Spin-to-OAM Converter: 2D Phase-Wheel & Beam View

Input Photon (spin qubit)

Weight toward |L⟩, |α|² 50%

q-Plate Converter

Topological charge q 1

Retardance δ 180°

Live Readouts

|ψ⟩_in = 0.71|L,ℓ=0⟩ + 0.71|R,ℓ=0⟩

Theory conversion sin²(δ/2)

100.0%

Measured (n trials)

— (0)

Output OAM ℓ this trial

—

Spin–orbit coupling S

0.00

How it works

A q-plate is a birefringent liquid-crystal disc whose optical axis winds azimuthally, α(φ) = qφ. It couples a photon's spin angular momentum (circular polarization, |L⟩/|R⟩) to its orbital angular momentum (a helical wavefront e^iℓφ), converting one quantum degree of freedom into another.

|L, ℓ=0⟩ → cos(δ/2)|L,0⟩ + i sin(δ/2)|R,+2q⟩
|R, ℓ=0⟩ → cos(δ/2)|R,0⟩ + i sin(δ/2)|L,-2q⟩
conversion probability = sin²(δ/2)

δ is the plate's retardance. At δ = 180° (a half-wave q-plate) every photon converts: the polarization qubit's amplitudes α, β are transferred deterministically onto the OAM + flipped-handedness degree of freedom. At 0 < δ < 180° each photon is only partly converted, so a superposition input emerges spin–orbit correlated. The "spin–orbit coupling" readout, S = sin(δ)·2√(|α|²|β|²), is a simplified visibility of that correlation — it vanishes for pure |L⟩/|R⟩ input and at δ = 0°/180°, and peaks at δ = 90° with a balanced input (verified numerically: at pL=pR=0.5, δ=90°, S evaluates to exactly 1, the theoretical maximum).

Beam-path panel (top-left) — the photon travels from source to the q-plate, then either straight through (unconverted, ℓ=0) or spirals outward with q full twists (converted, ℓ=±2q). Faint ghost trails from earlier trials persist so the split between the two outcomes accumulates visually.
OAM phase-wheel (top-right) — a genuine rendering of the helical wavefront's transverse phase, e^iℓφ, color-mapped around the ring; |ℓ| bright/dark wedges spin clockwise for +ℓ and counter-clockwise for −ℓ. Drag it to rotate your viewing angle — the physical mode doesn't change, only which side you're looking from.
Polarization bar (bottom-left) — the current |α|², |β|² split as a stacked bar, so you can see the input qubit's mixture at a glance.
Outcome histogram (bottom-right) — running tally of unconverted vs. converted-positive vs. converted-negative trials, compared against the theoretical split.

Real-world relevance: q-plates are the standard tool for generating and manipulating orbital-angular-momentum light in quantum optics labs — used to encode high-dimensional "qudits" for quantum communication and to create hybrid spin-orbit entangled (vector vortex) beams, exactly the partial-conversion regime shown here.

|L⟩ left-circular |R⟩ right-circular q-plate

drag the phase wheel to rotate view