (2015). From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation. Phys. Rev. Lett. 115: 020502. Devitt, S.J., Fowler, A.G., Stephens, A.M. et al. (2009). Architectural design for a topological cluster state quantum computer. New J. Phys. 11: 083032. Lund, A.P., Ralph, T.C., and Haselgrove, H.L. (2008). Fault-tolerant linear optical quantum computing with small-amplitude coherent states. Phys. Rev. Lett. 100: 030503. Menicucci, N.C. (2014). Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett. 112: 120504. Ofek, N., Petrenko, A., Heeres, R. et al. (2016). Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536: 441. Lund, A.P., Laing, A., Rahimi-Keshari, S. et al. (2014). Boson sampling from a Gaussian state. Phys. Rev. Lett. 113: 100502. Wang, J., Paesani, S., Ding, Y. et al. (2018). Multidimensional quantum entanglement with large-scale integrated optics. Science 360: 285.

Further Reading Nielsen, M. and Chuang, I. (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. Bruss, D. and Leuchs, G. (2019). Quantum Information: from Foundations to Quantum Technology Applications 2nd edition. Wiley.

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14 The Future: From Q-demonstrations to Q-technologies 14.1 Demonstrating Quantum Effects The idea of optical quanta, conceived by Max Planck at the start of the twentieth century, has opened the way to a remarkable range of effects that defy classical interpretation. Over the last five decades, since the first demonstration of the laser, a multitude of optical and photonics technologies have emerged that make use of the effects of stimulated emission, optical gain, coherence, non-linear optics, and integrated optics. The benefits to many parts of our daily lives have been significant, as the many recent celebrations of the lasers and its achievements have shown1 and still continue to grow. This is a remarkable demonstration of the unpredictability of research and resulting technological advances, of the success of fundamental research combined with engineering ingenuity and determination, and of the desire to create commercial opportunities. Optics in the visible regime has been an outstanding testing ground for the concepts of quantum physics. Alongside atomic physics it was the field where the theoretical concepts were developed to explain physical observations and principal questions. Optics was a field where the debate on wave–particle duality could be explored with direct experiments. However, it was the invention of the laser that led to experimental demonstrations that clearly distinguish between the quantum and classical description of light and established that the concept of quantum optics is necessary to explain the properties of light. The first quantum revolution, which used the duality of particles and waves, first-order correlations, and coherence functions, has led to reliable tools in engineering. The second quantum revolution, which looks at higher-order correlations, fluctuations, and entanglement, was also inspired by optics. Early experiments, such as the Hanbury Brown and Twiss (HBT) experiments and their observations of bunching, explored some of the effects under debate. Again it was the laser, non-linear optics, and the refined techniques for controlling specific systems and individual particles that provided the experimental tools. Through optics the details of the quantum theory have now been thoroughly tested, and there remain no doubt and hardly any loopholes that would allow conflicting interpretations. In Figure 14.1 we see on the left-hand side the progression of the experiments that started with HBT, anti-bunching experiments, photon 1 Webpage for 100 years of OSA A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

14 The Future: From Q-demonstrations to Q-technologies

Light Interference Young/Fresnel

Photon correlations HBT 1955 Laser 1960 Single photon 1964

X(2) photon pairs 1970s Bell inequalities with radiative cascade 1972, 1982 Beyond SQL squeezing 1985

Time

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HOM with X(2) pairs 1987 Bell inequality with X(2) pairs 1989 –1998

Atoms Interference

1990

BEC 1995 Single atoms 2002 Atom correlations HBT 2005

Photon on-demand generation and counting

X(3) photon pairs 2007 Beyond SQL 2010

Bell inequality with x(2) pairs 2015

HOM with X(3) pairs 2014 Bell inequality with molecular dissociation?

The future

Bell inequality tests with X (3) pairs?

Figure 14.1 Comparison of the evolution of the demonstration of quantum effects at the basis of the second quantum revolution. Both for photons and light and atoms. Source: From A. Aspect 2016.

pairs, and the first tests of the Bell inequality. In the last 30 years, optics in the visible spectrum was the regime where suitable physical parameters could be reached first where individual photons, the statistics of many photons, and the corresponding quantum states could be easily investigated, background noise could be suppressed even at room temperature, and non-linear effects were quickly developed. Quantum optics and the investigation of photons in the visible spectrum, as described in this book, paved the way. A significant consequence of the quantum nature of light in optical instruments is the presence of quantum noise in optical measurements. This noise limits the range of signals that can be used and signal-to-noise ratios (SNRs) that can be achieved. In addition, the performance of the system is different for very large signals, or modulations, and for small signals close to the quantum noise level. For example, feedback controllers, optical amplifiers, and even passive cavities

14.2 Matter Waves and Atoms

show quite different properties near the quantum noise level. However, all these effects can be described in one simple, consistent approach: quantum transfer functions can be used that describe the complete system and include all quantum effects. This engineer’s approach is as easy to use as conventional classical transfer functions. With this tool, optical systems can be determined and optimized – the quantum effects can be fully integrated into the optical and electronic design of new devices.

14.2 Matter Waves and Atoms The exploration of quantum ideas was extended to other physical particles and systems. In particular, we are now capable of manipulating atoms to such a degree that we can repeat many of the tests and demonstrations in optics, which we have described in this book, to the equivalent experiments with atoms. It has become possible to trap, cool, and manipulate individual atoms and ions to build detectors that can carry out the required correlation experiments and to analyse the results in the equivalent way to quantum optics. The right-hand side of Figure 14.1, inspired by talks given by A. Aspect and others, shows the sequences of demonstration that have been possible. As we can see, almost all these demonstrations have now been completed, with a systematic time delay. It is now quite realistic to consider the full range of quantum statistical effects. At the core of the atom experiments is the ability to access the complex wave function of atoms, in particular to the quadratures, the real and imaginary part of the wave function, the phase between two wave functions. We would like a situation where we can interfere matter waves, in a way comparable to the interference of light, and we want to access this interference, as we can access the interference fringes that are the signature of the probability wave function for light. This was demonstrated with electrons in 1953 by L. Marton et al. [1] and neutrons in 1974 by H. Rauch et al. [2]. Specific proposals for atom interferometers had been made since 1985 by many, including V. Chebotayev et al. [3], J.F. Clauser [4], C.J. Borde [5], M. Kasevich, and S. Chu [6]. The first experimental demonstrations of the coherent nature of atom matter waves are experiments in 1991 and 1992 with very small structures that generate double-slit interference of the matter wave, first with helium by O. Carnal and J. Mlynek in Konstanz [7] and with neon atoms by F. Shimizu et al. in Tokyo [8]. It is possible to build components that are fully equivalent to optical components. As key example, the concept of a beamsplitter can be generalized and can be applied to a wide variety of situations. This can be extended to all forms of electromagnetic waves, including X-rays or waves propagating inside materials. It is possible to build beamsplitters for an atomic wave function, for example, by propagating the atoms transversely through a standing wave of light, as shown in Figure 14.2. After interaction with the light, the atoms acquire additional momentum, and the wave function of the atoms will be split into two components, each propagating into a different direction. Careful optimization of the apparatus and parameters results in a beamsplitter with excellent quality. There is a full analogy

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p

p + ħk

Figure 14.2 Schematic diagram of a beamsplitter for atoms. Atoms, also described to be an input matter wave, are crossing a standing wave pattern. The exchange of photons creates two outputs, atoms propagating with different momentum and thus direction, equivalent to two coherent matter waves. Source: After B. Barett et al. [9].

between the properties of a beamsplitter, as described in Section 5.1 for light and electromagnetic waves, and the case of atomic wave functions [9] and reference therein. In thermal gases this seems to be difficult since the atoms are quite dilute and moving fast, and there is a high probability of collisions. However, when we use coherent light to excite an ensemble of atoms, we can create coherent ensembles of atoms, and we can drive the excitation and create Rabi oscillation of the time dynamics of the atom population with well-defined frequencies and phase. Using a combination of laser beams with precisely tuned intensity and frequency, it is possible to create signals that are directly linked to the wave function of many atoms. Using phase-coherent detection techniques such as for the excitation between different energy states of the atoms, such as Ramsey fringes, it is possible to measure frequencies and phase precisely. These techniques have developed to create many types of high precision instruments, for example, reviewed by D. Cronin et al. who are themselves pioneers in this research field [10]. This is one of the key techniques for atomic clocks where the precise locking of microwave frequencies to well-known atomic transitions, most commonly a Cs transition, is the central approach to building and synchronizing atomic clocks. In many experiments the final step is an absorption image using a finely tuned laser beam, measuring a specific number of atoms. Once all other noise sources have been removed, the remaining noise is atom counting noise, which √ is equivalent to optical quantum noise and also scales with the square root Na of the number of atoms in the sample. Using non-linearities that affect the matter wave detection, it has been possible to reduce this noise level, equivalent to an intensity squeezed stream of photons. The atom’s environment produces the non-linear interaction, and there have been observations of squeezed atom density measurements. Squeezed atomic states of atoms have been produced utilizing the macroscopic spin of a coherent sample of atoms. The interaction can be described by the coherent evolution of the atomic density function, which is elegantly described by a vector on a three-dimensional surface, the Bloch sphere. Using laser pulses with well-defined detuning, intensity, and pulse length, the excitation, the vector pointing from the centre to a point on the Bloch sphere, can be systematically manipulated. This is mathematically equivalent to the polarization of a laser beam described by a Poincaré sphere. As the measurement of the polarization

14.3 Q-Technology Based on Optics

of a mode of light is limited by quantum noise, and the uncertainty is represented by a small three-dimensional ball on the surface of the Poincare sphere, so is the atom density measured in such a temporal atom interferometer. With sufficiently strong non-linear interactions, the measurement uncertainty can be manipulated, equivalent to the techniques of generating polarization squeezed light described in Figure 9.48. These experiments were pioneered by E. Polzik and his team first in Aarhus and then in Copenhagen and have produced impressive results with strong noise suppression and the generation of entanglement between two separate atomic samples. Thermal samples of atoms have also been used for spatial atom interferometers where atoms travel, with thermal speed, over considerable distances. The interaction with resonant laser beams serves as beamsplitters for the matter waves. One matter wave has taken up a different momentum due to the absorption of one more photon per atom, compared with the other matter wave. They travel in separate locations and can be recombined after a distance with another laser interaction ‘beamsplitter’. There will be a detectable phase shift due to the difference in the effect of gravitation on the two paths in the atom interferometer. Such machines have been built with ever-increasing precision. The technology has matured, for example, as described by B. Barlett et al. [9] and in the book edited by G.M. Tino and M.A. Kasevich [11]. These devices have been used to measure local gravity and relativistic effects with amazing precision. In addition it is possible to cool the atoms to enhance the interaction and ultimately to reach a point where the individual wave function of all the atoms interacts and for Bosonic atoms to combine to form one macroscopic wave function. This so-called Bose condensation, which was predicted by Bose and Einstein, was observed for the first time in 1995, when a Bose–Einstein condensate (BEC) was formed. This provided a direct access to a macroscopic atomic wave function; laser excitation can be used to imprint phase information, and the dynamics of the wave function can be observed with time-of-flight expansion and absorption techniques. The phase distribution of the matter wave can be imaged, and this is equivalent to interferometry. We can also move BECs and the equivalent of a fully coherent laser interferometer can be built. It is now possible to compare the performance of non-coherent, thermal, and coherent BEC interferometers [12]. Quantum-enhanced interferometers using non-linear interactions and squeezed matter waves are feasible and will appear in the future [13].

14.3 Q-Technology Based on Optics 14.3.1

Applications of Squeezed Light

Some of the limitations introduced by quantum noise can be avoided by using non-classical, or squeezed, states of light. The experimental techniques developed over the past 30 years have strongly improved [14], and reliable, strong noise suppression is now a reality. Noise suppression better than 15 dB, or more than a factor of 20, has been reported. Technology has now advanced to the point where fundamental quantum effects such as the fluctuations of the vacuum field and the

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discreteness of light can be demonstrated in undergraduate teaching laboratories, and there are a growing number of applications for quantum light, in particular in precision metrology and applications ranging from large-scale gravitational-wave (GW) detectors to imaging and precision clocks. What are the likely applications? The far-sighted ideas of pioneers such as R. Glauber, C. Caves, Braginsky, and others have certainly turned into reality. However, the early euphoria has calmed down and the emphasis has shifted. It is clear that applications of squeezed light will only occur when the instrument is already quantum noise limited and losses are extremely small. For example, in conventional optical communication systems, this is not yet the case, and presently the trend is towards coherent communication techniques and multimode approaches, which use a multitude of degrees of freedom, including temporal and spatial modes and angular momentum of the photons. The advantages of quantum cryptography have opened completely new opportunities for non-classical light and created significant scientific and even commercial interest. The situation with optical sensors is very promising – in laboratory instruments such as polarimeters or interferometers, the improvement of the SNR with squeezed light is well demonstrated. We can also use it in biological measurements where there are natural limits to the intensities and optical power that can be used [15]. The GW detector is the most visible example: since 2016 GW detection is working well in parallel, and the use of squeezed light is routine in some specific instruments such as GEO60. In a few years squeezed light will be a feature of future global networks of GW detectors, including the future LIGO and Virgo detectors. The squeezed light enhancement compares to the increase of the laser power by a factor of 3 or more, giving a more than 10-fold increase in the volume of space that can be explored. The combination of squeezed light and quantum control can push this advantage even further. Other specialized applications will be adopted. A steady development has already taken place, and it is reasonable to join into the optimism of R. Slusher and B. Yurke, some of the pioneers of experimental quantum optics who said in 1990: The field is still young and full of surprises. It is therefore highly likely that the most useful applications of squeezed light have not yet been imagined. Slusher and Yurke [16] At the same time there are more and more opportunities, and an ever-growing research community, for testing the quantum statistical concepts and those of quantum information in many different physical systems through a wide range of in-principle demonstrations. Presently this includes systems of individually controlled atoms and ions and coherent macroscopic quantum states. The future will include even better tests with optomechanical, superconducting circuit, and nanoscale solid-state devices. There is even speculation, but very scarce evidence and reasonable counterarguments, that biological systems might be suitable.

14.3 Q-Technology Based on Optics

14.3.2

Quantum Communication and Logic with Photons

The advent of quantum information has been the major driver in the last decade. The understanding of fundamental effects such as entanglement and superposition states and the concepts of quantum units of information, carried and processed by physical examples of qubits, qudits, etc., have influenced the way information technology will evolve. Secure quantum communication and the quantum internet are all areas of active research and increasingly areas of public and commercial investments. Now we are looking at the effect of the second quantum revolution, or quantum 2.0 in fashionable language, as stipulated 20 years ago by pioneers such as G. Milburn [17] and H.J. Kimble [18]. We have already extended our fundamental understanding of light and information and will in the future lead to more amazing leaps in information technologies. This is now well described and documented in many summaries and manifestos, which describe the state of the art and future opportunities for industrial applications, for example, in the Quantum Manifesto 2016 of the European Union [19]. There will be new applications emerging, in particular once we are able to extend the work to even larger non-linearities. Many applications need continuous non-Gaussian states. The combination of hybrid detection, single photons, and homodyne detection, discussed in Chapter 5, already allows us to use this resource. Much like with heralded photons, it will be useful to have only the best parts of the optical selected in order to build communication and logic devices. We have already seen the demonstration of negative Wigner functions for light, and this will also be possible in other systems. We have seen that at the level of single photons even many exotic effects appear: non-local effects, negative ‘probability’ distributions, and wave–particle duality. We have also seen how these effects may be harnessed to perform useful tasks such as the optical distribution of fundamentally secure secret keys. The work of E. Knill, R. Laflamme, and G. Milburn, showing that massive non-linearities can be conditionally induced by photon counting and then boosted to near-deterministic operation by teleportation, was a major step ahead. It is worth pointing out that there is no strict requirement for a purely deterministic process where the events occur at a precise time and every time, as long as there is a high probability that the desired event will take place within a reasonable time interval. Whilst this is slow and might take several trials, the impact of the logical process can be so powerful that is outweighs the disadvantages. An often quoted example is factorization, where one direction of the valuation is exceedingly hard whilst the other is easy and fast and can be used to verify the result quickly. Other examples are in the area of quantum simulations where there are no fast converging classical algorithms, and the quantum logic approach can have significant advantage [20]. For secure quantum communication, based on single photons and CW fluctuations, the scene has been set for commercial applications. This has been demonstrated in fibre-based applications and now awaits commercial investments. It has also taken the first step to space-based applications, for example, using a satellite launched in 2016 by China and similar systems that will follow. The aim would be for optical secure communication systems with global

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coverage, including satellite-based relay systems [21]. In addition many experts in cyber security are already planning for the time when quantum computers will be used to decypher classically encrypted messages. The search is one for much stronger forms of mathematical post quantum encryption that will withstand the use of the quantum computers. At the same time the advances in generating photons on demand, multiple entangled photons, photon number resolving detectors, and large-scale reliable multimode devices allow major improvements to quantum logic devices. From advances in photonics integration, we see the development of almost lossless optical waveguide systems that contain more and more complex structures extending the state of the art, shown in Figure 13.24, towards units that can take on real tasks. At the same time we can expect that even large non-linear interactions in new nanomaterials and nanostructures will become operational. Once these components are all integrated into the photonic chips, a wide range of optical quantum logic and processing will be possible. The role of such optical quantum computers is not yet clear, whilst the competition of different technologies continues. Scalability to many qubits is likely to remain a limitation for a while. It is presently a matter of debate that quantum computing technology will be most practical for the various applications. Large research investments are following up on this question. One can be confident that optics will capture a niche and is very likely the medium of choice to communicate quantum information, the equivalent of the traditional ‘information bus’ inside every computer. Finally, there will be a substantial demand for quantum memories inside a quantum computer, with the information likely to be transferred optically from the logic unit to a device, which can hold quantum states for a long lifetime, preferable approaching seconds. There has been extensive research about the various options of storing and retrieving quantum information. The most direct option could be a switchable low-loss cavity in which the photons are contained for extended times with minimal losses. Such optical delay lines have been used extensively in quantum logic demonstrations. Whilst one can think of possible resonator designs, so far no really good technical solutions have been found for the lossless storage and retrieval after significant times. In parallel even bigger applications of the quantum concept have emerged in the form of atomic and condensed matter physics, the understanding of the fundamental properties of matter, and the design of new materials that allowed the invention of solid-state electronics with its applications in computing and communication that touches every aspect of our lives. Combining the laser with all these technologies, our opportunities have rapidly expanded, very much so since the first edition of this guide was published. We now have access to a whole range of quantum technology platforms, which means different physical systems, such as single neutral atoms, single ions, superconducting circuits, mechanical systems and cryogenic temperatures, BECs, and degenerate Fermi gases. These will allow engineers to gain new access to quantum effects and to make the devices robust and reliable, complex and adaptive to the real needs, and, as always required, cost efficient. We will see all these platforms starting to overlap; electronic, optical, mechanical, and atomic devices will merge even more, and all forms of tailored devices will emerge (Figure 14.3).

Matter wave interferometer

Spin system

Q-enhanced clock

Atoms locked in optical crystal

Bose–Einstein condensate macroscopic wave function

Quantum memory

Atom in magnetic field Thermal gas/cold MOT

Spin system

Defect in crystal NV centre

Spin system

Quantum dot

Photon on demand Quantized light field X2

Propagating photons X1

Intra-cavity

Polarization

Q-sensors Q-logic

Cryogenic stripline and resonator microwave photons

Vibrations in opto-mechanical system quantized phonons

Line of ions vibrations between them Quantized phonons

Figure 14.3 Different physical systems that all have quantum properties equivalent to photons and laser beams. These can form the platforms for new devices.

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14.3.3

Cavity QED

Many researchers have now developed the skills and technology to custom-build mechanical systems that allow us to contain light in a controlled way, to build up the light resonantly, and to achieve very high enhancement factors, with Q’s of up to 107 or larger [22]. In such systems the losses are low, the containment is large, and the system encounters new forms of non-linearities. In particular any atoms in this field will experience dramatic effects due to the close link between the radiation field and the energy eigenstates of the atoms. A combination of cavity containment and atomic response can lead to the strong coupling regime where the properties of the photon states and atomic states cannot be separated, where new dynamics appears and a whole wealth of new quantum phenomena occurs. This has already been used in systems with ultracold atoms inside cryogenic cavities for detecting the emerging atoms as a quantum non-demolition recording of the light field, as discussed in Chapter 11, or where single atoms are used to implement quantum logic gates, as discussed in Chapter 13. Several cavities can be coupled, and this has been the a most productive system to build and explore macroscopic quantum systems and their entanglement. There is a whole wide research field emerging that will explore the opportunities in such specialized cavities. 14.3.4 Extending to Other Wavelengths: Microwaves and Cryogenic Circuits Many of the demonstrations of squeezed light can now be performed with microwaves, at typical frequencies of about 4-6 GHz. This is a great example of the opportunities opened up by new technologies and manufacturing processes. For this progress we require precision-made strip lines resonant with the microwaves, with sufficient low losses to allow long decoherence times allowing the transport over significant lengths. Couplers act as beamsplitters and all of this is operated at cryogenic temperatures of about 10 mK. For the non-linearities, many Josephson junctions, in some cases several hundred, are formed into a superconducting quantum interference device (SQID) that is assembled into a Josephson parametric amplifier (JPA) that provides the low-noise phase-sensitive gain required to manipulate the quadratures of the microwaves. Several groups have developed this technology in parallel in the last 10 years. For example, experiments in 2008 at JILA and NIST in Colorado by M.A. Castellanos-Beltran et al. [23] have shown significant squeezing, opening the field to many more experiments. It is now possible to produce entangled modes of microwaves and to demonstrate effects such as entanglement and teleportation [24]. 14.3.5

Quantum Optomechanics

A closely related topic is the performance and the quantization of the cavity structure itself. It is possible to quantize the mechanical parameters of the cavity, to consider and access mechanical modes, and to consider the mechanical

14.3 Q-Technology Based on Optics

eigenmodes and their excitation, phonons. It has been possible to make these systems more and more precise, to make them lighter, and to cool them sufficiently well that the modes are distinguishable and their excitation is distinct from the thermal background noise. We can treat them as modes with quadratures that carry information and quantum correlations. By matching the eigenfrequencies of the mechanical resonators to a cryogenic microwave circuit, it is possible to use the optomechanical effects, such as the light pressure acting on the very light nanoscale structures, as non-linear effects, equivalent to Kerr non-linearities. By careful engineering it is now possible to create complete circuits that can manipulate the quadratures of the microwave field, generate squeezing, and entanglement. Several groups are constantly developing this technology, and rapid progress is being made, which is well described in recent books on this topic [22, 25]. A combination of all these techniques, not available even 10 years ago, has allowed us to produce reliable sources of quantum states in new frequency regimes compatible with measurement devices and also the direct extension to quantum logic devices, which can be scaled to more and more extended and refined structures that in turn allow more and more complex quantum logic operations. This follows the path of integrated electronic circuits where refined production technology, in many cases lithography combined with nanometre-scale reproduction, has allowed the industry to produce remarkably complex devices in large numbers and at low cost. As experimentalists we have lost the direct access to the individual components and the need for ongoing alignment and tweaking. This has been replaced with a fully digital design and manufacturing approach. Quantum optics, on optical tables, will be replaced by various forms of precision integrated devices, built with the capabilities of nanoengineering. This is the logical next step for applications where only the performance matters, not the way it is built. 14.3.6 Transfer of Quantum Information Between Different Physical Systems A promising approach is to transfer the quantum information from one system, such as propagating photons, to another system, for example, into the excitation states of atoms or artificial atoms and two-level systems, such as vacancies in crystals or alternatively extremely small structures such as quantum dots. The progress is remarkable in regard to the quality of the systems and the control that can be achieved. Quantum systems of a similar nature can be imbedded in the material. Then the process of quantization can proceed, as described in Chapter 4, to both the light and the internal quantum systems, which will have internal excitation mechanisms, in the form of quantized spin waves, polarities, or plasmons, or phonons, depending on the nature of the quantum system. Provided we can create a close resonance between the system, most of the time by tuning the frequency of the optical field and the environment of the system, such as magnetic and electric fields, a transfer is possible. There is a wealth of experience, and these techniques are at the core and have been optimized in the various forms of spectroscopy for many years.

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14.3.7

Transferring and Storing Quantum States

The new requirement is to maintain the quantum nature of the information, for example, entanglement between the excitation of several modes. Entangled photons should transfer into entangled excitons and vice versa. The trick is to have an efficient way to excite these modes and turn photons into excitations of the internal quantum systems. Using cleverly optimized pulse sequences a close to 100% of excitation and information is possible. The internal modes could be excitons, polaritons, or phonons depending on the nature of the quantum system that is being used. Once these excitations have been transferred, they need to be protected from spontaneous decay and other losses. An active mechanism, controllable from the outside, is required to transfer the quantum excitation back to photons and thus retrieve the quantum information. There is a wealth of research in optimizing such systems, and this includes room temperature atomic gases, where the information is stored in the format of spin excitations in atoms in cleverly designed inhomogeneous magnetic fields [26], spin waves in atomic systems [27], and in crystals with nitrogen vacancies (NV centres) in crystals. The storage time has been extended to reach truly remarkable values of up to 1 s [28]. In parallel the time delay of entanglement for considerable time intervals and quality [29] has been demonstrated. It has been possible to achieve transfer efficiencies of up to 80% [26] with room temperature atoms. Considerable improvements are expected from cold atoms of high density, for example, atoms held and cooled in a magneto-optical trap (MOT) or even a BEC. Cleverly tailored laser pulse sequences are used to read the quantum information in and out. With the rapid technical development, it will be possible to combine the positive advantages of properties of all these individual systems and to build practical quantum memories that will expand the usefulness of quantum communication and logic [30, 31].

14.4 Outlook Does quantum optics have an attractive future? Most definitely, even without being able to predict the exact path, it will follow. Global secure communication based on quantum protocols is likely, and the details will depend on the commercial demand. In addition optics is attractive as a means for short-distance, for example, point-of-sale, secure communication. Optics is likely to play a role as part of quantum computing, maybe in processors, but surely as an internal bus and as part of the required quantum memory. For specialized tasks optical simulators are seen as the new approach. Another is going to be quantum-enhanced devices. There will be better timing devices and better optical clocks, and the inherent quantum limit will be suppressed in those applications where the best possible SNR is necessary and can be afforded. Atom and optical techniques will be combined to produce better sensors, with the military need for navigation without the present space based systems, such as the global position system (GPS), a big motivator and sponsor. For each application the best platform will be selected, and one would expect a future device to use a combination and integration of existing technologies, of optics, controlled atoms and ions, superconducting circuits, and optomechanics.

References

In all these systems design will have to take account of quantum statistics and other quantum effects, and quantum control will emerge with new engineering approaches to optimize the devices. There will be a major positive influence in the form of quantum-inspired technologies. These could be essentially classical devices that had exemplary improvements that were discovered by people involved and trained through quantum optics experiments. Many modern laser applications are in this category, so are many sensors that are essential parts of robotic and autonomous systems. Another example is the enhancement of communication systems through spatial multiplexing that uses the same tricks as used in multimode entanglement. In addition imaging systems in new wavelength regimes that make use of the same techniques as multi-wavelength entanglement. The skills learned in quantum optics are widely applicable. And finally there are still some unanswered major physics questions that remain open for study. Will quantum optics allow us to test the overlap between quantum physics and gravitation? Will we be able to show true non-classical effects in macroscopic objects? Can we extend our understanding of decoherence and the deeper link between space and time? Whilst for the moment teleportation is not a viable technology for transporting objects, many new insights and applications will be built on our understanding of quantum optics anno 2019.

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eter. Phys. Rev. 90: 490. 2 Rauch, H., Treimer, W., and Bonse, U. (1974). Test of a single crystal neutron

interferometer. Phys. Lett. 47: 369.

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3 Chebotayev, V., Dubetsky, B., Kasantsev, A., and Yakoloev, V. (1985). Interfer-

ence of atoms in separated optical fields. J. Opt. Soc. Am. B 2: 1791. 4 Clauser, J.F. (1988). Ultra high sensitivity accelerometers and gyroscopes using

neutral atom matter wave interferometry. Physica B & C 151: 262. 5 Borde, C.J. (1989). Atomic interferometry with internal state labelling. Phys.

Lett. A 140: 10. 6 Kasevich, M. and Chu, S. (1991). Atom interferometry using stimulated

Raman transitions. Phys. Rev. Lett. 67: 181. 7 Carnal, O. and Mlynek, J. (1991). Young’s double-slit experiment with atoms:

a simple atom interferometer. Phys. Rev. Lett. 66: 2689. 8 Shimizu, F., Shimizu, K., and Takuma, H. (1992). Double-slit interference with

ultra cold metastable neon atoms. Phys. Rev. A 46: R17. 9 Barrett, B., Bertoldi, A., and Bouyer, P. (2016). Inertial quantum sensors using

light and matter. Phys. Scr. 91: 053006. 10 Cronin, D., Schmiedmayer, J., and Pritchard, D.E. (2009). Optics and interfer-

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with Bose–Einstein condensates. In: Proceedings, International School of Physics “Enrico Fermi”, 657–689. IOS. ISBN: 978-1-61499-448-0. Gross, C. and Oberthaler, M.K. (2014). Matter wave interferometry beyond classical limits. In: Proceedings, International School of Physics “Enrico Fermi”, 743–768. IOS. ISBN: 978-1-61499-448-0. Andersen, U.L., Gehring, T., Marquardt, C., and Leuchs, G. (2016). 30 years of squeezed light generation. Phys. Scr. 91: 053001. Taylor, M., Janousek, J., Daria, V. et al. (2012). Biological measurement beyond the quantum limit. Nat. Photonics 7: 229. Slusher, R.E. and Yurke, B. (1990). Squeezed light for coherent communications. J. Lightwave Technol. 8: 466. https://doi.org/10.1109/50.50742. Milburn, G. (1996). Quantum Technology (Frontier of Science). Allen & Unwin. Kimble, H.J. (2008). The quantum internet. Nature 453: 1023. Calarco, T. et al. (2016). Quantum Manifesto. European Union. qurope.eu/ manisfesto. Lanyon, B.P., Whitfield, J.D., Gillet, G.G. et al. (2010). Towards quantum chemistry on a quantum computer. Nat. Chem. https://doi.org/10.1038/nchem .483. Young, R. China’s quantum satellite could make data breaches a thing of the past. https://phys.org/news/2016-10-china-quantum-satellite-breaches.html (accessed 12 October 2016). Aspelmeyer, M., Kippenberg, T.J., and Marquardt, F. (2014). Cavity Optomechanics. Springer-Verlag. ISBN: 978364255311. Casterlllanos-Beltran, M.A., Irwin, K.D., Hilton, G.C. et al. (2008). Amplification and squeezing of quantum noise with a tuneable Josephson meta material. Nat. Phys. 4: 928.

Further Reading

24 Fedoro, K.G., Zhong, L., Pogorzalek, S. et al. (2016). Displacement of propa-

gating squeezed microwave states, Phys. Rev. Lett. 117: 020502. 25 Bowen, W.P. and Milburn, G.J. (2016). Quantum Optomechanics. CCR Book. 26 Hétet, G., Longdell, J.J., Alexander, A.L. et al. (2008). Electro-optic quantum

memory for light using two-level atoms. Phys. Rev. Lett. 100: 023601. 27 Zhang, W., Ding, D.-S., Shi, S. et al. (2016). Storing a single photon as a spin

28

29

30 31

wave entangled with a flying photon in the telecommunication bandwidth. Phys. Rev. A 93: 022316. Ranˇci´c, M., Hedges, M.P., Ahlefeldt, R.L., and Sellars, M.J. (2018). Coherence time of over a second in a telecom-compatible quantum memory storage material. Nat. Phys. 14: 50. https://doi.org/10.1038/nphys4254. Ferguson, K.R., Beavan, S.E., Longdell, J.J., and Sellars, M.J. (2016). Generation of light with multimode time-delayed entanglement using storage in a solid-state spin-wave quantum memory. Phys. Rev. Lett. 117: 020501. Astner, T., Gugler, J., Angerer, A. et al. (2018). Solid-state electron spin lifetime limited by photonic vacuum modes. Nat. Mater. 17: 313. Kaczmarek, K.T., Ledingham, P.M., Brecht, B. et al. (2018). High-speed noise-free optical quantum memory. Phys. Rev. A97: 042316.

Further Reading Schleich, W.P., Ranade, K.S., Anton, C. et al. (2016). Quantum technology: from research to application. Appl. Phys. B 122: 130. Bruss, D. and Leuchs, G. (2019). Quantum Information: From Foundations to Quantum Technology Applications. 2nd Edition. Wiley.

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Appendices Appendix A: List of Quantum Operators, States, and Functions For completeness, most quantum operators, states, and functions used in quantum optics are listed here. The operators, individually or as a combination, describe the effect of optical ̃ A ̃ † for the annihisystems. The fundamental operator are the pairs â , â † and A, lation and creation of photons. Here hats indicate mode operators whilst tildes indicate continuum operators. Note that these operators are not Hermitian and thus do not represent observables. The second group of operators are combinations of these, such as the quadrature operators. These are Hermitian and thus do describe observable properties of a mode. The effect of optical processes such as modulation or squeezing can be described by unitary operators. Note that the phase operator is listed but cannot be defined in a closed form. It can be defined only in the limit of a series of operators. All quantum optical states can be expanded in terms of the number states |n⟩. The coherent state is generated by the displacement operator and the squeezed state by a combination of the squeezing and displacement operator. The description of the states contains exactly the minimum number of parameters that are required to describe the state uniquely: two parameters (one complex number) for a coherent state and four parameters for a minimum uncertainty squeezed state. The states are eigenfunctions of a certain operator, for example, the coherent state is eigenstate of the annihilation operator. The functions are used to evaluate the properties of quantum states. There are two groups. The first group contains functions that describe the statistical properties of the light. These are the variance of the quadratures and the variance of the photon number. Similarly, the correlation functions g (n) are used to describe the statistical and coherence properties of the light. The second group of functions provides two dimensional representations of the states. There is a whole continuum of such representations. In practice, however, only a limited number of functions are employed. In this text we only use the Q and Wigner functions.

A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

550

Appendices

Op

Name

Expression

Type

â

Annihilation

â

Non-Hermitian

â ̃ A

Creation

Non-Hermitian

̃† A ̂ D

Creation

â ̃ A ̃† A

Displacement

̂ D(𝛼) = exp(𝛼 â † − 𝛼 ∗ â † )

Unitary

̂† D

Displacement

̂ † (𝛼) = D ̂ −1 (𝛼) = D ̂ † (𝛼) D

Unitary

Ŝ

Squeezing

Unitary

Ŝ †

̂ S(𝜉) = exp( 12 𝜉 ∗ â 2 − 12 𝜉(̂a† )2 )

Squeezing

̂ Ŝ † (𝜉) = Ŝ −1 (𝜉) = S(−𝜉)

Unitary

n̂ ̃ N

Number

Hermitian

Number

â † â ̃ ̃ †A A

Phase

Only as limit of series

Hermitian

Quadrature

â e−i𝜃 + â † ei𝜃

Hermitian

Amplitude quadrature

â + â

Hermitian

Phase quadrature

i(̂a† − â )

Hermitian

†

̂ 𝚽 ̂ X𝟏(𝜃) ̂ X𝟏 ̂ X𝟏

†

Annihilation

Non-Hermitian Non-Hermitian

Hermitian

†

State

Name

Expression

Significance

|0⟩

Vacuum Fock

|𝛼⟩

Coherent

â |0⟩ = 0|0⟩ √ ̂ â † |n⟩ n + 1|n + 1⟩ n|n⟩ = |n⟩ ̂ ̂ |𝛼⟩ = D(𝛼)|0⟩ a|𝛼⟩ = 𝛼|𝛼⟩

Ground state

|n⟩ |𝛼, 𝜉⟩

Squeezed

̂ Ŝ † (𝜉)|0⟩ |𝛼, 𝜉⟩ = D(𝛼)

Minimum uncertainty

Complete, orthogonal Overcomplete, orthogonal

Function

Name

Expression

Vn

Variance of n

⟨(̂a† â )2 ⟩ − ⟨̂a† â ⟩2 ̂ 2⟩ ⟨|𝜹X𝟏|

V 1(Ω)

Noise spectrum

g (1) (𝜏)

First-order correlation

g (2) (𝜏)

Second-order correlation

̂ + 𝜏)⟩∕⟨X𝟏(t) ̂ ̂ ̂ † X𝟏(t ⟨X(t) + X𝟏(t)⟩ † † 2 ̂ 2 ̂ ̂ ̂ 2⟩ ⟨X𝟏 (t) X𝟏(t + 𝜏) ⟩∕⟨X𝟏 (t)2 X𝟏(t)

S(Ω)

Squeezing spectrum

̂ Var(X𝟏(𝜃) with Θ optimized for each Ω))

Q

Q function

̂ Q(𝛼) = 1∕𝜋⟨𝛼|𝝆|𝛼⟩

P

P function

𝜌 = ∫ P(𝛼)|𝛼⟩⟨𝛼|d2 𝛼

W

Wigner function

1∕𝜋 2 ∫ e𝜇

∗

𝛼−𝜇𝛼 ∗

X(𝜇)d2 𝜇

Appendices

Appendix B: Calculation of the Quantum Properties of a Feedback Loop An electronic intensity control system, or electronic noise eater, as discussed in Section 8.3.2, is described by the following set of equations: √ ̃ il = 𝜹A ̃ el ̃ las + g(Ω) 𝜹A 𝜹A √ √ ̃ f = 𝜖 𝜹A ̃ il + 1 − 𝜖 𝜹A ̃ vac1 𝜹A √ √ ̃ el = 𝜂 𝜹A ̃ f − 1 − 𝜂 𝜹A ̃ vac2 𝜹A √ √ ̃ o = 1 − 𝜖 𝜹A ̃ il − 𝜖 𝜹A ̃ vac1 𝜹A √ √ ̃ out = 𝜂 𝜹A ̃ o − 1 − 𝜂 𝜹A ̃ vac3 𝜹A (B.1) Here we are explicitly solving these equations to derive the transfer functions for the photocurrents inside the system (il, f , and el)) and for the output beam (out). Combining the different parts of Eq. (B.1), we obtain √ √ ̃ el = 𝜂 𝜹A ̃ f − 1 − 𝜂 𝜹A ̃ vac2 𝜹A √ √ √ ̃ il + 𝜂(1 − 𝜖) 𝜹A ̃ vac1 − 1 − 𝜂 𝜹A ̃ vac2 = 𝜂𝜖 𝜹A √ √ √ √ ̃ las + 𝜂𝜖 g(Ω) 𝜹A ̃ el + 𝜂(1 − 𝜖) 𝜹A ̃ vac1 − 1 − 𝜂 𝜹A ̃ vac2 = 𝜂𝜖𝜹A which leads to the equations for the quadrature: √ √ √ √ ̂ el = 𝜂𝜖 𝜹X𝟏 ̂ las + 𝜂𝜖 g(Ω) 𝜹X𝟏 ̂ el + 𝜂(1 − 𝜖) 𝜹X𝟏 ̂ vac1 − 1 − 𝜂 𝜹X𝟏 ̂ vac2 𝜹X𝟏 √ √ √ ̂ las + 𝜂(1 − 𝜖) 𝜹X𝟏 ̂ vac1 − 1 − 𝜂 𝜹X𝟏 ̂ vac2 𝜂𝜖 𝜹X𝟏 ̂ el = 𝜹X𝟏 (B.2) √ 1 − 𝜂𝜖 g(Ω) √ and we can now define the open loop gain h(Ω) = 𝜂𝜖 g(Ω) where we keep the frequency dependence explicitly as a reminder of the specific properties of the feedback system. The operators for the quadratures always contain the frequency dependence of the field. We can now derive the spectrum for the in-loop electric current as 𝜂𝜖V 1las (Ω) + 𝜂(1 − 𝜖) + (1 − 𝜂) V 1el (Ω) = √ |1 − 𝜂𝜖g(Ω)|2 𝜂𝜖(V 1las (Ω) − 1) + 1 = (B.3) |1 − h(Ω)|2 In a similar way we derive the spectrum of the photocurrent at the output of the apparatus. We start with the field amplitude: √ √ √ √ ̂ vac3 ̂ el + 𝜹X𝟏 ̂ las ) − 𝜂𝜖 𝜹X𝟏 ̂ vac1 − 1 − 𝜂 𝜹X𝟏 ̂ out = 𝜂(1 − 𝜖)( g(Ω) 𝜹X𝟏 𝜹X𝟏 (B.4)

551

552

Appendices

The spectrum of the outcoming laser beam can be derived by substituting Eq. (B.2) into Eq. (B.4) resulting in √ √ √ ̂ las + (𝜂 g(Ω) − 𝜂𝜖) 𝜹X𝟏 ̂ vac1 𝜂(1 − 𝜖) 𝜹X𝟏 ̂ 𝜹X𝟏out = 1 − h(Ω) √ √ ̂ vac2 + (1 − h(Ω)) 1 − 𝜂 𝜹X𝟏 ̂ vac3 𝜂(1 − 𝜖)(1 − 𝜂)g(Ω) 𝜹X𝟏 − 1 − h(Ω) which leads to V 1out (Ω) =

√ √ 𝜂(1 − 𝜖)V 1las (Ω) + |𝜂 g (Ω) − 𝜂𝜖|2

|1 − h(Ω)|2 𝜂(1 − 𝜂)(1 − 𝜖) g(Ω) + |1 − h(Ω)|2 (1 − 𝜂) + |1 − h(Ω)|2 2 1 − 𝜖 𝜂𝜖(V 1las (Ω) − 1) + |h(Ω)| =1+ 𝜖 |1 − h(Ω)|2

(B.5)

This spectrum is obviously, in most situations, above the QNL, never below. This result shows the effect of the feedback loop adding vacuum noise that is entering the system through the unused port of the beamsplitter and at the detectors. This is quite different from the predictions of a classical theory. The classical equations apply in the limit of large noise or modulations, that is, V 1las ≫ 1.

Appendix C: Detection of Signal and Noise with an ESA The response of the spectrum analyser to a single frequency harmonic input voltage signal is straightforward, if we can assume that the RBW is larger than the spectral width of the signal. In this case the function displayed will have the width of the RBW and the maximum value of 𝜇disp will be proportional to the square of 2 of the single harmonic input. This peak value displayed is the the amplitude Ssig only information of interest and is independent of RBW. However, the response of the ESA to a signal with a spectral distribution that is wider than RBW is different. In this case 𝜇disp (Ω) corresponds to the convolution stated in Eq. (7.6) for a fixed value of Ω = Ωdet . Consequently, details in the spectrum of the input signal narrower than RBW will not be detected. The exact shape of the function res(Ω) depends on the specific analyser used; a Gaussian shape is generally assumed, but you should check the manual of your analyser. A second, and very common, application is the detection of noise with a bandwidth much larger than RBW. Both harmonic signal and broadband noise will be displayed; however, the way the two influence 𝜇disp is quite different. We need to consider the details of the processing occurring inside the ESA to predict the distribution function of the displayed values and to interpret 𝜇disp . The input signal to the spectrum analyser is bipolar, which means positive and negative voltages are present. A single frequency harmonic signal will oscillate between the two extreme values ±Ssig . These maxima have no uncertainty during the time of measurement, which means the signal distribution function has effectively a

Appendices

Figure C.1 Input and display distribution functions for a spectrum analyser. The width 𝜎noise of the distribution at the input determines the peak value 𝜇disp of the distribution on the display.

Input voltage

Display values au

V

0 µdisp 0

negligible width, 𝜎sig = 0. In contrast, the noise has a Gaussian distribution, centred at zero, 𝜇noise = 0, with the width of the distribution given by 𝜎noise . The analyser produces the power spectrum, which means it is taking the square of the input signal, and it also performs an averaging process. Consequently it shows only positive values and the distribution at the output is asymmetric, similar to a Rayleigh function, as shown in Figure C.1. The mean value 𝜇disp of this distribution is close to but not exactly at the maximum value of the distribution, and it has a width 𝜎disp measured from full maximum to half maximum. We measure 𝜇disp and expect it to represent both the variance of the noise, given 2 2 , and the power of the input signal given by Ssig . The actual details are a by 𝜎noise little more complicated since we have to evaluate the full distribution function. The full mathematical details of calculating the Rayleigh distribution are given by D.A. Hill and D.P. Haworth [1], and for signals that are substantially larger than 2 2 the noise, the result is given by 𝜇disp = 𝜎noise + CS Ssig . This is frequently referred to as adding the signal and the noise in quadrature in the same way as the variance of two noisy signals with Gaussian distribution functions can simply be added. This is a remarkably simple result since the actual mathematical background is quite complex. Please note that most instruments have been calibrated accurately for signal rather than noise measurements and an absolute comparison between signal and noise, or determination of CS , can be difficult. This is well known to the designers of the ESA, and the details are given in their manuals, for example, the Hewlett Packard manual. These details are further elaborated in Ref. [1], but it is sufficient in most cases to work with a correction factor CS that corresponds to about 2.5 dB. In conclusion, we can add the square of the signal amplitude and the square of the RMS of the noise. The scaling factor CS is a consequence of the actual sampling process inside the ESA, and this is only important if we want to measure absolute signal and noise levels or determine absolute signal-to-noise ratios (SNR). In most of our experiments, the relative value, compared to a calibration trace measured with the same instrument, is sufficient. With increasing signal, the output value 𝜇disp rises gradually, since the ESA always measures (signal + noise) and not signal alone. Very small modulations cannot be detected; they disappear in the noise, and Hill and Haworth showed that CS can vary in the limit of a small signal with SNR ≃ 1. Since most interesting

553

554

Appendices

application in quantum optics will be exactly in this regime of small signals, we have to be aware of these properties of the ESA. However, once the signal is dominant, the display is proportional to the signal strength alone.

Reference 1 Accurate measurement of low signal-to-noise ratios using superheterodyne

spectrum analysers, D.A. Hill, D.P. Haworth, IEEE transactions on instrumentation and measurement, 19, 432 (1990).

Appendix D: An Example of Analogue Processing of Photocurrents As a practical example of how spectrum analysers are used, consider the case where we wish to measure the correlation of two fluctuating currents i1 (t) and i2 (t). The two currents are combined in a device called an RF splitter/combiner that has two outputs providing the sum and the difference of the two input signals and has the appropriate electrical impedance that matches the impedance of the cables. With the spectrum analyser, we can measure the four values i1 (t)2 , i2 (t)2 , [i1 (t) + i2 (t)]2 , and [i1 (t) − i2 (t)]2 using the two output ports of the splitter/combiner or by terminating the inputs individually. All four values have to be measured separately and are combined to calculate C. According to Eq. (4.135), the cross-correlation coefficient can be expressed as C(i1 , i2 ) =

[i1 (t) + i2 (t)]2 − [i1 (t) − i2 (t)]2 i (t)2 + i2 (t)2 × √1 2 2 [i1 (t) + i2 (t)] + [i1 (t) − i2 (t)] 2i1 (t)2 i2 (t)2

(D.1)

One extreme case is the balanced situation where the two currents have the same long-term average, which means i1 (t)2 = i2 (t)2 . Here Eq. (D.1) is greatly simplified to C= =

[i1 (t) + i2 (t)]2 − [i1 (t) − i2 (t)]2 [i1 (t) + i2 (t)]2 + [i1 (t) − i2 (t)]2

(D.2)

and in this special case only two measurements are required. For perfect cross-correlation, C = 1, we have i1 (t)2 = i2 (t)2 , and the minimum of the signal is equal to zero, which is easily detectable. For C = 0, which means completely uncorrelated signals, the cross terms i1 (t)i2 (t) all cancel, and we get the condition that [i1 (t) + i2 (t)]2 = [i1 (t) − i2 (t)]2 , which means there is no difference between these two measurements. Actually, the two signals could be added with any phase difference and the result remains constant. This is an important test: the total power of two uncorrelated signals is the sum of the individual powers and is the same for addition and subtraction. As mentioned above, this applies to noise signals as well as modulation signals, and the spectrum analyser can be used to evaluate the correlation of two noise

Appendices

sources. In other words, it can test the independence of the noise sources. If they are independent, the variances should simply add in quadrature, the variance of the sum or the difference should be equal to the sum of the variances. In practice to change from sum to difference can be more technically involved than it appears from the equations. The change in sign can be experimentally achieved by swapping the outputs of the electronic splitter/combiner, provided that there is no phase lag, or delay, between the two signals. This is difficult to achieve at higher frequencies Ω since the delay has to be much smaller than the inverse of Ω. A reliable test is to deliberately introduce a common modulation signal to the two currents. This provides a strong output, and we know that these signals are perfectly correlated and by measuring C in the way described above, the entire apparatus can be tested. An alternative technique is to systematically vary the phase lag Φ between the two signals. This can be done by introducing a fixed time delay Δ𝜏, for example, by sending one signal through an extra length of cable or through an RC time delay line, and by recording the spectra of both outputs of the splitter/combiner for range of frequencies, Ω1 to Ω2 . The delay Δ𝜏 should be such that Δ𝜏 > 1∕Ω1 . The phase lag for each detection frequency is Φ(Ω) = 2𝜋Δ𝜏∕Ω and Δ𝜏 should be chosen sufficiently long that Φ(Ω) varies by more than 2𝜋 in the range of detection frequencies used. In this way a spectrum is recorded that contains information of both [i1 (t) + i2 (t)]2 and [i1 (t) − i2 (t)]2 on one trace. The spectrum looks much like an interference fringe, the visibility of these ‘fringes’ can be measured, and the value of the correlation coefficient can be determined, as discussed in Section 2.3. If the signals are correlated, there will be a strong modulation of the trace, the minimum signal is close to zero, the visibility is close to 1, and the correlation function C is close to unity. Figure D.1 shows the layout for this technique with the results given in Figure D.2. Two signals from a classical noise source are fed into a splitter/combiner, and one signal is delayed by 1 μs in regard to the other. They are combined with a second splitter/combiner generating the sum Pi+ and difference Pi− signals. Ideally the trace should show perfect modulation. However, Figure D.2 shows that the two currents Pi+ and Pi− are almost out of phase, but not exactly. The splitter/combiner components are not perfect. In this case they provide a shift of about 170∘ , not the optimum 180∘ . These techniques are used extensively in the investigation of sub-Poissonian light sources (Section 9.1) in the QND experiments (Chapter 11) and in the investigation of optical entanglement as described in Chapter 13. Time delay

i1(t)

~

Signal

i2 (t)

+_

Figure D.1 Layout of experiment with splitter/combiner.

555

Appendices

2.5 Pi+

Pi–

2

1.5 Pi au

556

1

0.5

0 0

200

400 600 Frequency

800

1000

Figure D.2 Noise correlation for two classical noise sources added with a delay line.

Appendix E: Symbols and Abbreviations Nomenclature

Symbol

Electric field

E

Optical amplitude

𝛼

Optical phase

𝜙

Intensity

I

Intensity fluctuation

𝛿I(t)

Variance

ΔI(t)2 = VARi = [𝛿I(t)2 ] − [𝛿I(t)]2

Optical power

P(t)

Current

i(t)

RF noise power

pi (Ω)

Gaussian beam diameter

2W

Gaussian beam parameter

z0

Optical frequency (Hz)

𝜈

Detection frequency (Hz)

Ω

Mirror reflectivity

𝜖

Quantum efficiency

𝜂

Current junction division

𝜁

Appendices

Nomenclature

Symbol

Correlation function

C (E1 , E2 )

Visibility

VIS (E1 , E2 )

Normalized visibility



Signal-to-noise ratio

M(Ω)2 ∕ΔI(Ω)2

General operator

𝝌̂

Variance of operator

̂ = ⟨𝝌̂ † 𝝌⟩ ̂ − ⟨𝝌⟩ ̂ 2 Var (𝝌)

Creation and annihilation operator

â , â †

Travelling wave, time domain

̂ A ̂† A,

Travelling wave, frequency domain

̃ A ̃† A,

Number state

|n⟩

Coherent state (internal)

|𝛼⟩

Propagating coherent state

|Aout ⟩

Expectation value

⟨̂a⟩ = ⟨𝛼|̂a|𝛼⟩ = 𝛼

Operator for fluctuations, fr. domain

̃ 𝜹A ̃† 𝜹A,

Linearized operator

̃ â 1 = 𝛼 + 𝜹A

Photon number operator

n̂ = â † â

Photon flux operator

̃ ̃ †A ̃ =A N

Photon number

̂ 1 ⟩ = |𝛼1 |2 n𝛼1 = ⟨𝛼1 |n|𝛼

Variance of photon number

̂ evaluated for |𝛼1 ⟩ V n1 = Var (n)

Generalized quadrature

̂ X(𝜃) = â † exp(i𝜃) + â exp(−i𝜃)

Amplitude quadrature operator

̂ = X(0) ̂ X𝟏 = â + â †

Phase quadrature operator

̂ = X( ̂ 𝜋 ) = −i(̂a − â † ) X𝟐 2

Measured amplitude noise

̂ 2 ⟩ = Fourier transform [V (0)]| V 1out (Ω) = ⟨|𝜹X𝟏| out Ω

Squeezing parameter

𝜉 = rs exp(i2𝜃s )

Squeezing angle

𝜃s

Squeezing degree

rs

Squeezed quadrature amplitude

̂ = X(𝜃 ̂ s ) = â † exp(i𝜃s ) + â exp(−i𝜃s ) Y𝟏

Squeezing spectrum

̂ S(Ω) =Var (X(𝜃)) optimized for each Ω

Photon number probability

𝛼 (n)

Fano number

fAout = V NAout ∕NAout = V 1out

557

558

Appendices

Abbreviations

AM

amplitude modulation

AOM

acousto-optical modulator

APD

avalanche photodiode

CCD

charge-coupled device

CV

continuous variable

CW

continuous wave

EOM

electro-optic modulator

EPR

Einstein–Podolsky–Rosen

ESA

electronic spectrum analyser

FSR

free spectral range

HOM

Hong–Ou–Mandel

LCD

liquid crystal display

LED

light-emitting diode

OPA

optical parametric amplifier

OPO

optical parametric oscillator

PBS

polarizing beamsplitter

PDC

parametric down-converter

PDH

Pound–Drever–Hall

PID

proportional, integral, differential

PMT

photomultiplier tube

Qdot

quantum dot

QKD

quantum key distribution

QND

quantum non-demolition

QNL

quantum noise limit

RBW

resolution bandwidth

RF

radio frequency

RMS

root mean square

RRO

resonant relaxation oscillation

SHG

second harmonic generation

SLM

spatial light modulator

SNR

signal-to-noise ratio

SPAD

single photon avalanche diode

SSPM

solid-state photon multiplier

SQL

standard quantum limit

TTL

transistor–transistor logic

VBW

video bandwidth

4WM

four-wave mixing

559

Index a acousto-optical modulators (AOM) 192, 332 amplifier electronic 216 noise 252–254 optical 216 parametric 220–221, 333 saturation 272 amplitude modulator 191, 192, 225, 226, 386, 490 amplitude quadrature 25 operator 102–104 uncertainty 102 amplitude squeezed light 227, 312, 322, 339, 358–360, 365, 367, 380, 381 anti-bunching 4, 13, 84–88, 481, 533 applications, of quantum light 377–419 atoms 3, 4, 37, 38, 46, 47, 65, 68, 70, 71, 195, 203, 331, 332, 432, 433, 535–537 techniques 544 attenuation 101 effect on entanglement 459 effect on squeezed light 325

b beam divergence 22 shape 22 twin photons 365–366 waist 22

beamsplitter classical 140–142 phaseshift 141 photons 144–146 polarizing 79–81, 142 quantum model 143 beat signals 31, 32, 68, 171, 247, 258, 271, 279, 293, 298 Bell inequality 461–468, 534 measurement 493 state 493, 508 bit 255, 473, 474, 486, 493 Bohmian interpretation 446 Bose condensation 537 bunching 4, 13, 84–88

c canonicalquantization 94, 95, 119, 143, 156, 176, 181, 219 carrier frequency 30, 126, 164, 180, 255 Casimir force 96 cavity locking 290–296 phase response 171 power response 167 QED 542 reflection 167 stability 172 transfer function 178 transmission 167 chaotic light 46–49 charge coupled device (CCD) 7, 56, 66, 244, 446

A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

560

Index

classical beamsplitter 140–153 cavity 164–169 correlations 41, 42 fluctuations 37–39 interferometer 153–162 mode of light 26–27 quadratures 24–25 clone 9, 492 coherent state 6, 9, 13, 99–108, 112, 123, 127, 128, 149–151, 178, 228–230, 278, 311 coincidence imaging 412–414 communication, classical 384 commutator 95, 144, 286 complementarity 441, 445 controlled-NOT (CNOT) 474 experiment 508 theory 506 conventional optics 1, 186, 325, 389, 425, 538 correlation detection electronically 555 function, second order 45, 86 independent sources 76 covariance matrix 111–113, 130, 133–136, 459–461 cryogenic circuits 542 cryptography 6, 14, 135, 246, 474, 485, 538 current junction 151–152

d dark fringe 158, 159, 393, 398, 401, 407 decibel (dB) scale 261 dense coding 493 density operator 101, 108, 118, 146, 228, 382, 482, 483, 508 detection balanced 273–275 beats 247 delay 67 direct 269–273 electronic circuit 251 frequency 259 heterodyne 279–281 homodyne 275–279

process 65 in quantum optics 117–118 saturation 272 squeezed light 322–330 detector circuit 251–252, 254 materials 251 displacement 61–62, 99, 100, 111, 127, 129, 130, 134, 135, 149, 160, 371, 378, 409, 411, 416, 460, 496, 497, 549 dissipation 101 down conversion 219–221, 237, 413–415, 448, 451, 477, 479, 480, 501, 504, 508, 511, 517, 519

e efficiency 271 detection 271 detector 249 homodyne detector 279 electronic amplifier 252 noise 252 electronic spectrum analyser 247, 257–260 electro-optical modulator 189, 191, 366, 468 energy, zero point 96 entanglement 453 of formation 460–461 EPR paradox 6, 453, 461

f Fano factor 109 beamsplitter 148 feedback classical 282 quantum system 285 fidelity 230, 382, 383, 484, 492, 493, 498, 500, 501, 503, 509, 510, 514, 519, 520 finesse 94, 166, 167, 170, 175, 181, 204, 345, 347, 348, 481, 513 Fock state 97–99, 235, 236, 246, 389, 433, 475

Index

Fourier transform 39, 45, 54, 89, 122, 127, 168, 178, 182, 255, 258, 269, 349, 379 four wave mixing 307, 330 effect on noise 309 Franson interferometer 77 free spectral range 163, 202, 332, 521 frequency modulation 30–32 space 122 spectrum 115, 123

g gain electrical 252 noise 252 Gaussian beam 21–24, 26, 28, 57, 115, 173, 184, 185, 189, 202 Gaussian distribution 9, 10, 38, 104, 105, 107, 248, 309–311, 387, 490, 496, 553 gravitational waves (GW) 389 detector 538 guided acoustic wave Brillouin scattering (GAWBS) 195, 196, 346

h Hanbury–Brown and Twiss (HBT) experiments 85, 533 harmonic oscillator 69, 95–97 Heisenberg picture 115–117, 120, 121, 127, 128, 135, 176, 216, 462, 496 Heisenberg’s uncertainty principle 9, 13, 77 heterodyne detection 258, 278–280, 388, 399 holograms 185–186 homodyne detection 275 dual 496 Hong–Ou–Mandel (HOM) 450, 481, 494 hybrid detection 3, 539

i impedance matching 168 indistinguishability 78, 441, 446–453, 481, 482 information, classical 384 intensity noise 13 spectrum 12, 209 interferometer classical 153 configurations 393 ideal 394 Michelson 41, 396 noise sources 402 nonlinear 356–358 quantum model 155–156 realistic 401 Sagnac 353, 396 sensitivity limit 160 single photon 83–84, 442 squeezed light 405–411 stellar 41

j Josephson parametric amplifier (JPA) 542

k Kerr effect 343 classical 305 simulation 306 Kirchoff’s law 151, 152 KLM 510, 522

l laser argon ion 209–210 different classes 209 diode 213, 358–360 dye 209–210 Nd:YAG 210 noise suppression 358 quantum model 207–209 rate equations 203–207 threshold 201 lenses 23, 32, 115, 139, 174, 184–186, 279, 462

561

562

Index

linearization 121–124, 208, 215, 318, 341, 363 locking Pound–Drever–Hall 292–293 tilt locking 293–294 Log Negativity 459–460

m matter waves 535–537 mean field approximation 170 metamaterials 186 Michelson interferometer 396 stellar interferometer 41 micromaser 88, 89, 432 minimum uncertainty 102 observation 338 mirror, mounting 295–296 mixed state 101, 329, 338, 452, 458, 478, 479, 485 mode matching 161, 172–174, 193, 194, 238, 292, 366, 451, 478, 479 mode mismatch 155, 160–162, 173 modes classical 26 Hermite–Gaussian 23 matching 172–174 parameters 27 photon number 69 quantised 94–95 spatial 23 modulation 30 AM 30 amplitude 30 interferometers 399 sidebands 399 modulator 189 amplitude 191–193 phase 191–193 multimode lasers 32 multimode systems 36

n Nd:YAG laser 210–213 no-cloning limit 492, 500 noise adding of 262

amplifier 252–254 chaotic light 46 dark 252 in dB 261–262 eater 12, 13, 282 electronic 271 feedback control 282 intensity 12 Johnson 252 optical 195 quantum 9, 12 shot noise 248–249 spectrum 123 thermal 252 transfer function 126 noise spectrum laser 208 squeezed state 329 non-Gaussian states 106, 112, 135, 363, 388, 504, 539 nonlinear optics 1 non-linear processes 13, 109, 125, 135, 170, 195–196, 219, 225, 236, 263, 290, 307–309, 318, 319, 326, 330, 351, 352, 425 nonlocality 453 number state 78, 83, 97–100, 105, 106, 108, 132, 407, 478, 518–520

o operator amplitude quadrature 102–104 annihilation 97 creation 97 density 101 displacement 99 harmonic oscillator 97 linearization 123 ordering 67 parity 106 phase 101 phase quadrature 102 photon number 97 radiation field 97 unitary 115 optical fibres 6, 188–189, 195, 346, 431, 432, 451, 508

Index

optical parametric oscillation (OPO) 126, 195, 221, 380, 431 optical techniques 90, 257, 468, 544

state 81, 473, 482 pulsed lasers 33, 121, 199, 215, 482

q p pair production 221, 223–224 parametric amplification 125, 220–221, 230, 338, 415, 454, 455, 457, 458, 477 parity operator 106 partition noise 150, 185 P-function 108 phase classical 20 modulator 191–193 operator 101 quadrature 25 phase quadrature operator 102 uncertainty 102 phasor diagram 25 effect of attenuation 325 squeezed state 310 photocurrent 4, 10–12, 19, 25, 31, 36, 42, 44, 45, 55, 61, 88, 109, 117, 120, 225, 235, 247‘, 255–262, 280, 331, 349 photo-detection process 9 photon 67 bunching 4 generation process 13 interference 73–78 and laserbeams 533 photon number, distribution 317 photon pressure 159 photon statistics squeezed state 317–319 PID controller 294–295 Poissonian distribution 9, 72, 109, 113, 147, 148, 151, 248, 264, 313, 339 polarization 33 beamsplitter 79 cavity 174 mirrors 142 modulator 187 single photon 79–81 squeezing 367–368

Q-function 104 Gaussian 107 Q-technology based on optics cavity QED 542 cryogenic circuits 542 microwaves 542 quantum communication 539–541 quantum information transfer 543 quantum opto-mechanics 542–543 squeezed light application 537–538 transferring and storing quantum states 544 quadrature amplitude 24–25 classical 25 operator 102–104 phase 24 rotation 324 squeezing 316 quadrature squeezed light 313 detection 322–330 quantum communication 195, 234, 539–541, 544 quantum computation 235, 475, 505–525 quantum cryptography 6, 246, 474, 484, 538 quantum effects demonstration 533 quantum efficiency detector 249 material 250 quantum enhanced imaging 411–414 quantum enhanced metrology 2 quantum imaging 414 quantum information 14, 16, 135, 230, 235, 437, 459, 473–525 quantum key distribution (QKD) 6, 235, 474, 476, 484–492 quantum noise 9, 12, 269, 537 balance 394 calibration 269, 273–281 limited images 412 penalty in feedback 288

563

564

Index

quantum noise (contd.) properties 112 pulsed 349 representation 112–113 spectrum 123 tomography 361 quantum non-demolition (QND) 16, 89, 90, 160, 243, 349, 386, 425, 445 concept 425 noise transfer 426 results 430–432 single photon 432–437 quantum optics guide 14–16 motivation 7–12 origin and progress of 3–7 photon correlations consequences 12–14 quantum optomechanics 542–543 qubit 6, 234, 473–475, 482–484, 487, 493–495, 508, 510, 511, 513, 514

r Rayleigh length 22 recycling 397–398 reflectivity, matrix description 141 representation P 106 Q 106 resolution bandwidth 257, 270, 332, 409 Rydberg atoms 88, 89

s Sagnac interferometer 352, 353, 393, 396, 397, 405, 466 saturation, electronic 254 Schrödinger picture 115 second harmonic generation (SHG) 135, 170, 339–343 shot noise 4, 9, 13, 73, 248–249, 252–253, 356, 366 sidebands 123, 126 classical 30 correlation 308, 319 squeezed light 327

signal to noise ratio, interferometer 400 single mode approximation 213 single mode laser 5, 32, 123, 201, 202, 210, 214, 334 single photon QND 432–437 single photon Rabi frequency 88 soliton 346, 354–357, 415, 431, 432 spatial light modulators (SLM) 55, 186, 193–195 spatial modes 5, 23, 26, 29, 32, 36, 95, 164, 173, 182, 184, 185, 193, 201, 238, 239, 297, 413, 417, 449, 464, 475, 521, 538 spatial squeezing 414 spectrum analyser 39, 76, 163, 167, 258–261, 270, 272, 275, 324 spectrum noise 123 squeezed light direct detection 322 effect of attenuation 327 first observation 331, 332 generation 317–319 homodyne detection 322 phasor diagram 310 photon number distribution 317 propagation 325–330 Q-function 310 rotation of quadrature 324 squeezed state definition 314–317 different types 312–313 spectrum 329 vacuum 313 Wigner function 311 squeezing atomic systems 348–349 concept 303–313 diode laser 358–360 ellipse 310 formal derivation 314 four wave mixing 330–333 Kerr effect 343–349 nonlinear interferometers 356–358 polarization 367–368 pulsed 349–358 SHG 339–343

Index

sideband model 319–322 soliton 354–355 spatial properties 414 spectral filtering 355–356 spectrum OPO 335 spectrum SHG 354 twin beams 365, 366 two modes 319 standard quantum limit 160, 395 state Bell 493, 508 coherent 99–101 Fock 97–99 minimum uncertainty 102 mixed 101–102 number 97–99 polarization 473, 482 propagation 116 squeezed 303 sub-Poissonian definition 111 diode laser 358–360 superconducting quantum interference device (SQID) 243, 542 superposition 3, 6, 20, 23, 24, 100, 101, 113, 121, 145, 149, 344, 432, 434, 453, 474

t teleportation 492–505 thermal light 4, 37, 43, 46, 70, 73, 85, 86, 136, 235 threshold of laser 201 tomography 258, 325, 360–363, 366, 433, 478, 508 traditional physical optics 1 transfer coefficient 227, 228, 428, 500 transfer function 126

Bode plot 282, 283 cavity 177–180 diode laser 213 laser noise 208 Nyquist plot 282, 283 optical gain 217

u uncertainty area 107 coherent state 103 unitary operator 115, 116, 145

v vacuum state detection 322 squeezed 313 variance 94, 123 classical 39 of squeezed state 328 video bandwidth 260, 270, 332, 361 visibility 42, 44, 45, 58, 73, 74, 155, 161, 185, 279, 400, 401, 442, 444, 445, 451, 452, 464, 478, 495, 555

w waveplate 79 weirdness 2 Wigner function 106–108, 110, 128, 310, 311, 321, 326–328, 349, 360–362, 478, 480, 520

x X operation 493

z zero point energy 95–97 Z operation 492, 512

565