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arXiv:1911.12912 · Physical Review A

Data processing over single-port homodyne detection to realize superresolution and supersensitivity

Jia-Min Xu, Aixi Chen, Wen Yang, Guangri Jin

Published Dec 23, 2019• 12 citations
arXivHTMLPDF

Abstract

Performing homodyne detection at one port of a squeezed-state light interferometer and then binarizing the measurement data are important to achieve superresolving and supersensitive phase measurements. Here we propose an alternative data-processing technique by dividing the measurement quadrature into three bins (equivalent to a multi-outcome measurement), which leads to a higher improvement in the phase resolution and the phase sensitivity under realistic experimental conditions. Furthermore, we develop an alternative phase-estimation protocol based on a combination of the inversion estimators of each outcome and show that the estimator can saturate the Cram\'er-Rao lower bound, similar to the asymptotically unbiased maximum likelihood estimator.

DOI
10.1103/physreva.100.063839

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Performing homodyne detection at one port of a squeezed-state light interferometer and then binarizing the measurement data are important to achieve superresolving and supersensitive phase measurements. Here we propose an alternative data-processing technique by dividing the measurement quadrature into three bins (equivalent to a multi-outcome measurement), which leads to a higher improvement in the phase resolution and the phase sensitivity under realistic experimental conditions. Furthermore, we develop an alternative phase-estimation protocol based on a combination of the inversion estimators of each outcome and show that the estimator can saturate the Cram'er-Rao lower bound, similar to the asymptotically unbiased maximum likelihood estimator.
Performing homodyne detection at a single output port of a squeezed-state light interferometer and then separating the measurement quadrature into several bins can realize superresolving and supersensitive phase measurements. However, the phase resolution and the achievable phase sensitivity depend on the bin size that is adopted in the data processing. By maximizing classical Fisher information, we analytically derive an optimal value of the bin size and the associated best sensitivity for the case of three bins, which can be regarded as a three-outcome measurement. Our results indicate that both the resolution and the achievable sensitivity are better than that of the previous binary–outcome case. Finally, we present an approximate maximum Likelihood estimator to asymptotically saturate the ultimate lower bound of the phase sensitivity.
Abstract Performing homodyne detection at a single output port of a squeezed-state light interferometer and then separating the measurement quadrature into two intervals can realize super-resolving and super-sensitive phase measurements, which is equivalent to a binary-outcome measurement. Obviously, the single-port homodyne detection may lose almost part of the phase information, reducing the estimation precision. Here, we propose a data-processing technique over the double-port homodyne detection, where the two-dimensional measurement quadrature (p 1, p 2) has been divided into two regions. With such a binary-outcome measurement, we estimate the phase shift accumulated in the interferometer by inverting the output signal. By analyzing the full width at half maximum of the signal and the phase sensitivity, we show that both the resolution and the achievable sensitivity are better than that of the previous binary-outcome scheme.
As the simplest phase estimation protocol, the inversion estimator has been widely adopted in the quantum metrology, since its performance can be simply quantified by the error-propagation formula. For a general binary-outcome measurement, we show that both the inversion estimator and the maximum likelihood estimator are the same and can asymptotically saturate the Cramér-Rao lower bound. These observations are applied to the parity detection and the zero-nonzero photon counting that have achieved super-resolved phase measurement near the shot-noise phase sensitivity in a coherent-state light Mach–Zehnder interferometer. We numerically show that the phase uncertainty almost follows the theoretical prediction of the lower bound. At the optimal working point, the best sensitivities also reach their associated analytic results for the two binary-outcome measurement schemes.
The Cramér-Rao bound plays a central role in both classical and quantum parameter estimation, but finding the observable and the resulting inversion estimator that saturates this bound remains an open issue for general multi-outcome measurements. Here we consider multi-outcome homodyne detection in a coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cramér-Rao bound over the whole range of phase interval. This provides a clue on constructing optimal inversion estimators for phase estimation and other parameter estimation in any multi-outcome measurement.
Quantum metrology can beat the shot-noise limit (SNL) with nonclassical resources, yet simultaneously realizing sub-Rayleigh resolution and sub-SNL sensitivity in a deployable interferometer remains challenging. Here we experimentally validate a deterministic phase-measurement protocol that combines single-port balanced homodyne detection in a Mach-Zehnder interferometer with ternary-binning post-processing. For a coherent probe with a mean photon number of about 67, ternary binning preserves sub-Rayleigh resolution while mitigating the near-zero-phase ill-conditioning of binary-binning readout, yielding a 7.15-fold resolution enhancement relative to the unprocessed interferometer. With 5.99 ± 0.13 dB squeezed-state injection into the vacuum port, the resolution enhancement increases to 13.69-fold, and sub-SNL sensitivity is achieved; the minimum sensitivity improves by a factor of 1.78 compared with the coherent probe. The protocol is deterministic and experimentally accessible, offering a practical route to multiparameter estimation and quantum-enhanced microscopy.
Optimal measurement scheme with an efficient data processing is important in quantum-enhanced interferometry. Here we prove that for a general binary-outcome measurement, the simplest data processing based on inverting the average signal can saturate the Cram\'er-Rao bound. This idea is illustrated by binary-outcome homodyne detection, even-odd photon counting (i.e., parity detection), and zero-nonzero photon counting that have achieved super-resolved interferometric fringe and shot-noise limited sensitivity in coherent-light Mach-Zehnder interferometer. The roles of phase diffusion are investigated in these binary-outcome measurements. We find that the diffusion degrades the fringe resolution and the achievable phase sensitivity. Our analytical results confirm that the zero-nonzero counting can produce a slightly better sensitivity than that of the parity detection, as demonstrated in a recent experiment.