Figure 1

The results of a single numerical simulation of the stochastic Schrödinger equation (SSE), Eq. (5), with $M=1$, $\eta =1$, and $N=10$ spins initially aligned along the $x$ axis. (A) In a quantization axis perpendicular to the polarization, the level distribution of a coherent spin state (CSS) is Gaussian for large $N$. Under conditional measurements the state evolves at short times into a spin-squeezed state and, eventually, into a random eigenstate of $J_z$. (B) A map of the state’s angular distribution on the Bloch sphere in spherical coordinates. The uncertainty in the transverse direction to the measurement axis grows until there is no information about the perpendicular component direction. (C) At long times, the population is at most divided among two levels that compete to be the final winner, which in this case appears to be $m=1$. (D) All of the state information is obtained by properly filtering the noisy photocurrent.Reuse & Permissions

Figure 2

Many open-loop moment trajectories of the SSE, Eq. (5). The trajectory of Fig. 1 is darkened. (A) At short times, the evolution of the variance (shown on a log scale) is deterministic and given by ${⟨\Delta J_z^2⟩}_s\left(t\right)$. At long times, the variances become stochastic but bounded (above by $1∕4$ and below by $\exp [-2(Mt-1)]∕4$). The average of all 10 000 trajectories (only 10 are shown) gives. $E\left[⟨\Delta J_z^2⟩\left(t\right)\right]$. (B) The projective nature of the measurement is made clear by the evolution of 100 trajectories of $⟨J_z⟩\left(t\right)$. The distribution of the final results is given by the first histogram of Fig. 1(A). (C) The evolution of the 100 trajectories all starting in an $x$-polarized CSS. When $\eta =1$, certain regions of Hilbert space are forbidden by the evolution.Reuse & Permissions

Figure 3

One hundred closed-loop moment trajectories of the SSE with feedback law $b\left(t\right)=\lambda ⟨J_x{J}_z+J_z{J}_x⟩\left(t\right)∕2$ and $\lambda =10$ chosen from numerical considerations. (A), (B) If the control is successful, the quantity $⟨J_z^2⟩\left(t\right)$ should go to zero on every trial. For this controller the number of successful trajectories is increased significantly (from 25% to 90%), but the remaining fraction is attracted to neighboring fixed points, causing the mean $\mathrm{E}\left[⟨J_z^2⟩\left(t\right)\right]$ to saturate at a nonzero value. Although the successful fraction converges exponentially, the fastest converging trajectories are slower than in the open-loop case. This is evident in (C) as the converging trajectories have visibly not yet reached $⟨J_x⟩=0$ at time $t=5$.Reuse & Permissions

Figure 4

One hundred closed-loop moment trajectories of the SSE with feedback law $b\left(t\right)=\lambda ⟨J_z⟩\left(t\right)$ and $\lambda =10$ chosen from numerical considerations. (A) The average over 10 000 trajectories suggests that with this control law the mean $\mathrm{E}\left[⟨J_z^2⟩\left(t\right)\right]$ descends to zero exponentially and the target state is deterministically prepared. (B) Despite a number of early excursions, all 100 trajectories shown converge to the desired value of $m=0$. (C) Those trajectories that do not descend to the goal directly (about 10 of 100) are recycled and rotated back into the attractive region of the target state. Again, the control slightly compromises the best-case convergence rate and the trajectories have a nonzero (but still decreasing) $⟨J_x⟩$ at $t=5$.Reuse & Permissions