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I don't know if we have the technology yet.

Double slits, one electron at a time.

The moving electron should produce a magnetic field. Can we detect such a weak magnetic field and tell which slit has the stronger field => electron passed through that slit?

I suppose that begs the question whether responding to an electron's magnetic field constitutes a measurement.

EDIT: Similar question is here Electron double slit experiment and the electric field which refers to an experiment http://arstechnica.com/science/2012/05/disentangling-the-wave-particle-duality-in-the-double-slit-experiment/ saying interference remains even after knowing which-way information (for photons).

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    $\begingroup$ The measuring device usually used for this purpose is a SQUID (but not the mollusc), assuming that you want a so-called strong measurement. Weak measurements are a whole different game. $\endgroup$ – dmckee Nov 23 '15 at 0:16
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In effect, it's done all the time in a transmission electron microscope. Usually it's not a simple double slit but rather a multiple slit (in the form of a crystalline lattice). This is happening in the presence of a strong and rather inhomogeneous magnetic field, produced by the microscope's objective lens. The interaction (and remember, it is an interaction; the lens affects the electron, and the electron affects the lens, albeit by a tiny and practically immeasurable amount) between the electron and that static magnetic field doesn't generate any decoherence, though, so the interference remains and you can see both diffraction patterns and atomic resolution images. The variation of image contrast with defocus is consistent with wave interference and not simple mass-thickness contrast, so you're not just seeing the classical shadows of atoms. Specifically, as you vary the focus, the atoms can appear as either light spots or dark spots.

However, if the electromagnetic field of the fast-moving electron excites a particular material excitation in a way that produces a measurable net loss of energy by the electron, then that part of the electron wave won't interfere with the part that went through elastically. Not only is it entangled with degrees of freedom in the material, but also its temporal frequency is different so you'd be hard pressed to see interference anyway. In that case you can think of yourself as having used the electromagnetic properties of the material to measure the field of the passing electron.

Long story short: If the interaction results in decoherence, it counts as a measurement.

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If you would measure the electron at one of the slits, then the interference patterns would no longer be formed. That is because the pattern is produced by interference of an electron amplitudes diffraction from slits 1 and 2. If you know that electron is at slit 1, it is of course no longer at slit 2, and therefore you wouldn't get the interference pattern.

The measuring at slits could be called wave function collapse, but I would rather talk about quantum entanglement with environment. Let's say we have two slit states $\left|1\right>$ and $\left|2\right>$, which describe electron being at slits 1 and 2 respectively. Let's also say that we have two already diffracted states at the detected (whatever device used to measure the interference pattern) $\left|1'\right>$ and $\left|2'\right>$. The electrons propagate from the slit states to detector states in trivial manner, so that $\left|1\right>$ becomes $\left|1'\right>$ and $\left|2\right>$ becomes $\left|2'\right>$ as the electrons travel.

The initial wave function is $\left|1\right>+\left|2\right>$, and therefore the interference pattern is $\left|1'\right>+\left|2'\right>$.

Now, let's define environment states $\left|M1\right>$ and $\left|M2\right>$, which are corresponding the magnetic measurement system saying BEEP or BLIIP as the electron passes through slit 1 or slit 2.

If we hear BEEP, we know that our system is in state $\left|1\right>\left|M1\right>$, and if we hear BLIIP we know that our system is in state $\left|2\right>\left|M2\right>$. As the electron is only in either on of these two states, the electron will travel to the measurement device resulting into $\left|1'\right>\left|M1'\right>$ and $\left|2'\right>\left|M2'\right>$ respectively. Therefore the interference pattern vanishes.

If you believe in many worlds theory, you can even say that the world is in $\left|1\right>\left|M1\right> + \left|2\right>\left|M2\right>$ state, and then the electron travels to measuring device and the world is in $\left|1'\right>\left|M1'\right> + \left|2'\right>\left|M2'\right>$. But if you hear BEEP you are trapped in universe $M1'$ forever, and if you hear BLIIP you are trapped to $M2$ respectively.

If you don't measure the electrons at slits at all, you will encounter a wave function like $\big(\left|1'\right> + \left|2'\right>\big) \left|M\right>$, and you can then measure the interference pattern.

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  • $\begingroup$ Well, IF it constitutes a measurement, then it "collapses" the wave function into this $|1>|M1> + |2>|M2>$. If it doesn't do that, then it is a poor measuring device. If it would for example change the world wave function from $(|1>+|2>)|M>$ into $(|1>+|2>)|M1> + (|1>+|2>)|M2>$ keeping the interference, then it wouldn't actually measure anything, since the machine would BEEP and BLIIP randomly. There is really no way around this:) The article you linked uses two photons. One would have to do that thing I did with states all over again starting with entangled two-photon states. Not today :) $\endgroup$ – Mikael Kuisma Nov 23 '15 at 0:32
  • $\begingroup$ Sorry, don't know about experiments. But I'm sure that nobody has both measured the slit and obtained the interference pattern. Also, I would guess that measuring a magnetic field of an electron is very difficult. $\endgroup$ – Mikael Kuisma Nov 23 '15 at 8:59
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That is quite much the point, isn't it. Claus Jönsson of the University of Tübingen did this with electrons in 1961. In 1974 the Italian physicists Pier Giorgio Merli, Gian Franco Missiroli, and Giulio Pozzi repeated the experiment using single electrons, showing that each electron interferes with itself as predicted by quantum theory.

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