Ultrastrong coupling in two-resonator circuit QED

We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as $17.5\%$ of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings approximation. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional inductance provided by a Josephson junction.

Figure 1

Sample and measurement setup. (a) False-color image of the sample chip. Nb ground planes are shown in blue and feed lines in orange. The resonator signal lines reside along the ground plane edges. The green and red rectangles mark the areas shown on an enlarged scale in (b) and (c), respectively. (b) Coupling capacitor defining the resonators. (c) Resonator coupling area with signal lines (green) and flux qubit (red). Light/dark green stripes highlight Nb-Al overlap areas. The yellow rectangle marks the area shown in (d). (d) Flux qubit galvanically coupled to both resonators. The black rectangle marks the area shown in (e). (e) $\mathrm{Al}/{\mathrm{AlO}}_{\mathrm{x}}/\mathrm{Al}$ Josephson junction fabricated using shadow evaporation. (f) Sketch of the coupling mechanisms and measurement setup. The wiggly arrow symbolizes the input microwave line connected to resonator $\mathrm{A}$ and the black triangle denotes the corresponding output line featuring microwave amplifiers. The crosses intersecting one qubit branch symbolize the three Josephson junctions.

Figure 2

Resonant modes of the galvanically coupled qubit-resonator system. The arrows indicate in-phase and out-of-phase oscillating currents. (a) Antiparallel mode. (b) Parallel mode. (c) Transverse mode.

Figure 3

(a) Transmission measured through resonator $\mathrm{A}$ as a function of the applied magnetic flux with the qubit in the ground state. Green line: fit using the Hamiltonian of Eq. (1). The area shown in panel (b) is marked by the black rectangle. (b) Detail of (a). Solid green line: fit using the Hamiltonian of Eq. (1). Dashed black line: Jaynes-Cumming approximation obtained using the parameters producing the green line but neglecting the counter-rotating terms in the Hamiltonian of Eq. (1).

Figure 4

Breakdown of the Jaynes-Cummings approximation. (a) Transmission measured through resonator $\mathrm{A}$ as a function of the magnetic flux applied to the flux qubit with the qubit in the ground state. Data: detail from Fig. 3. Green line: fit using the quantum Rabi Hamiltonian of Eq. (1). Dashed black line: Jaynes-Cumming approximation using the parameters producing the green line but neglecting the counter-rotating terms in the Hamiltonian of Eq. (1). Dashed blue line: fit using the Jaynes-Cummings Hamiltonian of Eq. (3), where the transverse mode is treated as an independent phenomenon. Dashed white line: probe frequency ${\omega }_{\mathrm{probe}}$ used for two-tone spectroscopy. (b) Same as (a), but with the qubit driven by a strong microwave tone. (c) Two-tone spectroscopy. Green and dashed blue lines as in panel (a). ${\omega }_{\mathrm{s}}$ is the variable spectroscopy tone. Color code: transmission magnitude measured at ${\omega }_{\mathrm{probe}}$. (d),(e) Details from (c).

Figure 5

Panels from Fig. 3 and Fig. 3 including the data points (black dots) used for the fit.

Figure 6

Transmission measured through resonator $\mathrm{A}$ depending on the magnetic flux applied to the flux qubit. Dashed white line: transition between eigenstates of the Hamiltonian of Eq. (1) as described in the main text.