The Englert–Greenberger–Yasin duality relation (often called the Englert–Greenberger relation) relates the visibility,
Contents
Although it is treated as a single relation, it actually involves two separate relations, which mathematically look very similar. The first relationship was first experimentally shown by Greenberger and Yasin in 1988. It was later theoretically derived by Jaeger, Shimony, and Vaidman in 1995. This relation involves correctly guessing which of the two paths the particle would have taken, based on the initial preparation. Here
The significance of the relation is that it expresses quantitatively the complementarity of wave and particle viewpoints in double slit experiments. The complementarity principle in quantum mechanics, formulated by Nils Bohr, says that the wave and particle aspects of quantum objects cannot be observed at the same time. The Englert-Greenberger relation makes this more precise; an experiment can yield partial information about the wave and particle aspects of a photon simultaneously, but the more information a particular experiment gives about one, the less it will give about the other. The distinguishability
The mathematics of two-slit diffraction
This section reviews the mathematical formulation of the double-slit experiment. The formulation is in terms of the diffraction and interference of waves. The culmination of the development is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the Englert–Greenberger duality relation. The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave–particle duality. Of this experiment, Richard Feynman once said that it "has in it the heart of quantum mechanics. In reality it contains the only mystery."
The wave function in the Young double-aperture experiment can be written as
The function
is the wave function associated with the pinhole at A centered on
where
To distinguish which pinhole a photon passed through, one needs some measure of the distinguishability between pinholes. Such a measure is given by
where
Since the Born probability measure is given by
and
then we get:
We have in particular
where
The visibility of the fringes is defined by
where
Equivalently, this can be written as
And hence we get, for a single photon in a pure quantum state, the duality relation
There are two extremal cases with a straightforward intuitive interpretation: In a single hole experiment, the fringe visibility is zero (as there are no fringes). That is,
The above presentation was limited to a pure quantum state. More generally, for a mixture of quantum states, one will have
For the remainder of the development, we assume the light source is a laser, so that we can assume
Complementarity
The mathematical discussion presented above does not require quantum mechanics at its heart. In particular, the derivation is essentially valid for waves of any sort. With slight modifications to account for the squaring of amplitudes, the derivation could be applied to, for example, sound waves or water waves in a ripple tank.
For the relation to be a precise formulation of Bohr complementarity, one must introduce wave–particle duality in the discussion. This means one must consider both wave and particle behavior of light on an equal footing. Wave–particle duality implies that one must A) use the unitary evolution of the wave before the observation and B) consider the particle aspect after the detection (this is called the Heisenberg–von Neumann collapse postulate). Indeed since one could only observe the photon in one point of space (a photon can not be absorbed twice) this implies that the meaning of the wave function is essentially statistical and cannot be confused with a classical wave (such as those that occur in air or water).
In this context the direct observation of a photon in the aperture plane precludes the following recording of the same photon in the focal plane (F). Reciprocally the observation in (F) means that we did not absorb the photon before. If both holes are open this implies that we don't know where we would have detected the photon in the aperture plane.
A maximal value of distinguishability
Similarly, if
The above treatment formalizes wave particle duality for the double-slit experiment.