fluxonium.m

Fluxonium Hamiltonian in the truncated basis of the harmonic oscillator formed by the charging and the inductive energies of the circuit (Y. Lu, Optimal Control and Coherence Engineering for Superconducting Qubits, PhD thesis, Northwestern University, 2026, Eq. 2.30):

    H=4*ec*n^2-ej*cos(phi-phi_e)+(el/2)*phi^2,   [phi,n]=1i

where ec is the charging energy, ej is the Josephson energy, el=(Phi_0/2*pi)^2/L is the inductive energy of the shunt inductance L (Eq. 2.31, the factor 1/2 is explicit in the potential and is not absorbed into el), and phi_e=2*pi*Phi_e/Phi_0 is the reduced external flux (Eq. 2.32). Eq. 2.30 has the inductive term as (el/2)*(phi+phi_e)^2 and the Josephson term as -ej*cos(phi); the form above is its image under the translation of the phase origin to the minimum of the inductive potential, which does not change the spectrum. The offset charge of Eq. 2.30 is dropped because the unbounded phase variable makes it removable by a gauge transformation. Phase and charge are built from the ladder operators of the linear oscillator 4*ec*n^2+(el/2)*phi^2 with the plasma frequency sqrt(8*ec*el) as

         phi=phi_zpf*(b'+b),   n=1i*n_zpf*(b'-b)

    phi_zpf=(2*ec/el)^(1/4),   n_zpf=(el/(32*ec))^(1/4)

and the cosine is computed as a matrix function of the Hermitian phase operator using the matrix exponential. The ladder operators come from weyl.m, the real part of (U+U')/2 with U=expm(1i*(phi-phi_e)) drops the round-off in the cosine, and the Hamiltonian is returned in rad/s.

Syntax

     [H,n_op,phi_op]=fluxonium(ec,ej,el,phi_e,nlevels)

Parameters

  ec      - charging energy in Hz (energy over the
            Planck constant), a positive real number

  ej      - Josephson energy in Hz (energy over the
            Planck constant), a positive real number

  el      - inductive energy in Hz (energy over the
            Planck constant), a positive real number

  phi_e   - reduced external flux 2*pi*Phi_e/Phi_0
            in radians, a real number

  nlevels - number of oscillator basis states, a
            positive integer

Outputs

  H       - fluxonium Hamiltonian in rad/s (2*pi times
            the energy in Hz), a real symmetric matrix
            of dimension nlevels

  n_op    - charge operator (Cooper pair number) in the
            oscillator basis, a Hermitian matrix of di-
            mension nlevels

  phi_op  - phase operator in radians in the oscillator
            basis, a real symmetric matrix of dimension
            nlevels

Examples

See fluxonium_spectrum.m in examples/quantum_tech/circuit_qed directory.

Notes

The oscillator basis must be large enough for the lowest eigenstates to be converged; nlevels of the order of 30 to 60 is sufficient for ej/el=5 and ec/el=1, larger ej/el ratios need more states. Check the convergence by repeating the calculation with a bigger nlevels.

See also

weyl.m, oscillator.m, Kernel utilities

Version 2.13, authors: Ilya Kuprov