Asymptotic expressions for charge-matrix elements of the fluxonium circuit

In charge-coupled circuit QED systems, transition amplitudes and dispersive shifts are governed by the matrix elements of the charge operator. For the fluxonium circuit, these matrix elements are not limited to nearest-neighbor energy levels and are conveniently tunable by magnetic flux. Previously, their values were largely obtained numerically. Here, we present analytical expressions for the fluxonium charge matrix elements. We show that new selection rules emerge in the asymptotic limit of large Josephson energy and small inductive energy. We illustrate the usefulness of our expressions for the qualitative understanding of charge matrix elements in the parameter regime probed by previous experiments.

Figure 1

(a) Fluxonium energy levels (solid curves) as a function of external flux ${\Phi }_{\text{ext}}$, for the parameters realized experimentally.[1] Shaded regions in the background show position and width of the bands ${\epsilon }_s\left(p\right)$. The three $s=0$ persistent-current states with parabolic flux dependence are labeled by their quantum number $m$. (b) Fluxonium eigenfunctions for ${\Phi }_{\mathrm{ext}}/{\Phi }_0=0.4$ [vertical dashed line in (a)]. The bold black curve shows the fluxonium potential $V\left(\phi \right)$. Local minima are labeled by the quantum numbers $m$. The fluxonium wavefunctions (thin curves) are offset by their eigenenergies.

Figure 2

Comparison of numerical results and asymptotic predictions for charge matrix elements. Solid curves show numerical results for the magnitude of the charge matrix elements, $|\langle 0_l|\mathsf{N}|1_l\rangle |$, $|\langle 0_l|\mathsf{N}|2_l\rangle |$, and $|\langle 0_l|\mathsf{N}|3_l\rangle |$. Dashed curves show the asymptotic matrix elements between the ground state and the lowest metaplasmon state, namely $|\langle 0_l|\mathsf{N}|0_{\nu },1_s\rangle |$, and between the two low-lying persistent current states, $|\langle 0_m,0_s|\mathsf{N}|-1_m,0_s\rangle |$.