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.
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.
Hilbert-space ergodicity (HSE) provides a new framework for studying thermalization in driven quantum systems, complementing the eigenstate thermalization hypothesis, which is restricted to static systems. This ergodicity is hierarchical: by quantifying how randomly the dynamics explores the Hilbert space, one obtains a family of levels termed k-HSE. While HSE has been observed at the lowest and highest levels, finite-order HSE dynamics remains largely unexplored due to the difficulty of constructing such drives. Here, we explore this intermediate regime and uncover its distinctive physics. We first propose and prove that a family of m-tone drives on qubits realizes k-HSE up to k=2m−3, with drive parameters determined at O(k) cost. Using a single nitrogen-vacancy center in diamond, we verify this design by showing that a 3-tone drive realizes 3-HSE, with fourth-order statistics depending on the initial state. Further in-depth theoretical analysis shows that this initial-state dependence is generic across drives, demonstrating the possibility of recovering the initial state from higher-order statistics even when the dynamics is ergodic. Our work broadens the study of quantum ergodicity and reveals intriguing physics within its hierarchy.
Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry's conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity-generated by the Fibonacci word and its generalizations-for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.
Quantum evolution is generally expected to drive a quantum many-body system toward equilibrium. This expectation is often justified by the Hilbert-space ergodicity of generic quantum dynamics, namely, the idea that pure-state evolution explores Hilbert space uniformly up to physical constraints. Such a statement can be made rigorous by requiring the associated state ensemble to form the Haar-random ensemble, or its more structured generalization, the Scrooge ensemble. In this Letter, we report the emergence of exact Hilbert-space ergodicity in a continuously monitored quantum many-body system. For any target density matrix σ, we construct a continuously monitored system for which we rigorously prove that the Scrooge ensemble of σ is the unique late-time equilibrium distribution of quantum trajectories. Remarkably, this requires only that the jump operators in the monitoring form a deformed unitary 1-design, a seemingly much weaker condition than full ergodicity. We numerically demonstrate our predictions by simulating continuously monitored systems whose equilibrium states are thermal states. Our results establish a rigorous mechanism for the emergence of Hilbert-space ergodicity and provide a practical route for its investigation on quantum devices.
Systems reaching thermal equilibrium are ubiquitous. For classical systems, this phenomenon is typically understood statistically through ergodicity in phase space, but translating this to quantum systems is a long-standing problem of interest. Recently, a strong notion of quantum ergodicity has been proposed, namely, that isolated, global quantum states uniformly explore their available state space, dubbed . Here, we observe signatures of this process with an experimental Rydberg quantum simulator and various numerical models, before generalizing to the case of a local quantum system interacting with its environment. For a closed system, where the environment is a complementary subsystem, we predict and observe a smooth quantum-to-classical transition in that observables progress from large, quantum fluctuations to small, Gaussian fluctuations as the bath size grows. This transition exhibits universal properties on a quantitative level among a wide range of systems, including those at finite temperature, those with itinerant particles, and random circuits. For an open system, where the environment is uncontrolled, we predict the statistics of observables under largely arbitrary noise channels including those with correlated errors, allowing us to discriminate between candidate error models both for continuous Hamiltonian time evolution and for digital random circuits. This allows for computationally efficient experimental noise learning and, more broadly, is a new avenue for quantitatively classifying the behavior of noisy quantum systems. Ultimately, our results clarify the role of ergodicity in quantum dynamics, with fundamental and practical consequences.
Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society 2024
The maximum entropy principle is foundational for statistical analyses of complex dynamics. This principle has been challenged by the findings of a previous work [Phys. Rev. X 7, 031034 (2017)PRXHAE2160-330810.1103/PhysRevX.7.031034], where it was argued that a quantum system driven in time by a certain aperiodic sequence without any explicit symmetries, dubbed the Thue-Morse drive, gives rise to emergent nonergodic steady states which are underpinned by effective conserved quantities. Here, we resolve this apparent tension. We rigorously prove that the Thue-Morse drive achieves a very strong notion of quantum ergodicity in the long-time limit: The time evolution of any initial state uniformly visits every corner of its Hilbert space. On the other hand, we find the dynamics also approximates a Floquet drive for arbitrarily long, albeit finite periods of time with no characteristic timescale, resulting in a scale-free ergodic dynamics we call critically slow complete Hilbert-space ergodicity. Furthermore, numerical studies reveal that critically slow complete Hilbert-space ergodicity is not specific to the Thue-Morse drive and is, in fact, exhibited by many other aperiodic drives derived from morphic sequences, i.e., words derived from repeatedly applying substitution rules on basic characters. Our Letter presents a new class of dynamics in time-dependent quantum systems where full ergodicity is eventually attained but only after astronomically long times.