Phosphorus-31 Spin Qubits in Silicon: Electrical Detection and Magnetic Control — E8 Intelligence Research

FINDING: Phosphorus-31 nuclear spin (I=1/2) in silicon is a leading qubit platform, with electrical detection and magnetic-field control of spin states demonstrated — the core math is spin-1/2 Hilbert space and Zeeman/electron-nuclear hyperfine coupling. MATH: - Nuclear spin: \\( I = 1/2 \\) → two-level system, Hilbert space \\(\\mathbb{C}^2\\), Pauli matrices \\(\\sigma_x, \\sigma_y, \\sigma_z\\). - Zeeman Hamiltonian: \\( H_Z = -\\gamma_n \\hbar \\mathbf{B} \\cdot \\mathbf{I} \\) → energy splitting \\(\\Delta E = \\gamma_n \\hbar B\\) (linear in field). - Hyperfine coupling (P donor in Si): \\( H_{\\text{hf}} = A \\, \\mathbf{I} \\cdot \\mathbf{S} \\) with \\( A \\approx 117 \\, \\text{MHz} \\) for \\(^{31}\\text{P}\\) in bulk Si (known value). - Electron spin resonance (ESR) / nuclear magnetic resonance (NMR) frequencies: \\(\\omega = \\gamma B\\) — for \\(^{31}\\text{P}\\), \\(\\gamma_n \\approx 17.235 \\, \\text{MHz/T}\\). - Coherence time \\(T_2\\) — for \\(^{31}\\text{P}\\) in isotopically purified \\(^{28}\\text{Si}\\), \\(T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com