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arXiv:2302.11535

Optimal phase measurements in a lossy Mach-Zehnder interferometer

Wenfeng Huang, Xinyun Liang, Chun-Hua Yuan, Weiping Zhang, Liqing Chen

Published Feb 22, 2023• 0 citations
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Abstract

In this work, we discuss two phase-measurement methods for the Mach-Zehnder interferometer (MZI) in the presence of internal losses and give the corresponding optimum conditions. We find theoretically that when the core parameters (reflectivities, phase difference) are optimized, the phase sensitivity of the two methods can reach a generalized bound on precision: standard interferometric limit (SIL). In the experiment, we design an MZI with adjustable beam splitting ratios and losses to verify phase sensitivity optimization. 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 optimizing 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 interferometry.

DOI
10.48550/arxiv.2302.11535

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In this work, we discuss two phase-measurement methods for the Mach-Zehnder interferometer (MZI) in the presence of internal losses and give the corresponding optimum conditions. We find theoretically that when the core parameters (reflectivities, phase difference) are optimized, the phase sensitivity of the two methods can reach a generalized bound on precision: standard interferometric limit (SIL). In the experiment, we design an MZI with adjustable beam splitting ratios and losses to verify phase sensitivity optimization. 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 optimizing 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 interferometry.
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.
Improving interferometric phase sensitivity is crucial for high-precision measurements in rapidly developing quantum technologies. The Mach–Zehnder interferometer (MZI) is a versatile tool for analyzing this phenomenon. By splitting and recombining a light beam using beam splitters, the MZI allows for precise phase sensitivity analysis using tools like the quantum Cramér–Rao bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the phase sensitivity of an MZI in various scenarios using different detection schemes and input states. We compare the single- and two-parameter quantum estimation and their associated QCRB for three phase-shift situations: in both arms; only in the upper arm (asymmetric); and in both arms symmetrically. We then investigate the phase sensitivity under three detection schemes: intensity difference; single-mode intensity; and balanced homodyne. Additionally, we explore the use of Perelomov and Barut–Girardello coherent states, two types of SU(1,1) coherent states, in all scenarios. Notably, we demonstrate that, under optimal conditions, all detection schemes can achieve the QCRB by utilizing SU(1,1) coherent states as input states.
Interferometry is a widely used technique for precision measurements in both classical and quantum contexts. One way to increase the precision of phase measurements, for example, in a Mach-Zehnder interferometer (MZI), is to use high-intensity lasers. In this paper we study the phase sensitivity of a MZI in two detection setups (difference intensity detection and single-mode intensity detection) and for three input scenarios (coherent, double coherent, and coherent plus squeezed vacuum). For the coherent and double coherent input, both detection setups can reach the quantum Cram\'er-Rao bound, although at different values of the optimal phase shift. The double coherent input scenario has the unique advantage of changing the optimal phase shift by varying the input power ratio.
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.
Phase-shifting interferometry (PSI) promises excellent performance for quantitative phase measurement. However, the PSI is commonly realized by utilizing the Mach-Zehnder interferometer (MZI) fed with classical resources, and hence its phase sensitivity is hindered by the photon shot noise. In this paper, we propose to introduce specific phase-shifting algorithms into the well-established quantum-enhanced MZI, seeded by coherent and squeezed vacuum states and followed by single-mode intensity detection and difference intensity detection, to achieve quantitative phase measurement with sensitivity beyond that achieved by the corresponding Mach-Zehnder-type PSI with classical input. We show that when the average photon number of the coherent state is dominant, the optimal phase sensitivity of the Mach-Zehnder-type PSI with coherent plus squeezed vacuum input is better than the phase sensitivity of the Mach-Zehnder-type PSI with single coherent input under the same phase-sensing intensity condition in both of detection setups. We also show that for the typical MZI seeded with a high-power coherent state and a finitely squeezed vacuum state, the combination of intensity detection and phase-shifting algorithm can still improve the sensitivity of quantitative phase measurement. Furthermore, we derive the phase sensitivity of the fixed MZI as well as the quantum Cramér-Rao bound for two detection schemes. Finally, we explore the impacts of photon losses on the phase sensitivity enhancements of the Mach-Zehnder-type PSI and the fixed MZI in the high-intensity regime for each detection setup. The robustness to losses is compared between the Mach-Zehnder-type PSI and the fixed MZI. Our results may be helpful to promote the exploration of quantum-enhanced quantitative phase sensing.
An actively correlated Mach-Zehnder interferometer (ACMZI) renders a number of significant advantages over a conventional Mach-Zehnder interferometer (MZI) with a squeezed vacuum state input such as an active output readout, loss tolerant detection, and multiarms for multiparameter estimation. We theoretically studied the quantum Cram\'er-Rao bound of an ACMZI, where the quantum Fisher information obtained by the phase-averaging method can give the conclusive phase-sensing sensitivity without any external phase reference. We numerically calculate the phase sensitivities with the method of homodyne detection and intensity detection in the presence of losses. Under lossless and very low loss conditions, the ACMZI is operated in a balanced case to beat the standard quantum limit (SQL). As the loss increases, the reduction in sensitivity increases. Fortunately, within a certain range, we can adjust the gain parameters of the beam recombination process to reduce the reduction in sensitivity and realize the sensitivity can continue to beat the SQL in an unbalanced situation. Our scheme provides an optimization method of phase estimation in the presence of losses.