  Accessing sub-wavelength information about a scene from the far-field without
invasive near-field manipulations is a fundamental challenge in wave
engineering. Yet it is well understood that the dwell time of waves in complex
media sets the scale for the waves' sensitivity to perturbations. Modern
coded-aperture imagers leverage the degrees of freedom (DoF) offered by complex
media as natural multiplexor but do not recognize and reap the fundamental
difference between placing the object of interest outside or within the complex
medium. Here, we show that the precision of localizing a sub-wavelength object
can be improved by several orders of magnitude simply by enclosing it in its
far field with a reverberant chaotic cavity. We identify deep learning as
suitable noise-robust tool to extract sub-wavelength information encoded in
multiplexed measurements, achieving resolutions well beyond those available in
the training data. We demonstrate our finding in the microwave domain:
harnessing the configurational DoF of a simple programmable metasurface, we
localize a sub-wavelength object inside a chaotic cavity with a resolution of
$\lambda/76$ using intensity-only single-frequency single-pixel measurements.
Our results may have important applications in photoacoustic imaging as well as
human-machine interaction based on reverberating elastic waves, sound or
microwaves.
