|Oct10-04, 02:22 PM||#1|
Decoherence, Unitary Evolution and Information: Some ideas
This thread is based on some reading I've done on decoherence (a lot of it by Zurek) and recent experiments involving decoherence and complementarity. One interesting item is the recent experiment by Anton Zeilinger and team involving C60 molecules interacting to produce and interference pattern and the washing out of the interference pattern when the C60 molecules are heated which then causes the emission of photons as they travel.
Here's a link: http://www.quantum.univie.ac.at/rese...eco/index.html
This seems to be an example of complementarity - where there is a trade-off between which-way information and coherence, like the tradeoff between position information and momentum information that is quantified by the
Heisenberg uncertainty principle. My understanding is that the decoherence of the C60 molecules is caused by the emission of photons which can carry of information about the path that the C60 molecules were taking through the
equipment. If the C60 molecules travel across the gratings without any path information being measured, the wave function's behavior can be calculated by the usual Feynman sum-over-paths method where the particle takes all possible paths between the starting and endpoints, which results in an interference pattern here, as long as the particles are not observed in transit between start and end. The strange result of this experiment is that even though the photons which are being emitted by the C60 molecules may not be observed, the result is that if the particles are hot enough to emit photons, the interference pattern starts to disappear, exactly as if they were being observed by a detector (with low but varying efficiency). It doesn't seem to matter that the photons are just being randomly shot off into the surrounding environment and not directly observed - the simple fact that they *could* have been observed and thus used to determine path information on the C60 molecules is enough to cause the interference pattern (the wave functions superposition) to disappear.
It's tempting to want to interpret the meaning of this (at least for me) to mean that the environment is acting as an observer or measuring device, but this is rather vague and unquantitative. Rather, I'm trying to understand it in
mathematical terms of wavefunctions and unitary evolution. OK - let's start with the wavefunction of the C60 molecules. The experiment moves between two extremes: (1) Where the C60 molecules are cold enough that no photons are emitted along the way between start and end, and interference is prominent and (2) Where the C60 molecules are hot and emit photons along the way which are then absorbed by the environment. The wavefunctions describing the C60 molecules in each case is quite different. In the first case (1) we can basically ignore the environment (outside of the diffraction gratings, etc.) and just write the wavefuntion in terms of the C60 molecules alone and in (2) the wavefunction of the C60 molecules is different and it's evolution in time results in an entanglement with the environment.
What's really interesting here is that the wavefunction in case (1) is exhibiting strikingly quantum behavior and the wavefunction in case (2) is describing much more classical-like behavior (i.e. lack of interference), but the evolution of the wavefunction in each case (if we believe that quantum mechanics applies equally) is unitary. The lack of interference is one of the hallmarks of classical behavior, and is never observed when a wavefunction undergoes "collapse", so we have isolated a feature of the "collapse" process but somehow retained unitary evolution. This is where dechoerence comes in. In my understanding (rather limited) the fact that the interference pattern of the C60 molecules disappears as photons are emitted (as the temperature of the particles is increased) is caused by the entanglement of the C60 molecules with the photons which are being emitted along the way. In case (1) each C60 molecule passes through the diffraction grating, it's wavefunction spreading out across the diffraction grating and interfering with itself, with a *pure vacuum* (i.e no interactions) in between. In case (2) you can visualize the wavefunction as it interacts with photons as creating something like bumps or kinks in the space between the gratings and the scanning mask. Obviously, the wavefunction is no longer distributed coherently through space - there are random interactions with photons that shoot of in various directions. The key point here is that the coherence of the
wavefunction is not totally lost, but instead some of it is now entangled in the photons that are shooting off in random directions. If we continue to measure the pattern if interaction of the C60 molecules using the detectors at the end, we'll see that the interference pattern has begun to disappear - because some of the coherence information has been carried away by the photons which we are not measuring. In technical terms, we have taken a partial trace of the C60 molecules' density matrix with the detectors in which we are tracing out the state of the randomly emitted photons. If it were possible to recover the photons and retrieve the information they carried away, the interference pattern could be restored.
Another way to look at this is the Feynman sum-over-paths method, which uses the assumption that the wavefunction propagates over every possible path between endpoints and the phase of the wavefunction cancels out in all places except those most likely to be the path of the particle (or other object). For example, a free particle is most likely to travel from point A to point B in a straight line because the paths which go in loops or erratic patterns between these points tend to cancel out. The interesting thing here is that if we assume that the particle does take all paths between point A and point B, then suppose we have a free particle traveling from A to B in a straight line (say West to East on a compass to visalize), but there is another particle C at the North point of the compass. According to the um-over-paths method, the particle traveling from A-B is actually aware that the particle C is there - because it will interfere with some of the paths in that area. In the classical world this effect is so ridiculously small that it can be ignored for all practical purposes, but in the quantum world, this effect can be significant, and it is exactly what is affecting the C60 molecules which emit photons as they travel from start to end.
The next question that comes to mind is how to quantify all this interaction, coherence and decoherence taking place. It appears to me that when the interference pattern of the C60 molecules disappears, we are essentially losing information which is being carried away into the environment by the emitted photons. My guess is that this information (not sure how to quantify it) is being conserved, since it is possible (theoretically) to recover it. The reasoning is that since the experiment is governed by the laws of quantum mechanics, it is described by a unitary evolution, and this unitary evolution can be reversed (theoretically - the second law of thermodynamics makes it vanishingly unlikely) to run the experiment backwards. So, it seems that in some sense, unitary evolution preserves information, although that information can become entangled with the environment. In order to correctly describe the entire experiment however, we need to define the wavefunction of the entire experiment, including the interferometer, C60 molecules, diffraction grating detectors, etc.), and this will always be described by a unitary evolution. However, if we consider only the wavefunction of the C60 molecules themselves, we are making an approximation to their behavior because their wavefunction will be entangled with the wavefunction of the rest of the system. I imagine if it is a valid wavefunction that it will undergo a unitary evolution, but I wonder if the fact that an approximation is being made may cause it to appear that a non-unitary evolution may take place. What we see in the classical world certainly seems to be non-unitary behavior, but I wonder how much of that is due to series of approximations of the wavefunctions of large, strongly entangled systems.
Anyway, I hope a few individuals found this of interest. I would really like to understand how decoherence unitary evolution and information fit together.
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|Oct10-04, 03:01 PM||#2|
I found that VERY interestin, especially case of the side particle in the sum over path. This reminds me of another well known case where the environment cann affect the quantum amplitudes; the Aharonov-Bohm effect. Here the disturbing factor is that the space of paths is not simply connected, so again, you can't just add up the variations and get zero everywhere except on the zero-variance path.
|Oct10-04, 04:07 PM||#3|
A particle,(molocule) wave-function sits within a Spacetime, this is what Particles are?..their evolutionary make-up is governed not by the experiment, but the Wavefunction of the Universe at Large.
To measure something you have to have at minimum TWO locations, one for the detector and one for theobject ?
Detectors do not occupy the same location as the object , to measure "whole" particles, one must become detached/disconnected?..theobject thus becomes isolated from the observer/detector.
Displacement from Equilibrium (thermal), will invoke a background 'input'. Linear momentum (p), of the C60 in this experiment produces displacement from Equilibrium (q) via the appearence of Photons (unobserved) from outside of the 'whole' initial setup, the photons are an artifact of the Universe wavefunction , not from the Classical Particle,(C60) wavefunction?..one can almost state that the Universe's wavefunction has 'Lent' a photon to preserve
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