β = u·a + v·a† mixes creation and annihilation operators — Wigner function visualization
Wigner Function W(x,p)
Quadrature Distribution
Squeezing r0.70
Squeezing angle φ0.00
Displacement α0.0
Thermal n̄0.0
The Bogoliubov transformation β = u·a + v·a† (|u|²−|v|²=1) mixes positive and negative frequency modes.
The corresponding unitary is the squeezing operator S(ξ) = exp(ξ*a²/2 − ξa†²/2).
For a coherent state |α⟩, applying S(r,φ) gives a squeezed state with reduced noise in one quadrature:
ΔX₁ = e^{−r}/2, ΔX₂ = e^{r}/2 (uncertainty still saturated: ΔX₁ΔX₂ = 1/4).
The Wigner function W(x,p) shows the phase-space quasi-probability — squeezed states are ellipses.
Bogoliubov transformations appear in BCS superconductivity, Hawking radiation, and BEC physics.