Weakly Interacting Bose Gas

Bogoliubov (1947) showed that weak interactions transform a BEC: quasiparticle spectrum changes from quadratic (free) to linear at low k (phonon) and quadratic at high k. The crossover defines the healing length ξ = ℏ/√(2mgn).

Dispersion

Condensate Fraction

Phonon-Roton

Gas Parameters

Interaction (na³): 0.01 Temperature T/Tc: 0.3

Display

k range: 4 ξ⁻¹ Show free particle

Bogoliubov dispersion: E(k) = √[ (ℏ²k²/2m)² + 2gn·(ℏ²k²/2m) ]
Speed of sound: c = √(gn/m) | Healing length: ξ = ℏ/√(2mgn)
Quantum depletion: N₀/N = 1 − (8/3√π)√(na³)