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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Dynamical tunneling in a system with non-monotonous potential and impenetrable walls

Andrey T. Bagmanov, Andrey L. Sanin, A. A. Smirnovsky

Published Apr 3, 2006• 0 citations
Publisher

Abstract

Dynamical tunneling of a quantum wave packet in the system with distributed potential and impenetrable walls is discussed. This potential is specified as algebraic difference of quadratic and cubed functions. It is limited on finite length and incorporates local minimum and maximum. Potential distribution in the vicinity of local minimum forms asymmetric well, and barrier originates into domain with maximum. Initial Gaussian wave packet is located into the potential minimum and tunneling comes through potential maximum. Numerical integration of Schrodinger equation was carried out at zero conditions on system walls. For high barrier, the wave packet oscillates in the well for a long time and its small portions passes through the barrier. If the barrier height is small, tunneling takes place in a short time interval. Expectation coordinate and field velocity, probability density at separate points as function of time, control of uncertainty relation and normalization condition were calculated in detail.

DOI
10.1117/12.676299

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Dynamical tunneling of a quantum wave packet in the system with distributed potential and impenetrable walls is discussed. This potential is specified as algebraic difference of quadratic and cubed functions. It is limited on finite length and incorporates local minimum and maximum. Potential distribution in the vicinity of local minimum forms asymmetric well, and barrier originates into domain with maximum. Initial Gaussian wave packet is located into the potential minimum and tunneling comes through potential maximum. Numerical integration of Schrodinger equation was carried out at zero conditions on system walls. For high barrier, the wave packet oscillates in the well for a long time and its small portions passes through the barrier. If the barrier height is small, tunneling takes place in a short time interval. Expectation coordinate and field velocity, probability density at separate points as function of time, control of uncertainty relation and normalization condition were calculated in detail.
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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.
We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing Mathematica we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.