Wave packets, "negative times" and the elephant in the room

Abstract

Controversy surrounding the "tunnelling time problem" stems from the seeming inability of quantum mechanics to provide, in the usual way, a definition of the duration a particle is supposed to spend in a given region of space. For this reason, the problem is often approached from an "operational" angle. One such approach uses the position of the transmitted wave packet in order to infer the duration the particle spends in the barrier. Here we replace the barrier with a tuneable Mach-Zehnder interferometer (MZI). With this analogy one is able, at least in principle, to achieve any advance or delay of the wave packet sent to the chosen outgoing port. The Uncertainty Principle prevents one from combining the durations spent in each arm the MZI into a meaningful duration when both arms are engaged. There is no justification for invoking "superluminal" or "negative" times, since the particle is able to arrive at the same position (and with a higher probability) if the same initial state propagates through only one arm of the MZI. The same is true, we argue, in the case of tunnelling, where the transmitted wave packet results from destructive interference between multiple copies of the free state, delayed relative to the free propagation

Figures (4)

• Figure 1: a) A wave packet is divided into two parts which are recombined again while passing through a pair of beam splitters (BS). A delay by ${\tau}$ sec. occurs for the part passing through the right route. A wave packet travelling to detector $D_1$ consists of two parts (cf. Eq.(\ref{['2a']}) which may or may not overlap, depending on $v$, ${\tau}$, and the initial wave packet's width $\Delta x$. Their sum can result in a much smaller advanced Gaussian. b) A path diagram of the system in a) and the four amplitudes $\mathcal{A}_i$. The paths leading to different final states $D_1$ and $D_2$ never interfere. c) An equivalent quantum measurement problem (see Sect.IV).
• Figure 2: Contour plot of the probability density $P(x)=|\mathcal{A}_1G(x)+\mathcal{A}_2G(x+v{\tau})|^2$ vs. $x$ and $\Delta x$ in the case where the second WP is delayed. The path amplitudes are chosen so that $\mathcal{A}_1/\mathcal{A}_2=-1.5$. As the width $\Delta x$ increases, the maximum and the minimum, located at $x=-v{\tau}$ and $x=-v|{\tau}|/2$ for $\Delta x << v{\tau}$, disappear. The surviving maximum is pushed forward, and asymptotically approaches $\overline x=2$ [cf. (\ref{['4a']})], indicated by the dotted line. Also shown is the centre of mass (COM) of the density, whose position coincides with that of the density's peak only in the limit $\Delta x \to \infty$.
• Figure 3: Comparison between the normalised to unity density $|G_1(x)|^2$ in Eq.(\ref{['2a']}) and that obtained from Eq.(\ref{['3a']}) with $\overline x$ given by Eq.(\ref{['AA2']}) for $\Delta x/v{\tau}=5$. The amplitudes $\mathcal{A}_{1,2}$ are the same as in Fig.2, and the advancement of the peak of about $1.35v{\tau}$ is somewhat smaller than $2v{\tau}$, achieved for $\Delta x/v{\tau}\to \infty$.
• Figure 4: A comparison between the density $|G_1(x)|$ in Eq.(\ref{['2a']}) and the one obtained for the initial WP travelling along the left arm only. The parameters are the same as in Figs. 2 and 3.