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Molecular Physics

Quantum decay rates in dense media

Stefan G. Christov

Published Aug 10, 1992• 6 citations
Publisher

Abstract

A quantum theory of decay processes in condensed phases is presented. It is based on a general dynamic rate theory applied to a model in which a particle is bilinearly coupled to a system of harmonic oscillators. Both adiabatic and nonadiabatic processes are considered. The many-dimensional problem is reduced to a one-dimensional case making use of a separable (normal mode) coordinate correlated to the particle coordinate. The rate constant is calculated throughout the temperature range. The results of earlier work using stochastic theory are obtained in the limit of weak particle-medium coupling.

DOI
10.1080/00268979200102021

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A quantum theory of decay processes in condensed phases is presented. It is based on a general dynamic rate theory applied to a model in which a particle is bilinearly coupled to a system of harmonic oscillators. Both adiabatic and nonadiabatic processes are considered. The many-dimensional problem is reduced to a one-dimensional case making use of a separable (normal mode) coordinate correlated to the particle coordinate. The rate constant is calculated throughout the temperature range. The results of earlier work using stochastic theory are obtained in the limit of weak particle-medium coupling.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.
We extend the quantum theory of dissipation in the context of system-reservoir model, where the reservoir in question is kept in a nonequilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which depending on the relaxation of nonequilibrium bath is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the cavity is kept in contact with a nonequilibrium bath.
We extend the quantum theory of dissipation in the context of a system-reservoir model, where the reservoir in question is kept in a non-equilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which, depending on the relaxation of a non-equilibrium bath, is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the cavity is kept in contact with a non-equilibrium bath.
A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces is given. The present theory is primarily concerned with processes involving the fragmentation of two-particle systems and/or the ’’collision’’ of two particles with internal structure, i.e., electronic, spin, and nuclear degrees of freedom. A fully quantum mechanical joint energy space–quantum phase space master equation for two-particle systems is presented. A simpler form of this equation emerges in the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion. (This limit is equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2.) The simpler master equation is converted into a nonlinear quantum Fokker–Planck equation for nonadiabatic systems. Introducing simplifying assumptions, we obtain a quantun analog of the stochastic Liouville equation augmented with irreversible kinetic terms, which describes nonadiabatic transitions and the interruption of coherence in energy space at different points in quantum phase space. The final results are appropriately modified to incorporate the influence of applied time-dependent electric and magnetic fields. The present work provides a formal theoretical framework for examining the role of quantum effects that are neglected in ’’semiclassical’’ kinetic and ’’diffusional’’ theories for nonadiabatic processes.
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 maximum entropy approach to time-dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two-particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2. The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.