Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well

TL;DR

This paper introduces an analytical quantisation condition for hyperbolic double well potentials, enabling comprehensive spectral analysis and detailed insights into wave-packet dynamics, including tunneling and quantum interference effects.

Contribution

It develops a novel quantisation method based on the Heun confluent differential equation, extending beyond quasi-exact solvability for hyperbolic double wells.

Findings

Accurately computes the eigenspectrum and eigenstates.

Analyzes wave-packet dynamics with excellent agreement to numerical results.

Disentangles eigenfrequencies governing phase-space behavior.

Abstract

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schr\"odinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system's eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication [H. Chomet et al, New J. Phys. 21, 123004 (2019)]. Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.