Exploring the Quantum

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Exploring the Quantum

Atoms, Cavities and Photons

Serge HarocheProfesseur, Collège de France

Jean-Michel RaimondProfesseur, Université P. et M. Curie et Institut Universitaire de France

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Claudine, Julien and Judith

Fabienne, Yves and Marie

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Foreword

The counter-intuitive aspects of quantum physics were illustrated in the early days ofthe theory by famous thought experiments, from the Einstein and Bohr photon boxto Schrödinger’s cat. Modern versions of these experiments, involving single particles– electrons, atoms or photons – have now been actually realized in many laboratoriesaround the world. By manipulating these simple systems in a controlled environment,physicists directly unveil the strange features of the quantum. State superpositions,entanglement and complementarity define a novel quantum logic which can be har-nessed for information processing, raising great hopes for applications.

This book describes a class of such thought experiments which have come of age.Selecting among a vast and fast expanding domain of research, we have chosen toanalyse in detail experiments performed with atoms and photons in high-Q cavitiesas well as related ones, dealing with ions in traps or cold atoms in optical lattices. Inthese apparently disparate domains, the same underlying physics is at work: two-levelspin-like systems are interacting with quantum harmonic oscillators.

We believe that a description of these real ‘spin-spring’ experiments provides amore concrete illustration of quantum concepts than that given by abstract discus-sions about idealized experiments. Although the latter are simpler to analyse and arecertainly here to stay in introductory courses of quantum mechanics, real ‘thoughtexperiments’ should in our view become central in the teaching of modern quantumphysics at an intermediate or advanced level. The efforts to carry out these experimentsin laboratories have been largely triggered by the hopes placed in the developmentof quantum information for practical applications in communication and computing.Conversely, this fast expanding field of research is bound to have an increasing influ-ence on the teaching and learning of quantum concepts.

Dealing with real systems necessarily involves a description of the interaction ofthese systems with their unavoidable environment, in other words a discussion of re-laxation and decoherence. These phenomena are described by a formalism (densityoperator or stochastic Monte Carlo approach) which replaces the simple state de-scription of elementary quantum physics. Mastering this approach and understandingdecoherence provide a deep understanding of one important aspect of the quantum, itsrelation to classical physics. Thought experiments have been invented at the dawn ofthe quantum age to illustrate the puzzling features of the quantum–classical boundary.It is thus no surprise that an understanding of the modern version of these experimentsmust also address this important issue.

Starting from the simple goal to describe experiments illustrative of basic quantum

viii

laws, we have thus been led to widen our perspective, ending up with a book whichpresents a comprehensive discussion of many important issues in modern physics. Itcombines a fundamental approach, based on an analysis of quantum concepts andof useful theoretical tools, with a detailed analysis of experiments, including a briefoverview of the various technological developments which have made these experimentspossible. In balancing these theoretical and applied points of view, we have tried toconvey at the same time the strange beauties of the quantum and the difficulties whichhad to be overcome to unveil them, and possibly to harness them for achieving usefultasks.

This book is intended for students at the undergraduate or graduate level, withan elementary knowledge of quantum mechanics. We have not assumed that they hadbeen previously exposed to a detailed discussion of concepts such as entanglement,non-locality, decoherence or measurement theory, which we have chosen to exposefrom scratch. At the same time, we hope that our work will also be useful to teachersin the field of quantum optics and quantum information science. We have attemptedto present many examples of physical situations which are computed in detail, andwhich could be easily turned into instructive problem sets for students.

The many connexions and comparisons we make between atom–cavity, ion trapand cold atom experiments might also be useful to scientists working in these variousfields of quantum optics. Reading this book might suggest to them new perspectivesfor their work, as writing it has helped us to sharpen our understanding and to designnew experiments. Finally, theorists in quantum information science might learn hereabout some of the challenges that experimenters have to address in order to put theirbright ideas into practice.

The material of this book is based on the lectures on quantum information wehave respectively given at Collège de France from 2001 to 2006, and at Ecole NormaleSupérieure (ENS) from 2003 to 2006. The atom–cavity experiments which are at theheart of the book discussions have been carried out by our research team at ENS. Weare indebted to all the colleagues, students, postdocs and visitors who have workedwith us over the years. We should mention in particular Michel Brune, who has playedan essential role in all these experiments and who has provided precious advice toimprove the manuscript. Our ENS colleagues Jean Dalibard and Yvan Castin havebeen consulted on some aspects of cold atom physics and we are glad to acknowledgetheir illuminating input. Special thanks go also to Luiz Davidovich and Nicim Zagury,from the Federal University of Rio de Janeiro, whose theoretical insights have beenprecious to design new experiments.

We have devoted the last chapters of the book to the description of experimentsperformed in other laboratories, mainly in Boulder, Innsbruck, Munich and Mainz. Wethank D. Wineland, D. Leibfried, R. Blatt, C. Roos and I. Bloch for helpful discussionsand for critical reading of parts of the manuscript. We are of course responsible for anyapproximation or error remaining in the description of their work. Finally and fore-most, we thank Claudine and Fabienne for their constant support and encouragement.

Serge Haroche and Jean-Michel RaimondParis, May 2006

Contents

1 Unveiling the quantum 11.1 One century of quantum physics 41.2 Emergence of the microscopic world 101.3 Thought experiments coming of age 141.4 Aims and outline of this book 20

2 Strangeness and power of the quantum 252.1 The superposition principle and the wave function 262.2 Quantum interference and complementarity 342.3 Identical particles 422.4 Entanglement and non-locality 522.5 The quantum–classical boundary 682.6 Taming the quantum to process information 83

3 Of spins and springs 1013.1 The field oscillator 1063.2 Coupled field modes 1263.3 The spin system 1433.4 Coupling a spin and a spring: the Jaynes–Cummings model 151

4 The environment is watching 1634.1 Quantum description of open systems 1654.2 Quantum maps: the Kraus sum representation 1734.3 The Lindblad master equation 1784.4 Quantum Monte Carlo trajectories 1894.5 Damped spin–spring system: from Rabi to Purcell 2034.6 Kicking a spring with spins: the micromaser 2084.7 Collective coupling of N spins to a spring: superradiance 220

5 Photons in a box 2315.1 A short history of cavity QED 2325.2 Giant atom in a cavity: an ideal cavity QED situation 2515.3 Two experiments unveiling the quantum in a cavity 2725.4 An atom–photon entangling machine 278

6 Seeing light in subtle ways 2976.1 Complementarity at quantum–classical boundary 299

x

6.2 Non-destructive photon number measurement 3136.3 A quantum gate for multi-particle entanglement engineering 3266.4 The quantum analogue/digital converter 3346.5 Photon number parity and Wigner function measurements 348

7 Taming Schrödinger’s cat 3557.1 Representations of photonic cats 3587.2 A thought experiment to generate optical cats 3647.3 Dispersive cats in cavity QED 3697.4 Resonant cats in cavity QED 3857.5 Decoherence of cavity cats 4057.6 Non-local cats 430

8 Atoms in a box 4438.1 Ion trap physics 4468.2 Engineering ionic states of motion 4718.3 Ion relaxation and engineered environments 4788.4 Quantum logic with trapped ions: individual qubit addressing 4898.5 Quantum logic with trapped ions: collective qubit addressing 5018.6 Perspectives of ion traps for quantum information 513

9 Entangling matter waves 5179.1 Second quantization of matter waves 5209.2 Main features of Bose–Einstein condensation 5239.3 The phase in Bose–Einstein condensate interference 5269.4 Coherent collisions and cat-state generation 5349.5 Matter waves in periodical lattices 5469.6 Entangling collisions in a Bose–Einstein condensate 556

10 Conclusion 565

Appendix 569A.1 Characteristic functions 570A.2 The Wigner distribution 572A.3 The Husimi-Q distribution 579A.4 Phase-space representations of relaxation 582

Bibliography 587

Index 603

1

Unveiling the quantum

We never experiment with just one electron or atom or (small) molecule. In thought-experiments wesometimes assume that we do; this invariably entails ridiculous consequences...

E. Schrödinger,British Journal of the Philosophy of Sciences, 3, 1952.

Thought experiments are at the heart of quantum physics. The fathers of the theoryhave time and again imagined simple machines manipulating isolated atoms or pho-tons and discussed what quantum rules had to say about the outcome of such idealizedexperiments. These conceptual constructions have helped them to elaborate a coherenttheory. At the same time, they were convinced that such experiments would remainvirtual. As late as the middle of the last century, at a time when the quantum ideaswere firmly established, Erwin Schrödinger argued that manipulating an isolated atomin the real world would remain forever impossible. And yet, by the end of the twen-tieth century, those experiments came of age. Physicists can now trap single atomsor photons in a confined region of space, prepare these particles in well-defined statesand follow in real time their evolution. These experiments have been made possibleby powerful technologies born from quantum concepts, such as modern computersand lasers. Physics has thus come full circle. Thought experiments, considered as puredreams, have helped to develop the tools which have made these fantasies real.

This history is intellectually satisfying since it displays the deep consistency of ourunderstanding of Nature. By actually performing in the laboratory thought experi-ments, we are able to observe directly ‘in action’ the quantum laws that the foundingfathers had deduced from remarkable efforts of abstraction. Apart from this intellec-tual satisfaction, what is the rationale for performing such experiments which remaina technological challenge? Four incentives at least might be invoked.

First, many aspects of the quantum theory are so counter-intuitive that somephysicists – including Einstein – have never really accepted them. Entanglement isone of the most intriguing features of the quantum. After interacting, two microscopicquantum systems generally end up in a non-separable state. The properties of eachsystem cannot be described independently of those of the other. This occurs whateverthe distance between the components of the entangled state. Entanglement thus nat-urally leads to non-locality. This notion tells us that physics at one place cannot be

2 Unveiling the quantum

described independently of what goes on in another disconnected part of the Universe.This is certainly the quantum feature most difficult to admit by a classical mind. It isonly by investigating non-locality through real experiments that one can put Natureto test and find out whether quantum laws give the last word on this intriguing issue.

Our second motivation has to do with the exploration of the connexion betweenquantum and classical physics. One reason why non-locality is counter-intuitive isthat it is only observed in systems made of a few photons, electrons or atoms. Macro-scopic systems, directly accessible to our senses, never display non-locality. Nor dothey exhibit other strange aspects of the quantum such as state superposition andquantum interference. There seems to be a boundary between the microscopic world,directly displaying quantum laws, and the macro-world where these laws are veiled.By performing thought experiments on systems of increasing size, one might hope toexplore this frontier. The phenomenon which tends to blur quantum effects in themacroscopic world, to destroy quantum interference and quantum coherence, is called‘decoherence’. Performing ‘thought experiments’ is a direct test of our ability to min-imize decoherence in real systems and provides a simple and direct way to study thisphenomenon.

Our third incentive is more practical. The counter-intuitive features of quantumlaws can, in principle, be exploited to process information according to a non-classicallogic. Communicating or computing is a quite different state of affairs if one codesinformation into ensembles of individually addressable atoms or photons. Two quan-tum states of each of these particles, defined as a quantum bit, or simply a qubit, canbe associated to the 0 and 1 values of the usual binary code. These qubits not onlyexist as 0’s and 1’s, but can also evolve into entangled superpositions of these states.Secure cryptographic schemes can be elaborated with qubits, enabling two partnersto exchange information while being certain that no spy could eavesdrop. Quantumcomputers solving some problems such as the factoring of large numbers in a timemuch shorter than classical machines can be envisioned. These hopes have led to thefast development of a new field of research, at the frontier between physics and infor-mation science, the ‘physics of quantum information’. In this context, simple thoughtexperiments realized with a few atoms or photons are important to demonstrate thefeasibility of quantum information and to study its fundamental and technical limita-tions as well as the possible ways to overcome them.

The last – but not least – incentive for realizing thought experiments is pedagogical.In most textbooks, quantum physics is introduced by discussing early experiments,performed in the first part of the last century, which displayed only indirectly andimperfectly the quantum features. Now that experiments in which quantum laws areclearly apparent have become feasible, it is important to perform them as illustrationsof fundamental concepts. Even if quantum computers never come of age, students andresearchers studying these experiments will gain a deeper familiarity with quantumlaws, which will help them find novel ways to explore Nature and to develop powerfultechnological tools.

Most of the thought experiments realized recently are made of simple basic ele-ments. Inside a confined region of space, a ‘box’ as we will call it for short, a few atomsor photons evolve, largely impervious to what happens in the outside world. On thissimple stage, the laws of the game are the quantum postulates. State superpositions,

Unveiling the quantum 3

quantum interference and entanglement are directly displayed, illustrating as clearlyas can be the quantum concepts. In some realizations, the box is a simple configura-tion of electromagnetic fields trapping atomic particles, without material separationbetween the inside and the outside. In others, the box has real walls, for examplemirrors confining photons.

Simple tools are used to manipulate and detect the particles in the box. If theseparticles are atoms or ions, laser beams propagating across the box prepare them inwell-defined states and manipulate them. Photons scattered off these beams are usedto detect the atoms, providing direct sight inside the box. If photons are trapped,atomic beams are used instead to manipulate and probe them. In all cases, light–matter interaction processes are essential to build the box, to manipulate the particlesand to detect them.

This book is devoted to these atom–photon-box ‘thought experiments’ which havebecome real. Our goal is to describe them and to present the theoretical framework re-quired to understand their physics. Along the way, we will keep in mind our quadruplemotivation: the interest of these experiments as tests of the counter-intuitive aspectsof the theory, their importance in the exploration of the quantum–classical bound-ary, their role as elementary steps towards harnessing the quantum for informationprocessing and their utility as a pedagogical approach to quantum phenomena.

We have alluded several times to the strangeness of the quantum world. Sayingthat the theory is strange, we are in good company. Just recall the famous statement ofRichard Feynmann that ‘nobody really understands quantum physics’. This aphorismby one of the fathers of quantum field theory must be taken with a grain of salt. Thereis a pitfall to avoid when invoking the wonders of the quantum, their manifestationsin thought experiments and possible uses in quantum information. The danger is togive the feeling that quantum theory is mysterious or controversial, with promiseslooming only in the future. In fact, the quantum has already delivered a lot. Most ofthe fundamental and technological advances of the last century, which make our life sodifferent from our great-grandparents’, are due to the deep understanding of Naturebrought about by the quantum revolution. If the theory appears strange, it is mainlybecause we try to describe it with words of our everyday life, which are adaptedto the properties of macroscopic objects. Even if quantum concepts are necessaryto understand in depth the electric conductivity of metals, the superfluidity of liquidhelium or the colour of the sky, these macroscopic phenomena are not ‘strange’ becausethey can be described with usual words, which is not the case for an ion in a trap ora photon in a cavity.

A distinction must thus be made between the ‘microscopic’ quantum strangenessdirectly displayed in thought experiments and the apparent plainness of the macro-scopic physics, which do not violate our common sense in spite of its quantum sub-strate. The physics of quantum information has the ambition to use the quantum tocompute better, but we must not forget that the quantum is already present, albeit ina veiled form, in the physical processes making classical computers so successful. Thereis thus a ‘veiled’ and a ‘naked’ quantum which should be distinguished, even if the op-position between them is sometimes blurred. This book is about the naked quantum,its manifestations in realized thought experiments and its possible applications.

In this introductory chapter, we start (Section 1.1) by recalling some of the great

4 Unveiling the quantum

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Fig. 1.1 Quantum physics describes Nature at all scales. Pictorial representation of (a) a

superstring, (b) a nuclear reaction (fusion of deuterium and tritium), (c) a hydrogen atom,

(d) a biological molecule, (e) a human being, (f) a planet (image credit: NASA), (g) the

‘whirlpool’ galaxy (image credit: NASA/JPL), (h) the microwave blackbody radiation map

(image credit: NASA/WMAP Science Team).

advances of quantum theory in the twentieth century. These advances result from thesuccessful work of generations of scientists who have made use of quantum physics,spending only a small fraction of their time puzzling at its wonders and directingtheir efforts to apply it at explaining Nature. Any meaningful discussion about wherethe quantum can bring us in the future must be carried on with this background ofsuccesses in mind.

We then come back (1.2) to the elaboration of the theory, the long process duringwhich the strange and counter-intuitive manifestations of the quantum have emerged.It is during this process – mainly in the 1920s – that thought experiments were con-ceived. We recall (1.3) their main features and describe, on a few examples, how theyare now realized in the laboratory by exploiting modern technology. We stress thatthese experiments can be quite generally viewed as manipulations of quantum oscilla-tors and two-level spin-like particles in mutual interaction. We conclude the chapterby presenting the aims and outline of the book (1.4).

1.1 One century of quantum physicsFrom the very small to the very large, covering about sixty orders of magnitude indimensions, quantum theory is invoked to describe all phenomena, from the still mys-terious vibrations of the microscopic strings which might be the ultimate constituentsof Nature, to the fluctuations of the microwave radiation received from the outpostsof the known Universe. Between these extremes, we find all the objects of the worldaround us (Fig. 1.1). Add twenty zeros to the dimensions of the strings and you havethe size of the atomic nucleus, the centre of radioactivity and nuclear energy. Fivemore zeros and here is the atom, a nucleus bound to its swarm of electrons by elec-tromagnetic forces, and simple molecules, small ensembles of atoms which combineaccording to the laws of chemistry. Two or three orders of magnitude more and we

One century of quantum physics 5

reach bio-molecules, where life appears at the most elementary level. Count six tonine further orders of magnitude and we are in the range of the centimetres to tens ofmetres, the human scale with its variety of objects, solids, liquids and gases made ofhuge numbers of atoms. From our scale to the astronomical one, the size of planets andstars where gravitation is dominant, you must still add seven to nine zeros. And youfinally need eighteen more orders of magnitude to complete, through the explorationof galaxies, our voyage towards the horizon of our Universe.

1.1.1 Quantum theory as a unifying description of Nature

Along this vertiginous path, physics must account for a boundless variety of phenom-ena. Some have been empirically known for a long time, others discovered during thelast century thanks to the development of powerful methods of investigation. At thevery small and very large ends of the dimension scale, others are still raising manyquestions. Most of these phenomena require us, at least for part of their understand-ing, to invoke quantum theory whose success has been overwhelming. Theorists willinsist on the extreme precision of the description of the interactions between electronsand photons by quantum electrodynamics. They will note the remarkable agreementbetween theory and experiment in this field: the magnetic moment of the electron,measured in a particle trap experiment, has been found to be, in units of Bohr’s mag-neton g = 2× 1.00159652188, a value which agrees with theory within a few parts perbillion!

Particle physicists will also mention, as a proof of the success of quantum theory,the unification of three out of the four fundamental interactions – the electromagnetic,strong and weak forces – in the Standard Model, which reveals the deep symmetries ofNature. They will also note the promising attempts to include gravitation in a unifiedtheory of strings. And all physicists will certainly emphasize the universality of physics,a remarkable consequence of quantum laws. It is the quantum theory which explainsthe radiation spectrum of hydrogen in our laboratory lamps, but also in intergalacticspace. Quantum chemistry applies to the reactions in laboratory test tubes, but alsoto the processes in the interstellar dust where molecules are formed and destroyed,producing radiation detected by our telescopes after travelling billions of years throughspace. Let us finally evoke cosmology and the remarkable link between the infinitelysmall and large scales, underlined by the quantum theory. It emphasizes the similaritybetween the phenomena which occurred at the origin of the Universe, in a medium ofinconceivably large temperature and density, and those which happen in the violentcollisions between particles studied by large accelerators on Earth.

1.1.2 The transistor and the computer revolution

Quantum physics not only describes the structure of matter and the interactions be-tween particles. Through the deep knowledge it gives us about microscopic phenomena,it has provided us with tools to act, communicate, diagnose and compute with a powerand a precision which could not be imagined before its advent. Since some of thesetools are necessary for realizing the experiments described here, it is appropriate toreflect a little on these achievements of the quantum in our everyday life.

Let us start with the computer. The small desk machines which have become aubiquitous fixture of offices, homes and laboratories are based on a small processor,

6 Unveiling the quantum

(a) (b)

(c) (d)

Fig. 1.2 The history of the computer. (a) Pascal machine c© Musée des Arts et Métiers,reprinted with permission; (b) Babbage machine c© Science and Society Picture Library,reprinted with permission; (c) the ENIAC machine (US Army photo); (d) a modern laptop

(photograph by the authors).

made of a silicium chip the size of a stamp, on which a labyrinth of tiny electricalwires is printed. These wires link small layers of semiconductor materials making upmicroscopic transistors which behave as logical gates. The properties of these materialsare ruled by quantum laws (tunnel effect, exclusion principle,...). It is by exploitingthese laws that one can achieve extremely complex calculations. The principles ofthe computer have been known for a long time. The capacity to store informationand to compute dates back at least to Pascal’s machine in the seventeenth century(Fig. 1.2a). The idea of computer programming can be attributed to Babbage who,in the nineteenth century, had invented a mechanical device able to perform simplecomputation tasks (Fig. 1.2b).

The development of the electrical industry in the first part of the last centuryreplaced the mechanical parts of Pascal’s and Babbage’s machines by vacuum tubes,leading to the first modern computers. The machines, which were built during theSecond World War and immediately after, under the impulsion of physicists andmathematicians such as Turing, von Neumann and Brillouin, were huge assembliesof cupboard-size boxes, stuffed with tubes and linked by kilometres of wires, requiringthe assistance of an army of engineers (Fig. 1.2c). And their poorly reliable perfor-mances were by far short of those of the modern laptop computers, a thousand times

One century of quantum physics 7

smaller in weight and volume (Fig. 1.2d)! It is the advent of the transistor replacingthe bulky tubes, and the integration of a huge number of them in a chip, which hasmade all the applications of the modern computer possible, including the delicate ma-nipulation and control of individual atoms and photons described in this book. Thelogical principles of the computer could have been understood by a nineteenth centuryscientist, but its practical realization relies on a technology totally inconceivable to apre-quantum mind.

1.1.3 The laser and the optical revolution

The laser is another example of an invention based on a quantum idea, now ubiquitousin our everyday life. Laser beam shows are today commonplace but, until the 1960s,nobody had seen anything like it. Light had been forever made of random waves,difficult to direct, to focus or to force to oscillate at a well-defined frequency. The laserhas changed this state of affairs and has allowed us to tame radiation by exploitingthe properties of atomic stimulated emission, discovered by Einstein at the dawn ofthe quantum era. Lasers now have a huge variety of uses, from the very mundaneto the most sophisticated. Laser light travelling through fibres can transport hugeamounts of information over very long distances. Laser beams are used to print andread out information on compact disks, with applications for the reproduction ofsounds, pictures and movies.

In scientific research, the flexible properties of laser beams have countless appli-cations, some of which are essential to realize the manipulations of single particleswhich will be our main topic here. Experiments exploit their high intensity to studynon-linear optical processes. The extreme monochromaticity of lasers and their timecoherence is used for high-resolution spectroscopy of atoms and molecules. Combin-ing monochromaticity and high intensity has proved essential to trap and cool atomsto extremely low temperatures. Laser pulses of femtosecond duration probe very fastprocesses in molecules and solids and study chemical reactions in real time.

1.1.4 The atomic clock and the measurement of time

The precise measurement of time is another illustration of the importance of quantumprocesses in science and in society. Atoms emit and absorb radiation at well-defined,never changing frequencies, characteristic of each element. This fundamental quan-tum property has led physicists to base the measurement of time no longer on theperturbation-prone oscillations of mechanical or electrical pendulums, neither on thefluctuating motions of planets, but on the immutable oscillations of electrons in atoms.

This has led to atomic clocks built by locking a radiosource to the hyperfine radio-frequency transition of caesium atoms propagating in an atomic beam. The atomicresonance is observed by using an interferometric method which we will describe indetail later on. In short, the atoms interact successively with two microwave pulses asthey travel along the beam. By detecting downstream one of the hyperfine states andsweeping the microwave frequency, one obtains a modulated signal, called a ‘Ramseyfringe’ pattern. The centre of this pattern is determined with a precision dependingon the fringe spacing, which is inversely proportional to the time interval between themicrowave pulses. Recently, laser-cooled atoms have replaced the thermal atoms offormer clocks. The interval between the pulses has been considerably increased and

8 Unveiling the quantum

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of Springer Science and Business Media.

the clock’s precision improved by two orders of magnitude.Figure 1.3 shows the Ramsey fringe pattern of a cold-atom clock, with a fringe

spacing of less than 1 Hz, while the microwave frequency is 9.2 GHz. The large signal-to-noise ratio of the fringes makes it possible to determine the atomic frequency withinone part in 103 of the fringe interval. This clock has accuracy better than a secondover a geological time of three million years! The techniques developed for these clockshave much in common with the methods of atomic manipulation described later on.

Time measurements based on atomic clocks and on a related device, the hydrogenmaser, have been used in general and special relativity tests. The utility of these clocksis not restricted to fundamental science. Their extraordinary precision is exploited inthe Global Positioning System (GPS), which consists of a large set of clocks embarkedon satellites. They send signals to small and cheap receivers, not much bigger than awristwatch. Measuring the arrival times of these signals allows the receiver to deter-mine its location with a precision in the metre range. These devices, combined withnavigation computers, have become standard fixtures of planes, boats and cars and weare no longer surprised to be routinely guided from the sky with such an extraordinaryprecision.

1.1.5 Nuclear magnetic resonance and medical diagnosis

Nuclear Magnetic Resonance (NMR) is another technology based on quantum sciencewhich nowadays plays a major role in scientific research and in medical diagnosis.The nuclei of a variety of atoms carry magnetic moments attached to their intrinsicangular momentum or spin. The origin of this magnetism is fundamentally quantum.The quantization of the spin orientation in space, the fact that its projection on anydirection can take only discrete values, has been one of the smoking guns of thequantum theory, forcing physicists to renounce their classical views. The evolution of

One century of quantum physics 9

(b)

(a)

Fig. 1.4 Magnetic resonance imaging. (a) Picture of an MRI imager. Photograph by D.

Glutron CIERM–U2R2M. (b) MRI pictures of the brain exhibiting regions concerned by a

specific activity (light spots). Reprinted from Rangaswamy et al. (2004), with permission

from Elsevier.

spins in magnetic fields, both static and time-modulated, requires a quantum approachto be understood in depth. In a solid or liquid sample, the spins are affected by theirmagnetic environment and their evolution bears witness of their surroundings.

In a typical NMR experiment, one immerses the sample in a large magnetic fieldwhich superimposes its effects on the local microscopic field. One furthermore appliesto the sample sequences of tailored radio-frequency pulses. These pulses set the nucleiin gyration around the magnetic field and the dance of the spins is detected through themagnetic flux they induce in pick-up coils surrounding the sample. A huge amountof information is gained on the medium, on the local density of spins and on theirenvironment.

This information is widely exploited in chemistry and biology. It has also invadedour lives through the Magnetic Resonance Imaging (MRI), which has become a stan-dard technique of medical diagnosis. The MRI machines used in hospitals (Fig. 1.4a)are the result of three converging quantum technologies carried to a remarkable levelof sophistication. The large magnetic field in which the patient is placed is producedby coils cooled to liquid helium temperature and exhibiting superconductivity, a fun-damentally quantum effect. Only such coils can produce without heating the largefields required for MRI imaging. The choreography of the spins observed by the NMRmethod is also ruled by quantum laws. Finally the signals picked up in the detectioncoils are transformed into images by powerful computers which also exploit quantum

10 Unveiling the quantum

effects in their semiconductor circuits. Figure 1.4(b) shows images of the brain madeby MRI, exhibiting the regions which are concerned by a specific activity. The changein the magnetic environment produced by the blood flux increase in these regionsmodifies the NMR signals, and the brain activity – thought or motion command – canthus be directly witnessed.

Information gained in this way is coded into the spins of the protons which makeup the hydrogen atoms of our bodies. These spins can be considered, in the languageof information, as binary bits which take one value – say 0 – if the spin points alongthe direction of the field and the other value – say 1 – if it points in the oppositedirection. When the spins are rotated by the NMR pulses along a direction normalto the field, quantum laws tell us that they are put in a superposition of the 0 and1 states. In a way, we can thus say that an NMR machine exploits quantum logic,dealing with qubits in superposition of their ordinary classical binary values.

This simple idea has been carried further in quantum information physics. By per-forming complex NMR experiments on liquid samples made of organic molecules, sim-ple quantum computing operations have been achieved. The spins of these moleculesare manipulated by complex pulse sequences, according to techniques originally devel-oped by chemists, biologists and medical doctors in NMR and MRI. In these experi-ments, a huge number of molecules contribute to the signal. The situation is thus verydifferent from the manipulation of the simple quantum systems we will be consider-ing. Some of these methods do however apply. We will see that the atoms and ionswhich are individually manipulated in the modern realizations of thought experimentsare also two-level spin-like systems and that one applies to them sequences of pulsessimilar to those invented for NMR physics.

1.2 Emergence of the microscopic world

When reflecting on the wonderful tools that quantum science has offered us, it isimportant to remind ourselves that all these inventions have required a deep knowledgeof the microscopic structure of matter. The end product is generally a macroscopiccontraption whose intimate workings are concealed from the layman enjoying them.It is indeed striking that we are using these devices without wondering about the waythey work, most of the time oblivious or plainly unaware of the microscopic effectswhich are essential to their operation. To a nineteenth century physicist projected intoday’s world by a time machine, this modern technology would superficially appearfamiliar, a combination of boxes and keyboards. Its working would, however, appearas pure magic. Up to the beginning of the last century indeed, the microscopic worldwas a terra incognita, whose mere existence was ignored by many – among whom notthe least famous scientists of the time.

Even those who – like Lord Kelvin – firmly believed in atoms did not realize theimportance of the microscopic world for a deep understanding of Nature. His famousstatement that everything was understood about luminous and thermal effects, withthe possible exception of two small clouds still obscuring the otherwise clear sky ofphysics, is typical of the general belief of the end of nineteenth century scientists.Influenced by philosophers such as Ernst Mach, most of them thought that only thedirectly observable macroscopic quantities – electrical currents, temperature, pres-sure – really mattered. Once direct relations between these quantities had been found

Emergence of the microscopic world 11

Fig. 1.5 The Young double-slit experiment. A source S of particles collimated by a slit in

the left screen illuminates the screen at right through two narrow slits in the central screen.

The particle impacts on the detection screen form bright and dark fringes which reveal a

quantum interference process.

(Newton equations for mechanics, Maxwell for electrodynamics, Gibbs for thermody-namics), everything was supposed to be understood. The existence of an atomic realitybeneath the surface of objects was strongly advocated by some, Boltzmann notably.But many scientists believed that the atomic hypothesis was at best a convenient wayto describe things, which should not be given too much objective reality.

Remarkably though, Lord Kelvin’s allusion to the two small clouds hinted at somepossible trouble in this simple picture. One of these clouds referred to the ether, ahypothetical medium supposed to permeate everything, which Michelson’s experimenthad just shown to have inconsistent properties. The other cloud pointed to anomaliesin the distribution of energy in heated bodies and in the light they radiate (the so-called blackbody problem). In spite of his apparent triumphalism, Lord Kelvin hadclearly seen from where the changes were coming. The first cloud announced Einstein’srelativity and the second soon led Planck to the idea that the exchanges of energybetween matter and radiation occurred in discrete lumps, opening the quantum erawith all the consequences which were recalled above.

As we have noted, the main difficulty that the founders of the theory encounteredwas that the language of everyday life, built on the experience of the macroscopicworld, is inadequate to describe microscopic reality. A large object has to be ‘here’ or‘there’. An atom or a photon, on the contrary, can be ‘suspended’ between differentpositions. If it is sent across a double slit in a Young interferometer, as depicted inFig. 1.5, a drawing inspired by Bohr, it can cross the apparatus in a superpositionstate, passing in a way at the same time through the two openings.

Each state of the superposition is associated to a complex number, its probabilityamplitude. These amplitudes interfere constructively or destructively depending ontheir relative phases and the impacts of the particles on a detection screen behind thetwo slits form a pattern of ‘bright’ and ‘dark’ fringes. The latter correspond to pointswhere the particles are less likely to end up when the two slits are open than whenonly one is. This strange result, which will be discussed in much more detail lateron, imparts to the particle, atom or photon, a wavy character. This is strange to aclassical mind because, in mundane language, wave and particle aspects are mutually

12 Unveiling the quantum

exclusive. Quantum laws however make them in a way ‘coexist’. This is what Bohr hascalled ‘complementarity’. This concept is difficult to comprehend and explaining it tothe layman does not go without danger. To combine waves and particles in an appar-ently hybrid entity seems to evoke the monsters, half-man half-beast, which appearin bas-reliefs of Romanesque churches. It may lead to the wrong idea that quantumphysics is spooky and ill-defined. This idea may also be reinforced by a misunderstoodinterpretation of expressions such as ‘quantum uncertainties’ or ‘quantum indetermi-nacy’. Nothing is more remote from quantum physics which, far from being fuzzy,gives an extraordinarily accurate description of Nature, imparting to the natural en-tities – nuclei, photons, atoms or molecules – a structural stability and universalitywhich cannot be explained by classical physics.

In order to access this description, one must abandon inadequate images and im-merse oneself in the mathematical framework, which is elegant and simple. The notionof state superposition, so fuzzy in classical terms, is merely associated to the linear-ity which ascribes to each state of a quantum system a vector in an abstract space.These vectors add up and combine according to simple rules of linear algebra. Oncethese rules have been defined, the theory describes without any ambiguity micro-scopic phenomena. The concepts of superposition, interference, complementarity andentanglement, which remain vague to the layman, are mere consequences of the math-ematical formalism. A new logic, at odds with the classical one, emerges in a perfectlycoherent manner. To understand it, one needs to make an effort of abstraction. More-over, the relationship between theory and observation is less direct than in classicalphysics. The quantum concepts are generally veiled by decoherence phenomena. Thecombination of abstraction and apparent remoteness between direct observation andtheory makes quantum physics difficult to teach at an elementary level. It explainsalso the difficulties encountered by the theory when it emerged and the psychologicalresistance it still has to overcome.

The quantum laws have been revealed through a difficult and long path followedthrough a lot of turns and pitfalls. To understand the microscopic world, physicistshave carried out a complex investigation job, led by a few hints obtained by pickingup pieces of a complex puzzle. The spectra of simple atoms, the spatial quantizationof the atomic spin, were among the evidence of a veiled quantum world, impossibleto interpret as long as physicists stuck to classical concepts. The truth has comeprogressively, through some brilliant intuitions. Among them are the revelation ofmatter waves by de Broglie in 1923 and the guess of the exclusion principle by Pauliin 1925. The mathematical formalism evoked above was then elaborated, in 1925–26,by Heisenberg, Schrödinger and Dirac.

But the formalism was not enough. To understand its implications was also essen-tial. To do so, physicists have repeatedly used thought experiments. They imaginedsimple physical situations in which one important parameter or set of parameters takesan unusual value. It is chosen to make a specific effect, usually negligible because itis masked by the complication of the real world, play a dominant role. Besides thisexaggeration, the ‘experiment’ must be, in all other respects, realistic and obey therules of physics. In this way, the ‘experimenters’ try to find out whether the laws ofNature, supposed to be universal, do apply to situations in which the relevant param-eters take these unusual values. By analysing the expected behaviour, they hunt for

Emergence of the microscopic world 13

Fig. 1.6 Scheme of Bohr’s thought experiment with a moving slit.

possible inconsistencies and contradictions.Before quantum physics, the thought experiment test had already been used suc-

cessfully by Einstein in the elaboration of relativity. His ‘experiments’ with trainsgoing at velocities close to the speed of light, or with elevators falling in a gravita-tional field, belong to this category. In the fast trains he imagined, all physics wasrealistic. The yardsticks and the clocks were those of everyday life. Only the velocityof the carriages was unusual. Einstein analysed all the consequences of his theory andconcluded that if trains were able to go that fast, strange things would happen. Thesestrange things were, however, perfectly logical and, hence, Einstein concluded thatNature must behave in this odd way.

The quantum thought experiments conceived by Bohr and Einstein during theirdiscussions at the Solvay meetings of 1927 and 1930 were of a different kind. Theyinvolved isolated particles, electrons, atoms or photons. The unusual parameter wasthe sensitivity of the experimental equipment used to manipulate and detect theseparticles. The sketch in Fig. 1.6, showing an electron crossing a Young double-slitapparatus, is again a drawing inspired by Bohr. The upper slit is here supposed to belight enough to recoil appreciably under the kick of the electron when it is scatteredby its edges. And the stiffness of the spring to which the upper slit frame is attachedis supposed to be so small that it starts oscillating with observable amplitude whenthe electron follows the upper path.

This apparatus was conceived for the discussion of complementarity and we willcome back to it later. At this stage, let us only note that the issue was to understandthe relationship between the fringe visibility and the amount of information acquiredabout the path of the electron by observing the recoil of the slit. At the time ofthe Solvay meeting, the experiment was clearly impossible in any imaginable form.However, Bohr was careful to describe the situation in otherwise realistic terms, asshown by the numerous bolts and dials he liked to represent half jokingly in hisdrawings.

The original blueprint of another Einstein–Bohr experiment is exposed in a win-dow case at the Niels Bohr Institute in Copenhagen, together with a ‘semi-realistic’rendition of the apparatus which Gamow, then a postdoctoral visitor in Denmark,

14 Unveiling the quantum

Fig. 1.7 The photon-box thought experiment. Left: sketch inspired by Bohr (1949). Right:

model realized by Gamow. The box carries a blackboard with the formulas used by Bohr to

interpret the experiment. Note, on top, the spring by which the box is suspended and the

horizontal pointer on the left side which, by moving in front of a graduated ruler, yields the

box weight. With permission from the Niels Bohr Archive.

made in wood and metal and offered to Bohr as a token of his admiration for histwo mentors (Fig. 1.7). Here, the point was to refute a suggestion by Einstein thatone could determine with arbitrary precision the emission time of a photon and itsenergy, in violation of one of the Heisenberg uncertainty relations. Einstein and Bohrenvisioned a machine intended to measure the time of escape of a photon from a boxand to determine simultaneously the photon energy, through the equivalence principle.This photon box is of course unrealistic because here again, the spring to which it isattached is, in the real world, insensitive to the gravitational mass of a single photon.Making the assumption that such a spring, or the equivalent of it, could exist, Bohrtried to analyse all the consequences in order to show the consistency of quantumtheory. We will not analyse here this thought experiment, which is not devoid of dif-ficulties. The fact that Bohr had to invoke general relativity to show the coherence ofquantum physics is odd if we remark that gravitation is not yet explained by quan-tum theory. We mention here this problem only as an example of the semi-realisticconsiderations leading to the elaboration of thought experiments.

1.3 Thought experiments coming of age

At the end of the twentieth century, the status of thought experiments has changed.Whereas they mainly served as a conceptual aid to theoretical thinking, they have, bybecoming real, turned into fundamental tests of the theory. Einstein and Bohr usedthought experiments to show that the laws of Nature were logically coherent. Modernexperimenters perform realistic versions of these experiments, which up to now havealways confirmed that Nature does indeed obey these laws. These experiments havebecome feasible thanks to the development of modern technologies unimaginable atthe time when the quantum concepts were elaborated. This technology is based on

Thought experiments coming of age 15

Fig. 1.8 Tracks produced by particles after an energetic collision. This picture is one of the

first pieces of evidence of the W boson. c© CERN. See also Arnison et al. (1983).

our knowledge of relativity and of the quantum, so in a deep way, our understandingof Nature is fully consistent.

Relativity thought experiments were the first to come of age. Of course trains, eventhe French TGV, still do not rush by platforms at about the speed of light c. Since the1930s though, accelerators routinely prepare trains of particles travelling close to thatspeed in high-vacuum tunnels. The natural lifetime of some of these particles providesan accurate clock. That a particle in the accelerator lives longer than at rest has beenchecked with precision, yielding confirmation of Einstein’s idea about the relativityof time. Similar tests have been carried out in low-energy physics too, with particleshaving velocities a thousand to a million times smaller than c. One takes advantagethen of the extreme precision of time measurement in modern atomic physics. The timedilation effect is measured by embarking atomic clocks in planes and satellites. Andsuch experiments could be performed in fast trains too, since the square of the ratio oftheir velocity to c is of the order of 10−12, well within the sensitivity of atomic clocks.Relativity thought experiments have even become part of routine technology. Thecorrection for time dilation and gravitational effects are essential to the operation ofthe Global Positioning System (GPS). Satellites also provide modern versions of free-falling elevators of the general relativity thought experiments. Here again relativitytests have been performed or are being planned to check various aspects of Einstein’stheory.

Quantum thought experiments have also become real. The first requisite to performthem is to achieve single-particle detection sensitivity. In accelerator experiments theparticles are detected by their tracks or by the energy they deposit in bolometers (Fig.1.8 shows particle tracks following a collision event). A low-energy photon counter alsodiscriminates the arrival of single light quanta, yielding discrete ‘clicks’. During thesedetection events, though, the particles are destroyed as they are being observed. Asremarked by Schrödinger, the particles detected in this way are fossil signatures ofthe past and the physicist works as a palaeontologist reconstructing past history fromexperimental records.

Genuine atomic or photonic thought experiments have more to them than verysensitive detection. They must be able not only to detect single particles, but to do itwithout destroying them. The particles should be manipulated with an exquisite sen-sitivity in such a way that their repeated observation should be possible, as opposedto the static observations mentioned above. Repeating the measurements gives accessto correlations of various kinds. Quantum physics, in essence, describes and explains

16 Unveiling the quantum

(a) (b) (c)

Fig. 1.9 Scanning tunnelling microscopy. (a) a scanning tunnelling microscope image of a

nickel surface. Each peak corresponds to a single nickel atom. Reprinted with permission

from IBM’s Almaden visualization laboratory Web site www.almadem.ibm.com. (b) a fa-

mous company’s logo written with xenon atoms deposited on a nickel surface. Reprinted

by permission from Macmillan Publishers Ltd: Nature, Eigler and Schweizer (1990). (c) a

‘quantum corral’. A ring of deposited iron atoms confine the electrons on a copper surface.

The electronic stationary wave is clearly visible inside the corral. Reprinted with permission

from Crommie et al. (1993), c© AAAS.

correlations between events. It is thus essential that the thought experiments testingit have the ability to perform repeated measurements on single particles. The physi-cist should then be compared not to the palaeontologist but rather to the biologistoperating in vivo.

A first example of single-atom observation and manipulation is provided by Scan-ning Tunnelling Microscopy (STM). It consists in translating above the surface of asolid a tiny stylus, whose tip is made of a few atoms only. A voltage difference isapplied across the small gap between this tip and the surface. When the stylus is closeenough, an electron current tunnels through this gap. The current intensity decreasesexponentially with the tip-to-surface distance which is controlled by piezoelectricaldevices. Moving the stylus across the surface while locking the tunnel current to afixed value forces it to follow the surface at a fixed distance, visualizing each atom asa bump above the average plane. One thus reconstructs a three-dimensional atomicmap resolving single atoms. The tip feels individual atoms in a process similar toBraille reading by the blind, the tiny stylus replacing the finger tip. An example of anatomic map is shown in Fig. 1.9(a).

Large electric fields applied to the tip are used, not only to read out, but to pickup atoms one by one and redeposit them in chosen locations. One can thus writeartificial atomic patterns. The three letters of Fig. 1.9(b) have been drawn in thisway. Written at the same scale, all the books of the British Library fit on the surfaceof a stamp. One can also build small enclosures made of a single-atom line, so-called‘quantum corrals’. The electronic density inside the corral area can be recorded bySTM methods (Fig. 1.9c). The pattern of electronic waves constrained by the corralboundary is strikingly reminiscent of the water waves at the surface of a pond. Thecorral is a few tens of angströms across. This is a typical example of the astonishingachievements of nanotechnology. In these experiments, atoms can also be continuouslymonitored. The motion of atomic lines on surfaces under various physical or chemicalprocesses can be observed in real time. The STM images and the ones based on arelated technology, the atomic force microscope, are typical examples of experiments

www.almadem.ibm.com

Thought experiments coming of age 17

(a) (b)

Fig. 1.10 (a) Images of calcium ions in a Paul trap. Top: a single ion. Botton: string of

five ions. Reprinted from Nägerl et al. (1998a), with kind permission of Springer Science

and Business Media and courtesy of R. Blatt. (b) Quantum jumps of a single ion in a trap.

Laser-induced fluorescence of a barium ion on the |6P 〉 → |6S〉 transition versus time. Thefluorescence is suddenly interrupted by transitions from |6S〉 to the metastable |5D〉 levelinduced by additional lamp irradiation. Reprinted with permission from Nagourney et al.

(1986). c© American Physical Society.

which, until the 1980s, were deemed impossible by most physicists. They rely onextraordinary improvements in tip micro-fabrication and computer control as well ason a deep knowledge of quantum tunnelling properties.

Even though STM manipulations can be performed on single atoms, they are toocrude to permit quantum correlation experiments. The atoms on a solid surface arestrongly interacting with each other and perturbed by thermal vibrations (phonons)and other collective effects. Under these conditions, decoherence is an effective processand classical concepts can be used to understand most phenomena, even if they occurat the atomic scale. STM lacks one characteristic feature of a quantum thought ex-periment, an extreme conceptual simplicity. A thought experiment, to be illustrativeand demonstrative, should be described by a simple model, emphasizing the physicalconcepts without the complications coming from the environment, which tend to veilthe quantum.

This condition is realized in the quantum optics experiments which emerged at thetime of the birth of STM, in the late 1970s. In one class of experiments, single atomsare observed in vivo and can be continuously manipulated. Instead of being closelypacked and strongly interacting with each other, they are isolated, living so to speakalone in vacuum, only weakly perturbed by neighbouring particles. In another kind ofquantum optics experiments, photons instead of atoms are continuously observed andmanipulated, again under conditions in which the perturbation from the surroundingis kept minimal.

Single-atom manipulation experiments were first performed in ion traps. Theseexperiments were a direct generalization to charged atoms of the pioneering studiesperformed by Dehmelt’s group in Seattle on a single electron, which had culminated inthe extremely precise measurement of the electron magnetic moment recalled above. Aconfiguration of static and oscillating electric fields (Paul trap), or of static electric andmagnetic fields (Penning trap) keeps an ion in a confined region of space, away from

18 Unveiling the quantum

Fig. 1.11 A cavity-QED experiment in the

microwave domain. Atoms effusing from the

oven O are prepared in a very excited Ry-

dberg state in box B, using laser L. They

interact with the photons in the supercon-

ducting cavity C before being detected by

the state-selective detector D.

OB

C

DL

all material boundaries. The restoring force acting on the charge of the ion inducesits oscillations around equilibrium, with an amplitude which can be reduced down tozero-point fluctuations. Laser beams are used to manipulate the ion, to cool down itsmotion in the trap, and to detect it.

Observing the ion is a process quite similar to ordinary seeing. The light scatteredby the particle is focused by a lens into a photon detector or simply into the eye. Itappears as a small diffraction-limited spot, about a micron in diameter, of course muchlarger than the actual ionic size. The ion has a huge scattering cross-section for themonochromatic laser light, roughly equal to the square of the radiation wavelength.It scatters millions of photons per second from the laser beam, thousands of whichare collected by the lens or the naked eye. The first direct ‘seeings’ of single ionswere made in 1980. Figure 1.10(a) shows more recent images of an isolated Ca+ ion(top) and of a string of five ions (bottom) confined in a trap. When many ions arepresent the trap confining potential competes with the Coulomb repulsion betweenthe charged particles. If the temperature of the ions is low enough, a quasi-crystallineorder is observed.

Not only can the position of the particles be detected in real time with micrometreresolution, but their internal evolution can also be monitored. The ion has electroniclevels and the quantum jumps between them can be observed using laser light again,as shown in Fig. 1.10(b). The ion scatters light between two levels separated by atransition resonant with the laser. If a quantum jump induced by a lamp brings theion from one of these two levels into a third one, the resonant scattering is suddenlyinterrupted and the ion becomes invisible. The scattered light comes back as suddenlyas it vanishes when another spontaneous quantum jump brings the ion back into oneof the two levels pertaining to the laser-resonant transition. These jumps appear asa random blinking analogous to the twinkling of a faint star. The telegraphic signalsof single ions in traps have become routine in atomic clocks and other high-resolutionspectroscopy experiments. These sudden jumps are a direct quantum manifestation,which Schrödinger believed could not be observed. They are a typical example of athought experiment becoming real.

In another kind of experiment, dual in some way of ion traps, the roles of matterand radiation are exchanged. It is the field instead of the atom which is confined.Photons are stored in a cavity with highly reflecting walls and atoms, sent one by one

Thought experiments coming of age 19

� � � � � � � � �

�

�

����

��

� � � � � � � � � � � � �

� � �

Fig. 1.12 Fourier transform of the time-dependent atomic signal detected after the resonant

interaction of a single Rydberg atom with a small field stored in a superconducting cavity.

The discrete peaks at frequencies proportional to the square root of successive integers reveal

directly the graininess of the photon field. See Chapter 5 for more explanations. Reprinted

with permission from Brune et al. (1996b). c© American Physical Society.

across the cavity, interact with them. These experiments belong to Cavity QuantumElectrodynamics, ‘CQED’ for short, a subfield of quantum optics which emerged inthe 1970s. Its primary goal was to study how the radiative properties of atoms aremodified when they radiate close to boundaries. CQED experiments are performedwith optical or microwave fields. In one microwave version (Fig. 1.11), photons withwavelength in the millimetre range are bouncing between two mirrors made of highlyreflecting superconducting material. A large matter–field coupling is obtained withatoms prepared in very excited Rydberg states, which have a huge electric dipoleand interact strongly with microwaves. The Rydberg atoms leaving the cavity aredetected by a state-selective field-ionization detector and the atomic signal, a functionof the atom–photon interaction time, is used to get information about the cavity fieldevolution.

Figure 1.12 shows the Fourier transform of such a signal obtained in an experimentdescribed and analysed in a forthcoming chapter. Suffice it to say at this stage that thediscrete frequency peaks directly reveal the graininess of the photon field. Each Fouriercomponent corresponds to a photon number, 0, 1, 2 or 3 in this case. The experimentis clear evidence of field quantization in a box, the first quantum effect guessed byPlanck, which turned out to be the starting point of the quantum revolution.

These CQED experiments are modern renditions of the Einstein–Bohr photon box.The photons are not directly weighed by a pointer attached to a scale, as was assumedby Bohr, but rather by the atoms which interact with them and carry away, imprintedon their own quantum state, information about the field. In this way, photons canbe counted without being destroyed, as one could count marbles in a box. Or thecavity field can be prepared into a strange superposition state, displaying at the sametime two different phases or amplitudes. Such a situation reminds us of the famous catthat Schrödinger assumed, in another celebrated thought experiment, to be coherentlysuspended between life and death.

A striking feature of the ion and photon trap experiments is the simplicity of themodel accounting for them. Basically, they can be understood by describing the field

20 Unveiling the quantum

and the atoms as interacting quantum oscillators and spins, the most basic quantumsystems. This is obvious in the case of CQED. Only the two atomic levels pertainingto the transition resonant with the cavity are relevant and the atom then behavesto a good approximation as a two-level spin-1/2-like system. The field in the cavitymode is a quantum oscillator whose elementary excitations are the photons. The cou-pling between the spin and the oscillator is described by a very simple Hamiltonian,introduced by Jaynes and Cummings in the early days of quantum optics.

The spin and quantum oscillator ingredients are also present in the ion traps. Inorder to understand the scattering of laser light by an ion, it is convenient to de-scribe it as a two-level system interacting with a classical field. When the trappedion is moving, its coupling to the laser light is modified by the Doppler shift, mod-ulated by the ion’s oscillation. The laser light then couples the electronic degree offreedom of the ion, seen as a two-level spin-like system, to its centre-of-mass motion,viewed as a mechanical quantum oscillator. This coupling is again, under convenientapproximations, described by a Jaynes–Cummings Hamiltonian. In both situations,the evolution of spins and oscillators coupled together is fully analytical. The system’sevolution is simple enough so that the quantum laws manifest themselves withoutinessential complications.

We have neglected so far the perturbing effect of the environment, responsible fordecoherence. The thought experiments with ions and photons require keeping theseperturbations to a minimum. This is achieved by insulating the trapped particles fromother atoms or photons with a combination of techniques: high vacuum to reduce therate of collisions with background molecules, low temperature to minimize thermalphotons, highly reflecting cavity mirrors to keep photon loss as small as possible. Fora realistic description of the experiment, the residual effect of the environment mustbe accounted for. Here again, the spin–oscillator model is helpful. The ion spontaneousemission is due to its coupling to a continuum of field modes described as quantumoscillators. Similarly, photons in a cavity can be modelled as escaping in a continuumof outside modes, each of which is also a quantum oscillator.

1.4 Aims and outline of this book

Experiments which display quantum features and can be modelled by spins and oscil-lators are more than textbook illustration of quantum theory. They are also an idealtesting ground for quantum information science. A two-level atom acts as a qubit, andthe same is true of a quantum oscillator if its evolution is restricted to the groundand first excited states (no more than one photon or phonon). Thought experimentswith atoms and photons in boxes or with ions in traps have thus naturally evolvedinto demonstrations of quantum information procedures. Engineering atom–atom oratom–photon entanglement in a deterministic way, realizing quantum logic operationsof various kinds and performing simple quantum algorithms has become a flourishingfield of research. One of our goals is to describe some of these experiments, with abalanced theoretical and experimental approach.

As a general rule, the simpler the theoretical model the experiment can be reducedto, the more complex the experimental procedure. A kind of ‘complementarity’ rulecan be formulated. The product of the theoretical and practical complexity of thoughtexperiments becoming real has to be greater and, unfortunately, often much greater

Aims and outline of this book 21

than some fixed value. If one tries to force the atoms or ions to behave as exact two-level objects and the photons or phonons to be perfect excitations of ideal oscillators,one has to struggle hard with experimental difficulties which, in real life, conspireto blur these simple pictures. If one accepts dealing with complications coming fromthe environment, the experiment becomes easier, but the theory has to include sub-tle refinements, and becomes more intricate and less directly related to fundamentalconcepts.

In practice, a compromise has to be reached and the physicist must deal withsome level of complexity, both in theory and in experiment. To do justice to thisfield of research, it is necessary to detail somewhat these two complementary aspects.This is a challenging task, which must avoid the pitfalls of being either too sketchyon experimental details, giving the impression that one is again merely imagining avirtual experiment, or being too technical, thus burying the physics under engineeringprecision. We have chosen here the middle ground between these two dangers. In thisway, we hope to convey the beauty and the amazement of the physics and to give, atthe same time, an understanding of the difficulties which must be overcome to unveilthe quantum, and hopefully to harness it for achieving information science goals.

Another challenge is to choose between the many experiments, whose number keptrapidly increasing as the writing of this book progressed. Here again, a decision hadto be made between detailing some selected experiments or else making an exhaustivecatalogue, at the price of being sketchy on details. In order to be consistent with ourmiddle-ground approach, which requires a precise discussion of both experiment andtheory, we have adopted the selective approach.

We have chosen to focus primarily on microwave CQED photon-box experiments.One reason for this choice is of course that these are the ones we know best, since theybelong to our own field of research. A more objective argument is that the Jaynes–Cummings model directly corresponds to the microwave CQED situation. At the timeit was introduced, it was a gross simplification of practical situations in laser physics.In CQED, it is however the quasi-exact description of the atom–field coupling. Theevolution of the atom–field system in a high-Q cavity can thus be described from firstprinciples and with precision. The unavoidable decoherence is also well-understood. Itis due to the damping of the field in the cavity and to the spontaneous emission of theatoms. Those are well-known relaxation processes, which can be described accuratelyand quantitatively.

The understanding of microwave CQED experiments can also be very useful toanalyse experiments belonging to other fields of quantum optics. The similarities ofthe Hamiltonians provide, for instance, a natural link between CQED and ion trapexperiments. We use this link to describe some of these experiments too, even thoughit will be done in a less detailed way. An incursion will finally be made into a noveland very active field, the physics of Bose–Einstein condensates, to which some of theconcepts of CQED can also be applied.

This book is also intended to give the reader an opportunity to learn and re-flect about the quantum. We will assume that (s)he is familiar with the basics ofthis physics, but might not have had the opportunity to think too much about allits conceptual aspects. The strange features of the quantum are often presented innegative terms (uncertainty relations, impossibility theorems such as the ‘no-cloning’

22 Unveiling the quantum

one, non-separability, non-locality and so on). Quantum information is trying to turnthese negative points into pluses, showing that, at least in principle, they should allowone to do more than what is possible with classical physics. Learning about thoughtexperiments coming of age should be a good opportunity to think about these some-what paradoxical issues and to learn about the nascent quantum information science.At a more basic level, the experiments we describe here provide interesting problemsfor intermediate and advanced quantum physics courses. Tools such as the densityoperator and its master equation, the Monte Carlo approach to relaxation processesand the Wigner function representation will be described in detail. We hope that thisvolume will thus be useful to teaching in advanced quantum mechanics.

We start in Chapter 2 with a discussion of the quantum ideas incompatible witha classical description of Nature. This exploration of the strange microscopic worldemphasizes the quantum concepts which are illustrated by the experiments describedin the following chapters. The principle of state superposition and its consequencesare analysed. The ubiquitous phenomenon of quantum interference is described in thelight of the complementarity principle. We discuss the symmetrization postulate forthe description of identical particles. We then analyse entanglement and non-locality,as it is revealed by Bell’s inequality experiments. We next show how decoherencecontributes to explaining the quantum–classical boundary and to shed light on theimportant issue of measurement in quantum physics. The chapter ends by a reviewof quantum information, which introduces the principles of quantum cryptography,teleportation and quantum computing.

Chapter 3 is a short self-consistent introduction to quantum optics, viewed as astory about field oscillators (or ‘springs’) coherently coupled with two-level atoms(‘spins’). We recall the quantum description of a harmonic oscillator and introducephoton creation and annihilation operators and field quadratures. The description ofa coherent state as the closest approximation to a classical field is recalled. We thenanalyse the coupling of two oscillators, which provides a simple model for the beam-splitter, a basic tool in quantum optics which we will encounter time and again in thisbook. We next recall the properties of two-level systems seen as pseudo-spins, sim-ple models for atoms interacting with a field mode. We describe how these spins aremanipulated and prepared in arbitrary quantum states by classical radiation pulses,according to methods borrowed from NMR. We show how a sequence of pulses realizesa Ramsey interferometer, an essential tool in many of the experiments described lateron. After introducing the Jaynes–Cummings Hamiltonian describing the coherent in-teraction of an atom with a field mode, we analyse the resonant exchange of energybetween the spin and the spring described by this Hamiltonian, as well as their disper-sive interaction when they are off-resonant. These resonant and dispersive couplingsplay essential roles in the experiments described in subsequent chapters.

We will have been dealing so far with ideal spins and springs, evolving in splen-did isolation. The atoms and fields of real experiments, though, are ‘open quantumsystems’, unavoidably interacting with their environment. This leads to irreversibledissipative processes, which we analyse in Chapter 4. An open system is described byits density operator, a concept which replaces the wave function of elementary quan-tum physics. Under reasonable assumptions, this operator obeys a ‘master equation’which describes the system’s damping under the perturbing effect of the environment.

Aims and outline of this book 23

Using a method inspired from quantum information, we derive the master equationby analysing how the open system loses information through its entanglement withits surroundings. This equation can be cast in a form making explicit the elemen-tary ‘quantum jumps’ experienced by the system as it keeps losing information. Wesolve the master equation by the powerful Monte Carlo method, imagining thoughtexperiments in which the environment is continuously monitored, thus detecting thequantum jumps undergone by the system. We illustrate these general ideas by describ-ing the spontaneous emission of a two-level atom and the relaxation of a field storedin a cavity. With this master equation formalism at hand, we revisit the spin–springsystem introduced in Chapter 3 and analyse the effect of cavity damping on the coher-ent atom–field evolution. We describe a damped cavity field interacting with a streamof atoms crossing one by one the cavity, a device known as a ‘micromaser’. Finally, weconsider the collective coupling of many atoms to a damped cavity mode, which leadsus to say a few words about the superradiance phenomenon.

Chapter 5 then presents the essentials of cavity QED. After a brief history of thesubject, we present the main ingredients of a microwave cavity QED experiment, theRydberg atoms and the superconducting cavity. We then describe the basic methodsused in these experiments. Ramsey interferometry performed by subjecting the atomsto classical pulses before and after they cross the cavity is shown to yield precise infor-mation about the atom–cavity coupling. An experiment in which an atom resonantlyexchanges energy with the field, undergoing a ‘Rabi oscillation’, is described. Thisexperiment yields in particular the Fourier transform signals mentioned above (Fig.1.12), which clearly demonstrate the photon ‘graininess’ of the field in the cavity. Theuse of the Rabi oscillation as a tool to entangle atoms to photons and atoms to atomsis described in detail, as well as coherent cavity-assisted collisions in which two atomsare entangled via their common interaction with the cavity field.

Chapter 6 is devoted to experiments performed with a few atoms and photons ina cavity. We show how the photon-box experiment can be turned into a direct testof complementarity, very close in principle to the original Bohr thought experimentwith a moving slit (Fig. 1.6). We also show how the cavity photons can be measuredand continuously monitored without being destroyed, a striking illustration at thesingle-particle level of what is called a ‘quantum non-demolition’ measuring process.We present a simple version of this procedure applied to a field containing at mostone photon. The measurement then reduces to a determination of the photon numberparity and can be viewed as the operation of a ‘quantum gate’ coupling conditionally aphotonic and an atomic qubit. We show that, in general, this parity measurement canbe turned into a direct experimental determination of the field Wigner distribution,yielding a complete description of the non-classical field features.

Chapter 7 is about field state superpositions involving many photons in a cavity.After a brief reminder about the Schrödinger cat problem in quantum optics, wedescribe how a single atom interacting with a field made of many photons can leaveits quantum imprint on this field, bringing it into a superposition of states with distinctclassical attributes. We describe the preparation and detection of these ‘Schrödingercat’ states. We present simple decoherence models, which account for their extremefragility. We describe experimental studies of decoherence, which constitute directexplorations of the quantum–classical boundary. We discuss limitations to the size of

24 Unveiling the quantum

these cat states and ways to protect them efficiently against decoherence. Finally, wedescribe proposals to generate and study non-local cats, superpositions of field statesdelocalised in two cavities.

In the last two chapters, we leave the photon box and analyse other systems bearingstrong similarities with it. Chapter 8 is devoted to ion trap physics. We show howions can be individually confined, manipulated and detected. We then present theHamiltonian describing the interaction of a trapped ion with laser beams resonantor nearly-resonant with an electronic transition of the ion. We compare it with theJaynes–Cummings Hamiltonian of CQED. We describe experiments which are theion-trap version of the Rabi oscillations observed in a photon box. We show how laserbeams can be tailored to create artificial environments for the ion vibration, making itpossible to generate and study various states of motion. We finally describe quantumlogic operations in which ions act as qubits and we show how these operations can becombined to perform simple quantum algorithms and to build entangled superpositionsof several ions. We conclude this chapter by discussing proposals to implement logicat a larger scale with trapped ions.

The experiments described so far will have dealt with systems in which the com-plexity is built ‘from the bottom’, starting by the control of individual particles andlearning how to manipulate more and more of them in a coherent way. One can startinstead from a large system and try, by cooling it down, to reduce or suppress de-coherence, making the quantum behaviour appear ‘from the top’. This is the aim ofmesoscopic physics, a very active domain of condensed matter physics. We cannotdo justice to this wide field, but Chapter 9, dealing with Bose–Einstein condensates,makes a brief incursion into it. Bose–Einstein condensation is indeed a domain of re-search in which atomic and condensed matter physics meet. Ultra-cold atomic samplesconfined in a magnetic trap behave as giant matter waves bearing strong similaritieswith laser or microwave light fields. The collective behaviour of atoms in these matterwaves is also reminiscent of the physics of superfluidity or superconductivity. Manyanalogies relate the atoms in a Bose–Einstein condensate either to photons in a cavity,or to particles in a liquid or a solid. Some of the ideas of CQED can be generalized andadapted. Quantum logic operations can be performed with Bose–Einstein condensatestrapped in optical periodic potentials. Condensates involving two or more differentmatter wave modes can be combined by using tools reminiscent of the beam-splitterof quantum optics or the Josephson junction of solid state physics. Mesoscopic super-positions of matter waves containing large numbers of atoms could be prepared andstudied. These matter Schrödinger cats bear strong similarities with their photoniccousins of CQED.

The book ends with a short conclusion and a technical appendix about the descrip-tion of quantum states in phase space. In the conclusion, we evoke future prospects andmentioning the challenges ahead. How far will the industry of thought experiments becarried and the quantum classical boundary pushed back? What are the odds to beatdecoherence and make quantum information practical? What are the best systems toachieve these goals? Should they be reached from bottom-up, as in atomic physics, orfrom top-down, as in condensed matter physics? These are some of the open issues webriefly touch on in these final remarks.

2

Strangeness and power of thequantum

No one really understands quantum physics

R. Feynman

The epigraph to this chapter underlines the counter-intuitive nature of the laws whichrule the world at the microscopic scale. Paradoxically, these laws are expressed by amathematical formalism that undergraduate students can easily comprehend. Physi-cists have been using this formalism successfully without generally attempting to‘understand it’ in the way that Feynman had in mind. For a long time it was notwell-considered to ponder about the weirdness of the quantum. Those who venturedinto interpretations of the quantum postulates were often deemed to be lost for sci-ence. This perception has changed with the development of quantum information andthe advent of ‘thought experiments’ manipulating single particles in the laboratory.The strangeness of physics at the microscopic scale plays a central role in this newdomain, since its main goal is to take advantage of this strangeness for novel tasksin communication and computing. Rather than trying to ‘understand’ the quantumweirdness, physicists now work at defining and quantifying it. They study, theoret-ically and experimentally, entanglement, non-locality and decoherence in systems ofincreasing complexity. Meanwhile, they acquire a kind of operational familiarity withthese concepts. They develop a new kind of intuition, which allows them to guess theresult of an experiment before performing it or even simulating it by a calculation.Whether such quantum intuition is different from ‘understanding’ is a question whichwe leave to philosophers.

The aim of this chapter is to describe the non-classical aspects of the quantumtheory, which are essential in the experiments described later on. It is destined pri-marily for readers familiar with the quantum formalism, but who might have learnedit in textbooks which mainly apply this formalism to solve practical problems, withoutdwelling too much on conceptual discussions. In order to avoid the pitfalls of every-day language, we will base from the outset our discussion on the supposedly knownframework of the theory. Only after the mathematics has been made explicit, will wediscuss the concepts in qualitative terms.

26 Strangeness and power of the quantum

The strangeness of the quantum can be traced back to the superposition principlerooted in the linearity of quantum theory. We recall in the first section (2.1) howthis principle conditions the properties of the wave function of a quantum particle.We then discuss the ubiquitous quantum interference phenomenon and its relationswith complementarity (Section 2.2). The next section (2.3) is about the subtletiesof the superposition principle when applied to a system of identical particles. Weshow that the very notion of identity leads to deep consequences in the quantumrealm and we give a brief description of fermionic and bosonic systems. The lastthree sections relate the superposition principle to entanglement, non-locality andquantum information, our central topics. We show in Section 2.4 that entanglement isa direct consequence of the superposition principle applied to composite systems. Weana