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Journal of Physics A Mathematical and Theoretical

Quantum tunneling effect of a time-dependent inverted harmonic oscillator

Guangjie Guo, Zhongzhou Ren, Ju Guo-Xing, Xiaoyong Guo

Published Apr 5, 2011• 12 citations
Publisher

Abstract

An algebraic method to solve the time-dependent inverted harmonic oscillator (TDIHO) is proposed, and the analytical expressions of the wavefunctions obtained by this method are suitable for discussion of the quantum tunneling effect. With the Gaussian wave packet as the initial state, the general expressions of transmission and reflection probabilities, and dwell time of TDIHO are obtained. The quantum tunneling effect is further illustrated through two specific cases.

DOI
10.1088/1751-8113/44/18/185301

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An algebraic method to solve the time-dependent inverted harmonic oscillator (TDIHO) is proposed, and the analytical expressions of the wavefunctions obtained by this method are suitable for discussion of the quantum tunneling effect. With the Gaussian wave packet as the initial state, the general expressions of transmission and reflection probabilities, and dwell time of TDIHO are obtained. The quantum tunneling effect is further illustrated through two specific cases.
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.
The time evolution of a time-dependent inverted harmonic oscillator (TDIHO) of arbitrary dimensions is investigated. Using the algebraic method, we obtain the exact orthogonal basis of solutions of a TDIHO in which the dimensionality d and angular momentum l appear as parameters, and also discuss its properties. With the wavepacket as the initial state, the general expressions of sojourn time of a TDIHO are given. The method is also applied to the inverted Caldirola–Kanai harmonic oscillator. The results show that the sojourn time appears as an increasing function of the dissipation parameter in arbitrary dimensions, but a decreasing function of the dimensionality.
New exact and asymptotic results for a quantum inverted oscillator, driven by the variable external force, are presented. To illustrate the advantages of our approach, we applied the obtained propagator to the descriptions of evolution the initial Gaussian wave packet under arbitrary time-dependent force or a δ-pulse action. The alternative problem of tunneling a particle through a parabolic barrier under the influence of a low-frequency harmonic force we solve in the quasistatic approximation. Going to the Heisenberg picture in the Caldeira-Leggett model, we describe the influence of a time-dependent force on an open inverted oscillator. The resulting dynamics is presented in the form of combination the evolution of the average value of a coordinate operator and the stochastic diffusion, being induced the wave packet spreading and the Brownian motion.
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.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
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.