[Paper Review] Transmission through a potential barrier in quantum mechanics of multiple degrees of freedom: complex way to the top

Fedor Bezrukov, D. G. Levkov|ArXiv.org|Jan 7, 2003

Quantum chaos and dynamical systems6 citations

TL;DR

This paper develops a semiclassical method for calculating tunneling exponents in quantum systems with multiple degrees of freedom, introducing a regularization technique to select physically relevant classical solutions through bifurcation points. It shows that at high energies, tunneling occurs via a classical state near the barrier top, validated against exact quantum solutions in a two-degree-of-freedom model.

ABSTRACT

A semiclassical method for the calculation of tunneling exponent in systems with many degrees of freedom is developed. We find that corresponding classical solution as function of energy form several branches joint by bifurcation points. A regularization technique is proposed, which enables one to choose physically relevant branches of solutions everywhere in the classically forbidden region and also in the allowed region. At relatively high energy the physical branch describes tunneling via creation of a classical state, close to the top of the barrier. The method is checked against exact solutions of the Schrodinger equation in a quantum mechanical system of two degrees of freedom.

Motivation & Objective

• To develop a reliable semiclassical method for computing tunneling exponents in quantum systems with multiple degrees of freedom.
• To address the challenge of selecting physically relevant classical solutions in the Euclidean time path integral formalism when multiple branches emerge due to bifurcations.
• To introduce a regularization technique that consistently selects the correct solution branch in both classically allowed and forbidden regions.
• To demonstrate that at high energies, tunneling proceeds via a classical state near the barrier top, not through standard WKB-type paths.
• To validate the method against exact quantum mechanical solutions in a two-particle harmonic oscillator model with a barrier.

Proposed method

• Formulates a $T/\theta$ boundary value problem to describe tunneling transitions, where $T$ is the Euclidean time and $\theta$ is related to the oscillator excitation number.
• Uses a path integral representation of the transition amplitude, with the dominant contribution given by the saddle point of the action functional.
• Applies regularization via a small positive parameter $\epsilon$ to the functional $F[q] = F[q] + 2\epsilon T_{\mathrm{int}}[q]$, which selects the solution with minimal interaction time.
• Imposes boundary conditions in complex time, with real initial momentum and position, and a phase condition $v = u^* e^{\theta}$ linking oscillator amplitude and phase.
• Solves the classical equations of motion in Euclidean time, with constraints linking energy $E$, excitation number $N$, and the Lagrange multipliers $T$, $\theta$.
• Extremizes the exponent $F = -ET - N\theta + 2\mathrm{Im}\,S_0(T,\theta)$, where $S_0$ is the regularized action, to find the dominant tunneling path.

Experimental results

Research questions

• RQ1 How can one consistently select the physically relevant classical solution in a multi-degree-of-freedom tunneling problem when multiple solution branches emerge via bifurcations?
• RQ2 What is the role of the initial phase of the oscillator in determining the tunneling path, and how can it be optimized to find the dominant contribution?
• RQ3 How does the tunneling mechanism change at high energies, particularly when the system can classically access the barrier top?
• RQ4 Can a regularization technique be constructed that selects the correct solution branch in both classically allowed and forbidden regions?
• RQ5 To what extent does the $T/\theta$ boundary value problem reproduce exact quantum mechanical results in a two-degree-of-freedom system?

Key findings

• The regularization technique successfully selects the physically relevant solution branch in both classically allowed and forbidden regions by minimizing the interaction time $T_{\mathrm{int}}$.
• At high energies, the dominant tunneling path corresponds to a classical state that reaches the top of the barrier, rather than following a standard WKB-type trajectory.
• The $T/\theta$ boundary value problem yields a unique solution in the over-barrier region, corresponding to the classical path that minimizes $T_{\mathrm{int}}$ for fixed $E$ and $N$.
• The method reproduces the exact quantum mechanical results for the two-degree-of-freedom model, validating its accuracy.
• The solution branches are connected via bifurcation points, and the regularization procedure ensures continuity and physical consistency across these points.
• The action exponent $F$ is computed as $F = -ET - N\theta + 2\mathrm{Im}\,S_0(T,\theta)$, with $S_0$ derived from the regularized Euclidean action.