Non-quantum explanation for Mach-Zehnder interferometer effect

Question

The phenomenon of all photons being detected at only one detector seems quite reasonable to me, classically. If a photon gets deflected at the first beam splitter for whatever reason, it gets deflected at the second BS as well for the same reason. Similarly, if a photon passes through the first BS and it does the same at the second BS as well. This results in only one Detector receiving all photons.

This reasoning doesn't deal with superposition, probability amplitudes and quantum weirdness etc.

What's wrong with the above reasoning? Kindly correct me.

Answer 1

Classically one cannot talk about photons. Photons are particles in the sense that they are elementary exitations of electromagnetic field. However, our classical intuition about particles is heavily biased towards fermions, and any such discussion involving photons inevitably ends up thinking of them as electrons or little balls scattered by a beamsplitter.

A meaningful classical discussion of interferometry deals with electromagnetic waves, which, as all waves, exhibit the phenomena of superposition, interference, etc. On the other hand, being a coherent state of electromagnetic field, electromagnetic wave is a superposition of states with different photon numbers, i.e. number of photons in it is undefined.

Answer 2

This is a good question! Let us begin with the assumption that each photon has a property called brute. Brute can take on two values, $0$ if it reflects off a beam splitter, $1$ if it passes through a beam splitter. What this means is that brute $1$ photons always pass through a beam splitter and brute $0$ always gets reflected from one. Theories of this kind fall under the category of hidden-variable theory. Quantum mechanics has systematically eliminated such theories by providing counters (some harder than others) that can’t be explained by hidden variables unless you give up on locality.

But in this case, it is easy to dismiss. Because we have the brute property, if we send in one photon at a time, only one of the paths is taken. Either 3 or 4. But regardless, all photons will be detected at $D_1$. All well and good. And our brute hypothesis explains the behaviour.

However, if we make path $3$ a little bigger (as compared to the wavelength of light) than path $2$, an interference pattern emerges at the detector. The pattern occurs even if we send one photon at a time (albeit after a lot of times)!

The interference pattern is dependent on the path difference between the two paths. That means there is some cross talk between the photons going in the two paths. But that is impossible if we send in one photon at a time! We can only reconcile this if we give up on the fact that the photon took a unique path. This is the opposite of what our brute theory says.

So the fact that an interference pattern emerges even for the single photon case rules out the fact that the photon took a unique path.

Answer 3

If a photon gets deflected at the first beam splitter for whatever reason, it gets deflected at the second BS as well for the same reason.

The accepted term for this idea is that the photon has a "hidden variable", some aspect or state that we don't know about (and so can't test for and can't include in our frameworks of physics) but which determines whether the photon reflects from or passes through the beam splitters.

The problem is that there cannot be a consistent set of hidden variables underpinning quantum mechanical behaviour that are both a) counterfactual-definite, meaning that the classical result you observe "was there all along", and would have still been there if you hadn't made the measurement you did, b) "local", not involving influences travelling faster than light. That might not sound important, but in a Relativistic context, "travels faster than light" is equivalent to being able to travel backwards in time and most physicists want to avoid that notion if at all possible. For more detail on how we've ruled out hidden variables generally, the terms to look for are Bell's theorem and the EPR experiments that show Bell's theorem does not apply to reality.

As for why the interferometer in particular cannot be explained by hidden variables, there's an extension of the experiment you described that's nicknamed the "quantum bomb tester" and looks like this:

The experiment works like this: a "bomb" that is supposed to be triggered to explode by absorbing a photon is placed at point B. However, some of the bombs are broken in such a way that they do not absorb the photon and do not explode, letting the photon continue unimpeded. This means that a photon passing the bomb may or may not reach the second beam splitter depending on whether or not the bomb is functional. This in turn means (per quantum mechanics) that the probabilities of reaching each of the two final detectors depends on whether bomb is functional.

When you work through all the numbers, you find that the probability of detecting the photon at detector D is non-zero if and only if the bomb is functional. Therefore if you do find the photon there, you know that the bomb is functional - but also that your photon didn't interact with it, because it didn't explode either. (There is a seperate possibility it does explode, but you can make that arbitrarily unlikely with a more complex setup.) The bomb's state is influencing the photon despite not "really" interacting with it, i.e. not exploding, which cannot be explained by classical unique paths or hidden variables.