Direct Measurement of Time-Frequency Analogues of Sub-Planck Structures

TL;DR

This paper experimentally demonstrates sub-Planck structures in the time-frequency phase space using superpositions of electric pulses, revealing interference patterns with displacements smaller than uncertainty limits.

Contribution

It introduces a method to directly measure time-frequency sub-Planck structures using FROG, advancing understanding of quantum-like interference in classical optical fields.

Findings

Observed chessboard-like interference patterns with sub-uncertainty displacements.

Demonstrated phase space structures analogous to quantum sub-Planck features.

Validated the correspondence between optical phase space and quantum position-momentum structures.

Abstract

Exploiting the correspondence between Wigner distribution function and a frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate existence of the chessboard-like interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase space representation of compass states. Using superpositions of four electric pulses as realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for superposition of two Sch\"{o}dinger-cat-like states in a position-momentum phase space.