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Journal of the Optical Society of America B

Parity measurements versus dual-homodyne measurements in coherent-state Mach–Zehnder interferometry

Jeffrey H. Shapiro

Published Jul 25, 2017• 3 citations
Publisher

Abstract

Previous studies [J. Opt. Soc. Am. B27, A170 (2010)JOBPDE0740-322410.1364/JOSAB.27.00A170; J. Appl. Phys.114, 193102 (2013)JAPIAU0021-897910.1063/1.4829016] have asserted that a coherent-state Mach–Zehnder interferometer (MZI) with parity-based phase estimation can achieve superresolution at the shot-noise limit (SNL). We show that these studies have ignored the need for repeated measurements in order to achieve root-mean-square accuracy at the SNL. Consequently, their superresolution claims need major revision. For comparison, we present the performance of a coherent-state MZI that uses dual-homodyne detection and show that it can achieve SNL scaling in a single measurement, albeit without superresolution.

DOI
10.1364/josab.34.001699

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Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
The use of an interferometer to perform an ultra-precise parameter estimation under noisy conditions is a challenging task. Here we discuss nearly optimal measurement schemes for a well known, sensitive input state, squeezed vacuum and coherent light. We find that a single mode intensity measurement, while the simplest and able to beat the shot-noise limit, is outperformed by other measurement schemes in the low-power regime. However, at high powers, intensity measurement is only outperformed by a small factor. Specifically, we confirm, that an optimal measurement choice under lossless conditions is the parity measurement. In addition, we also discuss the performance of several other common measurement schemes when considering photon loss, detector efficiency, phase drift, and thermal photon noise. We conclude that, with noise considerations, homodyne remains near optimal in both the low and high power regimes. Surprisingly, some of the remaining investigated measurement schemes, including the previous optimal parity measurement, do not remain even near optimal when noise is introduced.
We study the sensitivity of phase estimation in a lossy Mach-Zehnder interferometer (MZI) using two general, and practical, resources generated by a laser and a nonlinear optical medium with passive optimal elements, which are readily available in the laboratory: One is a two-mode separable coherent and squeezed vacuum state at a beam splitter and the other is a two-mode squeezed vacuum state. In view of the ultimate precision given by quantum Fisher information, we show that the two-mode squeezed vacuum state can achieve a lower bound of estimation error than the coherent and squeezed vacuum state under a photon-loss channel. We further consider practical measurement schemes, homodyne detection and photon number resolving detection (PNRD), to characterize the accuracy of phase estimation in reality and find that the coherent and squeezed vacuum state largely achieves a lower bound than the two-mode squeezed vacuum in the lossy MZI while maintaining quantum enhancement over the shot-noise limit. By comparing homodyne detection and PNRD, we demonstrate that quadrature measurement with homodyne detection is more robust against photon loss than parity measurement with PNRD. We also show that double homodyne detection can provide a better tool for phase estimation than single homodyne detection against photon loss.
Abstract We investigate the phase sensitivities of three schemes with the input of the squeezed coherent states based on the Mach–Zehnder interferometer (MZI). With employing the balanced homodyne detection, the phase sensitivities of these schemes can surpass the shot noise limit (SNL) and reach the sub-Heisenberg limit. By adjusting the combination of the squeezing parameter and the photon number of the coherent state, both the phase sensitivities with using the balanced homodyne detection and the quantum Cramer–Rao bound can be improved. In the regime of low squeezing parameters, the scheme with the two-mode squeezed coherent state outperforms the other two schemes. The phase sensitivity can only reach the sub-SNL when the photon number is much larger. These schemes exhibit strong robustness against internal losses of the MZI and photon detector losses. This work will play an important role in the field of quantum precision measurement.