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Physical review. B./Physical review. B

Physical theory of two-stage thermalization

Cheryne Jonay, Tianci Zhou

Published Jul 22, 2024• 9 citations
Publisher

Abstract

Thermalization time marks the scale at which the subsystem entanglement reaches its thermal value. It was recently shown to exhibit two distinct stages in a generic class of quantum circuits with decay rates ${r}_{1}$ and ${r}_{2}$ before and after thermalization. The two rates are observed in the Floquet dynamics of the state purity, entanglement spectra, and (local) correlation functions, both in clean and disordered systems. Entanglement membrane theory interprets ${r}_{1}$ as domain-wall energy and ${r}_{2}$ as either a geometric effect (phantom eigenvalue) or a competition with an emergent magnon mode. In the case when the domain wall dominates, this offers a practical measure for entanglement growth via local correlations.

DOI
10.1103/physrevb.110.l020306

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Thermalization time marks the scale at which the subsystem entanglement reaches its thermal value. It was recently shown to exhibit two distinct stages in a generic class of quantum circuits with decay rates r1{r}_{1}r1​ and r2{r}_{2}r2​ before and after thermalization. The two rates are observed in the Floquet dynamics of the state purity, entanglement spectra, and (local) correlation functions, both in clean and disordered systems. Entanglement membrane theory interprets r1{r}_{1}r1​ as domain-wall energy and r2{r}_{2}r2​ as either a geometric effect (phantom eigenvalue) or a competition with an emergent magnon mode. In the case when the domain wall dominates, this offers a practical measure for entanglement growth via local correlations.
One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates r1r_1r1​ and r2r_2r2​ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with r1r_1r1​ corresponding to the domain wall free energy. Circuit geometry can lead to r1<r2r_1 < r_2r1​<r2​, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to r2<r1r_2 < r_1r2​<r1​ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.
Classical statistical mechanics relies on the ergodic hypothesis, which states that a system explores the entire allowed phase space during its time evolution. Pure quantum states are at odds with this concept as they remain in a single quantum state even if the system's parameters are suddenly altered. However, in a generic many-body system, coherent dynamics drive local thermalization of its subsystems. In this case, any local observable appears thermal and can be described by a statistical ensemble even though the full state remains pure. The key to understanding the phenomenon of quantum thermalization lies in the entanglement properties of the system: The presence of entanglement creates local entropy that validates the use of statistical physics.\nIn this thesis, we experimentally study the emergence of statistical mechanics in a quantum state and observe the fundamental role of quantum entanglement in facilitating this emergence. We perform microscopy on an evolving quantum system, and we see thermalization occur on a local scale while we measure that the full quantum state remains pure.\nThe only known robust exception from the paradigm of quantum thermalization is provided by disordered systems. In this case, the particle transport ceases through the system, preventing the formation of a sufficient amount of entanglement required for thermalization. However, if the many-body system is interacting, the state develops non-local quantum correlations that are inaccessible to any local observable. These correlations spread exponentially slowly in time throughout the entire system and are considered a hallmark feature of many-body-localized states.\nIn order to study the interplay between thermalization and localization in our system, we develop novel methods for high fidelity state preparation and readout. We achieved an exquisite isolation of our system from the environment, which allows us to monitor coherent many-body dynamics over multiple decades of time evolution.\nWe experimentally realize such a many-body localized system and observe the formation of non-local quantum correlations, while the particle transport is frozen in the system. Our work experimentally establishes many-body localization as a qualitatively distinct phenomenon from localization in non-interacting, disordered systems.
We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with three distinct timescales. First, on an o(1) timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free 1/f form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.
Abstract Understanding thermalization and irreversibility in open quantum systems requires char-acterizing the competition between coherent unitary dynamics and dissipative environmen-tal interactions. In this work, we present a systematic numerical study of thermalization dynamics in a two-qubit open quantum system governed by a completely positive and trace-preserving Lindblad master equation. The coherent evolution rate is controlled by a real scaling parameter α multiplying the Hamiltonian commutator, while decoherence is indepen-dently controlled by a dephasing rate γ . Although such a scaling of coherent dynamics can be absorbed into a rescaling of time in isolated systems, it becomes physically meaningful in open systems where decoherence introduces an absolute timescale. We investigate how the relative strength of coherent evolution and decoherence influences observable dynamics, purity decay, entropy production, and thermalization times. Using robust local and two-site observables, we show that varying α at fixed γ produces distinct transient trajectories, even though all evolutions converge to the same steady state. We further define an operational thermalization time τ ( γ, α ) based on a purity threshold and map the resulting thermalization-time surface across a broad parameter range. Increasing α generally accelerates oscillatory dynamics and modifies the approach to equilibrium, leading to measurable shifts in thermalization times without altering asymptotic states. These findings provide a clear and physically consistent characterization of how Hamil-tonian strength and decoherence rate jointly determine thermalization behavior in open quantum systems. The framework offers a practical parameterization for numerical mod-eling and experimental studies of coherent–dissipative competition in quantum information processing, quantum control, and open-system dynamics.
Abstract We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system’s size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits.
Energy-filtered quantum states are promising candidates for efficiently simulating thermal states. We explore a protocol designed to transition a product state into an eigenstate located in the middle of the spectrum; this is achieved by gradually reducing its energy variance, which allows us to comprehensively understand the crossover phenomenon and the subsequent convergence toward thermal behavior. We introduce and discuss three energy-filtering regimes (short, medium, and long), and we interpret them as stages of thermalization. We show that the properties of the filtered states are locally indistinguishable from those of time-averaged density matrices, routinely employed in the theory of thermalization. On the other hand, nonlocal quantum correlations are generated in the medium regimes and are witnessed by the Rényi entanglement entropies of subsystems, which we compute via replica methods. Specifically, two-point correlation functions break cluster decomposition, and the entanglement entropy of large regions scales as the logarithm of the volume during the medium filter time.