Operational approach to indirectly measuring the tunneling time

The tunneling time through an arbitrary bounded one-dimensional barrier is investigated using the dwell-time operator. We relate the tunneling time to the conditioned average of the dwell-time operator because of the natural postselection in the case of successful tunneling. We discuss an indirect measurement by timing the particle and show that we are able to reconstruct the conditioned average value of the dwell-time operator by applying the contextual values formalism for generalized measurements based on the physics of Larmor precession. The experimentally measurable tunneling time in the weak interaction limit is given by the weak value of the dwell-time operator plus a measurement-context-dependent disturbance term. We show how the expectation value and higher moments of the dwell-time operator can be extracted from measurement data of the particle's spin.

Figure 1

The one-dimensional tunneling configuration of our system: A wave packet is traveling toward the potential barrier from the left-hand side. After interaction with the barrier, the spin measurement is performed on the reflected and transmitted portion.

Figure 2

Contextual values times Larmor frequency ${\omega }_L$ are shown as a function of incident wave vector, for square (a), symmetric (b), and antisymmetric (c) potential barriers. The values of the parameters are $\hslash =1$, $m=1/2$, $d{k}_0=3\pi$, $a=k_0^2/d^2$, and $\epsilon =0.5k_0^2$. The plots correspond to the spin postselection in $\theta =\pi /2-\pi /8$ and $\varphi =\pi /4$ in Eq. (16).

Figure 3

The real part of the weak value and the conditioned average are compared for different shapes of the symmetric potential barrier. Panels (a) and (b) are for the square barrier, corresponding to the spin postselection, $(\theta ,\varphi )=(\pi /2-\pi /8,\pi /4)$ and $(\theta ,\varphi )=(\pi /2-\pi /200,\pi /4)$, respectively. Panel (c) is for the non-square-symmetric potential as in Fig. 2b when $(\theta ,\varphi )=(\pi /2-\pi /8,\pi /4)$. The values of the parameters are $\hslash =1$, $m=1/2$, $d{k}_0=3\pi$, and $a=k_0^2/d^2$.