Structures far below sub-Planck scale in quantum phase-space through superoscillations

TL;DR

This paper demonstrates that superoscillations can produce highly localized structures in quantum phase-space below the previously established minimum scale, without increasing interferometric sensitivity.

Contribution

It introduces a method using superoscillatory functions to create structures in Wigner distributions below Zurek's minimum scale, expanding understanding of quantum phase-space features.

Findings

Wigner distribution can have structures smaller than Zurek's scale

Superoscillations enable sub-scale structures without sensitivity increase

Structures are exponentially small in amplitude

Abstract

In 2001, Zurek derived the generic minimum scale $a_{Z}$ for the area of structures of Wigner's quantum phase distribution. Here we show by construction, using superoscillatory functions, that the Wigner distribution can locally show regular spotty structures on scales much below Zurek's scale $a_{Z}$. The price to pay for the presence of such structures is their exponential smallness. For the case we construct there is no increased interferometric sensitivity from the presence of patches with superoscillatory structure in phase-space.

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