After an outline of the quantum theory of decaying states on the basis of the resolvent method. a Hamiltonian is considered describing an ionic micro-crystal containing one electronic impurity center. The complete Hamiltonian of the coupled system micro-crystal plus radiation field can be decomposed into a diagonalized part and a “perturbation”, so that the perturbation leads to terms associated with nonradiative electronic transitions, anharmonic lattice transitions, radiative electronic transitions, and radiative lattice transitions. The state of the system micro-crystal plus radiation field is supposed to be given by a density operator in the representation where the unperturbed Hamiltonian is diagonal. By means of the assumption of initial random phases the quantum theory of decaying states yields an approximatively valid differential equation for the diagonal elements of the density operator describing the time-dependent radiative and nonradiative electronic processes in the micro-crystal. Under certain additional conditions the differential equation has the form of the Pauli-Master-Equation.
After an outline of the quantum theory of decaying states on the basis of the resolvent method. a Hamiltonian is considered describing an ionic micro-crystal containing one electronic impurity center. The complete Hamiltonian of the coupled system micro-crystal plus radiation field can be decomposed into a diagonalized part and a “perturbation”, so that the perturbation leads to terms associated with nonradiative electronic transitions, anharmonic lattice transitions, radiative electronic transitions, and radiative lattice transitions. The state of the system micro-crystal plus radiation field is supposed to be given by a density operator in the representation where the unperturbed Hamiltonian is diagonal. By means of the assumption of initial random phases the quantum theory of decaying states yields an approximatively valid differential equation for the diagonal elements of the density operator describing the time-dependent radiative and nonradiative electronic processes in the micro-crystal. Under certain additional conditions the differential equation has the form of the Pauli-Master-Equation.
In this chapter, we derive quantum master equations with explicit, microscopic coefficients, for the systems of interest of a superradiant semiconductor structure: the active electrons, the electromagnetic field, and the optical crystal vibrations. These vibrations determine an important retardation in the field propagation (refractive index), and a spectrum splitting (the Raman effect). For the active electrons we consider three environmental systems: the quasi-free electrons/holes of the conduction regions, the crystal lattice vibrations excited by electron transitions, and the free electromagnetic field. For the electromagnetic field, we consider the absorption by coupling to the conduction electrons/holes, and to the optical vibrations of the crystal, while these vibrations are damped by coupling with the valence electron transitions to thermally released states. For the electron-field coupling we consider the potential derived in chapter 2 from the Lorentz force, while the momentum difference is supposed to be taken by the crystal lattce. For the coupling of the crystal vibrations to the electromagnetic field and electron transitions we consider potentials obtained from the momentum conservation. For the active electrons, we find a quantum master equation with a Markovian term describing correlated transitions with the environmental particles, and a non-Markovian term given by the self-consistent field of the environmental particles. Since the dissipative environment of the electromagnetic field is contained inside the quantization volume of this field, which is taken as a unit volume, its quantum master equation includes a dissipative term of a space integral form. For the field mean values, the dissipation integral, of propagation through the dissipative environment, can be divided in two parts: an integral from the initial coordinate up to the boundary of the quantum uncertainty region, taken for a coherent wave, which describes dephasing, and an integral over the uncertainty region, which describes absorption. Similar equations are obtained for the optical vibration field. When the vibrational field is eliminated from these equations, we obtain a frequency splitting, corresponding to the Raman effect, and an absorption rate, including the absorption of the electromagnetic waves by conduction electrons/holes, and the absorption of the vibrational waves by the valence electrons, excited by the crystal deformations in the thermally released states.
A theory is developed to describe the interplay of vibrational relaxation and luminescence occurring simultaneously in a molecule, in terms of a master equation involving true sinks of probability. Specifically, the basic equation is the Montroll-Shuler equation augmented by the addition of sink terms which can be nonlinear as well as linear in the vibrational energy. These terms describe radiative and nonradiative decay and expressions for the former are derived explicitly in terms of Franck-Condon factors. Exact solutions in terms of known functions are obtained in the linear case for several physically motivated initial vibrational distributions, viz., $\ensuremath{\delta}$-function, Boltzmann, Poisson, and Laguerre. Two perturbation schemes are developed to analyze the nonlinear case, one of which is useful for small nonlinearities and the other for arbitrarily large nonlinearities but for low temperatures. Illustrative applications of the theory include the exact calculation of time- and frequency-resolved emission spectra (linear decay calculation) and the perturbation analysis of quantum yields in the presence of strongly energy-dependent nonradiative transitions (nonlinear decay calculation).
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.
A quantum-mechanical description of a vibronic multimode standing-wave transition-metal ion laser is presented. The impurity ion levels are coupled both by the radiation field and by the phonons of the host lattice. A dynamic Heisenberg–Langevin set of equations for the material system and the photon and phonon operators, including radiative and nonradiative damping terms and quantum-stochastic forces, has been derived. The numerical solutions of those equations show the crucial role of the phonons in the laser dynamics. The competition between the photons and the phonons leads to regular stable self-pulsation for certain sets of parameters.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
A theory describing the radiative decay of an autoionizing state under strong pumping by a coherent field is developed. The theory systematically takes into account the radiative decay of the unperturbed continuum. The problem at hand corresponds to the case of strongly coupled bound states decaying to electron and photon continua with the two continua also weakly coupled to each other. A master equation describing the time evolution of the atomic system is derived, and its exact solution under arbitrary initial conditions is given. The effect of radiative decay on the Fano profiles and photoelectron spectra is analyzed in detail. The time development of the system is also examined. The radiative decay of the autoionizing state and the unperturbed continuum changes the spectra in a significant way. The characteristics of the spectra are correlated with the dressed states (with complex energies) of the system. The changes in the structure of the dressed states as a function of the system parameters such as the spontaneous-emission rate and laser intensity are discussed in detail.