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Physical Review A

Cumulant Expansions and Pressure Broadening as an Example of Relaxation

Antoine Royer

Published Nov 1, 1972• 97 citations
Publisher

Abstract

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 {${\mathrm{Tr}}_{B}$ is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and ${H}^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: ${H}^{\mathrm{x}}( )=[H, ( )]$; $H$ is the total Hamiltonian for the system radiator plus bath}; $U(t)$ is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$, $L(t)={H}_{s}^{\mathrm{x}}+R(t)$ (${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R(t)$ added to the Liouvillian ${H}_{s}^{\mathrm{x}}$ of the unperturbed radiator. The operator $R(t)$ 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 $\overline{R}$, 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.

DOI
10.1103/physreva.6.1741

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Density Expansion of the Memory Operator in Pressure-Broadening Theory

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The motion of a radiating atom immersed in a gas of other atoms, the bath, may be described by means of a nonunitary time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\ifmmode\times\else\texttimes\fi{}}}$ acting in its Liouville space (TrB{\mathrm{Tr}}_{B}TrB​ is the trace over bath variables, $\ensuremath{\rho}$ is the density matrix, and ${H}^{\ifmmode\times\else\texttimes\fi{}}$ is the quantum-mechanical Liouvillian). In a previous paper, U(t)U(t)U(t) was written in the form $U(t)={\mathrm{exp}}_{\ensuremath{\rightarrow}}[i\ensuremath{\int}{0}^{t}{\mathrm{dt}}^{\ensuremath{'}}{L}_{1}({t}^{\ensuremath{'}})]$ (${\mathrm{exp}}_{\ensuremath{\rightarrow}}$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian L1(t){L}_{1}(t)L1​(t) was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U(\ensuremath{\omega})=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dt}{e}^{\ensuremath{-}i\ensuremath{\omega}t}U(t)$ is written in the form $iU(\ensuremath{\omega})={[\ensuremath{\omega}\ensuremath{-}{L}_{2}(\ensuremath{\omega})]}^{\ensuremath{-}1}$, and the frequency-dependent effective Liouvillian ${L}_{2}(\ensuremath{\omega})$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating ${L}_{2}(\ensuremath{\omega})$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on L1(t){L}_{1}(t)L1​(t), which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of ${L}_{2}(\ensuremath{\omega})$, one allows the radiator to interact simultaneously with two, three,... perturbers.
The theory of pressure broadening by foreign atoms is formulated within the framework of an adiabatic representation (which we distinguish from the adiabatic approximation). This allows us to treat all the electrons in the gas as indistinguishable, and thus include the possibility of electron exchange between atoms; this effect, which is responsible for much of the interatomic interaction, is neglected in all previous theories, apart from the adiabatic and 'nearest neighbor' theories which are of limited applicability. By means of projection operators, we define a spectral matrix, whose dimension is equal to the number of distinct frequencies characteristic of the isolated radiator. The spectrum is equal to the sum of all the elements of the spectral matrix; the diagonal elements correspond to the different lines of the spectrum, and the off diagonal elements represent quantum interference between overlapping lines. The Fourier transform of the spectral matrix is the correlation matrix, whose elements time correlate components of the dipole moment operator responsible for different lines of the spectrum. The time evolution of the zero perturber correlation matrix is given by eiΩτ, where Ω is a diagonal matrix whose elements are the different frequencies characteristic of the radiator. The perturbing gas causes the 'reduced Liouvillian' Ω to acquire a time dependent or frequency dependent non-Hermitian nondiagonal part. To obtain low density approximations, we treat that non-Hermitian 'perturbation' to first order in the gas density.
Within the framework of the adiabatic theory of pressure broadening, an expansion of the spectrum in powers of the density is obtained by making use of a technique of local averaging in the time domain. The first term of the expansion, ${I}_{c}(\ensuremath{\omega})$, is essentially the impact distribution; the second term is $n{J}_{w}(\ensuremath{\omega})$ folded into ${I}_{c}(\ensuremath{\omega})$, where ${J}_{w}(\ensuremath{\omega})$ is the wing of the one-perturber distribution, that is, the intensity distribution obtained if there is only one perturbing atom interacting with the radiating atom. Higher-order terms are successive convolution powers of $n{J}_{w}(\ensuremath{\omega})$, all folded into ${I}_{c}(\ensuremath{\omega})$. The expansion should be useful mostly for understanding and predicting qualitative features of the spectrum. In particular, the results imply that structure in the wings, such as satellite bands, is broadened and shifted in the same way as the line center under increasing pressure, in agreement with experimental observations.
The theoretical results of Baranger, Kolb, and Griem on pressure broadening are rederived by a more compact and flexible procedure directly applicable to other relaxation processes. Pressure broadening is worked out to first order in the pressure, including previously disregarded corrections. The procedure adapts the concepts and techniques of scattering theory to the Liouville representation of density matrices. Its key quantity is a frequency-dependent relaxation operator $〈{M}_{c}(\ensuremath{\omega})〉$ introduced by Zwanzig.
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.
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.