Gaussian wavepacket dynamics and quantum tunneling in asymmetric double-well systems

TL;DR

This study investigates quantum tunneling in asymmetric double-well systems using spectral methods, revealing how asymmetry affects tunneling probability, resonance conditions, and uncertainty relations, with implications for quantum dynamics understanding.

Contribution

It provides a detailed analysis of tunneling dynamics in asymmetric double-well potentials, highlighting the effects of asymmetry and initial conditions on tunneling behavior and quantum uncertainties.

Findings

Tunneling probability decreases with potential asymmetry.

Resonant tunneling occurs for specific asymmetry values when starting from the upper minimum.

Narrower Gaussian wavepackets experience less reduction in tunneling due to asymmetry.

Abstract

We have studied dynamical properties and quantum tunneling in asymmetric double-well (DW) systems, by solving Schr\"{o}dinger equation with the use of two kinds of spectral methods for initially squeezed Gaussian wavepackets. Time dependences of wavefunction, averages of position and momentum, the auto-correlation function, an uncertainty product and the tunneling probability have been calculated. Our calculations have shown that (i) the tunneling probability is considerably reduced by a potential asymmetry $\Delta U$, (ii) a resonant tunneling with $|\Delta U| \simeq \kappa \:\hbar \omega$ is realized for motion starting from upper minimum of asymmetric potential wells, but not for motion from lower minimum, ($\kappa=0,1,2,...$; $\omega$: oscillator frequency at minima), (iii) the reduction of the tunneling probability by an asymmetry is less significant for the Gaussian wavepacket with narrower width, and (iv) the uncertainty product $<\delta x^2> <\delta p^2>$ in the resonant tunneling state is larger than that in the non-resonant tunneling state. The item (ii) is in contrast with the earlier study [Mugnai {\it et al.}, Phys. Rev. A {\bf 38} (1987) 2182] which showed the symmetric result for motion starting from upper and lower minima.