Ferbo

The Fermionic-Bosonic (FerBo) qubit: a hybrid semiconductor-superconductor circuit coupling Andreev bound states in a weak link to the bosonic mode of an LC oscillator.

See Theory background for the physics background and the companion paper for a detailed treatment.

class HybridSuperQubits.ferbo.Ferbo(Ec, El, Ej, Gamma, delta_Gamma, er, phase, dimension, flux_grouping='ABS', Delta=40)[source]

Bases: QubitBase

Parameters:

flux_grouping (Literal['EL', 'ABS'])

PARAM_LABELS: dict[str, str] = {'Ec': '$E_C$', 'Ej': '$E_J$', 'El': '$E_L$', 'Gamma': '$\\Gamma$', 'delta_Gamma': '$\\delta \\Gamma$', 'er': '$\\epsilon_r$', 'phase': '$\\Phi_{\\mathrm{ext}} / \\Phi_0$'}
OPERATOR_LABELS: dict[str, str] = {'d_hamiltonian_d_EL': '\\partial \\hat{H} / \\partial E_L', 'd_hamiltonian_d_deltaGamma': '\\partial \\hat{H} / \\partial \\delta \\Gamma', 'd_hamiltonian_d_er': '\\partial \\hat{H} / \\partial \\epsilon_r', 'd_hamiltonian_d_ng': '\\partial \\hat{H} / \\partial n_g', 'd_hamiltonian_d_phase': '\\partial \\hat{H} / \\partial \\phi_{{ext}}', 'n_operator': '\\hat{n}', 'phase_operator': '\\hat{\\phi}'}
__init__(Ec, El, Ej, Gamma, delta_Gamma, er, phase, dimension, flux_grouping='ABS', Delta=40)[source]

Initializes the Ferbo class with the given parameters.

Parameters:

  • Ec (float) – Charging energy.

  • El (float) – Inductive energy.

  • Ej (float) – Josephson energy.

  • Gamma (float) – Coupling strength.

  • delta_Gamma (float) – Coupling strength difference.

  • er (float) – Energy relaxation rate.

  • phase (float) – External magnetic phase.

  • dimension (int) – Dimension of the Hilbert space.

  • flux_grouping (Literal["EL", "ABS"], optional) – Flux grouping (‘EL’ or ‘ABS’) (default is ‘ABS’).

  • Delta (float) – Superconducting gap.

property phase_zpf: float

Returns the zero-point fluctuation of the phase.

Returns:

Zero-point fluctuation of the phase.

Return type:

float

property n_zpf: float

Returns the zero-point fluctuation of the charge number.

Returns:

Zero-point fluctuation of the charge number.

Return type:

float

property lc_energy: float

Returns the plasma energy.

Returns:

Plasma energy.

Return type:

float

property transparency: float

Return the transparency of the weak link.

Returns:

Transparency of the weak link.

Return type:

float

phi_osc()[source]

Returns the oscillator length for the LC oscillator composed of the inductance and capacitance.

Returns:

Oscillator length.

Return type:

float

n_operator()[source]

Returns the charge number operator.

Returns:

The charge number operator.

Return type:

np.ndarray

phase_operator()[source]

Returns the total phase operator.

Returns:

The total phase operator.

Return type:

np.ndarray

jrl_potential()[source]

Returns the Josephson Resonance Level potential.

Returns:

The Josephson Resonance Level potential.

Return type:

np.ndarray

hamiltonian()[source]

Returns the Hamiltonian of the system.

Returns:

The Hamiltonian of the system.

Return type:

np.ndarray

d_hamiltonian_d_EC()[source]

Returns the derivative of the Hamiltonian with respect to the charging energy.

Returns:

The derivative of the Hamiltonian with respect to the charging energy.

Return type:

np.ndarray

d_hamiltonian_d_EL()[source]

Returns the derivative of the Hamiltonian with respect to the inductive energy.

Returns:

The derivative of the Hamiltonian with respect to the inductive energy.

Return type:

np.ndarray

d_hamiltonian_d_EJ()[source]

Returns the derivative of the Hamiltonian with respect to the Josephson energy.

Returns:

The derivative of the Hamiltonian with respect to the Josephson energy.

Return type:

np.ndarray

d_hamiltonian_d_Gamma()[source]

Returns the derivative of the Hamiltonian with respect to Gamma.

Returns:

The derivative of the Hamiltonian with respect to Gamma.

Return type:

np.ndarray

d_hamiltonian_d_er()[source]

Returns the derivative of the Hamiltonian with respect to the energy relaxation rate.

Returns:

The derivative of the Hamiltonian with respect to the energy relaxation rate.

Return type:

np.ndarray

d_hamiltonian_d_deltaGamma()[source]

Returns the derivative of the Hamiltonian with respect to the coupling strength difference.

Returns:

The derivative of the Hamiltonian with respect to the coupling strength difference.

Return type:

np.ndarray

d2_hamiltonian_d_Gamma2()[source]

Returns the second derivative of the Hamiltonian with respect to Gamma.

Returns:

The second derivative of the Hamiltonian with respect to Gamma.

Return type:

np.ndarray

d2_hamiltonian_d_deltaGamma2()[source]

Returns the second derivative of the Hamiltonian with respect to the coupling strength difference.

Returns:

The second derivative of the Hamiltonian with respect to the coupling strength difference.

Return type:

np.ndarray

d2_hamiltonian_d_er2()[source]

Returns the second derivative of the Hamiltonian with respect to the energy relaxation rate.

Returns:

The second derivative of the Hamiltonian with respect to the energy relaxation rate.

Return type:

np.ndarray

d_hamiltonian_d_ng()[source]

Returns the derivative of the Hamiltonian with respect to the number of charge offset.

Returns:

The derivative of the Hamiltonian with respect to the number of charge offset.

Return type:

np.ndarray

d2_hamiltonian_d_ng2()[source]

Returns the second derivative of the Hamiltonian with respect to the number of charge offset.

Returns:

The second derivative of the Hamiltonian with respect to the number of charge offset.

Return type:

np.ndarray

d_hamiltonian_d_phase()[source]

Returns the derivative of the Hamiltonian with respect to the external magnetic phase.

Returns:

The derivative of the Hamiltonian with respect to the external magnetic phase.

Return type:

np.ndarray

d2_hamiltonian_d_phase2()[source]

Returns the second derivative of the Hamiltonian with respect to the external magnetic phase.

Returns:

The second derivative of the Hamiltonian with respect to the external magnetic phase.

Return type:

np.ndarray

wigner(which=0, phi_grid=None, n_grid=None, esys=None)[source]

Computes the Wigner function for a given wavefunction.

Parameters:

  • which (int, optional) – Index of desired wavefunction (default is 0).

  • phi_grid (np.ndarray, optional) – Custom grid for phi; if None, a default grid is used.

  • n_grid (np.ndarray, optional) – Custom grid for n; if None, a default grid is used.

  • esys (Tuple[np.ndarray, np.ndarray], optional) – Precomputed eigenvalues and eigenvectors.

Returns:

The Wigner function.

Return type:

np.ndarray

reduced_density_matrix(which=0, esys=None, subsys=0)[source]

Computes the reduced density matrix for a given wavefunction.

Parameters:

  • which (int, optional) – Index of desired wavefunction (default is 0).

  • esys (Tuple[np.ndarray, np.ndarray], optional) – Precomputed eigenvalues and eigenvectors.

  • subsys (int, optional) – Subsystem to compute the reduced density matrix for (default is 0). 0 for the tracing out the Fock states, 1 for the Andreev states.

Returns:

The reduced density matrix.

Return type:

np.ndarray

wavefunction(which=0, phi_grid=None, esys=None, basis='phase', representation='ballistic')[source]

Returns a wave function in the phi basis.

Parameters:

  • which (int, optional) – Index of desired wave function (default is 0).

  • phi_grid (np.ndarray, optional) – Custom grid for phi; if None, a default grid is used.

  • basis (Literal["phase", "charge"], optional) – Basis in which to return the wavefunction (‘phase’ or ‘charge’) (default is ‘phase’).

  • representation (Literal["ballistic", "Andreev"], optional) – Representation used for the two-component wavefunction. Use ‘ballistic’ for the basis used to define the Hamiltonian matrix, or ‘Andreev’ for the local phi-dependent eigenbasis of the Andreev sector.

  • esys (tuple[ndarray, ndarray] | None)

Returns:

Wave function data containing basis labels, amplitudes, and energy.

Return type:

Dict[str, Any]

potential(phi, return_evecs=False, include_berry=True)[source]

Calculates the potential energy for given values of phi.

Parameters:

  • phi (Union[float, np.ndarray]) – The phase values at which to calculate the potential.

  • return_evecs (bool, optional) – If True, returns both eigenvalues and eigenvectors (default is False).

  • include_berry (bool, optional) – If True, includes the scalar Berry correction in the potential eigenvalues (default is True).

Returns:

If return_evecs is False: The potential energy values (eigenvalues). If return_evecs is True: A tuple of (eigenvalues, eigenvectors). Eigenvalues shape: (len(phi), 2) Eigenvectors shape: (len(phi), 2, 2) where each [i, :, :] is the rotation matrix at phi[i]

Return type:

Union[np.ndarray, Tuple[np.ndarray, np.ndarray]]

tphi_1_over_f_flux(A_noise=1e-06, esys=None, get_rate=False, **kwargs)[source]
Parameters:
Return type:

float

plot_wavefunction(which=0, phi_grid=None, esys=None, scaling=1, plot_potential=False, basis='phase', andreev_basis='ballistic', mode='abs', **kwargs)[source]

Plot the wave function in the phi basis.

Parameters:

  • which (Union[int, Iterable[int]], optional) – Index or indices of desired wave function(s) (default is 0).

  • phi_grid (np.ndarray, optional) – Custom grid for phi; if None, a default grid is used.

  • esys (Tuple[np.ndarray, np.ndarray], optional) – Precomputed eigenvalues and eigenvectors.

  • scaling (float, optional) – Scaling factor for the wavefunction (default is 1).

  • plot_potential (bool, optional) – Whether to plot the potential (default is False).

  • basis (Literal["phase", "charge"], optional) – Basis in which to return the wavefunction (‘phase’ or ‘charge’) (default is ‘phase’).

  • andreev_basis (Literal["ballistic", "Andreev"], optional) – Andreev basis used for plotting. Use ‘ballistic’ for the basis used to define the Hamiltonian matrix, or ‘Andreev’ for the local phi-dependent Andreev eigenbasis.

  • mode (Literal["abs", "abs2", "real", "imag"], optional) – Mode of the wavefunction (‘abs’, ‘abs2’, ‘real’, or ‘imag’) (default is ‘abs’).

  • **kwargs –

    Additional arguments for plotting. Can include: - fig_ax: Tuple[plt.Figure, plt.Axes], optional

    Figure and axes to use for plotting. If not provided, a new figure and axes are created.

Returns:

The figure and axes of the plot.

Return type:

Tuple[plt.Figure, plt.Axes]

plot_state(which=0, phi_grid=None, n_grid=None, wigner_func=False, esys=None, plot_bloch=False, **kwargs)[source]

Plot the Wigner function of the state and the Bloch sphere.

Parameters:

  • which (int, optional) – Index of desired wavefunction (default is 0).

  • phi_grid (np.ndarray, optional) – Custom grid for phi; if None, a default grid is used.

  • n_grid (np.ndarray, optional) – Custom grid for n; if None, a default grid is used.

  • wigner_func (bool, optional) – Precomputed wigner_func function (default is False).

  • esys (Tuple[np.ndarray, np.ndarray], optional) – Precomputed eigenvalues and eigenvectors.

  • plot_bloch (bool, optional) – Whether to plot the Bloch sphere (default is False).

  • **kwargs (dict, optional) –

    Additional arguments for plotting. Can include: - fig_ax: Tuple[plt.Figure, plt.Axes], optional

    Figure and axes to use for plotting. If not provided, a new figure and axes are created.

    • cmap: str, optional

      Colormap to use for the Wigner function (default is ‘seismic’).

    • bloch_view: Tuple[float, float], optional

      Tuple with (elevation, azimuth) for Bloch sphere view (default is (-30, 60)).

    • bloch_position: Tuple[float, float, float, float], optional

      Position of the Bloch sphere inset in figure coordinates (left, bottom, width, height). If not provided, a default position is calculated.

Returns:

The figure and axes of the plot.

Return type:

Tuple[plt.Figure, plt.Axes]