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:2002.04774 · Physical review. B./Physical review. B

Separable and entangled states in the high-spin XX central spin model

Ning Wu, Xi‐Wen Guan, Jon Links

Published Apr 28, 2020• 16 citations
arXivHTMLPDF

Abstract

It is shown in a recent preprint (arXiv:2001.10008) that the central spin model with $XX$-type qubit-bath coupling is integrable for a central spin ${s}_{0}=1/2$. Two types of eigenstates, separable states (dark states) and entangled states (bright states) between the central spin and the bath spins, are manifested. In this work we show by using an operator product state approach that the $XX$ central spin model with central spin ${s}_{0}>1/2$ and inhomogeneous coupling is partially solvable. That is, a subset of the eigenstates are obtained by the operator product state ansatz. These are the separable states and those entangled states in the single-spin-excitation subspace with respect to the fully polarized reference state. Due to the high degeneracy of the separable states, the resulting Bethe ansatz equations are found to be nonunique. In the case of ${s}_{0}=1/2$ we show that all the separable and entangled states can be written in terms of the operator product states, recovering the results in arXiv:2001.10008. Moreover, we also apply our method to the case of homogeneous coupling and derive the corresponding Bethe ansatz equations.

DOI
10.1103/physrevb.101.155145

Related papers

Integrability and dark states in an anisotropic central spin model

Tamiro Villazon, Anushya Chandran, Pieter W. Claeys

Central spin models describe a variety of quantum systems in which a spin-1 2 qubit interacts with a bath of surrounding spins, as realized in quantum dots and defect centers in diamond.We show that the fully anisotropic central spin Hamiltonian with (XX) Heisenberg interactions is integrable.Building on the class of integrable Richardson-Gaudin models, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz.These states divide into two exponentially large classes: bright states, where the qubit is entangled with the bath, and dark states, where it is not.We discuss how dark states limit qubit-assisted spin bath polarization and provide a robust long-lived quantum memory for qubit states.

arXiv:2001.10008•Physical Review Research•Aug 28, 2020•45 citations

Integrability and quench dynamics in the spin-1 central spin XX model

Long Tang, David M. Long, Anatoli Polkovnikov +2 more

Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.

arXiv:2212.04477•SciPost Physics•Jul 26, 2023•8 citations

Dynamics of the homogeneous two-qubit X X Z central spin model with the spin bath prepared in superpositions of symmetric Dicke states

Zejiang Li, Pei Yang, Wen‐Long You +1 more

We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the $XXZ$-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the $W$-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.

arXiv:2006.00550•Physical Review A•Sep 16, 2020•11 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

Non-Markovian master equation for the XX central spin model

Elena Ferraro, Heinz‐Peter Breuer, Anna Napoli +2 more

The non-Markovian correlated projection operator technique is applied to the model of a central spin coupled to a spin bath through non uniform XX Heisenberg coupling. The second order results of the Nakajima-Zwanzig and of the time-convolutionless methods are compared with the exact solution considering a fully polarized initial bath state.

ICTON Mediterranean Winter Conference (ICTON-MW)•Dec 1, 2008•1 citation
It is shown in a recent preprint (arXiv:2001.10008) that the central spin model with XXXXXX-type qubit-bath coupling is integrable for a central spin s0=1/2{s}_{0}=1/2s0​=1/2. Two types of eigenstates, separable states (dark states) and entangled states (bright states) between the central spin and the bath spins, are manifested. In this work we show by using an operator product state approach that the XXXXXX central spin model with central spin s0>1/2{s}_{0}>1/2s0​>1/2 and inhomogeneous coupling is partially solvable. That is, a subset of the eigenstates are obtained by the operator product state ansatz. These are the separable states and those entangled states in the single-spin-excitation subspace with respect to the fully polarized reference state. Due to the high degeneracy of the separable states, the resulting Bethe ansatz equations are found to be nonunique. In the case of s0=1/2{s}_{0}=1/2s0​=1/2 we show that all the separable and entangled states can be written in terms of the operator product states, recovering the results in arXiv:2001.10008. Moreover, we also apply our method to the case of homogeneous coupling and derive the corresponding Bethe ansatz equations.
Central spin models describe a variety of quantum systems in which a spin-1 2 qubit interacts with a bath of surrounding spins, as realized in quantum dots and defect centers in diamond.We show that the fully anisotropic central spin Hamiltonian with (XX) Heisenberg interactions is integrable.Building on the class of integrable Richardson-Gaudin models, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz.These states divide into two exponentially large classes: bright states, where the qubit is entangled with the bath, and dark states, where it is not.We discuss how dark states limit qubit-assisted spin bath polarization and provide a robust long-lived quantum memory for qubit states.
Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. We show that the family of models with a spin-1 at the center and XX interactions of arbitrary strength with surrounding spins is integrable. Specifically, we derive an extensive set of conserved quantities and obtain the exact eigenstates using the Bethe ansatz. As in the homogenous limit, the states divide into two exponentially large classes: bright states, in which the spin-1 is entangled with its surroundings, and dark states, in which it is not. On resonance, the bright states further break up into two classes depending on their weight on states with central spin polarization zero. These classes are probed in quench dynamics wherein they prevent the central spin from reaching thermal equilibrium. In the single spin-flip sector we explicitly construct the bright states and show that the central spin exhibits oscillatory dynamics as a consequence of the semilocalization of these eigenstates. We relate the integrability to the closely related class of integrable Richardson-Gaudin models, and conjecture that the spin-s central spin XX model is integrable for any s.
We obtain exact dynamics of a two-qubit central spin model (CSM) consisting of two interacting qubits homogeneously coupled to a spin bath via the XXZXXZXXZ-type coupling, with the bath initially prepared in linear superpositions of the symmetric Dicke states. Using the interaction picture Hamiltonian with respect to the non-spin-flipping part of the model, we derive a sequence of equations of motion within each magnetization sector satisfied by the probability amplitudes of the time-evolved state. These equations of motion admit analytical solutions for the single-qubit CSM in which one of the two central qubits decouples from the rest of the system. Based on this, we provide a quantitative interpretation to the observed collapse-revival phenomena in the single-qubit Rabi oscillations when the bath is prepared in the spin coherent state. We then study the disentanglement and coherence dynamics of two initially entangled noninteracting qubits when the two qubits interact with individual baths or with a common bath. For individual baths, the coherent dynamics is found to be positively correlated to the single-qubit purity dynamics, and entanglement sudden disappearance and revivals are observed in both cases. The entanglement creation of two initially separable qubits coupled to a common bath is also studied and collapse and revival behaviors in the entanglement dynamics are observed. Choosing the equally weighted state and the WWW-class states as the bath initial states, we finally study the dynamics of entanglement between two individual bath spins and demonstrate the entanglement sharing mechanism in such a system.
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.