In our preceding treatments of the interaction of radiation with matter, we assumed that the electric field is classical. In most cases, this assumption is validated by experiment. There are several instances, however, for which a classical field fails to give experimentally observed results, whereas a quantized field succeeds. This is true, for example, of spontaneous emission, a phenom enon which was described phenomenologically in our earlier work (see Chaps. 2 and 7). Another case involves the 2 and 2p \ states in the hydrogen atom, which are predicted to have the same energy not only by the simple theory of Chap. 1, but also by the more sophisticated Dirac theory. In both cases, a classical electric field was assumed. Experimentally, the two levels differ by approximately the Lamb shift, 1057 MHz. A fully quantized treatment of the field and atom system gives impressive agreement with the experimentally observed shift. Derivation of the fluctuations in intensity of a laser near threshold requires the quantum theory o f radiation. In addition there are other problems for which the concept of a “particle,” the photon, is either necessary or convenient. Despite a popular impression to the contrary, however the explanation of the photoelectric effect is not one of these (see Prob. 2-10).
In our preceding treatments of the interaction of radiation with matter, we assumed that the electric field is classical. In most cases, this assumption is validated by experiment. There are several instances, however, for which a classical field fails to give experimentally observed results, whereas a quantized field succeeds. This is true, for example, of spontaneous emission, a phenom enon which was described phenomenologically in our earlier work (see Chaps. 2 and 7). Another case involves the 2 and 2p \ states in the hydrogen atom, which are predicted to have the same energy not only by the simple theory of Chap. 1, but also by the more sophisticated Dirac theory. In both cases, a classical electric field was assumed. Experimentally, the two levels differ by approximately the Lamb shift, 1057 MHz. A fully quantized treatment of the field and atom system gives impressive agreement with the experimentally observed shift. Derivation of the fluctuations in intensity of a laser near threshold requires the quantum theory o f radiation. In addition there are other problems for which the concept of a “particle,” the photon, is either necessary or convenient. Despite a popular impression to the contrary, however the explanation of the photoelectric effect is not one of these (see Prob. 2-10).
The idea of the photon has stirred the imaginations of physicists ever since 1905 when Einstein originally proposed the use of light quanta to explain the photoelectric effect. This concept is formalized in the quantum theory of radiation, which has had unfailing success in explaining the interaction of electromagnetic radiation with matter, seemingly limited only by the ability of physicists to perform the indicated calculations. Nevertheless, it has its conceptual problems—various infinities and frequent misinterpretations. Consequently an increasing number of workers are asking, “to what extent is the quantized field really necessary and useful?” In fact the experimental results of the photoelectric effect were explained by G. Wentzel in 1927 without the quantum theory of radiation. Similarly most electro-optic phenomena such as stimulated emission, reaction of the emitted field on the emitting atom, resonance fluorescence, and so on, do not require the quantization of the field for their explanation. As we will see, these processes can all be quantitatively explained and physically understood in terms of the semiclassical theory of the matter–field interaction in which the electric field is treated classically while the atoms obey the laws of quantum mechanics. The quantized field is fundamentally required for accurate descriptions of certain processes involving fluctuations in the electromagnetic field: for example, spontaneous emission, the Lamb shift, the anomalous magnetic moment of the electron, and certain aspects of blackbody radiation. (The Compton effect also fits here, but see later under references 8b and c.) Here we will outline how the photon concept originated and developed, where it is not required and is often misused, and finally where it plays an essential role in the understanding of physical phenomena. In our discussion we will attempt to give a logically consistent definition of the word “photon”—a statement far more necessary than one might think, for so many contradictory uses exist of this elusive beast. In particular consider the original coining of the word by G. N. Lewis: “[because it appears to spend] only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains an important structural element within the atom…, I therefore take the liberty of proposing for this hypothetical new atom which is not light but plays an essential part in every process of radiation, the name photon!” (our exclamation point). Clearly the present usage of the word is very different.
Interactions of a quantized electromagnetic field In the preceding chapters we have studied the quantum properties of the electromagnetic field, but we have treated it as a free or non-interacting quantum system until now. However, both the emission and the absorption of light imply interactions with other quantum systems, and we now turn to the treatment of such interaction problems. It is true that, to a limited extent, we have already succeeded in treating some interactions of light in previous chapters, without invoking the full apparatus of the quantum theory of interacting systems. For example, in Chapter 9 the electromagnetic field was treated as a classical potential acting on an atomic quantum system, and in Sections 12.2 and 12.9 we invoked simple heuristic arguments to describe the operation of certain optical detectors. But these are limited applications, and, in any case, the validity of results obtained in this way needs to be confirmed. To describe the state of a quantized electromagnetic field (F) in interaction with some other quantum system (A), we evidently require an enlarged, or product Hilbert space, which encompasses both F and A, of which the Hilbert spaces of F and A are subspaces. We shall find it convenient to refer to the other system A as an ‘atomic system’, simply to give a name, without restricting the nature of A. The dynamical variables of the field Ô F( t ) and of the atomic system Ô A( t ) commute at the same time t , and at the beginning, when the interaction between them is assumed to be turned on, each operator acts only on the state vectors within its respective Hilbert space.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space. The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept. It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
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.