Frequency Conversion code

[jviatteau]

Added code to solve equations of motion for frequency conversion based on the 2013 paper by Andreas Christ, "Theory of quantum frequency conversion and type-II parametric down-conversion in the high-gain regime".

Created new folder FrequencyConversion containing new code files.

Added FC_solve_EoM.py which provides functions to calculate the numerical solution to the FC equations of motion.

Added test_solve_EoM.py and example_solve_EoM.ipynb as tests and examples for the functions provided by FC_solve_EoM.py.

Added test_fermionic_bloch_messiah.py which provides tests for the fermionic Bloch-Messiah decomposition (to be added later).

[MHoude2]

Few comments for the "Fc_solve_EoM.py".

import numpy as jnp
from scipy.linalg import svd
from scipy.linalg import expm

why jnp and not just np?
Numpy has its own SVD, why use the scipy one?

If you are done loading things to the PR, you need to add reviewers. I can add Nico if you would like.

We should also probably not include the Fermionic Bloch-Messiah since it fails and wait until we get it to work properly.

[jviatteau]

I changed the jnp back to np. That was an artifact from the time when I was asked to use jax in my code, so that you could run the code faster on GPUs. I switched back to ordinary numpy (without switching jnp to np) when I needed to work with double precision with the fermionic bloch-messiah function.

I also replaced the scipy svd by the numpy one.

I removed fermionic_bloch_messiah.py but kept the test file, so you have it somewhere when the time comes to test it.

Also, I may not have the permissions or something, but it seems I can't add reviewers myself. I would then be grateful if you could indeed add Nicolas as reviewer.

[nquesada]

Thsi removal of phases in incorrect. You need to multiply phases on both sides.

[MHoude2]

As Nico said, this is incorrect. The correct method is outlined in https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.033519, paragraph between Eq.65 and 66. I also do it in my "phases" function in 'propagator.py'. Note that these are for SPDC where the idler mode is ^\dagger. For F.C. one needs to take the opposite sign for that mode.

[jviatteau]

Yes, I remember thinking about that and looking at it when looking for how to remove the phases. But then, I thought that since I have my waveguide start at z=0 and finish at z=L I only had phases on one side, since my z_0 would be 0. Is that incorrect or is there a profound reason that I should have z_0 = -L/2, z_1 = L/2, or is it just a convention? In any case, I can change it for sure, just so it remains consistent with what you have in propagator.py.

[MHoude2]

Right ok, yeah we should keep it all consistent.

[MHoude2]

This way of generating the interaction matrix is incorrect. By combining everything into this 'D' you are omitting a \Delta \omega term which depends on the step size used. This term comes from discretizing the integral into a sum. (see Sec.4 of https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.033519, more precisely Eq. 59a for this off diagonal term). If you also look at my function "Hprop" in "propagator.py" I have a very convoluted way of expressing that \Delta \omega.

[MHoude2]

You will also need to divide the JCA by \Delta \omega. (See line 161 in propagator.py)

[MHoude2]

This seems to be L1 normalization, whereas we want to use L2 normalization. Denominator should be (pi*sig**2)^(1/4)