Figure 1

(a) Coplanar waveguide resonator coupled via finger capacitors $C_{\kappa }$ to input and output transmission lines. Two transmons are capacitively coupled to the resonator at its ends ($C_g$). Additional ac-signal lines are capacitively coupled to the qubits (not used in the experiments). (b) Optical micrograph of a transmon qubit. (c) Schematic of the measurement setup. The state of the qubit is determined by measuring the transmission of the RF signal through the transmission line cavity modeled as an LCR oscillator. When a spectroscopy signal (Spec) is resonant with a qubit transition, the resonance frequency of the cavity is shifted and the change in transmission amplitude is recorded at the analog-digital converter (ADC) after down-conversion with a local oscillator (LO) [34]. The qubit frequencies can be tuned independently with superconducting coils (SC Coil 1 or 2).Reuse & Permissions

Figure 2

(a) Energy level diagram of two transversely coupled qubits. (b) Spectroscopic measurement of the avoided level crossing in sample A as a function of normalized flux $\Phi /{\Phi }_0$ threading the first qubit loop with the second qubit at a fixed frequency. The solid lines indicate energy levels calculated from a diagonalization of the two-qubit Jaynes-Cummings Hamiltonian. (c) Experimentally extracted value of the coupling strength $J$ as a function of qubit frequency (circles). Lines indicate calculated values of $J({\omega }_{\mathit{ge}})$ for a model including one resonator mode with frequency-independent (thin solid gray line) or frequency-dependent coupling strength (dashed red line). Theoretical predictions including four (thick solid green line) and five resonator modes (dotted blue line). The thin solid black line comprises finite-size effects of the transmons using 60 resonator modes.Reuse & Permissions

Figure 3

(a) Spatial mode structure of a coplanar waveguide resonator. Qubits are positioned at opposite ends of the $\lambda /2$ resonator with coupling of alternating sign, $g_j^{(1)}=(-1){}^{j+1}{g}_j^{(2)}$ to the $j$th resonator mode at frequency ${\omega }_j$. (b) Energy level diagram and coupling scheme for two qubits with transition frequencies ${\omega }_{\mathit{ge}}^{(1)}$ and ${\omega }_{\mathit{ge}}^{(2)}$ coupled to a transmission line cavity with fundamental frequency ${\omega }_0$. Energy levels with a photon in the $j$th resonator mode ($\mathit{gg}{1}_j$) or a qubit excitation ($\mathrm{ge}0$ or $\mathit{eg}0$) are shown. The interaction strength $J$ depends on the detunings ${\Delta }_j^{(i)}$ and the coupling strengths $g_j^{(i)}$ of both qubits ($i=1,2$) to the resonator mode $j$.Reuse & Permissions

Figure 4

Qubit-qubit coupling strength $J$ versus qubit frequency in sample B. The vertical lines indicate the frequencies of the coplanar waveguide modes ${\omega }_j$. Experimental data (dots) confirm the expected increase of $J$ with increasing frequency. The solid red (dotted blue) line is the calculated $J$ including $N=6$ ($N=7$) resonator modes. The dashed green line is a fit to a model with additional resonances.Reuse & Permissions

Figure 5

(a) Energy levels of the qubit-resonator system when the qubits are degenerate and their transition frequencies are simultaneously swept across the resonator modes. The parameters are those of sample B (Table ). (b) Spectroscopic measurements of the anticrossing in the regions labeled A, B, C, and D in (a).Reuse & Permissions

Figure 6

(a) Coupling strength $g_j^{(2)}$ of qubit 2 to the $j$th harmonic mode in sample B. (b) Expectation value of the permutation operator $P$ for the eigenstates of the dispersive Jaynes-Cummings Hamiltonian as a function of qubit-qubit detuning ${\delta }_q$. (c) Observed shift $\left|{\delta }_q\right|$ of the dark state relative to qubit-qubit resonance. The calculated shift due to an asymmetric field distribution of the driven transmission line at the qubit positions is indicated by the solid line.Reuse & Permissions