Gaussian-state interferometry with passive and active elements

Abstract

We address precision of optical interferometers fed by Gaussian states and involving passive and/or active elements, such as beam splitters, photodetectors and optical parametric amplifiers. We first address the ultimate bounds to precision by discussing the behaviour of the quantum Fisher information. We then consider photodetection at the output and calculate the sensitivity of the interferometers taking into account the non unit quantum efficiency of the detectors. Our results show that in the ideal case of photon number detectors with unit quantum efficiency the best configuration is the symmetric one, namely, passive (active) interferometer with passive (active) detection stage: in this case one may achieve Heisenberg scaling of sensitivity by suitably optimizing over Gaussian states at the input. On the other hand, in the realistic case of detectors with non unit quantum efficiency, the performances of passive scheme are unavoidably degraded, whereas detectors involving optical parametric amplifiers allow to fully compensate the presence of loss in the detection stage, thus restoring the Heisenberg scaling.

Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers

“…The second stage is the measurement of some aspect of this quantum state, from which the phase shift is inferred. An important development in our understanding of quantum sensing is the realization that analysis of the first stage alone, independent of the second stage, sets limits on the potential phase sensitivity of the interferometer [10]. These limits on the best phase sensitivity of an interferometer can be quantified using the Fisher information.…”

“…We can also calculate a Fisher information associated with the entire apparatus including detection, F C , which is necessarily less than or equal to F Q . These are conventionally referred to as the quantum (F Q ) and classical (F C ) Fisher information [10].…”

Quantum optical metrology in the lossy SU(2) and SU(1,1) interferometers

“…We see that the QCRB in the presence of the external photon loss is always smaller than the ideal case. Only for some exceptional points, α = 0 (squeezed vacuum state) and r = 0 (coherent state), they can be equal to ∆φ = 1/(2α) and √ 2Y , respectively [13,14]. In other words, by increasing the amplifier gain, the effects of the detector inefficiency on the QCRB cannot be asymptotically canceled out for a generic coherent-squeezed state.…”

“…Now we investigate the photon loss at the detectors, due to imperfect detectors [13,14]. This external loss can also be modeled by a fictitious beam splitter as the internal loss, where the transmissivity parameter ξ k is related to the detection inefficiency.…”

“…We will find that the squeezed component of the input state in the lossy SU (1, 1) interferometer is effectively unnecessary. It will be also shown that by increasing the amplifier gain of the OPA before the final detection, the effects of the detector inefficiency on the phase sensitivity cannot be asymptotically removed for a generic coherentsqueezed state, except for the coherent state [13] and the squeezed vacuum state [14].…”