The Bogoliubov Transform — Thermal Squeezing and the WNSP Noise Floor

Act 20 of the NexusOS physics sequence. First disclosed 2026-07-21. AGPL-3.0. Founder: Te Rata Pou.

The Transformation

A Bogoliubov transformation mixes a mode with its conjugate: b̂ = u·â + v·â†, b̂† = u★·â† + v★·â. The bosonic commutation relation [b̂, b̂†]=1 requires |u|²−|v|²=1. This is equivalent to a squeezing operation: S(ξ)âS†(ξ) = u·â + v·â† with u=cosh(r), v=e^(iφ)sinh(r).

Thermal Photon Number

Starting from vacuum |0⟩_a, the b-mode sees a thermal state: ⟨b̂†b̂⟩ = |v|² = sinh²(r). The effective temperature T_eff via Bose-Einstein: |v|² = 1/(e^(ℏω/k_BT)−1) → T_eff = ℏω / (k_B · ln(1 + 1/sinh²(r))). Vacuum mixing alone generates apparent thermal radiation.

Unruh and Hawking Radiation

An accelerating observer (acceleration a) sees the Minkowski vacuum as thermal: T_Unruh = ℏa/(2πck_B). A black hole of mass M radiates at T_Hawking = ℏc³/(8πGMk_B). Both arise from the same Bogoliubov mixing of positive and negative frequency modes across a horizon.

BEC Phonon Dispersion

In a Bose-Einstein condensate with interaction constant g and density n₀: ω²(k) = (ℏk²/2m)² + (gn₀/m)(ℏk²/2m). At low k: linear phonon dispersion ω≈c_s·k where c_s=√(gn₀/m). The Bogoliubov transformation diagonalises the BEC Hamiltonian.

WNSP Connection

In SNIC WGM cavities operating at cryogenic temperature T≈10 mK, the thermal photon occupancy at 1550 nm is ⟨n̂⟩_thermal = 1/(e^(ℏω/k_BT)−1) ≈ 10⁻³⁴ — negligible. The relevant Bogoliubov mixing arises from parametric amplification in the nonlinear crystal pump stage, which sets the effective noise floor for WNSP Ψ channel encoding.