QUANTUM CONTRASTS / LIGHT IN SUPERPOSITION

Kerr Cats & Wigner Interference

Twist a coherent light field into two or three coherent components. Explore the signed Wigner landscape, move a phase-space probe and collect even-or-odd parity outcomes. Add phase noise to see interference disappear.

Ideal single-mode Kerr evolution in a rotating frame, with optional Gaussian phase noise applied after preparation. Forty Fock levels; no photon loss, drive imperfections or detector inefficiency. The Wigner surface is a model, not a measured reconstruction. Negative values are not negative probabilities.

POSITIVENEGATIVE

WIGNER

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Phase space · height is five times the signed Wigner value · drag to rotate

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ASK THE STATE. KEEP THE ANSWERS.

Can you record a negative value?

Each parity trial returns even or odd. A negative Wigner estimate means more odd outcomes.

Raw records

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Wigner · at probe

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Wigner · from parity

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Purity · model

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MOVE THE DISPLACEMENT PROBE

$\beta=\operatorname{Re}\beta+i\operatorname{Im}\beta$

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Background: model Wigner values on a fixed color scale. Outlined dots: recorded parity estimates. The gold cross is the probe; this is not a tomographic reconstruction.

EVEN OR ODD / RAW OUTCOMES

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Interval: 95% Wilson interval from the raw parity counts. It describes sampling uncertainty only. Every trial starts from a fresh identical field.

Model, units and what is actually measured

H=\frac{\hbar K}{2}\hat n(\hat n-1),\quad r=\frac{Kt}{2\pi}

Positive Kerr coefficient in a frame rotating with the linear cavity frequency. The initial coherent amplitude has adjustable magnitude and angle. Time is a fraction of the full revival. Gaussian phase noise acts after evolution; it is not a fitted decoherence rate or photon loss.

W(\beta)=\frac2\pi\operatorname{Tr}[D(-\beta)\rho D(\beta)(-1)^{\hat n}]

The plot axes are dimensionless real and imaginary coherent amplitudes. This Wigner convention integrates to one over that plane. The probe means displace by minus its coordinate, then measure parity. The samples are ideal binary readouts; the smooth surface always comes from the density matrix. Forty Fock levels; no drive, energy loss, ancilla pulse or detector model.

Kirchmair et al. · Kerr evolution and fractional revivals · Banaszek et al. · displaced parity

Start with the essentials

Focus question: One wave packet. Several places at once.
One-sentence intuition: Interference can make a Wigner function negative. The underlying parity outcomes remain ordinary positive-probability events.

Core mathematical model

Evolution

c_n(r)=e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}e^{-i\pi r n(n-1)}

Number-dependent phases shear the field and form fractional-revival superpositions. A full period restores every Fock phase.

Remove coherence

\rho_{mn}\longmapsto\rho_{mn}e^{-(m-n)^2\sigma_\phi^2/2}

This declared noise channel preserves the diagonal populations. It is not a model of energy loss.

What is recorded

\widehat W(\beta)=\frac2\pi\frac{N_{\rm even}-N_{\rm odd}}{N_{\rm even}+N_{\rm odd}}

Fresh identical states are displaced by minus the probe coordinate, then read out as even or odd. Wilson intervals describe finite-sample uncertainty.

Common difficulties

A model is not a detector

Typical misconceptionA smooth Wigner surface has been reconstructed from the displayed shots.

Better mental modelIdeal single-mode Kerr evolution in a rotating frame, with optional Gaussian phase noise applied after preparation. Forty Fock levels; no photon loss, drive imperfections or detector inefficiency. The Wigner surface is a model, not a measured reconstruction. Negative values are not negative probabilities.

Run the experiment

01
Find the cat
Start at Prepare, then choose Halfway or One third. Rotate the signed surface.
What to observe: Positive lobes coexist with negative interference valleys. A full revival returns a single coherent packet.
02
Collect evidence
Move the probe into a negative valley and collect 500 parities. Pin it and export the raw CSV.
What to observe: Odd events outnumber even ones; a sufficiently precise interval lies below zero. No individual event has a negative probability.
03
Remove the interference
Increase phase uncertainty, compare the curves and restore the pinned preparation.
What to observe: Noise reduces purity and changes the interference while keeping number populations fixed.