Figure 1

$1/\sqrt{F}$, the SNR resulting from Eq. (8), normalized to the ideal case of the QP limit ($F=1$). This plot assumes $\epsilon =0.1$. When few photons per particle can be detected, i.e., when $\epsilon /b_{\ell }\ll 1$ (far left of plot), cycling to very deep completion $(b_{\ell }rT\gg 1$) does not significantly affect the SNR. Even when one photon per particle can be detected on average, i.e., when $\epsilon /b_{\ell }=1$ (dashed red line), the SNR never exceeds roughly half its ideal value. By further closing the optical cycle, i.e., such that $\epsilon /b_{\ell }\gg 1$ (right of dashed red line), the SNR can be improved to near the optimal value given by the QP limit. However, to reach this optimal regime, the number of photons that would be scattered in the absence of dark states, $rT$, must be small compared to the average number that can be scattered before a particle exits the optical cycle, $1/b_{\ell }$. For example, with $1/b_{\ell }=1000$ (dashed green line) and $rT=100$ so that $b_{\ell }rT=0.1$ (lower circle), the SNR is more than 30% larger than in the case when $rT=10000$ and $b_{\ell }rT=10$ (upper circle).