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Russian Journal of Physical Chemistry B

Relaxation of excited molecules in a medium with memory at short and long times

A. S. Moskalenko, Stanislav Ya Umanskii

Published Feb 1, 2012• 1 citation
Publisher

Abstract

Two qualitatively different approaches to the dynamics of vibrational wave packets in the presence of medium-induced relaxation, with taking into account relaxation memory effects and without taking them into account (Makovian approximation) were compared for a molecular system with one vibrational degree of freedom (the Morse oscillator) in a medium. The time evolution of the populations of levels, mean system energy, and response of “pumping-probing” experiments were calculated. It was found that, as distinct from the Markovian approximation, the approach including memory effects can predict the evolution of a molecular system satisfying the detailed equilibrium principle. The two approaches specified also predict the existence of a qualitative difference of phase characteristics in the behavior of the system at short times.

DOI
10.1134/s1990793112010216

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Two qualitatively different approaches to the dynamics of vibrational wave packets in the presence of medium-induced relaxation, with taking into account relaxation memory effects and without taking them into account (Makovian approximation) were compared for a molecular system with one vibrational degree of freedom (the Morse oscillator) in a medium. The time evolution of the populations of levels, mean system energy, and response of “pumping-probing” experiments were calculated. It was found that, as distinct from the Markovian approximation, the approach including memory effects can predict the evolution of a molecular system satisfying the detailed equilibrium principle. The two approaches specified also predict the existence of a qualitative difference of phase characteristics in the behavior of the system at short times.
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 investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.
Effective descriptions accounting for the evolution of quantum systems that are acted on by a bath are desirable. As the number of bath degrees of freedom increases and full quantum simulations turn out computationally prohibitive, simpler models become essential to understand and gain an insight into the main physical mechanisms involved in the system dynamics. In this regard, vibrational decoherence of an I2 diatomics is tackled here within the framework of Markovian quantum state diffusion. The I2 dynamics are analyzed in terms of an effective decoherence rate, Λ, and the specific choice of the initial state, in particular, Gaussian wave packets and two-state superpositions. It is found that, for Markovian baths, the relevant quantity regarding decoherence is the product of friction (η) and temperature (T); there is no distinction between varying one or the other. It is also observed that decoherence becomes faster as the energy levels involved in the system state correspond to higher eigenvalues. This effect is due to a population redistribution during the dynamical process and an eventual irreversible loss of the initial coherence. These results have been compared with those available in the literature from more detailed semiclassical IVR simulations, finding a good agreement.
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.
We have studied by using quantum mechanics the dynamics of a driven damped Morse oscillator, whose classical solutions exhibit bistable and chaotic behavior. In this treatment the Morse Hamiltonian, expressed in terms of the generators of an su(2) algebra, is coupled to a bath of harmonic oscillators and driven by a classical field, simulating an infrared laser. We have derived the generalized master equation from the Liouville equation within the Markoffian and weak‐coupling approximation. The time dependence of the average energy of the Morse oscillator is obtained by solving the time evolution equations for the matrix elements numerically as a function of the relaxation times, laser amplitude and frequency. Bistability does not show up as a hysteresis loop in the quantum solution, but as a bimodal population distribution in vibrational energy levels. As the field amplitude increases the time series of the average Morse energy becomes increasingly irregular.
Population and phase relaxation in two molecular models is studied. The analysis is based on recent work by Lindblad, in which the time evolution of an N-level quantum system is described using the coresponding generator of a completely positive semigroup. One of the models studied corresponds to a diatomic molecule in a dense medium. New results are obtained for the influence of two quantum transitions on population and phase relaxation. The other model studied corresponds to the case of exchange dephasing in a polyatomic molecule embedded in a dense medium. The formalism permits inclusion of coupling terms neglected by Harris, Shelby, and Cornelius; it is shown that inclusion of these terms leads to a rich range of line shapes, which information is not available from the approximation used by Harris et al.