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:2609.18641

Resonance-Protected Pointer States: Stationary-Phase Analysis of Entanglement in an Environment-Coupled Two-Spin System

Kentaro Urasaki

quant-phSubmitted Sep 16, 2026• Announced Sep 17, 2026• 0 citations
v1 · Sep 16, 2026v2 · Sep 17, 2026
arXivHTMLPDF

Abstract

In this manuscript, we analyze the dynamics of a model in which two spin-1/2 systems are each coupled noncommutingly to a static environmental field and to a self-Hamiltonian, while also being mutually coupled through a $σ_y^{(1)}σ_y^{(2)}$ interaction. By constructing an exact solution that exploits parity symmetry, and by applying the stationary-phase approximation (the saddle-point method) in the continuum limit of the environmental field, we show that coherence associated with the ordinary isolated stationary point decays as $t^{-1}$, whereas, under the resonance condition at which the local effective fields of the two spins cancel each other, only the coherence between a pair of maximally entangled, Bell-type states- the eigenbasis of the inter-system coupling-survives, decaying via the anomalously slow power law $t^{-1/2}$. Through a comparison with the pointer-basis theory of W. H. Zurek and coworkers (the 1981 and 2005 models), we further reveal a mechanism not reducible to local coupling: in this system, the pointer observable is determined not by the individual system-environment couplings but by a statistical resonance condition between the two environments.

Related papers

Decoherence and pointer states in small antiferromagnets: A benchmark test

Hylke C. Donker, Hans A. De Raedt, Mikhail I. Katsnelson

We study the decoherence process of a four spin-1/2 antiferromagnet that is coupled to an environment of spin-1/2 particles. The preferred basis of the antiferromagnet is discussed in two limiting cases and we identify two exact pointer states. Decoherence near the two limits is examined whereby entropy is used to quantify the robustness of states against environmental coupling. We find that close to the quantum measurement limit, the self-Hamiltonian of the system of interest can become dynamically relevant on macroscopic timescales. We illustrate this point by explicitly constructing a state that is more robust than (generic) states diagonal in the system-environment interaction Hamiltonian.

arXiv:1612.03099•SciPost Physics•Mar 23, 2017•8 citations

Decoherence from spin environments

Fernando Cucchietti, Juan Pablo Paz, Wojciech Hubert Zurek

We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).

arXiv:quant-ph/0312207•Physical Review A•Nov 29, 2005•220 citations

Non-Markovian reduced dynamics and entanglement evolution of two coupled spins in a quantum spin environment

Xiao-Zhong Yuan, Hsi‐Sheng Goan, Ka-Di Zhu

The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg $XY$ spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.

arXiv:cond-mat/0612049•Physical Review B•Jan 19, 2007•100 citations

Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems

Stavros Mouslopoulos

In the ordered phase of a $\mathbb{Z}_2$-symmetric collective spin system, two natural bases -- localised pointer states $\{|P\rangle,|R\rangle\}$ and energy eigenstates $\{|E_0\rangle,|E_1\rangle\}$ -- yield Lindblad dephasing rates that differ by a factor approaching $2$ as $N\to\infty$ and reaching $2.42$ near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces $\langle E_i|\hat{J}_z|E_i\rangle=0$ exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: $η_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42$, where $m_*$ is the order parameter and $G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)$ (advantage over the classical mean-field estimate), and $η_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86$, where $J_{01}=\langle E_0|\hat{J}_z|E_1\rangle$ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the $\mathbb{Z}_2$ parity of the Lindblad jump operator.

quant-pharXiv:2605.00952•May 4, 2026•0 citations

Time-dependent pointer states of the generalized spin-boson model and consequences regarding the decoherence of the central system

Hoofar Daneshvar, G. W. F. Drake

We consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system ($\hat{H}_{\cal S}$) and the interaction with the environment ($\hat{H}_{\rm int}$) do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we obtain the time-evolution operator for our model. Using our time-evolution operator we calculate the time-dependent pointer states of the system and the environment (which are characterized by their ability not to entangle with each other) for the case that the environment initially is prepared in the coherent state. We show that our solution for the pointer states of the system and the environment is valid over a length of time which is proportional to $\bar{n}$, the average number of bosons in the environment. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a \emph{sinusoidal function} with a slow decaying envelope which is characterized by a decay time proportional to $\bar{n}$; while it will experience a much faster decoherence, when the system initially is \emph{not} prepared in one of its initial pointer states.

arXiv:1110.0072•Oct 1, 2011•1 citation

Quantum Decoherence and Pointer Basis: Dynamics in State Vectors

Kentaro Urasaki

It is well-known that the pointer basis of a quantum system satisfies the condition to diagonalize the interaction Hamiltonian between the subsystems. We show that this condition can be translated into the form $δΛ=0,$ where $Λ$, so-called the action, is the time integrated interaction energy: it is found out naturally in the phase of state vectors due to diagonal interaction. The careful treatment of a two states system demonstrates that the states of the total system branch into the states with different values of the action. Mathematically the pointer states are selected out by the saddle point condition on the phase $Λ$. This study helps us to understand the precise mechanism and the general dynamics of decoherence.

arXiv:0907.0150•Jul 1, 2009•0 citations
In this manuscript, we analyze the dynamics of a model in which two spin-1/2 systems are each coupled noncommutingly to a static environmental field and to a self-Hamiltonian, while also being mutually coupled through a σy(1)σy(2)σ_y^{(1)}σ_y^{(2)}σy(1)​σy(2)​ interaction. By constructing an exact solution that exploits parity symmetry, and by applying the stationary-phase approximation (the saddle-point method) in the continuum limit of the environmental field, we show that coherence associated with the ordinary isolated stationary point decays as t−1t^{-1}t−1, whereas, under the resonance condition at which the local effective fields of the two spins cancel each other, only the coherence between a pair of maximally entangled, Bell-type states- the eigenbasis of the inter-system coupling-survives, decaying via the anomalously slow power law t−1/2t^{-1/2}t−1/2. Through a comparison with the pointer-basis theory of W. H. Zurek and coworkers (the 1981 and 2005 models), we further reveal a mechanism not reducible to local coupling: in this system, the pointer observable is determined not by the individual system-environment couplings but by a statistical resonance condition between the two environments.
We study the decoherence process of a four spin-1/2 antiferromagnet that is coupled to an environment of spin-1/2 particles. The preferred basis of the antiferromagnet is discussed in two limiting cases and we identify two exact pointer states. Decoherence near the two limits is examined whereby entropy is used to quantify the robustness of states against environmental coupling. We find that close to the quantum measurement limit, the self-Hamiltonian of the system of interest can become dynamically relevant on macroscopic timescales. We illustrate this point by explicitly constructing a state that is more robust than (generic) states diagonal in the system-environment interaction Hamiltonian.
We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).
The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg XYXYXY spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.
In the ordered phase of a Z2\mathbb{Z}_2Z2​-symmetric collective spin system, two natural bases -- localised pointer states {∣P⟩,∣R⟩}\{|P\rangle,|R\rangle\}{∣P⟩,∣R⟩} and energy eigenstates {∣E0⟩,∣E1⟩}\{|E_0\rangle,|E_1\rangle\}{∣E0​⟩,∣E1​⟩} -- yield Lindblad dephasing rates that differ by a factor approaching 222 as N→∞N\to\inftyN→∞ and reaching 2.422.422.42 near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces ⟨Ei∣J^z∣Ei⟩=0\langle E_i|\hat{J}_z|E_i\rangle=0⟨Ei​∣J^z​∣Ei​⟩=0 exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: ηMF=(Nm∗)2/(2G01)≈2.42η_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42ηMF​=(Nm∗​)2/(2G01​)≈2.42, where m∗m_*m∗​ is the order parameter and G01=12(⟨E0∣J^z2∣E0⟩+⟨E1∣J^z2∣E1⟩)G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)G01​=21​(⟨E0​∣J^z2​∣E0​⟩+⟨E1​∣J^z2​∣E1​⟩) (advantage over the classical mean-field estimate), and ηexact=(G01+J012)/G01≈1.86η_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86ηexact​=(G01​+J012​)/G01​≈1.86, where J01=⟨E0∣J^z∣E1⟩J_{01}=\langle E_0|\hat{J}_z|E_1\rangleJ01​=⟨E0​∣J^z​∣E1​⟩ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the Z2\mathbb{Z}_2Z2​ parity of the Lindblad jump operator.
We consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system (H^S\hat{H}_{\cal S}H^S​) and the interaction with the environment (H^int\hat{H}_{\rm int}H^int​) do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we obtain the time-evolution operator for our model. Using our time-evolution operator we calculate the time-dependent pointer states of the system and the environment (which are characterized by their ability not to entangle with each other) for the case that the environment initially is prepared in the coherent state. We show that our solution for the pointer states of the system and the environment is valid over a length of time which is proportional to nˉ\bar{n}nˉ, the average number of bosons in the environment. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a \emph{sinusoidal function} with a slow decaying envelope which is characterized by a decay time proportional to nˉ\bar{n}nˉ; while it will experience a much faster decoherence, when the system initially is \emph{not} prepared in one of its initial pointer states.
It is well-known that the pointer basis of a quantum system satisfies the condition to diagonalize the interaction Hamiltonian between the subsystems. We show that this condition can be translated into the form δΛ=0,δΛ=0,δΛ=0, where ΛΛΛ, so-called the action, is the time integrated interaction energy: it is found out naturally in the phase of state vectors due to diagonal interaction. The careful treatment of a two states system demonstrates that the states of the total system branch into the states with different values of the action. Mathematically the pointer states are selected out by the saddle point condition on the phase ΛΛΛ. This study helps us to understand the precise mechanism and the general dynamics of decoherence.