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arXiv:2406.15577 · Physical Review Letters

Decoherence of Histories: Chaotic Versus Integrable Systems

Jiaozi Wang, Philipp Strasberg

Published May 14, 2025• 4 citations
arXivHTMLPDF

Abstract

We study the emergence of decoherent histories in isolated systems based on exact numerical integration of the Schrödinger equation for a Heisenberg chain. We reveal that the nature of the system, which we switch from (i) chaotic to (ii) interacting integrable to (iii) noninteracting integrable, strongly impacts decoherence of coarse spin observables. From a finite size scaling law we infer a strong exponential suppression of coherences for (i), a weak exponential suppression for (ii), and no exponential suppression for (iii) on a relevant short (nonequilibrium) timescale. Moreover, for longer times we find stronger decoherence for (i) but the opposite for (ii), hinting even at a possible power-law decay for (ii) at equilibrium timescales. This behavior is encoded in the multitime properties of the quantum histories and it can not be explained by environmentally induced decoherence. Our results suggest that chaoticity plays a crucial role in the emergence of classicality in finite size systems.

DOI
10.1103/m8vq-l449

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We study the emergence of decoherent histories in isolated systems based on exact numerical integration of the Schrödinger equation for a Heisenberg chain. We reveal that the nature of the system, which we switch from (i) chaotic to (ii) interacting integrable to (iii) noninteracting integrable, strongly impacts decoherence of coarse spin observables. From a finite size scaling law we infer a strong exponential suppression of coherences for (i), a weak exponential suppression for (ii), and no exponential suppression for (iii) on a relevant short (nonequilibrium) timescale. Moreover, for longer times we find stronger decoherence for (i) but the opposite for (ii), hinting even at a possible power-law decay for (ii) at equilibrium timescales. This behavior is encoded in the multitime properties of the quantum histories and it can not be explained by environmentally induced decoherence. Our results suggest that chaoticity plays a crucial role in the emergence of classicality in finite size systems.
Within the histories formalism the decoherence functional is a formal tool to investigate the emergence of classicality in isolated quantum systems, yet an explicit evaluation of it from first principles has not been reported. We provide such an evaluation for up to five-time histories based on exact numerical diagonalization of the Schrödinger equation. We find a robust emergence of decoherence for slow and coarse observables of a generic random matrix model and extract a finite-size scaling law by varying the Hilbert space dimension over 4 orders of magnitude. Specifically, we conjecture and observe an exponential suppression of coherent effects as a function of the particle number of the system. This suggests a solution to the preferred basis problem of the many-worlds interpretation (or the set selection problem of the histories formalism) within a minimal theoretical framework without relying on environmentally induced decoherence, quantum Darwinism, Markov approximations, low-entropy initial states, or ensemble averages. Published by the American Physical Society 2024
Answers to the question how a classical world emerges from underlying quantum physics are revisited, connected and extended as follows. First, three distinct concepts are compared: decoherence in open quantum systems, consistent/decoherent histories and Kolmogorov consistency. Second, the crucial role of quantum Markovianity (defined rigorously) to connect these concepts is established. Third, using a random matrix theory model, quantum effects are shown to be exponentially suppressed in the measurement statistics of slow and coarse observables despite the presence of large amount of coherences. This is also numerically exemplified, and it highlights the potential and importance of non-integrability and chaos for the emergence of classicality.
We study a finite-resolution route to decoherence-like behavior in quantum systems whose classical limit is chaotic. The mechanism does not require tracing over an external environment, adding phenomenological noise, or changing unitary quantum mechanics. Instead, chaotic phase-space transport drives Wigner-function interference fringes to progressively shorter wavelengths, while any observation, reconstruction, or deliberately coarse description has finite phase-space resolution. Modeling that resolution by Gaussian coarse-graining, we show that interference components transported below the resolution scale are exponentially attenuated in the coarse-grained phase-space description. The exact quantum state remains pure under unitary evolution; the suppression is, therefore, operational rather than objective. In the semiclassical pre-Ehrenfest window, the instability scale is set by finite-time Lyapunov exponents, while the dimensionless suppression factor is determined by the initial fringe geometry and the coarse-graining covariance. The same geometry gives a robustness criterion: interference components with strong projection onto covector-expanding Oseledets directions are suppressed first, while components avoiding those directions remain visible longer. A hyperbolic cat-map example illustrates the mechanism explicitly. We state the regime of validity, distinguish the mechanism from environment-induced decoherence, and give a practical extraction procedure for numerical or experimental tests.
Decoherence in quantum systems is conventionally modeled as the effect of interactions with an external environment. However, such a prescription excludes isolated many-body systems, which are also expected to display classical behavior at macroscopic scales. In isolated systems, decoherence must emerge internally from microscopic degrees of freedom that are invisible to the macroscopic description. Here we explicitly show that classicality can emerge in such a fashion. We consider a weakly disordered, 333-local chaotic kicked top of NNN qubits, where the collective spin sector serves as the macroscopic description, while the microscopic permutation sector acts as an internal bath, decohering the collective spin sector. Starting from closed unitary dynamics, we derive and numerically confirm an effective Lindblad equation for the collective spin variables. In the thermodynamic limit these reduced dynamics converge to a classical chaotic Fokker--Planck equation with vanishingly small diffusion on the spherical phase space, producing a quantum-classical correspondence beyond the Ehrenfest time. The chaotic dynamics evolve the pure many-body wavefunction into continuously branching components associated with distinct classical trajectories. These branches acquire nearly orthogonal microscopic records in the permutation sector, preventing quantum interferences and ensuring the corresponding histories remain consistent.
The problems of chaos and relaxation have a fundamental importance in the study of many-body classical and quantum systems. We investigate some of the issues related to these problems numerically in classical and quantum spin systems. New results reported in this thesis include: (i) A remarkably simple algorithm for discriminating chaotic from nonchaotic behavior in classical systems using a time series of one macroscopic observable. The effectiveness of this algorithm stems from the qualitative differences in the power spectra of chaotic and nonchaotic systems. (ii) A modified version of the Onsager regression relation applicable to pure quantum states. (iii) An efficient algorithm for computing the infinitetemperature time correlation functions in systems with large Hilbert spaces. (iv) Absence of exponential sensitivity to small perturbations in macroscopic nonintegrable systems of spins 1/2 . Such a behavior is contrasted with the exponential sensitivity to small perturbations in chaotic classical spin systems. (v) Accurate numerical investigations of free induction decay and spin diffusion in certain spin lattices. The consequences of these results have implications for the foundations of statistical mechanics or practical problems such as computing the long-time behavior of the free induction decay in solids.
The emergence of classical behavior and the origin of thermal equilibrium are two central problems in the foundations of physics. In the standard accounts, both phenomena are typically explained through interactions with an external environment: decoherence suppresses quantum interference, while coupling to a thermal bath drives relaxation toward equilibrium. Over the last decades, however, it has become clear that equilibration and thermalization can arise even in fully isolated quantum systems, in the operational sense that the expectation values of relevant observables remain close to equilibrium values for most of the time. Here, we ask whether the same intrinsic equilibration mechanism can also account for the emergence of classical behavior. Using rigorous bounds on equilibration in closed systems, we derive sufficient conditions under which a time-evolved pure state becomes, for most times, operationally indistinguishable from a classical mixture associated with a chosen physical property. We identify two complementary routes to such operational classicality: either the chosen property almost commutes with the system Hamiltonian or the observables used to probe the system lose access to the remaining coherence after equilibration. Our results show that classical behavior need not be confined to the energy basis and may emerge even when substantial coherence remains present in the equilibrium state. This establishes a direct connection between two foundational questions: the origin of thermalization in isolated quantum systems and the quantum-to-classical transition