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AIP conference proceedings

Dissipative molecular dynamics: Quantal versus classical treatment

Mingwhei Tung, Jian‐Min Yuan, James F. Heagy

Published Jan 1, 1987• 2 citations
Publisher

Abstract

We have studied by using quantum mechanics the dynamics of a driven damped Morse oscillator, whose classical solutions exhibit bistable and chaotic behavior. In this treatment the Morse Hamiltonian, expressed in terms of the generators of an su(2) algebra, is coupled to a bath of harmonic oscillators and driven by a classical field, simulating an infrared laser. We have derived the generalized master equation from the Liouville equation within the Markoffian and weak‐coupling approximation. The time dependence of the average energy of the Morse oscillator is obtained by solving the time evolution equations for the matrix elements numerically as a function of the relaxation times, laser amplitude and frequency. Bistability does not show up as a hysteresis loop in the quantum solution, but as a bimodal population distribution in vibrational energy levels. As the field amplitude increases the time series of the average Morse energy becomes increasingly irregular.

DOI
10.1063/1.36852

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We have studied by using quantum mechanics the dynamics of a driven damped Morse oscillator, whose classical solutions exhibit bistable and chaotic behavior. In this treatment the Morse Hamiltonian, expressed in terms of the generators of an su(2) algebra, is coupled to a bath of harmonic oscillators and driven by a classical field, simulating an infrared laser. We have derived the generalized master equation from the Liouville equation within the Markoffian and weak‐coupling approximation. The time dependence of the average energy of the Morse oscillator is obtained by solving the time evolution equations for the matrix elements numerically as a function of the relaxation times, laser amplitude and frequency. Bistability does not show up as a hysteresis loop in the quantum solution, but as a bimodal population distribution in vibrational energy levels. As the field amplitude increases the time series of the average Morse energy becomes increasingly irregular.
We have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.
We study quantum mechanical behavior of a Morse oscillator coupled to a bath of harmonic oscillators and driven by an infrared laser. The purpose is to compare quantum behavior of such a system with the classical and semiclassical solutions of a driven damped Morse oscillator. The general picture coming out of classical and semiclassical studies is that the total energy of the Morse oscillator in steady states or steady oscillations show jump and hysteretic behavior, as expected in a cusp catastrophe, when laser intensity or frequency is varied. Along with the bistable behavior we have also found period-doubling bifurcations leading to chaos on one or both branches. In the quantum treatment a generalized master equation is derived from the Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an SU(2) algebra. This algebra and the Markoffian approximation allows us to derive the equation of motion of the reduced density matrix, which is solved numerically. Bistable behavior does not exist in this quantum treatment, but solutions do show jump behavior and bifurcation phenomenon. Experimental implications of these studies are also discussed.
A new quantum statistical formulation of a damped classically driven Morse oscillator is presented. The theory is based on the realization of the Morse oscillator in terms of generators of an SU(2) Lie algebra, which allows us to construct the spin coherent states for the Morse oscillator. The c-number equivalents of the master equation in the form of the Fokker–Planck and Langevin equations have been derived and solved in the mean field limit to demonstrate the existence of multiple steady states and the associated molecular bistability. The nonstationary solution derived under adiabatic elimination of relevant variable and secular approximation is also presented. Some spectral characteristics such as shift and linewidth due to phase fluctuations have been calculated.
In this contribution, we discuss the statistical mechanical implications of the mean field Ehrenfest method of quantum/classical dynamics for a quantum harmonic oscillator in a classical heat bath using the Brownian motion Hamiltonian as a model. A mean field quantum/classical master equation is derived and compared to the corresponding Redfield master equation, and the deficiencies of the quantum/classical approach pointed out by analyzing the nature of energy/population relaxation and decoherence.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.