# What is the maximum differential group delay achievable by a space-time wave packet in free space?

## TL;DR

This paper investigates the maximum differential group delay achievable by space-time wave packets in free space, showing it is limited by spectral uncertainty and experimentally demonstrating delays of about 150 ps, far exceeding previous results.

## Contribution

It provides a theoretical and experimental analysis of the maximum achievable group delay of space-time wave packets, revealing spectral uncertainty as the limiting factor.

## Key findings

- Maximum differential group delay is limited by spectral uncertainty.
- Experimental delays of approximately ±150 ps were achieved.
- Propagation is bounded by a spectral-uncertainty-induced pilot envelope.

## Abstract

The group velocity of 'space-time' wave packets $-$ propagation-invariant pulsed beams endowed with tight spatio-temporal spectral correlations $-$ can take on arbitrary values in free space. Here we investigate theoretically and experimentally the maximum achievable group delay that realistic finite-energy space-time wave packets can achieve with respect to a reference pulse traveling at the speed of light. We find that this delay is determined solely by the spectral uncertainty in the association between the spatial frequencies and wavelengths underlying the wave packet spatio-temporal spectrum $-$ and not by the beam size, bandwidth, or pulse width. We show experimentally that the propagation of space-time wave packets is delimited by a spectral-uncertainty-induced `pilot envelope' that travels at a group velocity equal to the speed of light in vacuum. Temporal walk-off between the space-time wave packet and the pilot envelope limits the maximum achievable differential group delay to the width of the pilot envelope. Within this pilot envelope, the space-time wave packet can locally travel at an arbitrary group velocity and yet not violate relativistic causality because the leading or trailing edge of superluminal and subluminal space-time wave packets, respectively, are suppressed once they reach the envelope edge. Using pulses of width $\sim$4ps and a spectral uncertainty of $\sim$ 20 pm, we measure maximum differential group delays of approximately $\pm$ 150 ps, which exceed previously reported measurements by at least three orders of magnitude.