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Physical Review Letters

Path Decomposition for Multidimensional Tunneling

Assa Auerbach, Steven Allan Kivelson, Denis A. Nicole

Published Jul 30, 1984• 46 citations
Publisher

Abstract

In order to solve multidimensional tunneling problems that cannot be treated by the normal instanton techniques, we introduce the path decomposition expansion formalism, and show its usefulness by solving three generic examples: the symmetric and asymmetric double well, and the decay problem. The technique allows us to handle excited states and backscattering effects in nonseparable potentials.

DOI
10.1103/physrevlett.53.411

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In order to solve multidimensional tunneling problems that cannot be treated by the normal instanton techniques, we introduce the path decomposition expansion formalism, and show its usefulness by solving three generic examples: the symmetric and asymmetric double well, and the decay problem. The technique allows us to handle excited states and backscattering effects in nonseparable potentials.
Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.
As was pointed out in Section 3.1, there are some interesting features in multidimensional tunneling compared with the one-dimensional case [30,31,46]. These are the following: (1) There exist two types of tunneling, pure tunneling and mixed tunneling. (2) No tunneling path can be defined in the case of mixed tunneling. (3) Vibrational excitation can suppress tunneling because of mixed tunneling. In this section, these peculiar features are numerically demonstrated and analyzed [31] by taking tunneling splitting in symmetric double well potential as an example. Three typical two-dimensional model potentials are used.
Starting from trace formulas for the tunneling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.
Solving the tunneling problem in a double-well potential requires knowing the wave function not just of the ground state, but also of the first excited state. A method is described that allows one to construct potentials together with these two wave functions. No approximations are made. To illustrate the method several examples are given. These include potentials that either diverge for large \ensuremath{\Vert}x\ensuremath{\Vert} or become asymptotically flat, with one or two wells, which may be symmetric or asymmetric.