Calculating Kerr parameters

Qcircuit.kerr(**kwargs)

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.