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In Young's double slit experiment, we usually shine a beam of light over two (or more) slits, as a result of which an interference pattern is observed on the screen. This is because as a beam passes through a slit, it acts as a secondary source of light, thus radiating light in all directions, much like what we do in Huygen's principle. The light from these two slits can interfere either constructively or destructively (or neither), which explains the alternating dark and light fringes.

My doubt is, if we reduce the intensity of the incident beam to that of a single photon, what would happen?

Would we observe just two maxima on the screen, directly in front of the two slits, or would there still be an interference pattern (based on some quantum principles maybe)?

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Have a look at this single photon at a time experiment

snglephot

single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

Yes, one sees the quantum nature of photons, the classical interference pattern emerges from the quantum mechanical wave function probability distribution for the boundary conditions "photon scattering on double slits, given width and distance apart".

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This experiment was carried out in the early 1900's by the famous fluid mechanic Geoffrey Ingram Taylor and reported in his first published paper: G.I. Taylor, Interference fringes with feeble light, Proc. Camb. Phil. Soc. 15, 114-115 (1909).

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My doubt is, if we reduce the intensity of the incident beam to that of a single photon, what would happen?

This means that you want to understand the phenomena in terms of the photon. Photon follows a probability distribution to go to a particular position. Here's the link to a mathematical explanation of how an ensemble of photons end up making a pattern.

You can understand it with the following pictures, How an ensemble of photons makes a pattern? Coming one at a time. It's hard to see the pattern at first but as the ensemble becomes large, the pattern emerges.

enter image description here

Figure 3: Double-slit experiment with single photons

(source)

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