Pinpointing sweat spots for the Kerr-Cat Qubit

Introducing a detuning and a single photon drive to the Kerr-Cat Qubit alters its eigenstates to be not coherent states. 

However by displacing the original coherent states the eigenstates can be approximated in such a way that only a periodic deformation is present. 

If the period of the X rotation caused by the detuning is a multiple of the period of the deformation we can recover the original coherent state with high fidelity.

If however the deformation is maximal at the time we complete the X rotation the peak of our fidelity is cut off and only low fidelity rotations can be achieved.

Thus a careful choice of parameters is required.