Figure 1

(Color online) Proposed quantum feedback scheme adapted to a microwave cavity QED setup. $C$: high-$Q$ microwave cavity, $B$: box producing Rydberg atoms, $R_1$ and $R_2$: low-$Q$ Ramsey cavities, $D$: atomic field-ionization detector, $S$ and $S^{\prime }$: microwave sources coupled to $C$ and $R$’s, respectively. In a quantum filtering process, a real-time control system analyzes the results of QND measurements of the cavity field and computes the amplitude of a control injection pulse.Reuse & Permissions

Figure 2

(Color online) A single closed-loop quantum trajectory of an ideal measurement with $|n_{\text{tag}}⟩=|3⟩$. (a) Atomic states detected at each feedback cycle. (b) Control injections ${\alpha }_k$. (c) Evolution of photon-number probabilities $P\left(n<n_{\text{tag}}\right)$, $P\left(n_{\text{tag}}\right)$, and $P\left(n>n_{\text{tag}}\right)$ shown as dashed-dot (blue), solid (green), and dashed (red) curves, respectively. (d) Average over ${10}^4$ closed-loop quantum trajectories. Line styles are the same as in (c).Reuse & Permissions

Figure 3

(Color online) A single closed-loop quantum trajectory for realistic experimental parameters with $|n_{\text{tag}}⟩=|3⟩$. (a) Sequence of detected atomic states. (b) Control injections ${\alpha }_k$. (c) Evolution of the photon-number probabilities $P\left(n<n_{\text{tag}}\right)$, $P\left(n_{\text{tag}}\right)$, and $P\left(n>n_{\text{tag}}\right)$ of the field state ${\rho }^{\text{real}}$ deduced from the Monte Carlo simulation shown as dashed-dot (blue), solid (green), and dashed (red) curves, respectively. The thin (black) line gives the estimated state fidelity $F\left({\rho }_k\right)$. (d) Photon-number probabilities for $n=0–9$ in ${\rho }^{\text{real}}$. For better clarity, results of only one out of each ten feedback cycles are shown. (e) Average over ${10}^4$ closed-loop quantum trajectories. Line styles are the same as in (c). If the number of quantum trajectories is large enough (as is the case here), the averaged photon-number probabilities calculated from the real field states ${\rho }^{\text{real}}$ coincide with those averaged over the estimated states.Reuse & Permissions

Figure 4

(Color online) Feedback performance in the presence of a sudden photon-number loss. The fidelity of the actual cavity state ${\rho }_k^{\text{real}}$ (solid blue curve) and of the state estimated by the quantum filtering process ${\rho }_k$ (dashed red curve) are defined with respect to the target Fock state and are obtained by averaging over ${10}^4$ quantum trajectories. In each trajectory, the time origin is shifted to the time of a quantum jump.Reuse & Permissions

Figure 5

(Color online) Feedback convergence. (a) Histogram of convergence times with $F_{\text{conv}}=0.95$. The solid line gives the probability of the feedback convergence after a given time. The inset shows the average density matrix of the converged states. (b) Convergence for several different Fock states. Lower lines correspond to higher photon numbers. These curves result from an average over ${10}^4$ quantum trajectories.Reuse & Permissions