Robust Strong-Coupling Architecture in Circuit Quantum Electrodynamics

We report on a robust method to achieve strong coupling between a superconducting flux qubit and a high-quality quarter-wavelength coplanar waveguide resonator. We demonstrate the progression from the strong to ultrastrong coupling regime by varying the length of a shared inductive coupling element, ultimately achieving a qubit-resonator coupling strength of 655 MHz, $10\mathrm{\%}$ of the resonator frequency. We derive an analytical expression for the coupling strength in terms of circuit parameters and also discuss the maximum achievable coupling within this framework. We experimentally characterize flux qubits coupled to superconducting resonators using one- and two-tone spectroscopy methods, demonstrating excellent agreement with the proposed theoretical model.

Figure 1

The studied structure. (a) Microscopic view of the reported device. (b) Electron micrograph showing the capacitive coupling of a diagnostic resonator to a common feedline and an enlarged array of flux trapping holes in the inset. (c) Three-junction flux qubit with a $120$-$\mu \mathrm{m}$-long meander element producing the large coupling (shared edge highlighted with false color). The qubit is galvanically coupled to the resonator from one end and shunted to the ground plane from the other end. (d) Electron micrograph of two large identical junctions of a flux qubit. The additional line is an extended island between junction 1 and 2, terminating near the ground planes as shown in (c). The bigger island contributes to an additional capacitance ($C_{g2}$), hence lowering the charging energy of the qubit. (e) Electron micrograph of the test superconducting quantum interference device (SQUID) (highlighted with false color). The circuit diagram of the reported device is shown in Fig. 3.

Figure 2

(a) Employed measurement setup scheme. (b) Broad range $S_{21}$ transmission confirming the presence of seven superconducting resonators indicated by vertical arrows. (c) Fitted notch-type transmission of one of the superconducting resonators based on the transmission $S_{21}$ versus frequency. The red dashed line shows a fit to the experimental data using Eq. (1).

Figure 3

Circuit diagram of a three-junction flux qubit coupled to a resonator. The resonator is capacitively coupled to a transmission line with which a $S_{21}$ measurement is carried out. There are seven similar qubit-resonator systems coupled to a single transmission line (1-2) as shown in Fig. 1.

Figure 4

One-tone and two-tone spectroscopy of the qubits for (a) shared-coupling-element length $l=0.30\mu \mathrm{m}$ with $g/2\pi =180\mathrm{MHz}$ (b) shared-coupling-element length $120\mu \mathrm{m}$ and $g/2\pi =655\mathrm{MHz}$. The width of both coupling elements is $0.35\mu \mathrm{m}$. In (a),(b), the upper part of the figure in the left side shows a limited segment of the one-tone spectra where the $2g$ value is estimated based on experimental observation while figure in the bottom (framed in yellow and enlarged on the right side) represents two-tone spectra. The dotted lines show the theoretical model using Eq. (10). In (a), the experimental data in the green box are enlarged on the right side. In (a) we have $E_J/h=68.75\mathrm{GHz}$, $\alpha =0.5825$, ${\omega }_r/2\pi =6.170\mathrm{GHz}$ and in (b) $E_J/h=65.2\mathrm{GHz}$, $\alpha =0.547$, ${\omega }_r/2\pi =6.685\mathrm{GHz}$.

Figure 5

Measured coupling for the seven qubits together with the prediction from the theoretical model given in Eq. (8) with $\beta =1/4$. Qubits 3 and 7 are shown in Fig. 4.

Figure 6

In each row the figures from left to right represent one-tone spectra, two-tone spectra, and electron micrograph of qubits reported in Table 1.