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International Journal of Quantum Chemistry

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

Roberto C. Bochicchio, Horacio Grinberg

Published Jan 1, 2000• 1 citation
Publisher

Abstract

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

DOI
10.1002/1097-461x(2000)79:5<280::aid-qua3>3.0.co;2-w

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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
Abstract An explicit definition for the self‐energy field operator and the self‐energy fields obtained from an average of the associated operator within the algebraic formalism of superoperators is presented. It stems from the formal expansion of the many‐body propagator equations of motion hierachy in quantum many‐body systems within the scenario of the Liouvillian decoupling scheme developed in previous works. An essential theoretical property of such fields for the complete expansion of the propagator to any order in the interaction potential is shown. This states that the interaction potential to a given order only depends on one q‐reduced density matrix. The contraction order q of the density matrix depends on both the nature of the operators defining the propagator and the actual order of the expansion. This result is rigorous regarding infinite summation of the irreducible terms of the self‐energy fields and provides a direct way to estimate the extent to which many‐body effects are involved in successive approximations, i.e., truncation of the excitation level given by the reference state throughout the reduced density matrix and the expansion of the propagator. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002
Abstract Master equations for propagators in quantum open systems and their spectral resolutions are derived. The Zwanzig partitioning scheme along the superoperator algebra are used to derive equations of motion for partitioned operators in a Liouville space. The reservoir influence on the dynamical evolution of operators is shown to lead explicitly to dissipative effects arising from memory terms in the evolution equations of such operators. It is also shown that spectral representations may be written in a self‐consistent analytic way by means of the self‐energy fields for transition energies of the system by taking into account the lack of the complete knowledge about the reservoir. A kinematic fluid interpretation of the resultant equations is given and an explicit form of the “collision” superoperator is obtained. Finally, a simple example to illustrate the determination of self‐energy fields for the system–reservoir interaction corrections is given. © 1995 John Wiley & Sons, Inc.
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.
Liouville-space (reduced-density-operator) descriptions are developed for resonant and coherent electromagnetic interactions of quantized electronic systems, taking into account environmental decoherence and relaxation phenomena. Applications of interest include electromagnetically-induced transparency and related pump-probe optical phenomena in many-electron atomic systems (in electron-ion beam interactions, gases, and high-temperature plasmas) and semiconductor materials (bulk crystals and nanostructures). Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbationtheory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical description of the entire electromagnetic interaction is introduced. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear electromagnetic-response tensors occurring in a perturbation-theory treatment of the semiclassical electromagnetic interaction. These expressions can be evaluated for coherent initial electronic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Intense-field electromagnetic interactions are treated by means of an alternative, non-perturbative method, which is based on a Liouville-space Floquet-Fourier representation of the reduced density operator. Electron-electron quantum correlations are treated by the introduction of a cluster decomposition of the reduced density operator and a coupled hierarchy of reduced-density-operator equations.
A generalized time-dependent perturbation theory is derived for superoperators. Instead of using the ``standard'' breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phase-space-based semiclassical approximations. The theory is demonstrated for a model quartic potential.