“Superluminal” transmission via entanglement, superoscillations, and quasi-Dirac distributions

We analyze a system in which, due to entanglement between the spin and spatial degrees of freedom, the reduced transmitted state has the shape of the freely propagating pulse translated in the complex coordinate plane. In the case an apparently “superluminal” advancement of the pulse, the delay amplitude distribution is found to be a peculiar approximation to the Dirac $\delta$ function, and the transmission coefficient exhibits a well-defined superoscillatory window. Analogies with potential tunneling and Wheeler’s delayed choice experiment are highlighted.

Figure 1

The moments of the DAD (9) with $K=1$, $15$, and $30$ for $\mathrm{\alpha ̃}\equiv \alpha /(K\Delta x)=4.$

Figure 2

The DAD (9) for $K=30$ and $\alpha /\Delta x=-15$, $-15.5$, and $120$.

Figure 3

(a) The transmitted envelope [multiplied by the factor $1/\sqrt{P^{\mathrm{best}}}$ in Eq. (11)] for $K=30$, $\sigma /K\Delta x=2$, and $\mathrm{\alpha ̃}\equiv \alpha /(K\Delta x)=4.$ Also shown by the dashed line is $G(X-\alpha )$ in the right-hand side of Eq. (6); (b) same as (a) except for $\mathrm{\alpha ̃}=4.5$; (c) same as (a) except for $\mathrm{\alpha ̃}=3.5+2i$; (d) $\mathit{Re}[T(p)]$ vs. $p$ (solid) for the case shown in (a). Also shown are $\sin (-\alpha p)$ (dashed) and $A(p)$ normalized to a unit height (thick solid). Vertical lines and arrows indicate the boundaries of the superoscillatory window; (e) same as (d) but for the case shown in (b); (f) $|T(p)/\exp (-i{\alpha }_{1}p+{\alpha }_{2}p)|$ (solid) and $A(p)$ (thick solid) for the case shown in (c).