Description of detailed temporal excitation dyanmics for coherent excitation, such as is produced by idealized laser radiation, contrasts with evaluation of rate coefficients by means of generalized Golden Rule procedures; it requires an appropriate time-dependent Schroedinger equation. When the atom undergoing excitation is also affected by incoherent processes, such as collisions, this equation no longer suffices. The Heisenberg equations, or equivalent density-matrix equations, permit treatment in which coherence and incoherence play comparable roles in the excitation dynamics. Unlike rate equations, such equations must incorporate complexities that originate in the orientation degeneracy expressed by magnetic quantum numbers. In simple cases of coherent excitation, both for single-photon and multiphoton excitation, the sublevels merely require an average of 2J+1 independent Schroedinger equations. Relaxation couples the independent equations. It has been known for some time that appropriate state-multipole operators can simplify the description of many phenomena connected with optical pumping. This memo discusses application of these multipole operators to the description of Raman (or more general multiphoton) coherent excitation. In some simple limiting cases the equations simplify, but in general one has a hierarchy of coupled multipole polarizations and coherences in place of the populations and coherences that occur as variables in nondegenerate systems. 28 refs., 4 figs.
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.
A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.
Abstract The treatment of transient excitation of transitions between two atomic or molecular energy levels by monochromatic laser radiation under conditions in which phase-destroying collisions predominate is treated by means of numerical and approximate analytical solutions of the equations giving the rate of change of a modified density matrix. The analytical solutions are then used to obtain the transient values of the population of the excited state when Doppler broadening is taken into account. Numerical calculations are carried out for the ratio γ'/γ = 10, where 2γ is the damping constant for diagonal and γ' for non-diagonal elements of the density matrix. This extends the treatm ent given in two previous papers for the situation when γ' = γ, corresponding to very low pressures for which phase-destroying collisions are relatively unimportant.
We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.
Rate equations for the population dynamics of multilevel systems in laser-induced fluorescence spectroscopy are solved by the aid of perturbation theory. This leads to a general expression for the population of the level pumped by the laser radiation. Arbitrary laser pulse shapes as well as collisional processes are admitted. In the first-order approximation three functions depending on time and the properties of the laser pulse remain. These functions have been tabulated for a bell-shaped laser pulse. The results are used in several characteristic situations and are compared with the numerical solutions of the rate equations.