The Quantum World

The following sections are included:

• Preface

• Foreword

• Contents

At the end of the XIX^th century, physics was dominated by two main theories: classical (or Newtonian) mechanics and electromagnetism. To be entirely correct, we should add thermodynamics, which seemed to be grounded on different principles, but whose links with mechanics were progressively better understood thanks to the work of Maxwell and Boltzmann, among others. Classical mechanics, born with Galileo and Newton, claimed to explain the motion of lumps of matter under the action of forces. The paradigm for a lump of matter is a particle, or a corpuscle, which one can intuitively think of as a billiard ball of tiny dimensions, and which will be dubbed a micro-billiard ball in what follows. The second main component of XIX^th century physics, electromagnetism, is a theory of the electric and magnetic fields and also of optics, thanks to the synthesis between electromagnetism and optics performed by Maxwell, who understood that light waves are nothing other than a particular case of electromagnetic waves. We had, on the one hand, a mechanical theory where matter exhibiting a discrete character (particles) was carried along well localized trajectories and, on the other hand, a wave theory describing continuous phenomena which did not involve transport of matter. The two theories addressed different domains, the only obvious link being the law giving the force on a charged particle submitted to an electromagnetic field, or Lorentz force. In 1905, Einstein put an end to this dichotomic wave/particle view and launched two revolutions of physics: special relativity and quantum physics. First, he showed that Newton’s equations of motion must be modified when the particle velocities are not negligible with respect to that of light: this is the special relativity revolution, which introduces in mechanics a quantity characteristic of optics, the velocity of light. However, this is an aspect of the Einsteinian revolution which will not interest us directly, with the exception of Chapter 7. Then Einstein introduced the particle aspect of light: in modern language, he introduced the quantum properties of the electromagnetic field, epitomized by the concept of photon.

After briefly recalling the main properties of waves in classical physics, this chapter will lead us to the heart of the quantum world, elaborating on an example which is studied in some detail, the Mach–Zehnder interferometer. This apparatus is widely used today in physics laboratories, but we shall limit ourselves to a schematic description, at the level of what my experimental colleagues would call “a theorist’s version of an interferometer”.

The first practical application of quantum physics examined in this book is quantum cryptography. Quantum cryptography is a relatively recent invention (it dates back from the mid 1980s) but I chose it because it allows me to illustrate the fundamental principles with a minimum number of intermediate steps. I shall begin with a short summary of classical cryptography, reviewing briefly the two systems which are currently used today: the secret key system and the public key system. Quantum cryptography is not a new method for dissimulating the meaning of a message, but it allows one to be certain that no spy has accessed it. There exist many quantum cryptography protocols and various experimental devices have been proposed for implementing them. The simplest device is based on polarization, a concept which will be introduced first in the case of of light polarization, and then in that of photon polarization. The use of photon polarization gives the simplest implementation of the protocol proposed in 1984 by Bennett and Brassard, which is known by the acronym formed with their initials, the BB84 protocol.

The final form of quantum physics, in the particular case of wave mechanics, was established in the years 1925–1927 by Heisenberg, Schrödinger, Born and others, but the synthesis was the work of Bohr who gave an epistemological interpretation of all the technicalities built up over those years; this interpretation will be examined briefly in Chapter 10. Although Einstein acknowledged the success of quantum mechanics in atomic, molecular and solid state physics, he disagreed deeply with Bohr’s interpretation. For many years, he tried to find flaws in the formulation of quantum theory as it had been more or less accepted by a large majority of physicists, but his objections were brushed away by Bohr. However, in an article published in 1935 with Podolsky and Rosen, universally known under the acronym EPR, Einstein thought he had identified a difficulty in the by then standard interpretation. Bohr’s obscure, and in part beyond the point, answer showed that Einstein had hit a sensitive target. Nevertheless, until 1964, the so-called Bohr–Einstein debate stayed uniquely on a philosophical level, and it was actually forgotten by most physicists, as the few of them aware of it thought it had no practical implication. In 1964, the Northern Irish physicist John Bell realized that the assumptions contained in the EPR article could be tested experimentally. These assumptions led to inequalities, the Bell inequalities, which were in contradiction with quantum mechanical predictions: as we shall see later on, it is extremely likely that the assumptions of the EPR article are not consistent with experiment, which, on the contrary, vindicates the predictions of quantum physics. In Section 3.2, the origin of Bell’s inequalities will be explained with an intuitive example, then they will be compared with the predictions of quantum theory in Section 3.3, and finally their experimental status will be reviewed in Section 3.4.

The debate between Bohr and Einstein goes much beyond a simple controversy, which is after all almost eighty years old and has been settled today. In fact, the concept introduced in this debate, that of entanglement, lies at the heart of many very important developments of modern quantum physics, in particular all those linked to quantum information (Chapter 8). Moreover, we shall see that the phenomenon of non-local correlations compels us to revise in depth our space-time representation of quantum processes. These are the two reasons why a whole chapter is devoted to this debate.

Up to now, the spatial properties of quantum particles played no more than a secondary role: we only needed the de Broglie relation (1.4) which gives the quantum particles wavelength, and our discussion of the quantum properties of photons was based mainly on their polarization, which is an internal degree of freedom of the photon. The probability amplitudes which we used did not involve the positions or velocities of the particles, which are spatial, or external degrees of freedom. In the present chapter, we shall introduce spatial dependence by defining probability amplitudes that are functions of the position . In full generality, is a complex number, but we shall avoid this complication and discuss only cases where the probability amplitudes may be taken real. For simplicity, we also limit ourselves to particles propagating along a straight line, which we take as the Ox axis: x will define the position of the particle and the corresponding probability amplitude will be a function of x, a(x). In our discussion, we shall need to introduce the so-called potential well, where a particle travels back and forth between two points on the straight line. One important particular case is the infinite well, where the particle is confined between two infinitely high walls over which it cannot pass. This example is not at all academic, and we shall meet it again in Chapter 6 when explaining the design of a laser diode! Furthermore, it will allow us to introduce the notion of energy level, to write down the Heisenberg inequalities, to understand the interaction of a light wave with an atom and finally to explain schematically the principles of the laser.

This chapter and the following one address collective effects of quantum particles, that is, the effects which are observed when we put together a large number of identical particles, for example, electrons, helium-4 or rubidium-85 atoms. We shall see that quantum particles can be classified into two categories, bosons and fermions, whose collective behavior is radically different. Bosons have a tendency to pile up in the same quantum state, while fermions have a tendency to avoid each other. We say that bosons and fermions obey two different quantum statistics, the Bose–Einstein and the Fermi–Dirac statistics, respectively. Temperature is a collective effect, and in Section 5.1 we shall explain the concept of absolute temperature and its relation to the average kinetic energy of molecules. We shall describe in Section 5.2 how we can cool atoms down thanks to the Doppler effect, and explain how cold atoms can be used to improve the accuracy of atomic clocks by a factor of about 100. The effects of quantum statistics are prominent at low temperatures, and atom cooling will be used to obtain Bose–Einstein condensates at low enough temperatures, when the atoms are bosons.

In this chapter, we move on to the case of fermions, and we shall find out that fermions are no less interesting than bosons! In practice, electrons are the most important example of fermions, because they are responsible for electrical conductivity in metals and semiconductors. It is impossible to understand a phenomenon as familiar as electrical conductivity without appealing to quantum physics. Two properties play a fundamental role: first the propagation of electron waves in crystal lattices, and second the Pauli exclusion principle, which is a consequence of the fermionic character of the electrons. In Section 6.1, we introduce electron wave propagation in crystals which gives rise to the phenomenon of energy bands, and we describe the filling of these bands according to the Pauli principle. These results will be used in Section 6.2 to describe the electronic properties of semiconductors, on which almost all our modern technology (laser diodes, optical fiber communication, computers, smartphones and so forth) is grounded. Finally, in Sections 6.3 and 6.4, we shall describe the principles of light emitting diodes (LEDs) and laser diodes.

Up to now, we have only considered “non-relativistic” systems, where the velocities of massive particles such as electrons, protons, atoms, and so forth, are small with respect to the speed of light, c. Being massless, photons travel, of course, at the speed of light. This chapter is going to combine quantum physics with relativity, at first with special relativity (Einstein 1905) in Sections 7.1 and 7.2, and then in Section 7.3 with general relativity (Einstein 1915). In the first case, we obtain relativistic quantum field theory which is needed for elementary particle physics, and the first two sections will describe the present status of the theory which is known as the standard model of elementary particle physics, the standard model for short. The second case is that of quantum gravity and, with the example of length measurement, we shall try to understand the difficulties which this theory encounters and which have not yet been overcome today.

In everyday life, practically all the information which is processed, exchanged or stored is coded in the form of discrete entities called bits, which take two values only, by convention 0 and 1. With the present technology for computers and optical fibers, bits are carried by electrical currents and electromagnetic waves corresponding to macroscopic fluxes of electrons and photons, and they are stored in memories of various kinds, for example, magnetic memories. Although quantum physics is the basic physics which underlies the operation of a transistor (Chapter 6) or of a laser (Chapter 4), each exchanged or processed bit corresponds to a large number of elementary quantum systems, and its behavior can be described classically due to the strong interaction with the environment (Chapter 9). For about thirty years, physicists have learned to manipulate with great accuracy individual quantum systems: photons, electrons, neutrons, atoms, and so forth, which opens the way to using two-state quantum systems, such as the polarization states of a photon (Chapter 2) or the two energy levels of an atom or an ion (Chapter 4) in order to process, exchange or store information. In § 2.3.2, we used the two polarization states of a photon, vertical (V) and horizontal (H), to represent the values 0 and 1 of a bit and to exchange information. In what follows, it will be convenient to use Dirac’s notation (see Appendix A.2.2 for more details), where a vertical polarization state is denoted by |V〉 or |0〉 and a horizontal one by |H〉 or |1〉, while a state with arbitrary polarization will be denoted by |ψ〉. The polarization states of a photon give one possible realization of a quantum bit, or for short a qubit. Thanks to the properties of quantum physics, quantum computers using qubits, if they ever exist, would outperform classical computers for some specific, but very important, problems.

In Sections 8.1 and 8.2, we describe some typical quantum algorithms and, in order to do so, we shall not be able to avoid some technical developments. However, these two sections may be skipped in a first reading, as they are not necessary for understanding the more general considerations of Sections 8.3 and 8.4.

It is generally assumed that the fundamental laws of physics are those of quantum physics, and that classical physics is only an approximation valid under certain conditions. However, the passage from quantum-to-classical and its conditions of validity are still hotly debated today. We may, for example, ask the following question: in a quantum world, particles exhibit interference phenomena. Why is it that interferences are never observed in a classical world except, of course, with classical waves? Equivalently, why is it that we do not see linear superpositions of macroscopically distinguishable states? A possible answer relies on the concept of decoherence. Although implicit in previous work, this concept was introduced explicitly at the beginning of the 1980s, and after a rather modest start, it has become more and more popular in domains like quantum information (Section 8.4) and quantum measurement (Section 10.3), where it plays a major role. After having explained the concept of decoherence with the help of an elementary example, we shall describe experiments where decoherence is manifest at mesoscopic scales, intermediate between microscopic and macroscopic scales, that is, on the order of a few μm.

Although nobody can question the practical efficiency of quantum mechanics, there remains the serious question of its interpretation. As Valerio Scarani puts it, “We do not feel at ease with the indistinguishability principle (that is, the superposition principle) and some of its consequences.” Indeed, this principle which pervades the quantum world is in stark contradiction with our everyday experience. From the very beginning of quantum mechanics, a number of physicists—but not the majority of them!—have asked the question of its “interpretation”. One may simply deny that there is a problem: according to proponents of the minimalist interpretation, quantum mechanics is self-sufficient and needs no interpretation. The point of view held by a majority of physicists, that of the Copenhagen interpretation, will be examined in Section 10.1. The crux of the problem lies in the status of the state vector introduced in the preceding chapter to describe a quantum system, which is no more than a symbolic representation for the Copenhagen school of thought. Conversely, one may try to attribute some “external reality” to this state vector, that is, a correspondence between the mathematical description and the physical reality. In this latter case, it is the measurement problem which is brought to the fore. In 1932, von Neumann was first to propose a global approach, in an attempt to build a purely quantum theory of measurement examined in Section 10.2. This theory still underlies modern approaches, among them those grounded on decoherence theory, or on the macroscopic character of the measuring apparatus: see Section 10.3. Finally, there are non-standard interpretations such as Everett’s many worlds theory or the hidden variables theory of de Broglie and Bohm (Section 10.4). Note, however, that this variety of interpretations has no bearing whatsoever on the practical use of quantum mechanics. There is no controversy on the way we should use quantum mechanics!

The following sections are included:

• Appendices
  • Units
  • Electromagnetic waves and protons
  • Trigonometric functions and complex numbers
  • Theory of the Mach–Zehnder interferometer
  • RSA encryption
  • Mathematical description of polarization
  • Attenuated light
  • Proof of Bell’s inequality
  • Entangled states
  • The formalism of quantum theory
  • A quantum particle in a one-dimensional box
  • The de Broglie–Bohm theory

• References

• Index