We have leaned on the word interference since chapter 02 — it is the engine under every algorithm. Here is the apparatus where it becomes literally visible: a single photon, two paths, and a phase that decides where it lands.
Send a single photon into a Mach–Zehnder interferometer — one particle, indivisible — and it somehow explores both arms at once. At the far end the two possibilities recombine, and the photon reliably lands at one detector, dark at the other. It is not a coin flip: change the length of one arm by a hair and the bright and dark ports swap. Something is travelling both ways and comparing notes — that something is amplitude, and the comparison is interference.
Drop two stones in a pond and watch the ripples cross. Where two crests meet, the water heaves higher (constructive); where a crest meets a trough, the surface goes flat (destructive). Nudge one stone’s timing and the high and flat spots trade places. The photon’s two paths are those two ripples — and the phase φ is the timing nudge that decides which detector heaves and which goes flat.
Add a phase φ to the upper arm. With no phase, every photon exits the bright port — full constructive interference. Crank φ to 180° and they all swing to the dark port instead. The two detector probabilities trade off as cos²(φ/2) and sin²(φ/2).
Call the two arms |0⟩ (lower) and |1⟩ (upper). The whole device is a circuit you already know — H, phase, H — exactly the Deutsch interferometer from chapter 8b:
The phase was invisible in the middle — measure there and you’d get a coin flip with no trace of φ. The second beam splitter is what converts that hidden phase into a visible difference in where the photon lands. The same double slit, the same Born rule, the same interference you met in the Toolkit.
“The photon secretly takes one arm; we just don’t know which.” If that were true, blocking one arm couldn’t change where the other-arm photons land — but it does. Both amplitudes must be present to interfere; the moment you find out which path was taken, the interference vanishes and the ports go 50/50. Interference is the signature of unmeasured superposition.
That “blocking one arm changes everything” fact is not a footnote — it is a tool. In the next chapter we weaponise it to detect an object using photons that never touch it.