Cramér-Rao bound for time-continuous measurements in linear Gaussian quantum systems

We describe a compact and reliable method to calculate the Fisher information for the estimation of a dynamical parameter in a continuously measured linear Gaussian quantum system. Unlike previous methods in the literature, which involve the numerical integration of a stochastic master equation for the corresponding density operator in a Hilbert space of infinite dimension, the formulas here derived depend only on the evolution of first and second moments of the quantum states and thus can be easily evaluated without the need of any approximation. We also present some basic but physically meaningful examples where this result is exploited, calculating analytical and numerical bounds on the estimation of the squeezing parameter for a quantum parametric amplifier and of a constant force acting on a mechanical oscillator in a standard optomechanical scenario.

Figure 1

FI $F_t\left(\chi \right)$ for continuous homodyne and heterodyne detection as a function of time and for $\chi =-0.2\kappa$ (numerical evaluation with 2000 trajectories). Green dotted line: Homodyne detection of quadrature $\widehat{x}$. Red dashed line: Homodyne detection of quadrature $\widehat{p}$. Blue solid line: Heterodyne detection.

Figure 2

FI $F_t\left(\lambda \right)$ for continuous homodyne detection of the cavity field quadrature ${\widehat{p}}_c$ as a function of time and for different values of the cavity decay rate: $\kappa ={\omega }_m/2$, red solid line; and $\kappa ={\omega }_m/10$, green dashed line; $\kappa ={\omega }_m/20$, blue dotted line (the other parameters are chosen as $g={\omega }_m/2, \gamma ={\omega }_m/3$, and $\eta =1$). The inset shows the behavior of the FI at small times.