Zero-Pi Qubit (decoupled from zeta mode)

The Zero-Pi qubit [Brooks2013] [Dempster2014], when decoupled from the zeta mode, is described by the Hamiltonian

\[\begin{split}H &= -2E_\text{CJ}\partial_\phi^2+2E_{\text{C}\Sigma}(i\partial_\theta-n_g)^2 +2E_{C\Sigma}dC_J\,\partial_\phi\partial_\theta\\ &\qquad -2E_\text{J}\cos\theta\cos(\phi-\varphi_\text{ext}/2)+E_L\phi^2+2E_\text{J} + E_J dE_J \sin\theta\sin(\phi-\varphi_\text{ext}/2)\end{split}\]

expressed in phase basis. The definition of the relevant charging energies \(E_\text{CJ}\), \(E_{\text{C}\Sigma}\), Josephson energies \(E_\text{J}\), inductive energies \(E_\text{L}\), and relative amounts of disorder \(dC_\text{J}\), \(dE_\text{J}\) follows [Groszkowski2018].

Internally, the ZeroPi class formulates the Hamiltonian matrix by discretizing the phi variable, and using charge basis for the theta variable.

An instance of the Zero-Pi qubit is created as follows:

phi_grid = scqubits.Grid1d(-6*np.pi, 6*np.pi, 200)

zero_pi = scqubits.ZeroPi(grid = phi_grid,
                           EJ   = 0.25,
                           EL   = 10.0**(-2),
                           ECJ  = 0.5,
                           EC   = None,
                           ECS  = 10.0**(-3),
                           ng   = 0.1,
                           flux = 0.23,
                           ncut = 30)

Here, flux is given in units of the flux quantum, i.e., in the form \(\Phi_\text{ext}/\Phi_0\). In the above example, the disorder parameters dEJ and dCJ are not specified, and hence take on the default value zero (no disorder).

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

zero_pi = scqubits.ZeroPi.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

Implemented operators

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

Computation and visualization of matrix elements

Utility method for setting charging energies

Utility method for setting charging energies

Estimation of coherence times