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Journal of Physics A Mathematical and General

Dissipative tunnelling of the inverted Caldirola-Kanai oscillator

Sotirios Baskoutas, A. D. Jannussis, R Mignani

Published Mar 21, 1994• 32 citations
Publisher

Abstract

We discuss, in the phase time approach, quantum tunnelling in the presence of dissipation for an inverted oscillator with Caldirola-Kanai damping. The exact expressions of time delay, traversal time and effective tunnelling velocity are derived. Some paradoxical aspects of tunnelling related to the particle speed in crossing the barrier-such as the Hartmann-Fletcher effect-are briefly considered.

DOI
10.1088/0305-4470/27/6/039

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We discuss, in the phase time approach, quantum tunnelling in the presence of dissipation for an inverted oscillator with Caldirola-Kanai damping. The exact expressions of time delay, traversal time and effective tunnelling velocity are derived. Some paradoxical aspects of tunnelling related to the particle speed in crossing the barrier-such as the Hartmann-Fletcher effect-are briefly considered.
In order to study the behaviour of the inverted Caldirola-Kanai Hamiltonian in the quantum tunnelling effect, the authors consider a wavepacket as the initial state and they calculate exactly the probability density. They also obtain the transmission and reflection probability, the expectation value of the particle's energy and the sojourn time which appears to be an increasing function of the dissipation parameter gamma . A similar result is also obtained by considering a normalized plane wave as the initial state.
Using the evolution operator method, we derive the exact propagator of the generalized parametric oscillator in its more general form. This result is exploited to obtain the exact wavefunction of a damped and driven, inverted harmonic oscillator of the Caldirola-Kanai type, taking a Gaussian wavepacket as the initial state. We discuss the tunnelling process of such a system. The probability density and the persistence probability are evaluated. The expression for the sojourn time is derived for a small external force, and is the sum of two terms, whose explicit forms are obtained in the case of an extended wavepacket. The first term is an increasing function of the dissipation parameter gamma , whereas the second one is strictly due to the presence of the driving force.
We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.
Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.