A theory is developed to describe the interplay of vibrational relaxation and luminescence occurring simultaneously in a molecule, in terms of a master equation involving true sinks of probability. Specifically, the basic equation is the Montroll-Shuler equation augmented by the addition of sink terms which can be nonlinear as well as linear in the vibrational energy. These terms describe radiative and nonradiative decay and expressions for the former are derived explicitly in terms of Franck-Condon factors. Exact solutions in terms of known functions are obtained in the linear case for several physically motivated initial vibrational distributions, viz., $\ensuremath{\delta}$-function, Boltzmann, Poisson, and Laguerre. Two perturbation schemes are developed to analyze the nonlinear case, one of which is useful for small nonlinearities and the other for arbitrarily large nonlinearities but for low temperatures. Illustrative applications of the theory include the exact calculation of time- and frequency-resolved emission spectra (linear decay calculation) and the perturbation analysis of quantum yields in the presence of strongly energy-dependent nonradiative transitions (nonlinear decay calculation).
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 microscopic theory for radiative lifetimes and energy transfer rates in molecular aggregates is developed. Using an effective Hamiltonian, we show that the radiative decay rate of isolated small aggregates with N molecules is N\ensuremath{\gamma}, where \ensuremath{\gamma} is the radiative rate of a single molecule. This linear dependence on N saturates when the aggregate size becomes comparable with the wavelength of light associated with the optical transition, and for large N the rate attains a limiting value independent of N. Intermolecular dephasing processes are incorporated by introducing a thermal bath and solving for the time evolution of the density matrix with use of an effective Liouville operator. Intermolecular dephasing destroys the coherent interaction of the aggregate with the radiation field and the rate changes from N\ensuremath{\gamma} to \ensuremath{\gamma} as the dephasing rate increases. A coherence length which depends on the strength of intermolecular interactions and on the dephasing rate is defined and calculated.
We formulate the weak intramolecular coupling Förster resonance energy transfer theory in a form suitable for calculating the ultrafast non-linear response of molecular systems. This is done through a formally exact factorization of the time-dependent molecular statistical operator into the system and bath components. Combining this factorization with unperturbed environment evolution, we generalize the traditional Förster master equation for the state population probabilities into a complete master equation for the system's reduced statistical operator. The traditional Förster theory applies in the limit where the intermolecular coupling is weak and the system-bath coupling is strong. Our derivation explicitly yields a time-nonlocal Förster-type master equation that remains valid even in the limit of vanishing system-bath coupling. The theory predicts a rapid initial coherent evolution of populations arising from a transient initial coherence-dependent term, which induces a "slippage" of the initial condition that persists during subsequent rate-controlled transfer. Comparison with exact numerical results confirms the clear improvement of the present generalization over earlier formulations of the Förster theory and delineates its range of validity.
The approach to equilibrium of a quantum mechanical system in interaction with a bath is studied from a practical as well as a conceptual point of view. Explicit memory functions are derived for given models of bath couplings. If the system is a harmonic oscillator representing a molecule in interaction with a reservoir, the generalized master equation derived becomes an extension into the coherent domain of the well-known Montroll–Shuler equation for vibrational relaxation and unimolecular dissociation. A generalization of the Bethe–Teller result regarding energy relaxation is found for short times. The theory has obvious applications to relaxation dynamics at ultra-short times as in observations on the femtosecond time scale and to the investigation of quantum coherence at those short times. While vibrational relaxation in chemical physics is a primary target of the study, another system of interest in condensed matter physics, an electron or hole in a lattice subjected to a strong DC electric field that gives rise to well-known Wannier–Stark ladders, is naturally addressed with the theory. Specific system–bath interactions are explored to obtain explicit details of the dynamics. General phenomenological descriptions of the reservoir are considered rather than specific microscopic realizations.