Simple approach to computing tunneling time: Test cases

Abstract

ABSTRACT:We suggest a recipe for calculating the time taken by a particle to tunnel out from an initially localized state in one of the wells of symmetrical doublewell potential into the other well. We calculate average velocity ͗v ͘ of the tunneling particle by solving the Schrö dinger equation numerically for ⌿(x, t), then estimating ͗v(t)͘ ϭ (d/dt)͗x(t)͘ and time-averaging it. The time taken to tunnel is measured by calculating av ϭ l 0 /͗v ͘ where l 0 is an idealized estimate of the barrier width. We suggest an approximate partitioning of av into an intrinsic decay time ( d ) and a barrier crossing time ( b ), d being obtained from energy spread of the packet. av Ϫ d is shown to be close to the Wentzel-Brillouin-Kramers estimate of barrier crossing time sc . The response of av to an external driving field and nonzero temperatures are tested. We also apply the method to a purely barrier penetration problem as well as to the problem of tunneling through a fluctuating barrier.

Tunneling time in fluctuating symmetric double wells: Suppression and enhancement of tunneling by spatial symmetry-preserving perturbations

“…DE(t)s can help us in understanding the energetics of the processes. The tunneling time has been calculated by computing the average slope of the plot of hx(t)i against time to obtain average velocity hvi and dividing the idealized barrier width l 0 by hvi [28][29][30].…”

Tunneling time in driven double‐well system using entangled molecular dynamics method

“…Quantum tunneling time has significance in several physical phenomena, and it has some different definitions, we choose one of them suggested by Bhattacharyya and his coworkers . If the idealized width of the barrier is, l 0 , defined as distance between two minimum values of double well, quantum tunneling time is defined as where m is the mass of the tunneling particle.…”

“…τ d can be computed through the time energy uncertainty relation , where , and the barrier interaction time ( τ sc ) is defined as follow: …”

“…Tunneling time has implications in Josephson junction current, chiral tunneling in bilayer graphene, indeed, it has been observed experimentally . Several different definitions have been suggested, especially, Bhattacharyya and coworkers proposed a simple but fully quantum method to calculate tunneling time …”

“…[9][10][11] Several different definitions have been suggested, especially, Bhattacharyya and coworkers proposed a simple but fully quantum method to calculate tunneling time. [12,13] Entangled trajectory molecular dynamics method (ETMD) is a trajectory-based method, which can solve quantum Liouville equation through the evolution of trajectory ensemble in the phase space. [14][15][16][17] Different from other quantum trajectory methods, quantum effects arise in the formulation as a breakdown of the statistical independence of the ensemble.…”