Delayed choice: the maths
An ideal Mach–Zehnder interferometer sent one photon at a time; whether the second splitter is in can be chosen after the photon passes the first. Quantum probabilities, sampled photon by photon.
Equations
- P(D1) = cos²(φ/2)
- Second splitter in: the chance a photon reaches detector 1 at phase difference φ between the arms.
- P(D1) = ½
- Second splitter out: each detector sees one arm.
Constants
| φ | 0 to 2π | the phase difference, set by the slider |
Worked examples ✓ checked on every change
| Splitter in, φ = 0: chance of detector 1 | 100% | cos²(0) = 1 |
| Splitter in, φ = π | 0% | cos²(π/2) = 0 |
| Splitter in, φ = π/2 | 50% | cos²(π/4) = ½ |
What's simplified
- Ideal, lossless optics and perfect detectors.
- Photons are sampled one at a time with a random-number generator following the probabilities above.
Where it breaks
It doesn't, for single photons — the experiment has been done (Jacques et al. 2007) and matches.
Sources: J. A. Wheeler (1978); V. Jacques et al., Science 315, 966 (2007).