Abstract We develop the method of canonical quantisation for the case of the free electromagnetic field. We choose the Coulomb gauge, which has a simpler physical interpretation. We introduce the creation and annihilation operators in this framework. The formalism is applied to the problem of spontaneous emission of radiation from an excited atomic state at first order in the perturbation expansion. This allows us to obtain a concrete physical result, namely the computation of an excited state decay rate, and, at the same time, have a first look at abstract concepts, such as gauge invariance and renormalisation.
Abstract We develop the method of canonical quantisation for the case of the free electromagnetic field. We choose the Coulomb gauge, which has a simpler physical interpretation. We introduce the creation and annihilation operators in this framework. The formalism is applied to the problem of spontaneous emission of radiation from an excited atomic state at first order in the perturbation expansion. This allows us to obtain a concrete physical result, namely the computation of an excited state decay rate, and, at the same time, have a first look at abstract concepts, such as gauge invariance and renormalisation.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
In atomic, nuclear and particle physics we come across several situations where particles or radiation quanta (photons) are created or destroyed. For example in the beta decay of the neutron, a neutron is annihilated and a proton, electron and antineutrino are created. In some situations the interaction of radiation with matter can result in creation or annihilation of a pair of electron and positron. Such processes cannot be understood on the basis of (non-relativistic or relativistic) single particle wave equations because in such theories the probability ψ∗ψdτ (of finding the particle in the region of space dτ) when integrated over the whole space, results in unity at all times and so the existence of the particle is always guaranteed. Hence in the framework of single particle wave equation it is not possible to understand the creation or annihilation of particles. Even a simple process in which an atom undergoes a transition from an excited to the ground state by spontaneous emission of a photon in the absence of a perturbing electromagnetic field cannot be explained within this framework.
In this chapter, we discuss a purely quantum approach to the interaction between an atom and the electromagnetic field. In this treatment, the atom and electromagnetic field form a single quantum system, whose evolution is handled globally within a unified formalism. This will thus have the merit of being fully consistent from the theoretical standpoint. But the main advantage in an entirely quantum approach is that it can treat the full range of matter–radiation interaction phenomena. In particular, it provides a rigorous description of the spontaneous emission of light by an excited atom , something that falls outside the scope of the semi-classical framework applied in Chapters 2 and 3, where the lifetime of an excited atomic state had to be fed in phenomenologically. It can describe other phenomena of the same type, such as parametric fluorescence by a nonlinear crystal subjected to pumping radiation (see Chapter 7), which underlies many recent developments in quantum optics. The fully quantum approach also has the merit of allowing a simple interpretation in terms of photons for the various matter–radiation interaction processes, such as absorption, stimulated emission, scattering, and also the basic processes of nonlinear optics. Indeed, it provides a unified framework for both stimulated and spontaneous processes. Finally, it can be used to tackle completely new situations where matter and radiation interact, which lie outside the scope of any semi-classical description, such as cavity quantum electrodynamics or the production of single-photon or entangled-photons states .
Abstract The chapter first develops the wave equation in a rectangular cavity and gives the definition and number of density-of-modes within it. Prescription for converting the classical expressions of the EM field to quantized forms in terms of creation and annihilation operators is then outlined. After introducing the concept of photons, the interactions of an atomic system with photons are described in terms of the interaction Hamiltonian. The time-dependent perturbation theory, using the Fermi Golden Rule and the dipole approximation is employed to obtain expressions for the absorption and emission rates. The theory is then applied to the case of a two-level atomic system to obtain the expression for Einstein’s A-coefficient. Broadening of emission line is pointed out and the oscillator strength is defined.
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.
Many processes, including absorption and stimulated emission occurring in lasers, can be handled using a semi-classical model for the atom–radiation interaction, in which the matter is given a quantum description, but the radiation is represented as a classical electromagnetic field (see Chapter 2). There are other phenomena that cannot be adequately described without quantizing the radiation. For example, it has been known since the 1930s that spontaneous emission can only be treated correctly using a fully quantum framework for the interaction, in which both the matter and the radiation are quantized, as we shall see in Chapter 6. However, it was not until the 1970s that situations were found in which a free electromagnetic field , far from sources, exhibited properties and behaviour that could not be described by a classical field, but which could be perfectly well interpreted in terms of a quantized field. This chapter is devoted to the quantization of the free electromagnetic field, far from the charges and currents sourcing it. This free electromagnetic field will be called radiation , and in Chapter 6 we shall specify exactly what is meant by radiation when sources are present. The canonical quantization procedure used here starts from a description of the classical dynamics of the field in the framework of the Hamiltonian formalism, the basic features of which are discussed in Section 4.1.