# Calculating Kerr parameters¶

Qcircuit.kerr(**kwargs)[source]

Returns the Kerr parameters for the circuit normal modes.

The diagonal component K[m,m] of the returned matrix correspond to the anharmonicity (or self-Kerr) of mode m. An off-diagonal component K[m,n] corresponds to the cross-Kerr coupling between modes m and n. The modes are indexed with increasing normal mode frequencies, for example K[0,1] corresponds to the cross-Kerr interaction between the lowest frequency mode and next highest frequency mode. Kerr parameters are provided in units of Hertz, not in angular frequency.

Parameters

kwargs – Values for un-specified circuit components, ex: L=1e-9.

Returns

Kerr parameters

Return type

numpy.array of dimension 2

Notes

The Hamiltonian of a circuit in first order perturbation theory is given by

$$\hat{H} = \sum_m\sum_{n\ne m} (\hbar\omega_m-A_m-\frac{\chi_{mn}}{2})\hat{a}_m^\dagger\hat{a}_m -\frac{A_m}{2}\hat{a}_m^\dagger\hat{a}_m^\dagger\hat{a}_m\hat{a}_m -\chi_{mn}\hat{a}_m^\dagger\hat{a}_m\hat{a}_n^\dagger\hat{a}_n$$,

valid for weak anharmonicity $$\chi_{mn},A_m\ll \omega_m$$.

Here

• $$\omega_m$$ are the frequencies of the normal modes of the circuit where all junctions have been replaced with inductors characterized by their Josephson inductance

• $$A_m$$ is the anharmonicity of mode $$m$$ , the difference in frequency of the first two transitions of the mode

• $$\chi_{mn}$$ is the shift in mode $$m$$ that incurs if an excitation is created in mode $$n$$

This function returns the values of $$A_m$$ and $$\chi_{mn}$$ .

For more information on the underlying theory, see https://arxiv.org/pdf/1908.10342.pdf.