Elusive Interference

The classic demonstration of the wave-like behavior of light was Young’s two-slit experiment, in which light shining on a single slit produces a simple distribution of intensity on the screen, as shown in the left-hand image below, but when two slits are open the result is a wave pattern as depicted in the right hand image below.

The distribution of intensity at the collecting screen is what one would expect for a wave, because the secondary wavelets emanating from the two slits interfere with each other, sometimes constructively and sometimes destructively, producing the oscillating pattern. However, for a given characteristic frequency of the light source we can lower the intensity until just a single localized packet of electromagnetic energy is received at the collection screen each second, and yet the statistical distribution of the localized detections (with two slits) has the same oscillating pattern as shown. If a photon was a classical particle, each individual photon would have to pass through one slit or the other, and hence there would be no interference, so we conclude that photons are not classical particles.

One might think that we could determine which of the two slits each photon passes through. Indeed if we arrange to make such a determination, we find it passing through one or the other, but in that condition the interference pattern disappears. The interference occurs only when no interactions occur that would establish which of the two alternative slits the photon passed through. This applies even for very subtle interactions.

For example, suppose a photon is detected on the collector screen at a given location and time, and consider the two extremal paths that a classical particle moving at speed c could have taken, as shown below.

Since these paths have different lengths, a photon propagating at speed c must have been emitted at one of two different times from the source atom, depending on which path it follows, as depicted in the figure below.

In this simplified description, the amplitude of the emitting atom is proportional to e^iωt which has the phase angle ϕ = ωt. The photon can be regarded as carrying the phase angles of the source along every null path from the emission to the absorption. The two dominant paths are emitted at two different times (because the path lengths are different), so they carry amplitudes with two different phases, which are added together to give the amplitude of a photon striking that location. The different phases cause the alternating constructive and destructive interference.

Now, one might think we could closely monitor the emitting atom to determine when it recoils from emitting a photon. Depending on whether it recoils at time t₁ or time t₂ we might think that we could infer which path the photon followed to the given reception event. However, if we introduce measurement interactions to establish the time of emission, the interference pattern is destroyed. This should not be too surprising, because the interference requires that the quantum state of the source evolve in isolation to maintain quantum coherence. According to this view, the photon can be localized at the reception event, but the emission event can’t be temporally localized without destroying the interference.

Just as there are piece-wise null paths from the source atom to a given event of the receiving atom, so too are there piece-wise null paths from the receiving atom to/from a given event of the source atom. Conventional electrodynamics focuses only on the retarded intervals, and only on the intervals to a given event of the receiver to determine the probability of an interaction at that event. The phase of the receiving atom is also advancing, and the probability of a photon being emitted from a given event at the source atom to the receiving atom can be expressed in terms of the interference between the phases of the receiving atom (at the relevant frequency) at the two events corresponding to the two null paths. By symmetry, this suggests that a photon can’t be emitted from a given event of the source atom if the amplitudes of the two possible reception events (via the two piece-wise null paths) terminate at events on the receiver for which the phases cancel out (analogous to total destructive interference). If the situation is arranged so that we can determine which of the two events on the receiver is the reception event, then the interference effect disappears.

If we were able to arrange to determine the precise time of emission and the precise time of reception for a photon from a source atom to the receiving atom, we would be able to infer the path it must have taken, based on the time interval. However, with such an arrangement, the probability of happening to capture a transmission is zero.

Since we’ve referred to piece-wise null paths, one might think this applies only to massless energy (e.g., photons), whereas we know that massive particles such as electrons also exhibit quantum interference in two-slit experiments. An electron is a Dirac fermion, whose mass arises from the Higgs mechanism, i.e., by the components of the electron interacting with each other via the scalar Higgs field. The propagation of the components of the electron could be seen as consisting of piece-wise null (i.e., light-like) segments. Similar comments apply to light propagating through a refractive medium. In effect, the screen with two slits is like a primitive refractive medium.

Return to MathPages Main Menu