Diffraction in Wave Theory and Repeated Single Photon Events

Like Planck in his response to Einstein, I originally had trouble imagining how the photon nature of light could lead to the formation of a uniform long-range diffraction pattern. In the wave theory, the entire pattern forms at once and becomes uniformly brighter with increasing light intensity. This seemed to be confirmed in early studies when after some time the fully formed patterns were inspected by eye on a screen or on photographic film. Today we know that this is not how the pattern forms!

Modern diffraction experiments with area detectors like CCDs, composed of thousands of tiny light sensitive pixels, reveal that the pattern instead emerges gradually with time through an increasing number of pixels lighting up through hits by single photons. This is beautifully revealed in the experiment shown below. It shows the evolution of the double slit diffraction pattern recorded with an incident green laser beam that was attenuated so that at an instance of time only a single photon was in the vicinity of the slits.

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With time, represented by the addition of frames in the figure, a striped pattern expected from the wave theory begins to form point by point. The detected bright spots clearly originate from hits by single photons. While the question “what is a photon?” is still debated by fundamentalists today, the bright spots may have led Roy Glauber to joke “I don’t know anything about photons, but I know one when I see one”. The bright spots appear at different positions with different probabilities, the hallmark of quantum behavior. The tiny pixels do not get uniformly brighter but the number of bright spots per unit area, the pattern intensity, increases. After an accumulation of about a million photon bright spots, the pixelated pattern resembles a smooth diffracted intensity distribution as predicted by the wave theory.

The above figure supports the simplest quantum formulation of light and is the reason for Dirac’s famous and odd statement that a photon only interferes with itself. Dirac’s statement reflected the fact that at his time, light sources were so weak that they could be assumed to emit photons one-at-a-time, and therefore the interference probability of two or more photons was essentially zero. Correlations between photons are therefore neglected in Dirac’s quantum theory. Each tiny detector pixel remains either dark (destructive self-interference of a photon) or lights up (constructive self-interference).?

When I first set out to formulate diffraction in a photon picture, I only knew that Feynman in his “Lectures on Physics”, had given a quantum explanation of Young’s famous double slit experiment. He discussed the build-up of the diffraction pattern by spraying the slits with single electrons of the same energy, in close analogy to the figure above for photons. Feynman used his own formulation of quantum mechanics, based on all possible paths a particle can take from either slit to a detection point. In essence, he simply replaced the concept of a wave amplitude by a probability amplitude for each possible particle path. The coherent and chaotic cases in the wave theory similarly arose by replacing the birth phases of waves by birth phases in the particle amplitudes. In retrospect, Feynman’s clever quantum formulation directly maps on the wave formalism and it is no surprise that the pattern come out the same. It also turns out that for individual non-interacting electrons and photons (i.e. first order QED) the probability amplitudes are the same and thus the same diffraction patterns appear for non-interacting single electrons or photons.

In Glauber’s quantum optics, the diffraction problem is formulated in first order by considering the birth places of single photons. This is expressed through an operator that creates a single real photon out of the ZP state of darkness. Since we only consider single photons, the photon must be emitted? either from slit A or B. In each case the other slit emits 0 photons and is described by the zero-point state. Thus all allowed 1-photon quantum states emitted by the slits are linear combinations of two possible states, (1 in A and 0 in B) and (0 in A and 1 in B).? The emission of two photons from the same slit is forbidden and becomes allowed only in second order. As in the wave theory, the photon propagation paths from the two slits to a given detection point enter through geometric path lengths alone. In addition, Glauber’s theory specifies that the created (emitted) single photons are destroyed (detected) at the end points of possible geometric photon paths. ?

In Feynman’s probability amplitude formulation, the diffraction pattern depends explicitly on the difference of birth phases at the two slits and the geometric path length difference from the slits to a given detection point – as in the wave theory. In Glauber’s abstract operator formulation one first defines a diffraction probability operator which depends on the geometric paths from single photon creation to single photon destruction points. The creation and destruction operators associated with the start and end points of the paths do not contain any phase information themselves. They just create or destroy possible one-photon quantum states of light. The diffraction pattern for different quantum states of light is then calculated as the expectation value of the probability operator. ??

In my book I show that Feynman’s and Glauber’s formulations of quantum diffraction yield the same results. When a statistical average is formed over a large number of single photon events, the final wave pattern forms photon-by-photon as shown in the above figure. The beauty of Glauber’s quantum optics formulation is that different photon quantum states of light can be directly associated with different characteristic diffraction patterns. This becomes more apparent for the two-photon case which I will discuss in my next blog. Then certain two-photon patterns become allowed that cannot be explained by the wave theory.