The Mach-Zehnder Interferometer

A photon meets a half-silvered mirror. It does not choose a side. Both arms of the interferometer end up carrying half the probability — you can watch that on the table: the violet and teal beams stay equally bright whatever you change. Then the two arms meet again at a second splitter, and something strange happens to where the photon can come out.

Start with the table empty and the phase shifter still in the tray. Every photon leaves through D2. Not most of them, not 90% — all of them, and D1 counts zero. Two beams, each carrying half the probability, arriving at a detector that never clicks.

Now drag the phase shifter into an arm and pull φ up to π. The ports swap completely. You have changed nothing about how much light is in each arm; you have changed only the length of one path, by half a wavelength, and the photon has stopped being allowed to arrive where it was arriving before.

At the second splitter each detector can be reached two ways: straight through from one arm, or bounced from the other. Those two routes carry amplitudes, and amplitudes add before they are squared. At the dark port the two routes arrive exactly out of step and cancel; at the bright port they arrive in step and add to a probability of one.

The half-wavelength you added with the phase shifter puts the two routes out of step at the port that used to be bright and in step at the port that used to be dark. Slide φ slowly and the curve below the table is what you are tracing: sin²(φ/2) at D1 and cos²(φ/2) at D2, always summing to one, because the photon does arrive somewhere.

Turn the splitter reflectivity away from 50:50 and the dark port stops being dark. That is not a failure of the physics — cancellation needs two amplitudes of equal size, and a 70:30 splitter never delivers them. The visibility readout falls to 2RT/(R² + T²) and no amount of tuning φ recovers it.

Drag the beam block into either arm. The counts go to roughly 50/50 at both detectors, and they stay there as you sweep φ. The fringes are gone.

It is worth being precise about why, because the obvious explanation is the wrong one. The block did not disturb the photon in the other arm. What it did was make the two routes unequal — one of them now has zero amplitude — and one route cannot interfere with nothing. What is left is an ordinary beam splitter with one input: a coin flip.

Look at the count rate too. Only half the photons reach a detector at all; the rest end in the card. That is the tell that a block is a crude instrument. It destroys the interference by destroying the photon.

Now take the block off and switch on the polarisation marker instead. It is a half-wave plate in arm A that rotates that arm’s polarisation from horizontal to vertical. It absorbs nothing — the detected fraction stays at 100% — and yet the fringes vanish just as completely as they did under the card.

Nothing was blocked and nothing was measured. But the two routes to a detector no longer arrive in the same state: one is horizontal, one is vertical, and orthogonal states do not interfere. In principle, someone could put a polarising splitter in front of D1 and read off which arm every photon came from. The mere fact that the information is there to be read is enough. This is complementarity with the numbers attached: the visibility and the which-path distinguishability in the readout always satisfy V² + D² = 1.

Then switch on the 45° polariser in front of the detectors. Both the horizontal and the vertical route are projected onto the same diagonal state, the record of which arm the photon took is thrown away, and the fringes come back at full visibility — while the detected fraction drops to 50%. You paid for the interference with half your photons. The quantum eraser is what happens when you make that same choice with a second photon, after the first one has already been detected.

An interferometer converts a length into a click rate, and the conversion factor is a wavelength. That is why LIGO is a Michelson interferometer four kilometres on a side: a passing gravitational wave stretches one arm by a fraction of a proton’s width, and the dark port stops being dark.

It is also the shape of most of quantum computing. A Hadamard, some phases, another Hadamard — that is a Mach-Zehnder written in gates, and Grover’s search and the quantum Fourier transform are both large arrangements of exactly this move: put amplitude down every path, arrange the phases so the wrong answers cancel, and read out what is left.