Revisiting 1-Dimensional Double-Barrier Tunneling in Quantum Mechanics

TL;DR

This paper explores quantum tunneling through double barriers, drawing an analogy with optical interferometers, and provides analytical formulas and insights into resonance phenomena, phase time, and standing wave spectra.

Contribution

It introduces an analytical finesse formula for double-barrier tunneling based on optical analogy and validates it against numerical results, offering new perspectives on resonance and phase time.

Findings

Analytical finesse formula matches numerical results for deep tunneling resonances.

Resonance peak in phase time corresponds to resonance lifetime, consistent with uncertainty principle.

Phase time saturates for long barriers, similar to single barrier tunneling.

Abstract

This paper revisited quantum tunneling dynamics through a square double-barrier potential. We emphasized the similarity of tunneling dynamics through double-barrier and that of optical Fabry--P$\acute{e}$rot (FP) interferometer. Based on this similarity, we showed that the well-known resonant tunneling can also be interpreted as a result of matter multi-wave interference, analogous to that of FP interferometer. From this analogy, we also got an analytical finesse formula of double-barrier. Compared with that obtained numerically for a specific barrier configuration, we found that this formula works well for resonances at "deep tunneling region". Besides that, we also calculated standing wave spectrum inside the well of double barriers and phase time of double-barrier tunneling. The wave number spectrums of standing wave and phase time show another points of view on resonance. From semi-numerical calculations, we interpreted the peak of phase time at resonance as resonance life time, which coincides at least in order of magnitude with that obtained from uncertainty principle. Not to our surprise, phase time of double-barrier tunneling also saturates at long barrier length limit $l\rightarrow\infty$ as that of tunneling through a single barrier, and the limits are the same.