Generating QuTiP hamiltonian

Qcircuit.hamiltonian(modes='all', taylor=4, excitations=6, return_ops=False, **kwargs)[source]

Returns the circuits Hamiltonian for further analysis with QuTiP. The Hamiltonian is provided in units of frequency (not angular frequency), such that \(h=1\).

Parameters:

  • modes (array of integers, optional) – List of modes to consider, where the modes are ordered with increasing frequency, such that modes = [0,1] would lead to considering only the two lowest frequency modes of the circuit. By default all modes are considered.

  • taylor (integer, optional) – Order to which the potential of all josephson junctions should be taylor-expanded. Default is 4.

  • excitations (integer or array of integers, optional) – Number of energy levels considered for each junction. If one number is given, all modes have the same number of levels, if an array is given, its length should match the number of modes considered. For example if modes = [0,1] and excitations = [5,10], then we will consider 5 excitation levels for mode 0 and 10 for mode 1.

  • return_ops (Boolean, optional) – If set to True, a list of the annihilation operators will be returned along with the hamiltonian in the form <Hamiltonian>, <list of operators>. The form of the return is then H,[a_0,a_1,..] where a_i is the annihilation operator of the i-th considered mode, a QuTiP Qobj

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

Returns:

Hamiltonian of the circuit

Return type:

qutip.qobj

Notes

The Hamiltonian of the circuit, with the non-linearity of the Josephson junctions Taylor-expanded, is given in the limit of low dissipation by

\(\hat{H} = \sum_{m\in\text{modes}} \hbar \omega_m\hat{a}_m^\dagger\hat{a}_m + \sum_j\sum_{2n\le\text{taylor}}E_j\frac{(-1)^{n+1}}{(2n)!}\left[\frac{\phi_{zpf,m,j}}{\phi_0}(\hat{a}_m^\dagger+\hat{a}_m)\right]^{2n}\),

where \(\hat{a}_m\) is the annihilation operator of the m-th normal mode of the circuit, \(\omega_m\) is the frequency of the m-th normal mode, \(E_j\) is the Josephson energy of the j-th junction and \(\phi_0 = \hbar/2e\) and \(\phi_{zpf,m,j}\) is the zero point fluctuation of mode \(m\) through junction \(j\).

In the expression above, modes and taylor are arguments of the hamiltonian function.

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