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Kluwer Academic Publishers eBooks

Vibration-Induced Excited State Decay

Miroslav Menšı́k, Stanislav Nešpůrek

Published Jan 12, 2006• 0 citations
Publisher

Abstract

Time evolution of excited state population was studied theoretically for the system of two electronic levels coupled to two vibrational modes, which are also attached to a heat bath. In the excited state, the mutual coupling of the vibrational modes was included. It was demonstrated that excited state population relaxes faster if interacting vibrational mode dissipates its energy via vibrational mode of a smaller eigenfrequency. It was also demonstrated that fast component of excited state depopulation cannot be achieved via coherent mode-mode coupling, if the second mode is not directly coupled to the electronic inter-state transition.

DOI
10.1007/1-4020-2396-0_31

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Time evolution of excited state population was studied theoretically for the system of two electronic levels coupled to two vibrational modes, which are also attached to a heat bath. In the excited state, the mutual coupling of the vibrational modes was included. It was demonstrated that excited state population relaxes faster if interacting vibrational mode dissipates its energy via vibrational mode of a smaller eigenfrequency. It was also demonstrated that fast component of excited state depopulation cannot be achieved via coherent mode-mode coupling, if the second mode is not directly coupled to the electronic inter-state transition.
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.
Adsorbed species on a solid-state surface interact with the large number of substrate modes, which gives rise to thermal relaxation. Commonly, the temporal evolution of the quantum state of the adsorbate is described by a master equation for the level populations (vibrational bond, internal modes, etc). It is pointed out that this approach does not necessarily give a correct account of the coupling to the solid when the effective level widths become comparable to the level separations, or larger. It is shown that the evolution of the populations does not decouple any more from the time evolution of the coherences (off-diagonal matrix elements), which implies that a random-phase approximation cannot be justified, and that the density matrix of the adsorbate is not determined only by the golden-rule transition constants. Especially, the line profiles turn out to be very sensitive to the coherence-coherence couplings. Although the coherences vanish in thermal equilibrium, their time-regression operator, and hence their mutual couplings and their couplings to the populations, which determine the absorption profile, do not. This information is lost in a master-equation treatment of relaxation.
We present expressions for rates of radiationless transitions in two-level systems with arbitrary coupling to a set of damped vibrational modes; calculations are limited to transitions slower than vibrational relaxation. Explicit results are discussed within the harmonic approximation and Condon approximation; systematic methods are introduced for treating non-Condon and anharmonic effects. Results relevant to experiment are (1) significant deviations from the energy-gap law when a small number of vibrational modes is coupled to the transition, (2) dependence of transition rate on homogeneous and inhomogeneous vibrational linewidths, (3) nonexponential decay for transitions with vibrational frequency shifts, and (4) persistence of electronic coherence for long times either at low temperature or when frequency shifts are small. Our approach differs from previous work in the partitioning of the Hamiltonian and in the methods of evaluating operator averages; these differences are discussed in relation to the adiabatic approximation and the irreversibility of the transition.
We present an approach to laser cooling of internal molecular degrees of freedom for vibrationally hot molecules using a sequence of ultrashort laser pulses. It is assumed that the molecules initially occupy different vibrational states with a substantial portion of the molecules being in excited vibrational states. We show that the ultimate aim of increasing the vibrational ground-state population through a reduction of the system's entropy can be achieved through a multistep process. In the first step, we design an ultrashort laser pulse that selectively transfers most of the population of the excited vibrational states to an excited electronic surface; then the field is switched off and the system allowed to relax until most of the excited electronic state population has decayed due to spontaneous emission. By repeating this procedure a few times, the entropy of the system can be substantially reduced and the population of the vibrational ground state increased considerably, even if the lifetimes of the excited electronic states are much greater than the length of a control pulse, i.e., if dissipative effects are negligiable on the time scales of coherent control.
A quantum-mechanical treatment of the dynamics of relaxation of a twolevel mode coupled to a thermal bath is presented. The Zwanzig-Mori projection-operator formalism is employed to derive exact equations of motion for the correlation function and the excess population , where and are the coordinate and number operators associated with the two-level mode. In the van Hove weak-coupling limit it is shown that |G ν|2 and Δn decay exponentially with time constants τ and τ′, respectively. Explicit expressions for τ and τ′ are determined and the relationship between them clearly established. The application of the model to the specific problem of the effects of vibrational relaxation on isotropic Raman spectral lineshapes in dense media is emphasized.
A new general non-perturbative method for the solution of the generalized master equation for a simple two-level model of the electron-phonon system coupled to a thermodynamic reservoir is suggested. The model includes a two-level system interacting linearly with the single-mode phonons coupled to the thermodynamic bath. For the interaction of the bath with the vibrational mode we use a stochastic model generalized to finite temperatures. The results are compared with a direct numerical solution.