Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1405.7594 · Physical Review B

From quantum-mechanical to classical dynamics in the central-spin model

Daniel Stanek, Carsten Raas, Götz Silvester Uhrig

Published Aug 4, 2014• 35 citations
arXivHTMLPDF

Abstract

We discuss the semiclassical and classical character of the dynamics of a single spin $1/2$ coupled to a bath of noninteracting spins $1/2$. On the semiclassical level, we extend our previous approach presented in D. Stanek, C. Raas, and G. S. Uhrig, Phys. Rev. B 88, 155305 (2013) by the explicit consideration of the conservation of the total spin. On the classical level, we compare the results of the classical equations of motions in absence and presence of an external field to the full quantum result obtained by density-matrix renormalization (DMRG). We show that for large bath sizes and not too low magnetic field the classical dynamics, averaged over Gaussian distributed initial spin vectors, agrees quantitatively with the quantum-mechanical one. This observation paves the way for an efficient approach for certain parameter regimes.

DOI
10.1103/physrevb.90.064301

Related papers

Dynamics and decoherence in the central spin model in the low-field limit

Daniel Stanek, Carsten Raas, Götz Silvester Uhrig

We present a combination of analytic calculations and a powerful numerical method for large spin baths in the low-field limit. The hyperfine interaction between the central spin and the bath is fully captured by the density matrix renormalization group. The adoption of the density matrix renormalization group for the central spin model is presented and a proper method for calculating the real-time evolution at infinite temperature is identified. In addition, we study to which extent a semiclassical model, where a quantum spin-1/2 interacts with a bath of classical Gaussian fluctuations, can capture the physics of the central spin model. The model is treated by average Hamiltonian theory and by numerical simulation.

arXiv:1308.0242•Physical Review B•Oct 9, 2013•45 citations

Spin bath dynamics and dynamical renormalization group

Álvaro Gómez-León

We discuss the quantum dynamics of the central spin model in a regime where the central spin and bath are slaved to each other. The exact solution is found when the bath is static and is compared with the effect of an external field, finding that they are inequivalent due to the quantum nature of the environment. When the bath has dynamics, we analyze the differences between the numerical simulation using time-dependent perturbation theory and the equation of motion technique, which shows better accuracy. We demonstrate that the use of dynamical renormalization group (dRG), simultaneously with the equation of motion technique, provides a suitable analytical tool to understand the physics, to capture the main physical processes, and a powerful method to eliminate secular terms. In addition, this approach allows to separate classical nonlinear behavior from corrections due to quantum correlations.

arXiv:1902.07338•Physical review. B./Physical review. B•Sep 19, 2019•4 citations

Exact dynamics in the central-spin model

Michael Bortz

We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins $N_b$, the oscillation frequency is proportional to $N_b$, whereas the amplitude behaves like $1/N_b$, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.

•Dec 15, 2006•0 citations

Exact dynamics in the inhomogeneous central-spin model

Michael Bortz, Joachim Stolze

We study the dynamics of a single spin $1∕2$ coupled to a bath of spins $1∕2$ by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins ${N}_{b}$, the oscillation frequency is proportional to ${N}_{b}$, whereas the amplitude behaves as $1∕{N}_{b}$, to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.

arXiv:cond-mat/0612382•Physical Review B•Jul 26, 2007•112 citations

Quantum versus classical dynamics in spin models: Chains, ladders, and square lattices

Dennis Schubert, Jonas Richter, Fengping Jin +3 more

We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\"odinger equation in the quantum case or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-$1/2$ systems with up to $N=36$ lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semiclassical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number $S=1/2$ is small and far away from the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.

arXiv:2104.00472•Physical review. B./Physical review. B•Aug 11, 2021•31 citations

Efficient algorithms for the dynamics of large and infinite classical central spin models

Benedikt Fauseweh, Philipp Schering, Jan Hüdepohl +1 more

We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the long-time limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to ${10}^{5}\ensuremath{\hbar}/{J}_{\mathrm{Q}}$ where ${J}_{\mathrm{Q}}$ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.

arXiv:1705.10511•Physical review. B./Physical review. B•Aug 11, 2017•15 citations
We discuss the semiclassical and classical character of the dynamics of a single spin 1/21/21/2 coupled to a bath of noninteracting spins 1/21/21/2. On the semiclassical level, we extend our previous approach presented in D. Stanek, C. Raas, and G. S. Uhrig, Phys. Rev. B 88, 155305 (2013) by the explicit consideration of the conservation of the total spin. On the classical level, we compare the results of the classical equations of motions in absence and presence of an external field to the full quantum result obtained by density-matrix renormalization (DMRG). We show that for large bath sizes and not too low magnetic field the classical dynamics, averaged over Gaussian distributed initial spin vectors, agrees quantitatively with the quantum-mechanical one. This observation paves the way for an efficient approach for certain parameter regimes.
We present a combination of analytic calculations and a powerful numerical method for large spin baths in the low-field limit. The hyperfine interaction between the central spin and the bath is fully captured by the density matrix renormalization group. The adoption of the density matrix renormalization group for the central spin model is presented and a proper method for calculating the real-time evolution at infinite temperature is identified. In addition, we study to which extent a semiclassical model, where a quantum spin-1/2 interacts with a bath of classical Gaussian fluctuations, can capture the physics of the central spin model. The model is treated by average Hamiltonian theory and by numerical simulation.
We discuss the quantum dynamics of the central spin model in a regime where the central spin and bath are slaved to each other. The exact solution is found when the bath is static and is compared with the effect of an external field, finding that they are inequivalent due to the quantum nature of the environment. When the bath has dynamics, we analyze the differences between the numerical simulation using time-dependent perturbation theory and the equation of motion technique, which shows better accuracy. We demonstrate that the use of dynamical renormalization group (dRG), simultaneously with the equation of motion technique, provides a suitable analytical tool to understand the physics, to capture the main physical processes, and a powerful method to eliminate secular terms. In addition, this approach allows to separate classical nonlinear behavior from corrections due to quantum correlations.
We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins NbN_bNb​, the oscillation frequency is proportional to NbN_bNb​, whereas the amplitude behaves like 1/Nb1/N_b1/Nb​, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.
We study the dynamics of a single spin 1∕21∕21∕2 coupled to a bath of spins 1∕21∕21∕2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. For initially completely polarized bath spins and small magnetic field, we find persistent oscillations of the central spin about a nonzero mean value. For a large number of bath spins Nb{N}_{b}Nb​, the oscillation frequency is proportional to Nb{N}_{b}Nb​, whereas the amplitude behaves as 1∕Nb1∕{N}_{b}1∕Nb​, to leading order. No asymptotic decay of the oscillations due to the nonuniform couplings is observed, in contrast to some recent studies.
We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\"odinger equation in the quantum case or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-1/21/21/2 systems with up to N=36N=36N=36 lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semiclassical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number S=1/2S=1/2S=1/2 is small and far away from the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.
We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the long-time limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to ${10}^{5}\ensuremath{\hbar}/{J}_{\mathrm{Q}}$ where JQ{J}_{\mathrm{Q}}JQ​ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.