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Physical Review A

Improved Heisenberg equations-of-motion approach for nonequilibrium decay phenomena

C. F. Hart

Published Jul 1, 1983• 2 citations
Publisher

Abstract

It is pointed out that the usual Heisenberg equations-of-motion or Liouvillian superoperator perturbation-theory approaches to systems with arbitrary statistical ensembles can give spurious, nonconvergent results when attempts are made to calculate the expectation value for certain operators which commute with the zeroth-order Hamiltonian. The non-Markovian, long-time tail of the Weisskopf-Wigner spontaneous-emission decay of a two-level atom is presented as an example of this failure. The general reason for such failures is elucidated and a new method applicable to arbitrary equilibrium or nonequilibrium statistical ensembles is formulated. The new method is applied to the same example of spontaneous emission to derive the correct short- and long-time behavior. The physical model of a two-level atom is discussed and some mathematical techniques of general usefulness are presented. The present results are also briefly discussed in the context of previous work on nonexponential decay. A fully quantum-mechanical definition of time-dependent spectral line shape is given in terms of the response of a two-level---atom detector with variable frequency interacting with electromagnetic radiation. The non-Lorentzian line shape recorded by such a detector for the spontaneous emission of a source two-level atom is calculated as another example of a problem for which the usual methods give incorrect results. The extension of the present method to Liouvillian or master-equation approaches is discussed.

DOI
10.1103/physreva.28.186

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It is pointed out that the usual Heisenberg equations-of-motion or Liouvillian superoperator perturbation-theory approaches to systems with arbitrary statistical ensembles can give spurious, nonconvergent results when attempts are made to calculate the expectation value for certain operators which commute with the zeroth-order Hamiltonian. The non-Markovian, long-time tail of the Weisskopf-Wigner spontaneous-emission decay of a two-level atom is presented as an example of this failure. The general reason for such failures is elucidated and a new method applicable to arbitrary equilibrium or nonequilibrium statistical ensembles is formulated. The new method is applied to the same example of spontaneous emission to derive the correct short- and long-time behavior. The physical model of a two-level atom is discussed and some mathematical techniques of general usefulness are presented. The present results are also briefly discussed in the context of previous work on nonexponential decay. A fully quantum-mechanical definition of time-dependent spectral line shape is given in terms of the response of a two-level---atom detector with variable frequency interacting with electromagnetic radiation. The non-Lorentzian line shape recorded by such a detector for the spontaneous emission of a source two-level atom is calculated as another example of a problem for which the usual methods give incorrect results. The extension of the present method to Liouvillian or master-equation approaches is discussed.
Green's functions and transition self-energies of dissipative quantum-mechanical many-body systems are formulated in terms of operator-basis expansions of Heisenberg equations of motion in Liouville space. Matrix elements of the Liouvillian with respect to arbitrary nonorthogonal operator bases play a primary role in the analysis. The self-energies are expressed as functionals of initial statistical ensemble averages which are presumed known and not restricted to systems at or near equilibrium; this provides a first-principles approach to use of spectral line shapes as a diagnostic of the nonequilibrium state of a dense gas, plasma, and/or radiation field. A perturbation expansion of the self-energy is derived and the general term is interpreted in terms of a subtraction of reducible parts. A separation into resonant and nonresonant contributions plays a crucial role in the analysis. The first-order resonant contribution yields a generalized statistical Hartree-Fock approximation, whereas the second-order contribution generalizes the Weisskopf-Wigner theory of decaying states to include thermal and, more generally, nonequilibrium environmental effects on transition-energy shifts and widths. Off-diagonal contributions to the self-energy are shown to be essential for inclusion of environmental contributions.
Despite many years of research into Raman phenomena, the problem of how to include both spontaneous and stimulated Raman scattering into a unified set of partial differential equations persists. The issue is solved by formulating the quantum dynamics in the Heisenberg picture with a rigorous accounting for both time- and normal-ordering of the operators. It is shown how this can be done in a simple, straightforward way. Firstly, the technique is applied to a two-level Raman system, and comparison of analytical and numerical results verifies the approach. A connection to a fully time-dependent Langevin operator method is made for the spontaneous initiation of stimulated Raman scattering. Secondly, the technique is demonstrated for the much-studied two-level atom both in vacuum and in a lossy dielectric medium. It is shown to be fully consistent with accepted theories: using the rotating wave approximation, the Einstein A coefficient for the rate of spontaneous emission from a two-level atom can be derived in a manner parallel to the Weisskopf-Wigner approximation. The Lamb frequency shift is also calculated. It is shown throughout that field operators corresponding to spontaneous radiative terms do not commute with atomic/molecular operators. The approach may prove useful in many areas, including modeling the propagation of next-generation high-energy, high-intensity ultrafast laser pulses as well as spontaneous radiative processes in lossy media.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes possible a complete and gauge-invariant treatment of spontaneously decaying atomic state. All the shortcomings of the Weisskopf-Wigner gauge-dependent method (Markov approximation and the neglect of virtual transition, retardation and electron-spin effects) are removed. After deriving a gauge-invariant equation of motion for the decaying state, a new physical interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) is introduced. In this new interaction picture, apart from a natural solution of the renormalization problem, explicit finite analytic results for the whole non-Markovian Lyman- alpha spontaneous decay are derived. Furthermore, a quasi-Lorentzian expression for the radiative line shape containing the correct (complete) finite non-relativistic frequency shift, stemming from the Lamb shifts of both levels 1S and 2P, is calculated.
For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.
In this paper, a model is studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system. The model chosen for discussion is that of the Wigner-Weisskopf atom in interaction with a massless boson field, here 1-dimensional. The solution takes the form of a constant term plus a time-dependent one, expressed as the sum of residues at a series of poles along the real axis of a Laplace transform variable. Numerical calculations were performed on various aspects of the solution, and although these were too delicate to be quite certain around sensitive values of the time, the general picture is clear: After an initial decay to a value near zero, the series gave rise at fairly regular intervals to rapid and large fluctuations, the size of which never quite attains the initial value, but may nonetheless be large even after very long times. This result seems to be in agreement with the observation made by Zwanzig, namely, that for finite systems the master equation might demonstrate properties associated with the finite size of the system which would become important over certain long-time scales and would be ignored by the use of the customary ``thermodynamic limit'' of statistical mechanics. The relationship between this work and that of Montroll and Mazur and of Rubin is also discussed.