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

• URL: http://arxiv.org/abs/2605.00952v1
• Date: Fri, 01 May 2026 11:39:51 GMT
• Title: Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems
• Abstract summary: Lindblad dephasing rates differ by a factor approaching $2$ as $Ntoinfty$ and reaching $2.42$ near the quantum-critical crossover.<n>The discrepancy has a single algebraic origin: parity forces $langle E_i|hatJ_z|E_irangle=0$ exactly, eliminating the cross-term that doubles the localised-state rate.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: 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.

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