Squeezed Vacuum Generator (2D): Degenerate Optical Parametric Oscillator

Below threshold

Pump & Cavity

Pump power P/P_th 70%

Cavity detuning Δ/κ 0.00

Cavity linewidth κ 1.0

Local-oscillator / squeezing angle θ 0°

Live Readouts

Squeezed quadrature

0.0 dB

Anti-squeezed quadrature

0.0 dB

Intracavity mean photon number n̄

0.00

How it works

A degenerate optical parametric oscillator pumps a χ⁽²⁾ nonlinear crystal inside a resonant cavity with light at frequency 2ω. Pump photons split into pairs of identical "signal" photons at ω via spontaneous parametric down-conversion. Below the oscillation threshold the cavity builds up a vacuum field whose two quadratures — amplitude X₁ and phase X₂ — are squeezed asymmetrically instead of amplified together:

x = P_pump / P_threshold      (0 ≤ x < 1)
x_eff = x / √(1 + (Δ/κ)²)     (detuning weakens the parametric gain)

V₋(x_eff) = [(1 − x_eff)/(1 + x_eff)]²   squeezed quadrature variance
V₊(x_eff) = [(1 + x_eff)/(1 − x_eff)]²   anti-squeezed quadrature variance
n̄ = x_eff² / (1 − x_eff²)               intracavity mean photon number

Shot-noise unit: V = 1.  dB = 10·log₁₀(V)

Pump power — sets x. As x → 1 the squeezed quadrature's noise is crushed toward zero while the conjugate quadrature's noise diverges — the Heisenberg product V₋·V₊ = 1 stays fixed even as the ellipse stretches.
Cavity detuning — moving the cavity off two-photon resonance reduces the effective parametric interaction, so both squeezing and photon buildup weaken symmetrically for +Δ and −Δ.
Cavity linewidth κ — sets how fast intracavity fluctuations decorrelate; only changes the animation timescale here, not the steady-state variances.
LO / squeezing angle θ — a homodyne local-oscillator phase; rotates which quadrature you are looking at. The point cloud is the field's noise distribution in phase space — an ellipse, not a circle, is the signature of squeezing.
Drag the phase-space plot to pan, scroll (or pinch) to zoom.

Real-world relevance: sub-threshold degenerate OPOs are the standard laboratory source of continuous-variable squeezed light, used in gravitational-wave detectors (LIGO's frequency-dependent squeezing) and continuous-variable quantum key distribution to beat the standard quantum limit.

Note on this simulation's physics: the output-field squeezing variance used here is squared in x_eff, per the standard Collett–Gardiner input-output derivation for a sub-threshold degenerate OPO (verified below) — the 3D sibling of this simulation omits that square, which understates the squeezing by roughly a factor of two in dB at moderate pump powers.

Pump beam (2ω) χ⁽²⁾ crystal / cavity Signal noise cloud (ω)

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