Bayesian Feedback Control of a Two-Atom Spin-State in an Atom-Cavity System

We experimentally demonstrate real-time feedback control of the joint spin-state of two neutral cesium atoms inside a high finesse optical cavity. The quantum states are discriminated by their different cavity transmission levels. A Bayesian update formalism is used to estimate state occupation probabilities as well as transition rates. We stabilize the balanced two-atom mixed state, which is deterministically inaccessible, via feedback control and find very good agreement with Monte Carlo simulations. On average, the feedback loop achieves near optimal conditions by steering the system to the target state marginally exceeding the time to retrieve information about its state.

Figure 1

(a) Schematic of experimental setup, (b) simplified level scheme of Cs, (c) two-atom states and transition rates of probe ($R_{21}$,$R_{10}$), repumping ($R_{\mathrm{r}}$) and depumping ($R_d$) laser.

Figure 2

(a) Detected number of transmitted photons $n(t_i)$ through the cavity for continuous weak repumping for a bin time of $\Delta t=1\mathrm{ms}$. Right: The corresponding photon count histogram of $n(t_i)$ (black curve) is shown in comparison to separately measured photon count histograms $p[n|\alpha ]$ for $\alpha =0,1,2$ coupled atoms (shaded areas). (b) Estimated probabilities $p_{\alpha }^{\mathrm{post}}(t_i)$, which are computed in real time according to the Bayesian update formalism based on $n(t_i)$ as shown in (a). (c) Evolution of the Bayesian probabilities of the quantum jump rates starting from a uniform a priori distribution over 19 successive transmission measurements of 300 ms each. (a.u., arbitrary units.)

Figure 3

(a) Detected number of transmitted photons $n(t_i)$ through the cavity while feedback onto the two-atom state ${\widehat{\rho }}_{\alpha =1}$ is applied. The vertical lines indicate repumping and depumping feedback pulses, respectively. Note that the photon count histogram of $n(t_i)$ (black) resembles the histogram for exactly one coupled atom to a high degree as shown in the right figure. (b) Estimated Bayesian state probabilities $p_{\alpha }^{\mathrm{post}}(t_i)$ calculated in real time.

Figure 4

Comparison of mean probabilities for the case of feedback (a) and continuous weak repumping experiments for measured (c) and optimal repumping rate (b). The shaded bars indicate simulated results. (d) Analytical solution of the mean probabilities for different repumping rates $R_{\mathrm{r}}$.