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The Journal of Chemical Physics

Quantum statistical theory of damped, driven Morse oscillator: Bistable features and spectral characteristics

Deb Shankar Ray

Published Jan 15, 1990• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.458176

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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.
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.
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 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.
The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber PPP representation, the antinormal ordering QQQ representation and the Wigner WWW representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. 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. Illustrative examples for specific initial conditions of the density matrix are provided.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.