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Molecular Physics

Treatment of inelastic collisions of a particle with a quantum harmonic oscillator by density matrix evolution

Janez Mavri, Marc Ferdinand Lensink, Herman J. C. Berendsen

Published Aug 20, 1994• 12 citations
Publisher

Abstract

A density matrix evolution (DME) method (Berendsen, H. J. C., and Mavri, J., 1993, J. phys. Chem., 97, 13464) to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five level quantum harmonic oscillator. The results are compared with the exact quantum calculations of Secrest, D., and Johnson, B. R., 1966, J. chem. Phys., 45, 4556, and with the results of classical mechanics. The DME results are between the results of the full quantum treatment and classical mechanics but closer to the latter. Furthermore, we demonstrate the time reversibility of the DME equations of motion and discuss the importance for the time reversibility of the quantum phase described by the off-diagonal elements of the density matrix.

DOI
10.1080/00268979400100884

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A density matrix evolution (DME) method (Berendsen, H. J. C., and Mavri, J., 1993, J. phys. Chem., 97, 13464) to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five level quantum harmonic oscillator. The results are compared with the exact quantum calculations of Secrest, D., and Johnson, B. R., 1966, J. chem. Phys., 45, 4556, and with the results of classical mechanics. The DME results are between the results of the full quantum treatment and classical mechanics but closer to the latter. Furthermore, we demonstrate the time reversibility of the DME equations of motion and discuss the importance for the time reversibility of the quantum phase described by the off-diagonal elements of the density matrix.
A density matrix evolution(DME) method to simulate the dynamics of quantum systems embedded in a classical environment is presented. The method is applicable when the quantum dynamical degrees of freedom can be described in a Hilbert space of limited dimensionality. The method is applied to the case of proton-transfer reactions in a fluctuating double-well potential [details are given in the following paper in this issue] and compared to other analytical and numerical solutions. The embedding of the quantum system within the classical system by using consistent equations of motion for the classical system with-proper conservation properties is discussed and applied to the one-dimensional collision of a classical particle with a quantum oscillator.
A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville–von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge–Kutta integrator with an adaptive step size control fourth-order Runge–Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.
A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville-von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge-Kutta integrator with an adaptive step size control fourth-order Runge-Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.
The time evolution of the density matrix of the damped harmonic oscillator is studied within the Lindblad theory for open quantum systems. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation of the damped harmonic oscillator. Illustrative examples for specific initial conditions of the density matrix are provided. It is also shown that various master equations for the damped quantum oscillator, for damped collective modes in deep inelastic collisions of heavy ions and in different models of quantum optics are particular cases of the Lindblad equation and that only some of these equations satisfy the quantum mechanical constraints on the diffusion coefficients.
We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.
The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems.