Time scales of tunneling decay of a localized state

Abstract

Motivated by recent time domain experiments on ultrafast atom ionization, we analyze the transients and timescales that characterize, besides the relatively long lifetime, the decay by tunneling of a localized state. While the tunneling starts immediately, some time is required for the outgoing flux to develop. This short-term behavior depends strongly on the initial state. For the initial state tightly localized so that the initial transients are dominated by over-the-barrier motion, the timescale for the flux propagation through the barrier is close to the Büttiker-Landauer traversal time. Then a quasistationary, slowly decay process follows, which sets ideal conditions for observing diffraction in time at longer times and distances. To define operationally a tunnelling time at the barrier edge, we extrapolate backwards the propagation of the wave packet escaped from the potential. This extrapolated time is considerably longer than the timescale of the flux and density buildup at the barrier edge.

A Probability Distribution for Quantum Tunneling Times

“…On the other hand, the top plot of Figure 7 shows that for the thicker barrier the probability for superluminal times is negligible -the portion of the wave packet already inside the barrier at t = 0 is the same (∼ 27%), but the wave packet penetrates proportionally a smaller distance inside the barrier and, thus, it does not contribute in a significant way to the emergence of very small times in the clock readings. We note that the introduction of the cutoff t 0 , as in [36], would result in a time distribution similar to the truncated distributions shown in the top plots of Figures 6 and 7.…”

“…In [36], for t < 0, the particle is in an eigenstate of a semi-infinite square-well potential V 1 (x),…”

“…where N is a normalization constant, k 0 = √ E 0 , E 0 is the ground state energy, and q 0 = V 0 − k 2 0 . It is also assumed, as in [36], that immediately after the sudden deformation of the potential from V 1 (x) to V 2 (x), at t = 0, the wave function does not change. However, for t ≥ 0 the particle state, which is no longer an energy eigenstate, is given by a superposition of the energy eigenstates ψ k (x) (k = √ E) of the potential V 2 (x), i.e., [36] ψ(…”

“…where q = √ V 0 − k 2 and the coefficients A(k), C(k), D(k) and the phase Ω(k) are determined by the usual boundary conditions at x = a and x = b, and are such that the normalization ψ k (x), ψ k (x) = δ (k − k ) holds [36]. From the above expressions it follows that, without any loss of generality, we can take S(k) and all the eigenfunctions (20) to be real.…”

Virtual-detector approach to tunnel ionization and tunneling times

“…A related still open and actively studied problem of atomic physics is the question how long it takes to ionize an atom via an electron tunneling through the potential barrier [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] which is formed by the electron's binding energy and the Coulomb potential bent by the time-dependent electric field's potential; see Fig. 1.…”