Quantum Control of a Looping Preparation

[al onestone]

I'm working on a thought experiment and I have a question about trapping a pulse inside a looped preparation . If a pulsed particle beam is prepared with a definite energy(E), bandwidth(ΔE) and positional uncertainty/coherence length(Δx) is the pulse able to be trapped inside of a loop type preparation(a cavity) if the total length of the loop/cavity is shorter than the positional uncertainty of the particles.

Typical cavities in optics all have open input and output ports. I'm talking specifically about a loop which is enabled by a switch which opens and closes the loop. A simple example in optics would use a polarizing beam splitter as the entry point of the loop, only allowing in one polarization. Secondary rigid mirrors could be used to redirect the beam towards the back side of the beam splitter where the beam would be deflected because of its polarization. This would constitute a loop/cavity where the transmitted photons are trapped inside. In order to open the loop to release the photons you simply insert a wave plate which rotates the polarization and allows it to exit through the beam splitter. In total this would constitute a switched loop preparation which can trap photons and release them upon command of the experimenter. My question is, if the loop length is shorter than the uncertainty in position,Δx(the linewidth of the photon distribution) is it still possible for photons to become trapped inside or does the loop act as a forbidden region? If the photons are allowed to become trapped, is the pulse shape shortened to the loop length after release? This would violate Heisenberg uncertainty. If you think that they can become trapped and that it has no effect on the pulse shape, then you have to admit that the photons are still comming out of the loop at a time later than it would require light to traverse one length of the loop. This doesn't make sense. Either they don't become trapped or they do and they all exit within the amount of time needed to traverse one loop length. Please resolve.

 

 

[Schrödinger's hat]

The simplest answer is that putting the light into the loop will change both the energy bandwidth (uncertainty) and position uncertainty.

[al onestone]

Let me get this straight, you're saying that the photons which become trapped, photons with an already established energy bandwidth/uncertainty, have a reduced positional uncertainty and some how expand their bandwidth miraculously? I doubt it. In order for photons to enter a state with a larger bandwidth they have to be absorbed and re-emitted, which would make them differrent photons from the original ones. This is still not resolved.

[Schrödinger's hat]

Let me get this straight, you're saying that the photons which become trapped, photons with an already established energy bandwidth/uncertainty, have a reduced positional uncertainty and some how expand their bandwidth miraculously? I doubt it. In order for photons to enter a state with a larger bandwidth they have to be absorbed and re-emitted, which would make them differrent photons from the original ones. This is still not resolved.

 

Simply interrupting a beam to turn a (near) monochromatic wave into pulses is enough to change the bandwidth.

[al onestone]

Simply interrupting a beam to turn a (near) monochromatic wave into pulses is enough to change the bandwidth.

 

In order to change the eigenvalue spectrum of the description, what would be the mathematical model of a unitary operator which represents a "beam reshaping" process?

[Schrödinger's hat]

I haven't looked at this from a quantum perspective before, but classically you can do something along the lines of replacing the monochromatic wave:

[math]e^{ik(x-ct)}[/math]

With something multiplied by some step functions, ie.

[math]u(t0-t)u(t-t1)e^{ik(x-ct)}[/math]

 

If you want to recover the spectrum of the truncated just take the fourier transform of the new function.

 

The main potential problem I can see with this simplistic approach is the hard edge may introduce un-physical artefacts where -- were you to consider your shutter/barrier as a potential and treat it as a scattering problem --you'd have tunnelling.

Edited November 29, 2011 by Schrödinger's hat

[al onestone]

OK, lets assume that we can trap photons in a cavity with the end result of shortening its coherence length/pulse shape. And a consequence of this is the widening of its bandwidth. Let's also assume that we can find a unitary operator which produces the desired effect of changing the state description in the appropriate manner. My first problem with this is that in QM we require that all changes of state can be modelled by a unitary operator in the Von Neumann measurement scheme. If our preparation reduces the pulse width, this is like reducing the uncertainty in position which effectively is a measurement. In Von Neumann measurement we always have a measurement apparatus that is being "coupled to" and this projects the system onto a new basis. In our scheme we reshape the pulse with linear optics apparatii where no photons are coupling to anything physically real. It seems as though photons escape the typical QM Von Neumann description of measurement.

 

Anyway, assuming that our cavity can be idealized in a thought experiment as weve described so far, what if we use as a source an entangled pair of photon beams from downconversion. If we use polarization entangled beams and we subject one to a pulse reshaping preparation, which consequently shortens the pulse shape and widens the bandwidth, what effect does this have on the second entangled partner beam? Does the second beam have a similarly modified bandwidth? or does the second beam become disentangled from the first (as in teleportation, Bell state analysis)? Remember, not just the polarization is entangled in downconversion, the momentum/energy is entangled via wavevector conservation. So if the one beam's bandwidth is widened, then the energies of the second beam cant all match those of the first unless it is similarly modified? Please resolve.