Phase-space quasi-probability W(x, p) for a single-mode Gaussian quantum state: coherent |α⟩, displaced-squeezed D(α)S(ξ)|0⟩, and thermal. The 1σ ellipse rotates by half the squeeze phase θ/2 and shrinks below the vacuum floor along one quadrature — the basic picture of CV quantum optics.
About this model
This simulator plots the Wigner quasi-probability W(x, p) for a single-mode Gaussian quantum state of a harmonic oscillator. Three families are available: a coherent state |α⟩, a displaced-squeezed vacuum D(α)S(ξ)|0⟩, and a thermal state. For Gaussians, W itself is a Gaussian blob whose 1σ ellipse rotates by half the squeeze phase θ/2 and can shrink below the vacuum floor along one quadrature — the basic continuous-variable (CV) picture of squeezing. The model is strictly single-mode and Gaussian: no photon-number non-Gaussianity, no multimode entanglement, and no loss or detector inefficiency. You vary the coherent amplitude α, the squeeze magnitude and phase in ξ = r e^{iθ}, and the thermal occupation to see displacement, rotation, and elongation of the Wigner ellipse relative to the vacuum circle.
Who it's for: Advanced undergraduate and graduate students in quantum optics, continuous-variable quantum information, and physical chemistry of oscillator modes.
Key terms
Wigner function
Coherent state
Squeezed state
Phase space
Quasi-probability
Continuous-variable optics
How it works
Visualize the Wigner quasi-probability W(x, p) for a single-mode Gaussian state: coherent |α⟩, displaced-squeezed D(α)S(ξ)|0⟩ or thermal. The 1σ ellipse rotates by half the squeeze phase θ/2 and shrinks below the vacuum (red dashed) along one quadrature.
Frequently asked questions
Can the Wigner function be negative here?
Not for the states in this lab. Pure Gaussians (coherent, squeezed vacuum after displacement, and thermal mixtures of Gaussians) have non-negative Wigner functions. Negativity appears for non-Gaussian states such as Fock |n⟩ with n ≥ 1 or Schrödinger-cat superpositions, which are outside this single-mode Gaussian model.
Why does the squeeze ellipse rotate by θ/2 rather than θ?
The squeeze operator is written S(ξ) with ξ = r e^{iθ}. In the (x, p) plane the principal axes of the uncertainty ellipse are oriented at angle θ/2, so rotating the squeeze phase by θ turns the long and short axes by half that amount. A common misconception is to identify θ directly with the plotted ellipse angle.
What does shrinking below the vacuum floor mean?
Vacuum has equal quadrature variances at the Heisenberg floor. Squeezing reduces noise in one quadrature below that vacuum level while the conjugate quadrature is anti-squeezed so that the product still obeys the uncertainty principle. The 1σ contour therefore becomes an elongated ellipse that dips inside the vacuum circle along the squeezed axis.