Another inference/detection question

[gonzo]

I've been trying to think about what is really meant by "detection" in double-slit style experiments. I've had different explanations, most amounting to the effect of "if the path information could even in theory be known" (in regards to having detectors and not looking at the results, etc.).

So, I was thinking of a classic interferometer experiment with a beam splitter and photons. What happesn if you have some 100% (or near enough) mirrors in the path of one or both of them? Or maybe they use mirrors already to cause the photon to turn corners in the machines?

My question is, isn't it in theory possible to then detect which path a photon took because bouncing off a mirror will impart momentum to the mirror which could in theory be detected?

 

[humanino]

In order to have interferences, you must not measure which path was taken. If you have bouncing of the mirror, you have been interacting with the photon along one of the possible paths : superposition is lost. You would not see interferences anymore.

 

[vanesch]

In order to have interferences, you must not measure which path was taken. If you have bouncing of the mirror, you have been interacting with the photon along one of the possible paths : superposition is lost. You would not see interferences anymore.

The reason is in fact that in order to be able to measure this "bouncing", the position of the mirror must somehow be left free floating (on a rigid mirror, you have no way of finding out whether there was or not, a momentum transfer). If you then do all the math, you should normally find that this uncertainty in position on the mirror (treat the mirror as a quantum object: with a certain precision on its momentum which must be well below the momentum imparted by the photon must correspond also a certain uncertainty on its position) will result in a random phase shift in the interference pattern, exactly enough to wash it out.

cheers,
Patrick.

 

[marlon]

My question is, isn't it in theory possible to then detect which path a photon took because bouncing off a mirror will impart momentum to the mirror which could in theory be detected?

The interference patterns that you see at the detector when both slits are open are a direct result of the fact that the photon wavefunction is a superposition of the two possible ways the photon can pass. If you want to detect such a photon prior to detecting the actual interference pattern, you'll destroy this superposition, thus no interference patterns will be seen. So, YES you can detect a photon like this BUT you won't see no interference patterns.

In order to detect such a photon you will need to have entanglement between the photon states and the mirrorstates, just like in the Welcher Wech experiment

marlon

 

[gonzo]

Vanesch ... so there's no way to detect that some momentum has been imparted to a mirror if it isn't free floating? Won't there be a slight change in the conductance or something to that effect? And what does "free floating" mean when you are talking about a single photon bouncing off a single electron in a sea of free floating electrons (since I assume all the "perfect mirrors" are metals). I would think there would be some way to detect the collision in the mirror in theory.

I'm thinking some variation of the Mach-Zehnder interferometer would be a good experiment to discuss this in terms of, since it relies on two fully reflecting mirrors.

 

[vanesch]

Vanesch ... so there's no way to detect that some momentum has been imparted to a mirror if it isn't free floating? Won't there be a slight change in the conductance or something to that effect? And what does "free floating" mean when you are talking about a single photon bouncing off a single electron in a sea of free floating electrons (since I assume all the "perfect mirrors" are metals). I would think there would be some way to detect the collision in the mirror in theory.

Don't forget that you have only "mirror action" if you coherently interact with a lot of electrons ; otherwise you have scattering, which won't obey theta_in = theta_out. So you have to interact with a "large" body, whether it is a "sea of electrons", or the "mirror itself". Now, whatever it is you interact with, if you want to see an imparted momentum to it,through no matter what physical process, then that same physical process can also impart a momentum to the photon of the same order of magnitude. I took the simplest situation, in which we took the mirror as a massive body with no internal degrees of freedom, but with only a position: then the only way to have a physical process that is sensitive to a momentum is by letting that position "free" (or to a spring or whatever, but not rigidly fixed). Doing so then necessarily means that your position is uncertain in such a way that the second interference arm has an uncertain length.
You can, as you suggest, complicate the issue and consider the internal degrees of freedom of a mirror. But a similar effect then works: imagine that a photon "sends of a cloud of electrons deeper into the mirror, changing some resistivity" (or whatever). But that means that this cloud of electrons has some "freedom in position" and that the exact location of where the photon bounces off is unknown (or has to bounce off something in motion, which will give rise to a tiny doppler effect). I now claim that whenever such an effect will be observable, it will also do enough harm to the phase relationship between the two arms of the interferometer so as to destroy the interference pattern: or by changing the pathlength randomly by at least lambda/2, or by introducing a doppler effect that will amount to a phase shift of at least pi, or by a combination of both. Only the case of the "rigid mirror" is tractable easily with the math, of course, but I'm pretty sure all other ways you could devise will fail in similar ways.

This is not because I'm such a smart guy in solid state physics, but more because this is something that is similar like refuting a perpetuum mobile. You can complicate the issue and at the end of the day it will be hard to say exactly where the design goes wrong, but it MUST go wrong somewhere.


cheers,
Patrick.

 

[ZapperZ]

This is not exactly in line with the question being asked, but in case anyone missed it, Roger Penrose and Co. have proposed a quantum superposition demonstration using mirrors.[1] This is a further extension of the superposition of 10^9 electrons observed in SQUIDs experiments. If this proposal works, it would signify a superpositon of up to 10^14 atoms.

Zz.

[1] W. Marshall et al., PRL v. 91, p.130401 (2003), or http://arxiv.org/PS_cache/quant-ph/pdf/0210/0210001.pdf

 

[gonzo]

Thanks for that answer. I mostly follow your reasoning. But let me make sure.

So, you are saying that the mirrors they are using in these experiments are constructed in such a way that it is even in theory impossible to detect a collision with them, thus allowing the inteference pattern to show up.

You are also saying that if the mirror was changed slightly in a way that would make this theoretically possible, then the interference pattern will disappear (whether or not a "detector" is hooked up to the mirror).

Is this correct so far?

A few more clarifications. As I understand it, decoherence generally occurs when a "wave thingy" interacts with a suffienciently large number of other objects, where this number can often be quite small.

I guess my problem is that I don't really understand why a collision with the mirror doesn't ALWAYS cause decoherence. As you said, to refelect it has to interact with a large number of electrons.

I think it would be interesting to find the "limit" of these two situations. The smallest change you could make to the mirror set up to cause or remove the interference pattern. I don't even mean hooking up a detector, since it should be enough to just be theoretically able to hook up a detector, even if we don't have the right current technology to do it in just that experiment. Decoherence should only depend on the theoretical ability to get the information, right?

You mentioned degrees of freedom and the difference between a rigid and free mirror, where a rigid mirror wouldn't be able to retain the path information but a free one might. So, would there be some experimental way to adjust the mirror in ever so slight increments (as simple as loosening its screws?), to show the border between interference and non-interference?

Anyway, thanks for you replies.

 

[vanesch]

So, you are saying that the mirrors they are using in these experiments are constructed in such a way that it is even in theory impossible to detect a collision with them, thus allowing the inteference pattern to show up.

Well, don't overestimate that. A simple mirror like you use in your bathroom, solidly screwed onto a solid table, will do the trick :-)

You are also saying that if the mirror was changed slightly in a way that would make this theoretically possible, then the interference pattern will disappear (whether or not a "detector" is hooked up to the mirror).

Is this correct so far?

Yes. The "slight change" would be something like suspending the mirror on a spring, and try to see whether the photon impact set it to swing, for instance.

A few more clarifications. As I understand it, decoherence generally occurs when a "wave thingy" interacts with a suffienciently large number of other objects, where this number can often be quite small.

Or quite large. You can easily interact "coherently" with zillions of particles without inducing decoherence. Neutron or X-ray diffraction on crystals or soft matter is an example ; a mirror is another example. In fact, the trick is that after interaction, those zillions of particles should still occur in a product state with your initial system quantum state. If that's the case you do not induce decoherence (and your zillion of particles do not retain any information !). If you DO entangle your zillion of particles with different component states of your quantum system, then you DO induce decoherence (your zillion particles now know about your "which way information" and you destroy any interference ; it is just what we are discussing !).
But you can have very large quantum systems that do act "coherently". Zzapper has a lot of examples ready.

I guess my problem is that I don't really understand why a collision with the mirror doesn't ALWAYS cause decoherence. As you said, to refelect it has to interact with a large number of electrons.

Yes, but in such a way that all those electrons are left in the same state as they were before. From the moment they have a "memory" that the photon did pass (meaning: something got entangled with the photon state), you also should have decoherence.
You can complain that SOME momentum WAS transferred. Yes ! It was tranferred to the sea of electrons, which transferred it to the ions in the metallic layer of the mirror (assuming we have a metal coated mirror), which transferred it to the glass, which transferred it to the screw holding the mirror, which transferred it to the optical bench... and which transferred it to THE OTHER BRANCH of the interferrometer AND to the branch holding the photodetector. If all this is rigidly fixed to one another (the necessary condition to have branches of constant optical path length to obtain interference patterns) then these momenta compensate each other.
So, for a rigidly fixed mirror, the momentum transferred is not measurable, and with respect to the mirror, the electrons are again in an identical quantum state before and after the reflection of the photon. Again, if this quantum state is distinguishable, you HAVE decoherence.

I think it would be interesting to find the "limit" of these two situations. The smallest change you could make to the mirror set up to cause or remove the interference pattern.

In the case of a rigid mirror, you can do the calculation ! Give it a mass m, consider that it is also a quantum object, and look at its "momentum uncertainty" you obtain when the position is fixed within a fraction of the distance needed to shift the interference pattern by 180 degrees (wash it out). See now if this allows you to measure the delta-p of the photon bouncing. - this was the situation of the "rigidly fixed mirror with some momentum measurement".
Or, consider a mirror, suspended so that we know somehow its momentum (0) with a certain degree of precision which will allow us to find out the bounce of the photon. Now look at its position uncertainty, and the shift this will induce in the interference pattern.

You mentioned degrees of freedom and the difference between a rigid and free mirror, where a rigid mirror wouldn't be able to retain the path information but a free one might. So, would there be some experimental way to adjust the mirror in ever so slight increments (as simple as loosening its screws?), to show the border between interference and non-interference?

Well, do it for yourself as I suggested above, in a given interferometer setup.
The screw can be modeled by a spring with given spring constant (your mirror state is now given by a harmonic oscillator). A strong spring "fixes the position" but makes momentum (or energy) measurement difficult (you'd like to get at least one "energy step" between two levels). Also, don't forget that the energy will have to come from the photon, so you get a doppler-shifted photon coming out.
A weak screw "lets the position free" but makes the momentum (or energy) measurement easy. But now, you have an uncertainty on the optical pathlength.
I haven't done it explicitly myself in this case, but I'm pretty sure that you can work it out easily, and that it shows that each time, you're screwed :-)

cheers,
Patrick.

 

[gonzo]

Thanks. I think I have a better handle on it now. Of course it's still a little foggy in parts, but then again I think it was Feynman who said that if you aren't confused by QM you don't really understand it, so I don't feel so bad.