This is an independent, exact re-derivation of one-axis-twisting squeezing — not a flattened view of the paired 3D simulator's sphere, and not the same math. The 3D version linearizes the transverse spin components (Holstein–Primakoff) into a shear matrix. This 2D version instead time-evolves the real N+1-dimensional quantum state directly in the permutation-symmetric Dicke basis |j,m⟩, j = N/2:
|ψ(0)⟩ = Σ_m √C(N, j+m) · 2^(−N/2) |j,m⟩ (spin-coherent state along x)
|ψ(τ)⟩ = Σ_m c_m(0) · e^(−iτm²) |j,m⟩, τ = s/N, s = χNt
⟨Jz⟩=0 exactly (symmetry); ⟨Jx⟩, ⟨Jy⟩, ⟨Jy²⟩, ⟨{Jy,Jz}⟩ computed from
the exact ladder-operator matrix elements J±|j,m⟩ = √[j(j+1)∓m(m±1)] |j,m±1⟩ —
no small-angle approximation anywhere.
Diagonalizing the resulting exact (Jy, Jz) covariance matrix gives a shape-only squeezing parameter ξ² identical in definition to the 3D readout. But the true metrological figure of merit — the Wineland parameter — must also divide by the actual shrinking mean-spin length ⟨Jx⟩, not the ideal N/2 the linear model assumes:
ξ²_shape = λ_min(Cov) (SQL-normalized)
ξ²_W = ξ²_shape / (⟨Jx⟩/(N/2))² (Wineland 1992 — always ≥ ξ²_shape)
Three genuine differences from the linear model this reveals: (1) spin-length contraction — twisting always shortens ⟨Jx⟩ below N/2, a real decoherence-like effect invisible to a fixed-length shear; (2) the two models converge as N grows at fixed s, since Holstein–Primakoff becomes exact only in that limit — verified numerically below; (3) for small N and large shear, the exact ξ² rises back up (curvature/wrap and revivals) where the linear model keeps predicting ever-deeper squeezing without bound.
- Atom number N — sets the Dicke-basis dimension (N+1 states); the exact/linear gap shrinks as N grows.
- Shear s = χNt — same accumulated-twisting convention as the 3D sim, so the two are directly comparable at the same N and s.
- Readout axis ψ — which quadrature your sensor measures.
- The bottom strip plots ξ²(s) from this exact calculation (solid) against the 3D sim's linear-model prediction (dashed) over the full sweep range, so the breakdown is visible directly rather than asserted.
Real platforms: cold-atom clocks and Rydberg-atom magnetometers realize this exact collective-spin dynamics; the permutation symmetry that makes an exact N-atom simulation tractable in O(N) time (rather than the 2^N cost of a generic N-qubit state) is the same symmetry that makes these systems useful quantum sensors in the first place.