After Moreau et al., Sci. Adv. 2019 · SPDC in BBO · herald with a phase object, Fourier-filter the partner, image in coincidence · paper-faithful and lean builds
In short: A blue laser in a special crystal now and then makes two infrared photons that are entangled: generated together, with linked properties. One photon passes a hidden phase pattern and is caught by a fiber detector. Its partner passes a different filter and lands on a camera that only takes a picture when the first detector fires. The pictures show the hidden pattern, analysed by a filter that photon never met. The strength of that link (the Bell parameter S) beats anything a "pre-set instructions" explanation allows.
Each image counts only the photon-2 detections that arrive with a photon-1 detection. The dashed circles mark the ring read out for the Bell test. The last panel is photon 2 alone, with no trigger: it is the same blob whatever you put in arm 1. That is why nothing can be signalled.
Lobes rotate with the filter angle, following a cos² (Malus-like) law. The dashed lines are the CHSH read-out angles.
S = …
Photon budget
Alignment check
Simulation time —.
Type-II crystals give orthogonally polarized photons, so a PBS separates every pair (a 50:50 BS loses half). PPKTP is far brighter but long (narrow angular spread, longer correlation length).
Try
Lean build: no camera. A 50 µm probe fiber on two motorized stages is parked at 4 points on the ring for each of the 4 filter angles (16 measurements); a time tagger counts coincidences with the herald.
The question. Two photons are measured far apart, each with a setting you choose (here, a filter angle on one side and a position on the ring, which picks an edge angle of the phase object, on the other). Could the results be explained by each photon carrying a hidden "instruction sheet" set when it was generated, with nothing connecting them later? Any such local hidden-variable model obeys the CHSH inequality S ≤ 2. Quantum mechanics predicts up to 2√2 ≈ 2.83 for entangled photons. Measuring S > 2 rules out every local instruction-sheet model.
Why it can't send messages. Each photon on its own is completely random. Photon 2's image without the trigger (the last panel) is the same blob no matter what you put in arm 1. The pattern only appears when you compare the two detectors' records, and that comparison needs a classical signal (here, the trigger cable), which travels no faster than light. This is the no-signalling theorem: entanglement gives correlations, not communication.
Why it helps cryptography rather than blocking it. The same correlations are a resource. In entanglement-based quantum key distribution (Ekert's E91 protocol, 1991), two parties keep the outcomes as a shared secret key and use S to check for eavesdropping: an eavesdropper disturbs the correlations and pushes S toward 2. "Device-independent" QKD goes further and trusts only the measured S. What is limited is signalling, not secrecy.
Loopholes in this experiment. It is a demonstration, not a loophole-free test. Detection: only ≈ 50 % (SPAD) and ≈ 10 % (ICCD) of photons are detected, so a "fair sample" is assumed. Locality: the trigger cable connects the sides, and the settings are not switched fast and randomly. Loophole-free tests (Hensen et al. 2015; Giustina et al. and Shalm et al. 2015) closed these with fast random settings, space-like separation and high-efficiency detectors.
A simulation of the Glasgow "imaging Bell" experiment (Moreau et al., Sci. Adv. 2019), in two buildable versions: the paper-faithful one (two SLMs, gated ICCD, delay line) and a lean one (fixed phase plates, scanning probe fiber, time tagger). It teaches: - what spontaneous parametric down-conversion (SPDC) produces; - how a heralded coincidence image is formed; - what the CHSH Bell parameter S measures; - why the result cannot be used to signal; - which optics, alignment and detector choices make or break the measurement.
The lean build trades the camera for a scanning fiber: much cheaper, still a full Bell test, but no picture.
1. SPDC biphoton (near field). A pump photon splits into two photons at 2λp. In the image plane of a thin crystal both photons are generated at nearly the same point:
ψ(x₁, x₂) ∝ P((x₁ + x₂)/2) · C(x₁ − x₂)
Handled: closed form on an N × N grid (128² or 256²) in crystal-plane coordinates.
2. Heralding (arm 1). The crystal is imaged onto SLM 1 (magnification M₁ = f₁b/f₁a). SLM 1 shows a circular phase object of depth φ₀. The single-mode fiber behind a collimator of focal length f_c projects photon 1 onto a Gaussian mode of radius w = λ f_c / (π w_fiber) at the SLM. Photon 2's conditional field is:
ψ₂(x) = P(x) · [(G* · T₁) ⊛ C](x)
where T₁ = a·e^{iφ(x)} + b, and a, b come from the SLM fill factor, efficiency and polarization.
Handled: the convolution is done with FFTs. A fiber offset adds a tilt (phase ramp). Polarization along the liquid-crystal axis is modulated; the orthogonal part is not, and adds incoherently unless a polarizer recombines it.
3. Fourier filter (arm 2). A lens of focal length f₂ maps the crystal to SLM 2, where position x_F = λ f₂ κ / 2π (κ is the transverse wavevector). The π step at angle θ is drawn with a one-pixel edge blur; its centre can be offset. The SLM's finite active area clips high spatial frequencies. A second lens (f₃) re-images onto the camera.
Handled: FFT → multiply by the filter → inverse FFT. The image intensity is the incoherent sum over the polarization components of both arms.
4. Photon budget and noise. - Pairs per second ≈ 10⁵ × (d_eff/2)² × L(mm) × (1.66/n)³ × P(mW) × sinc²(2.783·θ_tilt/acc), with acc the FWHM angular acceptance (mrad·cm ÷ L) × cos²(pump polarization error) (ASSUMED scale). - Herald probability: the overlap ⟨G·T₁|ψ⟩, normalized. - A 50:50 BS sends both photons to the same arm half the time; a PBS (type-II) does not. - Arm transmissions include SLM efficiency and the delay line (0.99 per mirror, 0.995 per lens surface). - Camera: quantum efficiency (QE), dark counts per pixel per frame, and accidentals (singles × gate). - Lean build: SPAD QE, and the probe fiber's share of the ring.
Handled: Poisson sampling of every image or probe position. The uncertainty is the scatter of 20 sub-datasets, as in the paper.
5. CHSH Bell parameter. - For filter angle a (with a + 90°) and ring angle b (with b + 90°): E(a, b) = (N₊₊ + N₋₋ − N₊₋ − N₋₊) / ΣN. - S = |E(0°, 22.5°) − E(0°, 67.5°) + E(45°, 22.5°) + E(45°, 67.5°)|. Local hidden-variable models obey S ≤ 2; quantum mechanics allows up to 2√2. - Malus fit: each ring profile is fitted with c₀ + c₁ cos 2φ + c₂ sin 2φ, giving V = √(c₁² + c₂²)/c₀ and S = 2√2·V̄. The fit's R² ("Malus purity") checks the two-dimensional (ℓ = ±1) picture the analysis assumes.
Handled: exact formulas. The unknown rotation between the two arms' images is found from the noise-free pattern (in the lab: a calibration run).
6. No-signalling. Photon 2's own image (no trigger) is the pump-wide blob, whatever SLM 1 shows. The structure exists only in coincidences, which need the classical trigger.
Handled: shown as a separate panel.
src/bt_model.js: catalogue (pumps, crystals, SLMs, cameras, SPADs, collimators, fibers, lenses), 2D FFT, biphoton and herald projection, Fourier filtering, noise, ring analysis and CHSH. Pure functions; runs in Node.src/bt_app.js: the 3D layout (rebuilt when parts change), controls, images, ring plot, Bell explainer and parts lists.src/bt_shell.html + src/build.py: page layout; the build inlines everything into imaging_bell_test.html.cd src && python build.py.| Parameter | Value | Source |
|---|---|---|
| Pump (faithful / lean) | 355 nm Cobolt Zouk 20 mW / 405 nm OBIS 50 mW | paper (355 nm); catalogue |
| Pair wavelength | 710 nm / 810 nm (degenerate) | energy conservation |
| Crystal | BBO type-I 1 mm (default) | paper used BBO |
| Pump waist | 0.6 mm | ASSUMED |
| Arm 1 imaging | 4f, f = 100 → 200 mm (M₁ = 2) | ASSUMED |
| Phase object | π circle, radius 0.55 mm on SLM 1 | paper (shape), size ASSUMED |
| Herald collimator / fiber | PAF2S-11B, 780HP (MFD 5 µm) | catalogue |
| Fourier / imaging lenses | f = 250 mm / 250 mm | ASSUMED |
| Filter | π step at 0°, 45°, 90°, 135° | paper |
| ICCD | QE 0.12, 4 ns gate, dark 5×10⁻⁵ px⁻¹ frame⁻¹ | paper (efficiency, gate, dark) |
| Delay line | 20 m, 12 mirrors | paper (≈ 20 m) |
| Exposure | 40,000 × 1 s | paper |
| Lean dwell | 30 min per probe point, 16 points | ASSUMED |
| Pair brightness | 10⁵ s⁻¹ mW⁻¹ (1 mm BBO, 10 nm band) | ASSUMED order of magnitude |
Every mount, post, post holder, pedestal base and clamping fork drawn in the 3D view, counted for the build shown. Thorlabs metric part numbers (Ø1" optics, 12.7 mm posts, M6 table on 25 mm).
Catalogue parts from Thorlabs, HÜBNER (Cobolt), Coherent, HOLOEYE, Meadowlark, Hamamatsu, Andor, Teledyne Princeton Instruments, Amsterdam Scientific Instruments, Pi Imaging, Excelitas, Laser Components and Swabian Instruments; crystals and custom phase plates are quoted items. Confirm part numbers, coatings and current availability before ordering.