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Physical Review A

Density Expansion of the Memory Operator in Pressure-Broadening Theory

Antoine Royer

Published Mar 1, 1973• 22 citations
Publisher

Abstract

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 (${\mathrm{Tr}}_{B}$ 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, $U(t)$ 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 ${L}_{1}(t)$ 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 ${L}_{1}(t)$, 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.

DOI
10.1103/physreva.7.1078

Related papers

Cumulant Expansions and Pressure Broadening as an Example of Relaxation

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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.

Physical Review A•Nov 1, 1972•97 citations

Theory of Pressure Broadening in an Adiabatic Representation

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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.

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Dissipative quantum mechanics beyond the Bloch-Redfield approximation: A consistent weak-coupling expansion of the Ohmic spin boson model at arbitrary bias

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We study the time evolution of the reduced density matrix for the Ohmic spin boson model out of an uncorrelated but otherwise arbitrary initial state. We consider arbitrary bias $\ensuremath{\epsilon}$ and tunneling $\mathrm{\ensuremath{\Delta}}$ at zero temperature for a weak coupling $\ensuremath{\alpha}$ to the bosonic bath. Using the real-time renormalization group method, we present a consistent weak-coupling expansion one order beyond the Bloch-Redfield approximation within a renormalized perturbation theory with analytical results covering the whole crossover regime from small times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\ll}1$ to large times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\gg}1$, where $\mathrm{\ensuremath{\Omega}}=\sqrt{{\ensuremath{\epsilon}}^{2}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}^{2}}$ denotes the Rabi frequency in terms of the renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$. In addition, for exponentially small or large times, we perform a nonperturbative resummation of all logarithmic terms. We show that standard Born approximation schemes calculating the effective Liouvillian of the kinetic equation up to first order in $\ensuremath{\alpha}$ are not sufficient to account for various important corrections one order beyond the Bloch-Redfield solution. (1) The resummation of all secular terms $\ensuremath{\sim}{(\mathrm{\ensuremath{\Gamma}}t)}^{n}$ is necessary to obtain the correct exponential decay of all terms of the time evolution with decay rate $\mathrm{\ensuremath{\Gamma}}$ or $\mathrm{\ensuremath{\Gamma}}/2$, together with the correct pre-exponential functions. (2) The resummation of all logarithmic terms at high and low energies leads to a renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$ and to pre-exponential functions of logarithmic and power-law form. (3) The fact that two eigenvalues of $L(E)$ are close to each other by $O(\mathrm{\ensuremath{\Gamma}})$ requires degenerate perturbation theory for times $\mathrm{\ensuremath{\Gamma}}t\ensuremath{\sim}O(1)$, where certain terms of the Liouvillian in $O({\ensuremath{\alpha}}^{2})$ are needed to calculate the stationary state and the time evolution of the nonoscillating purely decaying modes up to $O(\ensuremath{\alpha})$. In contrast to the zero-bias case, we find two further interesting results for the time dynamics of the oscillating modes. (4) The terms of the pre-exponential functions with a strong time dependence show a leading long-time tail $\ensuremath{\sim}\ensuremath{\alpha}/(\mathrm{\ensuremath{\Omega}}t)$, besides other subleading terms $\ensuremath{\sim}\ensuremath{\alpha}/{(\mathrm{\ensuremath{\Omega}}t)}^{2}$ well-known from the zero-bias case. (5) The terms of the pre-exponential functions with a weak (logarithmic) time dependence vary according to a power law ${(\frac{1}{\mathrm{\ensuremath{\Omega}}t})}^{2\ensuremath{\alpha}\frac{{\ensuremath{\epsilon}}^{2}}{{\mathrm{\ensuremath{\Omega}}}^{2}}}$ for exponentially large times. The power-law exponent depends on the bias and has to be contrasted to the one at exponentially small times where it crosses over to the bias-independent result $2\ensuremath{\alpha}$. We discuss that the complexity to calculate one order beyond Bloch-Redfield approximation is rather generic and applies also to other models of dissipative quantum mechanics.

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Pressure Broadening as a Prototype of Relaxation

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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.

Physical Review•Jul 1, 1963•725 citations
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)=(TrB\ensuremathρ)\ensuremath−1TrB\ensuremathρeitH\ifmmode×\else\texttimes\fiU(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\ifmmode\times\else\texttimes\fi{}}}U(t)=(TrB​\ensuremathρ)\ensuremath−1TrB​\ensuremathρeitH\ifmmode×\else\texttimes\fi acting in its Liouville space (TrB{\mathrm{Tr}}_{B}TrB​ is the trace over bath variables, \ensuremathρ\ensuremath{\rho}\ensuremathρ is the density matrix, and H\ifmmode×\else\texttimes\fi{H}^{\ifmmode\times\else\texttimes\fi{}}H\ifmmode×\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)=exp\ensuremath→[i\ensuremath∫0tdt\ensuremath′L1(t\ensuremath′)]U(t)={\mathrm{exp}}_{\ensuremath{\rightarrow}}[i\ensuremath{\int}{0}^{t}{\mathrm{dt}}^{\ensuremath{'}}{L}_{1}({t}^{\ensuremath{'}})]U(t)=exp\ensuremath→​[i\ensuremath∫0tdt\ensuremath′L1​(t\ensuremath′)] (exp\ensuremath→{\mathrm{exp}}_{\ensuremath{\rightarrow}}exp\ensuremath→​ 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(\ensuremathω)=\ensuremath∫0\ensuremath∞dte\ensuremath−i\ensuremathωtU(t)U(\ensuremath{\omega})=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dt}{e}^{\ensuremath{-}i\ensuremath{\omega}t}U(t)U(\ensuremathω)=\ensuremath∫0\ensuremath∞dte\ensuremath−i\ensuremathωtU(t) is written in the form iU(\ensuremathω)=[\ensuremathω\ensuremath−L2(\ensuremathω)]\ensuremath−1iU(\ensuremath{\omega})={[\ensuremath{\omega}\ensuremath{-}{L}_{2}(\ensuremath{\omega})]}^{\ensuremath{-}1}iU(\ensuremathω)=[\ensuremathω\ensuremath−L2​(\ensuremathω)]\ensuremath−1, and the frequency-dependent effective Liouvillian L2(\ensuremathω){L}_{2}(\ensuremath{\omega})L2​(\ensuremathω), or "memory operator," is expanded in powers of the reduced density. It is argued that in treating L2(\ensuremathω){L}_{2}(\ensuremath{\omega})L2​(\ensuremathω) 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 L2(\ensuremathω){L}_{2}(\ensuremath{\omega})L2​(\ensuremathω), one allows the radiator to interact simultaneously with two, three,... perturbers.
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 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.
We study the time evolution of the reduced density matrix for the Ohmic spin boson model out of an uncorrelated but otherwise arbitrary initial state. We consider arbitrary bias $\ensuremath{\epsilon}$ and tunneling $\mathrm{\ensuremath{\Delta}}$ at zero temperature for a weak coupling $\ensuremath{\alpha}$ to the bosonic bath. Using the real-time renormalization group method, we present a consistent weak-coupling expansion one order beyond the Bloch-Redfield approximation within a renormalized perturbation theory with analytical results covering the whole crossover regime from small times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\ll}1$ to large times $\mathrm{\ensuremath{\Omega}}t\ensuremath{\gg}1$, where $\mathrm{\ensuremath{\Omega}}=\sqrt{{\ensuremath{\epsilon}}^{2}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}^{2}}$ denotes the Rabi frequency in terms of the renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$. In addition, for exponentially small or large times, we perform a nonperturbative resummation of all logarithmic terms. We show that standard Born approximation schemes calculating the effective Liouvillian of the kinetic equation up to first order in $\ensuremath{\alpha}$ are not sufficient to account for various important corrections one order beyond the Bloch-Redfield solution. (1) The resummation of all secular terms $\ensuremath{\sim}{(\mathrm{\ensuremath{\Gamma}}t)}^{n}$ is necessary to obtain the correct exponential decay of all terms of the time evolution with decay rate $\mathrm{\ensuremath{\Gamma}}$ or $\mathrm{\ensuremath{\Gamma}}/2$, together with the correct pre-exponential functions. (2) The resummation of all logarithmic terms at high and low energies leads to a renormalized tunneling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}$ and to pre-exponential functions of logarithmic and power-law form. (3) The fact that two eigenvalues of L(E)L(E)L(E) are close to each other by $O(\mathrm{\ensuremath{\Gamma}})$ requires degenerate perturbation theory for times $\mathrm{\ensuremath{\Gamma}}t\ensuremath{\sim}O(1)$, where certain terms of the Liouvillian in $O({\ensuremath{\alpha}}^{2})$ are needed to calculate the stationary state and the time evolution of the nonoscillating purely decaying modes up to $O(\ensuremath{\alpha})$. In contrast to the zero-bias case, we find two further interesting results for the time dynamics of the oscillating modes. (4) The terms of the pre-exponential functions with a strong time dependence show a leading long-time tail $\ensuremath{\sim}\ensuremath{\alpha}/(\mathrm{\ensuremath{\Omega}}t)$, besides other subleading terms $\ensuremath{\sim}\ensuremath{\alpha}/{(\mathrm{\ensuremath{\Omega}}t)}^{2}$ well-known from the zero-bias case. (5) The terms of the pre-exponential functions with a weak (logarithmic) time dependence vary according to a power law ${(\frac{1}{\mathrm{\ensuremath{\Omega}}t})}^{2\ensuremath{\alpha}\frac{{\ensuremath{\epsilon}}^{2}}{{\mathrm{\ensuremath{\Omega}}}^{2}}}$ for exponentially large times. The power-law exponent depends on the bias and has to be contrasted to the one at exponentially small times where it crosses over to the bias-independent result $2\ensuremath{\alpha}$. We discuss that the complexity to calculate one order beyond Bloch-Redfield approximation is rather generic and applies also to other models of dissipative quantum mechanics.
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.