Realizing Unitary k-Designs with a Single Quench

We present a single-quench protocol that generates unitary k-designs with minimal control. A system first evolves under a random Hamiltonian H_{1}; at a switch time t_{s}≥t_{Th} (the Thouless time), it is quenched to an independently drawn H_{2} from the same ensemble and then evolves under H_{2}. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary k-design using only a single control operation-far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Our analytical results are derived for Gaussian unitary random matrices, while our numerical demonstrations are carried out primarily in the Sachdev-Ye-Kitaev model. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of t_{Th} and provides a quantitative diagnostic of chaoticity. It further suggests symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and applications to randomized measurements, benchmarking, and tomography.