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 taylorexpanded. 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]
andexcitations = [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 thenH,[a_0,a_1,..]
wherea_i
is the annihilation operator of the ith considered mode, a QuTiP Qobjkwargs – Values for unspecified circuit components, ex:
L=1e9
.
 Returns
Hamiltonian of the circuit
 Return type
qutip.qobj
Notes
The Hamiltonian of the circuit, with the nonlinearity of the Josephson junctions Taylorexpanded, 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 mth normal mode of the circuit, \(\omega_m\) is the frequency of the mth normal mode, \(E_j\) is the Josephson energy of the jth 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
andtaylor
are arguments of thehamiltonian
function.For more details on the underlying theory, see https://arxiv.org/pdf/1908.10342.pdf