Two nondegenerate quantum levels coupled off-diagonally and linearly to a bath of quantum-mechanical harmonic oscillators are considered. In the weak-coupling limit one finds that the equations of motion for the reduced density-matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off-diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest two-component master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T1, defined as the sum of the ‘‘up’’ and ‘‘down’’ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the off-diagonal density-matrix elements is known as 1/T2. One finds that this satisfies the well-known relation 1/T2=1/2T1. In this paper the weak-coupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weak-coupling limit, but the rate constants are calculated to fourth order. For the population-relaxation rate constants this results in an extension to fourth order of Fermi’s golden rule. We find that these higher-order rate constants do indeed satisfy detailed balance. Comparing the dephasing and population-relaxation rate constants, we find that in fourth order 1/T2≠1/2T1.
Abstract The Golden Rule is a colloquial name for a standard formula in time-dependent quantum mechanics. It provides a way to calculate the rate of transition between quantum states. It is approximate and has limited applicability; however, it is the starting point for many treatments of rate processes. Later it will be used in the discussion of master equations. This section gives the usual derivation of the Golden Rule. The following section shows how it is used to relate optical absorption spectra to time correlation functions.
Spontaneous emission from a system of
N identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state
∣z0〉.
A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak-coupling solution of the generalized Prigogine-Résibois master equation for the model of the Wigner-Weisskopf atom in a one-dimensional radiation field. The Schrödinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak-coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincaré recurrence times for finite systems are made. It is found that considerable differences exist between the statistical-mechanical and quantum-mechanical results, but that both manifest nonanalyticity in the coupling parameter as it tends to zero. Suggestions are given for further work toward the resolution of these discrepancies and a better understanding of the weak-coupling limit.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
We construct generally applicable small-loss rate expansions for the density operator of an open system. Successive terms of those expansions yield characteristic loss rates for dissipation processes. Three applications are presented in order to give further insight into the context of those expansions. The first application, of a two-level atom coupling to a bosonic environment, shows the procedure and the advantage of the expansion, whereas the second application that consists of a single mode field in a cavity with linewidth $\ensuremath{\kappa}$ due to partial transmission through one mirror illustrates a practical use of those expansions in quantum measurements, and the third one, for an atom coupled to modes of a lossy cavity shows another use of the perturbative expansion.