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Physical Review A

Wave-packet localization in nonlinear external potentials with dissipation

Wei-Min Zhang, D.H. Feng

Published Aug 1, 1995• 8 citations
Publisher

Abstract

The spreading of a wave packet in various nonlinear external potentials with dissipation is studied. The dynamical description of this spreading is given by the use of the squeezed states and the semiquantal approach. We show that the position uncertainty of a Gaussian wave packet is dictated by quantum correlation and the behavior of its evolution can be vividly displayed. For various external potentials with dissipation, we find that the spreading of the wave packet is inevitably suppressed.

DOI
10.1103/physreva.52.1746

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The spreading of a wave packet in various nonlinear external potentials with dissipation is studied. The dynamical description of this spreading is given by the use of the squeezed states and the semiquantal approach. We show that the position uncertainty of a Gaussian wave packet is dictated by quantum correlation and the behavior of its evolution can be vividly displayed. For various external potentials with dissipation, we find that the spreading of the wave packet is inevitably suppressed.
Quantum dynamics of wave packet in one-dimensional system with distributed potential and dissipation is investigated in context of Schrodinger-Langevin-Kostin equation. Quadratic potential and double well potential bounded by external impenetrable walls are represented as examples. Numerical simulations are carried out for initial Gaussian packet at different values of dissipation parameters. For the quantum oscillator with dissipation the asymptotic dynamical limit in time is established. The mean values of coordinate and velocity tend to zero but uncertainty relation tends to minimal value 0.5 (in atomic units). Oscillations and resonances at external time-dependent action were studied. Dynamical properties of system with double well potential are investigated at different frequencies of external action. Fourier spectra of mean coordinate are characterized by sharply expressed singular components.
The quantum dynamics of a simplest dissipative system, a particle moving in a constant external field , is exactly studied by taking into account its interaction with a bath of Ohmic spectral density. We apply the main idea and methods developed in our recent work [1] to quantum dissipative system with constant external field. Quantizing the dissipative system we obtain the simple and exact solutions for the coordinate operator of the system in Heisenberg picture and the wave function of the composite system of system and bath in Schroedinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions with Heisenberg picture method and thereby the meaning of the wavefunction governed by it is clarified by analyzing the effect of the Brownian motion. Especially, the general effective Hamiltonian for the case with arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting fact that the dissipation suppresses the wave packet spreading.
We have studied the quantum dissipative problem of a Gaussian wave packet under the influence of a harmonic potential. A phenomenological approach to dissipation is adopted in the light of the well-known model in which the environment is composed of a bath of noninteracting harmonic oscillators. As one of the effects of the coupling to the bath is the evolution of an initially pure wave packet into a statistical mixture, we estimate the characteristic time elapsed for this to occur for different regimes of temperature, damping, and also different initial states.
We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr\"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.