Entangling Interactions Between Artificial Atoms Mediated by a Multimode Left-Handed Superconducting Ring Resonator

Superconducting metamaterial transmission lines implemented with lumped circuit elements can exhibit left-handed dispersion, where the group and phase velocity have opposite sign, in a frequency range relevant for superconducting artificial atoms. Forming such a metamaterial transmission line into a ring and coupling it to qubits at different points around the ring results in a multimode bus resonator with a compact footprint. Using flux-tunable qubits, we characterize and theoretically model the variation in the coupling strength between the two qubits and each of the ring-resonator modes. Although the qubits have negligible direct coupling between them, their interactions with the multimode ring resonator result in both a transverse exchange coupling and a higher-order $ZZ$ interaction between the qubits. As we vary the detuning between the qubits and their frequency relative to the ring-resonator modes, we observe significant variations in both of these interqubit interactions, including zero crossings and changes of sign. The ability to modulate interaction terms such as the $ZZ$ scale between zero and large values for small changes in qubit frequency provides a promising pathway for implementing entangling gates in a system capable of hosting many qubits.

Popular Summary

Implementing a fault-tolerant quantum processor requires the ability to couple together qubits for generating entanglement. Superconducting qubits are an attractive platform for quantum information processing. However, most typical approaches for coupling superconducting qubits only involve nearest-neighbor interactions, thus limiting the type of error-correction routines that can be performed. Here we demonstrate a multimode ring-resonator bus for entangling interactions between remote transmon qubits located at different points around the ring.

We form the ring-resonator bus from a metamaterial transmission line with left-handed wave dispersion, where the mode frequency is a falling function of wave number. This results in a dense frequency spectrum of standing-wave resonances near the qubit transition frequency range. The coupling of a qubit to a resonant mode depends on the standing-wave amplitude at the qubit location. We characterize the coupling strength of each qubit to the various ring-resonator modes through spectroscopic measurements and compare with a detailed theoretical model.

Coupling multiple qubits to a common resonant mode can result in a transverse exchange interaction between the qubits. For our multimode ring-resonator bus, the exchange coupling between the qubits depends on the contributions from each mode, which vary with the detuning of each qubit to the various modes and can be positive or negative. In addition, interactions between the higher excited states of each qubit and the bus resonator modes lead to a higher order ZZ interaction between the qubits, which also depends on the qubit detunings and can change sign. This produces a rich variation in the exchange coupling and $ZZ$ interactions between the qubits as we vary their transition frequencies, in close agreement with our theoretical modeling for the device. The ability to tune these entangling energy scales from large values through zero, and the possibility of extending this to more than two qubits around the ring, make this a promising platform for controlling entanglement in large qubit arrays.

Figure 1

(a) The mode structure for two pairs of degenerate ring-resonator modes. The ring resonator consists of 24 unit cells, shown with gray squares, with each cell containing a capacitor shunted to ground by two inductors. One pair of even (odd) modes with $k\mathrm{\Delta }x=10(2\pi /N)$ is shown in solid (dashed) orange lines, labeled $m_E^{(10)}$($m_O^{(10)}$); a second pair with $k\mathrm{\Delta }x=5(2\pi /N)$ is depicted with solid (dashed) green lines, labeled $m_E^{(5)}$($m_O^{(5)}$). The coupling strength of $Q_A$ and $Q_B$ to the mode is determined by the amplitude of the wave at the cell where each qubit is capacitively coupled to the resonator. (b) Theoretical $g$-coupling values for $Q_A$ and $Q_B$, coupled to a hypothetical 240-cell ring resonator with qubit separation $n_{AB}=6$. Simulated data points at which the coupling for $Q_A$, $g_i^A/2\pi$, and for $Q_B$, $g_i^B/2\pi$, share the same sign for a given mode are shown in purple and follow Eq. (3). Points at which the $g_i^A$ and $g_i^B$ coupling to $Q_A$ and $Q_B$, respectively, have opposite signs are shown in gray and are associated with Eq. (4).

Figure 2

(a) The chip layout for the metamaterial ring-resonator device. (b),(c) Optical micrographs of the device with false-color highlighting. (d) An SEM image of a Josephson junction in a qubit. The device has two flux-tunable transmon qubits, $Q_A$ and $Q_B$, coupled to a ring resonator at the positions indicated: $R_{\mathrm{in}}$/$R_{\mathrm{out}}$ connections to the upper feed line used to probe ring-resonator modes and $F_{\mathrm{in}}$/$F_{\mathrm{out}}$ connections used for measuring readout resonators coupled to each qubit. (e) The measured ring-resonator mode frequencies are shown with gray dashed lines. The theoretical mode frequencies are shown with green dashed-dotted lines when stray inductance due to wirebonds is not included and any degeneracy lifting is due to the qubits or the feed line. The solid blue lines show the large degeneracy-lifting effect of wirebonds on the ring-resonator mode frequencies.

Figure 3

Vacuum Rabi splittings measured via $R_{\mathrm{in}}$/$R_{\mathrm{out}}$, between three ring-resonator modes and (a) $Q_A$ and (b) $Q_B$. The dashed lines in red (blue) show fits to splitting data for $Q_A$ ($Q_B$) obtained from the reduced Hamiltonian, including one qubit and the three modes (see Appendix pp6). The horizontal dotted lines show the dressed mode frequencies. (c)–(f) Splittings and reduced-Hamiltonian fits for two single modes for each qubit. For all measurements, the spectator qubit is fixed at its upper flux-insensitive sweet spot (${\mathrm{\Phi }}_q/{\mathrm{\Phi }}_0=0$). As shown in (c), $Q_A$ has effectively zero coupling to the ring-resonator mode at 4.553 GHz, while (d) shows that $Q_B$ has a coupling strength of 26.6 MHz. (g) Extracted and theoretical $|g_i^q|/2\pi$ values for each ring-resonator mode $i$. The error bars on the experimental data are computed from 95% confidence intervals in $g$-coupling fits (see also Appendix pp6). The vertical dashed lines show bare ring-resonator mode frequencies.

Figure 4

(a),(b) The qubit spectroscopy for $Q_A$ as a function of flux when $Q_B$ is at a fixed frequency, denoted by a blue dotted line. The red dashed lines show fits to data obtained via minimization of the reduced Hamiltonian from which we determine $J/2\pi$ (see Appendix pp7). (c) The experimental and theoretical $J$-coupling values as a function of the frequency. The diamonds show experimental values for $J/2\pi$, with error bars computed from 95% confidence intervals in fits (see also Appendix pp7). The gray lines and purple crosses show theoretical $J/2\pi$ values calculated using Schrieffer-Wolff and Least Action, respectively. Dotted vertical lines show bare ring-resonator mode frequencies.

Figure 5

$\zeta /2\pi$ as a function of the dressed frequency of $Q_A$, $\widetilde{f_A}$, when the dressed frequency of $Q_B$, $\widetilde{f_B}$, is (a) 5.04 GHz and (b) 5.81 GHz (see Appendix pp9). The measured and theoretical values are shown as diamonds and gray lines, respectively. The vertical dashed lines in blue and gray show the location of $Q_B$ and the ring-resonator modes in frequency space, respectively. The insets show the Ramsey-oscillation-frequency fitting data as a function of the $Q_A$ drive frequency taken from Ramsey-fringe measurements. The red data points and fit lines are generated by performing a simple Ramsey pulse sequence (shown in the red box) on $Q_A$ at multiple detunings and extracting the oscillation frequency. The data resulting from the pulse sequence in which a $\pi$ pulse is applied to $Q_B$, followed by a Ramsey on $Q_A$ are shown in the purple box. The error bars are computed from 95% confidence intervals for both Ramsey-oscillation fits (see also Appendix pp9). The intersection of the fit lines where the Ramsey-oscillation frequency vanishes indicates the $Q_A$ frequency, from which we compute $\zeta /2\pi$.

Figure 6

The full physical-circuit schematic. Right-handed transmission lines labeled $R_{\mathrm{in}}$/$R_{\mathrm{out}}$ and $F_{\mathrm{in}}$/$F_{\mathrm{out}}$, in green and yellow, respectively, are used for microwave readout. The 24-cell ring-resonator circuit is shown in orange. The capacitively coupled qubits, $Q_A$ and $Q_B$, are shown in red and blue, respectively, and are separated by a distance of $n_{AB}=6$. The qubit readout cavities are comprised of quarter-wave coplanar waveguide resonators.

Figure 7

(a) An image of wirebonds used for grounding the device, as well as attachments to signal traces. (b) Device layout of the device layout, including approximate wirebond positions shown with white lines. (c) The theoretical wirebond model used for modeling mode resonances and coupling strengths.

Figure 8

The wiring schematic for room-temperature and cryogenic cabling down to the sample box. A standard heterodyne setup is used for measurement. All filters are low-pass unless otherwise indicated.

Figure 9

The measurement sequence for obtaining the $ZZ$-interaction strength. (a) A Ramsey pulse sequence is performed on $Q_A$ at variable drive frequencies while $Q_B$ is idle. (b) A second Ramsey-fringe measurement is performed on $Q_A$ after a $\pi$ pulse is applied to $Q_B$. (c) Horizontal slices of Ramsey-fringe measurements are fitted to obtain the Ramsey-oscillation frequency, $f_{\mathrm{Ramsey}}$, at each drive frequency. The $ZZ$ interaction, denoted $\zeta /2\pi$, is the change in the dressed transition frequency for $Q_A$, $\widetilde{f_A}$, depending on whether $Q_B$ is in the ground or excited state. (d)–(f) Three example plots of Ramsey frequency as a function of the $Q_A$ drive frequency ($\widetilde{f_A}$ drive) for different $Q_A$ bias points with linear fit lines for $ZZ$ measurements taken when $Q_B$ is at 5.04 GHz.

Figure 10

(a) The experimental and theoretical $ZZ$ interaction $\zeta /2\pi$ as a function of the $Q_A$ frequency $f_A$ shown with diamonds and gray lines, respectively. (b) The multiphoton energies as a function of $f_A$. The light gray regions in (a) and (b) denote a region of uncertainty in the theoretical $ZZ$ calculation. The discontinuity of the $ZZ$ interaction around $5.25$ GHz is caused by an anticrossing of the $|11⟩$ and $|20⟩$ states visible in (b). Since there are multiple other modes close to this region, the exact point of the frequency discontinuity strongly depends on various parameters, making accurate predictions challenging.