Quantum control of a looped beam.

[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?

 

[Cthugha]

Two comments:

1) From my experience with microcavities I can tell you that the quality factors of a cavity become pretty bad as soon as the cavity volume becomes close to the light wavelength. Cavities below 1 micron do not behave too well. Maybe photonic crystal cavities behave better. I am not sure about that.

2) Your idea seems somewhat analogous to Q-switching/cavity dumping. Do you know these techniques?
See http://en.wikipedia.org/wiki/Q-switching.

Yes, I know wikipedia is not a really good source, but that article is okish.

 

[al onestone]

Q-switching only applies to the case of a pump laser which feeds a gain medium (gas, crystal,etc) inside the cavity which produces stimulated emmision. My thought experiment only involves a cavity/loop, no lasing, no stimulated emmission. The only apparatus inserted into the loop is intended to release the light from the loop by switching its polarization (a quarter wave plate).
Your other point about the q-factor going down when the cavity is on the scale of the wavelength is not necessarily significant, as I am only addressing the consideration of the cavity/loop length being shorter than the uncertainty(which is usually much longer than the wavelength).

This thought experiment is meant to address the laws of quantum foundations, not to produce lasing.
Is a photon(or any particle) able to be trapped inside a loop/cavity which has total length less than the uncertainty in position in the direction of propogation of the particle?

 

[Cthugha]

Q-switching only applies to the case of a pump laser which feeds a gain medium (gas, crystal,etc) inside the cavity which produces stimulated emmision. My thought experiment only involves a cavity/loop, no lasing, no stimulated emmission. The only apparatus inserted into the loop is intended to release the light from the loop by switching its polarization (a quarter wave plate).

Cavity dumping does not really care whether you have a laser medium inside or not. The mechanism works independently of whether you directly insert lots of light or have the build-up take place inside the cavity.

This thought experiment is meant to address the laws of quantum foundations, not to produce lasing.
Is a photon(or any particle) able to be trapped inside a loop/cavity which has total length less than the uncertainty in position in the direction of propogation of the particle?

This does not make much sense as it is not directly possible to assign position uncertainty to a photon. If you mean coherence length instead, then of course that is possible. Coming back to lasers, already the cavities in which laser light is created are usually much smaller than the coherence length of the light contained inside. Here, it is the effective cavity length (cavity length times average number of round trips the light will perform on average) which is the quantity of interest.

 

[al onestone]

OK, let me get this straight with respect to my original gedankenexperiment. I'm thinking of the possibility that an ensemble of photons can be trapped inside a microcavity for a period of time shorter than the cavity lifetime. Unlike a typical cavity in optics, this cavity is aligned to trap the photons forever, but the experimenter flips a switch which opens the cavity and the ensemble is released. In cavity dumping, is it assumed that the photons have a lower probability of escape during the build up, or is the cavity completely closed? If its completely closed then cavity dumping is what I'm describing.
It is an important point in my thought experiment, you have to imajine that the cavity is completely closed and when it is suddenly reopened the light can take no longer to escape the cavity than the amount of time needed to traverse one length of the loop. Therfore its coherence length/uncertainty in position/correlation length/etc. is shortened to the length of the loop right?

 

[Cthugha]

Yes, I think you are at least thinking of something similar to cavity dumping.

In theoretical treatments one indeed takes two mirrors with 100 % reflectivity which form a perfect cavity. Now one adds another output which normally has an output of 0%. In practice polarizing beam splitters or acousto-optical modulators are used.

If you use a polarizing beam splitter, you just insert a Pockels cell into your cavity which is basically just a waveplate that can rotate the direction of the polarization of your light as a voltage is applied. So you can just switch the Pockels cell on, the polarization of the light in the cavity rotates and the light will leave the cavity via the polarizing beam splitter. Ideally within one round trip.
For an AOM the principle is similar. Here the light will be diffracted at the modulator while it is switched on and therefore be directed to some output coupler.

In real experiments one has of course to consider that the switching is not instantaneous and the output coupling may not be 100% efficient, but in terms of a Gedankenexperiment one can neglect these things.

 

[al onestone]

Ok, I think that my querry is resolved with respect to the photonic/optical version of my thought experiment. If I understand correctly, the microcavity with the preparation you have prescribed will produce an output with a shortened pulse length equal to the length of the cavity. Consequently, Heisenberg uncertainty determines that the output pulse has a longer bandwidth to compensate.

Question : In quantum mechanics we would consider this a change in the state description, which must have a mathematical model, a unitary operator which acts on the original description to produce the final one. We can assume that we can produce an operator which models this change of state. But in QM we always describe a real physical coupling which produces a change in state, with an associated unitary operator. This is the Von Neumann measurement scheme. Are photons simply different from typical particles in QM because they don't require a real physical coupling in order to change state? Since the photons never actually become absorbed and re-emitted (they only interact with the mirrors, the pockells cell and the beam splitter which otherwise don't effect the bandwidth of photons) then there is no real physical Von Neumann measurement. What is the measurement apparatus that is being coupled to in this change of state? Is it safe to say that my confusion arises from treating photons with a strict QM expectation?

None the less, my total thought experiment includes entangled beams produced from something like parametric downconversion with polarization and momentum entanglement. If one of the two entangled beams is subject to the cavity dumping preparation, does the bandwidth of the second beam become widened also? or does a portion of the second beam become unentangled(like teleportation protocol)? Has anyone tried this in the literature?