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.
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.
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 has been one of the fundamental concepts underpinning our understanding of thermalization in isolated systems since the first developments in classical statistical mechanics. Recently, a similar notion has been introduced for quantum systems, termed complete Hilbert space ergodicity (CHSE), in which the evolving quantum state explores all of the available Hilbert space. This contrasts with the eigenstate thermalization hypothesis (ETH), in which thermalization is formulated via the properties of matrix elements of local operators in the energy eigenbasis. In this work we explore how ETH-violation mechanisms, including quantum many-body scars and Hilbert space fragmentation, can affect complete Hilbert space ergodicity. We find that the presence of these mechanisms leads to CHSE in decoupled subspaces, a phenomenon we call Hilbert subspace ergodicity, and which represents a protocol for constructing t -designs in subspaces.
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.
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
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.
We identify a general mechanism for the finite-time thermalization of macroscopic observables, such as coarse-grained charge densities, in terms of elementary forms of the quantum dynamics of initial states: (1) aperiodicity, which provides a computable measure of (2) a dynamical partially ergodic exploration of the Hilbert space. Specifically, this mechanism predicts the equilibration of all (concentrated) macroscopic observables, in almost all states in an initial ensemble and almost all times within finite and longer intervals, given only the observable-independent information that the return probability of the ensemble of initial states is small over a finite time range. As a special case, it also accesses standard results on equilibration over infinitely long times in terms of (stronger versions of) the effective dimension of initial state delocalization in the energy eigenbasis. Our results incorporate macroscopic thermalization into the domain of operational quantum statistical mechanics, recently developed to provide finitely computable criteria for microscopic thermalization. We discuss an overall characterization of this approach as establishing connections between (1) the decay of a (theoretically or experimentally) computable probe indicating memorylessness, (2) a fundamental invariant mechanism in terms of the alignment of observables or states in the Hilbert space, and (3) predicting different natural forms of (classical and) quantum thermalization, most of which rigorously recover conventional eigenstate-based descriptions of infinite-time thermalization as a special case but provide stronger accessible predictions over finite observation times in the thermodynamic limit.