Figure 1

Schematic of circuit QED together with a microwave transmission measurement and a measurement-current-based feedback loop. The Cooper-pair box qubits are fabricated inside a superconducting transmission-line resonator and are capacitively coupled to the voltage standing wave.Reuse & Permissions

Figure 2

(a) Average concurrence over 500 trajectories based on Eq. (4) and a direct current feedback $H_{\mathrm{fb}}=u{I}_{\mathrm{hom}}(t)J_x$. (b, c) Concurrence obtained by a feedback $H_{\mathrm{fb}}=u\langle J_z\rangle {}_c(t)J_x$, i.e., removing the noise in the homodyne current. In (b), the average, and in (c), the individual quantum trajectory concurrences are illustrated, both showing considerable improvement over the result in (a). Parameters: ${\gamma }_p={\Gamma }_d$, $u=0.1$ in (a) and $u=1.0$ in (b) and (c). Here the choice of different feedback strength $u=0.1$ in (a) is from a rough numerical optimization since an increase of $u$ over this value will give a poorer result instead.Reuse & Permissions

Figure 3

(a) Average concurrence over 500 trajectories and (b) the concurrence of three representative individual trajectories by applying a filtered-current-based feedback control. Parameters: ${\gamma }_p={\Gamma }_d$, $u=10$, ${\gamma }_{\mathrm{ft}}=0.006{\Gamma }_d$, and $T=2000\hspace{0.5em}\mathit{dt}$.Reuse & Permissions

Figure 4

Environment-caused deterioration of the Bell state $|{\Phi }_{+}\rangle$ in the absence of feedback protection. (a) Average fidelity (over 1000 trajectories). (b) Fidelity of a representative trajectory (blue curves). The result with feedback is plotted here for comparison (red curves). The feedback parameters are the same as in Fig. 3.Reuse & Permissions

Figure 5

Justification of various approximations, including the JC model employed in Eq. (1), the dispersive effective Hamiltonian [Eq. (2)], and the adiabatic elimination of cavity photons leading to Eq. (3), through single-qubit decoherence evolution under measurements for (a) small and (b) large detuning, i.e., with $g/\Delta =0.5$ and 0.05, respectively. The result without rotating-wave approximation for qubit-cavity interaction is plotted by the solid (red) line, while the one from the JC model is plotted by the dotted (black) line. Also, by the dashed (blue) and dash-dotted (green) lines, we display the results without and with adiabatic elimination of the cavity photons, commonly based on the dispersive effective Hamiltonian [Eq. (2)]. Inset of (b) shows the entire landscape of $\mathrm{Re}{\rho }_{01}(t)$, showing the decoherence dynamics of qubit under dispersive measurement. Parameters (in units of $g$): ${\omega }_r={\omega }_m=2020$, $\Omega =2000$, $\epsilon =1$, and $\kappa =2$.Reuse & Permissions