Full Zero-Pi Qubit (incl. coupling to zeta mode)

../../_images/fullzeropi.png

The full Zero-Pi qubit [Brooks2013] [Dempster2014] includes the coupling of the primary qubit degrees of freedom to the zeta mode. It is described by the Hamiltonian \(H = H_{0-\pi} + H_\text{int} + H_\zeta\), where

\[\begin{split}H_{0-\pi} =& -2E_\text{CJ}\partial_\phi^2+2E_{\text{C}\Sigma}(i\partial_\theta-n_g)^2 +2E_{C\Sigma}dC_J\,\partial_\phi\partial_\theta\\ & -2E_\text{J}\cos\theta\cos(\phi-\varphi_\text{ext}/2)+E_L\phi^2+2E_\text{J} + E_J dE_J \sin\theta\sin(\phi-\varphi_\text{ext}/2) \\ H_\text{int} =& 2E_{C\Sigma}dC\,\partial_\theta\partial_\zeta + E_L dE_L \phi\,\zeta\\ H_\zeta =& \omega_\zeta a^\dagger a\end{split}\]

expressed in the 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 \(dC_\text{J}\), \(dE_\text{J}\), \(dC\), \(dE_\text{L}\) follows [Groszkowski2018]. Internally, the FullZeroPi class formulates the Hamiltonian matrix via the product basis of the decoupled Zero-Pi qubit (realized by ZeroPi) on one hand, and the zeta LC oscillator on the other hand.

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

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

zero_pi = scqubits.FullZeroPi(grid = phi_grid,
                             EJ   = 0.25,
                             EL   = 10.0**(-2),
                             ECJ  = 0.5,
                             EC   = 0.001,
                             dEJ  = 0.05,
                             dCJ  = 0.05,
                             dEL  = 0.01,
                             dC   = 0.08,
                             ng   = 0.1,
                             flux = 0.23,
                             ncut = 30,
                             zeropi_cutoff = 10,
                             zeta_cutoff = 30)

Here, flux is given in in terms 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, an instance of the disordered Zero-Pi qubit can alternatively be created with:

zero_pi = scqubits.ZeroPiFull.create()

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

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