Figure 1

Schematic of the experimental setup. (a) Optical analog. Three two-state atoms are identically coupled to a cavity mode with photon decay rate $\kappa$, atomic energy relaxation rate $\gamma$, and collective coupling strength $G_N$. (b) Schematic of the investigated system. The coplanar waveguide resonator is shown in gray (light blue), the transmon qubits $A$, $B$, and $C$ in dark gray (violet) and the first harmonic of the standing wave electric field in light gray (red).Reuse & Permissions

Figure 2

Circuit diagram and false color optical images of the sample. (a) Simplified electrical circuit diagram of the experimental setup. The waveguide resonator operated at a temperature of 20 mK, indicated as $LC$ oscillator with frequency ${\omega }_r$, is coupled to input and output leads with the capacitors $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$. Qubits $A$, $B$, and $C$ are controlled with external current biased coils ($I_{A,B,C}$) and coupled to the resonator via identical capacitors $C_g$. A transmission measurement is performed by applying a measurement tone ${\nu }_{\mathrm{rf}}$ to the input port of the resonator, amplifying the transmitted signal and digitizing it with an analog-to-digital converter (ADC) after down conversion with a local oscillator (LO) in a heterodyne detection scheme. (b) The coplanar microwave resonator is shown truncated in gray (light blue) on the substrate in black (dark green) and the locations of qubits $A$, $B$, and $C$ are indicated. (c) Top, magnified view of transmon qubit $B$ shown in dark gray (violet) embedded between ground plane and center conductor of the resonator. Bottom left, qubits $A$ and $C$, of same dimensions as qubit $B$, are shown at reduced scale. Bottom right, magnified view of SQUID loop of qubit $B$.Reuse & Permissions

Figure 3

Vacuum Rabi mode splitting with one, two, and three qubits. (a) Measured resonator transmission spectrum $T$ [gray (blue) corresponds to low and dark gray (red) to high transmission] versus normalized external flux bias ${\Phi }_A/{\Phi }_0$ of qubit $A$. Dash-dotted white lines indicate bare resonator ${\nu }_r$ and qubit ${\nu }_A$ frequencies and dashed white lines are calculated transition frequencies ${\nu }_{g0,Nn\pm }$ between $|g,0⟩$ and $|N,n\pm ⟩$. (b) Resonator transmission $T/T_{\max }$ at degeneracy normalized to the maximum resonator transmission $T_{\max }$ measured at ${\Phi }_{A,B,C}={\Phi }_0/2$ (not shown), as indicated with arrows in (a). Dark gray (red) line is a fit to two Lorentzians. (c) Resonator transmission spectrum $T/T_{\max }$ versus external flux bias ${\Phi }_C/{\Phi }_0$ of qubit $C$ with qubit $A$ degenerate with the resonator (${\nu }_A={\nu }_r$). (d) Transmission spectrum $T/T_{\max }$ at flux as indicated in (c). (e) Transmission spectrum versus flux ${\Phi }_B/{\Phi }_0$ with both qubits $A$ and $C$ at degeneracy (${\nu }_A={\nu }_C={\nu }_r$). The white dashed line at frequency ${\nu }_{g0,31d_{1,2}}={\nu }_r$ indicates the dark state occurring at degeneracy. (f) Transmission spectrum $T/T_{\max }$ at flux as indicated in (e).Reuse & Permissions

Figure 4

Level diagram representing the total energy of (a) one, (b) two, and (c) three qubits resonantly coupled to a single photon. Bare energy levels of the qubits $|g⟩$, $|e⟩$ and the cavity $|0⟩$, $|1⟩$ are shown in black. The bright dressed energy levels $|N,n\pm ⟩$, with $N$ qubits, $n$ excitations, and $\pm$ indicating the symmetry of the state, are shown as dark gray (blue) lines. The areas of the circles indicate the relative population of the bare states in the eigenstates $|N,n\pm ⟩$.Reuse & Permissions

Figure 5

Scaling of the collective dipole coupling strength. Measured coupling constants (dots) extracted from Fig. 3 and nine similar data sets and theoretical scaling (line).Reuse & Permissions