Photonic Circuit Idea (Does this already exist?)

I have had a few ideas for circuits that I would like to get feedback on (do such things already exist, what utility they serve, etc).

Any suggestions on how I might graphically render these?

Example: Here i is the input & o is the output. The slash directly to the right of the input is a 50/50 beam splitter. The remaining slashes are mirrors.

If a pulse were directed in, the output would first recieve 50% of the beam. The other half of the beam would pass through into the reflective chamber. Upon completion of the loop, the beam would again reach the splitter. Again, half the beam (25%) would pass to the output while the other half (25%) looped thru the circuit.

This process could be run thru $n$ times until some desired outcome is reached.

Does this already exist? What is the utility (if any)?

• That circuit diagram is pretty incomprehensible when not displayed inline. Please can you make an edit with a hand drawn version or sequential description of the circuit. – SLesslyTall Jul 1 '18 at 12:29
• @SLessyTall added a photo as per your request – meowzz Jul 1 '18 at 16:03
• This is very similar to the Elitzur-Vaidman bomb test when run multiple times (although, there, they changed the reflectivities of the beam splitters) – DaftWullie Jul 2 '18 at 5:19

To answer your first, general question: Optical circuits are usually drawn with a selection of conventional symbols, a directory of which to draw them can be found here.

With respect to that specific circuit, if a single photon was input, the output would produce a state in a decaying superposition of subsequent time bins. If you choose a temporal basis for your output photons $|t_k\rangle$, taking $|t_0\rangle$ as the time bin of your the first pass through the beamsplitter and $|t_1\rangle$ as the time bin of the next pass, etc, then the output state will be given by $$\sum_{k=0}^\infty \left(\frac{e^{-i\phi}}{\sqrt{2}}\right)^{k+1}|t_k\rangle,$$ where $\phi$ is the phase shift imparted on each photon after a single pass of the loop.

Alternatively, if you only care about the input and output photons, this can be more generally be represented by the Bogoliubov transformation on the optical mode operators $$a^\dagger_t \mapsto \sum_{k=0}^\infty \left(\frac{e^{-i\phi}}{\sqrt{2}}\right)^{k+1} b^\dagger_{t+k}$$

where $a_t^\dagger$ and $b^\dagger_t$ are the creation operators for photons in time bin $t$ in the optical input and output modes respectively.

Personally, I haven't seen this sort of state before and don't know any particular use for it. However, there may be some use for it in some sort of strange loop-based architecture for linear optical quantum computing, although I would doubt it.

• "strange loop-based architecture for linear optical quantum computing" yes!! thanks for the link as well! – meowzz Jul 1 '18 at 17:04
• any ideas on how to represent this as a matrix? – meowzz Jul 1 '18 at 18:40
• This question got me thinking and I realised I have assumed the phase imparted on the photon during each pass was 1, which may not be the case, so note I have updated the answer to reflect this. Because the bases of the input and output are infinite-dimensional, I do not think a matrix representation would be appropriate here. However, one general approach with linear optics is to define your input state as a series of mode creation operators and apply a Bogoliubov transformation to see what your final state is. To clarify this, I have added the relevant mode transformation you seek. – SLesslyTall Jul 1 '18 at 21:35