How Much Time Does a Tunneling Particle Spend in the Barrier Region?

Abstract

The question in the title may be answered by considering the outcome of a "weak measurement" in the sense of Aharonov et al. Various properties of the resulting time are discussed, including its close relation to the Larmor times. It is a universal description of a broad class of measurement interactions, and its physical implications are unambiguous.PACS numbers: 03.65. Bz,73.40.Gk The question posed in the title has remained controversial since the early days of quantum theory [1][2][3][4][5]. One commonly cited reason for this is the nonexistence of a quantum-mechanical time operator. However, it is quite possible to construct an operator Θ B which measures whether a particle is in the barrier region or not. Such a projection operator is Hermitian, and may correspond to a physical observable. It has eigenvalues 0 and 1, and its expectation value simply measures the integrated probability density over the region of interest-it is this expectation value divided by the incident flux which is referred to as the "dwell time" [6]. Thus the central problem is not the absence of an appropriate Hermitian operator, but rather the absence of well-defined histories (or trajectories) in standard quantum theory. For the dwell time measures a property of a wave function with both transmitted and reflected portions, and does not display a unique decomposition into portions corresponding to these individual scattering channels. Some workers point out that it can be fruitful to consider the expectation value for a particular outgoing channel rather than a particular incident state [7]. However, this approach answers the present title's question no better than does the usual dwell time; instead of discarding information about late times it discards information about early times. Other related approaches follow phase space trajectories [7], Bohm trajectories [4], or Feynman paths [8]. No consensus has been reached as to the validity and the relationship of these various approaches. Ideally, transmission and reflection times τ T and τ R would, when weighted by the transmission and reflection probabilities |t| 2 and |r| 2 , yield the dwell time τ d :this relation has served as one of the main criteria in a broad review of tunneling times [3], but has also been criticized [5]. In this paper I present an approach to tunneling times which adheres strictly to the standard formulation of quantum mechanics, defining the "dwell" time by explicitly considering a general von Neumann-style measurement interaction. The mean value of the measurement outcome may then be calculated for transmitted and reflected particles individually, and I show that it automatically satisfies the above equation. This time is in general complex, and is closely related to the Larmor times and to the complex time of Sokolovski, Baskin, and Connor [8]; the consideration of a measuring apparatus, however, makes interpretation of its real and imaginary parts straightforward, as will be demonstrated by examining a generalization of the Larmor time. Before presenti...

Quantum averages of weak values

“…The usefulness of this description has been demonstrated, both theoretically and experimentally, in a number of applications in which novel aspects of quantum processes have been uncovered when analyzed in terms of * Electronic address: abotero@uniandes.edu.co weak values. These include photon polarization interference [7,8,9,10,11], barrier tunnelling times [12,13,14], photon arrival times [15,16], anomalous pulse propagation [17,18,19], correlations in cavity QED experiments [20], complementarity in "which-way" experiments [21,22], non-classical aspects of light [23,24], communication protocols [25] and retrodiction "paradoxes" of quantum entanglement [26,27,28].…”

Extracting Joint Weak Values with Local, Single-Particle Measurements

“…where X fi is the expectation value ofX given a successful post-selection of the final state |f , andP (f ) is the projector |f f | onto the final state [9]. This amounts to finding the expectation value of the operator O x ≡ |f f |X and renormalizing.…”

“…They are therefore of special interest in the context of postselected subensembles. In particular, they have been used to "resolve" [2] Hardy's Paradox [3]; to interpret and extend novel cavity-QED experiments [4,5]; to reconcile diverging views on which-path experiments and uncertainty [6,7,8]; and to address the long-standing controversy over tunneling times [9]. Often, it is of interest to discuss the weak value of a nonlocal observable (especially, of course, in proposals to study locality [2,3,10]); unfortunately, it is usually difficult or impossible to carry out such measurements [2,11].…”

Larmor precession and barrier tunneling time of a neutral spinning particle

“…There are various approaches dealing with tunneling time [3], but there has been no clear-cut answer to this old question [4] [5]. Recently a number of experiments [6] [7] [8] indicating superluminal transmission of photons has renewed interest in this subject.…”