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From what I understand, it seems like you can only "add" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a local oscillator beam to mix with the signal beam so you can decipher the difference in phase change which would give you information about the system your signal beam is probing. Homodyne detection can give information regarding the position of an atom.

Say I wanted to detect the relative distance of two atoms. Each atom/ion is stuck inside an electric harmonic potential. The location of the harmonic potentials are known, (if this is possible) Now, let's say I turn off the harmonic potential trap and at the same time I use the homodyne measurement technique to determine the position of both of the atoms. This means shining a laser to detect the atom's position. (The light is absorbed and re-emitted?, or reflected?) In any case, light that comes out of the atom/ion cavity is related to the atom's position/momentum and it is combined with a local oscillator beam to determine the position and momentum of the atom. The light hits photodetectors which then integrate the signals together producing a photocurrent. The analysis of this photocurrent will allow one to figure out the atom's position relative to the center of the harmonic trap.

To figure out the relative position of the atoms from each other, I'll need to "subtract" the position of one atom with the other. If one atom's position is labeled x1 and the other is labeled x2, than wouldn't we simply do x2 - x1? If so, how is one to do this?! How does one "subtract" the position of one from the other using light beams?

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When two or more sources of light combine incoherently (not in any fixed phase relation), you can only "add", and that's intensity (power) - electromagnetic field "squared" and averaged, in the appropriate sense.

When you have control over phase relations between two beams, yeah, sure you "add" the two at some point. To "subtract" all you need to do is shift one beam by 180degrees. (You are dealing with monochromatic light?) Just add an optically transparent material sufficient to delay the beam by half a wavelength, and there you go.

Note that "subtract" and "add" may be meaningful only in a limited sense, in the change of phase relationship of two beams in a stable optical apparatus. That is, it's not in the phase relationship itself, but in how it changes by 180 degrees. This is practical and sufficient in many experiments. You don't really have control over things such as spacing of holes in an optical table down to 100nm accuracy, but whatever the distances are, you can be sure they won't change by much during the course of the experiment (<100nm).

If you have absolute geometric control to several-nanometer scale accuracy all the way from beam splitter to beam combiner (or detector) then you could compare the optical paths and say that the beams add, or subtract, or combine in an in-between phase relation, depending on the difference of optical paths being integer, half-integer or other number of wavelengths.

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So are you basically saying that we can use the homodyne measurement technique but to make one beam 180 degrees shifted before combining them together at a beam combiner? So this means I can get x2 - x1 – QEntanglement May 3 '13 at 5:22

You can subtract two beams if they have a well defined relative phase to each other using an interferometer (destructive interference). Of course, energy must be conserved, so the beams will constructively interfere somewhere else (usually a second port on the interferometer). If you don't have a well defined phase relation, then no, two beams of the the same wavelength can't be subtracted. But then again, they can't be added 100% as well (if the same wavelength). A "beam combiner" will have two output ports. Each will get half the summed power (assuming a 50%-50% reflector is used) in the later case.

Now for the measurement of the atom/ion. When a laser is tuned to a "cycling" transition in the ion, the ion will absorb photons from the laser and re-emit them via spontaneous emission. If you image the light onto a CCD, you will see the ion glow as long as you hold the ion in place with a potential (otherwise, the momentum kicks from the absorbed photons will cause the ion to accelerate very quickly). It is interesting to note that even though the detected light is from spontaneous emission, it does have a phase relation to the incoming light. So yes, if you collect that light, you could do a phase measurement of the emitted light by beating it against the same laser used to fluoresce the ion (if, as DarenW notes, you can stabilize the path length sufficiently). If the ion moves, the emitted light's phase at your interferometer/detector will change, either because the ion moves in the fluorescing laser (changing the phase of the light it sees and re-emits) or because of changes in the distance between the ion and your interferometer.

Something like this is done for what is called "remote entangling" of ions in ion traps. The two ions are both hit by a very short pulse of light derived from a common source. Each ion then emits a single photon (entangled with the ion internal state) which is collected into fibers. The photons are then interfered on a beam splitter and by measuring how they interfere, you can generate entanglement between the two ions. Single photons don't have phase information, but you could imagine using a similar setup with continuous fluorescent beams. If one ion moves with respect to the other, then you should see a change in the output powers from beam splitter. Lot of technical challenges, though.

You can also often see interference fringes in the image of two ions trapped in the same well if the halos around the ions imaged on the CCD overlap. Don't even need a beam splitter.

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