Quantum effects and quantum chaos in multidimensional tunneling

Abstract

The ground-state energy splitting due to tunneling in two-dimensional double wells of the form $V(x,y)=\frac{{({x}^{2}\ensuremath{-}{R}^{2})}^{2}}{8{R}^{2}}+\frac{{x}^{2}\ensuremath{-}{R}^{2}}{{R}^{2}}\ensuremath{\gamma}y+\frac{{\ensuremath{\omega}}^{2}}{2}{y}^{2}$ is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of ${R}^{\ensuremath{-}2}$, each term of the series being a finite polynomial in ${\ensuremath{\gamma}}^{2}$. We find an ascending sequence of the values of the parameter $\ensuremath{\gamma}$ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of ${R}^{\ensuremath{-}2}$ being an infinite series in powers of ${\overline{\ensuremath{\gamma}}}^{2}$, $\overline{\ensuremath{\gamma}}=\ensuremath{\gamma}/R$. Third, we find that the radius of convergence of the series is determined by the singularity at ${\overline{\ensuremath{\gamma}}}_{s}=\ensuremath{\omega}/2$. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of $\overline{\ensuremath{\gamma}}$, most likely of the form ${[1\ensuremath{-}{(\overline{\ensuremath{\gamma}}/{\overline{\ensuremath{\gamma}}}_{s})}^{2}]}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1/2$ and $\ensuremath{\alpha}=\ensuremath{-}1/2$ for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large $R$ the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter $\overline{\ensuremath{\gamma}}$ entering the Hamiltonian linearly.