Does Interference Between Entangled Photons Ever Occur?
“Interference between two different photons never occurs.” Paul Dirac [1]
Each time when we think we measure one of two entangled photons we actually measure a component of their joint entangled system that can not be an individual photon.
Here come some more considerations to this point.
A tourmaline crystal transmits only photons with polarization perpendicular to the optical axis (using Dirac’s terminology). It absorbs photons with parallel polarization. Photons with oblique polarization sometimes pass through tourmaline, adopting perpendicular polarization, and sometimes are absorbed, adopting parallel polarization. From a quantum mechanical perspective, photons with oblique polarization exist in a superposition of perpendicular and parallel polarizations. A tourmaline crystal measures them, forcing them to adopt either perpendicular or parallel polarization. This measurement does not change the state of the photon as a whole. It remains in a defined state, which Dirac calls translational, and which is determined by the wave function (! don’t mix with Schrodinger’s wave function) of classical wave optics. [1]
The peculiarity of the translational state of a photon in quantum mechanics is that when a beam of light is divided into components, each photon is also divided into parts. If the components are of equal luminosity, then all photons are divided in half. If we now want to measure the momentum of a single photon, the photon will jump as a whole from two components into one. If the photon was previously in a superposition of momentum states its momentum has now become defined, and its state has changed as a whole, not partially, as when observing polarization. The photon’s new state can no longer be represented by the wave function of classical optics. [1]
Dirac begins his description of the principle of superposition of states in his textbook Principles of Quantum Mechanics with these special cases. [1]
The difference between the two measurements is that polarization, as a mathematical quantity, obeys the commutation rules for multiplication, while position does not. Position is the canonical conjugate of momentum. This means that both quantities cannot be measured precisely simultaneously because measuring one quantity changes the state of the system so much that accurately measuring the other becomes impossible. That is, the quantities representing position and momentum are quantum numbers (q-numbers) as defined by Dirac, and the quantity representing polarization is a classical number (c-number). [2]
All of this is crucial for understanding quantum entanglement, if we consider it using the example of photon spin. Photon spin is intimately, but not canonically, related to polarization. Like polarization, spin is represented by a quantity that obeys commutation rules for multiplication. In other words, spin is a c-number, not a q-number.
Therefore, spin can be measured any number of times without disturbing the photon’s state. Unlike position, momentum, and other canonically conjugate parameters.
If we leave aside for now the question of where entangled photons come from and simply accept the fact that we have a pair of entangled photons, then, from the perspective of quantum mechanics, we have a single system in a determined state shared by both. We can only separate this system back into two separate photons by performing a measurement that destroys entanglement. [3] Therefore, our photons must travel in pairs everywhere until entanglement is removed. Then, they must both be in translational states and split into halves when the beam is split into equal components.
A beam of two entangled photons in this case, following Dirac’s explanation, will split not into two photons, but into two beams of two halves of the photons. Each photon will split in half, with one half going into one component of the beam and the other half into the other.
Now, if we measure the spin in one of the beam components, then, by analogy with polarization, the spin of both halves of the photons will be determined. Their spins will emerge from the superposition of states.
A photon’s spin can only be an integer. It cannot be split into halves. Therefore, a spin measurement must yield either 1 or -1, and the condition of quantum entanglement dictates that if the spin of one of the entangled photons is 1 after measurement, then the spin of the other will be -1 after measurement, and vice versa. That is, the photon spins must jump each into its own beam component entirely, while the entangled photons must remain in a superposition of states between the two beam components in terms of their position and momentum.
Having written all this, I thought it would be interesting to see what happens to photon polarization if we direct only one of the two components of a beam containing only half-photons at a tourmaline crystal. How will the other halves in the other component behave? Will their polarization come out of the superposition or not? Even without any entanglement, it should. Some halves should remain, while others disappear as a result of the tourmaline absorbing their halves in the other beam, because tourmaline cannot absorb half a photon. In other words, there will be a direct correlation between the absorption of a photon in one component and the disappearance of its half in the other. Measuring polarization can serve as an indirect measurement of spin, although I haven’t yet learned how this works. Offhand, it seems like spin can jump between beam components even without any entanglement. I think there’s some very interesting stuff lurking here.
Surely someone has done similar experiments before. I’ll definitely look for them, but I don’t think I’ll find a refutation of quantum mechanics that way.
I don’t know what the Vedanta, magic, and Castaneda have to say about this, but without their help, it’ll be very difficult, if not impossible, to figure it out, I think. After all Dirac himself wrote here: “Questions about what decides whether the Photon is to go through or not and how it changes its direction of polarization when it does go through cannot be investigated by experiment and should be regarded as outside the domain of science.” 🙃
References:
- Dirac, Paul (1930). The Principles of Quantum Mechanics. Oxford, Clarendon Press.
- Dirac, Paul, The physical interpretation of the quantum dynamics, Proc. R. Soc. Lond. A113621–641, 1927 https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0012
- Schrödinger, Erwin, The Present Status of Quantum Mechanics, Die Naturwissenschaften 1935. Volume 23, Issue 48.
