Advances in High Energy Physics

Volume 2018, Article ID 1372359, 11 pages

https://doi.org/10.1155/2018/1372359

## A Probability Distribution for Quantum Tunneling Times

^{1}Department of Mathematics & Statistics, State University of Ponta Grossa, Avenida Carlos Cavalcanti 4748, 84030-900 Ponta Grossa, PR, Brazil^{2}Department of Physics, Concordia College, 901 8th St. S., Moorhead, MN 56562, USA

Correspondence should be addressed to José T. Lunardi; moc.liamg@idranulttj

Received 29 June 2018; Accepted 3 August 2018; Published 19 August 2018

Academic Editor: Neelima G. Kelkar

Copyright © 2018 José T. Lunardi and Luiz A. Manzoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization.

#### 1. Introduction

The search for a proper definition of quantum tunneling times for massive particles, having well-behaved properties for a wide range of parameters, has remained an important and open* theoretical* problem since, essentially, the inception of quantum mechanics (see, e.g., [1, 2] and references therein). However, such tunneling times were beyond the experimental reach until recent advances in ultrafast physics have made possible measurements of time in the attosecond scale, opening up the experimental possibility of measuring electronic tunneling times through a classically forbidden region [3–6] and reigniting the discussion of tunneling times. Still, the intrinsic experimental difficulties associated with both the measurements and the interpretation of the results have, so far, prevented an elucidation of the problem and, in fact, contradictory results persist, with some experiments obtaining a finite nonzero result [3, 6] and others compatible with instantaneous tunneling [4]. It should be noticed that the similarity between Schrödinger and Helmholtz equations allows for analogies between quantum tunneling of massive particles and photons [7], and a noninstantaneous tunneling time is supported by this analogy and experiments measuring photonic tunneling times [8], as well as by many theoretical calculations based on both the Schrödinger (for reviews see, e.g., [1, 2]) and the Dirac equations (e.g., [9–16]).

The conceptual difficulty in obtaining an unambiguous and well-defined tunneling time is associated with the impossibility of obtaining a self-adjoint time operator in quantum mechanics [17], therefore leading to the need for operational definitions of time. Several such definitions exist, such as phase time [18], dwell time [19], the Larmor times [20–23], and the Salecker-Wigner-Peres (SWP) time [17, 24], and in some situations these lead to different, or even contradictory, results. This is not surprising, since by their own nature operational definitions can only describe limited aspects of the phenomena of tunneling, and it is unlikely that any one definition will be able to provide a unified description of the quantum tunneling times in a broad range of situations. Nevertheless, it remains an important task to obtain a well-defined and* real* time scale that accurately describes the recent experiments [3–6, 25–27].

It is important to notice that the time-independent approach to tunneling times (i.e., for incident particles with sharply defined energy), which comprises the vast majority of the literature, is ill-suited to accomplish the above-mentioned goal, since it ignores the essential role of localizability in defining a time scale [23, 28]; see, however, [29], which applies the time defined in [30] to investigate the half-life of -decaying nuclei. A few works (e.g., [23, 28, 31, 32]) address the issue of localizability and, consequently, arrive at a probabilistic definition of tunneling times (that is, an average time). In particular, in [28] the SWP clock was used to obtain an average tunneling time of transmission (reflection) for an incident wave packet, and such time was employed to investigate the Hartman effect [33] for a particle scattered off a square barrier and it was shown that it does not saturate in the opaque regime [28, 34].

The tunneling time scales considered in [23, 28, 31] involve taking an average over the spectral components of the transmitted wave packet and, thus, obscure the interpretation of the resulting average time. In this paper, we take as a starting point the real-valued average tunneling time obtained in [28], using the SWP quantum clock, and obtain a* probability distribution of transmission times*, by using a standard transformation between random variables. In addition to providing a more accurate time characterization of the tunneling process, this should provide a clearer connection with the experiments (which measure a distribution of tunneling times; see, e.g., Figure 4 in [3]). It is worth noting that some approaches using Feynman’s path integrals address the problem of obtaining a probabilistic distribution of the tunneling times (see, e.g., [35]). However, these methods in general result in a complex time (or, equivalently, multiple time scales), and some* arbitrary* procedure is needed to select the physically meaningful real time* a posteriori*.

After obtaining a general formula for the distribution of tunneling times, which is the main result of this work, we apply it to two specific cases. First, to illustrate the formalism in a simple scenario, we consider the situation of a particle tunneling through a rectangular barrier. Then, we consider a slight modification of the model proposed in [36] for the tunneling decay of an initially bound state, after the sudden deformation of the binding potential by the application of a strong external field; the modification considered here allows us to investigate the whole range of possibilities for the tunneling times, without having an “upper cutoff”, as is the case in the original model. Finally, some additional comments on the results are reserved for the last section.

#### 2. The SWP Clock’s Average Tunneling Time

We start by briefly reviewing the time-dependent application of the SWP clock to the scattering of a massive particle off a localized static potential barrier in one dimension (for details see [28]) which is appropriate, since it follows from the three-dimensional Schrödinger equation for this problem that the dynamics is essentially one-dimensional [3].

The SWP clock is a quantum rotor weakly coupled to the tunneling particle and that runs only when the particle is within the region in which , where is the potential energy. The Hamiltonian of the particle-clock system is given by (we use , where is the particle’s mass) [17]where if and zero otherwise. The clock’s Hamiltonian is , where the angle is the clock’s coordinate and is the clock’s angular frequency, with being a nonnegative integer or half-integer giving the clock’s total angular momentum, and is the clock’s resolution. The weak coupling condition amounts to assume that is large, in such a way that the clock’s energy eigenvalues, (), are very small compared to the barrier height and the particle’s energy. It is assumed that, at , well before it reaches the barrier, the particle is well-localized far to the left of the barrier and the wave function of the system is a product state of the formwhere is the particle’s initial state, represented by a wave packet centered around an energy , and the clock initial state is assumed to be “in the zero-th hour” [17]where are the clock’s eigenfunctions corresponding to the energy eigenvalues .

The state is strongly peaked at , thus allowing the interpretation of the angle as the clock’s hand, since for a freely running clock the peak evolves to , where is the time measured by the clock [17]. Since here clock and particle are coupled according to (1), when the particle passes through the region it becomes entangled with the clock, with the wave function for the entire system given bywhere is the incident particle’s energy, , and is the Fourier spectral decomposition of the initial wave packet in terms of the free particle eigenfunctions (we are assuming delta-normalized eigenfunctions). The functions satisfy a time-independent Schrödinger equation with a constant potential in the barrier region. Outside the potential barrier region and for a particle incident from the left, the (unnormalized) solution of the time-independent Schrödinger equation is given by [28]where [] stands for the transmission (reflection) coefficient, and it is assumed, without loss of generality, that the potential is located in the region . Considering only the transmitted solution in (5) and substituting it into the time-dependent solution (4), it can be shown that for weak couplingwhereis the stationary transmission clock time corresponding to the wave number component [17, 37]. The transmission coefficient corresponds to the stationary problem in the absence of the clock.

For* tunneling* times one is interested only in the clock’s reading for the* postselected* asymptotically transmitted wave packet. Thus, tracing out the particle’s degrees of freedom, the expectation value of the clock’s measurement can be defined, resulting in the average tunneling time [28]where is a normalization constant and is the probability density of finding the component in the transmitted wave packet. Similar expressions can be obtained for the reflection time.

#### 3. The Tunneling Times Distribution

An important aspect of the average tunneling time considered in the previous section is that it emphasizes the* probabilistic nature of the tunneling process*. However, since the average in (8) is over the time taken by the* spectral components* of the wave packet, it does not lend itself to an easy interpretation, given the spectral components of the wave packet tunnel with different times. Thus, instead of (8), one would rather obtain an average over (*real*) times of the formwhere stands for the probability density for observing a particular tunneling time for the asymptotically transmitted wave packet. This can easily be achieved by noticing that in probability theory (8) and (9), which must be equal, are related by a standard transformation between the two random variables and through a function . It follows that the probability distribution of times is given bywhich, in essence, is the statement that all the -components in the transmitted packet for which must contribute to the value of with a weight . Finally, using the properties of the Dirac delta function (specifically, we use the fact that , where is the set of zeros of the function and the prime indicates a derivative with respect to the independent variable), we obtainwhere is the set of zeros of the function and is the derivative of with respect to .

A similar definition of the distribution of tunneling times given in (10)-(11) can be obtained for any time scale which is probabilistic in nature, that is, of the form (8). Although several other probabilistic tunneling times exist in the literature (e.g., [23, 31, 32, 35]), the SWP clock has proven to yield well-behaved* real* times both in the time-independent [17, 37, 38] and time-dependent approaches [28, 34, 39] and it provides a simple procedure to* derive* the probabilistic expression (8). In addition, the role exerted by circularly polarized light in attoclock experiments [3, 25] seems to provide a natural possibility for interpretation in terms of the SWP clock.

As will be illustrated below, for the simple application of this formalism to the problem of a wave packet scattered off a rectangular potential barrier, the distribution of times (10)-(11) cannot, in general, be obtained analytically even for the simplest cases, except in trivial cases such as for a single Dirac delta potential barrier [40–42], in which case and .

It should also be noticed that, despite the fact that the derivation of the previous section leading to (8) and, thus (10)-(11), assumed a scattering situation, these expressions can be shown to be valid for any situation involving preselection of an initial state localized to the left of a potential “barrier” followed by postselection of an asymptotic transmitted wave packet. This allows us to obtain the distribution of times for a model that simulates the tunneling decay of an initially bound particle by ionization induced by the sudden application of a strong external field; the model considered below is a variant of that introduced in [36].

#### 4. The Distribution of Tunneling Times for a Rectangular Barrier

As a first illustration of the formalism developed above, let us consider a rectangular barrier of height located in the region . The particle’s initial state is assumed to be a Gaussian wave packet where the parameters , , and are chosen such that the wave packet is sharply peaked in a tunneling wave number and is initially well-localized around , far to the left of the barrier; in the calculations that follow we take , such that at the probability of finding the particle within or to the right of the barrier is negligible. The transmission coefficient and the spectral function are well-known and given bywhere . The stationary transmission clock time (7) is [22, 28]with tunneling times corresponding to real values of (i.e., ). Figure 1 shows a plot for the stationary transmission times , the distribution of wave numbers in the transmitted wave packet, and the distribution of wave numbers (momenta) in the incident packet, for two values of the barrier width. For the chosen parameters and barrier widths both the incident and the transmitted wave packets have an energy distribution very strongly peaked in a tunneling component (in the bottom plot of Figure 1 the barrier is much more opaque than that in the top plot and we can observe that—despite being with a negligible probability for the parameters chosen for this plot—in this situation some above-the-barrier components start to appear in the distribution of the transmitted wave packet. So, in order to consider mainly transmission by tunneling we must restrict the barrier widths to not too large ones). We also observe the very well-known fact that the transmitted wave packet “speeds up” when compared to the incident particle [28]. As a general rule, the larger is the barrier width (i.e., the more opaque is the barrier), the greater is the translation of the central component towards higher momenta. In what concerns the off-resonance stationary transmission time, it initially grows with the barrier width, and saturates for very opaque barriers (the Hartman effect); on the other hand, it presents peaks at resonant wave numbers that grow and narrow with the barrier width; for a detailed discussion see [28]).