We present a formalism to analyze the effects of collisions and intramolecular energy transfer and dephasing processes (
T1 and
T2 relaxation) on level-population dynamics and unimolecular decay of laser-driven gas molecules. Utilizing a Boltzmann or optical Bloch equation for the density matrix to describe laser excitation of a relaxing multilevel absorber, we develop a generalized master equation (GME) to describe the dynamics of level population evolution for multilevel absorbers having arbitrary numbers of quantum levels
N. Because the resulting level-population equations explicitly retain phase memory, the effect of complete or partial phase-coherent driving or incoherent driving by the laser field are all included within the GME solutions. The formalism provides straightforward means to calculate time-dependent solutions for arbitrary
N, where a single expression is valid on transient, post transient, approach to steady state, and steady-state time scales. We illustrate the behavior of these solutions for the special case of three- and eight-level systems, but the method is not restricted with regard to number of levels. In both this and the following paper, we use specific examples and general arguments to show that an ordinary master equation (OME) is invalid even under apparently favorable conditions including saturating field strengths. We demonstrate that, for an
N-level absorber with given coupling elements, there are
2N [or
2(N+1)] independent saturation solutions and that special conditions on the relaxation times are required to produce the ordinary equal-population solution. Explicit forms for the steady-state radiative-transition matrix $W(\ensuremath{\lambda}=0)$ are derived for arbitrary
N, and general arguments are given to show that the correct description of the dynamics given by the GME results from matrix singularities in $W(\ensuremath{\lambda}=0)$ which can never occur in an OME or similar rate equations.
We have recast a linear quantum-mechanical Boltzmann equation, describing the dynamics of a gas of
N-level atoms absorbing light between an arbitrary pair of levels, into the form of a generalized master equation. The atoms are permitted to undergo both phase-changing and energy-changing collisions with a heat bath of inert molecules. The possibility of unimolecular decay is incorporated into the Boltzmann equation. The solutions of the generalized master equation give an accurate description of the system for short as well as long times and for arbitrary light-field intensity. Eigenvalues obtained from the solution of the
N-level master equation are shown to have particular significance with regard to the scattered electromagnetic field, with the imaginary parts of the complex eigenvalues providing the location and the real parts of the eigenvalues giving the widths of scattered radiation bands. The dynamics and scattered field for a two-level system is discussed in detail. Two collision parameters giving the frequency of phase-changing and energy-changing collisions are important. For special values of these parameters, our results reduce to scattered field spectra calculated previously. When unimolecular decay is included, one identifies the smallest eigenvalue of the generalized master equation with the macroscopic rate constant. Observation of the scattered field for different incident field intensities would permit separation of radiative and thermal effects in an enhanced reaction rate, without requiring an accurate temperature measurement.
We consider a gas of molecules absorbing light from a monochromatic laser source and also undergoing both phase randomizing and thermalizing collisions with a heat bath of inert molecules. The generalized master equation formalism, used previously to study a system of this type with a two-level absorber, is extended to an absorber with N equally spaced levels. The possibility of unimolecular decay from any or all of the N levels is included in the formalism. The well-known relaxation model of Rubin and Shuler and the chemical reaction model of Montroll and Shuler are built into the formalism as special cases when the light source is turned off. Specific calculations for an eight-level model are presented and compared with results expected for an incoherent light source. A coherent effect which increases the upper level populations at steady state is found to exist. A corresponding increase in the macroscopic chemical rate constant is calculated for the coherent model when unimolecular decay from the top level is included. A particularly simple and general result is found for the power absorbed by a harmonic oscillator. It is also shown that, for the N-level model, one does not necessarily find equal populations in the limit of infinite light field intensity.
Using the generalized master equation (GME) formalism, we analyze several approximations for the solutions of Boltzmann or optical Bloch equations. Included in our analysis are the Wilcox-Lamb equations, ordinary master equation (OME), and a new approximation which we designate as the renormalized small-eigenvalue (RSE) approximation. We demonstrate that in multilevel systems, the GME formalism allows direct systematic calculation of radiative-transition coefficients required in the Wilcox-Lamb equations, including all multilevel phase-coherent transitions. However, serious limitations in the applicability of the Wilcox-Lamb approximation are noted both in the high-pressure domain, as an asymptotically exact equation, and for low pressures, as a coarse-grained approximation. The Wilcox-Lamb equations, generally assumed to be valid in a high-pressure domain, are shown to fail in the presence of very efficient inelastic (
T1) relaxation processes regardless of the pressure. Under similar conditions the presence of efficient inelastic processes is also shown to invalidate the use of an OME even when inelastic cross sections are somewhat smaller and the Wilcox-Lamb equations are valid. In particular, we show that when energy losses to the heat bath are severe, an ordinary master equation, traditionally believed to provide a valid description of radiative contributions to level-population evolution when the ratio of Rabi to elastic collision frequency is small, is not valid under any conditions. On this ground we strongly question the recent use of a multilevel OME description to explain the existence of an energy threshold for laser-induced decomposition of S
F6. Finally, we show that for low gas pressures where Rabi oscillations are a prominent feature of the exact solutions, neither the OME nor the Wilcox-Lamb approximation provides an adequate coarse-grained solution. We show that the Wilcox-Lamb solution is in error primarily because of a circumstance not previously recognized, namely, a systematic large error in the way exponential terms contributing to the solution are normalized. A closely related approximation, the RSE approximation, which we present here for the first time, is shown to provide an accurate coarse-grained approximation to the fully coherent GME solutions.
We consider the effects on the energy absorbed and the distribution of energy resulting from a phased response of a molecule to a driving laser field. Contrasting a Landau–Teller collision model with a strong collision model we compare the dynamic and steady-state solutions to our generalized master equation with solutions to an ordinary master equation, for truncated (finite-level) and nontruncated (infinite-level) harmonic oscillators. We find significant coherent enhancement in upper energy level populations which is most pronounced when an intense but below saturation laser field is present; these coherent effects, predicted for truncated oscillator level populations, are contrasted with the infinite level cases. We show that at steady state an ordinary master equation predicts the average energy of the harmonic oscillator correctly but fails to predict the correct population distribution of energy states.
The response of a collision-broadened gas sample to driving coherent radiation is studied theoretically, taking into account effects of coherent excitations of two or more atoms (or molecules). In analogy to the Bloch-type master equation for one-atom coherences, describing the motion of a single atom "dressed" by the relevant (incident and detected) field modes, a master equation is derived for two-atom coherences, including an effective interaction of the Bethe-Salpeter---type, accounting for the mutual interaction of the coherently driven pair. The master equation includes also all symmetrization effects owing to resonance exchange between identical atoms, and is limited to nonreactive gas atoms (or molecules) undergoing binary collisions, including otherwise all (internal and translational) relaxation effects. The self-energy kernels are expressed in a nonperturbative fashion, in terms of binary-collision scattering amplitudes, and include renormalization effects due to coincidence of radiative couplings with the collisions (optical and radiative collisions). The concept of two-atom coherences is generalized to higher coherence ranks by constructing a hierarchy of master equations, including vertex operators that upgrade or downgrade the coherence rank, as a prelude to a diagrammatic method for calculating continuous-wave spectra. This hierarchy is compared with prevalent Agarwal-type master equations, based on the Dicke pseudospin method, and used in the study of two-level atoms.