Bayesian parameter inference from continuously monitored quantum systems

We review the introduction of likelihood functions and Fisher information in classical estimation theory, and we show how they can be defined in a very similar manner within quantum measurement theory. We show that the stochastic master equations describing the dynamics of a quantum system subject to a definite set of measurements provides likelihood functions for unknown parameters in the system dynamics, and we show that the estimation error, given by the Fisher information, can be identified by stochastic master equation simulations. For large parameter spaces we describe and illustrate the efficient use of Markov chain Monte Carlo sampling of the likelihood function.

Figure 1

The top panel shows the three components of the Bloch vector for a two-level atom subject to laser excitation with Rabi frequency $\Omega =1.3$ and detuning $\Delta =1.43$ (dimensionless units). The atomic inversion is represented by the lower (blue) curve. The atom decays with a rate $\gamma =0.55$, leading to the observation of quantum jumps at the instants indicated by vertical dashed lines and the transient Bloch vector dynamics. The bottom panel shows the estimation of the detuning $\Delta$, treated as an unknown variable with a probability distribution, which is updated in a Bayesian manner, conditioned on the measurement record. See text.

Figure 2

The left panels show histograms of Markov chain Monte Carlo sampled distributions of the parameters $\Omega ,\Delta ,\gamma$ in our two-level atom model. The prior knowledge of the parameters assumes normal distributions, shown by the dashed lines, with mean values ${\mu }_{\Omega }=2.0$, ${\mu }_{\Delta }=3.0$, and ${\mu }_{\kappa }=1.0$ and standard deviations ${\sigma }_{\Omega }=0.8$, ${\sigma }_{\Delta }=1.0$, and ${\sigma }_{\kappa }=0.5$. The right panels display the correlations between the different pairs of sampled parameters.

Figure 3

Fisher Information matrix components for photodetection of a decaying two-level atom with decay rate $\gamma =0.55$ initially in the unexcited state up to $T=40$. The two top panels show the diagonal elements $I_{\Omega ,\Omega }$ and $I_{\Delta ,\Delta }$ and the bottom left panel shows the off-diagonal element $I_{\Omega ,\Delta }$ of the Fisher information matrix. The bottom right panel shows the relative entropy between the signal probability distribution $p$ and $p_0$, where $p_0$ is a Poissson process. Unlike the Fisher information shown in panels (a)–(c), the relative entropy does depend on the parameter $\lambda$, chosen for $p_0$, and it is shown here with the value of the stationary emission rate for the two-level atom ${\lambda }_{st}={\Omega }^2\gamma /({\gamma }^2+4{\Delta }^2+2{\Omega }^2)$ and $\gamma =0.55$. See text.

Figure 4

Simulated signal from a bimodal two-level atom, undergoing jumps and coherent evolution with two alternating sets of parameters, $a$ and $b$. The values used for this trajectory are ${\Omega }_a=1.1$, ${\Delta }_a=1.3$, ${\gamma }_a=1.6$, ${\Omega }_b=2.2$, ${\Delta }_b=0.2$, ${\gamma }_b=2.4$, and transition rates $W(a\to b)=0.03$ and $W(b\to a)=0.08$ (dimensionless units). The solid black curve is the (binned) observed signal while the red dashed curves show the mean expected counts for the two sets of parameters.

Figure 5

Marginal distributions for the eight unknown parameters in our bimodal two-level atomic system. All prior distributions were taken to be uniform on the shown intervals as indicated by the black dashed line. The estimation was based on the actual click events of the trajectory, represented in Fig. 4.