Physical Review A

Liouvillian Green's functions and self-energies for energy-shift and decay phenomena

Marvin D. Girardeau

Published Aug 1, 1983• 19 citations

Abstract

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.

DOI: 10.1103/physreva.28.1056

Related papers

Liouvillian Green's-function theory of spectral line shape

Marvin D. Girardeau, C. F. Hart

The general method of the preceding paper is applied to the theory of spectral line shape. Illustrative calculations of the transition-energy shift and width are carried out for a twolevel atom interacting with the quantized electromagnetic field. In the thermal-equilibrium case, explicit results are given in terms of the digamma function and Bose-Einstein distribution, generalizing the standard Lamb-shift and Weisskopf-Wigner natural width results to nonzero temperature. In the nonequilibrium case expressions are obtained which exhibit the influence of nonequilibrium photon distributions on the shift and width. A general theory of the time-dependent power radiated in the decay of an excitation is developed for systems not necessarily at or near equilibrium, in terms of appropriate generalizations of the Liouvillian Green's functions of the preceding paper. This approach is illustrated by calculation of the time-dependent radiated power for the same two-level model.

Physical Review AAug 1, 19837 citations

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

C. F. Hart

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.

Physical Review AJul 1, 19832 citations

Self-energy fields solutions to the generalized reduced Liouville equation in the perturbative approach

Roberto C. Bochicchio, Horacio Grinberg

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

International Journal of Quantum ChemistryJan 1, 20001 citation

Applications of nonequilibrium Green functions to time-dependent decaying systems

Songho Murmrgn Gim

We used nonequilibrium many-body Green techniques for the time varying resonant phenomena in few-body systems. By relating the transition energy shift and width to the self-energy and by applying the functional variational techniques to the transition energy shift and width, the corresponding initial state resonant wave functions are derived. Then we obtained time-dependent decaying state wave functions, evolving from these initial states by solving the time-dependent Schr\"odinger equation for the decaying system numerically. Finally, we observed the plausible time decaying behaviors of our model as predicted by real few-body systems.

Physical Review ADec 5, 20070 citations

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

C. F. Hart

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.

Physical Review ADec 1, 19840 citations

Thermal Green functions

C. Di Castro, Roberto Raimondi

In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.

Cambridge University Press eBooksAug 27, 20150 citations