Calculation of the tunneling time using the extended probability of the quantum histories approach

Highlights

• We reexamine decomposition of dwell time by quantum history approaches.

• The results are given as of derive of the transmission and reflection coefficient.

• Finally we show the limits case which is discussed by Steinberg.

Abstract

The dwell time of quantum tunneling has been derived by Steinberg (1995) [7] as a function of the relation between transmission and reflection times ${\tau }_t$ and ${\tau }_r$, weighted by the transmissivity and the reflectivity. In this paper, we reexamine the dwell time using the extended probability approach. The dwell time is calculated as the weighted average of three mutually exclusive events. We consider also the scattering process due to a resonance potential in the long-time limit. The results show that the dwell time can be expressed as the weighted sum of transmission, reflection and internal probabilities.

Introduction

Quantum tunneling is one of the most important quantum phenomena. Various tunneling times can be determined based on characterizing the time spent by a particle under the barrier. They are expressed in terms of the derivatives of the transmission coefficient $T=|T|e^{i{\theta }_t}$ or the reflection coefficient $R=|R|e^{i{\theta }_r}$. For example, the Larmor time was first introduced by Baz [1], [2] in a thought experiment designed to measure the time associated with scattering events. The Larmor times for transmission and reflection can be obtained respectively from

$${\tau }_t^{\mathit{LM}}=-ħ\frac{\partial }{\partial V}{\theta }_t\text{and}{\tau }_r^{\mathit{LM}}=-ħ\frac{\partial }{\partial V}{\theta }_r,$$

where V is the hight of a square potential barrier.

Another example is the Büttiker–Landauer time [3], [4] that invokes an oscillatory barrier to estimate the tunneling time. The original static barrier is augmented by small oscillations in the barrier height. The Büttiker–Landauer expressions for transmission and reflection times are given as

$${\tau }_t^{\mathit{BL}}=-ħ\frac{\partial }{\partial V}\ln |T|\text{and}{\tau }_r^{\mathit{BL}}=-ħ\frac{\partial }{\partial V}\ln |R|,$$

respectively.

Sokolovski and Baskin [5] presented an identity connecting the dwell time ${\tau }_d(k)$ and the complex time ${\tau }_{t(r)}^{\Omega }={\tau }_{t(r)}^{\mathit{LM}}-i{\tau }_{t(r)}^{\mathit{BL}}$ as

$${\tau }_d(k)={|T|}^2{\tau }_t^{\Omega }+{|R|}^2{\tau }_r^{\Omega },$$

where ${ħ}^2{k}^2/2m$ is an energy of the free particle.

Hauge and Støvneng [6] also proved Eq. (3) and suggested that the transmission and the reflection are mutually exclusive events. Steinberg [7] used conditional probability to define the dwell time. He proposed that the dwell time distribution consists of two parts, one from transmission and another from reflection. In spite of these results, we still do not know how the dwell time can be decomposed. In 2004, Yamada [8] derived four tunneling times in a unified manner without relying on any specific models by using the Gell-Mann and Hartle (GMH) decoherence functional [9] $D_{\mathit{GMH}}^{(\gamma )}(\tau ,{\tau }^{\prime })$ to define the following quantity

$$I[F]=\frac{1}{P({\Theta }_{\gamma })}\int d\tau \int d{\tau }^{\prime }F(\tau ,{\tau }^{\prime })D_{\mathit{GMH}}^{(\gamma )}(\tau ,{\tau }^{\prime }).$$

To understand the physical meaning of the quantity $I[F]$, it is necessary to explore the relationship between $D_{\mathit{GMH}}^{(\gamma )}(\tau ,{\tau }^{\prime })$ and the essential ideas of extended probability in quantum history [10], [11], [12].

In this paper, we discuss how the dwell time distribution is determined by the extended probabilities of alternative histories, which are decomposed into three elements. This formulation is compared with the GMH decoherence functional. The natural occurrence of the dwell time is given by the weighted average of three mutually exclusive events. We introduce a dwell time ${\tau }_{\mathit{in}}$ as the time for the particle to remain in a given region. Then we consider the dwell time for a resonance potential barrier. This gives a time ${\tau }_{\mathit{in}}$ that contributes to the dwell time in the long-time limit. Finally, we present our conclusions in Section 4.