Cumulant Expansions and Pressure Broadening as an Example of Relaxation

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)=(TrBρ)^-1×TrBρe^itHˣ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath {TrB is the trace over bath coordinates, ρ~e^-βH is the density matrix, and Hˣ is the quantum-mechanical Liouvillian: Hˣ( )=[H, ( )]; H is the total Hamiltonian for the system radiator plus bath}; $U(t)$ is written as U(t)=T_→exp[i∫0ᵗdt^'L(t^')], L(t)=Hₛˣ+R(t) (T_→ 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 Hₛˣ 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)=R̄+ ̃ \~R(t), where R̄≡lim_T→∞[(1/T)∫0TdtR(t)], the spectrum is written as the sum of its impact approximation, determined by R̄, plus a correction expanded in powers of ̃ \~R(t), which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.