Frequency Conversion code #11
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…onversion, with its test and example files
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Few comments for the "Fc_solve_EoM.py". import numpy as jnp why jnp and not just np? 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. |
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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 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. |
FrequencyConversion/FC_solve_EoM.py
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| Uc = np.diag(np.exp(1j * dw * kc * L)) @ Uc | ||
| Vc = np.diag(np.exp(1j * dw * kc * L)) @ Vc | ||
| Va = np.diag(np.exp(-1j * dw * ka * L)) @ Va | ||
| Ua = np.diag(np.exp(-1j * dw * ka * L)) @ Ua | ||
| return np.block([[Ua, Va], [Vc, Uc]]) | ||
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Thsi removal of phases in incorrect. You need to multiply phases on both sides.
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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.
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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.
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Right ok, yeah we should keep it all consistent.
FrequencyConversion/FC_solve_EoM.py
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| Kc = np.diag(1j * dw * kc) | ||
| # Bottom left and upper right (non-diagonal) blocks | ||
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| J = jnp.block([[Ka, -1j * jnp.conj(D) * pump], [-1j * D * pump.conj().T, Kc]]) |
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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.
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You will also need to divide the JCA by \Delta \omega. (See line 161 in propagator.py)
FrequencyConversion/FC_solve_EoM.py
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| Returns: | ||
| array: pulse shape | ||
| """ | ||
| gaussian = np.exp(-(dw**2) / (2 * sig**2)) / np.sqrt(2 * np.pi * sig**2) |
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This seems to be L1 normalization, whereas we want to use L2 normalization. Denominator should be (pi*sig**2)^(1/4)
… L2 normalization for pump pulse
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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
FrequencyConversioncontaining new code files.Added
FC_solve_EoM.pywhich provides functions to calculate the numerical solution to the FC equations of motion.Added
test_solve_EoM.pyandexample_solve_EoM.ipynbas tests and examples for the functions provided byFC_solve_EoM.py.Added
test_fermionic_bloch_messiah.pywhich provides tests for the fermionic Bloch-Messiah decomposition (to be added later).