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Journal of the Physical Society of Japan

Quantum Tunneling Rate through an Inverted Parabola with an Oscillating External Field – an Analysis Using the Exact Feynman Kernel –

Toshihiro Tanizawa

Published Mar 15, 1998• 0 citations
Publisher

Abstract

The quantum tunneling rate through an inverted parabola potential in an oscillating external field is studied. The tunneling rate is calculated within quasi-classical accuracy as a function of the frequency of the field. In this calculation, the complex-time path integral method, which has been previously applied to the case of tunneling through static potentials, is used after being appropriately extended for application to the case of tunneling through explicitly time-dependent potentials. Since the Feynman kernel for the case considered in the present paper can be exactly obtained, we are able to calculate the tunneling rate over a considerably wider range of frequencies of the oscillating field than in the case that this quantity is calculated through a perturbative treatment of the external field, as done in the previous studies. The present analysis reveals that the perturbative analysis for the present problem is insufficient even in the region where it has been considered to be appropriate. From the numerical calculation of the quasi-classical tunneling rate, we find that, as the frequency of the external field increases, all of the tunneling rates are initially enhanced by a significant amount, but beyond a certain value, all tunneling rates become rapidly decaying functions of this frequency. This fact may imply that the consideration beyond quasi-classical accuracy is needed in the high frequency regime. Tunneling processes that do not conserve energy are also considered. They cannot be neglected in this time-dependent problem, and we find that tunneling rates become large when tunneling particles acquire energy during tunneling.

DOI
10.1143/jpsj.67.799

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The quantum tunneling rate through an inverted parabola potential in an oscillating external field is studied. The tunneling rate is calculated within quasi-classical accuracy as a function of the frequency of the field. In this calculation, the complex-time path integral method, which has been previously applied to the case of tunneling through static potentials, is used after being appropriately extended for application to the case of tunneling through explicitly time-dependent potentials. Since the Feynman kernel for the case considered in the present paper can be exactly obtained, we are able to calculate the tunneling rate over a considerably wider range of frequencies of the oscillating field than in the case that this quantity is calculated through a perturbative treatment of the external field, as done in the previous studies. The present analysis reveals that the perturbative analysis for the present problem is insufficient even in the region where it has been considered to be appropriate. From the numerical calculation of the quasi-classical tunneling rate, we find that, as the frequency of the external field increases, all of the tunneling rates are initially enhanced by a significant amount, but beyond a certain value, all tunneling rates become rapidly decaying functions of this frequency. This fact may imply that the consideration beyond quasi-classical accuracy is needed in the high frequency regime. Tunneling processes that do not conserve energy are also considered. They cannot be neglected in this time-dependent problem, and we find that tunneling rates become large when tunneling particles acquire energy during tunneling.
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.
In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that the quasi-quantum fluctuation is expressed by the Lamé equation and was exactly solved. Then I have shown that the obtained kernel function is in agreement with exact solutions of the linear potential and the quadratic potential under certain limits as no time-development kernel function of the quartic potential has ever been found which contains the exact solution of the linear and the quadratic potential. It is natural because the classical motion in the quartic potential becomes those of the linear and the quadratic potential under the limits. Thus the obtained time-development kernel function also consists of the energy representation of the Green function of the quartic potential in the semiclassical path integral method given by Carlitz and Nicole (Ann. Phys.164 (1985) 411), which agrees with that of the WKB method in the operator formalism.
The effects of coupling to a harmonic oscillator on the quantum tunneling of a macroscopic motion are studied through the influence functional formalism of Feynman's path integral method for the general coupling form factor. As an example, we consider the model in which the potential barrier is parabolic and the coupling Hamiltonian is linear in both coordinates of the macroscopic motion and of the intrinsic harmonic oscillator. The results are then compared with the exact solution obtained through the canonical transformation into normal coordinates in the limiting cases when the normal coordinates reduce to the original coordinates. We found that: (1) In the adiabatic case, i.e., when the recurrence time $\frac{\ensuremath{\pi}}{\ensuremath{\omega}}$ of the oscillator is much shorter than the transmission time through the macroscopic potential barrier, the effect of oscillator coupling can be well represented by an effective potential. The coupling enhances the tunneling probability on the whole. (2) There exists a critical energy, above which the tunneling probability is reduced because of the linear oscillator coupling. In the weak coupling limit and when $\ensuremath{\omega}\ensuremath{\rightarrow}0$, the critical energy becomes $\ensuremath{-}\ensuremath{\infty}$, so that the coupling to the oscillator always reduces the tunneling probability.NUCLEAR REACTIONS Quantum tunneling, coupling of macroscopic motion to intrinsic oscillators, semiclassical method, heavy ion fusion.
A semiclassical method is applied to the problem of quantum tunneling in the presence of internal oscillator degrees of freedom. Comparison with coupled channel calculations verifies the accuracy of the semi-classical method. Providing the coupling has a physically reasonable form factor, the coupling produces an enhancement in the tunneling rate. The enhancement becomes larger when the tunneling degree of freedom couples with a harmonic oscillator with lower frequency. The semiclassical method is found to be quite useful in situations where the coupled channel method becomes numerically difficult.
The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result, tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is complex-valued. It is shown that RetpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the standard WPS result, tphaseT{\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}tphaseT​, for conditions similar to those employed in the conventional WPS analysis. Similarly, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.
The time evolution of quantum tunneling is studied when the tunneling system is immersed in a thermal medium. We analyze in detail the behavior of the system after integrating out the environment. An exact result for the inverted harmonic oscillator of the tunneling potential is derived and the barrier penetration factor is explicitly worked out as a function of time. The quantum mechanical formula for the case without environment is modified both by the potential renormalization effect and by a dynamical factor which may differ appreciably from the one previously obtained in the time range of 1/(curvature at the top of potential barrier).