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Physical Review C

Semiclassical theory of quantum tunneling in multidimensional systems

N. Takigawa, G. F. Bertsch

Published Jun 1, 1984• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.29.2358

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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 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.
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There has been a surprising degree of uncertainty about how, or even whether, the standard semiclassical theory of Van Vleck and Gutzwiller (VVG) can be used to treat the time-dependent dynamics of tunneling through barriers. At issue is how semiclassical theory describes the time evolution of tunneling. This uncertainty has hindered development of practical techniques for semiclassical calculations of multidimensional systems. It is shown here that the VVG theory can indeed be applied to barrier tunneling, but generally requires consideration of classical trajectories with both complex initial conditions and complex time paths, even for systems with one degree of freedom. The present approach is verified by applying it to the classically forbidden propagation of a wave packet through a one-dimensional barrier---a very basic problem that cannot be treated by the VVG theory without the modifications described here.
A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.
We extend a useful formalism to construct a wave function in a multi-dimensional tunneling system and to investigate the quantum excitations of an intrinsic degree of freedom, which is a harmonic oscillator linearly coupled to a tunneling degree of freedom. To demonstrate the usefulness of the formalism, we construct a wave function in an analytic form for a simple tunneling system with a cubic potential. Based on the wave function, we discuss interesting features of the multi-dimensional tunneling system in which the quantum excitations play an important role. It is shown that the quantum excitations of the linearly coupled harmonic oscillators increase the decay width in comparison with the case of no excitations.
The familiar Rabi model, comprising a two-level system coupled to a quantum harmonic oscillator, continues to produce rich and surprising physics when the coupling strength becomes comparable to the individual subsystem frequencies. We construct approximate solutions for the eigenstates and energies in the regime in which the oscillator frequency is small compared to that of the two-level system and the coupling strength matches or exceeds the oscillator frequency. The resulting oscillator dynamics closely resembles that of a particle tunneling in a classical double-well potential. Relating our calculation to an earlier semiclassical approximation in which coupling to the two-level system creates an effective potential for the oscillator, we examine the extent to which this picture is valid. We find that, for certain parameter regimes, the interpretation of the oscillator dynamics in terms of tunneling holds to a good approximation despite the fundamentally entangled nature of the joint system. We assess the prospects for observation of oscillator tunneling in the context of nano- or micromechanical experiments and find that it should be possible if suitably high coupling strengths can be engineered.