It is evident from the previous chapter that certain quantities that appear continuous at the macroscopic scale does actually exist in a discrete form at the microscopic scale. So, quantization of these entities makes sense and reveals a plethora of information at such length scales. In this book, our key concern lies with the properties of light, and rightfully so, we will explore different states of the optical radiation field. We will learn that light exists in many states that makes it so unique. Experimental evidence shows that light behaves as a particle at the microscopic scale (subject to specific conditions), although it behaves like a wave at the macroscopic scale. In this chapter, we will dwell deep and discuss many characteristics related to the particle nature of light. Specifically, we will consider two important states of light, the Fock and coherent states. These states can be regarded as the most fundamental states of light that enable complex interpretation of quantum features in the radiation field.
It is evident from the previous chapter that certain quantities that appear continuous at the macroscopic scale does actually exist in a discrete form at the microscopic scale. So, quantization of these entities makes sense and reveals a plethora of information at such length scales. In this book, our key concern lies with the properties of light, and rightfully so, we will explore different states of the optical radiation field. We will learn that light exists in many states that makes it so unique. Experimental evidence shows that light behaves as a particle at the microscopic scale (subject to specific conditions), although it behaves like a wave at the macroscopic scale. In this chapter, we will dwell deep and discuss many characteristics related to the particle nature of light. Specifically, we will consider two important states of light, the Fock and coherent states. These states can be regarded as the most fundamental states of light that enable complex interpretation of quantum features in the radiation field.
Light has long been perceived as a wave, or more specifically as an EM wave, and this is valid with respect to everyday observations. However, experiments involving light have produced surprising results that fall outside the realm of the wave nature of light. These experiments for the first time necessitated that light not be perceived as a wave but more like a particle. Some of these observations include photoelectric, Compton, and two-photon effects. In these experiments, light behaves more as an object filling a space just like a particle rather than a continuous entity like a wave. This is surprising given what we have learned in classical optics. In this chapter, we intend to analyze and understand the particle nature of light and its consequences.
Chapter 6 begins by transforming the optical wave itself into a quantized form, and the full implications of the wave‐particle duality of Chapter 2 become clear. The electric field may be pictured as an infinite ladder of discrete states and this structure allows optical interactions to be described photon by photon. The origin of spontaneous emission emerges from this treatment as a transition process that is stimulated by electric field fluctuations of the vacuum, and Weisskopf–Wigner theory explains why it is irreversible. The subject of coherent states is introduced and intriguing possibilities outlined regarding the noise properties of components of these states that form different quadratures in terms of the optical phase. Next, statistical analysis of light fields is considered to serve as a basis for more quantitative comparisons of fields and their noise properties. Methods of calculating the first and second degrees of coherence are presented, and the ranges of values corresponding to “classical” versus “quantum” light sources are discussed. The reduced density matrix is developed to justify analysis that focuses on only certain aspects of optical dynamics while ignoring others, as an important tool for simplifying complex problems. Finally, in view of its historical importance, the problem of calculating the fluorescence spectrum of resonantly excited two‐level atoms is considered using both a method of moments and dressed atom theory.
Understanding the particle nature of light and its consequences requires quantization of light. Quantization is a process of transforming a continuous system to a discrete system. In physical sciences, quantization is carried out for entities and quantities that are inherently discrete in nature. This refers to the graininess in the actual physical state (here, electromagnetic field) whose macroscopic ensemble manifestation (wave nature) is a large-number approximation. The microscopic grainy dynamics gives rise to delicate and tiny perturbations at the level of a single grain that sum up to produce large effects at the macroscopic level (continuous domain). Thus, quantization describes the physical state at the scale of a single grain while the continuous domain describes the average dynamics (macroscopic manifestation), which wipes out the properties of single grains. As an example, quantization of an electromagnetic field yields grainy particles called photons, the existence of which has been verified experimentally. These grains have well-defined quantized quantities such as energy. In this chapter, we discuss the grainy nature of light and its quantized quantities (energy, angular momentum, etc.) associated with the grains (photons) of light.
Classically, light is an electromagnetic phenomenon, described by Maxwell's equations. However, under certain conditions, such as low intensity or in the presence of certain nonlinear optical materials, light starts to behave differently, and we have to construct a ‘quantum theory of light’. We can exploit this quantum behaviour of light for quantum information processing, which is the subject of this book. In this chapter, we develop the quantum theory of the free electromagnetic quantum field. This means that we do not yet consider the interaction between light and matter; we postpone that to Chapter 7. We start from first principles, using the canonical quantization procedure in the Coulomb gauge: we derive the field equations of motion from the classical Lagrangian density for the vector potential, and promote the field and its canonical momentum to operators and impose the canonical commutation relations. This will lead to the well-known creation and annihilation operators, and ultimately to the concept of the photon. After quantization of the free electromagnetic field we consider transformations of the mode functions of the field. We will demonstrate the intimate relation between these linear mode transformations and the effect of beam splitters, phase shifters, and polarization rotations, and show how they naturally give rise to the concept of squeezing. Finally, we introduce coherent and squeezed states.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]
Have you ever tried to define light? If so, your definition is probably related to vision, the ability to see things. The Merriam-Webster English dictionary defines light as “something that makes vision possible”. However, the definition of the “nature of light” is a complex term already discussed by philosophers of Ancient Greece. In the 17th century, the debate on the matter focused on whether light was a particle or a wave, based on the different properties of light that had been gradually discovered over time. Isaac Newton (1642−1727) was a defender of the corpuscular theory of light, which considered light to be formed by particles, i.e., small pieces of matter, like dust or grains of sand. He was the author of Opticks (1706), the front cover of which appears in Fig. 1.1. Christian Huygens (1629–1695), on the other hand, supported the theory that light is a wave. These are two completely different definitions of the same physical concept. A range of experiments were developed to attempt to clarify this dilemma, but it was only until 1900 when Max Planck (1858-1947) introduced the basis of a theory that would go on to revolutionize scientific thinking, marking the beginning of modern physics: quantum theory. Developed during the 20th century thanks to advances made by scientists such as Niels Bohr, Born, Heisenberg, Schrödinger, Pauli, Dirac, Einstein or de Broglie, quantum theory clarified that light is neither a particle nor a wave, but that it has a dual nature: it behaves like a wave as it spreads but like a particle in its interactions with matter. In this book, we will specify where necessary whether light behaves as a particle or as a wave, although in most cases we refer to rays of light, which always represent the direction of propagation of light.