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Journal of Physics A Mathematical and General

Quantum tunnelling of a damped and driven, inverted harmonic oscillator

Sotirios Baskoutas, A. D. Jannussis, R Mignani

Published Dec 7, 1993• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/23/048

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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.
Using the algebraic method, we derive the exact wavefunction of a driven time-dependent inverted harmonic oscillator (TDIHO). With the Gaussian wave-packet as the initial state, the general expressions of the transmission probability and dwell time of the driven TDIHO are obtained. For a weak driving force, the dwell time can be separated into two terms. The first term is for the unperturbed part, and the second one is that from the perturbation, which is the effect from the driving force. Furthermore, with two examples, the effect of the driving force on the dwell time is explicitly indicated in detail.
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.
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.
We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.