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Cambridge University Press eBooks

Liouville Formalism

Peter Joseph Rayer

Published Jun 30, 2020• 0 citations
Publisher

Abstract

The Fano theory, described here, does not adopt the impact approximation, and is not confined to the line core. The main concession, implicit in an impact theory like that of Baranger or Anderson, is the neglect of initial correlations between the states of radiator and bath, allowing a separate average to be taken over the bath variables at the initial time. For Fano, the correlation function describes the linear response to the driving field and is governed dynamically by the Liouville operator, which, in Baranger’s line space, has a role similar to that of the Hamiltonian in the original state space. The response, in its Fourier transform, provides the line shape, and this is governed here by a relaxation operator that looks, formally, like a transition operator in quantum scattering. This is a very general theory that will, nevertheless, reduce to that of Baranger as soon as the impact approximation is imposed. Although the neglect of initial correlations will invalidate the fluctuation–dissipation theorem, this will only affect the line far-wing, where, unless remedied, it will cause an imbalance between the radiative processes induced by the field.

DOI
10.1017/9781108768825.007

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The Fano theory, described here, does not adopt the impact approximation, and is not confined to the line core. The main concession, implicit in an impact theory like that of Baranger or Anderson, is the neglect of initial correlations between the states of radiator and bath, allowing a separate average to be taken over the bath variables at the initial time. For Fano, the correlation function describes the linear response to the driving field and is governed dynamically by the Liouville operator, which, in Baranger’s line space, has a role similar to that of the Hamiltonian in the original state space. The response, in its Fourier transform, provides the line shape, and this is governed here by a relaxation operator that looks, formally, like a transition operator in quantum scattering. This is a very general theory that will, nevertheless, reduce to that of Baranger as soon as the impact approximation is imposed. Although the neglect of initial correlations will invalidate the fluctuation–dissipation theorem, this will only affect the line far-wing, where, unless remedied, it will cause an imbalance between the radiative processes induced by the field.
Based on a recently developed quantum dissipation formulation [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], we present a reduced Liouville-space approach to evaluate the response and correlation functions of dissipative systems. The weak system-bath interaction is treated properly for its effects on the initial state, the evolution, and the correlation between coherent driving and non-Markovian dissipation. Numerical demonstration shows this correlated effect cannot be neglected even in the calculation of linear response quantities that do not explicitly depend on external fields. Highlighted in this paper is also the proper choice of theory among various formulations in the weak system-bath interaction regime.
Pressure broadening by neutral atoms is treated in a time-dependent formalism making use of generalized cumulants due to Kubo. A thermal-equilibrium initial density matrix is assumed, unlike in previous theories by Baranger and Fano who neglect initial correlations between atoms; it is pointed out that the wings of the spectrum depend in an essential manner on these initial correlations. The treatment centers on a time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\mathrm{x}}}$ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath {TrB{\mathrm{Tr}}_{B}TrB​ is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and Hx{H}^{\mathrm{x}}Hx is the quantum-mechanical Liouvillian: Hx()=[H,()]{H}^{\mathrm{x}}( )=[H, ( )]Hx()=[H,()]; HHH is the total Hamiltonian for the system radiator plus bath}; U(t)U(t)U(t) is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$, L(t)=Hsx+R(t)L(t)={H}_{s}^{\mathrm{x}}+R(t)L(t)=Hsx​+R(t) (${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation R(t)R(t)R(t) added to the Liouvillian Hsx{H}_{s}^{\mathrm{x}}Hsx​ of the unperturbed radiator. The operator R(t)R(t)R(t) is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R(t)=\overline{R}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, where $\overline{R}\ensuremath{\equiv}{\mathrm{lim}}_{T\ensuremath{\rightarrow}\ensuremath{\infty}}[(\frac{1}{T})\ensuremath{\int}{0}^{T}\mathrm{dtR}(t)]$, the spectrum is written as the sum of its impact approximation, determined by R‾\overline{R}R, plus a correction expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.
The theory of Baranger is discussed, relating it to the approach taken by Anderson in the last chapter and that taken by Fano in the one to follow. Baranger is concerned to describe pressure broadening in a band of close, overlapping lines. His original concern was with line broadening by fast-moving electrons in a plasma, which allowed him to use the impact approximation, but not to assume that collisions may be associated with classical paths. For this reason, although matter here is in the form of neutral molecules, the use of Baranger’s theory in its most general form requires that collisions be treated in terms of quantum scattering theory. In forming the correlation function, Baranger sets the algebra in a product space where line vectors take the place of the energy states used by Anderson, and the optical cross-section that governs line broadening is replaced by the matrix of an operator in line space, with line widths and shifts on the diagonal and line coupling parameters for the other elements. In the case of isolated lines, Anderson’s theory may be regained, but the introduction of line space paved the way, later on, for a much more general viewpoint.