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The Journal of Chemical Physics

Semiclassical tunneling splittings for arbitrary vibrational states in multidimensional double wells

Kenneth G. Kay

Published Oct 9, 2018• 2 citations
Publisher

Abstract

, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.

DOI
10.1063/1.5047830

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, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.
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.
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.
Quantum tunneling rates through a barrier separating two-dimensional, symmetric, double-well potentials are shown to depend on the classical dynamics of the billiard trajectories in each well and, hence, on the shape of the wells. For shapes that lead to regular (integrable) classical dynamics the tunneling rates fluctuate greatly with eigenenergies of the states sometimes by over two orders of magnitude. Contrarily, shapes that lead to completely chaotic trajectories lead to tunneling rates whose fluctuations are greatly reduced, a phenomenon we call regularization of tunneling rates. We show that a random-plane-wave theory of tunneling accounts for the mean tunneling rates and the small fluctuation variances for the chaotic systems.
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.
A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.
Recent numerical experiments of Pecora et al. [Phys. Rev. E 83, 065201 (2011)] have investigated tunneling between two-dimensional symmetric double wells separated by a tunneling barrier. The wells were bounded by hard walls and by the potential barrier which was created by a step increase from the zero potential within a well to a uniform barrier potential within the barrier region, which is a situation potentially realizable in the context of quantum dots. Numerical results for the splitting of energy levels between symmetric and antisymmetric eigenstates were calculated. It was found that the splittings vary erratically from state to state, and the statistics of these variations were studied for different well shapes with the fluctuation levels being much less in chaotic wells than in comparable nonchaotic wells. Here we develop a quantitative theory for the statistics of the energy level splittings for chaotic wells. Our theory is based on the random plane wave hypothesis of Berry. While the fluctuation statistics are very different for chaotic and nonchaotic well dynamics, we show that the mean splittings of differently shaped wells, including integrable and chaotic wells, are the same if their well areas and barrier parameters are the same. We also consider the case of tunneling from a single well into a region with outgoing quantum waves.