Estimation of atomic interaction parameters by photon counting

Detection of radiation signals is at the heart of precision metrology and sensing. In this article we show how the fluctuations in photon-counting signals can be exploited to optimally extract information about the physical parameters that govern the dynamics of the emitter. For a simple two-level emitter subject to photon counting, we show that the Fisher information and the Cramér-Rao sensitivity bound based on the full detection record can be evaluated from the waiting-time distribution in the fluorescence signal which can, in turn, be calculated for both perfect and imperfect detectors by a quantum trajectory analysis. We provide an optimal estimator achieving that bound.

Figure 1

The top panel shows the excited-state population resulting from a quantum trajectory simulation with ${\Omega }_0=5\Gamma$ and $\delta =0$. The two middle panels show the evolution conditioned on the same detection events as in the top panel, but assuming ${\Omega }_0/2$ and $3{\Omega }_0/2$, respectively. The bottom panel shows the evolution of the probabilities $P\left(\Omega \right)$ for the candidate values $\Omega ={\Omega }_0$ (solid, black, top curve at $t=25{\Gamma }^{-1}$), $\Omega ={\Omega }_0/2$ (dashed, blue, middle curve at $t=25{\Gamma }^{-1}$), and $\Omega =3{\Omega }_0/2$ (dotted, red, bottom curve at $t=25{\Gamma }^{-1}$), conditioned on the measurement record.

Figure 2

The upper panel illustrates a simulated data record assuming ${\Omega }_0=3\Gamma$ and $\delta =2\Gamma$. The lower panel shows the resulting evolution of a quasicontinuous probability distribution for the Rabi frequency $\Omega$. The total number of detection events is 51.

Figure 3

The blue noisy dots show the distribution from a simulated data record of 10 000 detection events. The parameters are $\delta =0$ and $\Omega =5\Gamma$. The red curve is the corresponding theoretical waiting-time distribution $w(\tau ,\theta )$ for time intervals between detector clicks.

Figure 4

The blue curves show the delay function (16) for the two-level atom, calculated for $\delta =0$, $\Omega =5\Gamma$, and for different values, $\eta =1,0.7,0.4$, and $0.1$, of the detector efficiency. The waiting time distributions approach exponential functions for long times (18), dashed, red curves in the figures), when the detector is imperfect. Notice that due to the missed detection events, the waiting time distribution extends over longer time when $\eta$ decreases.

Figure 5

The uncertainty, scaled by $\sqrt{N}$, $a(\Omega ,\eta )=\Delta S(\Omega ,\eta )\sqrt{N}$, in estimating the Rabi frequency $\Omega$ for a two-level system, driven on resonance ($\delta =0$). The left panel shows the dependence on the detector efficiency $\eta$ for different Rabi frequencies. The right panel displays the smooth dependence of $a(\Omega ,\eta )$ on both parameters $\Omega$ and $\eta$.

Figure 6

The solid blue line shows the estimate of the Rabi frequency as a function of the number of simulated photo detection events included in the estimate. The dashed red lines indicate the CRB sensitivity bound, enclosing the actual value (dotted, red line). Results are shown for $100<N\le 10000$. The parameter values in the simulation are ${\Omega }_0=5\Gamma$ and $\delta =0$.

Figure 7

The Fisher Information per detection even for estimation of the laser-atom detuning $\delta$ in a two-level system by unit efficiency photon counting. All statistical properties of the counting signal, and hence $F\left(\delta \right)$, are even functions of $\delta$. Results are shown for three different values of the Rabi frequency, and it is seen that the information in each detector click is maximized for finite detuning.