Observation of the Three-State Dressed States in Circuit Quantum Electrodynamics

We have investigated the microwave response of a transmon qubit coupled directly to a transmission line. In a transmon qubit, owing to its weak anharmonicity, a single driving field may generate dressed states involving more than two bare states. We confirmed the formation of three-state dressed states by observing all of the six associated Rabi sidebands, which appear as either amplification or attenuation of the probe field. The experimental results are reproduced with good precision by a theoretical model incorporating the radiative coupling between the qubit and the microwave.

Figure 1

(a) Optical micrograph of a transmon qubit coupled to a microwave transmission line and (b) its schematic. (c) Theoretical model. A ladder-type three-level system is coupled directly to the transmission line. The arrows $a$ ($b$) denote the microwave fields propagating rightward (leftward).

Figure 2

(a) Level structure in the rotating frame of the bare states (left) and the dressed states (right). The drive frequency is set to ${\omega }_d={\omega }_{20}/2$. (b) Overlap between the bare and dressed states as a function of the Rabi frequency. (c) Stationary density matrix elements in the dressed-state basis: diagonal elements (left) and off-diagonal elements (right).

Figure 3

(a) Experimental and (b) theoretical plots of transmission $|t|$ as a function of the drive power and the probe frequency. The drive frequency ${\omega }_d/2\pi$ is set at 7.314 GHz [$=({\omega }_{20}/2)/2\pi$]. In the white region at ${\omega }_p/2\pi =7.558\mathrm{GHz}$, $|t|$ is less than 0.5 due to strong reflection. In (b), the drive power is expressed in ${\Omega }_{\mathrm{R},10}$. The transition frequencies between the dressed states are superposed, and they form six Rabi sidebands which appear symmetrically with respect to the drive frequency, ${\omega }_d/2\pi =7.314\mathrm{GHz}$. (c) Level structure of the dressed states. The $|m,N⟩\leftrightarrow |e,N+1⟩$ transition is indicated as an example. Population of the dressed states [strong drive region of Fig. 2c] is represented by the size of the circles.