# Calculating anharmonicities¶

Qcircuit.anharmonicities(**kwargs)[source]

Returns the anharmonicity of the circuit normal modes.

The array is ordered ordered with increasing normal mode frequencies such that the first element of the array corresponds to the anharmonicity of the lowest frequency mode. Anharmonicities are provided in units of Hertz, not in angular frequency.

Parameters

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

Returns

Normal mode anharmonicities

Return type

numpy.array

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 $$A_m$$.

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