Alternative perspective on quantum tunneling and instantons

Abstract

We present a new way to compute and interpret quantum tunneling in a 1-D double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of 10 −5 . In contrast to the classical potential, the quantum potential has a triple-well if the classical wells are deep enough. Its minima are located at the position of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instatons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semi-classical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such case is illustrated by an example.

Instanton solutions on the polymer harmonic oscillator

“…Otherwise, tunneling effect between each of the 'classical vacuums' gives rise to the band structure of the spectrum. In standard quantum mechanics, the tunneling effect can be realized by instanton and anti-instanton pseudoparticles in the long imaginary time regime [39][40][41][42] although it may be obtained within the deep quantum regime [43]. Instanton methods are a non-perturbative technique developed for handling phenomena in quantum field theory, and in particular in quantum chromodynamics, analogous to barrier penetration in quantum particle mechanics.…”

Chaotic instantons and enhancement of tunneling in double-well system with polychromatic perturbation

“…Alternative approach based on quantum instantons which are defined using an introduced notion of quantum action was suggested in Refs. [33,34].…”

Chaotic instantons and enhancement of tunneling in kicked double-well system with time-reversal symmetry

“…It will give additional support and pulse for the further development of analytical methods to investigate tunneling phenomenon in quantum systems with mixed classical dynamics. Alternative approach based on quantum instantons which are defined using an introduced notion of quantum action was suggested in [38,39]. Analytical approach to describe tunneling in perturbed systems based on nonlinear resonances consideration was developed in [25,40].…”