We introduce quantum optical dipole radiation fields defined in terms of photon creation and annihilation operators.These fields are identified through their spatial dependence, as the components of the total fields that survive infinitely far from the dipole source.We use these radiation fields to perturbatively evaluate the electromagnetic radiated energy-flux of the excited dipole.Our results indicate that the standard interpretation of a bare atom surrounded by a localised virtual photon cloud, is difficult to sustain, because the radiated energy-flux surviving infinitely far from the source contains virtual contributions.It follows that there is a clear distinction to be made between a radiative photon defined in terms of the radiation fields and a real photon, whose identification depends on whether or not a given process conserves the free energy.This free energy is represented by the difference between the total dipole-field Hamiltonian and its interaction component.
We introduce quantum optical dipole radiation fields defined in terms of photon creation and annihilation operators.These fields are identified through their spatial dependence, as the components of the total fields that survive infinitely far from the dipole source.We use these radiation fields to perturbatively evaluate the electromagnetic radiated energy-flux of the excited dipole.Our results indicate that the standard interpretation of a bare atom surrounded by a localised virtual photon cloud, is difficult to sustain, because the radiated energy-flux surviving infinitely far from the source contains virtual contributions.It follows that there is a clear distinction to be made between a radiative photon defined in terms of the radiation fields and a real photon, whose identification depends on whether or not a given process conserves the free energy.This free energy is represented by the difference between the total dipole-field Hamiltonian and its interaction component.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
It is common both in the classical and quantum optics to describe the optical field as a superposition of plane waves. However, it is well known that optically active materials emit photons vastly dominantly in the electric dipole approximation. The photons emitted in the electric dipole transitions are not plane waves, but spherical photon states corresponding to eigenstates J = 1 of the total angular momentum and M = ±1 of the z component of the total angular momentum. In addition, electric dipole photons are separated from the magnetic photons by the state index η = e for electric photons in distinction to η = m for magnetic photons. In this work, we study the far-field of light that is generated when a two-dimensional matrix of atoms emits electric dipole photons and compare this far-field at large distance from the emitting matrix with a plane wave. The goal of our work is to find out if the light emitted from the electric dipole transitions carry the memory of being electric dipole photons when they are far from the emitting atoms. In particular, it is well known that a plane wave includes all angular momentum components corresponding to quantum numbers J = 1, 2, 3, …, while the far field of the emitting matrix can include only the angular momentum component J = 1. Although the two-dimensional atomic matrix used as a light source in our simulations is certainly nontrivial to fabricate, it is nevertheless fully physical and we expect that with some modifications, the conclusions from the present simulations can be generalized to atomic light sources that are more easy to realize experimentally.
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.
In this chapter, we examine some of the principal attributes of electromagnetic radiation that feature in photons. At the outset, it is worth asserting that the photon concept owes its legitimacy to the quantum theory of light. As we shall see in later chapters, the photon is formally regarded as a quantum of excitation of any specific optical mode. In the literature, one can find occasional mentions of photons qualified by an adjective, such as dipole photons, electric photons, ballistic photons, etc.,* These should be regarded as potentially misleading because each alludes to a property that is more correctly associated with a particular kind of phenomenon. Even if the distinction between "real" and "virtual" photons represents one of the most widely used descriptors, it does not signify a clear-cut difference. We thus begin by listing the following photon properties, all of which can be involved in specific forms of optical interaction, with the exception of the first.
The problem of field quantization in a half space is discussed based on the detector modes, which provide a useful basis for the study of atomic or molecular interaction with radiation modes involving evanescent electromagnetic waves. The detector modes, each of which involves a single outgoing wave, are introduced in terms of the time-reversal and spatial-rotation transforms of the widely used triplet modes given by Carniglia and Mandel. The derivation of the orthogonal relation for the detector modes is therefore straightforward. We represent the creation and annihilation operators for the detector modes in terms of those for the triplet modes, and obtain the field operator, Hamiltonian, number operator, and pseudomomentum operator. Based on this formalism, we evaluate the differential radiation probability for atomic dipole radiation near a dielectric surface. The evaluation of the final-state mode density is straightforward for the detector modes involving single outgoing waves. With respect to the image dipole picture, the quantum-to-classical correspondence is clear in the detector-mode formalism.
The isolation usually encountered in optics between the part of a field that is of interest and its source motivates the consideration of a radiation mode weakly coupled to a quantum-mechanical source. After the introduction of some refinements into the quantum mechanics of a damped radiation mode, the field is expressed as the sum of two parts, one due to the source (the source field) and the other due to the loss mechanism (the "vacuum" field). The characteristic function for the field is calculated up to second order in perturbation theory. This function is then compared with the characteristic function for the field in the presence of a classical source. A method is exhibited by which a classical source can be found such that the two characteristic functions are identical when averaged over a half cycle. In particular, the two sources yield the same expectation values for the instantaneous amplitude and energy of the field. The description of the equivalent classical source must be given in statistical terms, in general, and requires only a knowledge of 〈S(0)(t)〉 and 〈S(0)(t1)S(0)(t2)〉, where S(0) is the dipole-moment operator of the quantum-mechanical source unperturbed by the mode under consideration (but otherwise arbitrarily complex, with the possibility of strong coupling to other modes). The theory is illustrated by a consideration of several simple sources---a two-level system, a harmonic oscillator, and a blackbody---for which equivalent classical sources are found. The two-time correlation functions for the field obtained with the two types of sources are compared and are shown to be the same up to first order in $\ensuremath{\xi}\ensuremath{\tau}$, where $\ensuremath{\tau}$ is the difference between the two times and $\ensuremath{\xi}$ is the inverse of the field relaxation time; the physical meaning of the second-order difference in the correlation functions is discussed. A limiting process, in which both the coupling to the source and the damping become small, is suggested as a method of adapting the results to free fields, but it is pointed out that for discussion of a single mode, a free field is physically less satisfactory than a damped field. It is concluded that, within a reasonable approximation scheme, the source field may be described classically (the "vacuum" field furnishing all the necessary quantum-mechanical properties of the total field).