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Observation of Hierarchy of Hilbert Space Ergodicities in the Quantum Dynamics of a Single Spin

Wenquan Liu1,*, Zou-Wei Pan1,*, Yue Fu2, Wen Wei Ho3,4,†, and Xing Rong1,5,‡

  • *These authors contributed equally to this work.
  • †Contact author: wenweiho@nus.edu.sg
  • ‡Contact author: xrong@ustc.edu.cn
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Phys. Rev. Lett. 136, 020401 – Published 13 January, 2026

DOI: https://doi.org/10.1103/6msb-cxbc

Abstract

Ergodicity, the property that all allowed configurations are explored over time, plays a pivotal role in explaining the equilibrium behavior of classical dynamical systems. Yet, such a property is typically precluded in quantum systems owing to stationary energy eigenstates. However, recent theoretical works have argued that ergodic explorations of the Hilbert space, occurring at varying levels as measured by statistical pseudorandomness of the time-evolved states, may happen for aperiodic driven quantum systems. Here, we experimentally investigate the hierarchy of Hilbert-space ergodicities (HSEs) achievable in the dynamics of a single spin. Through subjecting a nitrogen-vacancy center in diamond to various time-dependent modulations and continuously monitoring the spin trajectories with full state tomography, different degrees of HSE were observed, ranging from no HSE in a time-periodic drive, to partial HSE in a smoothly kicked time-quasiperiodic drive, to complete HSE in an aperiodic Fibonacci drive. We formulate a theoretical understanding of the increasing levels of HSE by attributing them to increasing levels of complexity associated with the drive sequences. Our Letter provides the first unambiguous experimental evidence of Hilbert space ergodicity, promoting deeper investigations into the mechanisms and fine-grained levels with which closed quantum systems reach equilibrium.

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Phys. Rev. Lett. 136, 020401– Published 13 January, 2026
  • Received 15 October 2025
  • Accepted 19 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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