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arXiv:2512.24451

Sub-Ensemble Correlations as a Covariance Geometry

Zuoxian Wang, Yuhao Zhang, Gaopu Hou, Zihua Liang, Gen Hu, Lu Liu, Yuan Sun, Feilong Xu, Mao Ye

physics.atom-phquant-phSubmitted Dec 30, 2025• Announced Jan 1, 2026• 0 citations
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Abstract

Conventional practice of spatially resolved detection in diffusion-coupled thermal atomic vapors implicitly treat localized responses as mutually independent. However, in this study, it is shown that observable correlations are governed by the intrinsic spatiotemporal covariance of a global spin-fluctuation field, such that spatial separation specifies only overlapping statistical projections rather than independent physical components. A unified field-theoretic description is established in which sub-ensembles are defined as measurement-induced statistical projections of a single stochastic field. Within this formulation, sub-ensemble correlations are determined by the covariance operator, inducing a natural geometry in which statistical independence corresponds to orthogonality of the measurement functionals. For collective spin fluctuations described by a diffusion-relaxation Ornstein-Uhlenbeck stochastic field, the covariance spectrum admits only a finite set of fluctuation modes in a bounded domain, imposing an intrinsic, field-level limit on the number of statistically distinguishable sub-ensembles. The loss of sub-ensemble independence is formalized through the notion of spatial sampling overlap, which quantifies the unavoidable statistical coupling arising from shared access to common low-order fluctuation modes. While multi-channel atomic magnetometry provides a concrete physical setting in which these constraints become explicit, the framework applies generically to diffusion-coupled stochastic fields.

Comments
Submitted to Quantum Science and Technology. This version corresponds to the submitted manuscript

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Conventional practice of spatially resolved detection in diffusion-coupled thermal atomic vapors implicitly treat localized responses as mutually independent. However, in this study, it is shown that observable correlations are governed by the intrinsic spatiotemporal covariance of a global spin-fluctuation field, such that spatial separation specifies only overlapping statistical projections rather than independent physical components. A unified field-theoretic description is established in which sub-ensembles are defined as measurement-induced statistical projections of a single stochastic field. Within this formulation, sub-ensemble correlations are determined by the covariance operator, inducing a natural geometry in which statistical independence corresponds to orthogonality of the measurement functionals. For collective spin fluctuations described by a diffusion-relaxation Ornstein-Uhlenbeck stochastic field, the covariance spectrum admits only a finite set of fluctuation modes in a bounded domain, imposing an intrinsic, field-level limit on the number of statistically distinguishable sub-ensembles. The loss of sub-ensemble independence is formalized through the notion of spatial sampling overlap, which quantifies the unavoidable statistical coupling arising from shared access to common low-order fluctuation modes. While multi-channel atomic magnetometry provides a concrete physical setting in which these constraints become explicit, the framework applies generically to diffusion-coupled stochastic fields.
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