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Basic principles( pink boxes) of a metrological protocol, using classical or quantum resources. The boxes and thought bubbles correspond to relevant actions of quantities which are specific to quantum protocols. The“ generation of the state” is considered as a black-box outputting a given quantum optical state( as an entangled or squeezed state). In“ Evolution”, a unitary operator generates a parameter depending evolution, and the resulting state now depends on the parameter to be estimated( τ in the example). The“ Measurement” step can also be formalized as the application of Positive Operator Valued Measurements( POVMs), and the probability of obtaining an outcome x depends on the parameter τ. In the post-processing, this probability can be used
to infer, by using an estimator K, the value of the parameter κ from the probability distribution P τ( x) obtained from the measurements. The precision δτ associated with this procedure on the obtained value of τ can be computed using the Cramér-Rao bound, where N is the number of repetitions of the experiment and F is the Fisher information( details in the main text). The Fisher information depends on the measurement performed( the POVM E^x). An optimization over all possible measurements leads to the quantum Fisher information and to the quantum Cramér-Rao bound, which can be up to quadratically smaller than the Cramér-Rao bound( depending on the chosen initial state), leading to a betterthan-classical precision. consequences of this fact are extreme: a finite visibility leads to the impossibility of reaching the QFI under any circumstances in this experiment. It is however possible to understand the approach to the maximal possible Fisher information in this set-up: the effects of a finite visibility highly depend on the spectrum of the entangled two-photon state( and not on its variance, as previously). In order to better understand these points, we recall the basic principles of the HOM experiment and interpret it as a metrological protocol.
In the HOM setup, two photons in two different propagation modes( as in the previous model) are injected into the two input ports of a 50 / 50 beam splitter. One of the modes is temporally delayed with respect to the other: this delay is the parameter to be measured, and the free propagation Hamiltonian is the time( parameter) dependent generator of the evolution.
After the beam-splitter, either both photons exit through the same port( bunching effect), or they exit through opposite ports( anti-bunching effect). Coincidence detection at the outputs determines whether the photons have bunched or not. The time delay in one path changes the distinguishability of the photonic modes, altering the coincidence detection probability and enabling measurements of time or path differences. The precision we compute is associated with this measurement, and the idea is to optimize the Fisher information associated with this quantity.
We have demonstrated in [ 7 ] how the spectral properties of the quantum state affect the scaling of precision with visibility. To explore this, we engineered various types of frequency entangled two-photon states and tuned the visibility of the HOM interference. The quantum source employed is an AlGaAs Bragg reflection waveguide, which generates polarization-entangled photon pairs via type-II spontaneous parametric down-conversion. As illustrated in Fig. 1, a continuous-wave laser with a wavelength of 772.42 nm pumps the waveguide, producing pairs of horizontally and vertically polarized photons that are collected in singlemode fibers and directed to a programmable filter. This filter enables the engineering of the spectral distribution( joint spectral amplitude) of the photons. This step corresponds to the“ Initial state” preparation in Box 1. At the filter output, the photon pairs are separated- the“ Evolution” step in Box 1- with horizontally and vertically polarized photons entering the HOM interferometer through arms 1 and 2, respectively. Precise control over polarization distinguishability— and consequently, HOM visibility— is achieved using two polarization controllers( FPC1 and
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