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arXiv:1008.3853 · Physical Review A

Time scales of tunneling decay of a localized state

Yue Ban, Eugene Ya. Sherman, Juan Gonzalo Muga, Μ. Büttiker

Published Dec 30, 2010• 22 citations
arXivHTMLPDF

Abstract

Motivated by recent time-domain experiments on ultrafast atom ionization, we analyze the transients and time scales that characterize, aside from the relatively long lifetime, the decay of a localized state by tunneling. 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 time scale for flux propagation through the barrier is close to the B\"uttiker-Landauer traversal time. Then a quasistationary, slow-decay process follows, which sets ideal conditions for observing diffraction in time at longer times and distances. To define operationally a tunneling time at the barrier edge, we extrapolate backward the propagation of the wave packet that escaped from the potential. This extrapolated time is considerably longer than the time scale of the flux and density buildup at the barrier edge.

DOI
10.1103/physreva.82.062121

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Motivated by recent time-domain experiments on ultrafast atom ionization, we analyze the transients and time scales that characterize, aside from the relatively long lifetime, the decay of a localized state by tunneling. 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 time scale for flux propagation through the barrier is close to the B"uttiker-Landauer traversal time. Then a quasistationary, slow-decay process follows, which sets ideal conditions for observing diffraction in time at longer times and distances. To define operationally a tunneling time at the barrier edge, we extrapolate backward the propagation of the wave packet that escaped from the potential. This extrapolated time is considerably longer than the time scale of the flux and density buildup at the barrier edge.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time τBLτ_{BL}τBL​ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein τBLτ_{BL}τBL​ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
We consider a unified analytical approach for scattering and decay processes in one-dimensional resonant tunneling systems using the formalism of resonant states to address the issue of the differences and similarities in the time evolution of decay between the decay of an arbitrary state prepared initially within a system and the formation and subsequent decay of a quasistationary state in the scattering of a Gaussian wavepacket on that system. We find three distinctive regimes. A first regime, which refers only to the quasistationary state, that is characterized by a buildup time of the probability density at a given position within the internal region of the potential. Here we find that the buildup time has a dependence on position. A second regime, dominated by the exponentially decaying terms, where the decay of the quasistationary state proceeds in an almost identical fashion as for the initially prepared decaying state. And finally, a third regime that involves the transition to nonexponential decay at long times and its ulterior behavior as an inverse power of time. Here we find that the time scale of the transition occurs at different times, which implies a dependence on the parameters of the initial state.
The notion of the first passage time is shown to offer a meaningful extension to quantum tunneling, providing a closed-integral-form analytical unification of the tunneling rate and the tunneling passage time. We demonstrate that, in suitable potential settings, the quantum first passage time, found as a solution to the Fokker-Planck and backward Kolmogorov's equations for the quantum probability density, recovers the hallmark results for the Kramers escape rate, the lifetime of tunneling quasi-stationary wave packets, leads to a classical, distance-over-speed passage time for a free-particle wave function, and offers useful insights into Keldysh's intimation on the electron barrier-traversal time in field-induced ionization.