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The Journal of Chemical Physics

Interference effects on relaxation in three-level systems: Breakdown of the rate equation description

Biman Bagchi, David W. Oxtoby

Published Aug 1, 1982• 22 citations
Publisher

Abstract

The stochastic Liouville equation and exact quantum mechanical simulations have been used to test the validity of rate law predictions for the vibrational energy relaxation in a three-level system. The model system consists of a low lying ground state which is radiatively coupled to only one of three closely spaced excited states. All three excited states are strongly coupled to baths of known statistical properties (either Gaussian or Poisson). We follow the energy exchange between the excited states after the light is turned off. Two different types of bath correlation have been considered: (A) The couplings between the different levels are the same and (B) the couplings are uncorrelated with each other. We find that relaxation differs for these two cases, in contradication to the prediction of the simple rate law description. We also present the first quantitative study of the interference between different pathways that can occur in such a three-level system. We find that the interference has a considerable effect on the decay of the middle level for the same bath (case A), even in the Markovian limit, while for independent baths (case B) interference becomes important only in the non-Markovian limit. These results are entirely different from the prediction of the conventional rate laws. We also discuss the effect of the excitation on the subsequent relaxation and the coherence transfer processes that lead to an oscillatory decay in the non-Markovian limit.

DOI
10.1063/1.443963

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The stochastic Liouville equation and exact quantum mechanical simulations have been used to test the validity of rate law predictions for the vibrational energy relaxation in a three-level system. The model system consists of a low lying ground state which is radiatively coupled to only one of three closely spaced excited states. All three excited states are strongly coupled to baths of known statistical properties (either Gaussian or Poisson). We follow the energy exchange between the excited states after the light is turned off. Two different types of bath correlation have been considered: (A) The couplings between the different levels are the same and (B) the couplings are uncorrelated with each other. We find that relaxation differs for these two cases, in contradication to the prediction of the simple rate law description. We also present the first quantitative study of the interference between different pathways that can occur in such a three-level system. We find that the interference has a considerable effect on the decay of the middle level for the same bath (case A), even in the Markovian limit, while for independent baths (case B) interference becomes important only in the non-Markovian limit. These results are entirely different from the prediction of the conventional rate laws. We also discuss the effect of the excitation on the subsequent relaxation and the coherence transfer processes that lead to an oscillatory decay in the non-Markovian limit.
The transient four-wave mixing signal from a three-level system coupled to a bath of harmonic oscillators is calculated in the short-pulse limit. This allows an examination of the non-Markovian effects on the relaxation processes in the excited levels between which beating occurs. In particular, it is possible to distinguish between oscillations which are due to oscillatory dephasing, a consequence of the memory effects, and oscillations due to the quantum beats. This also allows one to analyze how these two kinds of oscillations are mixed together. We show that we not only have an amplitude modulation of the beats as in the Markovian case but also deformations of the beats for short delays between the exciting light pulses. These deformations are a consequence of the nonequilibrium motion of the surroundings of the system under optical excitation. \textcopyright{} 1996 The American Physical Society.
We consider light scattering from a multilevel system (e.g., a molecule) interacting with several heat baths. The Liouville space approach is employed and weak incident radiation intensity is assumed, whereas the light scattering process is described in the lowest appropriate order in the interaction between the molecule and the incident and emitted photons. The interaction with other radiation field modes as well as with the thermal baths is treated more fully, such as to account for the correct damping terms and their temperature dependence. The formalism is suited for light scattering from an impurity molecule in both gas and dense phases. The thermal baths induce relaxation within the manifold of excited states as well as transitions to nonradiative channels. Damping due to such transitions (as well as radiative damping) corresponds to interactions with zero temperature baths. The physical interpretation of such a bath depends on the nature of the particular damping process. Thermal relaxation within the excited manifold is induced by interaction with the environment represented by a bath of temperature T. Part of this interaction which is diagonal in the molecular space corresponds to the so-called ’’quasielastic collisions’’ and induces modulations of the molecular energy levels and broadening of the relaxed emission. The remainder, nondiagonal part induces actual transitions between the excited molecular energy levels. The formalism yields in a single calculation expressions for the cross section for both the coherent (Raman) scattering process and the relaxed luminescence and enables us to discuss the effects of different factors such as temperature and quenching processes on the relative yields for these processes. The formalism also allows for interference between different levels of the excited manifold due to coupling of several levels to the same bath, and makes it possible to study the effect of thermal relaxation on these interference phenomena.
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.
The relaxation processes in a quantum system nonlinearly coupled to a harmonic Gaussian-Markovian heat bath are investigated by the quantum Fokker-Planck equation in the hierarchy form. This model describes frequency fluctuations in the quantum system with an arbitrary correlation time and thus bridges the gap between the Brownian oscillator model and the stochastic model by Anderson and Kubo. The effects of the finite correlation time and the system-bath coupling strength are studied for a harmonic model system by numerically integrating the equation of motion. The one-time correlation function of the system coordinate, which is measured in conventional Raman and infrared absorption experiments, already reflects the inhomogeneous character of the relaxation process. The finite correlation time of the frequency fluctuations, however, is directly evident only in the two- and three-time correlation function as probed by multidimensional spectroscopic techniques such as the Raman echo and the fifth-order 2D Raman experiment.
We focus on a dissipative three-level system as a case study to analytically demonstrate that strong system-bath coupling separates the relaxation dynamics of this dissipative quantum system into two distinct regimes: a short-time dynamics that, as expected, accelerates with increasing coupling to the environment, and a slow dynamics that, counterintuitively, becomes prolonged at sufficiently strong coupling. Using the reaction-coordinate polaron-transform mapping, we uncover the general mechanism behind this effect and derive accurate expressions for both relaxation timescales. Numerical simulations confirm our analytical predictions. From a practical perspective, our results suggest that strong coupling to a dissipative bath can autonomously generate and sustain long-lived quantum coherences, offering a promising strategy for bath-engineered quantum state preparation.
In a multi-level laser the pump process usually involves several intermediate levels as well as spontaneous relaxations and an external pump field. These details of the atomic excitation are often irrelevant and can be summarized by an effective heat-bath description and a two-level- laser model instead. Under certain circumstances, however, the heat-bath approximation will be too crude and it cannot account for phenomena such as intensity-noise squeezing and sub- Poissonian field statistics. We show how in the general situation the multi-level description can still be reduced to a two-level description in terms of a non-Markovian master equation. In steady state the atomic excitation is then described by a rate matrix rather than a single rate. The matrix or operator nature of the pump process gives rise to a correlation of the excitation rates with the statistical moments of the laser field. This correlation represents a correction to the standard laser model. In particular, it can be identified as the physical origin of the phenomenon of dynamic noise reduction in multi-level lasers.