Flux Qubit

The flux qubit [Orlando1999] is described by the Hamiltonian

\[\begin{split}H_\text{flux}=&(n_{i}-n_{gi})4(E_\text{C})_{ij}(n_{j}-n_{gj}) \\ -&E_{J}\cos\phi_{1}-E_{J}\cos\phi_{2} \\ -&\alpha E_{J}\cos(2\pi f + \phi_{1} - \phi_{2}),\end{split}\]

where \(i,j \in \{1,2\}, E_\text{C}=\tfrac{e^2}{2}C^{-1}\) and

\[\begin{split}C = \left(\begin{matrix} C_{J1}+C_{g1}+C_{J3} & -C_{J3} \\ -C_{J3} & C_{J2}+C_{g2}+C_{J3} \end{matrix}\right).\end{split}\]

\(C_{Ji}\) refers to the capacitance of the \(i^\text{th}\) junction and \(C_{gi}\) refers to the capacitance to ground of the \(i^\text{th}\) island. For simplicity, the Hamiltonian is written here in a mixed basis, however for the purposes of numerical diagonalization in the FluxQubit class, the charge basis is employed for both variables. The user must specify a charge-number cutoff ncut, chosen large enough so that convergence is achieved.

An instance of the flux qubit is initialized as follows:

EJ = 35.0
alpha = 0.6
fluxqubit = scqubits.FluxQubit(EJ1 = EJ,
                              EJ2 = EJ,
                              EJ3 = alpha*EJ,
                              ECJ1 = 1.0,
                              ECJ2 = 1.0,
                              ECJ3 = 1.0/alpha,
                              ECg1 = 50.0,
                              ECg2 = 50.0,
                              ng1 = 0.0,
                              ng2 = 0.0,
                              flux = 0.5,
                              ncut = 10)

From within Jupyter notebook, a flux qubit instance can alternatively be created with:

fluxqubit = scqubits.FluxQubit.create()

This functionality is enabled if the ipywidgets package is installed, and displays GUI forms prompting for the entry of the required parameters.

Calculational methods related to Hamiltonian and energy spectra

Wavefunctions and visualization of eigenstates and the potential

Implemented operators

The following operators are implemented for use in matrix element calculations.

Computation and visualization of matrix elements

Estimation of coherence times