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Annalen der Physik

Vibrational relaxation and decoherence in structured environments: a numerical investigation

Matteo Bonfanti, Keith H. Hughes, Irène Burghardt, Rocco Martinazzo

Published Jul 3, 2015• 11 citations
Publisher

Abstract

Vibrational relaxation is a key issue in chemical reaction dynamics in condensed phase and at the gas‐surface interface, where the environment is typically highly structured and cannot be expressed in terms of a simple friction coefficient. Rather, full knowledge of the coupling of the molecular oscillator to the environment is required, as typically subsumed in the spectral density of the environmental coupling. Here, we focus on harmonic Brownian motion and investigate the effectiveness of classical, canonical position autocorrelation functions to compute the spectral density of the coupling needed to describe vibrational relaxation in complex environments. Classical dynamics is performed on model systems, and several effects are investigated in detail, notably the presence of anharmonicity, the role of a high‐frequency “Debye” cutoff in the environment and the influence of the detailed structure of the latter. The spectral densities are then used in standard independent oscillator Hamiltonian models which are numerically solved at T = 0 K to investigate quantum relaxation and decoherence. image

DOI
10.1002/andp.201500144

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Vibrational relaxation is a key issue in chemical reaction dynamics in condensed phase and at the gas‐surface interface, where the environment is typically highly structured and cannot be expressed in terms of a simple friction coefficient. Rather, full knowledge of the coupling of the molecular oscillator to the environment is required, as typically subsumed in the spectral density of the environmental coupling. Here, we focus on harmonic Brownian motion and investigate the effectiveness of classical, canonical position autocorrelation functions to compute the spectral density of the coupling needed to describe vibrational relaxation in complex environments. Classical dynamics is performed on model systems, and several effects are investigated in detail, notably the presence of anharmonicity, the role of a high‐frequency “Debye” cutoff in the environment and the influence of the detailed structure of the latter. The spectral densities are then used in standard independent oscillator Hamiltonian models which are numerically solved at T = 0 K to investigate quantum relaxation and decoherence. image
The dynamics of a mixed quantum-classical system, in which the classical subsystem interacts with a dissipative bath, is investigated. This description of the dynamics will be appropriate if the details of the bath dynamics are unimportant but its presence plays an important role in the dissipation of energy to the environment. In this dynamical description, which can be simulated employing an ensemble of stochastic surface-hopping trajectories, the strength of the dissipation is controlled by a friction coefficient. We show that if decoherence, whose effects are controlled by the bath friction, is sufficiently rapid, the equation of motion can be reduced to a master equation. Thus, decoherence and the validity of master equation models may be explored as a function of bath friction. We use this framework to study the mechanism of decoherence in a simple model nonadiabatic chemical reaction.
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.
We consider a model of environment-induced dissipationless decoherence of a quantum system where the system is coupled to the bath degrees of freedom via the system Hamiltonian itself. We solve exactly for the reduced density operator of the system for an arbitrary spectral density of the thermal bath and also write down an exact master equation in the Lindblad form. We compare and contrast the above results with those obtained by considering the system frequencies to be randomly modulated as in stochastic models. We observe that a coupling to the bath as above necessarily induces a Kerr-like coherent contribution in the reduced dynamics of the system. This Kerr-like term is a reflection of the quantum nature of the bath and cannot be obtained from stochastic models. For the special case of a harmonic oscillator we consider the influence of decoherence and its relation with phase diffusion. Our numerical results exhibit oscillations in the evolution of system variables which overall is a signature of the quantum nature of the environment.
In open quantum systems, the environment is typically modelled as a collection of harmonic oscillators, whereas realistic environments often exhibit unique non-Markovian effects due to the anharmonicity of the environment. Here we present a framework for modelling anharmonic environments in open quantum systems numerically exactly, in which, via a matching criterion, we map the anharmonic environment onto an effective harmonic one. This results in an effective spectral density that is a collection of shifted bare spectra arising from the non-equidistant energy gaps of the anharmonic environment, along with a unique zero-frequency term that effectively acts as static disorder on the system. This is due to the non-zero variance of the diagonal of the bath coupling operator. In tandem, we develop a framework for investigating the effects of environments comprising a few damped anharmonic modes where the matching criterion fails to work due to the non-Gaussianity of the environment. For both continuous and discrete damped anharmonic environments, this work investigates their unique influence on the system's dynamics, including enhanced non-Markovianity compared to a harmonic approximation, novel effects in the absorption spectrum, and the potential for anharmonic environments to enhance energy transport.
The work is devoted to the investigation of the influence of a heat bath on the physical processes in a quantum system. We use the density matrix theory as one of the most powerfool tool for investigation of quantum relaxation. In the beginning of the work (chapter 2) we mention and recall the most important steps of derivation of the equation of motion for the reduced density matrix (master equation) for an arbitrary quantum system in diabatic representation interacting with the environment modeled by a set of independent harmonic oscillators. Chapter 3 deals with the question of the border between classical and quantum effects and reports on a study of the environmental influence on the time evolution of a coherent state or the superposition of two coherent states of a harmonic oscillator as a simple system displaying the peculiarities of the transition from quantum to classical regime. Chapters 4 and 5 concern the electron transfer (ET) problem, namely the mathematical description of the ET in molecular zinc-porphyrin-quinone complexes modeling artificial photosynthesis (chapter 4) and photoinduced processes in the porphyrin triad (chapter 5). Each chapter starts with an introduction and ends with a brief summary. The main achievements of the present work are summarized in the Conclusions.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.