Comparing experiments on quantum traversal time with the predictions of a space-time-symmetric formalism

The question of how long a particle takes to pass through a potential barrier is still a controversial topic in quantum mechanics. One of the main theoretical problems in obtaining estimates for measurable times is the fact that several previously defined time operators, which remained within the borders of standard quantum mechanics, present some kind of pathology. Recently, a time operator acting on an additional Hilbert space has been shown to support both Hermiticity and canonical relation with an energy observable. The theory is built in a framework that treats space and time as symmetrically as possible, in the nonrelativistic regime. In this work, we use this formalism to derive a closed analytical expression for the traversal time of a quantum particle impinging on a rectangular potential barrier. To do this, first we discuss the meaning of the expectation values in the additional Hilbert space and, then, we apply it to traversal times. We test our formalism in the specific experimental scenario of a realization by Ranfagni et al. [A. Ranfagni et al., Appl. Phys. Lett. 58, 774 (1991)]. The proposed approach displays a better performance in comparison with the Büttiker-Landauer and the phase-time approximations.

Figure 1

Delay-time measurements for a “potential barrier” of length $L$ vs frequency. The solid curve represents Eq. (23), the long dashed curve represents the PT model, and the short dashed curve illustrates the BL model. (a) Potential barrier of length $L=15$ cm. A bandwidth $\mathrm{\Lambda }=30$ MHz is used for our model. The dots represent the experimental data obtained by Ref. [8]. (b) Potential barrier of length $L=20$ cm. In this case, the bandwidth is $\mathrm{\Lambda }=50$ MHz. The bullets and open squares represent two different runs of measurements [8, 13]. The vertical line represents the cutoff frequency.