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The Journal of Chemical Physics

Comments on a semigroup formalism for the description of phase and population relaxation processes

Ronnie Kosloff, Stuart A. Rice

Published Apr 15, 1980• 41 citations
Publisher

Abstract

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.

DOI
10.1063/1.439701

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The Journal of Chemical Physics•Aug 15, 2014•10 citations
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.
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 study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open NNN-level systems. The Lindblad operators that govern the evolution of the system are expressed in terms of observable relaxation rates, explicit formulas for the decoherence rates due to population relaxation are derived, and it is shown that there are additional, nontrivial constraints on the pure dephasing rates for N>2N>2N>2. Explicit, experimentally testable inequality constraints for the decoherence rates are derived for three- and four-level systems, and the implications of the results are discussed for generic ladder, $\ensuremath{\Lambda}$, and V systems and transitions between degenerate energy levels.
Abstract The influence of intermolecular elastic (T2 processes) and inelastic (T1 processes) interactions on the population relaxation and phase relaxation of a specific molecular mode is studied in terms of a two mode model. Therein one mode (K‐level system) is assumed to undergo T1 and T2 relaxation processes as a consequence of its coupling to a thermal reservoir while the second mode (N‐level system) is not explicitly damped but is coupled to the first one. The quantum statistical equations of motion for the coupled two‐mode system are solved numerically with a special integration scheme in which the damping is considered in first order and the coupling of the modes in infinite order. Energy relaxation and dephasing of the N‐level system are studied as functions of the external relaxation behaviour and of the coupling between the modes. A dynamical decoupling of the two modes is found with decreasing intermolecular dephasing times T2.
The semigroup formalism permits a general description of relaxation processes in quantum mechanical systems coupled to Markovian baths. From its definition, the semigroup formalism should be expressed in terms of the dissipative and relaxation coupling of the eigenstates of the quantum system to the bath modes. Often, for convenience both in calculation and in physical interpretation, it is more straightforward to couple local states of the quantum mechanical system to the bath. Within a simple two-site system, we examine the generalities of these two different coupling schemes, noting situations in which they are the same and situations for which substantial differences occur.
In this paper we consider some features of vibrational relaxation of a guest molecule in a host matrix. The model system involves a harmonic molecule interacting with a harmonic medium. The molecule-medium coupling was handled by the rotating wave approximation considering linear terms in the intramolecular displacements and high terms in the medium displacements. Three specific models for the molecule-medium coupling were considered, which involve single phonon decay, vibron-phonon decay and multiphonon decay. Within the framework of the random phase approximation the Heisenberg equations of motion for the system could be expressed in terms of a unified scheme which is valid for both single phonon and multiphonon processes. Explicit solutions were derived utilizing the Wigner-Weisskopf approximation. This generalized formalism was applied for the study of the time evolution of the distribution, the cooling and the heating processes of the oscillator by a thermal field and for the coupling between vibrational relaxation and infra-red emission.
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.