Toolbox-for-Nonreciprocal-Dispersive-models-in-cQED

Codes to reproduce main results of arxiv.2312.08354. You will need access to Mathematica to reproduce the results.

Any comments or questions regarding the codes please reach Lautaro at labl2714@usherbrooke.ca

Example with Capacitive Filter

The responses were obtained with the Mathematica Notebook "Example-with-capacitor-filter-response-chirality-condition.nb". Also there, the chirality condition given by Eq. (9) was obtained leading to Eqs (22,23). Then, the analytical expressions were typed onto Python functions, and used to get all relevant matrices developed in appendices C and D to construct both the effective and exact Hamiltonians. In this case we found more convenient to use the admittance response, but any of the two is equally valid and are equivalent in the dispersive regime.

Example with resonator filter

The analytical form of the response was obtained with the Mathematica Notebook "Example-resonator-filter-analytical-response", saved as a .wbl file. Using this, the series decomposition Eq. (A1) is numerically obtained from Mathematica. We found the TransferFunction class of Mathematica more accurate than Sympy's equivalent. Then, the values are read and the quantum dynamics are obtained by truncating the Hilbert space to a maximum of 4 photons (5 to check convergence). The Impedance response was used, as it was found to be a convenient choice, due to the simplification of not having to account for poles at infinity, but once again, both approaches are valid and equivalent in the dispersive regime.

The workflow is: Input values in "Example-with-resonator-filter" and save them as a values file. Then, run "Example-resonator-filter-get-poles-As-Bs-wjchiral", which will also give you the frequency of the qubits $\overline{w}_i$ of Eq. (6) to get the desired hopping dynamics. Finally, run the whole of "Example-with-resonator-filter" and you will get the plots of the resulting dynamics.

Appendix-Scattering-Responses

Code to reproduce Fig. 13 in Appendix G5. As above, the impedance response is obtained in mathematica and then loaded onto the python. The workflow is: choose parameters in the python notebook, run the Mathematica Notebook "Example-resonator-filter-get-poles-As-Bs-wjchiral", then run all the python notebook.

To compare with the simplified effective response, we use Eq. (8) to get the effective coupling between the filter resonators. This is done computing the impedance seen from the resonators, which is in form equivalent to the one shown in Fig. 2 (i,b) with $Cg\to 0$ .