Finite-Size Effects in Quantum Metrology at Strong Coupling: Microscopic vs Phenomenological Approaches
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Abstract
We study the ultimate precision limits of a spin chain, strongly coupled to a heat bath, for measuring a general parameter and report the results for specific cases of magnetometry and thermometry. Employing a full polaron transform, we derive the effective Hamiltonian and obtain analytical expressions for the quantum Fisher information (QFI) of equilibrium states in both weak coupling (WC) and strong coupling (SC) regimes for a general parameter, explicitly accounting for finite-size (FS) effects. Furthermore, we utilize Hill's nanothermodynamics to calculate an effective QFI expression at SC. Our results reveal a potential advantage of SC for thermometry at low temperatures and demonstrate enhanced magnetometric precision through control of the anisotropy parameter. Crucially, we show that neglecting FS effects leads to considerable errors in QFI calculations. This work also highlights the inadequacy of phenomenological approaches in describing the metrological capability and thermodynamic behavior of systems at SC.