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A delayed choice quantum eraser, first performed by Yoon-Ho Kim, R. Yu, S.P. Kulik, Y.H. Shih, and Marlan O. Scully, and reported in early 1999, is an elaboration on a quantum eraser experiment involving the concepts considered in Wheeler's delayed choice experiment. It was designed to investigate peculiar consequences of the well-known double slit experiment in quantum mechanics, as well as the consequences of quantum entanglement.

Introduction
In the basic double slit experiment, a very narrow beam of coherent light from a source that is far enough away to have almost perfectly parallel wave fronts is directed perpendicularly towards a wall pierced by two parallel slit apertures. The widths of the slits and their separation are approximately the same size as the wavelength of the incident light.

If a detection screen (anything from a sheet of white paper to a digital camera) is put on the other side of the double slit wall, a pattern of light and dark fringes, called an interference pattern, will be observed.

Early in the history of this experiment, scientists discovered that, by decreasing the brightness of the light source sufficiently, individual particles of light that form the interference pattern are detectable. They next tried to discover by which slit a given unit of light (photon) had traveled.

Unexpectedly, the results discovered were that if anything is done to permit determination of which path the photon takes, the interference pattern disappears: there is no interference pattern. Each photon simply hits the detector by going through one of the two slits. Each slit yields a simple single pile of hits; there is no interference pattern.

It is counterintuitive that a different outcome results based on whether or not the photon is constrained to follow one or another path well after it goes through the slit but before it hits the detector.

Two inconsistent accounts of the nature of light have long contended. The discovery of light's interfering with itself seemed to prove that light could not be a particle. It seemed that it had to be a wave to explain the interference seen in the double-slit experiment (first devised by Thomas Young in his classic interference experiment of the eighteenth century).

In the early twentieth century, experiments with the photoelectric effect (the phenomenon that makes the light meters in cameras possible) gave equally strong evidence to support the idea that light is a particle phenomenon. Nothing is observable regarding it between the time a photon is emitted (which experimenters can at least locate in time by determining the time at which energy was supplied to the electron emitter) and the time it appears as the delivery of energy to some detector screen (such as a CCD or the emulsion of a film camera).

Nevertheless experimenters have tried to gain indirect information about which path a photon "really" takes when passing through the double-slit apparatus.

In the process they learned that constraining the path taken by one of a pair of entangled photons inevitably controls the path taken by the partner photon. Further, if the partner photon is sent through a double-slit device and thus interferes with itself, then very surprisingly the first photon will also behave in a way consistent with its having interfered with itself, even though there is no double-slit device in its way.

In a quantum eraser experiment, one arranges to detect which one of the slits the photon passes through, but also to construct the experiment in such a way that this information can be "erased" after the fact.

In practice, this "erasure" of path information frequently means removing the constraints that kept photons following two different paths separated from each other.

In one experiment, rather than splitting one photon or its probability wave between two slits, the photon is subjected to a beam splitter. If one thinks in terms of a stream of photons being randomly directed by such a beam splitter to go down two paths that are kept from interaction, it is clear that no photon can then interfere with any other or with itself.

If the rate of photon production is reduced so that only one photon is entering the apparatus at any one time, however, it becomes impossible to understand the photon as only moving through one path because when their outputs are redirected so that they coincide on a common detector then interference phenomena appear.

In the two diagrams to the right, photons are emitted one at a time from the yellow star. They each pass through a 50% beam splitter (green block) that reflects 1/2 of the photons, which travel along two possible paths, depicted by the red or blue lines.

In the top diagram, one can see that the trajectories of photons are clearly known — in the sense that if a photon emerges at the top of the apparatus it appears that it had to have come by the path that leads to that point (blue line), and if it emerges at the side of the apparatus it appears that it had to have come by way of the other path (red line).

Next, as shown in the bottom diagram, a second beam splitter is introduced at the top right. It can direct either beam towards either path; thus note that whatever emerges from each exit port may have come by way of either path.

It is in this sense that the path information has been "erased".

Note that total phase differences are introduced along the two paths because of the different effects of passing through a glass plate, being reflected off its first surface, or passing through the back surface of a semi-silvered beam splitter and being reflected by the back (inner side) of the reflective surface.

The result is that waves pass out of both the top upwards exit, and also the top-right exit. Specifically, waves passing out the top exit interfere destructively, whereas waves passing out the upper right side exit interfere constructively.

A more detailed explanation of the phase changes involved here can be found in the Mach-Zehnder interferometer article. Also, the experiment depicted above is reported in full in a reference.

If the second beam splitter in the lower diagram could be inserted or removed one might assert that a photon must have traveled by way of one path or the other if a photon were detected at the end of one path or the other. The appearance would be that the photon "chose" one path or the other at the only (bottom left) beam splitter, and therefore could only arrive at the respective path end.

The subjective assurance that the photon followed a single path is brought into question, however, if (after the photon has presumably "decided" which path to take) a second beam splitter then makes it impossible to say by which path the photon has traveled.

What once appeared to be a "black and white" issue now appears to be a "gray" issue. It is the mixture of two originally separated paths that constitutes what is colloquially referred to as "erasure." It is actually more like "a return to indeterminability."

The experiment


The experimental setup, described in detail in the original paper, is as follows. First, a photon is generated and passes through a double slit apparatus (vertical black line in the upper left hand corner of the diagram).

The photon goes through one (or both) of the two slits, whose paths are shown as red or light blue lines, indicating which slit the photon came through (red indicates slit A, light blue indicates slit B).

So far, the experiment is like a conventional two-slit experiment. However, after the slits a beta barium borate crystal (labeled as BBO) causes spontaneous parametric down conversion (SPDC), converting the photon (from either slit) into two identical entangled photons with 1/2 the frequency of the original photon. These photons are caused to diverge and follow two paths by the Glan-Thompson Prism.

One of these photons, referred to as the "signal" photon (look at the red and light-blue lines going upwards from the Glan-Thompson prism), continues to the target detector called D0. The positions where these "signal" photons detected by D0 occur can later be examined to discover if collectively those positions form an interference pattern.

The other entangled photon, referred to as the "idler" photon (look at the red and light-blue lines going downwards from the Glan-Thompson prism), is deflected by a prism that sends it along divergent paths depending on whether it came from slit A or slit B.

Somewhat beyond the path split, beam splitters (green blocks) are encountered that each have a 50% chance of allowing the idler to pass through and a 50% chance of causing it to be reflected. The gray blocks in the diagram are mirrors.

Because of the way the beam splitters are arranged, the idler can be detected by detectors labeled D1, D2, D3 and D4. Note that:

If it is recorded at detector D3, then it can only have come from slit B.

If it is recorded at detector D4 it can only have come from slit A.

If the idler is detected at detector D1 or D2, it might have come from either slit (A or B).

Thus, which detector receives the idler photon either reveals information, or specifically does not reveal information, about the path of the signal photon with which it is entangled.

If the idler is detected at either D1 or D2, the which-path information has been "erased", so there is no way of knowing whether it (and its entangled signal photon) came from slit A or slit B.

Whereas, if the idler is detected at D3 or D4, it is known that it (and its entangled signal photon) came from slit B or slit A, respectively.

By using a coincidence counter, the experimenters were able to isolate the entangled signal from the overwhelming photo-noise of the laboratory - recording only events where both signal and idler photons were detected.

When the experimenters looked only at the signal photons whose entangled idlers were detected at D1 or D2, they found an interference pattern.

However, when they looked at the signal photons whose entangled idlers were detected at D3 or similarly at D4, they found no interference.

This result is similar to that of the double-slit experiment, since interference is observed when it is not known which slit the photon went through, while no interference is observed when the path is known.

However, what makes this experiment possibly astonishing is that, unlike in the classic double-slit experiment, the choice of whether to preserve or erase the which-path information of the idler need not be made until after the position of the signal photon has already been measured by D0.

There is never any which-path information determined directly for the photons that are detected at D0, yet detection of which-path information by D3 or D4 means that no interference pattern is observed in the corresponding subset of signal photons at D0.

The results from Kim, et al. have shown that whether the idler photon is detected at a detector that preserves its which-path information (D3 or D4) or a detector that erases its which-path information (D1 or D2) determines whether interference is seen at D0, even though the idler photon is not observed until after the signal photon arrives at D0 due to the shorter optical path for the latter.

Some have interpreted this result to mean that the delayed choice to observe or not observe the path of the idler photon will change the outcome of an event in the past. However, an interference pattern may only be observed after the idlers have been detected (i.e., at D1 or D2).

Note that the total pattern of all signal photons at D0, whose entangled idlers went to multiple different detectors, will never show interference regardless of what happens to the idler photons. One can get an idea of how this works by looking carefully at both the graph of the subset of signal photons whose idlers went to detector D1 (fig. 3 in the paper ), and the graph of the subset of signal photons whose idlers went to detector D2 (fig. 4), and observing that the peaks of the first interference pattern line up with the troughs of the second and vice versa (noted in the paper as "a π phase shift between the two interference fringes"), so that the sum of the two will not show interference.

Explanation by physical optics
By examining the positions of signal photons at detector D0, which correspond to idler photons at D1 or D2, an interference pattern is found in the recorded data. There seems to be a misconception, where this pattern originates from. One might think, the signal photons shape the pattern on the way to detector D0 by interfering with each other. But at detector D0 only a blurred image of the double slit is projected by the converging lens, effectively smearing any direct interference patterns form the slits.

To understand the source of the derived interference pattern, one has to focus on the third beam splitter BSC, where the photon paths from both slits merge. At this point a phase difference exists between the merging paths, which is dependent on the different path lengths from slit A or B respectively to the splitter. Furthermore path length and phase difference depend in part on the deviation angle of the idler photon leaving the Glan-Thompson prism. While a fixed position on detector D0 is correlated to events at detector D1 or D2, only events at D1 or D2 are inspected, which share the same phase differences. After leaving the beam splitter BSC, each phase difference may lead to constructive or destructive interference on the paths from the splitter to the detectors D1 and D2, while also allowing any arbitrary intermediate values. But one has to note, that the combined probabilities for both paths behind BSC always add up to 1, as the difference of the phase shifts between that paths amounts to π or 180°.

By changing the detector D0 position, different phase shifts at detector D1 and D2 lead to a different statistic in observed correlated events: A derived interference pattern builds up in the data. Consequently the derived patterns from detectors D1 and D2 add up to the Gaussian distribution, which is observed with the detectors D3 and D4.

Data from events correlated with photons at detectors D3 and D4 show no derived interference patterns, because there is no interference of different photon paths from both slits in the first place.

Time relations among data
By noting which photons reaching Detector 0 correspond with photons reaching Detectors 1, 2, 3, and 4, it is possible to sort photon records collected by Detector 0 into four groups. Only then will it become possible to see interference patterns in two groups and only diffraction patterns in the other two groups. If there were no coincidence counter, then there would be no way to distinguish any photon that arrives at Detector 0 from any other photon that reaches it.

Photons will not reach detectors one through four in regular rotation, so the only way to sort out the photons that are entangled twins with the ones that reached each of those detectors is to group them according to which of those four detectors was activated when a photon reached Detector 0. Note that in the schematic diagrams the fringes or interference patterns imaged by Detector 1 and Detector 2 will add together to form a solid band. The addition of the diffraction patterns paired with the diffraction patterns seen by Detector 3 and Detector 4 will make the center area somewhat brighter than it would otherwise be, but would have no other influence on the confused picture produced by the raw data gathered at Detector 0.

It is impossible to know which group a photon appearing at Detector 0 at time T1 may belong to until after its entangled partner is found at one of the other detectors and their coincidence is measured at some slightly later time T2.

Problems with using retrocausality
This delayed choice quantum eraser experiment raises questions about time, time sequences, and thereby brings our usual ideas of time and causal sequence into question. If a determining factor in the complicated (lower) part of the apparatus determines an outcome in the simple part of the apparatus that consists of only a lens and a detection screen, then effect seems to precede cause. So if the light paths involved in the complicated part of the apparatus were greatly extended in order that, e.g., a year might go by before a photon showed up at D1, D2, D3, or D4, then when a photon showed up in one of these detectors it would cause the photon in the upper, simple part of the apparatus to have shown up in a certain mode a year earlier. Perhaps by re-routing light paths to the four detectors during that one year so that the number of possible outcomes is reduced to two or even perhaps to one, then the experimenter could send a signal back through time. Changing between the first possible arrangement and second possible arrangement of parts in the complicated part of the experiment would then function like the opening and closing of a telegraph key.

An objection that seems fatal is soon raised: The photons that show up in D1 through D4 do not follow some regular rotation. Therefore the photons that show up in D0 pile onto the same detection screen in random order. There is no way to tell, by simply looking at the time and place of each photon detected using D0, which of the other four detectors it corresponds to. So the result will be like trying to watch a motion picture screen on which four projectors are focused. The whole screen will be awash with light. In order to segregate the photons arriving at D0 into the ones that will form one or the other of two overlapping fringe patterns and also the two diffraction patterns, it will be necessary to know how to collect them into four sets. But to do that it is necessary to get messages from the second part of the experiment about which detector was involved with the detection of the entangled partner of each photon received at D0. To oversimplify a bit, the data collected at D0 would be like an encrypted message. However, it could only be decrypted when the key to the code was delivered by a message that could travel at no faster than the speed of light. This daunting obstacle to sending messages back in time has not, however, stopped all researchers from trying to find some way of getting around the stumbling block. The objection raised in regards to determining which photon is which at D0 is a reasonable objection; however this problem is solved by shooting a single photon at a time.

Details pertaining to retrocausality in the Kim experiment
In their paper, Kim, et al. explain that the concept of complementarity is one of the most basic principles of quantum mechanics. According to the Heisenberg Uncertainty Principle, it is not possible to precisely measure both the position and the momentum of a quantum particle at the same time. In other words, position and momentum are complementary. In 1927, Niels Bohr maintained that quantum particles have both "wave-like" behavior and "particle-like" behavior, but can exhibit only one kind of behavior under conditions that prevent exhibiting the complementary characteristics. This complementarity has come to be known as the wave-particle duality of quantum mechanics. Richard Feynman believed that the presence of these two aspects under conditions that prevent their simultaneous manifestation is the basic mystery of quantum mechanics.

According to Kim, et al., "The actual mechanisms that enforce complementarity vary from one experimental situation to another." In the double-slit experiment, the common wisdom is that complementarity makes it seemingly impossible to determine which slit the photon passes through without at the same time disturbing it enough to destroy the interference pattern. A 1982 paper by Scully and Drühl circumvented the issue of disturbance due to direct measurement of the photon, according to Kim, et al. Scully and Drühl "found a way around the position-momentum uncertainty obstacle and proposed a quantum eraser to obtain which-path or particle-like information without introducing large uncontrolled phase factors to disturb the interference."

Scully and Drühl found that there is no interference pattern when which-path information is obtained, even if this information was obtained without directly observing the original photon, but that if you somehow "erase" the which-path information, an interference pattern is again observed.

In the delayed choice quantum eraser discussed here, the pattern exists even if the which-path information is erased shortly later in time than the signal photons hit the primary detector. However, the interference pattern can only be seen retroactively once the idler photons have already been detected and the experimenter has obtained information about them, with the interference pattern being seen when the experimenter looks at particular subsets of signal photons that were matched with idlers that went to particular detectors.

The main stumbling block for using retrocausality to communicate information
The total pattern of signal photons at the primary detector never shows interference, so it is not possible to deduce what will happen to the idler photons by observing the signal photons alone, which would open up the possibility of gaining information faster-than-light (since one might deduce this information before there had been time for a message moving at the speed of light to travel from the idler detector to the signal photon detector) or even gaining information about the future (since as noted above, the signal photons may be detected at an earlier time than the idlers), both of which would qualify as violations of causality in physics. The apparatus under discussion here could not communicate information in a retro-causal manner because it takes another signal, one which must arrive via a process that can go no faster than the speed of light, to sort the superimposed data in the signal photons into four streams that reflect the states of the idler photons at their four distinct detection screens.

In fact, a theorem proved by Phillippe Eberhard shows that if the accepted equations of relativistic quantum field theory are correct, it should never be possible to experimentally violate causality using quantum effects (see reference for a treatment emphasizing the role of conditional probabilities).

Yet there are those who persevere in attempting to communicate retroactively
Some physicists have speculated about the possibility that these experiments might be changed in a way that would be consistent with previous experiments, yet which could allow for experimental causality violations.