Squeezed vacuum quadrature variance: (ΔX₁)² = e^(−2r)/4, (ΔX₂)² = e^(2r)/4; polar/Bloch decomp of S(r)

Layer 1 — Physicsin the quantum-sensing subtree

Squeezed-vacuum state construction and variance scaling. Setup: the squeeze operator S(r) = exp[(r/2)(a² − a†²)] acts on the vacuum to produce a squeezed state |r⟩ = S(r)|0⟩ with variance in one quadrature reduced by e^(2r) and the…

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