Fisher Information and the Quantum Cramér-Rao Sensitivity Limit of Continuous Measurements

Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramér-Rao sensitivity limit for parameter estimation by continuous measurements on an open quantum system. We illustrate our theory by its application to resonance fluorescence from a laser-driven two-state atom; we show that photon counting and homodyne detection records yield different sensitivity to the atomic parameters, while none of them exceed our general result.

Figure 1

A quantum system interacts with the quantized radiation field (the environment) through emission of photons. Under continuous probing until time $T$, the emitter quantum state evolves in a stochastic manner, while in the absence of observation, the full system and environment may at time $T$ be described by a pure quantum state $|\psi ,T⟩$.

Figure 2

Fisher information, divided by the total measurement time, for the probing of light emission by a two-level atom. The upper panel shows the Fisher information for estimation of the detuning $\mathrm{\Delta }$, the middle (lower) panel shows the Fisher information for estimation of the Rabi frequency $\mathrm{\Omega }$ (decay rate $\kappa$). Within each plot, the solid line shows the quantum Fisher information, which bounds any measurement strategy, while the long (short) dashed lines show the Fisher information for photon counting (homodyne detection).