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.
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.
The generalized master equation (GME) describing the time evolution of energy level populations for a laser-pumped molecular gas, suffering phase-changing and thermalizing collisions, is analyzed in the weak radiation field limit for a two-level absorber. We show that the GME reduces to an ordinary master equation (OME) when the absorber response is perturbed by collisions, by phase interruptions in the light field, or by a combination of both. The GME to OME reduction is possible only in the low intensity and weak collision limits thereby placing a stringent limitation on the use of ordinary master equations to obtain accurate estimates of the evolution of energy level populations.
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 have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.
We examine the effect of including phase-coherent coupling between a driving laser field and a gas of multilevel molecules (
N=20,40) on a nontransient time scale. Although multiphoton enhancement of upper energy-level populations is known to be negligible for systems containing a small number of levels (
N=3,4), such coherent enhancements increase strongly as a function of the number of consecutive radiatively coupled levels, and may play a significant role in recently observed laser-induced chemical reactions.