Circuit QED with fluxonium qubits: theory of the dispersive regime

TL;DR

This paper develops a theoretical framework for understanding dispersive interactions in circuit QED with fluxonium qubits, explaining large shifts and two-photon effects observed experimentally by considering multi-level virtual transitions without selection rules.

Contribution

It provides a general perturbation theory for multi-level qudits coupled to harmonic modes and applies it to fluxonium, revealing the significance of fourth-order effects and absence of selection rules.

Findings

Large dispersive shifts explained by virtual transitions without selection rules.

Prediction of two-photon vacuum Rabi splitting in fluxonium.

Good agreement between theory and experimental data across flux range.

Abstract

In circuit QED, protocols for quantum gates and readout of superconducting qubits often rely on the dispersive regime, reached when the qubit-photon detuning {\Delta} is large compared to their mutual coupling strength. For qubits including the Cooper-pair box and transmon, selection rules dramatically restrict the contributions to dispersive level shifts {\chi}. By contrast, without selection rules many virtual transitions contribute to {\chi} and can produce sizable dispersive shifts even at large detuning. We present theory for a generic multi-level qudit capacitively coupled to one or multiple harmonic modes, and give general expressions for the effective Hamiltonian in second and fourth order perturbation theory. Applying our results to the fluxonium system, we show that the absence of strong selection rules explains the surprisingly large dispersive shifts observed in experiments and also leads to the prediction of a two-photon vacuum Rabi splitting. Quantitative predictions from our theory are in good agreement with experimental data over a wide range of magnetic flux and reveal that fourth-order resonances are important for the phase modulation observed in fluxonium spectroscopy.