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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Bounces and the calculation of quantum tunneling effects

Liang Jiu-Qing, Harald J W Müller-Kirsten

Published Apr 15, 1992• 23 citations
Publisher

Abstract

The imaginary part of the energy of the metastable ground state for the inverted double-well potential is calculated by using the path-integral method. The tunneling process is dominated by bounces. It is shown that the evaluation of the determinant of the second variation of the action at the bounce can be avoided, and that the imaginary part of the energy results directly from characteristic properties of the bounce itself, namely, the antisymmetry of its first time derivative under time reversal. The imaginary part of the result is in exact agreement with that of the well-known WKB calculation of Bender and Wu.

DOI
10.1103/physrevd.45.2963

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The imaginary part of the energy of the metastable ground state for the inverted double-well potential is calculated by using the path-integral method. The tunneling process is dominated by bounces. It is shown that the evaluation of the determinant of the second variation of the action at the bounce can be avoided, and that the imaginary part of the energy results directly from characteristic properties of the bounce itself, namely, the antisymmetry of its first time derivative under time reversal. The imaginary part of the result is in exact agreement with that of the well-known WKB calculation of Bender and Wu.
We study tunneling in one-dimensional quantum mechanics using the path integral in real time, where solutions of the classical equation of motion live in the complex plane. Analyzing solutions with small (complex) energy, relevant for constructing the wave function after a long time, we unravel the analytic structure of the action, and show explicitly how the imaginary time bounce arises as a parameterization of the lowest order term in the energy expansion. The real time calculation naturally extends to describe the wave function in the free region of the potential, reproducing the usual WKB approximation. The extension of our analysis to the semiclassical correction due to fluctuations on the saddle is left for future work.
The complex-time formalism is developed in the framework of the path-integral formalism, to be used for analysis of the quantum tunneling phenomena. We show that subleading complex-time saddle-points do not account for the right WKB result. Instead, we develop a reduction formula, which enables us to construct Green functions from simple components of the potential, for which saddle-point method is applicable. This method leads us to the valid WKB result, which incorporates imaginary-time instantons and bounces, as well as the real-time boundary conditions.
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.
The path integral formulation of the traversal time in tunneling utilizes a restricted Feynman amplitude. We present a derivation based on the Fisher Lee formula to compute the average time and the integrated distribution for arbitrary potentials. The (complex) tunneling time can be expressed as a derivative with respect to a potential which surrounds the region over which the particle is “detected”, and a correspondence with other tunneling time scales which depend on energy derivatives of the transmission coefficient is found. A semiclassical treatment confirms that there are two time scales associated with this measurement procedure. The imaginary component is related to the multiple instanton bounces in the inverted potential. The issue of positive times is found to depend crucially on the behavior of the transmission coefficient for asymptotic limits of the measurement potential. The calculation of higher moments of the distribution allows to define a positive barrier interaction time, without appealing to any regularization procedure.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.
It is widely believed that the WKB approximation to quantum barrier penetration is described in the path-integral formalism by a saddle-point approximation based on an expansion about an imaginary-time semiclassical path. This article examines the relationship between tunneling amplitudes and imaginary-time paths. It is shown, using a simple example, that for some tunneling amplitudes the dominant semiclassical path cannot be chosen to have purely imaginary time if all other variables are chosen to be real.