Figure 1

Diagram of the model. A series of $N$ two-level atoms are coupled resonantly to a cavity and equally driven by an external laser field. Light that leaks from the cavity is detected in a photodetector, triggering a feedback pulse.Reuse & Permissions

Figure 2

$N$-concurrence for various kinds of states as a function of the number of atoms. Vertical bars indicate the range of the concurrence for the many-body singlets forming the subspace of pure steady states of Eq. (3) with ${\gamma }_i=0$. Concurrence for GHZ, W, and linear-cluster states are shown by dashed, solid, and dotted lines, respectively. Solid squares represent the stationary states of Eq. (3) in the absence of feedback, when all atoms start in the ground state.Reuse & Permissions

Figure 3

Transitions between subspaces (columns) induced by the feedback operation. The gray doublepointed arrow shows the interaction via the unitary $\widehat{U}$, the dotted arrow shows the effect of the ${\widehat{J}}_{-}$ operator, while the solid black arrow shows the net effect of the combined $\widehat{U}{\widehat{J}}_{-}$ operation. Diagrams show transformations that result in jumps that can go from the left to the right subspace (a) (one-way transitions), or in both ways (b) (two-way transitions).Reuse & Permissions

Figure 4

Diagram of the angular momentum basis states for a four-partite system. States in the same $J$ subspace are grouped together with the vertical ordering following the different values of $J_z$. Different columns within the same $J$ subspace are labeled with different ${\lambda }_J^k$.Reuse & Permissions

Figure 5

Schematics of two different feedback schemes. Diagrams (a) and (b) show the connections between states induced by $\widehat{U}$ while (c) and (d) represent the effect of $\widehat{U}{\widehat{J}}_{-}$. The coupling given by the driving Hamiltonian is schematically shown in (e). One scheme [(a) and (c)] has one-way transitions between $J$ subspaces, and the other [(b) and (d)] has two-way transitions that still provide a path to the target state.Reuse & Permissions

Figure 6

Simulations showing the effectiveness of the two feedbacks shown in Fig. 5 with a spontaneous emission rate of ${10}^{-3}\Gamma$. Each curve shows the overlap of the system state with a set of basis states. The thin lines represent the result of a single trajectory simulation and show the overlap with $J=2$ (solid gray line), $J=1$ (dotted blue line) and the target steady state (solid red line). The thick red line shows the overlap with the target steady state for a full master equation simulations. The one-way feedback (a) takes fewer jumps and shorter time to reach the steady state than the two-way feedback (b). The dips indicated by the arrows in each of these simulations represent spontaneous emission events.Reuse & Permissions

Figure 7

Overlap between the final state and the target steady state for the feedback of Eq. (15) with $a_1=a_2=A$ and $a_3=a_4=-A$ as a function of A. The simulations show results without spontaneous emission (dotted blue line) and with $\gamma =\Gamma \times {10}^{-3}$ (solid red line). For both of these simulations the system started with all atoms in the ground state. Spontaneous emission induces transitions to trapped states that do not connect to the target state, considerably decreasing the overlap.Reuse & Permissions

Figure 8

Overlap between the final state and the target steady state for the feedback in Eq. (17) as a function of $A$ and $\epsilon$. The spontaneous emission rates for these simulations is $\gamma =\Gamma \times {10}^{-3}$. As in Fig. 7, there are dips in the overlap when $A=k\pi$. For $\epsilon \ne 0$ the overlap gets closer to one as compared to the curve in Fig. 7.Reuse & Permissions