Quantization of inductively shunted superconducting circuits

We present a method for calculating the energy levels of superconducting circuits that contain highly anharmonic, inductively shunted modes with arbitrarily strong coupling. Our method starts by calculating the normal modes of the linearized circuit and proceeds with numerical diagonalization in this basis. As an example, we analyze the Hamiltonian of a fluxonium qubit inductively coupled to a readout resonator. While elementary, this simple example is nontrivial because it cannot be efficiently treated by the method known as “black-box quantization,” numerical diagonalization in the bare harmonic oscillator basis, or perturbation theory. Calculated spectra are compared to measured spectroscopy data, demonstrating excellent quantitative agreement between theory and experiment.

Figure 1

(a) Schematic diagram of cQED systems involving a multimode microwave environment on the left and various highly anharmonic, inductively shunted qubits on the right. (b) Schematic diagram of the fluxonium-resonator system. The fluxonium qubit (blue) is threaded by an external magnetic flux ${\mathrm{\Phi }}_{\text{ext}}$ and coupled to a dipole antenna used as a readout resonator (red) via an array of shared Josephson junctions (purple). (c) Circuit model of the system, obtained by replacing junction arrays with linear inductors.

Figure 2

(a) Energy levels $E_{n\mu }$ of the fluxonium-resonator system as a function of external flux ${\mathrm{\Phi }}_{\text{ext}}$. The zero-point energy is taken to be $E_{0g}$ and energies are given in frequency units, while flux is given in multiples of the magnetic flux quantum ${\mathrm{\Phi }}_0$. (b) Anticrossing between the $0\to 1$ readout transition and the $e\to h$ qubit transition, and the divergence of second-order perturbation theory.

Figure 3

(a), (b) Blue: qubit $g\to e$ transition frequency. (c), (d) Red: readout $0\to 1$ transition frequency. (e), (f) Purple: dispersive shift $\chi$. All results are plotted as a function of external flux ${\mathrm{\Phi }}_{\text{ext}}$ in units of ${\mathrm{\Phi }}_0$. Circles indicate data taken using two-tone spectroscopy (blue), single-tone spectroscopy (red), and single-tone spectroscopy preceded by qubit state preparation (purple). Solid lines indicate theoretical fits obtained from numerical diagonalization. Dashed lines indicate results from second-order perturbation theory.

Figure 4

Inherited anharmonicity of the readout resonator as a function of external flux ${\mathrm{\Phi }}_{\text{ext}}$, defined as the difference between the $1\to 2$ and $0\to 1$ readout transition frequencies. Results from numerical diagonalization are shown for the qubit in state $|g\rangle$ (black) and state $|e\rangle$ (orange).

Figure 5

Electrical circuit schematic for the fluxonium-resonator system (black), showing the decomposition into normal modes coupled by a single nonlinear inductive element. Transmission lines (dashed gray) are weakly coupled (solid gray) to these modes in order to treat the system as a two-port microwave device.

Figure 6

State-dependent scattering parameter magnitudes for the fluxonium-resonator system at ${\mathrm{\Phi }}_{\text{ext}}=0.5{\mathrm{\Phi }}_0$. (a) Reflection coefficient amplitude for the readout mode. (b) Transmission coefficient amplitude. (c) Reflection coefficient amplitude for the qubit mode. Amplitudes are plotted at frequencies near the $g\to e$ qubit transition (left-hand side) as well as those near the $0\to 1$ readout transition (right-hand side). Plots are layered from top to bottom in order of ascending energy.

Figure 7

State-dependent scattering parameter arguments for the fluxonium-resonator system at ${\mathrm{\Phi }}_{\text{ext}}=0.5{\mathrm{\Phi }}_0$. (a) Reflection coefficient phase for the readout mode. (b) Transmission coefficient phase. (c) Reflection coefficient phase for the qubit mode. Phases are plotted at frequencies near the $g\to e$ qubit transition (left-hand side) as well as those near the $0\to 1$ readout transition (right-hand side). Plots are layered from top to bottom in order of ascending energy.