Demonstrating $W$-type entanglement of Dicke states in resonant cavity quantum electrodynamics

Nonlinearity and entanglement are two important properties by which physical systems can be identified as nonclassical. We study the dynamics of the resonant interaction of up to $N=3$ two-level systems and a single mode of the electromagnetic field sharing a single excitation dynamically. We observe coherent vacuum Rabi oscillations and their nonlinear $\sqrt{N}$ speedup by tracking the populations of all qubits and the resonator in time. We use quantum state tomography to show explicitly that the dynamics generates maximally entangled states of the $W$ class in a time limited only by the collective interaction rate. We use an entanglement witness and the 3-tangle to characterize the state whose fidelity $F=78\%$ is limited in our experiments by crosstalk arising during the simultaneous qubit manipulations which is absent in a sequential approach with $F=91\%$.

Figure 1

(a) Optical microscope false color image of the sample with three qubits (A,B,C) capacitively coupled to a coplanar waveguide resonator (yellow). Each qubit is equipped with individual local charge (red) and magnetic flux-bias lines (green). (b) Enlarged view of the transmon qubit C. The island (blue) and the reservoir (lighter blue) are connected via a SQUID loop.

Figure 2

Dynamics of the collective vacuum Rabi oscillations. (a) Pulse sequence for the single and collective vacuum Rabi oscillation (QP, qubit preparation; Buf, buffer level for flux pulses; SRI, single resonant interaction; CRI, collective resonant interaction; ST, state tomography). Qubits not taking part in the collective interaction remain at their bias frequency (dashed green/blue). The duration $\tau$ of the collective interaction is varied. (b) Oscillations for $N=1$ (first row) to $N=3$ (third row). The population is shown for the excited states of qubit A (blue), B (green), C (orange), the $|ggg\rangle$ state (gray), and the resonator (violet). (c) Linear fit to the extracted and squared oscillation frequencies of the resonator population.

Figure 3

Real part of the density matrix of the $W$ state. (a) Obtained for the collective approach as shown in Fig. 2. (c) Generated by the sequential approach. (b), (d) Pauli sets for the collective and sequential approaches, respectively.