Bayesian parameter estimation by continuous homodyne detection

We simulate the process of continuous homodyne detection of the radiative emission from a quantum system, and we investigate how a Bayesian analysis can be employed to determine unknown parameters that govern the system evolution. Measurement backaction quenches the system dynamics at all times and we show that the ensuing transient evolution is more sensitive to system parameters than the steady state of the system. The parameter sensitivity can be quantified by the Fisher information, and we investigate numerically and analytically how the temporal noise correlations in the measurement signal contribute to the ultimate sensitivity limit of homodyne detection.

Figure 1

(a) Schematic experimental setup for balanced homodyne detection. The emission from the probed quantum system (here a two-level system in a cavity) is mixed with a strong local oscillator field with phase $\mathrm{\Phi }$ in a 50/50 beam splitter. The output ports are monitored by photo detectors and the homodyne current $J\left(t\right)$ is the difference between the two signals. (b) An example of a homodyne current obtained by simulating the monitoring of a two-level system according to Eqs. (2), (4) is shown as the noisy green curve. The mean signal is indicated by the dashed purple line. The signal shows nontrivial temporal correlations, e.g., between the different times connected with arrows in the figure. (c) The two-time homodyne current correlations Eq. (16) averaged over 5000 independent realizations of $J\left(t\right)$. The quantum regression theorem Eq. (17) yields the theoretical results shown with purple and red dashed curves. The inset shows the power spectrum of the homodyne current Eq. (18) for a Rabi frequency of ${\mathrm{\Omega }}_0=4\gamma$ for two choices of the local oscillator phase, $\mathrm{\Phi }=0$ (${\widehat{\sigma }}_{\mathrm{x}}$-probing) and $\mathrm{\Phi }=\pi /2$ (${\widehat{\sigma }}_{\mathrm{y}}$-probing).

Figure 2

The evolution of the quasicontinuous probability distribution $P\left(\mathrm{\Omega }\right|D_t)$ for the Rabi frequency of a resonantly driven two-level system, conditioned on noisy homodyne measurement records [Fig. 1]. The red lines track the most likely parameter value $S_{\mathrm{\Omega }}\left(D_t\right)={\max }_{\mathrm{\Omega }}\left[P\left(\mathrm{\Omega }\right|D_t)\right]$, and the dashed, black lines show the FWHM of $P\left(\mathrm{\Omega }\right|D_t)$. Results are shown for two choices of the local oscillator phase, (a) $\mathrm{\Phi }=\pi /2$ (${\widehat{\sigma }}_{\mathrm{y}}$ probing) and (b) $\mathrm{\Phi }=0$ (${\widehat{\sigma }}_{\mathrm{x}}$ probing). The true value ${\mathrm{\Omega }}_0=2\gamma$, assumed for the Rabi frequency, is gradually identified in each simulation.

Figure 3

Contributions to the Fisher information ${\mathcal{I}}_{\mathrm{\Phi }}\left(\mathrm{\Omega }\right)$ per time $T$ for Rabi frequency estimation in a resonantly driven two-level system. The Fisher information is upper bounded by the value $4T/\gamma$ of the QFI, which is independent of the Rabi frequency, and it is lower bounded by the information retrieved by the integrated (mean) signal ${\mathcal{I}}_{\mathrm{\Phi }}^{\left(1\right)}\left(\mathrm{\Omega }\right)$. ${\mathcal{I}}_{\mathrm{\Phi }}^{\left(2\right)}\left(\mathrm{\Omega }\right)$ describes the information provided by the two-time correlations in the homodyne current. Results are shown in (a) as a function of the local oscillator phase $\mathrm{\Phi }$ for $\mathrm{\Omega }=\gamma$ and in (b) as a function of the Rabi frequency $\mathrm{\Omega }$ for $\mathrm{\Phi }=\pi /2$. In (c), the value of ${\mathcal{I}}_{\mathrm{\Phi }}^{\left(1\right)}\left(\mathrm{\Omega }\right)+{\mathcal{I}}_{\mathrm{\Phi }}^{\left(2\right)}\left(\mathrm{\Omega }\right)$ per time $T$ is indicated by the color scheme as a function of $\mathrm{\Phi }$ and $\mathrm{\Omega }$, and the maximum value is tracked by the black line.

Figure 4

The full classical Fisher information for homodyne detection (markers) is shown for different values of the detector efficiency $\eta$ and compared to the combined information from the integrated signal and two-time correlations ${\mathcal{I}}_{\pi /2}^{\left(1\right)}\left(\mathrm{\Omega }\right)+{\mathcal{I}}_{\pi /2}^{\left(2\right)}\left(\mathrm{\Omega }\right)$ (lines in the same order). The results are shown as functions of the actual Rabi frequency $\mathrm{\Omega }$ for probing with the local oscillator phase $\mathrm{\Phi }=\pi /2$.