Abstract The Einstein A and B coefficients for atom–photon reactions are defined: they describe absorption, stimulated emission, and spontaneous emission. We calculate B and consequently find A by the Einstein trick. The procedure is validated by application of detailed balance. Separating out frequencies permits introduction of the spectral lineshape function (normalized line profile). A reformulation describes transitions involving single atomic states and single field modes. This points to a link with quantum electrodynamics.
Abstract The Einstein A and B coefficients for atom–photon reactions are defined: they describe absorption, stimulated emission, and spontaneous emission. We calculate B and consequently find A by the Einstein trick. The procedure is validated by application of detailed balance. Separating out frequencies permits introduction of the spectral lineshape function (normalized line profile). A reformulation describes transitions involving single atomic states and single field modes. This points to a link with quantum electrodynamics.
Recently, Ryu et al. showed that two broadened bands connected by a set of four Einstein-coefficient spectra for stimulated and spontaneous single-photon transitions will obey detailed balance at equilibrium if the spectra satisfy generalized Einstein relations. Here, quantum mechanical expressions for Einstein-coefficient spectra are obtained in the electric-dipole approximation using an intramolecular Boltzmann distribution and the quantized field operators in isotropic, dispersive media of Nienhuis and Alkemade [Physica B+C 81, 181-188 (1976)]. These expressions suggest relationships between Einstein-coefficient spectra and dipole-strength spectra. The electrodynamic relationship between the spectral density for electromagnetic energy and the spectral density for the square of the electric field is developed and used to define dipole-strength spectra in terms of conditional transition probabilities per unit time. These rigorously relate dipole-strength spectra to Einstein-coefficient spectra, thus establishing quantum formulas for dipole-strength spectra and new generalized Einstein relations between dipole-strength spectra. For transitions between two bands, the dipole-strength spectra depend on a single total dipole strength but replace Einstein's degeneracy ratio and transition frequency with a change in standard chemical potential and a single underlying lineshape that is manifested differently in the four spectra. At equilibrium, the relations specify the Stokes' shift between forward and reverse transitions. The relationships between dipole-strength spectra, spontaneous emission spectral densities, and stimulated transition cross sections depend on the refractive index, the dielectric constant, and the local field, but not on the derivative of the refractive index. The broadband relationships reduce to known relationships for narrow spectra inside materials and for line spectra in vacuum.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
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.
Recently, Ryu et al. [Proc. Natl. Acad. Sci. U. S. A. 121, e2410280121 (2024)] generalized Einstein's three coefficients for absorption, stimulated emission, and spontaneous emission between two quantum levels to a set of four spectra between two broadened bands. The spectra obey generalized Einstein relationships at thermal equilibrium; Einstein's relations are obtained as an approximation for line spectra. Here, the generalized Einstein relation between absorption and stimulated emission dipole-strength spectra is applied to investigate optical lineshape models. Lineshapes for the Bloch model, the stochastic model, and the semi-classical Brownian oscillator model do not obey the generalized Einstein relation and, therefore, fail to satisfy detailed balance with Planck blackbody radiation. The quantum Brownian oscillator model treats a harmonic quantum vibration that is bi-linearly coupled to a thermal bath of quantum harmonic oscillators, which generate damping and a random force. The two-state quantum Brownian oscillator lineshape model provides lineshapes for transitions between two displaced, but otherwise identical, harmonic potential energy surfaces on which the same quantum vibration is coupled to the same thermal bath of quantum harmonic oscillators. The absorption and stimulated emission lineshapes were calculated using the quantum Brownian oscillator model in under-damped, critically damped, and over-damped cases. The thermal and reorganization energies were each varied from values much smaller than the vibrational quantum of energy to values greater than it. All quantum Brownian oscillator lineshapes obey the generalized Einstein relation within the numerical precision of the calculation (14-30 digits), suggesting this lineshape model is compatible with detailed balance. The formula giving the electric-dipole transition cross section in terms of these lineshapes is presented.
We present Einstein coefficient spectra and a detailed-balance derivation of generalized Einstein relations between them that is based on the connection between spontaneous and stimulated emission. If two broadened levels or bands overlap in energy, transitions between them need not be purely absorptive or emissive. Consequently, spontaneous emission can occur in both transition directions, and four Einstein coefficient spectra replace the three Einstein coefficients for a line. At equilibrium, the four different spectra obey five pairwise relationships and one lineshape generates all four. These relationships are independent of molecular quantum statistics and predict the Stokes' shift between forward and reverse transitions required by equilibrium with blackbody radiation. For Boltzmann statistics, the relative strengths of forward and reverse transitions depend on the formal chemical potential difference between the initial and final bands, which becomes the standard chemical potential difference for ideal solutes. The formal chemical potential of a band replaces both the energy and degeneracy of a quantum level. Like the energies of quantum levels, the formal chemical potentials of bands obey the Rydberg-Ritz combination principle. Each stimulated Einstein coefficient spectrum gives a frequency-dependent transition cross-section. Transition cross-sections obey causality and a detailed-balance condition with spontaneous emission, but do not directly obey generalized Einstein relations. Even with an energetic width much less than the photon energy, a predominantly absorptive forward transition with an energetic width much greater than the thermal energy can have such an extreme Stokes' shift that its reverse transition cross-section becomes predominantly absorptive rather than emissive.