Fiber Optics Bandwidth Question

[gxiv]

Hi everyone :]

This is my first time here. I have this Quantum Optics question (though I don't believe there is anything QM about this question, so please let me know if I've incorrectly placed this thread) as part of a (nonmarked) assignment.

Here's the question:
"What is the maximal bandwidth that the quantum source may emit single photons, such that after traveling the long optical fiber of 100km the chromatic dispersion effects allow time-bin encoding with 3 ns delay, given the wavelength is at the transmission optimum of 1550 nm?"I am.. completely lost. A little back story: I study post quantum cryptography and am funded by a post quantum/QKD research group which is quite fond of assigning deliverables on tangentially related material. I have absolutely no background in in QM/QKD/really physics in general.

I would appreciate any and all advice :)

 

[Cthugha]

Ok, this indeed little QM and mostly standard optics. So if you do post quantum, is your background mainly in math?

In order to get to the question: If you want to do time-bin encoding, you want some kind of trade-off. Obviously you do not want a broad emission in time as this will not allow you to reach 3 ns time bins. Intuitively, the opposite approach might be to use emission pulses that are as short as possible to ensure that time bin encoding works. However, there is a problem. Time and energy form a Fourier transform pair. So obviously a temporally narrow pulse needs to be spectrally broad (have a large bandwidth). This is not a problem in itself, but fibers do show chromatic dispersion. This means that the different spectral components will travel through a fiber at different velocities. So if you use a bandwidth that is very narrow, the effects of dispersion will not be very prominent, but the pulse duration will be very long because it is related to the bandwidth by a Fourier transform. If the bandwidth is very broad, the initial pulse will be pretty narrow, but it will spread quickly due to chromatic dispersion. Somewhere in between, there will be some optimal point. The assignment essentially asks you to find this optimal point.

This will depend on the kind of fiber used (which might have been mentioned in the assignment). Usually the dispersion of the Fiber is given in ps/(nm km), which is the temporal spreading per spectral width and fiber length, see e.g. the following link for details:
http://www.hke-auditech.cz/download/files/White-Paper_Dispersion-in-Optical-Fibers_PMD_CD_Ltr.pdf

If you know the details of the fiber, it will be quite easy to find a function that relates the initial bandwidth to the temporal pulse width after 100 km of fiber. Set that to 3 ns and you should be done.
If you were not given any details on the fiber, the assignment might just ask you to find the bandwidth for a truly dispersionless fiber. This means, you would just have to think about the minimum spectral width of a pulse that yields a 3 ns pulse due to time-energy uncertainty: ΔtΔE≥ħ/2.

 

[bhobba]

Aside from Cthugha's reply I suggest you remedy your lack of knowledge of QM if you are studying that area.

I don't know your background. If it's as I suspect, math, the following would be be appropriate (in that order):
Sakurai - Modern Quantum Mechancis:
https://www.amazon.com/dp/9332519005/?tag=pfamazon01-20
Balentine - QM - A Modern Development
https://www.amazon.com/dp/9332519005/?tag=pfamazon01-20

And since your background is I suspect math, I also suggest, because physicists can be a bit sloppy in their math, to help you with that - QM For Mathematicians (but after the other two please - you need to get the physical concepts right first):
https://www.amazon.com/dp/0821846302/?tag=pfamazon01-20

Thanks
Bill