Preparation of subradiant states using local qubit control in circuit QED

Transitions between quantum states by photon absorption or emission are intimately related to the symmetries of the system which lead to selection rules and the formation of dark states. In a circuit quantum electrodynamics setup, in which two resonant superconducting qubits are coupled through an on-chip cavity and driven via the common cavity field, one single-excitation state remains dark. Here, we demonstrate that this dark state can be excited using local phase control of individual qubit drives to change the symmetry of the excitation field. We observe that the dark state decay via spontaneous emission into the cavity is suppressed, a characteristic signature of subradiance. This local control technique could be used to prepare and study highly correlated quantum states of cavity-coupled qubits.

Figure 1

(a) Schematic of a cavity QED setup with individual phase control of the driving field for each qubit. (b) Energy-level diagram of the two-qubit system coupled dispersively to a common cavity field. Symmetric states are indicated by thick solid (green) lines and the antisymmetric state is identified by the thick dashed (red) line.

Figure 2

Setup and micrograph of the sample with a transmission line resonator (green) and two transmon qubits (blue) addressable via local charge lines (yellow).

Figure 3

Spectroscopy of dark and bright states as a function of the relative phase $\phi$ between drives. (a) Normalized integrated transmission amplitudes $S_{\mathrm{n}}=S/S_{\max }$ of the antisymmetric state ${\psi }_a$ (black solid dots) and the symmetric state ${\psi }_s$ (red open diamonds) at the frequencies ${\omega }_a/2\pi =6.647\mathrm{GHz}$ and ${\omega }_s/2\pi =6.578\mathrm{GHz}$, respectively, as indicated by the arrows in (b). $S_{\mathrm{n}}$ is normalized to unity at the maximum value of both curves ($S_{\max }$). The lines indicate fits to the expected steady-state populations. (b) Transmission amplitude as a function of frequency ${\nu }_d$ and relative phase $\phi$ of the spectroscopic saturation pulse.

Figure 4

(a) Population of the antisymmetric dark (${\psi }_a$, black dots) and the symmetric bright (${\psi }_s$, red diamonds) state vs time at $\Delta /2\pi =-290\mathrm{MHz}$ with exponential fits to the data. The inset shows the energy relaxation of ${\psi }_a$ and ${\psi }_s$. Only the symmetric state ${\psi }_s$ is affected by Purcell decay ${\gamma }_{\kappa }$. (b) Measured decay times of ${\psi }_s$ (red diamonds), ${\psi }_a$ (black dots) and uncoupled qubit states $|eg\rangle$ (orange points) and $|ge\rangle$ (green crosses) as a function of detuning $\Delta$. Exponential fits to numerically simulated populations are shown for ${\psi }_s$ (dashed lines), ${\psi }_a$ (solid lines), and $|eg\rangle$ (dotted line) including dephasing (thick lines) and without dephasing (thin lines). The vertical gray line indicates the frequency of the $T_1$ measurement in (a) and the spectroscopic measurements shown in Fig. 3.