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 selfKerr) of modem
. An offdiagonal componentK[m,n]
corresponds to the crossKerr coupling between modesm
andn
. The modes are indexed with increasing normal mode frequencies, for exampleK[0,1]
corresponds to the crossKerr 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 unspecified circuit components, ex:
L=1e9
. Returns
Kerr parameters
 Return type
numpy.array of dimension 2
Notes
The Hamiltonian of a circuit in first order perturbation theory is given by
\(\hat{H} = \sum_m\sum_{n\ne m} (\hbar\omega_mA_m\frac{\chi_{mn}}{2})\hat{a}_m^\dagger\hat{a}_m \frac{A_m}{2}\hat{a}_m^\dagger\hat{a}_m^\dagger\hat{a}_m\hat{a}_m \chi_{mn}\hat{a}_m^\dagger\hat{a}_m\hat{a}_n^\dagger\hat{a}_n\),
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.