Hidden Markov model of atomic quantum jump dynamics in an optically probed cavity

We analyze the quantum jumps of an atom interacting with a cavity field, where strong coupling makes the cavity transmission depend on the time-dependent atomic state. In our analysis we employ a Bayesian approach that conditions the population of the atomic states at time $t$ on the cavity transmission observed both before and after $t$, and we show that the state assignment by this approach is more decisive than the usual conditional quantum states based on only earlier measurement data. We also provide an iterative protocol which, together with the atomic state populations, simultaneously estimates the atomic jump rates and the transmission signal distributions from the measurement data. Finally, we take into account technical fluctuations in the observed signal, e.g., due to spatial motion of the atom within the cavity, by representing atomic states by several hidden states, thereby significantly improving the state's recovery.

Figure 1

A single cesium atom is held by a dipole trap (DT) inside the mode of a high-finesse optical cavity. The transmission of a probe laser is observed with a single-photon counting module (SPCM).

Figure 2

Simplified level scheme of the cesium atom and the lasers used in the experiment.

Figure 3

(a) Detected cavity transmission of photons binned in 1 ms time intervals. In the shaded region I (II) the lower transmission level exhibits a slightly higher (lower) count rate (see Sec. 4). (b) The same data as in (a) binned in 50 $\mu$s time intervals. (c) Zoomed time region showing the individual detector clicks, experiencing a pronounced change in click rate as reflected also in (a) and (b).

Figure 4

(a) The time-dependent probability for the atom to be in $|F=4\rangle$, inferred from the forward Bayesian analysis of the measured data. The two states with slightly different transmission characteristics hidden in $|F=4\rangle$ are indicated with dark and light blue color. (b) The atomic state populations conditioned on the forward-backward Bayesian state estimation. In both figures, the state is conditioned on the data shown in Fig. 3.

Figure 5

State dependent photon count distribution for a time of 50 $\mu$s obtained after 200 iterations of the re-estimation algorithm described in the text. The photon count distribution with high mean value (middle, down hatched columns) is associated with the $|F=3\rangle$ state, while the HMM algorithm yields two states with low transmission that we ascribe to the $|F=4\rangle$ hyperfine state (left and right, shaded and up hatched columns).