Metrology with Unknown Detectors

The best possible precision is one of the key figures in metrology, but this is established by the exact response of the detection apparatus, which is often unknown. There exist techniques for detector characterization that have been introduced in the context of quantum technologies but apply as well for ordinary classical coherence; these techniques, though, rely on intense data processing. Here, we show that one can make use of the simpler approach of data fitting patterns in order to obtain an estimate of the Cramér-Rao bound allowed by an unknown detector, and we present applications in polarimetry. Further, we show how this formalism provides a useful calculation tool in an estimation problem involving a continuous-variable quantum state, i.e., a quantum harmonic oscillator.

Figure 1

Reconstructed Fisher information for two-outcome polarization measurements. The experimental points correspond to intensity measurements taken on a He:Ne laser with an optical power meter. The solid line indicates the prediction based on a model of the measurement. (Inset) Experimental setup, consisting of a half-wave plate and a polarizing beam splitter separating the horizonal and vertical components.

Figure 2

Reconstructed Fisher information for a four-outcome polarization measurement on either the $D/A$ (Pauli $X$) or $H/V$ (Pauli $Z$) basis: $\varphi$ is the scanned parameter, while $\chi$ is kept constant at the value 0. The experimental points correspond to intensity measurements taken on a He:Ne laser with an optical power meter: yellow, $F_{\chi ,\chi }$; blue, $F_{\varphi ,\varphi }$; green, $F_{\chi ,\varphi }$. The solid line indicates the prediction based on the reconstructed detector tomography [13]. (Inset) Experimental setup, consisting of a nonpolarizing beam splitter (BS) and two polarizers along different directions ($H/V$ in the upper arm, $D/A$ in the lower arm). The asymmetry in the Fisher information associated with the two parameters arises from the dependence of the splitting ration of BS on the polarization.

Figure 3

Fisher information associated with different outcomes of a weak-field homodyne for coherent states of real amplitude $\alpha$, for different local oscillators $\gamma$. The outcomes are in the order $(x_1,x_2)=(1,0)$, (0,1), (1,1), (2,1), (3,1), and (4,1).The (0,1) term is negligibly small, and so are other contributions in this regime.

Figure 4

Fisher information for a phase measurement ($\varphi =0.1$) with squeezed probe states of given energy ${\alpha }^2$ divided partly into the displacement with $r_d={\alpha }_0^2/{\alpha }^2$, and the rest into squeezing in the $P$ direction. We report $r_d=1$ in blue, $r_d=0.95$ in red, and $r_d=0.9$ in green.