Fluxonium

The Hamiltonian of the fluxonium qubit [Manucharyan2009] in phase basis representation is given by

\[H=-4E_\text{C}\partial_\phi^2-E_\text{J}\cos(\phi-\varphi_\text{ext}) +\frac{1}{2}E_L\phi^2.\]

Here, \(E_C\) is the charging energy, \(E_J\) the Josephson energy, \(E_L\) the inductive energy, and \(\varphi_\text{ext}=2\pi \Phi_\text{ext}/\Phi_0\) the external flux in dimensionless form. The Fluxonium class internally uses the \(E_C\)-\(E_L\) harmonic-oscillator basis [Zhu2013] with truncation level specified by cutoff.

An instance of the fluxonium qubit is created as follows:

fluxonium = scqubits.Fluxonium(EJ = 8.9,
                               EC = 2.5,
                               EL = 0.5,
                               flux = 0.33,
                               cutoff = 110)

Here, the flux threading the circuit loop is specified by flux which records the flux in units of the magnetic flux quantum, \(\Phi_\text{ext}/\Phi_0\).

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

fluxonium = scqubits.Fluxonium.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

scqubits.Fluxonium.hamiltonian([energy_esys])	Constructs Hamiltonian matrix in harmonic-oscillator, following Zhu et al., PRB 87, 024510 (2013), or eigenenergy basis.
scqubits.Fluxonium.eigenvals([evals_count, ...])	Calculates eigenvalues using scipy.linalg.eigh, returns numpy array of eigenvalues.
scqubits.Fluxonium.eigensys([evals_count, ...])	Calculates eigenvalues and corresponding eigenvectors using scipy.linalg.eigh.
scqubits.Fluxonium.get_spectrum_vs_paramvals(...)	Calculates eigenvalues/eigenstates for a varying system parameter, given an array of parameter values.

Wavefunctions and visualization of eigenstates

scqubits.Fluxonium.wavefunction(esys[, ...])	Returns a fluxonium wave function in phi basis
scqubits.Fluxonium.plot_wavefunction([...])	Plot 1d phase-basis wave function(s).

Implemented operators

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

scqubits.Fluxonium.n_operator([energy_esys])	Returns the \(n = - i d/d\phi\) operator in the LC harmonic oscillator or eigenenergy basis.
scqubits.Fluxonium.phi_operator([energy_esys])	Returns the phi operator in the LC harmonic oscillator or eigenenergy basis.
scqubits.Fluxonium.exp_i_phi_operator([...])	Returns the \(e^{i (\alpha \phi + \beta) }\) operator, with \(\alpha\) and \(\beta\) being numbers, in the LC harmonic oscillator or eigenenergy basis.
scqubits.Fluxonium.cos_phi_operator([alpha, ...])	Returns the \(\cos (\alpha \phi + \beta)\) operator with \(\alpha\) and \(\beta\) being numbers, in the LC harmonic oscillator or eigenenergy basis.
scqubits.Fluxonium.sin_phi_operator([alpha, ...])	Returns the \(\sin (\alpha \phi + \beta)\) operator with \(\alpha\) and \(\beta\) being numbers, in the LC harmonic oscillator or eigenenergy basis.

Computation and visualization of matrix elements

scqubits.Fluxonium.matrixelement_table(operator)	Returns table of matrix elements for operator with respect to the eigenstates of the qubit.
scqubits.Fluxonium.plot_matrixelements(operator)	Plots matrix elements for operator, given as a string referring to a class method that returns an operator matrix.
scqubits.Fluxonium.get_matelements_vs_paramvals(...)	Calculates matrix elements for a varying system parameter, given an array of parameter values.
scqubits.Fluxonium.plot_matelem_vs_paramvals(...)	Generates a simple plot of a set of eigenvalues as a function of one parameter.

Estimation of coherence times

scqubits.Fluxonium.plot_coherence_vs_paramvals(...)	Show plots of coherence for various channels supported by the qubit as they vary as a function of a changing parameter.
scqubits.Fluxonium.plot_t1_effective_vs_paramvals(...)	Plot effective \(T_1\) coherence time (rate) as a function of changing parameter.
scqubits.Fluxonium.plot_t2_effective_vs_paramvals(...)	Plot effective \(T_2\) coherence time (rate) as a function of changing parameter.
scqubits.Fluxonium.t1(i, j, noise_op, ...[, ...])	Calculate the transition time (or rate) using Fermi's Golden Rule due to a noise channel with a spectral density spectral_density and system noise operator noise_op.
scqubits.Fluxonium.t1_capacitive([i, j, ...])	\(T_1\) due to dielectric dissipation in the Josephson junction capacitances.
scqubits.Fluxonium.t1_charge_impedance([i, ...])	Noise due to charge coupling to an impedance (such as a transmission line).
scqubits.Fluxonium.t1_effective([...])	Calculate the effective \(T_1\) time (or rate).
scqubits.Fluxonium.t1_flux_bias_line([i, j, ...])	Noise due to a bias flux line.
scqubits.Fluxonium.t1_inductive([i, j, ...])	\(T_1\) due to inductive dissipation in a superinductor.
scqubits.Fluxonium.t1_quasiparticle_tunneling([...])	Noise due to quasiparticle tunneling across a Josephson junction.
scqubits.Fluxonium.t2_effective([...])	Calculate the effective \(T_2\) time (or rate).
scqubits.Fluxonium.tphi_1_over_f(A_noise, i, ...)	Calculate the 1/f dephasing time (or rate) due to arbitrary noise source.
scqubits.Fluxonium.tphi_1_over_f_cc([...])	Calculate the 1/f dephasing time (or rate) due to critical current noise.
scqubits.Fluxonium.tphi_1_over_f_flux([...])	Calculate the 1/f dephasing time (or rate) due to flux noise.