The theory of the line shape in coherent resonant absorption and emission in the infrared and optical region is developed. The Dicke spin model and the Lorentz oscillator model of a radiating molecule are discussed in detail, and the complications arising for degenerate levels are analyzed. The absorption and emission spectra are expressed in terms of the damped normal modes of the system of equivalent oscillators interacting via the retarded dipole–dipole interaction, and the results are shown to be identical with those derived using Heitler's quantum theory of radiation damping. The theory is applied to small clusters and linear chains of molecules. For a chain that is small compared with the wavelength, the natural line width is proportional to the total number of molecules, whereas for a long chain the width approaches a constant value. The discussion of three-dimensional systems, and the application to the zero-phonon lines in the infrared spectrum of solid hydrogen will be given in a separate paper.
The theory of the line shape in coherent resonant absorption and emission in the infrared and optical region is developed. The Dicke spin model and the Lorentz oscillator model of a radiating molecule are discussed in detail, and the complications arising for degenerate levels are analyzed. The absorption and emission spectra are expressed in terms of the damped normal modes of the system of equivalent oscillators interacting via the retarded dipole–dipole interaction, and the results are shown to be identical with those derived using Heitler's quantum theory of radiation damping. The theory is applied to small clusters and linear chains of molecules. For a chain that is small compared with the wavelength, the natural line width is proportional to the total number of molecules, whereas for a long chain the width approaches a constant value. The discussion of three-dimensional systems, and the application to the zero-phonon lines in the infrared spectrum of solid hydrogen will be given in a separate paper.
The theory of the line shape in coherent electric quadrupole absorption and emission processes involving N interacting molecules is developed, both quantum mechanically and with the use of a classical model. The theory is applied to small clusters and to linear chains of molecules. As in the dipole case, the effects of the interaction include in general a splitting of each molecular transition line into a number of components, a shift of each component from the molecular transition frequency, and a broadening or narrowing of each component relative to the width of the molecular line.
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.
The calculation of the retarded fourth-order resonance interaction between identical molecules with one shared quantum of excitation energy is described. The interaction is calculated in the electric dipole approximation for molecular transitions with the help of time-ordered graphs and a reduced Hamiltonian which treats molecule-photon and molecule-molecule interactions on the same footing. Interactions involving spins, valence forces and permanent electric or magnetic multipoles are not considered. In the near and wave zones the leading terms of the retarded interaction are proportional to R -6 and R -2 respectively, where R is the intermolecular separation.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by TrN, where N is the total number of atoms, T is the average time of coherent interaction of atoms and the radiation field, and r is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.
In this paper we apply the T matrix formalism of scattering theory to derive general expressions for the absorption cross sections, the cross sections for resonance fluorescence and the emission quantum yields from large molecules in the statistical limit. In the simple case of an isolated molecular resonance both the absorption line shape and the photon scattering cross section exhibit a Lorentzian distribution on the photon energy, the emission quantum yields are distributed among the ground state vibronic levels according to their radiative widths and, most important, the emission quantum yields are independent of the photon energy and of the spectral width of the exciting light. We were able to derive general expressions for the resonance scattering from a pair of overlapping resonances, including radiative corrections to infinite order. The absorption cross section does not vanish in the region of destructive interference but assumes a finite value which depends on the radiative widths. A sharp maximum in the partial and in the total emission quantum yields is exhibited in the destructive interference regions. This general scheme was applied to a pair of zero order discrete states, one of which is optically active, which interact with an optically inactive quasicontinuum. The energy dependent quantum yield depends on the total width of the radiatively impotent state and may exhibit a minimum. We have demonstrated that when interference effects are involved the decay characteristics of this system will differ for coherent and for narrow band excitation. The general formalism was utilized to derive approximate relations for the resonance fluorescence cross sections and for the quantum yield in the case of a Fano absorption line shape which are valid away from the interference region. Finally, we have applied the general theoretical scheme to the case of the direct photodissociation spectrum of molecules. We have demonstrated that a finite energy dependent emission quantum yield will be observed when a molecule is optically pumped into a dissociative continuum.
A large number of "molecules," which are taken to be angular momentum oscillators of frequency $\ensuremath{\nu}$, are considered to interact through their dipole moment with two electromagnetic cavity modes of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$, respectively, where $\ensuremath{\nu}={\ensuremath{\omega}}_{1}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\omega}}_{2}$. The behavior of the combined system is analyzed up to fourth order in perturbation theory. Phenomena known as two-photon emission or absorption, parametric amplification, various types of Raman emission, and frequency conversion are exhibited. The analytical method used is applicable to both a quantum-mechanical and a classical description of the modes and molecules; it displays explicitly the resulting differences. Expressions for the rate of energy change of the modes and molecules indicate resonance in second and fourth order. The second-order results are the same for the classical and quantum-mechanical treatments, while the fourth-order results show certain differences, which are examined in detail. The conditions under which each of the above phenomena occurs are discussed in a self-consistent manner, both classically and quantum-mechanically. It is shown that the second-order expression and the most significant part of the fourth-order expression can be derived very simply from a resonance interaction Hamiltonian that consists of a sum of products of three variables---two referring to the two modes, and the third to the molecules. The relationship between this resonance Hamiltonian and the fundamental dipole-moment Hamiltonian is examined, and the approximations by which the resonance Hamiltonian may be derived are considered. Arguments concerning the applicability of perturbation theory to steady-state situations are presented; it is pointed out that the second-order expressions for the energy transfer between molecules and field may be applicable to such a situation, in which case the classical and quantum-mechanical treatments are equivalent.