Previous work of the authors on a three-level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as integrals with well-behaved limits. Exact expressions are found, for a spontaneous emission problem, for the time evolution of the probabilities that the system be in (each) one of its three states and representative calculations of these probabilities are presented. The emergence of irreversible behavior upon constructing the thermodynamic limit of the problem is plainly seen and this demonstration permits for the first time a discussion of the relative effectiveness of competing decay channels in three-level quantum systems without imposing any ad hoc assumptions (such as exponential decay). Rather, the actual form of the decay emerges as a consequence of the structure and parameters of the Hamiltonian defining the model, and hence one can examine the variety of circumstances in which the evolution can reasonably be described as exponential. The results obtained should be of great use in clarifying certain outstanding conceptual problems in radiation physics, particularly those which deal with the universality of exponential decay in three- (and two-) level quantum systems in interaction with a radiation field and the conditions under which nonexponential or even nonergodic behavior can emerge in such dissipative quantum systems.
Previous work of the authors on a three-level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as integrals with well-behaved limits. Exact expressions are found, for a spontaneous emission problem, for the time evolution of the probabilities that the system be in (each) one of its three states and representative calculations of these probabilities are presented. The emergence of irreversible behavior upon constructing the thermodynamic limit of the problem is plainly seen and this demonstration permits for the first time a discussion of the relative effectiveness of competing decay channels in three-level quantum systems without imposing any ad hoc assumptions (such as exponential decay). Rather, the actual form of the decay emerges as a consequence of the structure and parameters of the Hamiltonian defining the model, and hence one can examine the variety of circumstances in which the evolution can reasonably be described as exponential. The results obtained should be of great use in clarifying certain outstanding conceptual problems in radiation physics, particularly those which deal with the universality of exponential decay in three- (and two-) level quantum systems in interaction with a radiation field and the conditions under which nonexponential or even nonergodic behavior can emerge in such dissipative quantum systems.
In this paper we investigate the dynamics of a model for a three-level atom in interaction with a radiation field. The exact solution to the spontaneous-emission problem is derived using methods developed earlier by the authors, and expressions are obtained for the probabilities of the atom’s being in the first or second excited states at any time t. For the special case that the strengths of the coupling between each of the excited states and the λth mode of the field are proportional, detailed conclusions can be drawn concerning the effects of such factors as system size, coupling function, and level splitting on the temporal evolution of the system. The evolution of excited quantum systems having one versus two modes of decay to the ground state is also compared, and similarities and differences in the temporal behavior are noted. Finally the relevance of the theory presented in this paper to experimental problems in radiation chemistry and physics is indicated in our concluding remarks.
In this paper we study the emission of a two-level atom in a radiation field in the case where one mode of the field is assumed to be excited initially, and where the system is assumed to be of infinite extent. (The restriction to a one-dimensional field, which has been made throughout this series, is not essential: It is made chiefly for ease of presentation of the mathematical methods.) An exact expression is obtained for the probability ρ (t) that the two-level quantum system is in the excited state at time t. This problem, previously unsolved in radiation theory, is tackled by reformulating the expression found in VII [J. Math. Phys. 16, 1013 (1975)] of this series for the time evolution of ρ (t) in a finite system in the presence of an extra photon, and then constructing the infinite-system limit. A quantitative assessment of the role of the extra photon and of the coupling constant in influencing the dynamics is obtained by studying numerically the expression derived for ρ (t) for a particular choice of initial condition. The study presented here casts light on the problem of time-reversal invariance and clarifies the sense in which exponential decay is universal; in particular, we find that: (1) It is the infinite-system limit which converts the time-reversible solutions of VII into the irreversible solution obtained here, and (2) it is the weak-coupling limit that imposes exponential form on the time dependence of the evolution of the system. The anticipated generalization of our methods to more complicated radiation-matter problems is discussed, and finally, several problems in radiation chemistry and physics, already accessible to exact analysis given the approach introduced here, are cited.
In pursuit of the study begun in the preceding paper of the spontaneous emission of a Wigner-Weisskopf atom in a one-dimensional radiation field, the exact solution of this problem in the thermodynamic limit of a system of infinite extent is examined. The analysis reveals the existence in general of three contributions to the probability of the atom's being excited at any moment: a constant contribution and two sorts of time-dependent ones, exponential and nonexponential. For a specific choice of coupling function, this probability is written down in closed form. The link between the constant contribution and the appearance of ghost states and ergodicity is investigated, and criteria given to determine when this contribution does not arise. A numerical study is made of the expressions obtained, and the nonexponential part of the solution compared with approximations for it which have been previously obtained by various means. It is concluded that the predictions of the weak-coupling Prigogine-Résibois master equation are inadequate except when the model is ergodic in a certain sense, and some suggestions are made concerning the relevance of these studies to nonequilibrium statistical mechanics.
In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
In a $\ensuremath{\Lambda}$ system with two nearly degenerate ground states and one excited state in an atom or quantum dot, spontaneous radiative decay can lead to a range of phenomena, including electron-photon entanglement, spontaneously generated coherence, and two-pathway decay. We show that a treatment of the radiative decay as a quantum evolution of a single physical system composed of a three-level electron subsystem and photons leads to a range of consequences depending on the electron-photon interaction and the measurement. Different treatments of the emitted photon channel the electron-photon system into a variety of final states. The theory is not restricted to the three-level system.