© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1101.2535 · Journal of Physics Conference Series

Entanglement, decoherence and thermal relaxation in exactly solvable models

Oleg Valentinovich Lychkovskiy

Published Jul 8, 2011• 2 citations

Abstract

Exactly solvable models provide an opportunity to study different aspects of reduced quantum dynamics in detail. We consider the reduced dynamics of a single spin in finite XX and XY spin 1/2 chains. First we introduce a general expression describing the evolution of the reduced density matrix. This expression proves to be tractable when the combined closed system (i.e. open system plus environment) is integrable. Then we focus on comparing decoherence and thermalization timescales in the XX chain. We find that for a single spin these timescales are comparable, in contrast to what should be expected for a macroscopic body. This indicates that the process of quantum relaxation of a system with few accessible states can not be separated in two distinct stages - decoherence and thermalization. Finally, we turn to finite-size effects in the time evolution of a single spin in the XY chain. We observe three consecutive stages of the evolution: regular evolution, partial revivals, irregular (apparently chaotic) evolution. The duration of the regular stage is proportional to the number of spins in the chain. We observe a "quiet and cold period" in the end of the regular stage, which breaks up abruptly at some threshold time.

DOI: 10.1088/1742-6596/306/1/012028

Related papers

Entanglement and relaxation in exactly solvable models

Oleg Valentinovich Lychkovskiy

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.

Optics and Spectroscopy•Nov 1, 2011•1 citation

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems I.

Sachiko Kitajima, Fumiaki Shibata

Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).

Journal of the Physical Society of Japan•Jan 1, 2000•14 citations

Evolution of Entanglement Spectra under Generic Quantum Dynamics

Po-Yao Chang, Xiao Chen, Sarang Gopalakrishnan +1 more

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.

arXiv:1811.00029•Physical Review Letters•Nov 6, 2019•30 citations

Thermalization in a simple spin-chain model

Peter Reimann, Christian Eidecker-Dunkel

We consider the common spin-1/2 XX model in one dimension with open boundary conditions and a large but finite number of spins. The system is in thermal equilibrium at times t < 0 , and is subject to a weak local perturbation (quantum quench) at t = 0 . Focusing mainly on single-spin perturbations and observables, we show that the system rethermalizes for sufficiently large times t > 0 without invoking any unproven assumptions besides the basic laws of quantum mechanics. Moreover, the time-dependent relaxation behavior is obtained in quantitative detail and is found to exhibit a wealth of interesting features.

arXiv:2502.19228•Physical review. B./Physical review. B•Feb 25, 2025•2 citations

Entanglement in an open Heisenberg spin chain

Subhendu Paul

We analyze the time evolution of a spin chain of three qubits with nearest-neighbour Heisenberg XX interaction, coupled to two thermal reservoirs. We use the ‘reduced density matrix approach’ to calculate the populations and correlations (concurrence) among the spins. The spin-reservoir interaction is of the energy conserving type. This allows us to analyze the evolution analytically without approximations – yielding explicit, yet complicated formulas for the spin dynamics. To analyze the latter and discuss it as a function of the various physical parameters and time (intial conditions, termperatures, coupling strenghts), we resort to numerical methods. Our main finding is that the external noise (coupling to reservoirs) causes the appearence of an asymptotic, stationary (time-periodic) regime for the dynamics of concurrence. We discuss the dependence of the relaxation time and the characteristics of the regime itself, on the various parameters.

•Sep 1, 2018•0 citations

Nonequilibrium thermal entanglement for simple spin chains

Ilya Sinayskiy, N. Pumulo, Francesco Petruccione

The dynamics of a chain of two and three spins coupled at both ends to separate bosonic baths at different temperatures is studied. An exact analytical solution of the quantum master equation in the Born-Markov approximation for the reduced density matrix of the chain is constructed. It is shown that for long times the reduced density matrix converges to the nonequilibrium steady-state. Dynamical and steady state properties of the concurrence between the first and the last spin are studied.

AIP conference proceedings•Jan 1, 2012•0 citations

Exactly solvable models provide an opportunity to study different aspects of reduced quantum dynamics in detail. We consider the reduced dynamics of a single spin in finite XX and XY spin 1/2 chains. First we introduce a general expression describing the evolution of the reduced density matrix. This expression proves to be tractable when the combined closed system (i.e. open system plus environment) is integrable. Then we focus on comparing decoherence and thermalization timescales in the XX chain. We find that for a single spin these timescales are comparable, in contrast to what should be expected for a macroscopic body. This indicates that the process of quantum relaxation of a system with few accessible states can not be separated in two distinct stages - decoherence and thermalization. Finally, we turn to finite-size effects in the time evolution of a single spin in the XY chain. We observe three consecutive stages of the evolution: regular evolution, partial revivals, irregular (apparently chaotic) evolution. The duration of the regular stage is proportional to the number of spins in the chain. We observe a "quiet and cold period" in the end of the regular stage, which breaks up abruptly at some threshold time.

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.

Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).

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.

We consider the common spin-1/2 XX model in one dimension with open boundary conditions and a large but finite number of spins. The system is in thermal equilibrium at times t < 0 , and is subject to a weak local perturbation (quantum quench) at t = 0 . Focusing mainly on single-spin perturbations and observables, we show that the system rethermalizes for sufficiently large times t > 0 without invoking any unproven assumptions besides the basic laws of quantum mechanics. Moreover, the time-dependent relaxation behavior is obtained in quantitative detail and is found to exhibit a wealth of interesting features.

We analyze the time evolution of a spin chain of three qubits with nearest-neighbour Heisenberg XX interaction, coupled to two thermal reservoirs. We use the ‘reduced density matrix approach’ to calculate the populations and correlations (concurrence) among the spins. The spin-reservoir interaction is of the energy conserving type. This allows us to analyze the evolution analytically without approximations – yielding explicit, yet complicated formulas for the spin dynamics. To analyze the latter and discuss it as a function of the various physical parameters and time (intial conditions, termperatures, coupling strenghts), we resort to numerical methods. Our main finding is that the external noise (coupling to reservoirs) causes the appearence of an asymptotic, stationary (time-periodic) regime for the dynamics of concurrence. We discuss the dependence of the relaxation time and the characteristics of the regime itself, on the various parameters.