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

a collective spin system in a symmetry-broken phase, there exist two natural bases: the localised states {|P⟩,|R⟩} that maximise the order parameter within the low-energy doublet, and the energy eigenstates {|E0⟩,|E1⟩} that diagonalise the Hamiltonian. Both bases yield well-controlled Lindblad dephasing rates for their respective coherences, and in the mesoscopic quantum regime, those rates differ. The exact di
agonal Redfield coefficient for ρPR is γϕ(G01 + J2 01); the mean-field (spin-coherent) approximation gives the larger rate γϕGloc where Gloc = (Nm∗)2/2, which governs the pure exponential decay of Re(ρPR). The energy-eigenstate coherence ρ01 decays at
rate γϕG01 where G01 = 1 2(⟨E0| ˆJ2 z|E0⟩ + ⟨E1| ˆ
J2 z|E1⟩). These two questions yield different answers in two distinct senses: the geometric ratio ηMF = Gloc/G01 approaches exactly 2 as N → ∞ (a model-independent statement about matrix elements, valid
even where the secular approximation fails); and at finite N in the LMG model, ηMF rises to a peak ηMF ≈ 2.42 within the mesoscopic secular window (2∆E · T2 ≫ 1, setting ℏ = 1), where both rates are simultaneously well-defined as exponential decay constants. The discrepancy in physical decay rates is therefore strictly a mesoscopic
inite-N effect; the universal ηMF = 2 is a geometric statement that survives the breakdown of the secular approximation in the thermodynamic limit. In the thermodynamic limit (N → ∞), the secular approximation fails, the doublet becomes degenerate, and the decaying components of both coherences converge exactly to the classical
macroscopic (pointer-basis) rate γϕGloc, while Re(ρ01) approaches a quasi-steady state (metastable plateau) within the doublet. At finite N, however, the secular approximation remains valid, and the two coherences decay at genuinely different rates. For quantum technologies — spin squeezing, quantum Fisher information, Leggett-Garg tests — this provides a quantitative quantum advantage conditional on the protocol
being sensitive to the eigenstate coherence ρ01 and remaining within the ground-state doublet. Two distinct protection factors are relevant: ηMF = Gloc/G01 ≈ 2.42 (peak), which quantifies the advantage over the classical mean-field pointer-state estimate
γϕGloc; and ηexact = (G01 +J2 01)/G01 ≈ 1.86, which is the true basis-dependent physical protection factor — the exact ratio of the pointer-state decay rate γϕ(G01 + J2 01) to the eigenstate rate γϕG01. We demonstrate this protected three-regime structure in the Lipkin-Meshkov-Glick model via exact diagonalisation and provide the precise
algebraic origin of the discrepancy via the Z2 parity of the Lindblad operator.