*This is part 5 in a lengthy series of posts attempting to explain the idea of quantum entanglement to a non-physics audience. Part 1 can be read here, Part 2 can be read here, Part 3 here,* and *Part 4 here*.

So at this point we have an idea of what entanglement is, and some reassurance that it doesn’t ruin all of physics with its existence! Now we turn to a very important question: how, in practice, do we produce entangled quantum particles?

In our discussions so far, we have imagined entanglement arising through the hypothetical decay of a neutral pion into an electron/positron pair, as illustrated below.

This is a fine idea in principle, but it is an utterly impractical method to reliably create entangled particles in a laboratory setting. First, let’s talk about the pion: a pion is a high-energy particle that we typically¹ only see as a product in high-energy particle physics experiments, such as those done at places like CERN. When produced, they are usually moving at relativistic speeds, and in a direction that is more or less random. They decay very quickly, on the order of 10 billionths of a billionth of a second when at rest, which means you can’t store them for future use. Even if you could store them, the direction of their decay products is random as well, which means we should revise our image above to appear as shown below.

In short: even if we managed to get a pion to sit still in one place, we wouldn’t know where to put our pair of detectors to spot the particles. We would miss the vast majority of entangled pairs.

So using pions as a source of entangled particles to do experiments is not practical. Fortunately, it turns out that we have a great source of particles available to us that are relatively easy to produce in an entangled state: photons, i.e. particles of light! By the use of a process known as spontaneous parameteric down conversion, we can reliably and easily produce photons with entangled polarization that appear in very predictable locations. Let’s see how!

We begin by noting that photons are fundamentally different than the particles we’ve been entangling so far, and the nature of their entanglement is somewhat different. Electrons and positrons are spin-1/2 particles, which we have seen can exist individually in a quantum superposition state of spin-up and spin-down.

Photons, however, are spin-1 particles, which can still be described by a superposition of two states, but the two states in question are horizontal and vertical polarization!

Here, I’ve used “V” and “H” for “vertical” and “horizontal.” In the classical wave description of light, the polarization of a light wave represents the way the electric field “waves”, or oscillates. It can oscillate along a line, vertically, horizontally, or along any angle, or it can oscillate in a circular path. The former case is known as linear polarization, the latter as circular polarization.

Just for some perspective, below is an image of what a vertically polarized electromagnetic wave would “look like” from the side: the electric field **E** wiggles up and down, the magnetic field **H** wiggles left to right, and the wave itself propagates in a direction perpendicular to both wiggling directions.

What we will soon is that we can entangle a pair of photons in such a way that if one is found to be horizontal, then the other is definitely vertical, with a quantum state that appears as follows.

To understand how one produces this state, however, we have to talk about *nonlinear optics*, so let us spend a little time explaining what this means.

One thing you may have noticed about beams of light under ordinary circumstances: they don’t interact with one another. If we send two beams of light on a collision course, they pass right through each other without effect. On a quantum level, we can say that the photons don’t interact with each other.²

This is only true, however, in vacuum. When light propagates in matter, the matter can “mediate” an interaction between two or more photons, causing photons to combine, split apart, or do even more unusual things. In order for these interactions to happen, however, photons have to be relatively densely packed or, equivalently, the beam of light itself has to be very intense. Such high-intensity beams only became possible with the invention of the laser in 1960, and the field of *nonlinear optics³*, describing the interactions of photons, was founded not long afterwards.

Two typical nonlinear processes are illustrated below. The first, called second-harmonic generation, involves the combination of two photons of frequency ω into a single photon of frequency 2ω. Because the energy of the photon is proportional to the frequency, this means that two photons combine into a single photon of twice the energy. If you’ve ever used a green laser pointer, you have probably seen second-harmonic generation in action: they typically achieve a green color by sending an infrared laser beam with wavelength 1064 nm through a nonlinear potassium titanyl phosphate (KTP) crystal , resulting in green output photons of wavelength 532 nm.

The second process pictured above is known as difference-frequency generation, and it involves a photon of frequency ω_{2} interacting with a photon of frequency ω_{1}. This causes the higher-frequency photon to split into a photon of frequency ω_{2}-ω_{1} and one of frequency ω_{1}, with the original photon of frequency ω_{1} remaining.

Both of these processes are known as *parametric* processes, which means that the net energy and momentum of the input and output photons are the same; no momentum or energy gets transferred to the nonlinear medium. This ends up being a bit of a problem, as in most materials the momentum of a photon of frequency 2ω is *larger* than the total momentum of two photons of frequency ω. This is solved by using crystal structures for which the *speed* of light depends on the *polarization* of light. Because there are two perpendicular polarization possible (think “horizontal” and “vertical” again), there are two speeds in such materials. This difference in speeds results in the phenomenon of double refraction, as seen in the photograph below. The two images result because the two different polarizations of light take slightly different paths through the crystal.

We don’t want to dive into all the complicated optics of crystals right now, but suffice to say that there are crystals for which the net momentum of two horizontal photons of frequency ω is equal to the momentum of one vertical photon of frequency 2ω, allowing second-harmonic generation to happen. Such a process is known as Type-I second-harmonic generation (there is also a Type-II).

But let’s think a bit more about the process of difference-frequency generation. It is what we refer to as a *stimulated* process, in that the additional photon of frequency ω_{1} is needed to “stimulate” the photon of frequency ω_{2} to split into two. A different type of stimulated process is in fact what makes a laser — “Light Amplification by *Stimulated* Emission of Radiation” — operate. In a laser, photons interact with excited atoms and cause those atoms to emit another photon, and this stimulated process results in the bright and directional beam of the laser.

However, we know that atoms can and will emit radiation without the need for being stimulated. Every ordinary light source — lightbulbs, lightning bugs, the sun — consists of a collection of atoms that *spontaneously* emit radiation. It is then natural to wonder: is it possible to have a photon spontaneously split into two in a nonlinear medium, as happens in difference-frequency generation but without the extra photon making it happen?

It can, in fact, in a process that is known as *spontaneous parametric down conversion (SPDC)*. “Spontaneous” refers to the fact that it can happen without a “helper” photon providing stimulation, while “parametric,” as mentioned earlier, indicates that it is a process that conserves energy and momentum among the photons. “Down conversion” is another way to refer to the specific case of difference-frequency generation in which one photon of frequency 2ω breaks into two photons of frequency ω: the frequency of the output photons goes down.

In Type-II SPDC, the two output photons always have perpendicular polarizations. The conservation of momentum results (in a way too difficult to explain here) in the output photons propagating along two intersecting conical surfaces. If one projects this output onto a screen, one will see two intersecting circles, as pictured below.^{4}

But let’s consider the positions of individual pairs of photons that are created. In the picture above, the original “pump” beam would be coming directly out of the page; because of momentum conservation, the pairs of photons must be on opposite sides of this central axis. This is illustrated below.

Part (a) of the figure gives a 3-dimensional perspective on the overlapping cones. In part (b), we visualize the positions of photon pairs as a series of discrete points: if one photon appears at position 5, then the other photon appears at position 5′. We imagine, for simplicity, that the photons on the upper circle are always vertically polarized, and the photons on the lower circle are always horizontally polarized.^{5 }It is impossible to predict where any individual pair will appear, thanks to the inherent randomness of quantum physics, but if one appears in an unprimed position, the other will appear in the corresponding primed position.

But now imagine what happens when the pair of photons appear in positions 1, 9′ and 1′, 9. It is possible for either a horizontal or vertical photon to appear in either place! We again have a situation where the law of physics tell us that the two photons must be perpendicular, but do not tell us whether the photon in 1, 9′, for instance, is horizontal or vertical. The combined quantum state of any pair of photons appearing in this pair of locations must therefore be the entangled state we wrote earlier!

There is one curious complication to this strategy that is illuminating and, when you ponder it for a while, quite astonishing. We have mentioned that photons propagating in an anisotropic crystal — like the one used to create SPDC — travel at different speeds depending on their polarization. This means that the pair of photons created in SPDC end up traveling at slightly different speeds through the crystal, which means that they end up arriving at the detector at slightly different times.

This means, however, that the photons are again *distinguishable* — we could tell which photon came from which location simply by timing when they arrive. This distinguishability effectively kills the appearance^{6} of entanglement between the photons, which means we won’t be able to do interesting experiments with them, like study wave interference.

This problem can be fixed the same way it was created: through the use of more anisotropic crystals! Suppose that the horizontal photon would arrive before the vertical one, because horizontally-polarized photons move faster than vertical ones in the SPDC crystal. We can correct this by putting anisotropic crystals in from of the two positions 1, 9′ and 1′, 9 in which horizontal photons move slower and vertical photons move faster! With the right choice of crystal, we can completely eliminate the time delay between the photons and make them indistinguishable again. This is illustrated below.

This is a simple example of a device referred to as a “quantum eraser,” which removes, or “erases,” information about a photon or photons that gives explicit information about their behavior.

I will hopefully have more to say about quantum erasers in a future post, but for now I would like to point out that this idea introduces another aspect of quantum physics that I have not addressed directly before: *the wave properties of quantum particles are intimately connected to the possible behaviors of that particle being indistinguishable*. The classic example of this is Young’s double slit experiment, which we used to introduce the wave properties of electrons in the first place:

We noted in part 2 of this series that even a single electron evolves as a wave through the double slit experiment — though it hits the screen at a definite point, the probability of it arriving at that point is dictated by the wave properties. When enough electrons accumulate at the screen *M*, the interference pattern becomes visible.

A different way to look at this is that the interference pattern appears because we have no way of *distinguishing* which slit the electron actually goes through. The wave itself essentially goes through both slits, and this is what gives us interference.

Let us imagine that we were to add a sensor to the lower slit in this experiment that would detect the presence of electrons passing through the slit. The two possible outcomes of the experiment for each electron are now distinguishable — we will know which slit the electron actually passed through. We would find that the interference pattern completely disappears! The wave effects are directly connected to our inability to tell which slit the electrons pass through.

This principle, that interference effects require indistinguishability, is pretty much a universal principle in quantum physics. We will likely encounter it again in later parts of this series.

In the next post, however, I am going to try and explain the revolutionary discovery of physicist John Stewart Bell, one that fundamentally changed the way we think about the meaning of quantum physics!

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¹ Pions are actually produced by the collision of high energy particles with the atmosphere, but they only appear at high altitudes. The original pion discoveries were done on mountaintops.

² This is only approximately true, in fact. Photons can interact with each other even in vacuum, though the effect is very weak and requires high-energy physics experiments to detect.

³ Why is it called “nonlinear optics”? In ordinary optics, the relationship between the polarization of a medium and the electric field of the illuminating light is a linear one. In processes like second-harmonic generation, the polarization is proportional to the *square* of the electric field — it is nonlinear with respect to the electric field. For our purposes, we can say that nonlinear optics involves the interaction of two or more photons.

^{4} Image taken from the paper by Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko, and Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75 (1995), 4337.

^{5} In practice, the relative orientations of the polarizations of the two photons varies on each circle, but the two created photons are always at 90 degrees from each other. We pretend that one is always vertical and the other always horizontal for simplicity.

^{6} I chose the word “appearance” here carefully, because a proper interpretation of what is going on is a bit more subtle.