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