The wave of a photon/Marking the wave, Quantum erasers

Mach-Zehnder interferometer
In a Mach-Zehnder interferometer a light beam is split by a 50% mirror in two beams (red and blue). Both beams are reflected by a mirror and cross each other. Finally each beam ends in a separate detector. In (1) each detector can detect photon from the red or blue path and no interference is measured. In (2) a second beam splitter mixes the red and blue beam. Now both detectors show interference, but cannot detect if the photon followed the red or blue path.

In the classical wave description the wave of the photon is split in two by the 50% mirror. In (1) each detector sees only one wave, so there is no interference with the other wave. In (2) both waves are mixed, so each detector sees both waves, causing interfere.

Particle-path detectors and entangled photons (Kim et al)
A high intensity laser radiates though the "red" and "blue" part of a beta barium borate crystal (BBO), which generates now and then two entangled photons with half the energy of the laser photons. These photons follow two paths by a Glan-Thompson Prism. One of the photons, the "signal" photon, goes upwards, through a lens, to the target detector D0. The other photon, the "idler" photon, goes downwards and is deflected by a prism that sends it along divergent paths, depending on whether it came from the red or blue BBO area. Beyond each path a 50% mirror acts as a beam splitter (green blocks), resulting in a 50% chance to pass through and a 50% chance to reflect to detectors D3 or D4. The photons which pass through are reflected by 100% mirrors (gray-green blocks) to the detectors D1 or D2. Where both beams cross, a 50% mirror is placed, which mixed both beams.Because of this arrangement, if the photon is recorded at detector D3 (resp. D4), it can only be a blue (resp. red) photon. If the photon is recorded at detector D1 or D2, it can be both a red or blue photon. A coincidence counter selects from D0 only those events which coincide with a chosen other detectors. This includes a delay of 8 ns to compensate for the 2.5 meter longer path to the other detectors. the result was:


 * When events were counted which coincided with D3 or D4, there was no interference.
 * When events were counted which coincided with D1 or D2, there was an interference pattern.

With classical wave physics a set of photons is emitted in the red or blue area of the BBO, no in both at the same time. There with ill be only one wave, to interfere with D3 or D4. But the interference with D1 or D2 cannot be explained classical, because of the absence of two waves.

Linear polarizers
In this experiment both waves are differently (perpendicular) polarized. Then no interference pattern is visible. A polariser in front of the detector, with its axis 45° to the other polarisers, erases this information, causing the interference pattern to reappear.

Classical this effect is know as the Fresnel and Araglaws laws, which state that perpendicular polarization does not interfere. The 45° polarisers force both perpendicular polarizations to become parallel polarizations, which can interfere. Also classical calculation gives the same result.

===Circular polarizers and entangled photons (Walborn et al) === A beta barium borate (BBO) crystal, radiated by a strong laser, will generating now and then two entangled photons. One photon (yellow) goes through a double slit to signal detector Ds, with rotating polarizer Q1 or Q2 in each path. Q1 and Q2 axis are pendicular, producing a opposite rotating polarization. The other photon (green) goes to detector Dp, with a linear polarizer cube POL in the path. Only photons in Ds are registered which coincidence with photons in Dp. Dp is situated closer to the BBO then Ds, so photons are first detected by Dp. The results were:


 * 1) Without Q1/Q2 and POL there was an interference pattern in Ds
 * 2) With Q1/Q2 there was no interference.
 * 3) With Q1/Q2 and POL, adjusted on the Q1 fast axis, there was interference.
 * 4) With Q1/Q2 and POL, adjusted on the Q2 fast axis, there was interference, but 180º shifted with 3.
 * 5) 1-4 with Dp on a larger distance then Ds gives the same result. This is called a delayed eraser, because the photon in Dp is detected later then in Ds

In the classical wave description Q1 and Q2 are quarter-wave plates which have perpendicular a "fast" and "slow" axis. The slow axis has a π/2 phase delay compared to the fast axis. Q1 and Q2 are mounted mutually perpendicular. In setup 2 the incoming photon has a random polarization (shown as red) which can be resolved into two waves, parallel (green) and perpendicular (blue) to the optical axis of the waveplate (see figure). Depending on the axis the wave on x or y are delayed π/2. At the detector the x and y vectors interfere (add up) and can be composed to one vector. Calculation shows that the probability is:

P = 0.5 - 0.5cos2αsinφ


 * A:  If α = 0 (incoming polarisation parallel to the fast axis of Q1) then P = 0.5 - 0.5sinφ
 * B:  If α = π/2 (incoming polarisation parallel to the fast axis of Q2) then P = 0.5 + 0.5sinφ
 * C:  If α = random then integrating α from 0 to 2π and normalised gives P = 1. This of course the same as A + B: P = 0.5 - 0.5sinφ + 0.5 + 0.5sinφ = 1

If the incoming polarization is parallel to a fast axis of a wave plate a φ interference pattern is visible on the detector. Between A and B there is a phase difference of π. If α is random then there is no φ pattern. C can be explained as having no interference pattern or being the sum of two interference pattern A and B which have a phase difference of π, which when added does not show an pattern. According classical rules both explanations are right. Also the Walborn article mentions that the result of setup 2 is the sum of setup 3 and 4.

In the experiment the incoming photon is the signal photon of a BBO. In setup 2 the polarization is random, so the measured result is as C above. The BBO emits an entanglement photon pair with mutually perpendicular polarization. According the article in set-up 3 POL was set to the angle of the fast axis of Q1. So the idler photons which passed through must have a signal photon with perpendicular polarisation, so parallel to the fast axis of Q2. These will show an interface pattern according B, which confirms with the measured result. In set-up 3 POL was rotated π/2, which gives the result of A. In other words, Dp detects only idler photons with a signal photon which polarization parallel to Q1 or Q2, which show an interference pattern according A or B. Dp selects those photons by coincidence from the output of Ds.

In above formulation the time the idler is detected in Dp is not important (as long as delays are used to compensate for path length difference). The "delayed" effect is in the classical wave explanation the effect that POL determines the polarisation of the signal wave before Q1/Q2, although POL is position on a much larger distance of the BBO then Q1/Q2.

Wheeler's astronomical experiment
Wheeler's delayed choice experiment is a thought experiment in which the type of detector (particle or wave) is changed after a photon passes the double slit. In a second version the scale is magnified to astronomical dimensions: a photon has originated from a star and its path is bent by an intervening galaxy, so that it could arrive at a detector on earth by two different paths images. The detector could be a screen (as wave detector) or two telescopes, each focused to either side of the black hole. In both versions Wheeler expected that the screen will measure an interference pattern, while two telescopes will observe the photon only in one of them, without interference. If the outputs of both telescopes are optically combined, the interference pattern would be back, but losing the information of the path of the photon.

This result follows the classical waves: interference is caused by adding two waves during detection. If a telescope is directed at one wave, so no interference. If both telescopes are optically combined there are two waves again, which will interfere. The experiment shows that a photon always keeps full wave- and particle properties, until they combine during absorption. . The first version has been confirmed in practice by Jacques e.a. with a Mach-Zender interferometer. The astronomical version is not yet done and probably too difficult in practice, because for an interference pattern the photons must be coherent when they arrive on earth, after such a long time.