Abstract The present chapter is devoted to a description of the procedure by which quantum mechanics is applied to the electromagnetic field. We obtain expressions for the operators that represent the field observables and we examine some of their properties. The quantization introduces characteristic quantum-mechanical effects into the properties of the radiation field, for example field operators that do not commute with each other and thus represent incompatible measured quantities. The Hamiltonian of the coupled radiation-atomic system is expressed wholly in terms of quantum-mechanical operators. We show that the full quantum theory of the interaction between the radiation field and atoms produces the same transition rates for absorption and stimulated emission as does the semiclassical theory. The rate of spontaneous emission confirms that obtained from the phenomenological Einstein theory. The treatment of absorption of radiation by atoms also identifies the quantum-mechanical operator that represents a measurement of the energy or intensity of a light beam, and this has important consequences for the descriptions of the experiments that can be performed in practice.
Abstract The present chapter is devoted to a description of the procedure by which quantum mechanics is applied to the electromagnetic field. We obtain expressions for the operators that represent the field observables and we examine some of their properties. The quantization introduces characteristic quantum-mechanical effects into the properties of the radiation field, for example field operators that do not commute with each other and thus represent incompatible measured quantities. The Hamiltonian of the coupled radiation-atomic system is expressed wholly in terms of quantum-mechanical operators. We show that the full quantum theory of the interaction between the radiation field and atoms produces the same transition rates for absorption and stimulated emission as does the semiclassical theory. The rate of spontaneous emission confirms that obtained from the phenomenological Einstein theory. The treatment of absorption of radiation by atoms also identifies the quantum-mechanical operator that represents a measurement of the energy or intensity of a light beam, and this has important consequences for the descriptions of the experiments that can be performed in practice.
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.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
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 quantum electrodynamic theory, based on the quantization of the electromagnetic field, introduces the concept of photon and the related photon creation operator, photon annihilation operator, and photon number operator. It is used to describe stimulated Raman scattering, stimulated Kerr scattering, and various multiphoton excitation processes. Within this theoretical regime the light (as a photon field) and molecules are treated as a combined quantum mechanical system. The light emission or absorption is always accompanied by the quantum transition of the whole system from its initial eigenstate to the final eigenstate. The molecular emission or absorption probability rate is determined by the corresponding transition matrix elements of the interaction Hamiltonian operator.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
In this chapter a quantum mechanical treatment of atomic response to the fields of an electromagnetic wave is presented. Absorption and emission of energy by atoms involves transition between their quantized energy levels driven by an interaction Hamiltonian. Schrodinger equation is used to derive the equations of motion for the density matrix of the atom, which is better suited to include the effects of mechanisms other than the interaction Hamiltonian – collisions, level decays, level shifts, etc. that also affect the state of the atom. Using the density matrix equation of motion and focusing on two atomic levels, induced dipole moment for a single two-level atom and polarization and susceptibility for a collection of two-level atoms, including the effects of level decays, and collisions, are calculated. Equations of motion for level populations are also derived. These expressions naturally predict stimulated emission as well as absorption. Expressed in terms of spontaneous emission rate they confirm the validity of equations derived in Chapter 5 using the dipole oscillator model of atomic response. The limit in which quantum mechanical equations for atomic response reduce to the classical dipole oscillator equations is also discussed.