The double-slit experiment
Quantum mechanics is a fundamental theory of nature that has some rather strange features. We can study these by looking at Young's double slit experiment. When we send a photon to a screen with two slits, quantum mechanics dictates that we see an interference pattern on the screen behind the two slits. At each run the photon is in a superposition of travelling through the right slit and the left slit:
Next, we can place a detector near each slit that tells us when a photon passed through that slit without destroying the photon. When a photon passes through the left slit, the left detector indicates this with a red light. Similarly, when the photon travels through the right slit, the right detector flashes its red light. Remarkably, extracting this information about the path of the photon destroys the interference pattern on the screen behind the slits:
In a real experiment with a bright laser, many photons pass through the slits simultaneously. However, photons do not see each other at all, and each will individually land on the screen according to the interference pattern shown in the first video. Having many photons doing this simultaneously will create the interference pattern much quicker. Here are two photons sent into the slits at the same time, creating an interference pattern twice as fast:
But what if the photons could interact with each other? If we can construct a special kind of two photon state such that the photons always go through the same slit (but we do not detect which slit), then we see something remarkable. The interference pattern becomes twice as narrow:
This is a consequence of entanglement between the paths of the two photons. Alternatively, you can understand this by considering that the two photons act as a single particle with twice the energy (since photons are energy packets of light). That means that the De Broglie wavelength of the bi-photon particle is half that of the original single photon, and consequently the interference pattern becomes twice as narrow (since the length scale of the interference pattern is determined by the wavelength of the particle).
This very experiment was performed in 2001 at the University of Baltimore, Maryland, and the experimenters found exactly this behaviour! In principle, we can repeat this for an arbitrary number of photons and obtain an arbitrary narrow interference pattern. This is the physical principle behind quantum lithography.
The light clock in special relativity
One of the counterintuitive predictions of Einstein's theory of special relativity is that clocks run at different speeds for different observers. One way to make this more intuitive is by using the light clock, in which a photon bouncing back and forth between two mirrors is used to mark the passage of time. Every time the photon hits the mirror we count as a "click" of the clock. We now place the clock in a moving train and look at it from two perspectives, namely the passenger on the train and the passenger on the platform:
The central postulate of special relativity says that light travels at the same speed for all observers. For the passenger on the platform, the photon travels diagonally and needs to cover a longer distance than the straight path seen by the passenger on the train. At a constant speed of light, this diagonal path takes longer than the straight path, and that is why the clock on the train seems to run slow to the passenger on the platform.