Approaching ultrastrong coupling in transmon circuit QED using a high-impedance resonator

In this experiment, we couple a superconducting transmon qubit to a high-impedance $645\mathrm{\Omega }$ microwave resonator. Doing so leads to a large qubit-resonator coupling rate $g$, measured through a large vacuum Rabi splitting of $2g\simeq 910$ MHz. The coupling is a significant fraction of the qubit and resonator oscillation frequencies $\omega$, placing our system close to the ultrastrong coupling regime ($\overline{g}=g/\omega =0.071$ on resonance). Combining this setup with a vacuum-gap transmon architecture shows the potential of reaching deep into the ultrastrong coupling $\overline{g}\sim 0.45$ with transmon qubits.

Figure 1

(a) Optical image of the device, showing the $\lambda /2$-microwave resonator formed by a meandering $1\mu \mathrm{m}$ wide stripline capacitively coupled to a $50\mathrm{\Omega }$ feedline on one end and a transmon qubit on the other end. (b) Zoom in on the transmon qubit, showing the SQUID loop formed by two Josephson junctions and the parallel plate vacuum-gap capacitor formed by an aluminum qubit as bottom electrode and graphite as top electrode. (c) Zoom in on the capacitive coupler between the resonator and the feedline. (d) Equivalent lumped element circuit of the device, as derived in the Supplemental Material [21].

Figure 2

(a) Color plot of the feedline transmission $|S_{21}|$ as a function of magnetic field and frequency, showing the coupling between the resonator and qubit. The magnetic flux penetrating the SQUID loop incorporated in the transmon qubit changes the Josephson energy, which results in the tunability of the $|g\rangle -|e\rangle$ transition energy of the qubit. Close to resonance the qubit and resonator show an avoided crossing centered at 6.23 GHz. Black dashed lines correspond to fits to this data. The horizontal (oblique) white dashed line corresponds to the bare resonator (transmon) frequency, ${\omega }_r/2\pi$ $[{\omega }_a\left(\varphi \right)/2\pi ]$. The lack of symmetry around the crossing point is due to the flux dependence of the coupling strength $g$. (b) Trace of the microwave response where the qubit and cavity are close to resonance $\varphi =0.63{\varphi }_0$ showing an anticrossing of $2g\simeq 2\pi \times 910$ MHz and a linewidth of 28 MHz.

Figure 3

(a) Color plot of the change in transmission ($\delta |S_{21}|$) of a microwave tone at resonance with the cavity at single-photon power as a function of frequency of a secondary qubit-drive tone and magnetic field. Due to the qubit-state dependent frequency shift of the cavity, the transmission changes as the drive tone excites the qubit, tracing the qubit $|g\rangle -|e\rangle$ transition frequency as a function of magnetic field. The secondary faint resonance corresponds to the $|e\rangle -|f\rangle$ transition of the thermally excited population in state $|e\rangle$. Power broadening is visible when the qubit transition frequency is close to the cavity. The power of the drive tone delivered to the feedline is constant, but close to the cavity a larger portion of the power is delivered to the qubit. Dashed lines correspond to fits to the data. (b) Line cut of the color plot where the qubit frequency is at ${\omega }_a=2\pi \times 3.586$ GHz with a linewidth of 38 MHz. The second peak corresponds to the first to second exited state transition, which corresponds to the (dressed) anharmonicity, a rough approximation of $E_c/h=370$ MHz.

Figure 4

Proposal for larger couplings which combines the use of a $\lambda /4$ high-impedance resonator and a vacuum gap transmon. The $\lambda /4$ can be coupled to a $50\mathrm{\Omega }$ feedline through the use of a shunt capacitor [44]. For $C_J=C_r=10$ fF and $C_c=200$ fF the system is expected to reach far into the USC regime $\overline{g}=0.45$ on resonance.