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AIP conference proceedings

Dissipative quantum noise in a parametric oscillator

Peter Hänggi A, Christine Zerbe

Published Jan 1, 1993• 4 citations
Publisher

Abstract

In this paper we investigate exact solutions for a parametric quantum oscillator. Without dissipation we focus on the propagator and the variances of momentum and position. In the presence of dissipation (Ohmic heat bath) we apply the influence‐functional method due to Feynman and Vernon to obtain exact expressions for the time evolution of the reduced density matrix. Knowing this density matrix we calculate and discuss the variances in presence of friction.

DOI
10.1063/1.44641

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In this paper we investigate exact solutions for a parametric quantum oscillator. Without dissipation we focus on the propagator and the variances of momentum and position. In the presence of dissipation (Ohmic heat bath) we apply the influence‐functional method due to Feynman and Vernon to obtain exact expressions for the time evolution of the reduced density matrix. Knowing this density matrix we calculate and discuss the variances in presence of friction.
We study the quantum fluctuational properties of a parametric oscillator with and without coupling to an Ohmic environment. After considering the momentum and coordinate variances as a function of initial squeezing for the undamped dynamics, we invoke the functional integral method to derive the fully exact reduced density matrix for parametric dissipative quantum Brownian motion, covering the whole temperature regime from T=0 up to the classical limit at room temperatures. Moreover, we present the exact result for the quantum master equation for both the density matrix and the corresponding Wigner function. The time evolution of the covariance matrix elements of damped quantum fluctuations is studied numerically. These variances undergo within the regime of global stability asymptotic, periodic oscillations. As an interesting result, we find that the minima of these oscillations fall below the corresponding thermal equilibrium values.
The exact form of a quantum propagator of a quantum oscillator, interacting with a bosonic bath consisting of N -distinguished quantum oscillators with different frequencies, is obtained in the Heisenberg picture. A reduced density matrix for the oscillator is obtained. The kernel or Greenʼs function connecting the initial density matrix of the oscillator to the density matrix in an arbitrary time is obtained, and its connection to the Feynman–Vernon influence functional is discussed. A weak coupling regime and squared mean values for position, momentum, and energy of the oscillator are obtained in equilibrium.
The microscopic origin of dissipation of a driven quantum many-body system is addressed in the framework of a parametric banded random matrix approach. We find noticeable violations of the fluctuation-dissipation theorem and we observe also that the energy diffusion has a markedly non-Gaussian character. Within the Feynman-Vernon path integral formalism and in the Markovian limit, we further consider the time evolution of a slow subsystem coupled to such a ‘bath’ of intrinsic degrees of freedom. We show that dissipation leads to qualitative modifications of the time evolution of the density matrix of the slow subsystem. In either the spatial, momentum or energy representation the density distribution acquires very long tails and tunneling is greatly enhanced.
Usually one finds that dissipation tends to make a quantum system more classical in nature. In this paper we study the effect of momentum dissipation on a quantum system. The momentum of the particle is coupled bilinearly to the momenta of a harmonic oscillator heat bath. For a harmonic oscillator system we find that the position and momentum variances for momentum coupling are respectively identical to momentum and position variances for spatial friction. This implies that momentum coupling leads to an increase in the fluctuations in position as the temperature is lowered, exactly the opposite of the classical like localization of the oscillator, found with spatial friction. For a parabolic barrier, momentum coupling causes an {\it increase} in the unstable normal mode barrier frequency, as compared to the lowering of the barrier frequency in the presence of purely spatial coupling. This increase in the frequency leads to an enhancement of the thermal tunneling flux, which below the crossover temperature becomes exponentially large. The crossover temperature between tunneling and thermal activation {\it increases} with momentum friction so that quantum effects in the escape are relevant at higher temperatures.
A test system is assumed to interact with a heat bath consisting of harmonic oscillators or an equivalent bath with a proper frequency spectrum producing a Gaussian-Markoffian random perturbation. The effect of reaction of the test system to the bath is considered in the high temperature approximation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression for the reduced density matrix of the test system. The result affords a basis to clarify the relationship between the stochastic and the dynamical approaches to treat the problem of partial destruction of quantum coherence of a system interacting with its environment.