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Physical Review A

Exact operator self-energies from resummation of the Heisenberg equations of motion

C. F. Hart

Published Dec 1, 1984• 0 citations
Publisher

Abstract

The Laplace-transformed Heisenberg equations of motion are resummed in such a way that the resulting iterative series expansion of any operator is given in terms of exact perturbed operator self-energies. We show how to incorporate this result into a recently introduced Liouvillian Green's-function and self-energy approach to nonequilibrium statistical mechanics. The spontaneous emission of a two-level atom is discussed as a simple example of this new resummation technique.

DOI
10.1103/physreva.30.3311

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The Laplace-transformed Heisenberg equations of motion are resummed in such a way that the resulting iterative series expansion of any operator is given in terms of exact perturbed operator self-energies. We show how to incorporate this result into a recently introduced Liouvillian Green's-function and self-energy approach to nonequilibrium statistical mechanics. The spontaneous emission of a two-level atom is discussed as a simple example of this new resummation technique.
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.
Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter $\ensuremath{\lambda}$ in the Hamiltonian $H={H}_{0}+\ensuremath{\lambda}{H}_{1}$. Here H0{H}_{0}H0​ is the exactly solvable part and H1{H}_{1}H1​ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order ${\ensuremath{\lambda}}^{2}$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.
A direct approach for the generalized reduced Liouville equation of motion decoupling problem associated with open quantum systems within the superoperator formalism is presented. The procedure is based on inversion of the perturbation series for the energy representation of the density operator so as to obtain one for the proper self-energy fields that emerge as a consequence of the analytic character of the associated spectral representation. Thus, the perturbation series that arises from the iteration of the energy-dependent matrix elements hierarchy involved in the statistical operator allows, upon further expansion of the inverse of such series, to get formally exact expressions for the corrections to all orders of the self-energy fields. The lower order corrections of these fields are discussed in terms of resonant and nonresonant contributions. The present approach provides matrix equations that show the close relation between the environment effects represented by the self-energy fields and the relaxation kernel that drives the system–reservoir interaction. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 280–290, 2000
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
In this paper we report a new method for resummation of operator expansions of the evolution operator. These resummation techniques are applied to the Feynman propagator and then to the derivation of an effective Schrödinger equation for general system–bath problems. This paper presents a significant advance over work previously reported by us [J. Chem. Phys. 96, 5952 (1992)], and it provides a highly accurate way to calculate quantum mechanical data for many dimensional systems. We then apply this new analytic resummation technique to the calculation of flux–flux correlation functions for two, three, and four dimensional problems in order to study the range of applicability of the approach. For moderate frequencies and coupling strengths, not only is the resummed operator calculational method essentially exact, it also requires a fraction of the computer time that the exact calculation consumes. For truly many dimensional problems this approach should provide the first accurate quantum mechanics of rate processes.