Quantum advantage of time-reversed ancilla-based metrology of absorption parameters

Quantum estimation of parameters defining open-system dynamics may be enhanced by using ancillas that are entangled with the probe but are not submitted to the dynamics. Here we consider the important problem of estimation of transmission of light by a sample, with losses due to absorption and scattering. We show, through the determination of the quantum Fisher information, that the ancilla strategy leads to the best possible precision in single-mode estimation, the one obtained for a Fock state input, through joint photon-counting of probe and ancilla, which are modes of a bimodal squeezed state produced by an optical parametric amplifier. This proposal overcomes the challenge of producing and detecting high photon-number Fock states, and it is quite robust against additional noise: we show that it is immune to phase noise and the precision does not change if the incoming state gets disentangled. Furthermore, the quantum gain is still present under moderate photon losses of the input beams. We also discuss an alternative to joint photon counting, which is readily implementable with present technology, and approaches the quantum Fisher information result for weak absorption, even with moderate photons losses of the input beams before the sample is probed: a time-reversal procedure, placing the sample between two optical parametric amplifiers, with the second undoing the squeezing produced by the first one. The precision of estimation of the loss parameter is obtained from the average outgoing total photon number and its variance. In both procedures, the state of the probe and the detection procedure are independent of the value of the parameter.