# Driven quantum dynamics: will it blend?

## TL;DR

This paper demonstrates how driven many-body quantum systems can generate Haar-random unitaries, with convergence properties analyzed through Bethe-Ansatz and mean-field techniques, revealing potential for physical randomness generation.

## Contribution

It introduces a method to produce Haar-uniform random operations in driven quantum systems using control and integrable models, and analyzes convergence times and spectral gaps.

## Key findings

- Any fully controllable system converges to a unitary q-design over time.
- The spectral gap of the driven spin chain's Liouvillean is independent of q.
- Bethe-Ansatz shows the gap's independence from q, suggesting a universal property.

## Abstract

Randomness is an essential tool in many disciplines of modern sciences, such as cryptography, black hole physics, random matrix theory and Monte Carlo sampling. In quantum systems, random operations can be obtained via random circuits thanks to so-called q-designs, and play a central role in the fast scrambling conjecture for black holes. Here we consider a more physically motivated way of generating random evolutions by exploiting the many-body dynamics of a quantum system driven with stochastic external pulses. We combine techniques from quantum control, open quantum systems and exactly solvable models (via the Bethe-Ansatz) to generate Haar-uniform random operations in driven many-body systems. We show that any fully controllable system converges to a unitary q-design in the long-time limit. Moreover, we study the convergence time of a driven spin chain by mapping its random evolution into a semigroup with an integrable Liouvillean and finding its gap. Remarkably, we find via Bethe-Ansatz techniques that the gap is independent of q. We use mean-field techniques to argue that this property may be typical for other controllable systems, although we explicitly construct counter-examples via symmetry breaking arguments to show that this is not always the case. Our findings open up new physical methods to transform classical randomness into quantum randomness, via a combination of quantum many-body dynamics and random driving.