Hello Eric,

Just my tuppenceworth...

Eric Cavalcanti wrote:

I think this discussion might have already took place
here, but I would like to take you opinions on this.

How do we define (de)coherence? What makes interference
happen or be lost?


First, these are two separate questions.

Decoherence is said to occur when two waves (or wavefunctions) which were initially in phase (or having a constant or well-defined phase difference) are no longer in phase. This can be for a variety of reasons. Typically, this is cited as occurring due to interactions with large numbers of particles (or a single interaction with one particle that goes on to affect a large number of particles). In this case the fact that these particles have a large number of internal states means that it is unlikely that the two waves remain in phase.

Another example of decoherence would be in light from a regular light-bulb. The polarization of the light is subject to rapid and random changes in direction (due to emission of individual photons by the bulb), so that while the horizontal and vertical components of the light are instantaneously coherent, they rapidly decohere from each other to give some other value of polarization. The fact that at any particular instant there is a well defined but random value of polarization for regular light is what allows us to do Young's Double Slits with unpolarized light, as at any point on the screen the light arriving from either slit shares the same polarization, even though the value of this polarization is subject to rapid fluctuations.

In answer to your second question, the loss of interference (at least in the Copenhagen Interpretation) is due to the collapse of the wavefunction, from a superposition of different possibilities to one actuality. The Copenhagen Interpretation really does not say anything about what causes this collapse (apart from the nebulously defined notion of observation). Decoherence has been invoked as one possible explanation for this loss of interference, specifically that once a large number of particles are involved in the quantum system, it is unlikely that any of them will be in phase enough for us to be able to see interference in practice.

In the Many Worlds Interpretation, it is not necessarily decoherence, but the linearity of the Schroedinger Wave Equation that makes interference disappear. Specifically, once an observer (or any other system for that matter) interacts with a superposed wavefunction, that system's wavefunction is also put into a superposition of relative states. The relative states are all separately solutions of the SWE, so linearity prevents them from directly interacting ( = exchanging energy) or subjectively noticing each other through interference, in the same way as ripples on a pond are capable of moving through each other.

Decoherence comes into the MWI explanation of (apparent) wavefunction collapse once a second observer (or system) interacts with the superposed system. Let's say our first observer/system has interacted with the particle on its way from the double slits to the screen in such a way that that observer/system knows (or has an unambiguous record of) which slit the particle went through. Now a second observer is going to record the position the photon strikes the screen. Under MWI, the particle is *still* in a superposition of states when it reaches the screen. However, it has also interacted with the first observer system, which for the sake of argument we shall assume consists of a large number of particles. Because of the interaction with the first observer, the second observer is not just interacting with the wavefunction of the particle that went through the slits, but also with the superposed relative state wavefunctions of the first observer(s). These two relative states are highly unlikely to be in phase because of the large number of particles involved. Therefore, the second observer is also highly unlikely to observe an interference pattern at the screen when the experiment is repeated many times.

Note that in MWI the second observer's wavefunction is also split into two relative states by watching the screen, and so she may obtain a result indicating that the particle went through either slit regardless of the first observer's result (who is actually in a superposition of having got both results). Linearity of the SWE ensures that the second observer's result will always agree with the first observer's result should they compare notes later in that particular branch of the multiverse.

This is also how the MWI preserves locality in the EPR paradox/Aspect experiments, which I think is an important experimental vindication of MWI.

Take the a double-slit-like experiment. A particle can take
two paths, A and B. We can in principle detect which path
the particle went through.

Suppose we can make the detecting apparatus 'non-interfering'
enough so that the particle is not grossly deflected by the
detection, but can still reach the screen. We know that the
result of this thought-experiment is that interference does not
happen.

The first answer is that the paths have 'decohered'. But what
exactly does that mean? In a MWI perspective, I like the
explanation that the two universes A and B are different by a
large number of particles: the electrons in a wire, which carry
the amplified pulse of the detector, which then reach a
computer, and such and such. Something of the order of 10^23
particles have changed state.

Now suppose we use some kind of very slow detector. The
detection is made by, say, a very slow process such that not
many particles (suppose only one particle, even though I don't
know how to make that detector) change their state before the
interfering particle reaches the screen. After that, we can amplify
this information and know which path the particle went through.
Again, I believe interference would not be possible. But it is a
little harder to say why.


I'm not sure I can answer this question with certainty. I think it depends on how many possible internal states the recording particle/system may have; the more states, the more decoherence, and the less likely it is that interference will be seen by the second observer. If the detector had only two possible internal states, I think it is indeed possible for the screen observer to see some interference if the experiment were repeated many times.

I don't think the Copenhagen Interpretation was designed to include single particles as observers; rather one would include them in the wavefunction of the total system. Consider as an example Helium. You could think of one electron as being the observer of the other electron; under CI both are included in the wavefunction nonetheless. I think that the CI would therefore make different predictions whether or not one assumes that the recording particle qualifies as an observer. I'm not aware of anything in the CI framework which would help you choose which assumption to make; rather you'd do this retroactively depending on which results you got. I know that the logical inconsistencies in the CI when more than one observer are included are exactly what led Everett to develop MWI in the first place; if anyone has any specific information about what these inconsistencies were, I'd be very excited to hear about it.

In the MWI, there is no distinction between observers and other systems, even single particles, and I'm pretty sure that it would predict that the second observer would see some interference in this case, with the amount of interference smoothly (and exponentially rapidly) decreasing with increasing number of internal states of the first observing system (due to decoherence).

Hope this helps,

Matt.


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When God plays dice with the Universe, He throws every number at once...

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