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arXiv:1605.03911 · EPJ Quantum Technology

Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum

Bryan Gard, Chenglong You, Devendra Kumar Mishra, R. Singh, Hwang Lee, T. R. Corbitt, Jonathan P. Dowling

Published Apr 7, 2017• 68 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1140/epjqt/s40507-017-0058-8

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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 Photon addition operations applied to squeezed states have been shown to significantly enhance phase sensitivity. In this work, we extend this approach by applying photon addition not only to coherent states but also within a Mach–Zehnder interferometer, using coherent and squeezed vacuum states as input. Both intensity‐difference and homodyne detection are used to evaluate photon addition schemes, and their phase sensitivities are compared under ideal and lossy conditions. We also analyze the quantum Fisher information of these two schemes. Results show that both schemes improve phase sensitivity, quantum Fisher information, and robustness against losses. In particular, photon addition within the interferometer performs better. Homodyne detection outperforms intensity difference detection under photon losses. Notably, the two schemes exhibit distinct parameter dependencies, rendering them suitable for different operational scenarios. When the squeezing parameter is small, photon addition employed at the coherent input with intensity difference detection can approach the Heisenberg limit in ideal conditions and can exceed the standard quantum limit in high‐loss conditions. The proposed scheme represents a valuable method for quantum precision measurements. PACS : 03.67.‐a, 05.30.‐d, 42.50,Dv, 03.65.Wj
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
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.
This work discusses the single-intensity and difference-intensity detection schemes for lossy Mach–Zehnder Interferometer (MZI) and gives the corresponding optimal conditions to achieve the best phase measurements. The interferometer is fed by a coherent state in one port and a vacuum state in the other. When the core parameters (reflectivities, phase difference) are optimal, the phase sensitivity of these two detection schemes can reach a generalized precision bound: the standard interferometric limit (SIL). In the experiment, we design an MZI with adjustable beam splitters to verify the phase sensitivity optimization compared to the conventional MZI with 50:50 beam splitters. The sensitivity improvements at loss rates from 0.4 to 0.998 are demonstrated based on difference-intensity detection, matching the theoretical results well. With a loss up to 0.998 in one arm, we achieve a sensitivity improvement of 2.5 dB by adjusting reflectivity, which equates to a 5.5 dB sensitivity improvement in single-intensity detection. Such optimal phase measurement methods provide practical solutions for the correct use of resources in lossy interferometers.
For the precision-based measurements, Mach-Zehnder interferometry is a widely used technique. There are various ways to enhance the precision of Mach-Zehnder interferometer (MZI), e.g., having a non-classical input state is one of the ways to enhance the precision of the phase estimation performed by MZI. The phase estimation performed by MZI is investigated here by considering that the input states of MZI are different combinations of photon added and subtracted squeezed coherent states (PASCS and PSSCS). Using quantum Fisher information, it is shown that the use of PASCS in both the input modes of MZI, provides the most precise estimate of the unknown phase. This system is also analyzed in two different measurement scenarios -- single intensity detection (SID) and intensity difference detection (IDD). Systematic analysis has established that the intensity measurement might not be an optimal measurement scheme for phase estimation in MZI as phase and intensity correspond to non-commuting observables. The impact of the photon loss on the MZI-based phase estimation setup is also studied and it is found that PASCS is robust against photon loss, when the loss in MZI is low.