We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.
We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.
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, 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 mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.
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.
This chapter considers an atom as a quantum mechanical system. But the electromagnetic radiation will be treated as classical waves described by the Maxwell equations. The treatment is called semi-classical. Quantum mechanics describes an atom as a quantized system with a set of discrete energy levels. Each energy level is designated by a set of quantum numbers like the principal quantum of the Rydberg level and the quantum numbers representing the orbital, spin and total angular momentum. A molecule will have a different set of quantum numbers arising from its rotational and vibrational motions. Atoms in a solid have energy levels in the form of a band. The observed spectrum due to spontaneous emission from a collection of atoms is not monochromatic, but it is spread over a certain range of frequencies. The absorption spectrum also shows that the atom does not absorb at a single frequency, but it is spread over a band of frequencies around the resonance frequency.
Light emission occurs as a result of interaction between light and matter. Let us assume a two-level atom with levels 1 (ground state) and 2 (excited state). The energies of the ground and excited states are E 1 and E 2 , respectively, and the energy difference is therefore given by the difference ΔE 12 =E 2 −E 1 . When light with photons of energy equal to this difference is absorbed by the atomic or molecular system, an electron will be excited from level 1 to level 2. The energy of the photons is given by ΔE 12 =E 12 =hν 12 (h is Planck's constant and 1→2 is the frequency of light resulting from the level 1→2 transition). This energy is absorbed by an atom or a molecule that has energy levels, separated by ΔE, where these energy levels are the ground and the excited states. This kind of “quantum jump” of an electron between two states occurs in atomic systems between electronic levels; it can be extended to molecular systems, where vibrational and translational energy levels participate in the quantum jump. The interaction between light and matter involves the transition of electrons between different states. This interaction results in the absorption of photons (stimulated absorption) as well as spontaneous and stimulated emission. These processes can be described using Einstein's A and B coefficients, as will be described in the next paragraphs.